LMFDB - Embedded newform 1840.4.a.q.1.5

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:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 55x^{3} + 104x^{2} + 255x + 72 $ x^5 - 2x^4 - 55x^3 + 104x^2 + 255x + 72
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 460)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$3.65968$ 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)

$f(q)$	$=$	$q+7.99061 q^{3} +5.00000 q^{5} -4.09549 q^{7} +36.8498 q^{9} +O(q^{10})$	$q+7.99061 q^{3} +5.00000 q^{5} -4.09549 q^{7} +36.8498 q^{9} +47.4497 q^{11} -7.45409 q^{13} +39.9530 q^{15} +58.5837 q^{17} -112.013 q^{19} -32.7255 q^{21} -23.0000 q^{23} +25.0000 q^{25} +78.7060 q^{27} +51.2016 q^{29} +124.263 q^{31} +379.152 q^{33} -20.4775 q^{35} +366.568 q^{37} -59.5627 q^{39} -339.503 q^{41} +497.620 q^{43} +184.249 q^{45} +609.046 q^{47} -326.227 q^{49} +468.119 q^{51} +120.509 q^{53} +237.248 q^{55} -895.048 q^{57} +309.004 q^{59} -76.7261 q^{61} -150.918 q^{63} -37.2704 q^{65} -502.624 q^{67} -183.784 q^{69} +347.930 q^{71} -99.0446 q^{73} +199.765 q^{75} -194.330 q^{77} +990.840 q^{79} -366.036 q^{81} -71.3340 q^{83} +292.918 q^{85} +409.132 q^{87} -914.981 q^{89} +30.5282 q^{91} +992.940 q^{93} -560.063 q^{95} -258.519 q^{97} +1748.51 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 7 q^{3} + 25 q^{5} + 20 q^{7} + 30 q^{9}+O(q^{10}) $ 5 * q + 7 * q^3 + 25 * q^5 + 20 * q^7 + 30 * q^9	$ 5 q + 7 q^{3} + 25 q^{5} + 20 q^{7} + 30 q^{9} + 63 q^{11} - 99 q^{13} + 35 q^{15} - 44 q^{17} + 199 q^{19} - 68 q^{21} - 115 q^{23} + 125 q^{25} + 28 q^{27} + 231 q^{29} + 518 q^{31} - 111 q^{33} + 100 q^{35} - 113 q^{37} + 974 q^{39} - 174 q^{41} + 298 q^{43} + 150 q^{45} + 360 q^{47} - 163 q^{49} + 1163 q^{51} + 217 q^{53} + 315 q^{55} - 684 q^{57} + 1551 q^{59} - 737 q^{61} + 1353 q^{63} - 495 q^{65} + 539 q^{67} - 161 q^{69} + 1736 q^{71} - 628 q^{73} + 175 q^{75} - 882 q^{77} + 2954 q^{79} - 495 q^{81} + 153 q^{83} - 220 q^{85} + 274 q^{87} - 1558 q^{89} + 37 q^{91} + 584 q^{93} + 995 q^{95} - 1375 q^{97} + 2487 q^{99}+O(q^{100}) $ 5 * q + 7 * q^3 + 25 * q^5 + 20 * q^7 + 30 * q^9 + 63 * q^11 - 99 * q^13 + 35 * q^15 - 44 * q^17 + 199 * q^19 - 68 * q^21 - 115 * q^23 + 125 * q^25 + 28 * q^27 + 231 * q^29 + 518 * q^31 - 111 * q^33 + 100 * q^35 - 113 * q^37 + 974 * q^39 - 174 * q^41 + 298 * q^43 + 150 * q^45 + 360 * q^47 - 163 * q^49 + 1163 * q^51 + 217 * q^53 + 315 * q^55 - 684 * q^57 + 1551 * q^59 - 737 * q^61 + 1353 * q^63 - 495 * q^65 + 539 * q^67 - 161 * q^69 + 1736 * q^71 - 628 * q^73 + 175 * q^75 - 882 * q^77 + 2954 * q^79 - 495 * q^81 + 153 * q^83 - 220 * q^85 + 274 * q^87 - 1558 * q^89 + 37 * q^91 + 584 * q^93 + 995 * q^95 - 1375 * q^97 + 2487 * 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
$3$	7.99061	1.53779	0.768897	−	0.639373i	$-0.220806\pi$
$3$	7.99061	1.53779	0.768897	+	0.639373i	$0.220806\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−4.09549	−0.221136	−0.110568	−	0.993869i	$-0.535267\pi$
$7$	−4.09549	−0.221136	−0.110568	+	0.993869i	$0.535267\pi$
$8$	0	0
$9$	36.8498	1.36481
$10$	0	0
$11$	47.4497	1.30060	0.650300	−	0.759677i	$-0.274643\pi$
$11$	47.4497	1.30060	0.650300	+	0.759677i	$0.274643\pi$
$12$	0	0
$13$	−7.45409	−0.159030	−0.0795151	−	0.996834i	$-0.525337\pi$
$13$	−7.45409	−0.159030	−0.0795151	+	0.996834i	$0.525337\pi$
$14$	0	0
$15$	39.9530	0.687722
$16$	0	0
$17$	58.5837	0.835802	0.417901	−	0.908493i	$-0.362766\pi$
$17$	58.5837	0.835802	0.417901	+	0.908493i	$0.362766\pi$
$18$	0	0
$19$	−112.013	−1.35250	−0.676248	−	0.736674i	$-0.736395\pi$
$19$	−112.013	−1.35250	−0.676248	+	0.736674i	$0.736395\pi$
$20$	0	0
$21$	−32.7255	−0.340061
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	78.7060	0.560999
$28$	0	0
$29$	51.2016	0.327859	0.163929	−	0.986472i	$-0.447583\pi$
$29$	51.2016	0.327859	0.163929	+	0.986472i	$0.447583\pi$
$30$	0	0
$31$	124.263	0.719947	0.359974	−	0.932963i	$-0.382786\pi$
$31$	124.263	0.719947	0.359974	+	0.932963i	$0.382786\pi$
$32$	0	0
$33$	379.152	2.00006
