LMFDB - Embedded newform 2112.4.a.bi.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2112,4,Mod(1,2112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2112.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2112.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,6,0,-10,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$124.612033932$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{97}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 24 $ x^2 - x - 24
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 66)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.42443$ of defining polynomial
Character	$\chi$	$=$	2112.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +4.84886 q^{5} -28.5466 q^{7} +9.00000 q^{9} -11.0000 q^{11} +61.9420 q^{13} +14.5466 q^{15} -20.3023 q^{17} +68.4886 q^{19} -85.6397 q^{21} +150.244 q^{23} -101.489 q^{25} +27.0000 q^{27} +108.907 q^{29} +93.5817 q^{31} -33.0000 q^{33} -138.418 q^{35} -409.163 q^{37} +185.826 q^{39} -302.373 q^{41} +161.209 q^{43} +43.6397 q^{45} +224.547 q^{47} +471.907 q^{49} -60.9069 q^{51} -649.710 q^{53} -53.3374 q^{55} +205.466 q^{57} +8.72056 q^{59} -593.942 q^{61} -256.919 q^{63} +300.348 q^{65} +361.582 q^{67} +450.733 q^{69} -618.733 q^{71} -260.327 q^{73} -304.466 q^{75} +314.012 q^{77} +1088.92 q^{79} +81.0000 q^{81} +336.141 q^{83} -98.4429 q^{85} +326.721 q^{87} +1175.58 q^{89} -1768.23 q^{91} +280.745 q^{93} +332.091 q^{95} +857.420 q^{97} -99.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 10 q^{5} + 2 q^{7} + 18 q^{9} - 22 q^{11} - 14 q^{13} - 30 q^{15} - 80 q^{17} - 60 q^{19} + 6 q^{21} + 202 q^{23} - 6 q^{25} + 54 q^{27} + 336 q^{29} - 128 q^{31} - 66 q^{33} - 592 q^{35}+ \cdots - 198 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 10 * q^5 + 2 * q^7 + 18 * q^9 - 22 * q^11 - 14 * q^13 - 30 * q^15 - 80 * q^17 - 60 * q^19 + 6 * q^21 + 202 * q^23 - 6 * q^25 + 54 * q^27 + 336 * q^29 - 128 * q^31 - 66 * q^33 - 592 * q^35 - 188 * q^37 - 42 * q^39 - 132 * q^41 + 480 * q^43 - 90 * q^45 + 390 * q^47 + 1062 * q^49 - 240 * q^51 - 610 * q^53 + 110 * q^55 - 180 * q^57 + 372 * q^59 - 1050 * q^61 + 18 * q^63 + 1428 * q^65 + 408 * q^67 + 606 * q^69 - 942 * q^71 + 740 * q^73 - 18 * q^75 - 22 * q^77 + 1646 * q^79 + 162 * q^81 - 352 * q^83 + 788 * q^85 + 1008 * q^87 + 2036 * q^89 - 4088 * q^91 - 384 * q^93 + 2240 * q^95 + 336 * q^97 - 198 * q^99

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
$5$	4.84886	0.433695	0.216848	−	0.976205i	$-0.430423\pi$
$5$	4.84886	0.433695	0.216848	+	0.976205i	$0.430423\pi$
$6$	0	0
$7$	−28.5466	−1.54137	−0.770685	−	0.637216i	$-0.780086\pi$
$7$	−28.5466	−1.54137	−0.770685	+	0.637216i	$0.780086\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	61.9420	1.32151	0.660755	−	0.750602i	$-0.270236\pi$
$13$	61.9420	1.32151	0.660755	+	0.750602i	$0.270236\pi$
$14$	0	0
$15$	14.5466	0.250394
$16$	0	0
$17$	−20.3023	−0.289649	−0.144824	−	0.989457i	$-0.546262\pi$
$17$	−20.3023	−0.289649	−0.144824	+	0.989457i	$0.546262\pi$
$18$	0	0
$19$	68.4886	0.826966	0.413483	−	0.910512i	$-0.364312\pi$
$19$	68.4886	0.826966	0.413483	+	0.910512i	$0.364312\pi$
$20$	0	0
$21$	−85.6397	−0.889910
$22$	0	0
$23$	150.244	1.36209	0.681046	−	0.732241i	$-0.261526\pi$
$23$	150.244	1.36209	0.681046	+	0.732241i	$0.261526\pi$
$24$	0	0
$25$	−101.489	−0.811909
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	108.907	0.697362	0.348681	−	0.937241i	$-0.386630\pi$
$29$	108.907	0.697362	0.348681	+	0.937241i	$0.386630\pi$
$30$	0	0
$31$	93.5817	0.542186	0.271093	−	0.962553i	$-0.412615\pi$
$31$	93.5817	0.542186	0.271093	+	0.962553i	$0.412615\pi$
$32$	0	0
$33$	−33.0000	−0.174078
$34$	0	0
$35$	−138.418	−0.668485
$36$	0	0
$37$	−409.163	−1.81800	−0.909001	−	0.416794i	$-0.863153\pi$
$37$	−409.163	−1.81800	−0.909001	+	0.416794i	$0.863153\pi$
$38$	0	0
$39$	185.826	0.762974
$40$	0	0
$41$	−302.373	−1.15177	−0.575886	−	0.817530i	$-0.695343\pi$
$41$	−302.373	−1.15177	−0.575886	+	0.817530i	$0.695343\pi$
$42$	0	0
$43$	161.209	0.571725	0.285862	−	0.958271i	$-0.407720\pi$
$43$	161.209	0.571725	0.285862	+	0.958271i	$0.407720\pi$
$44$	0	0
$45$	43.6397	0.144565
$46$	0	0
$47$	224.547	0.696883	0.348441	−	0.937331i	$-0.386711\pi$
$47$	224.547	0.696883	0.348441	+	0.937331i	$0.386711\pi$
$48$	0	0
$49$	471.907	1.37582
$50$	0	0
$51$	−60.9069	−0.167229
$52$	0	0
$53$	−649.710	−1.68386	−0.841930	−	0.539587i	$-0.818580\pi$
$53$	−649.710	−1.68386	−0.841930	+	0.539587i	$0.818580\pi$
$54$	0	0
$55$	−53.3374	−0.130764
$56$	0	0
$57$	205.466	0.477449
$58$	0	0
$59$	8.72056	0.0192427	0.00962136	−	0.999954i	$-0.496937\pi$
$59$	8.72056	0.0192427	0.00962136	+	0.999954i	$0.496937\pi$
$60$	0	0
$61$	−593.942	−1.24666	−0.623332	−	0.781957i	$-0.714221\pi$
$61$	−593.942	−1.24666	−0.623332	+	0.781957i	$0.714221\pi$
