LMFDB - Embedded newform 840.4.a.r.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [840,4,Mod(1,840)] 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("840.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	840.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,9,0,-15,0,21,0,27,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$49.5616044048$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.851417.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 125x - 404 $ x^3 - 125*x - 404
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.60762$ of defining polynomial
Character	$\chi$	$=$	840.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +64.1622 q^{11} +37.3012 q^{13} -15.0000 q^{15} -134.324 q^{17} +84.1622 q^{19} +21.0000 q^{21} +105.722 q^{23} +25.0000 q^{25} +27.0000 q^{27} -73.4633 q^{29} -236.811 q^{31} +192.487 q^{33} -35.0000 q^{35} +67.7220 q^{37} +111.904 q^{39} -390.112 q^{41} +119.676 q^{43} -45.0000 q^{45} +252.324 q^{47} +49.0000 q^{49} -402.973 q^{51} +460.340 q^{53} -320.811 q^{55} +252.487 q^{57} +397.012 q^{59} +520.946 q^{61} +63.0000 q^{63} -186.506 q^{65} +907.000 q^{67} +317.166 q^{69} +508.679 q^{71} -232.321 q^{73} +75.0000 q^{75} +449.135 q^{77} +718.583 q^{79} +81.0000 q^{81} +1246.94 q^{83} +671.622 q^{85} -220.390 q^{87} -485.019 q^{89} +261.108 q^{91} -710.433 q^{93} -420.811 q^{95} +142.892 q^{97} +577.460 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 9 q^{3} - 15 q^{5} + 21 q^{7} + 27 q^{9} + 14 q^{11} + 20 q^{13} - 45 q^{15} - 46 q^{17} + 74 q^{19} + 63 q^{21} + 144 q^{23} + 75 q^{25} + 81 q^{27} + 50 q^{29} + 182 q^{31} + 42 q^{33} - 105 q^{35}+ \cdots + 126 q^{99}+O(q^{100}) $ 3 * q + 9 * q^3 - 15 * q^5 + 21 * q^7 + 27 * q^9 + 14 * q^11 + 20 * q^13 - 45 * q^15 - 46 * q^17 + 74 * q^19 + 63 * q^21 + 144 * q^23 + 75 * q^25 + 81 * q^27 + 50 * q^29 + 182 * q^31 + 42 * q^33 - 105 * q^35 + 30 * q^37 + 60 * q^39 - 186 * q^41 + 716 * q^43 - 135 * q^45 + 400 * q^47 + 147 * q^49 - 138 * q^51 - 236 * q^53 - 70 * q^55 + 222 * q^57 + 272 * q^59 + 1098 * q^61 + 189 * q^63 - 100 * q^65 + 1044 * q^67 + 432 * q^69 + 1542 * q^71 + 996 * q^73 + 225 * q^75 + 98 * q^77 + 1896 * q^79 + 243 * q^81 + 604 * q^83 + 230 * q^85 + 150 * q^87 + 146 * q^89 + 140 * q^91 + 546 * q^93 - 370 * q^95 + 1072 * q^97 + 126 * 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
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	64.1622	1.75869	0.879346	−	0.476182i	$-0.157980\pi$
$11$	64.1622	1.75869	0.879346	+	0.476182i	$0.157980\pi$
$12$	0	0
$13$	37.3012	0.795806	0.397903	−	0.917427i	$-0.369738\pi$
$13$	37.3012	0.795806	0.397903	+	0.917427i	$0.369738\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	−134.324	−1.91638	−0.958189	−	0.286135i	$-0.907629\pi$
$17$	−134.324	−1.91638	−0.958189	+	0.286135i	$0.907629\pi$
$18$	0	0
$19$	84.1622	1.01622	0.508109	−	0.861293i	$-0.330345\pi$
$19$	84.1622	1.01622	0.508109	+	0.861293i	$0.330345\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	105.722	0.958459	0.479230	−	0.877690i	$-0.340916\pi$
$23$	105.722	0.958459	0.479230	+	0.877690i	$0.340916\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−73.4633	−0.470407	−0.235203	−	0.971946i	$-0.575576\pi$
$29$	−73.4633	−0.470407	−0.235203	+	0.971946i	$0.575576\pi$
$30$	0	0
$31$	−236.811	−1.37202	−0.686008	−	0.727594i	$-0.740638\pi$
$31$	−236.811	−1.37202	−0.686008	+	0.727594i	$0.740638\pi$
$32$	0	0
$33$	192.487	1.01538
$34$	0	0
$35$	−35.0000	−0.169031
$36$	0	0
$37$	67.7220	0.300903	0.150452	−	0.988617i	$-0.451927\pi$
$37$	67.7220	0.300903	0.150452	+	0.988617i	$0.451927\pi$
$38$	0	0
$39$	111.904	0.459459
$40$	0	0
$41$	−390.112	−1.48598	−0.742991	−	0.669301i	$-0.766594\pi$
$41$	−390.112	−1.48598	−0.742991	+	0.669301i	$0.766594\pi$
$42$	0	0
$43$	119.676	0.424427	0.212214	−	0.977223i	$-0.431933\pi$
$43$	119.676	0.424427	0.212214	+	0.977223i	$0.431933\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	252.324	0.783091	0.391546	−	0.920159i	$-0.371941\pi$
$47$	252.324	0.783091	0.391546	+	0.920159i	$0.371941\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−402.973	−1.10642
$52$	0	0
$53$	460.340	1.19307	0.596533	−	0.802588i	$-0.296544\pi$
$53$	460.340	1.19307	0.596533	+	0.802588i	$0.296544\pi$
$54$	0	0
$55$	−320.811	−0.786511
$56$	0	0
$57$	252.487	0.586713
$58$	0	0
$59$	397.012	0.876043	0.438021	−	0.898965i	$-0.355679\pi$
$59$	397.012	0.876043	0.438021	+	0.898965i	$0.355679\pi$
$60$	0	0
$61$	520.946	1.09345	0.546724	−	0.837313i	$-0.315875\pi$
$61$	520.946	1.09345	0.546724	+	0.837313i	$0.315875\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	−186.506	−0.355895
$66$	0	0
