LMFDB - Embedded newform 2009.4.a.j.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2009,4,Mod(1,2009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2009, 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("2009.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2009 = 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2009.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:	$118.534837202$
Analytic rank:	$1$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 287)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	2009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.68929 q^{2} -2.43988 q^{3} -5.14630 q^{4} +3.14907 q^{5} +4.12166 q^{6} +22.2079 q^{8} -21.0470 q^{9} +O(q^{10})$	\(q-1.68929 q^{2} -2.43988 q^{3} -5.14630 q^{4} +3.14907 q^{5} +4.12166 q^{6} +22.2079 q^{8} -21.0470 q^{9} -5.31970 q^{10} +30.9193 q^{11} +12.5563 q^{12} +31.4823 q^{13} -7.68335 q^{15} +3.65483 q^{16} -88.8707 q^{17} +35.5545 q^{18} +44.4699 q^{19} -16.2061 q^{20} -52.2316 q^{22} +92.8471 q^{23} -54.1845 q^{24} -115.083 q^{25} -53.1826 q^{26} +117.229 q^{27} -267.929 q^{29} +12.9794 q^{30} -95.7085 q^{31} -183.837 q^{32} -75.4392 q^{33} +150.128 q^{34} +108.314 q^{36} +148.796 q^{37} -75.1226 q^{38} -76.8128 q^{39} +69.9344 q^{40} +41.0000 q^{41} +168.504 q^{43} -159.120 q^{44} -66.2786 q^{45} -156.846 q^{46} -246.163 q^{47} -8.91734 q^{48} +194.409 q^{50} +216.833 q^{51} -162.017 q^{52} +454.822 q^{53} -198.033 q^{54} +97.3671 q^{55} -108.501 q^{57} +452.610 q^{58} +682.541 q^{59} +39.5408 q^{60} -371.262 q^{61} +161.679 q^{62} +281.316 q^{64} +99.1400 q^{65} +127.439 q^{66} -431.247 q^{67} +457.355 q^{68} -226.535 q^{69} +988.008 q^{71} -467.410 q^{72} -621.701 q^{73} -251.360 q^{74} +280.789 q^{75} -228.856 q^{76} +129.759 q^{78} -1.99994 q^{79} +11.5093 q^{80} +282.246 q^{81} -69.2609 q^{82} +565.566 q^{83} -279.860 q^{85} -284.652 q^{86} +653.714 q^{87} +686.652 q^{88} -839.802 q^{89} +111.964 q^{90} -477.819 q^{92} +233.517 q^{93} +415.840 q^{94} +140.039 q^{95} +448.540 q^{96} +880.803 q^{97} -650.758 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 5 q^{2} - 6 q^{3} + 117 q^{4} + 4 q^{5} - 12 q^{6} - 39 q^{8} + 236 q^{9}+O(q^{10}) $ 36 * q - 5 * q^2 - 6 * q^3 + 117 * q^4 + 4 * q^5 - 12 * q^6 - 39 * q^8 + 236 * q^9	\( 36 q - 5 q^{2} - 6 q^{3} + 117 q^{4} + 4 q^{5} - 12 q^{6} - 39 q^{8} + 236 q^{9} - 12 q^{10} - 140 q^{11} + 186 q^{12} - 72 q^{13} - 366 q^{15} - 15 q^{16} - 2 q^{17} - 212 q^{18} + 30 q^{19} - 334 q^{20} - 346 q^{22} - 314 q^{23} + 106 q^{24} + 570 q^{25} + 303 q^{26} - 204 q^{27} - 356 q^{29} - 357 q^{30} - 4 q^{31} - 532 q^{32} - 30 q^{33} - 364 q^{34} + 113 q^{36} - 1398 q^{37} - 264 q^{38} - 1348 q^{39} + 26 q^{40} + 1476 q^{41} - 1072 q^{43} - 1507 q^{44} - 1132 q^{45} - 1356 q^{46} - 622 q^{47} + 1724 q^{48} - 1426 q^{50} - 668 q^{51} - 877 q^{52} - 412 q^{53} - 1814 q^{54} + 1114 q^{55} - 4082 q^{57} - 1309 q^{58} - 620 q^{59} - 3724 q^{60} + 774 q^{61} + 1665 q^{62} - 3285 q^{64} - 1036 q^{65} - 1056 q^{66} - 2972 q^{67} - 1525 q^{68} + 3304 q^{69} - 3540 q^{71} - 821 q^{72} - 60 q^{73} - 2043 q^{74} + 450 q^{75} + 2171 q^{76} - 1136 q^{78} - 5190 q^{79} - 1564 q^{80} + 284 q^{81} - 205 q^{82} + 1656 q^{83} - 5064 q^{85} - 782 q^{86} - 1940 q^{87} - 4232 q^{88} - 1196 q^{89} + 8030 q^{90} - 4618 q^{92} + 698 q^{93} - 35 q^{94} - 1968 q^{95} - 7926 q^{96} + 3862 q^{97} - 5964 q^{99}+O(q^{100}) \) 36 * q - 5 * q^2 - 6 * q^3 + 117 * q^4 + 4 * q^5 - 12 * q^6 - 39 * q^8 + 236 * q^9 - 12 * q^10 - 140 * q^11 + 186 * q^12 - 72 * q^13 - 366 * q^15 - 15 * q^16 - 2 * q^17 - 212 * q^18 + 30 * q^19 - 334 * q^20 - 346 * q^22 - 314 * q^23 + 106 * q^24 + 570 * q^25 + 303 * q^26 - 204 * q^27 - 356 * q^29 - 357 * q^30 - 4 * q^31 - 532 * q^32 - 30 * q^33 - 364 * q^34 + 113 * q^36 - 1398 * q^37 - 264 * q^38 - 1348 * q^39 + 26 * q^40 + 1476 * q^41 - 1072 * q^43 - 1507 * q^44 - 1132 * q^45 - 1356 * q^46 - 622 * q^47 + 1724 * q^48 - 1426 * q^50 - 668 * q^51 - 877 * q^52 - 412 * q^53 - 1814 * q^54 + 1114 * q^55 - 4082 * q^57 - 1309 * q^58 - 620 * q^59 - 3724 * q^60 + 774 * q^61 + 1665 * q^62 - 3285 * q^64 - 1036 * q^65 - 1056 * q^66 - 2972 * q^67 - 1525 * q^68 + 3304 * q^69 - 3540 * q^71 - 821 * q^72 - 60 * q^73 - 2043 * q^74 + 450 * q^75 + 2171 * q^76 - 1136 * q^78 - 5190 * q^79 - 1564 * q^80 + 284 * q^81 - 205 * q^82 + 1656 * q^83 - 5064 * q^85 - 782 * q^86 - 1940 * q^87 - 4232 * q^88 - 1196 * q^89 + 8030 * q^90 - 4618 * q^92 + 698 * q^93 - 35 * q^94 - 1968 * q^95 - 7926 * q^96 + 3862 * q^97 - 5964 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.68929	−0.597254	−0.298627	−	0.954370i	$-0.596529\pi$
$2$	−1.68929	−0.597254	−0.298627	+	0.954370i	$0.596529\pi$
$3$	−2.43988	−0.469554	−0.234777	−	0.972049i	$-0.575436\pi$
$3$	−2.43988	−0.469554	−0.234777	+	0.972049i	$0.575436\pi$
$4$	−5.14630	−0.643288
$5$	3.14907	0.281662	0.140831	−	0.990034i	$-0.455023\pi$
$5$	3.14907	0.281662	0.140831	+	0.990034i	$0.455023\pi$
$6$	4.12166	0.280443
$7$	0	0
$7$	0	0
$8$	22.2079	0.981460
$9$	−21.0470	−0.779519
$10$	−5.31970	−0.168224
$11$	30.9193	0.847501	0.423750	−	0.905779i	$-0.360713\pi$
$11$	30.9193	0.847501	0.423750	+	0.905779i	$0.360713\pi$
$12$	12.5563	0.302059
$13$	31.4823	0.671662	0.335831	−	0.941922i	$-0.390983\pi$
$13$	31.4823	0.671662	0.335831	+	0.941922i	$0.390983\pi$
$14$	0	0
$15$	−7.68335	−0.132256
$16$	3.65483	0.0571068
$17$	−88.8707	−1.26790	−0.633950	−	0.773374i	$-0.718568\pi$
$17$	−88.8707	−1.26790	−0.633950	+	0.773374i	$0.718568\pi$
$18$	35.5545	0.465571
$19$	44.4699	0.536953	0.268476	−	0.963286i	$-0.413480\pi$
$19$	44.4699	0.536953	0.268476	+	0.963286i	$0.413480\pi$
$20$	−16.2061	−0.181190
$21$	0	0
$22$	−52.2316	−0.506173
$23$	92.8471	0.841737	0.420869	−	0.907122i	$-0.361725\pi$
$23$	92.8471	0.841737	0.420869	+	0.907122i	$0.361725\pi$
$24$	−54.1845	−0.460849
$25$	−115.083	−0.920667
$26$	−53.1826	−0.401153
$27$	117.229	0.835581
