LMFDB - Embedded newform 1045.4.a.i.1.10

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:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
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)

$f(q)$	$=$	$q-1.72202 q^{2} -1.27869 q^{3} -5.03464 q^{4} -5.00000 q^{5} +2.20194 q^{6} +3.57098 q^{7} +22.4459 q^{8} -25.3649 q^{9} +O(q^{10})$	\(q-1.72202 q^{2} -1.27869 q^{3} -5.03464 q^{4} -5.00000 q^{5} +2.20194 q^{6} +3.57098 q^{7} +22.4459 q^{8} -25.3649 q^{9} +8.61011 q^{10} +11.0000 q^{11} +6.43777 q^{12} +36.4227 q^{13} -6.14930 q^{14} +6.39347 q^{15} +1.62478 q^{16} +95.1214 q^{17} +43.6790 q^{18} +19.0000 q^{19} +25.1732 q^{20} -4.56619 q^{21} -18.9422 q^{22} -207.350 q^{23} -28.7015 q^{24} +25.0000 q^{25} -62.7206 q^{26} +66.9587 q^{27} -17.9786 q^{28} -186.149 q^{29} -11.0097 q^{30} -233.908 q^{31} -182.365 q^{32} -14.0656 q^{33} -163.801 q^{34} -17.8549 q^{35} +127.703 q^{36} -80.1636 q^{37} -32.7184 q^{38} -46.5735 q^{39} -112.230 q^{40} -45.4695 q^{41} +7.86308 q^{42} +91.9458 q^{43} -55.3811 q^{44} +126.825 q^{45} +357.061 q^{46} +634.785 q^{47} -2.07760 q^{48} -330.248 q^{49} -43.0505 q^{50} -121.631 q^{51} -183.375 q^{52} +48.6765 q^{53} -115.304 q^{54} -55.0000 q^{55} +80.1540 q^{56} -24.2952 q^{57} +320.552 q^{58} +348.391 q^{59} -32.1888 q^{60} -613.907 q^{61} +402.795 q^{62} -90.5777 q^{63} +301.039 q^{64} -182.113 q^{65} +24.2213 q^{66} +136.726 q^{67} -478.902 q^{68} +265.137 q^{69} +30.7465 q^{70} -601.964 q^{71} -569.340 q^{72} +503.855 q^{73} +138.043 q^{74} -31.9674 q^{75} -95.6582 q^{76} +39.2808 q^{77} +80.2005 q^{78} -294.513 q^{79} -8.12391 q^{80} +599.234 q^{81} +78.2995 q^{82} -897.143 q^{83} +22.9892 q^{84} -475.607 q^{85} -158.333 q^{86} +238.028 q^{87} +246.905 q^{88} +506.502 q^{89} -218.395 q^{90} +130.065 q^{91} +1043.93 q^{92} +299.097 q^{93} -1093.11 q^{94} -95.0000 q^{95} +233.190 q^{96} +1863.42 q^{97} +568.694 q^{98} -279.014 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 2 q^{2} + 9 q^{3} + 122 q^{4} - 125 q^{5} + 11 q^{6} - 15 q^{7} + 60 q^{8} + 300 q^{9}+O(q^{10}) $ 25 * q + 2 * q^2 + 9 * q^3 + 122 * q^4 - 125 * q^5 + 11 * q^6 - 15 * q^7 + 60 * q^8 + 300 * q^9	\( 25 q + 2 q^{2} + 9 q^{3} + 122 q^{4} - 125 q^{5} + 11 q^{6} - 15 q^{7} + 60 q^{8} + 300 q^{9} - 10 q^{10} + 275 q^{11} + 44 q^{12} + 53 q^{13} - 51 q^{14} - 45 q^{15} + 438 q^{16} + 153 q^{17} + 9 q^{18} + 475 q^{19} - 610 q^{20} + 259 q^{21} + 22 q^{22} - 7 q^{23} + 186 q^{24} + 625 q^{25} + 543 q^{26} + 495 q^{27} - 525 q^{28} + 169 q^{29} - 55 q^{30} + 102 q^{31} + 327 q^{32} + 99 q^{33} - 879 q^{34} + 75 q^{35} + 2293 q^{36} - 46 q^{37} + 38 q^{38} + 233 q^{39} - 300 q^{40} + 1190 q^{41} - 684 q^{42} - 408 q^{43} + 1342 q^{44} - 1500 q^{45} + 757 q^{46} + 1068 q^{47} + 715 q^{48} + 1930 q^{49} + 50 q^{50} + 1655 q^{51} - 94 q^{52} + 143 q^{53} + 1970 q^{54} - 1375 q^{55} - 1397 q^{56} + 171 q^{57} + 1366 q^{58} + 2945 q^{59} - 220 q^{60} + 1160 q^{61} + 194 q^{62} + 1804 q^{63} + 3000 q^{64} - 265 q^{65} + 121 q^{66} - 353 q^{67} + 5452 q^{68} + 3289 q^{69} + 255 q^{70} + 230 q^{71} + 196 q^{72} + 1357 q^{73} + 4379 q^{74} + 225 q^{75} + 2318 q^{76} - 165 q^{77} + 2008 q^{78} + 1266 q^{79} - 2190 q^{80} + 1709 q^{81} + 1010 q^{82} + 3856 q^{83} + 9354 q^{84} - 765 q^{85} + 6746 q^{86} + 3113 q^{87} + 660 q^{88} + 3562 q^{89} - 45 q^{90} - 833 q^{91} + 4276 q^{92} + 1312 q^{93} + 5124 q^{94} - 2375 q^{95} + 3828 q^{96} - 914 q^{97} + 2478 q^{98} + 3300 q^{99}+O(q^{100}) \) 25 * q + 2 * q^2 + 9 * q^3 + 122 * q^4 - 125 * q^5 + 11 * q^6 - 15 * q^7 + 60 * q^8 + 300 * q^9 - 10 * q^10 + 275 * q^11 + 44 * q^12 + 53 * q^13 - 51 * q^14 - 45 * q^15 + 438 * q^16 + 153 * q^17 + 9 * q^18 + 475 * q^19 - 610 * q^20 + 259 * q^21 + 22 * q^22 - 7 * q^23 + 186 * q^24 + 625 * q^25 + 543 * q^26 + 495 * q^27 - 525 * q^28 + 169 * q^29 - 55 * q^30 + 102 * q^31 + 327 * q^32 + 99 * q^33 - 879 * q^34 + 75 * q^35 + 2293 * q^36 - 46 * q^37 + 38 * q^38 + 233 * q^39 - 300 * q^40 + 1190 * q^41 - 684 * q^42 - 408 * q^43 + 1342 * q^44 - 1500 * q^45 + 757 * q^46 + 1068 * q^47 + 715 * q^48 + 1930 * q^49 + 50 * q^50 + 1655 * q^51 - 94 * q^52 + 143 * q^53 + 1970 * q^54 - 1375 * q^55 - 1397 * q^56 + 171 * q^57 + 1366 * q^58 + 2945 * q^59 - 220 * q^60 + 1160 * q^61 + 194 * q^62 + 1804 * q^63 + 3000 * q^64 - 265 * q^65 + 121 * q^66 - 353 * q^67 + 5452 * q^68 + 3289 * q^69 + 255 * q^70 + 230 * q^71 + 196 * q^72 + 1357 * q^73 + 4379 * q^74 + 225 * q^75 + 2318 * q^76 - 165 * q^77 + 2008 * q^78 + 1266 * q^79 - 2190 * q^80 + 1709 * q^81 + 1010 * q^82 + 3856 * q^83 + 9354 * q^84 - 765 * q^85 + 6746 * q^86 + 3113 * q^87 + 660 * q^88 + 3562 * q^89 - 45 * q^90 - 833 * q^91 + 4276 * q^92 + 1312 * q^93 + 5124 * q^94 - 2375 * q^95 + 3828 * q^96 - 914 * q^97 + 2478 * q^98 + 3300 * 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 $

$2$	−1.72202	−0.608826	−0.304413	−	0.952540i	$-0.598460\pi$
$2$	−1.72202	−0.608826	−0.304413	+	0.952540i	$0.598460\pi$
$3$	−1.27869	−0.246085	−0.123042	−	0.992401i	$-0.539265\pi$
$3$	−1.27869	−0.246085	−0.123042	+	0.992401i	$0.539265\pi$
$4$	−5.03464	−0.629330
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	2.20194	0.149823
$7$	3.57098	0.192815	0.0964074	−	0.995342i	$-0.469265\pi$
$7$	3.57098	0.192815	0.0964074	+	0.995342i	$0.469265\pi$
$8$	22.4459	0.991979
$9$	−25.3649	−0.939442
$10$	8.61011	0.272275
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	6.43777	0.154869
$13$	36.4227	0.777064	0.388532	−	0.921435i	$-0.372982\pi$
$13$	36.4227	0.777064	0.388532	+	0.921435i	$0.372982\pi$
$14$	−6.14930	−0.117391
$15$	6.39347	0.110052
$16$	1.62478	0.0253872
$17$	95.1214	1.35708	0.678539	−	0.734564i	$-0.262613\pi$
$17$	95.1214	1.35708	0.678539	+	0.734564i	$0.262613\pi$
$18$	43.6790	0.571957
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	25.1732	0.281445
$21$	−4.56619	−0.0474488
$22$	−18.9422	−0.183568
$23$	−207.350	−1.87980	−0.939901	−	0.341447i	$-0.889083\pi$
$23$	−207.350	−1.87980	−0.939901	+	0.341447i	$0.889083\pi$
$24$	−28.7015	−0.244111
$25$	25.0000	0.200000
$26$	−62.7206	−0.473097
$27$	66.9587	0.477267
$28$	−17.9786	−0.121344
$29$	−186.149	−1.19197	−0.595983	−	0.802997i	$-0.703237\pi$
$29$	−186.149	−1.19197	−0.595983	+	0.802997i	$0.703237\pi$
