LMFDB - Embedded newform 1045.4.a.i.1.18

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			1.18
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.11750 q^{2} +0.0197889 q^{3} +1.71881 q^{4} -5.00000 q^{5} +0.0616921 q^{6} -19.0525 q^{7} -19.5816 q^{8} -26.9996 q^{9} +O(q^{10})$	\(q+3.11750 q^{2} +0.0197889 q^{3} +1.71881 q^{4} -5.00000 q^{5} +0.0616921 q^{6} -19.0525 q^{7} -19.5816 q^{8} -26.9996 q^{9} -15.5875 q^{10} +11.0000 q^{11} +0.0340135 q^{12} -32.2831 q^{13} -59.3962 q^{14} -0.0989447 q^{15} -74.7962 q^{16} +51.1728 q^{17} -84.1713 q^{18} +19.0000 q^{19} -8.59407 q^{20} -0.377029 q^{21} +34.2925 q^{22} +106.507 q^{23} -0.387499 q^{24} +25.0000 q^{25} -100.643 q^{26} -1.06860 q^{27} -32.7477 q^{28} -56.5762 q^{29} -0.308460 q^{30} +281.187 q^{31} -76.5244 q^{32} +0.217678 q^{33} +159.531 q^{34} +95.2625 q^{35} -46.4073 q^{36} -97.5859 q^{37} +59.2325 q^{38} -0.638849 q^{39} +97.9080 q^{40} +227.781 q^{41} -1.17539 q^{42} +97.9642 q^{43} +18.9070 q^{44} +134.998 q^{45} +332.036 q^{46} +100.253 q^{47} -1.48014 q^{48} +19.9976 q^{49} +77.9375 q^{50} +1.01265 q^{51} -55.4887 q^{52} +660.636 q^{53} -3.33135 q^{54} -55.0000 q^{55} +373.078 q^{56} +0.375990 q^{57} -176.376 q^{58} -868.321 q^{59} -0.170068 q^{60} -0.00261696 q^{61} +876.601 q^{62} +514.410 q^{63} +359.805 q^{64} +161.416 q^{65} +0.678613 q^{66} -142.102 q^{67} +87.9565 q^{68} +2.10766 q^{69} +296.981 q^{70} -369.589 q^{71} +528.696 q^{72} -72.0949 q^{73} -304.224 q^{74} +0.494724 q^{75} +32.6575 q^{76} -209.577 q^{77} -1.99161 q^{78} +152.754 q^{79} +373.981 q^{80} +728.968 q^{81} +710.106 q^{82} +133.570 q^{83} -0.648043 q^{84} -255.864 q^{85} +305.404 q^{86} -1.11958 q^{87} -215.398 q^{88} +533.419 q^{89} +420.857 q^{90} +615.074 q^{91} +183.066 q^{92} +5.56439 q^{93} +312.539 q^{94} -95.0000 q^{95} -1.51434 q^{96} +92.7307 q^{97} +62.3427 q^{98} -296.996 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	3.11750	1.10220	0.551102	−	0.834438i	$-0.314208\pi$
$2$	3.11750	1.10220	0.551102	+	0.834438i	$0.314208\pi$
$3$	0.0197889	0.00380839	0.00190419	−	0.999998i	$-0.499394\pi$
$3$	0.0197889	0.00380839	0.00190419	+	0.999998i	$0.499394\pi$
$4$	1.71881	0.214852
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0.0616921	0.00419761
$7$	−19.0525	−1.02874	−0.514369	−	0.857569i	$-0.671974\pi$
$7$	−19.0525	−1.02874	−0.514369	+	0.857569i	$0.671974\pi$
$8$	−19.5816	−0.865393
$9$	−26.9996	−0.999985
$10$	−15.5875	−0.492920
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0.0340135	0.000818239 0
$13$	−32.2831	−0.688748	−0.344374	−	0.938833i	$-0.611909\pi$
$13$	−32.2831	−0.688748	−0.344374	+	0.938833i	$0.611909\pi$
$14$	−59.3962	−1.13388
$15$	−0.0989447	−0.00170316
$16$	−74.7962	−1.16869
$17$	51.1728	0.730071	0.365036	−	0.930994i	$-0.381057\pi$
$17$	51.1728	0.730071	0.365036	+	0.930994i	$0.381057\pi$
$18$	−84.1713	−1.10219
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−8.59407	−0.0960847
$21$	−0.377029	−0.00391783
$22$	34.2925	0.332327
$23$	106.507	0.965577	0.482788	−	0.875737i	$-0.339624\pi$
$23$	106.507	0.965577	0.482788	+	0.875737i	$0.339624\pi$
$24$	−0.387499	−0.00329575
$25$	25.0000	0.200000
$26$	−100.643	−0.759140
$27$	−1.06860	−0.00761671
$28$	−32.7477	−0.221026
$29$	−56.5762	−0.362274	−0.181137	−	0.983458i	$-0.557978\pi$
$29$	−56.5762	−0.362274	−0.181137	+	0.983458i	$0.557978\pi$
$30$	−0.308460	−0.00187723
$31$	281.187	1.62912	0.814559	−	0.580081i	$-0.196979\pi$
$31$	281.187	1.62912	0.814559	+	0.580081i	$0.196979\pi$
$32$	−76.5244	−0.422742
$33$	0.217678	0.00114827
$34$	159.531	0.804687
$35$	95.2625	0.460066
$36$	−46.4073	−0.214849
$37$	−97.5859	−0.433595	−0.216798	−	0.976217i	$-0.569561\pi$
$37$	−97.5859	−0.433595	−0.216798	+	0.976217i	$0.569561\pi$
$38$	59.2325	0.252863
$39$	−0.638849	−0.00262302
$40$	97.9080	0.387015
$41$	227.781	0.867642	0.433821	−	0.900999i	$-0.357165\pi$
$41$	227.781	0.867642	0.433821	+	0.900999i	$0.357165\pi$
$42$	−1.17539	−0.00431825
$43$	97.9642	0.347428	0.173714	−	0.984796i	$-0.444423\pi$
$43$	97.9642	0.347428	0.173714	+	0.984796i	$0.444423\pi$
$44$	18.9070	0.0647803
$45$	134.998	0.447207
$46$	332.036	1.06426
$47$	100.253	0.311137	0.155568	−	0.987825i	$-0.450279\pi$
$47$	100.253	0.311137	0.155568	+	0.987825i	$0.450279\pi$
$48$	−1.48014	−0.00445082
$49$	19.9976	0.0583021
$50$	77.9375	0.220441
$51$	1.01265	0.00278039
$52$	−55.4887	−0.147979
$53$	660.636	1.71218	0.856089	−	0.516829i	$-0.172888\pi$
$53$	660.636	1.71218	0.856089	+	0.516829i	$0.172888\pi$
$54$	−3.33135	−0.00839517
$55$	−55.0000	−0.134840
$56$	373.078	0.890263
$57$	0.375990	0.000873703 0
$58$	−176.376	−0.399299
$59$	−868.321	−1.91603	−0.958015	−	0.286718i	$-0.907436\pi$
$59$	−868.321	−1.91603	−0.958015	+	0.286718i	$0.907436\pi$
$60$	−0.170068	−0.000365927 0
$61$	−0.00261696	−5.49290e−6 0	−2.74645e−6	−	1.00000i	$-0.500001\pi$
$61$	−0.00261696	−5.49290e−6 0	−2.74645e−6		1.00000i	$0.500001\pi$
$62$	876.601	1.79562
$63$	514.410	1.02872
$64$	359.805	0.702743
$65$	161.416	0.308018
$66$	0.678613	0.00126563
$67$	−142.102	−0.259112	−0.129556	−	0.991572i	$-0.541355\pi$
$67$	−142.102	−0.259112	−0.129556	+	0.991572i	$0.541355\pi$
$68$	87.9565	0.156857
$69$	2.10766	0.00367729
$70$	296.981	0.507086
$71$	−369.589	−0.617776	−0.308888	−	0.951098i	$-0.599957\pi$
$71$	−369.589	−0.617776	−0.308888	+	0.951098i	$0.599957\pi$
$72$	528.696	0.865380
$73$	−72.0949	−0.115590	−0.0577950	−	0.998328i	$-0.518407\pi$
$73$	−72.0949	−0.115590	−0.0577950	+	0.998328i	$0.518407\pi$
$74$	−304.224	−0.477910
$75$	0.494724	0.000761677 0
$76$	32.6575	0.0492904
$77$	−209.577	−0.310176
$78$	−1.99161	−0.00289110
$79$	152.754	0.217546	0.108773	−	0.994067i	$-0.465308\pi$