$28$	0	0
$29$	−267.929	−1.71563	−0.857814	−	0.513960i	$-0.828178\pi$
$29$	−267.929	−1.71563	−0.857814	+	0.513960i	$0.828178\pi$
$30$	12.9794	0.0789901
$31$	−95.7085	−0.554508	−0.277254	−	0.960797i	$-0.589424\pi$
$31$	−95.7085	−0.554508	−0.277254	+	0.960797i	$0.589424\pi$
$32$	−183.837	−1.01557
$33$	−75.4392	−0.397948
$34$	150.128	0.757258
$35$	0	0
$36$	108.314	0.501455
$37$	148.796	0.661134	0.330567	−	0.943782i	$-0.392760\pi$
$37$	148.796	0.661134	0.330567	+	0.943782i	$0.392760\pi$
$38$	−75.1226	−0.320697
$39$	−76.8128	−0.315382
$40$	69.9344	0.276440
$41$	41.0000	0.156174
$41$	41.0000	0.156174
$42$	0	0
$43$	168.504	0.597596	0.298798	−	0.954316i	$-0.403414\pi$
$43$	168.504	0.597596	0.298798	+	0.954316i	$0.403414\pi$
$44$	−159.120	−0.545187
$45$	−66.2786	−0.219561
$46$	−156.846	−0.502731
$47$	−246.163	−0.763969	−0.381985	−	0.924169i	$-0.624759\pi$
$47$	−246.163	−0.763969	−0.381985	+	0.924169i	$0.624759\pi$
$48$	−8.91734	−0.0268147
$49$	0	0
$50$	194.409	0.549872
$51$	216.833	0.595348
$52$	−162.017	−0.432072
$53$	454.822	1.17877	0.589383	−	0.807853i	$-0.299371\pi$
$53$	454.822	1.17877	0.589383	+	0.807853i	$0.299371\pi$
$54$	−198.033	−0.499054
$55$	97.3671	0.238709
$56$	0	0
$57$	−108.501	−0.252129
$58$	452.610	1.02467
$59$	682.541	1.50609	0.753044	−	0.657970i	$-0.228585\pi$
$59$	682.541	1.50609	0.753044	+	0.657970i	$0.228585\pi$
$60$	39.5408	0.0850783
$61$	−371.262	−0.779266	−0.389633	−	0.920970i	$-0.627398\pi$
$61$	−371.262	−0.779266	−0.389633	+	0.920970i	$0.627398\pi$
$62$	161.679	0.331182
$63$	0	0
$64$	281.316	0.549445
$65$	99.1400	0.189182
$66$	127.439	0.237676
$67$	−431.247	−0.786346	−0.393173	−	0.919465i	$-0.628623\pi$
$67$	−431.247	−0.786346	−0.393173	+	0.919465i	$0.628623\pi$
$68$	457.355	0.815625
$69$	−226.535	−0.395241
$70$	0	0
$71$	988.008	1.65148	0.825739	−	0.564053i	$-0.190758\pi$
$71$	988.008	1.65148	0.825739	+	0.564053i	$0.190758\pi$
$72$	−467.410	−0.765067
$73$	−621.701	−0.996776	−0.498388	−	0.866954i	$-0.666074\pi$
$73$	−621.701	−0.996776	−0.498388	+	0.866954i	$0.666074\pi$
$74$	−251.360	−0.394865
$75$	280.789	0.432303
$76$	−228.856	−0.345415
$77$	0	0
$78$	129.759	0.188363
$79$	−1.99994	−0.00284824	−0.00142412	−	0.999999i	$-0.500453\pi$
$79$	−1.99994	−0.00284824	−0.00142412	+	0.999999i	$0.500453\pi$
$80$	11.5093	0.0160848
$81$	282.246	0.387168
$82$	−69.2609	−0.0932754
$83$	565.566	0.747938	0.373969	−	0.927441i	$-0.377997\pi$
$83$	565.566	0.747938	0.373969	+	0.927441i	$0.377997\pi$
$84$	0	0
$85$	−279.860	−0.357119
$86$	−284.652	−0.356917
$87$	653.714	0.805581
$88$	686.652	0.831788
$89$	−839.802	−1.00021	−0.500106	−	0.865964i	$-0.666705\pi$
$89$	−839.802	−1.00021	−0.500106	+	0.865964i	$0.666705\pi$
$90$	111.964	0.131133
$91$	0	0
$92$	−477.819	−0.541479
$93$	233.517	0.260372
$94$	415.840	0.456284
$95$	140.039	0.151239
$96$	448.540	0.476864
$97$	880.803	0.921979	0.460989	−	0.887406i	$-0.347495\pi$
$97$	880.803	0.921979	0.460989	+	0.887406i	$0.347495\pi$
$98$	0	0
$99$	−650.758	−0.660643
$100$	592.254	0.592254
$101$	1178.89	1.16143	0.580715	−	0.814107i	$-0.302773\pi$
$101$	1178.89	1.16143	0.580715	+	0.814107i	$0.302773\pi$
$102$	−366.294	−0.355574
$103$	−598.786	−0.572817	−0.286408	−	0.958108i	$-0.592461\pi$
$103$	−598.786	−0.572817	−0.286408	+	0.958108i	$0.592461\pi$
$104$	699.155	0.659209
$105$	0	0
$106$	−768.326	−0.704023
$107$	−114.469	−0.103421	−0.0517107	−	0.998662i	$-0.516467\pi$
$107$	−114.469	−0.103421	−0.0517107	+	0.998662i	$0.516467\pi$
$108$	−603.294	−0.537519
$109$	879.117	0.772515	0.386257	−	0.922391i	$-0.373768\pi$
$109$	879.117	0.772515	0.386257	+	0.922391i	$0.373768\pi$
$110$	−164.481	−0.142570
$111$	−363.045	−0.310438
$112$	0	0
$113$	−846.595	−0.704787	−0.352394	−	0.935852i	$-0.614632\pi$
$113$	−846.595	−0.704787	−0.352394	+	0.935852i	$0.614632\pi$
$114$	183.290	0.150585
$115$	292.382	0.237085
$116$	1378.85	1.10364
$117$	−662.607	−0.523573
$118$	−1153.01	−0.899517
$119$	0	0
$120$	−170.631	−0.129804
$121$	−374.999	−0.281742
$122$	627.169	0.465420
$123$	−100.035	−0.0733321
$124$	492.545	0.356708
$125$	−756.040	−0.540978
$126$	0	0
$127$	418.541	0.292437	0.146219	−	0.989252i	$-0.453290\pi$
$127$	418.541	0.292437	0.146219	+	0.989252i	$0.453290\pi$
$128$	995.475	0.687409
$129$	−411.129	−0.280604
$130$	−167.476	−0.112989
$131$	1143.15	0.762425	0.381213	−	0.924487i	$-0.375507\pi$
$131$	1143.15	0.762425	0.381213	+	0.924487i	$0.375507\pi$
$132$	388.233	0.255995
$133$	0	0
$134$	728.501	0.469648
$135$	369.162	0.235351
$136$	−1973.63	−1.24439
$137$	−1367.40	−0.852734	−0.426367	−	0.904550i	$-0.640207\pi$
$137$	−1367.40	−0.852734	−0.426367	+	0.904550i	$0.640207\pi$
$138$	382.684	0.236059
$139$	−922.992	−0.563217	−0.281608	−	0.959529i	$-0.590868\pi$
$139$	−922.992	−0.563217	−0.281608	+	0.959529i	$0.590868\pi$
$140$	0	0
$141$	600.607	0.358725
$142$	−1669.03	−0.986352
$143$	973.408	0.569234
$144$	−76.9233	−0.0445158
$145$	−843.730	−0.483227
$146$	1050.23	0.595328
$147$	0	0
$148$	−765.751	−0.425300
$149$	−2513.00	−1.38170	−0.690850	−	0.722998i	$-0.742763\pi$
$149$	−2513.00	−1.38170	−0.690850	+	0.722998i	$0.742763\pi$
$150$	−474.334	−0.258195
$151$	3372.62	1.81761	0.908807	−	0.417217i	$-0.136994\pi$
$151$	3372.62	1.81761	0.908807	+	0.417217i	$0.136994\pi$
$152$	987.584	0.526998
$153$	1870.46	0.988352
$154$	0	0
$155$	−301.393	−0.156184
$156$	395.302	0.202881
$157$	2162.49	1.09927	0.549635	−	0.835405i	$-0.314767\pi$
$157$	2162.49	1.09927	0.549635	+	0.835405i	$0.314767\pi$
$158$	3.37848	0.00170112
$159$	−1109.71	−0.553495
$160$	−578.918	−0.286047
$161$	0	0
$162$	−476.795	−0.231238
$163$	3439.36	1.65271	0.826353	−	0.563152i	$-0.190412\pi$
$163$	3439.36	1.65271	0.826353	+	0.563152i	$0.190412\pi$
$164$	−210.998	−0.100465
$165$	−237.564	−0.112087
$166$	−955.404	−0.446709
$167$	3151.34	1.46023	0.730113	−	0.683327i	$-0.239467\pi$
$167$	3151.34	1.46023	0.730113	+	0.683327i	$0.239467\pi$
$168$	0	0
$169$	−1205.87	−0.548870
$170$	472.765	0.213291