$30$	−11.0097	−0.0670028
$31$	−233.908	−1.35520	−0.677599	−	0.735432i	$-0.736979\pi$
$31$	−233.908	−1.35520	−0.677599	+	0.735432i	$0.736979\pi$
$32$	−182.365	−1.00744
$33$	−14.0656	−0.0741974
$34$	−163.801	−0.826225
$35$	−17.8549	−0.0862294
$36$	127.703	0.591220
$37$	−80.1636	−0.356184	−0.178092	−	0.984014i	$-0.556993\pi$
$37$	−80.1636	−0.356184	−0.178092	+	0.984014i	$0.556993\pi$
$38$	−32.7184	−0.139674
$39$	−46.5735	−0.191224
$40$	−112.230	−0.443627
$41$	−45.4695	−0.173199	−0.0865994	−	0.996243i	$-0.527600\pi$
$41$	−45.4695	−0.173199	−0.0865994	+	0.996243i	$0.527600\pi$
$42$	7.86308	0.0288881
$43$	91.9458	0.326084	0.163042	−	0.986619i	$-0.447869\pi$
$43$	91.9458	0.326084	0.163042	+	0.986619i	$0.447869\pi$
$44$	−55.3811	−0.189750
$45$	126.825	0.420131
$46$	357.061	1.14447
$47$	634.785	1.97006	0.985031	−	0.172377i	$-0.0551448\pi$
$47$	634.785	1.97006	0.985031	+	0.172377i	$0.0551448\pi$
$48$	−2.07760	−0.00624741
$49$	−330.248	−0.962822
$50$	−43.0505	−0.121765
$51$	−121.631	−0.333956
$52$	−183.375	−0.489030
$53$	48.6765	0.126155	0.0630776	−	0.998009i	$-0.479908\pi$
$53$	48.6765	0.126155	0.0630776	+	0.998009i	$0.479908\pi$
$54$	−115.304	−0.290573
$55$	−55.0000	−0.134840
$56$	80.1540	0.191268
$57$	−24.2952	−0.0564557
$58$	320.552	0.725700
$59$	348.391	0.768758	0.384379	−	0.923175i	$-0.374416\pi$
$59$	348.391	0.768758	0.384379	+	0.923175i	$0.374416\pi$
$60$	−32.1888	−0.0692594
$61$	−613.907	−1.28857	−0.644285	−	0.764786i	$-0.722845\pi$
$61$	−613.907	−1.28857	−0.644285	+	0.764786i	$0.722845\pi$
$62$	402.795	0.825080
$63$	−90.5777	−0.181138
$64$	301.039	0.587966
$65$	−182.113	−0.347514
$66$	24.2213	0.0451733
$67$	136.726	0.249310	0.124655	−	0.992200i	$-0.460218\pi$
$67$	136.726	0.249310	0.124655	+	0.992200i	$0.460218\pi$
$68$	−478.902	−0.854051
$69$	265.137	0.462591
$70$	30.7465	0.0524988
$71$	−601.964	−1.00620	−0.503099	−	0.864229i	$-0.667807\pi$
$71$	−601.964	−1.00620	−0.503099	+	0.864229i	$0.667807\pi$
$72$	−569.340	−0.931907
$73$	503.855	0.807832	0.403916	−	0.914796i	$-0.367649\pi$
$73$	503.855	0.807832	0.403916	+	0.914796i	$0.367649\pi$
$74$	138.043	0.216854
$75$	−31.9674	−0.0492170
$76$	−95.6582	−0.144378
$77$	39.2808	0.0581359
$78$	80.2005	0.116422
$79$	−294.513	−0.419434	−0.209717	−	0.977762i	$-0.567254\pi$
$79$	−294.513	−0.419434	−0.209717	+	0.977762i	$0.567254\pi$
$80$	−8.12391	−0.0113535
$81$	599.234	0.821994
$82$	78.2995	0.105448
$83$	−897.143	−1.18644	−0.593219	−	0.805041i	$-0.702143\pi$
$83$	−897.143	−1.18644	−0.593219	+	0.805041i	$0.702143\pi$
$84$	22.9892	0.0298610
$85$	−475.607	−0.606904
$86$	−158.333	−0.198528
$87$	238.028	0.293324
$88$	246.905	0.299093
$89$	506.502	0.603248	0.301624	−	0.953427i	$-0.402471\pi$
$89$	506.502	0.603248	0.301624	+	0.953427i	$0.402471\pi$
$90$	−218.395	−0.255787
$91$	130.065	0.149830
$92$	1043.93	1.18302
$93$	299.097	0.333494
$94$	−1093.11	−1.19943
$95$	−95.0000	−0.102598
$96$	233.190	0.247915
$97$	1863.42	1.95053	0.975265	−	0.221039i	$-0.0709449\pi$
$97$	1863.42	1.95053	0.975265	+	0.221039i	$0.0709449\pi$
$98$	568.694	0.586192
$99$	−279.014	−0.283253
$100$	−125.866	−0.125866
$101$	−585.793	−0.577115	−0.288557	−	0.957463i	$-0.593176\pi$
$101$	−585.793	−0.577115	−0.288557	+	0.957463i	$0.593176\pi$
$102$	209.451	0.203321
$103$	−570.748	−0.545995	−0.272997	−	0.962015i	$-0.588015\pi$
$103$	−570.748	−0.545995	−0.272997	+	0.962015i	$0.588015\pi$
$104$	817.541	0.770832
$105$	22.8310	0.0212198
$106$	−83.8219	−0.0768066
$107$	1720.26	1.55424	0.777120	−	0.629352i	$-0.216680\pi$
$107$	1720.26	1.55424	0.777120	+	0.629352i	$0.216680\pi$
$108$	−337.113	−0.300359
$109$	−788.808	−0.693157	−0.346579	−	0.938021i	$-0.612657\pi$
$109$	−788.808	−0.693157	−0.346579	+	0.938021i	$0.612657\pi$
$110$	94.7112	0.0820941
$111$	102.505	0.0876516
$112$	5.80206	0.00489503
$113$	−1086.15	−0.904214	−0.452107	−	0.891964i	$-0.649327\pi$
$113$	−1086.15	−0.904214	−0.452107	+	0.891964i	$0.649327\pi$
$114$	41.8368	0.0343717
$115$	1036.75	0.840673
$116$	937.193	0.750140
$117$	−923.859	−0.730007
$118$	−599.937	−0.468040
$119$	339.677	0.261665
$120$	143.507	0.109170
$121$	121.000	0.0909091
$122$	1057.16	0.784515
$123$	58.1416	0.0426216
$124$	1177.64	0.852867
$125$	−125.000	−0.0894427
$126$	155.977	0.110282
$127$	1319.59	0.922003	0.461002	−	0.887399i	$-0.347490\pi$
$127$	1319.59	0.922003	0.461002	+	0.887399i	$0.347490\pi$
$128$	940.528	0.649466
$129$	−117.571	−0.0802443
$130$	313.603	0.211576
$131$	−600.824	−0.400719	−0.200360	−	0.979722i	$-0.564211\pi$
$131$	−600.824	−0.400719	−0.200360	+	0.979722i	$0.564211\pi$
$132$	70.8155	0.0466947
$133$	67.8486	0.0442348
$134$	−235.446	−0.151787
$135$	−334.794	−0.213440
$136$	2135.09	1.34619
$137$	704.278	0.439201	0.219600	−	0.975590i	$-0.429525\pi$
$137$	704.278	0.439201	0.219600	+	0.975590i	$0.429525\pi$
$138$	−456.572	−0.281637
$139$	1979.99	1.20820	0.604102	−	0.796907i	$-0.293532\pi$
$139$	1979.99	1.20820	0.604102	+	0.796907i	$0.293532\pi$
$140$	89.8931	0.0542668
$141$	−811.696	−0.484802
$142$	1036.60	0.612600
$143$	400.650	0.234294
$144$	−41.2125	−0.0238498
$145$	930.745	0.533063
$146$	−867.649	−0.491830
$147$	422.286	0.236936
$148$	403.595	0.224158
$149$	2618.23	1.43956	0.719778	−	0.694205i	$-0.244244\pi$
$149$	2618.23	1.43956	0.719778	+	0.694205i	$0.244244\pi$
$150$	55.0485	0.0299646
$151$	741.681	0.399716	0.199858	−	0.979825i	$-0.435952\pi$
$151$	741.681	0.399716	0.199858	+	0.979825i	$0.435952\pi$
$152$	426.473	0.227576
$153$	−2412.75	−1.27490
$154$	−67.6424	−0.0353947
$155$	1169.54	0.606063
$156$	234.481	0.120343
$157$	−990.591	−0.503553	−0.251776	−	0.967785i	$-0.581015\pi$
$157$	−990.591	−0.503553	−0.251776	+	0.967785i	$0.581015\pi$
$158$	507.158	0.255363
$159$	−62.2423	−0.0310449
$160$	911.827	0.450539
$161$	−740.443	−0.362454
$162$	−1031.89	−0.500452
$163$	1838.21	0.883309	0.441655	−	0.897185i	$-0.354392\pi$
$163$	1838.21	0.883309	0.441655	+	0.897185i	$0.354392\pi$
$164$	228.923	0.108999
$165$	70.3282	0.0331821
$166$	1544.90	0.722334
$167$	−3096.07	−1.43462	−0.717309	−	0.696755i	$-0.754626\pi$
$167$	−3096.07	−1.43462	−0.717309	+	0.696755i	$0.754626\pi$
$168$	−102.492	−0.0470682
$169$	−870.388	−0.396171
$170$	819.005	0.369499
$171$	−481.934	−0.215523
$172$	−462.914	−0.205214
$173$	−108.616	−0.0477336	−0.0238668	−	0.999715i	$-0.507598\pi$
