LMFDB - Embedded newform 38.4.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,4,Mod(1,38)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("38.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 38 = 2 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	38.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:	$2.24207258022$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{73}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 18 $ x^2 - x - 18
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.77200$ of defining polynomial
Character	$\chi$	$=$	38.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +8.77200 q^{3} +4.00000 q^{4} -17.3160 q^{5} +17.5440 q^{6} -26.0880 q^{7} +8.00000 q^{8} +49.9480 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +8.77200 q^{3} +4.00000 q^{4} -17.3160 q^{5} +17.5440 q^{6} -26.0880 q^{7} +8.00000 q^{8} +49.9480 q^{9} -34.6320 q^{10} -4.22800 q^{11} +35.0880 q^{12} +64.0360 q^{13} -52.1760 q^{14} -151.896 q^{15} +16.0000 q^{16} -48.5440 q^{17} +99.8960 q^{18} +19.0000 q^{19} -69.2640 q^{20} -228.844 q^{21} -8.45600 q^{22} +92.0360 q^{23} +70.1760 q^{24} +174.844 q^{25} +128.072 q^{26} +201.300 q^{27} -104.352 q^{28} -88.2120 q^{29} -303.792 q^{30} -81.9681 q^{31} +32.0000 q^{32} -37.0880 q^{33} -97.0880 q^{34} +451.740 q^{35} +199.792 q^{36} -23.6161 q^{37} +38.0000 q^{38} +561.724 q^{39} -138.528 q^{40} +17.7200 q^{41} -457.688 q^{42} +368.404 q^{43} -16.9120 q^{44} -864.900 q^{45} +184.072 q^{46} -497.812 q^{47} +140.352 q^{48} +337.584 q^{49} +349.688 q^{50} -425.828 q^{51} +256.144 q^{52} -536.876 q^{53} +402.600 q^{54} +73.2120 q^{55} -208.704 q^{56} +166.668 q^{57} -176.424 q^{58} -36.7000 q^{59} -607.584 q^{60} +630.692 q^{61} -163.936 q^{62} -1303.04 q^{63} +64.0000 q^{64} -1108.85 q^{65} -74.1760 q^{66} +282.556 q^{67} -194.176 q^{68} +807.340 q^{69} +903.480 q^{70} +595.552 q^{71} +399.584 q^{72} -597.048 q^{73} -47.2321 q^{74} +1533.73 q^{75} +76.0000 q^{76} +110.300 q^{77} +1123.45 q^{78} +427.224 q^{79} -277.056 q^{80} +417.208 q^{81} +35.4400 q^{82} +493.768 q^{83} -915.376 q^{84} +840.588 q^{85} +736.808 q^{86} -773.796 q^{87} -33.8240 q^{88} -921.136 q^{89} -1729.80 q^{90} -1670.57 q^{91} +368.144 q^{92} -719.024 q^{93} -995.624 q^{94} -329.004 q^{95} +280.704 q^{96} +1082.74 q^{97} +675.168 q^{98} -211.180 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 9 q^{3} + 8 q^{4} - 9 q^{5} + 18 q^{6} - 18 q^{7} + 16 q^{8} + 23 q^{9}+O(q^{10}) $ 2 * q + 4 * q^2 + 9 * q^3 + 8 * q^4 - 9 * q^5 + 18 * q^6 - 18 * q^7 + 16 * q^8 + 23 * q^9	\( 2 q + 4 q^{2} + 9 q^{3} + 8 q^{4} - 9 q^{5} + 18 q^{6} - 18 q^{7} + 16 q^{8} + 23 q^{9} - 18 q^{10} - 17 q^{11} + 36 q^{12} + 17 q^{13} - 36 q^{14} - 150 q^{15} + 32 q^{16} - 80 q^{17} + 46 q^{18} + 38 q^{19} - 36 q^{20} - 227 q^{21} - 34 q^{22} + 73 q^{23} + 72 q^{24} + 119 q^{25} + 34 q^{26} + 189 q^{27} - 72 q^{28} + 3 q^{29} - 300 q^{30} + 212 q^{31} + 64 q^{32} - 40 q^{33} - 160 q^{34} + 519 q^{35} + 92 q^{36} + 192 q^{37} + 76 q^{38} + 551 q^{39} - 72 q^{40} - 50 q^{41} - 454 q^{42} + 677 q^{43} - 68 q^{44} - 1089 q^{45} + 146 q^{46} - 389 q^{47} + 144 q^{48} + 60 q^{49} + 238 q^{50} - 433 q^{51} + 68 q^{52} - 1219 q^{53} + 378 q^{54} - 33 q^{55} - 144 q^{56} + 171 q^{57} + 6 q^{58} - 287 q^{59} - 600 q^{60} + 313 q^{61} + 424 q^{62} - 1521 q^{63} + 128 q^{64} - 1500 q^{65} - 80 q^{66} + 1223 q^{67} - 320 q^{68} + 803 q^{69} + 1038 q^{70} + 200 q^{71} + 184 q^{72} + 378 q^{73} + 384 q^{74} + 1521 q^{75} + 152 q^{76} + 7 q^{77} + 1102 q^{78} + 1350 q^{79} - 144 q^{80} + 1142 q^{81} - 100 q^{82} - 670 q^{83} - 908 q^{84} + 579 q^{85} + 1354 q^{86} - 753 q^{87} - 136 q^{88} - 236 q^{89} - 2178 q^{90} - 2051 q^{91} + 292 q^{92} - 652 q^{93} - 778 q^{94} - 171 q^{95} + 288 q^{96} + 1294 q^{97} + 120 q^{98} + 133 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 9 * q^3 + 8 * q^4 - 9 * q^5 + 18 * q^6 - 18 * q^7 + 16 * q^8 + 23 * q^9 - 18 * q^10 - 17 * q^11 + 36 * q^12 + 17 * q^13 - 36 * q^14 - 150 * q^15 + 32 * q^16 - 80 * q^17 + 46 * q^18 + 38 * q^19 - 36 * q^20 - 227 * q^21 - 34 * q^22 + 73 * q^23 + 72 * q^24 + 119 * q^25 + 34 * q^26 + 189 * q^27 - 72 * q^28 + 3 * q^29 - 300 * q^30 + 212 * q^31 + 64 * q^32 - 40 * q^33 - 160 * q^34 + 519 * q^35 + 92 * q^36 + 192 * q^37 + 76 * q^38 + 551 * q^39 - 72 * q^40 - 50 * q^41 - 454 * q^42 + 677 * q^43 - 68 * q^44 - 1089 * q^45 + 146 * q^46 - 389 * q^47 + 144 * q^48 + 60 * q^49 + 238 * q^50 - 433 * q^51 + 68 * q^52 - 1219 * q^53 + 378 * q^54 - 33 * q^55 - 144 * q^56 + 171 * q^57 + 6 * q^58 - 287 * q^59 - 600 * q^60 + 313 * q^61 + 424 * q^62 - 1521 * q^63 + 128 * q^64 - 1500 * q^65 - 80 * q^66 + 1223 * q^67 - 320 * q^68 + 803 * q^69 + 1038 * q^70 + 200 * q^71 + 184 * q^72 + 378 * q^73 + 384 * q^74 + 1521 * q^75 + 152 * q^76 + 7 * q^77 + 1102 * q^78 + 1350 * q^79 - 144 * q^80 + 1142 * q^81 - 100 * q^82 - 670 * q^83 - 908 * q^84 + 579 * q^85 + 1354 * q^86 - 753 * q^87 - 136 * q^88 - 236 * q^89 - 2178 * q^90 - 2051 * q^91 + 292 * q^92 - 652 * q^93 - 778 * q^94 - 171 * q^95 + 288 * q^96 + 1294 * q^97 + 120 * q^98 + 133 * 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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	8.77200	1.68817	0.844086	−	0.536207i	$-0.180144\pi$
$3$	8.77200	1.68817	0.844086	+	0.536207i	$0.180144\pi$
$4$	4.00000	0.500000
$5$	−17.3160	−1.54879	−0.774395	−	0.632702i	$-0.781946\pi$
$5$	−17.3160	−1.54879	−0.774395	+	0.632702i	$0.781946\pi$
$6$	17.5440	1.19372
$7$	−26.0880	−1.40862	−0.704310	−	0.709893i	$-0.748743\pi$
$7$	−26.0880	−1.40862	−0.704310	+	0.709893i	$0.748743\pi$
$8$	8.00000	0.353553
$9$	49.9480	1.84993
$10$	−34.6320	−1.09516
$11$	−4.22800	−0.115890	−0.0579450	−	0.998320i	$-0.518455\pi$
$11$	−4.22800	−0.115890	−0.0579450	+	0.998320i	$0.518455\pi$
$12$	35.0880	0.844086
$13$	64.0360	1.36618	0.683092	−	0.730332i	$-0.260635\pi$
$13$	64.0360	1.36618	0.683092	+	0.730332i	$0.260635\pi$
$14$	−52.1760	−0.996045
$15$	−151.896	−2.61463
$16$	16.0000	0.250000
$17$	−48.5440	−0.692568	−0.346284	−	0.938130i	$-0.612557\pi$
$17$	−48.5440	−0.692568	−0.346284	+	0.938130i	$0.612557\pi$
$18$	99.8960	1.30810
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−69.2640	−0.774395
$21$	−228.844	−2.37799
$22$	−8.45600	−0.0819466
$23$	92.0360	0.834384	0.417192	−	0.908818i	$-0.363014\pi$
