LMFDB - Embedded newform 624.4.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(1,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("624.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.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:	$36.8171918436$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
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	$\chi$	$=$	624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +4.51669 q^{5} +7.48331 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +4.51669 q^{5} +7.48331 q^{7} +9.00000 q^{9} +66.8999 q^{11} -13.0000 q^{13} +13.5501 q^{15} +96.9666 q^{17} -31.4833 q^{19} +22.4499 q^{21} -183.600 q^{23} -104.600 q^{25} +27.0000 q^{27} +112.200 q^{29} +77.2831 q^{31} +200.700 q^{33} +33.7998 q^{35} +54.7664 q^{37} -39.0000 q^{39} +451.716 q^{41} +113.434 q^{43} +40.6502 q^{45} +42.2670 q^{47} -287.000 q^{49} +290.900 q^{51} -530.999 q^{53} +302.166 q^{55} -94.4499 q^{57} -219.666 q^{59} +822.865 q^{61} +67.3498 q^{63} -58.7169 q^{65} +872.082 q^{67} -550.799 q^{69} +100.299 q^{71} -165.634 q^{73} -313.799 q^{75} +500.633 q^{77} +545.266 q^{79} +81.0000 q^{81} +454.534 q^{83} +437.968 q^{85} +336.601 q^{87} -230.915 q^{89} -97.2831 q^{91} +231.849 q^{93} -142.200 q^{95} -1089.16 q^{97} +602.099 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 24 q^{5} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 + 24 * q^5 + 18 * q^9	$ 2 q + 6 q^{3} + 24 q^{5} + 18 q^{9} + 44 q^{11} - 26 q^{13} + 72 q^{15} + 164 q^{17} - 48 q^{19} - 8 q^{23} + 150 q^{25} + 54 q^{27} + 404 q^{29} - 40 q^{31} + 132 q^{33} - 112 q^{35} - 100 q^{37} - 78 q^{39} + 200 q^{41} + 616 q^{43} + 216 q^{45} + 324 q^{47} - 574 q^{49} + 492 q^{51} - 164 q^{53} - 144 q^{55} - 144 q^{57} - 140 q^{59} + 628 q^{61} - 312 q^{65} + 472 q^{67} - 24 q^{69} - 428 q^{71} - 900 q^{73} + 450 q^{75} + 672 q^{77} + 432 q^{79} + 162 q^{81} + 1388 q^{83} + 1744 q^{85} + 1212 q^{87} + 960 q^{89} - 120 q^{93} - 464 q^{95} - 532 q^{97} + 396 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 24 * q^5 + 18 * q^9 + 44 * q^11 - 26 * q^13 + 72 * q^15 + 164 * q^17 - 48 * q^19 - 8 * q^23 + 150 * q^25 + 54 * q^27 + 404 * q^29 - 40 * q^31 + 132 * q^33 - 112 * q^35 - 100 * q^37 - 78 * q^39 + 200 * q^41 + 616 * q^43 + 216 * q^45 + 324 * q^47 - 574 * q^49 + 492 * q^51 - 164 * q^53 - 144 * q^55 - 144 * q^57 - 140 * q^59 + 628 * q^61 - 312 * q^65 + 472 * q^67 - 24 * q^69 - 428 * q^71 - 900 * q^73 + 450 * q^75 + 672 * q^77 + 432 * q^79 + 162 * q^81 + 1388 * q^83 + 1744 * q^85 + 1212 * q^87 + 960 * q^89 - 120 * q^93 - 464 * q^95 - 532 * q^97 + 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 $

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	4.51669	0.403985	0.201992	−	0.979387i	$-0.435258\pi$
$5$	4.51669	0.403985	0.201992	+	0.979387i	$0.435258\pi$
$6$	0	0
$7$	7.48331	0.404061	0.202031	−	0.979379i	$-0.435246\pi$
$7$	7.48331	0.404061	0.202031	+	0.979379i	$0.435246\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	66.8999	1.83373	0.916867	−	0.399193i	$-0.130710\pi$
$11$	66.8999	1.83373	0.916867	+	0.399193i	$0.130710\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	13.5501	0.233241
$16$	0	0
$17$	96.9666	1.38340	0.691702	−	0.722183i	$-0.256861\pi$
$17$	96.9666	1.38340	0.691702	+	0.722183i	$0.256861\pi$
$18$	0	0
$19$	−31.4833	−0.380146	−0.190073	−	0.981770i	$-0.560872\pi$
$19$	−31.4833	−0.380146	−0.190073	+	0.981770i	$0.560872\pi$
$20$	0	0
$21$	22.4499	0.233285
$22$	0	0
$23$	−183.600	−1.66448	−0.832242	−	0.554412i	$-0.812943\pi$
$23$	−183.600	−1.66448	−0.832242	+	0.554412i	$0.812943\pi$
$24$	0	0
$25$	−104.600	−0.836796
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	112.200	0.718450	0.359225	−	0.933251i	$-0.383041\pi$
$29$	112.200	0.718450	0.359225	+	0.933251i	$0.383041\pi$
$30$	0	0
$31$	77.2831	0.447757	0.223878	−	0.974617i	$-0.428128\pi$
$31$	77.2831	0.447757	0.223878	+	0.974617i	$0.428128\pi$
$32$	0	0
$33$	200.700	1.05871
$34$	0	0
$35$	33.7998	0.163234
$36$	0	0
$37$	54.7664	0.243339	0.121669	−	0.992571i	$-0.461175\pi$
$37$	54.7664	0.243339	0.121669	+	0.992571i	$0.461175\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	451.716	1.72064	0.860319	−	0.509756i	$-0.170264\pi$
$41$	451.716	1.72064	0.860319	+	0.509756i	$0.170264\pi$
$42$	0	0
$43$	113.434	0.402291	0.201145	−	0.979561i	$-0.435534\pi$
$43$	113.434	0.402291	0.201145	+	0.979561i	$0.435534\pi$
$44$	0	0
$45$	40.6502	0.134662
$46$	0	0
$47$	42.2670	0.131176	0.0655880	−	0.997847i	$-0.479108\pi$
$47$	42.2670	0.131176	0.0655880	+	0.997847i	$0.479108\pi$
$48$	0	0
$49$	−287.000	−0.836735
$50$	0	0
$51$	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	$0.741554\pi$
$54$	0	0
$55$	302.166	0.740800
$56$	0	0
$57$	−94.4499	−0.219477
