LMFDB - Embedded newform 840.4.a.p.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,-6] 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:	$1$
Dimension:	$3$
Coefficient field:	3.3.821313.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 117x - 340 $ x^3 - 117*x - 340
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.18112$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +43.6126 q^{11} -33.6126 q^{13} +15.0000 q^{15} -86.6741 q^{17} +122.511 q^{19} +21.0000 q^{21} -25.4490 q^{23} +25.0000 q^{25} -27.0000 q^{27} +145.225 q^{29} -33.0616 q^{31} -130.838 q^{33} +35.0000 q^{35} +158.347 q^{37} +100.838 q^{39} +285.124 q^{41} -385.368 q^{43} -45.0000 q^{45} -225.776 q^{47} +49.0000 q^{49} +260.022 q^{51} +428.309 q^{53} -218.063 q^{55} -367.532 q^{57} -541.228 q^{59} +44.7775 q^{61} -63.0000 q^{63} +168.063 q^{65} -244.981 q^{67} +76.3470 q^{69} -182.491 q^{71} -988.435 q^{73} -75.0000 q^{75} -305.288 q^{77} -437.041 q^{79} +81.0000 q^{81} -634.752 q^{83} +433.371 q^{85} -435.675 q^{87} -1336.74 q^{89} +235.288 q^{91} +99.1847 q^{93} -612.553 q^{95} -418.956 q^{97} +392.513 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} - 15 q^{5} - 21 q^{7} + 27 q^{9} - 6 q^{11} + 36 q^{13} + 45 q^{15} + 90 q^{17} + 78 q^{19} + 63 q^{21} + 75 q^{25} - 81 q^{27} + 162 q^{29} + 114 q^{31} + 18 q^{33} + 105 q^{35} + 246 q^{37}+ \cdots - 54 q^{99}+O(q^{100}) $ 3 * q - 9 * q^3 - 15 * q^5 - 21 * q^7 + 27 * q^9 - 6 * q^11 + 36 * q^13 + 45 * q^15 + 90 * q^17 + 78 * q^19 + 63 * q^21 + 75 * q^25 - 81 * q^27 + 162 * q^29 + 114 * q^31 + 18 * q^33 + 105 * q^35 + 246 * q^37 - 108 * q^39 - 42 * q^41 - 348 * q^43 - 135 * q^45 - 480 * q^47 + 147 * q^49 - 270 * q^51 - 252 * q^53 + 30 * q^55 - 234 * q^57 - 408 * q^59 - 534 * q^61 - 189 * q^63 - 180 * q^65 - 372 * q^67 - 366 * q^71 - 60 * q^73 - 225 * q^75 + 42 * q^77 - 624 * q^79 + 243 * q^81 - 3228 * q^83 - 450 * q^85 - 486 * q^87 - 510 * q^89 - 252 * q^91 - 342 * q^93 - 390 * q^95 - 2304 * q^97 - 54 * 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
$11$	43.6126	1.19543	0.597713	−	0.801710i	$-0.296076\pi$
$11$	43.6126	1.19543	0.597713	+	0.801710i	$0.296076\pi$
$12$	0	0
$13$	−33.6126	−0.717112	−0.358556	−	0.933508i	$-0.616731\pi$
$13$	−33.6126	−0.717112	−0.358556	+	0.933508i	$0.616731\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−86.6741	−1.23656	−0.618281	−	0.785957i	$-0.712171\pi$
$17$	−86.6741	−1.23656	−0.618281	+	0.785957i	$0.712171\pi$
$18$	0	0
$19$	122.511	1.47926	0.739628	−	0.673016i	$-0.235002\pi$
$19$	122.511	1.47926	0.739628	+	0.673016i	$0.235002\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	−25.4490	−0.230717	−0.115358	−	0.993324i	$-0.536802\pi$
$23$	−25.4490	−0.230717	−0.115358	+	0.993324i	$0.536802\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	145.225	0.929918	0.464959	−	0.885332i	$-0.346069\pi$
$29$	145.225	0.929918	0.464959	+	0.885332i	$0.346069\pi$
$30$	0	0
$31$	−33.0616	−0.191549	−0.0957747	−	0.995403i	$-0.530533\pi$
$31$	−33.0616	−0.191549	−0.0957747	+	0.995403i	$0.530533\pi$
$32$	0	0
$33$	−130.838	−0.690180
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	158.347	0.703570	0.351785	−	0.936081i	$-0.385575\pi$
$37$	158.347	0.703570	0.351785	+	0.936081i	$0.385575\pi$
$38$	0	0
$39$	100.838	0.414025
$40$	0	0
$41$	285.124	1.08607	0.543036	−	0.839709i	$-0.317275\pi$
$41$	285.124	1.08607	0.543036	+	0.839709i	$0.317275\pi$
$42$	0	0
$43$	−385.368	−1.36670	−0.683350	−	0.730091i	$-0.739478\pi$
$43$	−385.368	−1.36670	−0.683350	+	0.730091i	$0.739478\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	−225.776	−0.700699	−0.350349	−	0.936619i	$-0.613937\pi$
$47$	−225.776	−0.700699	−0.350349	+	0.936619i	$0.613937\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	260.022	0.713930
$52$	0	0
$53$	428.309	1.11005	0.555026	−	0.831833i	$-0.312708\pi$
$53$	428.309	1.11005	0.555026	+	0.831833i	$0.312708\pi$
$54$	0	0
$55$	−218.063	−0.534611
$56$	0	0
$57$	−367.532	−0.854048
$58$	0	0
$59$	−541.228	−1.19427	−0.597134	−	0.802141i	$-0.703694\pi$
$59$	−541.228	−1.19427	−0.597134	+	0.802141i	$0.703694\pi$
$60$	0	0
$61$	44.7775	0.0939864	0.0469932	−	0.998895i	$-0.485036\pi$
$61$	44.7775	0.0939864	0.0469932	+	0.998895i	$0.485036\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	168.063	0.320702
$66$	0	0
$67$	−244.981	−0.446705	−0.223353	−	0.974738i	$-0.571700\pi$
$67$	−244.981	−0.446705	−0.223353	+	0.974738i	$0.571700\pi$
$68$	0	0
$69$	76.3470	0.133204
$70$	0	0
$71$	−182.491	−0.305038	−0.152519	−	0.988301i	$-0.548738\pi$
$71$	−182.491	−0.305038	−0.152519	+	0.988301i	$0.548738\pi$
$72$	0	0
