LMFDB - Embedded newform 507.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{14}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 14 $ x^2 - 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.74166$ of defining polynomial
Character	$\chi$	$=$	507.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.74166 q^{2} -3.00000 q^{3} +14.4833 q^{4} -4.51669 q^{5} +14.2250 q^{6} +7.48331 q^{7} -30.7417 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-4.74166 q^{2} -3.00000 q^{3} +14.4833 q^{4} -4.51669 q^{5} +14.2250 q^{6} +7.48331 q^{7} -30.7417 q^{8} +9.00000 q^{9} +21.4166 q^{10} +66.8999 q^{11} -43.4499 q^{12} -35.4833 q^{14} +13.5501 q^{15} +29.8999 q^{16} +96.9666 q^{17} -42.6749 q^{18} -31.4833 q^{19} -65.4166 q^{20} -22.4499 q^{21} -317.216 q^{22} +183.600 q^{23} +92.2250 q^{24} -104.600 q^{25} -27.0000 q^{27} +108.383 q^{28} +112.200 q^{29} -64.2497 q^{30} +77.2831 q^{31} +104.158 q^{32} -200.700 q^{33} -459.783 q^{34} -33.7998 q^{35} +130.350 q^{36} -54.7664 q^{37} +149.283 q^{38} +138.850 q^{40} -451.716 q^{41} +106.450 q^{42} -113.434 q^{43} +968.932 q^{44} -40.6502 q^{45} -870.566 q^{46} +42.2670 q^{47} -89.6997 q^{48} -287.000 q^{49} +495.975 q^{50} -290.900 q^{51} -530.999 q^{53} +128.025 q^{54} -302.166 q^{55} -230.049 q^{56} +94.4499 q^{57} -532.015 q^{58} -219.666 q^{59} +196.250 q^{60} +822.865 q^{61} -366.450 q^{62} +67.3498 q^{63} -733.082 q^{64} +951.649 q^{66} +872.082 q^{67} +1404.40 q^{68} -550.799 q^{69} +160.267 q^{70} +100.299 q^{71} -276.675 q^{72} +165.634 q^{73} +259.684 q^{74} +313.799 q^{75} -455.983 q^{76} +500.633 q^{77} -545.266 q^{79} -135.048 q^{80} +81.0000 q^{81} +2141.88 q^{82} +454.534 q^{83} -325.150 q^{84} -437.968 q^{85} +537.864 q^{86} -336.601 q^{87} -2056.61 q^{88} +230.915 q^{89} +192.749 q^{90} +2659.13 q^{92} -231.849 q^{93} -200.415 q^{94} +142.200 q^{95} -312.475 q^{96} +1089.16 q^{97} +1360.86 q^{98} +602.099 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 6 q^{3} + 14 q^{4} - 24 q^{5} + 6 q^{6} - 54 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 6 * q^3 + 14 * q^4 - 24 * q^5 + 6 * q^6 - 54 * q^8 + 18 * q^9	\( 2 q - 2 q^{2} - 6 q^{3} + 14 q^{4} - 24 q^{5} + 6 q^{6} - 54 q^{8} + 18 q^{9} - 32 q^{10} + 44 q^{11} - 42 q^{12} - 56 q^{14} + 72 q^{15} - 30 q^{16} + 164 q^{17} - 18 q^{18} - 48 q^{19} - 56 q^{20} - 380 q^{22} + 8 q^{23} + 162 q^{24} + 150 q^{25} - 54 q^{27} + 112 q^{28} + 404 q^{29} + 96 q^{30} - 40 q^{31} + 126 q^{32} - 132 q^{33} - 276 q^{34} + 112 q^{35} + 126 q^{36} + 100 q^{37} + 104 q^{38} + 592 q^{40} - 200 q^{41} + 168 q^{42} - 616 q^{43} + 980 q^{44} - 216 q^{45} - 1352 q^{46} + 324 q^{47} + 90 q^{48} - 574 q^{49} + 1194 q^{50} - 492 q^{51} - 164 q^{53} + 54 q^{54} + 144 q^{55} - 56 q^{56} + 144 q^{57} + 268 q^{58} - 140 q^{59} + 168 q^{60} + 628 q^{61} - 688 q^{62} - 194 q^{64} + 1140 q^{66} + 472 q^{67} + 1372 q^{68} - 24 q^{69} + 560 q^{70} - 428 q^{71} - 486 q^{72} + 900 q^{73} + 684 q^{74} - 450 q^{75} - 448 q^{76} + 672 q^{77} - 432 q^{79} + 1032 q^{80} + 162 q^{81} + 2832 q^{82} + 1388 q^{83} - 336 q^{84} - 1744 q^{85} - 840 q^{86} - 1212 q^{87} - 1524 q^{88} - 960 q^{89} - 288 q^{90} + 2744 q^{92} + 120 q^{93} + 572 q^{94} + 464 q^{95} - 378 q^{96} + 532 q^{97} + 574 q^{98} + 396 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 6 * q^3 + 14 * q^4 - 24 * q^5 + 6 * q^6 - 54 * q^8 + 18 * q^9 - 32 * q^10 + 44 * q^11 - 42 * q^12 - 56 * q^14 + 72 * q^15 - 30 * q^16 + 164 * q^17 - 18 * q^18 - 48 * q^19 - 56 * q^20 - 380 * q^22 + 8 * q^23 + 162 * q^24 + 150 * q^25 - 54 * q^27 + 112 * q^28 + 404 * q^29 + 96 * q^30 - 40 * q^31 + 126 * q^32 - 132 * q^33 - 276 * q^34 + 112 * q^35 + 126 * q^36 + 100 * q^37 + 104 * q^38 + 592 * q^40 - 200 * q^41 + 168 * q^42 - 616 * q^43 + 980 * q^44 - 216 * q^45 - 1352 * q^46 + 324 * q^47 + 90 * q^48 - 574 * q^49 + 1194 * q^50 - 492 * q^51 - 164 * q^53 + 54 * q^54 + 144 * q^55 - 56 * q^56 + 144 * q^57 + 268 * q^58 - 140 * q^59 + 168 * q^60 + 628 * q^61 - 688 * q^62 - 194 * q^64 + 1140 * q^66 + 472 * q^67 + 1372 * q^68 - 24 * q^69 + 560 * q^70 - 428 * q^71 - 486 * q^72 + 900 * q^73 + 684 * q^74 - 450 * q^75 - 448 * q^76 + 672 * q^77 - 432 * q^79 + 1032 * q^80 + 162 * q^81 + 2832 * q^82 + 1388 * q^83 - 336 * q^84 - 1744 * q^85 - 840 * q^86 - 1212 * q^87 - 1524 * q^88 - 960 * q^89 - 288 * q^90 + 2744 * q^92 + 120 * q^93 + 572 * q^94 + 464 * q^95 - 378 * q^96 + 532 * q^97 + 574 * q^98 + 396 * 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 $

$2$	−4.74166	−1.67643	−0.838215	−	0.545341i	$-0.816400\pi$
$2$	−4.74166	−1.67643	−0.838215	+	0.545341i	$0.816400\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	14.4833	1.81041
$5$	−4.51669	−0.403985	−0.201992	−	0.979387i	$-0.564742\pi$
$5$	−4.51669	−0.403985	−0.201992	+	0.979387i	$0.564742\pi$
$6$	14.2250	0.967887
$7$	7.48331	0.404061	0.202031	−	0.979379i	$-0.435246\pi$
$7$	7.48331	0.404061	0.202031	+	0.979379i	$0.435246\pi$
$8$	−30.7417	−1.35860
$9$	9.00000	0.333333
$10$	21.4166	0.677252
$11$	66.8999	1.83373	0.916867	−	0.399193i	$-0.130710\pi$
$11$	66.8999	1.83373	0.916867	+	0.399193i	$0.130710\pi$
$12$	−43.4499	−1.04524
$13$	0	0
$13$	0	0
$14$	−35.4833	−0.677380
$15$	13.5501	0.233241
$16$	29.8999	0.467186
$17$	96.9666	1.38340	0.691702	−	0.722183i	$-0.256861\pi$
$17$	96.9666	1.38340	0.691702	+	0.722183i	$0.256861\pi$
$18$	−42.6749	−0.558810
$19$	−31.4833	−0.380146	−0.190073	−	0.981770i	$-0.560872\pi$
$19$	−31.4833	−0.380146	−0.190073	+	0.981770i	$0.560872\pi$
$20$	−65.4166	−0.731380
$21$	−22.4499	−0.233285
$22$	−317.216	−3.07413
$23$	183.600	1.66448	0.832242	−	0.554412i	$-0.187057\pi$
$23$	183.600	1.66448	0.832242	+	0.554412i	$0.187057\pi$
$24$	92.2250	0.784389
$25$	−104.600	−0.836796
$26$	0	0
$27$	−27.0000	−0.192450
$28$	108.383	0.731518
$29$	112.200	0.718450	0.359225	−	0.933251i	$-0.383041\pi$
$29$	112.200	0.718450	0.359225	+	0.933251i	$0.383041\pi$
$30$	−64.2497	−0.391011
$31$	77.2831	0.447757	0.223878	−	0.974617i	$-0.428128\pi$
$31$	77.2831	0.447757	0.223878	+	0.974617i	$0.428128\pi$
$32$	104.158	0.575398
$33$	−200.700	−1.05871