$34$	0	0
$35$	−20.4775	−0.0988950
$36$	0	0
$37$	366.568	1.62874	0.814371	−	0.580344i	$-0.197082\pi$
$37$	366.568	1.62874	0.814371	+	0.580344i	$0.197082\pi$
$38$	0	0
$39$	−59.5627	−0.244556
$40$	0	0
$41$	−339.503	−1.29321	−0.646604	−	0.762826i	$-0.723811\pi$
$41$	−339.503	−1.29321	−0.646604	+	0.762826i	$0.723811\pi$
$42$	0	0
$43$	497.620	1.76480	0.882400	−	0.470500i	$-0.155926\pi$
$43$	497.620	1.76480	0.882400	+	0.470500i	$0.155926\pi$
$44$	0	0
$45$	184.249	0.610361
$46$	0	0
$47$	609.046	1.89018	0.945091	−	0.326807i	$-0.105973\pi$
$47$	609.046	1.89018	0.945091	+	0.326807i	$0.105973\pi$
$48$	0	0
$49$	−326.227	−0.951099
$50$	0	0
$51$	468.119	1.28529
$52$	0	0
$53$	120.509	0.312323	0.156161	−	0.987732i	$-0.450088\pi$
$53$	120.509	0.312323	0.156161	+	0.987732i	$0.450088\pi$
$54$	0	0
$55$	237.248	0.581646
$56$	0	0
$57$	−895.048	−2.07986
$58$	0	0
$59$	309.004	0.681846	0.340923	−	0.940091i	$-0.389260\pi$
$59$	309.004	0.681846	0.340923	+	0.940091i	$0.389260\pi$
$60$	0	0
$61$	−76.7261	−0.161045	−0.0805227	−	0.996753i	$-0.525659\pi$
$61$	−76.7261	−0.161045	−0.0805227	+	0.996753i	$0.525659\pi$
$62$	0	0
$63$	−150.918	−0.301808
$64$	0	0
$65$	−37.2704	−0.0711205
$66$	0	0
$67$	−502.624	−0.916497	−0.458249	−	0.888824i	$-0.651523\pi$
$67$	−502.624	−0.916497	−0.458249	+	0.888824i	$0.651523\pi$
$68$	0	0
$69$	−183.784	−0.320652
$70$	0	0
$71$	347.930	0.581574	0.290787	−	0.956788i	$-0.406083\pi$
$71$	347.930	0.581574	0.290787	+	0.956788i	$0.406083\pi$
$72$	0	0
$73$	−99.0446	−0.158799	−0.0793993	−	0.996843i	$-0.525300\pi$
$73$	−99.0446	−0.158799	−0.0793993	+	0.996843i	$0.525300\pi$
$74$	0	0
$75$	199.765	0.307559
$76$	0	0
$77$	−194.330	−0.287610
$78$	0	0
$79$	990.840	1.41112	0.705558	−	0.708652i	$-0.250696\pi$
$79$	990.840	1.41112	0.705558	+	0.708652i	$0.250696\pi$
$80$	0	0
$81$	−366.036	−0.502107
$82$	0	0
$83$	−71.3340	−0.0943364	−0.0471682	−	0.998887i	$-0.515020\pi$
$83$	−71.3340	−0.0943364	−0.0471682	+	0.998887i	$0.515020\pi$
$84$	0	0
$85$	292.918	0.373782
$86$	0	0
$87$	409.132	0.504179
$88$	0	0
$89$	−914.981	−1.08975	−0.544875	−	0.838517i	$-0.683423\pi$
$89$	−914.981	−1.08975	−0.544875	+	0.838517i	$0.683423\pi$
$90$	0	0
$91$	30.5282	0.0351673
$92$	0	0
$93$	992.940	1.10713
$94$	0	0
$95$	−560.063	−0.604855
$96$	0	0
$97$	−258.519	−0.270604	−0.135302	−	0.990804i	$-0.543200\pi$
$97$	−258.519	−0.270604	−0.135302	+	0.990804i	$0.543200\pi$
$98$	0	0
$99$	1748.51	1.77507
$100$	0	0
$101$	75.8444	0.0747208	0.0373604	−	0.999302i	$-0.488105\pi$
$101$	75.8444	0.0747208	0.0373604	+	0.999302i	$0.488105\pi$
$102$	0	0
$103$	−1196.34	−1.14446	−0.572228	−	0.820095i	$-0.693921\pi$
$103$	−1196.34	−1.14446	−0.572228	+	0.820095i	$0.693921\pi$
$104$	0	0
$105$	−163.627	−0.152080
$106$	0	0
$107$	736.778	0.665673	0.332837	−	0.942984i	$-0.391994\pi$
$107$	736.778	0.665673	0.332837	+	0.942984i	$0.391994\pi$
$108$	0	0
$109$	−153.186	−0.134610	−0.0673051	−	0.997732i	$-0.521440\pi$
$109$	−153.186	−0.134610	−0.0673051	+	0.997732i	$0.521440\pi$
$110$	0	0
$111$	2929.11	2.50467
$112$	0	0
$113$	1436.26	1.19568	0.597841	−	0.801614i	$-0.296025\pi$
$113$	1436.26	1.19568	0.597841	+	0.801614i	$0.296025\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	−274.682	−0.217046
$118$	0	0
$119$	−239.929	−0.184826
$120$	0	0
$121$	920.470	0.691563
$122$	0	0
$123$	−2712.84	−1.98869
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1896.14	1.32484	0.662421	−	0.749132i	$-0.269529\pi$
$127$	1896.14	1.32484	0.662421	+	0.749132i	$0.269529\pi$
$128$	0	0
$129$	3976.29	2.71390
$130$	0	0
$131$	−887.022	−0.591599	−0.295800	−	0.955250i	$-0.595586\pi$
$131$	−887.022	−0.591599	−0.295800	+	0.955250i	$0.595586\pi$
$132$	0	0
$133$	458.747	0.299085
$134$	0	0
$135$	393.530	0.250886
$136$	0	0
$137$	−940.922	−0.586776	−0.293388	−	0.955993i	$-0.594783\pi$
$137$	−940.922	−0.586776	−0.293388	+	0.955993i	$0.594783\pi$
$138$	0	0
$139$	1532.06	0.934876	0.467438	−	0.884026i	$-0.345177\pi$
$139$	1532.06	0.934876	0.467438	+	0.884026i	$0.345177\pi$
$140$	0	0
$141$	4866.65	2.90671