$62$	0	0
$63$	−256.919	−0.513790
$64$	0	0
$65$	300.348	0.573132
$66$	0	0
$67$	361.582	0.659317	0.329658	−	0.944100i	$-0.393066\pi$
$67$	361.582	0.659317	0.329658	+	0.944100i	$0.393066\pi$
$68$	0	0
$69$	450.733	0.786404
$70$	0	0
$71$	−618.733	−1.03423	−0.517113	−	0.855917i	$-0.672993\pi$
$71$	−618.733	−1.03423	−0.517113	+	0.855917i	$0.672993\pi$
$72$	0	0
$73$	−260.327	−0.417383	−0.208691	−	0.977982i	$-0.566920\pi$
$73$	−260.327	−0.417383	−0.208691	+	0.977982i	$0.566920\pi$
$74$	0	0
$75$	−304.466	−0.468756
$76$	0	0
$77$	314.012	0.464741
$78$	0	0
$79$	1088.92	1.55080	0.775399	−	0.631472i	$-0.217549\pi$
$79$	1088.92	1.55080	0.775399	+	0.631472i	$0.217549\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	336.141	0.444533	0.222266	−	0.974986i	$-0.428655\pi$
$83$	336.141	0.444533	0.222266	+	0.974986i	$0.428655\pi$
$84$	0	0
$85$	−98.4429	−0.125619
$86$	0	0
$87$	326.721	0.402622
$88$	0	0
$89$	1175.58	1.40013	0.700064	−	0.714080i	$-0.253155\pi$
$89$	1175.58	1.40013	0.700064	+	0.714080i	$0.253155\pi$
$90$	0	0
$91$	−1768.23	−2.03693
$92$	0	0
$93$	280.745	0.313031
$94$	0	0
$95$	332.091	0.358651
$96$	0	0
$97$	857.420	0.897503	0.448752	−	0.893657i	$-0.351869\pi$
$97$	857.420	0.897503	0.448752	+	0.893657i	$0.351869\pi$
$98$	0	0
$99$	−99.0000	−0.100504
$100$	0	0
$101$	1773.42	1.74715	0.873574	−	0.486692i	$-0.161797\pi$
$101$	1773.42	1.74715	0.873574	+	0.486692i	$0.161797\pi$
$102$	0	0
$103$	1584.17	1.51546	0.757730	−	0.652568i	$-0.226308\pi$
$103$	1584.17	1.51546	0.757730	+	0.652568i	$0.226308\pi$
$104$	0	0
$105$	−415.255	−0.385950
$106$	0	0
$107$	−934.468	−0.844284	−0.422142	−	0.906530i	$-0.638722\pi$
$107$	−934.468	−0.844284	−0.422142	+	0.906530i	$0.638722\pi$
$108$	0	0
$109$	−130.058	−0.114287	−0.0571436	−	0.998366i	$-0.518199\pi$
$109$	−130.058	−0.114287	−0.0571436	+	0.998366i	$0.518199\pi$
$110$	0	0
$111$	−1227.49	−1.04962
$112$	0	0
$113$	820.468	0.683036	0.341518	−	0.939875i	$-0.389059\pi$
$113$	820.468	0.683036	0.341518	+	0.939875i	$0.389059\pi$
$114$	0	0
$115$	728.513	0.590732
$116$	0	0
$117$	557.478	0.440503
$118$	0	0
$119$	579.561	0.446456
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−907.118	−0.664976
$124$	0	0
$125$	−1098.21	−0.785816
$126$	0	0
$127$	1391.99	0.972593	0.486296	−	0.873794i	$-0.338348\pi$
$127$	1391.99	0.972593	0.486296	+	0.873794i	$0.338348\pi$
$128$	0	0
$129$	483.627	0.330085
$130$	0	0
$131$	1306.33	0.871254	0.435627	−	0.900127i	$-0.356527\pi$
$131$	1306.33	0.871254	0.435627	+	0.900127i	$0.356527\pi$
$132$	0	0
$133$	−1955.11	−1.27466
$134$	0	0
$135$	130.919	0.0834646
$136$	0	0
$137$	1156.82	0.721412	0.360706	−	0.932680i	$-0.382536\pi$
$137$	1156.82	0.721412	0.360706	+	0.932680i	$0.382536\pi$
$138$	0	0
$139$	2687.70	1.64006	0.820028	−	0.572323i	$-0.193958\pi$
$139$	2687.70	1.64006	0.820028	+	0.572323i	$0.193958\pi$
$140$	0	0
$141$	673.640	0.402345
$142$	0	0
$143$	−681.362	−0.398450
$144$	0	0
$145$	528.074	0.302442
$146$	0	0
$147$	1415.72	0.794331
$148$	0	0
$149$	1197.77	0.658557	0.329278	−	0.944233i	$-0.393195\pi$
$149$	1197.77	0.658557	0.329278	+	0.944233i	$0.393195\pi$
$150$	0	0
$151$	575.710	0.310269	0.155134	−	0.987893i	$-0.450419\pi$
$151$	575.710	0.310269	0.155134	+	0.987893i	$0.450419\pi$
$152$	0	0
$153$	−182.721	−0.0965496
$154$	0	0
$155$	453.764	0.235143
$156$	0	0
$157$	2926.77	1.48778	0.743891	−	0.668301i	$-0.232978\pi$
$157$	2926.77	1.48778	0.743891	+	0.668301i	$0.232978\pi$
$158$	0	0
$159$	−1949.13	−0.972177
$160$	0	0
$161$	−4288.96	−2.09949
$162$	0	0
$163$	1331.07	0.639617	0.319809	−	0.947482i	$-0.396381\pi$
$163$	1331.07	0.639617	0.319809	+	0.947482i	$0.396381\pi$
$164$	0	0
$165$	−160.012	−0.0754966
$166$	0	0
$167$	3678.91	1.70468	0.852342	−	0.522984i	$-0.175181\pi$
$167$	3678.91	1.70468	0.852342	+	0.522984i	$0.175181\pi$
$168$	0	0
$169$	1639.81	0.746387
$170$	0	0
$171$	616.397	0.275655
$172$	0	0
$173$	208.253	0.0915213	0.0457607	−	0.998952i	$-0.485429\pi$
$173$	208.253	0.0915213	0.0457607	+	0.998952i	$0.485429\pi$
$174$	0	0
$175$	2897.15	1.25145
$176$	0	0
$177$	26.1617	0.0111098
$178$	0	0
$179$	3160.95	1.31989	0.659946	−	0.751313i	$-0.270579\pi$
$179$	3160.95	1.31989	0.659946	+	0.751313i	$0.270579\pi$
$180$	0	0
$181$	3914.68	1.60760	0.803800	−	0.594899i	$-0.202808\pi$
$181$	3914.68	1.60760	0.803800	+	0.594899i	$0.202808\pi$
$182$	0	0
$183$	−1781.83	−0.719762
$184$	0	0
$185$	−1983.98	−0.788458
$186$	0	0
$187$	223.325	0.0873324
$188$	0	0
$189$	−770.757	−0.296637
$190$	0	0
$191$	−4078.86	−1.54521	−0.772606	−	0.634885i	$-0.781047\pi$
$191$	−4078.86	−1.54521	−0.772606	+	0.634885i	$0.781047\pi$
$192$	0	0
$193$	3647.98	1.36056	0.680278	−	0.732954i	$-0.261859\pi$
$193$	3647.98	1.36056	0.680278	+	0.732954i	$0.261859\pi$
$194$	0	0
$195$	901.044	0.330898
$196$	0	0
