LMFDB - Embedded newform 1045.4.a.i.1.2

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			1.2
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)

$f(q)$	$=$	$q-5.23688 q^{2} -8.48150 q^{3} +19.4249 q^{4} -5.00000 q^{5} +44.4166 q^{6} -12.7409 q^{7} -59.8310 q^{8} +44.9358 q^{9} +O(q^{10})$	\(q-5.23688 q^{2} -8.48150 q^{3} +19.4249 q^{4} -5.00000 q^{5} +44.4166 q^{6} -12.7409 q^{7} -59.8310 q^{8} +44.9358 q^{9} +26.1844 q^{10} +11.0000 q^{11} -164.752 q^{12} -58.3414 q^{13} +66.7225 q^{14} +42.4075 q^{15} +157.928 q^{16} +17.8420 q^{17} -235.324 q^{18} +19.0000 q^{19} -97.1246 q^{20} +108.062 q^{21} -57.6057 q^{22} -33.9305 q^{23} +507.456 q^{24} +25.0000 q^{25} +305.527 q^{26} -152.123 q^{27} -247.491 q^{28} -115.427 q^{29} -222.083 q^{30} -62.4550 q^{31} -348.404 q^{32} -93.2965 q^{33} -93.4364 q^{34} +63.7044 q^{35} +872.875 q^{36} +137.364 q^{37} -99.5007 q^{38} +494.822 q^{39} +299.155 q^{40} +405.171 q^{41} -565.907 q^{42} +320.474 q^{43} +213.674 q^{44} -224.679 q^{45} +177.690 q^{46} +194.233 q^{47} -1339.47 q^{48} -180.670 q^{49} -130.922 q^{50} -151.327 q^{51} -1133.28 q^{52} -435.693 q^{53} +796.649 q^{54} -55.0000 q^{55} +762.299 q^{56} -161.149 q^{57} +604.476 q^{58} -106.247 q^{59} +823.762 q^{60} -474.068 q^{61} +327.069 q^{62} -572.522 q^{63} +561.123 q^{64} +291.707 q^{65} +488.583 q^{66} -398.070 q^{67} +346.579 q^{68} +287.782 q^{69} -333.613 q^{70} -1175.77 q^{71} -2688.55 q^{72} -86.5445 q^{73} -719.358 q^{74} -212.038 q^{75} +369.074 q^{76} -140.150 q^{77} -2591.33 q^{78} -1118.87 q^{79} -789.641 q^{80} +76.9624 q^{81} -2121.83 q^{82} -488.984 q^{83} +2099.09 q^{84} -89.2099 q^{85} -1678.28 q^{86} +978.992 q^{87} -658.141 q^{88} -729.362 q^{89} +1176.62 q^{90} +743.321 q^{91} -659.098 q^{92} +529.712 q^{93} -1017.17 q^{94} -95.0000 q^{95} +2954.99 q^{96} -549.764 q^{97} +946.146 q^{98} +494.294 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−5.23688	−1.85152	−0.925759	−	0.378115i	$-0.876572\pi$
$2$	−5.23688	−1.85152	−0.925759	+	0.378115i	$0.876572\pi$
$3$	−8.48150	−1.63227	−0.816133	−	0.577864i	$-0.803886\pi$
$3$	−8.48150	−1.63227	−0.816133	+	0.577864i	$0.803886\pi$
$4$	19.4249	2.42812
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	44.4166	3.02217
$7$	−12.7409	−0.687943	−0.343972	−	0.938980i	$-0.611772\pi$
$7$	−12.7409	−0.687943	−0.343972	+	0.938980i	$0.611772\pi$
$8$	−59.8310	−2.64418
$9$	44.9358	1.66429
$10$	26.1844	0.828024
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−164.752	−3.96333
$13$	−58.3414	−1.24469	−0.622346	−	0.782742i	$-0.713820\pi$
$13$	−58.3414	−1.24469	−0.622346	+	0.782742i	$0.713820\pi$
$14$	66.7225	1.27374
$15$	42.4075	0.729971
$16$	157.928	2.46763
$17$	17.8420	0.254548	0.127274	−	0.991868i	$-0.459377\pi$
$17$	17.8420	0.254548	0.127274	+	0.991868i	$0.459377\pi$
$18$	−235.324	−3.08146
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−97.1246	−1.08589
$21$	108.062	1.12291
$22$	−57.6057	−0.558253
$23$	−33.9305	−0.307609	−0.153805	−	0.988101i	$-0.549153\pi$
$23$	−33.9305	−0.307609	−0.153805	+	0.988101i	$0.549153\pi$
$24$	507.456	4.31600
$25$	25.0000	0.200000
$26$	305.527	2.30457
$27$	−152.123	−1.08430
$28$	−247.491	−1.67041
$29$	−115.427	−0.739111	−0.369555	−	0.929209i	$-0.620490\pi$
$29$	−115.427	−0.739111	−0.369555	+	0.929209i	$0.620490\pi$
$30$	−222.083	−1.35155
$31$	−62.4550	−0.361846	−0.180923	−	0.983497i	$-0.557909\pi$
$31$	−62.4550	−0.361846	−0.180923	+	0.983497i	$0.557909\pi$
$32$	−348.404	−1.92468
$33$	−93.2965	−0.492147
$34$	−93.4364	−0.471300
$35$	63.7044	0.307658
$36$	872.875	4.04109
$37$	137.364	0.610337	0.305168	−	0.952298i	$-0.401287\pi$
$37$	137.364	0.610337	0.305168	+	0.952298i	$0.401287\pi$
$38$	−99.5007	−0.424767
$39$	494.822	2.03167
$40$	299.155	1.18251
$41$	405.171	1.54334	0.771671	−	0.636021i	$-0.219421\pi$
$41$	405.171	1.54334	0.771671	+	0.636021i	$0.219421\pi$
$42$	−565.907	−2.07908
$43$	320.474	1.13655	0.568277	−	0.822837i	$-0.307610\pi$
$43$	320.474	1.13655	0.568277	+	0.822837i	$0.307610\pi$
$44$	213.674	0.732104
$45$	−224.679	−0.744293
$46$	177.690	0.569543
$47$	194.233	0.602803	0.301402	−	0.953497i	$-0.402546\pi$
$47$	194.233	0.602803	0.301402	+	0.953497i	$0.402546\pi$
$48$	−1339.47	−4.02782
$49$	−180.670	−0.526734
$50$	−130.922	−0.370303
$51$	−151.327	−0.415490
$52$	−1133.28	−3.02225
$53$	−435.693	−1.12919	−0.564595	−	0.825368i	$-0.690968\pi$
$53$	−435.693	−1.12919	−0.564595	+	0.825368i	$0.690968\pi$
$54$	796.649	2.00760
$55$	−55.0000	−0.134840
$56$	762.299	1.81905
$57$	−161.149	−0.374467
$58$	604.476	1.36848
$59$	−106.247	−0.234444	−0.117222	−	0.993106i	$-0.537399\pi$
$59$	−106.247	−0.234444	−0.117222	+	0.993106i	$0.537399\pi$
$60$	823.762	1.77245
$61$	−474.068	−0.995053	−0.497526	−	0.867449i	$-0.665758\pi$
$61$	−474.068	−0.995053	−0.497526	+	0.867449i	$0.665758\pi$
$62$	327.069	0.669965
$63$	−572.522	−1.14494
$64$	561.123	1.09594
$65$	291.707	0.556643
$66$	488.583	0.911218
$67$	−398.070	−0.725850	−0.362925	−	0.931818i	$-0.618222\pi$
$67$	−398.070	−0.725850	−0.362925	+	0.931818i	$0.618222\pi$
$68$	346.579	0.618072
$69$	287.782	0.502100
$70$	−333.613	−0.569633
$71$	−1175.77	−1.96532	−0.982661	−	0.185409i	$-0.940639\pi$
$71$	−1175.77	−1.96532	−0.982661	+	0.185409i	$0.940639\pi$
$72$	−2688.55	−4.40068
$73$	−86.5445	−0.138757	−0.0693785	−	0.997590i	$-0.522102\pi$
$73$	−86.5445	−0.138757	−0.0693785	+	0.997590i	$0.522102\pi$
$74$	−719.358	−1.13005
$75$	−212.038	−0.326453
$76$	369.074	0.557048
$77$	−140.150	−0.207423
$78$	−2591.33	−3.76167
