LMFDB - Embedded newform 1840.4.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,4,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1840.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1840.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$108.563514411$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 80x^{4} + 105x^{3} + 1328x^{2} - 816x - 4032 $ x^6 - 2x^5 - 80x^4 + 105x^3 + 1328x^2 - 816*x - 4032
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.61348$ of defining polynomial
Character	$\chi$	$=$	1840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.61348 q^{3} +5.00000 q^{5} -3.48289 q^{7} -13.9428 q^{9} +O(q^{10})$	$q-3.61348 q^{3} +5.00000 q^{5} -3.48289 q^{7} -13.9428 q^{9} -0.471534 q^{11} -47.8101 q^{13} -18.0674 q^{15} +47.8263 q^{17} +79.1946 q^{19} +12.5853 q^{21} -23.0000 q^{23} +25.0000 q^{25} +147.946 q^{27} -114.603 q^{29} +200.462 q^{31} +1.70388 q^{33} -17.4144 q^{35} +241.026 q^{37} +172.761 q^{39} -215.056 q^{41} +7.92520 q^{43} -69.7140 q^{45} +286.093 q^{47} -330.870 q^{49} -172.819 q^{51} -588.772 q^{53} -2.35767 q^{55} -286.168 q^{57} -27.0612 q^{59} +822.434 q^{61} +48.5612 q^{63} -239.050 q^{65} -298.821 q^{67} +83.1099 q^{69} +84.8368 q^{71} -82.0515 q^{73} -90.3369 q^{75} +1.64230 q^{77} +21.3201 q^{79} -158.143 q^{81} -139.552 q^{83} +239.131 q^{85} +414.115 q^{87} -716.047 q^{89} +166.517 q^{91} -724.364 q^{93} +395.973 q^{95} +214.063 q^{97} +6.57450 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} + 30 q^{5} + 14 q^{7} + 4 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 + 30 * q^5 + 14 * q^7 + 4 * q^9	$ 6 q + 4 q^{3} + 30 q^{5} + 14 q^{7} + 4 q^{9} + 23 q^{11} - 42 q^{13} + 20 q^{15} - 124 q^{17} - 53 q^{19} - 114 q^{21} - 138 q^{23} + 150 q^{25} + 103 q^{27} - 320 q^{29} + 229 q^{31} - 583 q^{33} + 70 q^{35} - 377 q^{37} - 37 q^{39} - 683 q^{41} + 168 q^{43} + 20 q^{45} - 211 q^{47} - 374 q^{49} - 777 q^{51} - 613 q^{53} + 115 q^{55} - 316 q^{57} - 1029 q^{59} - 1169 q^{61} - 183 q^{63} - 210 q^{65} + 1227 q^{67} - 92 q^{69} - 237 q^{71} - 1001 q^{73} + 100 q^{75} - 1498 q^{77} + 898 q^{79} - 838 q^{81} + 1281 q^{83} - 620 q^{85} + 695 q^{87} - 2780 q^{89} - 857 q^{91} - 1569 q^{93} - 265 q^{95} + 91 q^{97} + 1015 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 + 30 * q^5 + 14 * q^7 + 4 * q^9 + 23 * q^11 - 42 * q^13 + 20 * q^15 - 124 * q^17 - 53 * q^19 - 114 * q^21 - 138 * q^23 + 150 * q^25 + 103 * q^27 - 320 * q^29 + 229 * q^31 - 583 * q^33 + 70 * q^35 - 377 * q^37 - 37 * q^39 - 683 * q^41 + 168 * q^43 + 20 * q^45 - 211 * q^47 - 374 * q^49 - 777 * q^51 - 613 * q^53 + 115 * q^55 - 316 * q^57 - 1029 * q^59 - 1169 * q^61 - 183 * q^63 - 210 * q^65 + 1227 * q^67 - 92 * q^69 - 237 * q^71 - 1001 * q^73 + 100 * q^75 - 1498 * q^77 + 898 * q^79 - 838 * q^81 + 1281 * q^83 - 620 * q^85 + 695 * q^87 - 2780 * q^89 - 857 * q^91 - 1569 * q^93 - 265 * q^95 + 91 * q^97 + 1015 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.61348	−0.695414	−0.347707	−	0.937603i	$-0.613040\pi$
$3$	−3.61348	−0.695414	−0.347707	+	0.937603i	$0.613040\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−3.48289	−0.188058	−0.0940291	−	0.995569i	$-0.529975\pi$
$7$	−3.48289	−0.188058	−0.0940291	+	0.995569i	$0.529975\pi$
$8$	0	0
$9$	−13.9428	−0.516400
$10$	0	0
$11$	−0.471534	−0.0129248	−0.00646240	−	0.999979i	$-0.502057\pi$
$11$	−0.471534	−0.0129248	−0.00646240	+	0.999979i	$0.502057\pi$
$12$	0	0
$13$	−47.8101	−1.02001	−0.510005	−	0.860171i	$-0.670356\pi$
$13$	−47.8101	−1.02001	−0.510005	+	0.860171i	$0.670356\pi$
$14$	0	0
$15$	−18.0674	−0.310998
$16$	0	0
$17$	47.8263	0.682328	0.341164	−	0.940004i	$-0.389179\pi$
$17$	47.8263	0.682328	0.341164	+	0.940004i	$0.389179\pi$
$18$	0	0
$19$	79.1946	0.956236	0.478118	−	0.878296i	$-0.341319\pi$
$19$	79.1946	0.956236	0.478118	+	0.878296i	$0.341319\pi$
$20$	0	0
$21$	12.5853	0.130778
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	147.946	1.05453
$28$	0	0
$29$	−114.603	−0.733835	−0.366918	−	0.930253i	$-0.619587\pi$
$29$	−114.603	−0.733835	−0.366918	+	0.930253i	$0.619587\pi$
$30$	0	0
$31$	200.462	1.16142	0.580710	−	0.814111i	$-0.302775\pi$
$31$	200.462	1.16142	0.580710	+	0.814111i	$0.302775\pi$
$32$	0	0
$33$	1.70388	0.00898808
$34$	0	0
$35$	−17.4144	−0.0841022
$36$	0	0
$37$	241.026	1.07093	0.535465	−	0.844557i	$-0.320136\pi$
$37$	241.026	1.07093	0.535465	+	0.844557i	$0.320136\pi$
$38$	0	0
$39$	172.761	0.709329
$40$	0	0
$41$	−215.056	−0.819174	−0.409587	−	0.912271i	$-0.634327\pi$
$41$	−215.056	−0.819174	−0.409587	+	0.912271i	$0.634327\pi$
$42$	0	0
$43$	7.92520	0.0281065	0.0140533	−	0.999901i	$-0.495527\pi$
$43$	7.92520	0.0281065	0.0140533	+	0.999901i	$0.495527\pi$
$44$	0	0
$45$	−69.7140	−0.230941
$46$	0	0
$47$	286.093	0.887892	0.443946	−	0.896054i	$-0.353578\pi$
$47$	286.093	0.887892	0.443946	+	0.896054i	$0.353578\pi$
$48$	0	0