$142$	0	0
$143$	−353.694	−0.206835
$144$	0	0
$145$	256.008	0.146623
$146$	0	0
$147$	−2606.75	−1.46259
$148$	0	0
$149$	−876.152	−0.481726	−0.240863	−	0.970559i	$-0.577430\pi$
$149$	−876.152	−0.481726	−0.240863	+	0.970559i	$0.577430\pi$
$150$	0	0
$151$	2692.31	1.45097	0.725486	−	0.688237i	$-0.241615\pi$
$151$	2692.31	1.45097	0.725486	+	0.688237i	$0.241615\pi$
$152$	0	0
$153$	2158.80	1.14071
$154$	0	0
$155$	621.317	0.321970
$156$	0	0
$157$	2020.29	1.02699	0.513494	−	0.858093i	$-0.328351\pi$
$157$	2020.29	1.02699	0.513494	+	0.858093i	$0.328351\pi$
$158$	0	0
$159$	962.936	0.480288
$160$	0	0
$161$	94.1964	0.0461100
$162$	0	0
$163$	−1481.53	−0.711916	−0.355958	−	0.934502i	$-0.615845\pi$
$163$	−1481.53	−0.711916	−0.355958	+	0.934502i	$0.615845\pi$
$164$	0	0
$165$	1895.76	0.894452
$166$	0	0
$167$	−438.002	−0.202956	−0.101478	−	0.994838i	$-0.532357\pi$
$167$	−438.002	−0.202956	−0.101478	+	0.994838i	$0.532357\pi$
$168$	0	0
$169$	−2141.44	−0.974709
$170$	0	0
$171$	−4127.64	−1.84590
$172$	0	0
$173$	−4347.01	−1.91039	−0.955194	−	0.295981i	$-0.904353\pi$
$173$	−4347.01	−1.91039	−0.955194	+	0.295981i	$0.904353\pi$
$174$	0	0
$175$	−102.387	−0.0442272
$176$	0	0
$177$	2469.13	1.04854
$178$	0	0
$179$	−2424.48	−1.01237	−0.506185	−	0.862425i	$-0.668945\pi$
$179$	−2424.48	−1.01237	−0.506185	+	0.862425i	$0.668945\pi$
$180$	0	0
$181$	−2841.76	−1.16700	−0.583498	−	0.812115i	$-0.698316\pi$
$181$	−2841.76	−1.16700	−0.583498	+	0.812115i	$0.698316\pi$
$182$	0	0
$183$	−613.088	−0.247655
$184$	0	0
$185$	1832.84	0.728396
$186$	0	0
$187$	2779.78	1.08704
$188$	0	0
$189$	−322.340	−0.124057
$190$	0	0
$191$	1366.42	0.517649	0.258824	−	0.965924i	$-0.416665\pi$
$191$	1366.42	0.517649	0.258824	+	0.965924i	$0.416665\pi$
$192$	0	0
$193$	4697.44	1.75197	0.875983	−	0.482343i	$-0.160214\pi$
$193$	4697.44	1.75197	0.875983	+	0.482343i	$0.160214\pi$
$194$	0	0
$195$	−297.814	−0.109369
$196$	0	0
$197$	621.138	0.224641	0.112320	−	0.993672i	$-0.464172\pi$
$197$	621.138	0.224641	0.112320	+	0.993672i	$0.464172\pi$
$198$	0	0
$199$	2704.80	0.963508	0.481754	−	0.876307i	$-0.340000\pi$
$199$	2704.80	0.963508	0.481754	+	0.876307i	$0.340000\pi$
$200$	0	0
$201$	−4016.27	−1.40938
$202$	0	0
$203$	−209.696	−0.0725013
$204$	0	0
$205$	−1697.52	−0.578340
$206$	0	0
$207$	−847.546	−0.284582
$208$	0	0
$209$	−5314.95	−1.75906
$210$	0	0
$211$	2290.03	0.747166	0.373583	−	0.927597i	$-0.378129\pi$
$211$	2290.03	0.747166	0.373583	+	0.927597i	$0.378129\pi$
$212$	0	0
$213$	2780.18	0.894340
$214$	0	0
$215$	2488.10	0.789243
$216$	0	0
$217$	−508.920	−0.159206
$218$	0	0
$219$	−791.427	−0.244199
$220$	0	0
$221$	−436.688	−0.132918
$222$	0	0
$223$	−803.477	−0.241277	−0.120639	−	0.992697i	$-0.538494\pi$
$223$	−803.477	−0.241277	−0.120639	+	0.992697i	$0.538494\pi$
$224$	0	0
$225$	921.245	0.272962
$226$	0	0
$227$	3013.33	0.881064	0.440532	−	0.897737i	$-0.354790\pi$
$227$	3013.33	0.881064	0.440532	+	0.897737i	$0.354790\pi$
$228$	0	0
$229$	5037.73	1.45372	0.726862	−	0.686784i	$-0.240978\pi$
$229$	5037.73	1.45372	0.726862	+	0.686784i	$0.240978\pi$
$230$	0	0
$231$	−1552.81	−0.442284
$232$	0	0
$233$	−5751.24	−1.61707	−0.808533	−	0.588451i	$-0.799738\pi$
$233$	−5751.24	−1.61707	−0.808533	+	0.588451i	$0.799738\pi$
$234$	0	0
$235$	3045.23	0.845315
$236$	0	0
$237$	7917.41	2.17001
$238$	0	0
$239$	−1027.91	−0.278201	−0.139100	−	0.990278i	$-0.544421\pi$
$239$	−1027.91	−0.278201	−0.139100	+	0.990278i	$0.544421\pi$
$240$	0	0
$241$	2155.75	0.576199	0.288099	−	0.957600i	$-0.406977\pi$
$241$	2155.75	0.576199	0.288099	+	0.957600i	$0.406977\pi$
$242$	0	0
$243$	−5049.91	−1.33314
$244$	0	0
$245$	−1631.13	−0.425344
$246$	0	0
$247$	834.951	0.215088
$248$	0	0
$249$	−570.002	−0.145070
$250$	0	0
$251$	−3871.89	−0.973671	−0.486836	−	0.873494i	$-0.661849\pi$
$251$	−3871.89	−0.973671	−0.486836	+	0.873494i	$0.661849\pi$
$252$	0	0
$253$	−1091.34	−0.271194
$254$	0	0
$255$	2340.60	0.574799
$256$	0	0
$257$	696.880	0.169145	0.0845723	−	0.996417i	$-0.473048\pi$
$257$	696.880	0.169145	0.0845723	+	0.996417i	$0.473048\pi$