$58$	0	0
$59$	−219.666	−0.484714	−0.242357	−	0.970187i	$-0.577920\pi$
$59$	−219.666	−0.484714	−0.242357	+	0.970187i	$0.577920\pi$
$60$	0	0
$61$	822.865	1.72717	0.863583	−	0.504207i	$-0.168215\pi$
$61$	822.865	1.72717	0.863583	+	0.504207i	$0.168215\pi$
$62$	0	0
$63$	67.3498	0.134687
$64$	0	0
$65$	−58.7169	−0.112045
$66$	0	0
$67$	872.082	1.59018	0.795088	−	0.606495i	$-0.207425\pi$
$67$	872.082	1.59018	0.795088	+	0.606495i	$0.207425\pi$
$68$	0	0
$69$	−550.799	−0.960991
$70$	0	0
$71$	100.299	0.167653	0.0838263	−	0.996480i	$-0.473286\pi$
$71$	100.299	0.167653	0.0838263	+	0.996480i	$0.473286\pi$
$72$	0	0
$73$	−165.634	−0.265562	−0.132781	−	0.991145i	$-0.542391\pi$
$73$	−165.634	−0.265562	−0.132781	+	0.991145i	$0.542391\pi$
$74$	0	0
$75$	−313.799	−0.483125
$76$	0	0
$77$	500.633	0.740940
$78$	0	0
$79$	545.266	0.776547	0.388273	−	0.921544i	$-0.373072\pi$
$79$	545.266	0.776547	0.388273	+	0.921544i	$0.373072\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	454.534	0.601103	0.300552	−	0.953766i	$-0.402829\pi$
$83$	454.534	0.601103	0.300552	+	0.953766i	$0.402829\pi$
$84$	0	0
$85$	437.968	0.558874
$86$	0	0
$87$	336.601	0.414797
$88$	0	0
$89$	−230.915	−0.275022	−0.137511	−	0.990500i	$-0.543910\pi$
$89$	−230.915	−0.275022	−0.137511	+	0.990500i	$0.543910\pi$
$90$	0	0
$91$	−97.2831	−0.112066
$92$	0	0
$93$	231.849	0.258512
$94$	0	0
$95$	−142.200	−0.153573
$96$	0	0
$97$	−1089.16	−1.14008	−0.570041	−	0.821616i	$-0.693073\pi$
$97$	−1089.16	−1.14008	−0.570041	+	0.821616i	$0.693073\pi$
$98$	0	0
$99$	602.099	0.611245
$100$	0	0
$101$	−77.2336	−0.0760894	−0.0380447	−	0.999276i	$-0.512113\pi$
$101$	−77.2336	−0.0760894	−0.0380447	+	0.999276i	$0.512113\pi$
$102$	0	0
$103$	−1351.36	−1.29275	−0.646377	−	0.763018i	$-0.723717\pi$
$103$	−1351.36	−1.29275	−0.646377	+	0.763018i	$0.723717\pi$
$104$	0	0
$105$	101.399	0.0942434
$106$	0	0
$107$	−1133.67	−1.02426	−0.512129	−	0.858908i	$-0.671143\pi$
$107$	−1133.67	−1.02426	−0.512129	+	0.858908i	$0.671143\pi$
$108$	0	0
$109$	1017.66	0.894262	0.447131	−	0.894469i	$-0.352446\pi$
$109$	1017.66	0.894262	0.447131	+	0.894469i	$0.352446\pi$
$110$	0	0
$111$	164.299	0.140492
$112$	0	0
$113$	1570.06	1.30707	0.653536	−	0.756895i	$-0.273285\pi$
$113$	1570.06	1.30707	0.653536	+	0.756895i	$0.273285\pi$
$114$	0	0
$115$	−829.261	−0.672426
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	725.632	0.558979
$120$	0	0
$121$	3144.60	2.36258
$122$	0	0
$123$	1355.15	0.993411
$124$	0	0
$125$	−1037.03	−0.742037
$126$	0	0
$127$	1248.16	0.872099	0.436050	−	0.899923i	$-0.356377\pi$
$127$	1248.16	0.872099	0.436050	+	0.899923i	$0.356377\pi$
$128$	0	0
$129$	340.301	0.232263
$130$	0	0
$131$	1274.80	0.850227	0.425113	−	0.905140i	$-0.360234\pi$
$131$	1274.80	0.850227	0.425113	+	0.905140i	$0.360234\pi$
$132$	0	0
$133$	−235.600	−0.153602
$134$	0	0
$135$	121.951	0.0777469
$136$	0	0
$137$	874.915	0.545613	0.272807	−	0.962069i	$-0.412048\pi$
$137$	874.915	0.545613	0.272807	+	0.962069i	$0.412048\pi$
$138$	0	0
$139$	−310.334	−0.189368	−0.0946840	−	0.995507i	$-0.530184\pi$
$139$	−310.334	−0.189368	−0.0946840	+	0.995507i	$0.530184\pi$
$140$	0	0
$141$	126.801	0.0757345
$142$	0	0
$143$	−869.699	−0.508586
$144$	0	0
$145$	506.773	0.290243
$146$	0	0
$147$	−861.000	−0.483089
$148$	0	0
$149$	5.08064	0.00279344	0.00139672	−	0.999999i	$-0.499555\pi$
$149$	5.08064	0.00279344	0.00139672	+	0.999999i	$0.499555\pi$
$150$	0	0
$151$	−6.54894	−0.00352944	−0.00176472	−	0.999998i	$-0.500562\pi$
$151$	−6.54894	−0.00352944	−0.00176472	+	0.999998i	$0.500562\pi$
$152$	0	0
$153$	872.700	0.461135
$154$	0	0
$155$	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	$0.698503\pi$
$158$	0	0
$159$	−1593.00	−0.794546
$160$	0	0
$161$	−1373.93	−0.672553
$162$	0	0
$163$	1085.49	0.521606	0.260803	−	0.965392i	$-0.416013\pi$
$163$	1085.49	0.521606	0.260803	+	0.965392i	$0.416013\pi$
$164$	0	0
$165$	906.497	0.427701
$166$	0	0
$167$	109.066	0.0505374	0.0252687	−	0.999681i	$-0.491956\pi$
$167$	109.066	0.0505374	0.0252687	+	0.999681i	$0.491956\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−283.350	−0.126715
$172$	0	0
$173$	−1889.17	−0.830236	−0.415118	−	0.909768i	$-0.636260\pi$
$173$	−1889.17	−0.830236	−0.415118	+	0.909768i	$0.636260\pi$
$174$	0	0
$175$	−782.751	−0.338117
$176$	0	0
$177$	−658.999	−0.279850
$178$	0	0
$179$	−3427.86	−1.43134	−0.715672	−	0.698437i	$-0.753879\pi$
$179$	−3427.86	−1.43134	−0.715672	+	0.698437i	$0.753879\pi$
$180$	0	0
$181$	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
$184$	0	0
$185$	247.363	0.0983052
$186$	0	0
$187$	6487.06	2.53679
$188$	0	0
$189$	202.049	0.0777616
$190$	0	0
$191$	−957.735	−0.362824	−0.181412	−	0.983407i	$-0.558067\pi$
$191$	−957.735	−0.362824	−0.181412	+	0.983407i	$0.558067\pi$
$192$	0	0