$73$	−988.435	−1.58476	−0.792381	−	0.610027i	$-0.791159\pi$
$73$	−988.435	−1.58476	−0.792381	+	0.610027i	$0.791159\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	−305.288	−0.451829
$78$	0	0
$79$	−437.041	−0.622417	−0.311209	−	0.950342i	$-0.600734\pi$
$79$	−437.041	−0.622417	−0.311209	+	0.950342i	$0.600734\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−634.752	−0.839435	−0.419718	−	0.907655i	$-0.637871\pi$
$83$	−634.752	−0.839435	−0.419718	+	0.907655i	$0.637871\pi$
$84$	0	0
$85$	433.371	0.553008
$86$	0	0
$87$	−435.675	−0.536889
$88$	0	0
$89$	−1336.74	−1.59207	−0.796035	−	0.605250i	$-0.793073\pi$
$89$	−1336.74	−1.59207	−0.796035	+	0.605250i	$0.793073\pi$
$90$	0	0
$91$	235.288	0.271043
$92$	0	0
$93$	99.1847	0.110591
$94$	0	0
$95$	−612.553	−0.661543
$96$	0	0
$97$	−418.956	−0.438541	−0.219271	−	0.975664i	$-0.570368\pi$
$97$	−418.956	−0.438541	−0.219271	+	0.975664i	$0.570368\pi$
$98$	0	0
$99$	392.513	0.398475
$100$	0	0
$101$	−804.269	−0.792354	−0.396177	−	0.918174i	$-0.629663\pi$
$101$	−804.269	−0.792354	−0.396177	+	0.918174i	$0.629663\pi$
$102$	0	0
$103$	1958.38	1.87344	0.936722	−	0.350074i	$-0.113844\pi$
$103$	1958.38	1.87344	0.936722	+	0.350074i	$0.113844\pi$
$104$	0	0
$105$	−105.000	−0.0975900
$106$	0	0
$107$	−2000.92	−1.80782	−0.903908	−	0.427727i	$-0.859314\pi$
$107$	−2000.92	−1.80782	−0.903908	+	0.427727i	$0.859314\pi$
$108$	0	0
$109$	383.087	0.336634	0.168317	−	0.985733i	$-0.446167\pi$
$109$	383.087	0.336634	0.168317	+	0.985733i	$0.446167\pi$
$110$	0	0
$111$	−475.041	−0.406206
$112$	0	0
$113$	1080.84	0.899797	0.449898	−	0.893080i	$-0.351460\pi$
$113$	1080.84	0.899797	0.449898	+	0.893080i	$0.351460\pi$
$114$	0	0
$115$	127.245	0.103180
$116$	0	0
$117$	−302.513	−0.239037
$118$	0	0
$119$	606.719	0.467377
$120$	0	0
$121$	571.057	0.429043
$122$	0	0
$123$	−855.373	−0.627044
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	511.497	0.357386	0.178693	−	0.983905i	$-0.442813\pi$
$127$	511.497	0.357386	0.178693	+	0.983905i	$0.442813\pi$
$128$	0	0
$129$	1156.10	0.789064
$130$	0	0
$131$	−2881.89	−1.92207	−0.961036	−	0.276422i	$-0.910851\pi$
$131$	−2881.89	−1.92207	−0.961036	+	0.276422i	$0.910851\pi$
$132$	0	0
$133$	−857.574	−0.559106
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	541.985	0.337992	0.168996	−	0.985617i	$-0.445948\pi$
$137$	541.985	0.337992	0.168996	+	0.985617i	$0.445948\pi$
$138$	0	0
$139$	−1186.83	−0.724215	−0.362107	−	0.932136i	$-0.617943\pi$
$139$	−1186.83	−0.724215	−0.362107	+	0.932136i	$0.617943\pi$
$140$	0	0
$141$	677.328	0.404549
$142$	0	0
$143$	−1465.93	−0.857254
$144$	0	0
$145$	−726.126	−0.415872
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	−1963.28	−1.07945	−0.539726	−	0.841841i	$-0.681472\pi$
$149$	−1963.28	−1.07945	−0.539726	+	0.841841i	$0.681472\pi$
$150$	0	0
$151$	−920.523	−0.496100	−0.248050	−	0.968747i	$-0.579790\pi$
$151$	−920.523	−0.496100	−0.248050	+	0.968747i	$0.579790\pi$
$152$	0	0
$153$	−780.067	−0.412188
$154$	0	0
$155$	165.308	0.0856635
$156$	0	0
$157$	2892.70	1.47046	0.735230	−	0.677818i	$-0.237074\pi$
$157$	2892.70	1.47046	0.735230	+	0.677818i	$0.237074\pi$
$158$	0	0
$159$	−1284.93	−0.640889
$160$	0	0
$161$	178.143	0.0872027
$162$	0	0
$163$	−2408.56	−1.15738	−0.578691	−	0.815547i	$-0.696436\pi$
$163$	−2408.56	−1.15738	−0.578691	+	0.815547i	$0.696436\pi$
$164$	0	0
$165$	654.189	0.308658
$166$	0	0
$167$	−2682.53	−1.24300	−0.621498	−	0.783415i	$-0.713476\pi$
$167$	−2682.53	−1.24300	−0.621498	+	0.783415i	$0.713476\pi$
$168$	0	0
$169$	−1067.19	−0.485751
$170$	0	0
$171$	1102.60	0.493085
$172$	0	0
$173$	432.128	0.189908	0.0949541	−	0.995482i	$-0.469730\pi$
$173$	432.128	0.189908	0.0949541	+	0.995482i	$0.469730\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	1623.68	0.689511
$178$	0	0
$179$	511.575	0.213614	0.106807	−	0.994280i	$-0.465937\pi$
$179$	511.575	0.213614	0.106807	+	0.994280i	$0.465937\pi$
$180$	0	0
$181$	−2895.94	−1.18925	−0.594623	−	0.804004i	$-0.702699\pi$
$181$	−2895.94	−1.18925	−0.594623	+	0.804004i	$0.702699\pi$
$182$	0	0
$183$	−134.332	−0.0542630
$184$	0	0
$185$	−791.735	−0.314646
$186$	0	0
$187$	−3780.08	−1.47822
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	2617.00	0.991412	0.495706	−	0.868490i	$-0.334909\pi$
$191$	2617.00	0.991412	0.495706	+	0.868490i	$0.334909\pi$
$192$	0	0
$193$	138.928	0.0518147	0.0259073	−	0.999664i	$-0.491753\pi$
$193$	138.928	0.0518147	0.0259073	+	0.999664i	$0.491753\pi$
$194$	0	0
$195$	−504.189	−0.185157
$196$	0	0
$197$	−3767.74	−1.36264	−0.681322	−	0.731984i	$-0.738594\pi$
$197$	−3767.74	−1.36264	−0.681322	+	0.731984i	$0.738594\pi$
$198$	0	0