$173$	−108.616	−0.0477336	−0.0238668	+	0.999715i	$0.507598\pi$
$174$	−409.888	−0.178584
$175$	89.2745	0.0385630
$176$	17.8726	0.00765453
$177$	−445.486	−0.189180
$178$	−872.206	−0.367273
$179$	4275.79	1.78541	0.892703	−	0.450646i	$-0.148806\pi$
$179$	4275.79	1.78541	0.892703	+	0.450646i	$0.148806\pi$
$180$	−638.517	−0.264401
$181$	2381.14	0.977838	0.488919	−	0.872329i	$-0.337391\pi$
$181$	2381.14	0.977838	0.488919	+	0.872329i	$0.337391\pi$
$182$	−223.974	−0.0912202
$183$	784.999	0.317097
$184$	−4654.16	−1.86473
$185$	400.818	0.159290
$186$	−515.051	−0.203040
$187$	1046.34	0.409174
$188$	−3195.92	−1.23982
$189$	239.108	0.0920242
$190$	163.592	0.0624643
$191$	−1551.71	−0.587843	−0.293921	−	0.955830i	$-0.594960\pi$
$191$	−1551.71	−0.587843	−0.293921	+	0.955830i	$0.594960\pi$
$192$	−384.936	−0.144690
$193$	3054.44	1.13919	0.569594	−	0.821926i	$-0.307100\pi$
$193$	3054.44	1.13919	0.569594	+	0.821926i	$0.307100\pi$
$194$	−3208.84	−1.18753
$195$	232.867	0.0855179
$196$	1662.68	0.605933
$197$	−3249.18	−1.17510	−0.587550	−	0.809188i	$-0.699908\pi$
$197$	−3249.18	−1.17510	−0.587550	+	0.809188i	$0.699908\pi$
$198$	480.469	0.172452
$199$	3554.20	1.26608	0.633042	−	0.774117i	$-0.281806\pi$
$199$	3554.20	1.26608	0.633042	+	0.774117i	$0.281806\pi$
$200$	561.148	0.198396
$201$	−174.831	−0.0613515
$202$	1008.75	0.351363
$203$	−664.734	−0.229829
$204$	612.370	0.210169
$205$	227.348	0.0774568
$206$	982.839	0.332416
$207$	5259.42	1.76597
$208$	59.1789	0.0197275
$209$	209.000	0.0691714
$210$	−39.3154	−0.0129191
$211$	1978.45	0.645508	0.322754	−	0.946483i	$-0.395391\pi$
$211$	1978.45	0.645508	0.322754	+	0.946483i	$0.395391\pi$
$212$	−245.069	−0.0793933
$213$	769.728	0.247610
$214$	−2962.32	−0.946262
$215$	−459.729	−0.145829
$216$	1502.95	0.473439
$217$	−835.281	−0.261302
$218$	1358.34	0.422012
$219$	−644.276	−0.198795
$220$	276.905	0.0848589
$221$	3464.58	1.05454
$222$	−176.515	−0.0533646
$223$	−668.240	−0.200667	−0.100333	−	0.994954i	$-0.531991\pi$
$223$	−668.240	−0.200667	−0.100333	+	0.994954i	$0.531991\pi$
$224$	−651.223	−0.194249
$225$	−634.124	−0.187888
$226$	1870.37	0.550509
$227$	1777.16	0.519621	0.259811	−	0.965660i	$-0.416340\pi$
$227$	1777.16	0.519621	0.259811	+	0.965660i	$0.416340\pi$
$228$	122.318	0.0355293
$229$	−1758.18	−0.507354	−0.253677	−	0.967289i	$-0.581640\pi$
$229$	−1758.18	−0.507354	−0.253677	+	0.967289i	$0.581640\pi$
$230$	−1785.30	−0.511824
$231$	−50.2281	−0.0143064
$232$	−4178.29	−1.18240
$233$	3477.05	0.977636	0.488818	−	0.872386i	$-0.337428\pi$
$233$	3477.05	0.977636	0.488818	+	0.872386i	$0.337428\pi$
$234$	1590.91	0.444448
$235$	−3173.93	−0.881039
$236$	−1754.03	−0.483803
$237$	376.592	0.103216
$238$	−584.930	−0.159308
$239$	−630.688	−0.170694	−0.0853469	−	0.996351i	$-0.527200\pi$
$239$	−630.688	−0.170694	−0.0853469	+	0.996351i	$0.527200\pi$
$240$	10.3880	0.00279393
$241$	3688.46	0.985871	0.492935	−	0.870066i	$-0.335924\pi$
$241$	3688.46	0.985871	0.492935	+	0.870066i	$0.335924\pi$
$242$	−208.365	−0.0553479
$243$	−2574.12	−0.679548
$244$	3090.80	0.810936
$245$	1651.24	0.430587
$246$	−100.121	−0.0259491
$247$	692.031	0.178271
$248$	−5250.29	−1.34433
$249$	1147.17	0.291964
$250$	215.253	0.0544551
$251$	6221.19	1.56445	0.782227	−	0.622993i	$-0.214084\pi$
$251$	6221.19	1.56445	0.782227	+	0.622993i	$0.214084\pi$
$252$	456.027	0.113996
$253$	−2280.85	−0.566782
$254$	−2272.36	−0.561340
$255$	608.156	0.149350
$256$	−4027.92	−0.983379
$257$	828.503	0.201092	0.100546	−	0.994932i	$-0.467941\pi$
$257$	828.503	0.201092	0.100546	+	0.994932i	$0.467941\pi$
$258$	202.459	0.0488548
$259$	−286.263	−0.0686776
$260$	916.876	0.218701
$261$	4721.66	1.11978
$262$	1034.63	0.243968
$263$	705.502	0.165411	0.0827055	−	0.996574i	$-0.473644\pi$
$263$	705.502	0.165411	0.0827055	+	0.996574i	$0.473644\pi$
$264$	−315.716	−0.0736023
$265$	−243.382	−0.0564183
$266$	−116.837	−0.0269313
$267$	−647.661	−0.148450
$268$	−688.369	−0.156899
$269$	5698.29	1.29156	0.645782	−	0.763521i	$-0.276531\pi$
$269$	5698.29	1.29156	0.645782	+	0.763521i	$0.276531\pi$
$270$	576.522	0.129948
$271$	−5705.47	−1.27890	−0.639451	−	0.768832i	$-0.720838\pi$
$271$	−5705.47	−1.27890	−0.639451	+	0.768832i	$0.720838\pi$
$272$	154.551	0.0344524
$273$	−166.313	−0.0368708
$274$	−1212.78	−0.267397
$275$	275.000	0.0603023
$276$	−1334.87	−0.291122
$277$	−3052.76	−0.662175	−0.331088	−	0.943600i	$-0.607416\pi$
$277$	−3052.76	−0.662175	−0.331088	+	0.943600i	$0.607416\pi$
$278$	−3409.58	−0.735587
$279$	5933.07	1.27313
$280$	−400.770	−0.0855378
$281$	6732.48	1.42927	0.714637	−	0.699496i	$-0.246592\pi$
$281$	6732.48	1.42927	0.714637	+	0.699496i	$0.246592\pi$
$282$	1397.76	0.295160
$283$	−8983.01	−1.88687	−0.943436	−	0.331556i	$-0.892426\pi$
$283$	−8983.01	−1.88687	−0.943436	+	0.331556i	$0.892426\pi$
$284$	3030.68	0.633231
$285$	121.476	0.0252478
$286$	−689.927	−0.142644
$287$	−162.371	−0.0333953
$288$	4625.69	0.946428
$289$	4135.08	0.841661
$290$	−1602.76	−0.324543
$291$	−2382.74	−0.479996
$292$	−2536.73	−0.508393
$293$	2360.17	0.470589	0.235294	−	0.971924i	$-0.424395\pi$
$293$	2360.17	0.470589	0.235294	+	0.971924i	$0.424395\pi$
$294$	−727.186	−0.144253
$295$	−1741.96	−0.343799
$296$	−1799.35	−0.353328
$297$	736.546	0.143901
$298$	−4508.65	−0.876439
$299$	−7552.24	−1.46073
$300$	160.944	0.0309737
$301$	328.337	0.0628738
$302$	−1277.19	−0.243358
$303$	749.050	0.142019
$304$	30.8708	0.00582423
$305$	3069.53	0.576266
$306$	4154.80	0.776191
$307$	−7906.31	−1.46983	−0.734913	−	0.678161i	$-0.762777\pi$
$307$	−7906.31	−1.46983	−0.734913	+	0.678161i	$0.762777\pi$
$308$	−197.765	−0.0365867
$309$	729.812	0.134361
$310$	−2013.97	−0.368987
$311$	−6502.29	−1.18557	−0.592783	−	0.805362i	$-0.701971\pi$
$311$	−6502.29	−1.18557	−0.592783	+	0.805362i	$0.701971\pi$
$312$	−1045.39	−0.189690
$313$	9113.61	1.64579	0.822894	−	0.568195i	$-0.192358\pi$
$313$	9113.61	1.64579	0.822894	+	0.568195i	$0.192358\pi$
$314$	1705.82	0.306576
$315$	452.889	0.0810076
$316$	1482.77	0.263963
$317$	1257.98	0.222887	0.111444	−	0.993771i	$-0.464453\pi$
$317$	1257.98	0.222887	0.111444	+	0.993771i	$0.464453\pi$
$318$	107.183	0.0189009
$319$	−2047.64	−0.359391
$320$	−1505.19	−0.262947
$321$	−2199.68	−0.382475
$322$	1275.06	0.220671
$323$	1807.31	0.311335
