LMFDB - Embedded newform 471.4.a.c.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [471,4,Mod(1,471)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(471, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("471.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 471 = 3 \cdot 157 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	471.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:	$27.7898996127$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	471.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.56491 q^{2} -3.00000 q^{3} -5.55104 q^{4} +8.64342 q^{5} +4.69474 q^{6} +19.5992 q^{7} +21.2062 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-1.56491 q^{2} -3.00000 q^{3} -5.55104 q^{4} +8.64342 q^{5} +4.69474 q^{6} +19.5992 q^{7} +21.2062 q^{8} +9.00000 q^{9} -13.5262 q^{10} +57.0663 q^{11} +16.6531 q^{12} +4.89013 q^{13} -30.6710 q^{14} -25.9303 q^{15} +11.2224 q^{16} -106.298 q^{17} -14.0842 q^{18} +85.2925 q^{19} -47.9800 q^{20} -58.7976 q^{21} -89.3038 q^{22} -104.220 q^{23} -63.6187 q^{24} -50.2913 q^{25} -7.65264 q^{26} -27.0000 q^{27} -108.796 q^{28} +115.519 q^{29} +40.5786 q^{30} +261.543 q^{31} -187.212 q^{32} -171.199 q^{33} +166.347 q^{34} +169.404 q^{35} -49.9594 q^{36} -147.663 q^{37} -133.475 q^{38} -14.6704 q^{39} +183.294 q^{40} +247.930 q^{41} +92.0131 q^{42} +47.7139 q^{43} -316.777 q^{44} +77.7908 q^{45} +163.096 q^{46} -302.369 q^{47} -33.6673 q^{48} +41.1281 q^{49} +78.7016 q^{50} +318.894 q^{51} -27.1453 q^{52} +760.819 q^{53} +42.2527 q^{54} +493.248 q^{55} +415.625 q^{56} -255.878 q^{57} -180.778 q^{58} +523.883 q^{59} +143.940 q^{60} -407.939 q^{61} -409.293 q^{62} +176.393 q^{63} +203.191 q^{64} +42.2675 q^{65} +267.911 q^{66} -122.070 q^{67} +590.064 q^{68} +312.661 q^{69} -265.103 q^{70} +185.468 q^{71} +190.856 q^{72} -673.281 q^{73} +231.080 q^{74} +150.874 q^{75} -473.462 q^{76} +1118.45 q^{77} +22.9579 q^{78} +1250.33 q^{79} +97.0001 q^{80} +81.0000 q^{81} -387.989 q^{82} -104.222 q^{83} +326.388 q^{84} -918.777 q^{85} -74.6682 q^{86} -346.558 q^{87} +1210.16 q^{88} -233.008 q^{89} -121.736 q^{90} +95.8427 q^{91} +578.531 q^{92} -784.629 q^{93} +473.181 q^{94} +737.219 q^{95} +561.636 q^{96} +60.1127 q^{97} -64.3620 q^{98} +513.596 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9}+O(q^{10}) $ 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9} + 13 q^{10} + 61 q^{11} - 270 q^{12} + 4 q^{13} + 133 q^{14} - 96 q^{15} + 342 q^{16} + 308 q^{17} + 36 q^{18} + 32 q^{19} + 407 q^{20} + 12 q^{21} - 166 q^{22} + 53 q^{23} - 81 q^{24} + 746 q^{25} + 467 q^{26} - 594 q^{27} + 85 q^{28} + 634 q^{29} - 39 q^{30} - 163 q^{31} + 150 q^{32} - 183 q^{33} + 37 q^{34} + 782 q^{35} + 810 q^{36} - 2 q^{37} + 584 q^{38} - 12 q^{39} + 864 q^{40} + 1593 q^{41} - 399 q^{42} - 891 q^{43} + 2093 q^{44} + 288 q^{45} + 108 q^{46} + 1200 q^{47} - 1026 q^{48} + 2816 q^{49} + 4703 q^{50} - 924 q^{51} + 1866 q^{52} + 1182 q^{53} - 108 q^{54} + 970 q^{55} + 5362 q^{56} - 96 q^{57} + 1814 q^{58} + 2802 q^{59} - 1221 q^{60} + 2629 q^{61} + 2378 q^{62} - 36 q^{63} + 625 q^{64} + 2264 q^{65} + 498 q^{66} - 1074 q^{67} + 4383 q^{68} - 159 q^{69} + 4009 q^{70} + 3920 q^{71} + 243 q^{72} + 1086 q^{73} + 4904 q^{74} - 2238 q^{75} + 3750 q^{76} + 2966 q^{77} - 1401 q^{78} - 30 q^{79} + 7777 q^{80} + 1782 q^{81} + 2932 q^{82} + 1900 q^{83} - 255 q^{84} + 524 q^{85} + 3209 q^{86} - 1902 q^{87} - 100 q^{88} + 4488 q^{89} + 117 q^{90} - 818 q^{91} + 6210 q^{92} + 489 q^{93} + 3220 q^{94} + 3500 q^{95} - 450 q^{96} + 2178 q^{97} + 7629 q^{98} + 549 q^{99}+O(q^{100}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9 + 13 * q^10 + 61 * q^11 - 270 * q^12 + 4 * q^13 + 133 * q^14 - 96 * q^15 + 342 * q^16 + 308 * q^17 + 36 * q^18 + 32 * q^19 + 407 * q^20 + 12 * q^21 - 166 * q^22 + 53 * q^23 - 81 * q^24 + 746 * q^25 + 467 * q^26 - 594 * q^27 + 85 * q^28 + 634 * q^29 - 39 * q^30 - 163 * q^31 + 150 * q^32 - 183 * q^33 + 37 * q^34 + 782 * q^35 + 810 * q^36 - 2 * q^37 + 584 * q^38 - 12 * q^39 + 864 * q^40 + 1593 * q^41 - 399 * q^42 - 891 * q^43 + 2093 * q^44 + 288 * q^45 + 108 * q^46 + 1200 * q^47 - 1026 * q^48 + 2816 * q^49 + 4703 * q^50 - 924 * q^51 + 1866 * q^52 + 1182 * q^53 - 108 * q^54 + 970 * q^55 + 5362 * q^56 - 96 * q^57 + 1814 * q^58 + 2802 * q^59 - 1221 * q^60 + 2629 * q^61 + 2378 * q^62 - 36 * q^63 + 625 * q^64 + 2264 * q^65 + 498 * q^66 - 1074 * q^67 + 4383 * q^68 - 159 * q^69 + 4009 * q^70 + 3920 * q^71 + 243 * q^72 + 1086 * q^73 + 4904 * q^74 - 2238 * q^75 + 3750 * q^76 + 2966 * q^77 - 1401 * q^78 - 30 * q^79 + 7777 * q^80 + 1782 * q^81 + 2932 * q^82 + 1900 * q^83 - 255 * q^84 + 524 * q^85 + 3209 * q^86 - 1902 * q^87 - 100 * q^88 + 4488 * q^89 + 117 * q^90 - 818 * q^91 + 6210 * q^92 + 489 * q^93 + 3220 * q^94 + 3500 * q^95 - 450 * q^96 + 2178 * q^97 + 7629 * q^98 + 549 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.56491	−0.553281	−0.276640	−	0.960974i	$-0.589221\pi$
$2$	−1.56491	−0.553281	−0.276640	+	0.960974i	$0.589221\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−5.55104	−0.693880
$5$	8.64342	0.773091	0.386545	−	0.922270i	$-0.373668\pi$
$5$	8.64342	0.773091	0.386545	+	0.922270i	$0.373668\pi$
$6$	4.69474	0.319437
$7$	19.5992	1.05826	0.529128	−	0.848542i	$-0.322519\pi$
$7$	19.5992	1.05826	0.529128	+	0.848542i	$0.322519\pi$
$8$	21.2062	0.937191
$9$	9.00000	0.333333
$10$	−13.5262	−0.427736
$11$	57.0663	1.56419	0.782097	−	0.623157i	$-0.214150\pi$
$11$	57.0663	1.56419	0.782097	+	0.623157i	$0.214150\pi$
$12$	16.6531	0.400612
$13$	4.89013	0.104329	0.0521646	−	0.998639i	$-0.483388\pi$
$13$	4.89013	0.104329	0.0521646	+	0.998639i	$0.483388\pi$
$14$	−30.6710	−0.585513
$15$	−25.9303	−0.446344
$16$	11.2224	0.175350
$17$	−106.298	−1.51653	−0.758265	−	0.651946i	$-0.773953\pi$
$17$	−106.298	−1.51653	−0.758265	+	0.651946i	$0.773953\pi$
$18$	−14.0842	−0.184427
$19$	85.2925	1.02987	0.514933	−	0.857231i	$-0.327817\pi$
$19$	85.2925	1.02987	0.514933	+	0.857231i	$0.327817\pi$
$20$	−47.9800	−0.536433
$21$	−58.7976	−0.610985
$22$	−89.3038	−0.865438
$23$	−104.220	−0.944844	−0.472422	−	0.881373i	$-0.656620\pi$
$23$	−104.220	−0.944844	−0.472422	+	0.881373i	$0.656620\pi$
$24$	−63.6187	−0.541088
$25$	−50.2913	−0.402330
