LMFDB - Embedded newform 840.4.a.s.1.2

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 = [3,0,9,0,15,0,21,0,27,0,36] 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:	3.3.79853.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 67x + 228 $ x^3 - x^2 - 67*x + 228
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.75974$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +29.8155 q^{11} +77.8935 q^{13} +15.0000 q^{15} -28.0780 q^{17} +129.971 q^{19} +21.0000 q^{21} -185.496 q^{23} +25.0000 q^{25} +27.0000 q^{27} -187.496 q^{29} +16.6313 q^{31} +89.4466 q^{33} +35.0000 q^{35} -106.971 q^{37} +233.680 q^{39} +384.836 q^{41} -290.177 q^{43} +45.0000 q^{45} +530.333 q^{47} +49.0000 q^{49} -84.2339 q^{51} +167.603 q^{53} +149.078 q^{55} +389.914 q^{57} -75.7870 q^{59} -471.226 q^{61} +63.0000 q^{63} +389.467 q^{65} +138.078 q^{67} -556.488 q^{69} +272.999 q^{71} +706.304 q^{73} +75.0000 q^{75} +208.709 q^{77} +538.467 q^{79} +81.0000 q^{81} -648.721 q^{83} -140.390 q^{85} -562.488 q^{87} -626.623 q^{89} +545.254 q^{91} +49.8940 q^{93} +649.857 q^{95} +565.892 q^{97} +268.340 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 9 q^{3} + 15 q^{5} + 21 q^{7} + 27 q^{9} + 36 q^{11} + 50 q^{13} + 45 q^{15} + 46 q^{17} + 76 q^{19} + 63 q^{21} + 48 q^{23} + 75 q^{25} + 81 q^{27} + 42 q^{29} + 80 q^{31} + 108 q^{33} + 105 q^{35}+ \cdots + 324 q^{99}+O(q^{100}) $ 3 * q + 9 * q^3 + 15 * q^5 + 21 * q^7 + 27 * q^9 + 36 * q^11 + 50 * q^13 + 45 * q^15 + 46 * q^17 + 76 * q^19 + 63 * q^21 + 48 * q^23 + 75 * q^25 + 81 * q^27 + 42 * q^29 + 80 * q^31 + 108 * q^33 + 105 * q^35 + 130 * q^37 + 150 * q^39 + 206 * q^41 + 148 * q^43 + 135 * q^45 + 312 * q^47 + 147 * q^49 + 138 * q^51 + 82 * q^53 + 180 * q^55 + 228 * q^57 + 140 * q^59 + 186 * q^61 + 189 * q^63 + 250 * q^65 + 284 * q^67 + 144 * q^69 + 408 * q^71 + 526 * q^73 + 225 * q^75 + 252 * q^77 + 560 * q^79 + 243 * q^81 + 532 * q^83 + 230 * q^85 + 126 * q^87 - 290 * q^89 + 350 * q^91 + 240 * q^93 + 380 * q^95 + 966 * q^97 + 324 * 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
$11$	29.8155	0.817248	0.408624	−	0.912703i	$-0.366009\pi$
$11$	29.8155	0.817248	0.408624	+	0.912703i	$0.366009\pi$
$12$	0	0
$13$	77.8935	1.66183	0.830914	−	0.556401i	$-0.187818\pi$
$13$	77.8935	1.66183	0.830914	+	0.556401i	$0.187818\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−28.0780	−0.400583	−0.200291	−	0.979736i	$-0.564189\pi$
$17$	−28.0780	−0.400583	−0.200291	+	0.979736i	$0.564189\pi$
$18$	0	0
$19$	129.971	1.56934	0.784671	−	0.619913i	$-0.212832\pi$
$19$	129.971	1.56934	0.784671	+	0.619913i	$0.212832\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	−185.496	−1.68168	−0.840839	−	0.541285i	$-0.817938\pi$
$23$	−185.496	−1.68168	−0.840839	+	0.541285i	$0.817938\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−187.496	−1.20059	−0.600295	−	0.799778i	$-0.704950\pi$
$29$	−187.496	−1.20059	−0.600295	+	0.799778i	$0.704950\pi$
$30$	0	0
$31$	16.6313	0.0963573	0.0481787	−	0.998839i	$-0.484658\pi$
$31$	16.6313	0.0963573	0.0481787	+	0.998839i	$0.484658\pi$
$32$	0	0
$33$	89.4466	0.471838
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	−106.971	−0.475296	−0.237648	−	0.971351i	$-0.576376\pi$
$37$	−106.971	−0.475296	−0.237648	+	0.971351i	$0.576376\pi$
$38$	0	0
$39$	233.680	0.959457
$40$	0	0
$41$	384.836	1.46589	0.732943	−	0.680290i	$-0.238146\pi$
$41$	384.836	1.46589	0.732943	+	0.680290i	$0.238146\pi$
$42$	0	0
$43$	−290.177	−1.02911	−0.514553	−	0.857459i	$-0.672042\pi$
$43$	−290.177	−1.02911	−0.514553	+	0.857459i	$0.672042\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	530.333	1.64589	0.822946	−	0.568119i	$-0.192329\pi$
$47$	530.333	1.64589	0.822946	+	0.568119i	$0.192329\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−84.2339	−0.231276
$52$	0	0
$53$	167.603	0.434377	0.217188	−	0.976130i	$-0.430311\pi$
$53$	167.603	0.434377	0.217188	+	0.976130i	$0.430311\pi$
$54$	0	0
$55$	149.078	0.365484
$56$	0	0
$57$	389.914	0.906060
$58$	0	0
$59$	−75.7870	−0.167231	−0.0836155	−	0.996498i	$-0.526647\pi$
$59$	−75.7870	−0.167231	−0.0836155	+	0.996498i	$0.526647\pi$
$60$	0	0
$61$	−471.226	−0.989087	−0.494543	−	0.869153i	$-0.664665\pi$
$61$	−471.226	−0.989087	−0.494543	+	0.869153i	$0.664665\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	389.467	0.743192
$66$	0	0
$67$	138.078	0.251775	0.125887	−	0.992045i	$-0.459822\pi$
$67$	138.078	0.251775	0.125887	+	0.992045i	$0.459822\pi$
$68$	0	0
$69$	−556.488	−0.970917
$70$	0	0
$71$	272.999	0.456325	0.228162	−	0.973623i	$-0.426728\pi$
$71$	272.999	0.456325	0.228162	+	0.973623i	$0.426728\pi$
$72$	0	0