$23$	92.0360	0.834384	0.417192	+	0.908818i	$0.363014\pi$
$24$	70.1760	0.596859
$25$	174.844	1.39875
$26$	128.072	0.966038
$27$	201.300	1.43482
$28$	−104.352	−0.704310
$29$	−88.2120	−0.564847	−0.282424	−	0.959290i	$-0.591138\pi$
$29$	−88.2120	−0.564847	−0.282424	+	0.959290i	$0.591138\pi$
$30$	−303.792	−1.84882
$31$	−81.9681	−0.474900	−0.237450	−	0.971400i	$-0.576312\pi$
$31$	−81.9681	−0.474900	−0.237450	+	0.971400i	$0.576312\pi$
$32$	32.0000	0.176777
$33$	−37.0880	−0.195642
$34$	−97.0880	−0.489719
$35$	451.740	2.18166
$36$	199.792	0.924963
$37$	−23.6161	−0.104931	−0.0524656	−	0.998623i	$-0.516708\pi$
$37$	−23.6161	−0.104931	−0.0524656	+	0.998623i	$0.516708\pi$
$38$	38.0000	0.162221
$39$	561.724	2.30636
$40$	−138.528	−0.547580
$41$	17.7200	0.0674976	0.0337488	−	0.999430i	$-0.489255\pi$
$41$	17.7200	0.0674976	0.0337488	+	0.999430i	$0.489255\pi$
$42$	−457.688	−1.68150
$43$	368.404	1.30654	0.653268	−	0.757126i	$-0.273397\pi$
$43$	368.404	1.30654	0.653268	+	0.757126i	$0.273397\pi$
$44$	−16.9120	−0.0579450
$45$	−864.900	−2.86515
$46$	184.072	0.589999
$47$	−497.812	−1.54497	−0.772483	−	0.635036i	$-0.780985\pi$
$47$	−497.812	−1.54497	−0.772483	+	0.635036i	$0.780985\pi$
$48$	140.352	0.422043
$49$	337.584	0.984210
$50$	349.688	0.989067
$51$	−425.828	−1.16917
$52$	256.144	0.683092
$53$	−536.876	−1.39143	−0.695713	−	0.718320i	$-0.744911\pi$
$53$	−536.876	−1.39143	−0.695713	+	0.718320i	$0.744911\pi$
$54$	402.600	1.01457
$55$	73.2120	0.179489
$56$	−208.704	−0.498022
$57$	166.668	0.387293
$58$	−176.424	−0.399407
$59$	−36.7000	−0.0809818	−0.0404909	−	0.999180i	$-0.512892\pi$
$59$	−36.7000	−0.0809818	−0.0404909	+	0.999180i	$0.512892\pi$
$60$	−607.584	−1.30731
$61$	630.692	1.32380	0.661901	−	0.749592i	$-0.269750\pi$
$61$	630.692	1.32380	0.661901	+	0.749592i	$0.269750\pi$
$62$	−163.936	−0.335805
$63$	−1303.04	−2.60584
$64$	64.0000	0.125000
$65$	−1108.85	−2.11593
$66$	−74.1760	−0.138340
$67$	282.556	0.515219	0.257610	−	0.966249i	$-0.417065\pi$
$67$	282.556	0.515219	0.257610	+	0.966249i	$0.417065\pi$
$68$	−194.176	−0.346284
$69$	807.340	1.40858
$70$	903.480	1.54266
$71$	595.552	0.995480	0.497740	−	0.867326i	$-0.334163\pi$
$71$	595.552	0.995480	0.497740	+	0.867326i	$0.334163\pi$
$72$	399.584	0.654048
$73$	−597.048	−0.957250	−0.478625	−	0.878020i	$-0.658865\pi$
$73$	−597.048	−0.957250	−0.478625	+	0.878020i	$0.658865\pi$
$74$	−47.2321	−0.0741976
$75$	1533.73	2.36134
$76$	76.0000	0.114708
$77$	110.300	0.163245
$78$	1123.45	1.63084
$79$	427.224	0.608436	0.304218	−	0.952602i	$-0.401605\pi$
$79$	427.224	0.608436	0.304218	+	0.952602i	$0.401605\pi$
$80$	−277.056	−0.387198
$81$	417.208	0.572302
$82$	35.4400	0.0477280
$83$	493.768	0.652989	0.326495	−	0.945199i	$-0.394132\pi$
$83$	493.768	0.652989	0.326495	+	0.945199i	$0.394132\pi$
$84$	−915.376	−1.18900
$85$	840.588	1.07264
$86$	736.808	0.923861
$87$	−773.796	−0.953559
$88$	−33.8240	−0.0409733
$89$	−921.136	−1.09708	−0.548541	−	0.836124i	$-0.684816\pi$
$89$	−921.136	−1.09708	−0.548541	+	0.836124i	$0.684816\pi$
$90$	−1729.80	−2.02597
$91$	−1670.57	−1.92443
$92$	368.144	0.417192
$93$	−719.024	−0.801713
$94$	−995.624	−1.09246
$95$	−329.004	−0.355317
$96$	280.704	0.298430
$97$	1082.74	1.13336	0.566680	−	0.823938i	$-0.308227\pi$
$97$	1082.74	1.13336	0.566680	+	0.823938i	$0.308227\pi$
$98$	675.168	0.695942
$99$	−211.180	−0.214388
$100$	699.376	0.699376
$101$	−712.448	−0.701893	−0.350947	−	0.936395i	$-0.614140\pi$
$101$	−712.448	−0.701893	−0.350947	+	0.936395i	$0.614140\pi$
$102$	−851.656	−0.826731
$103$	−26.4797	−0.0253313	−0.0126656	−	0.999920i	$-0.504032\pi$
$103$	−26.4797	−0.0253313	−0.0126656	+	0.999920i	$0.504032\pi$
$104$	512.288	0.483019
$105$	3962.66	3.68301
$106$	−1073.75	−0.983887
$107$	−740.996	−0.669484	−0.334742	−	0.942310i	$-0.608649\pi$
$107$	−740.996	−0.669484	−0.334742	+	0.942310i	$0.608649\pi$
$108$	805.200	0.717411
$109$	−1983.08	−1.74261	−0.871304	−	0.490744i	$-0.836725\pi$
$109$	−1983.08	−1.74261	−0.871304	+	0.490744i	$0.836725\pi$
$110$	146.424	0.126918
$111$	−207.160	−0.177142
$112$	−417.408	−0.352155
$113$	−718.720	−0.598332	−0.299166	−	0.954201i	$-0.596708\pi$
$113$	−718.720	−0.598332	−0.299166	+	0.954201i	$0.596708\pi$
$114$	333.336	0.273858
$115$	−1593.70	−1.29229
$116$	−352.848	−0.282424
$117$	3198.47	2.52734
$118$	−73.3999	−0.0572628
$119$	1266.42	0.975565
$120$	−1215.17	−0.924410
$121$	−1313.12	−0.986570
$122$	1261.38	0.936069
$123$	155.440	0.113948
$124$	−327.872	−0.237450
$125$	−863.100	−0.617584
$126$	−2606.09	−1.84261
$127$	2610.72	1.82413	0.912063	−	0.410050i	$-0.134489\pi$
$127$	2610.72	1.82413	0.912063	+	0.410050i	$0.134489\pi$
$128$	128.000	0.0883883
$129$	3231.64	2.20566
$130$	−2217.70	−1.49619
$131$	−1216.69	−0.811472	−0.405736	−	0.913990i	$-0.632985\pi$
$131$	−1216.69	−0.811472	−0.405736	+	0.913990i	$0.632985\pi$
$132$	−148.352	−0.0978211
$133$	−495.672	−0.323160
$134$	565.112	0.364315
$135$	−3485.71	−2.22224
$136$	−388.352	−0.244860
$137$	1170.67	0.730053	0.365026	−	0.930997i	$-0.381060\pi$
$137$	1170.67	0.730053	0.365026	+	0.930997i	$0.381060\pi$
$138$	1614.68	0.996020
$139$	−271.083	−0.165417	−0.0827086	−	0.996574i	$-0.526357\pi$
$139$	−271.083	−0.165417	−0.0827086	+	0.996574i	$0.526357\pi$
$140$	1806.96	1.09083
$141$	−4366.81	−2.60817
$142$	1191.10	0.703910
$143$	−270.744	−0.158327
$144$	799.168	0.462482
$145$	1527.48	0.874830
$146$	−1194.10	−0.676878
$147$	2961.29	1.66152
$148$	−94.4642	−0.0524656
$149$	1841.19	1.01232	0.506161	−	0.862439i	$-0.331064\pi$
$149$	1841.19	1.01232	0.506161	+	0.862439i	$0.331064\pi$
$150$	3067.46	1.66972
$151$	3322.32	1.79051	0.895254	−	0.445557i	$-0.146994\pi$
$151$	3322.32	1.79051	0.895254	+	0.445557i	$0.146994\pi$
$152$	152.000	0.0811107
$153$	−2424.68	−1.28120
$154$	220.600	0.115432
$155$	1419.36	0.735521
$156$	2246.90	1.15318
$157$	243.616	0.123839	0.0619194	−	0.998081i	$-0.480278\pi$
$157$	243.616	0.123839	0.0619194	+	0.998081i	$0.480278\pi$
$158$	854.448	0.430229
$159$	−4709.48	−2.34897
$160$	−554.112	−0.273790
$161$	−2401.04	−1.17533
$162$	834.416	0.404678
$163$	−2598.11	−1.24847	−0.624233	−	0.781238i	$-0.714588\pi$
$163$	−2598.11	−1.24847	−0.624233	+	0.781238i	$0.714588\pi$
$164$	70.8801	0.0337488
$165$	642.216	0.303009
$166$	987.537	0.461733
$167$	−491.064	−0.227543	−0.113772	−	0.993507i	$-0.536293\pi$