$197$	−1650.21	−0.596817	−0.298408	−	0.954438i	$-0.596456\pi$
$197$	−1650.21	−0.596817	−0.298408	+	0.954438i	$0.596456\pi$
$198$	0	0
$199$	−586.583	−0.208954	−0.104477	−	0.994527i	$-0.533317\pi$
$199$	−586.583	−0.208954	−0.104477	+	0.994527i	$0.533317\pi$
$200$	0	0
$201$	1084.75	0.380657
$202$	0	0
$203$	−3108.92	−1.07489
$204$	0	0
$205$	−1466.16	−0.499518
$206$	0	0
$207$	1352.20	0.454030
$208$	0	0
$209$	−753.374	−0.249340
$210$	0	0
$211$	−3849.59	−1.25600	−0.628001	−	0.778212i	$-0.716127\pi$
$211$	−3849.59	−1.25600	−0.628001	+	0.778212i	$0.716127\pi$
$212$	0	0
$213$	−1856.20	−0.597111
$214$	0	0
$215$	781.680	0.247954
$216$	0	0
$217$	−2671.44	−0.835710
$218$	0	0
$219$	−780.981	−0.240976
$220$	0	0
$221$	−1257.56	−0.382773
$222$	0	0
$223$	−1175.63	−0.353033	−0.176516	−	0.984298i	$-0.556483\pi$
$223$	−1175.63	−0.353033	−0.176516	+	0.984298i	$0.556483\pi$
$224$	0	0
$225$	−913.397	−0.270636
$226$	0	0
$227$	2217.22	0.648290	0.324145	−	0.946007i	$-0.394923\pi$
$227$	2217.22	0.648290	0.324145	+	0.946007i	$0.394923\pi$
$228$	0	0
$229$	−1265.98	−0.365319	−0.182659	−	0.983176i	$-0.558471\pi$
$229$	−1265.98	−0.365319	−0.182659	+	0.983176i	$0.558471\pi$
$230$	0	0
$231$	942.037	0.268318
$232$	0	0
$233$	−293.859	−0.0826239	−0.0413120	−	0.999146i	$-0.513154\pi$
$233$	−293.859	−0.0826239	−0.0413120	+	0.999146i	$0.513154\pi$
$234$	0	0
$235$	1088.79	0.302235
$236$	0	0
$237$	3266.76	0.895353
$238$	0	0
$239$	−6989.10	−1.89158	−0.945789	−	0.324781i	$-0.894710\pi$
$239$	−6989.10	−1.89158	−0.945789	+	0.324781i	$0.894710\pi$
$240$	0	0
$241$	−2448.86	−0.654545	−0.327272	−	0.944930i	$-0.606129\pi$
$241$	−2448.86	−0.654545	−0.327272	+	0.944930i	$0.606129\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	2288.21	0.596687
$246$	0	0
$247$	4242.32	1.09284
$248$	0	0
$249$	1008.42	0.256651
$250$	0	0
$251$	1106.44	0.278239	0.139120	−	0.990276i	$-0.455573\pi$
$251$	1106.44	0.278239	0.139120	+	0.990276i	$0.455573\pi$
$252$	0	0
$253$	−1652.69	−0.410686
$254$	0	0
$255$	−295.329	−0.0725263
$256$	0	0
$257$	−5718.01	−1.38786	−0.693930	−	0.720043i	$-0.744122\pi$
$257$	−5718.01	−1.38786	−0.693930	+	0.720043i	$0.744122\pi$
$258$	0	0
$259$	11680.2	2.80221
$260$	0	0
$261$	980.162	0.232454
$262$	0	0
$263$	−261.764	−0.0613730	−0.0306865	−	0.999529i	$-0.509769\pi$
$263$	−261.764	−0.0613730	−0.0306865	+	0.999529i	$0.509769\pi$
$264$	0	0
$265$	−3150.35	−0.730281
$266$	0	0
$267$	3526.75	0.808364
$268$	0	0
$269$	−7286.28	−1.65150	−0.825748	−	0.564040i	$-0.809247\pi$
$269$	−7286.28	−1.65150	−0.825748	+	0.564040i	$0.809247\pi$
$270$	0	0
$271$	6886.53	1.54364	0.771821	−	0.635840i	$-0.219346\pi$
$271$	6886.53	1.54364	0.771821	+	0.635840i	$0.219346\pi$
$272$	0	0
$273$	−5304.70	−1.17602
$274$	0	0
$275$	1116.37	0.244800
$276$	0	0
$277$	−3891.57	−0.844121	−0.422061	−	0.906568i	$-0.638693\pi$
$277$	−3891.57	−0.844121	−0.422061	+	0.906568i	$0.638693\pi$
$278$	0	0
$279$	842.236	0.180729
$280$	0	0
$281$	6848.17	1.45383	0.726917	−	0.686725i	$-0.240952\pi$
$281$	6848.17	1.45383	0.726917	+	0.686725i	$0.240952\pi$
$282$	0	0
$283$	−4044.11	−0.849460	−0.424730	−	0.905320i	$-0.639631\pi$
$283$	−4044.11	−0.849460	−0.424730	+	0.905320i	$0.639631\pi$
$284$	0	0
$285$	996.274	0.207067
$286$	0	0
$287$	8631.70	1.77531
$288$	0	0
$289$	−4500.82	−0.916104
$290$	0	0
$291$	2572.26	0.518174
$292$	0	0
$293$	5940.00	1.18436	0.592181	−	0.805805i	$-0.298267\pi$
$293$	5940.00	1.18436	0.592181	+	0.805805i	$0.298267\pi$
$294$	0	0
$295$	42.2848	0.00834547
$296$	0	0
$297$	−297.000	−0.0580259
$298$	0	0
$299$	9306.43	1.80002
$300$	0	0
$301$	−4601.97	−0.881239
$302$	0	0
$303$	5320.26	1.00872
$304$	0	0
$305$	−2879.94	−0.540672
$306$	0	0
$307$	−8652.31	−1.60851	−0.804256	−	0.594283i	$-0.797436\pi$
$307$	−8652.31	−1.60851	−0.804256	+	0.594283i	$0.797436\pi$
$308$	0	0
$309$	4752.50	0.874952
$310$	0	0
$311$	−4922.55	−0.897531	−0.448766	−	0.893649i	$-0.648136\pi$
$311$	−4922.55	−0.897531	−0.448766	+	0.893649i	$0.648136\pi$
$312$	0	0
$313$	2148.33	0.387959	0.193979	−	0.981006i	$-0.437861\pi$
$313$	2148.33	0.387959	0.193979	+	0.981006i	$0.437861\pi$
$314$	0	0
$315$	−1245.76	−0.222828
$316$	0	0
$317$	10005.5	1.77276	0.886382	−	0.462954i	$-0.153210\pi$
$317$	10005.5	1.77276	0.886382	+	0.462954i	$0.153210\pi$
$318$	0	0
$319$	−1197.98	−0.210263
$320$	0	0
$321$	−2803.40	−0.487448
$322$	0	0
$323$	−1390.47	−0.239530
$324$	0	0
$325$	−6286.41	−1.07294
$326$	0	0
$327$	−390.174	−0.0659837
$328$	0	0
$329$	−6410.04	−1.07415
$330$	0	0
$331$	2913.27	0.483769	0.241885	−	0.970305i	$-0.422234\pi$
$331$	2913.27	0.483769	0.241885	+	0.970305i	$0.422234\pi$
$332$	0	0
$333$	−3682.47	−0.606001
$334$	0	0
$335$	1753.26	0.285942
$336$	0	0
$337$	6243.27	1.00918	0.504589	−	0.863360i	$-0.331644\pi$
$337$	6243.27	1.00918	0.504589	+	0.863360i	$0.331644\pi$
$338$	0	0
