LMFDB - Embedded newform 507.4.a.m.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, 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("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.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:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 25x^{2} + 24x + 78 $ x^4 - 2x^3 - 25x^2 + 24*x + 78
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$5.33039$ of defining polynomial
Character	$\chi$	$=$	507.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.33039 q^{2} -3.00000 q^{3} +20.4131 q^{4} -16.4131 q^{5} -15.9912 q^{6} +9.67968 q^{7} +66.1667 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+5.33039 q^{2} -3.00000 q^{3} +20.4131 q^{4} -16.4131 q^{5} -15.9912 q^{6} +9.67968 q^{7} +66.1667 q^{8} +9.00000 q^{9} -87.4882 q^{10} +27.5882 q^{11} -61.2393 q^{12} +51.5965 q^{14} +49.2393 q^{15} +189.390 q^{16} +107.928 q^{17} +47.9735 q^{18} -2.24723 q^{19} -335.042 q^{20} -29.0391 q^{21} +147.056 q^{22} +41.8090 q^{23} -198.500 q^{24} +144.390 q^{25} -27.0000 q^{27} +197.592 q^{28} +61.6213 q^{29} +262.465 q^{30} +191.932 q^{31} +480.187 q^{32} -82.7645 q^{33} +575.300 q^{34} -158.874 q^{35} +183.718 q^{36} +98.4236 q^{37} -11.9786 q^{38} -1086.00 q^{40} -30.7452 q^{41} -154.790 q^{42} +238.325 q^{43} +563.160 q^{44} -147.718 q^{45} +222.858 q^{46} -511.482 q^{47} -568.169 q^{48} -249.304 q^{49} +769.653 q^{50} -323.785 q^{51} +492.825 q^{53} -143.921 q^{54} -452.807 q^{55} +640.472 q^{56} +6.74170 q^{57} +328.466 q^{58} +484.179 q^{59} +1005.13 q^{60} -444.021 q^{61} +1023.07 q^{62} +87.1172 q^{63} +1044.47 q^{64} -441.167 q^{66} +190.114 q^{67} +2203.15 q^{68} -125.427 q^{69} -846.858 q^{70} +484.785 q^{71} +595.500 q^{72} -957.780 q^{73} +524.636 q^{74} -433.169 q^{75} -45.8729 q^{76} +267.045 q^{77} -375.216 q^{79} -3108.47 q^{80} +81.0000 q^{81} -163.884 q^{82} -715.765 q^{83} -592.777 q^{84} -1771.43 q^{85} +1270.37 q^{86} -184.864 q^{87} +1825.42 q^{88} -1038.15 q^{89} -787.394 q^{90} +853.451 q^{92} -575.796 q^{93} -2726.40 q^{94} +36.8840 q^{95} -1440.56 q^{96} +65.5636 q^{97} -1328.89 q^{98} +248.293 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 12 q^{3} + 22 q^{4} - 6 q^{5} - 6 q^{6} - 14 q^{7} + 54 q^{8} + 36 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 - 12 * q^3 + 22 * q^4 - 6 * q^5 - 6 * q^6 - 14 * q^7 + 54 * q^8 + 36 * q^9	\( 4 q + 2 q^{2} - 12 q^{3} + 22 q^{4} - 6 q^{5} - 6 q^{6} - 14 q^{7} + 54 q^{8} + 36 q^{9} - 62 q^{10} + 40 q^{11} - 66 q^{12} + 40 q^{14} + 18 q^{15} + 122 q^{16} + 98 q^{17} + 18 q^{18} + 124 q^{19} - 466 q^{20} + 42 q^{21} + 220 q^{22} + 104 q^{23} - 162 q^{24} - 58 q^{25} - 108 q^{27} - 144 q^{28} + 194 q^{29} + 186 q^{30} + 26 q^{31} + 654 q^{32} - 120 q^{33} + 1062 q^{34} + 88 q^{35} + 198 q^{36} + 102 q^{37} + 332 q^{38} - 998 q^{40} - 1054 q^{41} - 120 q^{42} + 450 q^{43} - 44 q^{44} - 54 q^{45} - 172 q^{46} - 96 q^{47} - 366 q^{48} + 1070 q^{49} + 996 q^{50} - 294 q^{51} + 262 q^{53} - 54 q^{54} + 204 q^{55} + 2164 q^{56} - 372 q^{57} + 722 q^{58} + 308 q^{59} + 1398 q^{60} - 928 q^{61} + 2780 q^{62} - 126 q^{63} + 1026 q^{64} - 660 q^{66} - 1134 q^{67} + 1786 q^{68} - 312 q^{69} - 2324 q^{70} + 1064 q^{71} + 486 q^{72} + 952 q^{73} + 1158 q^{74} + 174 q^{75} - 1708 q^{76} + 2508 q^{77} - 746 q^{79} - 2922 q^{80} + 324 q^{81} + 1734 q^{82} - 404 q^{83} + 432 q^{84} - 1394 q^{85} + 3168 q^{86} - 582 q^{87} + 3060 q^{88} + 1620 q^{89} - 558 q^{90} + 332 q^{92} - 78 q^{93} - 772 q^{94} + 2204 q^{95} - 1962 q^{96} + 2166 q^{97} - 1906 q^{98} + 360 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 12 * q^3 + 22 * q^4 - 6 * q^5 - 6 * q^6 - 14 * q^7 + 54 * q^8 + 36 * q^9 - 62 * q^10 + 40 * q^11 - 66 * q^12 + 40 * q^14 + 18 * q^15 + 122 * q^16 + 98 * q^17 + 18 * q^18 + 124 * q^19 - 466 * q^20 + 42 * q^21 + 220 * q^22 + 104 * q^23 - 162 * q^24 - 58 * q^25 - 108 * q^27 - 144 * q^28 + 194 * q^29 + 186 * q^30 + 26 * q^31 + 654 * q^32 - 120 * q^33 + 1062 * q^34 + 88 * q^35 + 198 * q^36 + 102 * q^37 + 332 * q^38 - 998 * q^40 - 1054 * q^41 - 120 * q^42 + 450 * q^43 - 44 * q^44 - 54 * q^45 - 172 * q^46 - 96 * q^47 - 366 * q^48 + 1070 * q^49 + 996 * q^50 - 294 * q^51 + 262 * q^53 - 54 * q^54 + 204 * q^55 + 2164 * q^56 - 372 * q^57 + 722 * q^58 + 308 * q^59 + 1398 * q^60 - 928 * q^61 + 2780 * q^62 - 126 * q^63 + 1026 * q^64 - 660 * q^66 - 1134 * q^67 + 1786 * q^68 - 312 * q^69 - 2324 * q^70 + 1064 * q^71 + 486 * q^72 + 952 * q^73 + 1158 * q^74 + 174 * q^75 - 1708 * q^76 + 2508 * q^77 - 746 * q^79 - 2922 * q^80 + 324 * q^81 + 1734 * q^82 - 404 * q^83 + 432 * q^84 - 1394 * q^85 + 3168 * q^86 - 582 * q^87 + 3060 * q^88 + 1620 * q^89 - 558 * q^90 + 332 * q^92 - 78 * q^93 - 772 * q^94 + 2204 * q^95 - 1962 * q^96 + 2166 * q^97 - 1906 * q^98 + 360 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	5.33039	1.88458	0.942289	−	0.334800i	$-0.108669\pi$
$2$	5.33039	1.88458	0.942289	+	0.334800i	$0.108669\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	20.4131	2.55164
$5$	−16.4131	−1.46803	−0.734016	−	0.679132i	$-0.762356\pi$
$5$	−16.4131	−1.46803	−0.734016	+	0.679132i	$0.762356\pi$
$6$	−15.9912	−1.08806
$7$	9.67968	0.522654	0.261327	−	0.965250i	$-0.415840\pi$
$7$	9.67968	0.522654	0.261327	+	0.965250i	$0.415840\pi$
$8$	66.1667	2.92418
$9$	9.00000	0.333333
$10$	−87.4882	−2.76662
$11$	27.5882	0.756195	0.378098	−	0.925766i	$-0.376578\pi$
$11$	27.5882	0.756195	0.378098	+	0.925766i	$0.376578\pi$
$12$	−61.2393	−1.47319
$13$	0	0
$13$	0	0
$14$	51.5965	0.984982
$15$	49.2393	0.847568
$16$	189.390	2.95921
$17$	107.928	1.53979	0.769895	−	0.638171i	$-0.220309\pi$
$17$	107.928	1.53979	0.769895	+	0.638171i	$0.220309\pi$
$18$	47.9735	0.628193
$19$	−2.24723	−0.0271342	−0.0135671	−	0.999908i	$-0.504319\pi$
$19$	−2.24723	−0.0271342	−0.0135671	+	0.999908i	$0.504319\pi$
$20$	−335.042	−3.74588
$21$	−29.0391	−0.301754
$22$	147.056	1.42511
$23$	41.8090	0.379034	0.189517	−	0.981877i	$-0.439308\pi$
$23$	41.8090	0.379034	0.189517	+	0.981877i	$0.439308\pi$
$24$	−198.500	−1.68828
$25$	144.390	1.15512
$26$	0	0
$27$	−27.0000	−0.192450
$28$	197.592	1.33362