$79$	−1118.87	−1.59346	−0.796728	−	0.604338i	$-0.793438\pi$
$79$	−1118.87	−1.59346	−0.796728	+	0.604338i	$0.793438\pi$
$80$	−789.641	−1.10356
$81$	76.9624	0.105573
$82$	−2121.83	−2.85753
$83$	−488.984	−0.646662	−0.323331	−	0.946286i	$-0.604803\pi$
$83$	−488.984	−0.646662	−0.323331	+	0.946286i	$0.604803\pi$
$84$	2099.09	2.72654
$85$	−89.2099	−0.113837
$86$	−1678.28	−2.10435
$87$	978.992	1.20642
$88$	−658.141	−0.797250
$89$	−729.362	−0.868677	−0.434338	−	0.900750i	$-0.643018\pi$
$89$	−729.362	−0.868677	−0.434338	+	0.900750i	$0.643018\pi$
$90$	1176.62	1.37807
$91$	743.321	0.856277
$92$	−659.098	−0.746910
$93$	529.712	0.590629
$94$	−1017.17	−1.11610
$95$	−95.0000	−0.102598
$96$	2954.99	3.14158
$97$	−549.764	−0.575465	−0.287732	−	0.957711i	$-0.592901\pi$
$97$	−549.764	−0.575465	−0.287732	+	0.957711i	$0.592901\pi$
$98$	946.146	0.975257
$99$	494.294	0.501803
$100$	485.623	0.485623
$101$	1053.86	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$101$	1053.86	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$102$	792.481	0.769287
$103$	−1586.51	−1.51770	−0.758852	−	0.651263i	$-0.774239\pi$
$103$	−1586.51	−1.51770	−0.758852	+	0.651263i	$0.774239\pi$
$104$	3490.62	3.29119
$105$	−540.309	−0.502179
$106$	2281.67	2.09071
$107$	1513.28	1.36723	0.683616	−	0.729842i	$-0.260406\pi$
$107$	1513.28	1.36723	0.683616	+	0.729842i	$0.260406\pi$
$108$	−2954.98	−2.63280
$109$	−104.313	−0.0916638	−0.0458319	−	0.998949i	$-0.514594\pi$
$109$	−104.313	−0.0916638	−0.0458319	+	0.998949i	$0.514594\pi$
$110$	288.028	0.249659
$111$	−1165.05	−0.996232
$112$	−2012.15	−1.69759
$113$	1017.29	0.846891	0.423446	−	0.905921i	$-0.360820\pi$
$113$	1017.29	0.846891	0.423446	+	0.905921i	$0.360820\pi$
$114$	843.916	0.693333
$115$	169.653	0.137567
$116$	−2242.16	−1.79465
$117$	−2621.62	−2.07153
$118$	556.403	0.434076
$119$	−227.323	−0.175115
$120$	−2537.28	−1.93018
$121$	121.000	0.0909091
$122$	2482.64	1.84236
$123$	−3436.46	−2.51915
$124$	−1213.18	−0.878605
$125$	−125.000	−0.0894427
$126$	2998.23	2.11987
$127$	−2216.20	−1.54847	−0.774237	−	0.632896i	$-0.781866\pi$
$127$	−2216.20	−1.54847	−0.774237	+	0.632896i	$0.781866\pi$
$128$	−151.303	−0.104480
$129$	−2718.10	−1.85516
$130$	−1527.63	−1.03063
$131$	1226.64	0.818108	0.409054	−	0.912510i	$-0.365859\pi$
$131$	1226.64	0.818108	0.409054	+	0.912510i	$0.365859\pi$
$132$	−1812.28	−1.19499
$133$	−242.077	−0.157825
$134$	2084.64	1.34392
$135$	760.614	0.484913
$136$	−1067.50	−0.673071
$137$	−1144.81	−0.713927	−0.356963	−	0.934118i	$-0.616188\pi$
$137$	−1144.81	−0.713927	−0.356963	+	0.934118i	$0.616188\pi$
$138$	−1507.08	−0.929646
$139$	−1210.57	−0.738702	−0.369351	−	0.929290i	$-0.620420\pi$
$139$	−1210.57	−0.738702	−0.369351	+	0.929290i	$0.620420\pi$
$140$	1237.45	0.747028
$141$	−1647.38	−0.983935
$142$	6157.35	3.63883
$143$	−641.755	−0.375289
$144$	7096.64	4.10685
$145$	577.134	0.330540
$146$	453.223	0.256911
$147$	1532.35	0.859770
$148$	2668.28	1.48197
$149$	−1670.58	−0.918519	−0.459260	−	0.888302i	$-0.651885\pi$
$149$	−1670.58	−0.918519	−0.459260	+	0.888302i	$0.651885\pi$
$150$	1110.42	0.604433
$151$	−1339.67	−0.721994	−0.360997	−	0.932567i	$-0.617564\pi$
$151$	−1339.67	−0.721994	−0.360997	+	0.932567i	$0.617564\pi$
$152$	−1136.79	−0.606616
$153$	801.745	0.423642
$154$	733.948	0.384047
$155$	312.275	0.161823
$156$	9611.89	4.93312
$157$	−2965.59	−1.50752	−0.753758	−	0.657152i	$-0.771761\pi$
$157$	−2965.59	−1.50752	−0.753758	+	0.657152i	$0.771761\pi$
$158$	5859.40	2.95031
$159$	3695.33	1.84314
$160$	1742.02	0.860741
$161$	432.305	0.211618
$162$	−403.043	−0.195470
$163$	1052.72	0.505860	0.252930	−	0.967485i	$-0.418606\pi$
$163$	1052.72	0.505860	0.252930	+	0.967485i	$0.418606\pi$
$164$	7870.41	3.74741
$165$	466.483	0.220095
$166$	2560.75	1.19731
$167$	2765.12	1.28127	0.640634	−	0.767847i	$-0.278672\pi$
$167$	2765.12	1.28127	0.640634	+	0.767847i	$0.278672\pi$
$168$	−6465.44	−2.96917
$169$	1206.72	0.549257
$170$	467.182	0.210772
$171$	853.781	0.381814
$172$	6225.18	2.75968
$173$	139.447	0.0612830	0.0306415	−	0.999530i	$-0.490245\pi$
$173$	139.447	0.0612830	0.0306415	+	0.999530i	$0.490245\pi$
$174$	−5126.86	−2.23372
$175$	−318.522	−0.137589
$176$	1737.21	0.744018
$177$	901.134	0.382674
$178$	3819.58	1.60837
$179$	−1643.56	−0.686285	−0.343143	−	0.939283i	$-0.611491\pi$
$179$	−1643.56	−0.686285	−0.343143	+	0.939283i	$0.611491\pi$
$180$	−4364.38	−1.80723
$181$	−3535.62	−1.45194	−0.725968	−	0.687728i	$-0.758608\pi$
$181$	−3535.62	−1.45194	−0.725968	+	0.687728i	$0.758608\pi$
$182$	−3892.68	−1.58541
$183$	4020.81	1.62419
$184$	2030.10	0.813374
$185$	−686.819	−0.272951
$186$	−2774.04	−1.09356
$187$	196.262	0.0767491
$188$	3772.95	1.46368
$189$	1938.18	0.745936
$190$	497.504	0.189962
$191$	4408.50	1.67009	0.835047	−	0.550178i	$-0.185440\pi$
$191$	4408.50	1.67009	0.835047	+	0.550178i	$0.185440\pi$
$192$	−4759.16	−1.78887
$193$	−671.571	−0.250470	−0.125235	−	0.992127i	$-0.539968\pi$
$193$	−671.571	−0.250470	−0.125235	+	0.992127i	$0.539968\pi$
$194$	2879.05	1.06548
$195$	−2474.11	−0.908589
$196$	−3509.50	−1.27897
$197$	−1002.05	−0.362403	−0.181202	−	0.983446i	$-0.557999\pi$
$197$	−1002.05	−0.362403	−0.181202	+	0.983446i	$0.557999\pi$
$198$	−2588.56	−0.929096
$199$	3215.27	1.14535	0.572675	−	0.819783i	$-0.305906\pi$
$199$	3215.27	1.14535	0.572675	+	0.819783i	$0.305906\pi$
$200$	−1495.77	−0.528836
$201$	3376.23	1.18478
$202$	−5518.93	−1.92233
$203$	1470.64	0.508466
$204$	−2939.51	−1.00886
$205$	−2025.85	−0.690204
$206$	8308.37	2.81005
