LMFDB - Embedded newform 2015.4.a.d.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2015,4,Mod(1,2015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2015, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2015.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2015 = 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2015.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:	$118.888848662$
Analytic rank:	$1$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	2015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.43322 q^{2} +0.799530 q^{3} -2.07942 q^{4} -5.00000 q^{5} -1.94544 q^{6} +19.2630 q^{7} +24.5255 q^{8} -26.3608 q^{9} +O(q^{10})$	\(q-2.43322 q^{2} +0.799530 q^{3} -2.07942 q^{4} -5.00000 q^{5} -1.94544 q^{6} +19.2630 q^{7} +24.5255 q^{8} -26.3608 q^{9} +12.1661 q^{10} +47.6676 q^{11} -1.66256 q^{12} -13.0000 q^{13} -46.8713 q^{14} -3.99765 q^{15} -43.0406 q^{16} +38.6301 q^{17} +64.1416 q^{18} -76.8062 q^{19} +10.3971 q^{20} +15.4014 q^{21} -115.986 q^{22} +34.3926 q^{23} +19.6089 q^{24} +25.0000 q^{25} +31.6319 q^{26} -42.6635 q^{27} -40.0560 q^{28} -150.318 q^{29} +9.72718 q^{30} +31.0000 q^{31} -91.4764 q^{32} +38.1117 q^{33} -93.9956 q^{34} -96.3152 q^{35} +54.8151 q^{36} +253.365 q^{37} +186.887 q^{38} -10.3939 q^{39} -122.627 q^{40} -340.750 q^{41} -37.4750 q^{42} -100.130 q^{43} -99.1211 q^{44} +131.804 q^{45} -83.6848 q^{46} +284.453 q^{47} -34.4123 q^{48} +28.0647 q^{49} -60.8306 q^{50} +30.8859 q^{51} +27.0325 q^{52} -181.624 q^{53} +103.810 q^{54} -238.338 q^{55} +472.435 q^{56} -61.4089 q^{57} +365.757 q^{58} -641.279 q^{59} +8.31281 q^{60} -348.265 q^{61} -75.4299 q^{62} -507.788 q^{63} +566.908 q^{64} +65.0000 q^{65} -92.7343 q^{66} +70.9732 q^{67} -80.3282 q^{68} +27.4979 q^{69} +234.356 q^{70} +1023.01 q^{71} -646.510 q^{72} +63.3094 q^{73} -616.494 q^{74} +19.9883 q^{75} +159.713 q^{76} +918.223 q^{77} +25.2907 q^{78} -0.0392350 q^{79} +215.203 q^{80} +677.629 q^{81} +829.122 q^{82} +485.101 q^{83} -32.0260 q^{84} -193.150 q^{85} +243.638 q^{86} -120.184 q^{87} +1169.07 q^{88} -1427.65 q^{89} -320.708 q^{90} -250.420 q^{91} -71.5166 q^{92} +24.7854 q^{93} -692.138 q^{94} +384.031 q^{95} -73.1382 q^{96} +1118.87 q^{97} -68.2877 q^{98} -1256.55 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 5 q^{2} - 17 q^{3} + 149 q^{4} - 200 q^{5} - 35 q^{6} - 20 q^{7} - 39 q^{8} + 247 q^{9}+O(q^{10}) $ 40 * q - 5 * q^2 - 17 * q^3 + 149 * q^4 - 200 * q^5 - 35 * q^6 - 20 * q^7 - 39 * q^8 + 247 * q^9	\( 40 q - 5 q^{2} - 17 q^{3} + 149 q^{4} - 200 q^{5} - 35 q^{6} - 20 q^{7} - 39 q^{8} + 247 q^{9} + 25 q^{10} + 127 q^{11} - 76 q^{12} - 520 q^{13} + 138 q^{14} + 85 q^{15} + 413 q^{16} - 264 q^{17} - 126 q^{18} - q^{19} - 745 q^{20} + 176 q^{21} - 191 q^{22} - 106 q^{23} + 31 q^{24} + 1000 q^{25} + 65 q^{26} - 344 q^{27} + 255 q^{28} + 107 q^{29} + 175 q^{30} + 1240 q^{31} - 372 q^{32} - 386 q^{33} - 6 q^{34} + 100 q^{35} + 790 q^{36} - 741 q^{37} - 318 q^{38} + 221 q^{39} + 195 q^{40} + 1232 q^{41} - 1180 q^{42} - 615 q^{43} - 152 q^{44} - 1235 q^{45} - 329 q^{46} - 784 q^{47} - 1089 q^{48} - 516 q^{49} - 125 q^{50} - 200 q^{51} - 1937 q^{52} - 1503 q^{53} + 1658 q^{54} - 635 q^{55} + 1518 q^{56} - 1704 q^{57} - 1035 q^{58} - 107 q^{59} + 380 q^{60} - 857 q^{61} - 155 q^{62} - 2636 q^{63} - 215 q^{64} + 2600 q^{65} - 1785 q^{66} - 2689 q^{67} - 2639 q^{68} + 2544 q^{69} - 690 q^{70} + 1554 q^{71} - 420 q^{72} - 1968 q^{73} - 27 q^{74} - 425 q^{75} - 110 q^{76} - 1040 q^{77} + 455 q^{78} - 3182 q^{79} - 2065 q^{80} - 1576 q^{81} - 386 q^{82} + 317 q^{83} - 617 q^{84} + 1320 q^{85} + 347 q^{86} - 216 q^{87} - 4081 q^{88} + 3610 q^{89} + 630 q^{90} + 260 q^{91} - 4965 q^{92} - 527 q^{93} - 2942 q^{94} + 5 q^{95} + 1002 q^{96} - 3318 q^{97} + 1659 q^{98} + 5943 q^{99}+O(q^{100}) \) 40 * q - 5 * q^2 - 17 * q^3 + 149 * q^4 - 200 * q^5 - 35 * q^6 - 20 * q^7 - 39 * q^8 + 247 * q^9 + 25 * q^10 + 127 * q^11 - 76 * q^12 - 520 * q^13 + 138 * q^14 + 85 * q^15 + 413 * q^16 - 264 * q^17 - 126 * q^18 - q^19 - 745 * q^20 + 176 * q^21 - 191 * q^22 - 106 * q^23 + 31 * q^24 + 1000 * q^25 + 65 * q^26 - 344 * q^27 + 255 * q^28 + 107 * q^29 + 175 * q^30 + 1240 * q^31 - 372 * q^32 - 386 * q^33 - 6 * q^34 + 100 * q^35 + 790 * q^36 - 741 * q^37 - 318 * q^38 + 221 * q^39 + 195 * q^40 + 1232 * q^41 - 1180 * q^42 - 615 * q^43 - 152 * q^44 - 1235 * q^45 - 329 * q^46 - 784 * q^47 - 1089 * q^48 - 516 * q^49 - 125 * q^50 - 200 * q^51 - 1937 * q^52 - 1503 * q^53 + 1658 * q^54 - 635 * q^55 + 1518 * q^56 - 1704 * q^57 - 1035 * q^58 - 107 * q^59 + 380 * q^60 - 857 * q^61 - 155 * q^62 - 2636 * q^63 - 215 * q^64 + 2600 * q^65 - 1785 * q^66 - 2689 * q^67 - 2639 * q^68 + 2544 * q^69 - 690 * q^70 + 1554 * q^71 - 420 * q^72 - 1968 * q^73 - 27 * q^74 - 425 * q^75 - 110 * q^76 - 1040 * q^77 + 455 * q^78 - 3182 * q^79 - 2065 * q^80 - 1576 * q^81 - 386 * q^82 + 317 * q^83 - 617 * q^84 + 1320 * q^85 + 347 * q^86 - 216 * q^87 - 4081 * q^88 + 3610 * q^89 + 630 * q^90 + 260 * q^91 - 4965 * q^92 - 527 * q^93 - 2942 * q^94 + 5 * q^95 + 1002 * q^96 - 3318 * q^97 + 1659 * q^98 + 5943 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.43322	−0.860274	−0.430137	−	0.902764i	$-0.641535\pi$
$2$	−2.43322	−0.860274	−0.430137	+	0.902764i	$0.641535\pi$
$3$	0.799530	0.153870	0.0769348	−	0.997036i	$-0.475487\pi$
$3$	0.799530	0.153870	0.0769348	+	0.997036i	$0.475487\pi$
$4$	−2.07942	−0.259928
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−1.94544	−0.132370
$7$	19.2630	1.04011	0.520053	−	0.854134i	$-0.325912\pi$
$7$	19.2630	1.04011	0.520053	+	0.854134i	$0.325912\pi$
$8$	24.5255	1.08388
$9$	−26.3608	−0.976324
$10$	12.1661	0.384726
$11$	47.6676	1.30658	0.653288	−	0.757110i	$-0.273389\pi$
$11$	47.6676	1.30658	0.653288	+	0.757110i	$0.273389\pi$
$12$	−1.66256	−0.0399950
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−46.8713	−0.894777
$15$	−3.99765	−0.0688126
$16$	−43.0406	−0.672510
$17$	38.6301	0.551128	0.275564	−	0.961283i	$-0.411135\pi$
$17$	38.6301	0.551128	0.275564	+	0.961283i	$0.411135\pi$
$18$	64.1416	0.839907
$19$	−76.8062	−0.927397	−0.463699	−	0.885993i	$-0.653478\pi$
$19$	−76.8062	−0.927397	−0.463699	+	0.885993i	$0.653478\pi$
$20$	10.3971	0.116243
$21$	15.4014	0.160041
$22$	−115.986	−1.12401
$23$	34.3926	0.311798	0.155899	−	0.987773i	$-0.450173\pi$