$49$	−330.870	−0.964634
$50$	0	0
$51$	−172.819	−0.474500
$52$	0	0
$53$	−588.772	−1.52592	−0.762962	−	0.646443i	$-0.776256\pi$
$53$	−588.772	−1.52592	−0.762962	+	0.646443i	$0.776256\pi$
$54$	0	0
$55$	−2.35767	−0.00578014
$56$	0	0
$57$	−286.168	−0.664980
$58$	0	0
$59$	−27.0612	−0.0597131	−0.0298566	−	0.999554i	$-0.509505\pi$
$59$	−27.0612	−0.0597131	−0.0298566	+	0.999554i	$0.509505\pi$
$60$	0	0
$61$	822.434	1.72626	0.863130	−	0.504981i	$-0.168501\pi$
$61$	822.434	1.72626	0.863130	+	0.504981i	$0.168501\pi$
$62$	0	0
$63$	48.5612	0.0971132
$64$	0	0
$65$	−239.050	−0.456162
$66$	0	0
$67$	−298.821	−0.544877	−0.272438	−	0.962173i	$-0.587830\pi$
$67$	−298.821	−0.544877	−0.272438	+	0.962173i	$0.587830\pi$
$68$	0	0
$69$	83.1099	0.145004
$70$	0	0
$71$	84.8368	0.141807	0.0709033	−	0.997483i	$-0.477412\pi$
$71$	84.8368	0.141807	0.0709033	+	0.997483i	$0.477412\pi$
$72$	0	0
$73$	−82.0515	−0.131553	−0.0657767	−	0.997834i	$-0.520953\pi$
$73$	−82.0515	−0.131553	−0.0657767	+	0.997834i	$0.520953\pi$
$74$	0	0
$75$	−90.3369	−0.139083
$76$	0	0
$77$	1.64230	0.00243061
$78$	0	0
$79$	21.3201	0.0303633	0.0151816	−	0.999885i	$-0.495167\pi$
$79$	21.3201	0.0303633	0.0151816	+	0.999885i	$0.495167\pi$
$80$	0	0
$81$	−158.143	−0.216931
$82$	0	0
$83$	−139.552	−0.184553	−0.0922763	−	0.995733i	$-0.529414\pi$
$83$	−139.552	−0.184553	−0.0922763	+	0.995733i	$0.529414\pi$
$84$	0	0
$85$	239.131	0.305146
$86$	0	0
$87$	414.115	0.510319
$88$	0	0
$89$	−716.047	−0.852818	−0.426409	−	0.904530i	$-0.640222\pi$
$89$	−716.047	−0.852818	−0.426409	+	0.904530i	$0.640222\pi$
$90$	0	0
$91$	166.517	0.191821
$92$	0	0
$93$	−724.364	−0.807667
$94$	0	0
$95$	395.973	0.427642
$96$	0	0
$97$	214.063	0.224071	0.112035	−	0.993704i	$-0.464263\pi$
$97$	214.063	0.224071	0.112035	+	0.993704i	$0.464263\pi$
$98$	0	0
$99$	6.57450	0.00667436
$100$	0	0
$101$	−1506.24	−1.48393	−0.741963	−	0.670441i	$-0.766105\pi$
$101$	−1506.24	−1.48393	−0.741963	+	0.670441i	$0.766105\pi$
$102$	0	0
$103$	−106.361	−0.101748	−0.0508738	−	0.998705i	$-0.516201\pi$
$103$	−106.361	−0.101748	−0.0508738	+	0.998705i	$0.516201\pi$
$104$	0	0
$105$	62.9266	0.0584858
$106$	0	0
$107$	204.554	0.184813	0.0924066	−	0.995721i	$-0.470544\pi$
$107$	204.554	0.184813	0.0924066	+	0.995721i	$0.470544\pi$
$108$	0	0
$109$	−639.512	−0.561965	−0.280982	−	0.959713i	$-0.590660\pi$
$109$	−639.512	−0.561965	−0.280982	+	0.959713i	$0.590660\pi$
$110$	0	0
$111$	−870.941	−0.744739
$112$	0	0
$113$	1658.55	1.38074	0.690368	−	0.723458i	$-0.257449\pi$
$113$	1658.55	1.38074	0.690368	+	0.723458i	$0.257449\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	666.606	0.526733
$118$	0	0
$119$	−166.573	−0.128317
$120$	0	0
$121$	−1330.78	−0.999833
$122$	0	0
$123$	777.101	0.569665
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	402.830	0.281460	0.140730	−	0.990048i	$-0.455055\pi$
$127$	402.830	0.281460	0.140730	+	0.990048i	$0.455055\pi$
$128$	0	0
$129$	−28.6375	−0.0195457
$130$	0	0
$131$	2451.00	1.63470	0.817348	−	0.576145i	$-0.195443\pi$
$131$	2451.00	1.63470	0.817348	+	0.576145i	$0.195443\pi$
$132$	0	0
$133$	−275.826	−0.179828
$134$	0	0
$135$	739.729	0.471598
$136$	0	0
$137$	400.136	0.249532	0.124766	−	0.992186i	$-0.460182\pi$
$137$	400.136	0.249532	0.124766	+	0.992186i	$0.460182\pi$
$138$	0	0
$139$	−1728.73	−1.05489	−0.527443	−	0.849591i	$-0.676849\pi$
$139$	−1728.73	−1.05489	−0.527443	+	0.849591i	$0.676849\pi$
$140$	0	0
$141$	−1033.79	−0.617452
$142$	0	0
$143$	22.5441	0.0131834
$144$	0	0
$145$	−573.014	−0.328181
$146$	0	0
$147$	1195.59	0.670820
$148$	0	0
$149$	1303.11	0.716477	0.358239	−	0.933630i	$-0.383377\pi$
$149$	1303.11	0.716477	0.358239	+	0.933630i	$0.383377\pi$
$150$	0	0
$151$	−187.501	−0.101050	−0.0505252	−	0.998723i	$-0.516090\pi$
$151$	−187.501	−0.101050	−0.0505252	+	0.998723i	$0.516090\pi$
$152$	0	0
$153$	−666.832	−0.352354
$154$	0	0
$155$	1002.31	0.519402
$156$	0	0
$157$	43.5636	0.0221449	0.0110725	−	0.999939i	$-0.496475\pi$
$157$	43.5636	0.0221449	0.0110725	+	0.999939i	$0.496475\pi$
$158$	0	0
$159$	2127.51	1.06115
$160$	0	0
$161$	80.1064	0.0392128
$162$	0	0
$163$	2659.59	1.27801	0.639003	−	0.769205i	$-0.279347\pi$
$163$	2659.59	1.27801	0.639003	+	0.769205i	$0.279347\pi$
$164$	0	0
$165$	8.51938	0.00401959
$166$	0	0
$167$	−3906.88	−1.81032	−0.905160	−	0.425071i	$-0.860249\pi$
$167$	−3906.88	−1.81032	−0.905160	+	0.425071i	$0.860249\pi$
$168$	0	0
$169$	88.8037	0.0404204
$170$	0	0
$171$	−1104.19	−0.493800
$172$	0	0
$173$	−4066.58	−1.78714	−0.893572	−	0.448919i	$-0.851809\pi$
$173$	−4066.58	−1.78714	−0.893572	+	0.448919i	$0.851809\pi$
$174$	0	0
$175$	−87.0722	−0.0376116
$176$	0	0
$177$	97.7851	0.0415253
$178$	0	0
$179$	−1429.97	−0.597099	−0.298550	−	0.954394i	$-0.596503\pi$
$179$	−1429.97	−0.597099	−0.298550	+	0.954394i	$0.596503\pi$
$180$	0	0
$181$	−1214.28	−0.498654	−0.249327	−	0.968419i	$-0.580209\pi$