$339$	2461.40	0.394351
$340$	0	0
$341$	−1029.40	−0.163475
$342$	0	0
$343$	−3679.85	−0.579280
$344$	0	0
$345$	2185.54	0.341059
$346$	0	0
$347$	2103.91	0.325486	0.162743	−	0.986668i	$-0.447966\pi$
$347$	2103.91	0.325486	0.162743	+	0.986668i	$0.447966\pi$
$348$	0	0
$349$	−2292.99	−0.351693	−0.175847	−	0.984418i	$-0.556266\pi$
$349$	−2292.99	−0.351693	−0.175847	+	0.984418i	$0.556266\pi$
$350$	0	0
$351$	1672.43	0.254325
$352$	0	0
$353$	−3624.83	−0.546545	−0.273273	−	0.961937i	$-0.588106\pi$
$353$	−3624.83	−0.546545	−0.273273	+	0.961937i	$0.588106\pi$
$354$	0	0
$355$	−3000.15	−0.448539
$356$	0	0
$357$	1738.68	0.257761
$358$	0	0
$359$	11451.6	1.68354	0.841770	−	0.539836i	$-0.181514\pi$
$359$	11451.6	1.68354	0.841770	+	0.539836i	$0.181514\pi$
$360$	0	0
$361$	−2168.31	−0.316127
$362$	0	0
$363$	363.000	0.0524864
$364$	0	0
$365$	−1262.29	−0.181017
$366$	0	0
$367$	4891.78	0.695773	0.347886	−	0.937537i	$-0.386900\pi$
$367$	4891.78	0.695773	0.347886	+	0.937537i	$0.386900\pi$
$368$	0	0
$369$	−2721.35	−0.383924
$370$	0	0
$371$	18547.0	2.59545
$372$	0	0
$373$	−8933.47	−1.24010	−0.620050	−	0.784562i	$-0.712888\pi$
$373$	−8933.47	−1.24010	−0.620050	+	0.784562i	$0.712888\pi$
$374$	0	0
$375$	−3294.63	−0.453691
$376$	0	0
$377$	6745.91	0.921570
$378$	0	0
$379$	−8587.65	−1.16390	−0.581950	−	0.813225i	$-0.697710\pi$
$379$	−8587.65	−1.16390	−0.581950	+	0.813225i	$0.697710\pi$
$380$	0	0
$381$	4175.97	0.561527
$382$	0	0
$383$	7708.99	1.02849	0.514244	−	0.857644i	$-0.328073\pi$
$383$	7708.99	1.02849	0.514244	+	0.857644i	$0.328073\pi$
$384$	0	0
$385$	1522.60	0.201556
$386$	0	0
$387$	1450.88	0.190575
$388$	0	0
$389$	−5883.52	−0.766854	−0.383427	−	0.923571i	$-0.625256\pi$
$389$	−5883.52	−0.766854	−0.383427	+	0.923571i	$0.625256\pi$
$390$	0	0
$391$	−3050.30	−0.394528
$392$	0	0
$393$	3918.98	0.503019
$394$	0	0
$395$	5280.01	0.672573
$396$	0	0
$397$	4941.81	0.624741	0.312370	−	0.949960i	$-0.398877\pi$
$397$	4941.81	0.624741	0.312370	+	0.949960i	$0.398877\pi$
$398$	0	0
$399$	−5865.34	−0.735926
$400$	0	0
$401$	8589.15	1.06963	0.534815	−	0.844969i	$-0.320381\pi$
$401$	8589.15	1.06963	0.534815	+	0.844969i	$0.320381\pi$
$402$	0	0
$403$	5796.64	0.716504
$404$	0	0
$405$	392.757	0.0481883
$406$	0	0
$407$	4500.80	0.548148
$408$	0	0
$409$	12631.2	1.52708	0.763538	−	0.645763i	$-0.223460\pi$
$409$	12631.2	1.52708	0.763538	+	0.645763i	$0.223460\pi$
$410$	0	0
$411$	3470.45	0.416507
$412$	0	0
$413$	−248.942	−0.0296601
$414$	0	0
$415$	1629.90	0.192792
$416$	0	0
$417$	8063.10	0.946887
$418$	0	0
$419$	−4800.62	−0.559727	−0.279864	−	0.960040i	$-0.590289\pi$
$419$	−4800.62	−0.559727	−0.279864	+	0.960040i	$0.590289\pi$
$420$	0	0
$421$	−6814.10	−0.788833	−0.394417	−	0.918932i	$-0.629053\pi$
$421$	−6814.10	−0.788833	−0.394417	+	0.918932i	$0.629053\pi$
$422$	0	0
$423$	2020.92	0.232294
$424$	0	0
$425$	2060.45	0.235168
$426$	0	0
$427$	16955.0	1.92157
$428$	0	0
$429$	−2044.09	−0.230045
$430$	0	0
$431$	10307.3	1.15194	0.575970	−	0.817471i	$-0.304624\pi$
$431$	10307.3	1.15194	0.575970	+	0.817471i	$0.304624\pi$
$432$	0	0
$433$	−8038.15	−0.892122	−0.446061	−	0.895003i	$-0.647174\pi$
$433$	−8038.15	−0.892122	−0.446061	+	0.895003i	$0.647174\pi$
$434$	0	0
$435$	1584.22	0.174615
$436$	0	0
$437$	10290.0	1.12640
$438$	0	0
$439$	−2905.28	−0.315858	−0.157929	−	0.987450i	$-0.550482\pi$
$439$	−2905.28	−0.315858	−0.157929	+	0.987450i	$0.550482\pi$
$440$	0	0
$441$	4247.16	0.458607
$442$	0	0
$443$	−17880.1	−1.91763	−0.958814	−	0.284034i	$-0.908327\pi$
$443$	−17880.1	−1.91763	−0.958814	+	0.284034i	$0.908327\pi$
$444$	0	0
$445$	5700.23	0.607229
$446$	0	0
$447$	3593.30	0.380218
$448$	0	0
$449$	3532.96	0.371338	0.185669	−	0.982612i	$-0.440555\pi$
$449$	3532.96	0.371338	0.185669	+	0.982612i	$0.440555\pi$
$450$	0	0
$451$	3326.10	0.347272
$452$	0	0
$453$	1727.13	0.179134
$454$	0	0
$455$	−8573.91	−0.883409
$456$	0	0
$457$	−12763.4	−1.30645	−0.653226	−	0.757163i	$-0.726585\pi$
$457$	−12763.4	−1.30645	−0.653226	+	0.757163i	$0.726585\pi$
$458$	0	0
$459$	−548.162	−0.0557429
$460$	0	0
$461$	9090.58	0.918418	0.459209	−	0.888328i	$-0.348133\pi$
$461$	9090.58	0.918418	0.459209	+	0.888328i	$0.348133\pi$
$462$	0	0
$463$	−2538.12	−0.254765	−0.127383	−	0.991854i	$-0.540658\pi$
$463$	−2538.12	−0.254765	−0.127383	+	0.991854i	$0.540658\pi$
$464$	0	0
$465$	1361.29	0.135760
$466$	0	0
$467$	−7540.05	−0.747135	−0.373567	−	0.927603i	$-0.621866\pi$
$467$	−7540.05	−0.747135	−0.373567	+	0.927603i	$0.621866\pi$
$468$	0	0
$469$	−10321.9	−1.01625
$470$	0	0
$471$	8780.31	0.858971
$472$	0	0
$473$	−1773.30	−0.172381
$474$	0	0
$475$	−6950.81	−0.671421
$476$	0	0
$477$	−5847.39	−0.561286
$478$	0	0
$479$	14344.5	1.36830	0.684151	−	0.729340i	$-0.260173\pi$
$479$	14344.5	1.36830	0.684151	+	0.729340i	$0.260173\pi$
$480$	0	0
$481$	−25344.4	−2.40251