$26$	−7.65264	−0.0577233
$27$	−27.0000	−0.192450
$28$	−108.796	−0.734303
$29$	115.519	0.739703	0.369851	−	0.929091i	$-0.379409\pi$
$29$	115.519	0.739703	0.369851	+	0.929091i	$0.379409\pi$
$30$	40.5786	0.246954
$31$	261.543	1.51531	0.757654	−	0.652657i	$-0.226346\pi$
$31$	261.543	1.51531	0.757654	+	0.652657i	$0.226346\pi$
$32$	−187.212	−1.03421
$33$	−171.199	−0.903087
$34$	166.347	0.839067
$35$	169.404	0.818129
$36$	−49.9594	−0.231293
$37$	−147.663	−0.656099	−0.328050	−	0.944660i	$-0.606391\pi$
$37$	−147.663	−0.656099	−0.328050	+	0.944660i	$0.606391\pi$
$38$	−133.475	−0.569805
$39$	−14.6704	−0.0602345
$40$	183.294	0.724534
$41$	247.930	0.944394	0.472197	−	0.881493i	$-0.343461\pi$
$41$	247.930	0.944394	0.472197	+	0.881493i	$0.343461\pi$
$42$	92.0131	0.338046
$43$	47.7139	0.169216	0.0846082	−	0.996414i	$-0.473036\pi$
$43$	47.7139	0.169216	0.0846082	+	0.996414i	$0.473036\pi$
$44$	−316.777	−1.08536
$45$	77.7908	0.257697
$46$	163.096	0.522764
$47$	−302.369	−0.938405	−0.469202	−	0.883091i	$-0.655459\pi$
$47$	−302.369	−0.938405	−0.469202	+	0.883091i	$0.655459\pi$
$48$	−33.6673	−0.101239
$49$	41.1281	0.119907
$50$	78.7016	0.222602
$51$	318.894	0.875569
$52$	−27.1453	−0.0723920
$53$	760.819	1.97182	0.985911	−	0.167271i	$-0.0534955\pi$
$53$	760.819	1.97182	0.985911	+	0.167271i	$0.0534955\pi$
$54$	42.2527	0.106479
$55$	493.248	1.20926
$56$	415.625	0.991789
$57$	−255.878	−0.594593
$58$	−180.778	−0.409263
$59$	523.883	1.15600	0.577998	−	0.816038i	$-0.303834\pi$
$59$	523.883	1.15600	0.577998	+	0.816038i	$0.303834\pi$
$60$	143.940	0.309709
$61$	−407.939	−0.856250	−0.428125	−	0.903719i	$-0.640826\pi$
$61$	−407.939	−0.856250	−0.428125	+	0.903719i	$0.640826\pi$
$62$	−409.293	−0.838390
$63$	176.393	0.352752
$64$	203.191	0.396858
$65$	42.2675	0.0806559
$66$	267.911	0.499661
$67$	−122.070	−0.222585	−0.111293	−	0.993788i	$-0.535499\pi$
$67$	−122.070	−0.222585	−0.111293	+	0.993788i	$0.535499\pi$
$68$	590.064	1.05229
$69$	312.661	0.545506
$70$	−265.103	−0.452655
$71$	185.468	0.310014	0.155007	−	0.987913i	$-0.450460\pi$
$71$	185.468	0.310014	0.155007	+	0.987913i	$0.450460\pi$
$72$	190.856	0.312397
$73$	−673.281	−1.07947	−0.539737	−	0.841834i	$-0.681476\pi$
$73$	−673.281	−1.07947	−0.539737	+	0.841834i	$0.681476\pi$
$74$	231.080	0.363007
$75$	150.874	0.232286
$76$	−473.462	−0.714603
$77$	1118.45	1.65532
$78$	22.9579	0.0333266
$79$	1250.33	1.78067	0.890334	−	0.455307i	$-0.150471\pi$
$79$	1250.33	1.78067	0.890334	+	0.455307i	$0.150471\pi$
$80$	97.0001	0.135562
$81$	81.0000	0.111111
$82$	−387.989	−0.522515
$83$	−104.222	−0.137829	−0.0689147	−	0.997623i	$-0.521954\pi$
$83$	−104.222	−0.137829	−0.0689147	+	0.997623i	$0.521954\pi$
$84$	326.388	0.423950
$85$	−918.777	−1.17242
$86$	−74.6682	−0.0936242
$87$	−346.558	−0.427068
$88$	1210.16	1.46595
$89$	−233.008	−0.277514	−0.138757	−	0.990326i	$-0.544311\pi$
$89$	−233.008	−0.277514	−0.138757	+	0.990326i	$0.544311\pi$
$90$	−121.736	−0.142579
$91$	95.8427	0.110407
$92$	578.531	0.655609
$93$	−784.629	−0.874863
$94$	473.181	0.519201
$95$	737.219	0.796180
$96$	561.636	0.597101
$97$	60.1127	0.0629229	0.0314614	−	0.999505i	$-0.489984\pi$
$97$	60.1127	0.0629229	0.0314614	+	0.999505i	$0.489984\pi$
$98$	−64.3620	−0.0663423
$99$	513.596	0.521398
$100$	279.169	0.279169
$101$	920.427	0.906791	0.453396	−	0.891309i	$-0.350213\pi$
$101$	920.427	0.906791	0.453396	+	0.891309i	$0.350213\pi$
$102$	−499.041	−0.484436
$103$	480.180	0.459355	0.229677	−	0.973267i	$-0.426233\pi$
$103$	480.180	0.459355	0.229677	+	0.973267i	$0.426233\pi$
$104$	103.701	0.0977764
$105$	−508.212	−0.472347
$106$	−1190.62	−1.09097
$107$	−181.687	−0.164153	−0.0820763	−	0.996626i	$-0.526155\pi$
$107$	−181.687	−0.164153	−0.0820763	+	0.996626i	$0.526155\pi$
$108$	149.878	0.133537
$109$	257.864	0.226596	0.113298	−	0.993561i	$-0.463859\pi$
$109$	257.864	0.226596	0.113298	+	0.993561i	$0.463859\pi$
$110$	−771.890	−0.669062
$111$	442.990	0.378799
$112$	219.950	0.185566
$113$	−586.752	−0.488469	−0.244234	−	0.969716i	$-0.578537\pi$
$113$	−586.752	−0.488469	−0.244234	+	0.969716i	$0.578537\pi$
$114$	400.426	0.328977
$115$	−900.819	−0.730450
$116$	−641.252	−0.513265
$117$	44.0112	0.0347764
$118$	−819.832	−0.639591
$119$	−2083.35	−1.60488
$120$	−549.883	−0.418310
$121$	1925.56	1.44670
$122$	638.390	0.473747
$123$	−743.790	−0.545246
$124$	−1451.84	−1.05144
$125$	−1515.12	−1.08413
$126$	−276.039	−0.195171
$127$	−1466.75	−1.02483	−0.512414	−	0.858739i	$-0.671249\pi$
$127$	−1466.75	−1.02483	−0.512414	+	0.858739i	$0.671249\pi$
$128$	1179.72	0.814636
$129$	−143.142	−0.0976972
$130$	−66.1450	−0.0446254
$131$	1785.62	1.19092	0.595459	−	0.803386i	$-0.296970\pi$
$131$	1785.62	1.19092	0.595459	+	0.803386i	$0.296970\pi$
$132$	950.332	0.626635
$133$	1671.66	1.08986
$134$	191.029	0.123152
$135$	−233.372	−0.148781
$136$	−2254.18	−1.42128
$137$	1670.80	1.04194	0.520972	−	0.853573i	$-0.325569\pi$
$137$	1670.80	1.04194	0.520972	+	0.853573i	$0.325569\pi$
$138$	−489.287	−0.301818
$139$	1686.42	1.02907	0.514535	−	0.857470i	$-0.327965\pi$
$139$	1686.42	1.02907	0.514535	+	0.857470i	$0.327965\pi$
$140$	−940.369	−0.567683
$141$	907.106	0.541788
$142$	−290.241	−0.171525
$143$	279.062	0.163191
$144$	101.002	0.0584501
$145$	998.481	0.571858
$146$	1053.63	0.597252
$147$	−123.384	−0.0692284
$148$	819.685	0.455254
$149$	2974.48	1.63543	0.817713	−	0.575626i	$-0.195242\pi$
$149$	2974.48	1.63543	0.817713	+	0.575626i	$0.195242\pi$
$150$	−236.105	−0.128519
$151$	−2785.52	−1.50121	−0.750605	−	0.660751i	$-0.770238\pi$
$151$	−2785.52	−1.50121	−0.750605	+	0.660751i	$0.770238\pi$
$152$	1808.73	0.965181
$153$	−956.681	−0.505510
$154$	−1750.28	−0.915856
$155$	2260.63	1.17147
$156$	81.4360	0.0417955
$157$	−157.000	−0.0798087
$157$	−157.000	−0.0798087
$158$	−1956.66	−0.985210
$159$	−2282.46	−1.13843
$160$	−1618.15	−0.799538
$161$	−2042.63	−0.999887
$162$	−126.758	−0.0614756
$163$	1888.03	0.907250	0.453625	−	0.891193i	$-0.350131\pi$
$163$	1888.03	0.907250	0.453625	+	0.891193i	$0.350131\pi$
$164$	−1376.27	−0.655296
$165$	−1479.74	−0.698169
$166$	163.098	0.0762584
$167$	2039.39	0.944988	0.472494	−	0.881334i	$-0.343354\pi$
$167$	2039.39	0.944988	0.472494	+	0.881334i	$0.343354\pi$
$168$	−1246.87	−0.572610
$169$	−2173.09	−0.989115
$170$	1437.81	0.648675