$207$	−1524.70	−0.511951
$208$	−9213.75	−3.07144
$209$	209.000	0.0691714
$210$	2829.53	0.929793
$211$	4290.52	1.39986	0.699932	−	0.714210i	$-0.253214\pi$
$211$	4290.52	1.39986	0.699932	+	0.714210i	$0.253214\pi$
$212$	−8463.30	−2.74180
$213$	9972.27	3.20793
$214$	−7924.84	−2.53145
$215$	−1602.37	−0.508282
$216$	9101.66	2.86708
$217$	795.731	0.248930
$218$	546.273	0.169717
$219$	734.027	0.226488
$220$	−1068.37	−0.327407
$221$	−1040.93	−0.316834
$222$	6101.23	1.84454
$223$	−513.087	−0.154076	−0.0770378	−	0.997028i	$-0.524546\pi$
$223$	−513.087	−0.154076	−0.0770378	+	0.997028i	$0.524546\pi$
$224$	4438.97	1.32407
$225$	1123.40	0.332858
$226$	−5327.43	−1.56803
$227$	1042.72	0.304881	0.152441	−	0.988313i	$-0.451287\pi$
$227$	1042.72	0.304881	0.152441	+	0.988313i	$0.451287\pi$
$228$	−3130.30	−0.909250
$229$	−1542.99	−0.445257	−0.222628	−	0.974903i	$-0.571464\pi$
$229$	−1542.99	−0.445257	−0.222628	+	0.974903i	$0.571464\pi$
$230$	−888.451	−0.254708
$231$	1188.68	0.338569
$232$	6906.09	1.95434
$233$	−2182.49	−0.613648	−0.306824	−	0.951766i	$-0.599266\pi$
$233$	−2182.49	−0.613648	−0.306824	+	0.951766i	$0.599266\pi$
$234$	13729.1	3.83547
$235$	−971.163	−0.269582
$236$	−2063.84	−0.569256
$237$	9489.72	2.60094
$238$	1190.46	0.324228
$239$	167.011	0.0452012	0.0226006	−	0.999745i	$-0.492805\pi$
$239$	167.011	0.0452012	0.0226006	+	0.999745i	$0.492805\pi$
$240$	6697.34	1.80130
$241$	−2055.77	−0.549477	−0.274738	−	0.961519i	$-0.588591\pi$
$241$	−2055.77	−0.549477	−0.274738	+	0.961519i	$0.588591\pi$
$242$	−633.663	−0.168320
$243$	3454.56	0.911976
$244$	−9208.74	−2.41610
$245$	903.349	0.235563
$246$	17996.3	4.66424
$247$	−1108.49	−0.285552
$248$	3736.74	0.956787
$249$	4147.31	1.05552
$250$	654.610	0.165605
$251$	6347.22	1.59615	0.798073	−	0.602560i	$-0.205853\pi$
$251$	6347.22	1.59615	0.798073	+	0.602560i	$0.205853\pi$
$252$	−11121.2	−2.78004
$253$	−373.236	−0.0927476
$254$	11606.0	2.86703
$255$	756.634	0.185813
$256$	−3696.62	−0.902496
$257$	−421.377	−0.102275	−0.0511377	−	0.998692i	$-0.516285\pi$
$257$	−421.377	−0.102275	−0.0511377	+	0.998692i	$0.516285\pi$
$258$	14234.4	3.43486
$259$	−1750.14	−0.419877
$260$	5666.38	1.35159
$261$	−5186.80	−1.23009
$262$	−6423.77	−1.51474
$263$	−384.024	−0.0900378	−0.0450189	−	0.998986i	$-0.514335\pi$
$263$	−384.024	−0.0900378	−0.0450189	+	0.998986i	$0.514335\pi$
$264$	5582.02	1.30132
$265$	2178.47	0.504989
$266$	1267.73	0.292216
$267$	6186.09	1.41791
$268$	−7732.48	−1.76245
$269$	5268.64	1.19418	0.597090	−	0.802174i	$-0.296323\pi$
$269$	5268.64	1.19418	0.597090	+	0.802174i	$0.296323\pi$
$270$	−3983.25	−0.897825
$271$	−337.080	−0.0755577	−0.0377789	−	0.999286i	$-0.512028\pi$
$271$	−337.080	−0.0755577	−0.0377789	+	0.999286i	$0.512028\pi$
$272$	2817.75	0.628130
$273$	−6304.48	−1.39767
$274$	5995.25	1.32185
$275$	275.000	0.0603023
$276$	5590.14	1.21916
$277$	319.320	0.0692638	0.0346319	−	0.999400i	$-0.488974\pi$
$277$	319.320	0.0692638	0.0346319	+	0.999400i	$0.488974\pi$
$278$	6339.63	1.36772
$279$	−2806.47	−0.602218
$280$	−3811.50	−0.813502
$281$	−2286.00	−0.485307	−0.242653	−	0.970113i	$-0.578018\pi$
$281$	−2286.00	−0.485307	−0.242653	+	0.970113i	$0.578018\pi$
$282$	8627.16	1.82177
$283$	−1983.67	−0.416669	−0.208334	−	0.978058i	$-0.566804\pi$
$283$	−1983.67	−0.416669	−0.208334	+	0.978058i	$0.566804\pi$
$284$	−22839.2	−4.77203
$285$	805.743	0.167467
$286$	3360.80	0.694853
$287$	−5162.24	−1.06173
$288$	−15655.8	−3.20322
$289$	−4594.66	−0.935205
$290$	−3022.38	−0.612001
$291$	4662.82	0.939311
$292$	−1681.12	−0.336918
$293$	−6110.31	−1.21832	−0.609161	−	0.793047i	$-0.708494\pi$
$293$	−6110.31	−1.21832	−0.609161	+	0.793047i	$0.708494\pi$
$294$	−8024.74	−1.59188
$295$	531.235	0.104846
$296$	−8218.60	−1.61384
$297$	−1673.35	−0.326928
$298$	8748.64	1.70065
$299$	1979.56	0.382878
$300$	−4118.81	−0.792666
$301$	−4083.12	−0.781885
$302$	7015.71	1.33678
$303$	−8938.29	−1.69469
$304$	3000.64	0.566113
$305$	2370.34	0.445001
$306$	−4198.64	−0.784380
$307$	5108.41	0.949683	0.474841	−	0.880071i	$-0.342505\pi$
$307$	5108.41	0.949683	0.474841	+	0.880071i	$0.342505\pi$
$308$	−2722.40	−0.503646
$309$	13456.0	2.47730
$310$	−1635.35	−0.299617
$311$	−2964.97	−0.540604	−0.270302	−	0.962776i	$-0.587124\pi$
$311$	−2964.97	−0.540604	−0.270302	+	0.962776i	$0.587124\pi$
$312$	−29605.7	−5.37209
$313$	−3880.64	−0.700787	−0.350394	−	0.936602i	$-0.613952\pi$
$313$	−3880.64	−0.700787	−0.350394	+	0.936602i	$0.613952\pi$
$314$	15530.4	2.79119
$315$	2862.61	0.512032
$316$	−21734.0	−3.86909
$317$	4339.29	0.768829	0.384415	−	0.923161i	$-0.374403\pi$
$317$	4339.29	0.768829	0.384415	+	0.923161i	$0.374403\pi$
$318$	−19352.0	−3.41260
$319$	−1269.69	−0.222850
$320$	−2805.61	−0.490120
$321$	−12834.8	−2.23169
$322$	−2263.93	−0.391813
$323$	338.998	0.0583973
$324$	1494.99	0.256343
$325$	−1458.53	−0.248938
$326$	−5512.95	−0.936608
$327$	884.729	0.149620
$328$	−24241.8	−4.08088
$329$	−2474.70	−0.414694
$330$	−2442.91	−0.407509
$331$	−11157.0	−1.85271	−0.926353	−	0.376657i	$-0.877073\pi$
$331$	−11157.0	−1.85271	−0.926353	+	0.376657i	$0.877073\pi$
$332$	−9498.47	−1.57017
$333$	6172.56	1.01578
$334$	−14480.6	−2.37229
$335$	1990.35	0.324610
$336$	17066.0	2.77091
$337$	6118.30	0.988977	0.494488	−	0.869184i	$-0.335355\pi$
$337$	6118.30	0.988977	0.494488	+	0.869184i	$0.335355\pi$
$338$	−6319.44	−1.01696
$339$	−8628.16	−1.38235
$340$	−1732.90	−0.276410
$341$	−687.004	−0.109101
$342$	−4471.15	−0.706936
$343$	6672.02	1.05031
$344$	−19174.3	−3.00525
$345$	−1438.91	−0.224546