$482$	0	0
$483$	−12866.9	−1.21214
$484$	0	0
$485$	4157.51	0.389243
$486$	0	0
$487$	−6686.65	−0.622179	−0.311090	−	0.950381i	$-0.600694\pi$
$487$	−6686.65	−0.622179	−0.311090	+	0.950381i	$0.600694\pi$
$488$	0	0
$489$	3993.22	0.369283
$490$	0	0
$491$	1779.34	0.163545	0.0817724	−	0.996651i	$-0.473942\pi$
$491$	1779.34	0.163545	0.0817724	+	0.996651i	$0.473942\pi$
$492$	0	0
$493$	−2211.06	−0.201990
$494$	0	0
$495$	−480.037	−0.0435880
$496$	0	0
$497$	17662.7	1.59413
$498$	0	0
$499$	−18944.9	−1.69958	−0.849790	−	0.527121i	$-0.823272\pi$
$499$	−18944.9	−1.69958	−0.849790	+	0.527121i	$0.823272\pi$
$500$	0	0
$501$	11036.7	0.984200
$502$	0	0
$503$	4021.33	0.356466	0.178233	−	0.983988i	$-0.442962\pi$
$503$	4021.33	0.356466	0.178233	+	0.983988i	$0.442962\pi$
$504$	0	0
$505$	8599.06	0.757729
$506$	0	0
$507$	4919.44	0.430927
$508$	0	0
$509$	3660.50	0.318760	0.159380	−	0.987217i	$-0.449050\pi$
$509$	3660.50	0.318760	0.159380	+	0.987217i	$0.449050\pi$
$510$	0	0
$511$	7431.44	0.643342
$512$	0	0
$513$	1849.19	0.159150
$514$	0	0
$515$	7681.39	0.657248
$516$	0	0
$517$	−2470.01	−0.210118
$518$	0	0
$519$	624.759	0.0528399
$520$	0	0
$521$	12236.4	1.02896	0.514480	−	0.857502i	$-0.327985\pi$
$521$	12236.4	1.02896	0.514480	+	0.857502i	$0.327985\pi$
$522$	0	0
$523$	1513.19	0.126515	0.0632573	−	0.997997i	$-0.479851\pi$
$523$	1513.19	0.126515	0.0632573	+	0.997997i	$0.479851\pi$
$524$	0	0
$525$	8691.45	0.722526
$526$	0	0
$527$	−1899.92	−0.157044
$528$	0	0
$529$	10406.3	0.855293
$530$	0	0
$531$	78.4850	0.00641424
$532$	0	0
$533$	−18729.6	−1.52208
$534$	0	0
$535$	−4531.10	−0.366162
$536$	0	0
$537$	9482.86	0.762040
$538$	0	0
$539$	−5190.98	−0.414826
$540$	0	0
$541$	1514.63	0.120368	0.0601840	−	0.998187i	$-0.480831\pi$
$541$	1514.63	0.120368	0.0601840	+	0.998187i	$0.480831\pi$
$542$	0	0
$543$	11744.0	0.928149
$544$	0	0
$545$	−630.633	−0.0495658
$546$	0	0
$547$	−1243.43	−0.0971940	−0.0485970	−	0.998818i	$-0.515475\pi$
$547$	−1243.43	−0.0971940	−0.0485970	+	0.998818i	$0.515475\pi$
$548$	0	0
$549$	−5345.48	−0.415555
$550$	0	0
$551$	7458.88	0.576695
$552$	0	0
$553$	−31084.9	−2.39035
$554$	0	0
$555$	−5951.93	−0.455217
$556$	0	0
$557$	−10134.4	−0.770932	−0.385466	−	0.922722i	$-0.625959\pi$
$557$	−10134.4	−0.770932	−0.385466	+	0.922722i	$0.625959\pi$
$558$	0	0
$559$	9985.62	0.755540
$560$	0	0
$561$	669.975	0.0504214
$562$	0	0
$563$	−6653.72	−0.498083	−0.249042	−	0.968493i	$-0.580116\pi$
$563$	−6653.72	−0.498083	−0.249042	+	0.968493i	$0.580116\pi$
$564$	0	0
$565$	3978.33	0.296229
$566$	0	0
$567$	−2312.27	−0.171263
$568$	0	0
$569$	11229.6	0.827360	0.413680	−	0.910422i	$-0.364243\pi$
$569$	11229.6	0.827360	0.413680	+	0.910422i	$0.364243\pi$
$570$	0	0
$571$	−2325.25	−0.170418	−0.0852089	−	0.996363i	$-0.527156\pi$
$571$	−2325.25	−0.170418	−0.0852089	+	0.996363i	$0.527156\pi$
$572$	0	0
$573$	−12236.6	−0.892129
$574$	0	0
$575$	−15248.1	−1.10589
$576$	0	0
$577$	12391.7	0.894058	0.447029	−	0.894519i	$-0.352482\pi$
$577$	12391.7	0.894058	0.447029	+	0.894519i	$0.352482\pi$
$578$	0	0
$579$	10943.9	0.785517
$580$	0	0
$581$	−9595.66	−0.685190
$582$	0	0
$583$	7146.81	0.507703
$584$	0	0
$585$	2703.13	0.191044
$586$	0	0
$587$	−3315.70	−0.233141	−0.116570	−	0.993182i	$-0.537190\pi$
$587$	−3315.70	−0.233141	−0.116570	+	0.993182i	$0.537190\pi$
$588$	0	0
$589$	6409.28	0.448370
$590$	0	0
$591$	−4950.64	−0.344572
$592$	0	0
$593$	11868.9	0.821918	0.410959	−	0.911654i	$-0.365194\pi$
$593$	11868.9	0.821918	0.410959	+	0.911654i	$0.365194\pi$
$594$	0	0
$595$	2810.21	0.193626
$596$	0	0
$597$	−1759.75	−0.120640
$598$	0	0
$599$	−5816.69	−0.396767	−0.198384	−	0.980124i	$-0.563569\pi$
$599$	−5816.69	−0.396767	−0.198384	+	0.980124i	$0.563569\pi$
$600$	0	0
$601$	−10796.6	−0.732782	−0.366391	−	0.930461i	$-0.619407\pi$
$601$	−10796.6	−0.732782	−0.366391	+	0.930461i	$0.619407\pi$
$602$	0	0
$603$	3254.24	0.219772
$604$	0	0
$605$	586.712	0.0394268
$606$	0	0
$607$	15422.0	1.03123	0.515616	−	0.856820i	$-0.327563\pi$
$607$	15422.0	1.03123	0.515616	+	0.856820i	$0.327563\pi$
$608$	0	0
$609$	−9326.75	−0.620590
$610$	0	0
$611$	13908.9	0.920937
$612$	0	0
$613$	−4389.04	−0.289187	−0.144593	−	0.989491i	$-0.546187\pi$
$613$	−4389.04	−0.289187	−0.144593	+	0.989491i	$0.546187\pi$
$614$	0	0
$615$	−4398.49	−0.288397
$616$	0	0
$617$	750.260	0.0489535	0.0244768	−	0.999700i	$-0.492208\pi$
$617$	750.260	0.0489535	0.0244768	+	0.999700i	$0.492208\pi$
$618$	0	0
$619$	13266.1	0.861402	0.430701	−	0.902495i	$-0.358266\pi$
$619$	13266.1	0.861402	0.430701	+	0.902495i	$0.358266\pi$
$620$	0	0
$621$	4056.60	0.262135
$622$	0	0
$623$	−33558.8	−2.15812
$624$	0	0
$625$	7361.00	0.471104
$626$	0	0
$627$	−2260.12	−0.143956
$628$	0	0
$629$	8306.95	0.526582
$630$	0	0
$631$	−6661.88	−0.420293	−0.210147	−	0.977670i	$-0.567394\pi$