$29$	61.6213	0.394579	0.197289	−	0.980345i	$-0.436786\pi$
$29$	61.6213	0.394579	0.197289	+	0.980345i	$0.436786\pi$
$30$	262.465	1.59731
$31$	191.932	1.11200	0.556000	−	0.831182i	$-0.312335\pi$
$31$	191.932	1.11200	0.556000	+	0.831182i	$0.312335\pi$
$32$	480.187	2.65269
$33$	−82.7645	−0.436589
$34$	575.300	2.90185
$35$	−158.874	−0.767272
$36$	183.718	0.850545
$37$	98.4236	0.437317	0.218659	−	0.975801i	$-0.429832\pi$
$37$	98.4236	0.437317	0.218659	+	0.975801i	$0.429832\pi$
$38$	−11.9786	−0.0511366
$39$	0	0
$40$	−1086.00	−4.29279
$41$	−30.7452	−0.117112	−0.0585561	−	0.998284i	$-0.518650\pi$
$41$	−30.7452	−0.117112	−0.0585561	+	0.998284i	$0.518650\pi$
$42$	−154.790	−0.568680
$43$	238.325	0.845216	0.422608	−	0.906313i	$-0.361115\pi$
$43$	238.325	0.845216	0.422608	+	0.906313i	$0.361115\pi$
$44$	563.160	1.92953
$45$	−147.718	−0.489344
$46$	222.858	0.714319
$47$	−511.482	−1.58739	−0.793695	−	0.608316i	$-0.791845\pi$
$47$	−511.482	−1.58739	−0.793695	+	0.608316i	$0.791845\pi$
$48$	−568.169	−1.70850
$49$	−249.304	−0.726833
$50$	769.653	2.17691
$51$	−323.785	−0.888998
$52$	0	0
$53$	492.825	1.27726	0.638630	−	0.769514i	$-0.279502\pi$
$53$	492.825	1.27726	0.638630	+	0.769514i	$0.279502\pi$
$54$	−143.921	−0.362687
$55$	−452.807	−1.11012
$56$	640.472	1.52833
$57$	6.74170	0.0156660
$58$	328.466	0.743615
$59$	484.179	1.06838	0.534192	−	0.845363i	$-0.320616\pi$
$59$	484.179	1.06838	0.534192	+	0.845363i	$0.320616\pi$
$60$	1005.13	2.16269
$61$	−444.021	−0.931985	−0.465993	−	0.884789i	$-0.654303\pi$
$61$	−444.021	−0.931985	−0.465993	+	0.884789i	$0.654303\pi$
$62$	1023.07	2.09565
$63$	87.1172	0.174218
$64$	1044.47	2.03998
$65$	0	0
$66$	−441.167	−0.822787
$67$	190.114	0.346658	0.173329	−	0.984864i	$-0.444548\pi$
$67$	190.114	0.346658	0.173329	+	0.984864i	$0.444548\pi$
$68$	2203.15	3.92898
$69$	−125.427	−0.218835
$70$	−846.858	−1.44598
$71$	484.785	0.810329	0.405164	−	0.914244i	$-0.367214\pi$
$71$	484.785	0.810329	0.405164	+	0.914244i	$0.367214\pi$
$72$	595.500	0.974727
$73$	−957.780	−1.53561	−0.767806	−	0.640683i	$-0.778651\pi$
$73$	−957.780	−1.53561	−0.767806	+	0.640683i	$0.778651\pi$
$74$	524.636	0.824159
$75$	−433.169	−0.666907
$76$	−45.8729	−0.0692367
$77$	267.045	0.395228
$78$	0	0
$79$	−375.216	−0.534368	−0.267184	−	0.963646i	$-0.586093\pi$
$79$	−375.216	−0.534368	−0.267184	+	0.963646i	$0.586093\pi$
$80$	−3108.47	−4.34422
$81$	81.0000	0.111111
$82$	−163.884	−0.220707
$83$	−715.765	−0.946571	−0.473286	−	0.880909i	$-0.656932\pi$
$83$	−715.765	−0.946571	−0.473286	+	0.880909i	$0.656932\pi$
$84$	−592.777	−0.769967
$85$	−1771.43	−2.26046
$86$	1270.37	1.59288
$87$	−184.864	−0.227810
$88$	1825.42	2.21125
$89$	−1038.15	−1.23645	−0.618224	−	0.786002i	$-0.712148\pi$
$89$	−1038.15	−1.23645	−0.618224	+	0.786002i	$0.712148\pi$
$90$	−787.394	−0.922207
$91$	0	0
$92$	853.451	0.967157
$93$	−575.796	−0.642013
$94$	−2726.40	−2.99156
$95$	36.8840	0.0398339
$96$	−1440.56	−1.53153
$97$	65.5636	0.0686286	0.0343143	−	0.999411i	$-0.489075\pi$
$97$	65.5636	0.0686286	0.0343143	+	0.999411i	$0.489075\pi$
$98$	−1328.89	−1.36977
$99$	248.293	0.252065
$100$	2947.44	2.94744
$101$	531.798	0.523920	0.261960	−	0.965079i	$-0.415631\pi$
$101$	531.798	0.523920	0.261960	+	0.965079i	$0.415631\pi$
$102$	−1725.90	−1.67539
$103$	−735.984	−0.704064	−0.352032	−	0.935988i	$-0.614509\pi$
$103$	−735.984	−0.704064	−0.352032	+	0.935988i	$0.614509\pi$
$104$	0	0
$105$	476.621	0.442985
$106$	2626.95	2.40710
$107$	−783.265	−0.707673	−0.353837	−	0.935307i	$-0.615123\pi$
$107$	−783.265	−0.707673	−0.353837	+	0.935307i	$0.615123\pi$
$108$	−551.153	−0.491063
$109$	−532.339	−0.467788	−0.233894	−	0.972262i	$-0.575147\pi$
$109$	−532.339	−0.467788	−0.233894	+	0.972262i	$0.575147\pi$
$110$	−2413.64	−2.09210
$111$	−295.271	−0.252485
$112$	1833.23	1.54664
$113$	−180.589	−0.150340	−0.0751699	−	0.997171i	$-0.523950\pi$
$113$	−180.589	−0.150340	−0.0751699	+	0.997171i	$0.523950\pi$
$114$	35.9359	0.0295237
$115$	−686.215	−0.556434
$116$	1257.88	1.00682
$117$	0	0
$118$	2580.86	2.01346
$119$	1044.71	0.804777
$120$	3258.00	2.47844
$121$	−569.893	−0.428169
$122$	−2366.81	−1.75640
$123$	92.2357	0.0676147
$124$	3917.92	2.83742
$125$	−318.242	−0.227716
$126$	464.369	0.328327
$127$	1431.63	1.00029	0.500146	−	0.865941i	$-0.333280\pi$
$127$	1431.63	1.00029	0.500146	+	0.865941i	$0.333280\pi$
$128$	1725.94	1.19182
$129$	−714.976	−0.487986
$130$	0	0
$131$	2067.32	1.37880	0.689400	−	0.724381i	$-0.257874\pi$
$131$	2067.32	1.37880	0.689400	+	0.724381i	$0.257874\pi$
$132$	−1689.48	−1.11402
$133$	−21.7525	−0.0141818
$134$	1013.38	0.653304
$135$	443.153	0.282523
$136$	7141.25	4.50262
$137$	−387.512	−0.241660	−0.120830	−	0.992673i	$-0.538556\pi$
$137$	−387.512	−0.241660	−0.120830	+	0.992673i	$0.538556\pi$
$138$	−668.575	−0.412412
$139$	752.568	0.459223	0.229611	−	0.973282i	$-0.426254\pi$
$139$	752.568	0.459223	0.229611	+	0.973282i	$0.426254\pi$
$140$	−3243.10	−1.95780
$141$	1534.45	0.916480
$142$	2584.09	1.52713
$143$	0	0
$144$	1704.51	0.986404
$145$	−1011.40	−0.579254
$146$	−5105.34	−2.89398
$147$	747.911	0.419637
$148$	2009.13	1.11587
$149$	−2636.72	−1.44972	−0.724862	−	0.688895i	$-0.758096\pi$
$149$	−2636.72	−1.44972	−0.724862	+	0.688895i	$0.758096\pi$
$150$	−2308.96	−1.25684
$151$	−3332.42	−1.79595	−0.897975	−	0.440046i	$-0.854962\pi$
$151$	−3332.42	−1.79595	−0.897975	+	0.440046i	$0.854962\pi$
$152$	−148.692	−0.0793454
$153$	971.354	0.513263
$154$	1423.45	0.744839
$155$	−3150.20	−1.63245
$156$	0	0
$157$	−1625.26	−0.826179	−0.413089	−	0.910690i	$-0.635550\pi$
$157$	−1625.26	−0.826179	−0.413089	+	0.910690i	$0.635550\pi$
$158$	−2000.05	−1.00706
$159$	−1478.48	−0.737426
$160$	−7881.36	−3.89423
$161$	404.698	0.198104
$162$	431.762	0.209398
$163$	−1835.37	−0.881944	−0.440972	−	0.897521i	$-0.645366\pi$
$163$	−1835.37	−0.881944	−0.440972	+	0.897521i	$0.645366\pi$
$164$	−627.605	−0.298828
$165$	1358.42	0.640927
$166$	−3815.31	−1.78389
$167$	1945.00	0.901248	0.450624	−	0.892714i	$-0.351202\pi$
$167$	1945.00	0.901248	0.450624	+	0.892714i	$0.351202\pi$
$168$	−1921.42	−0.882384
$169$	0	0
$170$	−9442.44	−4.26001
$171$	−20.2251	−0.00904474
$172$	4864.96	2.15668