$34$	−459.783	−2.31918
$35$	−33.7998	−0.163234
$36$	130.350	0.603471
$37$	−54.7664	−0.243339	−0.121669	−	0.992571i	$-0.538825\pi$
$37$	−54.7664	−0.243339	−0.121669	+	0.992571i	$0.538825\pi$
$38$	149.283	0.637287
$39$	0	0
$40$	138.850	0.548854
$41$	−451.716	−1.72064	−0.860319	−	0.509756i	$-0.829736\pi$
$41$	−451.716	−1.72064	−0.860319	+	0.509756i	$0.829736\pi$
$42$	106.450	0.391085
$43$	−113.434	−0.402291	−0.201145	−	0.979561i	$-0.564466\pi$
$43$	−113.434	−0.402291	−0.201145	+	0.979561i	$0.564466\pi$
$44$	968.932	3.31982
$45$	−40.6502	−0.134662
$46$	−870.566	−2.79039
$47$	42.2670	0.131176	0.0655880	−	0.997847i	$-0.479108\pi$
$47$	42.2670	0.131176	0.0655880	+	0.997847i	$0.479108\pi$
$48$	−89.6997	−0.269730
$49$	−287.000	−0.836735
$50$	495.975	1.40283
$51$	−290.900	−0.798708
$52$	0	0
$53$	−530.999	−1.37619	−0.688097	−	0.725618i	$-0.741554\pi$
$53$	−530.999	−1.37619	−0.688097	+	0.725618i	$0.741554\pi$
$54$	128.025	0.322629
$55$	−302.166	−0.740800
$56$	−230.049	−0.548958
$57$	94.4499	0.219477
$58$	−532.015	−1.20443
$59$	−219.666	−0.484714	−0.242357	−	0.970187i	$-0.577920\pi$
$59$	−219.666	−0.484714	−0.242357	+	0.970187i	$0.577920\pi$
$60$	196.250	0.422262
$61$	822.865	1.72717	0.863583	−	0.504207i	$-0.168215\pi$
$61$	822.865	1.72717	0.863583	+	0.504207i	$0.168215\pi$
$62$	−366.450	−0.750632
$63$	67.3498	0.134687
$64$	−733.082	−1.43180
$65$	0	0
$66$	951.649	1.77485
$67$	872.082	1.59018	0.795088	−	0.606495i	$-0.207425\pi$
$67$	872.082	1.59018	0.795088	+	0.606495i	$0.207425\pi$
$68$	1404.40	2.50453
$69$	−550.799	−0.960991
$70$	160.267	0.273651
$71$	100.299	0.167653	0.0838263	−	0.996480i	$-0.473286\pi$
$71$	100.299	0.167653	0.0838263	+	0.996480i	$0.473286\pi$
$72$	−276.675	−0.452867
$73$	165.634	0.265562	0.132781	−	0.991145i	$-0.457609\pi$
$73$	165.634	0.265562	0.132781	+	0.991145i	$0.457609\pi$
$74$	259.684	0.407941
$75$	313.799	0.483125
$76$	−455.983	−0.688221
$77$	500.633	0.740940
$78$	0	0
$79$	−545.266	−0.776547	−0.388273	−	0.921544i	$-0.626928\pi$
$79$	−545.266	−0.776547	−0.388273	+	0.921544i	$0.626928\pi$
$80$	−135.048	−0.188736
$81$	81.0000	0.111111
$82$	2141.88	2.88453
$83$	454.534	0.601103	0.300552	−	0.953766i	$-0.402829\pi$
$83$	454.534	0.601103	0.300552	+	0.953766i	$0.402829\pi$
$84$	−325.150	−0.422342
$85$	−437.968	−0.558874
$86$	537.864	0.674412
$87$	−336.601	−0.414797
$88$	−2056.61	−2.49132
$89$	230.915	0.275022	0.137511	−	0.990500i	$-0.456090\pi$
$89$	230.915	0.275022	0.137511	+	0.990500i	$0.456090\pi$
$90$	192.749	0.225751
$91$	0	0
$92$	2659.13	3.01341
$93$	−231.849	−0.258512
$94$	−200.415	−0.219907
$95$	142.200	0.153573
$96$	−312.475	−0.332206
$97$	1089.16	1.14008	0.570041	−	0.821616i	$-0.306927\pi$
$97$	1089.16	1.14008	0.570041	+	0.821616i	$0.306927\pi$
$98$	1360.86	1.40273
$99$	602.099	0.611245
$100$	−1514.95	−1.51495
$101$	−77.2336	−0.0760894	−0.0380447	−	0.999276i	$-0.512113\pi$
$101$	−77.2336	−0.0760894	−0.0380447	+	0.999276i	$0.512113\pi$
$102$	1379.35	1.33898
$103$	1351.36	1.29275	0.646377	−	0.763018i	$-0.276283\pi$
$103$	1351.36	1.29275	0.646377	+	0.763018i	$0.276283\pi$
$104$	0	0
$105$	101.399	0.0942434
$106$	2517.81	2.30709
$107$	1133.67	1.02426	0.512129	−	0.858908i	$-0.328857\pi$
$107$	1133.67	1.02426	0.512129	+	0.858908i	$0.328857\pi$
$108$	−391.049	−0.348414
$109$	−1017.66	−0.894262	−0.447131	−	0.894469i	$-0.647554\pi$
$109$	−1017.66	−0.894262	−0.447131	+	0.894469i	$0.647554\pi$
$110$	1432.77	1.24190
$111$	164.299	0.140492
$112$	223.750	0.188772
$113$	1570.06	1.30707	0.653536	−	0.756895i	$-0.273285\pi$
$113$	1570.06	1.30707	0.653536	+	0.756895i	$0.273285\pi$
$114$	−447.849	−0.367938
$115$	−829.261	−0.672426
$116$	1625.03	1.30069
$117$	0	0
$118$	1041.58	0.812588
$119$	725.632	0.558979
$120$	−416.551	−0.316881
$121$	3144.60	2.36258
$122$	−3901.75	−2.89547
$123$	1355.15	0.993411
$124$	1119.32	0.810625
$125$	1037.03	0.742037
$126$	−319.350	−0.225793
$127$	−1248.16	−0.872099	−0.436050	−	0.899923i	$-0.643623\pi$
$127$	−1248.16	−0.872099	−0.436050	+	0.899923i	$0.643623\pi$
$128$	2642.76	1.82491
$129$	340.301	0.232263
$130$	0	0
$131$	−1274.80	−0.850227	−0.425113	−	0.905140i	$-0.639766\pi$
$131$	−1274.80	−0.850227	−0.425113	+	0.905140i	$0.639766\pi$
$132$	−2906.80	−1.91670
$133$	−235.600	−0.153602
$134$	−4135.11	−2.66582
$135$	121.951	0.0777469
$136$	−2980.91	−1.87950
$137$	−874.915	−0.545613	−0.272807	−	0.962069i	$-0.587952\pi$
$137$	−874.915	−0.545613	−0.272807	+	0.962069i	$0.587952\pi$
$138$	2611.70	1.61103
$139$	310.334	0.189368	0.0946840	−	0.995507i	$-0.469816\pi$
$139$	310.334	0.189368	0.0946840	+	0.995507i	$0.469816\pi$
$140$	−489.533	−0.295522
$141$	−126.801	−0.0757345
$142$	−475.585	−0.281058
$143$	0	0
$144$	269.099	0.155729
$145$	−506.773	−0.290243
$146$	−785.380	−0.445195
$147$	861.000	0.483089
$148$	−793.199	−0.440544
$149$	−5.08064	−0.00279344	−0.00139672	−	0.999999i	$-0.500445\pi$
$149$	−5.08064	−0.00279344	−0.00139672	+	0.999999i	$0.500445\pi$
$150$	−1487.93	−0.809924
$151$	−6.54894	−0.00352944	−0.00176472	−	0.999998i	$-0.500562\pi$
$151$	−6.54894	−0.00352944	−0.00176472	+	0.999998i	$0.500562\pi$
$152$	967.849	0.516467
$153$	872.700	0.461135
$154$	−2373.83	−1.24213
$155$	−349.063	−0.180887
$156$	0	0
$157$	−2297.60	−1.16795	−0.583975	−	0.811772i	$-0.698503\pi$
$157$	−2297.60	−1.16795	−0.583975	+	0.811772i	$0.698503\pi$
$158$	2585.46	1.30183
$159$	1593.00	0.794546
$160$	−470.450	−0.232452
$161$	1373.93	0.672553
$162$	−384.074	−0.186270
$163$	1085.49	0.521606	0.260803	−	0.965392i	$-0.416013\pi$
$163$	1085.49	0.521606	0.260803	+	0.965392i	$0.416013\pi$
$164$	−6542.34	−3.11507
$165$	906.497	0.427701
$166$	−2155.24	−1.00771
$167$	109.066	0.0505374	0.0252687	−	0.999681i	$-0.491956\pi$
$167$	109.066	0.0505374	0.0252687	+	0.999681i	$0.491956\pi$
$168$	690.148	0.316941
$169$	0	0
$170$	2076.69	0.936912
$171$	−283.350	−0.126715
$172$	−1642.90	−0.728313
$173$	−1889.17	−0.830236	−0.415118	−	0.909768i	$-0.636260\pi$
$173$	−1889.17	−0.830236	−0.415118	+	0.909768i	$0.636260\pi$
$174$	1596.05	0.695379
$175$	−782.751	−0.338117
$176$	2000.30	0.856694
$177$	658.999	0.279850