$79$	152.754	0.217546	0.108773	+	0.994067i	$0.465308\pi$
$80$	373.981	0.522654
$81$	728.968	0.999956
$82$	710.106	0.956318
$83$	133.570	0.176641	0.0883205	−	0.996092i	$-0.471850\pi$
$83$	133.570	0.176641	0.0883205	+	0.996092i	$0.471850\pi$
$84$	−0.648043	−0.000841753 0
$85$	−255.864	−0.326498
$86$	305.404	0.382936
$87$	−1.11958	−0.00137968
$88$	−215.398	−0.260926
$89$	533.419	0.635306	0.317653	−	0.948207i	$-0.397105\pi$
$89$	533.419	0.635306	0.317653	+	0.948207i	$0.397105\pi$
$90$	420.857	0.492913
$91$	615.074	0.708542
$92$	183.066	0.207456
$93$	5.56439	0.00620431
$94$	312.539	0.342936
$95$	−95.0000	−0.102598
$96$	−1.51434	−0.00160996
$97$	92.7307	0.0970658	0.0485329	−	0.998822i	$-0.484545\pi$
$97$	92.7307	0.0970658	0.0485329	+	0.998822i	$0.484545\pi$
$98$	62.3427	0.0642608
$99$	−296.996	−0.301507
$100$	42.9704	0.0429704
$101$	−361.636	−0.356278	−0.178139	−	0.984005i	$-0.557008\pi$
$101$	−361.636	−0.356278	−0.178139	+	0.984005i	$0.557008\pi$
$102$	3.15695	0.00306456
$103$	−813.753	−0.778461	−0.389230	−	0.921140i	$-0.627259\pi$
$103$	−813.753	−0.778461	−0.389230	+	0.921140i	$0.627259\pi$
$104$	632.155	0.596038
$105$	1.88514	0.00175211
$106$	2059.53	1.88717
$107$	−526.749	−0.475913	−0.237957	−	0.971276i	$-0.576478\pi$
$107$	−526.749	−0.475913	−0.237957	+	0.971276i	$0.576478\pi$
$108$	−1.83672	−0.00163647
$109$	1739.64	1.52869	0.764345	−	0.644807i	$-0.223062\pi$
$109$	1739.64	1.52869	0.764345	+	0.644807i	$0.223062\pi$
$110$	−171.463	−0.148621
$111$	−1.93112	−0.00165130
$112$	1425.05	1.20228
$113$	−433.147	−0.360594	−0.180297	−	0.983612i	$-0.557706\pi$
$113$	−433.147	−0.360594	−0.180297	+	0.983612i	$0.557706\pi$
$114$	1.17215	0.000962999 0
$115$	−532.535	−0.431819
$116$	−97.2441	−0.0778352
$117$	871.632	0.688738
$118$	−2706.99	−2.11185
$119$	−974.969	−0.751052
$120$	1.93750	0.00147390
$121$	121.000	0.0909091
$122$	−0.00815837	−6.05429e−6 0
$123$	4.50754	0.00330432
$124$	483.308	0.350019
$125$	−125.000	−0.0894427
$126$	1603.67	1.13386
$127$	−1962.05	−1.37090	−0.685448	−	0.728121i	$-0.740394\pi$
$127$	−1962.05	−1.37090	−0.685448	+	0.728121i	$0.740394\pi$
$128$	1733.89	1.19731
$129$	1.93861	0.00132314
$130$	503.213	0.339498
$131$	−691.599	−0.461262	−0.230631	−	0.973041i	$-0.574079\pi$
$131$	−691.599	−0.461262	−0.230631	+	0.973041i	$0.574079\pi$
$132$	0.374149	0.000246708 0
$133$	−361.997	−0.236009
$134$	−443.002	−0.285594
$135$	5.34298	0.00340630
$136$	−1002.04	−0.631799
$137$	−902.000	−0.562504	−0.281252	−	0.959634i	$-0.590750\pi$
$137$	−902.000	−0.562504	−0.281252	+	0.959634i	$0.590750\pi$
$138$	6.57064	0.00405312
$139$	2355.28	1.43721	0.718606	−	0.695418i	$-0.244781\pi$
$139$	2355.28	1.43721	0.718606	+	0.695418i	$0.244781\pi$
$140$	163.739	0.0988460
$141$	1.98390	0.00118493
$142$	−1152.19	−0.680915
$143$	−355.114	−0.207665
$144$	2019.47	1.16867
$145$	282.881	0.162014
$146$	−224.756	−0.127404
$147$	0.395732	0.000222037 0
$148$	−167.732	−0.0931587
$149$	−318.651	−0.175201	−0.0876005	−	0.996156i	$-0.527920\pi$
$149$	−318.651	−0.175201	−0.0876005	+	0.996156i	$0.527920\pi$
$150$	1.54230	0.000839523 0
$151$	850.719	0.458480	0.229240	−	0.973370i	$-0.426376\pi$
$151$	850.719	0.458480	0.229240	+	0.973370i	$0.426376\pi$
$152$	−372.050	−0.198535
$153$	−1381.64	−0.730061
$154$	−653.358	−0.341877
$155$	−1405.93	−0.728564
$156$	−1.09806	−0.000563560 0
$157$	2632.06	1.33797	0.668986	−	0.743275i	$-0.266729\pi$
$157$	2632.06	1.33797	0.668986	+	0.743275i	$0.266729\pi$
$158$	476.210	0.239780
$159$	13.0733	0.00652063
$160$	382.622	0.189056
$161$	−2029.23	−0.993326
$162$	2272.56	1.10216
$163$	1953.44	0.938681	0.469340	−	0.883017i	$-0.344492\pi$
$163$	1953.44	0.938681	0.469340	+	0.883017i	$0.344492\pi$
$164$	391.513	0.186415
$165$	−1.08839	−0.000513523 0
$166$	416.404	0.194694
$167$	−2294.28	−1.06309	−0.531547	−	0.847028i	$-0.678389\pi$
$167$	−2294.28	−1.06309	−0.531547	+	0.847028i	$0.678389\pi$
$168$	7.38283	0.00339046
$169$	−1154.80	−0.525626
$170$	−797.656	−0.359867
$171$	−512.993	−0.229412
$172$	168.382	0.0746455
$173$	−251.126	−0.110363	−0.0551813	−	0.998476i	$-0.517574\pi$
$173$	−251.126	−0.110363	−0.0551813	+	0.998476i	$0.517574\pi$
$174$	−3.49030	−0.00152069
$175$	−476.312	−0.205748
$176$	−822.758	−0.352373
$177$	−17.1832	−0.00729698
$178$	1662.93	0.700237
$179$	929.159	0.387981	0.193991	−	0.981003i	$-0.437857\pi$
$179$	929.159	0.387981	0.193991	+	0.981003i	$0.437857\pi$
$180$	232.037	0.0960833
$181$	3959.26	1.62591	0.812953	−	0.582329i	$-0.197858\pi$
$181$	3959.26	1.62591	0.812953	+	0.582329i	$0.197858\pi$
$182$	1917.49	0.780957
$183$	−5.17868e−5 0	−2.09191e−8 0
$184$	−2085.58	−0.835603
$185$	487.929	0.193910
$186$	17.3470	0.00683841
$187$	562.900	0.220125
$188$	172.317	0.0668483
$189$	20.3594	0.00783561
$190$	−296.163	−0.113084
$191$	4660.59	1.76559	0.882797	−	0.469755i	$-0.155658\pi$
$191$	4660.59	1.76559	0.882797	+	0.469755i	$0.155658\pi$
$192$	7.12015	0.00267632
$193$	4152.29	1.54864	0.774321	−	0.632793i	$-0.218091\pi$
$193$	4152.29	1.54864	0.774321	+	0.632793i	$0.218091\pi$
$194$	289.088	0.106986
$195$	3.19425	0.00117305
$196$	34.3722	0.0125263
$197$	4744.11	1.71576	0.857878	−	0.513853i	$-0.171782\pi$
$197$	4744.11	1.71576	0.857878	+	0.513853i	$0.171782\pi$
$198$	−925.884	−0.332322
$199$	3356.52	1.19567	0.597833	−	0.801621i	$-0.296029\pi$
$199$	3356.52	1.19567	0.597833	+	0.801621i	$0.296029\pi$
$200$	−489.540	−0.173079
$201$	−2.81204	−0.000986797 0
$202$	−1127.40	−0.392691
$203$	1077.92	0.372685
$204$	1.74057	0.000597373 0
$205$	−1138.90	−0.388022
$206$	−2536.88	−0.858022
$207$	−2875.65	−0.965563
$208$	2414.65	0.804933
$209$	209.000	0.0691714
$210$	5.87694	0.00193118