$73$	706.304	1.13242	0.566210	−	0.824261i	$-0.308409\pi$
$73$	706.304	1.13242	0.566210	+	0.824261i	$0.308409\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	208.709	0.308891
$78$	0	0
$79$	538.467	0.766864	0.383432	−	0.923569i	$-0.374742\pi$
$79$	538.467	0.766864	0.383432	+	0.923569i	$0.374742\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−648.721	−0.857909	−0.428954	−	0.903326i	$-0.641118\pi$
$83$	−648.721	−0.857909	−0.428954	+	0.903326i	$0.641118\pi$
$84$	0	0
$85$	−140.390	−0.179146
$86$	0	0
$87$	−562.488	−0.693161
$88$	0	0
$89$	−626.623	−0.746313	−0.373157	−	0.927768i	$-0.621725\pi$
$89$	−626.623	−0.746313	−0.373157	+	0.927768i	$0.621725\pi$
$90$	0	0
$91$	545.254	0.628112
$92$	0	0
$93$	49.8940	0.0556319
$94$	0	0
$95$	649.857	0.701831
$96$	0	0
$97$	565.892	0.592347	0.296174	−	0.955134i	$-0.404289\pi$
$97$	565.892	0.592347	0.296174	+	0.955134i	$0.404289\pi$
$98$	0	0
$99$	268.340	0.272416
$100$	0	0
$101$	1169.19	1.15187	0.575934	−	0.817496i	$-0.304638\pi$
$101$	1169.19	1.15187	0.575934	+	0.817496i	$0.304638\pi$
$102$	0	0
$103$	1093.67	1.04624	0.523120	−	0.852259i	$-0.324768\pi$
$103$	1093.67	1.04624	0.523120	+	0.852259i	$0.324768\pi$
$104$	0	0
$105$	105.000	0.0975900
$106$	0	0
$107$	−876.898	−0.792270	−0.396135	−	0.918192i	$-0.629649\pi$
$107$	−876.898	−0.792270	−0.396135	+	0.918192i	$0.629649\pi$
$108$	0	0
$109$	−464.836	−0.408470	−0.204235	−	0.978922i	$-0.565471\pi$
$109$	−464.836	−0.408470	−0.204235	+	0.978922i	$0.565471\pi$
$110$	0	0
$111$	−320.914	−0.274412
$112$	0	0
$113$	−878.226	−0.731120	−0.365560	−	0.930788i	$-0.619122\pi$
$113$	−878.226	−0.731120	−0.365560	+	0.930788i	$0.619122\pi$
$114$	0	0
$115$	−927.480	−0.752069
$116$	0	0
$117$	701.041	0.553943
$118$	0	0
$119$	−196.546	−0.151406
$120$	0	0
$121$	−442.034	−0.332106
$122$	0	0
$123$	1154.51	0.846329
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1812.03	1.26607	0.633037	−	0.774122i	$-0.281808\pi$
$127$	1812.03	1.26607	0.633037	+	0.774122i	$0.281808\pi$
$128$	0	0
$129$	−870.530	−0.594154
$130$	0	0
$131$	1001.67	0.668065	0.334032	−	0.942562i	$-0.391590\pi$
$131$	1001.67	0.668065	0.334032	+	0.942562i	$0.391590\pi$
$132$	0	0
$133$	909.800	0.593155
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	−1166.21	−0.727272	−0.363636	−	0.931541i	$-0.618465\pi$
$137$	−1166.21	−0.727272	−0.363636	+	0.931541i	$0.618465\pi$
$138$	0	0
$139$	1178.17	0.718928	0.359464	−	0.933159i	$-0.382960\pi$
$139$	1178.17	0.718928	0.359464	+	0.933159i	$0.382960\pi$
$140$	0	0
$141$	1591.00	0.950257
$142$	0	0
$143$	2322.44	1.35813
$144$	0	0
$145$	−937.480	−0.536921
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	719.947	0.395841	0.197921	−	0.980218i	$-0.436581\pi$
$149$	719.947	0.395841	0.197921	+	0.980218i	$0.436581\pi$
$150$	0	0
$151$	−2391.00	−1.28859	−0.644294	−	0.764778i	$-0.722849\pi$
$151$	−2391.00	−1.28859	−0.644294	+	0.764778i	$0.722849\pi$
$152$	0	0
$153$	−252.702	−0.133528
$154$	0	0
$155$	83.1567	0.0430923
$156$	0	0
$157$	−14.9421	−0.00759560	−0.00379780	−	0.999993i	$-0.501209\pi$
$157$	−14.9421	−0.00759560	−0.00379780	+	0.999993i	$0.501209\pi$
$158$	0	0
$159$	502.808	0.250788
$160$	0	0
$161$	−1298.47	−0.635615
$162$	0	0
$163$	3289.95	1.58091	0.790455	−	0.612520i	$-0.209844\pi$
$163$	3289.95	1.58091	0.790455	+	0.612520i	$0.209844\pi$
$164$	0	0
$165$	447.233	0.211012
$166$	0	0
$167$	604.093	0.279917	0.139958	−	0.990157i	$-0.455303\pi$
$167$	604.093	0.279917	0.139958	+	0.990157i	$0.455303\pi$
$168$	0	0
$169$	3870.40	1.76167
$170$	0	0
$171$	1169.74	0.523114
$172$	0	0
$173$	−2457.72	−1.08010	−0.540048	−	0.841634i	$-0.681594\pi$
$173$	−2457.72	−1.08010	−0.540048	+	0.841634i	$0.681594\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	0	0
$177$	−227.361	−0.0965508
$178$	0	0
$179$	418.636	0.174806	0.0874031	−	0.996173i	$-0.472143\pi$
$179$	418.636	0.174806	0.0874031	+	0.996173i	$0.472143\pi$
$180$	0	0
$181$	1754.59	0.720538	0.360269	−	0.932848i	$-0.382685\pi$
$181$	1754.59	0.720538	0.360269	+	0.932848i	$0.382685\pi$
$182$	0	0
$183$	−1413.68	−0.571049
$184$	0	0
$185$	−534.856	−0.212559
$186$	0	0
$187$	−837.159	−0.327375
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	704.277	0.266805	0.133402	−	0.991062i	$-0.457410\pi$
$191$	704.277	0.266805	0.133402	+	0.991062i	$0.457410\pi$
$192$	0	0
$193$	−813.843	−0.303532	−0.151766	−	0.988416i	$-0.548496\pi$
$193$	−813.843	−0.303532	−0.151766	+	0.988416i	$0.548496\pi$
$194$	0	0
$195$	1168.40	0.429082
$196$	0	0
$197$	−3737.63	−1.35175	−0.675875	−	0.737016i	$-0.736234\pi$
$197$	−3737.63	−1.35175	−0.675875	+	0.737016i	$0.736234\pi$
$198$	0	0
$199$	−2276.16	−0.810817	−0.405408	−	0.914136i	$-0.632871\pi$