$23$	34.3926	0.311798	0.155899	+	0.987773i	$0.450173\pi$
$24$	19.6089	0.166777
$25$	25.0000	0.200000
$26$	31.6319	0.238597
$27$	−42.6635	−0.304096
$28$	−40.0560	−0.270353
$29$	−150.318	−0.962528	−0.481264	−	0.876576i	$-0.659822\pi$
$29$	−150.318	−0.962528	−0.481264	+	0.876576i	$0.659822\pi$
$30$	9.72718	0.0591977
$31$	31.0000	0.179605
$31$	31.0000	0.179605
$32$	−91.4764	−0.505341
$33$	38.1117	0.201042
$34$	−93.9956	−0.474121
$35$	−96.3152	−0.465150
$36$	54.8151	0.253774
$37$	253.365	1.12576	0.562878	−	0.826540i	$-0.309694\pi$
$37$	253.365	1.12576	0.562878	+	0.826540i	$0.309694\pi$
$38$	186.887	0.797816
$39$	−10.3939	−0.0426758
$40$	−122.627	−0.484728
$41$	−340.750	−1.29796	−0.648979	−	0.760806i	$-0.724804\pi$
$41$	−340.750	−1.29796	−0.648979	+	0.760806i	$0.724804\pi$
$42$	−37.4750	−0.137679
$43$	−100.130	−0.355108	−0.177554	−	0.984111i	$-0.556818\pi$
$43$	−100.130	−0.355108	−0.177554	+	0.984111i	$0.556818\pi$
$44$	−99.1211	−0.339615
$45$	131.804	0.436625
$46$	−83.6848	−0.268231
$47$	284.453	0.882803	0.441402	−	0.897310i	$-0.354481\pi$
$47$	284.453	0.882803	0.441402	+	0.897310i	$0.354481\pi$
$48$	−34.4123	−0.103479
$49$	28.0647	0.0818212
$50$	−60.8306	−0.172055
$51$	30.8859	0.0848018
$52$	27.0325	0.0720910
$53$	−181.624	−0.470717	−0.235358	−	0.971909i	$-0.575626\pi$
$53$	−181.624	−0.470717	−0.235358	+	0.971909i	$0.575626\pi$
$54$	103.810	0.261606
$55$	−238.338	−0.584318
$56$	472.435	1.12735
$57$	−61.4089	−0.142698
$58$	365.757	0.828038
$59$	−641.279	−1.41504	−0.707520	−	0.706693i	$-0.750186\pi$
$59$	−641.279	−1.41504	−0.707520	+	0.706693i	$0.750186\pi$
$60$	8.31281	0.0178863
$61$	−348.265	−0.730996	−0.365498	−	0.930812i	$-0.619101\pi$
$61$	−348.265	−0.730996	−0.365498	+	0.930812i	$0.619101\pi$
$62$	−75.4299	−0.154510
$63$	−507.788	−1.01548
$64$	566.908	1.10724
$65$	65.0000	0.124035
$66$	−92.7343	−0.172952
$67$	70.9732	0.129414	0.0647072	−	0.997904i	$-0.479389\pi$
$67$	70.9732	0.129414	0.0647072	+	0.997904i	$0.479389\pi$
$68$	−80.3282	−0.143253
$69$	27.4979	0.0479762
$70$	234.356	0.400156
$71$	1023.01	1.70999	0.854994	−	0.518638i	$-0.173561\pi$
$71$	1023.01	1.70999	0.854994	+	0.518638i	$0.173561\pi$
$72$	−646.510	−1.05822
$73$	63.3094	0.101504	0.0507521	−	0.998711i	$-0.483838\pi$
$73$	63.3094	0.101504	0.0507521	+	0.998711i	$0.483838\pi$
$74$	−616.494	−0.968459
$75$	19.9883	0.0307739
$76$	159.713	0.241056
$77$	918.223	1.35898
$78$	25.2907	0.0367129
$79$	−0.0392350	−5.58770e−5 0	−2.79385e−5	−	1.00000i	$-0.500009\pi$
$79$	−0.0392350	−5.58770e−5 0	−2.79385e−5		1.00000i	$0.500009\pi$
$80$	215.203	0.300756
$81$	677.629	0.929533
$82$	829.122	1.11660
$83$	485.101	0.641528	0.320764	−	0.947159i	$-0.396060\pi$
$83$	485.101	0.641528	0.320764	+	0.947159i	$0.396060\pi$
$84$	−32.0260	−0.0415991
$85$	−193.150	−0.246472
$86$	243.638	0.305490
$87$	−120.184	−0.148104
$88$	1169.07	1.41618
$89$	−1427.65	−1.70035	−0.850174	−	0.526502i	$-0.823503\pi$
$89$	−1427.65	−1.70035	−0.850174	+	0.526502i	$0.823503\pi$
$90$	−320.708	−0.375618
$91$	−250.420	−0.288474
$92$	−71.5166	−0.0810448
$93$	24.7854	0.0276358
$94$	−692.138	−0.759453
$95$	384.031	0.414745
$96$	−73.1382	−0.0777566
$97$	1118.87	1.17117	0.585585	−	0.810611i	$-0.300865\pi$
$97$	1118.87	1.17117	0.585585	+	0.810611i	$0.300865\pi$
$98$	−68.2877	−0.0703887
$99$	−1256.55	−1.27564
$100$	−51.9856	−0.0519856
$101$	−913.657	−0.900121	−0.450061	−	0.892998i	$-0.648598\pi$
$101$	−913.657	−0.900121	−0.450061	+	0.892998i	$0.648598\pi$
$102$	−75.1523	−0.0729528
$103$	690.972	0.661005	0.330502	−	0.943805i	$-0.392782\pi$
$103$	690.972	0.661005	0.330502	+	0.943805i	$0.392782\pi$
$104$	−318.831	−0.300615
$105$	−77.0069	−0.0715724
$106$	441.932	0.404946
$107$	168.148	0.151920	0.0759601	−	0.997111i	$-0.475798\pi$
$107$	168.148	0.151920	0.0759601	+	0.997111i	$0.475798\pi$
$108$	88.7155	0.0790431
$109$	−748.999	−0.658175	−0.329088	−	0.944299i	$-0.606741\pi$
$109$	−748.999	−0.658175	−0.329088	+	0.944299i	$0.606741\pi$
$110$	579.930	0.502674
$111$	202.573	0.173220
$112$	−829.093	−0.699482
$113$	1571.77	1.30849	0.654247	−	0.756281i	$-0.272986\pi$
$113$	1571.77	1.30849	0.654247	+	0.756281i	$0.272986\pi$
$114$	149.422	0.122760
$115$	−171.963	−0.139440
$116$	312.574	0.250188
$117$	342.690	0.270784
$118$	1560.37	1.21732
$119$	744.133	0.573231
$120$	−98.0444	−0.0745849
$121$	941.201	0.707138
$122$	847.407	0.628857
$123$	−272.440	−0.199716
$124$	−64.4621	−0.0466844
$125$	−125.000	−0.0894427
$126$	1235.56	0.873592
$127$	1116.37	0.780018	0.390009	−	0.920811i	$-0.372472\pi$
$127$	1116.37	0.780018	0.390009	+	0.920811i	$0.372472\pi$
$128$	−647.602	−0.447191
$129$	−80.0568	−0.0546404
$130$	−158.160	−0.106704
$131$	−173.370	−0.115629	−0.0578146	−	0.998327i	$-0.518413\pi$
$131$	−173.370	−0.115629	−0.0578146	+	0.998327i	$0.518413\pi$
$132$	−79.2503	−0.0522565
$133$	−1479.52	−0.964592
$134$	−172.694	−0.111332
$135$	213.318	0.135996
$136$	947.421	0.597358
$137$	−1424.34	−0.888246	−0.444123	−	0.895966i	$-0.646485\pi$
$137$	−1424.34	−0.888246	−0.444123	+	0.895966i	$0.646485\pi$
$138$	−66.9085	−0.0412727
$139$	2684.19	1.63791	0.818957	−	0.573854i	$-0.194552\pi$
$139$	2684.19	1.63791	0.818957	+	0.573854i	$0.194552\pi$
$140$	200.280	0.120905
$141$	227.429	0.135837
$142$	−2489.22	−1.47106
$143$	−619.679	−0.362379
$144$	1134.58	0.656587
$145$	751.589	0.430455
$146$	−154.046	−0.0873214
$147$	22.4386	0.0125898
$148$	−526.853	−0.292615
$149$	−1438.46	−0.790896	−0.395448	−	0.918488i	$-0.629411\pi$
$149$	−1438.46	−0.790896	−0.395448	+	0.918488i	$0.629411\pi$
$150$	−48.6359	−0.0264740
$151$	74.3539	0.0400718	0.0200359	−	0.999799i	$-0.493622\pi$
$151$	74.3539	0.0400718	0.0200359	+	0.999799i	$0.493622\pi$
$152$	−1883.71	−1.00519
$153$	−1018.32	−0.538079
$154$	−2234.24	−1.16909
$155$	−155.000	−0.0803219
$156$	21.6133	0.0110926
$157$	2699.15	1.37208	0.686038	−	0.727566i	$-0.259348\pi$
$157$	2699.15	1.37208	0.686038	+	0.727566i	$0.259348\pi$
$158$	0.0954676	4.80696e−5 0
$159$	−145.214	−0.0724290
$160$	457.382	0.225995
$161$	662.505	0.324303
$162$	−1648.82	−0.799653
$163$	−957.765	−0.460233	−0.230116	−	0.973163i	$-0.573911\pi$
$163$	−957.765	−0.460233	−0.230116	+	0.973163i	$0.573911\pi$
$164$	708.564	0.337375
$165$	−190.559	−0.0899088
$166$	−1180.36	−0.551890
$167$	−3403.78	−1.57720	−0.788600	−	0.614907i	$-0.789193\pi$