$171$	767.633	0.343288
$172$	−264.862	−0.117416
$173$	4500.46	1.97783	0.988913	−	0.148499i	$-0.0474441\pi$
$173$	4500.46	1.97783	0.988913	+	0.148499i	$0.0474441\pi$
$174$	542.333	0.236288
$175$	−985.669	−0.425769
$176$	640.422	0.274282
$177$	−1571.65	−0.667415
$178$	364.637	0.153543
$179$	2072.26	0.865298	0.432649	−	0.901562i	$-0.357579\pi$
$179$	2072.26	0.865298	0.432649	+	0.901562i	$0.357579\pi$
$180$	−431.820	−0.178811
$181$	−2851.43	−1.17097	−0.585483	−	0.810685i	$-0.699095\pi$
$181$	−2851.43	−1.17097	−0.585483	+	0.810685i	$0.699095\pi$
$182$	−149.986	−0.0610861
$183$	1223.82	0.494356
$184$	−2210.12	−0.885500
$185$	−1276.31	−0.507224
$186$	1227.88	0.484045
$187$	−6066.02	−2.37215
$188$	1678.46	0.651141
$189$	−529.178	−0.203662
$190$	−1153.68	−0.440511
$191$	−2526.75	−0.957223	−0.478611	−	0.878027i	$-0.658860\pi$
$191$	−2526.75	−0.957223	−0.478611	+	0.878027i	$0.658860\pi$
$192$	−609.574	−0.229126
$193$	−642.429	−0.239601	−0.119801	−	0.992798i	$-0.538226\pi$
$193$	−642.429	−0.239601	−0.119801	+	0.992798i	$0.538226\pi$
$194$	−94.0712	−0.0348140
$195$	−126.802	−0.0465667
$196$	−228.304	−0.0832011
$197$	−523.250	−0.189239	−0.0946194	−	0.995514i	$-0.530163\pi$
$197$	−523.250	−0.189239	−0.0946194	+	0.995514i	$0.530163\pi$
$198$	−803.734	−0.288479
$199$	−2455.22	−0.874602	−0.437301	−	0.899315i	$-0.644066\pi$
$199$	−2455.22	−0.874602	−0.437301	+	0.899315i	$0.644066\pi$
$200$	−1066.49	−0.377061
$201$	366.209	0.128510
$202$	−1440.39	−0.501710
$203$	2264.08	0.782796
$204$	−1770.19	−0.607540
$205$	2142.96	0.730102
$206$	−751.440	−0.254152
$207$	−937.982	−0.314948
$208$	54.8791	0.0182941
$209$	4867.33	1.61091
$210$	795.308	0.261340
$211$	−901.491	−0.294129	−0.147064	−	0.989127i	$-0.546982\pi$
$211$	−901.491	−0.294129	−0.147064	+	0.989127i	$0.546982\pi$
$212$	−4223.34	−1.36821
$213$	−556.403	−0.178987
$214$	284.324	0.0908225
$215$	412.412	0.130820
$216$	−572.568	−0.180363
$217$	5126.03	1.60358
$218$	−403.536	−0.125371
$219$	2019.84	0.623234
$220$	−2738.04	−0.839084
$221$	−519.811	−0.158218
$222$	−693.241	−0.209582
$223$	3487.33	1.04721	0.523607	−	0.851960i	$-0.324586\pi$
$223$	3487.33	1.04721	0.523607	+	0.851960i	$0.324586\pi$
$224$	−3669.20	−1.09446
$225$	−452.622	−0.134110
$226$	918.217	0.270260
$227$	−1393.79	−0.407528	−0.203764	−	0.979020i	$-0.565318\pi$
$227$	−1393.79	−0.407528	−0.203764	+	0.979020i	$0.565318\pi$
$228$	1420.39	0.412576
$229$	3591.02	1.03625	0.518125	−	0.855305i	$-0.326630\pi$
$229$	3591.02	1.03625	0.518125	+	0.855305i	$0.326630\pi$
$230$	1409.70	0.404144
$231$	−3355.36	−0.955698
$232$	2449.73	0.693243
$233$	2876.39	0.808750	0.404375	−	0.914593i	$-0.367489\pi$
$233$	2876.39	0.808750	0.404375	+	0.914593i	$0.367489\pi$
$234$	−68.8738	−0.0192411
$235$	−2613.50	−0.725472
$236$	−2908.10	−0.802123
$237$	−3750.98	−1.02807
$238$	3260.27	0.887948
$239$	−5987.87	−1.62060	−0.810299	−	0.586017i	$-0.800695\pi$
$239$	−5987.87	−1.62060	−0.810299	+	0.586017i	$0.800695\pi$
$240$	−291.000	−0.0782666
$241$	−4969.30	−1.32822	−0.664110	−	0.747635i	$-0.731189\pi$
$241$	−4969.30	−1.32822	−0.664110	+	0.747635i	$0.731189\pi$
$242$	−3013.33	−0.800432
$243$	−243.000	−0.0641500
$244$	2264.49	0.594135
$245$	355.487	0.0926990
$246$	1163.97	0.301674
$247$	417.092	0.107445
$248$	5546.34	1.42013
$249$	312.666	0.0795759
$250$	2371.03	0.599828
$251$	5481.15	1.37836	0.689178	−	0.724592i	$-0.257972\pi$
$251$	5481.15	1.37836	0.689178	+	0.724592i	$0.257972\pi$
$252$	−979.163	−0.244768
$253$	−5947.46	−1.47792
$254$	2295.34	0.567017
$255$	2756.33	0.676895
$256$	−3471.69	−0.847580
$257$	−4073.67	−0.988750	−0.494375	−	0.869249i	$-0.664603\pi$
$257$	−4073.67	−0.988750	−0.494375	+	0.869249i	$0.664603\pi$
$258$	224.005	0.0540540
$259$	−2894.08	−0.694321
$260$	−234.629	−0.0559656
$261$	1039.67	0.246568
$262$	−2794.34	−0.658912
$263$	−3276.10	−0.768110	−0.384055	−	0.923310i	$-0.625473\pi$
$263$	−3276.10	−0.768110	−0.384055	+	0.923310i	$0.625473\pi$
$264$	−3630.48	−0.846366
$265$	6576.08	1.52440
$266$	−2616.01	−0.603000
$267$	699.023	0.160223
$268$	677.615	0.154447
$269$	7184.53	1.62843	0.814217	−	0.580561i	$-0.197167\pi$
$269$	7184.53	1.62843	0.814217	+	0.580561i	$0.197167\pi$
$270$	365.208	0.0823179
$271$	−1889.78	−0.423602	−0.211801	−	0.977313i	$-0.567933\pi$
$271$	−1889.78	−0.423602	−0.211801	+	0.977313i	$0.567933\pi$
$272$	−1192.92	−0.265924
$273$	−287.528	−0.0637435
$274$	−2614.67	−0.576488
$275$	−2869.94	−0.629323
$276$	−1735.59	−0.378516
$277$	−297.952	−0.0646288	−0.0323144	−	0.999478i	$-0.510288\pi$
$277$	−297.952	−0.0646288	−0.0323144	+	0.999478i	$0.510288\pi$
$278$	−2639.11	−0.569364
$279$	2353.89	0.505102
$280$	3592.42	0.766743
$281$	−3974.84	−0.843840	−0.421920	−	0.906633i	$-0.638644\pi$
$281$	−3974.84	−0.843840	−0.421920	+	0.906633i	$0.638644\pi$
$282$	−1419.54	−0.299761
$283$	−1952.33	−0.410084	−0.205042	−	0.978753i	$-0.565733\pi$
$283$	−1952.33	−0.410084	−0.205042	+	0.978753i	$0.565733\pi$
$284$	−1029.54	−0.215112
$285$	−2211.66	−0.459675
$286$	−436.708	−0.0902904
$287$	4859.22	0.999411
$288$	−1684.91	−0.344736
$289$	6386.24	1.29986
$290$	−1562.54	−0.316398
$291$	−180.338	−0.0363285
$292$	3737.41	0.749026
$293$	4458.14	0.888899	0.444449	−	0.895804i	$-0.353399\pi$
$293$	4458.14	0.888899	0.444449	+	0.895804i	$0.353399\pi$
$294$	193.086	0.0383027
$295$	4528.14	0.893690
$296$	−3131.38	−0.614891
$297$	−1540.79	−0.301029
$298$	−4654.80	−0.904850
$299$	−509.651	−0.0985748
$300$	−837.508	−0.161178
$301$	935.154	0.179074
$302$	4359.11	0.830591
$303$	−2761.28	−0.523536
$304$	957.188	0.180587
$305$	−3525.99	−0.661959
$306$	1497.12	0.279689
$307$	−2883.98	−0.536148	−0.268074	−	0.963398i	$-0.586387\pi$
$307$	−2883.98	−0.536148	−0.268074	+	0.963398i	$0.586387\pi$
$308$	−6208.58	−1.14859
$309$	−1440.54	−0.265209
$310$	−3537.69	−0.648152
$311$	−5068.74	−0.924186	−0.462093	−	0.886831i	$-0.652901\pi$
$311$	−5068.74	−0.924186	−0.462093	+	0.886831i	$0.652901\pi$
$312$	−311.104	−0.0564512
$313$	7544.97	1.36251	0.681257	−	0.732045i	$-0.261434\pi$
$313$	7544.97	1.36251	0.681257	+	0.732045i	$0.261434\pi$
$314$	245.692	0.0441566
$315$	1524.64	0.272710
$316$	−6940.62	−1.23557
$317$	5140.14	0.910723	0.455362	−	0.890307i	$-0.349510\pi$
$317$	5140.14	0.910723	0.455362	+	0.890307i	$0.349510\pi$
