LMFDB - Embedded newform 39.4.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [39,4,Mod(1,39)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(39, 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("39.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 39 = 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	39.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:	$2.30107449022$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3144.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 8 $ x^3 - x^2 - 16*x - 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.526440$ of defining polynomial
Character	$\chi$	$=$	39.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.52644 q^{2} +3.00000 q^{3} -5.66998 q^{4} +19.3400 q^{5} +4.57932 q^{6} +4.84136 q^{7} -20.8664 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+1.52644 q^{2} +3.00000 q^{3} -5.66998 q^{4} +19.3400 q^{5} +4.57932 q^{6} +4.84136 q^{7} -20.8664 q^{8} +9.00000 q^{9} +29.5213 q^{10} -61.0728 q^{11} -17.0099 q^{12} +13.0000 q^{13} +7.39005 q^{14} +58.0199 q^{15} +13.5085 q^{16} -41.7885 q^{17} +13.7380 q^{18} -107.561 q^{19} -109.657 q^{20} +14.5241 q^{21} -93.2239 q^{22} +28.5138 q^{23} -62.5992 q^{24} +249.034 q^{25} +19.8437 q^{26} +27.0000 q^{27} -27.4504 q^{28} -89.8886 q^{29} +88.5639 q^{30} +183.108 q^{31} +187.551 q^{32} -183.218 q^{33} -63.7876 q^{34} +93.6318 q^{35} -51.0298 q^{36} +418.029 q^{37} -164.185 q^{38} +39.0000 q^{39} -403.555 q^{40} -142.674 q^{41} +22.1701 q^{42} -71.0935 q^{43} +346.281 q^{44} +174.060 q^{45} +43.5246 q^{46} +323.711 q^{47} +40.5256 q^{48} -319.561 q^{49} +380.136 q^{50} -125.365 q^{51} -73.7098 q^{52} -25.1047 q^{53} +41.2139 q^{54} -1181.15 q^{55} -101.022 q^{56} -322.683 q^{57} -137.210 q^{58} -684.508 q^{59} -328.972 q^{60} +308.125 q^{61} +279.503 q^{62} +43.5723 q^{63} +178.217 q^{64} +251.420 q^{65} -279.672 q^{66} +672.808 q^{67} +236.940 q^{68} +85.5413 q^{69} +142.923 q^{70} -326.837 q^{71} -187.798 q^{72} +24.3058 q^{73} +638.095 q^{74} +747.103 q^{75} +609.869 q^{76} -295.675 q^{77} +59.5311 q^{78} +166.810 q^{79} +261.255 q^{80} +81.0000 q^{81} -217.783 q^{82} -201.093 q^{83} -82.3513 q^{84} -808.188 q^{85} -108.520 q^{86} -269.666 q^{87} +1274.37 q^{88} +108.834 q^{89} +265.692 q^{90} +62.9377 q^{91} -161.673 q^{92} +549.323 q^{93} +494.126 q^{94} -2080.23 q^{95} +562.654 q^{96} +1157.95 q^{97} -487.791 q^{98} -549.655 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 9 q^{3} + 10 q^{4} + 4 q^{5} + 6 q^{6} + 30 q^{7} - 6 q^{8} + 27 q^{9}+O(q^{10}) $ 3 * q + 2 * q^2 + 9 * q^3 + 10 * q^4 + 4 * q^5 + 6 * q^6 + 30 * q^7 - 6 * q^8 + 27 * q^9	\( 3 q + 2 q^{2} + 9 q^{3} + 10 q^{4} + 4 q^{5} + 6 q^{6} + 30 q^{7} - 6 q^{8} + 27 q^{9} - 4 q^{10} - 16 q^{11} + 30 q^{12} + 39 q^{13} - 176 q^{14} + 12 q^{15} - 110 q^{16} - 146 q^{17} + 18 q^{18} + 94 q^{19} - 244 q^{20} + 90 q^{21} - 56 q^{22} - 48 q^{23} - 18 q^{24} + 145 q^{25} + 26 q^{26} + 81 q^{27} + 80 q^{28} - 2 q^{29} - 12 q^{30} + 302 q^{31} + 154 q^{32} - 48 q^{33} + 164 q^{34} + 80 q^{35} + 90 q^{36} + 374 q^{37} + 312 q^{38} + 117 q^{39} - 516 q^{40} + 480 q^{41} - 528 q^{42} - 260 q^{43} + 712 q^{44} + 36 q^{45} - 1104 q^{46} - 24 q^{47} - 330 q^{48} + 447 q^{49} + 814 q^{50} - 438 q^{51} + 130 q^{52} - 678 q^{53} + 54 q^{54} - 1552 q^{55} + 96 q^{56} + 282 q^{57} - 628 q^{58} - 1788 q^{59} - 732 q^{60} + 230 q^{61} + 1952 q^{62} + 270 q^{63} - 750 q^{64} + 52 q^{65} - 168 q^{66} + 74 q^{67} - 460 q^{68} - 144 q^{69} + 1216 q^{70} - 948 q^{71} - 54 q^{72} - 222 q^{73} + 1724 q^{74} + 435 q^{75} + 2392 q^{76} + 112 q^{77} + 78 q^{78} - 24 q^{79} + 1100 q^{80} + 243 q^{81} + 564 q^{82} - 796 q^{83} + 240 q^{84} - 248 q^{85} + 1800 q^{86} - 6 q^{87} + 1608 q^{88} + 1436 q^{89} - 36 q^{90} + 390 q^{91} - 1296 q^{92} + 906 q^{93} - 1920 q^{94} - 4032 q^{95} + 462 q^{96} + 3242 q^{97} - 5070 q^{98} - 144 q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 + 9 * q^3 + 10 * q^4 + 4 * q^5 + 6 * q^6 + 30 * q^7 - 6 * q^8 + 27 * q^9 - 4 * q^10 - 16 * q^11 + 30 * q^12 + 39 * q^13 - 176 * q^14 + 12 * q^15 - 110 * q^16 - 146 * q^17 + 18 * q^18 + 94 * q^19 - 244 * q^20 + 90 * q^21 - 56 * q^22 - 48 * q^23 - 18 * q^24 + 145 * q^25 + 26 * q^26 + 81 * q^27 + 80 * q^28 - 2 * q^29 - 12 * q^30 + 302 * q^31 + 154 * q^32 - 48 * q^33 + 164 * q^34 + 80 * q^35 + 90 * q^36 + 374 * q^37 + 312 * q^38 + 117 * q^39 - 516 * q^40 + 480 * q^41 - 528 * q^42 - 260 * q^43 + 712 * q^44 + 36 * q^45 - 1104 * q^46 - 24 * q^47 - 330 * q^48 + 447 * q^49 + 814 * q^50 - 438 * q^51 + 130 * q^52 - 678 * q^53 + 54 * q^54 - 1552 * q^55 + 96 * q^56 + 282 * q^57 - 628 * q^58 - 1788 * q^59 - 732 * q^60 + 230 * q^61 + 1952 * q^62 + 270 * q^63 - 750 * q^64 + 52 * q^65 - 168 * q^66 + 74 * q^67 - 460 * q^68 - 144 * q^69 + 1216 * q^70 - 948 * q^71 - 54 * q^72 - 222 * q^73 + 1724 * q^74 + 435 * q^75 + 2392 * q^76 + 112 * q^77 + 78 * q^78 - 24 * q^79 + 1100 * q^80 + 243 * q^81 + 564 * q^82 - 796 * q^83 + 240 * q^84 - 248 * q^85 + 1800 * q^86 - 6 * q^87 + 1608 * q^88 + 1436 * q^89 - 36 * q^90 + 390 * q^91 - 1296 * q^92 + 906 * q^93 - 1920 * q^94 - 4032 * q^95 + 462 * q^96 + 3242 * q^97 - 5070 * q^98 - 144 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.52644	0.539678	0.269839	−	0.962905i	$-0.413030\pi$
$2$	1.52644	0.539678	0.269839	+	0.962905i	$0.413030\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−5.66998	−0.708748
$5$	19.3400	1.72982	0.864909	−	0.501928i	$-0.167376\pi$
$5$	19.3400	1.72982	0.864909	+	0.501928i	$0.167376\pi$
$6$	4.57932	0.311583
$7$	4.84136	0.261409	0.130704	−	0.991421i	$-0.458276\pi$
$7$	4.84136	0.261409	0.130704	+	0.991421i	$0.458276\pi$
$8$	−20.8664	−0.922173
$9$	9.00000	0.333333
$10$	29.5213	0.933545
$11$	−61.0728	−1.67401	−0.837006	−	0.547194i	$-0.815696\pi$
$11$	−61.0728	−1.67401	−0.837006	+	0.547194i	$0.815696\pi$
$12$	−17.0099	−0.409196
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	7.39005	0.141077
$15$	58.0199	0.998711
$16$	13.5085	0.211071
$17$	−41.7885	−0.596188	−0.298094	−	0.954537i	$-0.596351\pi$
$17$	−41.7885	−0.596188	−0.298094	+	0.954537i	$0.596351\pi$
$18$	13.7380	0.179893
$19$	−107.561	−1.29875	−0.649374	−	0.760469i	$-0.724969\pi$
$19$	−107.561	−1.29875	−0.649374	+	0.760469i	$0.724969\pi$
$20$	−109.657	−1.22601
$21$	14.5241	0.150925
$22$	−93.2239	−0.903427
