LMFDB - Embedded newform 2005.4.a.b.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2005,4,Mod(1,2005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2005, 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("2005.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2005 = 5 \cdot 401 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2005.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:	$118.298829562$
Analytic rank:	$1$
Dimension:	$96$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	2005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.72165 q^{2} +1.10148 q^{3} +5.85064 q^{4} -5.00000 q^{5} -4.09933 q^{6} -29.5493 q^{7} +7.99915 q^{8} -25.7867 q^{9} +O(q^{10})$	\(q-3.72165 q^{2} +1.10148 q^{3} +5.85064 q^{4} -5.00000 q^{5} -4.09933 q^{6} -29.5493 q^{7} +7.99915 q^{8} -25.7867 q^{9} +18.6082 q^{10} +3.69510 q^{11} +6.44438 q^{12} +71.1903 q^{13} +109.972 q^{14} -5.50741 q^{15} -76.5751 q^{16} -5.28265 q^{17} +95.9691 q^{18} +1.34226 q^{19} -29.2532 q^{20} -32.5480 q^{21} -13.7519 q^{22} -82.1423 q^{23} +8.81092 q^{24} +25.0000 q^{25} -264.945 q^{26} -58.1437 q^{27} -172.882 q^{28} -91.8474 q^{29} +20.4966 q^{30} +181.214 q^{31} +220.992 q^{32} +4.07009 q^{33} +19.6601 q^{34} +147.746 q^{35} -150.869 q^{36} -349.282 q^{37} -4.99540 q^{38} +78.4149 q^{39} -39.9957 q^{40} -131.636 q^{41} +121.132 q^{42} +85.9577 q^{43} +21.6187 q^{44} +128.934 q^{45} +305.705 q^{46} -208.934 q^{47} -84.3462 q^{48} +530.160 q^{49} -93.0411 q^{50} -5.81874 q^{51} +416.509 q^{52} +269.414 q^{53} +216.390 q^{54} -18.4755 q^{55} -236.369 q^{56} +1.47847 q^{57} +341.823 q^{58} +452.870 q^{59} -32.2219 q^{60} +596.133 q^{61} -674.415 q^{62} +761.979 q^{63} -209.854 q^{64} -355.952 q^{65} -15.1474 q^{66} +394.251 q^{67} -30.9069 q^{68} -90.4783 q^{69} -549.860 q^{70} +459.602 q^{71} -206.272 q^{72} +536.862 q^{73} +1299.90 q^{74} +27.5371 q^{75} +7.85306 q^{76} -109.188 q^{77} -291.832 q^{78} -31.8399 q^{79} +382.876 q^{80} +632.198 q^{81} +489.902 q^{82} +1218.56 q^{83} -190.427 q^{84} +26.4132 q^{85} -319.904 q^{86} -101.168 q^{87} +29.5577 q^{88} +17.2164 q^{89} -479.845 q^{90} -2103.62 q^{91} -480.585 q^{92} +199.604 q^{93} +777.579 q^{94} -6.71128 q^{95} +243.419 q^{96} -1559.81 q^{97} -1973.07 q^{98} -95.2847 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 96 q - 9 q^{2} - 12 q^{3} + 355 q^{4} - 480 q^{5} - 40 q^{6} + 86 q^{7} - 135 q^{8} + 712 q^{9}+O(q^{10}) $ 96 * q - 9 * q^2 - 12 * q^3 + 355 * q^4 - 480 * q^5 - 40 * q^6 + 86 * q^7 - 135 * q^8 + 712 * q^9	\( 96 q - 9 q^{2} - 12 q^{3} + 355 q^{4} - 480 q^{5} - 40 q^{6} + 86 q^{7} - 135 q^{8} + 712 q^{9} + 45 q^{10} - 422 q^{11} - 78 q^{12} + 106 q^{13} - 204 q^{14} + 60 q^{15} + 1215 q^{16} - 214 q^{17} - 254 q^{18} - 510 q^{19} - 1775 q^{20} - 530 q^{21} + 182 q^{22} - 384 q^{23} - 577 q^{24} + 2400 q^{25} - 549 q^{26} - 288 q^{27} + 506 q^{28} - 952 q^{29} + 200 q^{30} - 400 q^{31} - 1294 q^{32} + 20 q^{33} - 304 q^{34} - 430 q^{35} + 1447 q^{36} + 10 q^{37} - 585 q^{38} - 834 q^{39} + 675 q^{40} - 674 q^{41} - 870 q^{42} - 1054 q^{43} - 3554 q^{44} - 3560 q^{45} - 855 q^{46} - 126 q^{47} - 584 q^{48} + 3536 q^{49} - 225 q^{50} - 3766 q^{51} + 859 q^{52} - 2534 q^{53} - 1528 q^{54} + 2110 q^{55} - 3412 q^{56} + 266 q^{57} + 759 q^{58} - 5260 q^{59} + 390 q^{60} - 1388 q^{61} - 1793 q^{62} + 1380 q^{63} + 2011 q^{64} - 530 q^{65} - 3106 q^{66} - 1592 q^{67} - 2308 q^{68} - 448 q^{69} + 1020 q^{70} - 3796 q^{71} - 3955 q^{72} + 2276 q^{73} - 5293 q^{74} - 300 q^{75} - 4399 q^{76} - 2420 q^{77} - 1850 q^{78} - 4246 q^{79} - 6075 q^{80} + 1416 q^{81} + 2608 q^{82} - 5752 q^{83} - 3460 q^{84} + 1070 q^{85} - 3851 q^{86} + 210 q^{87} + 2104 q^{88} - 3856 q^{89} + 1270 q^{90} - 4032 q^{91} - 4899 q^{92} + 1040 q^{93} - 4200 q^{94} + 2550 q^{95} - 4641 q^{96} + 1330 q^{97} - 3824 q^{98} - 9226 q^{99}+O(q^{100}) \) 96 * q - 9 * q^2 - 12 * q^3 + 355 * q^4 - 480 * q^5 - 40 * q^6 + 86 * q^7 - 135 * q^8 + 712 * q^9 + 45 * q^10 - 422 * q^11 - 78 * q^12 + 106 * q^13 - 204 * q^14 + 60 * q^15 + 1215 * q^16 - 214 * q^17 - 254 * q^18 - 510 * q^19 - 1775 * q^20 - 530 * q^21 + 182 * q^22 - 384 * q^23 - 577 * q^24 + 2400 * q^25 - 549 * q^26 - 288 * q^27 + 506 * q^28 - 952 * q^29 + 200 * q^30 - 400 * q^31 - 1294 * q^32 + 20 * q^33 - 304 * q^34 - 430 * q^35 + 1447 * q^36 + 10 * q^37 - 585 * q^38 - 834 * q^39 + 675 * q^40 - 674 * q^41 - 870 * q^42 - 1054 * q^43 - 3554 * q^44 - 3560 * q^45 - 855 * q^46 - 126 * q^47 - 584 * q^48 + 3536 * q^49 - 225 * q^50 - 3766 * q^51 + 859 * q^52 - 2534 * q^53 - 1528 * q^54 + 2110 * q^55 - 3412 * q^56 + 266 * q^57 + 759 * q^58 - 5260 * q^59 + 390 * q^60 - 1388 * q^61 - 1793 * q^62 + 1380 * q^63 + 2011 * q^64 - 530 * q^65 - 3106 * q^66 - 1592 * q^67 - 2308 * q^68 - 448 * q^69 + 1020 * q^70 - 3796 * q^71 - 3955 * q^72 + 2276 * q^73 - 5293 * q^74 - 300 * q^75 - 4399 * q^76 - 2420 * q^77 - 1850 * q^78 - 4246 * q^79 - 6075 * q^80 + 1416 * q^81 + 2608 * q^82 - 5752 * q^83 - 3460 * q^84 + 1070 * q^85 - 3851 * q^86 + 210 * q^87 + 2104 * q^88 - 3856 * q^89 + 1270 * q^90 - 4032 * q^91 - 4899 * q^92 + 1040 * q^93 - 4200 * q^94 + 2550 * q^95 - 4641 * q^96 + 1330 * q^97 - 3824 * q^98 - 9226 * 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 $

$2$	−3.72165	−1.31580	−0.657900	−	0.753105i	$-0.728555\pi$
$2$	−3.72165	−1.31580	−0.657900	+	0.753105i	$0.728555\pi$
$3$	1.10148	0.211980	0.105990	−	0.994367i	$-0.466199\pi$
$3$	1.10148	0.211980	0.105990	+	0.994367i	$0.466199\pi$
$4$	5.85064	0.731330
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−4.09933	−0.278924
$7$	−29.5493	−1.59551	−0.797756	−	0.602981i	$-0.793979\pi$
$7$	−29.5493	−1.59551	−0.797756	+	0.602981i	$0.793979\pi$
$8$	7.99915	0.353516
$9$	−25.7867	−0.955064
$10$	18.6082	0.588444
$11$	3.69510	0.101283	0.0506416	−	0.998717i	$-0.483873\pi$
$11$	3.69510	0.101283	0.0506416	+	0.998717i	$0.483873\pi$
$12$	6.44438	0.155028
$13$	71.1903	1.51882	0.759409	−	0.650613i	$-0.225488\pi$
$13$	71.1903	1.51882	0.759409	+	0.650613i	$0.225488\pi$
$14$	109.972	2.09937
$15$	−5.50741	−0.0948005
$16$	−76.5751	−1.19649
$17$	−5.28265	−0.0753665	−0.0376832	−	0.999290i	$-0.511998\pi$
$17$	−5.28265	−0.0753665	−0.0376832	+	0.999290i	$0.511998\pi$
$18$	95.9691	1.25667
$19$	1.34226	0.0162071	0.00810355	−	0.999967i	$-0.497421\pi$