$631$	−6661.88	−0.420293	−0.210147	+	0.977670i	$0.567394\pi$
$632$	0	0
$633$	−11548.8	−0.725154
$634$	0	0
$635$	6749.57	0.421809
$636$	0	0
$637$	29230.9	1.81816
$638$	0	0
$639$	−5568.60	−0.344742
$640$	0	0
$641$	−8201.93	−0.505393	−0.252696	−	0.967546i	$-0.581317\pi$
$641$	−8201.93	−0.505393	−0.252696	+	0.967546i	$0.581317\pi$
$642$	0	0
$643$	1625.36	0.0996860	0.0498430	−	0.998757i	$-0.484128\pi$
$643$	1625.36	0.0996860	0.0498430	+	0.998757i	$0.484128\pi$
$644$	0	0
$645$	2345.04	0.143156
$646$	0	0
$647$	−17134.8	−1.04117	−0.520586	−	0.853809i	$-0.674286\pi$
$647$	−17134.8	−1.04117	−0.520586	+	0.853809i	$0.674286\pi$
$648$	0	0
$649$	−95.9262	−0.00580190
$650$	0	0
$651$	−8014.31	−0.482497
$652$	0	0
$653$	28248.5	1.69288	0.846438	−	0.532488i	$-0.178743\pi$
$653$	28248.5	1.69288	0.846438	+	0.532488i	$0.178743\pi$
$654$	0	0
$655$	6334.19	0.377859
$656$	0	0
$657$	−2342.94	−0.139128
$658$	0	0
$659$	22515.5	1.33092	0.665462	−	0.746431i	$-0.268234\pi$
$659$	22515.5	1.33092	0.665462	+	0.746431i	$0.268234\pi$
$660$	0	0
$661$	−32919.0	−1.93706	−0.968532	−	0.248889i	$-0.919935\pi$
$661$	−32919.0	−1.93706	−0.968532	+	0.248889i	$0.919935\pi$
$662$	0	0
$663$	−3772.69	−0.220994
$664$	0	0
$665$	−9480.07	−0.552814
$666$	0	0
$667$	16362.6	0.949871
$668$	0	0
$669$	−3526.90	−0.203824
$670$	0	0
$671$	6533.36	0.375883
$672$	0	0
$673$	−19522.1	−1.11816	−0.559079	−	0.829114i	$-0.688845\pi$
$673$	−19522.1	−1.11816	−0.559079	+	0.829114i	$0.688845\pi$
$674$	0	0
$675$	−2740.19	−0.156252
$676$	0	0
$677$	1966.68	0.111648	0.0558239	−	0.998441i	$-0.482221\pi$
$677$	1966.68	0.111648	0.0558239	+	0.998441i	$0.482221\pi$
$678$	0	0
$679$	−24476.4	−1.38338
$680$	0	0
$681$	6651.65	0.374290
$682$	0	0
$683$	473.958	0.0265527	0.0132764	−	0.999912i	$-0.495774\pi$
$683$	473.958	0.0265527	0.0132764	+	0.999912i	$0.495774\pi$
$684$	0	0
$685$	5609.23	0.312873
$686$	0	0
$687$	−3797.93	−0.210917
$688$	0	0
$689$	−40244.3	−2.22524
$690$	0	0
$691$	−11717.4	−0.645081	−0.322541	−	0.946556i	$-0.604537\pi$
$691$	−11717.4	−0.645081	−0.322541	+	0.946556i	$0.604537\pi$
$692$	0	0
$693$	2826.11	0.154914
$694$	0	0
$695$	13032.3	0.711284
$696$	0	0
$697$	6138.85	0.333609
$698$	0	0
$699$	−881.578	−0.0477029
$700$	0	0
$701$	1044.41	0.0562723	0.0281361	−	0.999604i	$-0.491043\pi$
$701$	1044.41	0.0562723	0.0281361	+	0.999604i	$0.491043\pi$
$702$	0	0
$703$	−28023.0	−1.50343
$704$	0	0
$705$	3266.38	0.174495
$706$	0	0
$707$	−50625.1	−2.69300
$708$	0	0
$709$	−9454.73	−0.500818	−0.250409	−	0.968140i	$-0.580565\pi$
$709$	−9454.73	−0.500818	−0.250409	+	0.968140i	$0.580565\pi$
$710$	0	0
$711$	9800.27	0.516932
$712$	0	0
$713$	14060.1	0.738507
$714$	0	0
$715$	−3303.83	−0.172806
$716$	0	0
$717$	−20967.3	−1.09210
$718$	0	0
$719$	24494.8	1.27052	0.635258	−	0.772300i	$-0.280894\pi$
$719$	24494.8	1.27052	0.635258	+	0.772300i	$0.280894\pi$
$720$	0	0
$721$	−45222.5	−2.33589
$722$	0	0
$723$	−7346.59	−0.377901
$724$	0	0
$725$	−11052.8	−0.566194
$726$	0	0
$727$	−9498.52	−0.484568	−0.242284	−	0.970205i	$-0.577896\pi$
$727$	−9498.52	−0.484568	−0.242284	+	0.970205i	$0.577896\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−3272.91	−0.165599
$732$	0	0
$733$	25431.1	1.28147	0.640736	−	0.767761i	$-0.278629\pi$
$733$	25431.1	1.28147	0.640736	+	0.767761i	$0.278629\pi$
$734$	0	0
$735$	6864.63	0.344497
$736$	0	0
$737$	−3977.40	−0.198792
$738$	0	0
$739$	18092.4	0.900596	0.450298	−	0.892878i	$-0.351318\pi$
$739$	18092.4	0.900596	0.450298	+	0.892878i	$0.351318\pi$
$740$	0	0
$741$	12727.0	0.630953
$742$	0	0
$743$	−6191.22	−0.305698	−0.152849	−	0.988250i	$-0.548845\pi$
$743$	−6191.22	−0.305698	−0.152849	+	0.988250i	$0.548845\pi$
$744$	0	0
$745$	5807.81	0.285613
$746$	0	0
$747$	3025.27	0.148178
$748$	0	0
$749$	26675.8	1.30135
$750$	0	0
$751$	−24738.9	−1.20204	−0.601022	−	0.799233i	$-0.705239\pi$
$751$	−24738.9	−1.20204	−0.601022	+	0.799233i	$0.705239\pi$
$752$	0	0
$753$	3319.33	0.160642
$754$	0	0
$755$	2791.54	0.134562
$756$	0	0
$757$	−27356.0	−1.31344	−0.656718	−	0.754136i	$-0.728056\pi$
$757$	−27356.0	−1.31344	−0.656718	+	0.754136i	$0.728056\pi$
$758$	0	0
$759$	−4958.06	−0.237110
$760$	0	0
$761$	25360.9	1.20806	0.604028	−	0.796963i	$-0.293561\pi$
$761$	25360.9	1.20806	0.604028	+	0.796963i	$0.293561\pi$
$762$	0	0
$763$	3712.71	0.176159
$764$	0	0
$765$	−885.986	−0.0418731
$766$	0	0
$767$	540.169	0.0254294
$768$	0	0
$769$	5715.35	0.268012	0.134006	−	0.990981i	$-0.457216\pi$
$769$	5715.35	0.268012	0.134006	+	0.990981i	$0.457216\pi$
$770$	0	0
$771$	−17154.0	−0.801281
$772$	0	0
$773$	23280.3	1.08322	0.541612	−	0.840628i	$-0.317814\pi$
$773$	23280.3	1.08322	0.541612	+	0.840628i	$0.317814\pi$
$774$	0	0
$775$	−9497.48	−0.440206
$776$	0	0
$777$	35040.6	1.61786
$778$	0	0
$779$	−20709.1	−0.952477
$780$	0	0
$781$	6806.06	0.311831
$782$	0	0