$199$	−238.868	−0.0850900	−0.0425450	−	0.999095i	$-0.513547\pi$
$199$	−238.868	−0.0850900	−0.0425450	+	0.999095i	$0.513547\pi$
$200$	0	0
$201$	734.944	0.257905
$202$	0	0
$203$	−1016.58	−0.351476
$204$	0	0
$205$	−1425.62	−0.485706
$206$	0	0
$207$	−229.041	−0.0769055
$208$	0	0
$209$	5343.00	1.76834
$210$	0	0
$211$	−31.2397	−0.0101926	−0.00509628	−	0.999987i	$-0.501622\pi$
$211$	−31.2397	−0.0101926	−0.00509628	+	0.999987i	$0.501622\pi$
$212$	0	0
$213$	547.472	0.176114
$214$	0	0
$215$	1926.84	0.611207
$216$	0	0
$217$	231.431	0.0723989
$218$	0	0
$219$	2965.30	0.914962
$220$	0	0
$221$	2913.34	0.886754
$222$	0	0
$223$	2142.37	0.643334	0.321667	−	0.946853i	$-0.395757\pi$
$223$	2142.37	0.643334	0.321667	+	0.946853i	$0.395757\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	67.3453	0.0196910	0.00984551	−	0.999952i	$-0.496866\pi$
$227$	67.3453	0.0196910	0.00984551	+	0.999952i	$0.496866\pi$
$228$	0	0
$229$	4016.44	1.15901	0.579506	−	0.814968i	$-0.303246\pi$
$229$	4016.44	1.15901	0.579506	+	0.814968i	$0.303246\pi$
$230$	0	0
$231$	915.864	0.260863
$232$	0	0
$233$	2896.68	0.814455	0.407227	−	0.913327i	$-0.366496\pi$
$233$	2896.68	0.814455	0.407227	+	0.913327i	$0.366496\pi$
$234$	0	0
$235$	1128.88	0.313362
$236$	0	0
$237$	1311.12	0.359353
$238$	0	0
$239$	−5286.22	−1.43070	−0.715350	−	0.698766i	$-0.753733\pi$
$239$	−5286.22	−1.43070	−0.715350	+	0.698766i	$0.753733\pi$
$240$	0	0
$241$	4069.45	1.08770	0.543852	−	0.839181i	$-0.316965\pi$
$241$	4069.45	1.08770	0.543852	+	0.839181i	$0.316965\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	−4117.90	−1.06079
$248$	0	0
$249$	1904.26	0.484648
$250$	0	0
$251$	−6760.28	−1.70002	−0.850011	−	0.526766i	$-0.823405\pi$
$251$	−6760.28	−1.70002	−0.850011	+	0.526766i	$0.823405\pi$
$252$	0	0
$253$	−1109.90	−0.275805
$254$	0	0
$255$	−1300.11	−0.319279
$256$	0	0
$257$	−5479.56	−1.32998	−0.664991	−	0.746851i	$-0.731565\pi$
$257$	−5479.56	−1.32998	−0.664991	+	0.746851i	$0.731565\pi$
$258$	0	0
$259$	−1108.43	−0.265924
$260$	0	0
$261$	1307.03	0.309973
$262$	0	0
$263$	−557.611	−0.130737	−0.0653684	−	0.997861i	$-0.520822\pi$
$263$	−557.611	−0.130737	−0.0653684	+	0.997861i	$0.520822\pi$
$264$	0	0
$265$	−2141.55	−0.496431
$266$	0	0
$267$	4010.22	0.919183
$268$	0	0
$269$	−4911.11	−1.11314	−0.556571	−	0.830800i	$-0.687883\pi$
$269$	−4911.11	−1.11314	−0.556571	+	0.830800i	$0.687883\pi$
$270$	0	0
$271$	6301.62	1.41253	0.706266	−	0.707947i	$-0.250378\pi$
$271$	6301.62	1.41253	0.706266	+	0.707947i	$0.250378\pi$
$272$	0	0
$273$	−705.864	−0.156487
$274$	0	0
$275$	1090.31	0.239085
$276$	0	0
$277$	3649.72	0.791661	0.395831	−	0.918324i	$-0.370457\pi$
$277$	3649.72	0.791661	0.395831	+	0.918324i	$0.370457\pi$
$278$	0	0
$279$	−297.554	−0.0638498
$280$	0	0
$281$	578.282	0.122767	0.0613833	−	0.998114i	$-0.480449\pi$
$281$	578.282	0.122767	0.0613833	+	0.998114i	$0.480449\pi$
$282$	0	0
$283$	7265.87	1.52619	0.763094	−	0.646288i	$-0.223680\pi$
$283$	7265.87	1.52619	0.763094	+	0.646288i	$0.223680\pi$
$284$	0	0
$285$	1837.66	0.381942
$286$	0	0
$287$	−1995.87	−0.410497
$288$	0	0
$289$	2599.41	0.529088
$290$	0	0
$291$	1256.87	0.253192
$292$	0	0
$293$	3821.77	0.762015	0.381007	−	0.924572i	$-0.375577\pi$
$293$	3821.77	0.762015	0.381007	+	0.924572i	$0.375577\pi$
$294$	0	0
$295$	2706.14	0.534093
$296$	0	0
$297$	−1177.54	−0.230060
$298$	0	0
$299$	855.406	0.165450
$300$	0	0
$301$	2697.58	0.516564
$302$	0	0
$303$	2412.81	0.457466
$304$	0	0
$305$	−223.887	−0.0420320
$306$	0	0
$307$	−3833.93	−0.712749	−0.356375	−	0.934343i	$-0.615987\pi$
$307$	−3833.93	−0.712749	−0.356375	+	0.934343i	$0.615987\pi$
$308$	0	0
$309$	−5875.14	−1.08163
$310$	0	0
$311$	3434.32	0.626182	0.313091	−	0.949723i	$-0.398636\pi$
$311$	3434.32	0.626182	0.313091	+	0.949723i	$0.398636\pi$
$312$	0	0
$313$	6694.17	1.20887	0.604436	−	0.796654i	$-0.293399\pi$
$313$	6694.17	1.20887	0.604436	+	0.796654i	$0.293399\pi$
$314$	0	0
$315$	315.000	0.0563436
$316$	0	0
$317$	−3772.18	−0.668348	−0.334174	−	0.942511i	$-0.608457\pi$
$317$	−3772.18	−0.668348	−0.334174	+	0.942511i	$0.608457\pi$
$318$	0	0
$319$	6333.64	1.11165
$320$	0	0
$321$	6002.76	1.04374
$322$	0	0
$323$	−10618.5	−1.82919
$324$	0	0
$325$	−840.314	−0.143422
$326$	0	0
$327$	−1149.26	−0.194356
$328$	0	0
$329$	1580.43	0.264839
$330$	0	0
$331$	5474.12	0.909017	0.454508	−	0.890742i	$-0.349815\pi$
$331$	5474.12	0.909017	0.454508	+	0.890742i	$0.349815\pi$
$332$	0	0
$333$	1425.12	0.234523
$334$	0	0
$335$	1224.91	0.199773
$336$	0	0
$337$	−7222.50	−1.16746	−0.583731	−	0.811947i	$-0.698408\pi$
$337$	−7222.50	−1.16746	−0.583731	+	0.811947i	$0.698408\pi$
$338$	0	0
$339$	−3242.52	−0.519498