$167$	−3403.78	−1.57720	−0.788600	+	0.614907i	$0.789193\pi$
$168$	377.726	0.173466
$169$	169.000	0.0769231
$170$	469.978	0.212033
$171$	2024.67	0.905440
$172$	208.212	0.0923024
$173$	2994.53	1.31601	0.658004	−	0.753014i	$-0.271401\pi$
$173$	2994.53	1.31601	0.658004	+	0.753014i	$0.271401\pi$
$174$	292.434	0.127410
$175$	481.576	0.208021
$176$	−2051.64	−0.878684
$177$	−512.722	−0.217732
$178$	3473.80	1.46277
$179$	698.051	0.291479	0.145740	−	0.989323i	$-0.453444\pi$
$179$	698.051	0.291479	0.145740	+	0.989323i	$0.453444\pi$
$180$	−274.076	−0.113491
$181$	−3031.13	−1.24476	−0.622382	−	0.782714i	$-0.713835\pi$
$181$	−3031.13	−1.24476	−0.622382	+	0.782714i	$0.713835\pi$
$182$	609.327	0.248166
$183$	−278.448	−0.112478
$184$	843.494	0.337952
$185$	−1266.82	−0.503453
$186$	−60.3085	−0.0237744
$187$	1841.40	0.720089
$188$	−591.498	−0.229465
$189$	−821.829	−0.316293
$190$	−934.433	−0.356794
$191$	−4468.80	−1.69294	−0.846468	−	0.532439i	$-0.821276\pi$
$191$	−4468.80	−1.69294	−0.846468	+	0.532439i	$0.821276\pi$
$192$	453.260	0.170371
$193$	−951.809	−0.354988	−0.177494	−	0.984122i	$-0.556799\pi$
$193$	−951.809	−0.354988	−0.177494	+	0.984122i	$0.556799\pi$
$194$	−2722.45	−1.00753
$195$	51.9695	0.0190852
$196$	−58.3583	−0.0212676
$197$	1708.82	0.618011	0.309005	−	0.951060i	$-0.400004\pi$
$197$	1708.82	0.618011	0.309005	+	0.951060i	$0.400004\pi$
$198$	3057.48	1.09740
$199$	−5304.39	−1.88954	−0.944769	−	0.327736i	$-0.893714\pi$
$199$	−5304.39	−1.88954	−0.944769	+	0.327736i	$0.893714\pi$
$200$	613.137	0.216777
$201$	56.7452	0.0199129
$202$	2223.13	0.774351
$203$	−2895.58	−1.00113
$204$	−64.2249	−0.0220423
$205$	1703.75	0.580464
$206$	−1681.29	−0.568646
$207$	−906.614	−0.304415
$208$	559.528	0.186521
$209$	−3661.17	−1.21171
$210$	187.375	0.0615719
$211$	−1623.76	−0.529782	−0.264891	−	0.964278i	$-0.585336\pi$
$211$	−1623.76	−0.529782	−0.264891	+	0.964278i	$0.585336\pi$
$212$	377.673	0.122352
$213$	817.929	0.263115
$214$	−409.141	−0.130693
$215$	500.649	0.158809
$216$	−1046.34	−0.329605
$217$	597.154	0.186809
$218$	1822.48	0.566211
$219$	50.6178	0.0156184
$220$	495.605	0.151881
$221$	−502.191	−0.152855
$222$	−492.905	−0.149016
$223$	−3505.07	−1.05254	−0.526271	−	0.850317i	$-0.676410\pi$
$223$	−3505.07	−1.05254	−0.526271	+	0.850317i	$0.676410\pi$
$224$	−1762.11	−0.525608
$225$	−659.019	−0.195265
$226$	−3824.47	−1.12566
$227$	116.089	0.0339432	0.0169716	−	0.999856i	$-0.494598\pi$
$227$	116.089	0.0339432	0.0169716	+	0.999856i	$0.494598\pi$
$228$	127.695	0.0370913
$229$	−1023.35	−0.295305	−0.147653	−	0.989039i	$-0.547172\pi$
$229$	−1023.35	−0.295305	−0.147653	+	0.989039i	$0.547172\pi$
$230$	418.424	0.119957
$231$	734.147	0.209105
$232$	−3686.62	−1.04327
$233$	1531.21	0.430527	0.215263	−	0.976556i	$-0.430939\pi$
$233$	1531.21	0.430527	0.215263	+	0.976556i	$0.430939\pi$
$234$	−833.841	−0.232948
$235$	−1422.27	−0.394802
$236$	1333.49	0.367808
$237$	−0.0313696	−8.59778e−6 0
$238$	−1810.64	−0.493136
$239$	−4296.08	−1.16272	−0.581361	−	0.813646i	$-0.697480\pi$
$239$	−4296.08	−1.16272	−0.581361	+	0.813646i	$0.697480\pi$
$240$	172.061	0.0462772
$241$	−3084.55	−0.824453	−0.412227	−	0.911081i	$-0.635249\pi$
$241$	−3084.55	−0.824453	−0.412227	+	0.911081i	$0.635249\pi$
$242$	−2290.15	−0.608333
$243$	1693.70	0.447123
$244$	724.190	0.190006
$245$	−140.323	−0.0365916
$246$	662.908	0.171811
$247$	998.481	0.257214
$248$	760.290	0.194671
$249$	387.853	0.0987116
$250$	304.153	0.0769453
$251$	5879.10	1.47843	0.739214	−	0.673470i	$-0.235197\pi$
$251$	5879.10	1.47843	0.739214	+	0.673470i	$0.235197\pi$
$252$	1055.91	0.263952
$253$	1639.41	0.407387
$254$	−2716.39	−0.671029
$255$	−154.430	−0.0379245
$256$	−2959.50	−0.722535
$257$	420.443	0.102049	0.0510244	−	0.998697i	$-0.483751\pi$
$257$	420.443	0.102049	0.0510244	+	0.998697i	$0.483751\pi$
$258$	194.796	0.0470057
$259$	4880.58	1.17091
$260$	−135.162	−0.0322401
$261$	3962.49	0.939739
$262$	421.849	0.0994729
$263$	−5227.73	−1.22569	−0.612844	−	0.790204i	$-0.709975\pi$
$263$	−5227.73	−1.22569	−0.612844	+	0.790204i	$0.709975\pi$
$264$	934.708	0.217906
$265$	908.120	0.210511
$266$	3600.01	0.829814
$267$	−1141.45	−0.261632
$268$	−147.583	−0.0336384
$269$	2769.76	0.627790	0.313895	−	0.949458i	$-0.398366\pi$
$269$	2769.76	0.627790	0.313895	+	0.949458i	$0.398366\pi$
$270$	−519.050	−0.116994
$271$	−894.528	−0.200512	−0.100256	−	0.994962i	$-0.531966\pi$
$271$	−894.528	−0.200512	−0.100256	+	0.994962i	$0.531966\pi$
$272$	−1662.66	−0.370639
$273$	−200.218	−0.0443873
$274$	3465.74	0.764135
$275$	1191.69	0.261315
$276$	−57.1797	−0.0124703
$277$	372.530	0.0808057	0.0404028	−	0.999183i	$-0.487136\pi$
$277$	372.530	0.0808057	0.0404028	+	0.999183i	$0.487136\pi$
$278$	−6531.24	−1.40906
$279$	−817.183	−0.175353
$280$	−2362.18	−0.504168
$281$	−5334.65	−1.13252	−0.566261	−	0.824226i	$-0.691610\pi$
$281$	−5334.65	−1.13252	−0.566261	+	0.824226i	$0.691610\pi$
$282$	−553.385	−0.116857
$283$	−3700.27	−0.777238	−0.388619	−	0.921398i	$-0.627048\pi$
$283$	−3700.27	−0.777238	−0.388619	+	0.921398i	$0.627048\pi$
$284$	−2127.27	−0.444473
$285$	307.044	0.0638166
$286$	1507.82	0.311745
$287$	−6563.89	−1.35001
$288$	2411.39	0.493376
$289$	−3420.72	−0.696258
$290$	−1828.78	−0.370310
$291$	894.567	0.180208
$292$	−131.647	−0.0263837
$293$	7341.13	1.46373	0.731866	−	0.681449i	$-0.238650\pi$
$293$	7341.13	1.46373	0.731866	+	0.681449i	$0.238650\pi$
$294$	−54.5981	−0.0108307
$295$	3206.39	0.632825
$296$	6213.90	1.22019
$297$	−2033.67	−0.397325
$298$	3500.10	0.680388
$299$	−447.103	−0.0864771
$300$	−41.5640	−0.00799900
$301$	−1928.80	−0.369350
$302$	−180.920	−0.0344727
$303$	−730.496	−0.138501
$304$	3305.79	0.623684
$305$	1741.33	0.326911
$306$	2477.79	0.462896
$307$	5704.82	1.06056	0.530279	−	0.847823i	$-0.322087\pi$
$307$	5704.82	1.06056	0.530279	+	0.847823i	$0.322087\pi$
$308$	−1909.37	−0.353236
$309$	552.453	0.101709
$310$	377.150	0.0690989
$311$	2826.09	0.515283	0.257642	−	0.966241i	$-0.417055\pi$
$311$	2826.09	0.515283	0.257642	+	0.966241i	$0.417055\pi$
$312$	−254.915	−0.0462556
$313$	−1544.27	−0.278872	−0.139436	−	0.990231i	$-0.544529\pi$
$313$	−1544.27	−0.278872	−0.139436	+	0.990231i	$0.544529\pi$
$314$	−6567.64	−1.18036
$315$	2538.94	0.454137
$316$	0.0815862	1.45240e−5 0
$317$	−1915.82	−0.339442	−0.169721	−	0.985492i	$-0.554287\pi$
$317$	−1915.82	−0.339442	−0.169721	+	0.985492i	$0.554287\pi$