$258$	0	0
$259$	−1501.28	−0.360174
$260$	0	0
$261$	1886.77	0.447464
$262$	0	0
$263$	−3736.03	−0.875944	−0.437972	−	0.898989i	$-0.644303\pi$
$263$	−3736.03	−0.875944	−0.437972	+	0.898989i	$0.644303\pi$
$264$	0	0
$265$	602.543	0.139675
$266$	0	0
$267$	−7311.26	−1.67581
$268$	0	0
$269$	−1599.08	−0.362444	−0.181222	−	0.983442i	$-0.558005\pi$
$269$	−1599.08	−0.362444	−0.181222	+	0.983442i	$0.558005\pi$
$270$	0	0
$271$	−3830.39	−0.858596	−0.429298	−	0.903163i	$-0.641239\pi$
$271$	−3830.39	−0.858596	−0.429298	+	0.903163i	$0.641239\pi$
$272$	0	0
$273$	243.939	0.0540800
$274$	0	0
$275$	1186.24	0.260120
$276$	0	0
$277$	1381.98	0.299765	0.149882	−	0.988704i	$-0.452110\pi$
$277$	1381.98	0.299765	0.149882	+	0.988704i	$0.452110\pi$
$278$	0	0
$279$	4579.08	0.982590
$280$	0	0
$281$	8696.01	1.84612	0.923061	−	0.384653i	$-0.125679\pi$
$281$	8696.01	1.84612	0.923061	+	0.384653i	$0.125679\pi$
$282$	0	0
$283$	−2788.83	−0.585791	−0.292895	−	0.956145i	$-0.594619\pi$
$283$	−2788.83	−0.585791	−0.292895	+	0.956145i	$0.594619\pi$
$284$	0	0
$285$	−4475.24	−0.930142
$286$	0	0
$287$	1390.43	0.285974
$288$	0	0
$289$	−1480.95	−0.301435
$290$	0	0
$291$	−2065.72	−0.416133
$292$	0	0
$293$	−5644.32	−1.12541	−0.562704	−	0.826658i	$-0.690239\pi$
$293$	−5644.32	−1.12541	−0.562704	+	0.826658i	$0.690239\pi$
$294$	0	0
$295$	1545.02	0.304931
$296$	0	0
$297$	3734.57	0.729636
$298$	0	0
$299$	171.444	0.0331601
$300$	0	0
$301$	−2038.00	−0.390261
$302$	0	0
$303$	606.043	0.114905
$304$	0	0
$305$	−383.631	−0.0720217
$306$	0	0
$307$	4000.77	0.743766	0.371883	−	0.928280i	$-0.378712\pi$
$307$	4000.77	0.743766	0.371883	+	0.928280i	$0.378712\pi$
$308$	0	0
$309$	−9559.49	−1.75994
$310$	0	0
$311$	4019.54	0.732884	0.366442	−	0.930441i	$-0.380576\pi$
$311$	4019.54	0.732884	0.366442	+	0.930441i	$0.380576\pi$
$312$	0	0
$313$	−1716.06	−0.309896	−0.154948	−	0.987923i	$-0.549521\pi$
$313$	−1716.06	−0.309896	−0.154948	+	0.987923i	$0.549521\pi$
$314$	0	0
$315$	−754.591	−0.134973
$316$	0	0
$317$	−6737.77	−1.19379	−0.596895	−	0.802320i	$-0.703599\pi$
$317$	−6737.77	−1.19379	−0.596895	+	0.802320i	$0.703599\pi$
$318$	0	0
$319$	2429.50	0.426413
$320$	0	0
$321$	5887.31	1.02367
$322$	0	0
$323$	−6562.11	−1.13042
$324$	0	0
$325$	−186.352	−0.0318060
$326$	0	0
$327$	−1224.05	−0.207003
$328$	0	0
$329$	−2494.35	−0.417987
$330$	0	0
$331$	−7493.82	−1.24440	−0.622202	−	0.782857i	$-0.713762\pi$
$331$	−7493.82	−1.24440	−0.622202	+	0.782857i	$0.713762\pi$
$332$	0	0
$333$	13508.0	2.22292
$334$	0	0
$335$	−2513.12	−0.409870
$336$	0	0
$337$	8506.12	1.37495	0.687475	−	0.726208i	$-0.258719\pi$
$337$	8506.12	1.37495	0.687475	+	0.726208i	$0.258719\pi$
$338$	0	0
$339$	11476.6	1.83871
$340$	0	0
$341$	5896.25	0.936364
$342$	0	0
$343$	2740.81	0.431458
$344$	0	0
$345$	−918.920	−0.143400
$346$	0	0
$347$	9950.50	1.53940	0.769698	−	0.638408i	$-0.220407\pi$
$347$	9950.50	1.53940	0.769698	+	0.638408i	$0.220407\pi$
$348$	0	0
$349$	383.653	0.0588438	0.0294219	−	0.999567i	$-0.490633\pi$
$349$	383.653	0.0588438	0.0294219	+	0.999567i	$0.490633\pi$
$350$	0	0
$351$	−586.681	−0.0892158
$352$	0	0
$353$	−3407.37	−0.513756	−0.256878	−	0.966444i	$-0.582694\pi$
$353$	−3407.37	−0.513756	−0.256878	+	0.966444i	$0.582694\pi$
$354$	0	0
$355$	1739.65	0.260088
$356$	0	0
$357$	−1917.18	−0.284224
$358$	0	0
$359$	−2812.41	−0.413464	−0.206732	−	0.978398i	$-0.566283\pi$
$359$	−2812.41	−0.413464	−0.206732	+	0.978398i	$0.566283\pi$
$360$	0	0
$361$	5687.80	0.829246
$362$	0	0
$363$	7355.11	1.06348
$364$	0	0
$365$	−495.223	−0.0710169
$366$	0	0
$367$	−12141.8	−1.72696	−0.863482	−	0.504379i	$-0.831721\pi$
$367$	−12141.8	−1.72696	−0.863482	+	0.504379i	$0.831721\pi$
$368$	0	0
$369$	−12510.6	−1.76498
$370$	0	0
$371$	−493.542	−0.0690658
$372$	0	0
$373$	781.360	0.108465	0.0542323	−	0.998528i	$-0.482729\pi$
$373$	781.360	0.108465	0.0542323	+	0.998528i	$0.482729\pi$
$374$	0	0
$375$	998.826	0.137544
$376$	0	0
$377$	−381.661	−0.0521394
$378$	0	0
$379$	−14535.7	−1.97005	−0.985024	−	0.172415i	$-0.944843\pi$