$167$	−491.064	−0.227543	−0.113772	+	0.993507i	$0.536293\pi$
$168$	−1830.75	−0.840748
$169$	1903.61	0.866460
$170$	1681.18	0.758473
$171$	949.012	0.424402
$172$	1473.62	0.653268
$173$	1648.56	0.724496	0.362248	−	0.932082i	$-0.382009\pi$
$173$	1648.56	0.724496	0.362248	+	0.932082i	$0.382009\pi$
$174$	−1547.59	−0.674268
$175$	−4561.33	−1.97031
$176$	−67.6480	−0.0289725
$177$	−321.932	−0.136711
$178$	−1842.27	−0.775754
$179$	2326.81	0.971586	0.485793	−	0.874074i	$-0.338531\pi$
$179$	2326.81	0.971586	0.485793	+	0.874074i	$0.338531\pi$
$180$	−3459.60	−1.43257
$181$	−4637.46	−1.90442	−0.952208	−	0.305449i	$-0.901193\pi$
$181$	−4637.46	−1.90442	−0.952208	+	0.305449i	$0.901193\pi$
$182$	−3341.14	−1.36078
$183$	5532.43	2.23480
$184$	736.288	0.294999
$185$	408.936	0.162516
$186$	−1438.05	−0.566897
$187$	205.244	0.0802616
$188$	−1991.25	−0.772483
$189$	−5251.52	−2.02112
$190$	−658.008	−0.251247
$191$	5260.38	1.99281	0.996407	−	0.0846903i	$-0.0269901\pi$
$191$	5260.38	1.99281	0.996407	+	0.0846903i	$0.0269901\pi$
$192$	561.408	0.211022
$193$	16.1833	0.00603575	0.00301787	−	0.999995i	$-0.499039\pi$
$193$	16.1833	0.00603575	0.00301787	+	0.999995i	$0.499039\pi$
$194$	2165.49	0.801407
$195$	−9726.82	−3.57206
$196$	1350.34	0.492105
$197$	3784.71	1.36878	0.684390	−	0.729116i	$-0.260069\pi$
$197$	3784.71	1.36878	0.684390	+	0.729116i	$0.260069\pi$
$198$	−422.360	−0.151595
$199$	73.2079	0.0260783	0.0130391	−	0.999915i	$-0.495849\pi$
$199$	73.2079	0.0260783	0.0130391	+	0.999915i	$0.495849\pi$
$200$	1398.75	0.494534
$201$	2478.58	0.869779
$202$	−1424.90	−0.496313
$203$	2301.28	0.795655
$204$	−1703.31	−0.584587
$205$	−306.840	−0.104540
$206$	−52.9594	−0.0179119
$207$	4597.02	1.54355
$208$	1024.58	0.341546
$209$	−80.3320	−0.0265870
$210$	7925.33	2.60428
$211$	−2945.44	−0.961006	−0.480503	−	0.876993i	$-0.659546\pi$
$211$	−2945.44	−0.961006	−0.480503	+	0.876993i	$0.659546\pi$
$212$	−2147.50	−0.695713
$213$	5224.19	1.68054
$214$	−1481.99	−0.473397
$215$	−6379.29	−2.02355
$216$	1610.40	0.507286
$217$	2138.38	0.668954
$218$	−3966.15	−1.23221
$219$	−5237.31	−1.61600
$220$	292.848	0.0897446
$221$	−3108.57	−0.946175
$222$	−414.320	−0.125258
$223$	3125.30	0.938499	0.469250	−	0.883066i	$-0.344524\pi$
$223$	3125.30	0.938499	0.469250	+	0.883066i	$0.344524\pi$
$224$	−834.816	−0.249011
$225$	8733.11	2.58759
$226$	−1437.44	−0.423085
$227$	−3577.80	−1.04611	−0.523055	−	0.852299i	$-0.675208\pi$
$227$	−3577.80	−1.04611	−0.523055	+	0.852299i	$0.675208\pi$
$228$	666.672	0.193647
$229$	−4802.00	−1.38570	−0.692850	−	0.721082i	$-0.743645\pi$
$229$	−4802.00	−1.38570	−0.692850	+	0.721082i	$0.743645\pi$
$230$	−3187.39	−0.913785
$231$	967.552	0.275586
$232$	−705.696	−0.199704
$233$	5829.49	1.63907	0.819534	−	0.573031i	$-0.194232\pi$
$233$	5829.49	1.63907	0.819534	+	0.573031i	$0.194232\pi$
$234$	6396.94	1.78710
$235$	8620.12	2.39283
$236$	−146.800	−0.0404909
$237$	3747.61	1.02714
$238$	2532.83	0.689828
$239$	1364.33	0.369251	0.184625	−	0.982809i	$-0.440893\pi$
$239$	1364.33	0.369251	0.184625	+	0.982809i	$0.440893\pi$
$240$	−2430.34	−0.653656
$241$	−2647.22	−0.707563	−0.353782	−	0.935328i	$-0.615104\pi$
$241$	−2647.22	−0.707563	−0.353782	+	0.935328i	$0.615104\pi$
$242$	−2626.25	−0.697610
$243$	−1775.35	−0.468679
$244$	2522.77	0.661901
$245$	−5845.61	−1.52434
$246$	310.880	0.0805731
$247$	1216.68	0.313424
$248$	−655.745	−0.167903
$249$	4331.34	1.10236
$250$	−1726.20	−0.436698
$251$	1970.73	0.495582	0.247791	−	0.968814i	$-0.420295\pi$
$251$	1970.73	0.495582	0.247791	+	0.968814i	$0.420295\pi$
$252$	−5212.18	−1.30292
$253$	−389.128	−0.0966967
$254$	5221.44	1.28985
$255$	7373.64	1.81081
$256$	256.000	0.0625000
$257$	−7915.82	−1.92131	−0.960653	−	0.277752i	$-0.910411\pi$
$257$	−7915.82	−1.92131	−0.960653	+	0.277752i	$0.910411\pi$
$258$	6463.28	1.55964
$259$	616.096	0.147808
$260$	−4435.39	−1.05797
$261$	−4406.02	−1.04493
$262$	−2433.38	−0.573798
$263$	3287.96	0.770892	0.385446	−	0.922730i	$-0.374048\pi$
$263$	3287.96	0.770892	0.385446	+	0.922730i	$0.374048\pi$
$264$	−296.704	−0.0691700
$265$	9296.55	2.15503
$266$	−991.344	−0.228508
$267$	−8080.21	−1.85206
$268$	1130.22	0.257610
$269$	−4749.61	−1.07654	−0.538269	−	0.842773i	$-0.680922\pi$
$269$	−4749.61	−1.07654	−0.538269	+	0.842773i	$0.680922\pi$
$270$	−6971.43	−1.57136
$271$	242.661	0.0543933	0.0271967	−	0.999630i	$-0.491342\pi$
$271$	242.661	0.0543933	0.0271967	+	0.999630i	$0.491342\pi$
$272$	−776.704	−0.173142
$273$	−14654.3	−3.24878
$274$	2341.34	0.516225
$275$	−739.240	−0.162101
$276$	3229.36	0.704292
$277$	−4131.13	−0.896086	−0.448043	−	0.894012i	$-0.647879\pi$
$277$	−4131.13	−0.896086	−0.448043	+	0.894012i	$0.647879\pi$
$278$	−542.167	−0.116968
$279$	−4094.14	−0.878530
$280$	3613.92	0.771332
$281$	1007.19	0.213822	0.106911	−	0.994269i	$-0.465904\pi$
$281$	1007.19	0.213822	0.106911	+	0.994269i	$0.465904\pi$
$282$	−8733.62	−1.84425
$283$	2333.63	0.490176	0.245088	−	0.969501i	$-0.421183\pi$
$283$	2333.63	0.490176	0.245088	+	0.969501i	$0.421183\pi$
$284$	2382.21	0.497740
$285$	−2886.02	−0.599836
$286$	−541.488	−0.111954
$287$	−462.280	−0.0950785
$288$	1598.34	0.327024
$289$	−2556.48	−0.520350
$290$	3054.96	0.618598
$291$	9497.83	1.91331
$292$	−2388.19	−0.478625
$293$	−1588.68	−0.316763	−0.158381	−	0.987378i	$-0.550628\pi$
$293$	−1588.68	−0.316763	−0.158381	+	0.987378i	$0.550628\pi$
$294$	5922.58	1.17487
$295$	635.497	0.125424
$296$	−188.928	−0.0370988
$297$	−851.096	−0.166282
$298$	3682.38	0.715820
$299$	5893.62	1.13992
$300$	6134.93	1.18067
$301$	−9610.93	−1.84041
$302$	6644.64	1.26608
$303$	−6249.59	−1.18492
$304$	304.000	0.0573539
$305$	−10921.1	−2.05029
$306$	−4849.35	−0.905945
$307$	4057.46	0.754304	0.377152	−	0.926151i	$-0.376903\pi$
$307$	4057.46	0.754304	0.377152	+	0.926151i	$0.376903\pi$
$308$	441.200	0.0816224
$309$	−232.280	−0.0427636
$310$	2838.72	0.520092
$311$	2871.92	0.523638	0.261819	−	0.965117i	$-0.415678\pi$
$311$	2871.92	0.523638	0.261819	+	0.965117i	$0.415678\pi$
$312$	4493.79	0.815420
$313$	4322.67	0.780612	0.390306	−	0.920685i	$-0.372369\pi$
$313$	4322.67	0.780612	0.390306	+	0.920685i	$0.372369\pi$
$314$	487.232	0.0875672
$315$	22563.5	4.03591
$316$	1708.90	0.304218
$317$	2513.56	0.445349	0.222674	−	0.974893i	$-0.428521\pi$
$317$	2513.56	0.445349	0.222674	+	0.974893i	$0.428521\pi$
$318$	−9418.95	−1.66097
$319$	372.960	0.0654601
$320$	−1108.22	−0.193599