$67$	907.000	1.65385	0.826923	−	0.562315i	$-0.190089\pi$
$67$	907.000	1.65385	0.826923	+	0.562315i	$0.190089\pi$
$68$	0	0
$69$	317.166	0.553367
$70$	0	0
$71$	508.679	0.850270	0.425135	−	0.905130i	$-0.360227\pi$
$71$	508.679	0.850270	0.425135	+	0.905130i	$0.360227\pi$
$72$	0	0
$73$	−232.321	−0.372480	−0.186240	−	0.982504i	$-0.559630\pi$
$73$	−232.321	−0.372480	−0.186240	+	0.982504i	$0.559630\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	449.135	0.664723
$78$	0	0
$79$	718.583	1.02338	0.511689	−	0.859171i	$-0.329020\pi$
$79$	718.583	1.02338	0.511689	+	0.859171i	$0.329020\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1246.94	1.64903	0.824514	−	0.565842i	$-0.191449\pi$
$83$	1246.94	1.64903	0.824514	+	0.565842i	$0.191449\pi$
$84$	0	0
$85$	671.622	0.857031
$86$	0	0
$87$	−220.390	−0.271590
$88$	0	0
$89$	−485.019	−0.577662	−0.288831	−	0.957380i	$-0.593267\pi$
$89$	−485.019	−0.577662	−0.288831	+	0.957380i	$0.593267\pi$
$90$	0	0
$91$	261.108	0.300787
$92$	0	0
$93$	−710.433	−0.792134
$94$	0	0
$95$	−420.811	−0.454466
$96$	0	0
$97$	142.892	0.149572	0.0747859	−	0.997200i	$-0.476173\pi$
$97$	142.892	0.149572	0.0747859	+	0.997200i	$0.476173\pi$
$98$	0	0
$99$	577.460	0.586231
$100$	0	0
$101$	274.405	0.270340	0.135170	−	0.990822i	$-0.456842\pi$
$101$	274.405	0.270340	0.135170	+	0.990822i	$0.456842\pi$
$102$	0	0
$103$	−1722.86	−1.64814	−0.824069	−	0.566490i	$-0.808301\pi$
$103$	−1722.86	−1.64814	−0.824069	+	0.566490i	$0.808301\pi$
$104$	0	0
$105$	−105.000	−0.0975900
$106$	0	0
$107$	−1977.94	−1.78705	−0.893526	−	0.449012i	$-0.851776\pi$
$107$	−1977.94	−1.78705	−0.893526	+	0.449012i	$0.851776\pi$
$108$	0	0
$109$	489.888	0.430484	0.215242	−	0.976561i	$-0.430946\pi$
$109$	489.888	0.430484	0.215242	+	0.976561i	$0.430946\pi$
$110$	0	0
$111$	203.166	0.173727
$112$	0	0
$113$	−470.348	−0.391563	−0.195781	−	0.980648i	$-0.562724\pi$
$113$	−470.348	−0.391563	−0.195781	+	0.980648i	$0.562724\pi$
$114$	0	0
$115$	−528.610	−0.428636
$116$	0	0
$117$	335.711	0.265269
$118$	0	0
$119$	−940.270	−0.724323
$120$	0	0
$121$	2785.78	2.09300
$122$	0	0
$123$	−1170.34	−0.857932
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1836.70	1.28331	0.641656	−	0.766992i	$-0.278248\pi$
$127$	1836.70	1.28331	0.641656	+	0.766992i	$0.278248\pi$
$128$	0	0
$129$	359.027	0.245043
$130$	0	0
$131$	−2164.22	−1.44343	−0.721714	−	0.692191i	$-0.756646\pi$
$131$	−2164.22	−1.44343	−0.721714	+	0.692191i	$0.756646\pi$
$132$	0	0
$133$	589.135	0.384094
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	248.742	0.155120	0.0775600	−	0.996988i	$-0.475287\pi$
$137$	248.742	0.155120	0.0775600	+	0.996988i	$0.475287\pi$
$138$	0	0
$139$	2622.05	1.60000	0.799999	−	0.600001i	$-0.204833\pi$
$139$	2622.05	1.60000	0.799999	+	0.600001i	$0.204833\pi$
$140$	0	0
$141$	756.973	0.452118
$142$	0	0
$143$	2393.32	1.39958
$144$	0	0
$145$	367.317	0.210372
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	−28.8458	−0.0158600	−0.00792999	−	0.999969i	$-0.502524\pi$
$149$	−28.8458	−0.0158600	−0.00792999	+	0.999969i	$0.502524\pi$
$150$	0	0
$151$	991.355	0.534274	0.267137	−	0.963659i	$-0.413922\pi$
$151$	991.355	0.534274	0.267137	+	0.963659i	$0.413922\pi$
$152$	0	0
$153$	−1208.92	−0.638793
$154$	0	0
$155$	1184.05	0.613584
$156$	0	0
$157$	1727.31	0.878053	0.439026	−	0.898474i	$-0.355324\pi$
$157$	1727.31	0.878053	0.439026	+	0.898474i	$0.355324\pi$
$158$	0	0
$159$	1381.02	0.688818
$160$	0	0
$161$	740.054	0.362263
$162$	0	0
$163$	−531.081	−0.255199	−0.127600	−	0.991826i	$-0.540727\pi$
$163$	−531.081	−0.255199	−0.127600	+	0.991826i	$0.540727\pi$
$164$	0	0
$165$	−962.433	−0.454093
$166$	0	0
$167$	−1793.48	−0.831039	−0.415520	−	0.909584i	$-0.636400\pi$
$167$	−1793.48	−0.831039	−0.415520	+	0.909584i	$0.636400\pi$
$168$	0	0
$169$	−805.622	−0.366692
$170$	0	0
$171$	757.460	0.338739
$172$	0	0
$173$	3632.96	1.59658	0.798291	−	0.602272i	$-0.205738\pi$
$173$	3632.96	1.59658	0.798291	+	0.602272i	$0.205738\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	0	0
$177$	1191.04	0.505784
$178$	0	0
$179$	−2659.65	−1.11057	−0.555283	−	0.831662i	$-0.687390\pi$
$179$	−2659.65	−1.11057	−0.555283	+	0.831662i	$0.687390\pi$
$180$	0	0
$181$	−2504.85	−1.02864	−0.514320	−	0.857599i	$-0.671956\pi$
$181$	−2504.85	−1.02864	−0.514320	+	0.857599i	$0.671956\pi$
$182$	0	0
$183$	1562.84	0.631302
$184$	0	0
$185$	−338.610	−0.134568
$186$	0	0
$187$	−8618.54	−3.37032
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	700.765	0.265474	0.132737	−	0.991151i	$-0.457623\pi$
$191$	700.765	0.265474	0.132737	+	0.991151i	$0.457623\pi$
$192$	0	0
$193$	−77.4910	−0.0289012	−0.0144506	−	0.999896i	$-0.504600\pi$
$193$	−77.4910	−0.0289012	−0.0144506	+	0.999896i	$0.504600\pi$