$193$	−512.730	−0.191228	−0.0956142	−	0.995418i	$-0.530482\pi$
$193$	−512.730	−0.191228	−0.0956142	+	0.995418i	$0.530482\pi$
$194$	0	0
$195$	−176.151	−0.0646893
$196$	0	0
$197$	−3870.35	−1.39975	−0.699876	−	0.714265i	$-0.746761\pi$
$197$	−3870.35	−1.39975	−0.699876	+	0.714265i	$0.746761\pi$
$198$	0	0
$199$	−2305.83	−0.821388	−0.410694	−	0.911773i	$-0.634714\pi$
$199$	−2305.83	−0.821388	−0.410694	+	0.911773i	$0.634714\pi$
$200$	0	0
$201$	2616.25	0.918088
$202$	0	0
$203$	839.630	0.290298
$204$	0	0
$205$	2040.26	0.695111
$206$	0	0
$207$	−1652.40	−0.554828
$208$	0	0
$209$	−2106.23	−0.697086
$210$	0	0
$211$	3672.40	1.19819	0.599096	−	0.800677i	$-0.295527\pi$
$211$	3672.40	1.19819	0.599096	+	0.800677i	$0.295527\pi$
$212$	0	0
$213$	300.898	0.0967942
$214$	0	0
$215$	512.345	0.162519
$216$	0	0
$217$	578.334	0.180921
$218$	0	0
$219$	−496.902	−0.153322
$220$	0	0
$221$	−1260.57	−0.383687
$222$	0	0
$223$	−5087.05	−1.52760	−0.763798	−	0.645455i	$-0.776668\pi$
$223$	−5087.05	−1.52760	−0.763798	+	0.645455i	$0.776668\pi$
$224$	0	0
$225$	−941.396	−0.278932
$226$	0	0
$227$	2625.83	0.767763	0.383882	−	0.923382i	$-0.374587\pi$
$227$	2625.83	0.767763	0.383882	+	0.923382i	$0.374587\pi$
$228$	0	0
$229$	1678.73	0.484425	0.242213	−	0.970223i	$-0.422127\pi$
$229$	1678.73	0.484425	0.242213	+	0.970223i	$0.422127\pi$
$230$	0	0
$231$	1501.90	0.427782
$232$	0	0
$233$	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
$236$	0	0
$237$	1635.80	0.448340
$238$	0	0
$239$	−5219.69	−1.41269	−0.706347	−	0.707866i	$-0.749658\pi$
$239$	−5219.69	−1.41269	−0.706347	+	0.707866i	$0.749658\pi$
$240$	0	0
$241$	6103.56	1.63139	0.815695	−	0.578483i	$-0.196355\pi$
$241$	6103.56	1.63139	0.815695	+	0.578483i	$0.196355\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−1296.29	−0.338028
$246$	0	0
$247$	409.283	0.105433
$248$	0	0
$249$	1363.60	0.347047
$250$	0	0
$251$	−6423.40	−1.61530	−0.807652	−	0.589660i	$-0.799262\pi$
$251$	−6423.40	−1.61530	−0.807652	+	0.589660i	$0.799262\pi$
$252$	0	0
$253$	−12282.8	−3.05222
$254$	0	0
$255$	1313.90	0.322666
$256$	0	0
$257$	−1230.23	−0.298597	−0.149299	−	0.988792i	$-0.547702\pi$
$257$	−1230.23	−0.298597	−0.149299	+	0.988792i	$0.547702\pi$
$258$	0	0
$259$	409.834	0.0983238
$260$	0	0
$261$	1009.80	0.239483
$262$	0	0
$263$	514.992	0.120744	0.0603722	−	0.998176i	$-0.480771\pi$
$263$	514.992	0.120744	0.0603722	+	0.998176i	$0.480771\pi$
$264$	0	0
$265$	−2398.35	−0.555961
$266$	0	0
$267$	−692.745	−0.158784
$268$	0	0
$269$	−5132.60	−1.16335	−0.581673	−	0.813423i	$-0.697602\pi$
$269$	−5132.60	−1.16335	−0.581673	+	0.813423i	$0.697602\pi$
$270$	0	0
$271$	4300.00	0.963862	0.481931	−	0.876209i	$-0.339936\pi$
$271$	4300.00	0.963862	0.481931	+	0.876209i	$0.339936\pi$
$272$	0	0
$273$	−291.849	−0.0647015
$274$	0	0
$275$	−6997.70	−1.53446
$276$	0	0
$277$	−1812.80	−0.393215	−0.196607	−	0.980482i	$-0.562992\pi$
$277$	−1812.80	−0.393215	−0.196607	+	0.980482i	$0.562992\pi$
$278$	0	0
$279$	695.548	0.149252
$280$	0	0
$281$	−4073.08	−0.864696	−0.432348	−	0.901707i	$-0.642315\pi$
$281$	−4073.08	−0.864696	−0.432348	+	0.901707i	$0.642315\pi$
$282$	0	0
$283$	−6346.29	−1.33303	−0.666516	−	0.745491i	$-0.732215\pi$
$283$	−6346.29	−1.33303	−0.666516	+	0.745491i	$0.732215\pi$
$284$	0	0
$285$	−426.601	−0.0886654
$286$	0	0
$287$	3380.33	0.695243
$288$	0	0
$289$	4489.53	0.913806
$290$	0	0
$291$	−3267.49	−0.658226
$292$	0	0
$293$	−8390.97	−1.67306	−0.836529	−	0.547923i	$-0.815419\pi$
$293$	−8390.97	−1.67306	−0.836529	+	0.547923i	$0.815419\pi$
$294$	0	0
$295$	−992.164	−0.195817
$296$	0	0
$297$	1806.30	0.352902
$298$	0	0
$299$	2386.79	0.461645
$300$	0	0
$301$	848.861	0.162550
$302$	0	0
$303$	−231.701	−0.0439302
$304$	0	0
$305$	3716.62	0.697748
$306$	0	0
$307$	−4005.27	−0.744603	−0.372301	−	0.928112i	$-0.621431\pi$
$307$	−4005.27	−0.744603	−0.372301	+	0.928112i	$0.621431\pi$
$308$	0	0
$309$	−4054.09	−0.746372
$310$	0	0
$311$	−5836.53	−1.06418	−0.532088	−	0.846689i	$-0.678593\pi$
$311$	−5836.53	−1.06418	−0.532088	+	0.846689i	$0.678593\pi$
$312$	0	0
$313$	1763.19	0.318407	0.159204	−	0.987246i	$-0.449107\pi$
$313$	1763.19	0.318407	0.159204	+	0.987246i	$0.449107\pi$
$314$	0	0
$315$	304.198	0.0544115
$316$	0	0
$317$	−6106.35	−1.08191	−0.540957	−	0.841050i	$-0.681938\pi$
$317$	−6106.35	−1.08191	−0.540957	+	0.841050i	$0.681938\pi$
$318$	0	0
$319$	7506.18	1.31745
$320$	0	0
$321$	−3401.00	−0.591356
$322$	0	0
$323$	−3052.83	−0.525895
$324$	0	0
$325$	1359.79	0.232086
$326$	0	0
$327$	3052.99	0.516302
$328$	0	0
$329$	316.297	0.0530031
$330$	0	0
$331$	−7490.38	−1.24383	−0.621916	−	0.783084i	$-0.713646\pi$
$331$	−7490.38	−1.24383	−0.621916	+	0.783084i	$0.713646\pi$
$332$	0	0
$333$	492.898	0.0811130
$334$	0	0
$335$	3938.92	0.642406
$336$	0	0
$337$	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