$211$	−2336.84	−0.762439	−0.381220	−	0.924485i	$-0.624496\pi$
$211$	−2336.84	−0.762439	−0.381220	+	0.924485i	$0.624496\pi$
$212$	1135.51	0.367864
$213$	−7.31377	−0.00235273
$214$	−1642.14	−0.524553
$215$	−489.821	−0.155374
$216$	20.9248	0.00659145
$217$	−5357.31	−1.67594
$218$	5423.33	1.68493
$219$	−1.42668	−0.000440211 0
$220$	−94.5348	−0.0289706
$221$	−1652.02	−0.502835
$222$	−6.02027	−0.00182006
$223$	−4453.74	−1.33742	−0.668710	−	0.743523i	$-0.733153\pi$
$223$	−4453.74	−1.33742	−0.668710	+	0.743523i	$0.733153\pi$
$224$	1457.98	0.434890
$225$	−674.990	−0.199997
$226$	−1350.34	−0.397447
$227$	2318.12	0.677792	0.338896	−	0.940824i	$-0.389946\pi$
$227$	2318.12	0.677792	0.338896	+	0.940824i	$0.389946\pi$
$228$	0.646257	0.000187717 0
$229$	−5453.51	−1.57370	−0.786851	−	0.617143i	$-0.788290\pi$
$229$	−5453.51	−1.57370	−0.786851	+	0.617143i	$0.788290\pi$
$230$	−1660.18	−0.475952
$231$	−4.14732	−0.00118127
$232$	1107.85	0.313509
$233$	2689.35	0.756160	0.378080	−	0.925773i	$-0.376584\pi$
$233$	2689.35	0.756160	0.378080	+	0.925773i	$0.376584\pi$
$234$	2717.31	0.759129
$235$	−501.266	−0.139145
$236$	−1492.48	−0.411663
$237$	3.02283	0.000828499 0
$238$	−3039.47	−0.827812
$239$	475.570	0.128712	0.0643558	−	0.997927i	$-0.479501\pi$
$239$	475.570	0.128712	0.0643558	+	0.997927i	$0.479501\pi$
$240$	7.40069	0.00199047
$241$	1918.11	0.512682	0.256341	−	0.966586i	$-0.417483\pi$
$241$	1918.11	0.512682	0.256341	+	0.966586i	$0.417483\pi$
$242$	377.218	0.100200
$243$	43.2776	0.0114249
$244$	−0.00449806	−1.18016e−6 0
$245$	−99.9882	−0.0260735
$246$	14.0523	0.00364203
$247$	−613.379	−0.158010
$248$	−5506.09	−1.40983
$249$	2.64321	0.000672717 0
$250$	−389.688	−0.0985840
$251$	2007.61	0.504858	0.252429	−	0.967615i	$-0.418771\pi$
$251$	2007.61	0.504858	0.252429	+	0.967615i	$0.418771\pi$
$252$	884.175	0.221023
$253$	1171.58	0.291132
$254$	−6116.69	−1.51101
$255$	−5.06327	−0.00124343
$256$	2526.96	0.616933
$257$	−5052.48	−1.22632	−0.613162	−	0.789957i	$-0.710103\pi$
$257$	−5052.48	−1.22632	−0.613162	+	0.789957i	$0.710103\pi$
$258$	6.04361	0.00145837
$259$	1859.25	0.446056
$260$	277.444	0.0661781
$261$	1527.54	0.362269
$262$	−2156.06	−0.508404
$263$	3887.11	0.911368	0.455684	−	0.890142i	$-0.349395\pi$
$263$	3887.11	0.911368	0.455684	+	0.890142i	$0.349395\pi$
$264$	−4.26249	−0.000993706 0
$265$	−3303.18	−0.765709
$266$	−1128.53	−0.260130
$267$	10.5558	0.00241949
$268$	−244.246	−0.0556706
$269$	−2522.71	−0.571793	−0.285897	−	0.958260i	$-0.592291\pi$
$269$	−2522.71	−0.571793	−0.285897	+	0.958260i	$0.592291\pi$
$270$	16.6567	0.00375443
$271$	1582.83	0.354796	0.177398	−	0.984139i	$-0.443232\pi$
$271$	1582.83	0.354796	0.177398	+	0.984139i	$0.443232\pi$
$272$	−3827.53	−0.853228
$273$	12.1717	0.00269840
$274$	−2811.99	−0.619994
$275$	275.000	0.0603023
$276$	3.62268	0.000790072 0
$277$	1758.86	0.381515	0.190757	−	0.981637i	$-0.438906\pi$
$277$	1758.86	0.381515	0.190757	+	0.981637i	$0.438906\pi$
$278$	7342.59	1.58410
$279$	−7591.94	−1.62909
$280$	−1865.39	−0.398138
$281$	841.280	0.178600	0.0892999	−	0.996005i	$-0.471537\pi$
$281$	841.280	0.178600	0.0892999	+	0.996005i	$0.471537\pi$
$282$	6.18483	0.00130603
$283$	4053.90	0.851517	0.425758	−	0.904837i	$-0.360007\pi$
$283$	4053.90	0.851517	0.425758	+	0.904837i	$0.360007\pi$
$284$	−635.255	−0.132730
$285$	−1.87995	−0.000390732 0
$286$	−1107.07	−0.228889
$287$	−4339.79	−0.892577
$288$	2066.13	0.422736
$289$	−2294.35	−0.466996
$290$	881.882	0.178572
$291$	1.83504	0.000369664 0
$292$	−123.918	−0.0248347
$293$	1319.31	0.263055	0.131528	−	0.991313i	$-0.458012\pi$
$293$	1319.31	0.263055	0.131528	+	0.991313i	$0.458012\pi$
$294$	1.23370	0.000244730 0
$295$	4341.61	0.856875
$296$	1910.89	0.375230
$297$	−11.7546	−0.00229653
$298$	−993.396	−0.193107
$299$	−3438.38	−0.665039
$300$	0.850339	0.000163648 0
$301$	−1866.46	−0.357412
$302$	2652.12	0.505338
$303$	−7.15639	−0.00135685
$304$	−1421.13	−0.268116
$305$	0.0130848	2.45650e−6 0
$306$	−4307.28	−0.804675
$307$	3996.36	0.742945	0.371473	−	0.928444i	$-0.378853\pi$
$307$	3996.36	0.742945	0.371473	+	0.928444i	$0.378853\pi$
$308$	−360.225	−0.0666419
$309$	−16.1033	−0.00296468
$310$	−4383.00	−0.803025
$311$	3237.57	0.590307	0.295154	−	0.955450i	$-0.404629\pi$
$311$	3237.57	0.590307	0.295154	+	0.955450i	$0.404629\pi$
$312$	12.5097	0.00226994
$313$	−5718.32	−1.03265	−0.516323	−	0.856394i	$-0.672700\pi$
$313$	−5718.32	−1.03265	−0.516323	+	0.856394i	$0.672700\pi$
$314$	8205.46	1.47472
$315$	−2572.05	−0.460059
$316$	262.555	0.0467401
$317$	1128.83	0.200004	0.100002	−	0.994987i	$-0.468115\pi$
$317$	1128.83	0.200004	0.100002	+	0.994987i	$0.468115\pi$
$318$	40.7560	0.00718706
$319$	−622.338	−0.109230
$320$	−1799.02	−0.314276
$321$	−10.4238	−0.00181246
$322$	−6326.11	−1.09485
$323$	972.282	0.167490
$324$	1252.96	0.214843
$325$	−807.078	−0.137750
$326$	6089.84	1.03462
$327$	34.4256	0.00582184
$328$	−4460.31	−0.750852
$329$	−1910.07	−0.320078
$330$	−3.39306	−0.000566006 0
$331$	5231.71	0.868764	0.434382	−	0.900729i	$-0.356967\pi$
$331$	5231.71	0.868764	0.434382	+	0.900729i	$0.356967\pi$
$332$	229.582	0.0379516
$333$	2634.78	0.433589
$334$	−7152.43	−1.17175
$335$	710.508	0.115878
$336$	28.2003	0.00457873
$337$	−2371.33	−0.383307	−0.191654	−	0.981463i	$-0.561385\pi$
$337$	−2371.33	−0.383307	−0.191654	+	0.981463i	$0.561385\pi$
$338$	−3600.09	−0.579347
$339$	−8.57153	−0.00137328
$340$	−439.782	−0.0701487
$341$	3093.06	0.491198
$342$	−1599.26	−0.252859
$343$	6154.00	0.968760
$344$	−1918.30	−0.300662
$345$	−10.5383	−0.00164453
$346$	−782.884	−0.121642
$347$	2570.33	0.397644	0.198822	−	0.980036i	$-0.436288\pi$
