LMFDB - Embedded newform 471.4.a.c.1.7

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			1.7
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)

$f(q)$	$=$	$q-2.31678 q^{2} -3.00000 q^{3} -2.63252 q^{4} -1.90474 q^{5} +6.95035 q^{6} -14.6237 q^{7} +24.6332 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.31678 q^{2} -3.00000 q^{3} -2.63252 q^{4} -1.90474 q^{5} +6.95035 q^{6} -14.6237 q^{7} +24.6332 q^{8} +9.00000 q^{9} +4.41286 q^{10} -54.4365 q^{11} +7.89756 q^{12} -25.6201 q^{13} +33.8799 q^{14} +5.71421 q^{15} -36.0097 q^{16} -51.4869 q^{17} -20.8510 q^{18} +30.5572 q^{19} +5.01425 q^{20} +43.8711 q^{21} +126.117 q^{22} -68.6747 q^{23} -73.8997 q^{24} -121.372 q^{25} +59.3561 q^{26} -27.0000 q^{27} +38.4972 q^{28} -244.978 q^{29} -13.2386 q^{30} -123.281 q^{31} -113.639 q^{32} +163.309 q^{33} +119.284 q^{34} +27.8543 q^{35} -23.6927 q^{36} +77.4129 q^{37} -70.7943 q^{38} +76.8602 q^{39} -46.9198 q^{40} +23.0809 q^{41} -101.640 q^{42} -343.470 q^{43} +143.305 q^{44} -17.1426 q^{45} +159.104 q^{46} +565.545 q^{47} +108.029 q^{48} -129.147 q^{49} +281.192 q^{50} +154.461 q^{51} +67.4453 q^{52} +631.450 q^{53} +62.5531 q^{54} +103.687 q^{55} -360.229 q^{56} -91.6716 q^{57} +567.560 q^{58} -454.423 q^{59} -15.0428 q^{60} +262.319 q^{61} +285.615 q^{62} -131.613 q^{63} +551.355 q^{64} +48.7995 q^{65} -378.352 q^{66} -721.020 q^{67} +135.540 q^{68} +206.024 q^{69} -64.5323 q^{70} +531.598 q^{71} +221.699 q^{72} -498.073 q^{73} -179.349 q^{74} +364.116 q^{75} -80.4424 q^{76} +796.063 q^{77} -178.068 q^{78} +1050.86 q^{79} +68.5889 q^{80} +81.0000 q^{81} -53.4734 q^{82} +615.566 q^{83} -115.492 q^{84} +98.0689 q^{85} +795.745 q^{86} +734.933 q^{87} -1340.95 q^{88} +127.173 q^{89} +39.7157 q^{90} +374.660 q^{91} +180.788 q^{92} +369.843 q^{93} -1310.24 q^{94} -58.2034 q^{95} +340.918 q^{96} -87.9958 q^{97} +299.206 q^{98} -489.928 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.31678	−0.819106	−0.409553	−	0.912286i	$-0.634315\pi$
$2$	−2.31678	−0.819106	−0.409553	+	0.912286i	$0.634315\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−2.63252	−0.329065
$5$	−1.90474	−0.170365	−0.0851824	−	0.996365i	$-0.527147\pi$
$5$	−1.90474	−0.170365	−0.0851824	+	0.996365i	$0.527147\pi$
$6$	6.95035	0.472911
$7$	−14.6237	−0.789606	−0.394803	−	0.918766i	$-0.629187\pi$
$7$	−14.6237	−0.789606	−0.394803	+	0.918766i	$0.629187\pi$
$8$	24.6332	1.08865
$9$	9.00000	0.333333
$10$	4.41286	0.139547
$11$	−54.4365	−1.49211	−0.746055	−	0.665884i	$-0.768055\pi$
$11$	−54.4365	−1.49211	−0.746055	+	0.665884i	$0.768055\pi$
$12$	7.89756	0.189986
$13$	−25.6201	−0.546594	−0.273297	−	0.961930i	$-0.588114\pi$
$13$	−25.6201	−0.546594	−0.273297	+	0.961930i	$0.588114\pi$
$14$	33.8799	0.646771
$15$	5.71421	0.0983601
$16$	−36.0097	−0.562651
$17$	−51.4869	−0.734553	−0.367277	−	0.930112i	$-0.619710\pi$
$17$	−51.4869	−0.734553	−0.367277	+	0.930112i	$0.619710\pi$
$18$	−20.8510	−0.273035
$19$	30.5572	0.368963	0.184482	−	0.982836i	$-0.440939\pi$
$19$	30.5572	0.368963	0.184482	+	0.982836i	$0.440939\pi$
$20$	5.01425	0.0560611
$21$	43.8711	0.455879
$22$	126.117	1.22220
$23$	−68.6747	−0.622594	−0.311297	−	0.950313i	$-0.600763\pi$
$23$	−68.6747	−0.622594	−0.311297	+	0.950313i	$0.600763\pi$
$24$	−73.8997	−0.628530
$25$	−121.372	−0.970976
$26$	59.3561	0.447719
$27$	−27.0000	−0.192450
$28$	38.4972	0.259832
$29$	−244.978	−1.56866	−0.784332	−	0.620342i	$-0.786994\pi$
$29$	−244.978	−1.56866	−0.784332	+	0.620342i	$0.786994\pi$
$30$	−13.2386	−0.0805674
$31$	−123.281	−0.714255	−0.357128	−	0.934056i	$-0.616244\pi$
$31$	−123.281	−0.714255	−0.357128	+	0.934056i	$0.616244\pi$
$32$	−113.639	−0.627774
$33$	163.309	0.861471
$34$	119.284	0.601677
$35$	27.8543	0.134521
$36$	−23.6927	−0.109688
$37$	77.4129	0.343962	0.171981	−	0.985100i	$-0.444983\pi$
$37$	77.4129	0.343962	0.171981	+	0.985100i	$0.444983\pi$
$38$	−70.7943	−0.302220
$39$	76.8602	0.315576
$40$	−46.9198	−0.185467
$41$	23.0809	0.0879177	0.0439589	−	0.999033i	$-0.486003\pi$
$41$	23.0809	0.0879177	0.0439589	+	0.999033i	$0.486003\pi$
$42$	−101.640	−0.373413
$43$	−343.470	−1.21811	−0.609054	−	0.793129i	$-0.708451\pi$
$43$	−343.470	−1.21811	−0.609054	+	0.793129i	$0.708451\pi$
$44$	143.305	0.491001
$45$	−17.1426	−0.0567882
$46$	159.104	0.509971
$47$	565.545	1.75517	0.877587	−	0.479418i	$-0.159152\pi$
$47$	565.545	1.75517	0.877587	+	0.479418i	$0.159152\pi$
$48$	108.029	0.324847
$49$	−129.147	−0.376523
$50$	281.192	0.795332
$51$	154.461	0.424095
$52$	67.4453	0.179865
$53$	631.450	1.63654	0.818268	−	0.574837i	$-0.194935\pi$
$53$	631.450	1.63654	0.818268	+	0.574837i	$0.194935\pi$
$54$	62.5531	0.157637
$55$	103.687	0.254203
$56$	−360.229	−0.859601
$57$	−91.6716	−0.213021
$58$	567.560	1.28490
$59$	−454.423	−1.00273	−0.501363	−	0.865237i	$-0.667168\pi$
$59$	−454.423	−1.00273	−0.501363	+	0.865237i	$0.667168\pi$
$60$	−15.0428	−0.0323669
$61$	262.319	0.550598	0.275299	−	0.961359i	$-0.411223\pi$
$61$	262.319	0.550598	0.275299	+	0.961359i	$0.411223\pi$
$62$	285.615	0.585051
$63$	−131.613	−0.263202
$64$	551.355	1.07687
$65$	48.7995	0.0931204
$66$	−378.352	−0.705636
$67$	−721.020	−1.31472	−0.657362	−	0.753575i	$-0.728328\pi$
$67$	−721.020	−1.31472	−0.657362	+	0.753575i	$0.728328\pi$
$68$	135.540	0.241716
$69$	206.024	0.359455
$70$	−64.5323	−0.110187
$71$	531.598	0.888578	0.444289	−	0.895884i	$-0.353456\pi$
$71$	531.598	0.888578	0.444289	+	0.895884i	$0.353456\pi$
$72$	221.699	0.362882
$73$	−498.073	−0.798562	−0.399281	−	0.916829i	$-0.630740\pi$
$73$	−498.073	−0.798562	−0.399281	+	0.916829i	$0.630740\pi$