$340$	0	0
$341$	5170.23	0.821066
$342$	0	0
$343$	−4714.49	−0.742153
$344$	0	0
$345$	−2487.78	−0.388226
$346$	0	0
$347$	−11460.3	−1.77297	−0.886487	−	0.462753i	$-0.846862\pi$
$347$	−11460.3	−1.77297	−0.886487	+	0.462753i	$0.846862\pi$
$348$	0	0
$349$	−3673.96	−0.563503	−0.281751	−	0.959487i	$-0.590915\pi$
$349$	−3673.96	−0.563503	−0.281751	+	0.959487i	$0.590915\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	3388.65	0.510935	0.255467	−	0.966818i	$-0.417771\pi$
$353$	3388.65	0.510935	0.255467	+	0.966818i	$0.417771\pi$
$354$	0	0
$355$	453.020	0.0677290
$356$	0	0
$357$	2176.90	0.322727
$358$	0	0
$359$	9673.98	1.42221	0.711105	−	0.703086i	$-0.248195\pi$
$359$	9673.98	1.42221	0.711105	+	0.703086i	$0.248195\pi$
$360$	0	0
$361$	−5867.80	−0.855489
$362$	0	0
$363$	9433.79	1.36404
$364$	0	0
$365$	−748.117	−0.107283
$366$	0	0
$367$	8715.98	1.23970	0.619851	−	0.784720i	$-0.287193\pi$
$367$	8715.98	1.23970	0.619851	+	0.784720i	$0.287193\pi$
$368$	0	0
$369$	4065.44	0.573546
$370$	0	0
$371$	−3973.63	−0.556067
$372$	0	0
$373$	4667.99	0.647987	0.323994	−	0.946059i	$-0.394974\pi$
$373$	4667.99	0.647987	0.323994	+	0.946059i	$0.394974\pi$
$374$	0	0
$375$	−3111.09	−0.428416
$376$	0	0
$377$	−1458.60	−0.199262
$378$	0	0
$379$	−10862.7	−1.47225	−0.736123	−	0.676848i	$-0.763345\pi$
$379$	−10862.7	−1.47225	−0.736123	+	0.676848i	$0.763345\pi$
$380$	0	0
$381$	3744.49	0.503507
$382$	0	0
$383$	−10054.2	−1.34137	−0.670686	−	0.741742i	$-0.734000\pi$
$383$	−10054.2	−1.34137	−0.670686	+	0.741742i	$0.734000\pi$
$384$	0	0
$385$	2261.20	0.299329
$386$	0	0
$387$	1020.90	0.134097
$388$	0	0
$389$	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
$392$	0	0
$393$	3824.40	0.490879
$394$	0	0
$395$	2462.79	0.313713
$396$	0	0
$397$	−12019.9	−1.51955	−0.759775	−	0.650186i	$-0.774691\pi$
$397$	−12019.9	−1.51955	−0.759775	+	0.650186i	$0.774691\pi$
$398$	0	0
$399$	−706.799	−0.0886822
$400$	0	0
$401$	3599.80	0.448293	0.224147	−	0.974555i	$-0.428041\pi$
$401$	3599.80	0.448293	0.224147	+	0.974555i	$0.428041\pi$
$402$	0	0
$403$	−1004.68	−0.124185
$404$	0	0
$405$	365.852	0.0448872
$406$	0	0
$407$	3663.87	0.446219
$408$	0	0
$409$	−48.0968	−0.00581475	−0.00290737	−	0.999996i	$-0.500925\pi$
$409$	−48.0968	−0.00581475	−0.00290737	+	0.999996i	$0.500925\pi$
$410$	0	0
$411$	2624.74	0.315010
$412$	0	0
$413$	−1643.83	−0.195854
$414$	0	0
$415$	2052.99	0.242837
$416$	0	0
$417$	−931.001	−0.109332
$418$	0	0
$419$	−723.462	−0.0843518	−0.0421759	−	0.999110i	$-0.513429\pi$
$419$	−723.462	−0.0843518	−0.0421759	+	0.999110i	$0.513429\pi$
$420$	0	0
$421$	−14845.5	−1.71859	−0.859295	−	0.511481i	$-0.829097\pi$
$421$	−14845.5	−1.71859	−0.859295	+	0.511481i	$0.829097\pi$
$422$	0	0
$423$	380.403	0.0437253
$424$	0	0
$425$	−10142.7	−1.15763
$426$	0	0
$427$	6157.76	0.697880
$428$	0	0
$429$	−2609.10	−0.293632
$430$	0	0
$431$	−1103.11	−0.123283	−0.0616417	−	0.998098i	$-0.519634\pi$
$431$	−1103.11	−0.123283	−0.0616417	+	0.998098i	$0.519634\pi$
$432$	0	0
$433$	8893.53	0.987057	0.493528	−	0.869730i	$-0.335707\pi$
$433$	8893.53	0.987057	0.493528	+	0.869730i	$0.335707\pi$
$434$	0	0
$435$	1520.32	0.167572
$436$	0	0
$437$	5780.32	0.632747
$438$	0	0
$439$	10901.7	1.18521	0.592607	−	0.805492i	$-0.298099\pi$
$439$	10901.7	1.18521	0.592607	+	0.805492i	$0.298099\pi$
$440$	0	0
$441$	−2583.00	−0.278912
$442$	0	0
$443$	3781.37	0.405550	0.202775	−	0.979225i	$-0.435004\pi$
$443$	3781.37	0.405550	0.202775	+	0.979225i	$0.435004\pi$
$444$	0	0
$445$	−1042.97	−0.111105
$446$	0	0
$447$	15.2419	0.00161279
$448$	0	0
$449$	106.834	0.0112289	0.00561447	−	0.999984i	$-0.498213\pi$
$449$	106.834	0.0112289	0.00561447	+	0.999984i	$0.498213\pi$
$450$	0	0
$451$	30219.7	3.15519
$452$	0	0
$453$	−19.6468	−0.00203772
$454$	0	0
$455$	−439.397	−0.0452731
$456$	0	0
$457$	−1237.64	−0.126684	−0.0633419	−	0.997992i	$-0.520176\pi$
$457$	−1237.64	−0.126684	−0.0633419	+	0.997992i	$0.520176\pi$
$458$	0	0
$459$	2618.10	0.266236
$460$	0	0
$461$	8790.90	0.888141	0.444071	−	0.895992i	$-0.353534\pi$
$461$	8790.90	0.888141	0.444071	+	0.895992i	$0.353534\pi$
$462$	0	0
$463$	3861.55	0.387606	0.193803	−	0.981040i	$-0.437918\pi$
$463$	3861.55	0.387606	0.193803	+	0.981040i	$0.437918\pi$
$464$	0	0
$465$	1047.19	0.104435
$466$	0	0
$467$	8991.38	0.890945	0.445473	−	0.895296i	$-0.353036\pi$
$467$	8991.38	0.890945	0.445473	+	0.895296i	$0.353036\pi$
$468$	0	0
$469$	6526.06	0.642528
$470$	0	0
$471$	−6892.79	−0.674316
$472$	0	0
$473$	7588.71	0.737694
$474$	0	0
$475$	3293.14	0.318105
$476$	0	0
$477$	−4778.99	−0.458731
$478$	0	0
$479$	−4179.82	−0.398707	−0.199354	−	0.979928i	$-0.563884\pi$
$479$	−4179.82	−0.398707	−0.199354	+	0.979928i	$0.563884\pi$
$480$	0	0
$481$	−711.963	−0.0674901
$482$	0	0
$483$	−4121.80	−0.388299
$484$	0	0