$194$	0	0
$195$	−559.518	−0.205476
$196$	0	0
$197$	1452.73	0.525396	0.262698	−	0.964878i	$-0.415388\pi$
$197$	1452.73	0.525396	0.262698	+	0.964878i	$0.415388\pi$
$198$	0	0
$199$	−83.6214	−0.0297878	−0.0148939	−	0.999889i	$-0.504741\pi$
$199$	−83.6214	−0.0297878	−0.0148939	+	0.999889i	$0.504741\pi$
$200$	0	0
$201$	2721.00	0.954848
$202$	0	0
$203$	−514.243	−0.177797
$204$	0	0
$205$	1950.56	0.664551
$206$	0	0
$207$	951.498	0.319486
$208$	0	0
$209$	5400.03	1.78721
$210$	0	0
$211$	−2405.10	−0.784711	−0.392355	−	0.919814i	$-0.628340\pi$
$211$	−2405.10	−0.784711	−0.392355	+	0.919814i	$0.628340\pi$
$212$	0	0
$213$	1526.04	0.490903
$214$	0	0
$215$	−598.378	−0.189810
$216$	0	0
$217$	−1657.68	−0.518573
$218$	0	0
$219$	−696.962	−0.215052
$220$	0	0
$221$	−5010.46	−1.52507
$222$	0	0
$223$	4020.13	1.20721	0.603605	−	0.797283i	$-0.293730\pi$
$223$	4020.13	1.20721	0.603605	+	0.797283i	$0.293730\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	4082.33	1.19363	0.596814	−	0.802379i	$-0.296433\pi$
$227$	4082.33	1.19363	0.596814	+	0.802379i	$0.296433\pi$
$228$	0	0
$229$	−722.892	−0.208603	−0.104301	−	0.994546i	$-0.533261\pi$
$229$	−722.892	−0.208603	−0.104301	+	0.994546i	$0.533261\pi$
$230$	0	0
$231$	1347.41	0.383778
$232$	0	0
$233$	−775.992	−0.218184	−0.109092	−	0.994032i	$-0.534794\pi$
$233$	−775.992	−0.218184	−0.109092	+	0.994032i	$0.534794\pi$
$234$	0	0
$235$	−1261.62	−0.350209
$236$	0	0
$237$	2155.75	0.590848
$238$	0	0
$239$	−6340.95	−1.71616	−0.858080	−	0.513517i	$-0.828342\pi$
$239$	−6340.95	−1.71616	−0.858080	+	0.513517i	$0.828342\pi$
$240$	0	0
$241$	2444.73	0.653438	0.326719	−	0.945121i	$-0.394057\pi$
$241$	2444.73	0.653438	0.326719	+	0.945121i	$0.394057\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	3139.35	0.808712
$248$	0	0
$249$	3740.82	0.952066
$250$	0	0
$251$	−5451.97	−1.37102	−0.685508	−	0.728065i	$-0.740420\pi$
$251$	−5451.97	−1.37102	−0.685508	+	0.728065i	$0.740420\pi$
$252$	0	0
$253$	6783.35	1.68564
$254$	0	0
$255$	2014.87	0.494807
$256$	0	0
$257$	−2336.97	−0.567223	−0.283612	−	0.958939i	$-0.591533\pi$
$257$	−2336.97	−0.567223	−0.283612	+	0.958939i	$0.591533\pi$
$258$	0	0
$259$	474.054	0.113731
$260$	0	0
$261$	−661.170	−0.156802
$262$	0	0
$263$	1454.58	0.341039	0.170519	−	0.985354i	$-0.445455\pi$
$263$	1454.58	0.341039	0.170519	+	0.985354i	$0.445455\pi$
$264$	0	0
$265$	−2301.70	−0.533556
$266$	0	0
$267$	−1455.06	−0.333514
$268$	0	0
$269$	7702.27	1.74578	0.872891	−	0.487915i	$-0.162242\pi$
$269$	7702.27	1.74578	0.872891	+	0.487915i	$0.162242\pi$
$270$	0	0
$271$	−2505.75	−0.561672	−0.280836	−	0.959756i	$-0.590612\pi$
$271$	−2505.75	−0.561672	−0.280836	+	0.959756i	$0.590612\pi$
$272$	0	0
$273$	783.325	0.173659
$274$	0	0
$275$	1604.05	0.351739
$276$	0	0
$277$	−190.819	−0.0413906	−0.0206953	−	0.999786i	$-0.506588\pi$
$277$	−190.819	−0.0413906	−0.0206953	+	0.999786i	$0.506588\pi$
$278$	0	0
$279$	−2131.30	−0.457339
$280$	0	0
$281$	−1163.40	−0.246985	−0.123493	−	0.992345i	$-0.539410\pi$
$281$	−1163.40	−0.246985	−0.123493	+	0.992345i	$0.539410\pi$
$282$	0	0
$283$	−5727.95	−1.20315	−0.601574	−	0.798817i	$-0.705460\pi$
$283$	−5727.95	−1.20315	−0.601574	+	0.798817i	$0.705460\pi$
$284$	0	0
$285$	−1262.43	−0.262386
$286$	0	0
$287$	−2730.78	−0.561648
$288$	0	0
$289$	13130.0	2.67251
$290$	0	0
$291$	428.675	0.0863553
$292$	0	0
$293$	−2856.53	−0.569558	−0.284779	−	0.958593i	$-0.591920\pi$
$293$	−2856.53	−0.569558	−0.284779	+	0.958593i	$0.591920\pi$
$294$	0	0
$295$	−1985.06	−0.391778
$296$	0	0
$297$	1732.38	0.338461
$298$	0	0
$299$	3943.55	0.762748
$300$	0	0
$301$	837.730	0.160418
$302$	0	0
$303$	823.216	0.156081
$304$	0	0
$305$	−2604.73	−0.489005
$306$	0	0
$307$	3854.82	0.716632	0.358316	−	0.933600i	$-0.383351\pi$
$307$	3854.82	0.716632	0.358316	+	0.933600i	$0.383351\pi$
$308$	0	0
$309$	−5168.57	−0.951553
$310$	0	0
$311$	−7401.57	−1.34953	−0.674766	−	0.738032i	$-0.735755\pi$
$311$	−7401.57	−1.34953	−0.674766	+	0.738032i	$0.735755\pi$
$312$	0	0
$313$	997.193	0.180079	0.0900395	−	0.995938i	$-0.471301\pi$
$313$	997.193	0.180079	0.0900395	+	0.995938i	$0.471301\pi$
$314$	0	0
$315$	−315.000	−0.0563436
$316$	0	0
$317$	−10790.8	−1.91190	−0.955950	−	0.293531i	$-0.905170\pi$
$317$	−10790.8	−1.91190	−0.955950	+	0.293531i	$0.905170\pi$
$318$	0	0
$319$	−4713.57	−0.827301
$320$	0	0
$321$	−5933.82	−1.03175
$322$	0	0
$323$	−11305.0	−1.94746
$324$	0	0
$325$	932.529	0.159161
$326$	0	0
$327$	1469.66	0.248540
$328$	0	0
$329$	1766.27	0.295981
$330$	0	0
$331$	−1644.00	−0.272997	−0.136499	−	0.990640i	$-0.543585\pi$
$331$	−1644.00	−0.272997	−0.136499	+	0.990640i	$0.543585\pi$
$332$	0	0
$333$	609.498	0.100301
$334$	0	0