$171$	−935.959	−0.418565
$172$	−867.173	−0.384426
$173$	−2765.67	−1.21544	−0.607718	−	0.794153i	$-0.707915\pi$
$173$	−2765.67	−1.21544	−0.607718	+	0.794153i	$0.707915\pi$
$174$	−1104.31	−0.481136
$175$	0	0
$176$	113.005	0.0483980
$177$	−1665.31	−0.707190
$178$	1418.67	0.597380
$179$	−1738.20	−0.725807	−0.362903	−	0.931827i	$-0.618214\pi$
$179$	−1738.20	−0.725807	−0.362903	+	0.931827i	$0.618214\pi$
$180$	341.090	0.141241
$181$	1847.83	0.758829	0.379414	−	0.925227i	$-0.376125\pi$
$181$	1847.83	0.758829	0.379414	+	0.925227i	$0.376125\pi$
$182$	0	0
$183$	905.833	0.365908
$184$	2061.94	0.826132
$185$	468.571	0.186216
$186$	−394.478	−0.155508
$187$	−2747.82	−1.07455
$188$	1266.83	0.491452
$189$	0	0
$190$	−236.567	−0.0903282
$191$	−1177.08	−0.445920	−0.222960	−	0.974828i	$-0.571572\pi$
$191$	−1177.08	−0.445920	−0.222960	+	0.974828i	$0.571572\pi$
$192$	−686.376	−0.257994
$193$	−3950.31	−1.47331	−0.736657	−	0.676267i	$-0.763597\pi$
$193$	−3950.31	−1.47331	−0.736657	+	0.676267i	$0.763597\pi$
$194$	−1487.93	−0.550656
$195$	−241.889	−0.0888310
$196$	0	0
$197$	−3984.90	−1.44118	−0.720590	−	0.693362i	$-0.756129\pi$
$197$	−3984.90	−1.44118	−0.720590	+	0.693362i	$0.756129\pi$
$198$	1099.32	0.394572
$199$	−800.612	−0.285195	−0.142598	−	0.989781i	$-0.545545\pi$
$199$	−800.612	−0.285195	−0.142598	+	0.989781i	$0.545545\pi$
$200$	−2555.76	−0.903598
$201$	1052.19	0.369232
$202$	−1991.49	−0.693668
$203$	0	0
$204$	−1115.89	−0.382980
$205$	129.112	0.0439882
$206$	1011.52	0.342117
$207$	−1954.15	−0.656150
$208$	115.062	0.0383564
$209$	1374.98	0.455068
$210$	0	0
$211$	−320.362	−0.104524	−0.0522621	−	0.998633i	$-0.516643\pi$
$211$	−320.362	−0.104524	−0.0522621	+	0.998633i	$0.516643\pi$
$212$	−2340.65	−0.758286
$213$	−2410.62	−0.775458
$214$	193.371	0.0617689
$215$	530.632	0.168320
$216$	2603.40	0.820089
$217$	0	0
$218$	−1485.08	−0.461387
$219$	1516.87	0.468040
$220$	−501.080	−0.153558
$221$	−2797.85	−0.851600
$222$	613.287	0.185411
$223$	6154.08	1.84802	0.924008	−	0.382373i	$-0.124893\pi$
$223$	6154.08	1.84802	0.924008	+	0.382373i	$0.124893\pi$
$224$	0	0
$225$	2422.16	0.717677
$226$	1430.14	0.420937
$227$	−1151.15	−0.336582	−0.168291	−	0.985737i	$-0.553825\pi$
$227$	−1151.15	−0.336582	−0.168291	+	0.985737i	$0.553825\pi$
$228$	558.380	0.162191
$229$	−1025.45	−0.295910	−0.147955	−	0.988994i	$-0.547269\pi$
$229$	−1025.45	−0.295910	−0.147955	+	0.988994i	$0.547269\pi$
$230$	−493.919	−0.141600
$231$	0	0
$232$	−5950.15	−1.68382
$233$	−4846.25	−1.36261	−0.681306	−	0.731999i	$-0.738588\pi$
$233$	−4846.25	−1.36261	−0.681306	+	0.731999i	$0.738588\pi$
$234$	1119.34	0.312706
$235$	−775.185	−0.215181
$236$	−3512.56	−0.968848
$237$	4.87960	0.00133740
$238$	0	0
$239$	−56.3860	−0.0152607	−0.00763035	−	0.999971i	$-0.502429\pi$
$239$	−56.3860	−0.0152607	−0.00763035	+	0.999971i	$0.502429\pi$
$240$	−28.0814	−0.00755268
$241$	5160.49	1.37932	0.689661	−	0.724132i	$-0.257760\pi$
$241$	5160.49	1.37932	0.689661	+	0.724132i	$0.257760\pi$
$242$	633.482	0.168272
$243$	−3853.82	−1.01738
$244$	1910.63	0.501292
$245$	0	0
$246$	168.988	0.0437979
$247$	1400.01	0.360651
$248$	−2125.49	−0.544228
$249$	−1379.91	−0.351198
$250$	1277.17	0.323102
$251$	2112.47	0.531228	0.265614	−	0.964079i	$-0.414425\pi$
$251$	2112.47	0.531228	0.265614	+	0.964079i	$0.414425\pi$
$252$	0	0
$253$	2870.76	0.713373
$254$	−707.037	−0.174659
$255$	682.825	0.167687
$256$	−3932.17	−0.960003
$257$	2960.23	0.718499	0.359250	−	0.933241i	$-0.383033\pi$
$257$	2960.23	0.718499	0.359250	+	0.933241i	$0.383033\pi$
$258$	694.516	0.167592
$259$	0	0
$260$	−510.204	−0.121698
$261$	5639.11	1.33736
$262$	−1931.12	−0.455362
$263$	−375.879	−0.0881281	−0.0440641	−	0.999029i	$-0.514031\pi$
$263$	−375.879	−0.0881281	−0.0440641	+	0.999029i	$0.514031\pi$
$264$	−1675.35	−0.390570
$265$	1432.27	0.332014
$266$	0	0
$267$	2049.01	0.469653
$268$	2219.33	0.505847
$269$	−6616.45	−1.49967	−0.749837	−	0.661623i	$-0.769868\pi$
$269$	−6616.45	−1.49967	−0.749837	+	0.661623i	$0.769868\pi$
$270$	−623.621	−0.140564
$271$	−2417.18	−0.541820	−0.270910	−	0.962605i	$-0.587325\pi$
$271$	−2417.18	−0.541820	−0.270910	+	0.962605i	$0.587325\pi$
$272$	−324.807	−0.0724057
$273$	0	0
$274$	2309.93	0.509299
$275$	−3558.29	−0.780266
$276$	1165.82	0.254254
$277$	−2344.24	−0.508489	−0.254245	−	0.967140i	$-0.581827\pi$
$277$	−2344.24	−0.508489	−0.254245	+	0.967140i	$0.581827\pi$
$278$	1559.20	0.336384
$279$	2014.38	0.432250
$280$	0	0
$281$	−4543.98	−0.964666	−0.482333	−	0.875988i	$-0.660210\pi$
$281$	−4543.98	−0.964666	−0.482333	+	0.875988i	$0.660210\pi$
$282$	−1014.60	−0.214250
$283$	−1962.32	−0.412184	−0.206092	−	0.978533i	$-0.566075\pi$
$283$	−1962.32	−0.412184	−0.206092	+	0.978533i	$0.566075\pi$
$284$	−5084.58	−1.06238
$285$	−341.678	−0.0710150
$286$	−1644.37	−0.339977
$287$	0	0
$288$	3869.23	0.791654
$289$	2985.00	0.607571
$290$	1425.30	0.288609
$291$	−2149.05	−0.432919
$292$	3199.46	0.641214
$293$	−9747.58	−1.94355	−0.971775	−	0.235911i	$-0.924193\pi$
$293$	−9747.58	−1.94355	−0.971775	+	0.235911i	$0.924193\pi$
$294$	0	0
$295$	2149.37	0.424208
$296$	3304.46	0.648877
$297$	3624.63	0.708155
$298$	4245.19	0.825225
$299$	2923.04	0.565363
$300$	−1445.02	−0.278095
$301$	0	0
$302$	−5697.33	−1.08558
$303$	−2876.36	−0.545354
$304$	162.530	0.0306636
$305$	−1169.13	−0.219489
$306$	−3159.75	−0.590297
$307$	−6227.92	−1.15781	−0.578903	−	0.815397i	$-0.696519\pi$
$307$	−6227.92	−1.15781	−0.578903	+	0.815397i	$0.696519\pi$
$308$	0	0
$309$	1460.96	0.268969
$310$	509.140	0.0932814
$311$	−3386.20	−0.617408	−0.308704	−	0.951158i	$-0.599895\pi$
$311$	−3386.20	−0.617408	−0.308704	+	0.951158i	$0.599895\pi$
$312$	−1705.85	−0.309535
$313$	−2416.88	−0.436453	−0.218227	−	0.975898i	$-0.570027\pi$
$313$	−2416.88	−0.436453	−0.218227	+	0.975898i	$0.570027\pi$
$314$	−3653.07	−0.656544
$315$	0	0
$316$	10.2923	0.00183224
$317$	−9609.55	−1.70261	−0.851303	−	0.524675i	$-0.824187\pi$
$317$	−9609.55	−1.70261	−0.851303	+	0.524675i	$0.824187\pi$