$318$	3571.85	0.629873
$319$	6592.25	1.15704
$320$	1756.27	0.306807
$321$	545.061	0.0947736
$322$	3196.54	0.553218
$323$	−9066.41	−1.56182
$324$	−449.634	−0.0770978
$325$	−245.931	−0.0419748
$326$	−2954.60	−0.501964
$327$	−773.593	−0.130825
$328$	5257.66	0.885078
$329$	−5926.18	−0.993073
$330$	2315.67	0.386283
$331$	3018.10	0.501178	0.250589	−	0.968094i	$-0.419376\pi$
$331$	3018.10	0.501178	0.250589	+	0.968094i	$0.419376\pi$
$332$	578.540	0.0956371
$333$	−1328.97	−0.218700
$334$	−3191.48	−0.522844
$335$	−1055.10	−0.172078
$336$	−659.851	−0.107136
$337$	10695.3	1.72881	0.864406	−	0.502795i	$-0.167695\pi$
$337$	10695.3	1.72881	0.864406	+	0.502795i	$0.167695\pi$
$338$	3400.69	0.547259
$339$	1760.26	0.282018
$340$	5100.17	0.813516
$341$	14925.3	2.37023
$342$	−1201.28	−0.189935
$343$	−5916.44	−0.931364
$344$	1011.83	0.158588
$345$	2702.46	0.421726
$346$	−7042.84	−1.09429
$347$	−2681.80	−0.414889	−0.207445	−	0.978247i	$-0.566515\pi$
$347$	−2681.80	−0.414889	−0.207445	+	0.978247i	$0.566515\pi$
$348$	1923.76	0.296334
$349$	9395.17	1.44101	0.720504	−	0.693450i	$-0.243910\pi$
$349$	9395.17	1.44101	0.720504	+	0.693450i	$0.243910\pi$
$350$	1542.49	0.235570
$351$	−132.034	−0.0200782
$352$	−10683.5	−1.61770
$353$	1293.73	0.195066	0.0975331	−	0.995232i	$-0.468905\pi$
$353$	1293.73	0.195066	0.0975331	+	0.995232i	$0.468905\pi$
$354$	2459.50	0.369268
$355$	1603.08	0.239669
$356$	1293.44	0.192562
$357$	6250.05	0.926577
$358$	−3242.92	−0.478753
$359$	6925.79	1.01819	0.509093	−	0.860711i	$-0.329981\pi$
$359$	6925.79	1.01819	0.509093	+	0.860711i	$0.329981\pi$
$360$	1649.65	0.241511
$361$	415.812	0.0606229
$362$	4462.24	0.647873
$363$	−5776.68	−0.835253
$364$	−532.027	−0.0766093
$365$	−5819.45	−0.834531
$366$	−1915.17	−0.273518
$367$	−5721.01	−0.813717	−0.406858	−	0.913491i	$-0.633376\pi$
$367$	−5721.01	−0.813717	−0.406858	+	0.913491i	$0.633376\pi$
$368$	−1169.60	−0.165679
$369$	2231.37	0.314798
$370$	1997.32	0.280638
$371$	14911.4	2.08669
$372$	4355.51	0.607050
$373$	5246.31	0.728266	0.364133	−	0.931347i	$-0.381365\pi$
$373$	5246.31	0.728266	0.364133	+	0.931347i	$0.381365\pi$
$374$	9492.81	1.31246
$375$	4545.35	0.625922
$376$	−6412.10	−0.879465
$377$	564.905	0.0771726
$378$	828.118	0.112682
$379$	4232.28	0.573609	0.286804	−	0.957989i	$-0.407407\pi$
$379$	4232.28	0.573609	0.286804	+	0.957989i	$0.407407\pi$
$380$	−4092.33	−0.552453
$381$	4400.25	0.591684
$382$	3954.15	0.529613
$383$	2185.69	0.291602	0.145801	−	0.989314i	$-0.453424\pi$
$383$	2185.69	0.291602	0.145801	+	0.989314i	$0.453424\pi$
$384$	−3539.15	−0.470330
$385$	9667.25	1.27971
$386$	1005.35	0.132567
$387$	429.425	0.0564055
$388$	−333.688	−0.0436609
$389$	3512.07	0.457761	0.228881	−	0.973454i	$-0.426493\pi$
$389$	3512.07	0.457761	0.228881	+	0.973454i	$0.426493\pi$
$390$	198.435	0.0257645
$391$	11078.4	1.43288
$392$	872.172	0.112376
$393$	−5356.85	−0.687576
$394$	818.842	0.104702
$395$	10807.1	1.37662
$396$	−2851.00	−0.361788
$397$	5677.88	0.717796	0.358898	−	0.933377i	$-0.383153\pi$
$397$	5677.88	0.717796	0.358898	+	0.933377i	$0.383153\pi$
$398$	3842.21	0.483901
$399$	−5014.99	−0.629232
$400$	−564.390	−0.0705488
$401$	−3.69310	−0.000459912 0	−0.000229956	−	1.00000i	$-0.500073\pi$
$401$	−3.69310	−0.000459912 0	−0.000229956		1.00000i	$0.500073\pi$
$402$	−573.086	−0.0711019
$403$	1278.98	0.158091
$404$	−5109.33	−0.629204
$405$	700.117	0.0858990
$406$	−3543.10	−0.433106
$407$	−8426.59	−1.02627
$408$	6762.53	0.820576
$409$	−11692.6	−1.41360	−0.706801	−	0.707412i	$-0.749863\pi$
$409$	−11692.6	−1.41360	−0.706801	+	0.707412i	$0.749863\pi$
$410$	−3353.55	−0.403952
$411$	−5012.41	−0.601567
$412$	−2665.50	−0.318737
$413$	10267.7	1.22334
$414$	1467.86	0.174255
$415$	−900.834	−0.106555
$416$	−915.491	−0.107898
$417$	−5059.27	−0.594133
$418$	−7616.95	−0.891285
$419$	−260.881	−0.0304174	−0.0152087	−	0.999884i	$-0.504841\pi$
$419$	−260.881	−0.0304174	−0.0152087	+	0.999884i	$0.504841\pi$
$420$	2821.11	0.327752
$421$	−2827.24	−0.327295	−0.163648	−	0.986519i	$-0.552326\pi$
$421$	−2827.24	−0.327295	−0.163648	+	0.986519i	$0.552326\pi$
$422$	1410.76	0.162736
$423$	−2721.32	−0.312802
$424$	16134.1	1.84797
$425$	5345.86	0.610146
$426$	870.724	0.0990298
$427$	−7995.28	−0.906133
$428$	1008.55	0.113902
$429$	−837.185	−0.0942184
$430$	−645.389	−0.0723800
$431$	5544.47	0.619647	0.309823	−	0.950794i	$-0.399730\pi$
$431$	5544.47	0.619647	0.309823	+	0.950794i	$0.399730\pi$
$432$	−303.005	−0.0337462
$433$	−1060.30	−0.117678	−0.0588390	−	0.998267i	$-0.518740\pi$
$433$	−1060.30	−0.117678	−0.0588390	+	0.998267i	$0.518740\pi$
$434$	−8021.80	−0.887232
$435$	−2995.44	−0.330162
$436$	−1431.42	−0.157230
$437$	−8889.20	−0.973062
$438$	−3160.88	−0.344824
$439$	−17115.8	−1.86080	−0.930401	−	0.366544i	$-0.880541\pi$
$439$	−17115.8	−1.86080	−0.930401	+	0.366544i	$0.880541\pi$
$440$	10459.9	1.13331
$441$	370.153	0.0399690
$442$	813.459	0.0875392
$443$	−13983.7	−1.49975	−0.749873	−	0.661582i	$-0.769885\pi$
$443$	−13983.7	−1.49975	−0.749873	+	0.661582i	$0.769885\pi$
$444$	−2459.05	−0.262841
$445$	−2013.98	−0.214544
$446$	−5457.37	−0.579403
$447$	−8923.43	−0.944214
$448$	3982.38	0.419978
$449$	−10927.6	−1.14857	−0.574284	−	0.818656i	$-0.694719\pi$
$449$	−10927.6	−1.14857	−0.574284	+	0.818656i	$0.694719\pi$
$450$	708.314	0.0742006
$451$	14148.4	1.47721
$452$	3257.08	0.338939
$453$	8356.57	0.866724
$454$	2181.16	0.225478
$455$	828.408	0.0853547
$456$	−5426.20	−0.557248
$457$	−3534.21	−0.361758	−0.180879	−	0.983505i	$-0.557894\pi$
$457$	−3534.21	−0.361758	−0.180879	+	0.983505i	$0.557894\pi$
$458$	−5619.64	−0.573337
$459$	2870.04	0.291856
$460$	5000.48	0.506845
$461$	−7041.81	−0.711431	−0.355716	−	0.934594i	$-0.615763\pi$
$461$	−7041.81	−0.711431	−0.355716	+	0.934594i	$0.615763\pi$
$462$	5250.85	0.528770
$463$	1248.75	0.125344	0.0626719	−	0.998034i	$-0.480038\pi$
$463$	1248.75	0.125344	0.0626719	+	0.998034i	$0.480038\pi$
$464$	1296.41	0.129707
$465$	−6781.88	−0.676349
$466$	−4501.31	−0.447466
$467$	4042.94	0.400610	0.200305	−	0.979734i	$-0.435807\pi$
$467$	4042.94	0.400610	0.200305	+	0.979734i	$0.435807\pi$
$468$	−244.308	−0.0241307
$469$	−2392.47	−0.235552
$470$	4089.90	0.401390
$471$	471.000	0.0460776
$472$	11109.6	1.08339
$473$	2722.86	0.264687
$474$	5869.97	0.568811
