LMFDB - Embedded newform 2496.4.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,4,Mod(1,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, 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("2496.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2496.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:	$147.268767374$
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			1.2
Root			$3.74166$ of defining polynomial
Character	$\chi$	$=$	2496.1

$q$-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	$-0.564742\pi$
$5$	−4.51669	−0.403985	−0.201992	+	0.979387i	$0.564742\pi$
$6$	0	0
$7$	−7.48331	−0.404061	−0.202031	−	0.979379i	$-0.564754\pi$
$7$	−7.48331	−0.404061	−0.202031	+	0.979379i	$0.564754\pi$
$8$	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	$-0.187057\pi$
$23$	183.600	1.66448	0.832242	+	0.554412i	$0.187057\pi$
$24$	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	$-0.616959\pi$
$29$	−112.200	−0.718450	−0.359225	+	0.933251i	$0.616959\pi$
$30$	0	0
$31$	−77.2831	−0.447757	−0.223878	−	0.974617i	$-0.571872\pi$
$31$	−77.2831	−0.447757	−0.223878	+	0.974617i	$0.571872\pi$
$32$	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	$-0.538825\pi$
$37$	−54.7664	−0.243339	−0.121669	+	0.992571i	$0.538825\pi$
$38$	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	$-0.520892\pi$
$47$	−42.2670	−0.131176	−0.0655880	+	0.997847i	$0.520892\pi$
$48$	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	$-0.258446\pi$
$53$	530.999	1.37619	0.688097	+	0.725618i	$0.258446\pi$
$54$	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	$-0.831785\pi$
$61$	−822.865	−1.72717	−0.863583	+	0.504207i	$0.831785\pi$
$62$	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	$-0.526714\pi$
$71$	−100.299	−0.167653	−0.0838263	+	0.996480i	$0.526714\pi$
$72$	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	$-0.626928\pi$
$79$	−545.266	−0.776547	−0.388273	+	0.921544i	$0.626928\pi$
$80$	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	$-0.487887\pi$
$101$	77.2336	0.0760894	0.0380447	+	0.999276i	$0.487887\pi$
$102$	0	0
$103$	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
$106$	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	$-0.647554\pi$
$109$	−1017.66	−0.894262	−0.447131	+	0.894469i	$0.647554\pi$
$110$	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	$-0.643623\pi$
$127$	−1248.16	−0.872099	−0.436050	+	0.899923i	$0.643623\pi$
$128$	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	$-0.500445\pi$
$149$	−5.08064	−0.00279344	−0.00139672	+	0.999999i	$0.500445\pi$
$150$	0	0
$151$	6.54894	0.00352944	0.00176472	−	0.999998i	$-0.499438\pi$
$151$	6.54894	0.00352944	0.00176472	+	0.999998i	$0.499438\pi$
$152$	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	$-0.301497\pi$
$157$	2297.60	1.16795	0.583975	+	0.811772i	$0.301497\pi$
$158$	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	$-0.508044\pi$
$167$	−109.066	−0.0505374	−0.0252687	+	0.999681i	$0.508044\pi$
$168$	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	$-0.363740\pi$
$173$	1889.17	0.830236	0.415118	+	0.909768i	$0.363740\pi$
$174$	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	$-0.513625\pi$
$181$	−208.403	−0.0855826	−0.0427913	+	0.999084i	$0.513625\pi$
$182$	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	$-0.441933\pi$
$191$	957.735	0.362824	0.181412	+	0.983407i	$0.441933\pi$
$192$	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	$-0.253239\pi$
$197$	3870.35	1.39975	0.699876	+	0.714265i	$0.253239\pi$
$198$	0	0
$199$	2305.83	0.821388	0.410694	−	0.911773i	$-0.365286\pi$
$199$	2305.83	0.821388	0.410694	+	0.911773i	$0.365286\pi$
$200$	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	$-0.223332\pi$
$223$	5087.05	1.52760	0.763798	+	0.645455i	$0.223332\pi$
$224$	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	$-0.577873\pi$
$229$	−1678.73	−0.484425	−0.242213	+	0.970223i	$0.577873\pi$
$230$	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	$-0.250342\pi$
$239$	5219.69	1.41269	0.706347	+	0.707866i	$0.250342\pi$
$240$	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	$-0.519229\pi$
$263$	−514.992	−0.120744	−0.0603722	+	0.998176i	$0.519229\pi$
$264$	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	$-0.302398\pi$
$269$	5132.60	1.16335	0.581673	+	0.813423i	$0.302398\pi$