$321$	−6500.02	−1.13021
$322$	−4802.07	−0.831084
$323$	−922.336	−0.158886
$324$	1668.83	0.286151
$325$	11196.3	1.91095
$326$	−5196.22	−0.882798
$327$	−17395.6	−2.94182
$328$	141.760	0.0238640
$329$	12986.9	2.17627
$330$	1284.43	0.214260
$331$	−4573.78	−0.759509	−0.379754	−	0.925087i	$-0.623992\pi$
$331$	−4573.78	−0.759509	−0.379754	+	0.925087i	$0.623992\pi$
$332$	1975.07	0.326495
$333$	−1179.57	−0.194115
$334$	−982.129	−0.160897
$335$	−4892.74	−0.797967
$336$	−3661.50	−0.594498
$337$	9001.71	1.45506	0.727529	−	0.686077i	$-0.240669\pi$
$337$	9001.71	1.45506	0.727529	+	0.686077i	$0.240669\pi$
$338$	3807.22	0.612680
$339$	−6304.62	−1.01009
$340$	3362.35	0.536321
$341$	346.561	0.0550361
$342$	1898.02	0.300098
$343$	141.289	0.0222417
$344$	2947.23	0.461931
$345$	−13979.9	−2.18160
$346$	3297.12	0.512296
$347$	9358.68	1.44784	0.723920	−	0.689884i	$-0.242339\pi$
$347$	9358.68	1.44784	0.723920	+	0.689884i	$0.242339\pi$
$348$	−3095.18	−0.476780
$349$	5787.76	0.887712	0.443856	−	0.896098i	$-0.353610\pi$
$349$	5787.76	0.887712	0.443856	+	0.896098i	$0.353610\pi$
$350$	−9122.67	−1.39322
$351$	12890.5	1.96023
$352$	−135.296	−0.0204866
$353$	5784.59	0.872188	0.436094	−	0.899901i	$-0.356361\pi$
$353$	5784.59	0.872188	0.436094	+	0.899901i	$0.356361\pi$
$354$	−643.864	−0.0966695
$355$	−10312.6	−1.54179
$356$	−3684.55	−0.548541
$357$	11109.0	1.64692
$358$	4653.62	0.687015
$359$	−10132.3	−1.48959	−0.744796	−	0.667292i	$-0.767453\pi$
$359$	−10132.3	−1.48959	−0.744796	+	0.667292i	$0.767453\pi$
$360$	−6919.20	−1.01298
$361$	361.000	0.0526316
$362$	−9274.91	−1.34663
$363$	−11518.7	−1.66550
$364$	−6682.29	−0.962217
$365$	10338.5	1.48258
$366$	11064.9	1.58025
$367$	−6993.81	−0.994752	−0.497376	−	0.867535i	$-0.665703\pi$
$367$	−6993.81	−0.994752	−0.497376	+	0.867535i	$0.665703\pi$
$368$	1472.58	0.208596
$369$	885.080	0.124866
$370$	817.871	0.114917
$371$	14006.0	1.95999
$372$	−2876.10	−0.400857
$373$	6523.15	0.905512	0.452756	−	0.891634i	$-0.350441\pi$
$373$	6523.15	0.905512	0.452756	+	0.891634i	$0.350441\pi$
$374$	410.488	0.0567535
$375$	−7571.11	−1.04259
$376$	−3982.50	−0.546228
$377$	−5648.75	−0.771685
$378$	−10503.0	−1.42915
$379$	−9782.00	−1.32577	−0.662886	−	0.748720i	$-0.730669\pi$
$379$	−9782.00	−1.32577	−0.662886	+	0.748720i	$0.730669\pi$
$380$	−1316.02	−0.177658
$381$	22901.2	3.07944
$382$	10520.8	1.40913
$383$	9878.11	1.31788	0.658940	−	0.752196i	$-0.271005\pi$
$383$	9878.11	1.31788	0.658940	+	0.752196i	$0.271005\pi$
$384$	1122.82	0.149215
$385$	−1909.96	−0.252832
$386$	32.3666	0.00426792
$387$	18401.0	2.41700
$388$	4330.98	0.566680
$389$	−7891.25	−1.02854	−0.514270	−	0.857628i	$-0.671937\pi$
$389$	−7891.25	−1.02854	−0.514270	+	0.857628i	$0.671937\pi$
$390$	−19453.6	−2.52583
$391$	−4467.80	−0.577868
$392$	2700.67	0.347971
$393$	−10672.8	−1.36991
$394$	7569.43	0.967874
$395$	−7397.81	−0.942340
$396$	−844.720	−0.107194
$397$	−2787.84	−0.352437	−0.176219	−	0.984351i	$-0.556387\pi$
$397$	−2787.84	−0.352437	−0.176219	+	0.984351i	$0.556387\pi$
$398$	146.416	0.0184401
$399$	−4348.04	−0.545549
$400$	2797.50	0.349688
$401$	1264.42	0.157461	0.0787306	−	0.996896i	$-0.474913\pi$
$401$	1264.42	0.157461	0.0787306	+	0.996896i	$0.474913\pi$
$402$	4957.16	0.615027
$403$	−5248.91	−0.648801
$404$	−2849.79	−0.350947
$405$	−7224.37	−0.886375
$406$	4602.55	0.562613
$407$	99.8486	0.0121605
$408$	−3406.62	−0.413365
$409$	−8140.55	−0.984166	−0.492083	−	0.870548i	$-0.663764\pi$
$409$	−8140.55	−0.984166	−0.492083	+	0.870548i	$0.663764\pi$
$410$	−613.680	−0.0739207
$411$	10269.1	1.23246
$412$	−105.919	−0.0126656
$413$	957.429	0.114073
$414$	9194.03	1.09145
$415$	−8550.10	−1.01134
$416$	2049.15	0.241510
$417$	−2377.94	−0.279253
$418$	−160.664	−0.0187998
$419$	−9601.15	−1.11944	−0.559722	−	0.828680i	$-0.689092\pi$
$419$	−9601.15	−1.11944	−0.559722	+	0.828680i	$0.689092\pi$
$420$	15850.7	1.84151
$421$	5702.48	0.660147	0.330074	−	0.943955i	$-0.392926\pi$
$421$	5702.48	0.660147	0.330074	+	0.943955i	$0.392926\pi$
$422$	−5890.87	−0.679534
$423$	−24864.7	−2.85807
$424$	−4295.01	−0.491943
$425$	−8487.63	−0.968731
$426$	10448.4	1.18832
$427$	−16453.5	−1.86473
$428$	−2963.99	−0.334742
$429$	−2374.97	−0.267283
$430$	−12758.6	−1.43087
$431$	−4025.72	−0.449912	−0.224956	−	0.974369i	$-0.572224\pi$
$431$	−4025.72	−0.449912	−0.224956	+	0.974369i	$0.572224\pi$
$432$	3220.80	0.358706
$433$	−1347.10	−0.149510	−0.0747548	−	0.997202i	$-0.523817\pi$
$433$	−1347.10	−0.149510	−0.0747548	+	0.997202i	$0.523817\pi$
$434$	4276.77	0.473022
$435$	13399.1	1.47686
$436$	−7932.31	−0.871304
$437$	1748.68	0.191421
$438$	−10474.6	−1.14269
$439$	4109.36	0.446763	0.223381	−	0.974731i	$-0.428290\pi$
$439$	4109.36	0.446763	0.223381	+	0.974731i	$0.428290\pi$
$440$	585.696	0.0634590
$441$	16861.7	1.82072
$442$	−6217.13	−0.669047
$443$	6964.84	0.746974	0.373487	−	0.927635i	$-0.378162\pi$
$443$	6964.84	0.746974	0.373487	+	0.927635i	$0.378162\pi$
$444$	−828.640	−0.0885710
$445$	15950.4	1.69915
$446$	6250.59	0.663619
$447$	16150.9	1.70897
$448$	−1669.63	−0.176078
$449$	3041.21	0.319652	0.159826	−	0.987145i	$-0.448907\pi$
$449$	3041.21	0.319652	0.159826	+	0.987145i	$0.448907\pi$
$450$	17466.2	1.82970
$451$	−74.9202	−0.00782229
$452$	−2874.88	−0.299166
$453$	29143.4	3.02269
$454$	−7155.60	−0.739711
$455$	28927.6	2.98055
$456$	1333.34	0.136929
$457$	11984.3	1.22670	0.613352	−	0.789810i	$-0.289821\pi$
$457$	11984.3	1.22670	0.613352	+	0.789810i	$0.289821\pi$
$458$	−9604.01	−0.979838
$459$	−9771.91	−0.993712
$460$	−6374.79	−0.646143
$461$	−12126.7	−1.22515	−0.612577	−	0.790411i	$-0.709867\pi$
$461$	−12126.7	−1.22515	−0.612577	+	0.790411i	$0.709867\pi$
$462$	1935.10	0.194868
$463$	−6399.19	−0.642323	−0.321162	−	0.947024i	$-0.604073\pi$
$463$	−6399.19	−0.642323	−0.321162	+	0.947024i	$0.604073\pi$
$464$	−1411.39	−0.141212
$465$	12450.6	1.24169
$466$	11659.0	1.15900
$467$	993.366	0.0984315	0.0492157	−	0.998788i	$-0.484328\pi$
$467$	993.366	0.0984315	0.0492157	+	0.998788i	$0.484328\pi$
$468$	12793.9	1.26367
$469$	−7371.32	−0.725748
$470$	17240.2	1.69198
$471$	2137.00	0.209061
$472$	−293.600	−0.0286314
$473$	−1557.61	−0.151414
$474$	7495.22	0.726301
$475$	3322.04	0.320896
$476$	5065.67	0.487782
$477$	−26815.9	−2.57404
$478$	2728.65	0.261100
$479$	6639.36	0.633320	0.316660	−	0.948539i	$-0.397439\pi$