$475$	−4289.47	−0.414346
$476$	11564.8	1.11359
$477$	6847.37	0.657274
$478$	9370.50	0.896646
$479$	17579.7	1.67691	0.838453	−	0.544974i	$-0.183461\pi$
$479$	17579.7	1.67691	0.838453	+	0.544974i	$0.183461\pi$
$480$	4854.45	0.461613
$481$	−722.093	−0.0684503
$482$	7776.54	0.734879
$483$	6127.89	0.577285
$484$	−10688.9	−1.00384
$485$	519.579	0.0486451
$486$	380.274	0.0354930
$487$	4905.61	0.456457	0.228228	−	0.973608i	$-0.426707\pi$
$487$	4905.61	0.456457	0.228228	+	0.973608i	$0.426707\pi$
$488$	−8650.85	−0.802471
$489$	−5664.08	−0.523801
$490$	−556.307	−0.0512886
$491$	−7354.20	−0.675948	−0.337974	−	0.941155i	$-0.609742\pi$
$491$	−7354.20	−0.675948	−0.337974	+	0.941155i	$0.609742\pi$
$492$	4128.81	0.378336
$493$	−12279.4	−1.12178
$494$	−652.713	−0.0594473
$495$	4439.23	0.403088
$496$	2935.15	0.265710
$497$	3635.02	0.328074
$498$	−489.295	−0.0440278
$499$	16324.2	1.46448	0.732238	−	0.681049i	$-0.238476\pi$
$499$	16324.2	1.46448	0.732238	+	0.681049i	$0.238476\pi$
$500$	8410.47	0.752256
$501$	−6118.18	−0.545589
$502$	−8577.54	−0.762618
$503$	−17914.2	−1.58799	−0.793993	−	0.607927i	$-0.792001\pi$
$503$	−17914.2	−1.58799	−0.793993	+	0.607927i	$0.792001\pi$
$504$	3740.62	0.330596
$505$	7955.63	0.701032
$506$	9307.26	0.817704
$507$	6519.26	0.571066
$508$	8142.00	0.711108
$509$	−11701.3	−1.01896	−0.509482	−	0.860481i	$-0.670163\pi$
$509$	−11701.3	−1.01896	−0.509482	+	0.860481i	$0.670163\pi$
$510$	−4313.42	−0.374513
$511$	−13195.8	−1.14236
$512$	−4004.85	−0.345686
$513$	−2302.90	−0.198198
$514$	6374.95	0.547056
$515$	4150.40	0.355123
$516$	794.586	0.0677901
$517$	−17255.1	−1.46785
$518$	4528.98	0.384155
$519$	−13501.4	−1.14190
$520$	896.334	0.0755901
$521$	21310.5	1.79200	0.896000	−	0.444054i	$-0.146460\pi$
$521$	21310.5	1.79200	0.896000	+	0.444054i	$0.146460\pi$
$522$	−1627.00	−0.136421
$523$	4738.00	0.396134	0.198067	−	0.980188i	$-0.436534\pi$
$523$	4738.00	0.396134	0.198067	+	0.980188i	$0.436534\pi$
$524$	−9912.04	−0.826354
$525$	2957.01	0.245818
$526$	5126.81	0.424980
$527$	−27801.5	−2.29801
$528$	−1921.26	−0.158357
$529$	−1305.16	−0.107270
$530$	−10291.0	−0.843420
$531$	4714.95	0.385332
$532$	−9279.48	−0.756234
$533$	1212.41	0.0985278
$534$	−1093.91	−0.0886483
$535$	−1570.40	−0.126905
$536$	−2588.64	−0.208605
$537$	−6216.79	−0.499580
$538$	−11243.2	−0.900981
$539$	2347.03	0.187558
$540$	1295.46	0.103236
$541$	−7688.84	−0.611033	−0.305517	−	0.952187i	$-0.598829\pi$
$541$	−7688.84	−0.611033	−0.305517	+	0.952187i	$0.598829\pi$
$542$	2957.35	0.234371
$543$	8554.28	0.676057
$544$	19900.2	1.56841
$545$	2228.83	0.175179
$546$	449.957	0.0352681
$547$	−11116.0	−0.868899	−0.434449	−	0.900696i	$-0.643057\pi$
$547$	−11116.0	−0.868899	−0.434449	+	0.900696i	$0.643057\pi$
$548$	−9274.71	−0.722985
$549$	−3671.45	−0.285417
$550$	4491.21	0.348192
$551$	9852.93	0.761795
$552$	6630.35	0.511243
$553$	24505.4	1.88440
$554$	466.269	0.0357579
$555$	3828.94	0.292846
$556$	−9361.41	−0.714051
$557$	6800.82	0.517343	0.258672	−	0.965965i	$-0.416715\pi$
$557$	6800.82	0.517343	0.258672	+	0.965965i	$0.416715\pi$
$558$	−3683.63	−0.279463
$559$	233.328	0.0176542
$560$	1901.12	0.143459
$561$	18198.1	1.36956
$562$	6220.29	0.466881
$563$	−15783.4	−1.18151	−0.590755	−	0.806851i	$-0.701170\pi$
$563$	−15783.4	−1.18151	−0.590755	+	0.806851i	$0.701170\pi$
$564$	−5035.39	−0.375936
$565$	−5071.54	−0.377631
$566$	3055.23	0.226892
$567$	1587.53	0.117584
$568$	3933.07	0.290542
$569$	−3001.81	−0.221164	−0.110582	−	0.993867i	$-0.535271\pi$
$569$	−3001.81	−0.221164	−0.110582	+	0.993867i	$0.535271\pi$
$570$	3461.05	0.254329
$571$	−21819.2	−1.59913	−0.799566	−	0.600578i	$-0.794937\pi$
$571$	−21819.2	−1.59913	−0.799566	+	0.600578i	$0.794937\pi$
$572$	−1549.08	−0.113235
$573$	7580.26	0.552653
$574$	−7604.27	−0.552955
$575$	5241.37	0.380139
$576$	1828.72	0.132286
$577$	−2896.05	−0.208950	−0.104475	−	0.994528i	$-0.533316\pi$
$577$	−2896.05	−0.208950	−0.104475	+	0.994528i	$0.533316\pi$
$578$	−9993.91	−0.719190
$579$	1927.29	0.138334
$580$	−5542.61	−0.396801
$581$	−2042.66	−0.145859
$582$	282.214	0.0200999
$583$	43417.1	3.08431
$584$	−14277.7	−1.01167
$585$	380.407	0.0268853
$586$	−6976.61	−0.491811
$587$	−26866.7	−1.88911	−0.944556	−	0.328349i	$-0.893508\pi$
$587$	−26866.7	−1.88911	−0.944556	+	0.328349i	$0.893508\pi$
$588$	684.912	0.0480362
$589$	22307.7	1.56056
$590$	−7086.16	−0.494462
$591$	1569.75	0.109257
$592$	−1657.14	−0.115047
$593$	14107.8	0.976960	0.488480	−	0.872575i	$-0.337551\pi$
$593$	14107.8	0.976960	0.488480	+	0.872575i	$0.337551\pi$
$594$	2411.20	0.166554
$595$	−18007.3	−1.24072
$596$	−16511.4	−1.13479
$597$	7365.66	0.504952
$598$	797.560	0.0545395
$599$	−27302.2	−1.86233	−0.931167	−	0.364592i	$-0.881209\pi$
$599$	−27302.2	−1.86233	−0.931167	+	0.364592i	$0.881209\pi$
$600$	3199.47	0.217696
$601$	−22799.0	−1.54741	−0.773703	−	0.633549i	$-0.781598\pi$
$601$	−22799.0	−1.54741	−0.773703	+	0.633549i	$0.781598\pi$
$602$	−1463.44	−0.0990784
$603$	−1098.63	−0.0741950
$604$	15462.6	1.04166
$605$	16643.4	1.11843
$606$	4321.17	0.289662
$607$	10606.8	0.709251	0.354626	−	0.935008i	$-0.384608\pi$
$607$	10606.8	0.709251	0.354626	+	0.935008i	$0.384608\pi$
$608$	−15967.8	−1.06510
$609$	−6792.25	−0.451947
$610$	5517.87	0.366249
$611$	−1478.62	−0.0979030
$612$	5310.58	0.350764
$613$	−15314.4	−1.00904	−0.504521	−	0.863400i	$-0.668331\pi$
$613$	−15314.4	−1.00904	−0.504521	+	0.863400i	$0.668331\pi$
$614$	4513.18	0.296641
$615$	−6428.89	−0.421525
$616$	23718.2	1.55135
$617$	−10709.5	−0.698780	−0.349390	−	0.936977i	$-0.613611\pi$
$617$	−10709.5	−0.698780	−0.349390	+	0.936977i	$0.613611\pi$
$618$	2254.32	0.146735
$619$	−10109.8	−0.656457	−0.328228	−	0.944598i	$-0.606452\pi$
$619$	−10109.8	−0.656457	−0.328228	+	0.944598i	$0.606452\pi$
$620$	−12548.8	−0.812860
$621$	2813.94	0.181835
$622$	7932.15	0.511335
$623$	−4566.76	−0.293681
$624$	−164.637	−0.0105621
$625$	−6809.37	−0.435800
$626$	−11807.2	−0.753853
$627$	−14602.0	−0.930059
$628$	871.514	0.0553777
$629$	15696.3	0.994995
$630$	−2385.92	−0.150885
$631$	−25447.1	−1.60544	−0.802721	−	0.596355i	$-0.796615\pi$
$631$	−25447.1	−1.60544	−0.802721	+	0.596355i	$0.796615\pi$
$632$	26514.7	1.66883
$633$	2704.47	0.169815
$634$	−8043.89	−0.503886
$635$	−12677.7	−0.792285