$19$	1.34226	0.0162071	0.00810355	+	0.999967i	$0.497421\pi$
$20$	−29.2532	−0.327061
$21$	−32.5480	−0.338217
$22$	−13.7519	−0.133269
$23$	−82.1423	−0.744689	−0.372345	−	0.928095i	$-0.621446\pi$
$23$	−82.1423	−0.744689	−0.372345	+	0.928095i	$0.621446\pi$
$24$	8.81092	0.0749384
$25$	25.0000	0.200000
$26$	−264.945	−1.99846
$27$	−58.1437	−0.414435
$28$	−172.882	−1.16685
$29$	−91.8474	−0.588125	−0.294063	−	0.955786i	$-0.595007\pi$
$29$	−91.8474	−0.588125	−0.294063	+	0.955786i	$0.595007\pi$
$30$	20.4966	0.124739
$31$	181.214	1.04990	0.524952	−	0.851132i	$-0.324083\pi$
$31$	181.214	1.04990	0.524952	+	0.851132i	$0.324083\pi$
$32$	220.992	1.22082
$33$	4.07009	0.0214701
$34$	19.6601	0.0991672
$35$	147.746	0.713534
$36$	−150.869	−0.698467
$37$	−349.282	−1.55194	−0.775968	−	0.630773i	$-0.782738\pi$
$37$	−349.282	−1.55194	−0.775968	+	0.630773i	$0.782738\pi$
$38$	−4.99540	−0.0213253
$39$	78.4149	0.321960
$40$	−39.9957	−0.158097
$41$	−131.636	−0.501416	−0.250708	−	0.968063i	$-0.580663\pi$
$41$	−131.636	−0.501416	−0.250708	+	0.968063i	$0.580663\pi$
$42$	121.132	0.445026
$43$	85.9577	0.304847	0.152424	−	0.988315i	$-0.451292\pi$
$43$	85.9577	0.304847	0.152424	+	0.988315i	$0.451292\pi$
$44$	21.6187	0.0740715
$45$	128.934	0.427118
$46$	305.705	0.979862
$47$	−208.934	−0.648430	−0.324215	−	0.945983i	$-0.605100\pi$
$47$	−208.934	−0.648430	−0.324215	+	0.945983i	$0.605100\pi$
$48$	−84.3462	−0.253632
$49$	530.160	1.54566
$50$	−93.0411	−0.263160
$51$	−5.81874	−0.0159762
$52$	416.509	1.11076
$53$	269.414	0.698243	0.349121	−	0.937078i	$-0.386480\pi$
$53$	269.414	0.698243	0.349121	+	0.937078i	$0.386480\pi$
$54$	216.390	0.545314
$55$	−18.4755	−0.0452953
$56$	−236.369	−0.564038
$57$	1.47847	0.00343559
$58$	341.823	0.773855
$59$	452.870	0.999300	0.499650	−	0.866227i	$-0.333462\pi$
$59$	452.870	0.999300	0.499650	+	0.866227i	$0.333462\pi$
$60$	−32.2219	−0.0693305
$61$	596.133	1.25126	0.625632	−	0.780119i	$-0.284841\pi$
$61$	596.133	1.25126	0.625632	+	0.780119i	$0.284841\pi$
$62$	−674.415	−1.38146
$63$	761.979	1.52382
$64$	−209.854	−0.409871
$65$	−355.952	−0.679236
$66$	−15.1474	−0.0282503
$67$	394.251	0.718888	0.359444	−	0.933167i	$-0.382966\pi$
$67$	394.251	0.718888	0.359444	+	0.933167i	$0.382966\pi$
$68$	−30.9069	−0.0551178
$69$	−90.4783	−0.157860
$70$	−549.860	−0.938869
$71$	459.602	0.768235	0.384117	−	0.923284i	$-0.374506\pi$
$71$	459.602	0.768235	0.384117	+	0.923284i	$0.374506\pi$
$72$	−206.272	−0.337630
$73$	536.862	0.860752	0.430376	−	0.902650i	$-0.358381\pi$
$73$	536.862	0.860752	0.430376	+	0.902650i	$0.358381\pi$
$74$	1299.90	2.04204
$75$	27.5371	0.0423961
$76$	7.85306	0.0118527
$77$	−109.188	−0.161599
$78$	−291.832	−0.423635
$79$	−31.8399	−0.0453452	−0.0226726	−	0.999743i	$-0.507218\pi$
$79$	−31.8399	−0.0453452	−0.0226726	+	0.999743i	$0.507218\pi$
$80$	382.876	0.535085
$81$	632.198	0.867212
$82$	489.902	0.659764
$83$	1218.56	1.61149	0.805747	−	0.592260i	$-0.201764\pi$
$83$	1218.56	1.61149	0.805747	+	0.592260i	$0.201764\pi$
$84$	−190.427	−0.247348
$85$	26.4132	0.0337049
$86$	−319.904	−0.401118
$87$	−101.168	−0.124671
$88$	29.5577	0.0358052
$89$	17.2164	0.0205049	0.0102524	−	0.999947i	$-0.496736\pi$
$89$	17.2164	0.0205049	0.0102524	+	0.999947i	$0.496736\pi$
$90$	−479.845	−0.562002
$91$	−2103.62	−2.42329
$92$	−480.585	−0.544614
$93$	199.604	0.222559
$94$	777.579	0.853204
$95$	−6.71128	−0.00724803
$96$	243.419	0.258790
$97$	−1559.81	−1.63273	−0.816367	−	0.577534i	$-0.804015\pi$
$97$	−1559.81	−1.63273	−0.816367	+	0.577534i	$0.804015\pi$
$98$	−1973.07	−2.03377
$99$	−95.2847	−0.0967320
$100$	146.266	0.146266
$101$	650.026	0.640396	0.320198	−	0.947351i	$-0.396251\pi$
$101$	650.026	0.640396	0.320198	+	0.947351i	$0.396251\pi$
$102$	21.6553	0.0210215
$103$	−906.286	−0.866981	−0.433490	−	0.901158i	$-0.642718\pi$
$103$	−906.286	−0.866981	−0.433490	+	0.901158i	$0.642718\pi$
$104$	569.462	0.536926
$105$	162.740	0.151255
$106$	−1002.66	−0.918748
$107$	816.278	0.737501	0.368750	−	0.929528i	$-0.379786\pi$
$107$	816.278	0.737501	0.368750	+	0.929528i	$0.379786\pi$
$108$	−340.178	−0.303089
$109$	113.591	0.0998167	0.0499083	−	0.998754i	$-0.484107\pi$
$109$	113.591	0.0998167	0.0499083	+	0.998754i	$0.484107\pi$
$110$	68.7593	0.0595995
$111$	−384.728	−0.328980
$112$	2262.74	1.90901
$113$	226.326	0.188416	0.0942078	−	0.995553i	$-0.469968\pi$
$113$	226.326	0.188416	0.0942078	+	0.995553i	$0.469968\pi$
$114$	−5.50235	−0.00452055
$115$	410.712	0.333035
$116$	−537.366	−0.430114
$117$	−1835.77	−1.45057
$118$	−1685.42	−1.31488
$119$	156.098	0.120248
$120$	−44.0546	−0.0335135
$121$	−1317.35	−0.989742
$122$	−2218.60	−1.64641
$123$	−144.995	−0.106290
$124$	1060.22	0.767826
$125$	−125.000	−0.0894427
$126$	−2835.82	−2.00504
$127$	2068.71	1.44542	0.722709	−	0.691153i	$-0.242897\pi$
$127$	2068.71	1.44542	0.722709	+	0.691153i	$0.242897\pi$
$128$	−986.937	−0.681513
$129$	94.6809	0.0646216
$130$	1324.73	0.893739
$131$	−1492.35	−0.995320	−0.497660	−	0.867372i	$-0.665807\pi$
$131$	−1492.35	−0.995320	−0.497660	+	0.867372i	$0.665807\pi$
$132$	23.8127	0.0157017
$133$	−39.6627	−0.0258586
$134$	−1467.26	−0.945912
$135$	290.718	0.185341
$136$	−42.2567	−0.0266432
$137$	−344.928	−0.215103	−0.107552	−	0.994199i	$-0.534301\pi$
$137$	−344.928	−0.215103	−0.107552	+	0.994199i	$0.534301\pi$
$138$	336.728	0.207712
$139$	988.997	0.603494	0.301747	−	0.953388i	$-0.402430\pi$
$139$	988.997	0.603494	0.301747	+	0.953388i	$0.402430\pi$
$140$	864.411	0.521829
$141$	−230.137	−0.137454
$142$	−1710.47	−1.01084
$143$	263.056	0.153831
$144$	1974.62	1.14272
$145$	459.237	0.263018
$146$	−1998.01	−1.13258
$147$	583.962	0.327649
$148$	−2043.52	−1.13498
$149$	−1604.56	−0.882218	−0.441109	−	0.897454i	$-0.645415\pi$
$149$	−1604.56	−0.882218	−0.441109	+	0.897454i	$0.645415\pi$
$150$	−102.483	−0.0557848
$151$	−644.863	−0.347538	−0.173769	−	0.984786i	$-0.555595\pi$
$151$	−644.863	−0.347538	−0.173769	+	0.984786i	$0.555595\pi$
$152$	10.7369	0.00572946
$153$	136.222	0.0719798
$154$	406.358	0.212631
$155$	−906.071	−0.469531
$156$	458.777	0.235459
$157$	−340.491	−0.173084	−0.0865419	−	0.996248i	$-0.527582\pi$
$157$	−340.491	−0.173084	−0.0865419	+	0.996248i	$0.527582\pi$
$158$	118.497	0.0596652
$159$	296.755	0.148014
$160$	−1104.96	−0.545968
$161$	2427.25	1.18816
$162$	−2352.82	−1.14108
$163$	−1883.03	−0.904850	−0.452425	−	0.891802i	$-0.649441\pi$