$199$	−2276.16	−0.810817	−0.405408	+	0.914136i	$0.632871\pi$
$200$	0	0
$201$	414.234	0.145362
$202$	0	0
$203$	−1312.47	−0.453781
$204$	0	0
$205$	1924.18	0.655564
$206$	0	0
$207$	−1669.46	−0.560559
$208$	0	0
$209$	3875.17	1.28254
$210$	0	0
$211$	319.819	0.104347	0.0521736	−	0.998638i	$-0.483385\pi$
$211$	319.819	0.104347	0.0521736	+	0.998638i	$0.483385\pi$
$212$	0	0
$213$	818.998	0.263459
$214$	0	0
$215$	−1450.88	−0.460230
$216$	0	0
$217$	116.419	0.0364196
$218$	0	0
$219$	2118.91	0.653803
$220$	0	0
$221$	−2187.09	−0.665699
$222$	0	0
$223$	−2485.86	−0.746483	−0.373241	−	0.927734i	$-0.621754\pi$
$223$	−2485.86	−0.746483	−0.373241	+	0.927734i	$0.621754\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	3791.10	1.10848	0.554238	−	0.832358i	$-0.313010\pi$
$227$	3791.10	1.10848	0.554238	+	0.832358i	$0.313010\pi$
$228$	0	0
$229$	−5659.08	−1.63302	−0.816512	−	0.577329i	$-0.804095\pi$
$229$	−5659.08	−1.63302	−0.816512	+	0.577329i	$0.804095\pi$
$230$	0	0
$231$	626.126	0.178338
$232$	0	0
$233$	−5623.08	−1.58103	−0.790516	−	0.612441i	$-0.790188\pi$
$233$	−5623.08	−1.58103	−0.790516	+	0.612441i	$0.790188\pi$
$234$	0	0
$235$	2651.66	0.736066
$236$	0	0
$237$	1615.40	0.442749
$238$	0	0
$239$	−2675.51	−0.724119	−0.362059	−	0.932155i	$-0.617926\pi$
$239$	−2675.51	−0.724119	−0.362059	+	0.932155i	$0.617926\pi$
$240$	0	0
$241$	3532.97	0.944311	0.472155	−	0.881515i	$-0.343476\pi$
$241$	3532.97	0.944311	0.472155	+	0.881515i	$0.343476\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	245.000	0.0638877
$246$	0	0
$247$	10123.9	2.60798
$248$	0	0
$249$	−1946.16	−0.495314
$250$	0	0
$251$	6738.38	1.69451	0.847256	−	0.531185i	$-0.178253\pi$
$251$	6738.38	1.69451	0.847256	+	0.531185i	$0.178253\pi$
$252$	0	0
$253$	−5530.66	−1.37435
$254$	0	0
$255$	−421.169	−0.103430
$256$	0	0
$257$	2500.63	0.606946	0.303473	−	0.952840i	$-0.401854\pi$
$257$	2500.63	0.606946	0.303473	+	0.952840i	$0.401854\pi$
$258$	0	0
$259$	−748.798	−0.179645
$260$	0	0
$261$	−1687.46	−0.400197
$262$	0	0
$263$	2692.99	0.631395	0.315698	−	0.948860i	$-0.397761\pi$
$263$	2692.99	0.631395	0.315698	+	0.948860i	$0.397761\pi$
$264$	0	0
$265$	838.013	0.194259
$266$	0	0
$267$	−1879.87	−0.430884
$268$	0	0
$269$	−7265.79	−1.64685	−0.823426	−	0.567424i	$-0.807940\pi$
$269$	−7265.79	−1.64685	−0.823426	+	0.567424i	$0.807940\pi$
$270$	0	0
$271$	7991.87	1.79141	0.895704	−	0.444651i	$-0.146672\pi$
$271$	7991.87	1.79141	0.895704	+	0.444651i	$0.146672\pi$
$272$	0	0
$273$	1635.76	0.362641
$274$	0	0
$275$	745.388	0.163450
$276$	0	0
$277$	−546.762	−0.118598	−0.0592991	−	0.998240i	$-0.518887\pi$
$277$	−546.762	−0.118598	−0.0592991	+	0.998240i	$0.518887\pi$
$278$	0	0
$279$	149.682	0.0321191
$280$	0	0
$281$	−4387.55	−0.931457	−0.465729	−	0.884928i	$-0.654208\pi$
$281$	−4387.55	−0.931457	−0.465729	+	0.884928i	$0.654208\pi$
$282$	0	0
$283$	5565.87	1.16910	0.584552	−	0.811356i	$-0.301270\pi$
$283$	5565.87	1.16910	0.584552	+	0.811356i	$0.301270\pi$
$284$	0	0
$285$	1949.57	0.405202
$286$	0	0
$287$	2693.85	0.554053
$288$	0	0
$289$	−4124.63	−0.839534
$290$	0	0
$291$	1697.68	0.341992
$292$	0	0
$293$	−4355.58	−0.868449	−0.434224	−	0.900805i	$-0.642978\pi$
$293$	−4355.58	−0.868449	−0.434224	+	0.900805i	$0.642978\pi$
$294$	0	0
$295$	−378.935	−0.0747879
$296$	0	0
$297$	805.020	0.157279
$298$	0	0
$299$	−14448.9	−2.79466
$300$	0	0
$301$	−2031.24	−0.388965
$302$	0	0
$303$	3507.57	0.665032
$304$	0	0
$305$	−2356.13	−0.442333
$306$	0	0
$307$	2068.98	0.384634	0.192317	−	0.981333i	$-0.438400\pi$
$307$	2068.98	0.384634	0.192317	+	0.981333i	$0.438400\pi$
$308$	0	0
$309$	3281.02	0.604047
$310$	0	0
$311$	−5390.40	−0.982835	−0.491418	−	0.870924i	$-0.663521\pi$
$311$	−5390.40	−0.982835	−0.491418	+	0.870924i	$0.663521\pi$
$312$	0	0
$313$	−4070.92	−0.735149	−0.367575	−	0.929994i	$-0.619812\pi$
$313$	−4070.92	−0.735149	−0.367575	+	0.929994i	$0.619812\pi$
$314$	0	0
$315$	315.000	0.0563436
$316$	0	0
$317$	−4795.15	−0.849598	−0.424799	−	0.905288i	$-0.639655\pi$
$317$	−4795.15	−0.849598	−0.424799	+	0.905288i	$0.639655\pi$
$318$	0	0
$319$	−5590.29	−0.981180
$320$	0	0
$321$	−2630.69	−0.457417
$322$	0	0
$323$	−3649.33	−0.628651
$324$	0	0
$325$	1947.34	0.332366
$326$	0	0
$327$	−1394.51	−0.235830
$328$	0	0
$329$	3712.33	0.622089
$330$	0	0
$331$	3655.45	0.607015	0.303507	−	0.952829i	$-0.401842\pi$
$331$	3655.45	0.607015	0.303507	+	0.952829i	$0.401842\pi$
$332$	0	0
$333$	−962.741	−0.158432
$334$	0	0
$335$	690.390	0.112597
$336$	0	0
$337$	−10832.1	−1.75092	−0.875461	−	0.483290i	$-0.839442\pi$
$337$	−10832.1	−1.75092	−0.875461	+	0.483290i	$0.839442\pi$
$338$	0	0
$339$	−2634.68	−0.422112