$74$	−179.349	−0.281742
$75$	364.116	0.560593
$76$	−80.4424	−0.121413
$77$	796.063	1.17818
$78$	−178.068	−0.258491
$79$	1050.86	1.49659	0.748296	−	0.663365i	$-0.230872\pi$
$79$	1050.86	1.49659	0.748296	+	0.663365i	$0.230872\pi$
$80$	68.5889	0.0958559
$81$	81.0000	0.111111
$82$	−53.4734	−0.0720139
$83$	615.566	0.814062	0.407031	−	0.913414i	$-0.366564\pi$
$83$	615.566	0.814062	0.407031	+	0.913414i	$0.366564\pi$
$84$	−115.492	−0.150014
$85$	98.0689	0.125142
$86$	795.745	0.997760
$87$	734.933	0.905668
$88$	−1340.95	−1.62438
$89$	127.173	0.151464	0.0757320	−	0.997128i	$-0.475871\pi$
$89$	127.173	0.151464	0.0757320	+	0.997128i	$0.475871\pi$
$90$	39.7157	0.0465156
$91$	374.660	0.431594
$92$	180.788	0.204874
$93$	369.843	0.412376
$94$	−1310.24	−1.43767
$95$	−58.2034	−0.0628583
$96$	340.918	0.362446
$97$	−87.9958	−0.0921095	−0.0460548	−	0.998939i	$-0.514665\pi$
$97$	−87.9958	−0.0921095	−0.0460548	+	0.998939i	$0.514665\pi$
$98$	299.206	0.308412
$99$	−489.928	−0.497370
$100$	319.514	0.319514
$101$	1101.81	1.08549	0.542743	−	0.839899i	$-0.317386\pi$
$101$	1101.81	1.08549	0.542743	+	0.839899i	$0.317386\pi$
$102$	−357.852	−0.347378
$103$	−1144.90	−1.09525	−0.547625	−	0.836724i	$-0.684468\pi$
$103$	−1144.90	−1.09525	−0.547625	+	0.836724i	$0.684468\pi$
$104$	−631.105	−0.595048
$105$	−83.5628	−0.0776657
$106$	−1462.93	−1.34050
$107$	741.466	0.669909	0.334954	−	0.942234i	$-0.391279\pi$
$107$	741.466	0.669909	0.334954	+	0.942234i	$0.391279\pi$
$108$	71.0780	0.0633286
$109$	1568.58	1.37837	0.689187	−	0.724584i	$-0.257968\pi$
$109$	1568.58	1.37837	0.689187	+	0.724584i	$0.257968\pi$
$110$	−240.220	−0.208219
$111$	−232.239	−0.198587
$112$	526.595	0.444273
$113$	691.435	0.575617	0.287809	−	0.957688i	$-0.407073\pi$
$113$	691.435	0.575617	0.287809	+	0.957688i	$0.407073\pi$
$114$	212.383	0.174487
$115$	130.807	0.106068
$116$	644.909	0.516192
$117$	−230.581	−0.182198
$118$	1052.80	0.821339
$119$	752.929	0.580007
$120$	140.759	0.107079
$121$	1632.33	1.22639
$122$	−607.736	−0.450998
$123$	−69.2426	−0.0507593
$124$	324.540	0.235037
$125$	469.273	0.335785
$126$	304.919	0.215590
$127$	−1643.55	−1.14836	−0.574178	−	0.818730i	$-0.694678\pi$
$127$	−1643.55	−1.14836	−0.574178	+	0.818730i	$0.694678\pi$
$128$	−368.255	−0.254293
$129$	1030.41	0.703275
$130$	−113.058	−0.0762755
$131$	−948.031	−0.632289	−0.316144	−	0.948711i	$-0.602388\pi$
$131$	−948.031	−0.632289	−0.316144	+	0.948711i	$0.602388\pi$
$132$	−429.915	−0.283480
$133$	−446.859	−0.291335
$134$	1670.45	1.07690
$135$	51.4279	0.0327867
$136$	−1268.29	−0.799668
$137$	2147.37	1.33914	0.669570	−	0.742749i	$-0.266479\pi$
$137$	2147.37	1.33914	0.669570	+	0.742749i	$0.266479\pi$
$138$	−477.313	−0.294432
$139$	−1810.45	−1.10475	−0.552374	−	0.833596i	$-0.686278\pi$
$139$	−1810.45	−1.10475	−0.552374	+	0.833596i	$0.686278\pi$
$140$	−73.3270	−0.0442661
$141$	−1696.63	−1.01335
$142$	−1231.60	−0.727840
$143$	1394.67	0.815579
$144$	−324.087	−0.187550
$145$	466.618	0.267245
$146$	1153.93	0.654107
$147$	387.442	0.217386
$148$	−203.791	−0.113186
$149$	393.771	0.216503	0.108252	−	0.994124i	$-0.465475\pi$
$149$	393.771	0.216503	0.108252	+	0.994124i	$0.465475\pi$
$150$	−843.577	−0.459185
$151$	306.526	0.165197	0.0825984	−	0.996583i	$-0.473678\pi$
$151$	306.526	0.165197	0.0825984	+	0.996583i	$0.473678\pi$
$152$	752.722	0.401670
$153$	−463.382	−0.244851
$154$	−1844.30	−0.965054
$155$	234.818	0.121684
$156$	−202.336	−0.103845
$157$	−157.000	−0.0798087
$157$	−157.000	−0.0798087
$158$	−2434.61	−1.22587
$159$	−1894.35	−0.944854
$160$	216.453	0.106951
$161$	1004.28	0.491604
$162$	−187.659	−0.0910118
$163$	−1940.53	−0.932479	−0.466240	−	0.884658i	$-0.654392\pi$
$163$	−1940.53	−0.932479	−0.466240	+	0.884658i	$0.654392\pi$
$164$	−60.7609	−0.0289306
$165$	−311.061	−0.146764
$166$	−1426.13	−0.666803
$167$	−2409.31	−1.11640	−0.558198	−	0.829708i	$-0.688507\pi$
$167$	−2409.31	−1.11640	−0.558198	+	0.829708i	$0.688507\pi$
$168$	1080.69	0.496291
$169$	−1540.61	−0.701234
$170$	−227.204	−0.102505
$171$	275.015	0.122988
$172$	904.191	0.400837
$173$	−4258.73	−1.87159	−0.935795	−	0.352545i	$-0.885317\pi$
$173$	−4258.73	−1.87159	−0.935795	+	0.352545i	$0.885317\pi$
$174$	−1702.68	−0.741838
$175$	1774.91	0.766688
$176$	1960.24	0.839538
$177$	1363.27	0.578924
$178$	−294.632	−0.124065
$179$	−3921.83	−1.63760	−0.818802	−	0.574076i	$-0.805362\pi$
$179$	−3921.83	−1.63760	−0.818802	+	0.574076i	$0.805362\pi$
$180$	45.1283	0.0186870
$181$	511.777	0.210166	0.105083	−	0.994463i	$-0.466489\pi$
$181$	511.777	0.210166	0.105083	+	0.994463i	$0.466489\pi$
$182$	−868.006	−0.353521
$183$	−786.957	−0.317888
$184$	−1691.68	−0.677784
$185$	−147.451	−0.0585990
$186$	−856.846	−0.337779
$187$	2802.77	1.09603
$188$	−1488.81	−0.577566
$189$	394.840	0.151960
$190$	134.844	0.0514876
$191$	1851.23	0.701311	0.350656	−	0.936504i	$-0.385959\pi$
$191$	1851.23	0.701311	0.350656	+	0.936504i	$0.385959\pi$
$192$	−1654.06	−0.621728
$193$	1625.72	0.606329	0.303165	−	0.952938i	$-0.401957\pi$
$193$	1625.72	0.606329	0.303165	+	0.952938i	$0.401957\pi$
$194$	203.867	0.0754475
$195$	−146.398	−0.0537631
$196$	339.983	0.123901
$197$	4118.95	1.48966	0.744830	−	0.667255i	$-0.232531\pi$
$197$	4118.95	1.48966	0.744830	+	0.667255i	$0.232531\pi$
$198$	1135.06	0.407399
$199$	−265.672	−0.0946381	−0.0473190	−	0.998880i	$-0.515068\pi$
$199$	−265.672	−0.0946381	−0.0473190	+	0.998880i	$0.515068\pi$
$200$	−2989.78	−1.05705
$201$	2163.06	0.759057
$202$	−2552.65	−0.889128
$203$	3582.48	1.23863
$204$	−406.621	−0.139555
$205$	−43.9630	−0.0149781
$206$	2652.49	0.897127