$163$	−1883.03	−0.904850	−0.452425	+	0.891802i	$0.649441\pi$
$164$	−770.155	−0.366701
$165$	−20.3505	−0.00960171
$166$	−4535.03	−2.12040
$167$	3586.90	1.66205	0.831026	−	0.556233i	$-0.187754\pi$
$167$	3586.90	1.66205	0.831026	+	0.556233i	$0.187754\pi$
$168$	−260.356	−0.119565
$169$	2871.06	1.30681
$170$	−98.3007	−0.0443489
$171$	−34.6124	−0.0154788
$172$	502.908	0.222944
$173$	2937.45	1.29093	0.645464	−	0.763791i	$-0.276664\pi$
$173$	2937.45	1.29093	0.645464	+	0.763791i	$0.276664\pi$
$174$	376.512	0.164042
$175$	−738.732	−0.319102
$176$	−282.953	−0.121184
$177$	498.829	0.211832
$178$	−64.0733	−0.0269803
$179$	−294.582	−0.123006	−0.0615030	−	0.998107i	$-0.519589\pi$
$179$	−294.582	−0.123006	−0.0615030	+	0.998107i	$0.519589\pi$
$180$	754.345	0.312364
$181$	−168.628	−0.0692486	−0.0346243	−	0.999400i	$-0.511023\pi$
$181$	−168.628	−0.0692486	−0.0346243	+	0.999400i	$0.511023\pi$
$182$	7828.94	3.18857
$183$	656.630	0.265243
$184$	−657.068	−0.263259
$185$	1746.41	0.694046
$186$	−742.856	−0.292843
$187$	−19.5199	−0.00763336
$188$	−1222.40	−0.474216
$189$	1718.10	0.661236
$190$	24.9770	0.00953696
$191$	−1909.30	−0.723311	−0.361655	−	0.932312i	$-0.617788\pi$
$191$	−1909.30	−0.723311	−0.361655	+	0.932312i	$0.617788\pi$
$192$	−231.150	−0.0868846
$193$	5087.21	1.89733	0.948667	−	0.316276i	$-0.102432\pi$
$193$	5087.21	1.89733	0.948667	+	0.316276i	$0.102432\pi$
$194$	5805.08	2.14835
$195$	−392.074	−0.143985
$196$	3101.78	1.13038
$197$	1303.00	0.471241	0.235621	−	0.971845i	$-0.424288\pi$
$197$	1303.00	0.471241	0.235621	+	0.971845i	$0.424288\pi$
$198$	354.616	0.127280
$199$	−885.798	−0.315540	−0.157770	−	0.987476i	$-0.550431\pi$
$199$	−885.798	−0.315540	−0.157770	+	0.987476i	$0.550431\pi$
$200$	199.979	0.0707031
$201$	434.261	0.152390
$202$	−2419.17	−0.842633
$203$	2714.02	0.938360
$204$	−34.0434	−0.0116839
$205$	658.180	0.224240
$206$	3372.87	1.14077
$207$	2118.18	0.711226
$208$	−5451.41	−1.81725
$209$	4.95978	0.00164151
$210$	−605.661	−0.199022
$211$	−3206.19	−1.04608	−0.523041	−	0.852307i	$-0.675203\pi$
$211$	−3206.19	−1.04608	−0.523041	+	0.852307i	$0.675203\pi$
$212$	1576.24	0.510646
$213$	506.243	0.162851
$214$	−3037.90	−0.970404
$215$	−429.789	−0.136332
$216$	−465.100	−0.146509
$217$	−5354.75	−1.67513
$218$	−422.744	−0.131339
$219$	591.344	0.182463
$220$	−108.094	−0.0331258
$221$	−376.073	−0.114468
$222$	1431.82	0.432872
$223$	4609.36	1.38415	0.692076	−	0.721825i	$-0.256696\pi$
$223$	4609.36	1.38415	0.692076	+	0.721825i	$0.256696\pi$
$224$	−6530.16	−1.94783
$225$	−644.668	−0.191013
$226$	−842.305	−0.247917
$227$	2285.10	0.668139	0.334070	−	0.942548i	$-0.391578\pi$
$227$	2285.10	0.668139	0.334070	+	0.942548i	$0.391578\pi$
$228$	8.65001	0.00251255
$229$	−6093.85	−1.75849	−0.879243	−	0.476374i	$-0.841951\pi$
$229$	−6093.85	−1.75849	−0.879243	+	0.476374i	$0.841951\pi$
$230$	−1528.52	−0.438208
$231$	−120.268	−0.0342557
$232$	−734.700	−0.207911
$233$	−1408.33	−0.395979	−0.197989	−	0.980204i	$-0.563441\pi$
$233$	−1408.33	−0.395979	−0.197989	+	0.980204i	$0.563441\pi$
$234$	6832.07	1.90866
$235$	1044.67	0.289987
$236$	2649.58	0.730818
$237$	−35.0711	−0.00961229
$238$	−580.943	−0.158222
$239$	−3575.34	−0.967654	−0.483827	−	0.875164i	$-0.660754\pi$
$239$	−3575.34	−0.967654	−0.483827	+	0.875164i	$0.660754\pi$
$240$	421.731	0.113428
$241$	−2731.24	−0.730018	−0.365009	−	0.931004i	$-0.618934\pi$
$241$	−2731.24	−0.730018	−0.365009	+	0.931004i	$0.618934\pi$
$242$	4902.70	1.30230
$243$	2266.23	0.598267
$244$	3487.76	0.915087
$245$	−2650.80	−0.691238
$246$	539.619	0.139857
$247$	95.5557	0.0246156
$248$	1449.56	0.371157
$249$	1342.22	0.341605
$250$	465.206	0.117689
$251$	−6450.16	−1.62203	−0.811016	−	0.585023i	$-0.801085\pi$
$251$	−6450.16	−1.62203	−0.811016	+	0.585023i	$0.801085\pi$
$252$	4458.07	1.11441
$253$	−303.524	−0.0754246
$254$	−7698.99	−1.90188
$255$	29.0937	0.00714478
$256$	5351.86	1.30661
$257$	−2604.02	−0.632039	−0.316020	−	0.948753i	$-0.602347\pi$
$257$	−2604.02	−0.632039	−0.316020	+	0.948753i	$0.602347\pi$
$258$	−352.369	−0.0850292
$259$	10321.0	2.47613
$260$	−2082.55	−0.496746
$261$	2368.44	0.561697
$262$	5553.98	1.30964
$263$	−5780.61	−1.35531	−0.677657	−	0.735378i	$-0.737005\pi$
$263$	−5780.61	−1.35531	−0.677657	+	0.735378i	$0.737005\pi$
$264$	32.5573	0.00759001
$265$	−1347.07	−0.312264
$266$	147.611	0.0340248
$267$	18.9635	0.00434663
$268$	2306.62	0.525744
$269$	4701.86	1.06572	0.532858	−	0.846205i	$-0.321118\pi$
$269$	4701.86	1.06572	0.532858	+	0.846205i	$0.321118\pi$
$270$	−1081.95	−0.243872
$271$	−9.61911	−0.00215616	−0.00107808	−	0.999999i	$-0.500343\pi$
$271$	−9.61911	−0.00215616	−0.00107808	+	0.999999i	$0.500343\pi$
$272$	404.519	0.0901749
$273$	−2317.10	−0.513690
$274$	1283.70	0.283033
$275$	92.3776	0.0202567
$276$	−529.356	−0.115447
$277$	8371.13	1.81578	0.907892	−	0.419204i	$-0.137691\pi$
$277$	8371.13	1.81578	0.907892	+	0.419204i	$0.137691\pi$
$278$	−3680.70	−0.794077
$279$	−4672.92	−1.00273
$280$	1181.85	0.252246
$281$	−2890.60	−0.613661	−0.306831	−	0.951764i	$-0.599269\pi$
$281$	−2890.60	−0.613661	−0.306831	+	0.951764i	$0.599269\pi$
$282$	856.490	0.180863
$283$	2145.80	0.450723	0.225361	−	0.974275i	$-0.427644\pi$
$283$	2145.80	0.450723	0.225361	+	0.974275i	$0.427644\pi$
$284$	2688.97	0.561834
$285$	−7.39236	−0.00153644
$286$	−979.000	−0.202411
$287$	3889.75	0.800016
$288$	−5698.67	−1.16596
$289$	−4885.09	−0.994320
$290$	−1709.12	−0.346079
$291$	−1718.11	−0.346108
$292$	3140.99	0.629494
$293$	−3683.62	−0.734468	−0.367234	−	0.930129i	$-0.619695\pi$
$293$	−3683.62	−0.734468	−0.367234	+	0.930129i	$0.619695\pi$
$294$	−2173.30	−0.431120
$295$	−2264.35	−0.446900
$296$	−2793.96	−0.548633
$297$	−214.847	−0.0419754
$298$	5971.59	1.16082
$299$	−5847.74	−1.13105
$300$	161.110	0.0310055
$301$	−2539.99	−0.486387
$302$	2399.95	0.457291
$303$	715.992	0.135751
$304$	−102.783	−0.0193916
$305$	−2980.67	−0.559582
$306$	−506.971	−0.0947111
$307$	−5636.53	−1.04786	−0.523931	−	0.851761i	$-0.675535\pi$
$307$	−5636.53	−1.04786	−0.523931	+	0.851761i	$0.675535\pi$
$308$	−638.818	−0.118182
$309$	−998.258	−0.183783
$310$	3372.07	0.617809
$311$	2109.94	0.384706	0.192353	−	0.981326i	$-0.438388\pi$
$311$	2109.94	0.384706	0.192353	+	0.981326i	$0.438388\pi$
$312$	627.252	0.113818
$313$	7482.23	1.35118	0.675592	−	0.737276i	$-0.263888\pi$
$313$	7482.23	1.35118	0.675592	+	0.737276i	$0.263888\pi$
$314$	1267.19	0.227744