$347$	2570.33	0.397644	0.198822	+	0.980036i	$0.436288\pi$
$348$	−1.92436	−0.000296426 0
$349$	−155.258	−0.0238131	−0.0119065	−	0.999929i	$-0.503790\pi$
$349$	−155.258	−0.0238131	−0.0119065	+	0.999929i	$0.503790\pi$
$350$	−1484.90	−0.226776
$351$	34.4976	0.00524600
$352$	−841.769	−0.127461
$353$	−1845.69	−0.278289	−0.139145	−	0.990272i	$-0.544435\pi$
$353$	−1845.69	−0.278289	−0.139145	+	0.990272i	$0.544435\pi$
$354$	−53.5685	−0.00804275
$355$	1847.94	0.276278
$356$	916.848	0.136497
$357$	−19.2936	−0.00286030
$358$	2896.66	0.427634
$359$	−13553.8	−1.99260	−0.996301	−	0.0859372i	$-0.972612\pi$
$359$	−13553.8	−1.99260	−0.996301	+	0.0859372i	$0.972612\pi$
$360$	−2643.48	−0.387010
$361$	361.000	0.0526316
$362$	12343.0	1.79208
$363$	2.39446	0.000346217 0
$364$	1057.20	0.152231
$365$	360.475	0.0516934
$366$	−0.000161445 0	−2.30571e−8 0
$367$	12015.0	1.70893	0.854466	−	0.519508i	$-0.173885\pi$
$367$	12015.0	1.70893	0.854466	+	0.519508i	$0.173885\pi$
$368$	−7966.32	−1.12846
$369$	−6149.99	−0.867630
$370$	1521.12	0.213728
$371$	−12586.8	−1.76138
$372$	9.56416	0.00133301
$373$	−3968.80	−0.550929	−0.275465	−	0.961311i	$-0.588832\pi$
$373$	−3968.80	−0.550929	−0.275465	+	0.961311i	$0.588832\pi$
$374$	1754.84	0.242622
$375$	−2.47362	−0.000340632 0
$376$	−1963.12	−0.269255
$377$	1826.46	0.249515
$378$	63.4705	0.00863643
$379$	1556.37	0.210937	0.105469	−	0.994423i	$-0.466366\pi$
$379$	1556.37	0.210937	0.105469	+	0.994423i	$0.466366\pi$
$380$	−163.287	−0.0220433
$381$	−38.8269	−0.00522090
$382$	14529.4	1.94604
$383$	−3341.93	−0.445861	−0.222930	−	0.974834i	$-0.571562\pi$
$383$	−3341.93	−0.445861	−0.222930	+	0.974834i	$0.571562\pi$
$384$	34.3118	0.00455981
$385$	1047.89	0.138715
$386$	12944.8	1.70692
$387$	−2645.00	−0.347423
$388$	159.387	0.0208548
$389$	−4151.79	−0.541142	−0.270571	−	0.962700i	$-0.587212\pi$
$389$	−4151.79	−0.541142	−0.270571	+	0.962700i	$0.587212\pi$
$390$	9.95806	0.00129294
$391$	5450.26	0.704940
$392$	−391.586	−0.0504543
$393$	−13.6860	−0.00175666
$394$	14789.8	1.89111
$395$	−763.768	−0.0972895
$396$	−510.481	−0.0647793
$397$	−221.622	−0.0280174	−0.0140087	−	0.999902i	$-0.504459\pi$
$397$	−221.622	−0.0280174	−0.0140087	+	0.999902i	$0.504459\pi$
$398$	10464.0	1.31787
$399$	−7.16355	−0.000898812 0
$400$	−1869.90	−0.233738
$401$	14298.8	1.78066	0.890332	−	0.455313i	$-0.150473\pi$
$401$	14298.8	1.78066	0.890332	+	0.455313i	$0.150473\pi$
$402$	−8.76654	−0.00108765
$403$	−9077.59	−1.12205
$404$	−621.585	−0.0765471
$405$	−3644.84	−0.447194
$406$	3360.41	0.410775
$407$	−1073.44	−0.130734
$408$	−19.8294	−0.00240613
$409$	9665.23	1.16850	0.584248	−	0.811575i	$-0.301390\pi$
$409$	9665.23	1.16850	0.584248	+	0.811575i	$0.301390\pi$
$410$	−3550.53	−0.427679
$411$	−17.8496	−0.00214223
$412$	−1398.69	−0.167254
$413$	16543.7	1.97109
$414$	−8964.84	−1.06425
$415$	−667.850	−0.0789963
$416$	2470.45	0.291163
$417$	46.6085	0.00547345
$418$	651.558	0.0762410
$419$	−8285.63	−0.966061	−0.483030	−	0.875604i	$-0.660464\pi$
$419$	−8285.63	−0.966061	−0.483030	+	0.875604i	$0.660464\pi$
$420$	3.24021	0.000376444 0
$421$	8188.71	0.947965	0.473983	−	0.880534i	$-0.342816\pi$
$421$	8188.71	0.947965	0.473983	+	0.880534i	$0.342816\pi$
$422$	−7285.10	−0.840363
$423$	−2706.80	−0.311132
$424$	−12936.3	−1.48171
$425$	1279.32	0.146014
$426$	−22.8007	−0.00259319
$427$	0.0498596	5.65076e−6 0
$428$	−905.384	−0.102251
$429$	−7.02734	−0.000790870 0
$430$	−1527.02	−0.171254
$431$	−7050.08	−0.787913	−0.393956	−	0.919129i	$-0.628894\pi$
$431$	−7050.08	−0.787913	−0.393956	+	0.919129i	$0.628894\pi$
$432$	79.9269	0.00890158
$433$	−11054.0	−1.22684	−0.613421	−	0.789756i	$-0.710207\pi$
$433$	−11054.0	−1.22684	−0.613421	+	0.789756i	$0.710207\pi$
$434$	−16701.4	−1.84722
$435$	5.59792	0.000617011 0
$436$	2990.12	0.328442
$437$	2023.63	0.221518
$438$	−4.44769	−0.000485202 0
$439$	−9477.85	−1.03042	−0.515208	−	0.857065i	$-0.672285\pi$
$439$	−9477.85	−1.03042	−0.515208	+	0.857065i	$0.672285\pi$
$440$	1076.99	0.116690
$441$	−539.928	−0.0583013
$442$	−5150.16	−0.554227
$443$	−7746.73	−0.830831	−0.415415	−	0.909632i	$-0.636364\pi$
$443$	−7746.73	−0.830831	−0.415415	+	0.909632i	$0.636364\pi$
$444$	−3.31924	−0.000354784 0
$445$	−2667.09	−0.284118
$446$	−13884.5	−1.47411
$447$	−6.30578	−0.000667233 0
$448$	−6855.18	−0.722939
$449$	13131.5	1.38020	0.690102	−	0.723712i	$-0.257566\pi$
$449$	13131.5	1.38020	0.690102	+	0.723712i	$0.257566\pi$
$450$	−2104.28	−0.220437
$451$	2505.59	0.261604
$452$	−744.500	−0.0774742
$453$	16.8348	0.00174607
$454$	7226.74	0.747065
$455$	−3075.37	−0.316869
$456$	−7.36249	−0.000756097 0
$457$	8375.24	0.857281	0.428640	−	0.903475i	$-0.358993\pi$
$457$	8375.24	0.857281	0.428640	+	0.903475i	$0.358993\pi$
$458$	−17001.3	−1.73454
$459$	−54.6830	−0.00556075
$460$	−915.330	−0.0927771
$461$	−17737.0	−1.79196	−0.895981	−	0.444093i	$-0.853526\pi$
$461$	−17737.0	−1.79196	−0.895981	+	0.444093i	$0.853526\pi$
$462$	−12.9293	−0.00130200
$463$	12373.9	1.24204	0.621018	−	0.783796i	$-0.286719\pi$
$463$	12373.9	1.24204	0.621018	+	0.783796i	$0.286719\pi$
$464$	4231.69	0.423386
$465$	−27.8220	−0.00277465
$466$	8384.06	0.833442
$467$	4529.12	0.448785	0.224392	−	0.974499i	$-0.427960\pi$
$467$	4529.12	0.448785	0.224392	+	0.974499i	$0.427960\pi$
$468$	1498.17	0.147977
$469$	2707.39	0.266558
$470$	−1562.70	−0.153366
$471$	52.0858	0.00509551
$472$	17003.1	1.65812
$473$	1077.61	0.104753
$474$	9.42369	0.000913174 0
$475$	475.000	0.0458831
$476$	−1675.79	−0.161365
$477$	−17836.9	−1.71215
$478$	1482.59	0.141866
$479$	3675.03	0.350556	0.175278	−	0.984519i	$-0.443918\pi$
$479$	3675.03	0.350556	0.175278	+	0.984519i	$0.443918\pi$
$480$	7.57169	0.000719997 0
$481$	3150.38	0.298638