$783$	2940.49	0.134207
$784$	0	0
$785$	14191.5	0.645243
$786$	0	0
$787$	11393.8	0.516066	0.258033	−	0.966136i	$-0.416926\pi$
$787$	11393.8	0.516066	0.258033	+	0.966136i	$0.416926\pi$
$788$	0	0
$789$	−785.293	−0.0354337
$790$	0	0
$791$	−23421.5	−1.05281
$792$	0	0
$793$	−36790.0	−1.64748
$794$	0	0
$795$	−9451.05	−0.421628
$796$	0	0
$797$	−22736.7	−1.01051	−0.505255	−	0.862970i	$-0.668601\pi$
$797$	−22736.7	−1.01051	−0.505255	+	0.862970i	$0.668601\pi$
$798$	0	0
$799$	−4558.81	−0.201851
$800$	0	0
$801$	10580.2	0.466709
$802$	0	0
$803$	2863.60	0.125846
$804$	0	0
$805$	−20796.6	−0.910537
$806$	0	0
$807$	−21858.8	−0.953491
$808$	0	0
$809$	9102.96	0.395603	0.197802	−	0.980242i	$-0.436620\pi$
$809$	9102.96	0.395603	0.197802	+	0.980242i	$0.436620\pi$
$810$	0	0
$811$	5216.11	0.225847	0.112924	−	0.993604i	$-0.463978\pi$
$811$	5216.11	0.225847	0.112924	+	0.993604i	$0.463978\pi$
$812$	0	0
$813$	20659.6	0.891222
$814$	0	0
$815$	6454.18	0.277399
$816$	0	0
$817$	11041.0	0.472797
$818$	0	0
$819$	−15914.1	−0.678978
$820$	0	0
$821$	30452.7	1.29453	0.647264	−	0.762266i	$-0.275913\pi$
$821$	30452.7	1.29453	0.647264	+	0.762266i	$0.275913\pi$
$822$	0	0
$823$	−6518.45	−0.276086	−0.138043	−	0.990426i	$-0.544081\pi$
$823$	−6518.45	−0.276086	−0.138043	+	0.990426i	$0.544081\pi$
$824$	0	0
$825$	3349.12	0.141335
$826$	0	0
$827$	−10793.9	−0.453857	−0.226929	−	0.973911i	$-0.572868\pi$
$827$	−10793.9	−0.453857	−0.226929	+	0.973911i	$0.572868\pi$
$828$	0	0
$829$	−27828.9	−1.16591	−0.582953	−	0.812506i	$-0.698103\pi$
$829$	−27828.9	−1.16591	−0.582953	+	0.812506i	$0.698103\pi$
$830$	0	0
$831$	−11674.7	−0.487354
$832$	0	0
$833$	−9580.79	−0.398505
$834$	0	0
$835$	17838.5	0.739313
$836$	0	0
$837$	2526.71	0.104344
$838$	0	0
$839$	−25915.0	−1.06637	−0.533185	−	0.845999i	$-0.679005\pi$
$839$	−25915.0	−1.06637	−0.533185	+	0.845999i	$0.679005\pi$
$840$	0	0
$841$	−12528.3	−0.513686
$842$	0	0
$843$	20544.5	0.839372
$844$	0	0
$845$	7951.21	0.323704
$846$	0	0
$847$	−3454.14	−0.140125
$848$	0	0
$849$	−12132.3	−0.490436
$850$	0	0
$851$	−61474.5	−2.47628
$852$	0	0
$853$	29533.4	1.18547	0.592735	−	0.805398i	$-0.298048\pi$
$853$	29533.4	1.18547	0.592735	+	0.805398i	$0.298048\pi$
$854$	0	0
$855$	2988.82	0.119550
$856$	0	0
$857$	13227.6	0.527244	0.263622	−	0.964626i	$-0.415083\pi$
$857$	13227.6	0.527244	0.263622	+	0.964626i	$0.415083\pi$
$858$	0	0
$859$	−2633.40	−0.104599	−0.0522994	−	0.998631i	$-0.516655\pi$
$859$	−2633.40	−0.104599	−0.0522994	+	0.998631i	$0.516655\pi$
$860$	0	0
$861$	25895.1	1.02497
$862$	0	0
$863$	18661.1	0.736075	0.368037	−	0.929811i	$-0.380030\pi$
$863$	18661.1	0.736075	0.368037	+	0.929811i	$0.380030\pi$
$864$	0	0
$865$	1009.79	0.0396923
$866$	0	0
$867$	−13502.5	−0.528913
$868$	0	0
$869$	−11978.1	−0.467583
$870$	0	0
$871$	22397.1	0.871294
$872$	0	0
$873$	7716.78	0.299168
$874$	0	0
$875$	31350.2	1.21123
$876$	0	0
$877$	10303.7	0.396728	0.198364	−	0.980128i	$-0.436437\pi$
$877$	10303.7	0.396728	0.198364	+	0.980128i	$0.436437\pi$
$878$	0	0
$879$	17820.0	0.683792
$880$	0	0
$881$	−4209.76	−0.160988	−0.0804940	−	0.996755i	$-0.525650\pi$
$881$	−4209.76	−0.160988	−0.0804940	+	0.996755i	$0.525650\pi$
$882$	0	0
$883$	28123.1	1.07182	0.535910	−	0.844275i	$-0.319969\pi$
$883$	28123.1	1.07182	0.535910	+	0.844275i	$0.319969\pi$
$884$	0	0
$885$	126.854	0.00481826
$886$	0	0
$887$	7071.15	0.267673	0.133837	−	0.991003i	$-0.457270\pi$
$887$	7071.15	0.267673	0.133837	+	0.991003i	$0.457270\pi$
$888$	0	0
$889$	−39736.6	−1.49912
$890$	0	0
$891$	−891.000	−0.0335013
$892$	0	0
$893$	15378.9	0.576298
$894$	0	0
$895$	15327.0	0.572431
$896$	0	0
$897$	27919.3	1.03924
$898$	0	0
$899$	10191.7	0.378100
$900$	0	0
$901$	13190.6	0.487728
$902$	0	0
$903$	−13805.9	−0.508784
$904$	0	0
$905$	18981.7	0.697208
$906$	0	0
$907$	−38810.5	−1.42082	−0.710409	−	0.703790i	$-0.751490\pi$
$907$	−38810.5	−1.42082	−0.710409	+	0.703790i	$0.751490\pi$
$908$	0	0
$909$	15960.8	0.582382
$910$	0	0
$911$	−24183.5	−0.879512	−0.439756	−	0.898117i	$-0.644935\pi$
$911$	−24183.5	−0.879512	−0.439756	+	0.898117i	$0.644935\pi$
$912$	0	0
$913$	−3697.55	−0.134032
$914$	0	0
$915$	−8639.82	−0.312157
$916$	0	0
$917$	−37291.2	−1.34293
$918$	0	0
$919$	13999.6	0.502508	0.251254	−	0.967921i	$-0.419157\pi$
$919$	13999.6	0.502508	0.251254	+	0.967921i	$0.419157\pi$
$920$	0	0
$921$	−25956.9	−0.928675
$922$	0	0
$923$	−38325.6	−1.36674
$924$	0	0
$925$	41525.4	1.47605
$926$	0	0
$927$	14257.5	0.505154
$928$	0	0
$929$	−14390.9	−0.508234	−0.254117	−	0.967173i	$-0.581785\pi$
$929$	−14390.9	−0.508234	−0.254117	+	0.967173i	$0.581785\pi$
$930$	0	0
$931$	32320.2	1.13776
$932$	0	0
$933$	−14767.7	−0.518190
$934$	0	0
$935$	1082.87	0.0378756
$936$	0	0
$937$	43732.8	1.52475	0.762374	−	0.647137i	$-0.224034\pi$
$937$	43732.8	1.52475	0.762374	+	0.647137i	$0.224034\pi$