$315$	−3809.90	−0.681471
$316$	−186.284	−0.0331623
$317$	−8728.53	−1.54651	−0.773254	−	0.634096i	$-0.781372\pi$
$317$	−8728.53	−1.54651	−0.773254	+	0.634096i	$0.781372\pi$
$318$	−1104.42	−0.194757
$319$	−339.386	−0.0595672
$320$	1049.27	0.183300
$321$	899.116	0.156336
$322$	−9033.35	−1.56338
$323$	−7.09067	−0.00122147
$324$	3698.76	0.634219
$325$	1779.76	0.303764
$326$	7007.98	1.19060
$327$	125.118	0.0211592
$328$	−1052.97	−0.177259
$329$	6173.86	1.03458
$330$	75.7372	0.0126339
$331$	2793.87	0.463943	0.231971	−	0.972723i	$-0.425482\pi$
$331$	2793.87	0.463943	0.231971	+	0.972723i	$0.425482\pi$
$332$	7129.34	1.17853
$333$	9006.84	1.48220
$334$	−13349.2	−2.18693
$335$	−1971.26	−0.321496
$336$	2492.37	0.404672
$337$	3452.99	0.558149	0.279075	−	0.960269i	$-0.409972\pi$
$337$	3452.99	0.558149	0.279075	+	0.960269i	$0.409972\pi$
$338$	−10685.1	−1.71950
$339$	249.294	0.0399404
$340$	154.534	0.0246494
$341$	669.605	0.106338
$342$	128.815	0.0203670
$343$	−5530.44	−0.870599
$344$	687.589	0.107768
$345$	452.392	0.0705969
$346$	−10932.2	−1.69860
$347$	−6892.98	−1.06638	−0.533191	−	0.845995i	$-0.679007\pi$
$347$	−6892.98	−1.06638	−0.533191	+	0.845995i	$0.679007\pi$
$348$	−591.899	−0.0911757
$349$	−5595.05	−0.858155	−0.429077	−	0.903268i	$-0.641161\pi$
$349$	−5595.05	−0.858155	−0.429077	+	0.903268i	$0.641161\pi$
$350$	2749.30	0.419875
$351$	−4139.27	−0.629452
$352$	816.589	0.123649
$353$	−2532.08	−0.381782	−0.190891	−	0.981611i	$-0.561138\pi$
$353$	−2532.08	−0.381782	−0.190891	+	0.981611i	$0.561138\pi$
$354$	−1856.46	−0.278729
$355$	−2298.01	−0.343565
$356$	100.727	0.0149958
$357$	171.940	0.0254902
$358$	1096.33	0.161851
$359$	−7700.41	−1.13207	−0.566033	−	0.824382i	$-0.691523\pi$
$359$	−7700.41	−1.13207	−0.566033	+	0.824382i	$0.691523\pi$
$360$	1031.36	0.150993
$361$	−6857.20	−0.999737
$362$	627.572	0.0911173
$363$	−1451.03	−0.209806
$364$	−12307.5	−1.77223
$365$	−2684.31	−0.384940
$366$	−2443.75	−0.349007
$367$	−7882.26	−1.12112	−0.560559	−	0.828114i	$-0.689414\pi$
$367$	−7882.26	−1.12112	−0.560559	+	0.828114i	$0.689414\pi$
$368$	6290.06	0.891011
$369$	3394.46	0.478885
$370$	−6499.52	−0.913227
$371$	−7960.99	−1.11405
$372$	1167.81	0.162764
$373$	7707.74	1.06995	0.534975	−	0.844868i	$-0.320321\pi$
$373$	7707.74	1.06995	0.534975	+	0.844868i	$0.320321\pi$
$374$	72.6462	0.0100440
$375$	−137.685	−0.0189601
$376$	−1671.30	−0.229230
$377$	−6538.64	−0.893255
$378$	−6394.17	−0.870055
$379$	−5182.98	−0.702459	−0.351229	−	0.936289i	$-0.614236\pi$
$379$	−5182.98	−0.702459	−0.351229	+	0.936289i	$0.614236\pi$
$380$	−39.2653	−0.00530071
$381$	2278.64	0.306400
$382$	7105.75	0.951732
$383$	−9201.38	−1.22759	−0.613797	−	0.789464i	$-0.710359\pi$
$383$	−9201.38	−1.22759	−0.613797	+	0.789464i	$0.710359\pi$
$384$	−1087.09	−0.144467
$385$	545.938	0.0722691
$386$	−18932.8	−2.49651
$387$	−2216.57	−0.291149
$388$	−9125.92	−1.19407
$389$	2317.08	0.302006	0.151003	−	0.988533i	$-0.451750\pi$
$389$	2317.08	0.302006	0.151003	+	0.988533i	$0.451750\pi$
$390$	1459.16	0.189455
$391$	433.929	0.0561246
$392$	4240.83	0.546413
$393$	−1643.79	−0.210988
$394$	−4849.29	−0.620060
$395$	159.199	0.0202790
$396$	−557.477	−0.0707431
$397$	88.1257	0.0111408	0.00557040	−	0.999984i	$-0.498227\pi$
$397$	88.1257	0.0111408	0.00557040	+	0.999984i	$0.498227\pi$
$398$	3296.62	0.415188
$399$	−43.6878	−0.00548152
$400$	−1914.38	−0.239297
$401$	401.000	0.0499376
$401$	401.000	0.0499376
$402$	−1616.17	−0.200515
$403$	12900.7	1.59461
$404$	3803.07	0.468341
$405$	−3160.99	−0.387829
$406$	−10100.6	−1.23469
$407$	−1290.63	−0.157185
$408$	−46.5450	−0.00564784
$409$	13.9333	0.00168449	0.000842245	−	1.00000i	$-0.499732\pi$
$409$	13.9333	0.00168449	0.000842245		1.00000i	$0.499732\pi$
$410$	−2449.51	−0.295055
$411$	−379.932	−0.0455977
$412$	−5302.35	−0.634049
$413$	−13382.0	−1.59439
$414$	−7883.12	−0.935832
$415$	−6092.78	−0.720682
$416$	15732.5	1.85421
$417$	1089.36	0.127929
$418$	−18.4585	−0.00215990
$419$	−6252.42	−0.728999	−0.364499	−	0.931204i	$-0.618760\pi$
$419$	−6252.42	−0.728999	−0.364499	+	0.931204i	$0.618760\pi$
$420$	952.134	0.110618
$421$	6856.91	0.793790	0.396895	−	0.917864i	$-0.370088\pi$
$421$	6856.91	0.793790	0.396895	+	0.917864i	$0.370088\pi$
$422$	11932.3	1.37644
$423$	5387.73	0.619292
$424$	2155.08	0.246840
$425$	−132.066	−0.0150733
$426$	−1884.06	−0.214279
$427$	−17615.3	−1.99640
$428$	4775.75	0.539357
$429$	289.751	0.0326091
$430$	1599.52	0.179385
$431$	−15793.1	−1.76503	−0.882514	−	0.470286i	$-0.844151\pi$
$431$	−15793.1	−1.76503	−0.882514	+	0.470286i	$0.844151\pi$
$432$	4452.36	0.495866
$433$	7011.68	0.778198	0.389099	−	0.921196i	$-0.372786\pi$
$433$	7011.68	0.778198	0.389099	+	0.921196i	$0.372786\pi$
$434$	19928.5	2.20414
$435$	505.841	0.0557546
$436$	664.579	0.0729990
$437$	−110.256	−0.0120693
$438$	−2200.77	−0.240084
$439$	11157.1	1.21298	0.606492	−	0.795089i	$-0.292576\pi$
$439$	11157.1	1.21298	0.606492	+	0.795089i	$0.292576\pi$
$440$	−147.788	−0.0160126
$441$	−13671.1	−1.47620
$442$	1399.61	0.150617
$443$	−12931.3	−1.38688	−0.693438	−	0.720517i	$-0.743905\pi$
$443$	−12931.3	−1.38688	−0.693438	+	0.720517i	$0.743905\pi$
$444$	−2250.91	−0.240593
$445$	−86.0819	−0.00917005
$446$	−17154.4	−1.82127
$447$	−1767.39	−0.187013
$448$	6201.03	0.653953
$449$	6647.10	0.698655	0.349328	−	0.937001i	$-0.386410\pi$
$449$	6647.10	0.698655	0.349328	+	0.937001i	$0.386410\pi$
$450$	2399.23	0.251335
$451$	−486.408	−0.0507851
$452$	1324.15	0.137794
$453$	−710.306	−0.0736712
$454$	−8504.34	−0.879138
$455$	10518.1	1.08373
$456$	11.8265	0.00121453
$457$	133.861	0.0137019	0.00685094	−	0.999977i	$-0.497819\pi$
$457$	133.861	0.0137019	0.00685094	+	0.999977i	$0.497819\pi$
$458$	22679.2	2.31382
$459$	307.152	0.0312345
$460$	2402.93	0.243559
$461$	14821.8	1.49744	0.748720	−	0.662887i	$-0.230669\pi$
$461$	14821.8	1.49744	0.748720	+	0.662887i	$0.230669\pi$
$462$	447.596	0.0450737
$463$	7230.38	0.725754	0.362877	−	0.931837i	$-0.381794\pi$
$463$	7230.38	0.725754	0.362877	+	0.931837i	$0.381794\pi$
$464$	7033.22	0.703684
$465$	−998.021	−0.0995314
$466$	5241.32	0.521029
$467$	−10368.1	−1.02737	−0.513683	−	0.857980i	$-0.671719\pi$
$467$	−10368.1	−1.02737	−0.513683	+	0.857980i	$0.671719\pi$
$468$	−10740.4	−1.06085
$469$	−11649.8	−1.14699
$470$	−3887.90	−0.381564
$471$	−375.045	−0.0366904
$472$	3622.58	0.353268
$473$	317.623	0.0308759