$181$	−1214.28	−0.498654	−0.249327	+	0.968419i	$0.580209\pi$
$182$	0	0
$183$	−2971.85	−1.20047
$184$	0	0
$185$	1205.13	0.478934
$186$	0	0
$187$	−22.5517	−0.00881895
$188$	0	0
$189$	−515.278	−0.198312
$190$	0	0
$191$	847.793	0.321174	0.160587	−	0.987022i	$-0.448661\pi$
$191$	847.793	0.321174	0.160587	+	0.987022i	$0.448661\pi$
$192$	0	0
$193$	−2420.91	−0.902906	−0.451453	−	0.892295i	$-0.649094\pi$
$193$	−2420.91	−0.902906	−0.451453	+	0.892295i	$0.649094\pi$
$194$	0	0
$195$	863.803	0.317222
$196$	0	0
$197$	−757.003	−0.273778	−0.136889	−	0.990586i	$-0.543710\pi$
$197$	−757.003	−0.273778	−0.136889	+	0.990586i	$0.543710\pi$
$198$	0	0
$199$	−3639.49	−1.29647	−0.648233	−	0.761442i	$-0.724492\pi$
$199$	−3639.49	−1.29647	−0.648233	+	0.761442i	$0.724492\pi$
$200$	0	0
$201$	1079.78	0.378915
$202$	0	0
$203$	399.149	0.138004
$204$	0	0
$205$	−1075.28	−0.366346
$206$	0	0
$207$	320.684	0.107677
$208$	0	0
$209$	−37.3429	−0.0123592
$210$	0	0
$211$	1416.54	0.462172	0.231086	−	0.972933i	$-0.425772\pi$
$211$	1416.54	0.462172	0.231086	+	0.972933i	$0.425772\pi$
$212$	0	0
$213$	−306.556	−0.0986143
$214$	0	0
$215$	39.6260	0.0125696
$216$	0	0
$217$	−698.186	−0.218414
$218$	0	0
$219$	296.491	0.0914841
$220$	0	0
$221$	−2286.58	−0.695981
$222$	0	0
$223$	330.842	0.0993489	0.0496744	−	0.998765i	$-0.484182\pi$
$223$	330.842	0.0993489	0.0496744	+	0.998765i	$0.484182\pi$
$224$	0	0
$225$	−348.570	−0.103280
$226$	0	0
$227$	−1779.00	−0.520160	−0.260080	−	0.965587i	$-0.583749\pi$
$227$	−1779.00	−0.520160	−0.260080	+	0.965587i	$0.583749\pi$
$228$	0	0
$229$	−6143.57	−1.77283	−0.886416	−	0.462889i	$-0.846813\pi$
$229$	−6143.57	−1.77283	−0.886416	+	0.462889i	$0.846813\pi$
$230$	0	0
$231$	−5.93440	−0.00169028
$232$	0	0
$233$	−332.483	−0.0934836	−0.0467418	−	0.998907i	$-0.514884\pi$
$233$	−332.483	−0.0934836	−0.0467418	+	0.998907i	$0.514884\pi$
$234$	0	0
$235$	1430.46	0.397077
$236$	0	0
$237$	−77.0396	−0.0211150
$238$	0	0
$239$	5725.61	1.54962	0.774809	−	0.632195i	$-0.217846\pi$
$239$	5725.61	1.54962	0.774809	+	0.632195i	$0.217846\pi$
$240$	0	0
$241$	−4555.85	−1.21771	−0.608855	−	0.793281i	$-0.708371\pi$
$241$	−4555.85	−1.21771	−0.608855	+	0.793281i	$0.708371\pi$
$242$	0	0
$243$	−3423.09	−0.903668
$244$	0	0
$245$	−1654.35	−0.431397
$246$	0	0
$247$	−3786.30	−0.975370
$248$	0	0
$249$	504.269	0.128340
$250$	0	0
$251$	−4983.45	−1.25320	−0.626599	−	0.779342i	$-0.715554\pi$
$251$	−4983.45	−1.25320	−0.626599	+	0.779342i	$0.715554\pi$
$252$	0	0
$253$	10.8453	0.00269501
$254$	0	0
$255$	−864.095	−0.212203
$256$	0	0
$257$	1394.16	0.338387	0.169194	−	0.985583i	$-0.445884\pi$
$257$	1394.16	0.338387	0.169194	+	0.985583i	$0.445884\pi$
$258$	0	0
$259$	−839.466	−0.201397
$260$	0	0
$261$	1597.88	0.378952
$262$	0	0
$263$	−19.7969	−0.00464156	−0.00232078	−	0.999997i	$-0.500739\pi$
$263$	−19.7969	−0.00464156	−0.00232078	+	0.999997i	$0.500739\pi$
$264$	0	0
$265$	−2943.86	−0.682414
$266$	0	0
$267$	2587.42	0.593062
$268$	0	0
$269$	7027.21	1.59278	0.796388	−	0.604786i	$-0.206741\pi$
$269$	7027.21	1.59278	0.796388	+	0.604786i	$0.206741\pi$
$270$	0	0
$271$	−6958.79	−1.55984	−0.779919	−	0.625880i	$-0.784740\pi$
$271$	−6958.79	−1.55984	−0.779919	+	0.625880i	$0.784740\pi$
$272$	0	0
$273$	−601.705	−0.133395
$274$	0	0
$275$	−11.7883	−0.00258496
$276$	0	0
$277$	480.191	0.104158	0.0520792	−	0.998643i	$-0.483415\pi$
$277$	480.191	0.104158	0.0520792	+	0.998643i	$0.483415\pi$
$278$	0	0
$279$	−2795.00	−0.599757
$280$	0	0
$281$	−3026.73	−0.642561	−0.321280	−	0.946984i	$-0.604113\pi$
$281$	−3026.73	−0.642561	−0.321280	+	0.946984i	$0.604113\pi$
$282$	0	0
$283$	−4674.00	−0.981769	−0.490885	−	0.871225i	$-0.663326\pi$
$283$	−4674.00	−0.981769	−0.490885	+	0.871225i	$0.663326\pi$
$284$	0	0
$285$	−1430.84	−0.297388
$286$	0	0
$287$	749.017	0.154052
$288$	0	0
$289$	−2625.65	−0.534429
$290$	0	0
$291$	−773.513	−0.155822
$292$	0	0
$293$	256.216	0.0510864	0.0255432	−	0.999674i	$-0.491868\pi$
$293$	256.216	0.0510864	0.0255432	+	0.999674i	$0.491868\pi$
$294$	0	0
$295$	−135.306	−0.0267045
$296$	0	0
$297$	−69.7614	−0.0136295
$298$	0	0
$299$	1099.63	0.212687
$300$	0	0
$301$	−27.6026	−0.00528567
$302$	0	0
$303$	5442.76	1.03194
$304$	0	0
$305$	4112.17	0.772007
$306$	0	0
$307$	255.980	0.0475881	0.0237940	−	0.999717i	$-0.492425\pi$
$307$	255.980	0.0475881	0.0237940	+	0.999717i	$0.492425\pi$
$308$	0	0
$309$	384.331	0.0707567
$310$	0	0
$311$	−5042.65	−0.919430	−0.459715	−	0.888067i	$-0.652048\pi$
$311$	−5042.65	−0.919430	−0.459715	+	0.888067i	$0.652048\pi$
$312$	0	0
$313$	−937.802	−0.169354	−0.0846769	−	0.996408i	$-0.526986\pi$
$313$	−937.802	−0.169354	−0.0846769	+	0.996408i	$0.526986\pi$
$314$	0	0
$315$	242.806	0.0434303
$316$	0	0
$317$	−1848.54	−0.327521	−0.163760	−	0.986500i	$-0.552362\pi$
$317$	−1848.54	−0.327521	−0.163760	+	0.986500i	$0.552362\pi$
$318$	0	0
$319$	54.0391	0.00948467
$320$	0	0
$321$	−739.152	−0.128522
$322$	0	0