$346$	−730.267	−0.113467
$347$	−278.258	−0.0430480	−0.0215240	−	0.999768i	$-0.506852\pi$
$347$	−278.258	−0.0430480	−0.0215240	+	0.999768i	$0.506852\pi$
$348$	19016.8	2.92934
$349$	8872.28	1.36081	0.680405	−	0.732837i	$-0.261804\pi$
$349$	8872.28	1.36081	0.680405	+	0.732837i	$0.261804\pi$
$350$	1668.06	0.254748
$351$	8875.06	1.34962
$352$	−3832.44	−0.580312
$353$	−1957.89	−0.295207	−0.147604	−	0.989047i	$-0.547156\pi$
$353$	−1957.89	−0.295207	−0.147604	+	0.989047i	$0.547156\pi$
$354$	−4719.13	−0.708528
$355$	5878.84	0.878919
$356$	−14167.8	−2.10925
$357$	1928.04	0.285834
$358$	8607.10	1.27067
$359$	−10293.5	−1.51329	−0.756644	−	0.653827i	$-0.773163\pi$
$359$	−10293.5	−1.51329	−0.756644	+	0.653827i	$0.773163\pi$
$360$	13442.8	1.96805
$361$	361.000	0.0526316
$362$	18515.6	2.68829
$363$	−1026.26	−0.148388
$364$	14439.0	2.07914
$365$	432.722	0.0620540
$366$	−21056.5	−3.00722
$367$	−5917.09	−0.841607	−0.420803	−	0.907152i	$-0.638252\pi$
$367$	−5917.09	−0.841607	−0.420803	+	0.907152i	$0.638252\pi$
$368$	−5358.59	−0.759065
$369$	18206.7	2.56857
$370$	3596.79	0.505373
$371$	5551.12	0.776818
$372$	10289.6	1.43412
$373$	3567.34	0.495200	0.247600	−	0.968862i	$-0.420358\pi$
$373$	3567.34	0.495200	0.247600	+	0.968862i	$0.420358\pi$
$374$	−1027.80	−0.142102
$375$	1060.19	0.145994
$376$	−11621.1	−1.59392
$377$	6734.16	0.919965
$378$	−10150.0	−1.38111
$379$	7997.10	1.08386	0.541931	−	0.840423i	$-0.317693\pi$
$379$	7997.10	1.08386	0.541931	+	0.840423i	$0.317693\pi$
$380$	−1845.37	−0.249119
$381$	18796.7	2.52752
$382$	−23086.8	−3.09221
$383$	−455.049	−0.0607100	−0.0303550	−	0.999539i	$-0.509664\pi$
$383$	−455.049	−0.0607100	−0.0303550	+	0.999539i	$0.509664\pi$
$384$	1283.28	0.170539
$385$	700.749	0.0927622
$386$	3516.94	0.463750
$387$	14400.8	1.89156
$388$	−10679.1	−1.39729
$389$	11328.7	1.47658	0.738290	−	0.674484i	$-0.235634\pi$
$389$	11328.7	1.47658	0.738290	+	0.674484i	$0.235634\pi$
$390$	12956.6	1.68227
$391$	−605.388	−0.0783013
$392$	10809.6	1.39278
$393$	−10403.8	−1.33537
$394$	5247.64	0.670996
$395$	5594.36	0.712615
$396$	9601.63	1.21843
$397$	−11921.5	−1.50711	−0.753555	−	0.657385i	$-0.771663\pi$
$397$	−11921.5	−1.50711	−0.753555	+	0.657385i	$0.771663\pi$
$398$	−16838.0	−2.12063
$399$	2053.17	0.257612
$400$	3948.21	0.493526
$401$	−958.326	−0.119343	−0.0596715	−	0.998218i	$-0.519005\pi$
$401$	−958.326	−0.119343	−0.0596715	+	0.998218i	$0.519005\pi$
$402$	−17680.9	−2.19364
$403$	3643.71	0.450387
$404$	20471.1	2.52098
$405$	−384.812	−0.0472135
$406$	−7701.56	−0.941434
$407$	1511.00	0.184023
$408$	9054.03	1.09863
$409$	−3744.68	−0.452720	−0.226360	−	0.974044i	$-0.572683\pi$
$409$	−3744.68	−0.452720	−0.226360	+	0.974044i	$0.572683\pi$
$410$	10609.2	1.27792
$411$	9709.73	1.16532
$412$	−30817.8	−3.68516
$413$	1353.68	0.161284
$414$	7984.66	0.947886
$415$	2444.92	0.289196
$416$	20326.4	2.39563
$417$	10267.5	1.20576
$418$	−1094.51	−0.128072
$419$	14102.4	1.64427	0.822133	−	0.569296i	$-0.192784\pi$
$419$	14102.4	1.64427	0.822133	+	0.569296i	$0.192784\pi$
$420$	−10495.5	−1.21935
$421$	−9043.46	−1.04692	−0.523458	−	0.852052i	$-0.675358\pi$
$421$	−9043.46	−1.04692	−0.523458	+	0.852052i	$0.675358\pi$
$422$	−22468.9	−2.59187
$423$	8728.01	1.00324
$424$	26067.9	2.98578
$425$	446.050	0.0509096
$426$	−52223.6	−5.93953
$427$	6040.05	0.684540
$428$	29395.3	3.31980
$429$	5443.05	0.612571
$430$	8391.42	0.941094
$431$	−13023.0	−1.45544	−0.727720	−	0.685874i	$-0.759420\pi$
$431$	−13023.0	−1.45544	−0.727720	+	0.685874i	$0.759420\pi$
$432$	−24024.5	−2.67565
$433$	−3086.93	−0.342606	−0.171303	−	0.985218i	$-0.554798\pi$
$433$	−3086.93	−0.342606	−0.171303	+	0.985218i	$0.554798\pi$
$434$	−4167.15	−0.460898
$435$	−4894.96	−0.539530
$436$	−2026.27	−0.222570
$437$	−644.680	−0.0705704
$438$	−3844.01	−0.419347
$439$	9808.97	1.06642	0.533208	−	0.845984i	$-0.320986\pi$
$439$	9808.97	1.06642	0.533208	+	0.845984i	$0.320986\pi$
$440$	3290.70	0.356541
$441$	−8118.55	−0.876639
$442$	5451.21	0.586623
$443$	−7952.67	−0.852919	−0.426459	−	0.904507i	$-0.640239\pi$
$443$	−7952.67	−0.852919	−0.426459	+	0.904507i	$0.640239\pi$
$444$	−22631.0	−2.41897
$445$	3646.81	0.388484
$446$	2686.98	0.285274
$447$	14169.0	1.49927
$448$	−7149.20	−0.753946
$449$	9498.26	0.998331	0.499165	−	0.866507i	$-0.333640\pi$
$449$	9498.26	0.998331	0.499165	+	0.866507i	$0.333640\pi$
$450$	−5883.09	−0.616292
$451$	4456.88	0.465335
$452$	19760.8	2.05635
$453$	11362.4	1.17849
$454$	−5460.62	−0.564493
$455$	−3716.60	−0.382939
$456$	9641.67	0.990159
$457$	−3575.38	−0.365972	−0.182986	−	0.983116i	$-0.558576\pi$
$457$	−3575.38	−0.365972	−0.182986	+	0.983116i	$0.558576\pi$
$458$	8080.47	0.824400
$459$	−2714.17	−0.276006
$460$	3295.49	0.334028
$461$	−10867.5	−1.09794	−0.548970	−	0.835842i	$-0.684980\pi$
$461$	−10867.5	−1.09794	−0.548970	+	0.835842i	$0.684980\pi$
$462$	−6224.98	−0.626866
$463$	−253.532	−0.0254485	−0.0127242	−	0.999919i	$-0.504050\pi$
$463$	−253.532	−0.0254485	−0.0127242	+	0.999919i	$0.504050\pi$
$464$	−18229.1	−1.82385
$465$	−2648.56	−0.264138
$466$	11429.5	1.13618
$467$	10925.1	1.08256	0.541280	−	0.840843i	$-0.317940\pi$
$467$	10925.1	1.08256	0.541280	+	0.840843i	$0.317940\pi$
$468$	−50924.8	−5.02991
$469$	5071.76	0.499344
$470$	5085.87	0.499135
$471$	25152.7	2.46067
$472$	6356.86	0.619911
$473$	3525.21	0.342684
$474$	−49696.5	−4.81569
$475$	475.000	0.0458831
$476$	−4415.73	−0.425198
$477$	−19578.2	−1.87930
$478$	−874.619	−0.0836907
$479$	18830.8	1.79624	0.898122	−	0.439747i	$-0.144932\pi$