$178$	−1094.92	−0.461054
$179$	3427.86	1.43134	0.715672	−	0.698437i	$-0.246121\pi$
$179$	3427.86	1.43134	0.715672	+	0.698437i	$0.246121\pi$
$180$	−588.749	−0.243793
$181$	208.403	0.0855826	0.0427913	−	0.999084i	$-0.486375\pi$
$181$	208.403	0.0855826	0.0427913	+	0.999084i	$0.486375\pi$
$182$	0	0
$183$	−2468.60	−0.997180
$184$	−5644.15	−2.26137
$185$	247.363	0.0983052
$186$	1099.35	0.433378
$187$	6487.06	2.53679
$188$	612.166	0.237483
$189$	−202.049	−0.0777616
$190$	−674.265	−0.257454
$191$	957.735	0.362824	0.181412	−	0.983407i	$-0.441933\pi$
$191$	957.735	0.362824	0.181412	+	0.983407i	$0.441933\pi$
$192$	2199.25	0.826650
$193$	512.730	0.191228	0.0956142	−	0.995418i	$-0.469518\pi$
$193$	512.730	0.191228	0.0956142	+	0.995418i	$0.469518\pi$
$194$	−5164.45	−1.91127
$195$	0	0
$196$	−4156.71	−1.51484
$197$	3870.35	1.39975	0.699876	−	0.714265i	$-0.253239\pi$
$197$	3870.35	1.39975	0.699876	+	0.714265i	$0.253239\pi$
$198$	−2854.95	−1.02471
$199$	2305.83	0.821388	0.410694	−	0.911773i	$-0.365286\pi$
$199$	2305.83	0.821388	0.410694	+	0.911773i	$0.365286\pi$
$200$	3215.56	1.13687
$201$	−2616.25	−0.918088
$202$	366.215	0.127558
$203$	839.630	0.290298
$204$	−4213.19	−1.44599
$205$	2040.26	0.695111
$206$	−6407.70	−2.16721
$207$	1652.40	0.554828
$208$	0	0
$209$	−2106.23	−0.697086
$210$	−480.801	−0.157992
$211$	−3672.40	−1.19819	−0.599096	−	0.800677i	$-0.704473\pi$
$211$	−3672.40	−1.19819	−0.599096	+	0.800677i	$0.704473\pi$
$212$	−7690.62	−2.49148
$213$	−300.898	−0.0967942
$214$	−5375.46	−1.71710
$215$	512.345	0.162519
$216$	830.025	0.261463
$217$	578.334	0.180921
$218$	4825.41	1.49917
$219$	−496.902	−0.153322
$220$	−4376.36	−1.34116
$221$	0	0
$222$	−779.051	−0.235525
$223$	−5087.05	−1.52760	−0.763798	−	0.645455i	$-0.776668\pi$
$223$	−5087.05	−1.52760	−0.763798	+	0.645455i	$0.776668\pi$
$224$	779.449	0.232496
$225$	−941.396	−0.278932
$226$	−7444.71	−2.19122
$227$	2625.83	0.767763	0.383882	−	0.923382i	$-0.374587\pi$
$227$	2625.83	0.767763	0.383882	+	0.923382i	$0.374587\pi$
$228$	1367.95	0.397345
$229$	−1678.73	−0.484425	−0.242213	−	0.970223i	$-0.577873\pi$
$229$	−1678.73	−0.484425	−0.242213	+	0.970223i	$0.577873\pi$
$230$	3932.07	1.12727
$231$	−1501.90	−0.427782
$232$	−3449.22	−0.976088
$233$	648.506	0.182339	0.0911696	−	0.995835i	$-0.470939\pi$
$233$	648.506	0.182339	0.0911696	+	0.995835i	$0.470939\pi$
$234$	0	0
$235$	−190.907	−0.0529931
$236$	−3181.50	−0.877533
$237$	1635.80	0.448340
$238$	−3440.70	−0.937089
$239$	−5219.69	−1.41269	−0.706347	−	0.707866i	$-0.749658\pi$
$239$	−5219.69	−1.41269	−0.706347	+	0.707866i	$0.749658\pi$
$240$	405.145	0.108967
$241$	−6103.56	−1.63139	−0.815695	−	0.578483i	$-0.803645\pi$
$241$	−6103.56	−1.63139	−0.815695	+	0.578483i	$0.803645\pi$
$242$	−14910.6	−3.96070
$243$	−243.000	−0.0641500
$244$	11917.8	3.12689
$245$	1296.29	0.338028
$246$	−6425.64	−1.66538
$247$	0	0
$248$	−2375.81	−0.608323
$249$	−1363.60	−0.347047
$250$	−4917.24	−1.24397
$251$	6423.40	1.61530	0.807652	−	0.589660i	$-0.200738\pi$
$251$	6423.40	1.61530	0.807652	+	0.589660i	$0.200738\pi$
$252$	975.449	0.243839
$253$	12282.8	3.05222
$254$	5918.36	1.46201
$255$	1313.90	0.322666
$256$	−6666.39	−1.62754
$257$	−1230.23	−0.298597	−0.149299	−	0.988792i	$-0.547702\pi$
$257$	−1230.23	−0.298597	−0.149299	+	0.988792i	$0.547702\pi$
$258$	−1613.59	−0.389372
$259$	−409.834	−0.0983238
$260$	0	0
$261$	1009.80	0.239483
$262$	6044.66	1.42534
$263$	−514.992	−0.120744	−0.0603722	−	0.998176i	$-0.519229\pi$
$263$	−514.992	−0.120744	−0.0603722	+	0.998176i	$0.519229\pi$
$264$	6169.84	1.43836
$265$	2398.35	0.555961
$266$	1117.13	0.257503
$267$	−692.745	−0.158784
$268$	12630.6	2.87888
$269$	−5132.60	−1.16335	−0.581673	−	0.813423i	$-0.697602\pi$
$269$	−5132.60	−1.16335	−0.581673	+	0.813423i	$0.697602\pi$
$270$	−578.247	−0.130337
$271$	4300.00	0.963862	0.481931	−	0.876209i	$-0.339936\pi$
$271$	4300.00	0.963862	0.481931	+	0.876209i	$0.339936\pi$
$272$	2899.29	0.646306
$273$	0	0
$274$	4148.55	0.914682
$275$	−6997.70	−1.53446
$276$	−7977.39	−1.73979
$277$	−1812.80	−0.393215	−0.196607	−	0.980482i	$-0.562992\pi$
$277$	−1812.80	−0.393215	−0.196607	+	0.980482i	$0.562992\pi$
$278$	−1471.50	−0.317462
$279$	695.548	0.149252
$280$	1039.06	0.221771
$281$	4073.08	0.864696	0.432348	−	0.901707i	$-0.357685\pi$
$281$	4073.08	0.864696	0.432348	+	0.901707i	$0.357685\pi$
$282$	601.246	0.126963
$283$	6346.29	1.33303	0.666516	−	0.745491i	$-0.267785\pi$
$283$	6346.29	1.33303	0.666516	+	0.745491i	$0.267785\pi$
$284$	1452.67	0.303520
$285$	−426.601	−0.0886654
$286$	0	0
$287$	−3380.33	−0.695243
$288$	937.424	0.191799
$289$	4489.53	0.913806
$290$	2402.94	0.486572
$291$	−3267.49	−0.658226
$292$	2398.93	0.480777
$293$	8390.97	1.67306	0.836529	−	0.547923i	$-0.184581\pi$
$293$	8390.97	1.67306	0.836529	+	0.547923i	$0.184581\pi$
$294$	−4082.57	−0.809864
$295$	992.164	0.195817
$296$	1683.61	0.330601
$297$	−1806.30	−0.352902
$298$	24.0907	0.00468300
$299$	0	0
$300$	4544.84	0.874656
$301$	−848.861	−0.162550
$302$	31.0528	0.00591685
$303$	231.701	0.0439302
$304$	−941.348	−0.177599
$305$	−3716.62	−0.697748
$306$	−4138.04	−0.773059
$307$	−4005.27	−0.744603	−0.372301	−	0.928112i	$-0.621431\pi$
$307$	−4005.27	−0.744603	−0.372301	+	0.928112i	$0.621431\pi$
$308$	7250.82	1.34141
$309$	−4054.09	−0.746372
$310$	1655.14	0.303244
$311$	5836.53	1.06418	0.532088	−	0.846689i	$-0.321407\pi$
$311$	5836.53	1.06418	0.532088	+	0.846689i	$0.321407\pi$
$312$	0	0
$313$	1763.19	0.318407	0.159204	−	0.987246i	$-0.449107\pi$
$313$	1763.19	0.318407	0.159204	+	0.987246i	$0.449107\pi$
$314$	10894.4	1.95798
$315$	−304.198	−0.0544115
$316$	−7897.26	−1.40587
$317$	6106.35	1.08191	0.540957	−	0.841050i	$-0.318062\pi$
$317$	6106.35	1.08191	0.540957	+	0.841050i	$0.318062\pi$
$318$	−7553.44	−1.33200
$319$	7506.18	1.31745
$320$	3311.10	0.578425
$321$	−3401.00	−0.591356
$322$	−6514.72	−1.12749
$323$	−3052.83	−0.525895
$324$	1173.15	0.201157
$325$	0	0
$326$	−5147.00	−0.874436
$327$	3052.99	0.516302
$328$	13886.5	2.33766
$329$	316.297	0.0530031
$330$	−4298.30	−0.717011
$331$	−7490.38	−1.24383	−0.621916	−	0.783084i	$-0.713646\pi$