$318$	1874.62	0.330577
$319$	−8284.18	−1.45400
$320$	885.884	0.154758
$321$	279.289	0.0485620
$322$	0	0
$323$	−3952.07	−0.680803
$324$	−1452.52	−0.249061
$325$	−3623.08	−0.618377
$326$	−5810.07	−0.987085
$327$	−2144.94	−0.362738
$328$	910.524	0.153278
$329$	0	0
$330$	401.314	0.0669442
$331$	1737.05	0.288450	0.144225	−	0.989545i	$-0.453931\pi$
$331$	1737.05	0.288450	0.144225	+	0.989545i	$0.453931\pi$
$332$	−2910.57	−0.481140
$333$	−3131.72	−0.515367
$334$	−5323.52	−0.872126
$335$	−1358.03	−0.221484
$336$	0	0
$337$	−11068.8	−1.78918	−0.894591	−	0.446887i	$-0.852533\pi$
$337$	−11068.8	−1.78918	−0.894591	+	0.446887i	$0.852533\pi$
$338$	2037.06	0.327815
$339$	2065.59	0.330936
$340$	1440.25	0.229730
$341$	−2959.24	−0.469946
$342$	1581.11	0.249990
$343$	0	0
$344$	3742.12	0.586517
$345$	−713.377	−0.111324
$346$	4672.03	0.725924
$347$	3247.91	0.502469	0.251235	−	0.967926i	$-0.419163\pi$
$347$	3247.91	0.502469	0.251235	+	0.967926i	$0.419163\pi$
$348$	−3364.21	−0.518220
$349$	−412.528	−0.0632726	−0.0316363	−	0.999499i	$-0.510072\pi$
$349$	−412.528	−0.0632726	−0.0316363	+	0.999499i	$0.510072\pi$
$350$	0	0
$351$	3690.62	0.561228
$352$	−5684.11	−0.860694
$353$	6264.56	0.944558	0.472279	−	0.881449i	$-0.343431\pi$
$353$	6264.56	0.944558	0.472279	+	0.881449i	$0.343431\pi$
$354$	2813.20	0.422372
$355$	3111.31	0.465158
$356$	4321.87	0.643424
$357$	0	0
$358$	2936.33	0.433491
$359$	−1539.53	−0.226333	−0.113166	−	0.993576i	$-0.536099\pi$
$359$	−1539.53	−0.226333	−0.113166	+	0.993576i	$0.536099\pi$
$360$	−1471.91	−0.215490
$361$	−4881.42	−0.711682
$362$	−3121.52	−0.453213
$363$	914.951	0.132293
$364$	0	0
$365$	−1957.78	−0.280754
$366$	−1530.21	−0.218540
$367$	7303.35	1.03878	0.519389	−	0.854538i	$-0.326159\pi$
$367$	7303.35	1.03878	0.519389	+	0.854538i	$0.326159\pi$
$368$	339.341	0.0480689
$369$	−862.927	−0.121740
$370$	−791.552	−0.111218
$371$	0	0
$372$	−1201.75	−0.167494
$373$	−5959.70	−0.827296	−0.413648	−	0.910437i	$-0.635746\pi$
$373$	−5959.70	−0.827296	−0.413648	+	0.910437i	$0.635746\pi$
$374$	4641.86	0.641777
$375$	1844.64	0.254019
$376$	−5466.76	−0.749805
$377$	−8435.02	−1.15232
$378$	0	0
$379$	−9369.35	−1.26984	−0.634922	−	0.772576i	$-0.718968\pi$
$379$	−9369.35	−1.26984	−0.634922	+	0.772576i	$0.718968\pi$
$380$	−720.684	−0.0972903
$381$	−1021.19	−0.137315
$382$	1988.43	0.266327
$383$	820.810	0.109508	0.0547538	−	0.998500i	$-0.482563\pi$
$383$	820.810	0.109508	0.0547538	+	0.998500i	$0.482563\pi$
$384$	−2428.83	−0.322776
$385$	0	0
$386$	6673.22	0.879943
$387$	−3546.51	−0.465837
$388$	−4532.88	−0.593098
$389$	2529.60	0.329707	0.164853	−	0.986318i	$-0.447285\pi$
$389$	2529.60	0.329707	0.164853	+	0.986318i	$0.447285\pi$
$390$	408.621	0.0530547
$391$	−8251.38	−1.06724
$392$	0	0
$393$	−2789.15	−0.358000
$394$	6731.65	0.860750
$395$	−6.29796	−0.000802240 0
$396$	3349.00	0.424983
$397$	−4374.51	−0.553024	−0.276512	−	0.961011i	$-0.589178\pi$
$397$	−4374.51	−0.553024	−0.276512	+	0.961011i	$0.589178\pi$
$398$	1352.46	0.170334
$399$	0	0
$400$	−420.610	−0.0525763
$401$	3055.07	0.380456	0.190228	−	0.981740i	$-0.439077\pi$
$401$	3055.07	0.380456	0.190228	+	0.981740i	$0.439077\pi$
$402$	−1777.45	−0.220525
$403$	−3013.12	−0.372442
$404$	−6066.95	−0.747133
$405$	888.813	0.109051
$406$	0	0
$407$	4600.67	0.560312
$408$	4815.42	0.584310
$409$	7461.57	0.902080	0.451040	−	0.892504i	$-0.351053\pi$
$409$	7461.57	0.902080	0.451040	+	0.892504i	$0.351053\pi$
$410$	−218.108	−0.0262721
$411$	3336.28	0.400405
$412$	3081.53	0.368486
$413$	0	0
$414$	3301.13	0.391888
$415$	1781.01	0.210666
$416$	−5787.61	−0.682118
$417$	2251.99	0.264461
$418$	−2322.74	−0.271791
$419$	10294.7	1.20030	0.600152	−	0.799886i	$-0.295107\pi$
$419$	10294.7	1.20030	0.600152	+	0.799886i	$0.295107\pi$
$420$	0	0
$421$	−15953.1	−1.84681	−0.923404	−	0.383830i	$-0.874605\pi$
$421$	−15953.1	−1.84681	−0.923404	+	0.383830i	$0.874605\pi$
$422$	541.183	0.0624275
$423$	5180.99	0.595528
$424$	10100.7	1.15691
$425$	10227.5	1.16731
$426$	4072.23	0.463146
$427$	0	0
$428$	589.090	0.0665298
$429$	−2374.99	−0.267286
$430$	−896.391	−0.100530
$431$	−3756.52	−0.419826	−0.209913	−	0.977720i	$-0.567318\pi$
$431$	−3756.52	−0.419826	−0.209913	+	0.977720i	$0.567318\pi$
$432$	428.451	0.0477173
$433$	−4310.02	−0.478352	−0.239176	−	0.970976i	$-0.576877\pi$
$433$	−4310.02	−0.478352	−0.239176	+	0.970976i	$0.576877\pi$
$434$	0	0
$435$	2058.60	0.226901
$436$	−4524.20	−0.496949
$437$	4128.91	0.451973
$438$	−2562.44	−0.279539
$439$	3261.02	0.354533	0.177266	−	0.984163i	$-0.443275\pi$
$439$	3261.02	0.354533	0.177266	+	0.984163i	$0.443275\pi$
$440$	2162.32	0.234283
$441$	0	0
$442$	4726.38	0.508622
$443$	−3661.52	−0.392696	−0.196348	−	0.980534i	$-0.562908\pi$
$443$	−3661.52	−0.392696	−0.196348	+	0.980534i	$0.562908\pi$
$444$	1868.34	0.199701
$445$	−2644.60	−0.281721
$446$	−10396.0	−1.10373
$447$	6131.41	0.648783
$448$	0	0
$449$	−7347.50	−0.772272	−0.386136	−	0.922442i	$-0.626190\pi$
$449$	−7347.50	−0.772272	−0.386136	+	0.922442i	$0.626190\pi$
$450$	−4091.73	−0.428635
$451$	1267.69	0.132357
$452$	4356.83	0.453381
$453$	−8228.77	−0.853468
$454$	1944.62	0.201025
$455$	0	0
$456$	−2409.58	−0.247454
$457$	467.894	0.0478932	0.0239466	−	0.999713i	$-0.492377\pi$
$457$	467.894	0.0478932	0.0239466	+	0.999713i	$0.492377\pi$
$458$	1732.28	0.176733
$459$	−10418.2	−1.05943
$460$	−1504.69	−0.152514
$461$	10249.8	1.03554	0.517769	−	0.855521i	$-0.326763\pi$
$461$	10249.8	1.03554	0.517769	+	0.855521i	$0.326763\pi$
$462$	0	0
$463$	−10382.5	−1.04215	−0.521077	−	0.853510i	$-0.674469\pi$
$463$	−10382.5	−1.04215	−0.521077	+	0.853510i	$0.674469\pi$
$464$	−979.237	−0.0979740
$465$	735.362	0.0733368
$466$	8186.72	0.813825
$467$	7058.57	0.699425	0.349713	−	0.936857i	$-0.386279\pi$
$467$	7058.57	0.699425	0.349713	+	0.936857i	$0.386279\pi$
$468$	3409.98	0.336808
$469$	0	0
$470$	1309.51	0.128518
$471$	−5276.21	−0.516167
$472$	15157.8	1.47817
$473$	5210.02	0.506463
$474$	−8.24306	−0.000798769 0