$324$	−3016.93	−0.517306
$325$	910.567	0.155413
$326$	−3165.43	−0.537782
$327$	1008.64	0.170575
$328$	−1020.61	−0.171810
$329$	2266.81	0.379857
$330$	−121.107	−0.0202021
$331$	5510.45	0.915051	0.457526	−	0.889196i	$-0.348736\pi$
$331$	5510.45	0.915051	0.457526	+	0.889196i	$0.348736\pi$
$332$	4516.80	0.746661
$333$	2033.35	0.334615
$334$	5331.50	0.873433
$335$	−683.632	−0.111495
$336$	−7.41907	−0.00120459
$337$	7901.34	1.27719	0.638596	−	0.769542i	$-0.279516\pi$
$337$	7901.34	1.27719	0.638596	+	0.769542i	$0.279516\pi$
$338$	1498.83	0.241199
$339$	1388.85	0.222513
$340$	2394.51	0.381943
$341$	−2572.99	−0.408608
$342$	829.900	0.131216
$343$	−2404.16	−0.378461
$344$	2063.81	0.323468
$345$	−1325.69	−0.206877
$346$	187.039	0.0290615
$347$	8100.35	1.25317	0.626584	−	0.779354i	$-0.284452\pi$
$347$	8100.35	1.25317	0.626584	+	0.779354i	$0.284452\pi$
$348$	−1198.38	−0.184598
$349$	3313.40	0.508201	0.254101	−	0.967178i	$-0.418221\pi$
$349$	3313.40	0.508201	0.254101	+	0.967178i	$0.418221\pi$
$350$	−153.733	−0.0234782
$351$	2438.82	0.370867
$352$	−2006.02	−0.303753
$353$	3024.10	0.455968	0.227984	−	0.973665i	$-0.426787\pi$
$353$	3024.10	0.455968	0.227984	+	0.973665i	$0.426787\pi$
$354$	767.137	0.115178
$355$	3009.82	0.449985
$356$	−2550.05	−0.379642
$357$	−434.343	−0.0643917
$358$	−7363.00	−1.08700
$359$	9743.46	1.43242	0.716212	−	0.697883i	$-0.245874\pi$
$359$	9743.46	1.43242	0.716212	+	0.697883i	$0.245874\pi$
$360$	2846.70	0.416762
$361$	361.000	0.0526316
$362$	−4100.37	−0.595334
$363$	−154.722	−0.0223713
$364$	−654.830	−0.0942923
$365$	−2519.27	−0.361274
$366$	−1351.79	−0.193057
$367$	12793.4	1.81965	0.909826	−	0.414990i	$-0.136215\pi$
$367$	12793.4	1.81965	0.909826	+	0.414990i	$0.136215\pi$
$368$	−336.898	−0.0477229
$369$	1153.33	0.162710
$370$	−690.217	−0.0969802
$371$	173.823	0.0243246
$372$	−1505.85	−0.209878
$373$	−8747.19	−1.21424	−0.607121	−	0.794609i	$-0.707676\pi$
$373$	−8747.19	−1.21424	−0.607121	+	0.794609i	$0.707676\pi$
$374$	−1801.81	−0.249116
$375$	159.837	0.0220105
$376$	14248.3	1.95426
$377$	−6780.04	−0.926234
$378$	−411.750	−0.0560268
$379$	−11692.5	−1.58471	−0.792356	−	0.610059i	$-0.791146\pi$
$379$	−11692.5	−1.58471	−0.792356	+	0.610059i	$0.791146\pi$
$380$	478.291	0.0645679
$381$	−1687.35	−0.226891
$382$	2672.08	0.357894
$383$	−5655.53	−0.754528	−0.377264	−	0.926106i	$-0.623135\pi$
$383$	−5655.53	−0.754528	−0.377264	+	0.926106i	$0.623135\pi$
$384$	−1202.65	−0.159824
$385$	−196.404	−0.0259992
$386$	−5259.81	−0.693568
$387$	−2332.20	−0.306337
$388$	−9381.64	−1.22753
$389$	−4877.83	−0.635774	−0.317887	−	0.948129i	$-0.602973\pi$
$389$	−4877.83	−0.635774	−0.317887	+	0.948129i	$0.602973\pi$
$390$	−401.003	−0.0520655
$391$	−19723.4	−2.55104
$392$	−7412.73	−0.955100
$393$	768.270	0.0986109
$394$	5595.16	0.715432
$395$	1472.57	0.187577
$396$	1404.74	0.178259
$397$	−4558.39	−0.576270	−0.288135	−	0.957590i	$-0.593035\pi$
$397$	−4558.39	−0.576270	−0.288135	+	0.957590i	$0.593035\pi$
$398$	−6120.42	−0.770826
$399$	−86.7577	−0.0108855
$400$	40.6195	0.00507744
$401$	4245.77	0.528737	0.264368	−	0.964422i	$-0.414837\pi$
$401$	4245.77	0.528737	0.264368	+	0.964422i	$0.414837\pi$
$402$	301.063	0.0373524
$403$	−8519.56	−1.05308
$404$	2949.26	0.363196
$405$	−2996.17	−0.367607
$406$	1144.69	0.139926
$407$	−881.800	−0.107394
$408$	−2730.12	−0.331278
$409$	−5803.88	−0.701670	−0.350835	−	0.936437i	$-0.614102\pi$
$409$	−5803.88	−0.701670	−0.350835	+	0.936437i	$0.614102\pi$
$410$	−391.498	−0.0471578
$411$	−900.556	−0.108081
$412$	2873.51	0.343611
$413$	1244.10	0.148228
$414$	−9056.83	−1.07517
$415$	4485.72	0.530591
$416$	−6642.24	−0.782842
$417$	−2531.80	−0.297321
$418$	−359.902	−0.0421134
$419$	−3978.06	−0.463821	−0.231910	−	0.972737i	$-0.574498\pi$
$419$	−3978.06	−0.463821	−0.231910	+	0.972737i	$0.574498\pi$
$420$	−114.946	−0.0133542
$421$	−7855.12	−0.909347	−0.454674	−	0.890658i	$-0.650244\pi$
$421$	−7855.12	−0.909347	−0.454674	+	0.890658i	$0.650244\pi$
$422$	−3406.93	−0.393002
$423$	−16101.3	−1.85076
$424$	1092.59	0.125143
$425$	2378.03	0.271416
$426$	−1325.49	−0.150751
$427$	−2192.25	−0.248455
$428$	−8660.89	−0.978131
$429$	−512.308	−0.0576561
$430$	791.663	0.0887846
$431$	14658.5	1.63823	0.819113	−	0.573632i	$-0.194466\pi$
$431$	14658.5	1.63823	0.819113	+	0.573632i	$0.194466\pi$
$432$	108.793	0.0121165
$433$	−331.959	−0.0368428	−0.0184214	−	0.999830i	$-0.505864\pi$
$433$	−331.959	−0.0368428	−0.0184214	+	0.999830i	$0.505864\pi$
$434$	1438.37	0.159088
$435$	−1190.14	−0.131179
$436$	3971.37	0.436225
$437$	−3939.65	−0.431256
$438$	1109.46	0.121032
$439$	−13362.9	−1.45279	−0.726395	−	0.687277i	$-0.758806\pi$
$439$	−13362.9	−1.45279	−0.726395	+	0.687277i	$0.758806\pi$
$440$	−1234.53	−0.133758
$441$	8376.72	0.904516
$442$	−5966.08	−0.642030
$443$	−5924.63	−0.635413	−0.317706	−	0.948189i	$-0.602913\pi$
$443$	−5924.63	−0.635413	−0.317706	+	0.948189i	$0.602913\pi$
$444$	−516.075	−0.0551618
$445$	−2532.51	−0.269781
$446$	1150.72	0.122171
$447$	−3347.91	−0.354253
$448$	1075.00	0.113369
$449$	6267.62	0.658769	0.329385	−	0.944196i	$-0.393159\pi$
$449$	6267.62	0.658769	0.329385	+	0.944196i	$0.393159\pi$
$450$	1091.97	0.114391
$451$	−500.165	−0.0522214
$452$	5468.37	0.569049
$453$	−948.383	−0.0983640
$454$	−3060.30	−0.316359
$455$	−650.324	−0.0670058
$456$	−545.328	−0.0560029
$457$	−14005.2	−1.43355	−0.716777	−	0.697303i	$-0.754383\pi$
$457$	−14005.2	−1.43355	−0.716777	+	0.697303i	$0.754383\pi$
$458$	3027.63	0.308890
$459$	6369.21	0.647689
$460$	−5219.67	−0.529061
$461$	2811.59	0.284054	0.142027	−	0.989863i	$-0.454638\pi$
$461$	2811.59	0.284054	0.142027	+	0.989863i	$0.454638\pi$
$462$	86.4939	0.00871009
$463$	16084.2	1.61446	0.807230	−	0.590238i	$-0.200966\pi$
$463$	16084.2	1.61446	0.807230	+	0.590238i	$0.200966\pi$
$464$	−302.451	−0.0302607
$465$	−1495.48	−0.149143
$466$	−5987.56	−0.595211
$467$	14938.6	1.48024	0.740122	−	0.672472i	$-0.234768\pi$
$467$	14938.6	1.48024	0.740122	+	0.672472i	$0.234768\pi$
$468$	4651.30	0.459416
$469$	488.247	0.0480707
$470$	5465.57	0.536399
$471$	1266.66	0.123917
$472$	7819.97	0.762592
$473$	1011.40	0.0983180
$474$	−648.500	−0.0628409
$475$	475.000	0.0458831
$476$	−1710.15	−0.164674
$477$	−1234.68	−0.118516
$478$	1086.06	0.103923
$479$	7807.68	0.744764	0.372382	−	0.928079i	$-0.378541\pi$