$323$	3787.58	0.652466
$324$	0	0
$325$	−1195.25	−0.204002
$326$	0	0
$327$	2310.86	0.390798
$328$	0	0
$329$	−996.428	−0.166975
$330$	0	0
$331$	−1004.65	−0.166830	−0.0834149	−	0.996515i	$-0.526583\pi$
$331$	−1004.65	−0.166830	−0.0834149	+	0.996515i	$0.526583\pi$
$332$	0	0
$333$	−3360.57	−0.553028
$334$	0	0
$335$	−1494.10	−0.243676
$336$	0	0
$337$	−2771.03	−0.447915	−0.223958	−	0.974599i	$-0.571898\pi$
$337$	−2771.03	−0.447915	−0.223958	+	0.974599i	$0.571898\pi$
$338$	0	0
$339$	−5993.13	−0.960183
$340$	0	0
$341$	−94.5245	−0.0150111
$342$	0	0
$343$	2347.01	0.369466
$344$	0	0
$345$	415.550	0.0648477
$346$	0	0
$347$	−6229.95	−0.963808	−0.481904	−	0.876224i	$-0.660055\pi$
$347$	−6229.95	−0.963808	−0.481904	+	0.876224i	$0.660055\pi$
$348$	0	0
$349$	−1468.38	−0.225217	−0.112608	−	0.993639i	$-0.535921\pi$
$349$	−1468.38	−0.225217	−0.112608	+	0.993639i	$0.535921\pi$
$350$	0	0
$351$	−7073.30	−1.07563
$352$	0	0
$353$	−9490.40	−1.43094	−0.715471	−	0.698642i	$-0.753788\pi$
$353$	−9490.40	−1.43094	−0.715471	+	0.698642i	$0.753788\pi$
$354$	0	0
$355$	424.184	0.0634179
$356$	0	0
$357$	601.909	0.0892336
$358$	0	0
$359$	−11274.8	−1.65755	−0.828777	−	0.559579i	$-0.810963\pi$
$359$	−11274.8	−1.65755	−0.828777	+	0.559579i	$0.810963\pi$
$360$	0	0
$361$	−587.217	−0.0856126
$362$	0	0
$363$	4808.73	0.695298
$364$	0	0
$365$	−410.258	−0.0588325
$366$	0	0
$367$	−1545.02	−0.219753	−0.109877	−	0.993945i	$-0.535046\pi$
$367$	−1545.02	−0.219753	−0.109877	+	0.993945i	$0.535046\pi$
$368$	0	0
$369$	2998.49	0.423022
$370$	0	0
$371$	2050.62	0.286963
$372$	0	0
$373$	2582.90	0.358546	0.179273	−	0.983799i	$-0.442626\pi$
$373$	2582.90	0.358546	0.179273	+	0.983799i	$0.442626\pi$
$374$	0	0
$375$	−451.684	−0.0621997
$376$	0	0
$377$	5479.17	0.748519
$378$	0	0
$379$	−7206.37	−0.976693	−0.488346	−	0.872650i	$-0.662400\pi$
$379$	−7206.37	−0.976693	−0.488346	+	0.872650i	$0.662400\pi$
$380$	0	0
$381$	−1455.62	−0.195731
$382$	0	0
$383$	−7009.90	−0.935220	−0.467610	−	0.883935i	$-0.654885\pi$
$383$	−7009.90	−0.935220	−0.467610	+	0.883935i	$0.654885\pi$
$384$	0	0
$385$	8.21149	0.00108700
$386$	0	0
$387$	−110.499	−0.0145142
$388$	0	0
$389$	1787.27	0.232951	0.116475	−	0.993194i	$-0.462840\pi$
$389$	1787.27	0.232951	0.116475	+	0.993194i	$0.462840\pi$
$390$	0	0
$391$	−1100.00	−0.142275
$392$	0	0
$393$	−8856.64	−1.13679
$394$	0	0
$395$	106.600	0.0135789
$396$	0	0
$397$	9281.73	1.17339	0.586696	−	0.809807i	$-0.300428\pi$
$397$	9281.73	1.17339	0.586696	+	0.809807i	$0.300428\pi$
$398$	0	0
$399$	996.690	0.125055
$400$	0	0
$401$	−653.779	−0.0814169	−0.0407085	−	0.999171i	$-0.512961\pi$
$401$	−653.779	−0.0814169	−0.0407085	+	0.999171i	$0.512961\pi$
$402$	0	0
$403$	−9584.09	−1.18466
$404$	0	0
$405$	−790.715	−0.0970147
$406$	0	0
$407$	−113.652	−0.0138416
$408$	0	0
$409$	1453.62	0.175738	0.0878688	−	0.996132i	$-0.471994\pi$
$409$	1453.62	0.175738	0.0878688	+	0.996132i	$0.471994\pi$
$410$	0	0
$411$	−1445.88	−0.173528
$412$	0	0
$413$	94.2512	0.0112295
$414$	0	0
$415$	−697.762	−0.0825344
$416$	0	0
$417$	6246.73	0.733582
$418$	0	0
$419$	−7623.31	−0.888838	−0.444419	−	0.895819i	$-0.646590\pi$
$419$	−7623.31	−0.888838	−0.444419	+	0.895819i	$0.646590\pi$
$420$	0	0
$421$	13854.1	1.60382	0.801909	−	0.597446i	$-0.203818\pi$
$421$	13854.1	1.60382	0.801909	+	0.597446i	$0.203818\pi$
$422$	0	0
$423$	−3988.93	−0.458507
$424$	0	0
$425$	1195.66	0.136466
$426$	0	0
$427$	−2864.44	−0.324637
$428$	0	0
$429$	−81.4624	−0.00916793
$430$	0	0
$431$	−8971.33	−1.00263	−0.501315	−	0.865265i	$-0.667150\pi$
$431$	−8971.33	−1.00263	−0.501315	+	0.865265i	$0.667150\pi$
$432$	0	0
$433$	5591.55	0.620584	0.310292	−	0.950641i	$-0.399573\pi$
$433$	5591.55	0.620584	0.310292	+	0.950641i	$0.399573\pi$
$434$	0	0
$435$	2070.57	0.228222
$436$	0	0
$437$	−1821.48	−0.199389
$438$	0	0
$439$	−5295.36	−0.575704	−0.287852	−	0.957675i	$-0.592941\pi$
$439$	−5295.36	−0.575704	−0.287852	+	0.957675i	$0.592941\pi$
$440$	0	0
$441$	4613.25	0.498137
$442$	0	0
$443$	11257.3	1.20734	0.603671	−	0.797234i	$-0.293704\pi$
$443$	11257.3	1.20734	0.603671	+	0.797234i	$0.293704\pi$
$444$	0	0
$445$	−3580.24	−0.381392
$446$	0	0
$447$	−4708.76	−0.498248
$448$	0	0
$449$	−11550.1	−1.21399	−0.606997	−	0.794704i	$-0.707626\pi$
$449$	−11550.1	−1.21399	−0.606997	+	0.794704i	$0.707626\pi$
$450$	0	0
$451$	101.406	0.0105877
$452$	0	0
$453$	677.529	0.0702718
$454$	0	0
$455$	832.585	0.0857851
$456$	0	0
$457$	13408.8	1.37251	0.686253	−	0.727363i	$-0.259254\pi$
$457$	13408.8	1.37251	0.686253	+	0.727363i	$0.259254\pi$
$458$	0	0
$459$	7075.69	0.719532
$460$	0	0
$461$	18833.9	1.90278	0.951390	−	0.307990i	$-0.0996564\pi$
$461$	18833.9	1.90278	0.951390	+	0.307990i	$0.0996564\pi$
$462$	0	0
$463$	8469.58	0.850140	0.425070	−	0.905161i	$-0.360250\pi$
$463$	8469.58	0.850140	0.425070	+	0.905161i	$0.360250\pi$
$464$	0	0
$465$	−3621.82	−0.361200
$466$	0	0