$479$	6639.36	0.633320	0.316660	+	0.948539i	$0.397439\pi$
$480$	−4860.67	−0.462205
$481$	−1512.28	−0.143355
$482$	−5294.45	−0.500323
$483$	−21061.9	−1.98416
$484$	−5252.50	−0.493285
$485$	−18748.8	−1.75534
$486$	−3550.70	−0.331406
$487$	−11088.8	−1.03179	−0.515894	−	0.856652i	$-0.672540\pi$
$487$	−11088.8	−1.03179	−0.515894	+	0.856652i	$0.672540\pi$
$488$	5045.54	0.468034
$489$	−22790.6	−2.10762
$490$	−11691.2	−1.07787
$491$	−13215.2	−1.21465	−0.607324	−	0.794454i	$-0.707757\pi$
$491$	−13215.2	−1.21465	−0.607324	+	0.794454i	$0.707757\pi$
$492$	621.760	0.0569738
$493$	4282.17	0.391195
$494$	2433.37	0.221624
$495$	3656.80	0.332042
$496$	−1311.49	−0.118725
$497$	−15536.8	−1.40225
$498$	8662.67	0.779485
$499$	410.640	0.0368393	0.0184196	−	0.999830i	$-0.494137\pi$
$499$	410.640	0.0368393	0.0184196	+	0.999830i	$0.494137\pi$
$500$	−3452.40	−0.308792
$501$	−4307.62	−0.384132
$502$	3941.45	0.350429
$503$	−9407.88	−0.833950	−0.416975	−	0.908918i	$-0.636910\pi$
$503$	−9407.88	−0.833950	−0.416975	+	0.908918i	$0.636910\pi$
$504$	−10424.4	−0.921305
$505$	12336.7	1.08709
$506$	−778.256	−0.0683749
$507$	16698.5	1.46273
$508$	10442.9	0.912063
$509$	10482.2	0.912803	0.456402	−	0.889774i	$-0.349138\pi$
$509$	10482.2	0.912803	0.456402	+	0.889774i	$0.349138\pi$
$510$	14747.3	1.28043
$511$	15575.8	1.34840
$512$	512.000	0.0441942
$513$	3824.70	0.329171
$514$	−15831.6	−1.35857
$515$	458.523	0.0392329
$516$	12926.6	1.10283
$517$	2104.75	0.179046
$518$	1232.19	0.104516
$519$	14461.2	1.22307
$520$	−8870.79	−0.748096
$521$	−3181.02	−0.267492	−0.133746	−	0.991016i	$-0.542701\pi$
$521$	−3181.02	−0.267492	−0.133746	+	0.991016i	$0.542701\pi$
$522$	−8812.03	−0.738874
$523$	−4360.12	−0.364541	−0.182270	−	0.983248i	$-0.558345\pi$
$523$	−4360.12	−0.364541	−0.182270	+	0.983248i	$0.558345\pi$
$524$	−4866.77	−0.405736
$525$	−40012.0	−3.32622
$526$	6575.93	0.545103
$527$	3979.06	0.328900
$528$	−593.408	−0.0489106
$529$	−3696.37	−0.303803
$530$	18593.1	1.52383
$531$	−1833.09	−0.149810
$532$	−1982.69	−0.161580
$533$	1134.72	0.0922142
$534$	−16160.4	−1.30961
$535$	12831.1	1.03689
$536$	2260.45	0.182158
$537$	20410.8	1.64020
$538$	−9499.22	−0.761227
$539$	−1427.31	−0.114060
$540$	−13942.9	−1.11112
$541$	−23681.2	−1.88195	−0.940973	−	0.338481i	$-0.890087\pi$
$541$	−23681.2	−1.88195	−0.940973	+	0.338481i	$0.890087\pi$
$542$	485.322	0.0384619
$543$	−40679.8	−3.21498
$544$	−1553.41	−0.122430
$545$	34339.0	2.69894
$546$	−29308.5	−2.29723
$547$	−7373.25	−0.576339	−0.288169	−	0.957579i	$-0.593047\pi$
$547$	−7373.25	−0.576339	−0.288169	+	0.957579i	$0.593047\pi$
$548$	4682.69	0.365026
$549$	31501.8	2.44893
$550$	−1478.48	−0.114623
$551$	−1676.03	−0.129585
$552$	6458.72	0.498010
$553$	−11145.4	−0.857055
$554$	−8262.27	−0.633628
$555$	3587.18	0.274356
$556$	−1084.33	−0.0827086
$557$	4772.14	0.363020	0.181510	−	0.983389i	$-0.441902\pi$
$557$	4772.14	0.363020	0.181510	+	0.983389i	$0.441902\pi$
$558$	−8188.29	−0.621215
$559$	23591.1	1.78497
$560$	7227.84	0.545414
$561$	1800.40	0.135495
$562$	2014.38	0.151195
$563$	7276.49	0.544702	0.272351	−	0.962198i	$-0.412199\pi$
$563$	7276.49	0.544702	0.272351	+	0.962198i	$0.412199\pi$
$564$	−17467.2	−1.30408
$565$	12445.4	0.926691
$566$	4667.26	0.346607
$567$	−10884.1	−0.806156
$568$	4764.42	0.351955
$569$	−10685.1	−0.787245	−0.393622	−	0.919272i	$-0.628778\pi$
$569$	−10685.1	−0.787245	−0.393622	+	0.919272i	$0.628778\pi$
$570$	−5772.05	−0.424148
$571$	14856.1	1.08881	0.544404	−	0.838823i	$-0.316756\pi$
$571$	14856.1	1.08881	0.544404	+	0.838823i	$0.316756\pi$
$572$	−1082.98	−0.0791635
$573$	46144.0	3.36421
$574$	−924.560	−0.0672306
$575$	16092.0	1.16710
$576$	3196.67	0.231241
$577$	3212.67	0.231794	0.115897	−	0.993261i	$-0.463026\pi$
$577$	3212.67	0.231794	0.115897	+	0.993261i	$0.463026\pi$
$578$	−5112.96	−0.367943
$579$	141.960	0.0101894
$580$	6109.92	0.437415
$581$	−12881.4	−0.919814
$582$	18995.7	1.35291
$583$	2269.91	0.161252
$584$	−4776.39	−0.338439
$585$	−55384.8	−3.91432
$586$	−3177.35	−0.223985
$587$	−22321.1	−1.56949	−0.784745	−	0.619818i	$-0.787206\pi$
$587$	−22321.1	−1.56949	−0.784745	+	0.619818i	$0.787206\pi$
$588$	11845.2	0.830758
$589$	−1557.39	−0.108950
$590$	1270.99	0.0886881
$591$	33199.5	2.31074
$592$	−377.857	−0.0262328
$593$	−8202.50	−0.568021	−0.284010	−	0.958821i	$-0.591665\pi$
$593$	−8202.50	−0.568021	−0.284010	+	0.958821i	$0.591665\pi$
$594$	−1702.19	−0.117579
$595$	−21929.3	−1.51095
$596$	7364.75	0.506161
$597$	642.180	0.0440246
$598$	11787.2	0.806047
$599$	10583.3	0.721906	0.360953	−	0.932584i	$-0.382452\pi$
$599$	10583.3	0.721906	0.360953	+	0.932584i	$0.382452\pi$
$600$	12269.9	0.834858
$601$	−9051.94	−0.614370	−0.307185	−	0.951650i	$-0.599387\pi$
$601$	−9051.94	−0.614370	−0.307185	+	0.951650i	$0.599387\pi$
$602$	−19221.9	−1.30137
$603$	14113.1	0.953118
$604$	13289.3	0.895254
$605$	22738.1	1.52799
$606$	−12499.2	−0.837863
$607$	8123.48	0.543199	0.271599	−	0.962410i	$-0.412447\pi$
$607$	8123.48	0.543199	0.271599	+	0.962410i	$0.412447\pi$
$608$	608.000	0.0405554
$609$	20186.8	1.34320
$610$	−21842.1	−1.44977
$611$	−31877.9	−2.11071
$612$	−9698.71	−0.640600
$613$	22384.7	1.47490	0.737448	−	0.675404i	$-0.236031\pi$
$613$	22384.7	1.47490	0.737448	+	0.675404i	$0.236031\pi$
$614$	8114.91	0.533373
$615$	−2691.60	−0.176481
$616$	882.400	0.0577158
$617$	11349.1	0.740517	0.370259	−	0.928929i	$-0.379269\pi$
$617$	11349.1	0.740517	0.370259	+	0.928929i	$0.379269\pi$
$618$	−464.560	−0.0302384
$619$	−9106.25	−0.591294	−0.295647	−	0.955297i	$-0.595535\pi$
$619$	−9106.25	−0.591294	−0.295647	+	0.955297i	$0.595535\pi$
$620$	5677.44	0.367760
$621$	18526.9	1.19719
$622$	5743.84	0.370268
$623$	24030.6	1.54537
$624$	8987.59	0.576589
$625$	−6910.06	−0.442244
$626$	8645.34	0.551976
$627$	−704.672	−0.0448834
$628$	974.464	0.0619194
$629$	1146.42	0.0726720
$630$	45127.0	2.85382
$631$	−27784.2	−1.75289	−0.876444	−	0.481505i	$-0.840090\pi$
$631$	−27784.2	−1.75289	−0.876444	+	0.481505i	$0.840090\pi$
$632$	3417.79	0.215115
$633$	−25837.4	−1.62234
$634$	5027.12	0.314909
$635$	−45207.3	−2.82519
$636$	−18837.9	−1.17448
$637$	21617.5	1.34461
$638$	745.921	0.0462873
$639$	29746.7	1.84156
$640$	−2216.45	−0.136895
$641$	−16958.3	−1.04495	−0.522476	−	0.852654i	$-0.674992\pi$
$641$	−16958.3	−1.04495	−0.522476	+	0.852654i	$0.674992\pi$
$642$	−13000.0	−0.799176
$643$	4754.37	0.291592	0.145796	−	0.989315i	$-0.453426\pi$