$340$	0	0
$341$	−1441.90	−0.228983
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−381.735	−0.0595708
$346$	0	0
$347$	5472.91	0.846689	0.423345	−	0.905969i	$-0.360856\pi$
$347$	5472.91	0.846689	0.423345	+	0.905969i	$0.360856\pi$
$348$	0	0
$349$	−5256.06	−0.806161	−0.403081	−	0.915164i	$-0.632061\pi$
$349$	−5256.06	−0.806161	−0.403081	+	0.915164i	$0.632061\pi$
$350$	0	0
$351$	907.540	0.138008
$352$	0	0
$353$	1045.17	0.157588	0.0787940	−	0.996891i	$-0.474893\pi$
$353$	1045.17	0.157588	0.0787940	+	0.996891i	$0.474893\pi$
$354$	0	0
$355$	912.454	0.136417
$356$	0	0
$357$	−1820.16	−0.269840
$358$	0	0
$359$	2640.87	0.388245	0.194122	−	0.980977i	$-0.437814\pi$
$359$	2640.87	0.388245	0.194122	+	0.980977i	$0.437814\pi$
$360$	0	0
$361$	8149.84	1.18820
$362$	0	0
$363$	−1713.17	−0.247708
$364$	0	0
$365$	4942.17	0.708727
$366$	0	0
$367$	−166.143	−0.0236311	−0.0118155	−	0.999930i	$-0.503761\pi$
$367$	−166.143	−0.0236311	−0.0118155	+	0.999930i	$0.503761\pi$
$368$	0	0
$369$	2566.12	0.362024
$370$	0	0
$371$	−2998.16	−0.419560
$372$	0	0
$373$	12311.1	1.70896	0.854480	−	0.519484i	$-0.173876\pi$
$373$	12311.1	1.70896	0.854480	+	0.519484i	$0.173876\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	−4881.39	−0.666855
$378$	0	0
$379$	4943.54	0.670007	0.335004	−	0.942217i	$-0.391262\pi$
$379$	4943.54	0.670007	0.335004	+	0.942217i	$0.391262\pi$
$380$	0	0
$381$	−1534.49	−0.206337
$382$	0	0
$383$	−11086.7	−1.47912	−0.739562	−	0.673088i	$-0.764967\pi$
$383$	−11086.7	−1.47912	−0.739562	+	0.673088i	$0.764967\pi$
$384$	0	0
$385$	1526.44	0.202064
$386$	0	0
$387$	−3468.31	−0.455567
$388$	0	0
$389$	−272.697	−0.0355431	−0.0177716	−	0.999842i	$-0.505657\pi$
$389$	−272.697	−0.0355431	−0.0177716	+	0.999842i	$0.505657\pi$
$390$	0	0
$391$	2205.77	0.285296
$392$	0	0
$393$	8645.66	1.10971
$394$	0	0
$395$	2185.20	0.278353
$396$	0	0
$397$	−14255.5	−1.80218	−0.901088	−	0.433637i	$-0.857230\pi$
$397$	−14255.5	−1.80218	−0.901088	+	0.433637i	$0.857230\pi$
$398$	0	0
$399$	2572.72	0.322800
$400$	0	0
$401$	13445.0	1.67434	0.837171	−	0.546941i	$-0.184208\pi$
$401$	13445.0	1.67434	0.837171	+	0.546941i	$0.184208\pi$
$402$	0	0
$403$	1111.28	0.137362
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	6905.92	0.841066
$408$	0	0
$409$	−764.564	−0.0924334	−0.0462167	−	0.998931i	$-0.514716\pi$
$409$	−764.564	−0.0924334	−0.0462167	+	0.998931i	$0.514716\pi$
$410$	0	0
$411$	−1625.95	−0.195140
$412$	0	0
$413$	3788.59	0.451391
$414$	0	0
$415$	3173.76	0.375407
$416$	0	0
$417$	3560.50	0.418125
$418$	0	0
$419$	−5682.20	−0.662514	−0.331257	−	0.943540i	$-0.607473\pi$
$419$	−5682.20	−0.662514	−0.331257	+	0.943540i	$0.607473\pi$
$420$	0	0
$421$	2443.20	0.282836	0.141418	−	0.989950i	$-0.454834\pi$
$421$	2443.20	0.282836	0.141418	+	0.989950i	$0.454834\pi$
$422$	0	0
$423$	−2031.99	−0.233566
$424$	0	0
$425$	−2166.85	−0.247313
$426$	0	0
$427$	−313.442	−0.0355235
$428$	0	0
$429$	4397.79	0.494936
$430$	0	0
$431$	8799.01	0.983373	0.491686	−	0.870772i	$-0.336381\pi$
$431$	8799.01	0.983373	0.491686	+	0.870772i	$0.336381\pi$
$432$	0	0
$433$	9790.83	1.08665	0.543323	−	0.839524i	$-0.317166\pi$
$433$	9790.83	1.08665	0.543323	+	0.839524i	$0.317166\pi$
$434$	0	0
$435$	2178.38	0.240104
$436$	0	0
$437$	−3117.77	−0.341289
$438$	0	0
$439$	−17875.3	−1.94337	−0.971687	−	0.236271i	$-0.924075\pi$
$439$	−17875.3	−1.94337	−0.971687	+	0.236271i	$0.924075\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−10407.7	−1.11622	−0.558110	−	0.829767i	$-0.688473\pi$
$443$	−10407.7	−1.11622	−0.558110	+	0.829767i	$0.688473\pi$
$444$	0	0
$445$	6683.71	0.711996
$446$	0	0
$447$	5889.84	0.623222
$448$	0	0
$449$	877.278	0.0922078	0.0461039	−	0.998937i	$-0.485319\pi$
$449$	877.278	0.0922078	0.0461039	+	0.998937i	$0.485319\pi$
$450$	0	0
$451$	12435.0	1.29832
$452$	0	0
$453$	2761.57	0.286423
$454$	0	0
$455$	−1176.44	−0.121214
$456$	0	0
$457$	6739.70	0.689869	0.344934	−	0.938627i	$-0.387901\pi$
$457$	6739.70	0.689869	0.344934	+	0.938627i	$0.387901\pi$
$458$	0	0
$459$	2340.20	0.237977
$460$	0	0
$461$	4053.51	0.409525	0.204762	−	0.978812i	$-0.434358\pi$
$461$	4053.51	0.409525	0.204762	+	0.978812i	$0.434358\pi$
$462$	0	0
$463$	−7688.40	−0.771728	−0.385864	−	0.922556i	$-0.626097\pi$
$463$	−7688.40	−0.771728	−0.385864	+	0.922556i	$0.626097\pi$
$464$	0	0
$465$	−495.924	−0.0494579
$466$	0	0
$467$	−7130.29	−0.706532	−0.353266	−	0.935523i	$-0.614929\pi$
$467$	−7130.29	−0.706532	−0.353266	+	0.935523i	$0.614929\pi$
$468$	0	0
$469$	1714.87	0.168839
$470$	0	0
$471$	−8678.09	−0.848970
$472$	0	0
$473$	−16806.9	−1.63379
$474$	0	0
$475$	3062.76	0.295851
$476$	0	0
$477$	3854.78	0.370018