$23$	28.5138	0.258502	0.129251	−	0.991612i	$-0.458743\pi$
$23$	28.5138	0.258502	0.129251	+	0.991612i	$0.458743\pi$
$24$	−62.5992	−0.532417
$25$	249.034	1.99227
$26$	19.8437	0.149680
$27$	27.0000	0.192450
$28$	−27.4504	−0.185273
$29$	−89.8886	−0.575583	−0.287791	−	0.957693i	$-0.592921\pi$
$29$	−89.8886	−0.575583	−0.287791	+	0.957693i	$0.592921\pi$
$30$	88.5639	0.538983
$31$	183.108	1.06087	0.530437	−	0.847724i	$-0.322028\pi$
$31$	183.108	1.06087	0.530437	+	0.847724i	$0.322028\pi$
$32$	187.551	1.03608
$33$	−183.218	−0.966491
$34$	−63.7876	−0.321750
$35$	93.6318	0.452190
$36$	−51.0298	−0.236249
$37$	418.029	1.85739	0.928696	−	0.370843i	$-0.120931\pi$
$37$	418.029	1.85739	0.928696	+	0.370843i	$0.120931\pi$
$38$	−164.185	−0.700905
$39$	39.0000	0.160128
$40$	−403.555	−1.59519
$41$	−142.674	−0.543460	−0.271730	−	0.962373i	$-0.587596\pi$
$41$	−142.674	−0.543460	−0.271730	+	0.962373i	$0.587596\pi$
$42$	22.1701	0.0814506
$43$	−71.0935	−0.252132	−0.126066	−	0.992022i	$-0.540235\pi$
$43$	−71.0935	−0.252132	−0.126066	+	0.992022i	$0.540235\pi$
$44$	346.281	1.18645
$45$	174.060	0.576606
$46$	43.5246	0.139508
$47$	323.711	1.00464	0.502321	−	0.864681i	$-0.332480\pi$
$47$	323.711	1.00464	0.502321	+	0.864681i	$0.332480\pi$
$48$	40.5256	0.121862
$49$	−319.561	−0.931665
$50$	380.136	1.07519
$51$	−125.365	−0.344209
$52$	−73.7098	−0.196571
$53$	−25.1047	−0.0650641	−0.0325321	−	0.999471i	$-0.510357\pi$
$53$	−25.1047	−0.0650641	−0.0325321	+	0.999471i	$0.510357\pi$
$54$	41.2139	0.103861
$55$	−1181.15	−2.89574
$56$	−101.022	−0.241064
$57$	−322.683	−0.749832
$58$	−137.210	−0.310629
$59$	−684.508	−1.51043	−0.755215	−	0.655477i	$-0.772467\pi$
$59$	−684.508	−1.51043	−0.755215	+	0.655477i	$0.772467\pi$
$60$	−328.972	−0.707834
$61$	308.125	0.646744	0.323372	−	0.946272i	$-0.395184\pi$
$61$	308.125	0.646744	0.323372	+	0.946272i	$0.395184\pi$
$62$	279.503	0.572531
$63$	43.5723	0.0871363
$64$	178.217	0.348081
$65$	251.420	0.479765
$66$	−279.672	−0.521594
$67$	672.808	1.22681	0.613407	−	0.789767i	$-0.289798\pi$
$67$	672.808	1.22681	0.613407	+	0.789767i	$0.289798\pi$
$68$	236.940	0.422547
$69$	85.5413	0.149246
$70$	142.923	0.244037
$71$	−326.837	−0.546315	−0.273158	−	0.961969i	$-0.588068\pi$
$71$	−326.837	−0.546315	−0.273158	+	0.961969i	$0.588068\pi$
$72$	−187.798	−0.307391
$73$	24.3058	0.0389695	0.0194847	−	0.999810i	$-0.493797\pi$
$73$	24.3058	0.0389695	0.0194847	+	0.999810i	$0.493797\pi$
$74$	638.095	1.00239
$75$	747.103	1.15024
$76$	609.869	0.920484
$77$	−295.675	−0.437602
$78$	59.5311	0.0864176
$79$	166.810	0.237565	0.118783	−	0.992920i	$-0.462101\pi$
$79$	166.810	0.237565	0.118783	+	0.992920i	$0.462101\pi$
$80$	261.255	0.365115
$81$	81.0000	0.111111
$82$	−217.783	−0.293294
$83$	−201.093	−0.265938	−0.132969	−	0.991120i	$-0.542451\pi$
$83$	−201.093	−0.265938	−0.132969	+	0.991120i	$0.542451\pi$
$84$	−82.3513	−0.106967
$85$	−808.188	−1.03130
$86$	−108.520	−0.136070
$87$	−269.666	−0.332313
$88$	1274.37	1.54373
$89$	108.834	0.129622	0.0648109	−	0.997898i	$-0.479356\pi$
$89$	108.834	0.129622	0.0648109	+	0.997898i	$0.479356\pi$
$90$	265.692	0.311182
$91$	62.9377	0.0725018
$92$	−161.673	−0.183212
$93$	549.323	0.612496
$94$	494.126	0.542183
$95$	−2080.23	−2.24660
$96$	562.654	0.598183
$97$	1157.95	1.21208	0.606041	−	0.795434i	$-0.292757\pi$
$97$	1157.95	1.21208	0.606041	+	0.795434i	$0.292757\pi$
$98$	−487.791	−0.502799
$99$	−549.655	−0.558004
$100$	−1412.02	−1.41202
$101$	1702.75	1.67752	0.838761	−	0.544500i	$-0.183281\pi$
$101$	1702.75	1.67752	0.838761	+	0.544500i	$0.183281\pi$
$102$	−191.363	−0.185762
$103$	−1455.14	−1.39203	−0.696015	−	0.718027i	$-0.745045\pi$
$103$	−1455.14	−1.39203	−0.696015	+	0.718027i	$0.745045\pi$
$104$	−271.263	−0.255765
$105$	280.895	0.261072
$106$	−38.3209	−0.0351137
$107$	−822.762	−0.743359	−0.371679	−	0.928361i	$-0.621218\pi$
$107$	−822.762	−0.743359	−0.371679	+	0.928361i	$0.621218\pi$
$108$	−153.090	−0.136399
$109$	457.264	0.401816	0.200908	−	0.979610i	$-0.435611\pi$
$109$	457.264	0.401816	0.200908	+	0.979610i	$0.435611\pi$
$110$	−1802.95	−1.56277
$111$	1254.09	1.07237
$112$	65.3998	0.0551759
$113$	−381.693	−0.317758	−0.158879	−	0.987298i	$-0.550788\pi$
$113$	−381.693	−0.317758	−0.158879	+	0.987298i	$0.550788\pi$
$114$	−492.556	−0.404668
$115$	551.456	0.447161
$116$	509.667	0.407943
$117$	117.000	0.0924500
$118$	−1044.86	−0.815146
$119$	−202.313	−0.155849
$120$	−1210.67	−0.920985
$121$	2398.88	1.80232
$122$	470.334	0.349033
$123$	−428.021	−0.313767
$124$	−1038.22	−0.751892
$125$	2398.82	1.71645
$126$	66.5104	0.0470255
$127$	−1129.09	−0.788905	−0.394452	−	0.918916i	$-0.629066\pi$
$127$	−1129.09	−0.788905	−0.394452	+	0.918916i	$0.629066\pi$
$128$	−1228.37	−0.848232
$129$	−213.281	−0.145568
$130$	383.777	0.258919
$131$	−852.761	−0.568749	−0.284374	−	0.958713i	$-0.591786\pi$
$131$	−852.761	−0.568749	−0.284374	+	0.958713i	$0.591786\pi$
$132$	1038.84	0.684999
$133$	−520.742	−0.339504
$134$	1027.00	0.662085
$135$	522.179	0.332904
$136$	871.975	0.549789
$137$	−488.903	−0.304889	−0.152445	−	0.988312i	$-0.548715\pi$
$137$	−488.903	−0.304889	−0.152445	+	0.988312i	$0.548715\pi$
$138$	130.574	0.0805447
$139$	407.123	0.248430	0.124215	−	0.992255i	$-0.460359\pi$
$139$	407.123	0.248430	0.124215	+	0.992255i	$0.460359\pi$
$140$	−530.890	−0.320489
$141$	971.134	0.580030
$142$	−498.897	−0.294834
$143$	−793.946	−0.464287
$144$	121.577	0.0703570
$145$	−1738.44	−0.995654
$146$	37.1013	0.0210310
$147$	−958.684	−0.537897
$148$	−2370.21	−1.31642
$149$	1717.63	0.944388	0.472194	−	0.881495i	$-0.343462\pi$
$149$	1717.63	0.944388	0.472194	+	0.881495i	$0.343462\pi$
$150$	1140.41	0.620759
$151$	1341.79	0.723133	0.361567	−	0.932346i	$-0.382242\pi$
$151$	1341.79	0.723133	0.361567	+	0.932346i	$0.382242\pi$
$152$	2244.41	1.19767
$153$	−376.096	−0.198729
$154$	−451.331	−0.236164
$155$	3541.30	1.83512
$156$	−221.129	−0.113490
$157$	−760.546	−0.386612	−0.193306	−	0.981138i	$-0.561921\pi$
$157$	−760.546	−0.386612	−0.193306	+	0.981138i	$0.561921\pi$
$158$	254.626	0.128209
$159$	−75.3142	−0.0375648
$160$	3627.23	1.79224
$161$	138.046	0.0675746
$162$	123.642	0.0599642
$163$	2712.09	1.30323	0.651616	−	0.758549i	$-0.274091\pi$
$163$	2712.09	1.30323	0.651616	+	0.758549i	$0.274091\pi$
$164$	808.957	0.385176
$165$	−3543.44	−1.67185
$166$	−306.957	−0.143521
$167$	1551.69	0.719004	0.359502	−	0.933144i	$-0.382947\pi$
$167$	1551.69	0.719004	0.359502	+	0.933144i	$0.382947\pi$