$643$	4754.37	0.291592	0.145796	+	0.989315i	$0.453426\pi$
$644$	−9604.15	−0.587665
$645$	−55959.1	−3.41611
$646$	−1844.67	−0.112349
$647$	−11254.0	−0.683831	−0.341916	−	0.939731i	$-0.611076\pi$
$647$	−11254.0	−0.683831	−0.341916	+	0.939731i	$0.611076\pi$
$648$	3337.66	0.202339
$649$	155.167	0.00938498
$650$	22392.6	1.35125
$651$	18757.9	1.12931
$652$	−10392.4	−0.624233
$653$	−15515.1	−0.929793	−0.464896	−	0.885365i	$-0.653908\pi$
$653$	−15515.1	−0.929793	−0.464896	+	0.885365i	$0.653908\pi$
$654$	−34791.1	−2.08018
$655$	21068.2	1.25680
$656$	283.520	0.0168744
$657$	−29821.4	−1.77084
$658$	25973.9	1.53885
$659$	17203.2	1.01691	0.508453	−	0.861090i	$-0.330217\pi$
$659$	17203.2	1.01691	0.508453	+	0.861090i	$0.330217\pi$
$660$	2568.86	0.151504
$661$	2305.65	0.135672	0.0678361	−	0.997696i	$-0.478390\pi$
$661$	2305.65	0.135672	0.0678361	+	0.997696i	$0.478390\pi$
$662$	−9147.55	−0.537054
$663$	−27268.3	−1.59731
$664$	3950.15	0.230867
$665$	8583.06	0.500507
$666$	−2359.15	−0.137260
$667$	−8118.69	−0.471299
$668$	−1964.26	−0.113772
$669$	27415.1	1.58435
$670$	−9785.48	−0.564248
$671$	−2666.57	−0.153415
$672$	−7323.01	−0.420374
$673$	−14242.8	−0.815782	−0.407891	−	0.913031i	$-0.633736\pi$
$673$	−14242.8	−0.815782	−0.407891	+	0.913031i	$0.633736\pi$
$674$	18003.4	1.02888
$675$	35196.1	2.00696
$676$	7614.45	0.433230
$677$	−13480.0	−0.765256	−0.382628	−	0.923902i	$-0.624981\pi$
$677$	−13480.0	−0.765256	−0.382628	+	0.923902i	$0.624981\pi$
$678$	−12609.2	−0.714240
$679$	−28246.6	−1.59647
$680$	6724.71	0.379236
$681$	−31384.5	−1.76601
$682$	693.122	0.0389164
$683$	27626.1	1.54771	0.773854	−	0.633365i	$-0.218327\pi$
$683$	27626.1	1.54771	0.773854	+	0.633365i	$0.218327\pi$
$684$	3796.05	0.212201
$685$	−20271.4	−1.13070
$686$	282.578	0.0157272
$687$	−42123.2	−2.33930
$688$	5894.46	0.326634
$689$	−34379.4	−1.90094
$690$	−27959.8	−1.54263
$691$	17419.7	0.959009	0.479505	−	0.877539i	$-0.340816\pi$
$691$	17419.7	0.959009	0.479505	+	0.877539i	$0.340816\pi$
$692$	6594.24	0.362248
$693$	5509.27	0.301991
$694$	18717.4	1.02378
$695$	4694.08	0.256197
$696$	−6190.37	−0.337134
$697$	−860.201	−0.0467467
$698$	11575.5	0.627707
$699$	51136.3	2.76703
$700$	−18245.3	−0.985155
$701$	5069.39	0.273136	0.136568	−	0.990631i	$-0.456393\pi$
$701$	5069.39	0.273136	0.136568	+	0.990631i	$0.456393\pi$
$702$	25780.9	1.38609
$703$	−448.705	−0.0240729
$704$	−270.592	−0.0144862
$705$	75615.7	4.03951
$706$	11569.2	0.616730
$707$	18586.3	0.988701
$708$	−1287.73	−0.0683556
$709$	−16758.9	−0.887719	−0.443860	−	0.896096i	$-0.646391\pi$
$709$	−16758.9	−0.887719	−0.443860	+	0.896096i	$0.646391\pi$
$710$	−20625.2	−1.09021
$711$	21339.0	1.12556
$712$	−7369.09	−0.387877
$713$	−7544.02	−0.396249
$714$	22218.0	1.16455
$715$	4688.21	0.245215
$716$	9307.24	0.485793
$717$	11967.9	0.623359
$718$	−20264.6	−1.05330
$719$	−3885.84	−0.201554	−0.100777	−	0.994909i	$-0.532133\pi$
$719$	−3885.84	−0.201554	−0.100777	+	0.994909i	$0.532133\pi$
$720$	−13838.4	−0.716287
$721$	690.803	0.0356822
$722$	722.000	0.0372161
$723$	−23221.5	−1.19449
$724$	−18549.8	−0.952208
$725$	−15423.4	−0.790081
$726$	−23037.5	−1.17769
$727$	6468.37	0.329984	0.164992	−	0.986295i	$-0.447240\pi$
$727$	6468.37	0.329984	0.164992	+	0.986295i	$0.447240\pi$
$728$	−13364.6	−0.680390
$729$	−26838.0	−1.36351
$730$	20677.0	1.04834
$731$	−17883.8	−0.904865
$732$	22129.7	1.11740
$733$	25245.5	1.27212	0.636059	−	0.771640i	$-0.280563\pi$
$733$	25245.5	1.27212	0.636059	+	0.771640i	$0.280563\pi$
$734$	−13987.6	−0.703396
$735$	−51277.7	−2.57334
$736$	2945.15	0.147500
$737$	−1194.65	−0.0597087
$738$	1770.16	0.0882933
$739$	3229.28	0.160746	0.0803728	−	0.996765i	$-0.474389\pi$
$739$	3229.28	0.160746	0.0803728	+	0.996765i	$0.474389\pi$
$740$	1635.74	0.0812582
$741$	10672.8	0.529114
$742$	28012.0	1.38592
$743$	18876.2	0.932033	0.466016	−	0.884776i	$-0.345689\pi$
$743$	18876.2	0.932033	0.466016	+	0.884776i	$0.345689\pi$
$744$	−5752.19	−0.283448
$745$	−31882.0	−1.56788
$746$	13046.3	0.640294
$747$	24662.8	1.20798
$748$	820.976	0.0401308
$749$	19331.1	0.943049
$750$	−15142.2	−0.737221
$751$	24895.8	1.20967	0.604833	−	0.796352i	$-0.293240\pi$
$751$	24895.8	1.20967	0.604833	+	0.796352i	$0.293240\pi$
$752$	−7964.99	−0.386241
$753$	17287.2	0.836628
$754$	−11297.5	−0.545664
$755$	−57529.3	−2.77312
$756$	−21006.1	−1.01056
$757$	−36203.2	−1.73821	−0.869107	−	0.494624i	$-0.835306\pi$
$757$	−36203.2	−1.73821	−0.869107	+	0.494624i	$0.835306\pi$
$758$	−19564.0	−0.937462
$759$	−3413.43	−0.163241
$760$	−2632.03	−0.125624
$761$	11417.5	0.543868	0.271934	−	0.962316i	$-0.412337\pi$
$761$	11417.5	0.543868	0.271934	+	0.962316i	$0.412337\pi$
$762$	45802.5	2.17749
$763$	51734.5	2.45467
$764$	21041.5	0.996407
$765$	41985.7	1.98431
$766$	19756.2	0.931881
$767$	−2350.12	−0.110636
$768$	2245.63	0.105511
$769$	39414.5	1.84828	0.924138	−	0.382058i	$-0.124785\pi$
$769$	39414.5	1.84828	0.924138	+	0.382058i	$0.124785\pi$
$770$	−3819.91	−0.178779
$771$	−69437.6	−3.24350
$772$	64.7332	0.00301787
$773$	−14268.5	−0.663910	−0.331955	−	0.943295i	$-0.607708\pi$
$773$	−14268.5	−0.663910	−0.331955	+	0.943295i	$0.607708\pi$
$774$	36802.1	1.70908
$775$	−14331.6	−0.664268
$776$	8661.95	0.400704
$777$	5404.39	0.249526
$778$	−15782.5	−0.727288
$779$	336.680	0.0154850
$780$	−38907.3	−1.78603
$781$	−2517.99	−0.115366
$782$	−8935.59	−0.408614
$783$	−17757.1	−0.810455
$784$	5401.35	0.246053
$785$	−4218.46	−0.191800
$786$	−21345.7	−0.968669
$787$	−2922.28	−0.132361	−0.0661804	−	0.997808i	$-0.521081\pi$
$787$	−2922.28	−0.132361	−0.0661804	+	0.997808i	$0.521081\pi$
$788$	15138.9	0.684390
$789$	28842.0	1.30140
$790$	−14795.6	−0.666335
$791$	18750.0	0.842823
$792$	−1689.44	−0.0757976
$793$	40387.0	1.80856
$794$	−5575.67	−0.249211
$795$	81549.3	3.63806
$796$	292.832	0.0130391
$797$	−7724.25	−0.343296	−0.171648	−	0.985158i	$-0.554909\pi$
$797$	−7724.25	−0.343296	−0.171648	+	0.985158i	$0.554909\pi$
$798$	−8696.07	−0.385762
$799$	24165.8	1.06999
$800$	5595.01	0.247267
$801$	−46008.9	−2.02952
$802$	2528.83	0.111342
$803$	2524.32	0.110936
$804$	9914.32	0.434890
$805$	41576.4	1.82034
$806$	−10497.8	−0.458772
$807$	−41663.6	−1.81738
$808$	−5699.58	−0.248157
$809$	42980.8	1.86789	0.933947	−	0.357412i	$-0.116341\pi$
$809$	42980.8	1.86789	0.933947	+	0.357412i	$0.116341\pi$
$810$	−14448.7	−0.626762