$207$	−618.072	−0.207531
$208$	922.570	0.307542
$209$	−1663.43	−0.550534
$210$	193.597	0.0636165
$211$	3086.11	1.00690	0.503451	−	0.864024i	$-0.332063\pi$
$211$	3086.11	1.00690	0.503451	+	0.864024i	$0.332063\pi$
$212$	−1662.31	−0.538527
$213$	−1594.79	−0.513021
$214$	−1717.82	−0.548726
$215$	654.219	0.207523
$216$	−665.097	−0.209510
$217$	1802.82	0.563980
$218$	−3634.06	−1.12903
$219$	1494.22	0.461050
$220$	−272.958	−0.0836493
$221$	1319.10	0.401503
$222$	538.047	0.162664
$223$	−1738.38	−0.522019	−0.261009	−	0.965336i	$-0.584055\pi$
$223$	−1738.38	−0.522019	−0.261009	+	0.965336i	$0.584055\pi$
$224$	1661.83	0.495694
$225$	−1092.35	−0.323659
$226$	−1601.90	−0.471492
$227$	−331.688	−0.0969819	−0.0484909	−	0.998824i	$-0.515441\pi$
$227$	−331.688	−0.0969819	−0.0484909	+	0.998824i	$0.515441\pi$
$228$	241.327	0.0700977
$229$	−1928.01	−0.556359	−0.278180	−	0.960529i	$-0.589731\pi$
$229$	−1928.01	−0.556359	−0.278180	+	0.960529i	$0.589731\pi$
$230$	−303.052	−0.0868810
$231$	−2388.19	−0.680222
$232$	−6034.60	−1.70772
$233$	5467.10	1.53717	0.768587	−	0.639746i	$-0.220960\pi$
$233$	5467.10	1.53717	0.768587	+	0.639746i	$0.220960\pi$
$234$	534.205	0.149240
$235$	−1077.21	−0.299020
$236$	1196.28	0.329962
$237$	−3152.58	−0.864058
$238$	−1744.37	−0.475088
$239$	−4418.06	−1.19573	−0.597867	−	0.801595i	$-0.703985\pi$
$239$	−4418.06	−1.19573	−0.597867	+	0.801595i	$0.703985\pi$
$240$	−205.767	−0.0553424
$241$	−903.648	−0.241531	−0.120766	−	0.992681i	$-0.538535\pi$
$241$	−903.648	−0.241531	−0.120766	+	0.992681i	$0.538535\pi$
$242$	−3781.76	−1.00455
$243$	−243.000	−0.0641500
$244$	−690.560	−0.181183
$245$	245.992	0.0641462
$246$	160.420	0.0415773
$247$	−782.877	−0.201673
$248$	−3036.81	−0.777571
$249$	−1846.70	−0.469999
$250$	−1087.20	−0.275043
$251$	−5735.33	−1.44227	−0.721137	−	0.692792i	$-0.756380\pi$
$251$	−5735.33	−1.44227	−0.721137	+	0.692792i	$0.756380\pi$
$252$	346.475	0.0866105
$253$	3738.41	0.928980
$254$	3807.74	0.940626
$255$	−294.207	−0.0722507
$256$	−3557.67	−0.868572
$257$	1337.14	0.324547	0.162273	−	0.986746i	$-0.448117\pi$
$257$	1337.14	0.324547	0.162273	+	0.986746i	$0.448117\pi$
$258$	−2387.23	−0.576057
$259$	−1132.06	−0.271595
$260$	−128.466	−0.0306427
$261$	−2204.80	−0.522888
$262$	2196.38	0.517912
$263$	−3729.98	−0.874526	−0.437263	−	0.899334i	$-0.644052\pi$
$263$	−3729.98	−0.874526	−0.437263	+	0.899334i	$0.644052\pi$
$264$	4022.84	0.937836
$265$	−1202.75	−0.278808
$266$	1035.28	0.238635
$267$	−381.518	−0.0874477
$268$	1898.10	0.432630
$269$	7938.41	1.79931	0.899654	−	0.436604i	$-0.143819\pi$
$269$	7938.41	1.79931	0.899654	+	0.436604i	$0.143819\pi$
$270$	−119.147	−0.0268558
$271$	−3509.89	−0.786754	−0.393377	−	0.919377i	$-0.628693\pi$
$271$	−3509.89	−0.786754	−0.393377	+	0.919377i	$0.628693\pi$
$272$	1854.03	0.413297
$273$	−1123.98	−0.249181
$274$	−4974.98	−1.09690
$275$	6607.06	1.44880
$276$	−542.363	−0.118284
$277$	−8933.04	−1.93767	−0.968835	−	0.247708i	$-0.920323\pi$
$277$	−8933.04	−1.93767	−0.968835	+	0.247708i	$0.920323\pi$
$278$	4194.41	0.904906
$279$	−1109.53	−0.238085
$280$	686.141	0.146446
$281$	5270.43	1.11889	0.559444	−	0.828868i	$-0.311015\pi$
$281$	5270.43	1.11889	0.559444	+	0.828868i	$0.311015\pi$
$282$	3930.73	0.830041
$283$	5725.66	1.20267	0.601335	−	0.798997i	$-0.294636\pi$
$283$	5725.66	1.20267	0.601335	+	0.798997i	$0.294636\pi$
$284$	−1399.44	−0.292400
$285$	174.610	0.0362913
$286$	−3231.14	−0.668046
$287$	−337.528	−0.0694203
$288$	−1022.75	−0.209258
$289$	−2262.10	−0.460431
$290$	−1081.05	−0.218902
$291$	263.987	0.0531794
$292$	1311.19	0.262779
$293$	110.134	0.0219595	0.0109797	−	0.999940i	$-0.496505\pi$
$293$	110.134	0.0219595	0.0109797	+	0.999940i	$0.496505\pi$
$294$	−897.619	−0.178062
$295$	865.556	0.170829
$296$	1906.93	0.374453
$297$	1469.79	0.287157
$298$	−912.283	−0.177339
$299$	1759.45	0.340307
$300$	−958.543	−0.184472
$301$	5022.80	0.961825
$302$	−710.153	−0.135314
$303$	−3305.43	−0.626706
$304$	−1100.35	−0.207598
$305$	−499.648	−0.0938025
$306$	1073.56	0.200559
$307$	−1613.11	−0.299886	−0.149943	−	0.988695i	$-0.547909\pi$
$307$	−1613.11	−0.299886	−0.149943	+	0.988695i	$0.547909\pi$
$308$	−2095.65	−0.387698
$309$	3434.71	0.632343
$310$	−544.021	−0.0996721
$311$	5537.12	1.00959	0.504793	−	0.863240i	$-0.331569\pi$
$311$	5537.12	1.00959	0.504793	+	0.863240i	$0.331569\pi$
$312$	1893.32	0.343551
$313$	1460.33	0.263714	0.131857	−	0.991269i	$-0.457906\pi$
$313$	1460.33	0.263714	0.131857	+	0.991269i	$0.457906\pi$
$314$	363.735	0.0653718
$315$	250.689	0.0448403
$316$	−2766.41	−0.492476
$317$	−3903.72	−0.691655	−0.345828	−	0.938298i	$-0.612402\pi$
$317$	−3903.72	−0.691655	−0.345828	+	0.938298i	$0.612402\pi$
$318$	4388.80	0.773936
$319$	13335.7	2.34062
$320$	−1050.19	−0.183460
$321$	−2224.40	−0.386772
$322$	−2326.69	−0.402676
$323$	−1573.29	−0.271023
$324$	−213.234	−0.0365628
$325$	3109.56	0.530730
$326$	4495.79	0.763799
$327$	−4705.74	−0.795804
$328$	568.556	0.0957112
$329$	−8270.36	−1.38590
$330$	720.661	0.120215
$331$	6922.37	1.14951	0.574755	−	0.818325i	$-0.305097\pi$
$331$	6922.37	1.14951	0.574755	+	0.818325i	$0.305097\pi$
$332$	−1620.49	−0.267879
$333$	696.716	0.114654
$334$	5581.85	0.914447
$335$	1373.35	0.223983
$336$	−1579.78	−0.256501
$337$	−4549.66	−0.735418	−0.367709	−	0.929941i	$-0.619858\pi$
$337$	−4549.66	−0.735418	−0.367709	+	0.929941i	$0.619858\pi$
$338$	3569.26	0.574385
$339$	−2074.31	−0.332333
$340$	−258.168	−0.0411798
$341$	6710.98	1.06575
$342$	−637.149	−0.100740
$343$	6904.54	1.08691
$344$	−8460.77	−1.32609
$345$	−392.422	−0.0612384
$346$	9866.55	1.53303