$318$	353.338	0.0623089
$319$	−7165.29	−1.25761
$320$	−2834.54	−0.495173
$321$	134.439	0.0233759
$322$	−1612.02	−0.278989
$323$	−2967.03	−0.511114
$324$	−1409.08	−0.241611
$325$	−325.000	−0.0554700
$326$	2330.46	0.395927
$327$	−598.848	−0.101273
$328$	−8357.07	−1.40684
$329$	5479.43	0.918209
$330$	463.671	0.0773463
$331$	−3346.86	−0.555771	−0.277885	−	0.960614i	$-0.589634\pi$
$331$	−3346.86	−0.555771	−0.277885	+	0.960614i	$0.589634\pi$
$332$	−1008.73	−0.166751
$333$	−6678.89	−1.09910
$334$	8282.15	1.35682
$335$	−354.866	−0.0578758
$336$	−662.885	−0.107629
$337$	5386.87	0.870746	0.435373	−	0.900250i	$-0.356617\pi$
$337$	5386.87	0.870746	0.435373	+	0.900250i	$0.356617\pi$
$338$	−411.215	−0.0661750
$339$	1256.68	0.201338
$340$	401.641	0.0640649
$341$	1477.70	0.234668
$342$	−4926.47	−0.778927
$343$	−6066.61	−0.955004
$344$	−2455.73	−0.384896
$345$	−137.489	−0.0214556
$346$	−7286.35	−1.13213
$347$	5128.05	0.793338	0.396669	−	0.917962i	$-0.370166\pi$
$347$	5128.05	0.793338	0.396669	+	0.917962i	$0.370166\pi$
$348$	249.912	0.0384963
$349$	6800.04	1.04297	0.521487	−	0.853260i	$-0.325378\pi$
$349$	6800.04	1.04297	0.521487	+	0.853260i	$0.325378\pi$
$350$	−1171.78	−0.178955
$351$	554.626	0.0843412
$352$	−4360.46	−0.660266
$353$	−5261.30	−0.793288	−0.396644	−	0.917972i	$-0.629825\pi$
$353$	−5261.30	−0.793288	−0.396644	+	0.917972i	$0.629825\pi$
$354$	1247.57	0.187309
$355$	−5115.06	−0.764730
$356$	2968.69	0.441967
$357$	594.957	0.0882029
$358$	−1698.52	−0.250752
$359$	−3606.08	−0.530144	−0.265072	−	0.964229i	$-0.585396\pi$
$359$	−3606.08	−0.530144	−0.265072	+	0.964229i	$0.585396\pi$
$360$	3232.55	0.473251
$361$	−959.808	−0.139934
$362$	7375.42	1.07084
$363$	752.519	0.108807
$364$	520.728	0.0749823
$365$	−316.547	−0.0453940
$366$	677.527	0.0967621
$367$	−7814.75	−1.11152	−0.555758	−	0.831344i	$-0.687572\pi$
$367$	−7814.75	−1.11152	−0.555758	+	0.831344i	$0.687572\pi$
$368$	−1480.28	−0.209687
$369$	8982.44	1.26723
$370$	3082.47	0.433108
$371$	−3498.63	−0.489595
$372$	−51.5394	−0.00718332
$373$	−9778.42	−1.35739	−0.678696	−	0.734419i	$-0.737455\pi$
$373$	−9778.42	−1.35739	−0.678696	+	0.734419i	$0.737455\pi$
$374$	−4480.55	−0.619475
$375$	−99.9413	−0.0137625
$376$	6976.35	0.956856
$377$	1954.13	0.266957
$378$	1999.69	0.272098
$379$	−11839.0	−1.60456	−0.802278	−	0.596951i	$-0.796379\pi$
$379$	−11839.0	−1.60456	−0.802278	+	0.596951i	$0.796379\pi$
$380$	−798.563	−0.107804
$381$	892.575	0.120021
$382$	10873.6	1.45639
$383$	2471.78	0.329770	0.164885	−	0.986313i	$-0.447275\pi$
$383$	2471.78	0.329770	0.164885	+	0.986313i	$0.447275\pi$
$384$	−517.777	−0.0688091
$385$	−4591.12	−0.607753
$386$	2315.96	0.305387
$387$	2639.50	0.346701
$388$	−2326.59	−0.304420
$389$	−3913.62	−0.510099	−0.255049	−	0.966928i	$-0.582092\pi$
$389$	−3913.62	−0.510099	−0.255049	+	0.966928i	$0.582092\pi$
$390$	−126.453	−0.0164185
$391$	1328.59	0.171840
$392$	688.300	0.0886847
$393$	−138.615	−0.0177918
$394$	−4157.93	−0.531659
$395$	0.196175	2.49890e−5 0
$396$	2612.91	0.331574
$397$	−12911.7	−1.63229	−0.816144	−	0.577849i	$-0.803892\pi$
$397$	−12911.7	−1.63229	−0.816144	+	0.577849i	$0.803892\pi$
$398$	12906.8	1.62552
$399$	−1182.92	−0.148421
$400$	−1076.02	−0.134502
$401$	−540.973	−0.0673689	−0.0336844	−	0.999433i	$-0.510724\pi$
$401$	−540.973	−0.0673689	−0.0336844	+	0.999433i	$0.510724\pi$
$402$	−138.074	−0.0171306
$403$	−403.000	−0.0498135
$404$	1899.88	0.233967
$405$	−3388.15	−0.415700
$406$	7045.58	0.861248
$407$	12077.3	1.47088
$408$	757.492	0.0919153
$409$	−5675.37	−0.686135	−0.343067	−	0.939311i	$-0.611466\pi$
$409$	−5675.37	−0.686135	−0.343067	+	0.939311i	$0.611466\pi$
$410$	−4145.61	−0.499359
$411$	−1138.80	−0.136674
$412$	−1436.82	−0.171813
$413$	−12353.0	−1.47179
$414$	2205.99	0.261881
$415$	−2425.51	−0.286900
$416$	1189.19	0.140156
$417$	2146.09	0.252025
$418$	8908.44	1.04241
$419$	11386.8	1.32764	0.663819	−	0.747894i	$-0.268935\pi$
$419$	11386.8	1.32764	0.663819	+	0.747894i	$0.268935\pi$
$420$	160.130	0.0186037
$421$	−6068.07	−0.702469	−0.351235	−	0.936287i	$-0.614238\pi$
$421$	−6068.07	−0.702469	−0.351235	+	0.936287i	$0.614238\pi$
$422$	3950.96	0.455758
$423$	−7498.40	−0.861902
$424$	−4454.42	−0.510202
$425$	965.752	0.110226
$426$	−1990.20	−0.226351
$427$	−6708.64	−0.760314
$428$	−349.650	−0.0394883
$429$	−495.452	−0.0557591
$430$	−1218.19	−0.136619
$431$	−8095.56	−0.904755	−0.452377	−	0.891827i	$-0.649424\pi$
$431$	−8095.56	−0.904755	−0.452377	+	0.891827i	$0.649424\pi$
$432$	1836.27	0.204508
$433$	−2638.53	−0.292840	−0.146420	−	0.989223i	$-0.546775\pi$
$433$	−2638.53	−0.292840	−0.146420	+	0.989223i	$0.546775\pi$
$434$	−1453.01	−0.160707
$435$	600.918	0.0662340
$436$	1557.49	0.171078
$437$	−2641.56	−0.289160
$438$	−123.164	−0.0134361
$439$	632.411	0.0687547	0.0343774	−	0.999409i	$-0.489055\pi$
$439$	632.411	0.0687547	0.0343774	+	0.999409i	$0.489055\pi$
$440$	−5845.36	−0.633333
$441$	−739.806	−0.0798841
$442$	1221.94	0.131497
$443$	−5775.92	−0.619463	−0.309732	−	0.950824i	$-0.600239\pi$
$443$	−5775.92	−0.619463	−0.309732	+	0.950824i	$0.600239\pi$
$444$	−421.235	−0.0450246
$445$	7138.26	0.760418
$446$	8528.62	0.905475
$447$	−1150.10	−0.121695
$448$	10920.4	1.15165
$449$	17050.2	1.79209	0.896045	−	0.443964i	$-0.146428\pi$
$449$	17050.2	1.79209	0.896045	+	0.443964i	$0.146428\pi$
$450$	1603.54	0.167981
$451$	−16242.8	−1.69588
$452$	−3268.38	−0.340114
$453$	59.4482	0.00616583
$454$	−282.471	−0.0292005
$455$	1252.10	0.129009
$456$	−1506.08	−0.154668
$457$	10840.8	1.10966	0.554829	−	0.831965i	$-0.312784\pi$
$457$	10840.8	1.10966	0.554829	+	0.831965i	$0.312784\pi$
$458$	2490.04	0.254044
$459$	−1648.10	−0.167596
$460$	357.583	0.0362444
$461$	−1780.14	−0.179847	−0.0899233	−	0.995949i	$-0.528662\pi$
$461$	−1780.14	−0.179847	−0.0899233	+	0.995949i	$0.528662\pi$
$462$	−1786.34	−0.179888
$463$	−14032.0	−1.40847	−0.704235	−	0.709967i	$-0.748710\pi$
$463$	−14032.0	−1.40847	−0.704235	+	0.709967i	$0.748710\pi$
$464$	6469.77	0.647309
$465$	−123.927	−0.0123591
$466$	−3725.77	−0.370371
$467$	−19815.7	−1.96351	−0.981755	−	0.190150i	$-0.939103\pi$
$467$	−19815.7	−1.96351	−0.981755	+	0.190150i	$0.939103\pi$
$468$	−712.597	−0.0703842
$469$	1367.16	0.134605
$470$	3460.69	0.339638
$471$	2158.06	0.211121
$472$	−15727.7	−1.53374
$473$	−4772.95	−0.463975
$474$	0.0763292	7.39645e−6 0