$173$	2531.63	1.11258	0.556289	−	0.830989i	$-0.312225\pi$
$173$	2531.63	1.11258	0.556289	+	0.830989i	$0.312225\pi$
$174$	−985.397	−0.429326
$175$	1397.65	0.603726
$176$	5224.91	2.23774
$177$	−1452.54	−0.616832
$178$	−5533.76	−2.33018
$179$	4263.01	1.78007	0.890035	−	0.455892i	$-0.150680\pi$
$179$	4263.01	1.78007	0.890035	+	0.455892i	$0.150680\pi$
$180$	−3015.38	−1.24863
$181$	3944.61	1.61989	0.809946	−	0.586504i	$-0.199496\pi$
$181$	3944.61	1.61989	0.809946	+	0.586504i	$0.199496\pi$
$182$	0	0
$183$	1332.06	0.538082
$184$	2766.36	1.10836
$185$	−1615.43	−0.641995
$186$	−3069.22	−1.20992
$187$	2977.54	1.16438
$188$	−10440.9	−4.05044
$189$	−261.351	−0.100585
$190$	196.606	0.0750701
$191$	−214.109	−0.0811119	−0.0405559	−	0.999177i	$-0.512913\pi$
$191$	−214.109	−0.0811119	−0.0405559	+	0.999177i	$0.512913\pi$
$192$	−3133.42	−1.17779
$193$	−1207.19	−0.450234	−0.225117	−	0.974332i	$-0.572276\pi$
$193$	−1207.19	−0.450234	−0.225117	+	0.974332i	$0.572276\pi$
$194$	349.480	0.129336
$195$	0	0
$196$	−5089.06	−1.85461
$197$	−927.631	−0.335487	−0.167744	−	0.985831i	$-0.553648\pi$
$197$	−927.631	−0.335487	−0.167744	+	0.985831i	$0.553648\pi$
$198$	1323.50	0.475036
$199$	478.951	0.170613	0.0853064	−	0.996355i	$-0.472813\pi$
$199$	478.951	0.170613	0.0853064	+	0.996355i	$0.472813\pi$
$200$	9553.77	3.37777
$201$	−570.341	−0.200143
$202$	2834.69	0.987368
$203$	596.474	0.206228
$204$	−6609.44	−2.26840
$205$	504.624	0.171924
$206$	−3923.08	−1.32686
$207$	376.281	0.126345
$208$	0	0
$209$	−61.9970	−0.0205188
$210$	2540.58	0.834840
$211$	−1450.95	−0.473402	−0.236701	−	0.971583i	$-0.576066\pi$
$211$	−1450.95	−0.473402	−0.236701	+	0.971583i	$0.576066\pi$
$212$	10060.1	3.25910
$213$	−1454.35	−0.467843
$214$	−4175.11	−1.33367
$215$	−3911.66	−1.24080
$216$	−1786.50	−0.562759
$217$	1857.84	0.581191
$218$	−2837.58	−0.881583
$219$	2873.34	0.886586
$220$	−9243.19	−2.83262
$221$	0	0
$222$	−1573.91	−0.475828
$223$	−2059.79	−0.618536	−0.309268	−	0.950975i	$-0.600084\pi$
$223$	−2059.79	−0.618536	−0.309268	+	0.950975i	$0.600084\pi$
$224$	4648.06	1.38644
$225$	1299.51	0.385039
$226$	−962.612	−0.283327
$227$	4482.46	1.31062	0.655311	−	0.755359i	$-0.272537\pi$
$227$	4482.46	1.31062	0.655311	+	0.755359i	$0.272537\pi$
$228$	137.619	0.0399738
$229$	−1630.39	−0.470477	−0.235239	−	0.971938i	$-0.575587\pi$
$229$	−1630.39	−0.470477	−0.235239	+	0.971938i	$0.575587\pi$
$230$	−3657.80	−1.04864
$231$	−801.134	−0.228185
$232$	4077.27	1.15382
$233$	−1903.69	−0.535258	−0.267629	−	0.963522i	$-0.586240\pi$
$233$	−1903.69	−0.535258	−0.267629	+	0.963522i	$0.586240\pi$
$234$	0	0
$235$	8395.00	2.33034
$236$	9883.58	2.72613
$237$	1125.65	0.308517
$238$	5568.72	1.51667
$239$	3763.79	1.01866	0.509328	−	0.860572i	$-0.329894\pi$
$239$	3763.79	1.01866	0.509328	+	0.860572i	$0.329894\pi$
$240$	9325.40	2.50813
$241$	−3614.74	−0.966166	−0.483083	−	0.875575i	$-0.660483\pi$
$241$	−3614.74	−0.966166	−0.483083	+	0.875575i	$0.660483\pi$
$242$	−3037.75	−0.806918
$243$	−243.000	−0.0641500
$244$	−9063.85	−2.37809
$245$	4091.84	1.06701
$246$	491.652	0.127425
$247$	0	0
$248$	12699.5	3.25169
$249$	2147.30	0.546503
$250$	−1696.36	−0.429148
$251$	−5729.77	−1.44088	−0.720438	−	0.693520i	$-0.756059\pi$
$251$	−5729.77	−1.44088	−0.720438	+	0.693520i	$0.756059\pi$
$252$	1778.33	0.444541
$253$	1153.43	0.286624
$254$	7631.18	1.88513
$255$	5314.30	1.30508
$256$	844.191	0.206101
$257$	−5525.79	−1.34120	−0.670602	−	0.741818i	$-0.733964\pi$
$257$	−5525.79	−1.34120	−0.670602	+	0.741818i	$0.733964\pi$
$258$	−3811.10	−0.919647
$259$	952.709	0.228565
$260$	0	0
$261$	554.591	0.131526
$262$	11019.6	2.59846
$263$	5223.21	1.22463	0.612313	−	0.790615i	$-0.290239\pi$
$263$	5223.21	1.22463	0.612313	+	0.790615i	$0.290239\pi$
$264$	−5476.25	−1.27667
$265$	−8088.78	−1.87506
$266$	−115.949	−0.0267267
$267$	3114.46	0.713864
$268$	3880.81	0.884545
$269$	7203.88	1.63282	0.816410	−	0.577473i	$-0.195961\pi$
$269$	7203.88	1.63282	0.816410	+	0.577473i	$0.195961\pi$
$270$	2362.18	0.532436
$271$	−8577.69	−1.92272	−0.961360	−	0.275293i	$-0.911225\pi$
$271$	−8577.69	−1.92272	−0.961360	+	0.275293i	$0.911225\pi$
$272$	20440.5	4.55656
$273$	0	0
$274$	−2065.59	−0.455427
$275$	3983.44	0.873493
$276$	−2560.35	−0.558388
$277$	7169.19	1.55507	0.777536	−	0.628838i	$-0.216469\pi$
$277$	7169.19	1.55507	0.777536	+	0.628838i	$0.216469\pi$
$278$	4011.48	0.865442
$279$	1727.39	0.370667
$280$	−10512.1	−2.24364
$281$	849.157	0.180272	0.0901360	−	0.995929i	$-0.471270\pi$
$281$	849.157	0.180272	0.0901360	+	0.995929i	$0.471270\pi$
$282$	8179.20	1.72718
$283$	1115.37	0.234283	0.117141	−	0.993115i	$-0.462627\pi$
$283$	1115.37	0.234283	0.117141	+	0.993115i	$0.462627\pi$
$284$	9895.95	2.06766
$285$	−110.652	−0.0229981
$286$	0	0
$287$	−297.604	−0.0612091
$288$	4321.69	0.884229
$289$	6735.49	1.37095
$290$	−5391.14	−1.09165
$291$	−196.691	−0.0396227
$292$	−19551.2	−3.91832
$293$	−1863.53	−0.371565	−0.185782	−	0.982591i	$-0.559482\pi$
$293$	−1863.53	−0.371565	−0.185782	+	0.982591i	$0.559482\pi$
$294$	3986.66	0.790839
$295$	−7946.87	−1.56842
$296$	6512.36	1.27879
$297$	−744.880	−0.145530
$298$	−14054.8	−2.73212
$299$	0	0
$300$	−8842.31	−1.70170
$301$	2306.91	0.441755
$302$	−17763.1	−3.38461
$303$	−1595.39	−0.302485
$304$	−425.602	−0.0802959
$305$	7287.76	1.36818
$306$	5177.70	0.967285
$307$	−6387.50	−1.18747	−0.593736	−	0.804660i	$-0.702348\pi$
$307$	−6387.50	−1.18747	−0.593736	+	0.804660i	$0.702348\pi$
$308$	5451.21	1.00848
$309$	2207.95	0.406492
$310$	−16791.8	−3.07648
$311$	−3492.59	−0.636806	−0.318403	−	0.947955i	$-0.603147\pi$
$311$	−3492.59	−0.636806	−0.318403	+	0.947955i	$0.603147\pi$
$312$	0	0
$313$	−5912.01	−1.06762	−0.533812	−	0.845603i	$-0.679241\pi$
$313$	−5912.01	−1.06762	−0.533812	+	0.845603i	$0.679241\pi$
$314$	−8663.29	−1.55700
$315$	−1429.86	−0.255757
$316$	−7659.31	−1.36351
$317$	−1677.54	−0.297224	−0.148612	−	0.988896i	$-0.547481\pi$
$317$	−1677.54	−0.297224	−0.148612	+	0.988896i	$0.547481\pi$
$318$	−7880.86	−1.38974
$319$	1700.02	0.298378
$320$	−17143.0	−2.99476
$321$	2349.79	0.408575
$322$	2157.20	0.373342
$323$	−242.540	−0.0417810
$324$	1653.46	0.283515
$325$	0	0
$326$	−9783.22	−1.66209