$467$	−3284.01	−0.325408	−0.162704	−	0.986675i	$-0.552022\pi$
$467$	−3284.01	−0.325408	−0.162704	+	0.986675i	$0.552022\pi$
$468$	0	0
$469$	1040.76	0.102469
$470$	0	0
$471$	−157.416	−0.0153999
$472$	0	0
$473$	−3.73700	−0.000363271 0
$474$	0	0
$475$	1979.86	0.191247
$476$	0	0
$477$	8209.12	0.787987
$478$	0	0
$479$	−11552.7	−1.10200	−0.550999	−	0.834506i	$-0.685753\pi$
$479$	−11552.7	−1.10200	−0.550999	+	0.834506i	$0.685753\pi$
$480$	0	0
$481$	−11523.5	−1.09236
$482$	0	0
$483$	−289.462	−0.0272691
$484$	0	0
$485$	1070.32	0.100207
$486$	0	0
$487$	−9838.12	−0.915416	−0.457708	−	0.889103i	$-0.651329\pi$
$487$	−9838.12	−0.915416	−0.457708	+	0.889103i	$0.651329\pi$
$488$	0	0
$489$	−9610.35	−0.888742
$490$	0	0
$491$	5243.78	0.481973	0.240986	−	0.970528i	$-0.422529\pi$
$491$	5243.78	0.481973	0.240986	+	0.970528i	$0.422529\pi$
$492$	0	0
$493$	−5481.03	−0.500716
$494$	0	0
$495$	32.8725	0.00298487
$496$	0	0
$497$	−295.477	−0.0266679
$498$	0	0
$499$	6660.15	0.597493	0.298747	−	0.954332i	$-0.403431\pi$
$499$	6660.15	0.597493	0.298747	+	0.954332i	$0.403431\pi$
$500$	0	0
$501$	14117.4	1.25892
$502$	0	0
$503$	10129.4	0.897908	0.448954	−	0.893555i	$-0.351797\pi$
$503$	10129.4	0.897908	0.448954	+	0.893555i	$0.351797\pi$
$504$	0	0
$505$	−7531.20	−0.663632
$506$	0	0
$507$	−320.890	−0.0281089
$508$	0	0
$509$	12787.7	1.11357	0.556783	−	0.830658i	$-0.312035\pi$
$509$	12787.7	1.11357	0.556783	+	0.830658i	$0.312035\pi$
$510$	0	0
$511$	285.776	0.0247397
$512$	0	0
$513$	11716.5	1.00838
$514$	0	0
$515$	−531.803	−0.0455030
$516$	0	0
$517$	−134.902	−0.0114758
$518$	0	0
$519$	14694.5	1.24280
$520$	0	0
$521$	1798.69	0.151252	0.0756258	−	0.997136i	$-0.475905\pi$
$521$	1798.69	0.151252	0.0756258	+	0.997136i	$0.475905\pi$
$522$	0	0
$523$	17902.2	1.49677	0.748383	−	0.663267i	$-0.230831\pi$
$523$	17902.2	1.49677	0.748383	+	0.663267i	$0.230831\pi$
$524$	0	0
$525$	314.633	0.0261556
$526$	0	0
$527$	9587.34	0.792469
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	377.309	0.0308358
$532$	0	0
$533$	10281.9	0.835566
$534$	0	0
$535$	1022.77	0.0826509
$536$	0	0
$537$	5167.15	0.415231
$538$	0	0
$539$	156.016	0.0124677
$540$	0	0
$541$	8628.86	0.685737	0.342869	−	0.939383i	$-0.388601\pi$
$541$	8628.86	0.685737	0.342869	+	0.939383i	$0.388601\pi$
$542$	0	0
$543$	4387.75	0.346771
$544$	0	0
$545$	−3197.56	−0.251318
$546$	0	0
$547$	22747.6	1.77809	0.889046	−	0.457817i	$-0.151369\pi$
$547$	22747.6	1.77809	0.889046	+	0.457817i	$0.151369\pi$
$548$	0	0
$549$	−11467.0	−0.891441
$550$	0	0
$551$	−9075.93	−0.701720
$552$	0	0
$553$	−74.2555	−0.00571006
$554$	0	0
$555$	−4354.71	−0.333058
$556$	0	0
$557$	16101.0	1.22482	0.612408	−	0.790542i	$-0.290201\pi$
$557$	16101.0	1.22482	0.612408	+	0.790542i	$0.290201\pi$
$558$	0	0
$559$	−378.904	−0.0286690
$560$	0	0
$561$	81.4900	0.00613282
$562$	0	0
$563$	−21197.9	−1.58683	−0.793415	−	0.608681i	$-0.791699\pi$
$563$	−21197.9	−1.58683	−0.793415	+	0.608681i	$0.791699\pi$
$564$	0	0
$565$	8292.75	0.617484
$566$	0	0
$567$	550.794	0.0407957
$568$	0	0
$569$	−12951.3	−0.954214	−0.477107	−	0.878845i	$-0.658315\pi$
$569$	−12951.3	−0.954214	−0.477107	+	0.878845i	$0.658315\pi$
$570$	0	0
$571$	−20923.3	−1.53347	−0.766735	−	0.641964i	$-0.778120\pi$
$571$	−20923.3	−1.53347	−0.766735	+	0.641964i	$0.778120\pi$
$572$	0	0
$573$	−3063.48	−0.223349
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	18557.0	1.33889	0.669444	−	0.742863i	$-0.266533\pi$
$577$	18557.0	1.33889	0.669444	+	0.742863i	$0.266533\pi$
$578$	0	0
$579$	8747.90	0.627893
$580$	0	0
$581$	486.045	0.0347066
$582$	0	0
$583$	277.626	0.0197223
$584$	0	0
$585$	3333.03	0.235562
$586$	0	0
$587$	6992.85	0.491697	0.245848	−	0.969308i	$-0.420933\pi$
$587$	6992.85	0.491697	0.245848	+	0.969308i	$0.420933\pi$
$588$	0	0
$589$	15875.5	1.11059
$590$	0	0
$591$	2735.41	0.190389
$592$	0	0
$593$	−18032.5	−1.24875	−0.624373	−	0.781126i	$-0.714646\pi$
$593$	−18032.5	−1.24875	−0.624373	+	0.781126i	$0.714646\pi$
$594$	0	0
$595$	−832.867	−0.0573852
$596$	0	0
$597$	13151.2	0.901581
$598$	0	0
$599$	−22838.8	−1.55787	−0.778937	−	0.627102i	$-0.784241\pi$
$599$	−22838.8	−1.55787	−0.778937	+	0.627102i	$0.784241\pi$
$600$	0	0
$601$	−19813.2	−1.34476	−0.672378	−	0.740208i	$-0.734727\pi$
$601$	−19813.2	−1.34476	−0.672378	+	0.740208i	$0.734727\pi$
$602$	0	0
$603$	4166.39	0.281374
$604$	0	0
$605$	−6653.89	−0.447139
$606$	0	0
$607$	9761.11	0.652704	0.326352	−	0.945248i	$-0.394181\pi$
$607$	9761.11	0.652704	0.326352	+	0.945248i	$0.394181\pi$
$608$	0	0
$609$	−1442.31	−0.0959697
$610$	0	0
$611$	−13678.1	−0.905658
$612$	0	0
$613$	−14241.8	−0.938369	−0.469185	−	0.883100i	$-0.655452\pi$
$613$	−14241.8	−0.938369	−0.469185	+	0.883100i	$0.655452\pi$
$614$	0	0
$615$	3885.50	0.254762
$616$	0	0
$617$	−16478.7	−1.07522	−0.537609	−	0.843195i	$-0.680672\pi$
$617$	−16478.7	−1.07522	−0.537609	+	0.843195i	$0.680672\pi$
$618$	0	0
$619$	12967.4	0.842011	0.421005	−	0.907058i	$-0.361677\pi$