$478$	0	0
$479$	−9476.48	−0.903949	−0.451974	−	0.892031i	$-0.649280\pi$
$479$	−9476.48	−0.903949	−0.451974	+	0.892031i	$0.649280\pi$
$480$	0	0
$481$	−5322.45	−0.504538
$482$	0	0
$483$	−534.429	−0.0503465
$484$	0	0
$485$	2094.78	0.196122
$486$	0	0
$487$	14935.3	1.38970	0.694850	−	0.719155i	$-0.255471\pi$
$487$	14935.3	1.38970	0.694850	+	0.719155i	$0.255471\pi$
$488$	0	0
$489$	7225.69	0.668214
$490$	0	0
$491$	−13636.7	−1.25340	−0.626698	−	0.779262i	$-0.715594\pi$
$491$	−13636.7	−1.25340	−0.626698	+	0.779262i	$0.715594\pi$
$492$	0	0
$493$	−12587.3	−1.14990
$494$	0	0
$495$	−1962.57	−0.178204
$496$	0	0
$497$	1277.44	0.115293
$498$	0	0
$499$	548.198	0.0491798	0.0245899	−	0.999698i	$-0.492172\pi$
$499$	548.198	0.0491798	0.0245899	+	0.999698i	$0.492172\pi$
$500$	0	0
$501$	8047.59	0.717645
$502$	0	0
$503$	−4734.56	−0.419689	−0.209844	−	0.977735i	$-0.567296\pi$
$503$	−4734.56	−0.419689	−0.209844	+	0.977735i	$0.567296\pi$
$504$	0	0
$505$	4021.34	0.354351
$506$	0	0
$507$	3201.58	0.280448
$508$	0	0
$509$	15606.1	1.35899	0.679495	−	0.733680i	$-0.262199\pi$
$509$	15606.1	1.35899	0.679495	+	0.733680i	$0.262199\pi$
$510$	0	0
$511$	6919.04	0.598983
$512$	0	0
$513$	−3307.79	−0.284683
$514$	0	0
$515$	−9791.89	−0.837830
$516$	0	0
$517$	−9846.68	−0.837633
$518$	0	0
$519$	−1296.39	−0.109644
$520$	0	0
$521$	3248.19	0.273139	0.136570	−	0.990630i	$-0.456392\pi$
$521$	3248.19	0.273139	0.136570	+	0.990630i	$0.456392\pi$
$522$	0	0
$523$	−8129.71	−0.679708	−0.339854	−	0.940478i	$-0.610378\pi$
$523$	−8129.71	−0.679708	−0.339854	+	0.940478i	$0.610378\pi$
$524$	0	0
$525$	525.000	0.0436436
$526$	0	0
$527$	2865.58	0.236863
$528$	0	0
$529$	−11519.3	−0.946770
$530$	0	0
$531$	−4871.05	−0.398090
$532$	0	0
$533$	−9583.77	−0.778835
$534$	0	0
$535$	10004.6	0.808480
$536$	0	0
$537$	−1534.73	−0.123330
$538$	0	0
$539$	2137.02	0.170775
$540$	0	0
$541$	−2709.72	−0.215342	−0.107671	−	0.994187i	$-0.534339\pi$
$541$	−2709.72	−0.215342	−0.107671	+	0.994187i	$0.534339\pi$
$542$	0	0
$543$	8687.82	0.686612
$544$	0	0
$545$	−1915.44	−0.150547
$546$	0	0
$547$	−1018.76	−0.0796326	−0.0398163	−	0.999207i	$-0.512677\pi$
$547$	−1018.76	−0.0796326	−0.0398163	+	0.999207i	$0.512677\pi$
$548$	0	0
$549$	402.997	0.0313288
$550$	0	0
$551$	17791.6	1.37559
$552$	0	0
$553$	3059.29	0.235252
$554$	0	0
$555$	2375.20	0.181661
$556$	0	0
$557$	−5045.16	−0.383789	−0.191894	−	0.981416i	$-0.561463\pi$
$557$	−5045.16	−0.383789	−0.191894	+	0.981416i	$0.561463\pi$
$558$	0	0
$559$	12953.2	0.980076
$560$	0	0
$561$	11340.2	0.853450
$562$	0	0
$563$	6370.11	0.476853	0.238426	−	0.971161i	$-0.423368\pi$
$563$	6370.11	0.476853	0.238426	+	0.971161i	$0.423368\pi$
$564$	0	0
$565$	−5404.21	−0.402401
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−18225.0	−1.34276	−0.671382	−	0.741111i	$-0.734299\pi$
$569$	−18225.0	−1.34276	−0.671382	+	0.741111i	$0.734299\pi$
$570$	0	0
$571$	−2289.86	−0.167824	−0.0839121	−	0.996473i	$-0.526741\pi$
$571$	−2289.86	−0.167824	−0.0839121	+	0.996473i	$0.526741\pi$
$572$	0	0
$573$	−7851.01	−0.572392
$574$	0	0
$575$	−636.225	−0.0461433
$576$	0	0
$577$	3758.40	0.271169	0.135584	−	0.990766i	$-0.456709\pi$
$577$	3758.40	0.271169	0.135584	+	0.990766i	$0.456709\pi$
$578$	0	0
$579$	−416.783	−0.0299152
$580$	0	0
$581$	4443.27	0.317277
$582$	0	0
$583$	18679.7	1.32699
$584$	0	0
$585$	1512.57	0.106901
$586$	0	0
$587$	−5374.98	−0.377937	−0.188969	−	0.981983i	$-0.560514\pi$
$587$	−5374.98	−0.377937	−0.188969	+	0.981983i	$0.560514\pi$
$588$	0	0
$589$	−4050.39	−0.283351
$590$	0	0
$591$	11303.2	0.786722
$592$	0	0
$593$	−22616.9	−1.56621	−0.783106	−	0.621888i	$-0.786366\pi$
$593$	−22616.9	−1.56621	−0.783106	+	0.621888i	$0.786366\pi$
$594$	0	0
$595$	−3033.60	−0.209017
$596$	0	0
$597$	716.604	0.0491267
$598$	0	0
$599$	−5098.46	−0.347775	−0.173888	−	0.984766i	$-0.555633\pi$
$599$	−5098.46	−0.347775	−0.173888	+	0.984766i	$0.555633\pi$
$600$	0	0
$601$	16140.6	1.09549	0.547745	−	0.836645i	$-0.315486\pi$
$601$	16140.6	1.09549	0.547745	+	0.836645i	$0.315486\pi$
$602$	0	0
$603$	−2204.83	−0.148902
$604$	0	0
$605$	−2855.28	−0.191874
$606$	0	0
$607$	24797.5	1.65815	0.829077	−	0.559135i	$-0.188867\pi$
$607$	24797.5	1.65815	0.829077	+	0.559135i	$0.188867\pi$
$608$	0	0
$609$	3049.73	0.202925
$610$	0	0
$611$	7588.92	0.502479
$612$	0	0
$613$	14367.2	0.946632	0.473316	−	0.880893i	$-0.343057\pi$
$613$	14367.2	0.946632	0.473316	+	0.880893i	$0.343057\pi$
$614$	0	0
$615$	4276.87	0.280423
$616$	0	0
$617$	−1486.16	−0.0969699	−0.0484849	−	0.998824i	$-0.515439\pi$
$617$	−1486.16	−0.0969699	−0.0484849	+	0.998824i	$0.515439\pi$
$618$	0	0