$340$	0	0
$341$	495.872	0.0787478
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−2782.44	−0.434207
$346$	0	0
$347$	−9468.42	−1.46482	−0.732408	−	0.680866i	$-0.761604\pi$
$347$	−9468.42	−1.46482	−0.732408	+	0.680866i	$0.761604\pi$
$348$	0	0
$349$	−10596.8	−1.62532	−0.812659	−	0.582739i	$-0.801981\pi$
$349$	−10596.8	−1.62532	−0.812659	+	0.582739i	$0.801981\pi$
$350$	0	0
$351$	2103.12	0.319819
$352$	0	0
$353$	9564.40	1.44210	0.721050	−	0.692883i	$-0.243660\pi$
$353$	9564.40	1.44210	0.721050	+	0.692883i	$0.243660\pi$
$354$	0	0
$355$	1365.00	0.204075
$356$	0	0
$357$	−589.637	−0.0874143
$358$	0	0
$359$	−9841.20	−1.44679	−0.723396	−	0.690433i	$-0.757420\pi$
$359$	−9841.20	−1.44679	−0.723396	+	0.690433i	$0.757420\pi$
$360$	0	0
$361$	10033.6	1.46283
$362$	0	0
$363$	−1326.10	−0.191742
$364$	0	0
$365$	3531.52	0.506434
$366$	0	0
$367$	567.255	0.0806825	0.0403413	−	0.999186i	$-0.487155\pi$
$367$	567.255	0.0806825	0.0403413	+	0.999186i	$0.487155\pi$
$368$	0	0
$369$	3463.53	0.488628
$370$	0	0
$371$	1173.22	0.164179
$372$	0	0
$373$	1388.26	0.192712	0.0963560	−	0.995347i	$-0.469281\pi$
$373$	1388.26	0.192712	0.0963560	+	0.995347i	$0.469281\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	−14604.7	−1.99518
$378$	0	0
$379$	13163.5	1.78407	0.892036	−	0.451965i	$-0.149277\pi$
$379$	13163.5	1.78407	0.892036	+	0.451965i	$0.149277\pi$
$380$	0	0
$381$	5436.08	0.730968
$382$	0	0
$383$	−14445.1	−1.92718	−0.963589	−	0.267386i	$-0.913840\pi$
$383$	−14445.1	−1.92718	−0.963589	+	0.267386i	$0.913840\pi$
$384$	0	0
$385$	1043.54	0.138140
$386$	0	0
$387$	−2611.59	−0.343035
$388$	0	0
$389$	12118.3	1.57949	0.789747	−	0.613433i	$-0.210212\pi$
$389$	12118.3	1.57949	0.789747	+	0.613433i	$0.210212\pi$
$390$	0	0
$391$	5208.35	0.673651
$392$	0	0
$393$	3005.02	0.385707
$394$	0	0
$395$	2692.34	0.342952
$396$	0	0
$397$	−14256.7	−1.80233	−0.901164	−	0.433477i	$-0.857286\pi$
$397$	−14256.7	−1.80233	−0.901164	+	0.433477i	$0.857286\pi$
$398$	0	0
$399$	2729.40	0.342458
$400$	0	0
$401$	9.50036	0.00118311	0.000591553	−	1.00000i	$-0.499812\pi$
$401$	9.50036	0.00118311	0.000591553		1.00000i	$0.499812\pi$
$402$	0	0
$403$	1295.47	0.160129
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	−3189.40	−0.388435
$408$	0	0
$409$	−2879.25	−0.348092	−0.174046	−	0.984737i	$-0.555684\pi$
$409$	−2879.25	−0.348092	−0.174046	+	0.984737i	$0.555684\pi$
$410$	0	0
$411$	−3498.64	−0.419891
$412$	0	0
$413$	−530.509	−0.0632073
$414$	0	0
$415$	−3243.61	−0.383668
$416$	0	0
$417$	3534.51	0.415073
$418$	0	0
$419$	−5527.10	−0.644431	−0.322215	−	0.946666i	$-0.604428\pi$
$419$	−5527.10	−0.644431	−0.322215	+	0.946666i	$0.604428\pi$
$420$	0	0
$421$	−1747.81	−0.202335	−0.101167	−	0.994869i	$-0.532258\pi$
$421$	−1747.81	−0.202335	−0.101167	+	0.994869i	$0.532258\pi$
$422$	0	0
$423$	4772.99	0.548631
$424$	0	0
$425$	−701.949	−0.0801165
$426$	0	0
$427$	−3298.58	−0.373840
$428$	0	0
$429$	6967.31	0.784114
$430$	0	0
$431$	−6288.80	−0.702833	−0.351416	−	0.936219i	$-0.614300\pi$
$431$	−6288.80	−0.702833	−0.351416	+	0.936219i	$0.614300\pi$
$432$	0	0
$433$	−10816.9	−1.20053	−0.600263	−	0.799803i	$-0.704937\pi$
$433$	−10816.9	−1.20053	−0.600263	+	0.799803i	$0.704937\pi$
$434$	0	0
$435$	−2812.44	−0.309991
$436$	0	0
$437$	−24109.2	−2.63913
$438$	0	0
$439$	8418.06	0.915198	0.457599	−	0.889159i	$-0.348709\pi$
$439$	8418.06	0.915198	0.457599	+	0.889159i	$0.348709\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−8854.55	−0.949644	−0.474822	−	0.880082i	$-0.657488\pi$
$443$	−8854.55	−0.949644	−0.474822	+	0.880082i	$0.657488\pi$
$444$	0	0
$445$	−3133.11	−0.333761
$446$	0	0
$447$	2159.84	0.228539
$448$	0	0
$449$	3742.98	0.393412	0.196706	−	0.980462i	$-0.436975\pi$
$449$	3742.98	0.393412	0.196706	+	0.980462i	$0.436975\pi$
$450$	0	0
$451$	11474.1	1.19799
$452$	0	0
$453$	−7173.00	−0.743967
$454$	0	0
$455$	2726.27	0.280900
$456$	0	0
$457$	−7611.40	−0.779095	−0.389548	−	0.921006i	$-0.627369\pi$
$457$	−7611.40	−0.779095	−0.389548	+	0.921006i	$0.627369\pi$
$458$	0	0
$459$	−758.105	−0.0770922
$460$	0	0
$461$	−10901.1	−1.10133	−0.550665	−	0.834727i	$-0.685626\pi$
$461$	−10901.1	−1.10133	−0.550665	+	0.834727i	$0.685626\pi$
$462$	0	0
$463$	−14958.9	−1.50151	−0.750755	−	0.660581i	$-0.770310\pi$
$463$	−14958.9	−1.50151	−0.750755	+	0.660581i	$0.770310\pi$
$464$	0	0
$465$	249.470	0.0248794
$466$	0	0
$467$	−391.313	−0.0387748	−0.0193874	−	0.999812i	$-0.506172\pi$
$467$	−391.313	−0.0387748	−0.0193874	+	0.999812i	$0.506172\pi$
$468$	0	0
$469$	966.546	0.0951619
$470$	0	0
$471$	−44.8263	−0.00438532
$472$	0	0
$473$	−8651.78	−0.841034
$474$	0	0
$475$	3249.29	0.313868
$476$	0	0