$331$	−7490.38	−1.24383	−0.621916	+	0.783084i	$0.713646\pi$
$332$	6583.16	1.08825
$333$	−492.898	−0.0811130
$334$	−517.152	−0.0847224
$335$	−3938.92	−0.642406
$336$	−671.251	−0.108987
$337$	9462.46	1.52953	0.764767	−	0.644307i	$-0.222854\pi$
$337$	9462.46	1.52953	0.764767	+	0.644307i	$0.222854\pi$
$338$	0	0
$339$	−4710.19	−0.754639
$340$	−6343.22	−1.01179
$341$	5170.23	0.821066
$342$	1343.55	0.212429
$343$	−4714.49	−0.742153
$344$	3487.14	0.546553
$345$	2487.78	0.388226
$346$	8957.79	1.39183
$347$	11460.3	1.77297	0.886487	−	0.462753i	$-0.153138\pi$
$347$	11460.3	1.77297	0.886487	+	0.462753i	$0.153138\pi$
$348$	−4875.09	−0.750955
$349$	3673.96	0.563503	0.281751	−	0.959487i	$-0.409085\pi$
$349$	3673.96	0.563503	0.281751	+	0.959487i	$0.409085\pi$
$350$	3711.54	0.566829
$351$	0	0
$352$	6968.17	1.05513
$353$	−3388.65	−0.510935	−0.255467	−	0.966818i	$-0.582229\pi$
$353$	−3388.65	−0.510935	−0.255467	+	0.966818i	$0.582229\pi$
$354$	−3124.75	−0.469148
$355$	−453.020	−0.0677290
$356$	3344.41	0.497903
$357$	−2176.90	−0.322727
$358$	−16253.7	−2.39955
$359$	9673.98	1.42221	0.711105	−	0.703086i	$-0.248195\pi$
$359$	9673.98	1.42221	0.711105	+	0.703086i	$0.248195\pi$
$360$	1249.65	0.182951
$361$	−5867.80	−0.855489
$362$	−988.174	−0.143473
$363$	−9433.79	−1.36404
$364$	0	0
$365$	−748.117	−0.107283
$366$	11705.2	1.67170
$367$	−8715.98	−1.23970	−0.619851	−	0.784720i	$-0.712807\pi$
$367$	−8715.98	−1.23970	−0.619851	+	0.784720i	$0.712807\pi$
$368$	5489.61	0.777624
$369$	−4065.44	−0.573546
$370$	−1172.91	−0.164802
$371$	−3973.63	−0.556067
$372$	−3357.95	−0.468014
$373$	4667.99	0.647987	0.323994	−	0.946059i	$-0.394974\pi$
$373$	4667.99	0.647987	0.323994	+	0.946059i	$0.394974\pi$
$374$	−30759.4	−4.25276
$375$	−3111.09	−0.428416
$376$	−1299.36	−0.178216
$377$	0	0
$378$	958.049	0.130362
$379$	−10862.7	−1.47225	−0.736123	−	0.676848i	$-0.763345\pi$
$379$	−10862.7	−1.47225	−0.736123	+	0.676848i	$0.763345\pi$
$380$	2059.53	0.278031
$381$	3744.49	0.503507
$382$	−4541.25	−0.608248
$383$	−10054.2	−1.34137	−0.670686	−	0.741742i	$-0.734000\pi$
$383$	−10054.2	−1.34137	−0.670686	+	0.741742i	$0.734000\pi$
$384$	−7928.27	−1.05361
$385$	−2261.20	−0.299329
$386$	−2431.19	−0.320581
$387$	−1020.90	−0.134097
$388$	15774.7	2.06402
$389$	6418.50	0.836584	0.418292	−	0.908313i	$-0.362629\pi$
$389$	6418.50	0.836584	0.418292	+	0.908313i	$0.362629\pi$
$390$	0	0
$391$	17803.0	2.30265
$392$	8822.86	1.13679
$393$	3824.40	0.490879
$394$	−18351.9	−2.34658
$395$	2462.79	0.313713
$396$	8720.39	1.10661
$397$	12019.9	1.51955	0.759775	−	0.650186i	$-0.225309\pi$
$397$	12019.9	1.51955	0.759775	+	0.650186i	$0.225309\pi$
$398$	−10933.5	−1.37700
$399$	706.799	0.0886822
$400$	−3127.52	−0.390939
$401$	−3599.80	−0.448293	−0.224147	−	0.974555i	$-0.571959\pi$
$401$	−3599.80	−0.448293	−0.224147	+	0.974555i	$0.571959\pi$
$402$	12405.3	1.53911
$403$	0	0
$404$	−1118.60	−0.137753
$405$	−365.852	−0.0448872
$406$	−3981.24	−0.486664
$407$	−3663.87	−0.446219
$408$	8942.74	1.08513
$409$	48.0968	0.00581475	0.00290737	−	0.999996i	$-0.499075\pi$
$409$	48.0968	0.00581475	0.00290737	+	0.999996i	$0.499075\pi$
$410$	−9674.20	−1.16530
$411$	2624.74	0.315010
$412$	19572.2	2.34042
$413$	−1643.83	−0.195854
$414$	−7835.10	−0.930130
$415$	−2052.99	−0.242837
$416$	0	0
$417$	−931.001	−0.109332
$418$	9987.02	1.16862
$419$	723.462	0.0843518	0.0421759	−	0.999110i	$-0.486571\pi$
$419$	723.462	0.0843518	0.0421759	+	0.999110i	$0.486571\pi$
$420$	1468.60	0.170620
$421$	14845.5	1.71859	0.859295	−	0.511481i	$-0.170903\pi$
$421$	14845.5	1.71859	0.859295	+	0.511481i	$0.170903\pi$
$422$	17413.3	2.00868
$423$	380.403	0.0437253
$424$	16323.8	1.86970
$425$	−10142.7	−1.15763
$426$	1426.75	0.162269
$427$	6157.76	0.697880
$428$	16419.2	1.85433
$429$	0	0
$430$	−2429.36	−0.272452
$431$	−1103.11	−0.123283	−0.0616417	−	0.998098i	$-0.519634\pi$
$431$	−1103.11	−0.123283	−0.0616417	+	0.998098i	$0.519634\pi$
$432$	−807.297	−0.0899099
$433$	8893.53	0.987057	0.493528	−	0.869730i	$-0.335707\pi$
$433$	8893.53	0.987057	0.493528	+	0.869730i	$0.335707\pi$
$434$	−2742.26	−0.303301
$435$	1520.32	0.167572
$436$	−14739.1	−1.61898
$437$	−5780.32	−0.632747
$438$	2356.14	0.257034
$439$	−10901.7	−1.18521	−0.592607	−	0.805492i	$-0.701901\pi$
$439$	−10901.7	−1.18521	−0.592607	+	0.805492i	$0.701901\pi$
$440$	9289.08	1.00645
$441$	−2583.00	−0.278912
$442$	0	0
$443$	−3781.37	−0.405550	−0.202775	−	0.979225i	$-0.564996\pi$
$443$	−3781.37	−0.405550	−0.202775	+	0.979225i	$0.564996\pi$
$444$	2379.60	0.254348
$445$	−1042.97	−0.111105
$446$	24121.0	2.56091
$447$	15.2419	0.00161279
$448$	−5485.88	−0.578535
$449$	−106.834	−0.0112289	−0.00561447	−	0.999984i	$-0.501787\pi$
$449$	−106.834	−0.0112289	−0.00561447	+	0.999984i	$0.501787\pi$
$450$	4463.78	0.467610
$451$	−30219.7	−3.15519
$452$	22739.7	2.36634
$453$	19.6468	0.00203772
$454$	−12450.8	−1.28710
$455$	0	0
$456$	−2903.55	−0.298182
$457$	1237.64	0.126684	0.0633419	−	0.997992i	$-0.479824\pi$
$457$	1237.64	0.126684	0.0633419	+	0.997992i	$0.479824\pi$
$458$	7959.95	0.812105
$459$	−2618.10	−0.266236
$460$	−12010.5	−1.21737
$461$	−8790.90	−0.888141	−0.444071	−	0.895992i	$-0.646466\pi$
$461$	−8790.90	−0.888141	−0.444071	+	0.895992i	$0.646466\pi$
$462$	7121.49	0.717146
$463$	3861.55	0.387606	0.193803	−	0.981040i	$-0.437918\pi$
$463$	3861.55	0.387606	0.193803	+	0.981040i	$0.437918\pi$
$464$	3354.77	0.335650
$465$	1047.19	0.104435
$466$	−3074.99	−0.305679
$467$	−8991.38	−0.890945	−0.445473	−	0.895296i	$-0.646964\pi$
$467$	−8991.38	−0.890945	−0.445473	+	0.895296i	$0.646964\pi$
$468$	0	0
$469$	6526.06	0.642528
$470$	905.214	0.0888391
$471$	6892.79	0.674316
$472$	6752.91	0.658533
$473$	−7588.71	−0.737694
$474$	−7756.39	−0.751609
$475$	3293.14	0.318105
$476$	10509.6	1.01198
$477$	−4778.99	−0.458731
$478$	24750.0	2.36828
$479$	−4179.82	−0.398707	−0.199354	−	0.979928i	$-0.563884\pi$
$479$	−4179.82	−0.398707	−0.199354	+	0.979928i	$0.563884\pi$
$480$	1411.35	0.134206
$481$	0	0
$482$	28941.0	2.73491
$483$	−4121.80	−0.388299
$484$	45544.2	4.27725
$485$	−4919.41	−0.460575
$486$	1152.22	0.107543