$335$	−4535.00	−0.739622
$336$	0	0
$337$	1432.59	0.231567	0.115783	−	0.993274i	$-0.463062\pi$
$337$	1432.59	0.231567	0.115783	+	0.993274i	$0.463062\pi$
$338$	0	0
$339$	−1411.04	−0.226069
$340$	0	0
$341$	−15194.3	−2.41295
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−1585.83	−0.247473
$346$	0	0
$347$	8712.12	1.34781	0.673906	−	0.738817i	$-0.264615\pi$
$347$	8712.12	1.34781	0.673906	+	0.738817i	$0.264615\pi$
$348$	0	0
$349$	9774.41	1.49917	0.749587	−	0.661906i	$-0.230252\pi$
$349$	9774.41	1.49917	0.749587	+	0.661906i	$0.230252\pi$
$350$	0	0
$351$	1007.13	0.153153
$352$	0	0
$353$	−13113.2	−1.97718	−0.988590	−	0.150629i	$-0.951870\pi$
$353$	−13113.2	−1.97718	−0.988590	+	0.150629i	$0.951870\pi$
$354$	0	0
$355$	−2543.40	−0.380252
$356$	0	0
$357$	−2820.81	−0.418188
$358$	0	0
$359$	−9439.68	−1.38776	−0.693882	−	0.720088i	$-0.744101\pi$
$359$	−9439.68	−1.38776	−0.693882	+	0.720088i	$0.744101\pi$
$360$	0	0
$361$	224.271	0.0326973
$362$	0	0
$363$	8357.35	1.20839
$364$	0	0
$365$	1161.60	0.166578
$366$	0	0
$367$	10579.2	1.50472	0.752359	−	0.658754i	$-0.228916\pi$
$367$	10579.2	1.50472	0.752359	+	0.658754i	$0.228916\pi$
$368$	0	0
$369$	−3511.01	−0.495327
$370$	0	0
$371$	3222.38	0.450937
$372$	0	0
$373$	1105.36	0.153440	0.0767202	−	0.997053i	$-0.475555\pi$
$373$	1105.36	0.153440	0.0767202	+	0.997053i	$0.475555\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	−2740.27	−0.374353
$378$	0	0
$379$	−4108.67	−0.556855	−0.278428	−	0.960457i	$-0.589813\pi$
$379$	−4108.67	−0.556855	−0.278428	+	0.960457i	$0.589813\pi$
$380$	0	0
$381$	5510.10	0.740921
$382$	0	0
$383$	−5483.47	−0.731572	−0.365786	−	0.930699i	$-0.619200\pi$
$383$	−5483.47	−0.731572	−0.365786	+	0.930699i	$0.619200\pi$
$384$	0	0
$385$	−2245.68	−0.297273
$386$	0	0
$387$	1077.08	0.141476
$388$	0	0
$389$	2363.93	0.308113	0.154056	−	0.988062i	$-0.450766\pi$
$389$	2363.93	0.308113	0.154056	+	0.988062i	$0.450766\pi$
$390$	0	0
$391$	−14201.0	−1.83677
$392$	0	0
$393$	−6492.67	−0.833364
$394$	0	0
$395$	−3592.91	−0.457669
$396$	0	0
$397$	3084.33	0.389920	0.194960	−	0.980811i	$-0.437542\pi$
$397$	3084.33	0.389920	0.194960	+	0.980811i	$0.437542\pi$
$398$	0	0
$399$	1767.41	0.221757
$400$	0	0
$401$	−9721.19	−1.21061	−0.605303	−	0.795995i	$-0.706948\pi$
$401$	−9721.19	−1.21061	−0.605303	+	0.795995i	$0.706948\pi$
$402$	0	0
$403$	−8833.32	−1.09186
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	4345.19	0.529197
$408$	0	0
$409$	−4918.07	−0.594579	−0.297290	−	0.954787i	$-0.596083\pi$
$409$	−4918.07	−0.594579	−0.297290	+	0.954787i	$0.596083\pi$
$410$	0	0
$411$	746.225	0.0895586
$412$	0	0
$413$	2779.08	0.331113
$414$	0	0
$415$	−6234.69	−0.737467
$416$	0	0
$417$	7866.16	0.923759
$418$	0	0
$419$	−5134.19	−0.598619	−0.299310	−	0.954156i	$-0.596756\pi$
$419$	−5134.19	−0.598619	−0.299310	+	0.954156i	$0.596756\pi$
$420$	0	0
$421$	−5483.03	−0.634742	−0.317371	−	0.948301i	$-0.602800\pi$
$421$	−5483.03	−0.634742	−0.317371	+	0.948301i	$0.602800\pi$
$422$	0	0
$423$	2270.92	0.261030
$424$	0	0
$425$	−3358.11	−0.383276
$426$	0	0
$427$	3646.62	0.413284
$428$	0	0
$429$	7179.97	0.808047
$430$	0	0
$431$	−5786.67	−0.646715	−0.323358	−	0.946277i	$-0.604812\pi$
$431$	−5786.67	−0.646715	−0.323358	+	0.946277i	$0.604812\pi$
$432$	0	0
$433$	4120.24	0.457289	0.228645	−	0.973510i	$-0.426571\pi$
$433$	4120.24	0.457289	0.228645	+	0.973510i	$0.426571\pi$
$434$	0	0
$435$	1101.95	0.121459
$436$	0	0
$437$	8897.79	0.974003
$438$	0	0
$439$	−2138.78	−0.232525	−0.116262	−	0.993219i	$-0.537091\pi$
$439$	−2138.78	−0.232525	−0.116262	+	0.993219i	$0.537091\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	1510.82	0.162034	0.0810171	−	0.996713i	$-0.474183\pi$
$443$	1510.82	0.162034	0.0810171	+	0.996713i	$0.474183\pi$
$444$	0	0
$445$	2425.10	0.258338
$446$	0	0
$447$	−86.5373	−0.00915677
$448$	0	0
$449$	−11961.0	−1.25718	−0.628589	−	0.777738i	$-0.716367\pi$
$449$	−11961.0	−1.25718	−0.628589	+	0.777738i	$0.716367\pi$
$450$	0	0
$451$	−25030.4	−2.61339
$452$	0	0
$453$	2974.07	0.308463
$454$	0	0
$455$	−1305.54	−0.134516
$456$	0	0
$457$	−16355.0	−1.67408	−0.837042	−	0.547139i	$-0.815717\pi$
$457$	−16355.0	−1.67408	−0.837042	+	0.547139i	$0.815717\pi$
$458$	0	0
$459$	−3626.76	−0.368807
$460$	0	0
$461$	8224.60	0.830928	0.415464	−	0.909610i	$-0.363619\pi$
$461$	8224.60	0.830928	0.415464	+	0.909610i	$0.363619\pi$
$462$	0	0
$463$	14108.2	1.41612	0.708058	−	0.706155i	$-0.249572\pi$
$463$	14108.2	1.41612	0.708058	+	0.706155i	$0.249572\pi$
$464$	0	0
$465$	3552.16	0.354253
$466$	0	0
$467$	−19708.3	−1.95287	−0.976437	−	0.215801i	$-0.930764\pi$
$467$	−19708.3	−1.95287	−0.976437	+	0.215801i	$0.930764\pi$
$468$	0	0
$469$	6349.00	0.625095