$482$	5979.71	0.565079
$483$	−40.1562	−0.00378297
$484$	207.977	0.0195320
$485$	−463.654	−0.0434091
$486$	134.918	0.0125926
$487$	−14072.5	−1.30941	−0.654707	−	0.755883i	$-0.727208\pi$
$487$	−14072.5	−1.30941	−0.654707	+	0.755883i	$0.727208\pi$
$488$	0.0512442	4.75352e−6 0
$489$	38.6565	0.00357486
$490$	−311.713	−0.0287383
$491$	10765.0	0.989442	0.494721	−	0.869052i	$-0.335270\pi$
$491$	10765.0	0.989442	0.494721	+	0.869052i	$0.335270\pi$
$492$	7.74762	0.000709939 0
$493$	−2895.16	−0.264486
$494$	−1912.21	−0.174159
$495$	1484.98	0.134838
$496$	−21031.7	−1.90394
$497$	7041.59	0.635530
$498$	8.24021	0.000741471 0
$499$	−20259.6	−1.81752	−0.908761	−	0.417317i	$-0.862970\pi$
$499$	−20259.6	−1.81752	−0.908761	+	0.417317i	$0.862970\pi$
$500$	−214.852	−0.0192169
$501$	−45.4014	−0.00404867
$502$	6258.74	0.556456
$503$	−5921.88	−0.524938	−0.262469	−	0.964940i	$-0.584537\pi$
$503$	−5921.88	−0.524938	−0.262469	+	0.964940i	$0.584537\pi$
$504$	−10073.0	−0.890250
$505$	1808.18	0.159333
$506$	3652.40	0.320887
$507$	−22.8523	−0.00200179
$508$	−3372.40	−0.294540
$509$	15058.7	1.31132	0.655662	−	0.755055i	$-0.272390\pi$
$509$	15058.7	1.31132	0.655662	+	0.755055i	$0.272390\pi$
$510$	−15.7848	−0.00137051
$511$	1373.59	0.118912
$512$	−5993.30	−0.517322
$513$	−20.3033	−0.00174739
$514$	−15751.1	−1.35166
$515$	4068.76	0.348138
$516$	3.33211	0.000284279 0
$517$	1102.78	0.0938113
$518$	5796.23	0.491644
$519$	−4.96951	−0.000420303 0
$520$	−3160.78	−0.266556
$521$	1480.35	0.124483	0.0622413	−	0.998061i	$-0.480175\pi$
$521$	1480.35	0.124483	0.0622413	+	0.998061i	$0.480175\pi$
$522$	4762.10	0.399294
$523$	16576.2	1.38590	0.692951	−	0.720985i	$-0.256310\pi$
$523$	16576.2	1.38590	0.692951	+	0.720985i	$0.256310\pi$
$524$	−1188.73	−0.0991030
$525$	−9.42572	−0.000783566 0
$526$	12118.1	1.00451
$527$	14389.1	1.18937
$528$	−16.2815	−0.00134197
$529$	−823.240	−0.0676617
$530$	−10297.7	−0.843967
$531$	23444.3	1.91600
$532$	−622.207	−0.0507069
$533$	−7353.47	−0.597587
$534$	32.9077	0.00266677
$535$	2633.74	0.212835
$536$	2782.58	0.224233
$537$	18.3871	0.00147758
$538$	−7864.56	−0.630232
$539$	219.974	0.0175788
$540$	9.18359	0.000731850 0
$541$	3340.25	0.265450	0.132725	−	0.991153i	$-0.457627\pi$
$541$	3340.25	0.265450	0.132725	+	0.991153i	$0.457627\pi$
$542$	4934.46	0.391058
$543$	78.3495	0.00619208
$544$	−3915.96	−0.308632
$545$	−8698.20	−0.683651
$546$	37.9452	0.00297418
$547$	−3973.70	−0.310609	−0.155305	−	0.987867i	$-0.549636\pi$
$547$	−3973.70	−0.310609	−0.155305	+	0.987867i	$0.549636\pi$
$548$	−1550.37	−0.120855
$549$	0.0706568	5.49282e−6 0
$550$	857.313	0.0664654
$551$	−1074.95	−0.0831113
$552$	−41.2714	−0.00318230
$553$	−2910.34	−0.223798
$554$	5483.25	0.420507
$555$	9.65561	0.000738482 0
$556$	4048.29	0.308788
$557$	−7394.41	−0.562497	−0.281249	−	0.959635i	$-0.590749\pi$
$557$	−7394.41	−0.562497	−0.281249	+	0.959635i	$0.590749\pi$
$558$	−23667.9	−1.79559
$559$	−3162.59	−0.239290
$560$	−7125.27	−0.537674
$561$	11.1392	0.000838320 0
$562$	2622.69	0.196853
$563$	952.501	0.0713022	0.0356511	−	0.999364i	$-0.488649\pi$
$563$	952.501	0.0713022	0.0356511	+	0.999364i	$0.488649\pi$
$564$	3.40996	0.000254584 0
$565$	2165.74	0.161262
$566$	12638.0	0.938545
$567$	−13888.7	−1.02869
$568$	7237.14	0.534619
$569$	17923.3	1.32054	0.660268	−	0.751030i	$-0.270443\pi$
$569$	17923.3	1.32054	0.660268	+	0.751030i	$0.270443\pi$
$570$	−5.86075	−0.000430666 0
$571$	26202.4	1.92038	0.960191	−	0.279345i	$-0.0901173\pi$
$571$	26202.4	1.92038	0.960191	+	0.279345i	$0.0901173\pi$
$572$	−610.376	−0.0446173
$573$	92.2282	0.00672406
$574$	−13529.3	−0.983801
$575$	2662.68	0.193115
$576$	−9714.58	−0.702733
$577$	12075.4	0.871243	0.435621	−	0.900130i	$-0.356529\pi$
$577$	12075.4	0.871243	0.435621	+	0.900130i	$0.356529\pi$
$578$	−7152.64	−0.514724
$579$	82.1694	0.00589783
$580$	486.220	0.0348090
$581$	−2544.84	−0.181717
$582$	5.72075	0.000407445 0
$583$	7267.00	0.516241
$584$	1411.73	0.100031
$585$	−4358.16	−0.308013
$586$	4112.96	0.289940
$587$	398.688	0.0280334	0.0140167	−	0.999902i	$-0.495538\pi$
$587$	398.688	0.0280334	0.0140167	+	0.999902i	$0.495538\pi$
$588$	0.680190	4.77051e−5 0
$589$	5342.55	0.373745
$590$	13535.0	0.944450
$591$	93.8810	0.00653426
$592$	7299.05	0.506738
$593$	1918.64	0.132866	0.0664328	−	0.997791i	$-0.478838\pi$
$593$	1918.64	0.132866	0.0664328	+	0.997791i	$0.478838\pi$
$594$	−36.6448	−0.00253124
$595$	4874.84	0.335881
$596$	−547.703	−0.0376423
$597$	66.4220	0.00455356
$598$	−10719.2	−0.733008
$599$	13786.3	0.940389	0.470194	−	0.882563i	$-0.344184\pi$
$599$	13786.3	0.940389	0.470194	+	0.882563i	$0.344184\pi$
$600$	−9.68748	−0.000659150 0
$601$	−6050.61	−0.410665	−0.205332	−	0.978692i	$-0.565828\pi$
$601$	−6050.61	−0.410665	−0.205332	+	0.978692i	$0.565828\pi$
$602$	−5818.70	−0.393941
$603$	3836.69	0.259108
$604$	1462.23	0.0985053
$605$	−605.000	−0.0406558
$606$	−22.3101	−0.00149552
$607$	−14976.5	−1.00144	−0.500722	−	0.865608i	$-0.666932\pi$
$607$	−14976.5	−1.00144	−0.500722	+	0.865608i	$0.666932\pi$
$608$	−1453.96	−0.0969836
$609$	21.3309	0.00141933
$610$	0.0407918	2.70756e−6 0
$611$	−3236.48	−0.214295
$612$	−2374.79	−0.156855
$613$	−27233.5	−1.79437	−0.897186	−	0.441654i	$-0.854392\pi$
$613$	−27233.5	−1.79437	−0.897186	+	0.441654i	$0.854392\pi$
$614$	12458.6	0.818876
$615$	−22.5377	−0.00147774
$616$	4103.86	0.268424
$617$	−15064.0	−0.982905	−0.491452	−	0.870904i	$-0.663534\pi$
$617$	−15064.0	−0.982905	−0.491452	+	0.870904i	$0.663534\pi$
$618$	−50.2021	−0.00326768
$619$	578.639	0.0375726	0.0187863	−	0.999824i	$-0.494020\pi$
$619$	578.639	0.0375726	0.0187863	+	0.999824i	$0.494020\pi$
$620$	−2416.54	−0.156533