$270$	0	0
$271$	−4300.00	−0.963862	−0.481931	−	0.876209i	$-0.660064\pi$
$271$	−4300.00	−0.963862	−0.481931	+	0.876209i	$0.660064\pi$
$272$	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	$-0.437008\pi$
$277$	1812.80	0.393215	0.196607	+	0.980482i	$0.437008\pi$
$278$	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	$-0.184581\pi$
$293$	8390.97	1.67306	0.836529	+	0.547923i	$0.184581\pi$
$294$	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	$-0.321407\pi$
$311$	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	$0.449107\pi$
$314$	0	0
$315$	304.198	0.0544115
$316$	0	0
$317$	6106.35	1.08191	0.540957	−	0.841050i	$-0.318062\pi$
$317$	6106.35	1.08191	0.540957	+	0.841050i	$0.318062\pi$
$318$	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	$-0.409085\pi$
$349$	3673.96	0.563503	0.281751	+	0.959487i	$0.409085\pi$
$350$	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	$-0.751805\pi$
$359$	−9673.98	−1.42221	−0.711105	+	0.703086i	$0.751805\pi$
$360$	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	$-0.712807\pi$
$367$	−8715.98	−1.23970	−0.619851	+	0.784720i	$0.712807\pi$
$368$	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	$-0.605026\pi$
$373$	−4667.99	−0.647987	−0.323994	+	0.946059i	$0.605026\pi$
$374$	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	$-0.266000\pi$
$383$	10054.2	1.34137	0.670686	+	0.741742i	$0.266000\pi$
$384$	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	$-0.637371\pi$
$389$	−6418.50	−0.836584	−0.418292	+	0.908313i	$0.637371\pi$
$390$	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	$-0.225309\pi$
$397$	12019.9	1.51955	0.759775	+	0.650186i	$0.225309\pi$
$398$	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	$-0.170903\pi$
$421$	14845.5	1.71859	0.859295	+	0.511481i	$0.170903\pi$
$422$	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	$-0.480366\pi$
$431$	1103.11	0.123283	0.0616417	+	0.998098i	$0.480366\pi$
$432$	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	$-0.701901\pi$
$439$	−10901.7	−1.18521	−0.592607	+	0.805492i	$0.701901\pi$
$440$	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	$-0.646466\pi$
$461$	−8790.90	−0.888141	−0.444071	+	0.895992i	$0.646466\pi$
$462$	0	0
$463$	−3861.55	−0.387606	−0.193803	−	0.981040i	$-0.562082\pi$
$463$	−3861.55	−0.387606	−0.193803	+	0.981040i	$0.562082\pi$
$464$	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	$-0.436116\pi$
$479$	4179.82	0.398707	0.199354	+	0.979928i	$0.436116\pi$
$480$	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	$-0.828342\pi$
$487$	−18443.8	−1.71616	−0.858078	+	0.513519i	$0.828342\pi$
$488$	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	$-0.634620\pi$
$503$	−9260.11	−0.820851	−0.410425	+	0.911894i	$0.634620\pi$
$504$	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	$-0.356661\pi$
$509$	9996.40	0.870497	0.435248	+	0.900310i	$0.356661\pi$
$510$	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	$-0.530554\pi$
$541$	−2411.99	−0.191681	−0.0958406	+	0.995397i	$0.530554\pi$
$542$	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	$-0.268363\pi$
$557$	17488.0	1.33032	0.665160	+	0.746701i	$0.268363\pi$
$558$	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	$-0.496028\pi$
$599$	365.860	0.0249560	0.0124780	+	0.999922i	$0.496028\pi$
$600$	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	$-0.475232\pi$
$607$	2324.97	0.155465	0.0777327	+	0.996974i	$0.475232\pi$
$608$	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	$-0.506640\pi$
$613$	−633.133	−0.0417162	−0.0208581	+	0.999782i	$0.506640\pi$
$614$	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	$-0.443858\pi$
$631$	5562.30	0.350922	0.175461	+	0.984486i	$0.443858\pi$
$632$	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	$-0.485590\pi$
$647$	1489.51	0.0905083	0.0452542	+	0.998976i	$0.485590\pi$
$648$	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	$-0.603570\pi$
$653$	−10668.2	−0.639327	−0.319663	+	0.947531i	$0.603570\pi$
$654$	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	$-0.477977\pi$
$661$	2349.69	0.138264	0.0691320	+	0.997608i	$0.477977\pi$
$662$	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	$-0.490315\pi$
$677$	1071.74	0.0608421	0.0304211	+	0.999537i	$0.490315\pi$
$678$	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	$-0.366007\pi$