$347$	5372.48	0.831152	0.415576	−	0.909558i	$-0.363580\pi$
$347$	5372.48	0.831152	0.415576	+	0.909558i	$0.363580\pi$
$348$	−1934.73	−0.298024
$349$	7340.28	1.12583	0.562917	−	0.826513i	$-0.309679\pi$
$349$	7340.28	1.12583	0.562917	+	0.826513i	$0.309679\pi$
$350$	−4112.07	−0.627999
$351$	691.742	0.105192
$352$	6186.12	0.936709
$353$	1945.00	0.293263	0.146631	−	0.989191i	$-0.453157\pi$
$353$	1945.00	0.293263	0.146631	+	0.989191i	$0.453157\pi$
$354$	−3158.40	−0.474200
$355$	−1012.55	−0.151382
$356$	−334.785	−0.0498415
$357$	−2258.79	−0.334867
$358$	9086.02	1.34137
$359$	12716.3	1.86947	0.934737	−	0.355339i	$-0.115635\pi$
$359$	12716.3	1.86947	0.934737	+	0.355339i	$0.115635\pi$
$360$	−422.278	−0.0618223
$361$	−5925.26	−0.863866
$362$	−1185.68	−0.172149
$363$	−4896.99	−0.708059
$364$	−986.301	−0.142023
$365$	948.697	0.136047
$366$	1823.21	0.260384
$367$	−1450.81	−0.206354	−0.103177	−	0.994663i	$-0.532901\pi$
$367$	−1450.81	−0.206354	−0.103177	+	0.994663i	$0.532901\pi$
$368$	2472.95	0.350303
$369$	207.728	0.0293059
$370$	341.612	0.0479988
$371$	−9234.14	−1.29222
$372$	−973.619	−0.135698
$373$	941.162	0.130648	0.0653238	−	0.997864i	$-0.479192\pi$
$373$	941.162	0.130648	0.0653238	+	0.997864i	$0.479192\pi$
$374$	−6493.40	−0.897769
$375$	−1407.82	−0.193865
$376$	13931.2	1.91076
$377$	6276.35	0.857423
$378$	−914.758	−0.124471
$379$	1844.44	0.249981	0.124990	−	0.992158i	$-0.460110\pi$
$379$	1844.44	0.249981	0.124990	+	0.992158i	$0.460110\pi$
$380$	153.221	0.0206845
$381$	4930.64	0.663004
$382$	−4288.90	−0.574448
$383$	2889.26	0.385468	0.192734	−	0.981251i	$-0.438264\pi$
$383$	2889.26	0.385468	0.192734	+	0.981251i	$0.438264\pi$
$384$	1104.76	0.146816
$385$	−1516.29	−0.200720
$386$	−3766.43	−0.496648
$387$	−3091.23	−0.406036
$388$	231.651	0.0303100
$389$	−10035.4	−1.30800	−0.654002	−	0.756492i	$-0.726911\pi$
$389$	−10035.4	−1.30800	−0.654002	+	0.756492i	$0.726911\pi$
$390$	339.173	0.0440377
$391$	3535.85	0.457329
$392$	−3181.32	−0.409900
$393$	2844.09	0.365052
$394$	−9542.70	−1.22019
$395$	−2001.61	−0.254967
$396$	1289.75	0.163667
$397$	−5283.65	−0.667957	−0.333978	−	0.942581i	$-0.608391\pi$
$397$	−5283.65	−0.667957	−0.333978	+	0.942581i	$0.608391\pi$
$398$	615.504	0.0775186
$399$	1340.58	0.168203
$400$	4370.57	0.546321
$401$	−12323.0	−1.53462	−0.767310	−	0.641277i	$-0.778405\pi$
$401$	−12323.0	−1.53462	−0.767310	+	0.641277i	$0.778405\pi$
$402$	−5011.34	−0.621748
$403$	3158.47	0.390408
$404$	−2900.53	−0.357195
$405$	−154.284	−0.0189294
$406$	−8299.83	−1.01457
$407$	−4214.09	−0.513230
$408$	3804.87	0.461689
$409$	8458.79	1.02264	0.511321	−	0.859390i	$-0.329156\pi$
$409$	8458.79	1.02264	0.511321	+	0.859390i	$0.329156\pi$
$410$	101.853	0.0122686
$411$	−6442.11	−0.773152
$412$	3013.98	0.360409
$413$	6645.35	0.791758
$414$	1431.94	0.169990
$415$	−1172.49	−0.138687
$416$	2911.45	0.343138
$417$	5431.34	0.637827
$418$	3853.80	0.450946
$419$	11275.4	1.31466	0.657329	−	0.753604i	$-0.271686\pi$
$419$	11275.4	1.31466	0.657329	+	0.753604i	$0.271686\pi$
$420$	219.981	0.0255571
$421$	10959.5	1.26872	0.634362	−	0.773036i	$-0.281263\pi$
$421$	10959.5	1.26872	0.634362	+	0.773036i	$0.281263\pi$
$422$	−7149.84	−0.824760
$423$	5089.90	0.585058
$424$	15554.7	1.78161
$425$	6249.07	0.713234
$426$	3694.79	0.420218
$427$	−3836.07	−0.434755
$428$	−1951.92	−0.220444
$429$	−4184.00	−0.470875
$430$	−1515.68	−0.169983
$431$	12070.8	1.34902	0.674511	−	0.738265i	$-0.264355\pi$
$431$	12070.8	1.34902	0.674511	+	0.738265i	$0.264355\pi$
$432$	972.261	0.108282
$433$	16098.4	1.78670	0.893348	−	0.449366i	$-0.148350\pi$
$433$	16098.4	1.78670	0.893348	+	0.449366i	$0.148350\pi$
$434$	−4176.75	−0.461960
$435$	−1399.85	−0.154294
$436$	−4129.32	−0.453574
$437$	−2098.51	−0.229714
$438$	−3461.78	−0.377649
$439$	−8047.84	−0.874949	−0.437475	−	0.899231i	$-0.644127\pi$
$439$	−8047.84	−0.874949	−0.437475	+	0.899231i	$0.644127\pi$
$440$	2554.15	0.276737
$441$	−1162.33	−0.125508
$442$	−3056.06	−0.328873
$443$	144.746	0.0155239	0.00776193	−	0.999970i	$-0.497529\pi$
$443$	144.746	0.0155239	0.00776193	+	0.999970i	$0.497529\pi$
$444$	611.373	0.0653479
$445$	−242.230	−0.0258041
$446$	4027.44	0.427589
$447$	−1181.31	−0.124998
$448$	−8062.85	−0.850299
$449$	635.665	0.0668127	0.0334063	−	0.999442i	$-0.489364\pi$
$449$	635.665	0.0668127	0.0334063	+	0.999442i	$0.489364\pi$
$450$	2530.73	0.265111
$451$	−1256.44	−0.131183
$452$	−1820.22	−0.189416
$453$	−919.577	−0.0953764
$454$	768.448	0.0794385
$455$	−713.629	−0.0735284
$456$	−2258.17	−0.231904
$457$	−3246.74	−0.332333	−0.166166	−	0.986098i	$-0.553139\pi$
$457$	−3246.74	−0.332333	−0.166166	+	0.986098i	$0.553139\pi$
$458$	4466.77	0.455717
$459$	1390.15	0.141365
$460$	−344.353	−0.0349033
$461$	−17825.4	−1.80089	−0.900444	−	0.434971i	$-0.856759\pi$
$461$	−17825.4	−1.80089	−0.900444	+	0.434971i	$0.856759\pi$
$462$	5532.91	0.557174
$463$	−2349.60	−0.235843	−0.117921	−	0.993023i	$-0.537623\pi$
$463$	−2349.60	−0.235843	−0.117921	+	0.993023i	$0.537623\pi$
$464$	8821.57	0.882610
$465$	−704.453	−0.0702543
$466$	−12666.1	−1.25911
$467$	−17508.3	−1.73488	−0.867441	−	0.497541i	$-0.834236\pi$
$467$	−17508.3	−1.73488	−0.867441	+	0.497541i	$0.834236\pi$
$468$	607.008	0.0599550
$469$	10544.0	1.03811
$470$	2495.67	0.244929
$471$	471.000	0.0460776
$472$	−11193.9	−1.09161
$473$	18697.3	1.81755
$474$	7303.83	0.707755
$475$	−3708.79	−0.358254
$476$	−1982.10	−0.190860
$477$	5683.05	0.545512
$478$	10235.7	0.979434
$479$	8802.39	0.839649	0.419824	−	0.907605i	$-0.362092\pi$
$479$	8802.39	0.839649	0.419824	+	0.907605i	$0.362092\pi$
$480$	−649.358	−0.0617480