$619$	4047.27	0.262801	0.131400	−	0.991329i	$-0.458053\pi$
$619$	4047.27	0.262801	0.131400	+	0.991329i	$0.458053\pi$
$620$	0	0
$621$	687.123	0.0444014
$622$	0	0
$623$	9357.19	0.601746
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−16029.0	−1.02095
$628$	0	0
$629$	−13724.6	−0.870008
$630$	0	0
$631$	−8304.27	−0.523911	−0.261956	−	0.965080i	$-0.584367\pi$
$631$	−8304.27	−0.523911	−0.261956	+	0.965080i	$0.584367\pi$
$632$	0	0
$633$	93.7192	0.00588468
$634$	0	0
$635$	−2557.48	−0.159828
$636$	0	0
$637$	−1647.02	−0.102445
$638$	0	0
$639$	−1642.42	−0.101679
$640$	0	0
$641$	19916.6	1.22724	0.613618	−	0.789603i	$-0.289714\pi$
$641$	19916.6	1.22724	0.613618	+	0.789603i	$0.289714\pi$
$642$	0	0
$643$	−2751.60	−0.168760	−0.0843799	−	0.996434i	$-0.526891\pi$
$643$	−2751.60	−0.168760	−0.0843799	+	0.996434i	$0.526891\pi$
$644$	0	0
$645$	−5780.52	−0.352880
$646$	0	0
$647$	−20003.7	−1.21550	−0.607750	−	0.794129i	$-0.707928\pi$
$647$	−20003.7	−1.21550	−0.607750	+	0.794129i	$0.707928\pi$
$648$	0	0
$649$	−23604.3	−1.42766
$650$	0	0
$651$	−694.293	−0.0417995
$652$	0	0
$653$	20024.2	1.20001	0.600005	−	0.799996i	$-0.295165\pi$
$653$	20024.2	1.20001	0.600005	+	0.799996i	$0.295165\pi$
$654$	0	0
$655$	14409.4	0.859577
$656$	0	0
$657$	−8895.91	−0.528254
$658$	0	0
$659$	−29213.6	−1.72686	−0.863430	−	0.504469i	$-0.831688\pi$
$659$	−29213.6	−1.72686	−0.863430	+	0.504469i	$0.831688\pi$
$660$	0	0
$661$	−26257.8	−1.54510	−0.772548	−	0.634957i	$-0.781018\pi$
$661$	−26257.8	−1.54510	−0.772548	+	0.634957i	$0.781018\pi$
$662$	0	0
$663$	−8740.02	−0.511967
$664$	0	0
$665$	4287.87	0.250040
$666$	0	0
$667$	−3695.83	−0.214548
$668$	0	0
$669$	−6427.10	−0.371429
$670$	0	0
$671$	1952.86	0.112354
$672$	0	0
$673$	−31083.6	−1.78036	−0.890181	−	0.455606i	$-0.849423\pi$
$673$	−31083.6	−1.78036	−0.890181	+	0.455606i	$0.849423\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	−7341.96	−0.416801	−0.208401	−	0.978044i	$-0.566826\pi$
$677$	−7341.96	−0.416801	−0.208401	+	0.978044i	$0.566826\pi$
$678$	0	0
$679$	2932.69	0.165753
$680$	0	0
$681$	−202.036	−0.0113686
$682$	0	0
$683$	18541.4	1.03875	0.519377	−	0.854545i	$-0.326164\pi$
$683$	18541.4	1.03875	0.519377	+	0.854545i	$0.326164\pi$
$684$	0	0
$685$	−2709.92	−0.151155
$686$	0	0
$687$	−12049.3	−0.669156
$688$	0	0
$689$	−14396.6	−0.796032
$690$	0	0
$691$	−4576.60	−0.251957	−0.125978	−	0.992033i	$-0.540207\pi$
$691$	−4576.60	−0.251957	−0.125978	+	0.992033i	$0.540207\pi$
$692$	0	0
$693$	−2747.59	−0.150610
$694$	0	0
$695$	5934.16	0.323879
$696$	0	0
$697$	−24712.9	−1.34300
$698$	0	0
$699$	−8690.05	−0.470226
$700$	0	0
$701$	−11071.0	−0.596499	−0.298249	−	0.954488i	$-0.596403\pi$
$701$	−11071.0	−0.596499	−0.298249	+	0.954488i	$0.596403\pi$
$702$	0	0
$703$	19399.2	1.04076
$704$	0	0
$705$	−3386.64	−0.180920
$706$	0	0
$707$	5629.88	0.299482
$708$	0	0
$709$	12475.9	0.660852	0.330426	−	0.943832i	$-0.392808\pi$
$709$	12475.9	0.660852	0.330426	+	0.943832i	$0.392808\pi$
$710$	0	0
$711$	−3933.37	−0.207472
$712$	0	0
$713$	841.384	0.0441936
$714$	0	0
$715$	7329.66	0.383376
$716$	0	0
$717$	15858.7	0.826015
$718$	0	0
$719$	−24824.7	−1.28763	−0.643815	−	0.765181i	$-0.722649\pi$
$719$	−24824.7	−1.28763	−0.643815	+	0.765181i	$0.722649\pi$
$720$	0	0
$721$	−13708.6	−0.708095
$722$	0	0
$723$	−12208.4	−0.627986
$724$	0	0
$725$	3630.63	0.185984
$726$	0	0
$727$	−36455.6	−1.85979	−0.929893	−	0.367829i	$-0.880101\pi$
$727$	−36455.6	−1.85979	−0.929893	+	0.367829i	$0.880101\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	33401.5	1.69001
$732$	0	0
$733$	−14124.5	−0.711732	−0.355866	−	0.934537i	$-0.615814\pi$
$733$	−14124.5	−0.711732	−0.355866	+	0.934537i	$0.615814\pi$
$734$	0	0
$735$	735.000	0.0368856
$736$	0	0
$737$	−10684.3	−0.534003
$738$	0	0
$739$	−29272.5	−1.45711	−0.728557	−	0.684986i	$-0.759808\pi$
$739$	−29272.5	−1.45711	−0.728557	+	0.684986i	$0.759808\pi$
$740$	0	0
$741$	12353.7	0.612448
$742$	0	0
$743$	10705.4	0.528589	0.264294	−	0.964442i	$-0.414861\pi$
$743$	10705.4	0.528589	0.264294	+	0.964442i	$0.414861\pi$
$744$	0	0
$745$	9816.41	0.482745
$746$	0	0
$747$	−5712.77	−0.279812
$748$	0	0
$749$	14006.4	0.683290
$750$	0	0
$751$	13921.9	0.676457	0.338228	−	0.941064i	$-0.390172\pi$
$751$	13921.9	0.676457	0.338228	+	0.941064i	$0.390172\pi$
$752$	0	0
$753$	20280.9	0.981508
$754$	0	0
$755$	4602.61	0.221863
$756$	0	0
$757$	−30160.8	−1.44810	−0.724050	−	0.689747i	$-0.757722\pi$
$757$	−30160.8	−1.44810	−0.724050	+	0.689747i	$0.757722\pi$
$758$	0	0
$759$	3329.69	0.159236
$760$	0	0
$761$	12882.8	0.613670	0.306835	−	0.951763i	$-0.400730\pi$
$761$	12882.8	0.613670	0.306835	+	0.951763i	$0.400730\pi$
$762$	0	0