$477$	1508.42	0.144792
$478$	0	0
$479$	−3512.61	−0.335063	−0.167531	−	0.985867i	$-0.553580\pi$
$479$	−3512.61	−0.335063	−0.167531	+	0.985867i	$0.553580\pi$
$480$	0	0
$481$	−8332.36	−0.789860
$482$	0	0
$483$	−3895.42	−0.366972
$484$	0	0
$485$	2829.46	0.264906
$486$	0	0
$487$	−12027.8	−1.11917	−0.559583	−	0.828775i	$-0.689039\pi$
$487$	−12027.8	−1.11917	−0.559583	+	0.828775i	$0.689039\pi$
$488$	0	0
$489$	9869.84	0.912739
$490$	0	0
$491$	6430.72	0.591067	0.295534	−	0.955332i	$-0.404503\pi$
$491$	6430.72	0.591067	0.295534	+	0.955332i	$0.404503\pi$
$492$	0	0
$493$	5264.50	0.480936
$494$	0	0
$495$	1341.70	0.121828
$496$	0	0
$497$	1910.99	0.172474
$498$	0	0
$499$	−15084.4	−1.35325	−0.676625	−	0.736327i	$-0.736558\pi$
$499$	−15084.4	−1.35325	−0.676625	+	0.736327i	$0.736558\pi$
$500$	0	0
$501$	1812.28	0.161610
$502$	0	0
$503$	13464.9	1.19358	0.596788	−	0.802399i	$-0.296443\pi$
$503$	13464.9	1.19358	0.596788	+	0.802399i	$0.296443\pi$
$504$	0	0
$505$	5845.95	0.515131
$506$	0	0
$507$	11611.2	1.01710
$508$	0	0
$509$	−13625.6	−1.18653	−0.593263	−	0.805008i	$-0.702161\pi$
$509$	−13625.6	−1.18653	−0.593263	+	0.805008i	$0.702161\pi$
$510$	0	0
$511$	4944.13	0.428014
$512$	0	0
$513$	3509.23	0.302020
$514$	0	0
$515$	5468.36	0.467893
$516$	0	0
$517$	15812.2	1.34510
$518$	0	0
$519$	−7373.15	−0.623594
$520$	0	0
$521$	14367.8	1.20818	0.604092	−	0.796914i	$-0.293536\pi$
$521$	14367.8	1.20818	0.604092	+	0.796914i	$0.293536\pi$
$522$	0	0
$523$	1548.22	0.129443	0.0647216	−	0.997903i	$-0.479384\pi$
$523$	1548.22	0.129443	0.0647216	+	0.997903i	$0.479384\pi$
$524$	0	0
$525$	525.000	0.0436436
$526$	0	0
$527$	−466.974	−0.0385991
$528$	0	0
$529$	22241.8	1.82804
$530$	0	0
$531$	−682.083	−0.0557436
$532$	0	0
$533$	29976.2	2.43605
$534$	0	0
$535$	−4384.49	−0.354314
$536$	0	0
$537$	1255.91	0.100924
$538$	0	0
$539$	1460.96	0.116750
$540$	0	0
$541$	19041.9	1.51327	0.756633	−	0.653839i	$-0.226843\pi$
$541$	19041.9	1.51327	0.756633	+	0.653839i	$0.226843\pi$
$542$	0	0
$543$	5263.76	0.416003
$544$	0	0
$545$	−2324.18	−0.182673
$546$	0	0
$547$	−19812.1	−1.54863	−0.774317	−	0.632798i	$-0.781906\pi$
$547$	−19812.1	−1.54863	−0.774317	+	0.632798i	$0.781906\pi$
$548$	0	0
$549$	−4241.03	−0.329696
$550$	0	0
$551$	−24369.1	−1.88414
$552$	0	0
$553$	3769.27	0.289848
$554$	0	0
$555$	−1604.57	−0.122721
$556$	0	0
$557$	11494.2	0.874375	0.437188	−	0.899370i	$-0.355975\pi$
$557$	11494.2	0.874375	0.437188	+	0.899370i	$0.355975\pi$
$558$	0	0
$559$	−22602.9	−1.71020
$560$	0	0
$561$	−2511.48	−0.189010
$562$	0	0
$563$	−1776.27	−0.132968	−0.0664839	−	0.997787i	$-0.521178\pi$
$563$	−1776.27	−0.132968	−0.0664839	+	0.997787i	$0.521178\pi$
$564$	0	0
$565$	−4391.13	−0.326967
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	−4941.21	−0.364054	−0.182027	−	0.983294i	$-0.558266\pi$
$569$	−4941.21	−0.364054	−0.182027	+	0.983294i	$0.558266\pi$
$570$	0	0
$571$	17039.1	1.24880	0.624398	−	0.781106i	$-0.285344\pi$
$571$	17039.1	1.24880	0.624398	+	0.781106i	$0.285344\pi$
$572$	0	0
$573$	2112.83	0.154040
$574$	0	0
$575$	−4637.40	−0.336336
$576$	0	0
$577$	−10568.8	−0.762536	−0.381268	−	0.924465i	$-0.624512\pi$
$577$	−10568.8	−0.762536	−0.381268	+	0.924465i	$0.624512\pi$
$578$	0	0
$579$	−2441.53	−0.175244
$580$	0	0
$581$	−4541.05	−0.324259
$582$	0	0
$583$	4997.16	0.354993
$584$	0	0
$585$	3505.21	0.247731
$586$	0	0
$587$	−26447.8	−1.85965	−0.929826	−	0.367999i	$-0.880043\pi$
$587$	−26447.8	−1.85965	−0.929826	+	0.367999i	$0.880043\pi$
$588$	0	0
$589$	2161.60	0.151218
$590$	0	0
$591$	−11212.9	−0.780433
$592$	0	0
$593$	2986.35	0.206804	0.103402	−	0.994640i	$-0.467027\pi$
$593$	2986.35	0.206804	0.103402	+	0.994640i	$0.467027\pi$
$594$	0	0
$595$	−982.728	−0.0677108
$596$	0	0
$597$	−6828.47	−0.468125
$598$	0	0
$599$	6665.58	0.454672	0.227336	−	0.973816i	$-0.426998\pi$
$599$	6665.58	0.454672	0.227336	+	0.973816i	$0.426998\pi$
$600$	0	0
$601$	16493.8	1.11946	0.559729	−	0.828675i	$-0.310905\pi$
$601$	16493.8	1.11946	0.559729	+	0.828675i	$0.310905\pi$
$602$	0	0
$603$	1242.70	0.0839249
$604$	0	0
$605$	−2210.17	−0.148523
$606$	0	0
$607$	−13215.3	−0.883679	−0.441839	−	0.897094i	$-0.645674\pi$
$607$	−13215.3	−0.883679	−0.441839	+	0.897094i	$0.645674\pi$
$608$	0	0
$609$	−3937.42	−0.261990
$610$	0	0
$611$	41309.5	2.73519
$612$	0	0
$613$	−16667.9	−1.09822	−0.549111	−	0.835750i	$-0.685033\pi$
$613$	−16667.9	−1.09822	−0.549111	+	0.835750i	$0.685033\pi$
$614$	0	0
$615$	5772.54	0.378490
$616$	0	0
$617$	15660.8	1.02185	0.510923	−	0.859627i	$-0.329304\pi$
$617$	15660.8	1.02185	0.510923	+	0.859627i	$0.329304\pi$
$618$	0	0
$619$	29769.4	1.93301	0.966505	−	0.256648i	$-0.0826182\pi$