$621$	−113.813	−0.00735452
$622$	10093.1	0.650639
$623$	−10163.0	−0.653564
$624$	47.7835	0.00306550
$625$	625.000	0.0400000
$626$	−17826.9	−1.13819
$627$	4.13589	0.000263432 0
$628$	4524.03	0.287466
$629$	−4993.74	−0.316555
$630$	−8018.37	−0.507079
$631$	−7181.36	−0.453068	−0.226534	−	0.974003i	$-0.572739\pi$
$631$	−7181.36	−0.453068	−0.226534	+	0.974003i	$0.572739\pi$
$632$	−2991.16	−0.188263
$633$	−46.2436	−0.00290366
$634$	3519.12	0.220445
$635$	9810.25	0.613083
$636$	22.4706	0.00140097
$637$	−645.586	−0.0401555
$638$	−1940.14	−0.120393
$639$	9978.75	0.617767
$640$	−8669.43	−0.535452
$641$	8915.25	0.549347	0.274673	−	0.961538i	$-0.411430\pi$
$641$	8915.25	0.549347	0.274673	+	0.961538i	$0.411430\pi$
$642$	−32.4962	−0.00199770
$643$	19784.9	1.21343	0.606717	−	0.794918i	$-0.292486\pi$
$643$	19784.9	1.21343	0.606717	+	0.794918i	$0.292486\pi$
$644$	−3487.86	−0.213418
$645$	−9.69304	−0.000591726 0
$646$	3031.09	0.184608
$647$	−14967.2	−0.909459	−0.454729	−	0.890630i	$-0.650264\pi$
$647$	−14967.2	−0.909459	−0.454729	+	0.890630i	$0.650264\pi$
$648$	−14274.4	−0.865355
$649$	−9551.53	−0.577705
$650$	−2516.07	−0.151828
$651$	−106.016	−0.00638261
$652$	3357.60	0.201677
$653$	−10680.8	−0.640079	−0.320039	−	0.947404i	$-0.603696\pi$
$653$	−10680.8	−0.640079	−0.320039	+	0.947404i	$0.603696\pi$
$654$	107.322	0.00641685
$655$	3458.00	0.206283
$656$	−17037.1	−1.01401
$657$	1946.53	0.115588
$658$	−5954.66	−0.352791
$659$	−23851.7	−1.40991	−0.704955	−	0.709252i	$-0.749033\pi$
$659$	−23851.7	−1.40991	−0.704955	+	0.709252i	$0.749033\pi$
$660$	−1.87074	−0.000110331 0
$661$	19891.6	1.17049	0.585245	−	0.810857i	$-0.300998\pi$
$661$	19891.6	1.17049	0.585245	+	0.810857i	$0.300998\pi$
$662$	16309.9	0.957554
$663$	−32.6917	−0.00191499
$664$	−2615.51	−0.152864
$665$	1809.99	0.105546
$666$	8213.93	0.477903
$667$	−6025.77	−0.349803
$668$	−3943.45	−0.228408
$669$	−88.1349	−0.00509341
$670$	2215.01	0.127721
$671$	−0.0287865	−1.65617e−6 0
$672$	28.8519	0.00165623
$673$	−25126.3	−1.43915	−0.719576	−	0.694414i	$-0.755664\pi$
$673$	−25126.3	−1.43915	−0.719576	+	0.694414i	$0.755664\pi$
$674$	−7392.63	−0.422483
$675$	−26.7149	−0.00152334
$676$	−1984.89	−0.112932
$677$	−21610.9	−1.22685	−0.613424	−	0.789754i	$-0.710208\pi$
$677$	−21610.9	−1.22685	−0.613424	+	0.789754i	$0.710208\pi$
$678$	−26.7218	−0.00151363
$679$	−1766.75	−0.0998553
$680$	5010.22	0.282549
$681$	45.8731	0.00258129
$682$	9642.61	0.541400
$683$	27429.1	1.53667	0.768334	−	0.640050i	$-0.221086\pi$
$683$	27429.1	1.53667	0.768334	+	0.640050i	$0.221086\pi$
$684$	−881.739	−0.0492897
$685$	4510.00	0.251559
$686$	19185.1	1.06777
$687$	−107.919	−0.00599326
$688$	−7327.35	−0.406036
$689$	−21327.4	−1.17926
$690$	−32.8532	−0.00181261
$691$	30106.7	1.65747	0.828736	−	0.559639i	$-0.189060\pi$
$691$	30106.7	1.65747	0.828736	+	0.559639i	$0.189060\pi$
$692$	−431.638	−0.0237116
$693$	5658.51	0.310172
$694$	8013.01	0.438285
$695$	−11776.4	−0.642740
$696$	21.9233	0.00119396
$697$	11656.2	0.633441
$698$	−484.016	−0.0262468
$699$	53.2195	0.00287975
$700$	−818.693	−0.0442053
$701$	1317.24	0.0709722	0.0354861	−	0.999370i	$-0.488702\pi$
$701$	1317.24	0.0709722	0.0354861	+	0.999370i	$0.488702\pi$
$702$	107.546	0.00578216
$703$	−1854.13	−0.0994735
$704$	3957.85	0.211885
$705$	−9.91952	−0.000529916 0
$706$	−5753.94	−0.306732
$707$	6890.07	0.366517
$708$	−29.5347	−0.00156777
$709$	3288.97	0.174217	0.0871086	−	0.996199i	$-0.472237\pi$
$709$	3288.97	0.174217	0.0871086	+	0.996199i	$0.472237\pi$
$710$	5760.97	0.304514
$711$	−4124.29	−0.217543
$712$	−10445.2	−0.549790
$713$	29948.4	1.57304
$714$	−60.1478	−0.00315263
$715$	1775.57	0.0928708
$716$	1597.05	0.0833585
$717$	9.41104	0.000490184 0
$718$	−42254.1	−2.19625
$719$	−3661.83	−0.189935	−0.0949675	−	0.995480i	$-0.530275\pi$
$719$	−3661.83	−0.189935	−0.0949675	+	0.995480i	$0.530275\pi$
$720$	−10097.3	−0.522647
$721$	15504.0	0.800832
$722$	1125.42	0.0580107
$723$	37.9574	0.00195249
$724$	6805.23	0.349329
$725$	−1414.41	−0.0724548
$726$	7.46474	0.000381601 0
$727$	11805.0	0.602235	0.301118	−	0.953587i	$-0.402640\pi$
$727$	11805.0	0.602235	0.301118	+	0.953587i	$0.402640\pi$
$728$	−12044.1	−0.613167
$729$	−19681.3	−0.999913
$730$	1123.78	0.0569767
$731$	5013.10	0.253647
$732$	−8.90120e−5 0	−4.49450e−9 0
$733$	1342.28	0.0676376	0.0338188	−	0.999428i	$-0.489233\pi$
$733$	1342.28	0.0676376	0.0338188	+	0.999428i	$0.489233\pi$
$734$	37456.8	1.88359
$735$	−1.97866	−9.92980e−5 0
$736$	−8150.39	−0.408189
$737$	−1563.12	−0.0781251
$738$	−19172.6	−0.956304
$739$	−28516.8	−1.41949	−0.709747	−	0.704457i	$-0.751191\pi$
$739$	−28516.8	−1.41949	−0.709747	+	0.704457i	$0.751191\pi$
$740$	838.660	0.0416618
$741$	−12.1381	−0.000601762 0
$742$	−39239.3	−1.94140
$743$	9312.46	0.459813	0.229906	−	0.973213i	$-0.426158\pi$
$743$	9312.46	0.459813	0.229906	+	0.973213i	$0.426158\pi$
$744$	−108.960	−0.00536916
$745$	1593.26	0.0783523
$746$	−12372.7	−0.607236
$747$	−3606.34	−0.176638
$748$	967.521	0.0472942
$749$	10035.9	0.489590
$750$	−7.71151	−0.000375446 0
$751$	1404.41	0.0682394	0.0341197	−	0.999418i	$-0.489137\pi$
$751$	1404.41	0.0682394	0.0341197	+	0.999418i	$0.489137\pi$
$752$	−7498.55	−0.363623
$753$	39.7286	0.00192269
$754$	5693.98	0.275017
$755$	−4253.59	−0.205039
$756$	34.9941	0.00168349
$757$	−9484.37	−0.455370	−0.227685	−	0.973735i	$-0.573116\pi$
$757$	−9484.37	−0.455370	−0.227685	+	0.973735i	$0.573116\pi$
$758$	4851.98	0.232496
$759$	23.1843	0.00110874
$760$	1860.25	0.0887874
$761$	−6241.45	−0.297309	−0.148655	−	0.988889i	$-0.547494\pi$
$761$	−6241.45	−0.297309	−0.148655	+	0.988889i	$0.547494\pi$
$762$	−121.043	−0.00575449
$763$	−33144.5	−1.57262