$636$	12670.0	0.789936
$637$	201.122	0.0125098
$638$	−10316.3	−0.640167
$639$	1669.21	0.103338
$640$	10196.8	0.629787
$641$	21197.9	1.30619	0.653093	−	0.757278i	$-0.273471\pi$
$641$	21197.9	1.30619	0.653093	+	0.757278i	$0.273471\pi$
$642$	−852.973	−0.0524364
$643$	−11922.9	−0.731250	−0.365625	−	0.930762i	$-0.619145\pi$
$643$	−11922.9	−0.731250	−0.365625	+	0.930762i	$0.619145\pi$
$644$	11338.7	0.693802
$645$	−1237.23	−0.0755288
$646$	14188.2	0.864126
$647$	8031.46	0.488021	0.244010	−	0.969773i	$-0.421537\pi$
$647$	8031.46	0.488021	0.244010	+	0.969773i	$0.421537\pi$
$648$	1717.70	0.104132
$649$	29896.1	1.80820
$650$	384.861	0.0232239
$651$	−15378.1	−0.925829
$652$	−10480.5	−0.629523
$653$	−13145.9	−0.787807	−0.393903	−	0.919152i	$-0.628876\pi$
$653$	−13145.9	−0.787807	−0.393903	+	0.919152i	$0.628876\pi$
$654$	1210.61	0.0723830
$655$	15433.8	0.920687
$656$	2782.37	0.165600
$657$	−6059.53	−0.359825
$658$	9273.97	0.549448
$659$	27319.8	1.61492	0.807458	−	0.589925i	$-0.200843\pi$
$659$	27319.8	1.61492	0.807458	+	0.589925i	$0.200843\pi$
$660$	8214.12	0.484446
$661$	−5800.79	−0.341338	−0.170669	−	0.985328i	$-0.554593\pi$
$661$	−5800.79	−0.341338	−0.170669	+	0.985328i	$0.554593\pi$
$662$	−4723.08	−0.277292
$663$	1559.43	0.0913474
$664$	−2210.15	−0.129173
$665$	14448.9	0.842562
$666$	2079.72	0.121002
$667$	−12039.4	−0.698904
$668$	−11320.8	−0.655708
$669$	−10462.0	−0.604609
$670$	1651.14	0.0952077
$671$	−23279.6	−1.33934
$672$	11007.6	0.631886
$673$	−27685.5	−1.58573	−0.792867	−	0.609394i	$-0.791413\pi$
$673$	−27685.5	−1.58573	−0.792867	+	0.609394i	$0.791413\pi$
$674$	−16737.2	−0.956518
$675$	1357.87	0.0774285
$676$	12062.9	0.686328
$677$	−23279.0	−1.32155	−0.660773	−	0.750586i	$-0.729771\pi$
$677$	−23279.0	−1.32155	−0.660773	+	0.750586i	$0.729771\pi$
$678$	−2754.65	−0.156035
$679$	1178.16	0.0665885
$680$	−19483.8	−1.09878
$681$	4181.36	0.235287
$682$	−23356.8	−1.31140
$683$	−19228.5	−1.07725	−0.538623	−	0.842547i	$-0.681055\pi$
$683$	−19228.5	−1.07725	−0.538623	+	0.842547i	$0.681055\pi$
$684$	−4261.16	−0.238201
$685$	14441.5	0.805518
$686$	9258.73	0.515306
$687$	−10773.1	−0.598279
$688$	535.466	0.0296722
$689$	3720.51	0.205719
$690$	−4229.11	−0.233333
$691$	−12087.7	−0.665465	−0.332732	−	0.943021i	$-0.607971\pi$
$691$	−12087.7	−0.665465	−0.332732	+	0.943021i	$0.607971\pi$
$692$	−24982.3	−1.37237
$693$	10066.1	0.551773
$694$	4196.79	0.229550
$695$	14576.5	0.795564
$696$	−7349.18	−0.400244
$697$	−26354.4	−1.43220
$698$	−14702.6	−0.797283
$699$	−8629.18	−0.466932
$700$	5471.49	0.295433
$701$	23662.5	1.27492	0.637461	−	0.770483i	$-0.279985\pi$
$701$	23662.5	1.27492	0.637461	+	0.770483i	$0.279985\pi$
$702$	206.621	0.0111089
$703$	−12594.6	−0.675694
$704$	11595.4	0.620763
$705$	7840.50	0.418852
$706$	−2024.58	−0.107926
$707$	18039.6	0.959618
$708$	8724.30	0.463106
$709$	−34696.7	−1.83789	−0.918943	−	0.394391i	$-0.870956\pi$
$709$	−34696.7	−1.83789	−0.918943	+	0.394391i	$0.870956\pi$
$710$	−2508.68	−0.132604
$711$	11252.9	0.593556
$712$	−4941.21	−0.260084
$713$	−27258.1	−1.43173
$714$	−9780.80	−0.512657
$715$	2412.05	0.126161
$716$	−11503.2	−0.600413
$717$	17963.6	0.935653
$718$	−10838.3	−0.563343
$719$	−1001.65	−0.0519545	−0.0259772	−	0.999663i	$-0.508270\pi$
$719$	−1001.65	−0.0519545	−0.0259772	+	0.999663i	$0.508270\pi$
$720$	873.001	0.0451872
$721$	9411.13	0.486115
$722$	−650.711	−0.0335415
$723$	14907.9	0.766848
$724$	15828.4	0.812510
$725$	−5809.61	−0.297605
$726$	9040.00	0.462130
$727$	−12373.2	−0.631219	−0.315610	−	0.948889i	$-0.602209\pi$
$727$	−12373.2	−0.631219	−0.315610	+	0.948889i	$0.602209\pi$
$728$	2032.46	0.103473
$729$	729.000	0.0370370
$730$	9106.94	0.461730
$731$	−5071.89	−0.256622
$732$	−6793.46	−0.343024
$733$	−16739.8	−0.843520	−0.421760	−	0.906708i	$-0.638588\pi$
$733$	−16739.8	−0.843520	−0.421760	+	0.906708i	$0.638588\pi$
$734$	8952.88	0.450214
$735$	−1066.46	−0.0535198
$736$	19511.3	0.977166
$737$	−6966.07	−0.348166
$738$	−3491.90	−0.174172
$739$	13740.6	0.683975	0.341987	−	0.939705i	$-0.388900\pi$
$739$	13740.6	0.683975	0.341987	+	0.939705i	$0.388900\pi$
$740$	7084.88	0.351953
$741$	−1251.28	−0.0620334
$742$	−23335.1	−1.15453
$743$	27120.2	1.33909	0.669545	−	0.742771i	$-0.266489\pi$
$743$	27120.2	1.33909	0.669545	+	0.742771i	$0.266489\pi$
$744$	−16639.0	−0.819914
$745$	25709.6	1.26433
$746$	−8210.02	−0.402936
$747$	−937.997	−0.0459431
$748$	33672.7	1.64599
$749$	−3560.91	−0.173716
$750$	−7113.08	−0.346311
$751$	21401.2	1.03987	0.519934	−	0.854206i	$-0.325956\pi$
$751$	21401.2	1.03987	0.519934	+	0.854206i	$0.325956\pi$
$752$	−3393.31	−0.164550
$753$	−16443.5	−0.795794
$754$	−884.027	−0.0426981
$755$	−24076.5	−1.16057
$756$	2937.49	0.141317
$757$	8846.12	0.424726	0.212363	−	0.977191i	$-0.431884\pi$
$757$	8846.12	0.424726	0.212363	+	0.977191i	$0.431884\pi$
$758$	−6623.16	−0.317367
$759$	17842.4	0.853277
$760$	15633.6	0.746173
$761$	18697.0	0.890624	0.445312	−	0.895375i	$-0.353093\pi$
$761$	18697.0	0.890624	0.445312	+	0.895375i	$0.353093\pi$
$762$	−6886.02	−0.327368
$763$	5053.93	0.239796
$764$	14026.1	0.664198
$765$	−8268.99	−0.390805
$766$	−3420.42	−0.161338
$767$	2561.86	0.120604
$768$	10415.1	0.489351
$769$	−30311.1	−1.42139	−0.710693	−	0.703502i	$-0.751619\pi$
$769$	−30311.1	−1.42139	−0.710693	+	0.703502i	$0.751619\pi$
$770$	−15128.4	−0.708040
$771$	12221.0	0.570855
$772$	3566.15	0.166255
$773$	14549.2	0.676971	0.338486	−	0.940972i	$-0.390085\pi$
$773$	14549.2	0.676971	0.338486	+	0.940972i	$0.390085\pi$
$774$	−672.014	−0.0312081
$775$	−13153.3	−0.609654
$776$	1274.76	0.0589708
$777$	8682.23	0.400867
$778$	−5496.09	−0.253271
$779$	21146.6	0.972599
$780$	703.886	0.0323117
$781$	10584.0	0.484921
$782$	−17336.7	−0.792788
$783$	−3119.02	−0.142356
$784$	461.557	0.0210257
$785$	−1357.02	−0.0616994
$786$	8383.02	0.380423
$787$	−27056.5	−1.22549	−0.612745	−	0.790281i	$-0.709935\pi$
$787$	−27056.5	−1.22549	−0.612745	+	0.790281i	$0.709935\pi$
$788$	2904.58	0.131309
$789$	9828.29	0.443468
$790$	−16912.2	−0.761657
$791$	−11499.9	−0.516925
$792$	10891.4	0.488650
$793$	−1994.88	−0.0893319
$794$	−8885.40	−0.397142
$795$	−19728.2	−0.880111
$796$	13629.0	0.606869
$797$	−39099.7	−1.73774	−0.868872	−	0.495036i	$-0.835155\pi$