$168$	−303.065	−0.139179
$169$	169.000	0.0769231
$170$	−1233.65	−0.556568
$171$	−968.050	−0.432916
$172$	403.099	0.178698
$173$	−3970.26	−1.74482	−0.872409	−	0.488777i	$-0.837443\pi$
$173$	−3970.26	−1.74482	−0.872409	+	0.488777i	$0.837443\pi$
$174$	−411.629	−0.179342
$175$	1205.66	0.520798
$176$	−825.004	−0.353335
$177$	−2053.52	−0.872047
$178$	166.128	0.0699540
$179$	−2690.95	−1.12364	−0.561818	−	0.827261i	$-0.689898\pi$
$179$	−2690.95	−1.12364	−0.561818	+	0.827261i	$0.689898\pi$
$180$	−986.915	−0.408668
$181$	−4371.10	−1.79503	−0.897517	−	0.440980i	$-0.854631\pi$
$181$	−4371.10	−1.79503	−0.897517	+	0.440980i	$0.854631\pi$
$182$	96.0706	0.0391276
$183$	924.375	0.373398
$184$	−594.980	−0.238383
$185$	8084.66	3.21295
$186$	838.508	0.330551
$187$	2552.14	0.998026
$188$	−1835.44	−0.712038
$189$	130.717	0.0503082
$190$	−3175.34	−1.21244
$191$	1408.47	0.533578	0.266789	−	0.963755i	$-0.414037\pi$
$191$	1408.47	0.533578	0.266789	+	0.963755i	$0.414037\pi$
$192$	534.652	0.200964
$193$	−4131.69	−1.54096	−0.770481	−	0.637463i	$-0.779984\pi$
$193$	−4131.69	−1.54096	−0.770481	+	0.637463i	$0.779984\pi$
$194$	1767.54	0.654134
$195$	754.259	0.276993
$196$	1811.91	0.660316
$197$	−3401.23	−1.23009	−0.615045	−	0.788492i	$-0.710862\pi$
$197$	−3401.23	−1.23009	−0.615045	+	0.788492i	$0.710862\pi$
$198$	−839.015	−0.301142
$199$	−3520.74	−1.25416	−0.627081	−	0.778954i	$-0.715751\pi$
$199$	−3520.74	−1.25416	−0.627081	+	0.778954i	$0.715751\pi$
$200$	−5196.45	−1.83722
$201$	2018.42	0.708302
$202$	2599.14	0.905321
$203$	−435.183	−0.150463
$204$	710.820	0.243958
$205$	−2759.30	−0.940088
$206$	−2221.18	−0.751248
$207$	256.624	0.0861672
$208$	175.611	0.0585406
$209$	6569.05	2.17412
$210$	428.770	0.140895
$211$	−2245.22	−0.732545	−0.366272	−	0.930508i	$-0.619366\pi$
$211$	−2245.22	−0.732545	−0.366272	+	0.930508i	$0.619366\pi$
$212$	142.343	0.0461141
$213$	−980.510	−0.315415
$214$	−1255.90	−0.401174
$215$	−1374.95	−0.436142
$216$	−563.393	−0.177472
$217$	886.490	0.277322
$218$	697.986	0.216851
$219$	72.9173	0.0224990
$220$	6697.07	2.05235
$221$	−543.250	−0.165353
$222$	1914.29	0.578732
$223$	3431.26	1.03038	0.515188	−	0.857077i	$-0.327722\pi$
$223$	3431.26	1.03038	0.515188	+	0.857077i	$0.327722\pi$
$224$	908.003	0.270842
$225$	2241.31	0.664091
$226$	−582.631	−0.171487
$227$	−4757.91	−1.39116	−0.695581	−	0.718448i	$-0.744853\pi$
$227$	−4757.91	−1.39116	−0.695581	+	0.718448i	$0.744853\pi$
$228$	1829.61	0.531442
$229$	−4368.93	−1.26073	−0.630364	−	0.776300i	$-0.717094\pi$
$229$	−4368.93	−1.26073	−0.630364	+	0.776300i	$0.717094\pi$
$230$	841.764	0.241323
$231$	−887.026	−0.252649
$232$	1875.65	0.530787
$233$	−3642.00	−1.02401	−0.512007	−	0.858981i	$-0.671098\pi$
$233$	−3642.00	−1.02401	−0.512007	+	0.858981i	$0.671098\pi$
$234$	178.593	0.0498932
$235$	6260.57	1.73785
$236$	3881.15	1.07051
$237$	500.431	0.137158
$238$	−308.819	−0.0841082
$239$	2236.17	0.605213	0.302606	−	0.953116i	$-0.402143\pi$
$239$	2236.17	0.605213	0.302606	+	0.953116i	$0.402143\pi$
$240$	783.764	0.210799
$241$	6538.78	1.74772	0.873858	−	0.486181i	$-0.161610\pi$
$241$	6538.78	1.74772	0.873858	+	0.486181i	$0.161610\pi$
$242$	3661.75	0.972670
$243$	243.000	0.0641500
$244$	−1747.06	−0.458378
$245$	−6180.30	−1.61161
$246$	−653.348	−0.169333
$247$	−1398.29	−0.360208
$248$	−3820.80	−0.978310
$249$	−603.280	−0.153539
$250$	3661.65	0.926332
$251$	2507.12	0.630470	0.315235	−	0.949014i	$-0.397917\pi$
$251$	2507.12	0.630470	0.315235	+	0.949014i	$0.397917\pi$
$252$	−247.054	−0.0617577
$253$	−1741.42	−0.432735
$254$	−1723.49	−0.425755
$255$	−2424.56	−0.595420
$256$	−3300.77	−0.805853
$257$	−808.131	−0.196147	−0.0980735	−	0.995179i	$-0.531268\pi$
$257$	−808.131	−0.196147	−0.0980735	+	0.995179i	$0.531268\pi$
$258$	−325.560	−0.0785600
$259$	2023.83	0.485539
$260$	−1425.54	−0.340033
$261$	−808.998	−0.191861
$262$	−1301.69	−0.306941
$263$	2940.70	0.689472	0.344736	−	0.938700i	$-0.387968\pi$
$263$	2940.70	0.689472	0.344736	+	0.938700i	$0.387968\pi$
$264$	3823.11	0.891273
$265$	−485.525	−0.112549
$266$	−794.881	−0.183223
$267$	326.501	0.0748371
$268$	−3814.81	−0.869502
$269$	7111.50	1.61188	0.805940	−	0.591997i	$-0.201660\pi$
$269$	7111.50	1.61188	0.805940	+	0.591997i	$0.201660\pi$
$270$	797.075	0.179661
$271$	2034.96	0.456145	0.228072	−	0.973644i	$-0.426758\pi$
$271$	2034.96	0.456145	0.228072	+	0.973644i	$0.426758\pi$
$272$	−564.502	−0.125838
$273$	188.813	0.0418589
$274$	−746.281	−0.164542
$275$	−15209.2	−3.33509
$276$	−485.018	−0.105778
$277$	2723.20	0.590689	0.295345	−	0.955391i	$-0.404566\pi$
$277$	2723.20	0.590689	0.295345	+	0.955391i	$0.404566\pi$
$278$	621.449	0.134072
$279$	1647.97	0.353625
$280$	−1953.76	−0.416998
$281$	3265.56	0.693263	0.346632	−	0.938001i	$-0.387325\pi$
$281$	3265.56	0.693263	0.346632	+	0.938001i	$0.387325\pi$
$282$	1482.38	0.313030
$283$	1144.02	0.240299	0.120150	−	0.992756i	$-0.461663\pi$
$283$	1144.02	0.240299	0.120150	+	0.992756i	$0.461663\pi$
$284$	1853.16	0.387200
$285$	−6240.68	−1.29707
$286$	−1211.91	−0.250566
$287$	−690.735	−0.142065
$288$	1687.96	0.345361
$289$	−3166.72	−0.644560
$290$	−2653.63	−0.537333
$291$	3473.85	0.699796
$292$	−137.813	−0.0276195
$293$	−1677.35	−0.334444	−0.167222	−	0.985919i	$-0.553480\pi$
$293$	−1677.35	−0.334444	−0.167222	+	0.985919i	$0.553480\pi$
$294$	−1463.37	−0.290291
$295$	−13238.4	−2.61277
$296$	−8722.75	−1.71284
$297$	−1648.96	−0.322164
$298$	2621.86	0.509666
$299$	370.679	0.0716954
$300$	−4236.06	−0.815230
$301$	−344.190	−0.0659095
$302$	2048.16	0.390259
$303$	5108.24	0.968518
$304$	−1452.99	−0.274128
$305$	5959.13	1.11875
$306$	−574.088	−0.107250
$307$	7207.70	1.33995	0.669975	−	0.742383i	$-0.266305\pi$
$307$	7207.70	1.33995	0.669975	+	0.742383i	$0.266305\pi$
$308$	1676.47	0.310149
$309$	−4365.42	−0.803689
$310$	5405.57	0.990374
$311$	412.963	0.0752958	0.0376479	−	0.999291i	$-0.488013\pi$
$311$	412.963	0.0752958	0.0376479	+	0.999291i	$0.488013\pi$
$312$	−813.790	−0.147666
$313$	2936.39	0.530270	0.265135	−	0.964211i	$-0.414583\pi$
$313$	2936.39	0.530270	0.265135	+	0.964211i	$0.414583\pi$
$314$	−1160.93	−0.208646
$315$	842.686	0.150730
$316$	−945.812	−0.168374
$317$	377.956	0.0669657	0.0334828	−	0.999439i	$-0.489340\pi$
$317$	377.956	0.0669657	0.0334828	+	0.999439i	$0.489340\pi$
$318$	−114.963	−0.0202729
$319$	5489.75	0.963533
$320$	3446.71	0.602116
$321$	−2468.28	−0.429178