$475$	−1920.15	−0.185479
$476$	−1547.37	−0.148999
$477$	4787.75	0.459572
$478$	10453.3	1.00026
$479$	−8334.49	−0.795016	−0.397508	−	0.917599i	$-0.630125\pi$
$479$	−8334.49	−0.795016	−0.397508	+	0.917599i	$0.630125\pi$
$480$	365.691	0.0347738
$481$	−3293.74	−0.312228
$482$	7505.40	0.709256
$483$	529.693	0.0499003
$484$	−1957.15	−0.183805
$485$	−5594.33	−0.523763
$486$	−4121.15	−0.384649
$487$	−13117.4	−1.22054	−0.610271	−	0.792192i	$-0.708940\pi$
$487$	−13117.4	−1.22054	−0.610271	+	0.792192i	$0.708940\pi$
$488$	−8541.37	−0.792315
$489$	−765.762	−0.0708159
$490$	341.438	0.0314788
$491$	−6355.06	−0.584114	−0.292057	−	0.956401i	$-0.594340\pi$
$491$	−6355.06	−0.584114	−0.292057	+	0.956401i	$0.594340\pi$
$492$	566.518	0.0519118
$493$	−5806.78	−0.530475
$494$	−2429.53	−0.221274
$495$	6282.77	0.570484
$496$	−1334.26	−0.120786
$497$	19706.3	1.77857
$498$	−943.734	−0.0849191
$499$	−15223.0	−1.36568	−0.682840	−	0.730568i	$-0.739255\pi$
$499$	−15223.0	−1.36568	−0.682840	+	0.730568i	$0.739255\pi$
$500$	259.928	0.0232486
$501$	−2721.42	−0.242683
$502$	−14305.2	−1.27185
$503$	−3963.43	−0.351334	−0.175667	−	0.984450i	$-0.556208\pi$
$503$	−3963.43	−0.351334	−0.175667	+	0.984450i	$0.556208\pi$
$504$	−12453.8	−1.10066
$505$	4568.28	0.402546
$506$	−3989.05	−0.350465
$507$	135.121	0.0118361
$508$	−2321.41	−0.202748
$509$	16046.2	1.39732	0.698660	−	0.715454i	$-0.253780\pi$
$509$	16046.2	1.39732	0.698660	+	0.715454i	$0.253780\pi$
$510$	375.762	0.0326255
$511$	1219.53	0.105575
$512$	12381.9	1.06877
$513$	3276.82	0.282018
$514$	−1023.03	−0.0877900
$515$	−3454.86	−0.295610
$516$	166.472	0.0142025
$517$	13559.2	1.15345
$518$	−11875.5	−1.00730
$519$	2394.21	0.202494
$520$	1594.16	0.134439
$521$	7719.46	0.649128	0.324564	−	0.945864i	$-0.394782\pi$
$521$	7719.46	0.649128	0.324564	+	0.945864i	$0.394782\pi$
$522$	−9641.62	−0.808433
$523$	−3106.98	−0.259768	−0.129884	−	0.991529i	$-0.541460\pi$
$523$	−3106.98	−0.259768	−0.129884	+	0.991529i	$0.541460\pi$
$524$	360.510	0.0300553
$525$	385.035	0.0320082
$526$	12720.2	1.05443
$527$	1197.53	0.0989854
$528$	−1640.35	−0.135203
$529$	−10984.2	−0.902782
$530$	−2209.66	−0.181097
$531$	16904.6	1.38154
$532$	3076.55	0.250724
$533$	4429.76	0.359989
$534$	2777.41	0.225075
$535$	−840.739	−0.0679408
$536$	1740.65	0.140270
$537$	558.113	0.0448498
$538$	−6739.46	−0.540072
$539$	1337.78	0.106906
$540$	−443.578	−0.0353491
$541$	−17982.6	−1.42908	−0.714540	−	0.699595i	$-0.753364\pi$
$541$	−17982.6	−1.42908	−0.714540	+	0.699595i	$0.753364\pi$
$542$	2176.59	0.172495
$543$	−2423.48	−0.191531
$544$	−3533.74	−0.278507
$545$	3745.00	0.294345
$546$	487.175	0.0381853
$547$	3742.00	0.292498	0.146249	−	0.989248i	$-0.453280\pi$
$547$	3742.00	0.292498	0.146249	+	0.989248i	$0.453280\pi$
$548$	2961.81	0.230880
$549$	9180.53	0.713689
$550$	−2899.65	−0.224803
$551$	11545.3	0.892646
$552$	674.399	0.0520006
$553$	−0.755786	−5.81180e−5 0
$554$	−906.450	−0.0695151
$555$	−1012.86	−0.0774662
$556$	−5581.57	−0.425740
$557$	−6001.59	−0.456545	−0.228272	−	0.973597i	$-0.573308\pi$
$557$	−6001.59	−0.456545	−0.228272	+	0.973597i	$0.573308\pi$
$558$	1988.39	0.150852
$559$	1301.69	0.0984892
$560$	4145.47	0.312818
$561$	1472.26	0.110800
$562$	12980.4	0.974279
$563$	−16503.5	−1.23541	−0.617707	−	0.786409i	$-0.711938\pi$
$563$	−16503.5	−1.23541	−0.617707	+	0.786409i	$0.711938\pi$
$564$	−472.921	−0.0353077
$565$	−7858.86	−0.585177
$566$	9003.59	0.668638
$567$	13053.2	0.966813
$568$	25089.9	1.85343
$569$	5734.98	0.422536	0.211268	−	0.977428i	$-0.432241\pi$
$569$	5734.98	0.422536	0.211268	+	0.977428i	$0.432241\pi$
$570$	−747.108	−0.0548998
$571$	23012.3	1.68658	0.843288	−	0.537461i	$-0.180617\pi$
$571$	23012.3	1.68658	0.843288	+	0.537461i	$0.180617\pi$
$572$	1288.57	0.0941923
$573$	−3572.94	−0.260492
$574$	15971.4	1.16138
$575$	859.814	0.0623595
$576$	−14944.1	−1.08103
$577$	−5831.18	−0.420719	−0.210360	−	0.977624i	$-0.567463\pi$
$577$	−5831.18	−0.420719	−0.210360	+	0.977624i	$0.567463\pi$
$578$	8323.37	0.598973
$579$	−761.000	−0.0546219
$580$	−1562.87	−0.111887
$581$	9344.53	0.667257
$582$	−2176.68	−0.155028
$583$	−8657.59	−0.615027
$584$	1552.69	0.110019
$585$	−1713.45	−0.121098
$586$	−17862.6	−1.25921
$587$	−20527.7	−1.44339	−0.721695	−	0.692211i	$-0.756637\pi$
$587$	−20527.7	−1.44339	−0.721695	+	0.692211i	$0.756637\pi$
$588$	−46.6593	−0.00327244
$589$	−2380.99	−0.166565
$590$	−7801.87	−0.544403
$591$	1366.25	0.0950931
$592$	−10905.0	−0.757081
$593$	1828.87	0.126648	0.0633242	−	0.997993i	$-0.479830\pi$
$593$	1828.87	0.126648	0.0633242	+	0.997993i	$0.479830\pi$
$594$	4948.37	0.341808
$595$	−3720.66	−0.256357
$596$	2991.17	0.205576
$597$	−4241.02	−0.290743
$598$	1087.90	0.0743940
$599$	9127.58	0.622609	0.311305	−	0.950310i	$-0.399234\pi$
$599$	9127.58	0.622609	0.311305	+	0.950310i	$0.399234\pi$
$600$	490.222	0.0333554
$601$	−10796.3	−0.732761	−0.366380	−	0.930465i	$-0.619403\pi$
$601$	−10796.3	−0.732761	−0.366380	+	0.930465i	$0.619403\pi$
$602$	4693.21	0.317742
$603$	−1870.91	−0.126350
$604$	−154.613	−0.0104158
$605$	−4706.01	−0.316242
$606$	1777.46	0.119149
$607$	−25025.3	−1.67339	−0.836694	−	0.547671i	$-0.815514\pi$
$607$	−25025.3	−1.67339	−0.836694	+	0.547671i	$0.815514\pi$
$608$	7025.96	0.468652
$609$	−2315.10	−0.154044
$610$	−4237.03	−0.281234
$611$	−3697.89	−0.244846
$612$	2117.51	0.139862
$613$	−9752.40	−0.642571	−0.321285	−	0.946982i	$-0.604115\pi$
$613$	−9752.40	−0.642571	−0.321285	+	0.946982i	$0.604115\pi$
$614$	−13881.1	−0.912371
$615$	1362.20	0.0893159
$616$	22519.9	1.47297
$617$	21293.3	1.38936	0.694679	−	0.719320i	$-0.255546\pi$
$617$	21293.3	1.38936	0.694679	+	0.719320i	$0.255546\pi$
$618$	−1344.24	−0.0874973
$619$	7368.13	0.478433	0.239217	−	0.970966i	$-0.423109\pi$
$619$	7368.13	0.478433	0.239217	+	0.970966i	$0.423109\pi$
$620$	322.310	0.0208779
$621$	−1467.31	−0.0948165
$622$	−6876.52	−0.443285
$623$	−27500.9	−1.76854
$624$	447.360	0.0286999
$625$	625.000	0.0400000
$626$	3757.54	0.239907
$627$	−2927.21	−0.186446
$628$	−5612.68	−0.356641
$629$	9787.51	0.620435
$630$	−6177.81	−0.390682
$631$	9791.01	0.617709	0.308854	−	0.951109i	$-0.400054\pi$
$631$	9791.01	0.617709	0.308854	+	0.951109i	$0.400054\pi$
$632$	−0.962258	−6.05642e−5 0
$633$	−1298.24	−0.0815174
$634$	4661.61	0.292013
$635$	−5581.87	−0.348834
$636$	301.961	0.0188263
$637$	−364.841	−0.0226931