$481$	−1983.32	−0.188008
$482$	2093.56	0.197840
$483$	−3012.84	−0.283828
$484$	−4297.14	−0.403563
$485$	167.609	0.0156922
$486$	562.978	0.0525457
$487$	−6611.11	−0.615150	−0.307575	−	0.951524i	$-0.599518\pi$
$487$	−6611.11	−0.615150	−0.307575	+	0.951524i	$0.599518\pi$
$488$	6461.76	0.599406
$489$	5821.59	0.538367
$490$	−569.909	−0.0525426
$491$	−7422.92	−0.682264	−0.341132	−	0.940015i	$-0.610810\pi$
$491$	−7422.92	−0.682264	−0.341132	+	0.940015i	$0.610810\pi$
$492$	182.283	0.0167031
$493$	12613.1	1.15227
$494$	1813.76	0.165192
$495$	933.184	0.0847343
$496$	4439.31	0.401877
$497$	−7773.92	−0.701626
$498$	4278.40	0.384979
$499$	−6862.02	−0.615604	−0.307802	−	0.951450i	$-0.599593\pi$
$499$	−6862.02	−0.615604	−0.307802	+	0.951450i	$0.599593\pi$
$500$	−1235.37	−0.110495
$501$	7227.94	0.644552
$502$	13287.5	1.18138
$503$	−10967.1	−0.972163	−0.486081	−	0.873914i	$-0.661574\pi$
$503$	−10967.1	−0.972163	−0.486081	+	0.873914i	$0.661574\pi$
$504$	−3242.06	−0.286534
$505$	−2098.65	−0.184928
$506$	−8661.08	−0.760933
$507$	4621.84	0.404858
$508$	4326.67	0.377884
$509$	13615.2	1.18562	0.592812	−	0.805341i	$-0.298018\pi$
$509$	13615.2	1.18562	0.592812	+	0.805341i	$0.298018\pi$
$510$	681.613	0.0591810
$511$	7283.67	0.630549
$512$	11188.4	0.965746
$513$	−825.044	−0.0710070
$514$	−3097.86	−0.265838
$515$	2180.74	0.186592
$516$	−2712.57	−0.231423
$517$	−30786.3	−2.61891
$518$	2622.74	0.222465
$519$	12776.2	1.08056
$520$	1202.09	0.101375
$521$	−4695.85	−0.394873	−0.197436	−	0.980316i	$-0.563262\pi$
$521$	−4695.85	−0.394873	−0.197436	+	0.980316i	$0.563262\pi$
$522$	5108.04	0.428301
$523$	−13781.1	−1.15221	−0.576105	−	0.817376i	$-0.695428\pi$
$523$	−13781.1	−1.15221	−0.576105	+	0.817376i	$0.695428\pi$
$524$	2495.71	0.208064
$525$	−5324.72	−0.442648
$526$	8641.55	0.716330
$527$	6347.36	0.524659
$528$	−5880.72	−0.484707
$529$	−7450.78	−0.612376
$530$	2786.50	0.228373
$531$	−4089.81	−0.334242
$532$	1176.37	0.0958683
$533$	−591.333	−0.0480553
$534$	883.895	0.0716290
$535$	−1412.30	−0.114129
$536$	−17761.0	−1.43127
$537$	11765.5	0.945471
$538$	−18391.6	−1.47382
$539$	7030.33	0.561814
$540$	−135.385	−0.0107890
$541$	15172.6	1.20577	0.602885	−	0.797828i	$-0.294018\pi$
$541$	15172.6	1.20577	0.602885	+	0.797828i	$0.294018\pi$
$542$	8131.64	0.644435
$543$	−1535.33	−0.121340
$544$	5850.93	0.461134
$545$	−2987.73	−0.234826
$546$	2604.02	0.204106
$547$	−12590.5	−0.984153	−0.492076	−	0.870552i	$-0.663762\pi$
$547$	−12590.5	−0.984153	−0.492076	+	0.870552i	$0.663762\pi$
$548$	−5652.99	−0.440664
$549$	2360.87	0.183533
$550$	−15307.1	−1.18672
$551$	−7485.83	−0.578779
$552$	5075.04	0.391319
$553$	−15367.4	−1.18172
$554$	20695.9	1.58716
$555$	442.353	0.0338322
$556$	4766.04	0.363534
$557$	−20380.8	−1.55038	−0.775192	−	0.631725i	$-0.782347\pi$
$557$	−20380.8	−1.55038	−0.775192	+	0.631725i	$0.782347\pi$
$558$	2570.54	0.195017
$559$	8799.72	0.665811
$560$	−1003.02	−0.0756884
$561$	−8408.30	−0.632796
$562$	−12210.4	−0.916489
$563$	−20972.3	−1.56994	−0.784970	−	0.619534i	$-0.787322\pi$
$563$	−20972.3	−1.56994	−0.784970	+	0.619534i	$0.787322\pi$
$564$	4466.42	0.333458
$565$	−1317.00	−0.0980649
$566$	−13265.1	−0.985114
$567$	−1184.52	−0.0877340
$568$	13095.0	0.967346
$569$	11420.2	0.841403	0.420701	−	0.907199i	$-0.361784\pi$
$569$	11420.2	0.841403	0.420701	+	0.907199i	$0.361784\pi$
$570$	−404.533	−0.0297264
$571$	14171.6	1.03864	0.519319	−	0.854580i	$-0.326186\pi$
$571$	14171.6	1.03864	0.519319	+	0.854580i	$0.326186\pi$
$572$	−3671.49	−0.268379
$573$	−5553.70	−0.404902
$574$	781.978	0.0568626
$575$	8335.19	0.604524
$576$	4962.19	0.358955
$577$	−21307.5	−1.53734	−0.768668	−	0.639648i	$-0.779080\pi$
$577$	−21307.5	−1.53734	−0.768668	+	0.639648i	$0.779080\pi$
$578$	5240.79	0.377142
$579$	−4877.15	−0.350064
$580$	−1228.38	−0.0879409
$581$	−9001.85	−0.642788
$582$	−611.601	−0.0435596
$583$	−34373.9	−2.44189
$584$	−12269.1	−0.869351
$585$	439.195	0.0310401
$586$	−255.157	−0.0179871
$587$	18542.3	1.30379	0.651893	−	0.758311i	$-0.273975\pi$
$587$	18542.3	1.30379	0.651893	+	0.758311i	$0.273975\pi$
$588$	−1019.95	−0.0715340
$589$	−3767.12	−0.263534
$590$	−2005.30	−0.139927
$591$	−12356.8	−0.860055
$592$	−2787.61	−0.193531
$593$	−4436.28	−0.307211	−0.153606	−	0.988132i	$-0.549089\pi$
$593$	−4436.28	−0.307211	−0.153606	+	0.988132i	$0.549089\pi$
$594$	−3405.17	−0.235212
$595$	−1434.13	−0.0988128
$596$	−1036.61	−0.0712437
$597$	797.015	0.0546393
$598$	−4076.26	−0.278747
$599$	10831.5	0.738834	0.369417	−	0.929264i	$-0.379557\pi$
$599$	10831.5	0.738834	0.369417	+	0.929264i	$0.379557\pi$
$600$	8969.35	0.610287
$601$	25567.5	1.73531	0.867653	−	0.497170i	$-0.165628\pi$
$601$	25567.5	1.73531	0.867653	+	0.497170i	$0.165628\pi$
$602$	−11636.7	−0.787837
$603$	−6489.18	−0.438242
$604$	−806.935	−0.0543605
$605$	−3109.16	−0.208934
$606$	7657.95	0.513338
$607$	−12556.0	−0.839595	−0.419798	−	0.907618i	$-0.637899\pi$
$607$	−12556.0	−0.839595	−0.419798	+	0.907618i	$0.637899\pi$
$608$	−3472.50	−0.231626
$609$	−10747.4	−0.715121
$610$	1157.58	0.0768342
$611$	−14489.3	−0.959368
$612$	1219.86	0.0805719
$613$	−3218.26	−0.212046	−0.106023	−	0.994364i	$-0.533812\pi$
$613$	−3218.26	−0.212046	−0.106023	+	0.994364i	$0.533812\pi$
$614$	3737.22	0.245638
$615$	131.889	0.00864760
$616$	19609.6	1.28262
$617$	−11047.4	−0.720827	−0.360413	−	0.932793i	$-0.617364\pi$
$617$	−11047.4	−0.720827	−0.360413	+	0.932793i	$0.617364\pi$
$618$	−7957.48	−0.517956
$619$	17202.6	1.11701	0.558507	−	0.829500i	$-0.311375\pi$
$619$	17202.6	1.11701	0.558507	+	0.829500i	$0.311375\pi$