$379$	−14535.7	−1.97005	−0.985024	+	0.172415i	$0.944843\pi$
$380$	0	0
$381$	15151.3	2.03733
$382$	0	0
$383$	4963.41	0.662189	0.331095	−	0.943598i	$-0.392582\pi$
$383$	4963.41	0.662189	0.331095	+	0.943598i	$0.392582\pi$
$384$	0	0
$385$	−971.649	−0.128623
$386$	0	0
$387$	18337.2	2.40861
$388$	0	0
$389$	−7549.87	−0.984045	−0.492022	−	0.870583i	$-0.663742\pi$
$389$	−7549.87	−0.984045	−0.492022	+	0.870583i	$0.663742\pi$
$390$	0	0
$391$	−1347.42	−0.174277
$392$	0	0
$393$	−7087.85	−0.909757
$394$	0	0
$395$	4954.20	0.631070
$396$	0	0
$397$	3402.49	0.430141	0.215071	−	0.976598i	$-0.431002\pi$
$397$	3402.49	0.430141	0.215071	+	0.976598i	$0.431002\pi$
$398$	0	0
$399$	3665.66	0.459932
$400$	0	0
$401$	3236.55	0.403057	0.201528	−	0.979483i	$-0.435409\pi$
$401$	3236.55	0.403057	0.201528	+	0.979483i	$0.435409\pi$
$402$	0	0
$403$	−926.270	−0.114493
$404$	0	0
$405$	−1830.18	−0.224549
$406$	0	0
$407$	17393.5	2.11834
$408$	0	0
$409$	−3115.77	−0.376687	−0.188343	−	0.982103i	$-0.560312\pi$
$409$	−3115.77	−0.376687	−0.188343	+	0.982103i	$0.560312\pi$
$410$	0	0
$411$	−7518.54	−0.902341
$412$	0	0
$413$	−1265.52	−0.150781
$414$	0	0
$415$	−356.670	−0.0421885
$416$	0	0
$417$	12242.1	1.43765
$418$	0	0
$419$	10744.1	1.25271	0.626354	−	0.779538i	$-0.284546\pi$
$419$	10744.1	1.25271	0.626354	+	0.779538i	$0.284546\pi$
$420$	0	0
$421$	−15606.0	−1.80663	−0.903316	−	0.428976i	$-0.858875\pi$
$421$	−15606.0	−1.80663	−0.903316	+	0.428976i	$0.858875\pi$
$422$	0	0
$423$	22443.2	2.57974
$424$	0	0
$425$	1464.59	0.167160
$426$	0	0
$427$	314.231	0.0356129
$428$	0	0
$429$	−2826.23	−0.318069
$430$	0	0
$431$	11091.0	1.23952	0.619762	−	0.784790i	$-0.287229\pi$
$431$	11091.0	1.23952	0.619762	+	0.784790i	$0.287229\pi$
$432$	0	0
$433$	−9260.91	−1.02783	−0.513915	−	0.857841i	$-0.671805\pi$
$433$	−9260.91	−1.02783	−0.513915	+	0.857841i	$0.671805\pi$
$434$	0	0
$435$	2045.66	0.225476
$436$	0	0
$437$	2576.29	0.282015
$438$	0	0
$439$	−3716.39	−0.404041	−0.202020	−	0.979381i	$-0.564751\pi$
$439$	−3716.39	−0.404041	−0.202020	+	0.979381i	$0.564751\pi$
$440$	0	0
$441$	−12021.4	−1.29807
$442$	0	0
$443$	15339.9	1.64520	0.822598	−	0.568623i	$-0.192524\pi$
$443$	15339.9	1.64520	0.822598	+	0.568623i	$0.192524\pi$
$444$	0	0
$445$	−4574.91	−0.487351
$446$	0	0
$447$	−7000.99	−0.740795
$448$	0	0
$449$	−9439.98	−0.992206	−0.496103	−	0.868264i	$-0.665236\pi$
$449$	−9439.98	−0.992206	−0.496103	+	0.868264i	$0.665236\pi$
$450$	0	0
$451$	−16109.3	−1.68195
$452$	0	0
$453$	21513.2	2.23129
$454$	0	0
$455$	152.641	0.0157273
$456$	0	0
$457$	1375.90	0.140835	0.0704177	−	0.997518i	$-0.477567\pi$
$457$	1375.90	0.140835	0.0704177	+	0.997518i	$0.477567\pi$
$458$	0	0
$459$	4610.89	0.468884
$460$	0	0
$461$	−5253.55	−0.530764	−0.265382	−	0.964143i	$-0.585498\pi$
$461$	−5253.55	−0.530764	−0.265382	+	0.964143i	$0.585498\pi$
$462$	0	0
$463$	13847.3	1.38993	0.694966	−	0.719042i	$-0.255419\pi$
$463$	13847.3	1.38993	0.694966	+	0.719042i	$0.255419\pi$
$464$	0	0
$465$	4964.70	0.495123
$466$	0	0
$467$	−1546.89	−0.153279	−0.0766397	−	0.997059i	$-0.524419\pi$
$467$	−1546.89	−0.153279	−0.0766397	+	0.997059i	$0.524419\pi$
$468$	0	0
$469$	2058.49	0.202670
$470$	0	0
$471$	16143.4	1.57929
$472$	0	0
$473$	23611.9	2.29530
$474$	0	0
$475$	−2800.31	−0.270499
$476$	0	0
$477$	4440.72	0.426261
$478$	0	0
$479$	2649.92	0.252772	0.126386	−	0.991981i	$-0.459662\pi$
$479$	2649.92	0.252772	0.126386	+	0.991981i	$0.459662\pi$
$480$	0	0
$481$	−2732.43	−0.259019
$482$	0	0
$483$	752.686	0.0709077
$484$	0	0
$485$	−1292.59	−0.121018
$486$	0	0
$487$	18373.2	1.70959	0.854793	−	0.518969i	$-0.173684\pi$
$487$	18373.2	1.70959	0.854793	+	0.518969i	$0.173684\pi$
$488$	0	0
$489$	−11838.3	−1.09478
$490$	0	0
$491$	12551.2	1.15362	0.576809	−	0.816879i	$-0.304298\pi$
$491$	12551.2	1.15362	0.576809	+	0.816879i	$0.304298\pi$
$492$	0	0
$493$	2999.58	0.274025
$494$	0	0
$495$	8742.55	0.793836
$496$	0	0
$497$	−1424.95	−0.128607
$498$	0	0
$499$	−4376.15	−0.392592	−0.196296	−	0.980545i	$-0.562891\pi$
$499$	−4376.15	−0.392592	−0.196296	+	0.980545i	$0.562891\pi$