$474$	130.522	0.0126478
$475$	33.5564	0.00324142
$476$	913.276	0.0879410
$477$	−6947.31	−0.666867
$478$	13306.1	1.27324
$479$	−2518.22	−0.240209	−0.120105	−	0.992761i	$-0.538323\pi$
$479$	−2518.22	−0.240209	−0.120105	+	0.992761i	$0.538323\pi$
$480$	−1217.10	−0.115734
$481$	−24865.5	−2.35711
$482$	10164.7	0.960558
$483$	2673.57	0.251867
$484$	−7707.32	−0.723828
$485$	7799.07	0.730181
$486$	−8434.12	−0.787200
$487$	−13444.1	−1.25095	−0.625474	−	0.780245i	$-0.715094\pi$
$487$	−13444.1	−1.25095	−0.625474	+	0.780245i	$0.715094\pi$
$488$	4768.56	0.442341
$489$	−2074.13	−0.191810
$490$	9865.33	0.909531
$491$	14176.5	1.30301	0.651504	−	0.758645i	$-0.274138\pi$
$491$	14176.5	1.30301	0.651504	+	0.758645i	$0.274138\pi$
$492$	−848.312	−0.0777334
$493$	485.197	0.0443249
$494$	−355.624	−0.0323893
$495$	476.423	0.0432599
$496$	−13876.5	−1.25620
$497$	−13580.9	−1.22573
$498$	−4995.26	−0.449484
$499$	5303.44	0.475781	0.237891	−	0.971292i	$-0.423544\pi$
$499$	5303.44	0.475781	0.237891	+	0.971292i	$0.423544\pi$
$500$	−731.330	−0.0654122
$501$	3950.91	0.352323
$502$	24005.2	2.13427
$503$	−8015.90	−0.710560	−0.355280	−	0.934760i	$-0.615614\pi$
$503$	−8015.90	−0.710560	−0.355280	+	0.934760i	$0.615614\pi$
$504$	6095.19	0.538693
$505$	−3250.13	−0.286394
$506$	1129.61	0.0992437
$507$	3162.42	0.277018
$508$	12103.3	1.05708
$509$	20535.2	1.78822	0.894111	−	0.447846i	$-0.147809\pi$
$509$	20535.2	1.78822	0.894111	+	0.447846i	$0.147809\pi$
$510$	−108.276	−0.00940110
$511$	−15863.9	−1.37334
$512$	−12022.2	−1.03772
$513$	−78.0437	−0.00671679
$514$	9691.23	0.831638
$515$	4531.43	0.387726
$516$	553.944	0.0472598
$517$	−772.034	−0.0656751
$518$	−38411.2	−3.25809
$519$	3235.55	0.273651
$520$	−2847.31	−0.240121
$521$	−23684.8	−1.99165	−0.995826	−	0.0912672i	$-0.970908\pi$
$521$	−23684.8	−1.99165	−0.995826	+	0.0912672i	$0.970908\pi$
$522$	−8814.51	−0.739081
$523$	3101.90	0.259343	0.129672	−	0.991557i	$-0.458608\pi$
$523$	3101.90	0.259343	0.129672	+	0.991557i	$0.458608\pi$
$524$	−8731.19	−0.727908
$525$	−813.700	−0.0676434
$526$	21513.4	1.78332
$527$	−957.290	−0.0791275
$528$	−311.668	−0.0256886
$529$	−5419.64	−0.445438
$530$	5013.32	0.410876
$531$	−11678.0	−0.954395
$532$	−232.052	−0.0189112
$533$	−9371.20	−0.761561
$534$	−70.5756	−0.00571930
$535$	−4081.39	−0.329820
$536$	3153.67	0.254138
$537$	−324.477	−0.0260749
$538$	−17498.7	−1.40227
$539$	1959.00	0.156549
$540$	1700.89	0.135546
$541$	−16746.3	−1.33084	−0.665418	−	0.746471i	$-0.731746\pi$
$541$	−16746.3	−1.33084	−0.665418	+	0.746471i	$0.731746\pi$
$542$	35.7989	0.00283708
$543$	−185.740	−0.0146793
$544$	−1167.42	−0.0920090
$545$	−567.954	−0.0446394
$546$	8623.44	0.675914
$547$	1967.93	0.153825	0.0769127	−	0.997038i	$-0.475494\pi$
$547$	1967.93	0.153825	0.0769127	+	0.997038i	$0.475494\pi$
$548$	−2018.05	−0.157312
$549$	−15372.3	−1.19504
$550$	−343.797	−0.0266537
$551$	−123.283	−0.00953180
$552$	−723.749	−0.0558058
$553$	940.846	0.0723487
$554$	−31154.4	−2.38921
$555$	1923.64	0.147124
$556$	5786.27	0.441353
$557$	3663.42	0.278679	0.139339	−	0.990245i	$-0.455502\pi$
$557$	3663.42	0.278679	0.139339	+	0.990245i	$0.455502\pi$
$558$	17391.0	1.31939
$559$	6119.36	0.463008
$560$	−11313.7	−0.853734
$561$	−21.5009	−0.00161812
$562$	10757.8	0.807456
$563$	−14165.6	−1.06040	−0.530202	−	0.847872i	$-0.677884\pi$
$563$	−14165.6	−1.06040	−0.530202	+	0.847872i	$0.677884\pi$
$564$	−1346.45	−0.100525
$565$	−1131.63	−0.0842620
$566$	−7985.91	−0.593061
$567$	−18681.0	−1.38365
$568$	3676.42	0.271583
$569$	−1820.05	−0.134096	−0.0670480	−	0.997750i	$-0.521358\pi$
$569$	−1820.05	−0.134096	−0.0670480	+	0.997750i	$0.521358\pi$
$570$	27.5117	0.00202165
$571$	−15607.5	−1.14388	−0.571938	−	0.820297i	$-0.693808\pi$
$571$	−15607.5	−1.14388	−0.571938	+	0.820297i	$0.693808\pi$
$572$	1539.04	0.112501
$573$	−2103.06	−0.153328
$574$	−14476.3	−1.05266
$575$	−2053.56	−0.148938
$576$	5411.44	0.391453
$577$	1920.65	0.138575	0.0692874	−	0.997597i	$-0.477927\pi$
$577$	1920.65	0.138575	0.0692874	+	0.997597i	$0.477927\pi$
$578$	18180.6	1.30833
$579$	5603.48	0.402198
$580$	2686.83	0.192353
$581$	−36007.5	−2.57116
$582$	6394.19	0.455408
$583$	995.513	0.0707203
$584$	4294.44	0.304289
$585$	9178.83	0.648714
$586$	13709.1	0.966413
$587$	−6257.76	−0.440009	−0.220005	−	0.975499i	$-0.570607\pi$
$587$	−6257.76	−0.440009	−0.220005	+	0.975499i	$0.570607\pi$
$588$	3416.55	0.239619
$589$	243.236	0.0170159
$590$	8427.11	0.588032
$591$	1435.23	0.0998940
$592$	26746.3	1.85687
$593$	2773.52	0.192065	0.0960326	−	0.995378i	$-0.469385\pi$
$593$	2773.52	0.192065	0.0960326	+	0.995378i	$0.469385\pi$
$594$	799.584	0.0552312
$595$	−780.492	−0.0537765
$596$	−9387.69	−0.645193
$597$	−975.691	−0.0668884
$598$	21763.2	1.48823
$599$	−20875.7	−1.42397	−0.711986	−	0.702193i	$-0.752204\pi$
$599$	−20875.7	−1.42397	−0.711986	+	0.702193i	$0.752204\pi$
$600$	220.273	0.0149877
$601$	−5012.50	−0.340207	−0.170103	−	0.985426i	$-0.554410\pi$
$601$	−5012.50	−0.340207	−0.170103	+	0.985426i	$0.554410\pi$
$602$	9452.94	0.639988
$603$	−10166.5	−0.686584
$604$	−3772.87	−0.254165
$605$	6586.73	0.442626
$606$	−2664.67	−0.178622
$607$	8637.32	0.577559	0.288779	−	0.957396i	$-0.406751\pi$
$607$	8637.32	0.577559	0.288779	+	0.957396i	$0.406751\pi$
$608$	296.628	0.0197860
$609$	2989.45	0.198914
$610$	11093.0	0.736298
$611$	−14874.1	−0.984847
$612$	796.987	0.0526410
$613$	−21823.8	−1.43794	−0.718969	−	0.695043i	$-0.755386\pi$
$613$	−21823.8	−1.43794	−0.718969	+	0.695043i	$0.755386\pi$
$614$	20977.2	1.37878
$615$	724.973	0.0475345
$616$	−873.408	−0.0571276
$617$	7376.81	0.481327	0.240664	−	0.970609i	$-0.422635\pi$
$617$	7376.81	0.481327	0.240664	+	0.970609i	$0.422635\pi$
$618$	3715.16	0.241822
$619$	2000.90	0.129924	0.0649621	−	0.997888i	$-0.479307\pi$
$619$	2000.90	0.129924	0.0649621	+	0.997888i	$0.479307\pi$
$620$	−5301.09	−0.343382
$621$	4776.05	0.308626
$622$	−7852.44	−0.506196
$623$	−508.732	−0.0327157
$624$	−6004.63	−0.385220
$625$	625.000	0.0400000
$626$	−27846.2	−1.77789
$627$	5.46311	0.000347967 0
$628$	−1992.09	−0.126581
$629$	1845.13	0.116964
$630$	14179.1	0.896680
$631$	12184.8	0.768731	0.384365	−	0.923181i	$-0.374420\pi$
$631$	12184.8	0.768731	0.384365	+	0.923181i	$0.374420\pi$
$632$	−254.692	−0.0160302
$633$	−3531.56	−0.221749
$634$	32484.5	2.03490
$635$	−10343.5	−0.646410
$636$	1736.21	0.108247
$637$	37742.2	2.34757