$763$	−2681.61	−0.127236
$764$	0	0
$765$	3900.34	0.184336
$766$	0	0
$767$	18192.1	0.856424
$768$	0	0
$769$	−41394.6	−1.94113	−0.970564	−	0.240842i	$-0.922576\pi$
$769$	−41394.6	−1.94113	−0.970564	+	0.240842i	$0.922576\pi$
$770$	0	0
$771$	16438.7	0.767866
$772$	0	0
$773$	11180.8	0.520239	0.260119	−	0.965576i	$-0.416238\pi$
$773$	11180.8	0.520239	0.260119	+	0.965576i	$0.416238\pi$
$774$	0	0
$775$	−826.539	−0.0383099
$776$	0	0
$777$	3325.29	0.153532
$778$	0	0
$779$	34930.8	1.60658
$780$	0	0
$781$	−7958.89	−0.364650
$782$	0	0
$783$	−3921.08	−0.178963
$784$	0	0
$785$	−14463.5	−0.657610
$786$	0	0
$787$	33954.3	1.53792	0.768959	−	0.639298i	$-0.220775\pi$
$787$	33954.3	1.53792	0.768959	+	0.639298i	$0.220775\pi$
$788$	0	0
$789$	1672.83	0.0754809
$790$	0	0
$791$	−7565.89	−0.340091
$792$	0	0
$793$	−1505.09	−0.0673987
$794$	0	0
$795$	6424.64	0.286614
$796$	0	0
$797$	−20436.1	−0.908263	−0.454131	−	0.890935i	$-0.650050\pi$
$797$	−20436.1	−0.908263	−0.454131	+	0.890935i	$0.650050\pi$
$798$	0	0
$799$	19569.0	0.866458
$800$	0	0
$801$	−12030.7	−0.530690
$802$	0	0
$803$	−43108.2	−1.89446
$804$	0	0
$805$	−890.715	−0.0389982
$806$	0	0
$807$	14733.3	0.642673
$808$	0	0
$809$	4624.34	0.200968	0.100484	−	0.994939i	$-0.467961\pi$
$809$	4624.34	0.200968	0.100484	+	0.994939i	$0.467961\pi$
$810$	0	0
$811$	14943.4	0.647023	0.323511	−	0.946224i	$-0.395137\pi$
$811$	14943.4	0.647023	0.323511	+	0.946224i	$0.395137\pi$
$812$	0	0
$813$	−18904.9	−0.815526
$814$	0	0
$815$	12042.8	0.517597
$816$	0	0
$817$	−47211.7	−2.02170
$818$	0	0
$819$	2117.59	0.0903476
$820$	0	0
$821$	−15212.7	−0.646684	−0.323342	−	0.946282i	$-0.604806\pi$
$821$	−15212.7	−0.646684	−0.323342	+	0.946282i	$0.604806\pi$
$822$	0	0
$823$	4450.00	0.188478	0.0942390	−	0.995550i	$-0.469958\pi$
$823$	4450.00	0.188478	0.0942390	+	0.995550i	$0.469958\pi$
$824$	0	0
$825$	−3270.94	−0.138036
$826$	0	0
$827$	−12702.4	−0.534107	−0.267054	−	0.963682i	$-0.586050\pi$
$827$	−12702.4	−0.534107	−0.267054	+	0.963682i	$0.586050\pi$
$828$	0	0
$829$	18908.1	0.792164	0.396082	−	0.918215i	$-0.370370\pi$
$829$	18908.1	0.792164	0.396082	+	0.918215i	$0.370370\pi$
$830$	0	0
$831$	−10949.1	−0.457066
$832$	0	0
$833$	−4247.03	−0.176652
$834$	0	0
$835$	13412.7	0.555885
$836$	0	0
$837$	892.662	0.0368637
$838$	0	0
$839$	−3192.93	−0.131385	−0.0656925	−	0.997840i	$-0.520926\pi$
$839$	−3192.93	−0.131385	−0.0656925	+	0.997840i	$0.520926\pi$
$840$	0	0
$841$	−3298.65	−0.135252
$842$	0	0
$843$	−1734.85	−0.0708794
$844$	0	0
$845$	5335.97	0.217234
$846$	0	0
$847$	−3997.40	−0.162163
$848$	0	0
$849$	−21797.6	−0.881145
$850$	0	0
$851$	−4029.77	−0.162325
$852$	0	0
$853$	17579.9	0.705655	0.352827	−	0.935688i	$-0.385220\pi$
$853$	17579.9	0.705655	0.352827	+	0.935688i	$0.385220\pi$
$854$	0	0
$855$	−5512.98	−0.220514
$856$	0	0
$857$	−44839.3	−1.78726	−0.893629	−	0.448807i	$-0.851849\pi$
$857$	−44839.3	−1.78726	−0.893629	+	0.448807i	$0.851849\pi$
$858$	0	0
$859$	−33893.1	−1.34624	−0.673118	−	0.739535i	$-0.735046\pi$
$859$	−33893.1	−1.34624	−0.673118	+	0.739535i	$0.735046\pi$
$860$	0	0
$861$	5987.61	0.237000
$862$	0	0
$863$	−46411.4	−1.83066	−0.915332	−	0.402700i	$-0.868072\pi$
$863$	−46411.4	−1.83066	−0.915332	+	0.402700i	$0.868072\pi$
$864$	0	0
$865$	−2160.64	−0.0849295
$866$	0	0
$867$	−7798.22	−0.305469
$868$	0	0
$869$	−19060.5	−0.744054
$870$	0	0
$871$	8234.46	0.320338
$872$	0	0
$873$	−3770.60	−0.146180
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	−38925.3	−1.49876	−0.749382	−	0.662138i	$-0.769649\pi$
$877$	−38925.3	−1.49876	−0.749382	+	0.662138i	$0.769649\pi$
$878$	0	0
$879$	−11465.3	−0.439950
$880$	0	0
$881$	27018.9	1.03324	0.516622	−	0.856213i	$-0.327189\pi$
$881$	27018.9	1.03324	0.516622	+	0.856213i	$0.327189\pi$
$882$	0	0
$883$	−17707.9	−0.674880	−0.337440	−	0.941347i	$-0.609561\pi$
$883$	−17707.9	−0.674880	−0.337440	+	0.941347i	$0.609561\pi$
$884$	0	0
$885$	−8118.42	−0.308359
$886$	0	0
$887$	−3078.09	−0.116519	−0.0582594	−	0.998301i	$-0.518555\pi$
$887$	−3078.09	−0.116519	−0.0582594	+	0.998301i	$0.518555\pi$
$888$	0	0
$889$	−3580.48	−0.135079
$890$	0	0
$891$	3532.62	0.132825
$892$	0	0
$893$	−27660.0	−1.03651
$894$	0	0
$895$	−2557.88	−0.0955311
$896$	0	0
$897$	−2566.22	−0.0955224
$898$	0	0
$899$	−4801.37	−0.178125
$900$	0	0
$901$	−37123.3	−1.37265
$902$	0	0
$903$	−8092.73	−0.298238
$904$	0	0
$905$	14479.7	0.531847
$906$	0	0
$907$	−29103.8	−1.06547	−0.532733	−	0.846284i	$-0.678835\pi$
$907$	−29103.8	−1.06547	−0.532733	+	0.846284i	$0.678835\pi$
$908$	0	0
$909$	−7238.42	−0.264118
$910$	0	0
$911$	18644.6	0.678073	0.339037	−	0.940773i	$-0.389899\pi$