$619$	29769.4	1.93301	0.966505	+	0.256648i	$0.0826182\pi$
$620$	0	0
$621$	−5008.39	−0.323639
$622$	0	0
$623$	−4386.36	−0.282080
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	11625.5	0.740475
$628$	0	0
$629$	3003.53	0.190395
$630$	0	0
$631$	1789.65	0.112908	0.0564538	−	0.998405i	$-0.482021\pi$
$631$	1789.65	0.112908	0.0564538	+	0.998405i	$0.482021\pi$
$632$	0	0
$633$	959.457	0.0602448
$634$	0	0
$635$	9060.13	0.566205
$636$	0	0
$637$	3816.78	0.237404
$638$	0	0
$639$	2456.99	0.152108
$640$	0	0
$641$	30542.7	1.88200	0.941001	−	0.338405i	$-0.109887\pi$
$641$	30542.7	1.88200	0.941001	+	0.338405i	$0.109887\pi$
$642$	0	0
$643$	30530.7	1.87249	0.936246	−	0.351346i	$-0.114276\pi$
$643$	30530.7	1.87249	0.936246	+	0.351346i	$0.114276\pi$
$644$	0	0
$645$	−4352.65	−0.265714
$646$	0	0
$647$	−5584.47	−0.339332	−0.169666	−	0.985502i	$-0.554269\pi$
$647$	−5584.47	−0.339332	−0.169666	+	0.985502i	$0.554269\pi$
$648$	0	0
$649$	−2259.63	−0.136669
$650$	0	0
$651$	349.258	0.0210269
$652$	0	0
$653$	28563.9	1.71178	0.855890	−	0.517158i	$-0.173010\pi$
$653$	28563.9	1.71178	0.855890	+	0.517158i	$0.173010\pi$
$654$	0	0
$655$	5008.36	0.298768
$656$	0	0
$657$	6356.74	0.377473
$658$	0	0
$659$	−27049.9	−1.59896	−0.799480	−	0.600693i	$-0.794892\pi$
$659$	−27049.9	−1.59896	−0.799480	+	0.600693i	$0.794892\pi$
$660$	0	0
$661$	14059.7	0.827319	0.413660	−	0.910432i	$-0.364250\pi$
$661$	14059.7	0.827319	0.413660	+	0.910432i	$0.364250\pi$
$662$	0	0
$663$	−6561.27	−0.384342
$664$	0	0
$665$	4549.00	0.265267
$666$	0	0
$667$	34779.8	2.01901
$668$	0	0
$669$	−7457.59	−0.430982
$670$	0	0
$671$	−14049.9	−0.808329
$672$	0	0
$673$	−11466.4	−0.656754	−0.328377	−	0.944547i	$-0.606502\pi$
$673$	−11466.4	−0.656754	−0.328377	+	0.944547i	$0.606502\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	−29429.1	−1.67068	−0.835340	−	0.549734i	$-0.814729\pi$
$677$	−29429.1	−1.67068	−0.835340	+	0.549734i	$0.814729\pi$
$678$	0	0
$679$	3961.25	0.223886
$680$	0	0
$681$	11373.3	0.639979
$682$	0	0
$683$	−8962.58	−0.502114	−0.251057	−	0.967972i	$-0.580778\pi$
$683$	−8962.58	−0.502114	−0.251057	+	0.967972i	$0.580778\pi$
$684$	0	0
$685$	−5831.06	−0.325246
$686$	0	0
$687$	−16977.2	−0.942827
$688$	0	0
$689$	13055.1	0.721860
$690$	0	0
$691$	−429.183	−0.0236279	−0.0118140	−	0.999930i	$-0.503761\pi$
$691$	−429.183	−0.0236279	−0.0118140	+	0.999930i	$0.503761\pi$
$692$	0	0
$693$	1878.38	0.102964
$694$	0	0
$695$	5890.85	0.321514
$696$	0	0
$697$	−10805.4	−0.587208
$698$	0	0
$699$	−16869.2	−0.912809
$700$	0	0
$701$	−597.896	−0.0322143	−0.0161071	−	0.999870i	$-0.505127\pi$
$701$	−597.896	−0.0322143	−0.0161071	+	0.999870i	$0.505127\pi$
$702$	0	0
$703$	−13903.2	−0.745902
$704$	0	0
$705$	7954.99	0.424968
$706$	0	0
$707$	8184.33	0.435365
$708$	0	0
$709$	17195.6	0.910850	0.455425	−	0.890274i	$-0.349487\pi$
$709$	17195.6	0.910850	0.455425	+	0.890274i	$0.349487\pi$
$710$	0	0
$711$	4846.20	0.255621
$712$	0	0
$713$	−3085.05	−0.162042
$714$	0	0
$715$	11612.2	0.607372
$716$	0	0
$717$	−8026.53	−0.418070
$718$	0	0
$719$	4278.40	0.221916	0.110958	−	0.993825i	$-0.464608\pi$
$719$	4278.40	0.221916	0.110958	+	0.993825i	$0.464608\pi$
$720$	0	0
$721$	7655.71	0.395442
$722$	0	0
$723$	10598.9	0.545198
$724$	0	0
$725$	−4687.40	−0.240118
$726$	0	0
$727$	10680.1	0.544845	0.272423	−	0.962178i	$-0.412175\pi$
$727$	10680.1	0.544845	0.272423	+	0.962178i	$0.412175\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	8147.57	0.412242
$732$	0	0
$733$	26337.1	1.32712	0.663562	−	0.748121i	$-0.269044\pi$
$733$	26337.1	1.32712	0.663562	+	0.748121i	$0.269044\pi$
$734$	0	0
$735$	735.000	0.0368856
$736$	0	0
$737$	4116.87	0.205762
$738$	0	0
$739$	−1170.60	−0.0582696	−0.0291348	−	0.999575i	$-0.509275\pi$
$739$	−1170.60	−0.0582696	−0.0291348	+	0.999575i	$0.509275\pi$
$740$	0	0
$741$	30371.8	1.50572
$742$	0	0
$743$	−21396.9	−1.05649	−0.528247	−	0.849091i	$-0.677150\pi$
$743$	−21396.9	−1.05649	−0.528247	+	0.849091i	$0.677150\pi$
$744$	0	0
$745$	3599.74	0.177026
$746$	0	0
$747$	−5838.49	−0.285970
$748$	0	0
$749$	−6138.28	−0.299450
$750$	0	0
$751$	24731.8	1.20170	0.600850	−	0.799362i	$-0.294829\pi$
$751$	24731.8	1.20170	0.600850	+	0.799362i	$0.294829\pi$
$752$	0	0
$753$	20215.1	0.978327
$754$	0	0
$755$	−11955.0	−0.576274
$756$	0	0
$757$	−1375.15	−0.0660249	−0.0330124	−	0.999455i	$-0.510510\pi$
$757$	−1375.15	−0.0660249	−0.0330124	+	0.999455i	$0.510510\pi$
$758$	0	0
$759$	−16592.0	−0.793480
$760$	0	0
$761$	−2821.55	−0.134404	−0.0672018	−	0.997739i	$-0.521407\pi$
$761$	−2821.55	−0.134404	−0.0672018	+	0.997739i	$0.521407\pi$
$762$	0	0
$763$	−3253.85	−0.154387
$764$	0	0
$765$	−1263.51	−0.0597153
$766$	0	0