$638$	1263.07	0.0783786
$639$	−11851.6	−0.733714
$640$	4934.68	0.304782
$641$	21774.3	1.34171	0.670853	−	0.741590i	$-0.265928\pi$
$641$	21774.3	1.34171	0.670853	+	0.741590i	$0.265928\pi$
$642$	−3346.19	−0.205707
$643$	4089.23	0.250798	0.125399	−	0.992106i	$-0.459979\pi$
$643$	4089.23	0.250798	0.125399	+	0.992106i	$0.459979\pi$
$644$	14200.9	0.868937
$645$	−473.405	−0.0288997
$646$	26.3889	0.00160721
$647$	6986.98	0.424554	0.212277	−	0.977210i	$-0.431912\pi$
$647$	6986.98	0.424554	0.212277	+	0.977210i	$0.431912\pi$
$648$	5057.04	0.306573
$649$	1673.40	0.101212
$650$	−6623.63	−0.399692
$651$	−5898.16	−0.355095
$652$	−11017.0	−0.661744
$653$	19559.1	1.17214	0.586071	−	0.810260i	$-0.300674\pi$
$653$	19559.1	1.17214	0.586071	+	0.810260i	$0.300674\pi$
$654$	−465.646	−0.0278413
$655$	7461.73	0.445121
$656$	10080.0	0.599938
$657$	−13843.9	−0.822074
$658$	−22976.9	−1.36130
$659$	−255.462	−0.0151007	−0.00755036	−	0.999971i	$-0.502403\pi$
$659$	−255.462	−0.0151007	−0.00755036	+	0.999971i	$0.502403\pi$
$660$	−119.063	−0.00702202
$661$	5409.71	0.318326	0.159163	−	0.987252i	$-0.449121\pi$
$661$	5409.71	0.318326	0.159163	+	0.987252i	$0.449121\pi$
$662$	−10397.8	−0.610456
$663$	−414.238	−0.0242650
$664$	9747.41	0.569688
$665$	198.314	0.0115643
$666$	−33520.3	−1.95028
$667$	7544.55	0.437970
$668$	20985.7	1.21551
$669$	5077.13	0.293413
$670$	7336.32	0.423025
$671$	2202.77	0.126732
$672$	−7192.86	−0.412903
$673$	18634.3	1.06731	0.533655	−	0.845702i	$-0.320818\pi$
$673$	18634.3	1.06731	0.533655	+	0.845702i	$0.320818\pi$
$674$	−12850.8	−0.734413
$675$	−1453.59	−0.0828871
$676$	16797.5	0.955709
$677$	−204.825	−0.0116279	−0.00581393	−	0.999983i	$-0.501851\pi$
$677$	−204.825	−0.0116279	−0.00581393	+	0.999983i	$0.501851\pi$
$678$	−927.785	−0.0525536
$679$	46091.4	2.60504
$680$	211.283	0.0119152
$681$	2517.00	0.141632
$682$	−2492.03	−0.139919
$683$	−21746.4	−1.21830	−0.609152	−	0.793054i	$-0.708490\pi$
$683$	−21746.4	−1.21830	−0.609152	+	0.793054i	$0.708490\pi$
$684$	−202.505	−0.0113201
$685$	1724.64	0.0961972
$686$	20582.3	1.14554
$687$	−6712.27	−0.372764
$688$	−6582.22	−0.364746
$689$	19179.7	1.06050
$690$	−1683.64	−0.0928915
$691$	21065.5	1.15972	0.579862	−	0.814714i	$-0.303106\pi$
$691$	21065.5	1.15972	0.579862	+	0.814714i	$0.303106\pi$
$692$	17186.0	0.944094
$693$	2815.59	0.154337
$694$	25653.2	1.40315
$695$	−4944.99	−0.269891
$696$	−809.260	−0.0440731
$697$	695.386	0.0377900
$698$	20822.8	1.12916
$699$	−1551.25	−0.0839397
$700$	−4322.06	−0.233369
$701$	6693.97	0.360667	0.180334	−	0.983605i	$-0.442282\pi$
$701$	6693.97	0.360667	0.180334	+	0.983605i	$0.442282\pi$
$702$	15404.9	0.828233
$703$	−468.826	−0.0251524
$704$	−775.432	−0.0415130
$705$	1150.69	0.0614715
$706$	9423.50	0.502349
$707$	−19207.8	−1.02176
$708$	2918.47	0.154919
$709$	−29630.7	−1.56954	−0.784769	−	0.619788i	$-0.787219\pi$
$709$	−29630.7	−1.56954	−0.784769	+	0.619788i	$0.787219\pi$
$710$	8552.37	0.452063
$711$	821.047	0.0433075
$712$	137.716	0.00724879
$713$	−14885.3	−0.781852
$714$	−639.898	−0.0335400
$715$	−1315.28	−0.0687953
$716$	−1723.49	−0.0899580
$717$	−3938.17	−0.205124
$718$	28658.2	1.48957
$719$	−22773.5	−1.18124	−0.590619	−	0.806951i	$-0.701116\pi$
$719$	−22773.5	−1.18124	−0.590619	+	0.806951i	$0.701116\pi$
$720$	−9873.11	−0.511041
$721$	26780.1	1.38328
$722$	25520.1	1.31545
$723$	−3008.41	−0.154750
$724$	−986.580	−0.0506436
$725$	−2296.18	−0.117625
$726$	5400.23	0.276063
$727$	−17306.5	−0.882894	−0.441447	−	0.897287i	$-0.645535\pi$
$727$	−17306.5	−0.882894	−0.441447	+	0.897287i	$0.645535\pi$
$728$	−16827.2	−0.856671
$729$	−14573.1	−0.740391
$730$	9990.04	0.506504
$731$	−454.084	−0.0229753
$732$	3841.71	0.193980
$733$	8970.31	0.452013	0.226007	−	0.974126i	$-0.427433\pi$
$733$	8970.31	0.452013	0.226007	+	0.974126i	$0.427433\pi$
$734$	29335.0	1.47517
$735$	−2919.81	−0.146529
$736$	−18152.8	−0.909133
$737$	1456.80	0.0728113
$738$	−12633.0	−0.630117
$739$	−24518.4	−1.22046	−0.610232	−	0.792223i	$-0.708924\pi$
$739$	−24518.4	−1.22046	−0.610232	+	0.792223i	$0.708924\pi$
$740$	10217.6	0.507577
$741$	105.253	0.00521803
$742$	29628.0	1.46587
$743$	19771.5	0.976238	0.488119	−	0.872777i	$-0.337683\pi$
$743$	19771.5	0.976238	0.488119	+	0.872777i	$0.337683\pi$
$744$	1596.66	0.0786781
$745$	8022.79	0.394540
$746$	−28685.5	−1.40784
$747$	−31422.6	−1.53908
$748$	−114.204	−0.00558251
$749$	−24120.4	−1.17669
$750$	512.416	0.0249477
$751$	14781.5	0.718220	0.359110	−	0.933295i	$-0.383080\pi$
$751$	14781.5	0.718220	0.359110	+	0.933295i	$0.383080\pi$
$752$	15999.2	0.775837
$753$	−7104.73	−0.343839
$754$	24334.5	1.17535
$755$	3224.32	0.155424
$756$	10052.0	0.483582
$757$	22838.5	1.09654	0.548270	−	0.836301i	$-0.315287\pi$
$757$	22838.5	1.09654	0.548270	+	0.836301i	$0.315287\pi$
$758$	19289.2	0.924296
$759$	−334.327	−0.0159885
$760$	−53.6845	−0.00256229
$761$	−11111.8	−0.529307	−0.264653	−	0.964344i	$-0.585258\pi$
$761$	−11111.8	−0.529307	−0.264653	+	0.964344i	$0.585258\pi$
$762$	−8480.31	−0.403162
$763$	−3356.52	−0.159259
$764$	−11170.6	−0.528979
$765$	−681.111	−0.0321904
$766$	34244.3	1.61527
$767$	32240.0	1.51775
$768$	5894.98	0.276975
$769$	2896.51	0.135827	0.0679133	−	0.997691i	$-0.478366\pi$
$769$	2896.51	0.135827	0.0679133	+	0.997691i	$0.478366\pi$
$770$	−2031.79	−0.0950917
$771$	−2868.28	−0.133980
$772$	29763.5	1.38758
$773$	−22691.1	−1.05581	−0.527907	−	0.849302i	$-0.677023\pi$
$773$	−22691.1	−1.05581	−0.527907	+	0.849302i	$0.677023\pi$
$774$	8249.28	0.383094
$775$	4530.35	0.209981
$776$	−12477.2	−0.577197
$777$	11368.4	0.524891
$778$	−8623.34	−0.397380
$779$	−176.689	−0.00812651
$780$	−2293.89	−0.105300
$781$	1698.28	0.0778093
$782$	−1614.93	−0.0738488
$783$	5340.34	0.243740
$784$	−40597.1	−1.84936
$785$	1702.46	0.0774054
$786$	6117.62	0.277619
$787$	19583.3	0.887000	0.443500	−	0.896274i	$-0.353737\pi$
$787$	19583.3	0.887000	0.443500	+	0.896274i	$0.353737\pi$
$788$	7623.36	0.344633
$789$	−6367.24	−0.287300
$790$	−592.484	−0.0266831
$791$	−6687.77	−0.300619
$792$	−762.196	−0.0341963
$793$	42438.9	1.90044
$794$	−327.972	−0.0146591
$795$	−1483.77	−0.0661938
$796$	−5182.49	−0.230764
$797$	17263.5	0.767257	0.383628	−	0.923488i	$-0.374674\pi$
$797$	17263.5	0.767257	0.383628	+	0.923488i	$0.374674\pi$
$798$	162.590	0.00721258
$799$	1103.73	0.0488699
$800$	5524.81	0.244164
$801$	−443.954	−0.0195835