$911$	18644.6	0.678073	0.339037	+	0.940773i	$0.389899\pi$
$912$	0	0
$913$	−27683.2	−1.00348
$914$	0	0
$915$	671.662	0.0242672
$916$	0	0
$917$	20173.2	0.726475
$918$	0	0
$919$	44196.5	1.58641	0.793203	−	0.608957i	$-0.208412\pi$
$919$	44196.5	1.58641	0.793203	+	0.608957i	$0.208412\pi$
$920$	0	0
$921$	11501.8	0.411506
$922$	0	0
$923$	6133.98	0.218746
$924$	0	0
$925$	3958.67	0.140714
$926$	0	0
$927$	17625.4	0.624481
$928$	0	0
$929$	5307.06	0.187426	0.0937132	−	0.995599i	$-0.470126\pi$
$929$	5307.06	0.187426	0.0937132	+	0.995599i	$0.470126\pi$
$930$	0	0
$931$	6003.02	0.211322
$932$	0	0
$933$	−10303.0	−0.361526
$934$	0	0
$935$	18900.4	0.661080
$936$	0	0
$937$	15158.6	0.528505	0.264253	−	0.964454i	$-0.414875\pi$
$937$	15158.6	0.528505	0.264253	+	0.964454i	$0.414875\pi$
$938$	0	0
$939$	−20082.5	−0.697942
$940$	0	0
$941$	39704.6	1.37549	0.687743	−	0.725954i	$-0.258602\pi$
$941$	39704.6	1.37549	0.687743	+	0.725954i	$0.258602\pi$
$942$	0	0
$943$	−7256.13	−0.250575
$944$	0	0
$945$	−945.000	−0.0325300
$946$	0	0
$947$	40502.9	1.38983	0.694913	−	0.719093i	$-0.255443\pi$
$947$	40502.9	1.38983	0.694913	+	0.719093i	$0.255443\pi$
$948$	0	0
$949$	33223.8	1.13645
$950$	0	0
$951$	11316.5	0.385871
$952$	0	0
$953$	53381.6	1.81448	0.907239	−	0.420615i	$-0.138186\pi$
$953$	53381.6	1.81448	0.907239	+	0.420615i	$0.138186\pi$
$954$	0	0
$955$	−13085.0	−0.443373
$956$	0	0
$957$	−19000.9	−0.641811
$958$	0	0
$959$	−3793.89	−0.127749
$960$	0	0
$961$	−28697.9	−0.963309
$962$	0	0
$963$	−18008.3	−0.602605
$964$	0	0
$965$	−694.638	−0.0231722
$966$	0	0
$967$	43794.1	1.45638	0.728192	−	0.685373i	$-0.240361\pi$
$967$	43794.1	1.45638	0.728192	+	0.685373i	$0.240361\pi$
$968$	0	0
$969$	31855.5	1.05608
$970$	0	0
$971$	−32515.3	−1.07463	−0.537315	−	0.843382i	$-0.680561\pi$
$971$	−32515.3	−1.07463	−0.537315	+	0.843382i	$0.680561\pi$
$972$	0	0
$973$	8307.83	0.273727
$974$	0	0
$975$	2520.94	0.0828049
$976$	0	0
$977$	41204.2	1.34927	0.674636	−	0.738150i	$-0.264301\pi$
$977$	41204.2	1.34927	0.674636	+	0.738150i	$0.264301\pi$
$978$	0	0
$979$	−58298.7	−1.90320
$980$	0	0
$981$	3447.79	0.112211
$982$	0	0
$983$	31333.5	1.01667	0.508333	−	0.861161i	$-0.330262\pi$
$983$	31333.5	1.01667	0.508333	+	0.861161i	$0.330262\pi$
$984$	0	0
$985$	18838.7	0.609393
$986$	0	0
$987$	−4741.30	−0.152905
$988$	0	0
$989$	9807.23	0.315320
$990$	0	0
$991$	−6807.75	−0.218219	−0.109110	−	0.994030i	$-0.534800\pi$
$991$	−6807.75	−0.218219	−0.109110	+	0.994030i	$0.534800\pi$
$992$	0	0
$993$	−16422.3	−0.524821
$994$	0	0
$995$	1194.34	0.0380534
$996$	0	0
$997$	−33301.6	−1.05785	−0.528923	−	0.848670i	$-0.677404\pi$
$997$	−33301.6	−1.05785	−0.528923	+	0.848670i	$0.677404\pi$
$998$	0	0
$999$	−4275.37	−0.135402

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.4.a.p.1.3	✓	3	1.1	even	1	trivial
1680.4.a.bs.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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +43.6126 q^{11} -33.6126 q^{13} +15.0000 q^{15} -86.6741 q^{17} +122.511 q^{19} +21.0000 q^{21} -25.4490 q^{23} +25.0000 q^{25} -27.0000 q^{27} +145.225 q^{29} -33.0616 q^{31} -130.838 q^{33} +35.0000 q^{35} +158.347 q^{37} +100.838 q^{39} +285.124 q^{41} -385.368 q^{43} -45.0000 q^{45} -225.776 q^{47} +49.0000 q^{49} +260.022 q^{51} +428.309 q^{53} -218.063 q^{55} -367.532 q^{57} -541.228 q^{59} +44.7775 q^{61} -63.0000 q^{63} +168.063 q^{65} -244.981 q^{67} +76.3470 q^{69} -182.491 q^{71} -988.435 q^{73} -75.0000 q^{75} -305.288 q^{77} -437.041 q^{79} +81.0000 q^{81} -634.752 q^{83} +433.371 q^{85} -435.675 q^{87} -1336.74 q^{89} +235.288 q^{91} +99.1847 q^{93} -612.553 q^{95} -418.956 q^{97} +392.513 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} - 15 q^{5} - 21 q^{7} + 27 q^{9} - 6 q^{11} + 36 q^{13} + 45 q^{15} + 90 q^{17} + 78 q^{19} + 63 q^{21} + 75 q^{25} - 81 q^{27} + 162 q^{29} + 114 q^{31} + 18 q^{33} + 105 q^{35} + 246 q^{37}+ \cdots - 54 q^{99}+O(q^{100}) \) 3 * q - 9 * q^3 - 15 * q^5 - 21 * q^7 + 27 * q^9 - 6 * q^11 + 36 * q^13 + 45 * q^15 + 90 * q^17 + 78 * q^19 + 63 * q^21 + 75 * q^25 - 81 * q^27 + 162 * q^29 + 114 * q^31 + 18 * q^33 + 105 * q^35 + 246 * q^37 - 108 * q^39 - 42 * q^41 - 348 * q^43 - 135 * q^45 - 480 * q^47 + 147 * q^49 - 270 * q^51 - 252 * q^53 + 30 * q^55 - 234 * q^57 - 408 * q^59 - 534 * q^61 - 189 * q^63 - 180 * q^65 - 372 * q^67 - 366 * q^71 - 60 * q^73 - 225 * q^75 + 42 * q^77 - 624 * q^79 + 243 * q^81 - 3228 * q^83 - 450 * q^85 - 486 * q^87 - 510 * q^89 - 252 * q^91 - 342 * q^93 - 390 * q^95 - 2304 * q^97 - 54 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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