$701$	15168.3	0.817259	0.408629	+	0.912700i	$0.366007\pi$
$702$	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	$-0.423491\pi$
$709$	8988.17	0.476104	0.238052	+	0.971252i	$0.423491\pi$
$710$	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	$-0.429688\pi$
$719$	8448.18	0.438198	0.219099	+	0.975703i	$0.429688\pi$
$720$	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	$-0.570600\pi$
$727$	−8624.18	−0.439963	−0.219982	+	0.975504i	$0.570600\pi$
$728$	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	$-0.786913\pi$
$733$	−31124.2	−1.56835	−0.784174	+	0.620541i	$0.786913\pi$
$734$	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	$-0.676542\pi$
$743$	−21331.1	−1.05325	−0.526623	+	0.850099i	$0.676542\pi$
$744$	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	$-0.408152\pi$
$751$	11712.9	0.569122	0.284561	+	0.958658i	$0.408152\pi$
$752$	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	$-0.369438\pi$
$757$	16610.9	0.797537	0.398768	+	0.917052i	$0.369438\pi$
$758$	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	$-0.601631\pi$
$773$	−13491.8	−0.627772	−0.313886	+	0.949461i	$0.601631\pi$
$774$	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	$-0.662080\pi$
$797$	−21936.4	−0.974938	−0.487469	+	0.873140i	$0.662080\pi$
$798$	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	$-0.430763\pi$
$821$	10153.2	0.431608	0.215804	+	0.976437i	$0.430763\pi$
$822$	0	0
$823$	3282.93	0.139047	0.0695235	−	0.997580i	$-0.477852\pi$
$823$	3282.93	0.139047	0.0695235	+	0.997580i	$0.477852\pi$
$824$	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	$-0.795070\pi$
$829$	−38181.5	−1.59964	−0.799818	+	0.600243i	$0.795070\pi$
$830$	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	$-0.141273\pi$
$839$	43895.2	1.80623	0.903117	+	0.429395i	$0.141273\pi$
$840$	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	$-0.368837\pi$
$853$	19955.2	0.800998	0.400499	+	0.916297i	$0.368837\pi$
$854$	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	$-0.512627\pi$
$863$	−2010.93	−0.0793195	−0.0396597	+	0.999213i	$0.512627\pi$
$864$	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	$-0.249671\pi$
$877$	36767.3	1.41567	0.707836	+	0.706376i	$0.249671\pi$
$878$	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	$-0.714268\pi$
$887$	−32939.3	−1.24689	−0.623447	+	0.781866i	$0.714268\pi$
$888$	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	$-0.383206\pi$
$911$	19728.2	0.717481	0.358740	+	0.933437i	$0.383206\pi$
$912$	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	$-0.383186\pi$
$919$	19992.0	0.717602	0.358801	+	0.933414i	$0.383186\pi$
$920$	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	$-0.561264\pi$
$941$	−11043.0	−0.382563	−0.191282	+	0.981535i	$0.561264\pi$
$942$	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	$-0.528145\pi$
$967$	−5310.75	−0.176610	−0.0883052	+	0.996093i	$0.528145\pi$
$968$	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	$-0.220877\pi$
$983$	47385.7	1.53751	0.768753	+	0.639545i	$0.220877\pi$
$984$	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	$-0.545804\pi$
$991$	−8947.33	−0.286802	−0.143401	+	0.989665i	$0.545804\pi$
$992$	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	$-0.423904\pi$
$997$	14908.6	0.473582	0.236791	+	0.971561i	$0.423904\pi$
$998$	0	0
$999$	−1478.69	−0.0468306

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.4.a.bc.1.2		2
4.3	odd	2		2496.4.a.s.1.2		2
8.3	odd	2		624.4.a.r.1.1		2
8.5	even	2		39.4.a.b.1.2	✓	2
24.5	odd	2		117.4.a.c.1.1		2
24.11	even	2		1872.4.a.t.1.2		2
40.29	even	2		975.4.a.j.1.1		2
56.13	odd	2		1911.4.a.h.1.2		2
104.5	odd	4		507.4.b.f.337.1		4
104.21	odd	4		507.4.b.f.337.4		4
104.77	even	2		507.4.a.f.1.1		2
312.77	odd	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	8.5	even	2
117.4.a.c.1.1		2	24.5	odd	2
507.4.a.f.1.1		2	104.77	even	2
507.4.b.f.337.1		4	104.5	odd	4
507.4.b.f.337.4		4	104.21	odd	4
624.4.a.r.1.1		2	8.3	odd	2
975.4.a.j.1.1		2	40.29	even	2
1521.4.a.s.1.2		2	312.77	odd	2
1872.4.a.t.1.2		2	24.11	even	2
1911.4.a.h.1.2		2	56.13	odd	2
2496.4.a.s.1.2		2	4.3	odd	2
2496.4.a.bc.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2496.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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