$619$	12967.4	0.842011	0.421005	+	0.907058i	$0.361677\pi$
$620$	0	0
$621$	−3402.75	−0.219884
$622$	0	0
$623$	2493.91	0.160379
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	134.938	0.00859473
$628$	0	0
$629$	11527.4	0.730725
$630$	0	0
$631$	−27606.5	−1.74168	−0.870839	−	0.491568i	$-0.836424\pi$
$631$	−27606.5	−1.74168	−0.870839	+	0.491568i	$0.836424\pi$
$632$	0	0
$633$	−5118.62	−0.321401
$634$	0	0
$635$	2014.15	0.125873
$636$	0	0
$637$	15818.9	0.983936
$638$	0	0
$639$	−1182.86	−0.0732289
$640$	0	0
$641$	−5349.70	−0.329642	−0.164821	−	0.986324i	$-0.552705\pi$
$641$	−5349.70	−0.329642	−0.164821	+	0.986324i	$0.552705\pi$
$642$	0	0
$643$	20462.9	1.25502	0.627510	−	0.778609i	$-0.284074\pi$
$643$	20462.9	1.25502	0.627510	+	0.778609i	$0.284074\pi$
$644$	0	0
$645$	−143.188	−0.00874109
$646$	0	0
$647$	22106.4	1.34326	0.671632	−	0.740885i	$-0.265594\pi$
$647$	22106.4	1.34326	0.671632	+	0.740885i	$0.265594\pi$
$648$	0	0
$649$	12.7603	0.000771780 0
$650$	0	0
$651$	2522.88	0.151888
$652$	0	0
$653$	−23241.9	−1.39284	−0.696422	−	0.717632i	$-0.745226\pi$
$653$	−23241.9	−1.39284	−0.696422	+	0.717632i	$0.745226\pi$
$654$	0	0
$655$	12255.0	0.731058
$656$	0	0
$657$	1144.03	0.0679342
$658$	0	0
$659$	10646.2	0.629312	0.314656	−	0.949206i	$-0.398111\pi$
$659$	10646.2	0.629312	0.314656	+	0.949206i	$0.398111\pi$
$660$	0	0
$661$	−7132.80	−0.419718	−0.209859	−	0.977732i	$-0.567301\pi$
$661$	−7132.80	−0.419718	−0.209859	+	0.977732i	$0.567301\pi$
$662$	0	0
$663$	8262.49	0.483995
$664$	0	0
$665$	−1379.13	−0.0804215
$666$	0	0
$667$	2635.87	0.153015
$668$	0	0
$669$	−1195.49	−0.0690886
$670$	0	0
$671$	−387.805	−0.0223116
$672$	0	0
$673$	−4360.21	−0.249738	−0.124869	−	0.992173i	$-0.539851\pi$
$673$	−4360.21	−0.249738	−0.124869	+	0.992173i	$0.539851\pi$
$674$	0	0
$675$	3698.64	0.210905
$676$	0	0
$677$	4731.69	0.268617	0.134308	−	0.990940i	$-0.457119\pi$
$677$	4731.69	0.268617	0.134308	+	0.990940i	$0.457119\pi$
$678$	0	0
$679$	−745.558	−0.0421383
$680$	0	0
$681$	6428.38	0.361727
$682$	0	0
$683$	−18390.3	−1.03029	−0.515143	−	0.857105i	$-0.672261\pi$
$683$	−18390.3	−1.03029	−0.515143	+	0.857105i	$0.672261\pi$
$684$	0	0
$685$	2000.68	0.111594
$686$	0	0
$687$	22199.6	1.23285
$688$	0	0
$689$	28149.2	1.55646
$690$	0	0
$691$	−4601.17	−0.253309	−0.126655	−	0.991947i	$-0.540424\pi$
$691$	−4601.17	−0.253309	−0.126655	+	0.991947i	$0.540424\pi$
$692$	0	0
$693$	−22.8982	−0.00125517
$694$	0	0
$695$	−8643.66	−0.471759
$696$	0	0
$697$	−10285.3	−0.558945
$698$	0	0
$699$	1201.42	0.0650098
$700$	0	0
$701$	26193.8	1.41131	0.705653	−	0.708557i	$-0.250654\pi$
$701$	26193.8	1.41131	0.705653	+	0.708557i	$0.250654\pi$
$702$	0	0
$703$	19087.9	1.02406
$704$	0	0
$705$	−5168.95	−0.276133
$706$	0	0
$707$	5246.07	0.279065
$708$	0	0
$709$	−27391.6	−1.45094	−0.725468	−	0.688256i	$-0.758376\pi$
$709$	−27391.6	−1.45094	−0.725468	+	0.688256i	$0.758376\pi$
$710$	0	0
$711$	−297.262	−0.0156796
$712$	0	0
$713$	−4610.62	−0.242173
$714$	0	0
$715$	112.720	0.00589581
$716$	0	0
$717$	−20689.4	−1.07763
$718$	0	0
$719$	22997.2	1.19284	0.596420	−	0.802672i	$-0.296589\pi$
$719$	22997.2	1.19284	0.596420	+	0.802672i	$0.296589\pi$
$720$	0	0
$721$	370.442	0.0191345
$722$	0	0
$723$	16462.5	0.846812
$724$	0	0
$725$	−2865.07	−0.146767
$726$	0	0
$727$	20689.0	1.05545	0.527725	−	0.849415i	$-0.323045\pi$
$727$	20689.0	1.05545	0.527725	+	0.849415i	$0.323045\pi$
$728$	0	0
$729$	16639.1	0.845355
$730$	0	0
$731$	379.033	0.0191779
$732$	0	0
$733$	−30073.0	−1.51538	−0.757689	−	0.652615i	$-0.773672\pi$
$733$	−30073.0	−1.51538	−0.757689	+	0.652615i	$0.773672\pi$
$734$	0	0
$735$	5977.94	0.300000
$736$	0	0
$737$	140.904	0.00704242
$738$	0	0
$739$	19586.0	0.974945	0.487473	−	0.873138i	$-0.337919\pi$
$739$	19586.0	0.974945	0.487473	+	0.873138i	$0.337919\pi$
$740$	0	0
$741$	13681.7	0.678286
$742$	0	0
$743$	3890.73	0.192109	0.0960544	−	0.995376i	$-0.469378\pi$
$743$	3890.73	0.192109	0.0960544	+	0.995376i	$0.469378\pi$
$744$	0	0
$745$	6515.56	0.320418
$746$	0	0
$747$	1945.75	0.0953029
$748$	0	0
$749$	−712.439	−0.0347556
$750$	0	0
$751$	−10344.1	−0.502611	−0.251305	−	0.967908i	$-0.580860\pi$
$751$	−10344.1	−0.502611	−0.251305	+	0.967908i	$0.580860\pi$
$752$	0	0
$753$	18007.6	0.871491
$754$	0	0
$755$	−937.504	−0.0451911
$756$	0	0
$757$	−34437.8	−1.65345	−0.826727	−	0.562603i	$-0.809800\pi$
$757$	−34437.8	−1.65345	−0.826727	+	0.562603i	$0.809800\pi$
$758$	0	0
$759$	−39.1891	−0.00187414
$760$	0	0
$761$	−6532.05	−0.311152	−0.155576	−	0.987824i	$-0.549723\pi$
$761$	−6532.05	−0.311152	−0.155576	+	0.987824i	$0.549723\pi$
$762$	0	0
$763$	2227.35	0.105682
$764$	0	0
$765$	−3334.16	−0.157577
$766$	0	0
$767$	1293.80	0.0609080
$768$	0	0
$769$	−10863.9	−0.509444	−0.254722	−	0.967014i	$-0.581984\pi$
$769$	−10863.9	−0.509444	−0.254722	+	0.967014i	$0.581984\pi$
$770$	0	0
$771$	−5037.77	−0.235319
$772$	0	0