$500$	0	0
$501$	−3499.90	−0.312104
$502$	0	0
$503$	−12395.5	−1.09878	−0.549392	−	0.835565i	$-0.685140\pi$
$503$	−12395.5	−1.09878	−0.549392	+	0.835565i	$0.685140\pi$
$504$	0	0
$505$	379.222	0.0334162
$506$	0	0
$507$	−17111.4	−1.49890
$508$	0	0
$509$	−1670.81	−0.145496	−0.0727479	−	0.997350i	$-0.523177\pi$
$509$	−1670.81	−0.145496	−0.0727479	+	0.997350i	$0.523177\pi$
$510$	0	0
$511$	405.637	0.0351161
$512$	0	0
$513$	−8816.05	−0.758749
$514$	0	0
$515$	−5981.70	−0.511816
$516$	0	0
$517$	28899.0	2.45837
$518$	0	0
$519$	−34735.2	−2.93778
$520$	0	0
$521$	8045.24	0.676523	0.338261	−	0.941052i	$-0.390161\pi$
$521$	8045.24	0.676523	0.338261	+	0.941052i	$0.390161\pi$
$522$	0	0
$523$	−16389.8	−1.37032	−0.685160	−	0.728393i	$-0.740268\pi$
$523$	−16389.8	−1.37032	−0.685160	+	0.728393i	$0.740268\pi$
$524$	0	0
$525$	−818.137	−0.0680123
$526$	0	0
$527$	7279.81	0.601733
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	11386.7	0.930589
$532$	0	0
$533$	2530.69	0.205659
$534$	0	0
$535$	3683.89	0.297698
$536$	0	0
$537$	−19373.1	−1.55681
$538$	0	0
$539$	−15479.4	−1.23700
$540$	0	0
$541$	−7259.35	−0.576902	−0.288451	−	0.957495i	$-0.593140\pi$
$541$	−7259.35	−0.576902	−0.288451	+	0.957495i	$0.593140\pi$
$542$	0	0
$543$	−22707.4	−1.79460
$544$	0	0
$545$	−765.928	−0.0601995
$546$	0	0
$547$	2400.11	0.187607	0.0938037	−	0.995591i	$-0.470097\pi$
$547$	2400.11	0.187607	0.0938037	+	0.995591i	$0.470097\pi$
$548$	0	0
$549$	−2827.34	−0.219796
$550$	0	0
$551$	−5735.22	−0.443428
$552$	0	0
$553$	−4057.98	−0.312048
$554$	0	0
$555$	14645.5	1.12012
$556$	0	0
$557$	−4678.92	−0.355929	−0.177964	−	0.984037i	$-0.556951\pi$
$557$	−4678.92	−0.355929	−0.177964	+	0.984037i	$0.556951\pi$
$558$	0	0
$559$	−3709.31	−0.280657
$560$	0	0
$561$	22212.1	1.67165
$562$	0	0
$563$	−24347.0	−1.82257	−0.911283	−	0.411780i	$-0.864907\pi$
$563$	−24347.0	−1.82257	−0.911283	+	0.411780i	$0.864907\pi$
$564$	0	0
$565$	7181.31	0.534726
$566$	0	0
$567$	1499.10	0.111034
$568$	0	0
$569$	−21448.3	−1.58024	−0.790122	−	0.612950i	$-0.789983\pi$
$569$	−21448.3	−1.58024	−0.790122	+	0.612950i	$0.789983\pi$
$570$	0	0
$571$	−15363.0	−1.12596	−0.562980	−	0.826470i	$-0.690345\pi$
$571$	−15363.0	−1.12596	−0.562980	+	0.826470i	$0.690345\pi$
$572$	0	0
$573$	10918.6	0.796037
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	−7342.34	−0.529750	−0.264875	−	0.964283i	$-0.585331\pi$
$577$	−7342.34	−0.529750	−0.264875	+	0.964283i	$0.585331\pi$
$578$	0	0
$579$	37535.4	2.69416
$580$	0	0
$581$	292.148	0.0208612
$582$	0	0
$583$	5718.09	0.406207
$584$	0	0
$585$	−1373.41	−0.0970658
$586$	0	0
$587$	−8972.98	−0.630927	−0.315464	−	0.948938i	$-0.602160\pi$
$587$	−8972.98	−0.630927	−0.315464	+	0.948938i	$0.602160\pi$
$588$	0	0
$589$	−13919.1	−0.973726
$590$	0	0
$591$	4963.27	0.345451
$592$	0	0
$593$	−15585.1	−1.07926	−0.539631	−	0.841902i	$-0.681436\pi$
$593$	−15585.1	−1.07926	−0.539631	+	0.841902i	$0.681436\pi$
$594$	0	0
$595$	−1199.65	−0.0826566
$596$	0	0
$597$	21613.0	1.48168
$598$	0	0
$599$	−4314.40	−0.294293	−0.147147	−	0.989115i	$-0.547009\pi$
$599$	−4314.40	−0.294293	−0.147147	+	0.989115i	$0.547009\pi$
$600$	0	0
$601$	22589.4	1.53318	0.766588	−	0.642139i	$-0.221953\pi$
$601$	22589.4	1.53318	0.766588	+	0.642139i	$0.221953\pi$
$602$	0	0
$603$	−18521.6	−1.25084
$604$	0	0
$605$	4602.35	0.309276
$606$	0	0
$607$	19308.0	1.29109	0.645543	−	0.763724i	$-0.276631\pi$
$607$	19308.0	1.29109	0.645543	+	0.763724i	$0.276631\pi$
$608$	0	0
$609$	−1675.60	−0.111492
$610$	0	0
$611$	−4539.89	−0.300596
$612$	0	0
$613$	7921.30	0.521922	0.260961	−	0.965349i	$-0.415961\pi$
$613$	7921.30	0.521922	0.260961	+	0.965349i	$0.415961\pi$
$614$	0	0
$615$	−13564.2	−0.889367
$616$	0	0
$617$	−5410.15	−0.353005	−0.176503	−	0.984300i	$-0.556478\pi$
$617$	−5410.15	−0.353005	−0.176503	+	0.984300i	$0.556478\pi$
$618$	0	0
$619$	−24669.2	−1.60184	−0.800919	−	0.598772i	$-0.795655\pi$
$619$	−24669.2	−1.60184	−0.800919	+	0.598772i	$0.795655\pi$
$620$	0	0
$621$	−1810.24	−0.116976
$622$	0	0
$623$	3747.30	0.240983