$811$	28749.5	1.24480	0.622398	−	0.782701i	$-0.286158\pi$
$811$	28749.5	1.24480	0.622398	+	0.782701i	$0.286158\pi$
$812$	9205.11	0.397827
$813$	2128.62	0.0918253
$814$	199.697	0.00859875
$815$	44988.9	1.93361
$816$	−6813.25	−0.292293
$817$	6999.68	0.299740
$818$	−16281.1	−0.695911
$819$	−83441.8	−3.56006
$820$	−1227.36	−0.0522698
$821$	−30274.8	−1.28696	−0.643482	−	0.765461i	$-0.722511\pi$
$821$	−30274.8	−1.28696	−0.643482	+	0.765461i	$0.722511\pi$
$822$	20538.3	0.871477
$823$	17296.1	0.732568	0.366284	−	0.930503i	$-0.380630\pi$
$823$	17296.1	0.732568	0.366284	+	0.930503i	$0.380630\pi$
$824$	−211.838	−0.00895596
$825$	−6484.62	−0.273655
$826$	1914.86	0.0806615
$827$	−2022.80	−0.0850541	−0.0425271	−	0.999095i	$-0.513541\pi$
$827$	−2022.80	−0.0850541	−0.0425271	+	0.999095i	$0.513541\pi$
$828$	18388.1	0.771775
$829$	−43239.0	−1.81152	−0.905762	−	0.423786i	$-0.860701\pi$
$829$	−43239.0	−1.81152	−0.905762	+	0.423786i	$0.860701\pi$
$830$	−17100.2	−0.715128
$831$	−36238.3	−1.51275
$832$	4098.31	0.170773
$833$	−16387.7	−0.681632
$834$	−4755.89	−0.197462
$835$	8503.27	0.352417
$836$	−321.328	−0.0132935
$837$	−16500.2	−0.681397
$838$	−19202.3	−0.791567
$839$	27435.9	1.12895	0.564477	−	0.825449i	$-0.309078\pi$
$839$	27435.9	1.12895	0.564477	+	0.825449i	$0.309078\pi$
$840$	31701.3	1.30214
$841$	−16607.6	−0.680948
$842$	11405.0	0.466795
$843$	8835.09	0.360969
$844$	−11781.7	−0.480503
$845$	−32963.0	−1.34196
$846$	−49729.5	−2.02096
$847$	34256.8	1.38970
$848$	−8590.02	−0.347857
$849$	20470.6	0.827502
$850$	−16975.3	−0.684996
$851$	−2173.53	−0.0875530
$852$	20896.7	0.840271
$853$	−20978.4	−0.842071	−0.421035	−	0.907044i	$-0.638333\pi$
$853$	−20978.4	−0.842071	−0.421035	+	0.907044i	$0.638333\pi$
$854$	−32907.0	−1.31857
$855$	−16433.1	−0.657310
$856$	−5927.97	−0.236698
$857$	−30822.4	−1.22856	−0.614279	−	0.789089i	$-0.710553\pi$
$857$	−30822.4	−1.22856	−0.614279	+	0.789089i	$0.710553\pi$
$858$	−4749.94	−0.188998
$859$	−39267.6	−1.55971	−0.779856	−	0.625959i	$-0.784708\pi$
$859$	−39267.6	−1.55971	−0.779856	+	0.625959i	$0.784708\pi$
$860$	−25517.1	−1.01178
$861$	−4055.12	−0.160509
$862$	−8051.45	−0.318136
$863$	−24131.3	−0.951842	−0.475921	−	0.879488i	$-0.657885\pi$
$863$	−24131.3	−0.951842	−0.475921	+	0.879488i	$0.657885\pi$
$864$	6441.60	0.253643
$865$	−28546.5	−1.12209
$866$	−2694.21	−0.105719
$867$	−22425.4	−0.878441
$868$	8553.54	0.334477
$869$	−1806.30	−0.0705116
$870$	26798.1	1.04430
$871$	18093.8	0.703885
$872$	−15864.6	−0.616105
$873$	54080.9	2.09663
$874$	3497.37	0.135355
$875$	22516.6	0.869941
$876$	−20949.2	−0.808001
$877$	39380.6	1.51629	0.758147	−	0.652084i	$-0.226105\pi$
$877$	39380.6	1.51629	0.758147	+	0.652084i	$0.226105\pi$
$878$	8218.71	0.315909
$879$	−13935.9	−0.534750
$880$	1171.39	0.0448723
$881$	30887.5	1.18119	0.590595	−	0.806968i	$-0.298893\pi$
$881$	30887.5	1.18119	0.590595	+	0.806968i	$0.298893\pi$
$882$	33723.3	1.28744
$883$	28191.9	1.07444	0.537221	−	0.843441i	$-0.319474\pi$
$883$	28191.9	1.07444	0.537221	+	0.843441i	$0.319474\pi$
$884$	−12434.3	−0.473088
$885$	5574.58	0.211737
$886$	13929.7	0.528190
$887$	2760.58	0.104500	0.0522498	−	0.998634i	$-0.483361\pi$
$887$	2760.58	0.104500	0.0522498	+	0.998634i	$0.483361\pi$
$888$	−1657.28	−0.0626292
$889$	−68108.5	−2.56950
$890$	31900.8	1.20148
$891$	−1763.95	−0.0663240
$892$	12501.2	0.469250
$893$	−9458.43	−0.354439
$894$	32301.8	1.20843
$895$	−40291.0	−1.50478
$896$	−3339.26	−0.124506
$897$	51698.9	1.92439
$898$	6082.42	0.226028
$899$	7230.57	0.268246
$900$	34932.5	1.29379
$901$	26062.1	0.963657
$902$	−149.840	−0.00553120
$903$	−84307.1	−3.10694
$904$	−5749.76	−0.211542
$905$	80302.2	2.94954
$906$	58286.8	2.13736
$907$	−18969.1	−0.694443	−0.347222	−	0.937783i	$-0.612875\pi$
$907$	−18969.1	−0.694443	−0.347222	+	0.937783i	$0.612875\pi$
$908$	−14311.2	−0.523055
$909$	−35585.4	−1.29845
$910$	57855.3	2.10756
$911$	−48732.9	−1.77233	−0.886164	−	0.463371i	$-0.846640\pi$
$911$	−48732.9	−1.77233	−0.886164	+	0.463371i	$0.846640\pi$
$912$	2666.69	0.0968233
$913$	−2087.65	−0.0756749
$914$	23968.7	0.867411
$915$	−95799.6	−3.46124
$916$	−19208.0	−0.692850
$917$	31741.1	1.14306
$918$	−19543.8	−0.702660
$919$	−35850.4	−1.28683	−0.643414	−	0.765518i	$-0.722483\pi$
$919$	−35850.4	−1.28683	−0.643414	+	0.765518i	$0.722483\pi$
$920$	−12749.6	−0.456892
$921$	35592.0	1.27340
$922$	−24253.4	−0.866315
$923$	38136.8	1.36001
$924$	3870.21	0.137793
$925$	−4129.13	−0.146773
$926$	−12798.4	−0.454191
$927$	−1322.61	−0.0468610
$928$	−2822.79	−0.0998518
$929$	22936.8	0.810044	0.405022	−	0.914307i	$-0.367264\pi$
$929$	22936.8	0.810044	0.405022	+	0.914307i	$0.367264\pi$
$930$	24901.3	0.878005
$931$	6414.10	0.225793
$932$	23318.0	0.819534
$933$	25192.5	0.883992
$934$	1986.73	0.0696016
$935$	−3554.01	−0.124308
$936$	25587.8	0.893550
$937$	47925.4	1.67092	0.835462	−	0.549548i	$-0.185200\pi$
$937$	47925.4	1.67092	0.835462	+	0.549548i	$0.185200\pi$
$938$	−14742.6	−0.513181
$939$	37918.5	1.31781
$940$	34480.5	1.19641
$941$	−25842.2	−0.895251	−0.447626	−	0.894221i	$-0.647730\pi$
$941$	−25842.2	−0.895251	−0.447626	+	0.894221i	$0.647730\pi$
$942$	4274.00	0.147829
$943$	1630.88	0.0563189
$944$	−587.199	−0.0202455
$945$	90935.3	3.13029
$946$	−3115.22	−0.107066
$947$	36562.8	1.25463	0.627314	−	0.778766i	$-0.284154\pi$
$947$	36562.8	1.25463	0.627314	+	0.778766i	$0.284154\pi$
$948$	14990.4	0.513572
$949$	−38232.6	−1.30778
$950$	6644.07	0.226908
$951$	22048.9	0.751825
$952$	10131.3	0.344914
$953$	29813.1	1.01337	0.506684	−	0.862132i	$-0.330871\pi$
$953$	29813.1	1.01337	0.506684	+	0.862132i	$0.330871\pi$
$954$	−53631.8	−1.82012
$955$	−91088.7	−3.08645
$956$	5457.31	0.184625
$957$	3271.61	0.110508
$958$	13278.7	0.447825
$959$	−30540.5	−1.02837
$960$	−9721.35	−0.326828
$961$	−23072.2	−0.774470
$962$	−3024.56	−0.101368
$963$	−37011.3	−1.23850
$964$	−10588.9	−0.353782
$965$	−280.230	−0.00934811
$966$	−42123.8	−1.40301
$967$	30315.5	1.00815	0.504075	−	0.863660i	$-0.331833\pi$
$967$	30315.5	1.00815	0.504075	+	0.863660i	$0.331833\pi$
$968$	−10505.0	−0.348805
$969$	−8090.73	−0.268227
$970$	−37497.6	−1.24121
$971$	26455.6	0.874357	0.437178	−	0.899375i	$-0.355978\pi$
$971$	26455.6	0.874357	0.437178	+	0.899375i	$0.355978\pi$
$972$	−7101.41	−0.234339
$973$	7072.03	0.233010
$974$	−22177.6	−0.729584
$975$	98214.1	3.22602
$976$	10091.1	0.330950