$767$	−5903.31	−0.277909
$768$	0	0
$769$	−28343.6	−1.32912	−0.664562	−	0.747233i	$-0.731382\pi$
$769$	−28343.6	−1.32912	−0.664562	+	0.747233i	$0.731382\pi$
$770$	0	0
$771$	7501.89	0.350420
$772$	0	0
$773$	−17071.8	−0.794348	−0.397174	−	0.917743i	$-0.630009\pi$
$773$	−17071.8	−0.794348	−0.397174	+	0.917743i	$0.630009\pi$
$774$	0	0
$775$	415.783	0.0192715
$776$	0	0
$777$	−2246.39	−0.103718
$778$	0	0
$779$	50017.7	2.30048
$780$	0	0
$781$	8139.62	0.372930
$782$	0	0
$783$	−5062.39	−0.231054
$784$	0	0
$785$	−74.7105	−0.00339686
$786$	0	0
$787$	21482.7	0.973029	0.486515	−	0.873672i	$-0.338268\pi$
$787$	21482.7	0.973029	0.486515	+	0.873672i	$0.338268\pi$
$788$	0	0
$789$	8078.98	0.364536
$790$	0	0
$791$	−6147.58	−0.276337
$792$	0	0
$793$	−36705.4	−1.64369
$794$	0	0
$795$	2514.04	0.112156
$796$	0	0
$797$	−16836.2	−0.748269	−0.374134	−	0.927375i	$-0.622060\pi$
$797$	−16836.2	−0.748269	−0.374134	+	0.927375i	$0.622060\pi$
$798$	0	0
$799$	−14890.7	−0.659316
$800$	0	0
$801$	−5639.60	−0.248771
$802$	0	0
$803$	21058.8	0.925467
$804$	0	0
$805$	−6492.36	−0.284255
$806$	0	0
$807$	−21797.4	−0.950811
$808$	0	0
$809$	−5794.57	−0.251825	−0.125912	−	0.992041i	$-0.540186\pi$
$809$	−5794.57	−0.251825	−0.125912	+	0.992041i	$0.540186\pi$
$810$	0	0
$811$	−11502.9	−0.498052	−0.249026	−	0.968497i	$-0.580110\pi$
$811$	−11502.9	−0.498052	−0.249026	+	0.968497i	$0.580110\pi$
$812$	0	0
$813$	23975.6	1.03427
$814$	0	0
$815$	16449.7	0.707005
$816$	0	0
$817$	−37714.7	−1.61502
$818$	0	0
$819$	4907.29	0.209371
$820$	0	0
$821$	−11280.7	−0.479537	−0.239768	−	0.970830i	$-0.577071\pi$
$821$	−11280.7	−0.479537	−0.239768	+	0.970830i	$0.577071\pi$
$822$	0	0
$823$	39386.7	1.66820	0.834102	−	0.551610i	$-0.185986\pi$
$823$	39386.7	1.66820	0.834102	+	0.551610i	$0.185986\pi$
$824$	0	0
$825$	2236.17	0.0943676
$826$	0	0
$827$	−3916.64	−0.164686	−0.0823428	−	0.996604i	$-0.526240\pi$
$827$	−3916.64	−0.164686	−0.0823428	+	0.996604i	$0.526240\pi$
$828$	0	0
$829$	40515.5	1.69742	0.848710	−	0.528858i	$-0.177380\pi$
$829$	40515.5	1.69742	0.848710	+	0.528858i	$0.177380\pi$
$830$	0	0
$831$	−1640.29	−0.0684728
$832$	0	0
$833$	−1375.82	−0.0572261
$834$	0	0
$835$	3020.46	0.125183
$836$	0	0
$837$	449.046	0.0185440
$838$	0	0
$839$	−31999.4	−1.31674	−0.658369	−	0.752695i	$-0.728753\pi$
$839$	−31999.4	−1.31674	−0.658369	+	0.752695i	$0.728753\pi$
$840$	0	0
$841$	10765.8	0.441418
$842$	0	0
$843$	−13162.7	−0.537777
$844$	0	0
$845$	19352.0	0.787844
$846$	0	0
$847$	−3094.24	−0.125524
$848$	0	0
$849$	16697.6	0.674983
$850$	0	0
$851$	19842.7	0.799295
$852$	0	0
$853$	−31615.9	−1.26906	−0.634529	−	0.772899i	$-0.718806\pi$
$853$	−31615.9	−1.26906	−0.634529	+	0.772899i	$0.718806\pi$
$854$	0	0
$855$	5848.71	0.233944
$856$	0	0
$857$	31652.3	1.26164	0.630818	−	0.775931i	$-0.282719\pi$
$857$	31652.3	1.26164	0.630818	+	0.775931i	$0.282719\pi$
$858$	0	0
$859$	−12761.4	−0.506886	−0.253443	−	0.967350i	$-0.581563\pi$
$859$	−12761.4	−0.506886	−0.253443	+	0.967350i	$0.581563\pi$
$860$	0	0
$861$	8081.56	0.319882
$862$	0	0
$863$	−27328.3	−1.07794	−0.538972	−	0.842323i	$-0.681187\pi$
$863$	−27328.3	−1.07794	−0.538972	+	0.842323i	$0.681187\pi$
$864$	0	0
$865$	−12288.6	−0.483034
$866$	0	0
$867$	−12373.9	−0.484705
$868$	0	0
$869$	16054.7	0.626718
$870$	0	0
$871$	10755.4	0.418406
$872$	0	0
$873$	5093.03	0.197449
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	−16663.9	−0.641619	−0.320810	−	0.947144i	$-0.603955\pi$
$877$	−16663.9	−0.641619	−0.320810	+	0.947144i	$0.603955\pi$
$878$	0	0
$879$	−13066.7	−0.501399
$880$	0	0
$881$	11100.5	0.424499	0.212250	−	0.977215i	$-0.431921\pi$
$881$	11100.5	0.424499	0.212250	+	0.977215i	$0.431921\pi$
$882$	0	0
$883$	−48077.0	−1.83230	−0.916150	−	0.400835i	$-0.868720\pi$
$883$	−48077.0	−1.83230	−0.916150	+	0.400835i	$0.868720\pi$
$884$	0	0
$885$	−1136.80	−0.0431788
$886$	0	0
$887$	−35284.9	−1.33568	−0.667841	−	0.744304i	$-0.732781\pi$
$887$	−35284.9	−1.33568	−0.667841	+	0.744304i	$0.732781\pi$
$888$	0	0
$889$	12684.2	0.478531
$890$	0	0
$891$	2415.06	0.0908053
$892$	0	0
$893$	68928.1	2.58297
$894$	0	0
$895$	2093.18	0.0781757
$896$	0	0
$897$	−43346.8	−1.61350
$898$	0	0
$899$	−3118.31	−0.115686
$900$	0	0
$901$	−4705.94	−0.174004
$902$	0	0
$903$	−6093.71	−0.224569
$904$	0	0
$905$	8772.93	0.322234
$906$	0	0
$907$	35871.1	1.31321	0.656605	−	0.754235i	$-0.271992\pi$
$907$	35871.1	1.31321	0.656605	+	0.754235i	$0.271992\pi$
$908$	0	0
$909$	10522.7	0.383956
$910$	0	0
$911$	19485.1	0.708638	0.354319	−	0.935125i	$-0.384713\pi$
$911$	19485.1	0.708638	0.354319	+	0.935125i	$0.384713\pi$
$912$	0	0
$913$	−19342.0	−0.701124
$914$	0	0
$915$	−7068.39	−0.255381