$485$	−4919.41	−0.460575
$486$	0	0
$487$	18443.8	1.71616	0.858078	−	0.513519i	$-0.171658\pi$
$487$	18443.8	1.71616	0.858078	+	0.513519i	$0.171658\pi$
$488$	0	0
$489$	3256.46	0.301149
$490$	0	0
$491$	−8093.26	−0.743877	−0.371939	−	0.928257i	$-0.621307\pi$
$491$	−8093.26	−0.743877	−0.371939	+	0.928257i	$0.621307\pi$
$492$	0	0
$493$	10879.7	0.993907
$494$	0	0
$495$	2719.49	0.246933
$496$	0	0
$497$	750.571	0.0677418
$498$	0	0
$499$	10941.6	0.981591	0.490796	−	0.871275i	$-0.336706\pi$
$499$	10941.6	0.981591	0.490796	+	0.871275i	$0.336706\pi$
$500$	0	0
$501$	327.197	0.0291778
$502$	0	0
$503$	9260.11	0.820851	0.410425	−	0.911894i	$-0.365380\pi$
$503$	9260.11	0.820851	0.410425	+	0.911894i	$0.365380\pi$
$504$	0	0
$505$	−348.840	−0.0307389
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	−9996.40	−0.870497	−0.435248	−	0.900310i	$-0.643339\pi$
$509$	−9996.40	−0.870497	−0.435248	+	0.900310i	$0.643339\pi$
$510$	0	0
$511$	−1239.49	−0.107303
$512$	0	0
$513$	−850.049	−0.0731591
$514$	0	0
$515$	−6103.68	−0.522253
$516$	0	0
$517$	2827.66	0.240542
$518$	0	0
$519$	−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	$0.340460\pi$
$522$	0	0
$523$	4810.47	0.402193	0.201097	−	0.979571i	$-0.435549\pi$
$523$	4810.47	0.402193	0.201097	+	0.979571i	$0.435549\pi$
$524$	0	0
$525$	−2348.25	−0.195212
$526$	0	0
$527$	7493.88	0.619428
$528$	0	0
$529$	21541.8	1.77051
$530$	0	0
$531$	−1977.00	−0.161571
$532$	0	0
$533$	−5872.31	−0.477219
$534$	0	0
$535$	−5120.41	−0.413785
$536$	0	0
$537$	−10283.6	−0.826386
$538$	0	0
$539$	−19200.3	−1.53435
$540$	0	0
$541$	2411.99	0.191681	0.0958406	−	0.995397i	$-0.469446\pi$
$541$	2411.99	0.191681	0.0958406	+	0.995397i	$0.469446\pi$
$542$	0	0
$543$	625.208	0.0494111
$544$	0	0
$545$	4596.47	0.361268
$546$	0	0
$547$	4396.34	0.343646	0.171823	−	0.985128i	$-0.445034\pi$
$547$	4396.34	0.343646	0.171823	+	0.985128i	$0.445034\pi$
$548$	0	0
$549$	7405.79	0.575722
$550$	0	0
$551$	−3532.43	−0.273116
$552$	0	0
$553$	4080.40	0.313772
$554$	0	0
$555$	742.088	0.0567565
$556$	0	0
$557$	−17488.0	−1.33032	−0.665160	−	0.746701i	$-0.731637\pi$
$557$	−17488.0	−1.33032	−0.665160	+	0.746701i	$0.731637\pi$
$558$	0	0
$559$	−1474.64	−0.111575
$560$	0	0
$561$	19461.2	1.46462
$562$	0	0
$563$	6881.77	0.515154	0.257577	−	0.966258i	$-0.417076\pi$
$563$	6881.77	0.515154	0.257577	+	0.966258i	$0.417076\pi$
$564$	0	0
$565$	7091.49	0.528037
$566$	0	0
$567$	606.148	0.0448957
$568$	0	0
$569$	14733.5	1.08552	0.542758	−	0.839889i	$-0.317380\pi$
$569$	14733.5	1.08552	0.542758	+	0.839889i	$0.317380\pi$
$570$	0	0
$571$	−4488.51	−0.328964	−0.164482	−	0.986380i	$-0.552595\pi$
$571$	−4488.51	−0.328964	−0.164482	+	0.986380i	$0.552595\pi$
$572$	0	0
$573$	−2873.21	−0.209476
$574$	0	0
$575$	19204.4	1.39284
$576$	0	0
$577$	−10552.2	−0.761338	−0.380669	−	0.924711i	$-0.624306\pi$
$577$	−10552.2	−0.761338	−0.380669	+	0.924711i	$0.624306\pi$
$578$	0	0
$579$	−1538.19	−0.110406
$580$	0	0
$581$	3401.42	0.242882
$582$	0	0
$583$	−35523.8	−2.52357
$584$	0	0
$585$	−528.452	−0.0373484
$586$	0	0
$587$	1637.20	0.115118	0.0575591	−	0.998342i	$-0.481668\pi$
$587$	1637.20	0.115118	0.0575591	+	0.998342i	$0.481668\pi$
$588$	0	0
$589$	−2433.13	−0.170213
$590$	0	0
$591$	−11611.1	−0.808147
$592$	0	0
$593$	14024.0	0.971155	0.485577	−	0.874194i	$-0.338609\pi$
$593$	14024.0	0.971155	0.485577	+	0.874194i	$0.338609\pi$
$594$	0	0
$595$	3277.45	0.225819
$596$	0	0
$597$	−6917.50	−0.474229
$598$	0	0
$599$	−365.860	−0.0249560	−0.0124780	−	0.999922i	$-0.503972\pi$
$599$	−365.860	−0.0249560	−0.0124780	+	0.999922i	$0.503972\pi$
$600$	0	0
$601$	−10128.9	−0.687465	−0.343733	−	0.939068i	$-0.611691\pi$
$601$	−10128.9	−0.687465	−0.343733	+	0.939068i	$0.611691\pi$
$602$	0	0
$603$	7848.74	0.530058
$604$	0	0
$605$	14203.1	0.954446
$606$	0	0
$607$	−2324.97	−0.155465	−0.0777327	−	0.996974i	$-0.524768\pi$
$607$	−2324.97	−0.155465	−0.0777327	+	0.996974i	$0.524768\pi$
$608$	0	0
$609$	2518.89	0.167603
$610$	0	0
$611$	−549.471	−0.0363817
$612$	0	0
$613$	633.133	0.0417162	0.0208581	−	0.999782i	$-0.493360\pi$
$613$	633.133	0.0417162	0.0208581	+	0.999782i	$0.493360\pi$
$614$	0	0
$615$	6120.77	0.401323
$616$	0	0
$617$	−2981.85	−0.194562	−0.0972810	−	0.995257i	$-0.531015\pi$
$617$	−2981.85	−0.194562	−0.0972810	+	0.995257i	$0.531015\pi$
$618$	0	0
$619$	−15158.6	−0.984292	−0.492146	−	0.870513i	$-0.663787\pi$
$619$	−15158.6	−0.984292	−0.492146	+	0.870513i	$0.663787\pi$
$620$	0	0
$621$	−4957.19	−0.320330
$622$	0	0
$623$	−1728.01	−0.111126
$624$	0	0
$625$	8391.01	0.537025
$626$	0	0
$627$	−6318.69	−0.402463
$628$	0	0
$629$	5310.51	0.336636
$630$	0	0
$631$	−5562.30	−0.350922	−0.175461	−	0.984486i	$-0.556142\pi$
$631$	−5562.30	−0.350922	−0.175461	+	0.984486i	$0.556142\pi$
$632$	0	0
$633$	11017.2	0.691776