$470$	0	0
$471$	5181.93	0.506944
$472$	0	0
$473$	7678.65	0.746437
$474$	0	0
$475$	2104.05	0.203243
$476$	0	0
$477$	4143.06	0.397689
$478$	0	0
$479$	2640.71	0.251894	0.125947	−	0.992037i	$-0.459803\pi$
$479$	2640.71	0.251894	0.125947	+	0.992037i	$0.459803\pi$
$480$	0	0
$481$	2526.11	0.239461
$482$	0	0
$483$	2220.16	0.209153
$484$	0	0
$485$	−714.459	−0.0668905
$486$	0	0
$487$	9088.39	0.845656	0.422828	−	0.906210i	$-0.361038\pi$
$487$	9088.39	0.845656	0.422828	+	0.906210i	$0.361038\pi$
$488$	0	0
$489$	−1593.24	−0.147339
$490$	0	0
$491$	16740.6	1.53869	0.769343	−	0.638837i	$-0.220584\pi$
$491$	16740.6	1.53869	0.769343	+	0.638837i	$0.220584\pi$
$492$	0	0
$493$	9867.92	0.901478
$494$	0	0
$495$	−2887.30	−0.262170
$496$	0	0
$497$	3560.76	0.321372
$498$	0	0
$499$	−10407.4	−0.933669	−0.466834	−	0.884345i	$-0.654606\pi$
$499$	−10407.4	−0.933669	−0.466834	+	0.884345i	$0.654606\pi$
$500$	0	0
$501$	−5380.44	−0.479801
$502$	0	0
$503$	−10912.5	−0.967325	−0.483663	−	0.875254i	$-0.660694\pi$
$503$	−10912.5	−0.967325	−0.483663	+	0.875254i	$0.660694\pi$
$504$	0	0
$505$	−1372.03	−0.120900
$506$	0	0
$507$	−2416.87	−0.211710
$508$	0	0
$509$	−19663.0	−1.71228	−0.856138	−	0.516747i	$-0.827143\pi$
$509$	−19663.0	−1.71228	−0.856138	+	0.516747i	$0.827143\pi$
$510$	0	0
$511$	−1626.24	−0.140784
$512$	0	0
$513$	2272.38	0.195571
$514$	0	0
$515$	8614.29	0.737070
$516$	0	0
$517$	16189.7	1.37722
$518$	0	0
$519$	10898.9	0.921787
$520$	0	0
$521$	6146.99	0.516899	0.258449	−	0.966025i	$-0.416788\pi$
$521$	6146.99	0.516899	0.258449	+	0.966025i	$0.416788\pi$
$522$	0	0
$523$	−4531.94	−0.378906	−0.189453	−	0.981890i	$-0.560671\pi$
$523$	−4531.94	−0.378906	−0.189453	+	0.981890i	$0.560671\pi$
$524$	0	0
$525$	525.000	0.0436436
$526$	0	0
$527$	31809.5	2.62930
$528$	0	0
$529$	−989.860	−0.0813561
$530$	0	0
$531$	3573.11	0.292014
$532$	0	0
$533$	−14551.6	−1.18255
$534$	0	0
$535$	9889.69	0.799194
$536$	0	0
$537$	−7978.94	−0.641185
$538$	0	0
$539$	3143.95	0.251242
$540$	0	0
$541$	−9434.26	−0.749742	−0.374871	−	0.927077i	$-0.622313\pi$
$541$	−9434.26	−0.749742	−0.374871	+	0.927077i	$0.622313\pi$
$542$	0	0
$543$	−7514.54	−0.593885
$544$	0	0
$545$	−2449.44	−0.192518
$546$	0	0
$547$	22550.6	1.76270	0.881349	−	0.472467i	$-0.156636\pi$
$547$	22550.6	1.76270	0.881349	+	0.472467i	$0.156636\pi$
$548$	0	0
$549$	4688.51	0.364483
$550$	0	0
$551$	−6182.83	−0.478036
$552$	0	0
$553$	5030.08	0.386801
$554$	0	0
$555$	−1015.83	−0.0776929
$556$	0	0
$557$	−18831.6	−1.43253	−0.716266	−	0.697827i	$-0.754150\pi$
$557$	−18831.6	−1.43253	−0.716266	+	0.697827i	$0.754150\pi$
$558$	0	0
$559$	4464.04	0.337762
$560$	0	0
$561$	−25855.6	−1.94586
$562$	0	0
$563$	−17888.6	−1.33910	−0.669552	−	0.742765i	$-0.733514\pi$
$563$	−17888.6	−1.33910	−0.669552	+	0.742765i	$0.733514\pi$
$564$	0	0
$565$	2351.74	0.175112
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	21384.1	1.57552	0.787759	−	0.615984i	$-0.211241\pi$
$569$	21384.1	1.57552	0.787759	+	0.615984i	$0.211241\pi$
$570$	0	0
$571$	4342.21	0.318242	0.159121	−	0.987259i	$-0.449134\pi$
$571$	4342.21	0.318242	0.159121	+	0.987259i	$0.449134\pi$
$572$	0	0
$573$	2102.29	0.153272
$574$	0	0
$575$	2643.05	0.191692
$576$	0	0
$577$	−6822.39	−0.492235	−0.246118	−	0.969240i	$-0.579155\pi$
$577$	−6822.39	−0.492235	−0.246118	+	0.969240i	$0.579155\pi$
$578$	0	0
$579$	−232.473	−0.0166861
$580$	0	0
$581$	8728.57	0.623274
$582$	0	0
$583$	29536.4	2.09824
$584$	0	0
$585$	−1678.55	−0.118632
$586$	0	0
$587$	−14838.0	−1.04332	−0.521660	−	0.853154i	$-0.674687\pi$
$587$	−14838.0	−1.04332	−0.521660	+	0.853154i	$0.674687\pi$
$588$	0	0
$589$	−19930.5	−1.39427
$590$	0	0
$591$	4358.20	0.303337
$592$	0	0
$593$	−14718.8	−1.01927	−0.509635	−	0.860390i	$-0.670220\pi$
$593$	−14718.8	−1.01927	−0.509635	+	0.860390i	$0.670220\pi$
$594$	0	0
$595$	4701.35	0.323927
$596$	0	0
$597$	−250.864	−0.0171980
$598$	0	0
$599$	8013.13	0.546590	0.273295	−	0.961930i	$-0.411886\pi$
$599$	8013.13	0.546590	0.273295	+	0.961930i	$0.411886\pi$
$600$	0	0
$601$	−7175.55	−0.487016	−0.243508	−	0.969899i	$-0.578298\pi$
$601$	−7175.55	−0.487016	−0.243508	+	0.969899i	$0.578298\pi$
$602$	0	0
$603$	8163.00	0.551282
$604$	0	0
$605$	−13928.9	−0.936018
$606$	0	0
$607$	1315.22	0.0879458	0.0439729	−	0.999033i	$-0.485998\pi$
$607$	1315.22	0.0879458	0.0439729	+	0.999033i	$0.485998\pi$
$608$	0	0
$609$	−1542.73	−0.102651
$610$	0	0
$611$	9411.99	0.623189
$612$	0	0
$613$	2688.90	0.177168	0.0885838	−	0.996069i	$-0.471766\pi$
$613$	2688.90	0.177168	0.0885838	+	0.996069i	$0.471766\pi$
$614$	0	0
$615$	5851.68	0.383679
$616$	0	0
$617$	13505.4	0.881212	0.440606	−	0.897701i	$-0.354764\pi$
$617$	13505.4	0.881212	0.440606	+	0.897701i	$0.354764\pi$