$479$	18830.8	1.79624	0.898122	+	0.439747i	$0.144932\pi$
$480$	−14774.9	−1.40496
$481$	−8013.99	−0.759681
$482$	10765.8	1.01737
$483$	−3666.60	−0.345416
$484$	2350.42	0.220738
$485$	2748.82	0.257356
$486$	−18091.1	−1.68854
$487$	15469.7	1.43943	0.719713	−	0.694272i	$-0.244273\pi$
$487$	15469.7	1.43943	0.719713	+	0.694272i	$0.244273\pi$
$488$	28364.0	2.63110
$489$	−8928.61	−0.825697
$490$	−4730.73	−0.436148
$491$	12275.1	1.12824	0.564120	−	0.825693i	$-0.309215\pi$
$491$	12275.1	1.12824	0.564120	+	0.825693i	$0.309215\pi$
$492$	−66752.9	−6.11678
$493$	−2059.44	−0.188139
$494$	5805.01	0.528704
$495$	−2471.47	−0.224413
$496$	−9863.40	−0.892903
$497$	14980.3	1.35203
$498$	−21719.0	−1.95432
$499$	950.614	0.0852812	0.0426406	−	0.999090i	$-0.486423\pi$
$499$	950.614	0.0852812	0.0426406	+	0.999090i	$0.486423\pi$
$500$	−2428.12	−0.217177
$501$	−23452.4	−2.09137
$502$	−33239.6	−2.95529
$503$	−21242.2	−1.88299	−0.941493	−	0.337033i	$-0.890577\pi$
$503$	−21242.2	−1.88299	−0.941493	+	0.337033i	$0.890577\pi$
$504$	34254.6	3.02742
$505$	−5269.29	−0.464317
$506$	1954.59	0.171724
$507$	−10234.8	−0.896533
$508$	−43049.6	−3.75987
$509$	−18882.4	−1.64430	−0.822150	−	0.569271i	$-0.807225\pi$
$509$	−18882.4	−1.64430	−0.822150	+	0.569271i	$0.807225\pi$
$510$	−3962.40	−0.344036
$511$	1102.65	0.0954570
$512$	20569.2	1.77547
$513$	−2890.33	−0.248755
$514$	2206.70	0.189365
$515$	7932.55	0.678738
$516$	−52798.9	−4.50454
$517$	2136.56	0.181752
$518$	9165.25	0.777409
$519$	−1182.72	−0.100030
$520$	−17453.1	−1.47186
$521$	−19046.7	−1.60164	−0.800819	−	0.598907i	$-0.795602\pi$
$521$	−19046.7	−1.60164	−0.800819	+	0.598907i	$0.795602\pi$
$522$	27162.6	2.27754
$523$	7156.45	0.598336	0.299168	−	0.954200i	$-0.403291\pi$
$523$	7156.45	0.598336	0.299168	+	0.954200i	$0.403291\pi$
$524$	23827.4	1.98646
$525$	2701.55	0.224581
$526$	2011.09	0.166706
$527$	−1114.32	−0.0921073
$528$	−14734.2	−1.21443
$529$	−11015.7	−0.905377
$530$	−11408.4	−0.934995
$531$	−4774.30	−0.390182
$532$	−4702.32	−0.383217
$533$	−23638.2	−1.92099
$534$	−32395.8	−2.62529
$535$	−7566.38	−0.611445
$536$	23816.9	1.91928
$537$	13939.8	1.12020
$538$	−27591.2	−2.21105
$539$	−1987.37	−0.158816
$540$	14774.9	1.17742
$541$	20033.2	1.59204	0.796020	−	0.605270i	$-0.206935\pi$
$541$	20033.2	1.59204	0.796020	+	0.605270i	$0.206935\pi$
$542$	1765.25	0.139896
$543$	29987.4	2.36995
$544$	−6216.21	−0.489923
$545$	521.564	0.0409933
$546$	33015.8	2.58781
$547$	11074.3	0.865635	0.432818	−	0.901481i	$-0.357519\pi$
$547$	11074.3	0.865635	0.432818	+	0.901481i	$0.357519\pi$
$548$	−22237.9	−1.73350
$549$	−21302.7	−1.65606
$550$	−1440.14	−0.111651
$551$	−2193.11	−0.169564
$552$	−17218.3	−1.32764
$553$	14255.4	1.09621
$554$	−1672.24	−0.128243
$555$	5825.25	0.445528
$556$	−23515.3	−1.79365
$557$	8939.48	0.680032	0.340016	−	0.940420i	$-0.389567\pi$
$557$	8939.48	0.680032	0.340016	+	0.940420i	$0.389567\pi$
$558$	14697.1	1.11502
$559$	−18696.9	−1.41466
$560$	10060.7	0.759185
$561$	−1664.59	−0.125275
$562$	11971.5	0.898554
$563$	5793.35	0.433678	0.216839	−	0.976207i	$-0.430425\pi$
$563$	5793.35	0.433678	0.216839	+	0.976207i	$0.430425\pi$
$564$	−32000.3	−2.38911
$565$	−5086.46	−0.378741
$566$	10388.3	0.771469
$567$	−980.570	−0.0726280
$568$	70347.3	5.19667
$569$	−2104.66	−0.155065	−0.0775326	−	0.996990i	$-0.524704\pi$
$569$	−2104.66	−0.155065	−0.0775326	+	0.996990i	$0.524704\pi$
$570$	−4219.58	−0.310068
$571$	14455.6	1.05945	0.529726	−	0.848169i	$-0.322295\pi$
$571$	14455.6	1.05945	0.529726	+	0.848169i	$0.322295\pi$
$572$	−12466.0	−0.911244
$573$	−37390.7	−2.72604
$574$	27034.0	1.96582
$575$	−848.264	−0.0615218
$576$	25214.5	1.82397
$577$	937.849	0.0676658	0.0338329	−	0.999428i	$-0.489229\pi$
$577$	937.849	0.0676658	0.0338329	+	0.999428i	$0.489229\pi$
$578$	24061.7	1.73155
$579$	5695.93	0.408834
$580$	11210.8	0.802590
$581$	6230.08	0.444866
$582$	−24418.7	−1.73915
$583$	−4792.62	−0.340463
$584$	5178.04	0.366899
$585$	13108.1	0.926416
$586$	31999.0	2.25574
$587$	16377.0	1.15153	0.575767	−	0.817614i	$-0.304704\pi$
$587$	16377.0	1.15153	0.575767	+	0.817614i	$0.304704\pi$
$588$	29765.8	2.08762
$589$	−1186.64	−0.0830133
$590$	−2782.01	−0.194125
$591$	8498.92	0.591538
$592$	21693.6	1.50608
$593$	17309.3	1.19866	0.599331	−	0.800501i	$-0.295433\pi$
$593$	17309.3	1.19866	0.599331	+	0.800501i	$0.295433\pi$
$594$	8763.14	0.605313
$595$	1136.61	0.0783136
$596$	−32450.9	−2.23027
$597$	−27270.3	−1.86951
$598$	−10366.7	−0.708906
$599$	11575.1	0.789558	0.394779	−	0.918776i	$-0.370821\pi$
$599$	11575.1	0.789558	0.394779	+	0.918776i	$0.370821\pi$
$600$	12686.4	0.863201
$601$	11699.2	0.794046	0.397023	−	0.917809i	$-0.370043\pi$
$601$	11699.2	0.794046	0.397023	+	0.917809i	$0.370043\pi$
$602$	21382.8	1.44767
$603$	−17887.6	−1.20803
$604$	−26023.1	−1.75308
$605$	−605.000	−0.0406558
$606$	46808.8	3.13775
$607$	26464.5	1.76962	0.884812	−	0.465948i	$-0.154287\pi$
$607$	26464.5	1.76962	0.884812	+	0.465948i	$0.154287\pi$
$608$	−6619.67	−0.441551
$609$	−12473.2	−0.829952
$610$	−12413.2	−0.823927
$611$	−11331.8	−0.750304
$612$	15573.8	1.02865
$613$	3297.27	0.217252	0.108626	−	0.994083i	$-0.465355\pi$
$613$	3297.27	0.217252	0.108626	+	0.994083i	$0.465355\pi$
$614$	−26752.2	−1.75835
$615$	17182.3	1.12660
$616$	8385.29	0.548463
$617$	−11829.6	−0.771864	−0.385932	−	0.922527i	$-0.626120\pi$
$617$	−11829.6	−0.771864	−0.385932	+	0.922527i	$0.626120\pi$
$618$	−70467.4	−4.58676
$619$	−3569.59	−0.231783	−0.115892	−	0.993262i	$-0.536973\pi$
$619$	−3569.59	−0.231783	−0.115892	+	0.993262i	$0.536973\pi$