$475$	−5117.75	−0.494355
$476$	0	0
$477$	−9572.65	−0.918871
$478$	95.2523	0.00911451
$479$	−12450.9	−1.18768	−0.593838	−	0.804584i	$-0.702388\pi$
$479$	−12450.9	−1.18768	−0.593838	+	0.804584i	$0.702388\pi$
$480$	1412.49	0.134314
$481$	4684.44	0.444059
$482$	−8717.57	−0.823806
$483$	0	0
$484$	1929.86	0.181241
$485$	2773.71	0.259686
$486$	6510.22	0.607633
$487$	−6239.35	−0.580558	−0.290279	−	0.956942i	$-0.593748\pi$
$487$	−6239.35	−0.580558	−0.290279	+	0.956942i	$0.593748\pi$
$488$	−8244.95	−0.764818
$489$	−8391.60	−0.776035
$490$	0	0
$491$	19789.8	1.81894	0.909471	−	0.415766i	$-0.136487\pi$
$491$	19789.8	1.81894	0.909471	+	0.415766i	$0.136487\pi$
$492$	514.810	0.0471736
$493$	23811.1	2.17525
$494$	−2365.03	−0.215400
$495$	−2049.29	−0.186078
$496$	−349.799	−0.0316662
$497$	0	0
$498$	2331.07	0.209754
$499$	−6244.64	−0.560218	−0.280109	−	0.959968i	$-0.590371\pi$
$499$	−6244.64	−0.560218	−0.280109	+	0.959968i	$0.590371\pi$
$500$	3890.81	0.348005
$501$	−7688.87	−0.685655
$502$	−3568.58	−0.317278
$503$	17641.9	1.56384	0.781921	−	0.623378i	$-0.214240\pi$
$503$	17641.9	1.56384	0.781921	+	0.623378i	$0.214240\pi$
$504$	0	0
$505$	3712.43	0.327130
$506$	−4849.55	−0.426065
$507$	2942.17	0.257724
$508$	−2153.94	−0.188121
$509$	−8303.68	−0.723093	−0.361546	−	0.932354i	$-0.617751\pi$
$509$	−8303.68	−0.723093	−0.361546	+	0.932354i	$0.617751\pi$
$510$	−1153.49	−0.100152
$511$	0	0
$512$	−1321.22	−0.114044
$513$	5213.15	0.448667
$514$	−5000.69	−0.429127
$515$	−1885.62	−0.161341
$516$	2115.79	0.180509
$517$	−7611.17	−0.647464
$518$	0	0
$519$	6747.90	0.570713
$520$	2201.69	0.185674
$521$	3965.57	0.333464	0.166732	−	0.986002i	$-0.446679\pi$
$521$	3965.57	0.333464	0.166732	+	0.986002i	$0.446679\pi$
$522$	−9526.09	−0.798746
$523$	21295.8	1.78050	0.890249	−	0.455474i	$-0.150530\pi$
$523$	21295.8	1.78050	0.890249	+	0.455474i	$0.150530\pi$
$524$	−5883.01	−0.490459
$525$	0	0
$526$	634.968	0.0526349
$527$	8505.68	0.703061
$528$	−275.717	−0.0227255
$529$	−3546.42	−0.291478
$530$	−2419.52	−0.198296
$531$	−14365.4	−1.17402
$532$	0	0
$533$	1290.77	0.104896
$534$	−3461.37	−0.280502
$535$	−360.470	−0.0291299
$536$	−9577.09	−0.771767
$537$	4241.00	0.340806
$538$	11177.1	0.895686
$539$	0	0
$540$	−1899.82	−0.151399
$541$	7730.61	0.614353	0.307177	−	0.951653i	$-0.400616\pi$
$541$	7730.61	0.614353	0.307177	+	0.951653i	$0.400616\pi$
$542$	4083.32	0.323604
$543$	−4508.47	−0.356311
$544$	16337.7	1.28764
$545$	2768.40	0.217588
$546$	0	0
$547$	−17761.6	−1.38835	−0.694177	−	0.719804i	$-0.744232\pi$
$547$	−17761.6	−1.38835	−0.694177	+	0.719804i	$0.744232\pi$
$548$	7037.04	0.548553
$549$	7813.95	0.607452
$550$	6010.98	0.466017
$551$	−11914.8	−0.921212
$552$	−5030.88	−0.387914
$553$	0	0
$554$	3960.10	0.303697
$555$	−1143.25	−0.0874387
$556$	4749.99	0.362310
$557$	5504.34	0.418719	0.209359	−	0.977839i	$-0.432862\pi$
$557$	5504.34	0.418719	0.209359	+	0.977839i	$0.432862\pi$
$558$	−3402.87	−0.258163
$559$	5304.89	0.401383
$560$	0	0
$561$	6704.33	0.504558
$562$	7676.09	0.576150
$563$	−25944.4	−1.94215	−0.971073	−	0.238784i	$-0.923251\pi$
$563$	−25944.4	−1.94215	−0.971073	+	0.238784i	$0.923251\pi$
$564$	−3090.90	−0.230763
$565$	−2665.99	−0.198512
$566$	3314.93	0.246178
$567$	0	0
$568$	21941.6	1.62086
$569$	600.070	0.0442113	0.0221057	−	0.999756i	$-0.492963\pi$
$569$	600.070	0.0442113	0.0221057	+	0.999756i	$0.492963\pi$
$570$	577.193	0.0424140
$571$	898.195	0.0658289	0.0329144	−	0.999458i	$-0.489521\pi$
$571$	898.195	0.0658289	0.0329144	+	0.999458i	$0.489521\pi$
$572$	−5009.45	−0.366181
$573$	2871.93	0.209384
$574$	0	0
$575$	−10685.2	−0.774959
$576$	−5920.85	−0.428303
$577$	−18984.3	−1.36972	−0.684859	−	0.728675i	$-0.740136\pi$
$577$	−18984.3	−1.36972	−0.684859	+	0.728675i	$0.740136\pi$
$578$	−5042.52	−0.362874
$579$	9638.27	0.691801
$580$	4342.09	0.310854
$581$	0	0
$582$	3630.37	0.258563
$583$	14062.8	0.999006
$584$	−13806.7	−0.978296
$585$	−2086.60	−0.147471
$586$	16466.5	1.16079
$587$	−13253.8	−0.931930	−0.465965	−	0.884803i	$-0.654293\pi$
$587$	−13253.8	−0.931930	−0.465965	+	0.884803i	$0.654293\pi$
$588$	0	0
$589$	−4256.15	−0.297745
$590$	−3630.91	−0.253360
$591$	9722.66	0.676712
$592$	543.826	0.0377552
$593$	4020.79	0.278439	0.139219	−	0.990262i	$-0.455541\pi$
$593$	4020.79	0.278439	0.139219	+	0.990262i	$0.455541\pi$
$594$	−6123.04	−0.422949
$595$	0	0
$596$	12932.7	0.888830
$597$	1953.39	0.133915
$598$	−4937.85	−0.337665
$599$	−16551.6	−1.12901	−0.564507	−	0.825428i	$-0.690934\pi$
$599$	−16551.6	−1.12901	−0.564507	+	0.825428i	$0.690934\pi$
$600$	6235.74	0.424288
$601$	20266.4	1.37551	0.687757	−	0.725941i	$-0.258596\pi$
$601$	20266.4	1.37551	0.687757	+	0.725941i	$0.258596\pi$
$602$	0	0
$603$	9076.45	0.612971
$604$	−17356.5	−1.16925
$605$	−1180.90	−0.0793561
$606$	4859.00	0.325715
$607$	−26773.8	−1.79031	−0.895153	−	0.445760i	$-0.852934\pi$
$607$	−26773.8	−1.79031	−0.895153	+	0.445760i	$0.852934\pi$
$608$	−8175.24	−0.545312
$609$	0	0
$610$	1975.00	0.131091
$611$	−7749.76	−0.513129
$612$	−9625.96	−0.635795
$613$	−4354.71	−0.286925	−0.143462	−	0.989656i	$-0.545824\pi$
$613$	−4354.71	−0.286925	−0.143462	+	0.989656i	$0.545824\pi$
$614$	10520.8	0.691504
$615$	−315.017	−0.0206548
$616$	0	0
$617$	−23574.0	−1.53817	−0.769087	−	0.639144i	$-0.779289\pi$
$617$	−23574.0	−1.53817	−0.769087	+	0.639144i	$0.779289\pi$
$618$	−2467.99	−0.160643
$619$	−25975.9	−1.68669	−0.843343	−	0.537376i	$-0.819415\pi$
$619$	−25975.9	−1.68669	−0.843343	+	0.537376i	$0.819415\pi$
$620$	1551.06	0.100471
$621$	10884.3	0.703339
$622$	5720.27	0.368749
$623$	0	0
$624$	−280.738	−0.0180104
$625$	12004.6	0.768294
$626$	4082.80	0.260673
$627$	−3354.78	−0.213679
$628$	−11128.8	−0.707147
$629$	−13223.6	−0.838252
$630$	0	0
$631$	−13625.7	−0.859635	−0.429817	−	0.902916i	$-0.641422\pi$
$631$	−13625.7	−0.859635	−0.429817	+	0.902916i	$0.641422\pi$
$632$	−44.4145	−0.00279543
$633$	781.643	0.0490798
$634$	16233.3	1.01689
$635$	1318.02	0.0823684