$764$	8010.69	0.379341
$765$	6908.22	0.326493
$766$	−10418.5	−0.491429
$767$	28032.1	1.31966
$768$	50.0058	0.00234952
$769$	−3856.63	−0.180850	−0.0904249	−	0.995903i	$-0.528823\pi$
$769$	−3856.63	−0.180850	−0.0904249	+	0.995903i	$0.528823\pi$
$770$	3266.79	0.152892
$771$	−99.9833	−0.00467031
$772$	7137.01	0.332729
$773$	2515.69	0.117054	0.0585271	−	0.998286i	$-0.481360\pi$
$773$	2515.69	0.117054	0.0585271	+	0.998286i	$0.481360\pi$
$774$	−8245.78	−0.382931
$775$	7029.67	0.325824
$776$	−1815.82	−0.0840000
$777$	36.7927	0.00169875
$778$	−12943.2	−0.596448
$779$	4327.83	0.199051
$780$	5.49032	0.000252032 0
$781$	−4065.48	−0.186267
$782$	16991.2	0.776987
$783$	60.4571	0.00275934
$784$	−1495.75	−0.0681372
$785$	−13160.3	−0.598359
$786$	−42.6662	−0.00193620
$787$	23334.1	1.05689	0.528444	−	0.848968i	$-0.322776\pi$
$787$	23334.1	1.05689	0.528444	+	0.848968i	$0.322776\pi$
$788$	8154.25	0.368633
$789$	76.9219	0.00347084
$790$	−2381.05	−0.107233
$791$	8252.54	0.370956
$792$	5815.65	0.260922
$793$	0.0844835	3.78323e−6 0
$794$	−690.908	−0.0308809
$795$	−65.3665	−0.00291611
$796$	5769.24	0.256891
$797$	−15092.3	−0.670759	−0.335380	−	0.942083i	$-0.608865\pi$
$797$	−15092.3	−0.670759	−0.335380	+	0.942083i	$0.608865\pi$
$798$	−22.3324	−0.000990674 0
$799$	5130.23	0.227152
$800$	−1913.11	−0.0845483
$801$	−14402.1	−0.635297
$802$	44576.4	1.96265
$803$	−793.044	−0.0348517
$804$	−4.83338	−0.000212015 0
$805$	10146.1	0.444229
$806$	−28299.4	−1.23673
$807$	−49.9218	−0.00217761
$808$	7081.41	0.308321
$809$	20278.2	0.881264	0.440632	−	0.897688i	$-0.354754\pi$
$809$	20278.2	0.881264	0.440632	+	0.897688i	$0.354754\pi$
$810$	−11362.8	−0.492899
$811$	27731.0	1.20070	0.600350	−	0.799738i	$-0.295028\pi$
$811$	27731.0	1.20070	0.600350	+	0.799738i	$0.295028\pi$
$812$	1852.74	0.0800720
$813$	31.3225	0.00135120
$814$	−3346.46	−0.144095
$815$	−9767.18	−0.419791
$816$	−75.7427	−0.00324942
$817$	1861.32	0.0797054
$818$	30131.4	1.28792
$819$	−16606.8	−0.708531
$820$	−1957.56	−0.0833671
$821$	−39541.1	−1.68087	−0.840434	−	0.541914i	$-0.817700\pi$
$821$	−39541.1	−1.68087	−0.840434	+	0.541914i	$0.817700\pi$
$822$	−55.6462	−0.00236117
$823$	5846.71	0.247635	0.123817	−	0.992305i	$-0.460486\pi$
$823$	5846.71	0.247635	0.123817	+	0.992305i	$0.460486\pi$
$824$	15934.6	0.673674
$825$	5.44196	0.000229654 0
$826$	51575.0	2.17255
$827$	23335.3	0.981196	0.490598	−	0.871386i	$-0.336778\pi$
$827$	23335.3	0.981196	0.490598	+	0.871386i	$0.336778\pi$
$828$	−4942.71	−0.207453
$829$	5572.91	0.233480	0.116740	−	0.993162i	$-0.462756\pi$
$829$	5572.91	0.233480	0.116740	+	0.993162i	$0.462756\pi$
$830$	−2082.02	−0.0870699
$831$	34.8060	0.00145296
$832$	−11615.6	−0.484013
$833$	1023.33	0.0425647
$834$	145.302	0.00603286
$835$	11471.4	0.475431
$836$	359.232	0.0148616
$837$	−300.475	−0.0124085
$838$	−25830.5	−1.06480
$839$	39517.6	1.62610	0.813051	−	0.582193i	$-0.197805\pi$
$839$	39517.6	1.62610	0.813051	+	0.582193i	$0.197805\pi$
$840$	−36.9142	−0.00151626
$841$	−21188.1	−0.868758
$842$	25528.3	1.04485
$843$	16.6480	0.000680177 0
$844$	−4016.59	−0.163811
$845$	5774.00	0.235067
$846$	−8438.44	−0.342931
$847$	−2305.35	−0.0935216
$848$	−49413.1	−2.00101
$849$	80.2224	0.00324290
$850$	3988.28	0.160937
$851$	−10393.6	−0.418669
$852$	−12.5710	−0.000505488 0
$853$	−7771.95	−0.311966	−0.155983	−	0.987760i	$-0.549854\pi$
$853$	−7771.95	−0.311966	−0.155983	+	0.987760i	$0.549854\pi$
$854$	0.155437	6.22828e−6 0
$855$	2564.96	0.102596
$856$	10314.6	0.411852
$857$	28954.5	1.15410	0.577052	−	0.816707i	$-0.304203\pi$
$857$	28954.5	1.15410	0.577052	+	0.816707i	$0.304203\pi$
$858$	−21.9077	−0.000871699 0
$859$	−14073.6	−0.559004	−0.279502	−	0.960145i	$-0.590169\pi$
$859$	−14073.6	−0.559004	−0.279502	+	0.960145i	$0.590169\pi$
$860$	−841.912	−0.0333825
$861$	−85.8798	−0.00339928
$862$	−21978.6	−0.868440
$863$	−26941.7	−1.06270	−0.531348	−	0.847153i	$-0.678314\pi$
$863$	−26941.7	−1.06270	−0.531348	+	0.847153i	$0.678314\pi$
$864$	81.7736	0.00321990
$865$	1255.63	0.0493556
$866$	−34461.0	−1.35223
$867$	−45.4028	−0.00177850
$868$	−9208.23	−0.360078
$869$	1680.29	0.0655926
$870$	17.4515	0.000680071 0
$871$	4587.48	0.178463
$872$	−34064.9	−1.32292
$873$	−2503.69	−0.0970644
$874$	6308.68	0.244158
$875$	2381.56	0.0920131
$876$	−2.45220	−9.45802e−5 0
$877$	−15097.7	−0.581316	−0.290658	−	0.956827i	$-0.593874\pi$
$877$	−15097.7	−0.581316	−0.290658	+	0.956827i	$0.593874\pi$
$878$	−29547.2	−1.13573
$879$	26.1078	0.00100181
$880$	4113.79	0.157586
$881$	27289.1	1.04358	0.521790	−	0.853074i	$-0.325265\pi$
$881$	27289.1	1.04358	0.521790	+	0.853074i	$0.325265\pi$
$882$	−1683.23	−0.0642599
$883$	23558.7	0.897862	0.448931	−	0.893566i	$-0.351805\pi$
$883$	23558.7	0.897862	0.448931	+	0.893566i	$0.351805\pi$
$884$	−2839.51	−0.108035
$885$	85.9158	0.00326331
$886$	−24150.4	−0.915744
$887$	4964.68	0.187934	0.0939671	−	0.995575i	$-0.470045\pi$
$887$	4964.68	0.187934	0.0939671	+	0.995575i	$0.470045\pi$
$888$	37.8145	0.00142902
$889$	37382.0	1.41029
$890$	−8314.67	−0.313155
$891$	8018.65	0.301498
$892$	−7655.16	−0.287347
$893$	1904.81	0.0713797
$894$	−19.6583	−0.000735426 0
$895$	−4645.80	−0.173510
$896$	−33034.9	−1.23172
$897$	−68.0419	−0.00253273
$898$	40937.3	1.52127
$899$	−15908.5	−0.590187
$900$	−1160.18	−0.0429697
$901$	33806.6	1.25001
$902$	7811.17	0.288341
$903$	−36.9353	−0.00136116
$904$	8481.72	0.312055
$905$	−19796.3	−0.727128
$906$	52.4826	0.00192452
$907$	49602.3	1.81589	0.907947	−	0.419084i	$-0.137649\pi$
$907$	49602.3	1.81589	0.907947	+	0.419084i	$0.137649\pi$
$908$	3984.42	0.145625
$909$	9764.03	0.356273
$910$	−9587.47	−0.349254
$911$	2786.37	0.101335	0.0506677	−	0.998716i	$-0.483865\pi$