$797$	−39099.7	−1.73774	−0.868872	+	0.495036i	$0.835155\pi$
$798$	7848.03	0.348142
$799$	32141.2	1.42312
$800$	9415.13	0.416094
$801$	−2097.07	−0.0925048
$802$	5.77938	0.000254460 0
$803$	−38421.6	−1.68851
$804$	−2032.84	−0.0891702
$805$	−17655.3	−0.773004
$806$	−2001.50	−0.0874686
$807$	−21553.6	−0.940176
$808$	19518.8	0.849837
$809$	−34492.8	−1.49901	−0.749506	−	0.661997i	$-0.769709\pi$
$809$	−34492.8	−1.49901	−0.749506	+	0.661997i	$0.769709\pi$
$810$	−1095.62	−0.0475263
$811$	22617.9	0.979310	0.489655	−	0.871916i	$-0.337123\pi$
$811$	22617.9	0.979310	0.489655	+	0.871916i	$0.337123\pi$
$812$	−12568.0	−0.543166
$813$	5669.35	0.244567
$814$	13186.9	0.567813
$815$	16319.0	0.701387
$816$	3578.76	0.153531
$817$	4069.64	0.174270
$818$	18298.0	0.782119
$819$	862.584	0.0368023
$820$	−11895.7	−0.506604
$821$	31239.2	1.32796	0.663981	−	0.747750i	$-0.268866\pi$
$821$	31239.2	1.32796	0.663981	+	0.747750i	$0.268866\pi$
$822$	7844.00	0.332836
$823$	24454.4	1.03575	0.517877	−	0.855455i	$-0.326723\pi$
$823$	24454.4	1.03575	0.517877	+	0.855455i	$0.326723\pi$
$824$	10182.8	0.430503
$825$	8609.81	0.363340
$826$	−16068.0	−0.676851
$827$	−36311.2	−1.52680	−0.763400	−	0.645927i	$-0.776471\pi$
$827$	−36311.2	−1.52680	−0.763400	+	0.645927i	$0.776471\pi$
$828$	5206.78	0.218536
$829$	−20247.4	−0.848277	−0.424139	−	0.905597i	$-0.639423\pi$
$829$	−20247.4	−0.848277	−0.424139	+	0.905597i	$0.639423\pi$
$830$	1409.73	0.0589547
$831$	893.855	0.0373134
$832$	993.633	0.0414039
$833$	−4371.83	−0.181843
$834$	7917.33	0.328723
$835$	17627.3	0.730561
$836$	−27018.7	−1.11778
$837$	−7061.66	−0.291621
$838$	408.257	0.0168293
$839$	13884.2	0.571317	0.285659	−	0.958331i	$-0.407788\pi$
$839$	13884.2	0.571317	0.285659	+	0.958331i	$0.407788\pi$
$840$	−10777.3	−0.442679
$841$	−11044.3	−0.452840
$842$	4424.39	0.181086
$843$	11924.5	0.487191
$844$	5004.22	0.204090
$845$	−18782.9	−0.764676
$846$	4258.63	0.173067
$847$	37739.4	1.53098
$848$	8538.23	0.345760
$849$	5856.99	0.236762
$850$	−8365.81	−0.337582
$851$	15389.5	0.619911
$852$	3088.62	0.124195
$853$	713.901	0.0286559	0.0143280	−	0.999897i	$-0.495439\pi$
$853$	713.901	0.0286559	0.0143280	+	0.999897i	$0.495439\pi$
$854$	12511.9	0.501346
$855$	6634.97	0.265393
$856$	−3852.89	−0.153842
$857$	37402.9	1.49085	0.745425	−	0.666589i	$-0.232246\pi$
$857$	37402.9	1.49085	0.745425	+	0.666589i	$0.232246\pi$
$858$	1310.12	0.0521292
$859$	14199.6	0.564008	0.282004	−	0.959413i	$-0.409001\pi$
$859$	14199.6	0.564008	0.282004	+	0.959413i	$0.409001\pi$
$860$	−2289.31	−0.0907732
$861$	−14577.7	−0.577010
$862$	−8676.62	−0.342839
$863$	43895.3	1.73142	0.865709	−	0.500547i	$-0.166868\pi$
$863$	43895.3	1.73142	0.865709	+	0.500547i	$0.166868\pi$
$864$	5054.72	0.199034
$865$	38899.4	1.52904
$866$	1659.27	0.0651090
$867$	−19158.7	−0.750477
$868$	−28454.8	−1.11270
$869$	71351.5	2.78531
$870$	4687.61	0.182672
$871$	−596.938	−0.0232221
$872$	5468.33	0.212363
$873$	541.014	0.0209743
$874$	13910.8	0.538377
$875$	−29695.0	−1.14729
$876$	−11212.2	−0.432450
$877$	−13907.7	−0.535495	−0.267748	−	0.963489i	$-0.586279\pi$
$877$	−13907.7	−0.535495	−0.267748	+	0.963489i	$0.586279\pi$
$878$	26784.7	1.02955
$879$	−13374.4	−0.513206
$880$	5535.43	0.212045
$881$	4931.24	0.188579	0.0942893	−	0.995545i	$-0.469942\pi$
$881$	4931.24	0.188579	0.0942893	+	0.995545i	$0.469942\pi$
$882$	−579.258	−0.0221141
$883$	5548.54	0.211465	0.105732	−	0.994395i	$-0.466281\pi$
$883$	5548.54	0.211465	0.105732	+	0.994395i	$0.466281\pi$
$884$	2885.49	0.109785
$885$	−13584.4	−0.515972
$886$	21883.4	0.829781
$887$	25049.4	0.948227	0.474114	−	0.880464i	$-0.342769\pi$
$887$	25049.4	0.948227	0.474114	+	0.880464i	$0.342769\pi$
$888$	9394.13	0.355007
$889$	−28747.1	−1.08453
$890$	3151.71	0.118703
$891$	4622.37	0.173799
$892$	−19358.3	−0.726641
$893$	−25789.8	−0.966431
$894$	13964.4	0.522415
$895$	17911.4	0.668954
$896$	23121.5	0.862093
$897$	1528.95	0.0569122
$898$	17100.8	0.635481
$899$	30213.3	1.12088
$900$	2512.52	0.0930564
$901$	−80873.5	−2.99033
$902$	−22141.1	−0.817315
$903$	−2805.46	−0.103389
$904$	−12442.8	−0.457789
$905$	−24646.1	−0.905263
$906$	−13077.3	−0.479542
$907$	−28811.4	−1.05476	−0.527380	−	0.849629i	$-0.676826\pi$
$907$	−28811.4	−1.05476	−0.527380	+	0.849629i	$0.676826\pi$
$908$	7736.98	0.282776
$909$	8283.84	0.302264
$910$	−1296.39	−0.0472251
$911$	11132.7	0.404877	0.202439	−	0.979295i	$-0.435113\pi$
$911$	11132.7	0.404877	0.202439	+	0.979295i	$0.435113\pi$
$912$	−2871.56	−0.104262
$913$	−5947.56	−0.215592
$914$	5530.74	0.200154
$915$	10578.0	0.382182
$916$	−19933.9	−0.719034
$917$	34996.7	1.26030
$918$	−4491.37	−0.161479
$919$	18945.6	0.680041	0.340020	−	0.940418i	$-0.389566\pi$
$919$	18945.6	0.680041	0.340020	+	0.940418i	$0.389566\pi$
$920$	−19103.0	−0.684572
$921$	8651.94	0.309545
$922$	11019.8	0.393621
$923$	906.962	0.0323435
$924$	18625.7	0.663140
$925$	7426.17	0.263969
$926$	−1954.18	−0.0693503
$927$	4321.62	0.153118
$928$	−21626.6	−0.765008
$929$	−2928.55	−0.103426	−0.0517129	−	0.998662i	$-0.516468\pi$
$929$	−2928.55	−0.103426	−0.0517129	+	0.998662i	$0.516468\pi$
$930$	10613.1	0.374211
$931$	3507.92	0.123488
$932$	−15967.0	−0.561176
$933$	15206.2	0.533579
$934$	−6326.86	−0.221650
$935$	−52431.2	−1.83389
$936$	933.311	0.0325921
$937$	8878.84	0.309562	0.154781	−	0.987949i	$-0.450533\pi$
$937$	8878.84	0.309562	0.154781	+	0.987949i	$0.450533\pi$
$938$	3744.01	0.130326
$939$	−22634.9	−0.786648
$940$	14507.7	0.503391
$941$	−22362.1	−0.774690	−0.387345	−	0.921935i	$-0.626608\pi$
$941$	−22362.1	−0.774690	−0.387345	+	0.921935i	$0.626608\pi$
$942$	−737.075	−0.0254938
$943$	−25839.3	−0.892305
$944$	5879.24	0.202704
$945$	−4573.91	−0.157449
$946$	−4261.04	−0.146446
$947$	−23044.0	−0.790739	−0.395369	−	0.918522i	$-0.629383\pi$
$947$	−23044.0	−0.790739	−0.395369	+	0.918522i	$0.629383\pi$
$948$	20821.9	0.713357
$949$	−3292.43	−0.112621
$950$	6712.66	0.229250
$951$	−15420.4	−0.525806
$952$	−44180.0	−1.50408
$953$	−11627.9	−0.395239	−0.197620	−	0.980279i	$-0.563321\pi$
$953$	−11627.9	−0.395239	−0.197620	+	0.980279i	$0.563321\pi$
$954$	−10715.6	−0.363657
$955$	−21839.8	−0.740020
$956$	33238.9	1.12450
$957$	−19776.8	−0.668016
$958$	−27510.7	−0.927800
$959$	32746.4	1.10265
$960$	−5268.80	−0.177135
$961$	38613.8	1.29616
$962$	1130.01	0.0378722