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−42469.7	−2.70507
$628$	0	0
$629$	21474.9	1.36131
$630$	0	0
$631$	25558.4	1.61246	0.806231	−	0.591601i	$-0.201504\pi$
$631$	25558.4	1.61246	0.806231	+	0.591601i	$0.201504\pi$
$632$	0	0
$633$	18298.7	1.14899
$634$	0	0
$635$	9480.68	0.592487
$636$	0	0
$637$	2431.72	0.151253
$638$	0	0
$639$	12821.2	0.793737
$640$	0	0
$641$	−26198.6	−1.61432	−0.807162	−	0.590330i	$-0.798998\pi$
$641$	−26198.6	−1.61432	−0.807162	+	0.590330i	$0.798998\pi$
$642$	0	0
$643$	−19379.6	−1.18858	−0.594290	−	0.804251i	$-0.702567\pi$
$643$	−19379.6	−1.18858	−0.594290	+	0.804251i	$0.702567\pi$
$644$	0	0
$645$	19881.4	1.21369
$646$	0	0
$647$	−13631.7	−0.828308	−0.414154	−	0.910207i	$-0.635923\pi$
$647$	−13631.7	−0.828308	−0.414154	+	0.910207i	$0.635923\pi$
$648$	0	0
$649$	14662.1	0.886810
$650$	0	0
$651$	−4066.58	−0.244826
$652$	0	0
$653$	11260.7	0.674831	0.337415	−	0.941356i	$-0.390447\pi$
$653$	11260.7	0.674831	0.337415	+	0.941356i	$0.390447\pi$
$654$	0	0
$655$	−4435.11	−0.264571
$656$	0	0
$657$	−3649.78	−0.216730
$658$	0	0
$659$	14027.9	0.829212	0.414606	−	0.910001i	$-0.363919\pi$
$659$	14027.9	0.829212	0.414606	+	0.910001i	$0.363919\pi$
$660$	0	0
$661$	6828.08	0.401788	0.200894	−	0.979613i	$-0.435615\pi$
$661$	6828.08	0.401788	0.200894	+	0.979613i	$0.435615\pi$
$662$	0	0
$663$	−3489.40	−0.204400
$664$	0	0
$665$	2293.73	0.133755
$666$	0	0
$667$	−1177.64	−0.0683633
$668$	0	0
$669$	−6420.27	−0.371034
$670$	0	0
$671$	−3640.63	−0.209456
$672$	0	0
$673$	1763.77	0.101023	0.0505115	−	0.998723i	$-0.483915\pi$
$673$	1763.77	0.101023	0.0505115	+	0.998723i	$0.483915\pi$
$674$	0	0
$675$	1967.65	0.112200
$676$	0	0
$677$	−2484.00	−0.141016	−0.0705079	−	0.997511i	$-0.522462\pi$
$677$	−2484.00	−0.141016	−0.0705079	+	0.997511i	$0.522462\pi$
$678$	0	0
$679$	1058.76	0.0598403
$680$	0	0
$681$	24078.3	1.35489
$682$	0	0
$683$	−2363.24	−0.132396	−0.0661982	−	0.997806i	$-0.521087\pi$
$683$	−2363.24	−0.132396	−0.0661982	+	0.997806i	$0.521087\pi$
$684$	0	0
$685$	−4704.61	−0.262414
$686$	0	0
$687$	40254.5	2.23553
$688$	0	0
$689$	−898.281	−0.0496688
$690$	0	0
$691$	21226.8	1.16861	0.584303	−	0.811535i	$-0.301368\pi$
$691$	21226.8	1.16861	0.584303	+	0.811535i	$0.301368\pi$
$692$	0	0
$693$	−7161.02	−0.392532
$694$	0	0
$695$	7660.30	0.418089
$696$	0	0
$697$	−19889.3	−1.08086
$698$	0	0
$699$	−45955.9	−2.48671
$700$	0	0
$701$	6943.83	0.374130	0.187065	−	0.982348i	$-0.440103\pi$
$701$	6943.83	0.374130	0.187065	+	0.982348i	$0.440103\pi$
$702$	0	0
$703$	−41060.3	−2.20287
$704$	0	0
$705$	24333.3	1.29992
$706$	0	0
$707$	−310.620	−0.0165235
$708$	0	0
$709$	23968.7	1.26963	0.634813	−	0.772666i	$-0.281077\pi$
$709$	23968.7	1.26963	0.634813	+	0.772666i	$0.281077\pi$
$710$	0	0
$711$	36512.3	1.92590
$712$	0	0
$713$	−2858.06	−0.150119
$714$	0	0
$715$	−1768.47	−0.0924993
$716$	0	0
$717$	−8213.62	−0.427815
$718$	0	0
$719$	−20474.8	−1.06201	−0.531003	−	0.847370i	$-0.678185\pi$
$719$	−20474.8	−1.06201	−0.531003	+	0.847370i	$0.678185\pi$
$720$	0	0
$721$	4899.60	0.253080
$722$	0	0
$723$	17225.7	0.886075
$724$	0	0
$725$	1280.04	0.0655717
$726$	0	0
$727$	−26684.2	−1.36129	−0.680647	−	0.732611i	$-0.738301\pi$
$727$	−26684.2	−1.36129	−0.680647	+	0.732611i	$0.738301\pi$
$728$	0	0
$729$	−30468.9	−1.54798
$730$	0	0
$731$	29152.4	1.47502
$732$	0	0
$733$	5756.71	0.290080	0.145040	−	0.989426i	$-0.453669\pi$
$733$	5756.71	0.290080	0.145040	+	0.989426i	$0.453669\pi$
$734$	0	0
$735$	−13033.8	−0.654092
$736$	0	0
$737$	−23849.3	−1.19200
$738$	0	0
$739$	−9207.98	−0.458351	−0.229175	−	0.973385i	$-0.573603\pi$
$739$	−9207.98	−0.458351	−0.229175	+	0.973385i	$0.573603\pi$
$740$	0	0
$741$	6671.77	0.330760
$742$	0	0
$743$	1728.39	0.0853412	0.0426706	−	0.999089i	$-0.486413\pi$
$743$	1728.39	0.0853412	0.0426706	+	0.999089i	$0.486413\pi$
$744$	0	0
$745$	−4380.76	−0.215434
$746$	0	0
$747$	−2628.64	−0.128751
$748$	0	0
$749$	−3017.47	−0.147204
$750$	0	0
$751$	−12539.5	−0.609283	−0.304642	−	0.952467i	$-0.598537\pi$