$322$	210.718	0.0364685
$323$	4494.81	0.774298
$324$	−459.269	−0.0787497
$325$	3237.44	0.552557
$326$	4139.83	0.703326
$327$	1371.79	0.231989
$328$	2977.09	0.501165
$329$	1567.20	0.262622
$330$	−5408.84	−0.902263
$331$	−4428.17	−0.735330	−0.367665	−	0.929958i	$-0.619843\pi$
$331$	−4428.17	−0.735330	−0.367665	+	0.929958i	$0.619843\pi$
$332$	1140.20	0.188483
$333$	3762.26	0.619130
$334$	2368.57	0.388031
$335$	13012.1	2.12217
$336$	196.199	0.0318558
$337$	−1768.76	−0.285907	−0.142953	−	0.989729i	$-0.545660\pi$
$337$	−1768.76	−0.285907	−0.142953	+	0.989729i	$0.545660\pi$
$338$	257.968	0.0415137
$339$	−1145.08	−0.183458
$340$	4582.41	0.730930
$341$	−11182.9	−1.77592
$342$	−1477.67	−0.233635
$343$	−3207.70	−0.504955
$344$	1483.47	0.232509
$345$	1654.37	0.258168
$346$	−6060.36	−0.941639
$347$	2412.97	0.373300	0.186650	−	0.982426i	$-0.440237\pi$
$347$	2412.97	0.373300	0.186650	+	0.982426i	$0.440237\pi$
$348$	1529.00	0.235526
$349$	−9967.45	−1.52878	−0.764392	−	0.644752i	$-0.776961\pi$
$349$	−9967.45	−1.52878	−0.764392	+	0.644752i	$0.776961\pi$
$350$	1840.37	0.281063
$351$	351.000	0.0533761
$352$	−11454.3	−1.73442
$353$	4516.30	0.680959	0.340479	−	0.940252i	$-0.389411\pi$
$353$	4516.30	0.680959	0.340479	+	0.940252i	$0.389411\pi$
$354$	−3134.58	−0.470625
$355$	−6321.01	−0.945027
$356$	−617.084	−0.0918691
$357$	−606.939	−0.0899794
$358$	−4107.57	−0.606401
$359$	12159.8	1.78767	0.893833	−	0.448400i	$-0.148006\pi$
$359$	12159.8	1.78767	0.893833	+	0.448400i	$0.148006\pi$
$360$	−3632.00	−0.531731
$361$	4710.38	0.686745
$362$	−6672.22	−0.968740
$363$	7196.65	1.04057
$364$	−356.856	−0.0513855
$365$	470.072	0.0674102
$366$	1411.00	0.201514
$367$	−2674.25	−0.380367	−0.190183	−	0.981749i	$-0.560908\pi$
$367$	−2674.25	−0.380367	−0.190183	+	0.981749i	$0.560908\pi$
$368$	385.180	0.0545622
$369$	−1284.06	−0.181153
$370$	12340.7	1.73396
$371$	−121.541	−0.0170083
$372$	−3114.65	−0.434105
$373$	9601.74	1.33287	0.666433	−	0.745564i	$-0.267820\pi$
$373$	9601.74	1.33287	0.666433	+	0.745564i	$0.267820\pi$
$374$	3895.69	0.538613
$375$	7196.45	0.990995
$376$	−6754.69	−0.926454
$377$	−1168.55	−0.159638
$378$	199.531	0.0271502
$379$	9019.65	1.22245	0.611225	−	0.791457i	$-0.290677\pi$
$379$	9019.65	1.22245	0.611225	+	0.791457i	$0.290677\pi$
$380$	11794.9	1.59227
$381$	−3387.28	−0.455474
$382$	2149.95	0.287960
$383$	−4015.34	−0.535703	−0.267852	−	0.963460i	$-0.586314\pi$
$383$	−4015.34	−0.535703	−0.267852	+	0.963460i	$0.586314\pi$
$384$	−3685.12	−0.489727
$385$	−5718.35	−0.756972
$386$	−6306.78	−0.831623
$387$	−639.842	−0.0840439
$388$	−6565.55	−0.859060
$389$	−2725.35	−0.355221	−0.177610	−	0.984101i	$-0.556837\pi$
$389$	−2725.35	−0.355221	−0.177610	+	0.984101i	$0.556837\pi$
$390$	1151.33	0.149487
$391$	−1191.55	−0.154115
$392$	6668.09	0.859157
$393$	−2558.28	−0.328367
$394$	−5191.78	−0.663853
$395$	3226.11	0.410945
$396$	3116.53	0.395484
$397$	−4391.59	−0.555182	−0.277591	−	0.960699i	$-0.589536\pi$
$397$	−4391.59	−0.555182	−0.277591	+	0.960699i	$0.589536\pi$
$398$	−5374.19	−0.676844
$399$	−1562.23	−0.196013
$400$	3364.09	0.420511
$401$	3762.48	0.468552	0.234276	−	0.972170i	$-0.424728\pi$
$401$	3762.48	0.468552	0.234276	+	0.972170i	$0.424728\pi$
$402$	3081.00	0.382255
$403$	2380.40	0.294234
$404$	−9654.55	−1.18894
$405$	1566.54	0.192202
$406$	−664.281	−0.0812013
$407$	−25530.2	−3.10930
$408$	2615.93	0.317421
$409$	−6797.81	−0.821833	−0.410917	−	0.911673i	$-0.634791\pi$
$409$	−6797.81	−0.821833	−0.410917	+	0.911673i	$0.634791\pi$
$410$	−4211.91	−0.507345
$411$	−1466.71	−0.176028
$412$	8250.61	0.986599
$413$	−3313.95	−0.394840
$414$	391.721	0.0465025
$415$	−3889.14	−0.460025
$416$	2438.17	0.287358
$417$	1221.37	0.143431
$418$	10027.3	1.17332
$419$	−12594.2	−1.46841	−0.734207	−	0.678925i	$-0.762446\pi$
$419$	−12594.2	−1.46841	−0.734207	+	0.678925i	$0.762446\pi$
$420$	−1592.67	−0.185034
$421$	6888.04	0.797393	0.398697	−	0.917083i	$-0.369463\pi$
$421$	6888.04	0.797393	0.398697	+	0.917083i	$0.369463\pi$
$422$	−3427.19	−0.395338
$423$	2913.40	0.334881
$424$	523.845	0.0600004
$425$	−10406.8	−1.18777
$426$	−1496.69	−0.170223
$427$	1491.74	0.169065
$428$	4665.04	0.526854
$429$	−2381.84	−0.268056
$430$	−2098.77	−0.235376
$431$	−7384.53	−0.825291	−0.412645	−	0.910892i	$-0.635395\pi$
$431$	−7384.53	−0.825291	−0.412645	+	0.910892i	$0.635395\pi$
$432$	364.731	0.0406206
$433$	9068.33	1.00646	0.503229	−	0.864153i	$-0.332145\pi$
$433$	9068.33	1.00646	0.503229	+	0.864153i	$0.332145\pi$
$434$	1353.17	0.149665
$435$	−5215.33	−0.574841
$436$	−2592.68	−0.284786
$437$	−3066.97	−0.335728
$438$	111.304	0.0121422
$439$	16875.4	1.83466	0.917331	−	0.398125i	$-0.130339\pi$
$439$	16875.4	1.83466	0.917331	+	0.398125i	$0.130339\pi$
$440$	24646.3	2.67037
$441$	−2876.05	−0.310555
$442$	−829.239	−0.0892373
$443$	−6766.18	−0.725668	−0.362834	−	0.931854i	$-0.618191\pi$
$443$	−6766.18	−0.725668	−0.362834	+	0.931854i	$0.618191\pi$
$444$	−7110.64	−0.760037
$445$	2104.84	0.224222
$446$	5237.61	0.556072
$447$	5152.90	0.545243
$448$	862.814	0.0909914
$449$	140.944	0.0148141	0.00740706	−	0.999973i	$-0.497642\pi$
$449$	140.944	0.0148141	0.00740706	+	0.999973i	$0.497642\pi$
$450$	3421.22	0.358395
$451$	8713.47	0.909759
$452$	2164.19	0.225210
$453$	4025.36	0.417501
$454$	−7262.67	−0.750779
$455$	1217.21	0.125415
$456$	6733.24	0.691475
$457$	−17733.1	−1.81514	−0.907571	−	0.419898i	$-0.862066\pi$
$457$	−17733.1	−1.81514	−0.907571	+	0.419898i	$0.862066\pi$
$458$	−6668.90	−0.680387
$459$	−1128.29	−0.114736
$460$	−3126.74	−0.316924
$461$	2293.37	0.231699	0.115849	−	0.993267i	$-0.463041\pi$
$461$	2293.37	0.231699	0.115849	+	0.993267i	$0.463041\pi$
$462$	−1353.99	−0.136349
$463$	13770.9	1.38226	0.691129	−	0.722731i	$-0.257113\pi$
$463$	13770.9	1.38226	0.691129	+	0.722731i	$0.257113\pi$
$464$	−1214.27	−0.121489
$465$	10623.9	1.05951
$466$	−5559.29	−0.552638
$467$	−3477.37	−0.344568	−0.172284	−	0.985047i	$-0.555115\pi$
$467$	−3477.37	−0.344568	−0.172284	+	0.985047i	$0.555115\pi$
$468$	−663.388	−0.0655237
$469$	3257.31	0.320700
$470$	9556.38	0.937879
$471$	−2281.64	−0.223211
$472$	14283.2	1.39288
$473$	4341.88	0.422072
$474$	763.878	0.0740213
$475$	−26786.4	−2.58746
$476$	1147.11	0.110458
$477$	−225.943	−0.0216880
$478$	3413.38	0.326620
$479$	−3137.39	−0.299271	−0.149636	−	0.988741i	$-0.547810\pi$
$479$	−3137.39	−0.299271	−0.149636	+	0.988741i	$0.547810\pi$
$480$	10881.7	1.03475