$638$	17434.7	1.08189
$639$	−26967.3	−1.66950
$640$	3238.01	0.199990
$641$	−3140.27	−0.193500	−0.0967498	−	0.995309i	$-0.530845\pi$
$641$	−3140.27	−0.193500	−0.0967498	+	0.995309i	$0.530845\pi$
$642$	−327.121	−0.0201097
$643$	12720.5	0.780170	0.390085	−	0.920779i	$-0.372446\pi$
$643$	12720.5	0.780170	0.390085	+	0.920779i	$0.372446\pi$
$644$	−1377.63	−0.0842953
$645$	400.284	0.0244359
$646$	7219.44	0.439699
$647$	−19812.7	−1.20389	−0.601946	−	0.798537i	$-0.705608\pi$
$647$	−19812.7	−1.20389	−0.601946	+	0.798537i	$0.705608\pi$
$648$	16619.2	1.00751
$649$	−30568.2	−1.84886
$650$	790.798	0.0477194
$651$	477.443	0.0287442
$652$	1991.60	0.119627
$653$	−12431.9	−0.745017	−0.372509	−	0.928029i	$-0.621502\pi$
$653$	−12431.9	−0.745017	−0.372509	+	0.928029i	$0.621502\pi$
$654$	1457.13	0.0871228
$655$	866.851	0.0517110
$656$	14666.1	0.872889
$657$	−1668.88	−0.0991009
$658$	−13332.7	−0.789912
$659$	−20850.0	−1.23247	−0.616237	−	0.787561i	$-0.711344\pi$
$659$	−20850.0	−1.23247	−0.616237	+	0.787561i	$0.711344\pi$
$660$	396.252	0.0233698
$661$	−7285.01	−0.428675	−0.214337	−	0.976760i	$-0.568759\pi$
$661$	−7285.01	−0.428675	−0.214337	+	0.976760i	$0.568759\pi$
$662$	8143.66	0.478115
$663$	−401.517	−0.0235198
$664$	11897.3	0.695341
$665$	7397.60	0.431379
$666$	16251.2	0.945530
$667$	−5169.81	−0.300114
$668$	7077.89	0.409958
$669$	−2802.41	−0.161954
$670$	863.469	0.0497891
$671$	−16601.0	−0.955101
$672$	−1408.86	−0.0808752
$673$	−6137.91	−0.351559	−0.175779	−	0.984430i	$-0.556244\pi$
$673$	−6137.91	−0.351559	−0.175779	+	0.984430i	$0.556244\pi$
$674$	−13107.5	−0.749080
$675$	−1066.59	−0.0608193
$676$	−351.422	−0.0199944
$677$	20241.3	1.14910	0.574548	−	0.818471i	$-0.305178\pi$
$677$	20241.3	1.14910	0.574548	+	0.818471i	$0.305178\pi$
$678$	−3057.78	−0.173206
$679$	21552.7	1.21814
$680$	−4737.11	−0.267147
$681$	92.8168	0.00522283
$682$	−3595.56	−0.201879
$683$	−11318.1	−0.634080	−0.317040	−	0.948412i	$-0.602689\pi$
$683$	−11318.1	−0.634080	−0.317040	+	0.948412i	$0.602689\pi$
$684$	−4210.14	−0.235349
$685$	7121.70	0.397235
$686$	14761.4	0.821565
$687$	−818.200	−0.0454385
$688$	4309.65	0.238814
$689$	2361.11	0.130553
$690$	334.543	0.0184577
$691$	11761.4	0.647503	0.323751	−	0.946142i	$-0.395056\pi$
$691$	11761.4	0.647503	0.323751	+	0.946142i	$0.395056\pi$
$692$	−6226.88	−0.342067
$693$	−24205.1	−1.32680
$694$	−12477.7	−0.682488
$695$	−13421.0	−0.732498
$696$	−2947.56	−0.160527
$697$	−13163.2	−0.715340
$698$	−16546.0	−0.897243
$699$	1224.25	0.0662450
$700$	−1001.40	−0.0540705
$701$	15473.1	0.833679	0.416840	−	0.908980i	$-0.363138\pi$
$701$	15473.1	0.833679	0.416840	+	0.908980i	$0.363138\pi$
$702$	−1349.53	−0.0725565
$703$	−19460.0	−1.04402
$704$	27023.1	1.44669
$705$	−1137.14	−0.0607480
$706$	12801.9	0.682446
$707$	−17599.8	−0.936222
$708$	1066.16	0.0565945
$709$	26562.3	1.40701	0.703504	−	0.710691i	$-0.251618\pi$
$709$	26562.3	1.40701	0.703504	+	0.710691i	$0.251618\pi$
$710$	12446.1	0.657877
$711$	1.03426	5.45541e−5 0
$712$	−35013.9	−1.84298
$713$	1066.17	0.0560005
$714$	−1447.66	−0.0758787
$715$	3098.39	0.162061
$716$	−1451.54	−0.0757636
$717$	−3434.85	−0.178908
$718$	8774.40	0.456069
$719$	−29186.8	−1.51389	−0.756943	−	0.653481i	$-0.773308\pi$
$719$	−29186.8	−1.51389	−0.756943	+	0.653481i	$0.773308\pi$
$720$	−5672.92	−0.293635
$721$	13310.2	0.687515
$722$	2335.43	0.120382
$723$	−2466.19	−0.126858
$724$	6303.00	0.323549
$725$	−3757.94	−0.192506
$726$	−1831.05	−0.0936040
$727$	17142.8	0.874540	0.437270	−	0.899330i	$-0.355945\pi$
$727$	17142.8	0.874540	0.437270	+	0.899330i	$0.355945\pi$
$728$	−6141.66	−0.312672
$729$	−16941.8	−0.860734
$730$	770.229	0.0390513
$731$	−3868.02	−0.195710
$732$	579.012	0.0292362
$733$	14732.9	0.742393	0.371196	−	0.928554i	$-0.378948\pi$
$733$	14732.9	0.742393	0.371196	+	0.928554i	$0.378948\pi$
$734$	19015.0	0.956209
$735$	−112.193	−0.00563033
$736$	−3146.11	−0.157564
$737$	3383.12	0.169090
$738$	−21856.3	−1.09016
$739$	−26808.1	−1.33444	−0.667220	−	0.744860i	$-0.732516\pi$
$739$	−26808.1	−1.33444	−0.667220	+	0.744860i	$0.732516\pi$
$740$	2634.26	0.130861
$741$	798.316	0.0395774
$742$	8512.95	0.421187
$743$	5539.16	0.273502	0.136751	−	0.990605i	$-0.456334\pi$
$743$	5539.16	0.273502	0.136751	+	0.990605i	$0.456334\pi$
$744$	607.875	0.0299540
$745$	7192.32	0.353700
$746$	23793.1	1.16773
$747$	−12787.6	−0.626339
$748$	−3829.05	−0.187171
$749$	3239.04	0.158013
$750$	243.180	0.0118395
$751$	28660.5	1.39259	0.696295	−	0.717756i	$-0.254830\pi$
$751$	28660.5	1.39259	0.696295	+	0.717756i	$0.254830\pi$
$752$	−12243.0	−0.593694
$753$	4700.52	0.227485
$754$	−4754.84	−0.229656
$755$	−371.770	−0.0179206
$756$	1708.93	0.0822132
$757$	−3078.10	−0.147788	−0.0738940	−	0.997266i	$-0.523543\pi$
$757$	−3078.10	−0.147788	−0.0738940	+	0.997266i	$0.523543\pi$
$758$	28806.8	1.38036
$759$	1310.76	0.0626845
$760$	9418.55	0.449535
$761$	30928.2	1.47325	0.736626	−	0.676300i	$-0.236418\pi$
$761$	30928.2	1.47325	0.736626	+	0.676300i	$0.236418\pi$
$762$	−2171.84	−0.103251
$763$	−14428.0	−0.684572
$764$	9292.52	0.440041
$765$	5091.59	0.240636
$766$	−6014.39	−0.283693
$767$	8336.62	0.392461
$768$	−2366.21	−0.111176
$769$	−216.326	−0.0101442	−0.00507211	−	0.999987i	$-0.501615\pi$
$769$	−216.326	−0.0101442	−0.00507211	+	0.999987i	$0.501615\pi$
$770$	11171.2	0.522834
$771$	336.157	0.0157022
$772$	1979.21	0.0922713
$773$	33390.8	1.55367	0.776833	−	0.629707i	$-0.216825\pi$
$773$	33390.8	1.55367	0.776833	+	0.629707i	$0.216825\pi$
$774$	−6422.48	−0.298258
$775$	775.000	0.0359211
$776$	27440.7	1.26941
$777$	3902.17	0.180167
$778$	9522.71	0.438825
$779$	26171.7	1.20372
$780$	−108.066	−0.00496077
$781$	48764.5	2.23423
$782$	−3232.75	−0.147830
$783$	6413.09	0.292701
$784$	−1207.92	−0.0550256
$785$	−13495.8	−0.613611
$786$	337.281	0.0153059
$787$	20399.0	0.923949	0.461974	−	0.886893i	$-0.347141\pi$
$787$	20399.0	0.923949	0.461974	+	0.886893i	$0.347141\pi$
$788$	−3553.35	−0.160638
$789$	−4179.73	−0.188596
$790$	−0.477338	−2.14974e−5 0
$791$	30277.1	1.36097
$792$	−30817.6	−1.38265
$793$	4527.45	0.202742
$794$	31417.0	1.40422
$795$	726.070	0.0323913
$796$	11030.1	0.491144
$797$	−24558.5	−1.09148	−0.545738	−	0.837956i	$-0.683751\pi$
$797$	−24558.5	−1.09148	−0.545738	+	0.837956i	$0.683751\pi$
$798$	2878.31	0.127683
$799$	10988.4	0.486537
$800$	−2286.91	−0.101068