$620$	−618.162	−0.0400419
$621$	1854.22	0.119818
$622$	−12828.3	−0.826959
$623$	−1859.74	−0.119597
$624$	−2767.71	−0.177559
$625$	14277.7	0.913770
$626$	−3383.26	−0.216010
$627$	4990.28	0.317851
$628$	413.306	0.0262622
$629$	−3985.75	−0.252659
$630$	−580.791	−0.0367290
$631$	−13710.4	−0.864978	−0.432489	−	0.901639i	$-0.642365\pi$
$631$	−13710.4	−0.864978	−0.432489	+	0.901639i	$0.642365\pi$
$632$	25886.0	1.62926
$633$	−9258.33	−0.581336
$634$	9044.06	0.566539
$635$	3130.52	0.195639
$636$	4986.92	0.310918
$637$	3308.76	0.205805
$638$	−30896.0	−1.91722
$639$	4784.38	0.296193
$640$	701.428	0.0433225
$641$	22511.0	1.38710	0.693551	−	0.720407i	$-0.256045\pi$
$641$	22511.0	1.38710	0.693551	+	0.720407i	$0.256045\pi$
$642$	5153.45	0.316807
$643$	−26563.0	−1.62915	−0.814573	−	0.580061i	$-0.803029\pi$
$643$	−26563.0	−1.62915	−0.814573	+	0.580061i	$0.803029\pi$
$644$	−2643.78	−0.161770
$645$	−1962.66	−0.119813
$646$	3644.98	0.221997
$647$	−9805.81	−0.595837	−0.297918	−	0.954591i	$-0.596292\pi$
$647$	−9805.81	−0.595837	−0.297918	+	0.954591i	$0.596292\pi$
$648$	1995.29	0.120961
$649$	24737.2	1.49618
$650$	−7204.17	−0.434724
$651$	−5408.47	−0.325614
$652$	5108.49	0.306846
$653$	−9999.59	−0.599256	−0.299628	−	0.954056i	$-0.596863\pi$
$653$	−9999.59	−0.599256	−0.299628	+	0.954056i	$0.596863\pi$
$654$	10902.2	0.651848
$655$	1805.75	0.107720
$656$	−831.135	−0.0494670
$657$	−4482.66	−0.266187
$658$	19160.6	1.13520
$659$	−9399.82	−0.555637	−0.277819	−	0.960634i	$-0.589611\pi$
$659$	−9399.82	−0.555637	−0.277819	+	0.960634i	$0.589611\pi$
$660$	818.875	0.0482950
$661$	20448.6	1.20326	0.601632	−	0.798773i	$-0.294517\pi$
$661$	20448.6	1.20326	0.601632	+	0.798773i	$0.294517\pi$
$662$	−16037.6	−0.941571
$663$	−3957.29	−0.231808
$664$	15163.4	0.886225
$665$	851.148	0.0496333
$666$	−1614.14	−0.0939139
$667$	16823.8	0.976641
$668$	6342.56	0.367367
$669$	5215.13	0.301388
$670$	−3181.76	−0.183466
$671$	−14279.7	−0.821553
$672$	−4985.48	−0.286189
$673$	8770.58	0.502349	0.251175	−	0.967942i	$-0.419183\pi$
$673$	8770.58	0.502349	0.251175	+	0.967942i	$0.419183\pi$
$674$	10540.6	0.602386
$675$	3277.04	0.186864
$676$	4055.69	0.230752
$677$	−17089.9	−0.970191	−0.485095	−	0.874461i	$-0.661215\pi$
$677$	−17089.9	−0.970191	−0.485095	+	0.874461i	$0.661215\pi$
$678$	4805.71	0.272216
$679$	1286.82	0.0727302
$680$	2415.75	0.136235
$681$	995.063	0.0559925
$682$	−15547.9	−0.872961
$683$	−9267.12	−0.519175	−0.259588	−	0.965720i	$-0.583587\pi$
$683$	−9267.12	−0.519175	−0.259588	+	0.965720i	$0.583587\pi$
$684$	−723.982	−0.0404710
$685$	−4090.17	−0.228142
$686$	−15996.3	−0.890295
$687$	5784.02	0.321214
$688$	12368.2	0.685370
$689$	−16177.8	−0.894521
$690$	909.155	0.0501608
$691$	−15209.5	−0.837335	−0.418667	−	0.908140i	$-0.637503\pi$
$691$	−15209.5	−0.837335	−0.418667	+	0.908140i	$0.637503\pi$
$692$	11211.2	0.615875
$693$	7164.57	0.392726
$694$	−12446.9	−0.680802
$695$	3448.42	0.188210
$696$	18103.8	0.985952
$697$	−1188.36	−0.0645802
$698$	−17005.8	−0.922178
$699$	−16401.3	−0.887487
$700$	−4672.48	−0.252290
$701$	−8515.46	−0.458808	−0.229404	−	0.973331i	$-0.573678\pi$
$701$	−8515.46	−0.458808	−0.229404	+	0.973331i	$0.573678\pi$
$702$	−1602.62	−0.0861636
$703$	2365.52	0.126909
$704$	−30013.8	−1.60680
$705$	3231.64	0.172639
$706$	−4506.14	−0.240214
$707$	−16112.5	−0.857106
$708$	−3588.83	−0.190504
$709$	−3113.77	−0.164937	−0.0824684	−	0.996594i	$-0.526280\pi$
$709$	−3113.77	−0.164937	−0.0824684	+	0.996594i	$0.526280\pi$
$710$	2345.86	0.123998
$711$	9457.73	0.498864
$712$	3132.68	0.164890
$713$	8466.29	0.444691
$714$	5233.12	0.274292
$715$	−2656.47	−0.138946
$716$	10324.3	0.538878
$717$	13254.2	0.690358
$718$	−29460.9	−1.53130
$719$	−783.357	−0.0406318	−0.0203159	−	0.999794i	$-0.506467\pi$
$719$	−783.357	−0.0406318	−0.0203159	+	0.999794i	$0.506467\pi$
$720$	617.300	0.0319520
$721$	16742.7	0.864816
$722$	13727.5	0.707598
$723$	2710.94	0.139448
$724$	−1347.26	−0.0691584
$725$	29733.4	1.52313
$726$	11345.3	0.579976
$727$	−12719.2	−0.648871	−0.324436	−	0.945908i	$-0.605174\pi$
$727$	−12719.2	−0.648871	−0.324436	+	0.945908i	$0.605174\pi$
$728$	9229.09	0.469853
$729$	729.000	0.0370370
$730$	−2197.93	−0.111437
$731$	17684.2	0.894766
$732$	2071.68	0.104606
$733$	34607.9	1.74389	0.871945	−	0.489604i	$-0.162859\pi$
$733$	34607.9	1.74389	0.871945	+	0.489604i	$0.162859\pi$
$734$	3361.22	0.169026
$735$	−737.975	−0.0370348
$736$	7804.15	0.390849
$737$	39249.8	1.96172
$738$	−481.260	−0.0240046
$739$	−29220.8	−1.45454	−0.727269	−	0.686353i	$-0.759211\pi$
$739$	−29220.8	−1.45454	−0.727269	+	0.686353i	$0.759211\pi$
$740$	388.168	0.0192829
$741$	2348.63	0.116436
$742$	21393.5	1.05846
$743$	−24890.4	−1.22899	−0.614494	−	0.788921i	$-0.710640\pi$
$743$	−24890.4	−1.22899	−0.614494	+	0.788921i	$0.710640\pi$
$744$	9110.43	0.448931
$745$	−750.030	−0.0368845
$746$	−2180.47	−0.107014
$747$	5540.09	0.271354
$748$	−7378.34	−0.360667
$749$	−10843.0	−0.528964
$750$	3261.61	0.158796
$751$	3167.31	0.153897	0.0769486	−	0.997035i	$-0.475482\pi$
$751$	3167.31	0.153897	0.0769486	+	0.997035i	$0.475482\pi$
$752$	−20365.1	−0.987551
$753$	17206.0	0.832697
$754$	−14540.9	−0.702320
$755$	−583.850	−0.0281437
$756$	−1039.42	−0.0500046
$757$	−25764.0	−1.23700	−0.618501	−	0.785784i	$-0.712260\pi$
$757$	−25764.0	−1.23700	−0.618501	+	0.785784i	$0.712260\pi$
$758$	−4273.18	−0.204761
$759$	−11215.2	−0.536347
$760$	−1433.74	−0.0684304
$761$	20396.8	0.971596	0.485798	−	0.874071i	$-0.338529\pi$
$761$	20396.8	0.971596	0.485798	+	0.874071i	$0.338529\pi$
$762$	−11423.2	−0.543071
$763$	−22938.4	−1.08837