$479$	7807.68	0.744764	0.372382	+	0.928079i	$0.378541\pi$
$480$	−1165.95	−0.110871
$481$	−2919.78	−0.276778
$482$	−6351.61	−0.600224
$483$	946.800	0.0891944
$484$	−609.192	−0.0572119
$485$	−9317.09	−0.872303
$486$	4432.69	0.413726
$487$	8589.98	0.799279	0.399640	−	0.916672i	$-0.369135\pi$
$487$	8589.98	0.799279	0.399640	+	0.916672i	$0.369135\pi$
$488$	−13779.7	−1.27823
$489$	−2350.50	−0.217369
$490$	−2843.47	−0.262153
$491$	1525.60	0.140223	0.0701114	−	0.997539i	$-0.477665\pi$
$491$	1525.60	0.140223	0.0701114	+	0.997539i	$0.477665\pi$
$492$	−292.722	−0.0268231
$493$	−17706.7	−1.61759
$494$	−1191.69	−0.108536
$495$	1395.07	0.126674
$496$	−380.050	−0.0344047
$497$	−2149.60	−0.194010
$498$	−1975.45	−0.177755
$499$	5806.88	0.520945	0.260473	−	0.965481i	$-0.416122\pi$
$499$	5806.88	0.520945	0.260473	+	0.965481i	$0.416122\pi$
$500$	629.330	0.0562890
$501$	3958.93	0.353038
$502$	−10713.0	−0.952481
$503$	−17785.0	−1.57653	−0.788264	−	0.615338i	$-0.789020\pi$
$503$	−17785.0	−1.57653	−0.788264	+	0.615338i	$0.789020\pi$
$504$	−2033.10	−0.179686
$505$	2928.96	0.258093
$506$	3927.67	0.345072
$507$	1112.96	0.0974916
$508$	−6643.65	−0.580245
$509$	−10937.5	−0.952447	−0.476224	−	0.879324i	$-0.657995\pi$
$509$	−10937.5	−0.952447	−0.476224	+	0.879324i	$0.657995\pi$
$510$	−1047.26	−0.0909281
$511$	1799.26	0.155762
$512$	−588.062	−0.0507596
$513$	1272.22	0.109493
$514$	−1426.70	−0.122430
$515$	2853.74	0.244176
$516$	591.926	0.0505002
$517$	6982.64	0.593996
$518$	492.951	0.0418128
$519$	138.886	0.0117465
$520$	−4087.71	−0.344726
$521$	−3619.48	−0.304361	−0.152181	−	0.988353i	$-0.548630\pi$
$521$	−3619.48	−0.304361	−0.152181	+	0.988353i	$0.548630\pi$
$522$	−8130.79	−0.681753
$523$	8706.94	0.727969	0.363985	−	0.931405i	$-0.381416\pi$
$523$	8706.94	0.727969	0.363985	+	0.931405i	$0.381416\pi$
$524$	3024.93	0.252185
$525$	−114.155	−0.00948976
$526$	−1214.89	−0.100707
$527$	−22249.7	−1.83911
$528$	−22.8536	−0.00188366
$529$	30827.0	2.53366
$530$	419.109	0.0343490
$531$	−8836.93	−0.722204
$532$	−341.594	−0.0278383
$533$	−1656.12	−0.134587
$534$	1115.29	0.0903804
$535$	−8601.29	−0.695077
$536$	3068.95	0.247311
$537$	−5467.43	−0.439361
$538$	−9812.58	−0.786339
$539$	−3632.73	−0.290302
$540$	1685.57	0.134325
$541$	15297.7	1.21571	0.607853	−	0.794049i	$-0.292031\pi$
$541$	15297.7	1.21571	0.607853	+	0.794049i	$0.292031\pi$
$542$	9824.94	0.778630
$543$	−3044.75	−0.240631
$544$	−17346.8	−1.36717
$545$	3944.04	0.309989
$546$	286.395	0.0224479
$547$	−11225.4	−0.877450	−0.438725	−	0.898621i	$-0.644570\pi$
$547$	−11225.4	−0.877450	−0.438725	+	0.898621i	$0.644570\pi$
$548$	−3545.79	−0.276402
$549$	15571.7	1.21054
$550$	−473.556	−0.0367136
$551$	−3536.83	−0.273456
$552$	5951.25	0.458881
$553$	−1051.70	−0.0808732
$554$	5256.91	0.403150
$555$	−512.524	−0.0391990
$556$	−9968.53	−0.760360
$557$	16733.7	1.27294	0.636472	−	0.771300i	$-0.280393\pi$
$557$	16733.7	1.27294	0.636472	+	0.771300i	$0.280393\pi$
$558$	−10216.9	−0.775115
$559$	3348.91	0.253388
$560$	−29.0103	−0.00218912
$561$	−1337.94	−0.100692
$562$	−11593.5	−0.870180
$563$	14407.5	1.07851	0.539256	−	0.842142i	$-0.318705\pi$
$563$	14407.5	1.07851	0.539256	+	0.842142i	$0.318705\pi$
$564$	4086.60	0.305101
$565$	5430.74	0.404377
$566$	15468.9	1.14878
$567$	2139.85	0.158493
$568$	−13511.6	−0.998127
$569$	14089.3	1.03806	0.519030	−	0.854756i	$-0.326293\pi$
$569$	14089.3	1.03806	0.519030	+	0.854756i	$0.326293\pi$
$570$	−209.184	−0.0153715
$571$	11704.0	0.857790	0.428895	−	0.903354i	$-0.358903\pi$
$571$	11704.0	0.857790	0.428895	+	0.903354i	$0.358903\pi$
$572$	−2017.13	−0.147448
$573$	1984.17	0.144659
$574$	279.606	0.0203319
$575$	−5183.75	−0.375960
$576$	−7635.83	−0.552360
$577$	6589.19	0.475410	0.237705	−	0.971337i	$-0.423605\pi$
$577$	6589.19	0.475410	0.237705	+	0.971337i	$0.423605\pi$
$578$	−7120.70	−0.512425
$579$	−3905.70	−0.280337
$580$	−4685.97	−0.335473
$581$	−3203.68	−0.228763
$582$	4103.13	0.292234
$583$	535.441	0.0380372
$584$	11309.5	0.801353
$585$	4619.30	0.326469
$586$	−4064.26	−0.286507
$587$	10644.9	0.748485	0.374243	−	0.927331i	$-0.377903\pi$
$587$	10644.9	0.748485	0.374243	+	0.927331i	$0.377903\pi$
$588$	−2126.06	−0.149111
$589$	−4444.25	−0.310904
$590$	2999.69	0.209314
$591$	4154.71	0.289174
$592$	−130.248	−0.00904253
$593$	−27400.5	−1.89748	−0.948738	−	0.316063i	$-0.897639\pi$
$593$	−27400.5	−1.89748	−0.948738	+	0.316063i	$0.897639\pi$
$594$	−1268.35	−0.0876110
$595$	−1698.38	−0.117020
$596$	−13181.9	−0.905956
$597$	−4544.74	−0.311564
$598$	13005.1	0.889329
$599$	−17631.9	−1.20271	−0.601353	−	0.798984i	$-0.705371\pi$
$599$	−17631.9	−1.20271	−0.601353	+	0.798984i	$0.705371\pi$
$600$	−717.537	−0.0488222
$601$	−9665.65	−0.656023	−0.328012	−	0.944674i	$-0.606379\pi$
$601$	−9665.65	−0.656023	−0.328012	+	0.944674i	$0.606379\pi$
$602$	−565.403	−0.0382792
$603$	−3468.06	−0.234213
$604$	−3734.10	−0.251553
$605$	−605.000	−0.0406558
$606$	−1289.88	−0.0864650
$607$	2865.46	0.191607	0.0958034	−	0.995400i	$-0.469458\pi$
$607$	2865.46	0.191607	0.0958034	+	0.995400i	$0.469458\pi$
$608$	−3464.94	−0.231122
$609$	849.992	0.0565573
$610$	−5285.80	−0.350846
$611$	23120.6	1.53087
$612$	12147.3	0.802331
$613$	−417.176	−0.0274871	−0.0137435	−	0.999906i	$-0.504375\pi$
$613$	−417.176	−0.0274871	−0.0137435	+	0.999906i	$0.504375\pi$
$614$	13614.8	0.894869
$615$	−290.708	−0.0190610
$616$	881.694	0.0576696
$617$	19496.4	1.27211	0.636057	−	0.771642i	$-0.280564\pi$
$617$	19496.4	1.27211	0.636057	+	0.771642i	$0.280564\pi$
$618$	−1256.75	−0.0818025
$619$	−11978.6	−0.777805	−0.388903	−	0.921279i	$-0.627146\pi$
$619$	−11978.6	−0.777805	−0.388903	+	0.921279i	$0.627146\pi$
$620$	−5888.22	−0.381414
$621$	−13883.9	−0.897168
$622$	11197.1	0.721804
$623$	1808.71	0.116315
$624$	−75.6717	−0.00485464
$625$	625.000	0.0400000
$626$	−15693.8	−1.00200
$627$	−267.247	−0.0170220
$628$	4987.27	0.316901
$629$	−7625.28	−0.483370
$630$	−779.884	−0.0493195
$631$	−16619.0	−1.04848	−0.524241	−	0.851570i	$-0.675651\pi$
$631$	−16619.0	−1.04848	−0.524241	+	0.851570i	$0.675651\pi$
$632$	−6610.62	−0.416070
$633$	−2529.83	−0.158850
$634$	−2166.27	−0.135700
$635$	−6597.94	−0.412332
$636$	313.368	0.0195375
$637$	−12028.5	−0.748175
$638$	3526.08	0.218807
$639$	15268.8	0.945264