$618$	0	0
$619$	4610.97	0.299403	0.149702	−	0.988731i	$-0.452169\pi$
$619$	4610.97	0.299403	0.149702	+	0.988731i	$0.452169\pi$
$620$	0	0
$621$	2854.49	0.184456
$622$	0	0
$623$	−3395.14	−0.218336
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	16200.1	1.03185
$628$	0	0
$629$	−9096.71	−0.576645
$630$	0	0
$631$	−16967.6	−1.07047	−0.535237	−	0.844702i	$-0.679778\pi$
$631$	−16967.6	−1.07047	−0.535237	+	0.844702i	$0.679778\pi$
$632$	0	0
$633$	−7215.30	−0.453053
$634$	0	0
$635$	−9183.50	−0.573915
$636$	0	0
$637$	1827.76	0.113687
$638$	0	0
$639$	4578.12	0.283423
$640$	0	0
$641$	−17874.6	−1.10141	−0.550704	−	0.834700i	$-0.685641\pi$
$641$	−17874.6	−1.10141	−0.550704	+	0.834700i	$0.685641\pi$
$642$	0	0
$643$	2800.34	0.171749	0.0858746	−	0.996306i	$-0.472632\pi$
$643$	2800.34	0.171749	0.0858746	+	0.996306i	$0.472632\pi$
$644$	0	0
$645$	−1795.13	−0.109587
$646$	0	0
$647$	30486.5	1.85247	0.926235	−	0.376946i	$-0.123026\pi$
$647$	30486.5	1.85247	0.926235	+	0.376946i	$0.123026\pi$
$648$	0	0
$649$	25473.1	1.54069
$650$	0	0
$651$	−4973.03	−0.299398
$652$	0	0
$653$	21920.8	1.31367	0.656834	−	0.754035i	$-0.271895\pi$
$653$	21920.8	1.31367	0.656834	+	0.754035i	$0.271895\pi$
$654$	0	0
$655$	10821.1	0.645521
$656$	0	0
$657$	−2090.88	−0.124160
$658$	0	0
$659$	−8017.07	−0.473901	−0.236951	−	0.971522i	$-0.576148\pi$
$659$	−8017.07	−0.473901	−0.236951	+	0.971522i	$0.576148\pi$
$660$	0	0
$661$	6149.43	0.361854	0.180927	−	0.983497i	$-0.442090\pi$
$661$	6149.43	0.361854	0.180927	+	0.983497i	$0.442090\pi$
$662$	0	0
$663$	−15031.4	−0.880498
$664$	0	0
$665$	−2945.68	−0.171772
$666$	0	0
$667$	−7766.69	−0.450866
$668$	0	0
$669$	12060.4	0.696983
$670$	0	0
$671$	33425.0	1.92304
$672$	0	0
$673$	−21943.3	−1.25684	−0.628419	−	0.777875i	$-0.716298\pi$
$673$	−21943.3	−1.25684	−0.628419	+	0.777875i	$0.716298\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	22975.6	1.30432	0.652161	−	0.758081i	$-0.273863\pi$
$677$	22975.6	1.30432	0.652161	+	0.758081i	$0.273863\pi$
$678$	0	0
$679$	1000.24	0.0565328
$680$	0	0
$681$	12247.0	0.689142
$682$	0	0
$683$	24792.7	1.38897	0.694486	−	0.719506i	$-0.255632\pi$
$683$	24792.7	1.38897	0.694486	+	0.719506i	$0.255632\pi$
$684$	0	0
$685$	−1243.71	−0.0693718
$686$	0	0
$687$	−2168.68	−0.120437
$688$	0	0
$689$	17171.2	0.949450
$690$	0	0
$691$	−10258.6	−0.564767	−0.282383	−	0.959302i	$-0.591125\pi$
$691$	−10258.6	−0.564767	−0.282383	+	0.959302i	$0.591125\pi$
$692$	0	0
$693$	4042.22	0.221574
$694$	0	0
$695$	−13110.3	−0.715541
$696$	0	0
$697$	52401.5	2.84770
$698$	0	0
$699$	−2327.98	−0.125969
$700$	0	0
$701$	−26568.8	−1.43151	−0.715756	−	0.698350i	$-0.753918\pi$
$701$	−26568.8	−1.43151	−0.715756	+	0.698350i	$0.753918\pi$
$702$	0	0
$703$	5699.63	0.305783
$704$	0	0
$705$	−3784.87	−0.202193
$706$	0	0
$707$	1920.84	0.102179
$708$	0	0
$709$	22638.3	1.19915	0.599575	−	0.800319i	$-0.295336\pi$
$709$	22638.3	1.19915	0.599575	+	0.800319i	$0.295336\pi$
$710$	0	0
$711$	6467.25	0.341126
$712$	0	0
$713$	−25036.1	−1.31502
$714$	0	0
$715$	−11966.6	−0.625911
$716$	0	0
$717$	−19022.9	−0.990825
$718$	0	0
$719$	27763.3	1.44005	0.720026	−	0.693947i	$-0.244130\pi$
$719$	27763.3	1.44005	0.720026	+	0.693947i	$0.244130\pi$
$720$	0	0
$721$	−12060.0	−0.622938
$722$	0	0
$723$	7334.18	0.377263
$724$	0	0
$725$	−1836.58	−0.0940814
$726$	0	0
$727$	−26880.4	−1.37130	−0.685652	−	0.727930i	$-0.740483\pi$
$727$	−26880.4	−1.37130	−0.685652	+	0.727930i	$0.740483\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−16075.4	−0.813363
$732$	0	0
$733$	−10606.3	−0.534450	−0.267225	−	0.963634i	$-0.586107\pi$
$733$	−10606.3	−0.534450	−0.267225	+	0.963634i	$0.586107\pi$
$734$	0	0
$735$	−735.000	−0.0368856
$736$	0	0
$737$	58195.1	2.90861
$738$	0	0
$739$	23156.4	1.15267	0.576334	−	0.817215i	$-0.304483\pi$
$739$	23156.4	1.15267	0.576334	+	0.817215i	$0.304483\pi$
$740$	0	0
$741$	9418.04	0.466910
$742$	0	0
$743$	5393.39	0.266305	0.133152	−	0.991096i	$-0.457490\pi$
$743$	5393.39	0.266305	0.133152	+	0.991096i	$0.457490\pi$
$744$	0	0
$745$	144.229	0.00709280
$746$	0	0
$747$	11222.4	0.549676
$748$	0	0
$749$	−13845.6	−0.675442
$750$	0	0
$751$	−15724.7	−0.764049	−0.382024	−	0.924152i	$-0.624773\pi$
$751$	−15724.7	−0.764049	−0.382024	+	0.924152i	$0.624773\pi$
$752$	0	0
$753$	−16355.9	−0.791557
$754$	0	0
$755$	−4956.78	−0.238935
$756$	0	0
$757$	−15875.1	−0.762206	−0.381103	−	0.924533i	$-0.624456\pi$
$757$	−15875.1	−0.762206	−0.381103	+	0.924533i	$0.624456\pi$
$758$	0	0
$759$	20350.1	0.973202
$760$	0	0
$761$	−15321.4	−0.729828	−0.364914	−	0.931041i	$-0.618902\pi$
$761$	−15321.4	−0.729828	−0.364914	+	0.931041i	$0.618902\pi$
$762$	0	0
$763$	3429.22	0.162708
$764$	0	0