$916$	0	0
$917$	7011.71	0.252505
$918$	0	0
$919$	−37981.6	−1.36333	−0.681663	−	0.731666i	$-0.738743\pi$
$919$	−37981.6	−1.36333	−0.681663	+	0.731666i	$0.738743\pi$
$920$	0	0
$921$	6206.93	0.222069
$922$	0	0
$923$	21264.9	0.758333
$924$	0	0
$925$	−2674.28	−0.0950592
$926$	0	0
$927$	9843.05	0.348747
$928$	0	0
$929$	−21714.2	−0.766867	−0.383434	−	0.923568i	$-0.625258\pi$
$929$	−21714.2	−0.766867	−0.383434	+	0.923568i	$0.625258\pi$
$930$	0	0
$931$	6368.60	0.224192
$932$	0	0
$933$	−16171.2	−0.567440
$934$	0	0
$935$	−4185.80	−0.146407
$936$	0	0
$937$	2540.53	0.0885758	0.0442879	−	0.999019i	$-0.485898\pi$
$937$	2540.53	0.0885758	0.0442879	+	0.999019i	$0.485898\pi$
$938$	0	0
$939$	−12212.7	−0.424439
$940$	0	0
$941$	50111.1	1.73600	0.868000	−	0.496565i	$-0.165406\pi$
$941$	50111.1	1.73600	0.868000	+	0.496565i	$0.165406\pi$
$942$	0	0
$943$	−71385.6	−2.46515
$944$	0	0
$945$	945.000	0.0325300
$946$	0	0
$947$	−14172.8	−0.486331	−0.243165	−	0.969985i	$-0.578186\pi$
$947$	−14172.8	−0.486331	−0.243165	+	0.969985i	$0.578186\pi$
$948$	0	0
$949$	55016.5	1.88189
$950$	0	0
$951$	−14385.5	−0.490516
$952$	0	0
$953$	−2767.29	−0.0940621	−0.0470310	−	0.998893i	$-0.514976\pi$
$953$	−2767.29	−0.0940621	−0.0470310	+	0.998893i	$0.514976\pi$
$954$	0	0
$955$	3521.38	0.119319
$956$	0	0
$957$	−16770.9	−0.566484
$958$	0	0
$959$	−8163.49	−0.274883
$960$	0	0
$961$	−29514.4	−0.990715
$962$	0	0
$963$	−7892.08	−0.264090
$964$	0	0
$965$	−4069.22	−0.135744
$966$	0	0
$967$	4657.55	0.154888	0.0774441	−	0.996997i	$-0.475324\pi$
$967$	4657.55	0.154888	0.0774441	+	0.996997i	$0.475324\pi$
$968$	0	0
$969$	−10948.0	−0.362952
$970$	0	0
$971$	27887.3	0.921675	0.460837	−	0.887485i	$-0.347549\pi$
$971$	27887.3	0.921675	0.460837	+	0.887485i	$0.347549\pi$
$972$	0	0
$973$	8247.18	0.271729
$974$	0	0
$975$	5842.01	0.191891
$976$	0	0
$977$	15830.8	0.518396	0.259198	−	0.965824i	$-0.416542\pi$
$977$	15830.8	0.518396	0.259198	+	0.965824i	$0.416542\pi$
$978$	0	0
$979$	−18683.1	−0.609922
$980$	0	0
$981$	−4183.53	−0.136157
$982$	0	0
$983$	47949.9	1.55581	0.777906	−	0.628380i	$-0.216282\pi$
$983$	47949.9	1.55581	0.777906	+	0.628380i	$0.216282\pi$
$984$	0	0
$985$	−18688.1	−0.604521
$986$	0	0
$987$	11137.0	0.359163
$988$	0	0
$989$	53826.6	1.73062
$990$	0	0
$991$	13279.2	0.425657	0.212829	−	0.977090i	$-0.431732\pi$
$991$	13279.2	0.425657	0.212829	+	0.977090i	$0.431732\pi$
$992$	0	0
$993$	10966.4	0.350460
$994$	0	0
$995$	−11380.8	−0.362608
$996$	0	0
$997$	−23018.4	−0.731193	−0.365596	−	0.930773i	$-0.619135\pi$
$997$	−23018.4	−0.731193	−0.365596	+	0.930773i	$0.619135\pi$
$998$	0	0
$999$	−2888.22	−0.0914708

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.4.a.s.1.2	✓	3
4.3	odd	2		1680.4.a.br.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.4.a.s.1.2	✓	3	1.1	even	1	trivial
1680.4.a.br.1.2		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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +29.8155 q^{11} +77.8935 q^{13} +15.0000 q^{15} -28.0780 q^{17} +129.971 q^{19} +21.0000 q^{21} -185.496 q^{23} +25.0000 q^{25} +27.0000 q^{27} -187.496 q^{29} +16.6313 q^{31} +89.4466 q^{33} +35.0000 q^{35} -106.971 q^{37} +233.680 q^{39} +384.836 q^{41} -290.177 q^{43} +45.0000 q^{45} +530.333 q^{47} +49.0000 q^{49} -84.2339 q^{51} +167.603 q^{53} +149.078 q^{55} +389.914 q^{57} -75.7870 q^{59} -471.226 q^{61} +63.0000 q^{63} +389.467 q^{65} +138.078 q^{67} -556.488 q^{69} +272.999 q^{71} +706.304 q^{73} +75.0000 q^{75} +208.709 q^{77} +538.467 q^{79} +81.0000 q^{81} -648.721 q^{83} -140.390 q^{85} -562.488 q^{87} -626.623 q^{89} +545.254 q^{91} +49.8940 q^{93} +649.857 q^{95} +565.892 q^{97} +268.340 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 9 q^{3} + 15 q^{5} + 21 q^{7} + 27 q^{9} + 36 q^{11} + 50 q^{13} + 45 q^{15} + 46 q^{17} + 76 q^{19} + 63 q^{21} + 48 q^{23} + 75 q^{25} + 81 q^{27} + 42 q^{29} + 80 q^{31} + 108 q^{33} + 105 q^{35}+ \cdots + 324 q^{99}+O(q^{100}) \) 3 * q + 9 * q^3 + 15 * q^5 + 21 * q^7 + 27 * q^9 + 36 * q^11 + 50 * q^13 + 45 * q^15 + 46 * q^17 + 76 * q^19 + 63 * q^21 + 48 * q^23 + 75 * q^25 + 81 * q^27 + 42 * q^29 + 80 * q^31 + 108 * q^33 + 105 * q^35 + 130 * q^37 + 150 * q^39 + 206 * q^41 + 148 * q^43 + 135 * q^45 + 312 * q^47 + 147 * q^49 + 138 * q^51 + 82 * q^53 + 180 * q^55 + 228 * q^57 + 140 * q^59 + 186 * q^61 + 189 * q^63 + 250 * q^65 + 284 * q^67 + 144 * q^69 + 408 * q^71 + 526 * q^73 + 225 * q^75 + 252 * q^77 + 560 * q^79 + 243 * q^81 + 532 * q^83 + 230 * q^85 + 126 * q^87 - 290 * q^89 + 350 * q^91 + 240 * q^93 + 380 * q^95 + 966 * q^97 + 324 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more