$481$	5434.37	0.515148
$482$	9981.05	0.943204
$483$	414.137	0.0390142
$484$	−13601.6	−1.27739
$485$	22394.7	2.09668
$486$	370.925	0.0346204
$487$	5996.52	0.557964	0.278982	−	0.960296i	$-0.410003\pi$
$487$	5996.52	0.557964	0.278982	+	0.960296i	$0.410003\pi$
$488$	−6429.46	−0.596410
$489$	8136.26	0.752422
$490$	−9433.86	−0.869752
$491$	−9401.49	−0.864121	−0.432060	−	0.901845i	$-0.642213\pi$
$491$	−9401.49	−0.864121	−0.432060	+	0.901845i	$0.642213\pi$
$492$	2426.87	0.222382
$493$	3756.31	0.343156
$494$	−2134.41	−0.194396
$495$	−10630.3	−0.965246
$496$	2473.52	0.223920
$497$	−1582.34	−0.142812
$498$	−920.871	−0.0828619
$499$	−5052.33	−0.453253	−0.226626	−	0.973982i	$-0.572770\pi$
$499$	−5052.33	−0.453253	−0.226626	+	0.973982i	$0.572770\pi$
$500$	−13601.2	−1.21653
$501$	4655.08	0.415117
$502$	3826.96	0.340251
$503$	8184.02	0.725462	0.362731	−	0.931894i	$-0.381844\pi$
$503$	8184.02	0.725462	0.362731	+	0.931894i	$0.381844\pi$
$504$	−909.196	−0.0803548
$505$	32931.1	2.90181
$506$	−2658.17	−0.233537
$507$	507.000	0.0444116
$508$	6401.94	0.559134
$509$	6039.12	0.525892	0.262946	−	0.964811i	$-0.415306\pi$
$509$	6039.12	0.525892	0.262946	+	0.964811i	$0.415306\pi$
$510$	−3700.95	−0.321335
$511$	117.673	0.0101870
$512$	4788.54	0.413331
$513$	−2904.15	−0.249944
$514$	−1233.56	−0.105856
$515$	−28142.3	−2.40796
$516$	1209.30	0.103171
$517$	−19770.0	−1.68178
$518$	3089.25	0.262035
$519$	−11910.8	−1.00737
$520$	−5246.22	−0.442427
$521$	−14602.5	−1.22792	−0.613960	−	0.789337i	$-0.710424\pi$
$521$	−14602.5	−1.22792	−0.613960	+	0.789337i	$0.710424\pi$
$522$	−1234.89	−0.103543
$523$	8910.70	0.745005	0.372502	−	0.928031i	$-0.378500\pi$
$523$	8910.70	0.745005	0.372502	+	0.928031i	$0.378500\pi$
$524$	4835.14	0.403099
$525$	3616.99	0.300683
$526$	4488.80	0.372093
$527$	−7651.79	−0.632481
$528$	−2475.01	−0.203998
$529$	−11354.0	−0.933177
$530$	−741.124	−0.0607403
$531$	−6160.57	−0.503477
$532$	2952.60	0.240623
$533$	−1854.76	−0.150729
$534$	498.384	0.0403879
$535$	−15912.2	−1.28588
$536$	−14039.1	−1.13134
$537$	−8072.84	−0.648731
$538$	10855.3	0.869897
$539$	19516.5	1.55962
$540$	−2960.75	−0.235945
$541$	−13313.6	−1.05803	−0.529017	−	0.848611i	$-0.677439\pi$
$541$	−13313.6	−1.05803	−0.529017	+	0.848611i	$0.677439\pi$
$542$	3106.25	0.246171
$543$	−13113.3	−1.03636
$544$	−7837.48	−0.617701
$545$	8843.47	0.695069
$546$	288.212	0.0225903
$547$	−4116.94	−0.321806	−0.160903	−	0.986970i	$-0.551441\pi$
$547$	−4116.94	−0.321806	−0.160903	+	0.986970i	$0.551441\pi$
$548$	2772.07	0.216090
$549$	2773.12	0.215581
$550$	−23215.9	−1.79987
$551$	9668.52	0.747537
$552$	−1784.94	−0.137631
$553$	807.590	0.0621016
$554$	4156.79	0.318782
$555$	24254.0	1.85500
$556$	−2308.38	−0.176074
$557$	−6888.37	−0.524003	−0.262002	−	0.965067i	$-0.584383\pi$
$557$	−6888.37	−0.524003	−0.262002	+	0.965067i	$0.584383\pi$
$558$	2515.53	0.190844
$559$	−924.216	−0.0699288
$560$	1264.83	0.0954443
$561$	7656.42	0.576211
$562$	4984.68	0.374139
$563$	10537.1	0.788782	0.394391	−	0.918943i	$-0.370955\pi$
$563$	10537.1	0.788782	0.394391	+	0.918943i	$0.370955\pi$
$564$	−5506.31	−0.411095
$565$	−7381.92	−0.549664
$566$	1746.27	0.129684
$567$	392.150	0.0290454
$568$	6819.91	0.503798
$569$	26930.1	1.98413	0.992065	−	0.125722i	$-0.0401248\pi$
$569$	26930.1	1.98413	0.992065	+	0.125722i	$0.0401248\pi$
$570$	−9526.02	−0.700002
$571$	−3125.60	−0.229076	−0.114538	−	0.993419i	$-0.536539\pi$
$571$	−3125.60	−0.229076	−0.114538	+	0.993419i	$0.536539\pi$
$572$	4501.66	0.329063
$573$	4225.42	0.308062
$574$	−1054.36	−0.0766696
$575$	7100.91	0.515006
$576$	1603.96	0.116027
$577$	4787.13	0.345391	0.172696	−	0.984975i	$-0.444752\pi$
$577$	4787.13	0.345391	0.172696	+	0.984975i	$0.444752\pi$
$578$	−4833.81	−0.347855
$579$	−12395.1	−0.889675
$580$	9856.94	0.705668
$581$	−973.566	−0.0695186
$582$	5302.62	0.377664
$583$	1533.22	0.108918
$584$	−507.174	−0.0359366
$585$	2262.78	0.159922
$586$	−2560.38	−0.180492
$587$	18380.8	1.29243	0.646214	−	0.763156i	$-0.276351\pi$
$587$	18380.8	1.29243	0.646214	+	0.763156i	$0.276351\pi$
$588$	5435.72	0.381233
$589$	−19695.3	−1.37781
$590$	−20207.6	−1.41005
$591$	−10203.7	−0.710193
$592$	5646.96	0.392042
$593$	−13831.7	−0.957843	−0.478922	−	0.877858i	$-0.658972\pi$
$593$	−13831.7	−0.957843	−0.478922	+	0.877858i	$0.658972\pi$
$594$	−2517.05	−0.173865
$595$	−3912.73	−0.269590
$596$	−9738.94	−0.669333
$597$	−10562.2	−0.724091
$598$	565.819	0.0386924
$599$	12248.6	0.835502	0.417751	−	0.908562i	$-0.362818\pi$
$599$	12248.6	0.835502	0.417751	+	0.908562i	$0.362818\pi$
$600$	−15589.3	−1.06072
$601$	9719.56	0.659682	0.329841	−	0.944036i	$-0.393005\pi$
$601$	9719.56	0.659682	0.329841	+	0.944036i	$0.393005\pi$
$602$	−525.385	−0.0355699
$603$	6055.27	0.408938
$604$	−7607.91	−0.512519
$605$	46394.3	3.11768
$606$	7797.42	0.522688
$607$	1607.83	0.107512	0.0537560	−	0.998554i	$-0.482881\pi$
$607$	1607.83	0.107512	0.0537560	+	0.998554i	$0.482881\pi$
$608$	−20173.2	−1.34561
$609$	−1305.55	−0.0868696
$610$	9096.25	0.603764
$611$	4208.25	0.278638
$612$	2132.46	0.140849
$613$	14731.1	0.970610	0.485305	−	0.874345i	$-0.338709\pi$
$613$	14731.1	0.970610	0.485305	+	0.874345i	$0.338709\pi$
$614$	11002.1	0.723142
$615$	−8277.91	−0.542760
$616$	6169.68	0.403545
$617$	27951.8	1.82382	0.911909	−	0.410392i	$-0.134608\pi$
$617$	27951.8	1.82382	0.911909	+	0.410392i	$0.134608\pi$
$618$	−6663.55	−0.433733
$619$	16200.2	1.05192	0.525961	−	0.850509i	$-0.323706\pi$
$619$	16200.2	1.05192	0.525961	+	0.850509i	$0.323706\pi$
$620$	−20079.1	−1.30064
$621$	769.872	0.0497486
$622$	630.364	0.0406355
$623$	526.903	0.0338843
$624$	526.833	0.0337984
$625$	15263.8	0.976880
$626$	4482.22	0.286175
$627$	19707.2	1.25523
$628$	4312.28	0.274011
$629$	−17468.8	−1.10735
$630$	1286.31	0.0813457
$631$	12731.8	0.803239	0.401619	−	0.915807i	$-0.368447\pi$
$631$	12731.8	0.803239	0.401619	+	0.915807i	$0.368447\pi$
$632$	−3480.73	−0.219076
$633$	−6735.65	−0.422935
$634$	576.927	0.0361399
$635$	−21836.6	−1.36466
$636$	427.030	0.0266240
$637$	−4154.30	−0.258397
$638$	8379.77	0.519997
$639$	−2941.53	−0.182105
$640$	−23756.7	−1.46729
$641$	−11556.9	−0.712119	−0.356059	−	0.934463i	$-0.615880\pi$
$641$	−11556.9	−0.712119	−0.356059	+	0.934463i	$0.615880\pi$
$642$	−3767.69	−0.231618
$643$	−9181.25	−0.563100	−0.281550	−	0.959547i	$-0.590848\pi$
$643$	−9181.25	−0.563100	−0.281550	+	0.959547i	$0.590848\pi$