$963$	−1635.18	−0.0547175
$964$	27584.8	0.921626
$965$	−5552.78	−0.185233
$966$	−9589.63	−0.319401
$967$	−27021.2	−0.898598	−0.449299	−	0.893381i	$-0.648326\pi$
$967$	−27021.2	−0.898598	−0.449299	+	0.893381i	$0.648326\pi$
$968$	40833.8	1.35584
$969$	27199.2	0.901719
$970$	−813.097	−0.0269144
$971$	−36133.1	−1.19420	−0.597099	−	0.802167i	$-0.703680\pi$
$971$	−36133.1	−1.19420	−0.597099	+	0.802167i	$0.703680\pi$
$972$	1348.90	0.0445124
$973$	33052.5	1.08902
$974$	−7676.86	−0.252549
$975$	737.794	0.0242342
$976$	−4578.06	−0.150144
$977$	−22487.3	−0.736369	−0.368184	−	0.929753i	$-0.620020\pi$
$977$	−22487.3	−0.736369	−0.368184	+	0.929753i	$0.620020\pi$
$978$	8863.81	0.289809
$979$	−13296.9	−0.434086
$980$	−1973.33	−0.0643220
$981$	2320.78	0.0755319
$982$	11508.7	0.373989
$983$	−37567.1	−1.21893	−0.609463	−	0.792814i	$-0.708615\pi$
$983$	−37567.1	−1.21893	−0.609463	+	0.792814i	$0.708615\pi$
$984$	−15773.0	−0.511000
$985$	−4522.67	−0.146299
$986$	19216.3	0.620660
$987$	17778.5	0.573351
$988$	−2315.29	−0.0745540
$989$	−4972.75	−0.159883
$990$	−6947.01	−0.223021
$991$	53576.7	1.71738	0.858689	−	0.512497i	$-0.171280\pi$
$991$	53576.7	1.71738	0.858689	+	0.512497i	$0.171280\pi$
$992$	−48964.0	−1.56714
$993$	−9054.31	−0.289355
$994$	−5688.49	−0.181517
$995$	−21221.5	−0.676147
$996$	−1735.62	−0.0552161
$997$	60993.2	1.93749	0.968743	−	0.248067i	$-0.0797955\pi$
$997$	60993.2	1.93749	0.968743	+	0.248067i	$0.0797955\pi$
$998$	−25546.0	−0.810266
$999$	3986.91	0.126266

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.a.c.1.8	✓	22
3.2	odd	2		1413.4.a.e.1.15		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.a.c.1.8	✓	22	1.1	even	1	trivial
1413.4.a.e.1.15		22	3.2	odd	2

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:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	471.a (trivial)

\(f(q)\)	\(=\)	\(q-1.56491 q^{2} -3.00000 q^{3} -5.55104 q^{4} +8.64342 q^{5} +4.69474 q^{6} +19.5992 q^{7} +21.2062 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-1.56491 q^{2} -3.00000 q^{3} -5.55104 q^{4} +8.64342 q^{5} +4.69474 q^{6} +19.5992 q^{7} +21.2062 q^{8} +9.00000 q^{9} -13.5262 q^{10} +57.0663 q^{11} +16.6531 q^{12} +4.89013 q^{13} -30.6710 q^{14} -25.9303 q^{15} +11.2224 q^{16} -106.298 q^{17} -14.0842 q^{18} +85.2925 q^{19} -47.9800 q^{20} -58.7976 q^{21} -89.3038 q^{22} -104.220 q^{23} -63.6187 q^{24} -50.2913 q^{25} -7.65264 q^{26} -27.0000 q^{27} -108.796 q^{28} +115.519 q^{29} +40.5786 q^{30} +261.543 q^{31} -187.212 q^{32} -171.199 q^{33} +166.347 q^{34} +169.404 q^{35} -49.9594 q^{36} -147.663 q^{37} -133.475 q^{38} -14.6704 q^{39} +183.294 q^{40} +247.930 q^{41} +92.0131 q^{42} +47.7139 q^{43} -316.777 q^{44} +77.7908 q^{45} +163.096 q^{46} -302.369 q^{47} -33.6673 q^{48} +41.1281 q^{49} +78.7016 q^{50} +318.894 q^{51} -27.1453 q^{52} +760.819 q^{53} +42.2527 q^{54} +493.248 q^{55} +415.625 q^{56} -255.878 q^{57} -180.778 q^{58} +523.883 q^{59} +143.940 q^{60} -407.939 q^{61} -409.293 q^{62} +176.393 q^{63} +203.191 q^{64} +42.2675 q^{65} +267.911 q^{66} -122.070 q^{67} +590.064 q^{68} +312.661 q^{69} -265.103 q^{70} +185.468 q^{71} +190.856 q^{72} -673.281 q^{73} +231.080 q^{74} +150.874 q^{75} -473.462 q^{76} +1118.45 q^{77} +22.9579 q^{78} +1250.33 q^{79} +97.0001 q^{80} +81.0000 q^{81} -387.989 q^{82} -104.222 q^{83} +326.388 q^{84} -918.777 q^{85} -74.6682 q^{86} -346.558 q^{87} +1210.16 q^{88} -233.008 q^{89} -121.736 q^{90} +95.8427 q^{91} +578.531 q^{92} -784.629 q^{93} +473.181 q^{94} +737.219 q^{95} +561.636 q^{96} +60.1127 q^{97} -64.3620 q^{98} +513.596 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9}+O(q^{10}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9} + 13 q^{10} + 61 q^{11} - 270 q^{12} + 4 q^{13} + 133 q^{14} - 96 q^{15} + 342 q^{16} + 308 q^{17} + 36 q^{18} + 32 q^{19} + 407 q^{20} + 12 q^{21} - 166 q^{22} + 53 q^{23} - 81 q^{24} + 746 q^{25} + 467 q^{26} - 594 q^{27} + 85 q^{28} + 634 q^{29} - 39 q^{30} - 163 q^{31} + 150 q^{32} - 183 q^{33} + 37 q^{34} + 782 q^{35} + 810 q^{36} - 2 q^{37} + 584 q^{38} - 12 q^{39} + 864 q^{40} + 1593 q^{41} - 399 q^{42} - 891 q^{43} + 2093 q^{44} + 288 q^{45} + 108 q^{46} + 1200 q^{47} - 1026 q^{48} + 2816 q^{49} + 4703 q^{50} - 924 q^{51} + 1866 q^{52} + 1182 q^{53} - 108 q^{54} + 970 q^{55} + 5362 q^{56} - 96 q^{57} + 1814 q^{58} + 2802 q^{59} - 1221 q^{60} + 2629 q^{61} + 2378 q^{62} - 36 q^{63} + 625 q^{64} + 2264 q^{65} + 498 q^{66} - 1074 q^{67} + 4383 q^{68} - 159 q^{69} + 4009 q^{70} + 3920 q^{71} + 243 q^{72} + 1086 q^{73} + 4904 q^{74} - 2238 q^{75} + 3750 q^{76} + 2966 q^{77} - 1401 q^{78} - 30 q^{79} + 7777 q^{80} + 1782 q^{81} + 2932 q^{82} + 1900 q^{83} - 255 q^{84} + 524 q^{85} + 3209 q^{86} - 1902 q^{87} - 100 q^{88} + 4488 q^{89} + 117 q^{90} - 818 q^{91} + 6210 q^{92} + 489 q^{93} + 3220 q^{94} + 3500 q^{95} - 450 q^{96} + 2178 q^{97} + 7629 q^{98} + 549 q^{99}+O(q^{100}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9 + 13 * q^10 + 61 * q^11 - 270 * q^12 + 4 * q^13 + 133 * q^14 - 96 * q^15 + 342 * q^16 + 308 * q^17 + 36 * q^18 + 32 * q^19 + 407 * q^20 + 12 * q^21 - 166 * q^22 + 53 * q^23 - 81 * q^24 + 746 * q^25 + 467 * q^26 - 594 * q^27 + 85 * q^28 + 634 * q^29 - 39 * q^30 - 163 * q^31 + 150 * q^32 - 183 * q^33 + 37 * q^34 + 782 * q^35 + 810 * q^36 - 2 * q^37 + 584 * q^38 - 12 * q^39 + 864 * q^40 + 1593 * q^41 - 399 * q^42 - 891 * q^43 + 2093 * q^44 + 288 * q^45 + 108 * q^46 + 1200 * q^47 - 1026 * q^48 + 2816 * q^49 + 4703 * q^50 - 924 * q^51 + 1866 * q^52 + 1182 * q^53 - 108 * q^54 + 970 * q^55 + 5362 * q^56 - 96 * q^57 + 1814 * q^58 + 2802 * q^59 - 1221 * q^60 + 2629 * q^61 + 2378 * q^62 - 36 * q^63 + 625 * q^64 + 2264 * q^65 + 498 * q^66 - 1074 * q^67 + 4383 * q^68 - 159 * q^69 + 4009 * q^70 + 3920 * q^71 + 243 * q^72 + 1086 * q^73 + 4904 * q^74 - 2238 * q^75 + 3750 * q^76 + 2966 * q^77 - 1401 * q^78 - 30 * q^79 + 7777 * q^80 + 1782 * q^81 + 2932 * q^82 + 1900 * q^83 - 255 * q^84 + 524 * q^85 + 3209 * q^86 - 1902 * q^87 - 100 * q^88 + 4488 * q^89 + 117 * q^90 - 818 * q^91 + 6210 * q^92 + 489 * q^93 + 3220 * q^94 + 3500 * q^95 - 450 * q^96 + 2178 * q^97 + 7629 * q^98 + 549 * q^99

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