$801$	37634.0	1.66009
$802$	1316.31	0.0579557
$803$	3017.81	0.132623
$804$	−117.997	−0.00517593
$805$	−3312.53	−0.145033
$806$	980.589	0.0428533
$807$	2214.51	0.0965979
$808$	−22407.9	−0.975627
$809$	−10557.3	−0.458807	−0.229404	−	0.973331i	$-0.573678\pi$
$809$	−10557.3	−0.458807	−0.229404	+	0.973331i	$0.573678\pi$
$810$	8244.12	0.357616
$811$	−5531.15	−0.239488	−0.119744	−	0.992805i	$-0.538207\pi$
$811$	−5531.15	−0.239488	−0.119744	+	0.992805i	$0.538207\pi$
$812$	6021.13	0.260222
$813$	−715.202	−0.0308527
$814$	−29386.8	−1.26536
$815$	4788.83	0.205822
$816$	−1329.35	−0.0570300
$817$	7690.59	0.329326
$818$	13809.5	0.590264
$819$	6601.25	0.281644
$820$	−3542.82	−0.150879
$821$	2698.59	0.114716	0.0573579	−	0.998354i	$-0.481732\pi$
$821$	2698.59	0.114716	0.0573579	+	0.998354i	$0.481732\pi$
$822$	2770.96	0.117577
$823$	18564.3	0.786284	0.393142	−	0.919478i	$-0.371388\pi$
$823$	18564.3	0.786284	0.393142	+	0.919478i	$0.371388\pi$
$824$	16946.4	0.716452
$825$	952.793	0.0402085
$826$	30057.6	1.26615
$827$	27976.3	1.17634	0.588169	−	0.808738i	$-0.299849\pi$
$827$	27976.3	1.17634	0.588169	+	0.808738i	$0.299849\pi$
$828$	1885.23	0.0791260
$829$	32664.1	1.36848	0.684241	−	0.729256i	$-0.260134\pi$
$829$	32664.1	1.36848	0.684241	+	0.729256i	$0.260134\pi$
$830$	5901.80	0.246813
$831$	297.849	0.0124335
$832$	−7369.80	−0.307094
$833$	1084.14	0.0450939
$834$	−5221.92	−0.216811
$835$	17018.9	0.705345
$836$	7613.11	0.314958
$837$	−1322.57	−0.0546173
$838$	−27706.6	−1.14213
$839$	−3015.89	−0.124100	−0.0620502	−	0.998073i	$-0.519764\pi$
$839$	−3015.89	−0.124100	−0.0620502	+	0.998073i	$0.519764\pi$
$840$	−1888.63	−0.0775762
$841$	−1793.58	−0.0735406
$842$	14765.0	0.604316
$843$	−4265.21	−0.174261
$844$	3376.48	0.137705
$845$	−845.000	−0.0344010
$846$	18245.3	0.741473
$847$	18130.4	0.735499
$848$	7817.21	0.316562
$849$	−2958.48	−0.119593
$850$	−2349.89	−0.0948242
$851$	8713.87	0.351008
$852$	−1700.82	−0.0683910
$853$	29897.5	1.20008	0.600041	−	0.799969i	$-0.295151\pi$
$853$	29897.5	1.20008	0.600041	+	0.799969i	$0.295151\pi$
$854$	16323.6	0.654079
$855$	−10123.3	−0.404925
$856$	4123.91	0.164664
$857$	29029.5	1.15709	0.578546	−	0.815650i	$-0.303620\pi$
$857$	29029.5	1.15709	0.578546	+	0.815650i	$0.303620\pi$
$858$	1205.55	0.0479681
$859$	34912.4	1.38672	0.693362	−	0.720589i	$-0.256129\pi$
$859$	34912.4	1.38672	0.693362	+	0.720589i	$0.256129\pi$
$860$	−1041.06	−0.0412789
$861$	−5248.03	−0.207726
$862$	19698.3	0.778337
$863$	−6875.02	−0.271180	−0.135590	−	0.990765i	$-0.543293\pi$
$863$	−6875.02	−0.271180	−0.135590	+	0.990765i	$0.543293\pi$
$864$	3902.71	0.153672
$865$	−14972.6	−0.588537
$866$	6420.13	0.251923
$867$	−2734.97	−0.107133
$868$	−1241.74	−0.0485568
$869$	−1.87024	−7.30075e−5 0
$870$	−1462.17	−0.0569795
$871$	−922.652	−0.0358931
$872$	−18369.6	−0.713385
$873$	−29494.1	−1.14344
$874$	6427.51	0.248757
$875$	−2407.88	−0.0930299
$876$	−105.256	−0.00405966
$877$	−2271.63	−0.0874656	−0.0437328	−	0.999043i	$-0.513925\pi$
$877$	−2271.63	−0.0874656	−0.0437328	+	0.999043i	$0.513925\pi$
$878$	−1538.80	−0.0591479
$879$	5869.46	0.225224
$880$	10258.2	0.392960
$881$	28161.0	1.07692	0.538460	−	0.842651i	$-0.319006\pi$
$881$	28161.0	1.07692	0.538460	+	0.842651i	$0.319006\pi$
$882$	1800.11	0.0687222
$883$	−13317.7	−0.507560	−0.253780	−	0.967262i	$-0.581674\pi$
$883$	−13317.7	−0.507560	−0.253780	+	0.967262i	$0.581674\pi$
$884$	1044.27	0.0397313
$885$	2563.61	0.0973726
$886$	14054.1	0.532908
$887$	−33376.1	−1.26343	−0.631714	−	0.775202i	$-0.717648\pi$
$887$	−33376.1	−1.26343	−0.631714	+	0.775202i	$0.717648\pi$
$888$	4968.20	0.187750
$889$	21504.8	0.811301
$890$	−17369.0	−0.654169
$891$	32301.0	1.21450
$892$	7288.52	0.273585
$893$	−21847.8	−0.818710
$894$	2798.44	0.104691
$895$	−3490.26	−0.130354
$896$	−12474.8	−0.465126
$897$	−357.473	−0.0133062
$898$	−41486.9	−1.54169
$899$	−4659.85	−0.172875
$900$	1370.38	0.0507548
$901$	−7016.15	−0.259425
$902$	39522.3	1.45892
$903$	−1542.14	−0.0568318
$904$	38548.5	1.41826
$905$	15155.7	0.556675
$906$	−144.651	−0.00530431
$907$	559.717	0.0204907	0.0102454	−	0.999948i	$-0.496739\pi$
$907$	559.717	0.0204907	0.0102454	+	0.999948i	$0.496739\pi$
$908$	−241.398	−0.00882279
$909$	24084.7	0.878810
$910$	−3046.63	−0.110983
$911$	−28644.3	−1.04174	−0.520871	−	0.853635i	$-0.674393\pi$
$911$	−28644.3	−1.04174	−0.520871	+	0.853635i	$0.674393\pi$
$912$	2643.08	0.0959660
$913$	23123.6	0.838204
$914$	−26378.2	−0.954610
$915$	1392.24	0.0503018
$916$	2127.98	0.0767581
$917$	−3339.64	−0.120267
$918$	4010.18	0.144178
$919$	−9574.99	−0.343688	−0.171844	−	0.985124i	$-0.554973\pi$
$919$	−9574.99	−0.343688	−0.171844	+	0.985124i	$0.554973\pi$
$920$	−4217.47	−0.151137
$921$	4561.18	0.163188
$922$	4331.47	0.154717
$923$	−13299.1	−0.474265
$924$	−1526.60	−0.0543523
$925$	6334.12	0.225151
$926$	34143.0	1.21167
$927$	−18214.5	−0.645355
$928$	13750.5	0.486404
$929$	−83.2801	−0.00294115	−0.00147058	−	0.999999i	$-0.500468\pi$
$929$	−83.2801	−0.00294115	−0.00147058	+	0.999999i	$0.500468\pi$
$930$	301.543	0.0106322
$931$	−2155.54	−0.0758808
$932$	−3184.03	−0.111906
$933$	2259.55	0.0792865
$934$	48215.9	1.68916
$935$	−9207.02	−0.322034
$936$	8404.63	0.293498
$937$	−2981.84	−0.103962	−0.0519810	−	0.998648i	$-0.516554\pi$
$937$	−2981.84	−0.103962	−0.0519810	+	0.998648i	$0.516554\pi$
$938$	−3326.61	−0.115797
$939$	−1234.69	−0.0429100
$940$	2957.49	0.102620
$941$	47048.4	1.62990	0.814949	−	0.579533i	$-0.196765\pi$
$941$	47048.4	1.62990	0.814949	+	0.579533i	$0.196765\pi$
$942$	−5251.03	−0.181622
$943$	−11719.3	−0.404700
$944$	27601.0	0.951628
$945$	4109.15	0.141450
$946$	11613.6	0.399146
$947$	−7043.70	−0.241699	−0.120850	−	0.992671i	$-0.538562\pi$
$947$	−7043.70	−0.241699	−0.120850	+	0.992671i	$0.538562\pi$
$948$	0.0652306	2.23480e−6 0
$949$	−823.022	−0.0281522
$950$	4672.17	0.159563
$951$	−1531.75	−0.0522298
$952$	18250.2	0.621316
$953$	−15780.3	−0.536385	−0.268192	−	0.963365i	$-0.586426\pi$
$953$	−15780.3	−0.536385	−0.268192	+	0.963365i	$0.586426\pi$
$954$	−11649.7	−0.395358
$955$	22344.0	0.757104
$956$	8933.37	0.302224
$957$	−5728.86	−0.193509
$958$	20279.7	0.683932
$959$	−27437.1	−0.923870
$960$	−2266.30	−0.0761922
$961$	961.000	0.0322581
$962$	8014.42	0.268602
$963$	−4432.50	−0.148323
$964$	6414.08	0.214298
$965$	4759.05	0.158756
$966$	−1288.86	−0.0429280