$802$	−1492.38	−0.0657079
$803$	1983.76	0.0871798
$804$	2540.71	0.111447
$805$	−12136.2	−0.531361
$806$	−48011.8	−2.09819
$807$	5179.02	0.225911
$808$	5199.65	0.226390
$809$	−32439.9	−1.40980	−0.704899	−	0.709308i	$-0.749008\pi$
$809$	−32439.9	−1.40980	−0.704899	+	0.709308i	$0.749008\pi$
$810$	11764.1	0.510306
$811$	27648.5	1.19713	0.598564	−	0.801075i	$-0.295738\pi$
$811$	27648.5	1.19713	0.598564	+	0.801075i	$0.295738\pi$
$812$	15878.8	0.686251
$813$	−10.5953	−0.000457064 0
$814$	4803.28	0.206824
$815$	9415.17	0.404661
$816$	445.571	0.0191153
$817$	115.377	0.00494069
$818$	−51.8547	−0.00221645
$819$	54245.6	2.31440
$820$	3850.77	0.163994
$821$	27037.7	1.14936	0.574679	−	0.818379i	$-0.305127\pi$
$821$	27037.7	1.14936	0.574679	+	0.818379i	$0.305127\pi$
$822$	1413.97	0.0599975
$823$	23748.5	1.00586	0.502929	−	0.864328i	$-0.332256\pi$
$823$	23748.5	1.00586	0.502929	+	0.864328i	$0.332256\pi$
$824$	−7249.51	−0.306491
$825$	101.752	0.00429401
$826$	49803.0	2.09790
$827$	−1468.83	−0.0617609	−0.0308804	−	0.999523i	$-0.509831\pi$
$827$	−1468.83	−0.0617609	−0.0308804	+	0.999523i	$0.509831\pi$
$828$	12392.7	0.520141
$829$	−8032.91	−0.336544	−0.168272	−	0.985741i	$-0.553819\pi$
$829$	−8032.91	−0.336544	−0.168272	+	0.985741i	$0.553819\pi$
$830$	22675.2	0.948273
$831$	9220.65	0.384911
$832$	−14939.6	−0.622519
$833$	−2800.65	−0.116491
$834$	−4054.22	−0.168329
$835$	−17934.5	−0.743292
$836$	29.0179	0.00120048
$837$	−10536.5	−0.435117
$838$	23269.3	0.959217
$839$	−41503.9	−1.70783	−0.853917	−	0.520410i	$-0.825779\pi$
$839$	−41503.9	−1.70783	−0.853917	+	0.520410i	$0.825779\pi$
$840$	1301.78	0.0534711
$841$	−15953.1	−0.654109
$842$	−25519.0	−1.04447
$843$	−3183.95	−0.130084
$844$	−18758.3	−0.765032
$845$	−14355.3	−0.584423
$846$	−20051.2	−0.814865
$847$	38926.6	1.57914
$848$	−20630.4	−0.835438
$849$	2363.56	0.0955444
$850$	491.503	0.0198334
$851$	28690.8	1.15571
$852$	2961.85	0.119098
$853$	−10940.8	−0.439161	−0.219581	−	0.975594i	$-0.570469\pi$
$853$	−10940.8	−0.439161	−0.219581	+	0.975594i	$0.570469\pi$
$854$	65557.9	2.62687
$855$	173.062	0.00692234
$856$	6529.53	0.260718
$857$	11054.9	0.440641	0.220320	−	0.975428i	$-0.429290\pi$
$857$	11054.9	0.440641	0.220320	+	0.975428i	$0.429290\pi$
$858$	−1078.35	−0.0429071
$859$	31340.7	1.24485	0.622427	−	0.782677i	$-0.286147\pi$
$859$	31340.7	1.24485	0.622427	+	0.782677i	$0.286147\pi$
$860$	−2514.54	−0.0997036
$861$	4284.49	0.169588
$862$	58776.3	2.32242
$863$	−21938.4	−0.865346	−0.432673	−	0.901551i	$-0.642429\pi$
$863$	−21938.4	−0.865346	−0.432673	+	0.901551i	$0.642429\pi$
$864$	−12849.3	−0.505951
$865$	−14687.3	−0.577320
$866$	−26095.0	−1.02395
$867$	−5380.85	−0.210776
$868$	−31328.7	−1.22508
$869$	−117.652	−0.00459271
$870$	−1882.56	−0.0733619
$871$	28066.9	1.09186
$872$	908.629	0.0352868
$873$	40222.5	1.55937
$874$	410.334	0.0158807
$875$	3693.66	0.142707
$876$	3459.74	0.133440
$877$	22796.4	0.877743	0.438872	−	0.898550i	$-0.355378\pi$
$877$	22796.4	0.877743	0.438872	+	0.898550i	$0.355378\pi$
$878$	−41522.8	−1.59604
$879$	−4057.44	−0.155693
$880$	1414.77	0.0541951
$881$	−134.512	−0.00514395	−0.00257197	−	0.999997i	$-0.500819\pi$
$881$	−134.512	−0.00514395	−0.00257197	+	0.999997i	$0.500819\pi$
$882$	50879.0	1.94238
$883$	−3216.60	−0.122590	−0.0612952	−	0.998120i	$-0.519523\pi$
$883$	−3216.60	−0.122590	−0.0612952	+	0.998120i	$0.519523\pi$
$884$	−2200.27	−0.0837139
$885$	−2494.14	−0.0947341
$886$	48125.8	1.82485
$887$	−1597.48	−0.0604714	−0.0302357	−	0.999543i	$-0.509626\pi$
$887$	−1597.48	−0.0604714	−0.0302357	+	0.999543i	$0.509626\pi$
$888$	−3077.50	−0.116300
$889$	−61128.8	−2.30618
$890$	320.366	0.0120660
$891$	2336.04	0.0878341
$892$	26967.7	1.01227
$893$	−280.443	−0.0105092
$894$	6577.60	0.246072
$895$	1472.91	0.0550099
$896$	29163.3	1.08736
$897$	−6441.18	−0.239760
$898$	−24738.2	−0.919290
$899$	−16644.0	−0.617475
$900$	−3771.72	−0.139693
$901$	−1423.22	−0.0526241
$902$	1810.24	0.0668230
$903$	−2797.75	−0.103105
$904$	1810.42	0.0666079
$905$	843.138	0.0309689
$906$	2643.51	0.0969366
$907$	14007.0	0.512784	0.256392	−	0.966573i	$-0.417466\pi$
$907$	14007.0	0.512784	0.256392	+	0.966573i	$0.417466\pi$
$908$	13369.3	0.488631
$909$	−16762.0	−0.611619
$910$	−39144.7	−1.42597
$911$	19108.7	0.694951	0.347476	−	0.937689i	$-0.387039\pi$
$911$	19108.7	0.694951	0.347476	+	0.937689i	$0.387039\pi$
$912$	−113.214	−0.00411063
$913$	4502.69	0.163217
$914$	−498.183	−0.0180289
$915$	−3283.15	−0.118620
$916$	−35653.0	−1.28603
$917$	44097.8	1.58804
$918$	−1143.11	−0.0410984
$919$	38410.6	1.37872	0.689362	−	0.724417i	$-0.257891\pi$
$919$	38410.6	1.37872	0.689362	+	0.724417i	$0.257891\pi$
$920$	3285.34	0.117733
$921$	−6208.54	−0.222126
$922$	−55161.4	−1.97033
$923$	32719.2	1.16681
$924$	−703.647	−0.0250523
$925$	−8732.05	−0.310387
$926$	−26908.9	−0.954948
$927$	23370.2	0.828022
$928$	−20297.6	−0.717996
$929$	−45521.1	−1.60764	−0.803821	−	0.594872i	$-0.797203\pi$
$929$	−45521.1	−1.60764	−0.803821	+	0.594872i	$0.797203\pi$
$930$	3714.28	0.130963
$931$	711.611	0.0250506
$932$	−8239.66	−0.289591
$933$	2324.06	0.0815501
$934$	38586.5	1.35181
$935$	97.5996	0.00341374
$936$	−14684.6	−0.512799
$937$	−22424.1	−0.781817	−0.390908	−	0.920430i	$-0.627839\pi$
$937$	−22424.1	−0.781817	−0.390908	+	0.920430i	$0.627839\pi$
$938$	43356.6	1.50921
$939$	8241.54	0.286424
$940$	6112.00	0.212076
$941$	−23504.5	−0.814266	−0.407133	−	0.913369i	$-0.633471\pi$
$941$	−23504.5	−0.814266	−0.407133	+	0.913369i	$0.633471\pi$
$942$	1395.79	0.0482772
$943$	10812.9	0.373400
$944$	−34678.6	−1.19565
$945$	−8590.52	−0.295714
$946$	−1182.08	−0.0406265
$947$	51650.8	1.77236	0.886181	−	0.463339i	$-0.153349\pi$
$947$	51650.8	1.77236	0.886181	+	0.463339i	$0.153349\pi$
$948$	−205.188	−0.00702976
$949$	38219.3	1.30733
$950$	−124.885	−0.00426506
$951$	−9614.32	−0.327829
$952$	1248.65	0.0425096
$953$	−37231.6	−1.26553	−0.632764	−	0.774344i	$-0.718080\pi$
$953$	−37231.6	−1.26553	−0.632764	+	0.774344i	$0.718080\pi$
$954$	25855.4	0.877463
$955$	9546.52	0.323474
$956$	−20918.0	−0.707675
$957$	−373.827	−0.0126271
$958$	9371.91	0.316068
$959$	10192.4	0.343200
$960$	1155.75	0.0388560
$961$	3047.55	0.102298
$962$	92540.5	3.10148
$963$	−21049.1	−0.704361
$964$	−15979.5	−0.533884
$965$	−25436.1	−0.848514
$966$	−9950.07	−0.331406
$967$	−33598.6	−1.11733	−0.558664	−	0.829394i	$-0.688686\pi$