$327$	1597.02	0.270077
$328$	−2034.31	−0.342457
$329$	−4950.98	−0.829655
$330$	7240.92	1.20788
$331$	2010.31	0.333827	0.166913	−	0.985972i	$-0.446620\pi$
$331$	2010.31	0.333827	0.166913	+	0.985972i	$0.446620\pi$
$332$	−14611.0	−2.41531
$333$	885.812	0.145772
$334$	10367.6	1.69847
$335$	−3120.35	−0.508905
$336$	−5499.69	−0.892955
$337$	7139.24	1.15400	0.577002	−	0.816743i	$-0.304222\pi$
$337$	7139.24	1.15400	0.577002	+	0.816743i	$0.304222\pi$
$338$	0	0
$339$	541.768	0.0867988
$340$	−36160.5	−5.76787
$341$	5295.05	0.840889
$342$	−107.808	−0.0170455
$343$	−5733.31	−0.902536
$344$	15769.2	2.47156
$345$	2058.64	0.321257
$346$	13494.6	2.09674
$347$	1.13990	0.000176349 0	8.81743e−5	−	1.00000i	$-0.499972\pi$
$347$	1.13990	0.000176349 0	8.81743e−5		1.00000i	$0.499972\pi$
$348$	−3773.64	−0.581289
$349$	12199.1	1.87107	0.935535	−	0.353235i	$-0.114918\pi$
$349$	12199.1	1.87107	0.935535	+	0.353235i	$0.114918\pi$
$350$	7450.00	1.13777
$351$	0	0
$352$	13247.5	2.00595
$353$	−10892.3	−1.64232	−0.821160	−	0.570698i	$-0.806673\pi$
$353$	−10892.3	−1.64232	−0.821160	+	0.570698i	$0.806673\pi$
$354$	−7742.59	−1.16247
$355$	−7956.81	−1.18959
$356$	−21191.9	−3.15497
$357$	−3134.13	−0.464638
$358$	22723.5	3.35468
$359$	−3525.78	−0.518339	−0.259169	−	0.965832i	$-0.583449\pi$
$359$	−3525.78	−0.518339	−0.259169	+	0.965832i	$0.583449\pi$
$360$	−9773.99	−1.43093
$361$	−6853.95	−0.999264
$362$	21026.3	3.05281
$363$	1709.68	0.247204
$364$	0	0
$365$	15720.1	2.25433
$366$	7100.42	1.01406
$367$	2383.75	0.339049	0.169525	−	0.985526i	$-0.445777\pi$
$367$	2383.75	0.339049	0.169525	+	0.985526i	$0.445777\pi$
$368$	7918.19	1.12164
$369$	−276.707	−0.0390374
$370$	−8610.90	−1.20989
$371$	4770.39	0.667564
$372$	−11753.8	−1.63818
$373$	−13282.2	−1.84377	−0.921885	−	0.387463i	$-0.873352\pi$
$373$	−13282.2	−1.84377	−0.921885	+	0.387463i	$0.873352\pi$
$374$	15871.5	2.19437
$375$	954.727	0.131472
$376$	−33843.1	−4.64181
$377$	0	0
$378$	−1393.11	−0.189560
$379$	−4436.73	−0.601318	−0.300659	−	0.953732i	$-0.597207\pi$
$379$	−4436.73	−0.601318	−0.300659	+	0.953732i	$0.597207\pi$
$380$	752.917	0.101642
$381$	−4294.90	−0.577519
$382$	−1141.28	−0.152862
$383$	−810.412	−0.108120	−0.0540602	−	0.998538i	$-0.517216\pi$
$383$	−810.412	−0.108120	−0.0540602	+	0.998538i	$0.517216\pi$
$384$	−5177.83	−0.688100
$385$	−4383.03	−0.580207
$386$	−6434.78	−0.848501
$387$	2144.93	0.281739
$388$	1338.35	0.175115
$389$	3463.79	0.451469	0.225734	−	0.974189i	$-0.427522\pi$
$389$	3463.79	0.451469	0.225734	+	0.974189i	$0.427522\pi$
$390$	0	0
$391$	4512.37	0.583633
$392$	−16495.6	−2.12539
$393$	−6201.96	−0.796050
$394$	−4944.64	−0.632252
$395$	6158.45	0.784469
$396$	5068.44	0.643178
$397$	425.405	0.0537796	0.0268898	−	0.999638i	$-0.491440\pi$
$397$	425.405	0.0537796	0.0268898	+	0.999638i	$0.491440\pi$
$398$	2553.00	0.321533
$399$	65.2575	0.00818787
$400$	27345.9	3.41823
$401$	−1186.85	−0.147801	−0.0739007	−	0.997266i	$-0.523545\pi$
$401$	−1186.85	−0.147801	−0.0739007	+	0.997266i	$0.523545\pi$
$402$	−3040.14	−0.377185
$403$	0	0
$404$	10855.6	1.33685
$405$	−1329.46	−0.163115
$406$	3179.44	0.388653
$407$	2715.33	0.330697
$408$	−21423.7	−2.59959
$409$	8007.42	0.968071	0.484036	−	0.875048i	$-0.339170\pi$
$409$	8007.42	0.968071	0.484036	+	0.875048i	$0.339170\pi$
$410$	2689.84	0.324005
$411$	1162.54	0.139522
$412$	−15023.7	−1.79652
$413$	4686.70	0.558395
$414$	2005.73	0.238106
$415$	11747.9	1.38960
$416$	0	0
$417$	−2257.70	−0.265132
$418$	−330.468	−0.0386692
$419$	6832.46	0.796629	0.398314	−	0.917249i	$-0.369595\pi$
$419$	6832.46	0.796629	0.398314	+	0.917249i	$0.369595\pi$
$420$	9729.30	1.13034
$421$	10739.6	1.24326	0.621632	−	0.783309i	$-0.286470\pi$
$421$	10739.6	1.24326	0.621632	+	0.783309i	$0.286470\pi$
$422$	−7734.16	−0.892163
$423$	−4603.34	−0.529130
$424$	32608.6	3.73494
$425$	15583.7	1.77864
$426$	−7752.28	−0.881688
$427$	−4297.99	−0.487106
$428$	−15988.9	−1.80573
$429$	0	0
$430$	−20850.7	−2.33839
$431$	5214.45	0.582763	0.291382	−	0.956607i	$-0.405885\pi$
$431$	5214.45	0.582763	0.291382	+	0.956607i	$0.405885\pi$
$432$	−5113.52	−0.569501
$433$	8642.24	0.959168	0.479584	−	0.877496i	$-0.340788\pi$
$433$	8642.24	0.959168	0.479584	+	0.877496i	$0.340788\pi$
$434$	9903.02	1.09530
$435$	3034.19	0.334432
$436$	−10866.7	−1.19362
$437$	−93.9545	−0.0102848
$438$	15316.0	1.67084
$439$	−13026.2	−1.41619	−0.708097	−	0.706116i	$-0.750446\pi$
$439$	−13026.2	−1.41619	−0.708097	+	0.706116i	$0.750446\pi$
$440$	−29960.7	−3.24619
$441$	−2243.73	−0.242278
$442$	0	0
$443$	−11533.0	−1.23690	−0.618450	−	0.785824i	$-0.712239\pi$
$443$	−11533.0	−1.23690	−0.618450	+	0.785824i	$0.712239\pi$
$444$	−6027.39	−0.644250
$445$	17039.3	1.81515
$446$	−10979.5	−1.16568
$447$	7910.17	0.836998
$448$	10110.2	1.06621
$449$	9882.75	1.03874	0.519372	−	0.854548i	$-0.326166\pi$
$449$	9882.75	1.03874	0.519372	+	0.854548i	$0.326166\pi$
$450$	6926.88	0.725636
$451$	−848.204	−0.0885596
$452$	−3686.38	−0.383613
$453$	9997.26	1.03689
$454$	23893.3	2.46997
$455$	0	0
$456$	446.075	0.0458101
$457$	−15628.1	−1.59967	−0.799836	−	0.600218i	$-0.795080\pi$
$457$	−15628.1	−1.59967	−0.799836	+	0.600218i	$0.795080\pi$
$458$	−8690.63	−0.886652
$459$	−2914.06	−0.296333
$460$	−14007.8	−1.41982
$461$	−7747.46	−0.782723	−0.391361	−	0.920237i	$-0.627996\pi$
$461$	−7747.46	−0.782723	−0.391361	+	0.920237i	$0.627996\pi$
$462$	−4270.36	−0.430033
$463$	−333.422	−0.0334675	−0.0167337	−	0.999860i	$-0.505327\pi$
$463$	−333.422	−0.0334675	−0.0167337	+	0.999860i	$0.505327\pi$
$464$	11670.4	1.16764
$465$	9450.59	0.942496
$466$	−10147.4	−1.00874
$467$	8198.33	0.812363	0.406182	−	0.913792i	$-0.366860\pi$
$467$	8198.33	0.812363	0.406182	+	0.913792i	$0.366860\pi$
$468$	0	0
$469$	1840.24	0.181182
$470$	44748.7	4.39171
$471$	4875.79	0.476995
$472$	32036.5	3.12415
$473$	6574.96	0.639148
$474$	6000.14	0.581425
$475$	−324.477	−0.0313432
$476$	21325.8	2.05350
$477$	4435.43	0.425753
$478$	20062.5	1.91974
$479$	−6435.88	−0.613910	−0.306955	−	0.951724i	$-0.599310\pi$
$479$	−6435.88	−0.613910	−0.306955	+	0.951724i	$0.599310\pi$
$480$	23644.1	2.24833
$481$	0	0
$482$	−19268.0	−1.82082
$483$	−1214.09	−0.114375
$484$	−11633.3	−1.09253