$640$	−4702.64	−0.290450
$641$	−5978.94	−0.368415	−0.184207	−	0.982887i	$-0.558972\pi$
$641$	−5978.94	−0.368415	−0.184207	+	0.982887i	$0.558972\pi$
$642$	3787.90	0.232861
$643$	3759.36	0.230567	0.115284	−	0.993333i	$-0.463222\pi$
$643$	3759.36	0.230567	0.115284	+	0.993333i	$0.463222\pi$
$644$	3727.87	0.228103
$645$	587.853	0.0358863
$646$	−3112.22	−0.189549
$647$	23493.9	1.42758	0.713788	−	0.700361i	$-0.246978\pi$
$647$	23493.9	1.42758	0.713788	+	0.700361i	$0.246978\pi$
$648$	13450.4	0.815401
$649$	3832.31	0.231789
$650$	−1568.02	−0.0946195
$651$	1068.07	0.0643025
$652$	−9254.71	−0.555894
$653$	−9101.87	−0.545457	−0.272729	−	0.962091i	$-0.587926\pi$
$653$	−9101.87	−0.545457	−0.272729	+	0.962091i	$0.587926\pi$
$654$	−1736.91	−0.103851
$655$	3004.12	0.179207
$656$	−73.8781	−0.00439703
$657$	−12780.2	−0.758912
$658$	−3903.49	−0.231267
$659$	7835.90	0.463192	0.231596	−	0.972812i	$-0.425605\pi$
$659$	7835.90	0.463192	0.231596	+	0.972812i	$0.425605\pi$
$660$	−354.077	−0.0208825
$661$	28376.8	1.66979	0.834895	−	0.550409i	$-0.185528\pi$
$661$	28376.8	1.66979	0.834895	+	0.550409i	$0.185528\pi$
$662$	−9489.12	−0.557107
$663$	−4430.13	−0.259506
$664$	−20137.2	−1.17692
$665$	−339.243	−0.0197824
$666$	−3501.46	−0.203722
$667$	38598.0	2.24066
$668$	15587.6	0.902849
$669$	854.475	0.0493810
$670$	1177.23	0.0678811
$671$	−6752.98	−0.388518
$672$	832.715	0.0478016
$673$	2341.70	0.134125	0.0670623	−	0.997749i	$-0.478637\pi$
$673$	2341.70	0.134125	0.0670623	+	0.997749i	$0.478637\pi$
$674$	−13606.3	−0.777588
$675$	1673.97	0.0954535
$676$	4382.09	0.249322
$677$	1712.17	0.0971993	0.0485996	−	0.998818i	$-0.484524\pi$
$677$	1712.17	0.0971993	0.0485996	+	0.998818i	$0.484524\pi$
$678$	−2391.63	−0.135472
$679$	6654.23	0.376091
$680$	−10675.4	−0.602036
$681$	−2272.44	−0.127871
$682$	4430.74	0.248771
$683$	6397.02	0.358382	0.179191	−	0.983814i	$-0.442652\pi$
$683$	6397.02	0.358382	0.179191	+	0.983814i	$0.442652\pi$
$684$	2426.37	0.135635
$685$	−3521.39	−0.196417
$686$	4140.01	0.230417
$687$	2248.18	0.124852
$688$	149.392	0.00827836
$689$	1772.93	0.0980307
$690$	2282.86	0.125952
$691$	−6444.46	−0.354788	−0.177394	−	0.984140i	$-0.556767\pi$
$691$	−6444.46	−0.354788	−0.177394	+	0.984140i	$0.556767\pi$
$692$	546.842	0.0300402
$693$	−996.355	−0.0546153
$694$	−13949.0	−0.762962
$695$	−9899.94	−0.540325
$696$	5342.75	0.290972
$697$	−4325.13	−0.235044
$698$	−5705.75	−0.309406
$699$	−4446.09	−0.240581
$700$	−449.465	−0.0242689
$701$	22306.4	1.20186	0.600928	−	0.799303i	$-0.294798\pi$
$701$	22306.4	1.20186	0.600928	+	0.799303i	$0.294798\pi$
$702$	−4199.70	−0.225794
$703$	−1523.11	−0.0817143
$704$	3311.43	0.177279
$705$	4058.48	0.216810
$706$	−5207.57	−0.277605
$707$	−2091.86	−0.111276
$708$	2242.86	0.119056
$709$	−1216.49	−0.0644375	−0.0322188	−	0.999481i	$-0.510257\pi$
$709$	−1216.49	−0.0644375	−0.0322188	+	0.999481i	$0.510257\pi$
$710$	−5182.98	−0.273963
$711$	7470.31	0.394034
$712$	11368.9	0.598409
$713$	48500.8	2.54750
$714$	747.947	0.0392034
$715$	−2003.25	−0.104779
$716$	−21527.1	−1.12361
$717$	806.457	0.0420051
$718$	−16778.4	−0.872097
$719$	23263.2	1.20664	0.603319	−	0.797500i	$-0.293845\pi$
$719$	23263.2	1.20664	0.603319	+	0.797500i	$0.293845\pi$
$720$	206.062	0.0106660
$721$	−2038.13	−0.105276
$722$	−621.650	−0.0320435
$723$	−4716.42	−0.242608
$724$	−11988.2	−0.615383
$725$	−4653.72	−0.238393
$726$	266.435	0.0136203
$727$	−19442.0	−0.991834	−0.495917	−	0.868370i	$-0.665168\pi$
$727$	−19442.0	−0.991834	−0.495917	+	0.868370i	$0.665168\pi$
$728$	2919.42	0.148628
$729$	−12887.8	−0.654768
$730$	4338.24	0.219953
$731$	8746.01	0.442521
$732$	−3952.19	−0.199559
$733$	−27719.3	−1.39678	−0.698388	−	0.715719i	$-0.746099\pi$
$733$	−27719.3	−1.39678	−0.698388	+	0.715719i	$0.746099\pi$
$734$	−22030.6	−1.10785
$735$	−2111.43	−0.105961
$736$	37813.4	1.89378
$737$	1503.99	0.0751699
$738$	−1986.06	−0.0990623
$739$	16217.0	0.807244	0.403622	−	0.914926i	$-0.367751\pi$
$739$	16217.0	0.807244	0.403622	+	0.914926i	$0.367751\pi$
$740$	−2017.98	−0.100246
$741$	−884.896	−0.0438697
$742$	−299.326	−0.0148095
$743$	14891.0	0.735258	0.367629	−	0.929973i	$-0.380170\pi$
$743$	14891.0	0.735258	0.367629	+	0.929973i	$0.380170\pi$
$744$	6713.51	0.330819
$745$	−13091.1	−0.643789
$746$	15062.9	0.739263
$747$	22756.0	1.11459
$748$	−5267.93	−0.257506
$749$	6143.01	0.299681
$750$	−275.242	−0.0134006
$751$	6649.49	0.323094	0.161547	−	0.986865i	$-0.448352\pi$
$751$	6649.49	0.323094	0.161547	+	0.986865i	$0.448352\pi$
$752$	1031.39	0.0500144
$753$	−7955.00	−0.384989
$754$	11675.4	0.563915
$755$	−3708.40	−0.178758
$756$	−1203.83	−0.0579136
$757$	−40543.3	−1.94659	−0.973297	−	0.229548i	$-0.926275\pi$
$757$	−40543.3	−1.94659	−0.973297	+	0.229548i	$0.926275\pi$
$758$	20134.8	0.964814
$759$	2916.51	0.139476
$760$	−2132.36	−0.101775
$761$	32158.9	1.53188	0.765940	−	0.642912i	$-0.222274\pi$
$761$	32158.9	1.53188	0.765940	+	0.642912i	$0.222274\pi$
$762$	2905.65	0.138137
$763$	−2816.82	−0.133651
$764$	7812.32	0.369947
$765$	12063.7	0.570151
$766$	9738.95	0.459376
$767$	12689.4	0.597374
$768$	5150.48	0.241995
$769$	−2793.35	−0.130989	−0.0654946	−	0.997853i	$-0.520863\pi$
$769$	−2793.35	−0.130989	−0.0654946	+	0.997853i	$0.520863\pi$
$770$	338.212	0.0158290
$771$	−1059.40	−0.0494857
$772$	−15378.0	−0.716926
$773$	5078.69	0.236310	0.118155	−	0.992995i	$-0.462302\pi$
$773$	5078.69	0.236310	0.118155	+	0.992995i	$0.462302\pi$
$774$	4016.10	0.186506
$775$	−5847.70	−0.271040
$776$	41826.1	1.93489
$777$	366.043	0.0169005
$778$	8399.73	0.387076
$779$	−863.921	−0.0397345
$780$	−1172.40	−0.0538190
$781$	−6621.61	−0.303380
$782$	33964.1	1.55314
$783$	−12464.3	−0.568886
$784$	−536.581	−0.0244434
$785$	4952.96	0.225196
$786$	−1322.98	−0.0600369
$787$	24726.8	1.11997	0.559985	−	0.828503i	$-0.310807\pi$
$787$	24726.8	1.11997	0.559985	+	0.828503i	$0.310807\pi$
$788$	16358.5	0.739527
$789$	−902.121	−0.0407051
$790$	−2535.79	−0.114202
$791$	−3878.61	−0.174346
$792$	−6262.74	−0.280981
$793$	−22360.1	−1.00130
$794$	7849.65	0.350848
$795$	311.211	0.0138837
$796$	−17894.2	−0.796786
$797$	−9905.25	−0.440228	−0.220114	−	0.975474i	$-0.570643\pi$
$797$	−9905.25	−0.440228	−0.220114	+	0.975474i	$0.570643\pi$
$798$	149.399	0.00662738
$799$	60381.6	2.67353
$800$	−4559.13	−0.201487