$620$	6065.91	0.392924
$621$	5161.61	0.333540
$622$	15527.2	1.00094
$623$	9292.72	0.597600
$624$	78146.4	5.01340
$625$	625.000	0.0400000
$626$	20322.4	1.29752
$627$	−1772.63	−0.112906
$628$	−57606.4	−3.66042
$629$	2450.84	0.155360
$630$	−14991.2	−0.948035
$631$	27922.3	1.76160	0.880798	−	0.473492i	$-0.157007\pi$
$631$	27922.3	1.76160	0.880798	+	0.473492i	$0.157007\pi$
$632$	66943.2	4.21338
$633$	−36390.0	−2.28495
$634$	−22724.4	−1.42350
$635$	11081.0	0.692499
$636$	71781.5	4.47535
$637$	10540.5	0.655622
$638$	6649.24	0.412611
$639$	−52834.1	−3.27087
$640$	756.517	0.0467249
$641$	14897.6	0.917971	0.458986	−	0.888444i	$-0.348213\pi$
$641$	14897.6	0.917971	0.458986	+	0.888444i	$0.348213\pi$
$642$	67214.5	4.13200
$643$	−22604.2	−1.38635	−0.693175	−	0.720769i	$-0.743789\pi$
$643$	−22604.2	−1.38635	−0.693175	+	0.720769i	$0.743789\pi$
$644$	8397.50	0.513832
$645$	13590.5	0.829652
$646$	−1775.29	−0.108124
$647$	5130.34	0.311738	0.155869	−	0.987778i	$-0.450182\pi$
$647$	5130.34	0.311738	0.155869	+	0.987778i	$0.450182\pi$
$648$	−4604.74	−0.279153
$649$	−1168.72	−0.0706874
$650$	7638.17	0.460914
$651$	−6749.00	−0.406320
$652$	20448.9	1.22829
$653$	−12532.2	−0.751030	−0.375515	−	0.926816i	$-0.622534\pi$
$653$	−12532.2	−0.751030	−0.375515	+	0.926816i	$0.622534\pi$
$654$	−4633.22	−0.277023
$655$	−6133.20	−0.365869
$656$	63987.9	3.80840
$657$	−3888.95	−0.230932
$658$	12959.7	0.767814
$659$	−15314.6	−0.905268	−0.452634	−	0.891696i	$-0.649516\pi$
$659$	−15314.6	−0.905268	−0.452634	+	0.891696i	$0.649516\pi$
$660$	9061.39	0.534415
$661$	21086.2	1.24078	0.620392	−	0.784292i	$-0.286973\pi$
$661$	21086.2	1.24078	0.620392	+	0.784292i	$0.286973\pi$
$662$	58428.0	3.43032
$663$	8828.62	0.517157
$664$	29256.4	1.70989
$665$	1210.38	0.0705815
$666$	−32324.9	−1.88073
$667$	3916.49	0.227357
$668$	53712.3	3.11107
$669$	4351.75	0.251492
$670$	−10423.2	−0.601021
$671$	−5214.75	−0.300020
$672$	−37649.1	−2.16123
$673$	−24699.4	−1.41470	−0.707349	−	0.706864i	$-0.750109\pi$
$673$	−24699.4	−1.41470	−0.707349	+	0.706864i	$0.750109\pi$
$674$	−32040.8	−1.83111
$675$	−3803.07	−0.216860
$676$	23440.4	1.33366
$677$	2342.60	0.132989	0.0664943	−	0.997787i	$-0.478819\pi$
$677$	2342.60	0.132989	0.0664943	+	0.997787i	$0.478819\pi$
$678$	45184.6	2.55945
$679$	7004.48	0.395887
$680$	5337.52	0.301006
$681$	−8843.87	−0.497647
$682$	3597.76	0.202002
$683$	−27629.7	−1.54791	−0.773955	−	0.633241i	$-0.781724\pi$
$683$	−27629.7	−1.54791	−0.773955	+	0.633241i	$0.781724\pi$
$684$	16584.6	0.927090
$685$	5724.06	0.319278
$686$	−34940.6	−1.94466
$687$	13086.9	0.726777
$688$	50611.9	2.80459
$689$	25418.9	1.40549
$690$	7535.40	0.415750
$691$	8699.77	0.478951	0.239475	−	0.970902i	$-0.423025\pi$
$691$	8699.77	0.478951	0.239475	+	0.970902i	$0.423025\pi$
$692$	2708.75	0.148802
$693$	−6297.75	−0.345212
$694$	1457.20	0.0797041
$695$	6052.87	0.330357
$696$	−58574.0	−3.19000
$697$	7229.05	0.392855
$698$	−46463.1	−2.51956
$699$	18510.8	1.00164
$700$	−6187.27	−0.334081
$701$	−14275.8	−0.769171	−0.384585	−	0.923089i	$-0.625656\pi$
$701$	−14275.8	−0.769171	−0.384585	+	0.923089i	$0.625656\pi$
$702$	−46477.6	−2.49884
$703$	2609.91	0.140021
$704$	6172.35	0.330439
$705$	8236.92	0.440029
$706$	10253.2	0.546581
$707$	−13427.1	−0.714253
$708$	17504.4	0.929177
$709$	−35825.1	−1.89766	−0.948828	−	0.315792i	$-0.897730\pi$
$709$	−35825.1	−1.89766	−0.948828	+	0.315792i	$0.897730\pi$
$710$	−30786.8	−1.62733
$711$	−50277.5	−2.65197
$712$	43638.4	2.29694
$713$	2119.13	0.111307
$714$	−10096.9	−0.529226
$715$	3208.78	0.167834
$716$	−31925.9	−1.66638
$717$	−1416.51	−0.0737803
$718$	53905.9	2.80188
$719$	−9818.87	−0.509294	−0.254647	−	0.967034i	$-0.581959\pi$
$719$	−9818.87	−0.509294	−0.254647	+	0.967034i	$0.581959\pi$
$720$	−35483.2	−1.83664
$721$	20213.5	1.04409
$722$	−1890.51	−0.0974483
$723$	17436.0	0.896892
$724$	−68679.1	−3.52547
$725$	−2885.67	−0.147822
$726$	5374.41	0.274742
$727$	17942.0	0.915314	0.457657	−	0.889129i	$-0.348689\pi$
$727$	17942.0	0.915314	0.457657	+	0.889129i	$0.348689\pi$
$728$	−44473.6	−2.26415
$729$	−31377.8	−1.59416
$730$	−2266.12	−0.114894
$731$	5717.89	0.289308
$732$	78103.9	3.94372
$733$	31022.6	1.56323	0.781615	−	0.623762i	$-0.214396\pi$
$733$	31022.6	1.56323	0.781615	+	0.623762i	$0.214396\pi$
$734$	30987.1	1.55825
$735$	−7661.76	−0.384501
$736$	11821.5	0.592048
$737$	−4378.77	−0.218852
$738$	−95346.3	−4.75575
$739$	27630.6	1.37539	0.687693	−	0.726002i	$-0.258624\pi$
$739$	27630.6	1.37539	0.687693	+	0.726002i	$0.258624\pi$
$740$	−13341.4	−0.662756
$741$	9401.63	0.466096
$742$	−29070.5	−1.43829
$743$	−37914.1	−1.87205	−0.936025	−	0.351933i	$-0.885524\pi$
$743$	−37914.1	−1.87205	−0.936025	+	0.351933i	$0.885524\pi$
$744$	−31693.2	−1.56173
$745$	8352.91	0.410774
$746$	−18681.7	−0.916872
$747$	−21972.9	−1.07623
$748$	3812.37	0.186356
$749$	−19280.5	−0.940578
$750$	−5552.08	−0.270311
$751$	−26438.9	−1.28465	−0.642323	−	0.766434i	$-0.722029\pi$
$751$	−26438.9	−1.28465	−0.642323	+	0.766434i	$0.722029\pi$
$752$	30674.8	1.48749
$753$	−53833.9	−2.60534
$754$	−35266.0	−1.70333
$755$	6698.37	0.322886
$756$	37649.0	1.81122
$757$	−7154.92	−0.343527	−0.171764	−	0.985138i	$-0.554947\pi$
$757$	−7154.92	−0.343527	−0.171764	+	0.985138i	$0.554947\pi$
$758$	−41879.9	−2.00679
$759$	3165.60	0.151389
$760$	5683.94	0.271287
$761$	−3920.72	−0.186762	−0.0933811	−	0.995630i	$-0.529767\pi$
$761$	−3920.72	−0.186762	−0.0933811	+	0.995630i	$0.529767\pi$
$762$	−98436.2	−4.67975
$763$	1329.04	0.0630595
$764$	85634.8	4.05518