$636$	5710.90	0.356057
$637$	0	0
$638$	13994.4	0.868405
$639$	−20794.6	−1.28736
$640$	3134.82	0.193617
$641$	14845.7	0.914772	0.457386	−	0.889268i	$-0.348786\pi$
$641$	14845.7	0.914772	0.457386	+	0.889268i	$0.348786\pi$
$642$	−471.800	−0.0290039
$643$	−14609.6	−0.896030	−0.448015	−	0.894026i	$-0.647869\pi$
$643$	−14609.6	−0.896030	−0.448015	+	0.894026i	$0.647869\pi$
$644$	0	0
$645$	−1294.68	−0.0790354
$646$	6676.20	0.406612
$647$	−3694.79	−0.224509	−0.112254	−	0.993679i	$-0.535807\pi$
$647$	−3694.79	−0.224509	−0.112254	+	0.993679i	$0.535807\pi$
$648$	6268.09	0.379990
$649$	21103.7	1.27641
$650$	6120.43	0.369328
$651$	0	0
$652$	−17700.0	−1.06317
$653$	−16775.5	−1.00532	−0.502662	−	0.864483i	$-0.667646\pi$
$653$	−16775.5	−1.00532	−0.502662	+	0.864483i	$0.667646\pi$
$654$	3623.42	0.216646
$655$	3599.87	0.214746
$656$	149.848	0.00891858
$657$	13084.9	0.777005
$658$	0	0
$659$	2029.04	0.119939	0.0599697	−	0.998200i	$-0.480900\pi$
$659$	2029.04	0.119939	0.0599697	+	0.998200i	$0.480900\pi$
$660$	1222.57	0.0721040
$661$	−6719.58	−0.395403	−0.197701	−	0.980262i	$-0.563348\pi$
$661$	−6719.58	−0.395403	−0.197701	+	0.980262i	$0.563348\pi$
$662$	−2934.38	−0.172278
$663$	6826.40	0.399873
$664$	12560.0	0.734072
$665$	0	0
$666$	5290.38	0.307805
$667$	−24876.5	−1.44411
$668$	−16217.7	−0.939345
$669$	−15015.2	−0.867744
$670$	2294.10	0.132282
$671$	−11479.1	−0.660428
$672$	0	0
$673$	24199.9	1.38609	0.693045	−	0.720894i	$-0.256269\pi$
$673$	24199.9	1.38609	0.693045	+	0.720894i	$0.256269\pi$
$674$	18698.3	1.06860
$675$	−13491.1	−0.769291
$676$	6205.76	0.353081
$677$	17159.3	0.974128	0.487064	−	0.873366i	$-0.338068\pi$
$677$	17159.3	0.974128	0.487064	+	0.873366i	$0.338068\pi$
$678$	−3489.37	−0.197653
$679$	0	0
$680$	−6215.11	−0.350498
$681$	2808.65	0.158044
$682$	4999.01	0.280677
$683$	2440.06	0.136700	0.0683501	−	0.997661i	$-0.478226\pi$
$683$	2440.06	0.136700	0.0683501	+	0.997661i	$0.478226\pi$
$684$	4816.73	0.269258
$685$	−4306.03	−0.240183
$686$	0	0
$687$	2501.96	0.138946
$688$	615.854	0.0341268
$689$	14318.8	0.791733
$690$	1205.10	0.0664889
$691$	−20546.8	−1.13117	−0.565583	−	0.824691i	$-0.691349\pi$
$691$	−20546.8	−1.13117	−0.565583	+	0.824691i	$0.691349\pi$
$692$	14233.0	0.781875
$693$	0	0
$694$	−5486.66	−0.300102
$695$	−2906.57	−0.158637
$696$	14517.6	0.790645
$697$	−3643.70	−0.198013
$698$	696.879	0.0377898
$699$	11824.3	0.639820
$700$	0	0
$701$	−7033.05	−0.378937	−0.189468	−	0.981887i	$-0.560676\pi$
$701$	−7033.05	−0.378937	−0.189468	+	0.981887i	$0.560676\pi$
$702$	−6234.53	−0.335196
$703$	6616.97	0.354998
$704$	8698.08	0.465655
$705$	1891.36	0.101039
$706$	−10582.7	−0.564141
$707$	0	0
$708$	8570.21	0.454927
$709$	4233.67	0.224258	0.112129	−	0.993694i	$-0.464233\pi$
$709$	4233.67	0.224258	0.112129	+	0.993694i	$0.464233\pi$
$710$	−5255.90	−0.277818
$711$	42.0927	0.00222025
$712$	−18650.2	−0.981667
$713$	−8886.26	−0.466750
$714$	0	0
$715$	3065.33	0.160332
$716$	8945.32	0.466902
$717$	137.575	0.00716573
$718$	2600.72	0.135178
$719$	6827.15	0.354117	0.177058	−	0.984200i	$-0.443342\pi$
$719$	6827.15	0.354117	0.177058	+	0.984200i	$0.443342\pi$
$720$	−242.237	−0.0125384
$721$	0	0
$722$	8246.14	0.425055
$723$	−12591.0	−0.647667
$724$	−9509.48	−0.488145
$725$	30834.2	1.57952
$726$	−1545.62	−0.0790127
$727$	24222.3	1.23570	0.617851	−	0.786295i	$-0.288003\pi$
$727$	24222.3	1.23570	0.617851	+	0.786295i	$0.288003\pi$
$728$	0	0
$729$	1782.21	0.0905456
$730$	3307.26	0.167681
$731$	−14975.1	−0.757692
$732$	−4661.69	−0.235384
$733$	33556.0	1.69088	0.845442	−	0.534067i	$-0.179337\pi$
$733$	33556.0	1.69088	0.845442	+	0.534067i	$0.179337\pi$
$734$	−12337.5	−0.620415
$735$	0	0
$736$	−17068.8	−0.854841
$737$	−13333.8	−0.666429
$738$	1457.73	0.0727099
$739$	27588.9	1.37331	0.686653	−	0.726985i	$-0.259079\pi$
$739$	27588.9	1.37331	0.686653	+	0.726985i	$0.259079\pi$
$740$	−2411.41	−0.119791
$741$	−3415.86	−0.169345
$742$	0	0
$743$	−7503.10	−0.370474	−0.185237	−	0.982694i	$-0.559305\pi$
$743$	−7503.10	−0.370474	−0.185237	+	0.982694i	$0.559305\pi$
$744$	5185.92	0.255544
$745$	−7913.63	−0.389172
$746$	10067.7	0.494106
$747$	−11903.5	−0.583032
$748$	14141.1	0.691242
$749$	0	0
$750$	−3116.14	−0.151714
$751$	−14203.4	−0.690132	−0.345066	−	0.938578i	$-0.612143\pi$
$751$	−14203.4	−0.690132	−0.345066	+	0.938578i	$0.612143\pi$
$752$	−899.684	−0.0436278
$753$	−5154.17	−0.249440
$754$	14249.2	0.688229
$755$	10620.6	0.511952
$756$	0	0
$757$	−15469.6	−0.742739	−0.371370	−	0.928485i	$-0.621112\pi$
$757$	−15469.6	−0.742739	−0.371370	+	0.928485i	$0.621112\pi$
$758$	15827.5	0.758420
$759$	−7004.31	−0.334967
$760$	3109.98	0.148435
$761$	25298.7	1.20509	0.602547	−	0.798083i	$-0.294153\pi$
$761$	25298.7	1.20509	0.602547	+	0.798083i	$0.294153\pi$
$762$	1725.08	0.0820120
$763$	0	0
$764$	6057.62	0.286855
$765$	5890.22	0.278381
$766$	−1386.59	−0.0654039
$767$	21487.9	1.01158
$768$	9594.01	0.450773
$769$	−37278.1	−1.74809	−0.874046	−	0.485843i	$-0.838513\pi$
$769$	−37278.1	−1.74809	−0.874046	+	0.485843i	$0.838513\pi$
$770$	0	0
$771$	−7222.60	−0.337374
$772$	20329.5	0.947765
$773$	37946.9	1.76566	0.882830	−	0.469692i	$-0.155635\pi$
$773$	37946.9	1.76566	0.882830	+	0.469692i	$0.155635\pi$
$774$	5991.08	0.278223
$775$	11014.5	0.510517
$776$	19560.8	0.904886
$777$	0	0
$778$	−4273.23	−0.196919
$779$	1823.27	0.0838580
$780$	1244.83	0.0571439
$781$	30548.5	1.39963
$782$	13939.0	0.637413
$783$	−31409.0	−1.43355
$784$	0	0
$785$	6809.84	0.309623
$786$	4711.68	0.213817
$787$	−29896.1	−1.35410	−0.677052	−	0.735935i	$-0.736743\pi$
$787$	−29896.1	−1.35410	−0.677052	+	0.735935i	$0.736743\pi$
$788$	20507.5	0.927093
$789$	917.098	0.0413809
$790$	10.6391	0.000479141 0
$791$	0	0
$792$	−14452.0	−0.648394
$793$	−11688.2	−0.523403
$794$	7389.81	0.330296
$795$	−3494.56	−0.155898
$796$	4120.19	0.183463
$797$	−2482.95	−0.110352	−0.0551760	−	0.998477i	$-0.517572\pi$
$797$	−2482.95	−0.110352	−0.0551760	+	0.998477i	$0.517572\pi$