$485$	−1076.10	−0.100749
$486$	−1295.29	−0.120896
$487$	8095.37	0.753257	0.376629	−	0.926364i	$-0.377083\pi$
$487$	8095.37	0.753257	0.376629	+	0.926364i	$0.377083\pi$
$488$	−29379.4	−2.72529
$489$	5506.10	0.509191
$490$	21811.1	2.01087
$491$	5116.46	0.470270	0.235135	−	0.971963i	$-0.424447\pi$
$491$	5116.46	0.470270	0.235135	+	0.971963i	$0.424447\pi$
$492$	1882.82	0.172528
$493$	6650.67	0.607568
$494$	0	0
$495$	−4075.26	−0.370039
$496$	36349.9	3.29064
$497$	4692.56	0.423521
$498$	11445.9	1.02993
$499$	−18050.7	−1.61936	−0.809682	−	0.586870i	$-0.800360\pi$
$499$	−18050.7	−1.61936	−0.809682	+	0.586870i	$0.800360\pi$
$500$	−6496.31	−0.581047
$501$	−5834.99	−0.520336
$502$	−30541.9	−2.71544
$503$	10531.1	0.933512	0.466756	−	0.884386i	$-0.345423\pi$
$503$	10531.1	0.933512	0.466756	+	0.884386i	$0.345423\pi$
$504$	5764.25	0.509445
$505$	−8728.45	−0.769131
$506$	6148.25	0.540165
$507$	0	0
$508$	29224.1	2.55238
$509$	−1963.31	−0.170967	−0.0854834	−	0.996340i	$-0.527243\pi$
$509$	−1963.31	−0.170967	−0.0854834	+	0.996340i	$0.527243\pi$
$510$	28327.3	2.45952
$511$	−9271.00	−0.802593
$512$	−9307.69	−0.803409
$513$	60.6753	0.00522198
$514$	−29454.6	−2.52760
$515$	12079.8	1.03359
$516$	−14594.9	−1.24516
$517$	−14110.9	−1.20038
$518$	5078.31	0.430750
$519$	−7594.89	−0.642348
$520$	0	0
$521$	−7044.93	−0.592407	−0.296203	−	0.955125i	$-0.595721\pi$
$521$	−7044.93	−0.592407	−0.296203	+	0.955125i	$0.595721\pi$
$522$	2956.19	0.247872
$523$	−3213.29	−0.268657	−0.134328	−	0.990937i	$-0.542888\pi$
$523$	−3213.29	−0.268657	−0.134328	+	0.990937i	$0.542888\pi$
$524$	42200.4	3.51819
$525$	−4192.94	−0.348561
$526$	27841.7	2.30790
$527$	20714.9	1.71225
$528$	−15674.7	−1.29196
$529$	−10419.0	−0.856333
$530$	−43116.4	−3.53369
$531$	4357.61	0.356128
$532$	−444.036	−0.0361868
$533$	0	0
$534$	16601.3	1.34533
$535$	12855.8	1.03889
$536$	12579.2	1.01369
$537$	−12789.0	−1.02772
$538$	38399.5	3.07718
$539$	−6877.83	−0.549627
$540$	9046.13	0.720895
$541$	11251.4	0.894150	0.447075	−	0.894497i	$-0.352466\pi$
$541$	11251.4	0.894150	0.447075	+	0.894497i	$0.352466\pi$
$542$	−45722.4	−3.62352
$543$	−11833.8	−0.935245
$544$	51825.8	4.08458
$545$	8737.33	0.686727
$546$	0	0
$547$	1533.54	0.119871	0.0599353	−	0.998202i	$-0.480911\pi$
$547$	1533.54	0.119871	0.0599353	+	0.998202i	$0.480911\pi$
$548$	−7910.31	−0.616628
$549$	−3996.19	−0.310662
$550$	21233.3	1.64617
$551$	−138.477	−0.0107066
$552$	−8299.08	−0.639914
$553$	−3631.97	−0.279289
$554$	38214.6	2.93066
$555$	4846.30	0.370656
$556$	15362.2	1.17177
$557$	16845.7	1.28146	0.640731	−	0.767766i	$-0.278631\pi$
$557$	16845.7	1.28146	0.640731	+	0.767766i	$0.278631\pi$
$558$	9207.65	0.698550
$559$	0	0
$560$	−30089.0	−2.27052
$561$	−8932.62	−0.672256
$562$	4526.34	0.339737
$563$	−20820.1	−1.55855	−0.779273	−	0.626685i	$-0.784411\pi$
$563$	−20820.1	−1.55855	−0.779273	+	0.626685i	$0.784411\pi$
$564$	31322.8	2.33852
$565$	2964.03	0.220704
$566$	5945.37	0.441524
$567$	784.054	0.0580726
$568$	32076.6	2.36955
$569$	23636.6	1.74147	0.870735	−	0.491752i	$-0.163643\pi$
$569$	23636.6	1.74147	0.870735	+	0.491752i	$0.163643\pi$
$570$	−589.819	−0.0433417
$571$	−26955.1	−1.97554	−0.987771	−	0.155913i	$-0.950168\pi$
$571$	−26955.1	−1.97554	−0.987771	+	0.155913i	$0.950168\pi$
$572$	0	0
$573$	642.326	0.0468300
$574$	−1586.35	−0.115353
$575$	6036.78	0.437828
$576$	9400.25	0.679995
$577$	23499.8	1.69551	0.847755	−	0.530388i	$-0.177954\pi$
$577$	23499.8	1.69551	0.847755	+	0.530388i	$0.177954\pi$
$578$	35902.8	2.58367
$579$	3621.56	0.259943
$580$	−20645.7	−1.47805
$581$	−6928.38	−0.494729
$582$	−1048.44	−0.0746721
$583$	13596.1	0.965857
$584$	−63373.1	−4.49040
$585$	0	0
$586$	−9933.33	−0.700243
$587$	4637.50	0.326082	0.163041	−	0.986619i	$-0.447870\pi$
$587$	4637.50	0.326082	0.163041	+	0.986619i	$0.447870\pi$
$588$	15267.2	1.07076
$589$	−431.316	−0.0301733
$590$	−42359.9	−2.95582
$591$	2782.89	0.193694
$592$	18640.4	1.29411
$593$	12633.5	0.874869	0.437434	−	0.899250i	$-0.355887\pi$
$593$	12633.5	0.874869	0.437434	+	0.899250i	$0.355887\pi$
$594$	−3970.51	−0.274262
$595$	−17146.9	−1.18144
$596$	−53823.7	−3.69917
$597$	−1436.85	−0.0985033
$598$	0	0
$599$	−18757.1	−1.27946	−0.639730	−	0.768600i	$-0.720954\pi$
$599$	−18757.1	−1.27946	−0.639730	+	0.768600i	$0.720954\pi$
$600$	−28661.3	−1.95016
$601$	−3632.98	−0.246576	−0.123288	−	0.992371i	$-0.539344\pi$
$601$	−3632.98	−0.246576	−0.123288	+	0.992371i	$0.539344\pi$
$602$	12296.8	0.832522
$603$	1711.02	0.115553
$604$	−68025.0	−4.58261
$605$	9353.71	0.628566
$606$	−8504.08	−0.570057
$607$	−12700.0	−0.849219	−0.424610	−	0.905377i	$-0.639589\pi$
$607$	−12700.0	−0.849219	−0.424610	+	0.905377i	$0.639589\pi$
$608$	−1079.09	−0.0719786
$609$	−1789.42	−0.119066
$610$	38846.6	2.57845
$611$	0	0
$612$	19828.3	1.30966
$613$	21640.1	1.42584	0.712918	−	0.701248i	$-0.247373\pi$
$613$	21640.1	1.42584	0.712918	+	0.701248i	$0.247373\pi$
$614$	−34047.9	−2.23788
$615$	−1513.87	−0.0992605
$616$	17669.5	1.15572
$617$	−16541.7	−1.07933	−0.539663	−	0.841881i	$-0.681448\pi$
$617$	−16541.7	−1.07933	−0.539663	+	0.841881i	$0.681448\pi$
$618$	11769.2	0.766066
$619$	−21138.9	−1.37261	−0.686303	−	0.727316i	$-0.740767\pi$
$619$	−21138.9	−1.37261	−0.686303	+	0.727316i	$0.740767\pi$
$620$	−64305.2	−4.16542
$621$	−1128.84	−0.0729451
$622$	−18616.9	−1.20011
$623$	−10049.0	−0.646235
$624$	0	0
$625$	−12825.4	−0.820823
$626$	−31513.3	−2.01202
$627$	185.991	0.0118465
$628$	−33176.6	−2.10811
$629$	10622.7	0.673377
$630$	−7621.73	−0.481995
$631$	5489.80	0.346348	0.173174	−	0.984891i	$-0.444598\pi$
$631$	5489.80	0.346348	0.173174	+	0.984891i	$0.444598\pi$
$632$	−24826.8	−1.56259
$633$	4352.86	0.273319
$634$	−8941.94	−0.560141
$635$	−23497.5	−1.46846
$636$	−30180.3	−1.88164
$637$	0	0
$638$	9061.76	0.562318
$639$	4363.06	0.270110
$640$	−28328.1	−1.74963
$641$	4297.04	0.264778	0.132389	−	0.991198i	$-0.457735\pi$
$641$	4297.04	0.264778	0.132389	+	0.991198i	$0.457735\pi$
$642$	12525.3	0.769993
$643$	25696.9	1.57603	0.788016	−	0.615655i	$-0.211108\pi$
$643$	25696.9	1.57603	0.788016	+	0.615655i	$0.211108\pi$
$644$	8261.14	0.505488
$645$	11735.0	0.716378
$646$	−1292.83	−0.0787396