$634$	0	0
$635$	5637.56	0.352315
$636$	0	0
$637$	3731.00	0.232068
$638$	0	0
$639$	902.693	0.0558842
$640$	0	0
$641$	−24140.7	−1.48752	−0.743761	−	0.668446i	$-0.766960\pi$
$641$	−24140.7	−1.48752	−0.743761	+	0.668446i	$0.766960\pi$
$642$	0	0
$643$	1749.69	0.107311	0.0536555	−	0.998560i	$-0.482913\pi$
$643$	1749.69	0.107311	0.0536555	+	0.998560i	$0.482913\pi$
$644$	0	0
$645$	1537.03	0.0938305
$646$	0	0
$647$	−1489.51	−0.0905083	−0.0452542	−	0.998976i	$-0.514410\pi$
$647$	−1489.51	−0.0905083	−0.0452542	+	0.998976i	$0.514410\pi$
$648$	0	0
$649$	−14695.7	−0.888836
$650$	0	0
$651$	1735.00	0.104455
$652$	0	0
$653$	10668.2	0.639327	0.319663	−	0.947531i	$-0.396430\pi$
$653$	10668.2	0.639327	0.319663	+	0.947531i	$0.396430\pi$
$654$	0	0
$655$	5757.86	0.343478
$656$	0	0
$657$	−1490.71	−0.0885205
$658$	0	0
$659$	13933.1	0.823608	0.411804	−	0.911272i	$-0.364899\pi$
$659$	13933.1	0.823608	0.411804	+	0.911272i	$0.364899\pi$
$660$	0	0
$661$	−2349.69	−0.138264	−0.0691320	−	0.997608i	$-0.522023\pi$
$661$	−2349.69	−0.138264	−0.0691320	+	0.997608i	$0.522023\pi$
$662$	0	0
$663$	−3781.70	−0.221522
$664$	0	0
$665$	−1064.13	−0.0620529
$666$	0	0
$667$	−20599.9	−1.19585
$668$	0	0
$669$	−15261.1	−0.881958
$670$	0	0
$671$	55049.6	3.16716
$672$	0	0
$673$	−32421.0	−1.85697	−0.928483	−	0.371374i	$-0.878887\pi$
$673$	−32421.0	−1.85697	−0.928483	+	0.371374i	$0.878887\pi$
$674$	0	0
$675$	−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	$0.509685\pi$
$678$	0	0
$679$	−8150.56	−0.460663
$680$	0	0
$681$	7877.48	0.443268
$682$	0	0
$683$	305.487	0.0171144	0.00855721	−	0.999963i	$-0.497276\pi$
$683$	305.487	0.0171144	0.00855721	+	0.999963i	$0.497276\pi$
$684$	0	0
$685$	3951.72	0.220419
$686$	0	0
$687$	5036.18	0.279683
$688$	0	0
$689$	6902.99	0.381688
$690$	0	0
$691$	−2180.81	−0.120061	−0.0600303	−	0.998197i	$-0.519120\pi$
$691$	−2180.81	−0.120061	−0.0600303	+	0.998197i	$0.519120\pi$
$692$	0	0
$693$	4505.70	0.246980
$694$	0	0
$695$	−1401.68	−0.0765018
$696$	0	0
$697$	43801.4	2.38034
$698$	0	0
$699$	1945.52	0.105274
$700$	0	0
$701$	−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
$704$	0	0
$705$	572.720	0.0305956
$706$	0	0
$707$	−577.963	−0.0307448
$708$	0	0
$709$	−8988.17	−0.476104	−0.238052	−	0.971252i	$-0.576509\pi$
$709$	−8988.17	−0.476104	−0.238052	+	0.971252i	$0.576509\pi$
$710$	0	0
$711$	4907.39	0.258849
$712$	0	0
$713$	−14189.1	−0.745284
$714$	0	0
$715$	−3928.15	−0.205461
$716$	0	0
$717$	−15659.1	−0.815619
$718$	0	0
$719$	−8448.18	−0.438198	−0.219099	−	0.975703i	$-0.570312\pi$
$719$	−8448.18	−0.438198	−0.219099	+	0.975703i	$0.570312\pi$
$720$	0	0
$721$	−10112.7	−0.522352
$722$	0	0
$723$	18310.7	0.941883
$724$	0	0
$725$	−11736.1	−0.601197
$726$	0	0
$727$	8624.18	0.439963	0.219982	−	0.975504i	$-0.429400\pi$
$727$	8624.18	0.439963	0.219982	+	0.975504i	$0.429400\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	10999.3	0.556530
$732$	0	0
$733$	31124.2	1.56835	0.784174	−	0.620541i	$-0.213087\pi$
$733$	31124.2	1.56835	0.784174	+	0.620541i	$0.213087\pi$
$734$	0	0
$735$	−3888.87	−0.195161
$736$	0	0
$737$	58342.2	2.91596
$738$	0	0
$739$	17671.1	0.879626	0.439813	−	0.898089i	$-0.355045\pi$
$739$	17671.1	0.879626	0.439813	+	0.898089i	$0.355045\pi$
$740$	0	0
$741$	1227.85	0.0608720
$742$	0	0
$743$	21331.1	1.05325	0.526623	−	0.850099i	$-0.323458\pi$
$743$	21331.1	1.05325	0.526623	+	0.850099i	$0.323458\pi$
$744$	0	0
$745$	22.9477	0.00112851
$746$	0	0
$747$	4090.81	0.200368
$748$	0	0
$749$	−8483.58	−0.413863
$750$	0	0
$751$	−11712.9	−0.569122	−0.284561	−	0.958658i	$-0.591848\pi$
$751$	−11712.9	−0.569122	−0.284561	+	0.958658i	$0.591848\pi$
$752$	0	0
$753$	−19270.2	−0.932596
$754$	0	0
$755$	−29.5795	−0.00142584
$756$	0	0
$757$	−16610.9	−0.797537	−0.398768	−	0.917052i	$-0.630562\pi$
$757$	−16610.9	−0.797537	−0.398768	+	0.917052i	$0.630562\pi$
$758$	0	0
$759$	−36848.4	−1.76220
$760$	0	0
$761$	−29365.5	−1.39882	−0.699408	−	0.714723i	$-0.746553\pi$
$761$	−29365.5	−1.39882	−0.699408	+	0.714723i	$0.746553\pi$
$762$	0	0
$763$	7615.50	0.361336
$764$	0	0
$765$	3941.71	0.186291
$766$	0	0
$767$	2855.66	0.134435
$768$	0	0
$769$	−28599.9	−1.34114	−0.670572	−	0.741844i	$-0.733951\pi$
$769$	−28599.9	−1.34114	−0.670572	+	0.741844i	$0.733951\pi$
$770$	0	0
$771$	−3690.68	−0.172395
$772$	0	0
$773$	13491.8	0.627772	0.313886	−	0.949461i	$-0.398369\pi$
$773$	13491.8	0.627772	0.313886	+	0.949461i	$0.398369\pi$
$774$	0	0
$775$	−8083.78	−0.374681
$776$	0	0
$777$	1229.50	0.0567673
$778$	0	0
$779$	−14221.5	−0.654093
$780$	0	0
$781$	6710.01	0.307430
$782$	0	0
$783$	3029.41	0.138266
$784$	0	0
$785$	−10377.5	−0.471834
$786$	0	0
$787$	−25876.7	−1.17205	−0.586025	−	0.810293i	$-0.699308\pi$
$787$	−25876.7	−1.17205	−0.586025	+	0.810293i	$0.699308\pi$
$788$	0	0
$789$	1544.98	0.0697118