$773$	−13784.0	−0.641367	−0.320684	−	0.947186i	$-0.603913\pi$
$773$	−13784.0	−0.641367	−0.320684	+	0.947186i	$0.603913\pi$
$774$	0	0
$775$	5011.54	0.232284
$776$	0	0
$777$	3033.39	0.140054
$778$	0	0
$779$	−17031.3	−0.783324
$780$	0	0
$781$	−40.0034	−0.00183282
$782$	0	0
$783$	−16955.0	−0.773848
$784$	0	0
$785$	217.818	0.00990351
$786$	0	0
$787$	32850.3	1.48791	0.743956	−	0.668228i	$-0.232947\pi$
$787$	32850.3	1.48791	0.743956	+	0.668228i	$0.232947\pi$
$788$	0	0
$789$	71.5356	0.00322780
$790$	0	0
$791$	−5776.54	−0.259659
$792$	0	0
$793$	−39320.6	−1.76080
$794$	0	0
$795$	10637.6	0.474560
$796$	0	0
$797$	22602.3	1.00453	0.502267	−	0.864712i	$-0.332499\pi$
$797$	22602.3	1.00453	0.502267	+	0.864712i	$0.332499\pi$
$798$	0	0
$799$	13682.7	0.605833
$800$	0	0
$801$	9983.70	0.440395
$802$	0	0
$803$	38.6901	0.00170030
$804$	0	0
$805$	400.532	0.0175365
$806$	0	0
$807$	−25392.7	−1.10764
$808$	0	0
$809$	1381.30	0.0600296	0.0300148	−	0.999549i	$-0.490445\pi$
$809$	1381.30	0.0600296	0.0300148	+	0.999549i	$0.490445\pi$
$810$	0	0
$811$	−33739.7	−1.46086	−0.730431	−	0.682986i	$-0.760681\pi$
$811$	−33739.7	−1.46086	−0.730431	+	0.682986i	$0.760681\pi$
$812$	0	0
$813$	25145.4	1.08473
$814$	0	0
$815$	13297.9	0.571541
$816$	0	0
$817$	627.633	0.0268765
$818$	0	0
$819$	−2321.71	−0.0990564
$820$	0	0
$821$	36482.0	1.55083	0.775414	−	0.631453i	$-0.217541\pi$
$821$	36482.0	1.55083	0.775414	+	0.631453i	$0.217541\pi$
$822$	0	0
$823$	30169.7	1.27782	0.638912	−	0.769280i	$-0.279385\pi$
$823$	30169.7	1.27782	0.638912	+	0.769280i	$0.279385\pi$
$824$	0	0
$825$	42.5969	0.00179762
$826$	0	0
$827$	7436.40	0.312683	0.156342	−	0.987703i	$-0.450030\pi$
$827$	7436.40	0.312683	0.156342	+	0.987703i	$0.450030\pi$
$828$	0	0
$829$	−1804.21	−0.0755884	−0.0377942	−	0.999286i	$-0.512033\pi$
$829$	−1804.21	−0.0755884	−0.0377942	+	0.999286i	$0.512033\pi$
$830$	0	0
$831$	−1735.16	−0.0724332
$832$	0	0
$833$	−15824.3	−0.658197
$834$	0	0
$835$	−19534.4	−0.809600
$836$	0	0
$837$	29657.5	1.22475
$838$	0	0
$839$	17516.1	0.720768	0.360384	−	0.932804i	$-0.382646\pi$
$839$	17516.1	0.720768	0.360384	+	0.932804i	$0.382646\pi$
$840$	0	0
$841$	−11255.2	−0.461486
$842$	0	0
$843$	10937.0	0.446846
$844$	0	0
$845$	444.019	0.0180766
$846$	0	0
$847$	4634.95	0.188027
$848$	0	0
$849$	16889.4	0.682736
$850$	0	0
$851$	−5543.59	−0.223304
$852$	0	0
$853$	20054.4	0.804981	0.402491	−	0.915424i	$-0.368145\pi$
$853$	20054.4	0.804981	0.402491	+	0.915424i	$0.368145\pi$
$854$	0	0
$855$	−5520.97	−0.220834
$856$	0	0
$857$	−19271.7	−0.768153	−0.384077	−	0.923301i	$-0.625480\pi$
$857$	−19271.7	−0.768153	−0.384077	+	0.923301i	$0.625480\pi$
$858$	0	0
$859$	10280.1	0.408325	0.204162	−	0.978937i	$-0.434553\pi$
$859$	10280.1	0.408325	0.204162	+	0.978937i	$0.434553\pi$
$860$	0	0
$861$	−2706.55	−0.107130
$862$	0	0
$863$	27096.6	1.06881	0.534403	−	0.845230i	$-0.320536\pi$
$863$	27096.6	1.06881	0.534403	+	0.845230i	$0.320536\pi$
$864$	0	0
$865$	−20332.9	−0.799235
$866$	0	0
$867$	9487.72	0.371649
$868$	0	0
$869$	−10.0531	−0.000392439 0
$870$	0	0
$871$	14286.6	0.555780
$872$	0	0
$873$	−2984.64	−0.115710
$874$	0	0
$875$	−435.361	−0.0168204
$876$	0	0
$877$	28836.5	1.11031	0.555154	−	0.831748i	$-0.312659\pi$
$877$	28836.5	1.11031	0.555154	+	0.831748i	$0.312659\pi$
$878$	0	0
$879$	−925.830	−0.0355262
$880$	0	0
$881$	−30241.1	−1.15647	−0.578234	−	0.815871i	$-0.696258\pi$
$881$	−30241.1	−1.15647	−0.578234	+	0.815871i	$0.696258\pi$
$882$	0	0
$883$	33389.5	1.27253	0.636267	−	0.771469i	$-0.280478\pi$
$883$	33389.5	1.27253	0.636267	+	0.771469i	$0.280478\pi$
$884$	0	0
$885$	488.926	0.0185707
$886$	0	0
$887$	−47808.2	−1.80974	−0.904871	−	0.425685i	$-0.860033\pi$
$887$	−47808.2	−1.80974	−0.904871	+	0.425685i	$0.860033\pi$
$888$	0	0
$889$	−1403.01	−0.0529308
$890$	0	0
$891$	74.5698	0.00280380
$892$	0	0
$893$	22657.0	0.849034
$894$	0	0
$895$	−7149.84	−0.267031
$896$	0	0
$897$	−3973.49	−0.147905
$898$	0	0
$899$	−22973.5	−0.852290
$900$	0	0
$901$	−28158.7	−1.04118
$902$	0	0
$903$	99.7412	0.00367572
$904$	0	0
$905$	−6071.38	−0.223005
$906$	0	0
$907$	−546.579	−0.0200098	−0.0100049	−	0.999950i	$-0.503185\pi$
$907$	−546.579	−0.0200098	−0.0100049	+	0.999950i	$0.503185\pi$
$908$	0	0
$909$	21001.2	0.766299
$910$	0	0
$911$	−6770.01	−0.246213	−0.123107	−	0.992393i	$-0.539286\pi$
$911$	−6770.01	−0.246213	−0.123107	+	0.992393i	$0.539286\pi$
$912$	0	0
$913$	65.8036	0.00238530
$914$	0	0
$915$	−14859.2	−0.536864
$916$	0	0
$917$	−8536.56	−0.307418
$918$	0	0
$919$	−1720.53	−0.0617575	−0.0308788	−	0.999523i	$-0.509831\pi$
$919$	−1720.53	−0.0617575	−0.0308788	+	0.999523i	$0.509831\pi$
$920$	0	0
$921$	−924.977	−0.0330934
$922$	0	0
$923$	−4056.05	−0.144644
$924$	0	0
$925$	6025.65	0.214186
$926$	0	0
$927$	1482.96	0.0525425
$928$	0	0
$929$	10509.9	0.371171	0.185586	−	0.982628i	$-0.440582\pi$
$929$	10509.9	0.371171	0.185586	+	0.982628i	$0.440582\pi$