$487$	18443.8	1.71616	0.858078	−	0.513519i	$-0.171658\pi$
$487$	18443.8	1.71616	0.858078	+	0.513519i	$0.171658\pi$
$488$	−25296.2	−2.34653
$489$	−3256.46	−0.301149
$490$	−6146.56	−0.566680
$491$	8093.26	0.743877	0.371939	−	0.928257i	$-0.378693\pi$
$491$	8093.26	0.743877	0.371939	+	0.928257i	$0.378693\pi$
$492$	19627.0	1.79849
$493$	10879.7	0.993907
$494$	0	0
$495$	−2719.49	−0.246933
$496$	2310.76	0.209185
$497$	750.571	0.0677418
$498$	6465.73	0.581800
$499$	10941.6	0.981591	0.490796	−	0.871275i	$-0.336706\pi$
$499$	10941.6	0.981591	0.490796	+	0.871275i	$0.336706\pi$
$500$	15019.6	1.34340
$501$	−327.197	−0.0291778
$502$	−30457.5	−2.70794
$503$	−9260.11	−0.820851	−0.410425	−	0.911894i	$-0.634620\pi$
$503$	−9260.11	−0.820851	−0.410425	+	0.911894i	$0.634620\pi$
$504$	−2070.45	−0.182986
$505$	348.840	0.0307389
$506$	−58240.8	−5.11684
$507$	0	0
$508$	−18077.5	−1.57886
$509$	9996.40	0.870497	0.435248	−	0.900310i	$-0.356661\pi$
$509$	9996.40	0.870497	0.435248	+	0.900310i	$0.356661\pi$
$510$	−6230.08	−0.540927
$511$	1239.49	0.107303
$512$	10467.7	0.903538
$513$	850.049	0.0731591
$514$	5833.32	0.500577
$515$	−6103.68	−0.522253
$516$	4928.69	0.420491
$517$	2827.66	0.240542
$518$	1943.29	0.164833
$519$	5667.51	0.479337
$520$	0	0
$521$	11427.9	0.960973	0.480486	−	0.877002i	$-0.340460\pi$
$521$	11427.9	0.960973	0.480486	+	0.877002i	$0.340460\pi$
$522$	−4788.14	−0.401477
$523$	−4810.47	−0.402193	−0.201097	−	0.979571i	$-0.564451\pi$
$523$	−4810.47	−0.402193	−0.201097	+	0.979571i	$0.564451\pi$
$524$	−18463.3	−1.53926
$525$	2348.25	0.195212
$526$	2441.92	0.202419
$527$	7493.88	0.619428
$528$	−6000.90	−0.494613
$529$	21541.8	1.77051
$530$	−11372.2	−0.932030
$531$	−1977.00	−0.161571
$532$	−3412.26	−0.278083
$533$	0	0
$534$	3284.76	0.266190
$535$	−5120.41	−0.413785
$536$	−26809.2	−2.16042
$537$	−10283.6	−0.826386
$538$	24337.0	1.95027
$539$	−19200.3	−1.53435
$540$	1766.25	0.140754
$541$	−2411.99	−0.191681	−0.0958406	−	0.995397i	$-0.530554\pi$
$541$	−2411.99	−0.191681	−0.0958406	+	0.995397i	$0.530554\pi$
$542$	−20389.1	−1.61585
$543$	−625.208	−0.0494111
$544$	10099.9	0.796008
$545$	4596.47	0.361268
$546$	0	0
$547$	−4396.34	−0.343646	−0.171823	−	0.985128i	$-0.554966\pi$
$547$	−4396.34	−0.343646	−0.171823	+	0.985128i	$0.554966\pi$
$548$	−12671.7	−0.987786
$549$	7405.79	0.575722
$550$	33180.7	2.57242
$551$	−3532.43	−0.273116
$552$	16932.5	1.30560
$553$	−4080.40	−0.313772
$554$	8595.67	0.659197
$555$	−742.088	−0.0567565
$556$	4494.66	0.342835
$557$	17488.0	1.33032	0.665160	−	0.746701i	$-0.268363\pi$
$557$	17488.0	1.33032	0.665160	+	0.746701i	$0.268363\pi$
$558$	−3298.05	−0.250211
$559$	0	0
$560$	−1010.61	−0.0762608
$561$	−19461.2	−1.46462
$562$	−19313.2	−1.44960
$563$	−6881.77	−0.515154	−0.257577	−	0.966258i	$-0.582924\pi$
$563$	−6881.77	−0.515154	−0.257577	+	0.966258i	$0.582924\pi$
$564$	−1836.50	−0.137111
$565$	−7091.49	−0.528037
$566$	−30092.0	−2.23473
$567$	606.148	0.0448957
$568$	−3083.36	−0.227773
$569$	14733.5	1.08552	0.542758	−	0.839889i	$-0.317380\pi$
$569$	14733.5	1.08552	0.542758	+	0.839889i	$0.317380\pi$
$570$	2022.79	0.148641
$571$	4488.51	0.328964	0.164482	−	0.986380i	$-0.447405\pi$
$571$	4488.51	0.328964	0.164482	+	0.986380i	$0.447405\pi$
$572$	0	0
$573$	−2873.21	−0.209476
$574$	16028.4	1.16553
$575$	−19204.4	−1.39284
$576$	−6597.74	−0.477267
$577$	10552.2	0.761338	0.380669	−	0.924711i	$-0.375694\pi$
$577$	10552.2	0.761338	0.380669	+	0.924711i	$0.375694\pi$
$578$	−21287.8	−1.53193
$579$	−1538.19	−0.110406
$580$	−7339.75	−0.525460
$581$	3401.42	0.242882
$582$	15493.3	1.10347
$583$	−35523.8	−2.52357
$584$	−5091.86	−0.360793
$585$	0	0
$586$	−39787.1	−2.80476
$587$	1637.20	0.115118	0.0575591	−	0.998342i	$-0.481668\pi$
$587$	1637.20	0.115118	0.0575591	+	0.998342i	$0.481668\pi$
$588$	12470.1	0.874591
$589$	−2433.13	−0.170213
$590$	−4704.50	−0.328273
$591$	−11611.1	−0.808147
$592$	−1637.51	−0.113685
$593$	−14024.0	−0.971155	−0.485577	−	0.874194i	$-0.661391\pi$
$593$	−14024.0	−0.971155	−0.485577	+	0.874194i	$0.661391\pi$
$594$	8564.84	0.591616
$595$	−3277.45	−0.225819
$596$	−73.5845	−0.00505728
$597$	−6917.50	−0.474229
$598$	0	0
$599$	365.860	0.0249560	0.0124780	−	0.999922i	$-0.496028\pi$
$599$	365.860	0.0249560	0.0124780	+	0.999922i	$0.496028\pi$
$600$	−9646.69	−0.656374
$601$	−10128.9	−0.687465	−0.343733	−	0.939068i	$-0.611691\pi$
$601$	−10128.9	−0.687465	−0.343733	+	0.939068i	$0.611691\pi$
$602$	4025.01	0.272503
$603$	7848.74	0.530058
$604$	−94.8504	−0.00638975
$605$	−14203.1	−0.954446
$606$	−1098.65	−0.0736459
$607$	2324.97	0.155465	0.0777327	−	0.996974i	$-0.475232\pi$
$607$	2324.97	0.155465	0.0777327	+	0.996974i	$0.475232\pi$
$608$	−3279.25	−0.218735
$609$	−2518.89	−0.167603
$610$	17623.0	1.16973
$611$	0	0
$612$	12639.6	0.834845
$613$	−633.133	−0.0417162	−0.0208581	−	0.999782i	$-0.506640\pi$
$613$	−633.133	−0.0417162	−0.0208581	+	0.999782i	$0.506640\pi$
$614$	18991.6	1.24827
$615$	−6120.77	−0.401323
$616$	−15390.3	−1.00664
$617$	2981.85	0.194562	0.0972810	−	0.995257i	$-0.468985\pi$
$617$	2981.85	0.194562	0.0972810	+	0.995257i	$0.468985\pi$
$618$	19223.1	1.25124
$619$	−15158.6	−0.984292	−0.492146	−	0.870513i	$-0.663787\pi$
$619$	−15158.6	−0.984292	−0.492146	+	0.870513i	$0.663787\pi$
$620$	−5055.60	−0.327480
$621$	−4957.19	−0.320330
$622$	−27674.8	−1.78402
$623$	1728.01	0.111126
$624$	0	0
$625$	8391.01	0.537025
$626$	−8360.45	−0.533787
$627$	6318.69	0.402463
$628$	−33276.8	−2.11447
$629$	−5310.51	−0.336636
$630$	1442.40	0.0912170
$631$	−5562.30	−0.350922	−0.175461	−	0.984486i	$-0.556142\pi$
$631$	−5562.30	−0.350922	−0.175461	+	0.984486i	$0.556142\pi$
$632$	16762.4	1.05502
$633$	11017.2	0.691776
$634$	−28954.2	−1.81375
$635$	5637.56	0.352315
$636$	23071.9	1.43846
$637$	0	0
$638$	−35591.7	−2.20861
$639$	902.693	0.0558842
$640$	−11936.5	−0.737237
$641$	−24140.7	−1.48752	−0.743761	−	0.668446i	$-0.766960\pi$
$641$	−24140.7	−1.48752	−0.743761	+	0.668446i	$0.766960\pi$
$642$	16126.4	0.991366
$643$	1749.69	0.107311	0.0536555	−	0.998560i	$-0.482913\pi$
$643$	1749.69	0.107311	0.0536555	+	0.998560i	$0.482913\pi$
$644$	19899.1	1.21760
$645$	−1537.03	−0.0938305