$764$	−4873.41	−0.230777
$765$	882.620	0.0417140
$766$	−6693.79	−0.315740
$767$	11642.3	0.548085
$768$	10673.0	0.501471
$769$	14658.6	0.687391	0.343696	−	0.939081i	$-0.388321\pi$
$769$	14658.6	0.687391	0.343696	+	0.939081i	$0.388321\pi$
$770$	3512.91	0.164411
$771$	−4011.42	−0.187377
$772$	−4279.73	−0.199522
$773$	177.655	0.00826623	0.00413312	−	0.999991i	$-0.498684\pi$
$773$	177.655	0.00826623	0.00413312	+	0.999991i	$0.498684\pi$
$774$	7161.70	0.332587
$775$	14962.9	0.693525
$776$	−2167.62	−0.100275
$777$	3396.19	0.156805
$778$	23249.8	1.07139
$779$	705.286	0.0324384
$780$	385.397	0.0176916
$781$	−28938.3	−1.32586
$782$	−8191.79	−0.374601
$783$	6614.40	0.301889
$784$	4650.55	0.211851
$785$	299.043	0.0135966
$786$	−6589.14	−0.299016
$787$	−3594.25	−0.162797	−0.0813985	−	0.996682i	$-0.525939\pi$
$787$	−3594.25	−0.162797	−0.0813985	+	0.996682i	$0.525939\pi$
$788$	−10843.2	−0.490195
$789$	11189.9	0.504908
$790$	4637.29	0.208845
$791$	−10111.3	−0.454511
$792$	−12068.5	−0.541460
$793$	−6720.63	−0.300954
$794$	12241.1	0.547127
$795$	3608.24	0.160970
$796$	699.386	0.0311421
$797$	−3415.52	−0.151799	−0.0758995	−	0.997115i	$-0.524183\pi$
$797$	−3415.52	−0.151799	−0.0758995	+	0.997115i	$0.524183\pi$
$798$	−3105.83	−0.137776
$799$	−29118.1	−1.28927
$800$	13792.6	0.609554
$801$	1144.55	0.0504880
$802$	28549.8	1.25702
$803$	27113.3	1.19154
$804$	−5694.30	−0.249779
$805$	−1912.88	−0.0837520
$806$	−7317.48	−0.319786
$807$	−23815.2	−1.03883
$808$	27141.1	1.18171
$809$	8274.46	0.359598	0.179799	−	0.983703i	$-0.442455\pi$
$809$	8274.46	0.359598	0.179799	+	0.983703i	$0.442455\pi$
$810$	357.441	0.0155052
$811$	4138.29	0.179180	0.0895901	−	0.995979i	$-0.471444\pi$
$811$	4138.29	0.179180	0.0895901	+	0.995979i	$0.471444\pi$
$812$	−9430.96	−0.407588
$813$	10529.7	0.454233
$814$	9763.12	0.420390
$815$	3696.20	0.158862
$816$	−5562.08	−0.238617
$817$	−10495.5	−0.449437
$818$	−19597.2	−0.837652
$819$	3371.94	0.143865
$820$	115.733	0.00492876
$821$	−36851.7	−1.56654	−0.783272	−	0.621679i	$-0.786451\pi$
$821$	−36851.7	−1.56654	−0.783272	+	0.621679i	$0.786451\pi$
$822$	14925.0	0.633294
$823$	220.320	0.00933154	0.00466577	−	0.999989i	$-0.498515\pi$
$823$	220.320	0.00933154	0.00466577	+	0.999989i	$0.498515\pi$
$824$	−28202.7	−1.19234
$825$	−19821.2	−0.836467
$826$	−15395.8	−0.648534
$827$	−14767.2	−0.620927	−0.310463	−	0.950585i	$-0.600484\pi$
$827$	−14767.2	−0.620927	−0.310463	+	0.950585i	$0.600484\pi$
$828$	1627.09	0.0682913
$829$	−23514.2	−0.985142	−0.492571	−	0.870272i	$-0.663943\pi$
$829$	−23514.2	−0.985142	−0.492571	+	0.870272i	$0.663943\pi$
$830$	2716.40	0.113600
$831$	26799.1	1.11871
$832$	−14125.8	−0.588609
$833$	6649.40	0.276576
$834$	−12583.2	−0.522448
$835$	4589.10	0.190195
$836$	4379.00	0.181161
$837$	3328.59	0.137459
$838$	−26122.7	−1.07684
$839$	17829.0	0.733641	0.366821	−	0.930292i	$-0.380446\pi$
$839$	17829.0	0.733641	0.366821	+	0.930292i	$0.380446\pi$
$840$	−2058.42	−0.0845504
$841$	35625.1	1.46070
$842$	−25390.7	−1.03922
$843$	−15811.3	−0.645991
$844$	−8124.24	−0.331337
$845$	2934.46	0.119466
$846$	−11792.2	−0.479225
$847$	−23870.7	−0.968368
$848$	−22738.3	−0.920799
$849$	−17177.0	−0.694361
$850$	−14477.7	−0.584214
$851$	−5316.31	−0.214149
$852$	4198.32	0.168817
$853$	40143.4	1.61135	0.805677	−	0.592355i	$-0.201802\pi$
$853$	40143.4	1.61135	0.805677	+	0.592355i	$0.201802\pi$
$854$	8887.34	0.356111
$855$	−523.830	−0.0209528
$856$	18264.7	0.729293
$857$	18194.3	0.725209	0.362604	−	0.931943i	$-0.381888\pi$
$857$	18194.3	0.725209	0.362604	+	0.931943i	$0.381888\pi$
$858$	9693.42	0.385697
$859$	−2983.80	−0.118517	−0.0592584	−	0.998243i	$-0.518874\pi$
$859$	−2983.80	−0.118517	−0.0592584	+	0.998243i	$0.518874\pi$
$860$	−1722.25	−0.0682885
$861$	1012.58	0.0400798
$862$	−27965.3	−1.10499
$863$	−12345.0	−0.486939	−0.243470	−	0.969909i	$-0.578286\pi$
$863$	−12345.0	−0.486939	−0.243470	+	0.969909i	$0.578286\pi$
$864$	3068.26	0.120815
$865$	8111.75	0.318853
$866$	−37296.5	−1.46349
$867$	6786.30	0.265830
$868$	−4745.97	−0.185586
$869$	−57205.0	−2.23308
$870$	3243.16	0.126383
$871$	18472.6	0.718621
$872$	38639.2	1.50056
$873$	−791.962	−0.0307032
$874$	4861.78	0.188160
$875$	−6862.51	−0.265138
$876$	−3933.56	−0.151715
$877$	28438.7	1.09499	0.547495	−	0.836809i	$-0.315581\pi$
$877$	28438.7	1.09499	0.547495	+	0.836809i	$0.315581\pi$
$878$	18645.1	0.716676
$879$	−330.403	−0.0126783
$880$	−3733.74	−0.143028
$881$	45604.4	1.74398	0.871992	−	0.489520i	$-0.162828\pi$
$881$	45604.4	1.74398	0.871992	+	0.489520i	$0.162828\pi$
$882$	2692.86	0.102804
$883$	24825.6	0.946146	0.473073	−	0.881023i	$-0.343145\pi$
$883$	24825.6	0.946146	0.473073	+	0.881023i	$0.343145\pi$
$884$	−3472.55	−0.132121
$885$	−2596.67	−0.0986282
$886$	−335.344	−0.0127157
$887$	−32159.4	−1.21737	−0.608685	−	0.793412i	$-0.708303\pi$
$887$	−32159.4	−1.21737	−0.608685	+	0.793412i	$0.708303\pi$
$888$	−5720.79	−0.216191
$889$	24034.8	0.906749
$890$	561.195	0.0211363
$891$	−4409.36	−0.165790
$892$	4576.31	0.171778
$893$	17281.5	0.647594
$894$	2736.85	0.102387
$895$	7470.05	0.278990
$896$	5385.25	0.200791
$897$	−5278.35	−0.196476
$898$	−1472.70	−0.0547267
$899$	30201.1	1.12043
$900$	2875.63	0.106505
$901$	−32511.4	−1.20212
$902$	2910.90	0.107453
$903$	−15068.4	−0.555310
$904$	17032.3	0.626643
$905$	−974.801	−0.0358049
$906$	2130.46	0.0781234
$907$	−9280.59	−0.339754	−0.169877	−	0.985465i	$-0.554337\pi$
$907$	−9280.59	−0.339754	−0.169877	+	0.985465i	$0.554337\pi$
$908$	873.175	0.0319133
$909$	9916.28	0.361829
$910$	1653.32	0.0602276
$911$	37.9622	0.00138062	0.000690310	−	1.00000i	$-0.499780\pi$