$801$	−12847.4	−0.566716
$802$	−7311.30	−0.321909
$803$	5542.40	0.243571
$804$	880.213	0.0386103
$805$	3702.21	0.162094
$806$	14670.9	0.641141
$807$	−7286.37	−0.317835
$808$	−13148.7	−0.572486
$809$	8836.19	0.384010	0.192005	−	0.981394i	$-0.438501\pi$
$809$	8836.19	0.384010	0.192005	+	0.981394i	$0.438501\pi$
$810$	5159.46	0.223809
$811$	−35059.9	−1.51803	−0.759014	−	0.651075i	$-0.774318\pi$
$811$	−35059.9	−1.51803	−0.759014	+	0.651075i	$0.774318\pi$
$812$	3346.70	0.144638
$813$	7295.55	0.314718
$814$	1518.48	0.0653841
$815$	−9191.03	−0.395028
$816$	−197.624	−0.00847822
$817$	1746.97	0.0748088
$818$	9994.40	0.427195
$819$	−3299.08	−0.140756
$820$	−1144.61	−0.0487459
$821$	2374.14	0.100923	0.0504617	−	0.998726i	$-0.483931\pi$
$821$	2374.14	0.100923	0.0504617	+	0.998726i	$0.483931\pi$
$822$	1550.78	0.0658024
$823$	33056.6	1.40010	0.700048	−	0.714096i	$-0.253162\pi$
$823$	33056.6	1.40010	0.700048	+	0.714096i	$0.253162\pi$
$824$	−12811.0	−0.541615
$825$	−351.641	−0.0148395
$826$	−2142.37	−0.0902451
$827$	39553.1	1.66311	0.831557	−	0.555440i	$-0.187450\pi$
$827$	39553.1	1.66311	0.831557	+	0.555440i	$0.187450\pi$
$828$	−26479.3	−1.11138
$829$	33133.5	1.38815	0.694073	−	0.719905i	$-0.255815\pi$
$829$	33133.5	1.38815	0.694073	+	0.719905i	$0.255815\pi$
$830$	−7724.50	−0.323038
$831$	3903.54	0.162951
$832$	10964.6	0.456888
$833$	−31413.7	−1.30663
$834$	4359.81	0.181017
$835$	15480.4	0.641581
$836$	−1052.24	−0.0435317
$837$	−15662.2	−0.646792
$838$	6850.30	0.282386
$839$	−6572.41	−0.270447	−0.135223	−	0.990815i	$-0.543175\pi$
$839$	−6572.41	−0.270447	−0.135223	+	0.990815i	$0.543175\pi$
$840$	512.462	0.0210496
$841$	10262.4	0.420781
$842$	13526.7	0.553635
$843$	−8608.78	−0.351723
$844$	−9960.79	−0.406238
$845$	4351.94	0.177173
$846$	27726.8	1.12679
$847$	432.089	0.0175286
$848$	79.0886	0.00320273
$849$	11486.5	0.464330
$850$	−4095.03	−0.165245
$851$	16621.9	0.669556
$852$	−3875.31	−0.155828
$853$	−35701.1	−1.43304	−0.716519	−	0.697567i	$-0.754266\pi$
$853$	−35701.1	−1.43304	−0.716519	+	0.697567i	$0.754266\pi$
$854$	3775.10	0.151266
$855$	2409.67	0.0963847
$856$	38612.8	1.54177
$857$	19225.7	0.766321	0.383160	−	0.923682i	$-0.374836\pi$
$857$	19225.7	0.766321	0.383160	+	0.923682i	$0.374836\pi$
$858$	882.206	0.0351026
$859$	−16641.3	−0.660993	−0.330496	−	0.943807i	$-0.607216\pi$
$859$	−16641.3	−0.660993	−0.330496	+	0.943807i	$0.607216\pi$
$860$	2314.57	0.0917747
$861$	207.623	0.00821807
$862$	−25242.3	−0.997396
$863$	4047.31	0.159643	0.0798216	−	0.996809i	$-0.474565\pi$
$863$	4047.31	0.159643	0.0798216	+	0.996809i	$0.474565\pi$
$864$	−12211.0	−0.480816
$865$	543.079	0.0213471
$866$	571.641	0.0224309
$867$	−5287.50	−0.207120
$868$	4205.34	0.164445
$869$	−3239.65	−0.126464
$870$	2049.44	0.0798651
$871$	4979.94	0.193730
$872$	−17705.5	−0.687597
$873$	−47265.5	−1.83241
$874$	6784.16	0.262560
$875$	−446.373	−0.0172459
$876$	3243.70	0.125108
$877$	−8760.86	−0.337324	−0.168662	−	0.985674i	$-0.553945\pi$
$877$	−8760.86	−0.337324	−0.168662	+	0.985674i	$0.553945\pi$
$878$	23011.1	0.884497
$879$	−3017.93	−0.115805
$880$	−89.3630	−0.00342321
$881$	−1754.87	−0.0671090	−0.0335545	−	0.999437i	$-0.510683\pi$
$881$	−1754.87	−0.0671090	−0.0335545	+	0.999437i	$0.510683\pi$
$882$	−14424.9	−0.550693
$883$	−12354.4	−0.470850	−0.235425	−	0.971893i	$-0.575648\pi$
$883$	−12354.4	−0.470850	−0.235425	+	0.971893i	$0.575648\pi$
$884$	−17442.9	−0.663652
$885$	2227.43	0.0846037
$886$	10202.3	0.386856
$887$	25801.2	0.976684	0.488342	−	0.872652i	$-0.337602\pi$
$887$	25801.2	0.976684	0.488342	+	0.872652i	$0.337602\pi$
$888$	2300.81	0.0869485
$889$	4712.22	0.177776
$890$	4361.03	0.164250
$891$	6591.57	0.247841
$892$	3364.35	0.126286
$893$	12060.9	0.451963
$894$	5765.18	0.215678
$895$	−21379.0	−0.798458
$896$	3358.61	0.125227
$897$	9657.01	0.359463
$898$	−10793.0	−0.401076
$899$	43541.7	1.61535
$900$	3192.59	0.118244
$901$	4630.17	0.171202
$902$	861.295	0.0317938
$903$	−419.842	−0.0154723
$904$	−24379.6	−0.896962
$905$	−11905.7	−0.437303
$906$	1633.14	0.0598866
$907$	−36427.6	−1.33358	−0.666791	−	0.745245i	$-0.732332\pi$
$907$	−36427.6	−1.33358	−0.666791	+	0.745245i	$0.732332\pi$
$908$	−8947.35	−0.327013
$909$	14858.6	0.542166
$910$	1119.87	0.0407949
$911$	52755.8	1.91864	0.959318	−	0.282327i	$-0.0911063\pi$
$911$	52755.8	1.91864	0.959318	+	0.282327i	$0.0911063\pi$
$912$	−39.4744	−0.00143325
$913$	−9868.58	−0.357724
$914$	24117.2	0.872785
$915$	−3925.00	−0.141810
$916$	8851.82	0.319293
$917$	−2145.53	−0.0772646
$918$	−10967.9	−0.394330
$919$	17657.5	0.633807	0.316904	−	0.948458i	$-0.397357\pi$
$919$	17657.5	0.633807	0.316904	+	0.948458i	$0.397357\pi$
$920$	23270.8	0.833930
$921$	10109.7	0.361702
$922$	−4841.62	−0.172939
$923$	−21925.2	−0.781880
$924$	252.881	0.00900342
$925$	−2004.09	−0.0712369
$926$	−27697.3	−0.982925
$927$	14477.0	0.512930
$928$	33947.1	1.20083
$929$	14341.8	0.506501	0.253251	−	0.967401i	$-0.418500\pi$
$929$	14341.8	0.506501	0.253251	+	0.967401i	$0.418500\pi$
$930$	2575.26	0.0908021
$931$	−6274.71	−0.220887
$932$	−17505.7	−0.615256
$933$	8314.44	0.291750
$934$	−25724.5	−0.901212
$935$	−5231.68	−0.182988
$936$	−20736.9	−0.724152
$937$	−1567.58	−0.0546538	−0.0273269	−	0.999627i	$-0.508700\pi$
$937$	−1567.58	−0.0546538	−0.0273269	+	0.999627i	$0.508700\pi$
$938$	−840.772	−0.0292667
$939$	−11653.5	−0.405003
$940$	15979.6	0.554464
$941$	25916.1	0.897813	0.448907	−	0.893579i	$-0.351814\pi$
$941$	25916.1	0.897813	0.448907	+	0.893579i	$0.351814\pi$
$942$	−2181.22	−0.0754438
$943$	9428.11	0.325579
$944$	566.060	0.0195166
$945$	−1195.54	−0.0411545
$946$	−1741.66	−0.0598586
$947$	8936.21	0.306640	0.153320	−	0.988177i	$-0.451004\pi$
$947$	8936.21	0.306640	0.153320	+	0.988177i	$0.451004\pi$
$948$	−1896.01	−0.0649573
$949$	18351.7	0.627738
$950$	−817.960	−0.0279349
$951$	−1608.57	−0.0548492
$952$	7624.36	0.259566
$953$	−46561.7	−1.58267	−0.791334	−	0.611384i	$-0.790613\pi$
$953$	−46561.7	−1.58267	−0.791334	+	0.611384i	$0.790613\pi$
$954$	2126.14	0.0721554
$955$	7758.56	0.262891
$956$	3175.29	0.107423
$957$	2618.30	0.0884407
$958$	−13445.0	−0.453432
$959$	2514.96	0.0846844
$960$	1924.68	0.0647071
$961$	24922.0	0.836562
$962$	5027.91	0.168510
$963$	−43634.3	−1.46012
$964$	−18570.1	−0.620439
$965$	−15272.2	−0.509461