$647$	2174.98	0.132160	0.0660798	−	0.997814i	$-0.478951\pi$
$647$	2174.98	0.132160	0.0660798	+	0.997814i	$0.478951\pi$
$648$	5359.50	0.324909
$649$	13357.6	0.807907
$650$	0	0
$651$	−5573.52	−0.335551
$652$	−37465.5	−2.25040
$653$	−15454.5	−0.926160	−0.463080	−	0.886316i	$-0.653256\pi$
$653$	−15454.5	−0.926160	−0.463080	+	0.886316i	$0.653256\pi$
$654$	8512.73	0.508982
$655$	−33931.1	−2.02412
$656$	−5822.82	−0.346560
$657$	−8620.02	−0.511870
$658$	−26390.7	−1.56355
$659$	−3148.77	−0.186129	−0.0930643	−	0.995660i	$-0.529666\pi$
$659$	−3148.77	−0.186129	−0.0930643	+	0.995660i	$0.529666\pi$
$660$	27729.6	1.63541
$661$	−2099.70	−0.123553	−0.0617767	−	0.998090i	$-0.519677\pi$
$661$	−2099.70	−0.123553	−0.0617767	+	0.998090i	$0.519677\pi$
$662$	10715.7	0.629122
$663$	0	0
$664$	−47359.8	−2.76795
$665$	357.026	0.0208193
$666$	4721.73	0.274720
$667$	2576.32	0.149559
$668$	39703.4	2.29966
$669$	6179.36	0.357112
$670$	−16632.7	−0.959071
$671$	−12249.7	−0.704763
$672$	−13944.2	−0.800460
$673$	30970.8	1.77390	0.886950	−	0.461865i	$-0.152819\pi$
$673$	30970.8	1.77390	0.886950	+	0.461865i	$0.152819\pi$
$674$	38055.0	2.17481
$675$	−3898.52	−0.222302
$676$	0	0
$677$	14640.6	0.831141	0.415570	−	0.909561i	$-0.363582\pi$
$677$	14640.6	0.831141	0.415570	+	0.909561i	$0.363582\pi$
$678$	2887.83	0.163579
$679$	634.635	0.0358690
$680$	−117210.	−6.60999
$681$	−13447.4	−0.756688
$682$	28224.7	1.58472
$683$	−6685.83	−0.374563	−0.187281	−	0.982306i	$-0.559968\pi$
$683$	−6685.83	−0.374563	−0.187281	+	0.982306i	$0.559968\pi$
$684$	−412.856	−0.0230789
$685$	6360.27	0.354764
$686$	−30560.8	−1.70090
$687$	4891.18	0.271630
$688$	45136.3	2.50117
$689$	0	0
$690$	10973.4	0.605434
$691$	−30194.1	−1.66228	−0.831141	−	0.556062i	$-0.812312\pi$
$691$	−30194.1	−1.66228	−0.831141	+	0.556062i	$0.812312\pi$
$692$	51678.4	2.83890
$693$	2403.40	0.131743
$694$	6.07611	0.000332343 0
$695$	−12352.0	−0.674154
$696$	−12231.8	−0.666158
$697$	−3318.28	−0.180328
$698$	65026.0	3.52618
$699$	5711.08	0.309031
$700$	28530.3	1.54049
$701$	−30300.9	−1.63260	−0.816298	−	0.577631i	$-0.803977\pi$
$701$	−30300.9	−1.63260	−0.816298	+	0.577631i	$0.803977\pi$
$702$	0	0
$703$	−221.181	−0.0118663
$704$	28815.1	1.54263
$705$	−25185.0	−1.34542
$706$	−58060.3	−3.09508
$707$	5147.64	0.273829
$708$	−29650.8	−1.57393
$709$	26123.2	1.38375	0.691875	−	0.722017i	$-0.256785\pi$
$709$	26123.2	1.38375	0.691875	+	0.722017i	$0.256785\pi$
$710$	−42412.9	−2.24187
$711$	−3376.94	−0.178123
$712$	−68691.1	−3.61560
$713$	8024.48	0.421486
$714$	−16706.2	−0.875647
$715$	0	0
$716$	87021.3	4.54209
$717$	−11291.4	−0.588122
$718$	−18793.8	−0.976850
$719$	19325.7	1.00240	0.501200	−	0.865331i	$-0.332892\pi$
$719$	19325.7	1.00240	0.501200	+	0.865331i	$0.332892\pi$
$720$	−27976.2	−1.44807
$721$	−7124.09	−0.367982
$722$	−36534.2	−1.88319
$723$	10844.2	0.557816
$724$	80521.7	4.13338
$725$	8897.47	0.455784
$726$	9113.26	0.465875
$727$	26065.8	1.32975	0.664875	−	0.746954i	$-0.268485\pi$
$727$	26065.8	1.32975	0.664875	+	0.746954i	$0.268485\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	83794.4	4.24845
$731$	25722.0	1.30145
$732$	27191.5	1.37299
$733$	1055.45	0.0531843	0.0265921	−	0.999646i	$-0.491534\pi$
$733$	1055.45	0.0531843	0.0265921	+	0.999646i	$0.491534\pi$
$734$	12706.4	0.638965
$735$	−12275.5	−0.616041
$736$	20076.2	1.00546
$737$	5244.89	0.262141
$738$	−1474.96	−0.0735690
$739$	9410.40	0.468426	0.234213	−	0.972185i	$-0.424749\pi$
$739$	9410.40	0.468426	0.234213	+	0.972185i	$0.424749\pi$
$740$	−32976.0	−1.63814
$741$	0	0
$742$	25428.1	1.25808
$743$	−7523.70	−0.371491	−0.185746	−	0.982598i	$-0.559470\pi$
$743$	−7523.70	−0.371491	−0.185746	+	0.982598i	$0.559470\pi$
$744$	−38098.5	−1.87736
$745$	43276.8	2.12824
$746$	−70799.4	−3.47473
$747$	−6441.89	−0.315524
$748$	60780.8	2.97108
$749$	−7581.76	−0.369868
$750$	5089.07	0.247769
$751$	−12984.1	−0.630886	−0.315443	−	0.948945i	$-0.602153\pi$
$751$	−12984.1	−0.630886	−0.315443	+	0.948945i	$0.602153\pi$
$752$	−96869.3	−4.69742
$753$	17189.3	0.831890
$754$	0	0
$755$	54695.3	2.63651
$756$	−5334.99	−0.256656
$757$	−27934.6	−1.34122	−0.670609	−	0.741811i	$-0.733967\pi$
$757$	−27934.6	−1.34122	−0.670609	+	0.741811i	$0.733967\pi$
$758$	−23649.5	−1.13323
$759$	−3460.30	−0.165482
$760$	2440.49	0.116482
$761$	−15519.3	−0.739255	−0.369627	−	0.929180i	$-0.620515\pi$
$761$	−15519.3	−0.739255	−0.369627	+	0.929180i	$0.620515\pi$
$762$	−22893.5	−1.08838
$763$	−5152.88	−0.244491
$764$	−4370.62	−0.206968
$765$	−15942.9	−0.753487
$766$	−4319.82	−0.203761
$767$	0	0
$768$	−2532.57	−0.118993
$769$	12885.2	0.604228	0.302114	−	0.953272i	$-0.402308\pi$
$769$	12885.2	0.604228	0.302114	+	0.953272i	$0.402308\pi$
$770$	−23363.3	−1.09345
$771$	16577.4	0.774344
$772$	−24642.4	−1.14883
$773$	5892.04	0.274155	0.137078	−	0.990560i	$-0.456229\pi$
$773$	5892.04	0.274155	0.137078	+	0.990560i	$0.456229\pi$
$774$	11433.3	0.530958
$775$	27713.0	1.28449
$776$	4338.12	0.200682
$777$	−2858.13	−0.131962
$778$	18463.4	0.850828
$779$	69.0916	0.00317775
$780$	0	0
$781$	13374.3	0.612766
$782$	24052.7	1.09990
$783$	−1663.77	−0.0759367
$784$	−47215.5	−2.15085
$785$	26675.6	1.21286
$786$	−33058.9	−1.50022
$787$	−21020.4	−0.952091	−0.476045	−	0.879421i	$-0.657930\pi$
$787$	−21020.4	−0.952091	−0.476045	+	0.879421i	$0.657930\pi$
$788$	−18935.8	−0.856042
$789$	−15669.6	−0.707038
$790$	32827.0	1.47839
$791$	−1748.05	−0.0785757
$792$	16428.7	0.737084
$793$	0	0
$794$	2267.58	0.101352
$795$	24266.4	1.08256
$796$	9776.87	0.435342
$797$	31355.6	1.39356	0.696782	−	0.717283i	$-0.254614\pi$
$797$	31355.6	1.39356	0.696782	+	0.717283i	$0.254614\pi$
$798$	347.848	0.0154307
$799$	−55203.3	−2.44425
$800$	69334.0	3.06416
$801$	−9343.37	−0.412150
$802$	−6326.37	−0.278543
$803$	−26423.4	−1.16122
$804$	−11642.4	−0.510692
$805$	−6642.34	−0.290822
$806$	0	0
$807$	−21611.7	−0.942709
$808$	35187.3	1.53204
$809$	−18132.5	−0.788017	−0.394009	−	0.919107i	$-0.628912\pi$
$809$	−18132.5	−0.788017	−0.394009	+	0.919107i	$0.628912\pi$
$810$	−7086.55	−0.307402
$811$	−24755.3	−1.07186	−0.535928	−	0.844263i	$-0.680038\pi$
$811$	−24755.3	−1.07186	−0.535928	+	0.844263i	$0.680038\pi$