$765$	6044.60	0.285677
$766$	0	0
$767$	14809.0	0.697161
$768$	0	0
$769$	23892.3	1.12039	0.560195	−	0.828361i	$-0.310726\pi$
$769$	23892.3	1.12039	0.560195	+	0.828361i	$0.310726\pi$
$770$	0	0
$771$	−7010.92	−0.327487
$772$	0	0
$773$	39844.3	1.85394	0.926972	−	0.375131i	$-0.122402\pi$
$773$	39844.3	1.85394	0.926972	+	0.375131i	$0.122402\pi$
$774$	0	0
$775$	−5920.27	−0.274403
$776$	0	0
$777$	1422.16	0.0656625
$778$	0	0
$779$	−32832.7	−1.51008
$780$	0	0
$781$	32638.0	1.49536
$782$	0	0
$783$	−1983.51	−0.0905299
$784$	0	0
$785$	−8636.54	−0.392677
$786$	0	0
$787$	3713.76	0.168210	0.0841050	−	0.996457i	$-0.473197\pi$
$787$	3713.76	0.168210	0.0841050	+	0.996457i	$0.473197\pi$
$788$	0	0
$789$	4363.74	0.196899
$790$	0	0
$791$	−3292.43	−0.147997
$792$	0	0
$793$	19431.9	0.870173
$794$	0	0
$795$	−6905.10	−0.308049
$796$	0	0
$797$	−38977.2	−1.73230	−0.866150	−	0.499784i	$-0.833413\pi$
$797$	−38977.2	−1.73230	−0.866150	+	0.499784i	$0.833413\pi$
$798$	0	0
$799$	−33893.3	−1.50070
$800$	0	0
$801$	−4365.17	−0.192554
$802$	0	0
$803$	−14906.2	−0.655078
$804$	0	0
$805$	−3700.27	−0.162009
$806$	0	0
$807$	23106.8	1.00793
$808$	0	0
$809$	−3120.01	−0.135592	−0.0677958	−	0.997699i	$-0.521597\pi$
$809$	−3120.01	−0.135592	−0.0677958	+	0.997699i	$0.521597\pi$
$810$	0	0
$811$	−29109.5	−1.26038	−0.630192	−	0.776439i	$-0.717024\pi$
$811$	−29109.5	−1.26038	−0.630192	+	0.776439i	$0.717024\pi$
$812$	0	0
$813$	−7517.24	−0.324282
$814$	0	0
$815$	2655.40	0.114129
$816$	0	0
$817$	10072.2	0.431310
$818$	0	0
$819$	2349.97	0.100262
$820$	0	0
$821$	−40513.0	−1.72218	−0.861092	−	0.508449i	$-0.830219\pi$
$821$	−40513.0	−1.72218	−0.861092	+	0.508449i	$0.830219\pi$
$822$	0	0
$823$	23901.9	1.01235	0.506177	−	0.862430i	$-0.331058\pi$
$823$	23901.9	1.01235	0.506177	+	0.862430i	$0.331058\pi$
$824$	0	0
$825$	4812.16	0.203076
$826$	0	0
$827$	15304.0	0.643498	0.321749	−	0.946825i	$-0.395729\pi$
$827$	15304.0	0.643498	0.321749	+	0.946825i	$0.395729\pi$
$828$	0	0
$829$	−11830.6	−0.495650	−0.247825	−	0.968805i	$-0.579716\pi$
$829$	−11830.6	−0.495650	−0.247825	+	0.968805i	$0.579716\pi$
$830$	0	0
$831$	−572.456	−0.0238969
$832$	0	0
$833$	−6581.89	−0.273768
$834$	0	0
$835$	8967.39	0.371652
$836$	0	0
$837$	−6393.89	−0.264045
$838$	0	0
$839$	23656.2	0.973425	0.486712	−	0.873562i	$-0.338196\pi$
$839$	23656.2	0.973425	0.486712	+	0.873562i	$0.338196\pi$
$840$	0	0
$841$	−18992.1	−0.778717
$842$	0	0
$843$	−3490.21	−0.142597
$844$	0	0
$845$	4028.11	0.163990
$846$	0	0
$847$	19500.5	0.791080
$848$	0	0
$849$	−17183.8	−0.694638
$850$	0	0
$851$	7159.70	0.288404
$852$	0	0
$853$	−36520.8	−1.46594	−0.732970	−	0.680260i	$-0.761867\pi$
$853$	−36520.8	−1.46594	−0.732970	+	0.680260i	$0.761867\pi$
$854$	0	0
$855$	−3787.30	−0.151489
$856$	0	0
$857$	18593.1	0.741106	0.370553	−	0.928811i	$-0.379168\pi$
$857$	18593.1	0.741106	0.370553	+	0.928811i	$0.379168\pi$
$858$	0	0
$859$	−37547.5	−1.49139	−0.745696	−	0.666287i	$-0.767883\pi$
$859$	−37547.5	−1.49139	−0.745696	+	0.666287i	$0.767883\pi$
$860$	0	0
$861$	−8192.35	−0.324268
$862$	0	0
$863$	27100.5	1.06896	0.534479	−	0.845182i	$-0.320508\pi$
$863$	27100.5	1.06896	0.534479	+	0.845182i	$0.320508\pi$
$864$	0	0
$865$	−18164.8	−0.714013
$866$	0	0
$867$	39390.1	1.54297
$868$	0	0
$869$	46105.8	1.79981
$870$	0	0
$871$	33832.2	1.31614
$872$	0	0
$873$	1286.03	0.0498573
$874$	0	0
$875$	−875.000	−0.0338062
$876$	0	0
$877$	18493.3	0.712057	0.356028	−	0.934475i	$-0.384131\pi$
$877$	18493.3	0.712057	0.356028	+	0.934475i	$0.384131\pi$
$878$	0	0
$879$	−8569.60	−0.328834
$880$	0	0
$881$	−22217.0	−0.849614	−0.424807	−	0.905284i	$-0.639658\pi$
$881$	−22217.0	−0.849614	−0.424807	+	0.905284i	$0.639658\pi$
$882$	0	0
$883$	25125.6	0.957581	0.478790	−	0.877929i	$-0.341075\pi$
$883$	25125.6	0.957581	0.478790	+	0.877929i	$0.341075\pi$
$884$	0	0
$885$	−5955.18	−0.226193
$886$	0	0
$887$	30683.1	1.16149	0.580743	−	0.814087i	$-0.302762\pi$
$887$	30683.1	1.16149	0.580743	+	0.814087i	$0.302762\pi$
$888$	0	0
$889$	12856.9	0.485047
$890$	0	0
$891$	5197.14	0.195410
$892$	0	0
$893$	21236.2	0.795791
$894$	0	0
$895$	13298.2	0.496660
$896$	0	0
$897$	11830.7	0.440373
$898$	0	0
$899$	17396.9	0.645406
$900$	0	0
$901$	−61834.9	−2.28637
$902$	0	0
$903$	2513.19	0.0926176
$904$	0	0
$905$	12524.2	0.460021
$906$	0	0
$907$	−5114.94	−0.187253	−0.0936266	−	0.995607i	$-0.529846\pi$
$907$	−5114.94	−0.187253	−0.0936266	+	0.995607i	$0.529846\pi$
$908$	0	0
$909$	2469.65	0.0901133
$910$	0	0
$911$	25031.4	0.910347	0.455174	−	0.890403i	$-0.349577\pi$
$911$	25031.4	0.910347	0.455174	+	0.890403i	$0.349577\pi$
$912$	0	0
$913$	80006.3	2.90013
$914$	0	0
$915$	−7814.19	−0.282327
$916$	0	0
$917$	−15149.6	−0.545565