$644$	−782.716	−0.0478934
$645$	−4124.84	−0.251807
$646$	6861.06	0.417871
$647$	5244.11	0.318651	0.159326	−	0.987226i	$-0.449068\pi$
$647$	5244.11	0.318651	0.159326	+	0.987226i	$0.449068\pi$
$648$	−1690.18	−0.102464
$649$	41804.8	2.52848
$650$	4941.76	0.298203
$651$	2659.47	0.160112
$652$	−15377.5	−0.923663
$653$	−16421.4	−0.984106	−0.492053	−	0.870565i	$-0.663753\pi$
$653$	−16421.4	−0.984106	−0.492053	+	0.870565i	$0.663753\pi$
$654$	2093.96	0.125199
$655$	−16492.4	−0.983832
$656$	−1927.31	−0.114709
$657$	218.752	0.0129898
$658$	2392.24	0.141732
$659$	1838.11	0.108653	0.0543266	−	0.998523i	$-0.482699\pi$
$659$	1838.11	0.108653	0.0543266	+	0.998523i	$0.482699\pi$
$660$	20091.2	1.18492
$661$	−5500.93	−0.323694	−0.161847	−	0.986816i	$-0.551745\pi$
$661$	−5500.93	−0.323694	−0.161847	+	0.986816i	$0.551745\pi$
$662$	−6759.34	−0.396841
$663$	−1629.75	−0.0954665
$664$	4196.10	0.245241
$665$	−10071.1	−0.587281
$666$	5742.86	0.334131
$667$	−2563.07	−0.148789
$668$	−8798.08	−0.509593
$669$	10293.8	0.594888
$670$	19862.2	1.14529
$671$	−18818.0	−1.08266
$672$	2724.01	0.156370
$673$	−25986.7	−1.48843	−0.744216	−	0.667939i	$-0.767177\pi$
$673$	−25986.7	−1.48843	−0.744216	+	0.667939i	$0.767177\pi$
$674$	−2699.91	−0.154298
$675$	6723.92	0.383413
$676$	−958.227	−0.0545191
$677$	−11691.3	−0.663714	−0.331857	−	0.943330i	$-0.607675\pi$
$677$	−11691.3	−0.663714	−0.331857	+	0.943330i	$0.607675\pi$
$678$	−1747.89	−0.0990080
$679$	5606.05	0.316849
$680$	16864.0	0.951035
$681$	−14273.7	−0.803188
$682$	−17070.0	−0.958423
$683$	−11111.5	−0.622501	−0.311251	−	0.950328i	$-0.600748\pi$
$683$	−11111.5	−0.622501	−0.311251	+	0.950328i	$0.600748\pi$
$684$	5488.82	0.306828
$685$	−9455.37	−0.527403
$686$	−4896.36	−0.272513
$687$	−13106.8	−0.727882
$688$	−960.370	−0.0532177
$689$	−326.361	−0.0180455
$690$	2525.29	0.139328
$691$	−7542.55	−0.415242	−0.207621	−	0.978209i	$-0.566572\pi$
$691$	−7542.55	−0.415242	−0.207621	+	0.978209i	$0.566572\pi$
$692$	22511.3	1.23664
$693$	−2661.08	−0.145867
$694$	3683.26	0.201462
$695$	7873.75	0.429738
$696$	5626.96	0.306450
$697$	5962.11	0.324005
$698$	−15214.7	−0.825051
$699$	−10926.0	−0.591215
$700$	−6836.10	−0.369114
$701$	−8231.17	−0.443491	−0.221745	−	0.975105i	$-0.571175\pi$
$701$	−8231.17	−0.443491	−0.221745	+	0.975105i	$0.571175\pi$
$702$	535.780	0.0288059
$703$	−44963.6	−2.41228
$704$	−10884.2	−0.582691
$705$	18781.7	1.00335
$706$	6893.86	0.367498
$707$	8243.61	0.438519
$708$	11643.4	0.618061
$709$	28044.6	1.48553	0.742764	−	0.669554i	$-0.233515\pi$
$709$	28044.6	1.48553	0.742764	+	0.669554i	$0.233515\pi$
$710$	−9648.64	−0.510010
$711$	1501.29	0.0791884
$712$	−2270.96	−0.119534
$713$	5221.09	0.274238
$714$	−926.457	−0.0485599
$715$	−15354.9	−0.803133
$716$	15257.6	0.796374
$717$	6708.51	0.349420
$718$	18561.3	0.964764
$719$	29686.4	1.53980	0.769901	−	0.638163i	$-0.220306\pi$
$719$	29686.4	1.53980	0.769901	+	0.638163i	$0.220306\pi$
$720$	2351.29	0.121705
$721$	−7044.86	−0.363889
$722$	7190.12	0.370621
$723$	19616.3	1.00904
$724$	24784.0	1.27223
$725$	−22385.3	−1.14672
$726$	10985.3	0.561572
$727$	−27654.5	−1.41080	−0.705398	−	0.708812i	$-0.749232\pi$
$727$	−27654.5	−1.41080	−0.705398	+	0.708812i	$0.749232\pi$
$728$	−1313.28	−0.0668592
$729$	729.000	0.0370370
$730$	717.537	0.0363798
$731$	2970.89	0.150318
$732$	−5241.19	−0.264645
$733$	−13077.5	−0.658975	−0.329488	−	0.944160i	$-0.606876\pi$
$733$	−13077.5	−0.658975	−0.329488	+	0.944160i	$0.606876\pi$
$734$	−4082.08	−0.205276
$735$	−18540.9	−0.930465
$736$	5347.79	0.267829
$737$	−41090.3	−2.05370
$738$	−1960.04	−0.0977645
$739$	−4218.33	−0.209978	−0.104989	−	0.994473i	$-0.533481\pi$
$739$	−4218.33	−0.209978	−0.104989	+	0.994473i	$0.533481\pi$
$740$	−45839.9	−2.27717
$741$	−4194.88	−0.207966
$742$	−185.525	−0.00917903
$743$	7725.54	0.381457	0.190728	−	0.981643i	$-0.438915\pi$
$743$	7725.54	0.381457	0.190728	+	0.981643i	$0.438915\pi$
$744$	−11462.4	−0.564828
$745$	33218.9	1.63362
$746$	14656.5	0.719319
$747$	−1809.84	−0.0886460
$748$	−14470.6	−0.707349
$749$	−3983.29	−0.194321
$750$	10984.9	0.534818
$751$	7506.12	0.364717	0.182358	−	0.983232i	$-0.441627\pi$
$751$	7506.12	0.364717	0.182358	+	0.983232i	$0.441627\pi$
$752$	4372.87	0.212051
$753$	7521.35	0.364002
$754$	−1783.72	−0.0861531
$755$	25950.1	1.25089
$756$	−741.162	−0.0356558
$757$	−2741.62	−0.131632	−0.0658162	−	0.997832i	$-0.520965\pi$
$757$	−2741.62	−0.131632	−0.0658162	+	0.997832i	$0.520965\pi$
$758$	13768.0	0.659729
$759$	−5224.25	−0.249839
$760$	43406.9	2.07175
$761$	29740.0	1.41666	0.708328	−	0.705883i	$-0.249450\pi$
$761$	29740.0	1.41666	0.708328	+	0.705883i	$0.249450\pi$
$762$	−5170.48	−0.245809
$763$	2213.78	0.105038
$764$	−7986.01	−0.378172
$765$	−7273.69	−0.343766
$766$	−6129.18	−0.289107
$767$	−8898.60	−0.418918
$768$	−9902.32	−0.465259
$769$	−19896.3	−0.933004	−0.466502	−	0.884520i	$-0.654486\pi$
$769$	−19896.3	−0.933004	−0.466502	+	0.884520i	$0.654486\pi$
$770$	−8728.72	−0.408521
$771$	−2424.39	−0.113246
$772$	23426.6	1.09215
$773$	−13601.3	−0.632866	−0.316433	−	0.948615i	$-0.602485\pi$
$773$	−13601.3	−0.632866	−0.316433	+	0.948615i	$0.602485\pi$
$774$	−976.680	−0.0453566
$775$	45600.1	2.11355
$776$	−24162.2	−1.11775
$777$	6071.48	0.280326
$778$	−4160.09	−0.191705
$779$	15346.1	0.705818
$780$	−4276.63	−0.196318
$781$	19960.8	0.914539
$782$	−1818.83	−0.0831727
$783$	−2426.99	−0.110771
$784$	−4316.81	−0.196648
$785$	−14708.9	−0.668770
$786$	−3905.06	−0.177212
$787$	−1498.29	−0.0678631	−0.0339315	−	0.999424i	$-0.510803\pi$
$787$	−1498.29	−0.0678631	−0.0339315	+	0.999424i	$0.510803\pi$
$788$	19284.9	0.871824
$789$	8822.09	0.398067
$790$	4924.46	0.221778
$791$	−1847.91	−0.0830647
$792$	11469.3	0.514577
$793$	4005.62	0.179374
$794$	−6703.49	−0.299620
$795$	−1456.57	−0.0649803
$796$	19962.5	0.888885
$797$	−3713.30	−0.165034	−0.0825168	−	0.996590i	$-0.526296\pi$
$797$	−3713.30	−0.165034	−0.0825168	+	0.996590i	$0.526296\pi$
$798$	−2384.64	−0.105784
$799$	−13527.4	−0.598955
$800$	46706.7	2.06416
$801$	979.502	0.0432072
$802$	5743.20	0.252867
$803$	−1484.42	−0.0652354
$804$	−11444.4	−0.502007
$805$	2669.80	0.116892
$806$	3633.54	0.158791
$807$	21334.5	0.930620
$808$	−35530.2	−1.54697
$809$	−34527.6	−1.50053	−0.750263	−	0.661139i	$-0.770073\pi$
$809$	−34527.6	−1.50053	−0.750263	+	0.661139i	$0.770073\pi$
$810$	2391.22	0.103727
$811$	−37279.2	−1.61412	−0.807059	−	0.590471i	$-0.798942\pi$