$812$	12175.9	0.526219
$813$	25733.1	1.11008
$814$	14473.8	0.623225
$815$	30124.0	1.29472
$816$	−61321.4	−2.63073
$817$	−535.572	−0.0229343
$818$	42682.7	1.82441
$819$	0	0
$820$	10300.9	0.438688
$821$	4082.65	0.173551	0.0867755	−	0.996228i	$-0.472344\pi$
$821$	4082.65	0.173551	0.0867755	+	0.996228i	$0.472344\pi$
$822$	6196.77	0.262941
$823$	−34327.0	−1.45391	−0.726954	−	0.686687i	$-0.759065\pi$
$823$	−34327.0	−1.45391	−0.726954	+	0.686687i	$0.759065\pi$
$824$	−48697.6	−2.05881
$825$	−11950.3	−0.504312
$826$	24981.9	1.05234
$827$	3228.87	0.135767	0.0678833	−	0.997693i	$-0.478375\pi$
$827$	3228.87	0.135767	0.0678833	+	0.997693i	$0.478375\pi$
$828$	7681.06	0.322386
$829$	−10452.4	−0.437908	−0.218954	−	0.975735i	$-0.570265\pi$
$829$	−10452.4	−0.437908	−0.218954	+	0.975735i	$0.570265\pi$
$830$	62621.0	2.61880
$831$	−21507.6	−0.897821
$832$	0	0
$833$	−26906.9	−1.11917
$834$	−12034.5	−0.499663
$835$	−31923.4	−1.32306
$836$	−1265.55	−0.0523564
$837$	−5182.16	−0.214004
$838$	36419.7	1.50131
$839$	−28289.0	−1.16406	−0.582028	−	0.813169i	$-0.697741\pi$
$839$	−28289.0	−1.16406	−0.582028	+	0.813169i	$0.697741\pi$
$840$	31536.4	1.29537
$841$	−20591.8	−0.844308
$842$	57246.1	2.34303
$843$	−2547.47	−0.104080
$844$	−29618.5	−1.20795
$845$	0	0
$846$	−24537.6	−0.997187
$847$	−5516.39	−0.223784
$848$	93335.9	3.77968
$849$	−3346.12	−0.135263
$850$	83067.2	3.35198
$851$	4114.99	0.165758
$852$	−29687.8	−1.19377
$853$	−26631.8	−1.06900	−0.534498	−	0.845170i	$-0.679499\pi$
$853$	−26631.8	−1.06900	−0.534498	+	0.845170i	$0.679499\pi$
$854$	−22910.0	−0.917989
$855$	331.956	0.0132780
$856$	−51826.0	−2.06936
$857$	11796.7	0.470209	0.235104	−	0.971970i	$-0.424457\pi$
$857$	11796.7	0.470209	0.235104	+	0.971970i	$0.424457\pi$
$858$	0	0
$859$	−22672.8	−0.900567	−0.450283	−	0.892886i	$-0.648677\pi$
$859$	−22672.8	−0.900567	−0.450283	+	0.892886i	$0.648677\pi$
$860$	−79849.0	−3.16608
$861$	892.812	0.0353391
$862$	27795.0	1.09826
$863$	21421.1	0.844940	0.422470	−	0.906377i	$-0.361163\pi$
$863$	21421.1	0.844940	0.422470	+	0.906377i	$0.361163\pi$
$864$	−12965.1	−0.510510
$865$	−41551.9	−1.63330
$866$	46066.6	1.80763
$867$	−20206.5	−0.791520
$868$	37924.3	1.48299
$869$	−10351.5	−0.404086
$870$	16173.4	0.630264
$871$	0	0
$872$	−35223.1	−1.36790
$873$	590.072	0.0228762
$874$	−500.814	−0.0193825
$875$	−3080.48	−0.119016
$876$	58653.7	2.26224
$877$	−5155.20	−0.198493	−0.0992466	−	0.995063i	$-0.531643\pi$
$877$	−5155.20	−0.198493	−0.0992466	+	0.995063i	$0.531643\pi$
$878$	−69435.0	−2.66893
$879$	5590.58	0.214523
$880$	−85756.9	−3.28507
$881$	23692.2	0.906027	0.453013	−	0.891504i	$-0.350349\pi$
$881$	23692.2	0.906027	0.453013	+	0.891504i	$0.350349\pi$
$882$	−11960.0	−0.456591
$883$	−14591.5	−0.556108	−0.278054	−	0.960565i	$-0.589689\pi$
$883$	−14591.5	−0.556108	−0.278054	+	0.960565i	$0.589689\pi$
$884$	0	0
$885$	23840.6	0.905529
$886$	−61475.2	−2.33104
$887$	−9722.26	−0.368029	−0.184014	−	0.982924i	$-0.558909\pi$
$887$	−9722.26	−0.368029	−0.184014	+	0.982924i	$0.558909\pi$
$888$	−19537.1	−0.738312
$889$	13857.8	0.522806
$890$	90826.1	3.42078
$891$	2234.64	0.0840217
$892$	−42046.6	−1.57828
$893$	1149.42	0.0430726
$894$	42164.3	1.57739
$895$	−69969.2	−2.61320
$896$	16706.6	0.622911
$897$	0	0
$898$	52678.9	1.95759
$899$	11827.1	0.438771
$900$	26526.9	0.982479
$901$	53189.7	1.96671
$902$	−4521.26	−0.166898
$903$	−6920.74	−0.255048
$904$	−11949.0	−0.439621
$905$	−64743.2	−2.37805
$906$	53289.3	1.95411
$907$	11799.0	0.431951	0.215975	−	0.976399i	$-0.430707\pi$
$907$	11799.0	0.431951	0.215975	+	0.976399i	$0.430707\pi$
$908$	91500.9	3.34423
$909$	4786.18	0.174640
$910$	0	0
$911$	−43012.4	−1.56429	−0.782143	−	0.623099i	$-0.785873\pi$
$911$	−43012.4	−1.56429	−0.782143	+	0.623099i	$0.785873\pi$
$912$	1276.81	0.0463589
$913$	−19746.6	−0.715793
$914$	−83303.8	−3.01471
$915$	−21863.3	−0.789921
$916$	−33281.3	−1.20049
$917$	20011.0	0.720635
$918$	−15533.1	−0.558462
$919$	−4951.41	−0.177728	−0.0888639	−	0.996044i	$-0.528324\pi$
$919$	−4951.41	−0.177728	−0.0888639	+	0.996044i	$0.528324\pi$
$920$	−45404.5	−1.62711
$921$	19162.5	0.685587
$922$	−41297.0	−1.47510
$923$	0	0
$924$	−16353.6	−0.582245
$925$	14211.3	0.505152
$926$	−1777.27	−0.0630721
$927$	−6623.85	−0.234688
$928$	29589.8	1.04669
$929$	−8934.86	−0.315547	−0.157774	−	0.987475i	$-0.550432\pi$
$929$	−8934.86	−0.315547	−0.157774	+	0.987475i	$0.550432\pi$
$930$	50375.4	1.77621
$931$	560.243	0.0197221
$932$	−38860.2	−1.36578
$933$	10477.8	0.367660
$934$	43700.3	1.53096
$935$	−48870.6	−1.70935
$936$	0	0
$937$	−13182.8	−0.459620	−0.229810	−	0.973235i	$-0.573811\pi$
$937$	−13182.8	−0.459620	−0.229810	+	0.973235i	$0.573811\pi$
$938$	9809.20	0.341452
$939$	17736.0	0.616393
$940$	171368.	5.94618
$941$	−21693.7	−0.751536	−0.375768	−	0.926714i	$-0.622621\pi$
$941$	−21693.7	−0.751536	−0.375768	+	0.926714i	$0.622621\pi$
$942$	25989.9	0.898934
$943$	−1285.43	−0.0443895
$944$	91698.4	3.16158
$945$	4289.59	0.147662
$946$	35047.1	1.20452
$947$	−49790.0	−1.70851	−0.854254	−	0.519856i	$-0.825986\pi$
$947$	−49790.0	−1.70851	−0.854254	+	0.519856i	$0.825986\pi$
$948$	22977.9	0.787224
$949$	0	0
$950$	−1729.59	−0.0590687
$951$	5032.61	0.171602
$952$	69125.0	2.35331
$953$	4217.93	0.143371	0.0716853	−	0.997427i	$-0.477162\pi$
$953$	4217.93	0.143371	0.0716853	+	0.997427i	$0.477162\pi$
$954$	23642.6	0.802365
$955$	3514.19	0.119075
$956$	76830.5	2.59924
$957$	−5100.05	−0.172269
$958$	−34305.8	−1.15696
$959$	−3750.99	−0.126304
$960$	51429.0	1.72903
$961$	7046.87	0.236544
$962$	0	0
$963$	−7049.38	−0.235891
$964$	−73788.1	−2.46530
$965$	19813.7	0.660958
$966$	−6471.60	−0.215549
$967$	−40927.9	−1.36107	−0.680534	−	0.732717i	$-0.738252\pi$
$967$	−40927.9	−1.36107	−0.680534	+	0.732717i	$0.738252\pi$
$968$	−37707.9	−1.25204
$969$	727.619	0.0241223
$970$	−5736.04	−0.189869
$971$	−17114.8	−0.565645	−0.282822	−	0.959172i	$-0.591271\pi$
$971$	−17114.8	−0.565645	−0.282822	+	0.959172i	$0.591271\pi$
$972$	−4960.38	−0.163688
$973$	7284.62	0.240015
$974$	43151.5	1.41957
$975$	0	0
$976$	−84093.0	−2.75794
$977$	−118.470	−0.00387940	−0.00193970	−	0.999998i	$-0.500617\pi$