$977$	30207.7	0.989183	0.494591	−	0.869126i	$-0.335318\pi$
$977$	30207.7	0.989183	0.494591	+	0.869126i	$0.335318\pi$
$978$	−45581.3	−1.49032
$979$	3894.56	0.127141
$980$	−23382.4	−0.762168
$981$	−99050.7	−3.22370
$982$	−26430.3	−0.858886
$983$	−5878.48	−0.190737	−0.0953685	−	0.995442i	$-0.530403\pi$
$983$	−5878.48	−0.190737	−0.0953685	+	0.995442i	$0.530403\pi$
$984$	1243.52	0.0402866
$985$	−65536.1	−2.11995
$986$	8564.33	0.276616
$987$	113921.	3.67392
$988$	4866.74	0.156712
$989$	33906.4	1.09015
$990$	7313.59	0.234789
$991$	−42532.3	−1.36335	−0.681676	−	0.731654i	$-0.738749\pi$
$991$	−42532.3	−1.36335	−0.681676	+	0.731654i	$0.738749\pi$
$992$	−2622.98	−0.0839513
$993$	−40121.2	−1.28218
$994$	−31073.5	−0.991542
$995$	−1267.67	−0.0403898
$996$	17325.3	0.551179
$997$	6320.28	0.200767	0.100384	−	0.994949i	$-0.467993\pi$
$997$	6320.28	0.200767	0.100384	+	0.994949i	$0.467993\pi$
$998$	821.281	0.0260493
$999$	−4753.91	−0.150558

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.4.a.c.1.2	✓	2
3.2	odd	2		342.4.a.h.1.2		2
4.3	odd	2		304.4.a.c.1.1		2
5.2	odd	4		950.4.b.i.799.3		4
5.3	odd	4		950.4.b.i.799.2		4
5.4	even	2		950.4.a.e.1.1		2
7.6	odd	2		1862.4.a.e.1.1		2
8.3	odd	2		1216.4.a.p.1.2		2
8.5	even	2		1216.4.a.g.1.1		2
19.18	odd	2		722.4.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.a.c.1.2	✓	2	1.1	even	1	trivial
304.4.a.c.1.1		2	4.3	odd	2
342.4.a.h.1.2		2	3.2	odd	2
722.4.a.f.1.1		2	19.18	odd	2
950.4.a.e.1.1		2	5.4	even	2
950.4.b.i.799.2		4	5.3	odd	4
950.4.b.i.799.3		4	5.2	odd	4
1216.4.a.g.1.1		2	8.5	even	2
1216.4.a.p.1.2		2	8.3	odd	2
1862.4.a.e.1.1		2	7.6	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:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	38.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +8.77200 q^{3} +4.00000 q^{4} -17.3160 q^{5} +17.5440 q^{6} -26.0880 q^{7} +8.00000 q^{8} +49.9480 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +8.77200 q^{3} +4.00000 q^{4} -17.3160 q^{5} +17.5440 q^{6} -26.0880 q^{7} +8.00000 q^{8} +49.9480 q^{9} -34.6320 q^{10} -4.22800 q^{11} +35.0880 q^{12} +64.0360 q^{13} -52.1760 q^{14} -151.896 q^{15} +16.0000 q^{16} -48.5440 q^{17} +99.8960 q^{18} +19.0000 q^{19} -69.2640 q^{20} -228.844 q^{21} -8.45600 q^{22} +92.0360 q^{23} +70.1760 q^{24} +174.844 q^{25} +128.072 q^{26} +201.300 q^{27} -104.352 q^{28} -88.2120 q^{29} -303.792 q^{30} -81.9681 q^{31} +32.0000 q^{32} -37.0880 q^{33} -97.0880 q^{34} +451.740 q^{35} +199.792 q^{36} -23.6161 q^{37} +38.0000 q^{38} +561.724 q^{39} -138.528 q^{40} +17.7200 q^{41} -457.688 q^{42} +368.404 q^{43} -16.9120 q^{44} -864.900 q^{45} +184.072 q^{46} -497.812 q^{47} +140.352 q^{48} +337.584 q^{49} +349.688 q^{50} -425.828 q^{51} +256.144 q^{52} -536.876 q^{53} +402.600 q^{54} +73.2120 q^{55} -208.704 q^{56} +166.668 q^{57} -176.424 q^{58} -36.7000 q^{59} -607.584 q^{60} +630.692 q^{61} -163.936 q^{62} -1303.04 q^{63} +64.0000 q^{64} -1108.85 q^{65} -74.1760 q^{66} +282.556 q^{67} -194.176 q^{68} +807.340 q^{69} +903.480 q^{70} +595.552 q^{71} +399.584 q^{72} -597.048 q^{73} -47.2321 q^{74} +1533.73 q^{75} +76.0000 q^{76} +110.300 q^{77} +1123.45 q^{78} +427.224 q^{79} -277.056 q^{80} +417.208 q^{81} +35.4400 q^{82} +493.768 q^{83} -915.376 q^{84} +840.588 q^{85} +736.808 q^{86} -773.796 q^{87} -33.8240 q^{88} -921.136 q^{89} -1729.80 q^{90} -1670.57 q^{91} +368.144 q^{92} -719.024 q^{93} -995.624 q^{94} -329.004 q^{95} +280.704 q^{96} +1082.74 q^{97} +675.168 q^{98} -211.180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 9 q^{3} + 8 q^{4} - 9 q^{5} + 18 q^{6} - 18 q^{7} + 16 q^{8} + 23 q^{9}+O(q^{10}) \) 2 * q + 4 * q^2 + 9 * q^3 + 8 * q^4 - 9 * q^5 + 18 * q^6 - 18 * q^7 + 16 * q^8 + 23 * q^9	\( 2 q + 4 q^{2} + 9 q^{3} + 8 q^{4} - 9 q^{5} + 18 q^{6} - 18 q^{7} + 16 q^{8} + 23 q^{9} - 18 q^{10} - 17 q^{11} + 36 q^{12} + 17 q^{13} - 36 q^{14} - 150 q^{15} + 32 q^{16} - 80 q^{17} + 46 q^{18} + 38 q^{19} - 36 q^{20} - 227 q^{21} - 34 q^{22} + 73 q^{23} + 72 q^{24} + 119 q^{25} + 34 q^{26} + 189 q^{27} - 72 q^{28} + 3 q^{29} - 300 q^{30} + 212 q^{31} + 64 q^{32} - 40 q^{33} - 160 q^{34} + 519 q^{35} + 92 q^{36} + 192 q^{37} + 76 q^{38} + 551 q^{39} - 72 q^{40} - 50 q^{41} - 454 q^{42} + 677 q^{43} - 68 q^{44} - 1089 q^{45} + 146 q^{46} - 389 q^{47} + 144 q^{48} + 60 q^{49} + 238 q^{50} - 433 q^{51} + 68 q^{52} - 1219 q^{53} + 378 q^{54} - 33 q^{55} - 144 q^{56} + 171 q^{57} + 6 q^{58} - 287 q^{59} - 600 q^{60} + 313 q^{61} + 424 q^{62} - 1521 q^{63} + 128 q^{64} - 1500 q^{65} - 80 q^{66} + 1223 q^{67} - 320 q^{68} + 803 q^{69} + 1038 q^{70} + 200 q^{71} + 184 q^{72} + 378 q^{73} + 384 q^{74} + 1521 q^{75} + 152 q^{76} + 7 q^{77} + 1102 q^{78} + 1350 q^{79} - 144 q^{80} + 1142 q^{81} - 100 q^{82} - 670 q^{83} - 908 q^{84} + 579 q^{85} + 1354 q^{86} - 753 q^{87} - 136 q^{88} - 236 q^{89} - 2178 q^{90} - 2051 q^{91} + 292 q^{92} - 652 q^{93} - 778 q^{94} - 171 q^{95} + 288 q^{96} + 1294 q^{97} + 120 q^{98} + 133 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 9 * q^3 + 8 * q^4 - 9 * q^5 + 18 * q^6 - 18 * q^7 + 16 * q^8 + 23 * q^9 - 18 * q^10 - 17 * q^11 + 36 * q^12 + 17 * q^13 - 36 * q^14 - 150 * q^15 + 32 * q^16 - 80 * q^17 + 46 * q^18 + 38 * q^19 - 36 * q^20 - 227 * q^21 - 34 * q^22 + 73 * q^23 + 72 * q^24 + 119 * q^25 + 34 * q^26 + 189 * q^27 - 72 * q^28 + 3 * q^29 - 300 * q^30 + 212 * q^31 + 64 * q^32 - 40 * q^33 - 160 * q^34 + 519 * q^35 + 92 * q^36 + 192 * q^37 + 76 * q^38 + 551 * q^39 - 72 * q^40 - 50 * q^41 - 454 * q^42 + 677 * q^43 - 68 * q^44 - 1089 * q^45 + 146 * q^46 - 389 * q^47 + 144 * q^48 + 60 * q^49 + 238 * q^50 - 433 * q^51 + 68 * q^52 - 1219 * q^53 + 378 * q^54 - 33 * q^55 - 144 * q^56 + 171 * q^57 + 6 * q^58 - 287 * q^59 - 600 * q^60 + 313 * q^61 + 424 * q^62 - 1521 * q^63 + 128 * q^64 - 1500 * q^65 - 80 * q^66 + 1223 * q^67 - 320 * q^68 + 803 * q^69 + 1038 * q^70 + 200 * q^71 + 184 * q^72 + 378 * q^73 + 384 * q^74 + 1521 * q^75 + 152 * q^76 + 7 * q^77 + 1102 * q^78 + 1350 * q^79 - 144 * q^80 + 1142 * q^81 - 100 * q^82 - 670 * q^83 - 908 * q^84 + 579 * q^85 + 1354 * q^86 - 753 * q^87 - 136 * q^88 - 236 * q^89 - 2178 * q^90 - 2051 * q^91 + 292 * q^92 - 652 * q^93 - 778 * q^94 - 171 * q^95 + 288 * q^96 + 1294 * q^97 + 120 * q^98 + 133 * q^99

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