$798$	0	0
$799$	21876.7	0.968637
$800$	21156.6	0.934999
$801$	17675.3	0.779683
$802$	−5160.89	−0.227229
$803$	−19222.5	−0.844768
$804$	−5414.88	−0.237522
$805$	0	0
$806$	5090.03	0.222443
$807$	16143.3	0.704178
$808$	26180.8	1.13990
$809$	−13931.9	−0.605464	−0.302732	−	0.953076i	$-0.597899\pi$
$809$	−13931.9	−0.605464	−0.302732	+	0.953076i	$0.597899\pi$
$810$	−1501.46	−0.0651309
$811$	23457.0	1.01564	0.507822	−	0.861462i	$-0.330451\pi$
$811$	23457.0	1.01564	0.507822	+	0.861462i	$0.330451\pi$
$812$	0	0
$813$	5897.62	0.254414
$814$	−7771.87	−0.334648
$815$	10830.8	0.465504
$816$	792.490	0.0339984
$817$	7493.37	0.320881
$818$	−12604.7	−0.538771
$819$	0	0
$820$	−664.450	−0.0282971
$821$	−29634.9	−1.25976	−0.629881	−	0.776691i	$-0.716896\pi$
$821$	−29634.9	−1.25976	−0.629881	+	0.776691i	$0.716896\pi$
$822$	−5635.94	−0.239143
$823$	42873.9	1.81591	0.907953	−	0.419073i	$-0.137645\pi$
$823$	42873.9	1.81591	0.907953	+	0.419073i	$0.137645\pi$
$824$	−13297.8	−0.562197
$825$	8681.79	0.366377
$826$	0	0
$827$	32579.8	1.36990	0.684952	−	0.728589i	$-0.259823\pi$
$827$	32579.8	1.36990	0.684952	+	0.728589i	$0.259823\pi$
$828$	10056.7	0.422093
$829$	−36526.5	−1.53030	−0.765150	−	0.643852i	$-0.777335\pi$
$829$	−36526.5	−1.53030	−0.765150	+	0.643852i	$0.777335\pi$
$830$	−3008.64	−0.125821
$831$	5719.65	0.238763
$832$	8856.45	0.369041
$833$	0	0
$834$	−3804.25	−0.157950
$835$	9923.79	0.411290
$836$	−7076.05	−0.292740
$837$	−11219.8	−0.463336
$838$	−17390.7	−0.716886
$839$	−15136.4	−0.622845	−0.311423	−	0.950272i	$-0.600805\pi$
$839$	−15136.4	−0.622845	−0.311423	+	0.950272i	$0.600805\pi$
$840$	0	0
$841$	47397.1	1.94338
$842$	26949.4	1.10301
$843$	11086.7	0.452963
$844$	1648.68	0.0672391
$845$	−3797.37	−0.154596
$846$	−8752.19	−0.355682
$847$	0	0
$848$	1662.30	0.0673156
$849$	4787.83	0.193543
$850$	−17277.3	−0.697183
$851$	13815.3	0.556501
$852$	12405.8	0.498843
$853$	2912.85	0.116922	0.0584608	−	0.998290i	$-0.481381\pi$
$853$	2912.85	0.116922	0.0584608	+	0.998290i	$0.481381\pi$
$854$	0	0
$855$	−2947.41	−0.117894
$856$	−2542.11	−0.101504
$857$	7934.06	0.316245	0.158123	−	0.987419i	$-0.449456\pi$
$857$	7934.06	0.316245	0.158123	+	0.987419i	$0.449456\pi$
$858$	4012.05	0.159638
$859$	−17792.9	−0.706734	−0.353367	−	0.935485i	$-0.614963\pi$
$859$	−17792.9	−0.706734	−0.353367	+	0.935485i	$0.614963\pi$
$860$	−2730.79	−0.108278
$861$	0	0
$862$	6345.85	0.250743
$863$	−26295.2	−1.03719	−0.518597	−	0.855019i	$-0.673545\pi$
$863$	−26295.2	−1.03719	−0.518597	+	0.855019i	$0.673545\pi$
$864$	−21551.0	−0.848588
$865$	−8709.32	−0.342342
$866$	7280.87	0.285698
$867$	−7283.02	−0.285287
$868$	0	0
$869$	−61.8366	−0.00241388
$870$	−3477.56	−0.135518
$871$	−13576.6	−0.528159
$872$	19523.3	0.758192
$873$	−18538.3	−0.718700
$874$	−6974.92	−0.269943
$875$	0	0
$876$	−7806.29	−0.301085
$877$	31547.9	1.21471	0.607353	−	0.794432i	$-0.292231\pi$
$877$	31547.9	1.21471	0.607353	+	0.794432i	$0.292231\pi$
$878$	−5508.81	−0.211746
$879$	23782.9	0.912602
$880$	355.860	0.0136319
$881$	−19775.2	−0.756234	−0.378117	−	0.925758i	$-0.623428\pi$
$881$	−19775.2	−0.756234	−0.378117	+	0.925758i	$0.623428\pi$
$882$	0	0
$883$	7278.33	0.277390	0.138695	−	0.990335i	$-0.455709\pi$
$883$	7278.33	0.277390	0.138695	+	0.990335i	$0.455709\pi$
$884$	14398.6	0.547824
$885$	−5244.20	−0.199188
$886$	6185.37	0.234539
$887$	−2685.15	−0.101644	−0.0508221	−	0.998708i	$-0.516184\pi$
$887$	−2685.15	−0.101644	−0.0508221	+	0.998708i	$0.516184\pi$
$888$	−8062.46	−0.304683
$889$	0	0
$890$	4467.49	0.168259
$891$	8726.83	0.328125
$892$	−31670.7	−1.18881
$893$	−10946.8	−0.410215
$894$	−10357.7	−0.387488
$895$	−5473.73	−0.204432
$896$	0	0
$897$	−7131.84	−0.265469
$898$	12412.1	0.461243
$899$	25643.1	0.951330
$900$	−12465.2	−0.461673
$901$	−40420.4	−1.49456
$902$	−2141.49	−0.0790510
$903$	0	0
$904$	−18801.1	−0.691721
$905$	5818.95	0.213733
$906$	13900.8	0.509737
$907$	−35185.6	−1.28811	−0.644057	−	0.764977i	$-0.722750\pi$
$907$	−35185.6	−1.28811	−0.644057	+	0.764977i	$0.722750\pi$
$908$	5924.14	0.216519
$909$	−24812.2	−0.905356
$910$	0	0
$911$	10496.0	0.381720	0.190860	−	0.981617i	$-0.438872\pi$
$911$	10496.0	0.381720	0.190860	+	0.981617i	$0.438872\pi$
$912$	−396.553	−0.0143982
$913$	17486.9	0.633878
$914$	−790.409	−0.0286044
$915$	2852.54	0.103062
$916$	5277.26	0.190355
$917$	0	0
$918$	17599.3	0.632750
$919$	23701.1	0.850738	0.425369	−	0.905020i	$-0.360144\pi$
$919$	23701.1	0.850738	0.425369	+	0.905020i	$0.360144\pi$
$920$	6493.20	0.232690
$921$	15195.3	0.543652
$922$	−17314.9	−0.618479
$923$	31104.7	1.10923
$924$	0	0
$925$	−17124.0	−0.608684
$926$	17539.1	0.622430
$927$	12602.7	0.446522
$928$	49255.4	1.74234
$929$	56090.7	1.98092	0.990460	−	0.137800i	$-0.0440030\pi$
$929$	56090.7	1.98092	0.990460	+	0.137800i	$0.0440030\pi$
$930$	−1242.24	−0.0438007
$931$	0	0
$932$	24940.3	0.876551
$933$	8261.91	0.289907
$934$	−11924.0	−0.417735
$935$	−8653.08	−0.302659
$936$	−14715.1	−0.513866
$937$	−16422.6	−0.572577	−0.286288	−	0.958143i	$-0.592422\pi$
$937$	−16422.6	−0.572577	−0.286288	+	0.958143i	$0.592422\pi$
$938$	0	0
$939$	5896.88	0.204939
$940$	3989.34	0.138423
$941$	−47774.6	−1.65506	−0.827528	−	0.561424i	$-0.810254\pi$
$941$	−47774.6	−1.65506	−0.827528	+	0.561424i	$0.810254\pi$
$942$	8913.04	0.308283
$943$	3806.73	0.131457
$944$	2494.57	0.0860078
$945$	0	0
$946$	−8801.24	−0.302487
$947$	13672.4	0.469160	0.234580	−	0.972097i	$-0.424628\pi$
$947$	13672.4	0.469160	0.234580	+	0.972097i	$0.424628\pi$
$948$	−25.1119	−0.000860334 0
$949$	−19572.6	−0.669496
$950$	8645.36	0.295255
$951$	23446.1	0.799466
$952$	0	0
$953$	2821.39	0.0959012	0.0479506	−	0.998850i	$-0.484731\pi$
$953$	2821.39	0.0959012	0.0479506	+	0.998850i	$0.484731\pi$
$954$	16171.0	0.548799
$955$	−3706.72	−0.125599
$956$	290.179	0.00981702
$957$	20212.4	0.682730
$958$	21033.2	0.709345
$959$	0	0
$960$	−2161.45	−0.0726671
$961$	−20630.9	−0.692521
$962$	−7913.38	−0.265216
$963$	2409.22	0.0806190