$790$	0	0
$791$	11749.3	0.528137
$792$	0	0
$793$	−10697.3	−0.479030
$794$	0	0
$795$	−7195.06	−0.320984
$796$	0	0
$797$	21936.4	0.974938	0.487469	−	0.873140i	$-0.337920\pi$
$797$	21936.4	0.974938	0.487469	+	0.873140i	$0.337920\pi$
$798$	0	0
$799$	4098.49	0.181469
$800$	0	0
$801$	−2078.23	−0.0916739
$802$	0	0
$803$	−11080.9	−0.486969
$804$	0	0
$805$	−6205.62	−0.271701
$806$	0	0
$807$	−15397.8	−0.671658
$808$	0	0
$809$	5583.23	0.242640	0.121320	−	0.992613i	$-0.461287\pi$
$809$	5583.23	0.242640	0.121320	+	0.992613i	$0.461287\pi$
$810$	0	0
$811$	−12925.4	−0.559647	−0.279823	−	0.960052i	$-0.590276\pi$
$811$	−12925.4	−0.559647	−0.279823	+	0.960052i	$0.590276\pi$
$812$	0	0
$813$	12900.0	0.556486
$814$	0	0
$815$	4902.80	0.210721
$816$	0	0
$817$	−3571.27	−0.152929
$818$	0	0
$819$	−875.548	−0.0373555
$820$	0	0
$821$	−10153.2	−0.431608	−0.215804	−	0.976437i	$-0.569237\pi$
$821$	−10153.2	−0.431608	−0.215804	+	0.976437i	$0.569237\pi$
$822$	0	0
$823$	−3282.93	−0.139047	−0.0695235	−	0.997580i	$-0.522148\pi$
$823$	−3282.93	−0.139047	−0.0695235	+	0.997580i	$0.522148\pi$
$824$	0	0
$825$	−20993.1	−0.885922
$826$	0	0
$827$	17689.3	0.743795	0.371897	−	0.928274i	$-0.378707\pi$
$827$	17689.3	0.743795	0.371897	+	0.928274i	$0.378707\pi$
$828$	0	0
$829$	38181.5	1.59964	0.799818	−	0.600243i	$-0.204930\pi$
$829$	38181.5	1.59964	0.799818	+	0.600243i	$0.204930\pi$
$830$	0	0
$831$	−5438.40	−0.227023
$832$	0	0
$833$	−27829.4	−1.15754
$834$	0	0
$835$	492.615	0.0204163
$836$	0	0
$837$	2086.64	0.0861708
$838$	0	0
$839$	−43895.2	−1.80623	−0.903117	−	0.429395i	$-0.858727\pi$
$839$	−43895.2	−1.80623	−0.903117	+	0.429395i	$0.858727\pi$
$840$	0	0
$841$	−11800.1	−0.483829
$842$	0	0
$843$	−12219.2	−0.499233
$844$	0	0
$845$	763.320	0.0310757
$846$	0	0
$847$	23532.0	0.954627
$848$	0	0
$849$	−19038.9	−0.769626
$850$	0	0
$851$	−10055.1	−0.405034
$852$	0	0
$853$	−19955.2	−0.800998	−0.400499	−	0.916297i	$-0.631163\pi$
$853$	−19955.2	−0.800998	−0.400499	+	0.916297i	$0.631163\pi$
$854$	0	0
$855$	−1279.80	−0.0511910
$856$	0	0
$857$	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	$-0.139147\pi$
$859$	45617.4	1.81193	0.905964	+	0.423354i	$0.139147\pi$
$860$	0	0
$861$	10141.0	0.401399
$862$	0	0
$863$	2010.93	0.0793195	0.0396597	−	0.999213i	$-0.487373\pi$
$863$	2010.93	0.0793195	0.0396597	+	0.999213i	$0.487373\pi$
$864$	0	0
$865$	−8532.78	−0.335403
$866$	0	0
$867$	13468.6	0.527586
$868$	0	0
$869$	36478.2	1.42398
$870$	0	0
$871$	−11337.1	−0.441035
$872$	0	0
$873$	−9802.48	−0.380027
$874$	0	0
$875$	−7760.41	−0.299828
$876$	0	0
$877$	−36767.3	−1.41567	−0.707836	−	0.706376i	$-0.750329\pi$
$877$	−36767.3	−1.41567	−0.707836	+	0.706376i	$0.750329\pi$
$878$	0	0
$879$	−25172.9	−0.965940
$880$	0	0
$881$	35401.2	1.35380	0.676899	−	0.736076i	$-0.263323\pi$
$881$	35401.2	1.35380	0.676899	+	0.736076i	$0.263323\pi$
$882$	0	0
$883$	11928.0	0.454596	0.227298	−	0.973825i	$-0.427011\pi$
$883$	11928.0	0.454596	0.227298	+	0.973825i	$0.427011\pi$
$884$	0	0
$885$	−2976.49	−0.113055
$886$	0	0
$887$	32939.3	1.24689	0.623447	−	0.781866i	$-0.285732\pi$
$887$	32939.3	1.24689	0.623447	+	0.781866i	$0.285732\pi$
$888$	0	0
$889$	9340.40	0.352381
$890$	0	0
$891$	5418.89	0.203748
$892$	0	0
$893$	−1330.70	−0.0498660
$894$	0	0
$895$	−15482.6	−0.578241
$896$	0	0
$897$	7160.38	0.266531
$898$	0	0
$899$	8671.18	0.321691
$900$	0	0
$901$	−51489.2	−1.90383
$902$	0	0
$903$	2546.58	0.0938483
$904$	0	0
$905$	941.289	0.0345740
$906$	0	0
$907$	−1500.89	−0.0549461	−0.0274731	−	0.999623i	$-0.508746\pi$
$907$	−1500.89	−0.0549461	−0.0274731	+	0.999623i	$0.508746\pi$
$908$	0	0
$909$	−695.102	−0.0253631
$910$	0	0
$911$	−19728.2	−0.717481	−0.358740	−	0.933437i	$-0.616794\pi$
$911$	−19728.2	−0.717481	−0.358740	+	0.933437i	$0.616794\pi$
$912$	0	0
$913$	30408.3	1.10226
$914$	0	0
$915$	11149.9	0.402845
$916$	0	0
$917$	9539.72	0.343543
$918$	0	0
$919$	−19992.0	−0.717602	−0.358801	−	0.933414i	$-0.616814\pi$
$919$	−19992.0	−0.717602	−0.358801	+	0.933414i	$0.616814\pi$
$920$	0	0
$921$	−12015.8	−0.429897
$922$	0	0
$923$	−1303.89	−0.0464984
$924$	0	0
$925$	−5728.54	−0.203625
$926$	0	0
$927$	−12162.3	−0.430918
$928$	0	0
$929$	−1923.62	−0.0679354	−0.0339677	−	0.999423i	$-0.510814\pi$
$929$	−1923.62	−0.0679354	−0.0339677	+	0.999423i	$0.510814\pi$
$930$	0	0
$931$	9035.71	0.318081
$932$	0	0
$933$	−17509.6	−0.614403
$934$	0	0
$935$	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	$0.557200\pi$
$938$	0	0
$939$	5289.58	0.183833
$940$	0	0
$941$	11043.0	0.382563	0.191282	−	0.981535i	$-0.438736\pi$
$941$	11043.0	0.382563	0.191282	+	0.981535i	$0.438736\pi$
$942$	0	0
$943$	−82934.8	−2.86398
$944$	0	0
$945$	912.594	0.0314145
$946$	0	0
$947$	−32105.2	−1.10167	−0.550833	−	0.834615i	$-0.685690\pi$