$967$	−33598.6	−1.11733	−0.558664	+	0.829394i	$0.688686\pi$
$968$	−10537.6	−0.349889
$969$	−7.81025	−0.000258928 0
$970$	−29025.4	−0.960772
$971$	−23834.9	−0.787744	−0.393872	−	0.919165i	$-0.628865\pi$
$971$	−23834.9	−0.787744	−0.393872	+	0.919165i	$0.628865\pi$
$972$	13258.9	0.437531
$973$	−29224.2	−0.962881
$974$	50034.3	1.64600
$975$	1960.37	0.0643919
$976$	−45649.0	−1.49712
$977$	25525.9	0.835872	0.417936	−	0.908477i	$-0.362754\pi$
$977$	25525.9	0.835872	0.417936	+	0.908477i	$0.362754\pi$
$978$	7719.17	0.252384
$979$	63.6163	0.00207680
$980$	−15508.9	−0.505523
$981$	−2929.13	−0.0953313
$982$	−52760.0	−1.71450
$983$	−34740.2	−1.12720	−0.563601	−	0.826047i	$-0.690585\pi$
$983$	−34740.2	−1.12720	−0.563601	+	0.826047i	$0.690585\pi$
$984$	−1159.83	−0.0375753
$985$	−6514.98	−0.210746
$986$	−1805.73	−0.0583227
$987$	6800.40	0.219310
$988$	559.062	0.0180022
$989$	−7060.77	−0.227017
$990$	−1773.08	−0.0569214
$991$	−33920.7	−1.08731	−0.543657	−	0.839308i	$-0.682961\pi$
$991$	−33920.7	−1.08731	−0.543657	+	0.839308i	$0.682961\pi$
$992$	40046.9	1.28174
$993$	3077.40	0.0983468
$994$	50543.3	1.61281
$995$	4428.99	0.141114
$996$	7852.84	0.249826
$997$	10320.7	0.327842	0.163921	−	0.986473i	$-0.447586\pi$
$997$	10320.7	0.327842	0.163921	+	0.986473i	$0.447586\pi$
$998$	−19737.5	−0.626033
$999$	20308.5	0.643177

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2005.4.a.b.1.20	✓	96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2005.4.a.b.1.20	✓	96	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:	\( N \)	\(=\)	\( 2005 = 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2005.a (trivial)

\(f(q)\)	\(=\)	\(q-3.72165 q^{2} +1.10148 q^{3} +5.85064 q^{4} -5.00000 q^{5} -4.09933 q^{6} -29.5493 q^{7} +7.99915 q^{8} -25.7867 q^{9} +O(q^{10})\)	\(q-3.72165 q^{2} +1.10148 q^{3} +5.85064 q^{4} -5.00000 q^{5} -4.09933 q^{6} -29.5493 q^{7} +7.99915 q^{8} -25.7867 q^{9} +18.6082 q^{10} +3.69510 q^{11} +6.44438 q^{12} +71.1903 q^{13} +109.972 q^{14} -5.50741 q^{15} -76.5751 q^{16} -5.28265 q^{17} +95.9691 q^{18} +1.34226 q^{19} -29.2532 q^{20} -32.5480 q^{21} -13.7519 q^{22} -82.1423 q^{23} +8.81092 q^{24} +25.0000 q^{25} -264.945 q^{26} -58.1437 q^{27} -172.882 q^{28} -91.8474 q^{29} +20.4966 q^{30} +181.214 q^{31} +220.992 q^{32} +4.07009 q^{33} +19.6601 q^{34} +147.746 q^{35} -150.869 q^{36} -349.282 q^{37} -4.99540 q^{38} +78.4149 q^{39} -39.9957 q^{40} -131.636 q^{41} +121.132 q^{42} +85.9577 q^{43} +21.6187 q^{44} +128.934 q^{45} +305.705 q^{46} -208.934 q^{47} -84.3462 q^{48} +530.160 q^{49} -93.0411 q^{50} -5.81874 q^{51} +416.509 q^{52} +269.414 q^{53} +216.390 q^{54} -18.4755 q^{55} -236.369 q^{56} +1.47847 q^{57} +341.823 q^{58} +452.870 q^{59} -32.2219 q^{60} +596.133 q^{61} -674.415 q^{62} +761.979 q^{63} -209.854 q^{64} -355.952 q^{65} -15.1474 q^{66} +394.251 q^{67} -30.9069 q^{68} -90.4783 q^{69} -549.860 q^{70} +459.602 q^{71} -206.272 q^{72} +536.862 q^{73} +1299.90 q^{74} +27.5371 q^{75} +7.85306 q^{76} -109.188 q^{77} -291.832 q^{78} -31.8399 q^{79} +382.876 q^{80} +632.198 q^{81} +489.902 q^{82} +1218.56 q^{83} -190.427 q^{84} +26.4132 q^{85} -319.904 q^{86} -101.168 q^{87} +29.5577 q^{88} +17.2164 q^{89} -479.845 q^{90} -2103.62 q^{91} -480.585 q^{92} +199.604 q^{93} +777.579 q^{94} -6.71128 q^{95} +243.419 q^{96} -1559.81 q^{97} -1973.07 q^{98} -95.2847 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 9 q^{2} - 12 q^{3} + 355 q^{4} - 480 q^{5} - 40 q^{6} + 86 q^{7} - 135 q^{8} + 712 q^{9}+O(q^{10}) \) 96 * q - 9 * q^2 - 12 * q^3 + 355 * q^4 - 480 * q^5 - 40 * q^6 + 86 * q^7 - 135 * q^8 + 712 * q^9	\( 96 q - 9 q^{2} - 12 q^{3} + 355 q^{4} - 480 q^{5} - 40 q^{6} + 86 q^{7} - 135 q^{8} + 712 q^{9} + 45 q^{10} - 422 q^{11} - 78 q^{12} + 106 q^{13} - 204 q^{14} + 60 q^{15} + 1215 q^{16} - 214 q^{17} - 254 q^{18} - 510 q^{19} - 1775 q^{20} - 530 q^{21} + 182 q^{22} - 384 q^{23} - 577 q^{24} + 2400 q^{25} - 549 q^{26} - 288 q^{27} + 506 q^{28} - 952 q^{29} + 200 q^{30} - 400 q^{31} - 1294 q^{32} + 20 q^{33} - 304 q^{34} - 430 q^{35} + 1447 q^{36} + 10 q^{37} - 585 q^{38} - 834 q^{39} + 675 q^{40} - 674 q^{41} - 870 q^{42} - 1054 q^{43} - 3554 q^{44} - 3560 q^{45} - 855 q^{46} - 126 q^{47} - 584 q^{48} + 3536 q^{49} - 225 q^{50} - 3766 q^{51} + 859 q^{52} - 2534 q^{53} - 1528 q^{54} + 2110 q^{55} - 3412 q^{56} + 266 q^{57} + 759 q^{58} - 5260 q^{59} + 390 q^{60} - 1388 q^{61} - 1793 q^{62} + 1380 q^{63} + 2011 q^{64} - 530 q^{65} - 3106 q^{66} - 1592 q^{67} - 2308 q^{68} - 448 q^{69} + 1020 q^{70} - 3796 q^{71} - 3955 q^{72} + 2276 q^{73} - 5293 q^{74} - 300 q^{75} - 4399 q^{76} - 2420 q^{77} - 1850 q^{78} - 4246 q^{79} - 6075 q^{80} + 1416 q^{81} + 2608 q^{82} - 5752 q^{83} - 3460 q^{84} + 1070 q^{85} - 3851 q^{86} + 210 q^{87} + 2104 q^{88} - 3856 q^{89} + 1270 q^{90} - 4032 q^{91} - 4899 q^{92} + 1040 q^{93} - 4200 q^{94} + 2550 q^{95} - 4641 q^{96} + 1330 q^{97} - 3824 q^{98} - 9226 q^{99}+O(q^{100}) \) 96 * q - 9 * q^2 - 12 * q^3 + 355 * q^4 - 480 * q^5 - 40 * q^6 + 86 * q^7 - 135 * q^8 + 712 * q^9 + 45 * q^10 - 422 * q^11 - 78 * q^12 + 106 * q^13 - 204 * q^14 + 60 * q^15 + 1215 * q^16 - 214 * q^17 - 254 * q^18 - 510 * q^19 - 1775 * q^20 - 530 * q^21 + 182 * q^22 - 384 * q^23 - 577 * q^24 + 2400 * q^25 - 549 * q^26 - 288 * q^27 + 506 * q^28 - 952 * q^29 + 200 * q^30 - 400 * q^31 - 1294 * q^32 + 20 * q^33 - 304 * q^34 - 430 * q^35 + 1447 * q^36 + 10 * q^37 - 585 * q^38 - 834 * q^39 + 675 * q^40 - 674 * q^41 - 870 * q^42 - 1054 * q^43 - 3554 * q^44 - 3560 * q^45 - 855 * q^46 - 126 * q^47 - 584 * q^48 + 3536 * q^49 - 225 * q^50 - 3766 * q^51 + 859 * q^52 - 2534 * q^53 - 1528 * q^54 + 2110 * q^55 - 3412 * q^56 + 266 * q^57 + 759 * q^58 - 5260 * q^59 + 390 * q^60 - 1388 * q^61 - 1793 * q^62 + 1380 * q^63 + 2011 * q^64 - 530 * q^65 - 3106 * q^66 - 1592 * q^67 - 2308 * q^68 - 448 * q^69 + 1020 * q^70 - 3796 * q^71 - 3955 * q^72 + 2276 * q^73 - 5293 * q^74 - 300 * q^75 - 4399 * q^76 - 2420 * q^77 - 1850 * q^78 - 4246 * q^79 - 6075 * q^80 + 1416 * q^81 + 2608 * q^82 - 5752 * q^83 - 3460 * q^84 + 1070 * q^85 - 3851 * q^86 + 210 * q^87 + 2104 * q^88 - 3856 * q^89 + 1270 * q^90 - 4032 * q^91 - 4899 * q^92 + 1040 * q^93 - 4200 * q^94 + 2550 * q^95 - 4641 * q^96 + 1330 * q^97 - 3824 * q^98 - 9226 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more