$918$	0	0
$919$	−21211.8	−0.761384	−0.380692	−	0.924702i	$-0.624314\pi$
$919$	−21211.8	−0.761384	−0.380692	+	0.924702i	$0.624314\pi$
$920$	0	0
$921$	11564.5	0.413748
$922$	0	0
$923$	18974.3	0.676650
$924$	0	0
$925$	1693.05	0.0601807
$926$	0	0
$927$	−15505.7	−0.549379
$928$	0	0
$929$	−43267.9	−1.52807	−0.764034	−	0.645176i	$-0.776784\pi$
$929$	−43267.9	−1.52807	−0.764034	+	0.645176i	$0.776784\pi$
$930$	0	0
$931$	4123.95	0.145174
$932$	0	0
$933$	−22204.7	−0.779153
$934$	0	0
$935$	43092.7	1.50725
$936$	0	0
$937$	−48574.6	−1.69356	−0.846778	−	0.531946i	$-0.821461\pi$
$937$	−48574.6	−1.69356	−0.846778	+	0.531946i	$0.821461\pi$
$938$	0	0
$939$	2991.58	0.103969
$940$	0	0
$941$	5802.60	0.201020	0.100510	−	0.994936i	$-0.467953\pi$
$941$	5802.60	0.201020	0.100510	+	0.994936i	$0.467953\pi$
$942$	0	0
$943$	−41243.4	−1.42425
$944$	0	0
$945$	−945.000	−0.0325300
$946$	0	0
$947$	38033.8	1.30510	0.652551	−	0.757745i	$-0.273699\pi$
$947$	38033.8	1.30510	0.652551	+	0.757745i	$0.273699\pi$
$948$	0	0
$949$	−8665.83	−0.296422
$950$	0	0
$951$	−32372.4	−1.10384
$952$	0	0
$953$	10519.4	0.357563	0.178782	−	0.983889i	$-0.442784\pi$
$953$	10519.4	0.357563	0.178782	+	0.983889i	$0.442784\pi$
$954$	0	0
$955$	−3503.82	−0.118724
$956$	0	0
$957$	−14140.7	−0.477643
$958$	0	0
$959$	1741.19	0.0586298
$960$	0	0
$961$	26288.4	0.882427
$962$	0	0
$963$	−17801.4	−0.595684
$964$	0	0
$965$	387.455	0.0129250
$966$	0	0
$967$	−501.274	−0.0166700	−0.00833500	−	0.999965i	$-0.502653\pi$
$967$	−501.274	−0.0166700	−0.00833500	+	0.999965i	$0.502653\pi$
$968$	0	0
$969$	−33915.1	−1.12436
$970$	0	0
$971$	−53115.6	−1.75547	−0.877734	−	0.479148i	$-0.840946\pi$
$971$	−53115.6	−1.75547	−0.877734	+	0.479148i	$0.840946\pi$
$972$	0	0
$973$	18354.4	0.604742
$974$	0	0
$975$	2797.59	0.0918918
$976$	0	0
$977$	−48265.5	−1.58050	−0.790250	−	0.612784i	$-0.790049\pi$
$977$	−48265.5	−1.58050	−0.790250	+	0.612784i	$0.790049\pi$
$978$	0	0
$979$	−31119.9	−1.01593
$980$	0	0
$981$	4408.99	0.143495
$982$	0	0
$983$	−4660.45	−0.151216	−0.0756080	−	0.997138i	$-0.524090\pi$
$983$	−4660.45	−0.151216	−0.0756080	+	0.997138i	$0.524090\pi$
$984$	0	0
$985$	−7263.67	−0.234964
$986$	0	0
$987$	5298.81	0.170885
$988$	0	0
$989$	12652.3	0.406796
$990$	0	0
$991$	−8652.42	−0.277349	−0.138675	−	0.990338i	$-0.544284\pi$
$991$	−8652.42	−0.277349	−0.138675	+	0.990338i	$0.544284\pi$
$992$	0	0
$993$	−4931.99	−0.157615
$994$	0	0
$995$	418.107	0.0133215
$996$	0	0
$997$	−32335.4	−1.02715	−0.513577	−	0.858044i	$-0.671680\pi$
$997$	−32335.4	−1.02715	−0.513577	+	0.858044i	$0.671680\pi$
$998$	0	0
$999$	1828.49	0.0579089

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.4.a.r.1.3	✓	3
4.3	odd	2		1680.4.a.bq.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.4.a.r.1.3	✓	3	1.1	even	1	trivial
1680.4.a.bq.1.1		3	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	840.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +64.1622 q^{11} +37.3012 q^{13} -15.0000 q^{15} -134.324 q^{17} +84.1622 q^{19} +21.0000 q^{21} +105.722 q^{23} +25.0000 q^{25} +27.0000 q^{27} -73.4633 q^{29} -236.811 q^{31} +192.487 q^{33} -35.0000 q^{35} +67.7220 q^{37} +111.904 q^{39} -390.112 q^{41} +119.676 q^{43} -45.0000 q^{45} +252.324 q^{47} +49.0000 q^{49} -402.973 q^{51} +460.340 q^{53} -320.811 q^{55} +252.487 q^{57} +397.012 q^{59} +520.946 q^{61} +63.0000 q^{63} -186.506 q^{65} +907.000 q^{67} +317.166 q^{69} +508.679 q^{71} -232.321 q^{73} +75.0000 q^{75} +449.135 q^{77} +718.583 q^{79} +81.0000 q^{81} +1246.94 q^{83} +671.622 q^{85} -220.390 q^{87} -485.019 q^{89} +261.108 q^{91} -710.433 q^{93} -420.811 q^{95} +142.892 q^{97} +577.460 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 9 q^{3} - 15 q^{5} + 21 q^{7} + 27 q^{9} + 14 q^{11} + 20 q^{13} - 45 q^{15} - 46 q^{17} + 74 q^{19} + 63 q^{21} + 144 q^{23} + 75 q^{25} + 81 q^{27} + 50 q^{29} + 182 q^{31} + 42 q^{33} - 105 q^{35}+ \cdots + 126 q^{99}+O(q^{100}) \) 3 * q + 9 * q^3 - 15 * q^5 + 21 * q^7 + 27 * q^9 + 14 * q^11 + 20 * q^13 - 45 * q^15 - 46 * q^17 + 74 * q^19 + 63 * q^21 + 144 * q^23 + 75 * q^25 + 81 * q^27 + 50 * q^29 + 182 * q^31 + 42 * q^33 - 105 * q^35 + 30 * q^37 + 60 * q^39 - 186 * q^41 + 716 * q^43 - 135 * q^45 + 400 * q^47 + 147 * q^49 - 138 * q^51 - 236 * q^53 - 70 * q^55 + 222 * q^57 + 272 * q^59 + 1098 * q^61 + 189 * q^63 - 100 * q^65 + 1044 * q^67 + 432 * q^69 + 1542 * q^71 + 996 * q^73 + 225 * q^75 + 98 * q^77 + 1896 * q^79 + 243 * q^81 + 604 * q^83 + 230 * q^85 + 150 * q^87 + 146 * q^89 + 140 * q^91 + 546 * q^93 - 370 * q^95 + 1072 * q^97 + 126 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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