$947$	−32105.2	−1.10167	−0.550833	+	0.834615i	$0.685690\pi$
$948$	0	0
$949$	2153.24	0.0736535
$950$	0	0
$951$	−18319.0	−0.624643
$952$	0	0
$953$	−9473.37	−0.322007	−0.161003	−	0.986954i	$-0.551473\pi$
$953$	−9473.37	−0.322007	−0.161003	+	0.986954i	$0.551473\pi$
$954$	0	0
$955$	−4325.79	−0.146575
$956$	0	0
$957$	22518.5	0.760628
$958$	0	0
$959$	6547.26	0.220461
$960$	0	0
$961$	−23818.3	−0.799514
$962$	0	0
$963$	−10203.0	−0.341419
$964$	0	0
$965$	−2315.84	−0.0772534
$966$	0	0
$967$	5310.75	0.176610	0.0883052	−	0.996093i	$-0.471855\pi$
$967$	5310.75	0.176610	0.0883052	+	0.996093i	$0.471855\pi$
$968$	0	0
$969$	−9158.49	−0.303626
$970$	0	0
$971$	−24271.2	−0.802164	−0.401082	−	0.916042i	$-0.631366\pi$
$971$	−24271.2	−0.802164	−0.401082	+	0.916042i	$0.631366\pi$
$972$	0	0
$973$	−2322.32	−0.0765163
$974$	0	0
$975$	4079.38	0.133995
$976$	0	0
$977$	−49602.5	−1.62428	−0.812142	−	0.583460i	$-0.801699\pi$
$977$	−49602.5	−1.62428	−0.812142	+	0.583460i	$0.801699\pi$
$978$	0	0
$979$	−15448.2	−0.504317
$980$	0	0
$981$	9158.98	0.298087
$982$	0	0
$983$	−47385.7	−1.53751	−0.768753	−	0.639545i	$-0.779123\pi$
$983$	−47385.7	−1.53751	−0.768753	+	0.639545i	$0.779123\pi$
$984$	0	0
$985$	−17481.2	−0.565478
$986$	0	0
$987$	948.891	0.0306014
$988$	0	0
$989$	−20826.4	−0.669607
$990$	0	0
$991$	8947.33	0.286802	0.143401	−	0.989665i	$-0.454196\pi$
$991$	8947.33	0.286802	0.143401	+	0.989665i	$0.454196\pi$
$992$	0	0
$993$	−22471.1	−0.718127
$994$	0	0
$995$	−10414.7	−0.331828
$996$	0	0
$997$	−14908.6	−0.473582	−0.236791	−	0.971561i	$-0.576096\pi$
$997$	−14908.6	−0.473582	−0.236791	+	0.971561i	$0.576096\pi$
$998$	0	0
$999$	1478.69	0.0468306

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.a.r.1.1		2
3.2	odd	2		1872.4.a.t.1.2		2
4.3	odd	2		39.4.a.b.1.2	✓	2
8.3	odd	2		2496.4.a.bc.1.2		2
8.5	even	2		2496.4.a.s.1.2		2
12.11	even	2		117.4.a.c.1.1		2
20.19	odd	2		975.4.a.j.1.1		2
28.27	even	2		1911.4.a.h.1.2		2
52.31	even	4		507.4.b.f.337.1		4
52.47	even	4		507.4.b.f.337.4		4
52.51	odd	2		507.4.a.f.1.1		2
156.155	even	2		1521.4.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.b.1.2	✓	2	4.3	odd	2
117.4.a.c.1.1		2	12.11	even	2
507.4.a.f.1.1		2	52.51	odd	2
507.4.b.f.337.1		4	52.31	even	4
507.4.b.f.337.4		4	52.47	even	4
624.4.a.r.1.1		2	1.1	even	1	trivial
975.4.a.j.1.1		2	20.19	odd	2
1521.4.a.s.1.2		2	156.155	even	2
1872.4.a.t.1.2		2	3.2	odd	2
1911.4.a.h.1.2		2	28.27	even	2
2496.4.a.s.1.2		2	8.5	even	2
2496.4.a.bc.1.2		2	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +4.51669 q^{5} +7.48331 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +4.51669 q^{5} +7.48331 q^{7} +9.00000 q^{9} +66.8999 q^{11} -13.0000 q^{13} +13.5501 q^{15} +96.9666 q^{17} -31.4833 q^{19} +22.4499 q^{21} -183.600 q^{23} -104.600 q^{25} +27.0000 q^{27} +112.200 q^{29} +77.2831 q^{31} +200.700 q^{33} +33.7998 q^{35} +54.7664 q^{37} -39.0000 q^{39} +451.716 q^{41} +113.434 q^{43} +40.6502 q^{45} +42.2670 q^{47} -287.000 q^{49} +290.900 q^{51} -530.999 q^{53} +302.166 q^{55} -94.4499 q^{57} -219.666 q^{59} +822.865 q^{61} +67.3498 q^{63} -58.7169 q^{65} +872.082 q^{67} -550.799 q^{69} +100.299 q^{71} -165.634 q^{73} -313.799 q^{75} +500.633 q^{77} +545.266 q^{79} +81.0000 q^{81} +454.534 q^{83} +437.968 q^{85} +336.601 q^{87} -230.915 q^{89} -97.2831 q^{91} +231.849 q^{93} -142.200 q^{95} -1089.16 q^{97} +602.099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 24 q^{5} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 + 24 * q^5 + 18 * q^9	\( 2 q + 6 q^{3} + 24 q^{5} + 18 q^{9} + 44 q^{11} - 26 q^{13} + 72 q^{15} + 164 q^{17} - 48 q^{19} - 8 q^{23} + 150 q^{25} + 54 q^{27} + 404 q^{29} - 40 q^{31} + 132 q^{33} - 112 q^{35} - 100 q^{37} - 78 q^{39} + 200 q^{41} + 616 q^{43} + 216 q^{45} + 324 q^{47} - 574 q^{49} + 492 q^{51} - 164 q^{53} - 144 q^{55} - 144 q^{57} - 140 q^{59} + 628 q^{61} - 312 q^{65} + 472 q^{67} - 24 q^{69} - 428 q^{71} - 900 q^{73} + 450 q^{75} + 672 q^{77} + 432 q^{79} + 162 q^{81} + 1388 q^{83} + 1744 q^{85} + 1212 q^{87} + 960 q^{89} - 120 q^{93} - 464 q^{95} - 532 q^{97} + 396 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 24 * q^5 + 18 * q^9 + 44 * q^11 - 26 * q^13 + 72 * q^15 + 164 * q^17 - 48 * q^19 - 8 * q^23 + 150 * q^25 + 54 * q^27 + 404 * q^29 - 40 * q^31 + 132 * q^33 - 112 * q^35 - 100 * q^37 - 78 * q^39 + 200 * q^41 + 616 * q^43 + 216 * q^45 + 324 * q^47 - 574 * q^49 + 492 * q^51 - 164 * q^53 - 144 * q^55 - 144 * q^57 - 140 * q^59 + 628 * q^61 - 312 * q^65 + 472 * q^67 - 24 * q^69 - 428 * q^71 - 900 * q^73 + 450 * q^75 + 672 * q^77 + 432 * q^79 + 162 * q^81 + 1388 * q^83 + 1744 * q^85 + 1212 * q^87 + 960 * q^89 - 120 * q^93 - 464 * q^95 - 532 * q^97 + 396 * q^99

Introduction

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