$765$	−4008.72	−0.189458
$766$	2383.04	0.112406
$767$	6198.59	0.291810
$768$	31352.9	1.47311
$769$	12650.1	0.593207	0.296603	−	0.955001i	$-0.404146\pi$
$769$	12650.1	0.593207	0.296603	+	0.955001i	$0.404146\pi$
$770$	−3669.74	−0.171751
$771$	3573.91	0.166941
$772$	−13045.2	−0.608170
$773$	12149.5	0.565315	0.282658	−	0.959221i	$-0.408784\pi$
$773$	12149.5	0.565315	0.282658	+	0.959221i	$0.408784\pi$
$774$	−75415.1	−3.50225
$775$	−1561.37	−0.0723693
$776$	32892.9	1.52163
$777$	14843.8	0.685351
$778$	−59327.2	−2.73391
$779$	7698.25	0.354067
$780$	−48059.4	−2.20616
$781$	−12933.4	−0.592567
$782$	3170.35	0.144976
$783$	17559.1	0.801417
$784$	−28532.9	−1.29978
$785$	14828.0	0.674181
$786$	54483.2	2.47246
$787$	41527.8	1.88095	0.940473	−	0.339868i	$-0.110382\pi$
$787$	41527.8	1.88095	0.940473	+	0.339868i	$0.110382\pi$
$788$	−19464.8	−0.879957
$789$	3257.10	0.146966
$790$	−29297.0	−1.31942
$791$	−12961.2	−0.582613
$792$	−29574.1	−1.32686
$793$	27657.8	1.23853
$794$	62431.4	2.79044
$795$	−18476.7	−0.824276
$796$	62456.4	2.78104
$797$	12583.6	0.559266	0.279633	−	0.960107i	$-0.409787\pi$
$797$	12583.6	0.559266	0.279633	+	0.960107i	$0.409787\pi$
$798$	−10752.2	−0.476974
$799$	3465.50	0.153442
$800$	−8710.09	−0.384935
$801$	−32774.5	−1.44573
$802$	5018.64	0.220965
$803$	−951.989	−0.0418368
$804$	65583.0	2.87678
$805$	−2161.53	−0.0946382
$806$	−19081.7	−0.833900
$807$	−44686.0	−1.94922
$808$	−63053.3	−2.74531
$809$	−7569.24	−0.328950	−0.164475	−	0.986381i	$-0.552593\pi$
$809$	−7569.24	−0.328950	−0.164475	+	0.986381i	$0.552593\pi$
$810$	2015.22	0.0874166
$811$	16982.2	0.735298	0.367649	−	0.929965i	$-0.380163\pi$
$811$	16982.2	0.735298	0.367649	+	0.929965i	$0.380163\pi$
$812$	28567.0	1.23461
$813$	2858.94	0.123330
$814$	−7912.93	−0.340723
$815$	−5263.58	−0.226227
$816$	−23898.8	−1.02528
$817$	6089.01	0.260743
$818$	19610.5	0.838219
$819$	33401.8	1.42509
$820$	−39352.1	−1.67589
$821$	−23428.0	−0.995912	−0.497956	−	0.867202i	$-0.665916\pi$
$821$	−23428.0	−0.995912	−0.497956	+	0.867202i	$0.665916\pi$
$822$	−50848.7	−2.15761
$823$	−13062.9	−0.553273	−0.276637	−	0.960975i	$-0.589220\pi$
$823$	−13062.9	−0.553273	−0.276637	+	0.960975i	$0.589220\pi$
$824$	94922.4	4.01308
$825$	−2332.41	−0.0984293
$826$	−7089.06	−0.298620
$827$	−35018.5	−1.47245	−0.736223	−	0.676739i	$-0.763392\pi$
$827$	−35018.5	−1.47245	−0.736223	+	0.676739i	$0.763392\pi$
$828$	−29617.1	−1.24308
$829$	22553.8	0.944906	0.472453	−	0.881356i	$-0.343369\pi$
$829$	22553.8	0.944906	0.472453	+	0.881356i	$0.343369\pi$
$830$	−12803.7	−0.535451
$831$	−2708.31	−0.113057
$832$	−32736.7	−1.36411
$833$	−3223.51	−0.134079
$834$	−53769.6	−2.23248
$835$	−13825.6	−0.573000
$836$	4059.81	0.167956
$837$	9500.83	0.392350
$838$	−73852.6	−3.04439
$839$	5747.94	0.236521	0.118260	−	0.992983i	$-0.462268\pi$
$839$	5747.94	0.236521	0.118260	+	0.992983i	$0.462268\pi$
$840$	32327.2	1.32785
$841$	−11065.7	−0.453715
$842$	47359.5	1.93838
$843$	19388.7	0.792150
$844$	83342.9	3.39903
$845$	−6033.59	−0.245635
$846$	−45707.5	−1.85752
$847$	−1541.65	−0.0625403
$848$	−68808.2	−2.78642
$849$	16824.5	0.680114
$850$	−2335.91	−0.0942600
$851$	−4660.83	−0.187745
$852$	193711.	7.78922
$853$	−12691.2	−0.509423	−0.254712	−	0.967017i	$-0.581981\pi$
$853$	−12691.2	−0.509423	−0.254712	+	0.967017i	$0.581981\pi$
$854$	−31631.0	−1.26744
$855$	−4268.91	−0.170753
$856$	−90540.7	−3.61521
$857$	47052.4	1.87547	0.937737	−	0.347347i	$-0.112917\pi$
$857$	47052.4	1.87547	0.937737	+	0.347347i	$0.112917\pi$
$858$	−28504.6	−1.13418
$859$	8927.00	0.354581	0.177291	−	0.984159i	$-0.443267\pi$
$859$	8927.00	0.354581	0.177291	+	0.984159i	$0.443267\pi$
$860$	−31125.9	−1.23417
$861$	43783.5	1.73303
$862$	68199.8	2.69477
$863$	19816.9	0.781664	0.390832	−	0.920462i	$-0.372187\pi$
$863$	19816.9	0.781664	0.390832	+	0.920462i	$0.372187\pi$
$864$	53000.2	2.08692
$865$	−697.235	−0.0274066
$866$	16165.9	0.634340
$867$	38969.6	1.52650
$868$	15457.0	0.604430
$869$	−12307.6	−0.480445
$870$	25634.3	0.998948
$871$	23224.0	0.903460
$872$	6241.13	0.242375
$873$	−24704.1	−0.957741
$874$	3376.11	0.130662
$875$	1592.61	0.0615315
$876$	14258.4	0.549940
$877$	28135.4	1.08331	0.541656	−	0.840600i	$-0.317798\pi$
$877$	28135.4	1.08331	0.541656	+	0.840600i	$0.317798\pi$
$878$	−51368.4	−1.97449
$879$	51824.6	1.98862
$880$	−8686.05	−0.332735
$881$	−43068.9	−1.64702	−0.823512	−	0.567298i	$-0.807989\pi$
$881$	−43068.9	−1.64702	−0.823512	+	0.567298i	$0.807989\pi$
$882$	42515.9	1.62311
$883$	18453.2	0.703282	0.351641	−	0.936135i	$-0.385624\pi$
$883$	18453.2	0.703282	0.351641	+	0.936135i	$0.385624\pi$
$884$	−20219.9	−0.769309
$885$	−4505.67	−0.171137
$886$	41647.2	1.57919
$887$	−570.394	−0.0215918	−0.0107959	−	0.999942i	$-0.503437\pi$
$887$	−570.394	−0.0215918	−0.0107959	+	0.999942i	$0.503437\pi$
$888$	69706.1	2.63422
$889$	28236.4	1.06526
$890$	−19097.9	−0.719285
$891$	846.587	0.0318313
$892$	−9966.68	−0.374113
$893$	3690.42	0.138293
$894$	−74201.6	−2.77592
$895$	8217.78	0.306916
$896$	1927.74	0.0718764
$897$	−16789.6	−0.624959
$898$	−49741.3	−1.84843
$899$	7208.97	0.267445
$900$	21821.9	0.808218
$901$	−7773.63	−0.287433
$902$	−23340.1	−0.861576
$903$	34631.0	1.27624
$904$	−60865.5	−2.23933
$905$	17678.1	0.649326
$906$	−59503.8	−2.18199
$907$	−22819.4	−0.835398	−0.417699	−	0.908586i	$-0.637163\pi$
$907$	−22819.4	−0.835398	−0.417699	+	0.908586i	$0.637163\pi$
$908$	20254.8	0.740287
$909$	47356.0	1.72794
$910$	19463.4	0.709018
$911$	19077.1	0.693802	0.346901	−	0.937902i	$-0.387234\pi$