$646$	14475.5	0.881626
$647$	1489.51	0.0905083	0.0452542	−	0.998976i	$-0.485590\pi$
$647$	1489.51	0.0905083	0.0452542	+	0.998976i	$0.485590\pi$
$648$	−2490.07	−0.150956
$649$	−14695.7	−0.888836
$650$	0	0
$651$	−1735.00	−0.104455
$652$	15721.4	0.944323
$653$	10668.2	0.639327	0.319663	−	0.947531i	$-0.396430\pi$
$653$	10668.2	0.639327	0.319663	+	0.947531i	$0.396430\pi$
$654$	−14476.2	−0.865544
$655$	5757.86	0.343478
$656$	−13506.3	−0.803858
$657$	1490.71	0.0885205
$658$	−1499.77	−0.0888559
$659$	−13933.1	−0.823608	−0.411804	−	0.911272i	$-0.635101\pi$
$659$	−13933.1	−0.823608	−0.411804	+	0.911272i	$0.635101\pi$
$660$	13129.1	0.774317
$661$	2349.69	0.138264	0.0691320	−	0.997608i	$-0.477977\pi$
$661$	2349.69	0.138264	0.0691320	+	0.997608i	$0.477977\pi$
$662$	35516.8	2.08520
$663$	0	0
$664$	−13973.1	−0.816660
$665$	1064.13	0.0620529
$666$	2337.15	0.135980
$667$	20599.9	1.19585
$668$	1579.63	0.0914937
$669$	15261.1	0.881958
$670$	18677.0	1.07695
$671$	55049.6	3.16716
$672$	−2338.35	−0.134232
$673$	−32421.0	−1.85697	−0.928483	−	0.371374i	$-0.878887\pi$
$673$	−32421.0	−1.85697	−0.928483	+	0.371374i	$0.878887\pi$
$674$	−44867.7	−2.56415
$675$	2824.19	0.161042
$676$	0	0
$677$	−1071.74	−0.0608421	−0.0304211	−	0.999537i	$-0.509685\pi$
$677$	−1071.74	−0.0608421	−0.0304211	+	0.999537i	$0.509685\pi$
$678$	22334.1	1.26510
$679$	8150.56	0.460663
$680$	13463.9	0.759287
$681$	−7877.48	−0.443268
$682$	−24515.5	−1.37646
$683$	305.487	0.0171144	0.00855721	−	0.999963i	$-0.497276\pi$
$683$	305.487	0.0171144	0.00855721	+	0.999963i	$0.497276\pi$
$684$	−4103.84	−0.229407
$685$	3951.72	0.220419
$686$	22354.5	1.24417
$687$	5036.18	0.279683
$688$	−3391.66	−0.187944
$689$	0	0
$690$	−11796.2	−0.650833
$691$	−2180.81	−0.120061	−0.0600303	−	0.998197i	$-0.519120\pi$
$691$	−2180.81	−0.120061	−0.0600303	+	0.998197i	$0.519120\pi$
$692$	−27361.4	−1.50307
$693$	4505.70	0.246980
$694$	−54340.9	−2.97227
$695$	−1401.68	−0.0765018
$696$	10347.7	0.563545
$697$	−43801.4	−2.38034
$698$	−17420.6	−0.944672
$699$	−1945.52	−0.105274
$700$	−11336.8	−0.612132
$701$	−15168.3	−0.817259	−0.408629	−	0.912700i	$-0.633993\pi$
$701$	−15168.3	−0.817259	−0.408629	+	0.912700i	$0.633993\pi$
$702$	0	0
$703$	1724.23	0.0925043
$704$	−49043.1	−2.62554
$705$	572.720	0.0305956
$706$	16067.8	0.856545
$707$	−577.963	−0.0307448
$708$	9544.49	0.506644
$709$	8988.17	0.476104	0.238052	−	0.971252i	$-0.423491\pi$
$709$	8988.17	0.476104	0.238052	+	0.971252i	$0.423491\pi$
$710$	2148.07	0.113543
$711$	−4907.39	−0.258849
$712$	−7098.71	−0.373645
$713$	14189.1	0.745284
$714$	10322.1	0.541029
$715$	0	0
$716$	49646.8	2.59132
$717$	15659.1	0.815619
$718$	−45870.7	−2.38423
$719$	8448.18	0.438198	0.219099	−	0.975703i	$-0.429688\pi$
$719$	8448.18	0.438198	0.219099	+	0.975703i	$0.429688\pi$
$720$	−1215.44	−0.0629120
$721$	10112.7	0.522352
$722$	27823.1	1.43417
$723$	18310.7	0.941883
$724$	3018.36	0.154940
$725$	−11736.1	−0.601197
$726$	44731.8	2.28671
$727$	−8624.18	−0.439963	−0.219982	−	0.975504i	$-0.570600\pi$
$727$	−8624.18	−0.439963	−0.219982	+	0.975504i	$0.570600\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	3547.31	0.179852
$731$	−10999.3	−0.556530
$732$	−35753.5	−1.80531
$733$	−31124.2	−1.56835	−0.784174	−	0.620541i	$-0.786913\pi$
$733$	−31124.2	−1.56835	−0.784174	+	0.620541i	$0.786913\pi$
$734$	41328.2	2.07827
$735$	−3888.87	−0.195161
$736$	19123.4	0.957742
$737$	58342.2	2.91596
$738$	19276.9	0.961509
$739$	17671.1	0.879626	0.439813	−	0.898089i	$-0.355045\pi$
$739$	17671.1	0.879626	0.439813	+	0.898089i	$0.355045\pi$
$740$	3582.63	0.177973
$741$	0	0
$742$	18841.6	0.932206
$743$	21331.1	1.05325	0.526623	−	0.850099i	$-0.323458\pi$
$743$	21331.1	1.05325	0.526623	+	0.850099i	$0.323458\pi$
$744$	7127.43	0.351215
$745$	22.9477	0.00112851
$746$	−22134.0	−1.08630
$747$	4090.81	0.200368
$748$	93954.1	4.59265
$749$	8483.58	0.413863
$750$	14751.7	0.718208
$751$	11712.9	0.569122	0.284561	−	0.958658i	$-0.408152\pi$
$751$	11712.9	0.569122	0.284561	+	0.958658i	$0.408152\pi$
$752$	1263.78	0.0612835
$753$	−19270.2	−0.932596
$754$	0	0
$755$	29.5795	0.00142584
$756$	−2926.35	−0.140781
$757$	−16610.9	−0.797537	−0.398768	−	0.917052i	$-0.630562\pi$
$757$	−16610.9	−0.797537	−0.398768	+	0.917052i	$0.630562\pi$
$758$	51507.4	2.46812
$759$	−36848.4	−1.76220
$760$	−4371.47	−0.208645
$761$	29365.5	1.39882	0.699408	−	0.714723i	$-0.253447\pi$
$761$	29365.5	1.39882	0.699408	+	0.714723i	$0.253447\pi$
$762$	−17755.1	−0.844093
$763$	−7615.50	−0.361336
$764$	13871.2	0.656861
$765$	−3941.71	−0.186291
$766$	47673.5	2.24871
$767$	0	0
$768$	19999.2	0.939659
$769$	28599.9	1.34114	0.670572	−	0.741844i	$-0.266049\pi$
$769$	28599.9	1.34114	0.670572	+	0.741844i	$0.266049\pi$
$770$	10721.8	0.501803
$771$	3690.68	0.172395
$772$	7426.03	0.346203
$773$	−13491.8	−0.627772	−0.313886	−	0.949461i	$-0.601631\pi$
$773$	−13491.8	−0.627772	−0.313886	+	0.949461i	$0.601631\pi$
$774$	4840.78	0.224804
$775$	−8083.78	−0.374681
$776$	−33482.7	−1.54892
$777$	1229.50	0.0567673
$778$	−30434.3	−1.40247
$779$	14221.5	0.654093
$780$	0	0
$781$	6710.01	0.307430
$782$	−84415.9	−3.86024
$783$	−3029.41	−0.138266
$784$	−8581.27	−0.390911
$785$	10377.5	0.471834
$786$	−18134.0	−0.822923
$787$	−25876.7	−1.17205	−0.586025	−	0.810293i	$-0.699308\pi$
$787$	−25876.7	−1.17205	−0.586025	+	0.810293i	$0.699308\pi$
$788$	56055.5	2.53413
$789$	1544.98	0.0697118
$790$	−11677.7	−0.525918
$791$	11749.3	0.528137
$792$	−18509.5	−0.830438
$793$	0	0
$794$	−56994.2	−2.54742
$795$	−7195.06	−0.320984
$796$	33396.1	1.48705
$797$	21936.4	0.974938	0.487469	−	0.873140i	$-0.337920\pi$
$797$	21936.4	0.974938	0.487469	+	0.873140i	$0.337920\pi$
$798$	−3351.40	−0.148669
$799$	4098.49	0.181469
$800$	−10894.9	−0.481491
$801$	2078.23	0.0916739
$802$	17069.0	0.751531
$803$	11080.9	0.486969
$804$	−37891.9	−1.66212
$805$	−6205.62	−0.271701
$806$	0	0
$807$	15397.8	0.671658
$808$	2374.29	0.103375
$809$	5583.23	0.242640	0.121320	−	0.992613i	$-0.461287\pi$
$809$	5583.23	0.242640	0.121320	+	0.992613i	$0.461287\pi$
$810$	1734.74	0.0752502
$811$	−12925.4	−0.559647	−0.279823	−	0.960052i	$-0.590276\pi$