$811$	−37279.2	−1.61412	−0.807059	+	0.590471i	$0.798942\pi$
$812$	2467.48	0.106640
$813$	6104.89	0.263355
$814$	−38970.3	−1.67802
$815$	52451.6	2.25436
$816$	−1693.51	−0.0726526
$817$	7646.90	0.327455
$818$	−10376.4	−0.443525
$819$	566.439	0.0241673
$820$	15645.2	0.666285
$821$	13877.9	0.589943	0.294972	−	0.955506i	$-0.404690\pi$
$821$	13877.9	0.589943	0.294972	+	0.955506i	$0.404690\pi$
$822$	−2238.84	−0.0949983
$823$	18945.1	0.802410	0.401205	−	0.915988i	$-0.368592\pi$
$823$	18945.1	0.802410	0.401205	+	0.915988i	$0.368592\pi$
$824$	30363.5	1.28369
$825$	−45627.6	−1.92552
$826$	−5058.55	−0.213086
$827$	−7804.75	−0.328171	−0.164086	−	0.986446i	$-0.552467\pi$
$827$	−7804.75	−0.328171	−0.164086	+	0.986446i	$0.552467\pi$
$828$	−1455.05	−0.0610708
$829$	5784.85	0.242360	0.121180	−	0.992631i	$-0.461332\pi$
$829$	5784.85	0.242360	0.121180	+	0.992631i	$0.461332\pi$
$830$	−5936.54	−0.248265
$831$	8169.59	0.341035
$832$	2316.82	0.0965402
$833$	13354.0	0.555448
$834$	1864.35	0.0774065
$835$	30009.7	1.24375
$836$	−37246.4	−1.54090
$837$	4943.91	0.204165
$838$	−19224.3	−0.792471
$839$	−5011.42	−0.206214	−0.103107	−	0.994670i	$-0.532878\pi$
$839$	−5011.42	−0.206214	−0.103107	+	0.994670i	$0.532878\pi$
$840$	−5861.27	−0.240754
$841$	−16309.0	−0.668704
$842$	10514.2	0.430336
$843$	9796.68	0.400256
$844$	12730.3	0.519190
$845$	3268.45	0.133063
$846$	4447.13	0.180728
$847$	11613.9	0.471142
$848$	−339.128	−0.0137332
$849$	3432.05	0.138737
$850$	−15885.3	−0.641013
$851$	11919.6	0.480138
$852$	5559.48	0.223550
$853$	−22059.0	−0.885446	−0.442723	−	0.896659i	$-0.645987\pi$
$853$	−22059.0	−0.885446	−0.442723	+	0.896659i	$0.645987\pi$
$854$	2277.06	0.0912404
$855$	−18722.0	−0.748866
$856$	17168.1	0.685506
$857$	13956.2	0.556283	0.278141	−	0.960540i	$-0.410282\pi$
$857$	13956.2	0.556283	0.278141	+	0.960540i	$0.410282\pi$
$858$	−3635.73	−0.144664
$859$	12498.5	0.496442	0.248221	−	0.968703i	$-0.420154\pi$
$859$	12498.5	0.496442	0.248221	+	0.968703i	$0.420154\pi$
$860$	7795.92	0.309115
$861$	−2072.20	−0.0820215
$862$	−11272.0	−0.445391
$863$	−38631.2	−1.52378	−0.761890	−	0.647707i	$-0.775728\pi$
$863$	−38631.2	−1.52378	−0.761890	+	0.647707i	$0.775728\pi$
$864$	5063.88	0.199394
$865$	−76784.7	−3.01822
$866$	13842.3	0.543163
$867$	−9500.17	−0.372137
$868$	−5026.38	−0.196551
$869$	−10187.6	−0.397687
$870$	−7960.89	−0.310229
$871$	8746.51	0.340257
$872$	−9541.46	−0.370544
$873$	10421.5	0.404027
$874$	−4681.55	−0.181185
$875$	11613.5	0.448696
$876$	−413.440	−0.0159462
$877$	−856.756	−0.0329881	−0.0164941	−	0.999864i	$-0.505250\pi$
$877$	−856.756	−0.0329881	−0.0164941	+	0.999864i	$0.505250\pi$
$878$	25759.2	0.990127
$879$	−5032.06	−0.193091
$880$	−15955.6	−0.611206
$881$	33638.6	1.28640	0.643198	−	0.765700i	$-0.277607\pi$
$881$	33638.6	1.28640	0.643198	+	0.765700i	$0.277607\pi$
$882$	−4390.12	−0.167600
$883$	−31109.1	−1.18562	−0.592811	−	0.805342i	$-0.701982\pi$
$883$	−31109.1	−1.18562	−0.592811	+	0.805342i	$0.701982\pi$
$884$	3080.22	0.117193
$885$	−39715.1	−1.50848
$886$	−10328.2	−0.391627
$887$	26080.3	0.987248	0.493624	−	0.869675i	$-0.335672\pi$
$887$	26080.3	0.987248	0.493624	+	0.869675i	$0.335672\pi$
$888$	−26168.3	−0.988907
$889$	−5466.35	−0.206227
$890$	3212.91	0.121008
$891$	−4946.89	−0.186001
$892$	−19455.2	−0.730277
$893$	−34818.8	−1.30478
$894$	7865.58	0.294256
$895$	−52042.8	−1.94369
$896$	−5946.99	−0.221736
$897$	1112.04	0.0413934
$898$	215.142	0.00799486
$899$	−16459.3	−0.610621
$900$	−12708.2	−0.470673
$901$	1049.09	0.0387905
$902$	13300.6	0.490977
$903$	−1032.57	−0.0380529
$904$	7964.56	0.293028
$905$	−84536.9	−3.10508
$906$	6144.47	0.225316
$907$	20169.0	0.738369	0.369184	−	0.929356i	$-0.379637\pi$
$907$	20169.0	0.738369	0.369184	+	0.929356i	$0.379637\pi$
$908$	26977.3	0.985983
$909$	15324.7	0.559174
$910$	1858.00	0.0676837
$911$	19982.2	0.726716	0.363358	−	0.931650i	$-0.381630\pi$
$911$	19982.2	0.726716	0.363358	+	0.931650i	$0.381630\pi$
$912$	−4358.98	−0.158268
$913$	12281.3	0.445184
$914$	−27068.5	−0.979593
$915$	17877.4	0.645910
$916$	24771.7	0.893538
$917$	−4128.52	−0.148676
$918$	−1722.27	−0.0619207
$919$	38513.1	1.38241	0.691203	−	0.722661i	$-0.257081\pi$
$919$	38513.1	1.38241	0.691203	+	0.722661i	$0.257081\pi$
$920$	−11506.9	−0.412360
$921$	21623.1	0.773621
$922$	3500.70	0.125043
$923$	−4248.88	−0.151521
$924$	5029.42	0.179065
$925$	104103.	3.70043
$926$	21020.4	0.745975
$927$	−13096.3	−0.464010
$928$	−16858.7	−0.596352
$929$	23218.9	0.820009	0.410005	−	0.912083i	$-0.365527\pi$
$929$	23218.9	0.820009	0.410005	+	0.912083i	$0.365527\pi$
$930$	16216.7	0.571793
$931$	34372.3	1.21000
$932$	20650.1	0.725768
$933$	1238.89	0.0434721
$934$	−5307.99	−0.185956
$935$	49358.3	1.72640
$936$	−2441.37	−0.0852550
$937$	−11112.9	−0.387452	−0.193726	−	0.981056i	$-0.562057\pi$
$937$	−11112.9	−0.387452	−0.193726	+	0.981056i	$0.562057\pi$
$938$	4972.08	0.173075
$939$	8809.17	0.306152
$940$	−35497.3	−1.23170
$941$	45570.4	1.57869	0.789347	−	0.613947i	$-0.210419\pi$
$941$	45570.4	1.57869	0.789347	+	0.613947i	$0.210419\pi$
$942$	−3482.78	−0.120462
$943$	−4068.16	−0.140485
$944$	−9246.71	−0.318808
$945$	2528.06	0.0870240
$946$	6627.62	0.227783
$947$	34903.9	1.19770	0.598852	−	0.800860i	$-0.295624\pi$
$947$	34903.9	1.19770	0.598852	+	0.800860i	$0.295624\pi$
$948$	−2837.44	−0.0972106
$949$	315.975	0.0108082
$950$	−40887.8	−1.39640
$951$	1133.87	0.0386626
$952$	4221.55	0.143720
$953$	−2886.52	−0.0981151	−0.0490575	−	0.998796i	$-0.515622\pi$
$953$	−2886.52	−0.0981151	−0.0490575	+	0.998796i	$0.515622\pi$
$954$	−344.888	−0.0117046
$955$	27239.8	0.922994
$956$	−12679.0	−0.428943
$957$	16469.2	0.556296
$958$	−4789.03	−0.161510
$959$	−2366.96	−0.0797008
$960$	10340.1	0.347632
$961$	3737.42	0.125455
$962$	8295.24	0.278014
$963$	−7404.85	−0.247786
$964$	−37074.8	−1.23869
$965$	−79906.8	−2.66559
$966$	632.155	0.0210551
$967$	11593.8	0.385556	0.192778	−	0.981242i	$-0.438250\pi$
$967$	11593.8	0.385556	0.192778	+	0.981242i	$0.438250\pi$
$968$	−50056.1	−1.66205
$969$	13484.4	0.447041
$970$	34184.1	1.13153
$971$	−4952.12	−0.163667	−0.0818337	−	0.996646i	$-0.526078\pi$
$971$	−4952.12	−0.163667	−0.0818337	+	0.996646i	$0.526078\pi$
$972$	−1377.81	−0.0454662
$973$	1971.03	0.0649418
$974$	9153.33	0.301121
$975$	9712.33	0.319019
$976$	4162.32	0.136509
$977$	−19650.1	−0.643462	−0.321731	−	0.946831i	$-0.604265\pi$