$911$	2786.37	0.101335	0.0506677	+	0.998716i	$0.483865\pi$
$912$	−28.1226	−0.00102109
$913$	1469.27	0.0532593
$914$	26109.8	0.944898
$915$	0.000258934 0	9.35530e−9 0
$916$	−9373.57	−0.338113
$917$	13176.7	0.474518
$918$	−170.474	−0.00612907
$919$	54977.7	1.97339	0.986695	−	0.162582i	$-0.0519821\pi$
$919$	54977.7	1.97339	0.986695	+	0.162582i	$0.0519821\pi$
$920$	10427.9	0.373693
$921$	79.0837	0.00282942
$922$	−55295.1	−1.97511
$923$	11931.5	0.425492
$924$	−7.12847	−0.000253798 0
$925$	−2439.65	−0.0867190
$926$	38575.6	1.36898
$927$	21971.0	0.778449
$928$	4329.46	0.153148
$929$	29299.4	1.03475	0.517375	−	0.855759i	$-0.326909\pi$
$929$	29299.4	1.03475	0.517375	+	0.855759i	$0.326909\pi$
$930$	−86.7350	−0.00305823
$931$	379.955	0.0133754
$932$	4622.50	0.162462
$933$	64.0681	0.00224812
$934$	14119.5	0.494652
$935$	−2814.50	−0.0984428
$936$	−17067.9	−0.596029
$937$	−32653.4	−1.13846	−0.569232	−	0.822177i	$-0.692759\pi$
$937$	−32653.4	−1.13846	−0.569232	+	0.822177i	$0.692759\pi$
$938$	8440.29	0.293801
$939$	−113.159	−0.00393272
$940$	−861.583	−0.0298955
$941$	33834.4	1.17213	0.586063	−	0.810266i	$-0.300677\pi$
$941$	33834.4	1.17213	0.586063	+	0.810266i	$0.300677\pi$
$942$	162.378	0.00561629
$943$	24260.2	0.837775
$944$	64947.1	2.23925
$945$	−101.797	−0.00350419
$946$	3359.44	0.115460
$947$	−55812.6	−1.91517	−0.957585	−	0.288151i	$-0.906959\pi$
$947$	−55812.6	−1.91517	−0.957585	+	0.288151i	$0.906959\pi$
$948$	5.19569	0.000178004 0
$949$	2327.45	0.0796124
$950$	1480.81	0.0505726
$951$	22.3383	0.000761691 0
$952$	19091.5	0.649955
$953$	−31088.3	−1.05672	−0.528358	−	0.849022i	$-0.677192\pi$
$953$	−31088.3	−1.05672	−0.528358	+	0.849022i	$0.677192\pi$
$954$	−55606.6	−1.88714
$955$	−23302.9	−0.789597
$956$	817.417	0.0276539
$957$	−12.3154	−0.000415989 0
$958$	11456.9	0.386384
$959$	17185.3	0.578669
$960$	−35.6008	−0.00119689
$961$	49275.1	1.65403
$962$	9821.30	0.329160
$963$	14222.0	0.475906
$964$	3296.87	0.110151
$965$	−20761.4	−0.692574
$966$	−125.187	−0.00416960
$967$	−2479.26	−0.0824485	−0.0412242	−	0.999150i	$-0.513126\pi$
$967$	−2479.26	−0.0824485	−0.0412242	+	0.999150i	$0.513126\pi$
$968$	−2369.37	−0.0786721
$969$	19.2404	0.000637866 0
$970$	−1445.44	−0.0478457
$971$	37627.1	1.24358	0.621788	−	0.783186i	$-0.286407\pi$
$971$	37627.1	1.24358	0.621788	+	0.783186i	$0.286407\pi$
$972$	74.3862	0.00245467
$973$	−44874.0	−1.47851
$974$	−43871.0	−1.44324
$975$	−15.9712	−0.000524604 0
$976$	0.195738	6.41950e−6 0
$977$	54912.1	1.79815	0.899076	−	0.437793i	$-0.144240\pi$
$977$	54912.1	1.79815	0.899076	+	0.437793i	$0.144240\pi$
$978$	120.512	0.00394022
$979$	5867.61	0.191552
$980$	−171.861	−0.00560194
$981$	−46969.6	−1.52867
$982$	33559.8	1.09057
$983$	−13965.2	−0.453125	−0.226562	−	0.973997i	$-0.572749\pi$
$983$	−13965.2	−0.453125	−0.226562	+	0.973997i	$0.572749\pi$
$984$	−88.2648	−0.00285953
$985$	−23720.6	−0.767310
$986$	−9025.67	−0.291517
$987$	−37.7983	−0.00121898
$988$	−1054.29	−0.0339487
$989$	10433.9	0.335468
$990$	4629.42	0.148619
$991$	22538.4	0.722459	0.361229	−	0.932477i	$-0.382357\pi$
$991$	22538.4	0.722459	0.361229	+	0.932477i	$0.382357\pi$
$992$	−21517.7	−0.688696
$993$	103.530	0.00330859
$994$	21952.2	0.700483
$995$	−16782.6	−0.534718
$996$	4.54319	0.000144534 0
$997$	34427.1	1.09360	0.546799	−	0.837264i	$-0.315846\pi$
$997$	34427.1	1.09360	0.546799	+	0.837264i	$0.315846\pi$
$998$	−63159.3	−2.00328
$999$	104.280	0.00330257

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.i.1.18	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.i.1.18	✓	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)

\(f(q)\)	\(=\)	\(q+3.11750 q^{2} +0.0197889 q^{3} +1.71881 q^{4} -5.00000 q^{5} +0.0616921 q^{6} -19.0525 q^{7} -19.5816 q^{8} -26.9996 q^{9} +O(q^{10})\)	\(q+3.11750 q^{2} +0.0197889 q^{3} +1.71881 q^{4} -5.00000 q^{5} +0.0616921 q^{6} -19.0525 q^{7} -19.5816 q^{8} -26.9996 q^{9} -15.5875 q^{10} +11.0000 q^{11} +0.0340135 q^{12} -32.2831 q^{13} -59.3962 q^{14} -0.0989447 q^{15} -74.7962 q^{16} +51.1728 q^{17} -84.1713 q^{18} +19.0000 q^{19} -8.59407 q^{20} -0.377029 q^{21} +34.2925 q^{22} +106.507 q^{23} -0.387499 q^{24} +25.0000 q^{25} -100.643 q^{26} -1.06860 q^{27} -32.7477 q^{28} -56.5762 q^{29} -0.308460 q^{30} +281.187 q^{31} -76.5244 q^{32} +0.217678 q^{33} +159.531 q^{34} +95.2625 q^{35} -46.4073 q^{36} -97.5859 q^{37} +59.2325 q^{38} -0.638849 q^{39} +97.9080 q^{40} +227.781 q^{41} -1.17539 q^{42} +97.9642 q^{43} +18.9070 q^{44} +134.998 q^{45} +332.036 q^{46} +100.253 q^{47} -1.48014 q^{48} +19.9976 q^{49} +77.9375 q^{50} +1.01265 q^{51} -55.4887 q^{52} +660.636 q^{53} -3.33135 q^{54} -55.0000 q^{55} +373.078 q^{56} +0.375990 q^{57} -176.376 q^{58} -868.321 q^{59} -0.170068 q^{60} -0.00261696 q^{61} +876.601 q^{62} +514.410 q^{63} +359.805 q^{64} +161.416 q^{65} +0.678613 q^{66} -142.102 q^{67} +87.9565 q^{68} +2.10766 q^{69} +296.981 q^{70} -369.589 q^{71} +528.696 q^{72} -72.0949 q^{73} -304.224 q^{74} +0.494724 q^{75} +32.6575 q^{76} -209.577 q^{77} -1.99161 q^{78} +152.754 q^{79} +373.981 q^{80} +728.968 q^{81} +710.106 q^{82} +133.570 q^{83} -0.648043 q^{84} -255.864 q^{85} +305.404 q^{86} -1.11958 q^{87} -215.398 q^{88} +533.419 q^{89} +420.857 q^{90} +615.074 q^{91} +183.066 q^{92} +5.56439 q^{93} +312.539 q^{94} -95.0000 q^{95} -1.51434 q^{96} +92.7307 q^{97} +62.3427 q^{98} -296.996 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 2 q^{2} + 9 q^{3} + 122 q^{4} - 125 q^{5} + 11 q^{6} - 15 q^{7} + 60 q^{8} + 300 q^{9}+O(q^{10}) \) 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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