$911$	19077.1	0.693802	0.346901	+	0.937902i	$0.387234\pi$
$912$	−25449.9	−0.924046
$913$	−5378.82	−0.194976
$914$	18723.8	0.677603
$915$	−20104.1	−0.726360
$916$	−29972.5	−1.08113
$917$	−15628.5	−0.562812
$918$	14213.8	0.511030
$919$	13363.4	0.479671	0.239835	−	0.970814i	$-0.422907\pi$
$919$	13363.4	0.479671	0.239835	+	0.970814i	$0.422907\pi$
$920$	−10150.5	−0.363752
$921$	−43327.0	−1.55013
$922$	56911.9	2.03286
$923$	68595.9	2.44622
$924$	23090.0	0.822084
$925$	3434.09	0.122067
$926$	1327.72	0.0471183
$927$	−71291.2	−2.52590
$928$	40215.1	1.42255
$929$	−15508.4	−0.547700	−0.273850	−	0.961772i	$-0.588297\pi$
$929$	−15508.4	−0.547700	−0.273850	+	0.961772i	$0.588297\pi$
$930$	13870.2	0.489055
$931$	−3432.73	−0.120841
$932$	−42394.8	−1.49001
$933$	25147.4	0.882410
$934$	−57213.7	−2.00438
$935$	−981.309	−0.0343233
$936$	156854.	5.47749
$937$	−34580.4	−1.20565	−0.602823	−	0.797875i	$-0.705958\pi$
$937$	−34580.4	−1.20565	−0.602823	+	0.797875i	$0.705958\pi$
$938$	−26560.2	−0.924544
$939$	32913.6	1.14387
$940$	−18864.8	−0.654576
$941$	−15093.1	−0.522871	−0.261435	−	0.965221i	$-0.584196\pi$
$941$	−15093.1	−0.522871	−0.261435	+	0.965221i	$0.584196\pi$
$942$	−131721.	−4.55596
$943$	−13747.7	−0.474746
$944$	−16779.4	−0.578520
$945$	−9690.90	−0.333593
$946$	−18461.1	−0.634485
$947$	25277.1	0.867365	0.433683	−	0.901066i	$-0.357214\pi$
$947$	25277.1	0.867365	0.433683	+	0.901066i	$0.357214\pi$
$948$	184337.	6.31539
$949$	5049.13	0.172710
$950$	−2487.52	−0.0849534
$951$	−36803.7	−1.25493
$952$	13600.9	0.463035
$953$	−869.076	−0.0295405	−0.0147703	−	0.999891i	$-0.504702\pi$
$953$	−869.076	−0.0295405	−0.0147703	+	0.999891i	$0.504702\pi$
$954$	102529.	3.47955
$955$	−22042.5	−0.746889
$956$	3244.19	0.109754
$957$	10768.9	0.363751
$958$	−98614.6	−3.32578
$959$	14585.9	0.491141
$960$	23795.8	0.800007
$961$	−25890.4	−0.869067
$962$	41968.3	1.40656
$963$	68000.3	2.27547
$964$	−39933.2	−1.33419
$965$	3357.85	0.112014
$966$	19201.5	0.639544
$967$	−20773.9	−0.690843	−0.345422	−	0.938448i	$-0.612264\pi$
$967$	−20773.9	−0.690843	−0.345422	+	0.938448i	$0.612264\pi$
$968$	−7239.55	−0.240380
$969$	−2875.21	−0.0953200
$970$	−14395.2	−0.476498
$971$	10790.2	0.356616	0.178308	−	0.983975i	$-0.442938\pi$
$971$	10790.2	0.356616	0.178308	+	0.983975i	$0.442938\pi$
$972$	67104.6	2.21438
$973$	15423.8	0.508185
$974$	−81013.1	−2.66512
$975$	12370.6	0.406333
$976$	−74868.8	−2.45542
$977$	57782.2	1.89213	0.946067	−	0.323970i	$-0.105018\pi$
$977$	57782.2	1.89213	0.946067	+	0.323970i	$0.105018\pi$
$978$	46758.1	1.52879
$979$	−8022.98	−0.261916
$980$	17547.5	0.571973
$981$	−4687.38	−0.152555
$982$	−64283.1	−2.08896
$983$	8538.01	0.277030	0.138515	−	0.990360i	$-0.455767\pi$
$983$	8538.01	0.277030	0.138515	+	0.990360i	$0.455767\pi$
$984$	205607.	6.66107
$985$	5010.27	0.162072
$986$	10785.1	0.348343
$987$	20989.1	0.676891
$988$	−21532.3	−0.693353
$989$	−10873.9	−0.349614
$990$	12942.8	0.415504
$991$	−47486.0	−1.52214	−0.761071	−	0.648669i	$-0.775326\pi$
$991$	−47486.0	−1.52214	−0.761071	+	0.648669i	$0.775326\pi$
$992$	21759.5	0.696437
$993$	94628.3	3.02411
$994$	−78450.1	−2.50331
$995$	−16076.4	−0.512216
$996$	80561.3	2.56293
$997$	−50762.0	−1.61249	−0.806244	−	0.591584i	$-0.798503\pi$
$997$	−50762.0	−1.61249	−0.806244	+	0.591584i	$0.798503\pi$
$998$	−4978.25	−0.157900
$999$	−20896.2	−0.661787

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.i.1.2	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.i.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-5.23688 q^{2} -8.48150 q^{3} +19.4249 q^{4} -5.00000 q^{5} +44.4166 q^{6} -12.7409 q^{7} -59.8310 q^{8} +44.9358 q^{9} +O(q^{10})\)	\(q-5.23688 q^{2} -8.48150 q^{3} +19.4249 q^{4} -5.00000 q^{5} +44.4166 q^{6} -12.7409 q^{7} -59.8310 q^{8} +44.9358 q^{9} +26.1844 q^{10} +11.0000 q^{11} -164.752 q^{12} -58.3414 q^{13} +66.7225 q^{14} +42.4075 q^{15} +157.928 q^{16} +17.8420 q^{17} -235.324 q^{18} +19.0000 q^{19} -97.1246 q^{20} +108.062 q^{21} -57.6057 q^{22} -33.9305 q^{23} +507.456 q^{24} +25.0000 q^{25} +305.527 q^{26} -152.123 q^{27} -247.491 q^{28} -115.427 q^{29} -222.083 q^{30} -62.4550 q^{31} -348.404 q^{32} -93.2965 q^{33} -93.4364 q^{34} +63.7044 q^{35} +872.875 q^{36} +137.364 q^{37} -99.5007 q^{38} +494.822 q^{39} +299.155 q^{40} +405.171 q^{41} -565.907 q^{42} +320.474 q^{43} +213.674 q^{44} -224.679 q^{45} +177.690 q^{46} +194.233 q^{47} -1339.47 q^{48} -180.670 q^{49} -130.922 q^{50} -151.327 q^{51} -1133.28 q^{52} -435.693 q^{53} +796.649 q^{54} -55.0000 q^{55} +762.299 q^{56} -161.149 q^{57} +604.476 q^{58} -106.247 q^{59} +823.762 q^{60} -474.068 q^{61} +327.069 q^{62} -572.522 q^{63} +561.123 q^{64} +291.707 q^{65} +488.583 q^{66} -398.070 q^{67} +346.579 q^{68} +287.782 q^{69} -333.613 q^{70} -1175.77 q^{71} -2688.55 q^{72} -86.5445 q^{73} -719.358 q^{74} -212.038 q^{75} +369.074 q^{76} -140.150 q^{77} -2591.33 q^{78} -1118.87 q^{79} -789.641 q^{80} +76.9624 q^{81} -2121.83 q^{82} -488.984 q^{83} +2099.09 q^{84} -89.2099 q^{85} -1678.28 q^{86} +978.992 q^{87} -658.141 q^{88} -729.362 q^{89} +1176.62 q^{90} +743.321 q^{91} -659.098 q^{92} +529.712 q^{93} -1017.17 q^{94} -95.0000 q^{95} +2954.99 q^{96} -549.764 q^{97} +946.146 q^{98} +494.294 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 2 q^{2} + 9 q^{3} + 122 q^{4} - 125 q^{5} + 11 q^{6} - 15 q^{7} + 60 q^{8} + 300 q^{9}+O(q^{10}) \) 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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