$977$	−19650.1	−0.643462	−0.321731	+	0.946831i	$0.604265\pi$
$978$	12419.5	0.406065
$979$	−6646.77	−0.216988
$980$	35042.2	1.14223
$981$	4115.38	0.133939
$982$	−14350.8	−0.466347
$983$	56818.4	1.84357	0.921783	−	0.387707i	$-0.126733\pi$
$983$	56818.4	1.84357	0.921783	+	0.387707i	$0.126733\pi$
$984$	8931.26	0.289348
$985$	−65779.7	−2.12783
$986$	5733.78	0.185193
$987$	4701.61	0.151625
$988$	7928.30	0.255296
$989$	−2027.15	−0.0651764
$990$	−16226.5	−0.520922
$991$	−19120.4	−0.612897	−0.306448	−	0.951887i	$-0.599141\pi$
$991$	−19120.4	−0.612897	−0.306448	+	0.951887i	$0.599141\pi$
$992$	34342.1	1.09915
$993$	−13284.5	−0.424543
$994$	−2415.34	−0.0770723
$995$	−68090.9	−2.16947
$996$	3420.59	0.108821
$997$	38887.9	1.23530	0.617650	−	0.786453i	$-0.288085\pi$
$997$	38887.9	1.23530	0.617650	+	0.786453i	$0.288085\pi$
$998$	−7712.07	−0.244611
$999$	11286.8	0.357455

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.4.a.c.1.2	✓	3
3.2	odd	2		117.4.a.f.1.2		3
4.3	odd	2		624.4.a.t.1.3		3
5.4	even	2		975.4.a.l.1.2		3
7.6	odd	2		1911.4.a.k.1.2		3
8.3	odd	2		2496.4.a.bp.1.1		3
8.5	even	2		2496.4.a.bl.1.1		3
12.11	even	2		1872.4.a.bk.1.1		3
13.5	odd	4		507.4.b.g.337.3		6
13.8	odd	4		507.4.b.g.337.4		6
13.12	even	2		507.4.a.h.1.2		3
39.38	odd	2		1521.4.a.u.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.c.1.2	✓	3	1.1	even	1	trivial
117.4.a.f.1.2		3	3.2	odd	2
507.4.a.h.1.2		3	13.12	even	2
507.4.b.g.337.3		6	13.5	odd	4
507.4.b.g.337.4		6	13.8	odd	4
624.4.a.t.1.3		3	4.3	odd	2
975.4.a.l.1.2		3	5.4	even	2
1521.4.a.u.1.2		3	39.38	odd	2
1872.4.a.bk.1.1		3	12.11	even	2
1911.4.a.k.1.2		3	7.6	odd	2
2496.4.a.bl.1.1		3	8.5	even	2
2496.4.a.bp.1.1		3	8.3	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:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	39.a (trivial)

\(f(q)\)	\(=\)	\(q+1.52644 q^{2} +3.00000 q^{3} -5.66998 q^{4} +19.3400 q^{5} +4.57932 q^{6} +4.84136 q^{7} -20.8664 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+1.52644 q^{2} +3.00000 q^{3} -5.66998 q^{4} +19.3400 q^{5} +4.57932 q^{6} +4.84136 q^{7} -20.8664 q^{8} +9.00000 q^{9} +29.5213 q^{10} -61.0728 q^{11} -17.0099 q^{12} +13.0000 q^{13} +7.39005 q^{14} +58.0199 q^{15} +13.5085 q^{16} -41.7885 q^{17} +13.7380 q^{18} -107.561 q^{19} -109.657 q^{20} +14.5241 q^{21} -93.2239 q^{22} +28.5138 q^{23} -62.5992 q^{24} +249.034 q^{25} +19.8437 q^{26} +27.0000 q^{27} -27.4504 q^{28} -89.8886 q^{29} +88.5639 q^{30} +183.108 q^{31} +187.551 q^{32} -183.218 q^{33} -63.7876 q^{34} +93.6318 q^{35} -51.0298 q^{36} +418.029 q^{37} -164.185 q^{38} +39.0000 q^{39} -403.555 q^{40} -142.674 q^{41} +22.1701 q^{42} -71.0935 q^{43} +346.281 q^{44} +174.060 q^{45} +43.5246 q^{46} +323.711 q^{47} +40.5256 q^{48} -319.561 q^{49} +380.136 q^{50} -125.365 q^{51} -73.7098 q^{52} -25.1047 q^{53} +41.2139 q^{54} -1181.15 q^{55} -101.022 q^{56} -322.683 q^{57} -137.210 q^{58} -684.508 q^{59} -328.972 q^{60} +308.125 q^{61} +279.503 q^{62} +43.5723 q^{63} +178.217 q^{64} +251.420 q^{65} -279.672 q^{66} +672.808 q^{67} +236.940 q^{68} +85.5413 q^{69} +142.923 q^{70} -326.837 q^{71} -187.798 q^{72} +24.3058 q^{73} +638.095 q^{74} +747.103 q^{75} +609.869 q^{76} -295.675 q^{77} +59.5311 q^{78} +166.810 q^{79} +261.255 q^{80} +81.0000 q^{81} -217.783 q^{82} -201.093 q^{83} -82.3513 q^{84} -808.188 q^{85} -108.520 q^{86} -269.666 q^{87} +1274.37 q^{88} +108.834 q^{89} +265.692 q^{90} +62.9377 q^{91} -161.673 q^{92} +549.323 q^{93} +494.126 q^{94} -2080.23 q^{95} +562.654 q^{96} +1157.95 q^{97} -487.791 q^{98} -549.655 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 9 q^{3} + 10 q^{4} + 4 q^{5} + 6 q^{6} + 30 q^{7} - 6 q^{8} + 27 q^{9}+O(q^{10}) \) 3 * q + 2 * q^2 + 9 * q^3 + 10 * q^4 + 4 * q^5 + 6 * q^6 + 30 * q^7 - 6 * q^8 + 27 * q^9	\( 3 q + 2 q^{2} + 9 q^{3} + 10 q^{4} + 4 q^{5} + 6 q^{6} + 30 q^{7} - 6 q^{8} + 27 q^{9} - 4 q^{10} - 16 q^{11} + 30 q^{12} + 39 q^{13} - 176 q^{14} + 12 q^{15} - 110 q^{16} - 146 q^{17} + 18 q^{18} + 94 q^{19} - 244 q^{20} + 90 q^{21} - 56 q^{22} - 48 q^{23} - 18 q^{24} + 145 q^{25} + 26 q^{26} + 81 q^{27} + 80 q^{28} - 2 q^{29} - 12 q^{30} + 302 q^{31} + 154 q^{32} - 48 q^{33} + 164 q^{34} + 80 q^{35} + 90 q^{36} + 374 q^{37} + 312 q^{38} + 117 q^{39} - 516 q^{40} + 480 q^{41} - 528 q^{42} - 260 q^{43} + 712 q^{44} + 36 q^{45} - 1104 q^{46} - 24 q^{47} - 330 q^{48} + 447 q^{49} + 814 q^{50} - 438 q^{51} + 130 q^{52} - 678 q^{53} + 54 q^{54} - 1552 q^{55} + 96 q^{56} + 282 q^{57} - 628 q^{58} - 1788 q^{59} - 732 q^{60} + 230 q^{61} + 1952 q^{62} + 270 q^{63} - 750 q^{64} + 52 q^{65} - 168 q^{66} + 74 q^{67} - 460 q^{68} - 144 q^{69} + 1216 q^{70} - 948 q^{71} - 54 q^{72} - 222 q^{73} + 1724 q^{74} + 435 q^{75} + 2392 q^{76} + 112 q^{77} + 78 q^{78} - 24 q^{79} + 1100 q^{80} + 243 q^{81} + 564 q^{82} - 796 q^{83} + 240 q^{84} - 248 q^{85} + 1800 q^{86} - 6 q^{87} + 1608 q^{88} + 1436 q^{89} - 36 q^{90} + 390 q^{91} - 1296 q^{92} + 906 q^{93} - 1920 q^{94} - 4032 q^{95} + 462 q^{96} + 3242 q^{97} - 5070 q^{98} - 144 q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 + 9 * q^3 + 10 * q^4 + 4 * q^5 + 6 * q^6 + 30 * q^7 - 6 * q^8 + 27 * q^9 - 4 * q^10 - 16 * q^11 + 30 * q^12 + 39 * q^13 - 176 * q^14 + 12 * q^15 - 110 * q^16 - 146 * q^17 + 18 * q^18 + 94 * q^19 - 244 * q^20 + 90 * q^21 - 56 * q^22 - 48 * q^23 - 18 * q^24 + 145 * q^25 + 26 * q^26 + 81 * q^27 + 80 * q^28 - 2 * q^29 - 12 * q^30 + 302 * q^31 + 154 * q^32 - 48 * q^33 + 164 * q^34 + 80 * q^35 + 90 * q^36 + 374 * q^37 + 312 * q^38 + 117 * q^39 - 516 * q^40 + 480 * q^41 - 528 * q^42 - 260 * q^43 + 712 * q^44 + 36 * q^45 - 1104 * q^46 - 24 * q^47 - 330 * q^48 + 447 * q^49 + 814 * q^50 - 438 * q^51 + 130 * q^52 - 678 * q^53 + 54 * q^54 - 1552 * q^55 + 96 * q^56 + 282 * q^57 - 628 * q^58 - 1788 * q^59 - 732 * q^60 + 230 * q^61 + 1952 * q^62 + 270 * q^63 - 750 * q^64 + 52 * q^65 - 168 * q^66 + 74 * q^67 - 460 * q^68 - 144 * q^69 + 1216 * q^70 - 948 * q^71 - 54 * q^72 - 222 * q^73 + 1724 * q^74 + 435 * q^75 + 2392 * q^76 + 112 * q^77 + 78 * q^78 - 24 * q^79 + 1100 * q^80 + 243 * q^81 + 564 * q^82 - 796 * q^83 + 240 * q^84 - 248 * q^85 + 1800 * q^86 - 6 * q^87 + 1608 * q^88 + 1436 * q^89 - 36 * q^90 + 390 * q^91 - 1296 * q^92 + 906 * q^93 - 1920 * q^94 - 4032 * q^95 + 462 * q^96 + 3242 * q^97 - 5070 * q^98 - 144 * q^99

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