$938$	0	0
$939$	6445.00	0.223988
$940$	0	0
$941$	−28547.0	−0.988953	−0.494477	−	0.869191i	$-0.664640\pi$
$941$	−28547.0	−0.988953	−0.494477	+	0.869191i	$0.664640\pi$
$942$	0	0
$943$	−45429.8	−1.56882
$944$	0	0
$945$	−3737.29	−0.128650
$946$	0	0
$947$	11846.7	0.406512	0.203256	−	0.979126i	$-0.434848\pi$
$947$	11846.7	0.406512	0.203256	+	0.979126i	$0.434848\pi$
$948$	0	0
$949$	−16125.2	−0.551576
$950$	0	0
$951$	30016.6	1.02351
$952$	0	0
$953$	−278.267	−0.00945852	−0.00472926	−	0.999989i	$-0.501505\pi$
$953$	−278.267	−0.00945852	−0.00472926	+	0.999989i	$0.501505\pi$
$954$	0	0
$955$	−19777.8	−0.670151
$956$	0	0
$957$	−3593.93	−0.121395
$958$	0	0
$959$	−33023.1	−1.11196
$960$	0	0
$961$	−21033.5	−0.706034
$962$	0	0
$963$	−8410.21	−0.281428
$964$	0	0
$965$	17688.5	0.590066
$966$	0	0
$967$	22547.6	0.749826	0.374913	−	0.927060i	$-0.377673\pi$
$967$	22547.6	0.749826	0.374913	+	0.927060i	$0.377673\pi$
$968$	0	0
$969$	−4171.42	−0.138293
$970$	0	0
$971$	45119.0	1.49118	0.745591	−	0.666404i	$-0.232168\pi$
$971$	45119.0	1.49118	0.745591	+	0.666404i	$0.232168\pi$
$972$	0	0
$973$	−76724.7	−2.52793
$974$	0	0
$975$	−18859.2	−0.619465
$976$	0	0
$977$	24727.3	0.809721	0.404861	−	0.914378i	$-0.367320\pi$
$977$	24727.3	0.809721	0.404861	+	0.914378i	$0.367320\pi$
$978$	0	0
$979$	−12931.4	−0.422155
$980$	0	0
$981$	−1170.52	−0.0380957
$982$	0	0
$983$	28417.4	0.922050	0.461025	−	0.887387i	$-0.347482\pi$
$983$	28417.4	0.922050	0.461025	+	0.887387i	$0.347482\pi$
$984$	0	0
$985$	−8001.66	−0.258836
$986$	0	0
$987$	−19230.1	−0.620163
$988$	0	0
$989$	24220.8	0.778741
$990$	0	0
$991$	−27157.1	−0.870507	−0.435254	−	0.900308i	$-0.643341\pi$
$991$	−27157.1	−0.870507	−0.435254	+	0.900308i	$0.643341\pi$
$992$	0	0
$993$	8739.80	0.279304
$994$	0	0
$995$	−2844.26	−0.0906222
$996$	0	0
$997$	−23061.5	−0.732562	−0.366281	−	0.930504i	$-0.619369\pi$
$997$	−23061.5	−0.732562	−0.366281	+	0.930504i	$0.619369\pi$
$998$	0	0
$999$	−11047.4	−0.349875

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.4.a.bi.1.2		2
4.3	odd	2		2112.4.a.bb.1.2		2
8.3	odd	2		66.4.a.c.1.1	✓	2
8.5	even	2		528.4.a.n.1.1		2
24.5	odd	2		1584.4.a.ba.1.2		2
24.11	even	2		198.4.a.h.1.2		2
40.3	even	4		1650.4.c.u.199.1		4
40.19	odd	2		1650.4.a.s.1.1		2
40.27	even	4		1650.4.c.u.199.4		4
88.43	even	2		726.4.a.o.1.1		2
264.131	odd	2		2178.4.a.bf.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
66.4.a.c.1.1	✓	2	8.3	odd	2
198.4.a.h.1.2		2	24.11	even	2
528.4.a.n.1.1		2	8.5	even	2
726.4.a.o.1.1		2	88.43	even	2
1584.4.a.ba.1.2		2	24.5	odd	2
1650.4.a.s.1.1		2	40.19	odd	2
1650.4.c.u.199.1		4	40.3	even	4
1650.4.c.u.199.4		4	40.27	even	4
2112.4.a.bb.1.2		2	4.3	odd	2
2112.4.a.bi.1.2		2	1.1	even	1	trivial
2178.4.a.bf.1.2		2	264.131	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2112.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +4.84886 q^{5} -28.5466 q^{7} +9.00000 q^{9} -11.0000 q^{11} +61.9420 q^{13} +14.5466 q^{15} -20.3023 q^{17} +68.4886 q^{19} -85.6397 q^{21} +150.244 q^{23} -101.489 q^{25} +27.0000 q^{27} +108.907 q^{29} +93.5817 q^{31} -33.0000 q^{33} -138.418 q^{35} -409.163 q^{37} +185.826 q^{39} -302.373 q^{41} +161.209 q^{43} +43.6397 q^{45} +224.547 q^{47} +471.907 q^{49} -60.9069 q^{51} -649.710 q^{53} -53.3374 q^{55} +205.466 q^{57} +8.72056 q^{59} -593.942 q^{61} -256.919 q^{63} +300.348 q^{65} +361.582 q^{67} +450.733 q^{69} -618.733 q^{71} -260.327 q^{73} -304.466 q^{75} +314.012 q^{77} +1088.92 q^{79} +81.0000 q^{81} +336.141 q^{83} -98.4429 q^{85} +326.721 q^{87} +1175.58 q^{89} -1768.23 q^{91} +280.745 q^{93} +332.091 q^{95} +857.420 q^{97} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 10 q^{5} + 2 q^{7} + 18 q^{9} - 22 q^{11} - 14 q^{13} - 30 q^{15} - 80 q^{17} - 60 q^{19} + 6 q^{21} + 202 q^{23} - 6 q^{25} + 54 q^{27} + 336 q^{29} - 128 q^{31} - 66 q^{33} - 592 q^{35}+ \cdots - 198 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 10 * q^5 + 2 * q^7 + 18 * q^9 - 22 * q^11 - 14 * q^13 - 30 * q^15 - 80 * q^17 - 60 * q^19 + 6 * q^21 + 202 * q^23 - 6 * q^25 + 54 * q^27 + 336 * q^29 - 128 * q^31 - 66 * q^33 - 592 * q^35 - 188 * q^37 - 42 * q^39 - 132 * q^41 + 480 * q^43 - 90 * q^45 + 390 * q^47 + 1062 * q^49 - 240 * q^51 - 610 * q^53 + 110 * q^55 - 180 * q^57 + 372 * q^59 - 1050 * q^61 + 18 * q^63 + 1428 * q^65 + 408 * q^67 + 606 * q^69 - 942 * q^71 + 740 * q^73 - 18 * q^75 - 22 * q^77 + 1646 * q^79 + 162 * q^81 - 352 * q^83 + 788 * q^85 + 1008 * q^87 + 2036 * q^89 - 4088 * q^91 - 384 * q^93 + 2240 * q^95 + 336 * q^97 - 198 * q^99

Introduction

L-functions

Modular forms

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Representations

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Database

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