$811$	−12925.4	−0.559647	−0.279823	+	0.960052i	$0.590276\pi$
$812$	12160.6	0.525559
$813$	−12900.0	−0.556486
$814$	17372.8	0.748054
$815$	−4902.80	−0.210721
$816$	−8697.87	−0.373145
$817$	3571.27	0.152929
$818$	−228.058	−0.00974801
$819$	0	0
$820$	29549.7	1.25844
$821$	10153.2	0.431608	0.215804	−	0.976437i	$-0.430763\pi$
$821$	10153.2	0.431608	0.215804	+	0.976437i	$0.430763\pi$
$822$	−12445.6	−0.528092
$823$	3282.93	0.139047	0.0695235	−	0.997580i	$-0.477852\pi$
$823$	3282.93	0.139047	0.0695235	+	0.997580i	$0.477852\pi$
$824$	−41543.1	−1.75634
$825$	20993.1	0.885922
$826$	7794.49	0.328335
$827$	17689.3	0.743795	0.371897	−	0.928274i	$-0.378707\pi$
$827$	17689.3	0.743795	0.371897	+	0.928274i	$0.378707\pi$
$828$	23932.2	1.00447
$829$	38181.5	1.59964	0.799818	−	0.600243i	$-0.204930\pi$
$829$	38181.5	1.59964	0.799818	+	0.600243i	$0.204930\pi$
$830$	9734.56	0.407098
$831$	5438.40	0.227023
$832$	0	0
$833$	−27829.4	−1.15754
$834$	4414.49	0.183287
$835$	−492.615	−0.0204163
$836$	−30505.2	−1.26201
$837$	−2086.64	−0.0861708
$838$	−3430.41	−0.141410
$839$	−43895.2	−1.80623	−0.903117	−	0.429395i	$-0.858727\pi$
$839$	−43895.2	−1.80623	−0.903117	+	0.429395i	$0.858727\pi$
$840$	−3117.18	−0.128039
$841$	−11800.1	−0.483829
$842$	−70392.4	−2.88109
$843$	−12219.2	−0.499233
$844$	−53188.5	−2.16922
$845$	0	0
$846$	−1803.74	−0.0733024
$847$	23532.0	0.954627
$848$	−15876.8	−0.642938
$849$	−19038.9	−0.769626
$850$	48093.0	1.94068
$851$	−10055.1	−0.405034
$852$	−4358.00	−0.175238
$853$	19955.2	0.800998	0.400499	−	0.916297i	$-0.368837\pi$
$853$	19955.2	0.800998	0.400499	+	0.916297i	$0.368837\pi$
$854$	−29198.0	−1.16995
$855$	1279.80	0.0511910
$856$	−34850.8	−1.39156
$857$	26030.4	1.03755	0.518776	−	0.854910i	$-0.326388\pi$
$857$	26030.4	1.03755	0.518776	+	0.854910i	$0.326388\pi$
$858$	0	0
$859$	−45617.4	−1.81193	−0.905964	−	0.423354i	$-0.860853\pi$
$859$	−45617.4	−1.81193	−0.905964	+	0.423354i	$0.860853\pi$
$860$	7420.45	0.294227
$861$	10141.0	0.401399
$862$	5230.59	0.206676
$863$	2010.93	0.0793195	0.0396597	−	0.999213i	$-0.487373\pi$
$863$	2010.93	0.0793195	0.0396597	+	0.999213i	$0.487373\pi$
$864$	−2812.27	−0.110735
$865$	8532.78	0.335403
$866$	−42170.1	−1.65473
$867$	−13468.6	−0.527586
$868$	8376.19	0.327542
$869$	−36478.2	−1.42398
$870$	−7208.83	−0.280922
$871$	0	0
$872$	31284.7	1.21495
$873$	9802.48	0.380027
$874$	27408.3	1.06076
$875$	7760.41	0.299828
$876$	−7196.79	−0.277576
$877$	36767.3	1.41567	0.707836	−	0.706376i	$-0.249671\pi$
$877$	36767.3	1.41567	0.707836	+	0.706376i	$0.249671\pi$
$878$	51692.0	1.98693
$879$	−25172.9	−0.965940
$880$	−9034.72	−0.346091
$881$	35401.2	1.35380	0.676899	−	0.736076i	$-0.263323\pi$
$881$	35401.2	1.35380	0.676899	+	0.736076i	$0.263323\pi$
$882$	12247.7	0.467575
$883$	−11928.0	−0.454596	−0.227298	−	0.973825i	$-0.572989\pi$
$883$	−11928.0	−0.454596	−0.227298	+	0.973825i	$0.572989\pi$
$884$	0	0
$885$	−2976.49	−0.113055
$886$	17930.0	0.679875
$887$	−32939.3	−1.24689	−0.623447	−	0.781866i	$-0.714268\pi$
$887$	−32939.3	−1.24689	−0.623447	+	0.781866i	$0.714268\pi$
$888$	−5050.83	−0.190872
$889$	−9340.40	−0.352381
$890$	4945.41	0.186259
$891$	5418.89	0.203748
$892$	−73677.3	−2.76558
$893$	−1330.70	−0.0498660
$894$	−72.2720	−0.00270373
$895$	−15482.6	−0.578241
$896$	19776.6	0.737376
$897$	0	0
$898$	506.569	0.0188245
$899$	8671.18	0.321691
$900$	−13634.5	−0.504983
$901$	−51489.2	−1.90383
$902$	143292.	5.28946
$903$	2546.58	0.0938483
$904$	−48266.4	−1.77579
$905$	−941.289	−0.0345740
$906$	−93.1585	−0.00341610
$907$	1500.89	0.0549461	0.0274731	−	0.999623i	$-0.491254\pi$
$907$	1500.89	0.0549461	0.0274731	+	0.999623i	$0.491254\pi$
$908$	38030.7	1.38997
$909$	−695.102	−0.0253631
$910$	0	0
$911$	19728.2	0.717481	0.358740	−	0.933437i	$-0.383206\pi$
$911$	19728.2	0.717481	0.358740	+	0.933437i	$0.383206\pi$
$912$	2824.04	0.102537
$913$	30408.3	1.10226
$914$	−5868.48	−0.212376
$915$	11149.9	0.402845
$916$	−24313.5	−0.877011
$917$	−9539.72	−0.343543
$918$	12414.1	0.446326
$919$	19992.0	0.717602	0.358801	−	0.933414i	$-0.383186\pi$
$919$	19992.0	0.717602	0.358801	+	0.933414i	$0.383186\pi$
$920$	25492.9	0.913560
$921$	12015.8	0.429897
$922$	41683.4	1.48891
$923$	0	0
$924$	−21752.5	−0.774463
$925$	5728.54	0.203625
$926$	−18310.2	−0.649794
$927$	12162.3	0.430918
$928$	11686.6	0.413395
$929$	1923.62	0.0679354	0.0339677	−	0.999423i	$-0.489186\pi$
$929$	1923.62	0.0679354	0.0339677	+	0.999423i	$0.489186\pi$
$930$	−4965.42	−0.175078
$931$	9035.71	0.318081
$932$	9392.52	0.330110
$933$	−17509.6	−0.614403
$934$	42634.0	1.49361
$935$	−29300.0	−1.02483
$936$	0	0
$937$	−10252.9	−0.357468	−0.178734	−	0.983897i	$-0.557200\pi$
$937$	−10252.9	−0.357468	−0.178734	+	0.983897i	$0.557200\pi$
$938$	−30944.4	−1.07715
$939$	−5289.58	−0.183833
$940$	−2764.96	−0.0959394
$941$	−11043.0	−0.382563	−0.191282	−	0.981535i	$-0.561264\pi$
$941$	−11043.0	−0.382563	−0.191282	+	0.981535i	$0.561264\pi$
$942$	−32683.2	−1.13044
$943$	−82934.8	−2.86398
$944$	−6568.00	−0.226451
$945$	912.594	0.0314145
$946$	35983.1	1.23669
$947$	−32105.2	−1.10167	−0.550833	−	0.834615i	$-0.685690\pi$
$947$	−32105.2	−1.10167	−0.550833	+	0.834615i	$0.685690\pi$
$948$	23691.8	0.811680
$949$	0	0
$950$	−15614.9	−0.533280
$951$	−18319.0	−0.624643
$952$	−22307.1	−0.759431
$953$	−9473.37	−0.322007	−0.161003	−	0.986954i	$-0.551473\pi$
$953$	−9473.37	−0.322007	−0.161003	+	0.986954i	$0.551473\pi$
$954$	22660.3	0.769031
$955$	−4325.79	−0.146575
$956$	−75598.4	−2.55756
$957$	−22518.5	−0.760628
$958$	19819.3	0.668404
$959$	−6547.26	−0.220461
$960$	−9933.30	−0.333954
$961$	−23818.3	−0.799514
$962$	0	0
$963$	10203.0	0.341419
$964$	−88399.8	−2.95349
$965$	−2315.84	−0.0772534
$966$	19544.2	0.650956
$967$	5310.75	0.176610	0.0883052	−	0.996093i	$-0.471855\pi$
$967$	5310.75	0.176610	0.0883052	+	0.996093i	$0.471855\pi$
$968$	−96670.1	−3.20981
$969$	9158.49	0.303626
$970$	23326.2	0.772122
$971$	24271.2	0.802164	0.401082	−	0.916042i	$-0.368634\pi$
$971$	24271.2	0.802164	0.401082	+	0.916042i	$0.368634\pi$
$972$	−3519.45	−0.116138
$973$	2322.32	0.0765163
$974$	−87454.2	−2.87702