$930$	0	0
$931$	−26203.1	−0.922418
$932$	0	0
$933$	18221.5	0.639384
$934$	0	0
$935$	−112.758	−0.00394395
$936$	0	0
$937$	2556.90	0.0891467	0.0445733	−	0.999006i	$-0.485807\pi$
$937$	2556.90	0.0891467	0.0445733	+	0.999006i	$0.485807\pi$
$938$	0	0
$939$	3388.72	0.117771
$940$	0	0
$941$	4197.13	0.145401	0.0727006	−	0.997354i	$-0.476838\pi$
$941$	4197.13	0.145401	0.0727006	+	0.997354i	$0.476838\pi$
$942$	0	0
$943$	4946.30	0.170810
$944$	0	0
$945$	−2576.39	−0.0886879
$946$	0	0
$947$	2618.81	0.0898627	0.0449313	−	0.998990i	$-0.485693\pi$
$947$	2618.81	0.0898627	0.0449313	+	0.998990i	$0.485693\pi$
$948$	0	0
$949$	3922.89	0.134186
$950$	0	0
$951$	6679.64	0.227763
$952$	0	0
$953$	−29973.8	−1.01883	−0.509417	−	0.860520i	$-0.670139\pi$
$953$	−29973.8	−1.01883	−0.509417	+	0.860520i	$0.670139\pi$
$954$	0	0
$955$	4238.97	0.143633
$956$	0	0
$957$	−195.269	−0.00659577
$958$	0	0
$959$	−1393.63	−0.0469266
$960$	0	0
$961$	10393.9	0.348895
$962$	0	0
$963$	−2852.06	−0.0954375
$964$	0	0
$965$	−12104.5	−0.403792
$966$	0	0
$967$	22017.7	0.732203	0.366101	−	0.930575i	$-0.380692\pi$
$967$	22017.7	0.732203	0.366101	+	0.930575i	$0.380692\pi$
$968$	0	0
$969$	−13686.3	−0.453734
$970$	0	0
$971$	759.143	0.0250897	0.0125448	−	0.999921i	$-0.496007\pi$
$971$	759.143	0.0250897	0.0125448	+	0.999921i	$0.496007\pi$
$972$	0	0
$973$	6020.97	0.198380
$974$	0	0
$975$	4319.01	0.141866
$976$	0	0
$977$	33821.4	1.10751	0.553757	−	0.832678i	$-0.313194\pi$
$977$	33821.4	1.10751	0.553757	+	0.832678i	$0.313194\pi$
$978$	0	0
$979$	337.640	0.0110225
$980$	0	0
$981$	8916.59	0.290199
$982$	0	0
$983$	32640.5	1.05907	0.529537	−	0.848287i	$-0.322366\pi$
$983$	32640.5	1.05907	0.529537	+	0.848287i	$0.322366\pi$
$984$	0	0
$985$	−3785.02	−0.122437
$986$	0	0
$987$	3600.57	0.116117
$988$	0	0
$989$	−182.280	−0.00586062
$990$	0	0
$991$	−14390.6	−0.461284	−0.230642	−	0.973039i	$-0.574083\pi$
$991$	−14390.6	−0.461284	−0.230642	+	0.973039i	$0.574083\pi$
$992$	0	0
$993$	3630.28	0.116016
$994$	0	0
$995$	−18197.5	−0.579797
$996$	0	0
$997$	16412.9	0.521367	0.260683	−	0.965424i	$-0.416052\pi$
$997$	16412.9	0.521367	0.260683	+	0.965424i	$0.416052\pi$
$998$	0	0
$999$	35658.8	1.12932

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.u.1.2		6
4.3	odd	2		920.4.a.a.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.a.1.5	✓	6	4.3	odd	2
1840.4.a.u.1.2		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q-3.61348 q^{3} +5.00000 q^{5} -3.48289 q^{7} -13.9428 q^{9} +O(q^{10})\)	\(q-3.61348 q^{3} +5.00000 q^{5} -3.48289 q^{7} -13.9428 q^{9} -0.471534 q^{11} -47.8101 q^{13} -18.0674 q^{15} +47.8263 q^{17} +79.1946 q^{19} +12.5853 q^{21} -23.0000 q^{23} +25.0000 q^{25} +147.946 q^{27} -114.603 q^{29} +200.462 q^{31} +1.70388 q^{33} -17.4144 q^{35} +241.026 q^{37} +172.761 q^{39} -215.056 q^{41} +7.92520 q^{43} -69.7140 q^{45} +286.093 q^{47} -330.870 q^{49} -172.819 q^{51} -588.772 q^{53} -2.35767 q^{55} -286.168 q^{57} -27.0612 q^{59} +822.434 q^{61} +48.5612 q^{63} -239.050 q^{65} -298.821 q^{67} +83.1099 q^{69} +84.8368 q^{71} -82.0515 q^{73} -90.3369 q^{75} +1.64230 q^{77} +21.3201 q^{79} -158.143 q^{81} -139.552 q^{83} +239.131 q^{85} +414.115 q^{87} -716.047 q^{89} +166.517 q^{91} -724.364 q^{93} +395.973 q^{95} +214.063 q^{97} +6.57450 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} + 30 q^{5} + 14 q^{7} + 4 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 + 30 * q^5 + 14 * q^7 + 4 * q^9	\( 6 q + 4 q^{3} + 30 q^{5} + 14 q^{7} + 4 q^{9} + 23 q^{11} - 42 q^{13} + 20 q^{15} - 124 q^{17} - 53 q^{19} - 114 q^{21} - 138 q^{23} + 150 q^{25} + 103 q^{27} - 320 q^{29} + 229 q^{31} - 583 q^{33} + 70 q^{35} - 377 q^{37} - 37 q^{39} - 683 q^{41} + 168 q^{43} + 20 q^{45} - 211 q^{47} - 374 q^{49} - 777 q^{51} - 613 q^{53} + 115 q^{55} - 316 q^{57} - 1029 q^{59} - 1169 q^{61} - 183 q^{63} - 210 q^{65} + 1227 q^{67} - 92 q^{69} - 237 q^{71} - 1001 q^{73} + 100 q^{75} - 1498 q^{77} + 898 q^{79} - 838 q^{81} + 1281 q^{83} - 620 q^{85} + 695 q^{87} - 2780 q^{89} - 857 q^{91} - 1569 q^{93} - 265 q^{95} + 91 q^{97} + 1015 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 + 30 * q^5 + 14 * q^7 + 4 * q^9 + 23 * q^11 - 42 * q^13 + 20 * q^15 - 124 * q^17 - 53 * q^19 - 114 * q^21 - 138 * q^23 + 150 * q^25 + 103 * q^27 - 320 * q^29 + 229 * q^31 - 583 * q^33 + 70 * q^35 - 377 * q^37 - 37 * q^39 - 683 * q^41 + 168 * q^43 + 20 * q^45 - 211 * q^47 - 374 * q^49 - 777 * q^51 - 613 * q^53 + 115 * q^55 - 316 * q^57 - 1029 * q^59 - 1169 * q^61 - 183 * q^63 - 210 * q^65 + 1227 * q^67 - 92 * q^69 - 237 * q^71 - 1001 * q^73 + 100 * q^75 - 1498 * q^77 + 898 * q^79 - 838 * q^81 + 1281 * q^83 - 620 * q^85 + 695 * q^87 - 2780 * q^89 - 857 * q^91 - 1569 * q^93 - 265 * q^95 + 91 * q^97 + 1015 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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