$966$	−1630.41	−0.0543039
$967$	32630.8	1.08514	0.542572	−	0.840009i	$-0.317450\pi$
$967$	32630.8	1.08514	0.542572	+	0.840009i	$0.317450\pi$
$968$	2715.96	0.0901799
$969$	−2310.99	−0.0766148
$970$	16044.2	0.531081
$971$	31791.0	1.05069	0.525347	−	0.850888i	$-0.323936\pi$
$971$	31791.0	1.05069	0.525347	+	0.850888i	$0.323936\pi$
$972$	12959.8	0.427660
$973$	7070.50	0.232960
$974$	−14792.1	−0.486622
$975$	−1164.34	−0.0382447
$976$	−997.465	−0.0327132
$977$	−31448.7	−1.02982	−0.514910	−	0.857245i	$-0.672174\pi$
$977$	−31448.7	−1.02982	−0.514910	+	0.857245i	$0.672174\pi$
$978$	4047.62	0.132340
$979$	5571.52	0.181886
$980$	−8313.41	−0.270982
$981$	20008.1	0.651181
$982$	−2627.12	−0.0853714
$983$	44211.1	1.43450	0.717252	−	0.696814i	$-0.245400\pi$
$983$	44211.1	1.43450	0.717252	+	0.696814i	$0.245400\pi$
$984$	1305.04	0.0422797
$985$	16245.9	0.525521
$986$	30491.4	0.984831
$987$	−2898.55	−0.0934771
$988$	−3484.13	−0.112191
$989$	−19065.0	−0.612973
$990$	−2402.34	−0.0771227
$991$	−54591.7	−1.74991	−0.874956	−	0.484202i	$-0.839110\pi$
$991$	−54591.7	−1.74991	−0.874956	+	0.484202i	$0.839110\pi$
$992$	42656.7	1.36527
$993$	−7046.19	−0.225180
$994$	3701.66	0.118118
$995$	−17771.0	−0.566210
$996$	−5775.60	−0.183742
$997$	7630.69	0.242393	0.121197	−	0.992629i	$-0.461327\pi$
$997$	7630.69	0.242393	0.121197	+	0.992629i	$0.461327\pi$
$998$	−9999.57	−0.317165
$999$	−5367.66	−0.169995

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.i.1.10	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.i.1.10	✓	25	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)

\(f(q)\)	\(=\)	\(q-1.72202 q^{2} -1.27869 q^{3} -5.03464 q^{4} -5.00000 q^{5} +2.20194 q^{6} +3.57098 q^{7} +22.4459 q^{8} -25.3649 q^{9} +O(q^{10})\)	\(q-1.72202 q^{2} -1.27869 q^{3} -5.03464 q^{4} -5.00000 q^{5} +2.20194 q^{6} +3.57098 q^{7} +22.4459 q^{8} -25.3649 q^{9} +8.61011 q^{10} +11.0000 q^{11} +6.43777 q^{12} +36.4227 q^{13} -6.14930 q^{14} +6.39347 q^{15} +1.62478 q^{16} +95.1214 q^{17} +43.6790 q^{18} +19.0000 q^{19} +25.1732 q^{20} -4.56619 q^{21} -18.9422 q^{22} -207.350 q^{23} -28.7015 q^{24} +25.0000 q^{25} -62.7206 q^{26} +66.9587 q^{27} -17.9786 q^{28} -186.149 q^{29} -11.0097 q^{30} -233.908 q^{31} -182.365 q^{32} -14.0656 q^{33} -163.801 q^{34} -17.8549 q^{35} +127.703 q^{36} -80.1636 q^{37} -32.7184 q^{38} -46.5735 q^{39} -112.230 q^{40} -45.4695 q^{41} +7.86308 q^{42} +91.9458 q^{43} -55.3811 q^{44} +126.825 q^{45} +357.061 q^{46} +634.785 q^{47} -2.07760 q^{48} -330.248 q^{49} -43.0505 q^{50} -121.631 q^{51} -183.375 q^{52} +48.6765 q^{53} -115.304 q^{54} -55.0000 q^{55} +80.1540 q^{56} -24.2952 q^{57} +320.552 q^{58} +348.391 q^{59} -32.1888 q^{60} -613.907 q^{61} +402.795 q^{62} -90.5777 q^{63} +301.039 q^{64} -182.113 q^{65} +24.2213 q^{66} +136.726 q^{67} -478.902 q^{68} +265.137 q^{69} +30.7465 q^{70} -601.964 q^{71} -569.340 q^{72} +503.855 q^{73} +138.043 q^{74} -31.9674 q^{75} -95.6582 q^{76} +39.2808 q^{77} +80.2005 q^{78} -294.513 q^{79} -8.12391 q^{80} +599.234 q^{81} +78.2995 q^{82} -897.143 q^{83} +22.9892 q^{84} -475.607 q^{85} -158.333 q^{86} +238.028 q^{87} +246.905 q^{88} +506.502 q^{89} -218.395 q^{90} +130.065 q^{91} +1043.93 q^{92} +299.097 q^{93} -1093.11 q^{94} -95.0000 q^{95} +233.190 q^{96} +1863.42 q^{97} +568.694 q^{98} -279.014 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 2 q^{2} + 9 q^{3} + 122 q^{4} - 125 q^{5} + 11 q^{6} - 15 q^{7} + 60 q^{8} + 300 q^{9}+O(q^{10}) \) 25 * q + 2 * q^2 + 9 * q^3 + 122 * q^4 - 125 * q^5 + 11 * q^6 - 15 * q^7 + 60 * q^8 + 300 * q^9	\( 25 q + 2 q^{2} + 9 q^{3} + 122 q^{4} - 125 q^{5} + 11 q^{6} - 15 q^{7} + 60 q^{8} + 300 q^{9} - 10 q^{10} + 275 q^{11} + 44 q^{12} + 53 q^{13} - 51 q^{14} - 45 q^{15} + 438 q^{16} + 153 q^{17} + 9 q^{18} + 475 q^{19} - 610 q^{20} + 259 q^{21} + 22 q^{22} - 7 q^{23} + 186 q^{24} + 625 q^{25} + 543 q^{26} + 495 q^{27} - 525 q^{28} + 169 q^{29} - 55 q^{30} + 102 q^{31} + 327 q^{32} + 99 q^{33} - 879 q^{34} + 75 q^{35} + 2293 q^{36} - 46 q^{37} + 38 q^{38} + 233 q^{39} - 300 q^{40} + 1190 q^{41} - 684 q^{42} - 408 q^{43} + 1342 q^{44} - 1500 q^{45} + 757 q^{46} + 1068 q^{47} + 715 q^{48} + 1930 q^{49} + 50 q^{50} + 1655 q^{51} - 94 q^{52} + 143 q^{53} + 1970 q^{54} - 1375 q^{55} - 1397 q^{56} + 171 q^{57} + 1366 q^{58} + 2945 q^{59} - 220 q^{60} + 1160 q^{61} + 194 q^{62} + 1804 q^{63} + 3000 q^{64} - 265 q^{65} + 121 q^{66} - 353 q^{67} + 5452 q^{68} + 3289 q^{69} + 255 q^{70} + 230 q^{71} + 196 q^{72} + 1357 q^{73} + 4379 q^{74} + 225 q^{75} + 2318 q^{76} - 165 q^{77} + 2008 q^{78} + 1266 q^{79} - 2190 q^{80} + 1709 q^{81} + 1010 q^{82} + 3856 q^{83} + 9354 q^{84} - 765 q^{85} + 6746 q^{86} + 3113 q^{87} + 660 q^{88} + 3562 q^{89} - 45 q^{90} - 833 q^{91} + 4276 q^{92} + 1312 q^{93} + 5124 q^{94} - 2375 q^{95} + 3828 q^{96} - 914 q^{97} + 2478 q^{98} + 3300 q^{99}+O(q^{100}) \) 25 * q + 2 * q^2 + 9 * q^3 + 122 * q^4 - 125 * q^5 + 11 * q^6 - 15 * q^7 + 60 * q^8 + 300 * q^9 - 10 * q^10 + 275 * q^11 + 44 * q^12 + 53 * q^13 - 51 * q^14 - 45 * q^15 + 438 * q^16 + 153 * q^17 + 9 * q^18 + 475 * q^19 - 610 * q^20 + 259 * q^21 + 22 * q^22 - 7 * q^23 + 186 * q^24 + 625 * q^25 + 543 * q^26 + 495 * q^27 - 525 * q^28 + 169 * q^29 - 55 * q^30 + 102 * q^31 + 327 * q^32 + 99 * q^33 - 879 * q^34 + 75 * q^35 + 2293 * q^36 - 46 * q^37 + 38 * q^38 + 233 * q^39 - 300 * q^40 + 1190 * q^41 - 684 * q^42 - 408 * q^43 + 1342 * q^44 - 1500 * q^45 + 757 * q^46 + 1068 * q^47 + 715 * q^48 + 1930 * q^49 + 50 * q^50 + 1655 * q^51 - 94 * q^52 + 143 * q^53 + 1970 * q^54 - 1375 * q^55 - 1397 * q^56 + 171 * q^57 + 1366 * q^58 + 2945 * q^59 - 220 * q^60 + 1160 * q^61 + 194 * q^62 + 1804 * q^63 + 3000 * q^64 - 265 * q^65 + 121 * q^66 - 353 * q^67 + 5452 * q^68 + 3289 * q^69 + 255 * q^70 + 230 * q^71 + 196 * q^72 + 1357 * q^73 + 4379 * q^74 + 225 * q^75 + 2318 * q^76 - 165 * q^77 + 2008 * q^78 + 1266 * q^79 - 2190 * q^80 + 1709 * q^81 + 1010 * q^82 + 3856 * q^83 + 9354 * q^84 - 765 * q^85 + 6746 * q^86 + 3113 * q^87 + 660 * q^88 + 3562 * q^89 - 45 * q^90 - 833 * q^91 + 4276 * q^92 + 1312 * q^93 + 5124 * q^94 - 2375 * q^95 + 3828 * q^96 - 914 * q^97 + 2478 * q^98 + 3300 * q^99

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