$967$	36339.9	1.20849	0.604246	−	0.796798i	$-0.293474\pi$
$967$	36339.9	1.20849	0.604246	+	0.796798i	$0.293474\pi$
$968$	23083.4	0.766456
$969$	−2372.23	−0.0786450
$970$	13612.2	0.450580
$971$	−2145.90	−0.0709219	−0.0354609	−	0.999371i	$-0.511290\pi$
$971$	−2145.90	−0.0709219	−0.0354609	+	0.999371i	$0.511290\pi$
$972$	−3521.92	−0.116220
$973$	51705.7	1.70361
$974$	31917.5	1.05000
$975$	−259.847	−0.00853515
$976$	14989.5	0.491602
$977$	5613.46	0.183818	0.0919092	−	0.995767i	$-0.470703\pi$
$977$	5613.46	0.183818	0.0919092	+	0.995767i	$0.470703\pi$
$978$	1863.27	0.0609211
$979$	−68052.8	−2.22163
$980$	291.792	0.00951117
$981$	19744.2	0.642592
$982$	15463.3	0.502498
$983$	30263.3	0.981941	0.490970	−	0.871176i	$-0.336642\pi$
$983$	30263.3	0.981941	0.490970	+	0.871176i	$0.336642\pi$
$984$	−6681.73	−0.216469
$985$	−8544.08	−0.276383
$986$	14129.2	0.456355
$987$	4380.97	0.141285
$988$	−2076.26	−0.0668570
$989$	−3443.72	−0.110722
$990$	−15287.4	−0.490773
$991$	−42703.5	−1.36884	−0.684421	−	0.729087i	$-0.739945\pi$
$991$	−42703.5	−1.36884	−0.684421	+	0.729087i	$0.739945\pi$
$992$	−2835.77	−0.0907619
$993$	−2675.92	−0.0855163
$994$	−47949.9	−1.53006
$995$	26521.9	0.845027
$996$	−806.511	−0.0256579
$997$	25780.6	0.818935	0.409468	−	0.912325i	$-0.365714\pi$
$997$	25780.6	0.818935	0.409468	+	0.912325i	$0.365714\pi$
$998$	37040.9	1.17486
$999$	−10809.4	−0.342338

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2015.4.a.d.1.13	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2015.4.a.d.1.13	✓	40	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2015 = 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2015.a (trivial)

\(f(q)\)	\(=\)	\(q-2.43322 q^{2} +0.799530 q^{3} -2.07942 q^{4} -5.00000 q^{5} -1.94544 q^{6} +19.2630 q^{7} +24.5255 q^{8} -26.3608 q^{9} +O(q^{10})\)	\(q-2.43322 q^{2} +0.799530 q^{3} -2.07942 q^{4} -5.00000 q^{5} -1.94544 q^{6} +19.2630 q^{7} +24.5255 q^{8} -26.3608 q^{9} +12.1661 q^{10} +47.6676 q^{11} -1.66256 q^{12} -13.0000 q^{13} -46.8713 q^{14} -3.99765 q^{15} -43.0406 q^{16} +38.6301 q^{17} +64.1416 q^{18} -76.8062 q^{19} +10.3971 q^{20} +15.4014 q^{21} -115.986 q^{22} +34.3926 q^{23} +19.6089 q^{24} +25.0000 q^{25} +31.6319 q^{26} -42.6635 q^{27} -40.0560 q^{28} -150.318 q^{29} +9.72718 q^{30} +31.0000 q^{31} -91.4764 q^{32} +38.1117 q^{33} -93.9956 q^{34} -96.3152 q^{35} +54.8151 q^{36} +253.365 q^{37} +186.887 q^{38} -10.3939 q^{39} -122.627 q^{40} -340.750 q^{41} -37.4750 q^{42} -100.130 q^{43} -99.1211 q^{44} +131.804 q^{45} -83.6848 q^{46} +284.453 q^{47} -34.4123 q^{48} +28.0647 q^{49} -60.8306 q^{50} +30.8859 q^{51} +27.0325 q^{52} -181.624 q^{53} +103.810 q^{54} -238.338 q^{55} +472.435 q^{56} -61.4089 q^{57} +365.757 q^{58} -641.279 q^{59} +8.31281 q^{60} -348.265 q^{61} -75.4299 q^{62} -507.788 q^{63} +566.908 q^{64} +65.0000 q^{65} -92.7343 q^{66} +70.9732 q^{67} -80.3282 q^{68} +27.4979 q^{69} +234.356 q^{70} +1023.01 q^{71} -646.510 q^{72} +63.3094 q^{73} -616.494 q^{74} +19.9883 q^{75} +159.713 q^{76} +918.223 q^{77} +25.2907 q^{78} -0.0392350 q^{79} +215.203 q^{80} +677.629 q^{81} +829.122 q^{82} +485.101 q^{83} -32.0260 q^{84} -193.150 q^{85} +243.638 q^{86} -120.184 q^{87} +1169.07 q^{88} -1427.65 q^{89} -320.708 q^{90} -250.420 q^{91} -71.5166 q^{92} +24.7854 q^{93} -692.138 q^{94} +384.031 q^{95} -73.1382 q^{96} +1118.87 q^{97} -68.2877 q^{98} -1256.55 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 5 q^{2} - 17 q^{3} + 149 q^{4} - 200 q^{5} - 35 q^{6} - 20 q^{7} - 39 q^{8} + 247 q^{9}+O(q^{10}) \) 40 * q - 5 * q^2 - 17 * q^3 + 149 * q^4 - 200 * q^5 - 35 * q^6 - 20 * q^7 - 39 * q^8 + 247 * q^9	\( 40 q - 5 q^{2} - 17 q^{3} + 149 q^{4} - 200 q^{5} - 35 q^{6} - 20 q^{7} - 39 q^{8} + 247 q^{9} + 25 q^{10} + 127 q^{11} - 76 q^{12} - 520 q^{13} + 138 q^{14} + 85 q^{15} + 413 q^{16} - 264 q^{17} - 126 q^{18} - q^{19} - 745 q^{20} + 176 q^{21} - 191 q^{22} - 106 q^{23} + 31 q^{24} + 1000 q^{25} + 65 q^{26} - 344 q^{27} + 255 q^{28} + 107 q^{29} + 175 q^{30} + 1240 q^{31} - 372 q^{32} - 386 q^{33} - 6 q^{34} + 100 q^{35} + 790 q^{36} - 741 q^{37} - 318 q^{38} + 221 q^{39} + 195 q^{40} + 1232 q^{41} - 1180 q^{42} - 615 q^{43} - 152 q^{44} - 1235 q^{45} - 329 q^{46} - 784 q^{47} - 1089 q^{48} - 516 q^{49} - 125 q^{50} - 200 q^{51} - 1937 q^{52} - 1503 q^{53} + 1658 q^{54} - 635 q^{55} + 1518 q^{56} - 1704 q^{57} - 1035 q^{58} - 107 q^{59} + 380 q^{60} - 857 q^{61} - 155 q^{62} - 2636 q^{63} - 215 q^{64} + 2600 q^{65} - 1785 q^{66} - 2689 q^{67} - 2639 q^{68} + 2544 q^{69} - 690 q^{70} + 1554 q^{71} - 420 q^{72} - 1968 q^{73} - 27 q^{74} - 425 q^{75} - 110 q^{76} - 1040 q^{77} + 455 q^{78} - 3182 q^{79} - 2065 q^{80} - 1576 q^{81} - 386 q^{82} + 317 q^{83} - 617 q^{84} + 1320 q^{85} + 347 q^{86} - 216 q^{87} - 4081 q^{88} + 3610 q^{89} + 630 q^{90} + 260 q^{91} - 4965 q^{92} - 527 q^{93} - 2942 q^{94} + 5 q^{95} + 1002 q^{96} - 3318 q^{97} + 1659 q^{98} + 5943 q^{99}+O(q^{100}) \) 40 * q - 5 * q^2 - 17 * q^3 + 149 * q^4 - 200 * q^5 - 35 * q^6 - 20 * q^7 - 39 * q^8 + 247 * q^9 + 25 * q^10 + 127 * q^11 - 76 * q^12 - 520 * q^13 + 138 * q^14 + 85 * q^15 + 413 * q^16 - 264 * q^17 - 126 * q^18 - q^19 - 745 * q^20 + 176 * q^21 - 191 * q^22 - 106 * q^23 + 31 * q^24 + 1000 * q^25 + 65 * q^26 - 344 * q^27 + 255 * q^28 + 107 * q^29 + 175 * q^30 + 1240 * q^31 - 372 * q^32 - 386 * q^33 - 6 * q^34 + 100 * q^35 + 790 * q^36 - 741 * q^37 - 318 * q^38 + 221 * q^39 + 195 * q^40 + 1232 * q^41 - 1180 * q^42 - 615 * q^43 - 152 * q^44 - 1235 * q^45 - 329 * q^46 - 784 * q^47 - 1089 * q^48 - 516 * q^49 - 125 * q^50 - 200 * q^51 - 1937 * q^52 - 1503 * q^53 + 1658 * q^54 - 635 * q^55 + 1518 * q^56 - 1704 * q^57 - 1035 * q^58 - 107 * q^59 + 380 * q^60 - 857 * q^61 - 155 * q^62 - 2636 * q^63 - 215 * q^64 + 2600 * q^65 - 1785 * q^66 - 2689 * q^67 - 2639 * q^68 + 2544 * q^69 - 690 * q^70 + 1554 * q^71 - 420 * q^72 - 1968 * q^73 - 27 * q^74 - 425 * q^75 - 110 * q^76 - 1040 * q^77 + 455 * q^78 - 3182 * q^79 - 2065 * q^80 - 1576 * q^81 - 386 * q^82 + 317 * q^83 - 617 * q^84 + 1320 * q^85 + 347 * q^86 - 216 * q^87 - 4081 * q^88 + 3610 * q^89 + 630 * q^90 + 260 * q^91 - 4965 * q^92 - 527 * q^93 - 2942 * q^94 + 5 * q^95 + 1002 * q^96 - 3318 * q^97 + 1659 * q^98 + 5943 * q^99

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