$964$	−26557.5	−0.887301
$965$	−12439.8	−0.414976
$966$	0	0
$967$	−19660.0	−0.653798	−0.326899	−	0.945059i	$-0.606004\pi$
$967$	−19660.0	−0.653798	−0.326899	+	0.945059i	$0.606004\pi$
$968$	−8327.95	−0.276519
$969$	9642.57	0.319674
$970$	−4685.60	−0.155099
$971$	−52982.9	−1.75108	−0.875541	−	0.483143i	$-0.839495\pi$
$971$	−52982.9	−1.75108	−0.875541	+	0.483143i	$0.839495\pi$
$972$	19832.9	0.654466
$973$	0	0
$974$	10540.1	0.346741
$975$	8839.87	0.290361
$976$	−1356.90	−0.0445013
$977$	−5245.98	−0.171785	−0.0858925	−	0.996304i	$-0.527374\pi$
$977$	−5245.98	−0.171785	−0.0858925	+	0.996304i	$0.527374\pi$
$978$	14175.8	0.463490
$979$	−25966.0	−0.847680
$980$	0	0
$981$	−18502.8	−0.602190
$982$	−33430.7	−1.08637
$983$	39339.2	1.27643	0.638213	−	0.769860i	$-0.279674\pi$
$983$	39339.2	1.27643	0.638213	+	0.769860i	$0.279674\pi$
$984$	−2221.57	−0.0719725
$985$	−12548.7	−0.405925
$986$	−40223.8	−1.29917
$987$	0	0
$988$	−7204.89	−0.232002
$989$	15645.1	0.503019
$990$	3461.84	0.111136
$991$	2368.20	0.0759115	0.0379557	−	0.999279i	$-0.487915\pi$
$991$	2368.20	0.0759115	0.0379557	+	0.999279i	$0.487915\pi$
$992$	17594.8	0.563141
$993$	−4238.19	−0.135443
$994$	0	0
$995$	−2521.19	−0.0803286
$996$	7101.43	0.225921
$997$	14564.0	0.462634	0.231317	−	0.972878i	$-0.425697\pi$
$997$	14564.0	0.462634	0.231317	+	0.972878i	$0.425697\pi$
$998$	10549.0	0.334592
$999$	17443.2	0.552431

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.4.a.j.1.14		36
7.2	even	3		287.4.e.a.165.23	✓	72
7.4	even	3		287.4.e.a.247.23	yes	72
7.6	odd	2		2009.4.a.k.1.14		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.4.e.a.165.23	✓	72	7.2	even	3
287.4.e.a.247.23	yes	72	7.4	even	3
2009.4.a.j.1.14		36	1.1	even	1	trivial
2009.4.a.k.1.14		36	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2009 = 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2009.a (trivial)

\(f(q)\)	\(=\)	\(q-1.68929 q^{2} -2.43988 q^{3} -5.14630 q^{4} +3.14907 q^{5} +4.12166 q^{6} +22.2079 q^{8} -21.0470 q^{9} +O(q^{10})\)	\(q-1.68929 q^{2} -2.43988 q^{3} -5.14630 q^{4} +3.14907 q^{5} +4.12166 q^{6} +22.2079 q^{8} -21.0470 q^{9} -5.31970 q^{10} +30.9193 q^{11} +12.5563 q^{12} +31.4823 q^{13} -7.68335 q^{15} +3.65483 q^{16} -88.8707 q^{17} +35.5545 q^{18} +44.4699 q^{19} -16.2061 q^{20} -52.2316 q^{22} +92.8471 q^{23} -54.1845 q^{24} -115.083 q^{25} -53.1826 q^{26} +117.229 q^{27} -267.929 q^{29} +12.9794 q^{30} -95.7085 q^{31} -183.837 q^{32} -75.4392 q^{33} +150.128 q^{34} +108.314 q^{36} +148.796 q^{37} -75.1226 q^{38} -76.8128 q^{39} +69.9344 q^{40} +41.0000 q^{41} +168.504 q^{43} -159.120 q^{44} -66.2786 q^{45} -156.846 q^{46} -246.163 q^{47} -8.91734 q^{48} +194.409 q^{50} +216.833 q^{51} -162.017 q^{52} +454.822 q^{53} -198.033 q^{54} +97.3671 q^{55} -108.501 q^{57} +452.610 q^{58} +682.541 q^{59} +39.5408 q^{60} -371.262 q^{61} +161.679 q^{62} +281.316 q^{64} +99.1400 q^{65} +127.439 q^{66} -431.247 q^{67} +457.355 q^{68} -226.535 q^{69} +988.008 q^{71} -467.410 q^{72} -621.701 q^{73} -251.360 q^{74} +280.789 q^{75} -228.856 q^{76} +129.759 q^{78} -1.99994 q^{79} +11.5093 q^{80} +282.246 q^{81} -69.2609 q^{82} +565.566 q^{83} -279.860 q^{85} -284.652 q^{86} +653.714 q^{87} +686.652 q^{88} -839.802 q^{89} +111.964 q^{90} -477.819 q^{92} +233.517 q^{93} +415.840 q^{94} +140.039 q^{95} +448.540 q^{96} +880.803 q^{97} -650.758 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 5 q^{2} - 6 q^{3} + 117 q^{4} + 4 q^{5} - 12 q^{6} - 39 q^{8} + 236 q^{9}+O(q^{10}) \) 36 * q - 5 * q^2 - 6 * q^3 + 117 * q^4 + 4 * q^5 - 12 * q^6 - 39 * q^8 + 236 * q^9	\( 36 q - 5 q^{2} - 6 q^{3} + 117 q^{4} + 4 q^{5} - 12 q^{6} - 39 q^{8} + 236 q^{9} - 12 q^{10} - 140 q^{11} + 186 q^{12} - 72 q^{13} - 366 q^{15} - 15 q^{16} - 2 q^{17} - 212 q^{18} + 30 q^{19} - 334 q^{20} - 346 q^{22} - 314 q^{23} + 106 q^{24} + 570 q^{25} + 303 q^{26} - 204 q^{27} - 356 q^{29} - 357 q^{30} - 4 q^{31} - 532 q^{32} - 30 q^{33} - 364 q^{34} + 113 q^{36} - 1398 q^{37} - 264 q^{38} - 1348 q^{39} + 26 q^{40} + 1476 q^{41} - 1072 q^{43} - 1507 q^{44} - 1132 q^{45} - 1356 q^{46} - 622 q^{47} + 1724 q^{48} - 1426 q^{50} - 668 q^{51} - 877 q^{52} - 412 q^{53} - 1814 q^{54} + 1114 q^{55} - 4082 q^{57} - 1309 q^{58} - 620 q^{59} - 3724 q^{60} + 774 q^{61} + 1665 q^{62} - 3285 q^{64} - 1036 q^{65} - 1056 q^{66} - 2972 q^{67} - 1525 q^{68} + 3304 q^{69} - 3540 q^{71} - 821 q^{72} - 60 q^{73} - 2043 q^{74} + 450 q^{75} + 2171 q^{76} - 1136 q^{78} - 5190 q^{79} - 1564 q^{80} + 284 q^{81} - 205 q^{82} + 1656 q^{83} - 5064 q^{85} - 782 q^{86} - 1940 q^{87} - 4232 q^{88} - 1196 q^{89} + 8030 q^{90} - 4618 q^{92} + 698 q^{93} - 35 q^{94} - 1968 q^{95} - 7926 q^{96} + 3862 q^{97} - 5964 q^{99}+O(q^{100}) \) 36 * q - 5 * q^2 - 6 * q^3 + 117 * q^4 + 4 * q^5 - 12 * q^6 - 39 * q^8 + 236 * q^9 - 12 * q^10 - 140 * q^11 + 186 * q^12 - 72 * q^13 - 366 * q^15 - 15 * q^16 - 2 * q^17 - 212 * q^18 + 30 * q^19 - 334 * q^20 - 346 * q^22 - 314 * q^23 + 106 * q^24 + 570 * q^25 + 303 * q^26 - 204 * q^27 - 356 * q^29 - 357 * q^30 - 4 * q^31 - 532 * q^32 - 30 * q^33 - 364 * q^34 + 113 * q^36 - 1398 * q^37 - 264 * q^38 - 1348 * q^39 + 26 * q^40 + 1476 * q^41 - 1072 * q^43 - 1507 * q^44 - 1132 * q^45 - 1356 * q^46 - 622 * q^47 + 1724 * q^48 - 1426 * q^50 - 668 * q^51 - 877 * q^52 - 412 * q^53 - 1814 * q^54 + 1114 * q^55 - 4082 * q^57 - 1309 * q^58 - 620 * q^59 - 3724 * q^60 + 774 * q^61 + 1665 * q^62 - 3285 * q^64 - 1036 * q^65 - 1056 * q^66 - 2972 * q^67 - 1525 * q^68 + 3304 * q^69 - 3540 * q^71 - 821 * q^72 - 60 * q^73 - 2043 * q^74 + 450 * q^75 + 2171 * q^76 - 1136 * q^78 - 5190 * q^79 - 1564 * q^80 + 284 * q^81 - 205 * q^82 + 1656 * q^83 - 5064 * q^85 - 782 * q^86 - 1940 * q^87 - 4232 * q^88 - 1196 * q^89 + 8030 * q^90 - 4618 * q^92 + 698 * q^93 - 35 * q^94 - 1968 * q^95 - 7926 * q^96 + 3862 * q^97 - 5964 * q^99

Introduction

L-functions

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