$911$	37.9622	0.00138062	0.000690310		1.00000i	$0.499780\pi$
$912$	3301.06	0.119856
$913$	−33509.2	−1.21467
$914$	7521.99	0.272216
$915$	1498.94	0.0541569
$916$	5075.52	0.183078
$917$	13863.7	0.499259
$918$	−3220.67	−0.115793
$919$	28169.5	1.01113	0.505563	−	0.862790i	$-0.331285\pi$
$919$	28169.5	1.01113	0.505563	+	0.862790i	$0.331285\pi$
$920$	3222.20	0.115471
$921$	4839.33	0.173139
$922$	41297.5	1.47512
$923$	−13619.6	−0.485692
$924$	6286.95	0.223837
$925$	−9395.76	−0.333979
$926$	5443.52	0.193180
$927$	−10304.1	−0.365084
$928$	27839.1	0.984767
$929$	−27222.3	−0.961394	−0.480697	−	0.876887i	$-0.659616\pi$
$929$	−27222.3	−0.961394	−0.480697	+	0.876887i	$0.659616\pi$
$930$	1632.06	0.0575457
$931$	−3946.38	−0.138923
$932$	−14392.2	−0.505830
$933$	−16611.4	−0.582885
$934$	40563.0	1.42105
$935$	−5338.53	−0.186726
$936$	−5679.95	−0.198349
$937$	22880.9	0.797743	0.398872	−	0.917007i	$-0.369402\pi$
$937$	22880.9	0.797743	0.398872	+	0.917007i	$0.369402\pi$
$938$	−24428.1	−0.850326
$939$	−4380.98	−0.152255
$940$	2835.78	0.0983969
$941$	−10962.0	−0.379758	−0.189879	−	0.981808i	$-0.560810\pi$
$941$	−10962.0	−0.379758	−0.189879	+	0.981808i	$0.560810\pi$
$942$	−1091.20	−0.0377424
$943$	−1585.07	−0.0547371
$944$	16363.6	0.564185
$945$	−752.066	−0.0258886
$946$	−43317.6	−1.48877
$947$	47023.0	1.61356	0.806781	−	0.590851i	$-0.201208\pi$
$947$	47023.0	1.61356	0.806781	+	0.590851i	$0.201208\pi$
$948$	8299.22	0.284331
$949$	12760.7	0.436490
$950$	8592.45	0.293448
$951$	11711.2	0.399327
$952$	18547.1	0.631422
$953$	53270.6	1.81071	0.905354	−	0.424658i	$-0.139606\pi$
$953$	53270.6	1.81071	0.905354	+	0.424658i	$0.139606\pi$
$954$	−13166.4	−0.446832
$955$	−3526.11	−0.119479
$956$	11630.6	0.393475
$957$	−40007.2	−1.35136
$958$	−20393.2	−0.687761
$959$	−31402.5	−1.05739
$960$	3150.56	0.105921
$961$	−14592.8	−0.489839
$962$	4594.93	0.153998
$963$	6673.20	0.223303
$964$	2378.87	0.0794795
$965$	−3096.56	−0.103297
$966$	6980.08	0.232485
$967$	−9878.02	−0.328496	−0.164248	−	0.986419i	$-0.552520\pi$
$967$	−9878.02	−0.328496	−0.164248	+	0.986419i	$0.552520\pi$
$968$	40209.6	1.33511
$969$	4719.88	0.156475
$970$	−388.313	−0.0128536
$971$	25733.2	0.850481	0.425241	−	0.905080i	$-0.360189\pi$
$971$	25733.2	0.850481	0.425241	+	0.905080i	$0.360189\pi$
$972$	639.702	0.0211095
$973$	26475.4	0.872316
$974$	15316.5	0.503873
$975$	−9328.68	−0.306417
$976$	−9446.02	−0.309795
$977$	−13936.0	−0.456349	−0.228174	−	0.973620i	$-0.573276\pi$
$977$	−13936.0	−0.456349	−0.228174	+	0.973620i	$0.573276\pi$
$978$	−13487.4	−0.440980
$979$	−6922.84	−0.226001
$980$	−647.578	−0.0211083
$981$	14117.2	0.459458
$982$	17197.3	0.558847
$983$	30942.9	1.00399	0.501997	−	0.864869i	$-0.332599\pi$
$983$	30942.9	1.00399	0.501997	+	0.864869i	$0.332599\pi$
$984$	−1705.67	−0.0552589
$985$	−7845.50	−0.253785
$986$	−29221.9	−0.943829
$987$	24811.1	0.800147
$988$	2060.94	0.0663636
$989$	23587.7	0.758387
$990$	−2161.98	−0.0694064
$991$	−13275.5	−0.425539	−0.212769	−	0.977102i	$-0.568248\pi$
$991$	−13275.5	−0.425539	−0.212769	+	0.977102i	$0.568248\pi$
$992$	14009.6	0.448391
$993$	−20767.1	−0.663670
$994$	18010.5	0.574706
$995$	506.034	0.0161230
$996$	4861.47	0.154660
$997$	−26234.2	−0.833346	−0.416673	−	0.909056i	$-0.636804\pi$
$997$	−26234.2	−0.833346	−0.416673	+	0.909056i	$0.636804\pi$
$998$	15897.8	0.504245
$999$	−2090.15	−0.0661956

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.a.c.1.7	✓	22
3.2	odd	2		1413.4.a.e.1.16		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.a.c.1.7	✓	22	1.1	even	1	trivial
1413.4.a.e.1.16		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)

\(f(q)\)	\(=\)	\(q-2.31678 q^{2} -3.00000 q^{3} -2.63252 q^{4} -1.90474 q^{5} +6.95035 q^{6} -14.6237 q^{7} +24.6332 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.31678 q^{2} -3.00000 q^{3} -2.63252 q^{4} -1.90474 q^{5} +6.95035 q^{6} -14.6237 q^{7} +24.6332 q^{8} +9.00000 q^{9} +4.41286 q^{10} -54.4365 q^{11} +7.89756 q^{12} -25.6201 q^{13} +33.8799 q^{14} +5.71421 q^{15} -36.0097 q^{16} -51.4869 q^{17} -20.8510 q^{18} +30.5572 q^{19} +5.01425 q^{20} +43.8711 q^{21} +126.117 q^{22} -68.6747 q^{23} -73.8997 q^{24} -121.372 q^{25} +59.3561 q^{26} -27.0000 q^{27} +38.4972 q^{28} -244.978 q^{29} -13.2386 q^{30} -123.281 q^{31} -113.639 q^{32} +163.309 q^{33} +119.284 q^{34} +27.8543 q^{35} -23.6927 q^{36} +77.4129 q^{37} -70.7943 q^{38} +76.8602 q^{39} -46.9198 q^{40} +23.0809 q^{41} -101.640 q^{42} -343.470 q^{43} +143.305 q^{44} -17.1426 q^{45} +159.104 q^{46} +565.545 q^{47} +108.029 q^{48} -129.147 q^{49} +281.192 q^{50} +154.461 q^{51} +67.4453 q^{52} +631.450 q^{53} +62.5531 q^{54} +103.687 q^{55} -360.229 q^{56} -91.6716 q^{57} +567.560 q^{58} -454.423 q^{59} -15.0428 q^{60} +262.319 q^{61} +285.615 q^{62} -131.613 q^{63} +551.355 q^{64} +48.7995 q^{65} -378.352 q^{66} -721.020 q^{67} +135.540 q^{68} +206.024 q^{69} -64.5323 q^{70} +531.598 q^{71} +221.699 q^{72} -498.073 q^{73} -179.349 q^{74} +364.116 q^{75} -80.4424 q^{76} +796.063 q^{77} -178.068 q^{78} +1050.86 q^{79} +68.5889 q^{80} +81.0000 q^{81} -53.4734 q^{82} +615.566 q^{83} -115.492 q^{84} +98.0689 q^{85} +795.745 q^{86} +734.933 q^{87} -1340.95 q^{88} +127.173 q^{89} +39.7157 q^{90} +374.660 q^{91} +180.788 q^{92} +369.843 q^{93} -1310.24 q^{94} -58.2034 q^{95} +340.918 q^{96} -87.9958 q^{97} +299.206 q^{98} -489.928 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9}+O(q^{10}) \) 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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