$975$	0	0
$976$	24603.6	0.806907
$977$	49602.5	1.62428	0.812142	−	0.583460i	$-0.198301\pi$
$977$	49602.5	1.62428	0.812142	+	0.583460i	$0.198301\pi$
$978$	15441.0	0.504856
$979$	15448.2	0.504317
$980$	18774.6	0.611971
$981$	−9158.98	−0.298087
$982$	−38375.5	−1.24706
$983$	−47385.7	−1.53751	−0.768753	−	0.639545i	$-0.779123\pi$
$983$	−47385.7	−1.53751	−0.768753	+	0.639545i	$0.779123\pi$
$984$	−41659.5	−1.34965
$985$	−17481.2	−0.565478
$986$	−51587.7	−1.66621
$987$	−948.891	−0.0306014
$988$	0	0
$989$	−20826.4	−0.669607
$990$	12894.9	0.413966
$991$	−8947.33	−0.286802	−0.143401	−	0.989665i	$-0.545804\pi$
$991$	−8947.33	−0.286802	−0.143401	+	0.989665i	$0.545804\pi$
$992$	8049.67	0.257638
$993$	22471.1	0.718127
$994$	−3558.95	−0.113564
$995$	−10414.7	−0.331828
$996$	−19749.5	−0.628299
$997$	−14908.6	−0.473582	−0.236791	−	0.971561i	$-0.576096\pi$
$997$	−14908.6	−0.473582	−0.236791	+	0.971561i	$0.576096\pi$
$998$	−51881.4	−1.64557
$999$	1478.69	0.0468306

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.f.1.1		2
3.2	odd	2		1521.4.a.s.1.2		2
13.5	odd	4		507.4.b.f.337.4		4
13.8	odd	4		507.4.b.f.337.1		4
13.12	even	2		39.4.a.b.1.2	✓	2
39.38	odd	2		117.4.a.c.1.1		2
52.51	odd	2		624.4.a.r.1.1		2
65.64	even	2		975.4.a.j.1.1		2
91.90	odd	2		1911.4.a.h.1.2		2
104.51	odd	2		2496.4.a.s.1.2		2
104.77	even	2		2496.4.a.bc.1.2		2
156.155	even	2		1872.4.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.b.1.2	✓	2	13.12	even	2
117.4.a.c.1.1		2	39.38	odd	2
507.4.a.f.1.1		2	1.1	even	1	trivial
507.4.b.f.337.1		4	13.8	odd	4
507.4.b.f.337.4		4	13.5	odd	4
624.4.a.r.1.1		2	52.51	odd	2
975.4.a.j.1.1		2	65.64	even	2
1521.4.a.s.1.2		2	3.2	odd	2
1872.4.a.t.1.2		2	156.155	even	2
1911.4.a.h.1.2		2	91.90	odd	2
2496.4.a.s.1.2		2	104.51	odd	2
2496.4.a.bc.1.2		2	104.77	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

\(f(q)\)	\(=\)	\(q-4.74166 q^{2} -3.00000 q^{3} +14.4833 q^{4} -4.51669 q^{5} +14.2250 q^{6} +7.48331 q^{7} -30.7417 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-4.74166 q^{2} -3.00000 q^{3} +14.4833 q^{4} -4.51669 q^{5} +14.2250 q^{6} +7.48331 q^{7} -30.7417 q^{8} +9.00000 q^{9} +21.4166 q^{10} +66.8999 q^{11} -43.4499 q^{12} -35.4833 q^{14} +13.5501 q^{15} +29.8999 q^{16} +96.9666 q^{17} -42.6749 q^{18} -31.4833 q^{19} -65.4166 q^{20} -22.4499 q^{21} -317.216 q^{22} +183.600 q^{23} +92.2250 q^{24} -104.600 q^{25} -27.0000 q^{27} +108.383 q^{28} +112.200 q^{29} -64.2497 q^{30} +77.2831 q^{31} +104.158 q^{32} -200.700 q^{33} -459.783 q^{34} -33.7998 q^{35} +130.350 q^{36} -54.7664 q^{37} +149.283 q^{38} +138.850 q^{40} -451.716 q^{41} +106.450 q^{42} -113.434 q^{43} +968.932 q^{44} -40.6502 q^{45} -870.566 q^{46} +42.2670 q^{47} -89.6997 q^{48} -287.000 q^{49} +495.975 q^{50} -290.900 q^{51} -530.999 q^{53} +128.025 q^{54} -302.166 q^{55} -230.049 q^{56} +94.4499 q^{57} -532.015 q^{58} -219.666 q^{59} +196.250 q^{60} +822.865 q^{61} -366.450 q^{62} +67.3498 q^{63} -733.082 q^{64} +951.649 q^{66} +872.082 q^{67} +1404.40 q^{68} -550.799 q^{69} +160.267 q^{70} +100.299 q^{71} -276.675 q^{72} +165.634 q^{73} +259.684 q^{74} +313.799 q^{75} -455.983 q^{76} +500.633 q^{77} -545.266 q^{79} -135.048 q^{80} +81.0000 q^{81} +2141.88 q^{82} +454.534 q^{83} -325.150 q^{84} -437.968 q^{85} +537.864 q^{86} -336.601 q^{87} -2056.61 q^{88} +230.915 q^{89} +192.749 q^{90} +2659.13 q^{92} -231.849 q^{93} -200.415 q^{94} +142.200 q^{95} -312.475 q^{96} +1089.16 q^{97} +1360.86 q^{98} +602.099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 6 q^{3} + 14 q^{4} - 24 q^{5} + 6 q^{6} - 54 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 6 * q^3 + 14 * q^4 - 24 * q^5 + 6 * q^6 - 54 * q^8 + 18 * q^9	\( 2 q - 2 q^{2} - 6 q^{3} + 14 q^{4} - 24 q^{5} + 6 q^{6} - 54 q^{8} + 18 q^{9} - 32 q^{10} + 44 q^{11} - 42 q^{12} - 56 q^{14} + 72 q^{15} - 30 q^{16} + 164 q^{17} - 18 q^{18} - 48 q^{19} - 56 q^{20} - 380 q^{22} + 8 q^{23} + 162 q^{24} + 150 q^{25} - 54 q^{27} + 112 q^{28} + 404 q^{29} + 96 q^{30} - 40 q^{31} + 126 q^{32} - 132 q^{33} - 276 q^{34} + 112 q^{35} + 126 q^{36} + 100 q^{37} + 104 q^{38} + 592 q^{40} - 200 q^{41} + 168 q^{42} - 616 q^{43} + 980 q^{44} - 216 q^{45} - 1352 q^{46} + 324 q^{47} + 90 q^{48} - 574 q^{49} + 1194 q^{50} - 492 q^{51} - 164 q^{53} + 54 q^{54} + 144 q^{55} - 56 q^{56} + 144 q^{57} + 268 q^{58} - 140 q^{59} + 168 q^{60} + 628 q^{61} - 688 q^{62} - 194 q^{64} + 1140 q^{66} + 472 q^{67} + 1372 q^{68} - 24 q^{69} + 560 q^{70} - 428 q^{71} - 486 q^{72} + 900 q^{73} + 684 q^{74} - 450 q^{75} - 448 q^{76} + 672 q^{77} - 432 q^{79} + 1032 q^{80} + 162 q^{81} + 2832 q^{82} + 1388 q^{83} - 336 q^{84} - 1744 q^{85} - 840 q^{86} - 1212 q^{87} - 1524 q^{88} - 960 q^{89} - 288 q^{90} + 2744 q^{92} + 120 q^{93} + 572 q^{94} + 464 q^{95} - 378 q^{96} + 532 q^{97} + 574 q^{98} + 396 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 6 * q^3 + 14 * q^4 - 24 * q^5 + 6 * q^6 - 54 * q^8 + 18 * q^9 - 32 * q^10 + 44 * q^11 - 42 * q^12 - 56 * q^14 + 72 * q^15 - 30 * q^16 + 164 * q^17 - 18 * q^18 - 48 * q^19 - 56 * q^20 - 380 * q^22 + 8 * q^23 + 162 * q^24 + 150 * q^25 - 54 * q^27 + 112 * q^28 + 404 * q^29 + 96 * q^30 - 40 * q^31 + 126 * q^32 - 132 * q^33 - 276 * q^34 + 112 * q^35 + 126 * q^36 + 100 * q^37 + 104 * q^38 + 592 * q^40 - 200 * q^41 + 168 * q^42 - 616 * q^43 + 980 * q^44 - 216 * q^45 - 1352 * q^46 + 324 * q^47 + 90 * q^48 - 574 * q^49 + 1194 * q^50 - 492 * q^51 - 164 * q^53 + 54 * q^54 + 144 * q^55 - 56 * q^56 + 144 * q^57 + 268 * q^58 - 140 * q^59 + 168 * q^60 + 628 * q^61 - 688 * q^62 - 194 * q^64 + 1140 * q^66 + 472 * q^67 + 1372 * q^68 - 24 * q^69 + 560 * q^70 - 428 * q^71 - 486 * q^72 + 900 * q^73 + 684 * q^74 - 450 * q^75 - 448 * q^76 + 672 * q^77 - 432 * q^79 + 1032 * q^80 + 162 * q^81 + 2832 * q^82 + 1388 * q^83 - 336 * q^84 - 1744 * q^85 - 840 * q^86 - 1212 * q^87 - 1524 * q^88 - 960 * q^89 - 288 * q^90 + 2744 * q^92 + 120 * q^93 + 572 * q^94 + 464 * q^95 - 378 * q^96 + 532 * q^97 + 574 * q^98 + 396 * q^99

Introduction

L-functions

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