$977$	−118.470	−0.00387940	−0.00193970	+	0.999998i	$0.500617\pi$
$978$	29349.7	0.959610
$979$	−28640.7	−0.934996
$980$	83527.2	2.72263
$981$	−4791.05	−0.155929
$982$	27272.8	0.886261
$983$	26002.8	0.843705	0.421852	−	0.906665i	$-0.361380\pi$
$983$	26002.8	0.843705	0.421852	+	0.906665i	$0.361380\pi$
$984$	6102.93	0.197718
$985$	15225.3	0.492506
$986$	35450.7	1.14501
$987$	14853.0	0.479002
$988$	0	0
$989$	9964.14	0.320365
$990$	−21722.8	−0.697368
$991$	−16062.0	−0.514860	−0.257430	−	0.966297i	$-0.582876\pi$
$991$	−16062.0	−0.514860	−0.257430	+	0.966297i	$0.582876\pi$
$992$	92163.3	2.94979
$993$	−6030.93	−0.192735
$994$	25013.2	0.798159
$995$	−7861.07	−0.250465
$996$	43832.9	1.39448
$997$	1361.54	0.0432503	0.0216251	−	0.999766i	$-0.493116\pi$
$997$	1361.54	0.0432503	0.0216251	+	0.999766i	$0.493116\pi$
$998$	−96217.5	−3.05182
$999$	−2657.44	−0.0841617

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.m.1.4		4
3.2	odd	2		1521.4.a.v.1.1		4
13.3	even	3		39.4.e.c.22.1	yes	8
13.5	odd	4		507.4.b.h.337.1		8
13.8	odd	4		507.4.b.h.337.8		8
13.9	even	3		39.4.e.c.16.1	✓	8
13.12	even	2		507.4.a.i.1.1		4
39.29	odd	6		117.4.g.e.100.4		8
39.35	odd	6		117.4.g.e.55.4		8
39.38	odd	2		1521.4.a.bb.1.4		4
52.3	odd	6		624.4.q.i.529.1		8
52.35	odd	6		624.4.q.i.289.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.e.c.16.1	✓	8	13.9	even	3
39.4.e.c.22.1	yes	8	13.3	even	3
117.4.g.e.55.4		8	39.35	odd	6
117.4.g.e.100.4		8	39.29	odd	6
507.4.a.i.1.1		4	13.12	even	2
507.4.a.m.1.4		4	1.1	even	1	trivial
507.4.b.h.337.1		8	13.5	odd	4
507.4.b.h.337.8		8	13.8	odd	4
624.4.q.i.289.1		8	52.35	odd	6
624.4.q.i.529.1		8	52.3	odd	6
1521.4.a.v.1.1		4	3.2	odd	2
1521.4.a.bb.1.4		4	39.38	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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

\(f(q)\)	\(=\)	\(q+5.33039 q^{2} -3.00000 q^{3} +20.4131 q^{4} -16.4131 q^{5} -15.9912 q^{6} +9.67968 q^{7} +66.1667 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+5.33039 q^{2} -3.00000 q^{3} +20.4131 q^{4} -16.4131 q^{5} -15.9912 q^{6} +9.67968 q^{7} +66.1667 q^{8} +9.00000 q^{9} -87.4882 q^{10} +27.5882 q^{11} -61.2393 q^{12} +51.5965 q^{14} +49.2393 q^{15} +189.390 q^{16} +107.928 q^{17} +47.9735 q^{18} -2.24723 q^{19} -335.042 q^{20} -29.0391 q^{21} +147.056 q^{22} +41.8090 q^{23} -198.500 q^{24} +144.390 q^{25} -27.0000 q^{27} +197.592 q^{28} +61.6213 q^{29} +262.465 q^{30} +191.932 q^{31} +480.187 q^{32} -82.7645 q^{33} +575.300 q^{34} -158.874 q^{35} +183.718 q^{36} +98.4236 q^{37} -11.9786 q^{38} -1086.00 q^{40} -30.7452 q^{41} -154.790 q^{42} +238.325 q^{43} +563.160 q^{44} -147.718 q^{45} +222.858 q^{46} -511.482 q^{47} -568.169 q^{48} -249.304 q^{49} +769.653 q^{50} -323.785 q^{51} +492.825 q^{53} -143.921 q^{54} -452.807 q^{55} +640.472 q^{56} +6.74170 q^{57} +328.466 q^{58} +484.179 q^{59} +1005.13 q^{60} -444.021 q^{61} +1023.07 q^{62} +87.1172 q^{63} +1044.47 q^{64} -441.167 q^{66} +190.114 q^{67} +2203.15 q^{68} -125.427 q^{69} -846.858 q^{70} +484.785 q^{71} +595.500 q^{72} -957.780 q^{73} +524.636 q^{74} -433.169 q^{75} -45.8729 q^{76} +267.045 q^{77} -375.216 q^{79} -3108.47 q^{80} +81.0000 q^{81} -163.884 q^{82} -715.765 q^{83} -592.777 q^{84} -1771.43 q^{85} +1270.37 q^{86} -184.864 q^{87} +1825.42 q^{88} -1038.15 q^{89} -787.394 q^{90} +853.451 q^{92} -575.796 q^{93} -2726.40 q^{94} +36.8840 q^{95} -1440.56 q^{96} +65.5636 q^{97} -1328.89 q^{98} +248.293 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 12 q^{3} + 22 q^{4} - 6 q^{5} - 6 q^{6} - 14 q^{7} + 54 q^{8} + 36 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 - 12 * q^3 + 22 * q^4 - 6 * q^5 - 6 * q^6 - 14 * q^7 + 54 * q^8 + 36 * q^9	\( 4 q + 2 q^{2} - 12 q^{3} + 22 q^{4} - 6 q^{5} - 6 q^{6} - 14 q^{7} + 54 q^{8} + 36 q^{9} - 62 q^{10} + 40 q^{11} - 66 q^{12} + 40 q^{14} + 18 q^{15} + 122 q^{16} + 98 q^{17} + 18 q^{18} + 124 q^{19} - 466 q^{20} + 42 q^{21} + 220 q^{22} + 104 q^{23} - 162 q^{24} - 58 q^{25} - 108 q^{27} - 144 q^{28} + 194 q^{29} + 186 q^{30} + 26 q^{31} + 654 q^{32} - 120 q^{33} + 1062 q^{34} + 88 q^{35} + 198 q^{36} + 102 q^{37} + 332 q^{38} - 998 q^{40} - 1054 q^{41} - 120 q^{42} + 450 q^{43} - 44 q^{44} - 54 q^{45} - 172 q^{46} - 96 q^{47} - 366 q^{48} + 1070 q^{49} + 996 q^{50} - 294 q^{51} + 262 q^{53} - 54 q^{54} + 204 q^{55} + 2164 q^{56} - 372 q^{57} + 722 q^{58} + 308 q^{59} + 1398 q^{60} - 928 q^{61} + 2780 q^{62} - 126 q^{63} + 1026 q^{64} - 660 q^{66} - 1134 q^{67} + 1786 q^{68} - 312 q^{69} - 2324 q^{70} + 1064 q^{71} + 486 q^{72} + 952 q^{73} + 1158 q^{74} + 174 q^{75} - 1708 q^{76} + 2508 q^{77} - 746 q^{79} - 2922 q^{80} + 324 q^{81} + 1734 q^{82} - 404 q^{83} + 432 q^{84} - 1394 q^{85} + 3168 q^{86} - 582 q^{87} + 3060 q^{88} + 1620 q^{89} - 558 q^{90} + 332 q^{92} - 78 q^{93} - 772 q^{94} + 2204 q^{95} - 1962 q^{96} + 2166 q^{97} - 1906 q^{98} + 360 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 12 * q^3 + 22 * q^4 - 6 * q^5 - 6 * q^6 - 14 * q^7 + 54 * q^8 + 36 * q^9 - 62 * q^10 + 40 * q^11 - 66 * q^12 + 40 * q^14 + 18 * q^15 + 122 * q^16 + 98 * q^17 + 18 * q^18 + 124 * q^19 - 466 * q^20 + 42 * q^21 + 220 * q^22 + 104 * q^23 - 162 * q^24 - 58 * q^25 - 108 * q^27 - 144 * q^28 + 194 * q^29 + 186 * q^30 + 26 * q^31 + 654 * q^32 - 120 * q^33 + 1062 * q^34 + 88 * q^35 + 198 * q^36 + 102 * q^37 + 332 * q^38 - 998 * q^40 - 1054 * q^41 - 120 * q^42 + 450 * q^43 - 44 * q^44 - 54 * q^45 - 172 * q^46 - 96 * q^47 - 366 * q^48 + 1070 * q^49 + 996 * q^50 - 294 * q^51 + 262 * q^53 - 54 * q^54 + 204 * q^55 + 2164 * q^56 - 372 * q^57 + 722 * q^58 + 308 * q^59 + 1398 * q^60 - 928 * q^61 + 2780 * q^62 - 126 * q^63 + 1026 * q^64 - 660 * q^66 - 1134 * q^67 + 1786 * q^68 - 312 * q^69 - 2324 * q^70 + 1064 * q^71 + 486 * q^72 + 952 * q^73 + 1158 * q^74 + 174 * q^75 - 1708 * q^76 + 2508 * q^77 - 746 * q^79 - 2922 * q^80 + 324 * q^81 + 1734 * q^82 - 404 * q^83 + 432 * q^84 - 1394 * q^85 + 3168 * q^86 - 582 * q^87 + 3060 * q^88 + 1620 * q^89 - 558 * q^90 + 332 * q^92 - 78 * q^93 - 772 * q^94 + 2204 * q^95 - 1962 * q^96 + 2166 * q^97 - 1906 * q^98 + 360 * q^99

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