$751$	−12539.5	−0.609283	−0.304642	+	0.952467i	$0.598537\pi$
$752$	0	0
$753$	−30938.7	−1.49730
$754$	0	0
$755$	13461.5	0.648894
$756$	0	0
$757$	−29017.4	−1.39321	−0.696603	−	0.717457i	$-0.745306\pi$
$757$	−29017.4	−1.39321	−0.696603	+	0.717457i	$0.745306\pi$
$758$	0	0
$759$	−8720.49	−0.417040
$760$	0	0
$761$	−30059.0	−1.43185	−0.715925	−	0.698177i	$-0.753995\pi$
$761$	−30059.0	−1.43185	−0.715925	+	0.698177i	$0.753995\pi$
$762$	0	0
$763$	627.370	0.0297672
$764$	0	0
$765$	10794.0	0.510141
$766$	0	0
$767$	−2303.35	−0.108434
$768$	0	0
$769$	−2352.27	−0.110305	−0.0551527	−	0.998478i	$-0.517565\pi$
$769$	−2352.27	−0.110305	−0.0551527	+	0.998478i	$0.517565\pi$
$770$	0	0
$771$	5568.49	0.260109
$772$	0	0
$773$	−5656.93	−0.263216	−0.131608	−	0.991302i	$-0.542014\pi$
$773$	−5656.93	−0.263216	−0.131608	+	0.991302i	$0.542014\pi$
$774$	0	0
$775$	3106.58	0.143989
$776$	0	0
$777$	−11996.1	−0.553872
$778$	0	0
$779$	38028.6	1.74906
$780$	0	0
$781$	16509.2	0.756396
$782$	0	0
$783$	4029.87	0.183928
$784$	0	0
$785$	10101.5	0.459283
$786$	0	0
$787$	−13145.9	−0.595428	−0.297714	−	0.954655i	$-0.596224\pi$
$787$	−13145.9	−0.595428	−0.297714	+	0.954655i	$0.596224\pi$
$788$	0	0
$789$	−29853.1	−1.34702
$790$	0	0
$791$	−5882.20	−0.264408
$792$	0	0
$793$	571.923	0.0256111
$794$	0	0
$795$	4814.68	0.214791
$796$	0	0
$797$	12040.5	0.535127	0.267564	−	0.963540i	$-0.413781\pi$
$797$	12040.5	0.535127	0.267564	+	0.963540i	$0.413781\pi$
$798$	0	0
$799$	35680.2	1.57982
$800$	0	0
$801$	−33716.9	−1.48730
$802$	0	0
$803$	−4699.63	−0.206534
$804$	0	0
$805$	470.982	0.0206210
$806$	0	0
$807$	−12777.6	−0.557365
$808$	0	0
$809$	37967.8	1.65003	0.825017	−	0.565108i	$-0.191166\pi$
$809$	37967.8	1.65003	0.825017	+	0.565108i	$0.191166\pi$
$810$	0	0
$811$	−34040.5	−1.47389	−0.736943	−	0.675954i	$-0.763732\pi$
$811$	−34040.5	−1.47389	−0.736943	+	0.675954i	$0.763732\pi$
$812$	0	0
$813$	−30607.1	−1.32034
$814$	0	0
$815$	−7407.65	−0.318379
$816$	0	0
$817$	−55739.7	−2.38689
$818$	0	0
$819$	1124.96	0.0479966
$820$	0	0
$821$	−1974.57	−0.0839380	−0.0419690	−	0.999119i	$-0.513363\pi$
$821$	−1974.57	−0.0839380	−0.0419690	+	0.999119i	$0.513363\pi$
$822$	0	0
$823$	−38254.5	−1.62025	−0.810127	−	0.586254i	$-0.800602\pi$
$823$	−38254.5	−1.62025	−0.810127	+	0.586254i	$0.800602\pi$
$824$	0	0
$825$	9478.79	0.400011
$826$	0	0
$827$	16811.6	0.706890	0.353445	−	0.935455i	$-0.385010\pi$
$827$	16811.6	0.706890	0.353445	+	0.935455i	$0.385010\pi$
$828$	0	0
$829$	32015.2	1.34129	0.670647	−	0.741777i	$-0.266017\pi$
$829$	32015.2	1.34129	0.670647	+	0.741777i	$0.266017\pi$
$830$	0	0
$831$	11042.8	0.460976
$832$	0	0
$833$	−19111.6	−0.794930
$834$	0	0
$835$	−2190.01	−0.0907646
$836$	0	0
$837$	9780.27	0.403890
$838$	0	0
$839$	−41374.8	−1.70252	−0.851262	−	0.524741i	$-0.824162\pi$
$839$	−41374.8	−1.70252	−0.851262	+	0.524741i	$0.824162\pi$
$840$	0	0
$841$	−21767.4	−0.892509
$842$	0	0
$843$	69486.4	2.83895
$844$	0	0
$845$	−10707.2	−0.435903
$846$	0	0
$847$	−3769.78	−0.152929
$848$	0	0
$849$	−22284.5	−0.900825
$850$	0	0
$851$	−8431.08	−0.339616
$852$	0	0
$853$	−6906.10	−0.277210	−0.138605	−	0.990348i	$-0.544262\pi$
$853$	−6906.10	−0.277210	−0.138605	+	0.990348i	$0.544262\pi$
$854$	0	0
$855$	−20638.2	−0.825511
$856$	0	0
$857$	−16566.0	−0.660307	−0.330154	−	0.943927i	$-0.607101\pi$
$857$	−16566.0	−0.660307	−0.330154	+	0.943927i	$0.607101\pi$
$858$	0	0
$859$	−31187.1	−1.23875	−0.619377	−	0.785094i	$-0.712615\pi$
$859$	−31187.1	−1.23875	−0.619377	+	0.785094i	$0.712615\pi$
$860$	0	0
$861$	11110.4	0.439770
$862$	0	0
$863$	−20668.7	−0.815262	−0.407631	−	0.913147i	$-0.633645\pi$
$863$	−20668.7	−0.815262	−0.407631	+	0.913147i	$0.633645\pi$
$864$	0	0
$865$	−21735.0	−0.854351
$866$	0	0
$867$	−11833.7	−0.463545
$868$	0	0
$869$	47015.0	1.83530
$870$	0	0
$871$	3746.61	0.145751
$872$	0	0
$873$	−9526.37	−0.369323
$874$	0	0
$875$	−511.937	−0.0197790
$876$	0	0
$877$	7978.35	0.307195	0.153597	−	0.988134i	$-0.450914\pi$
$877$	7978.35	0.307195	0.153597	+	0.988134i	$0.450914\pi$
$878$	0	0
$879$	−45101.6	−1.73065
$880$	0	0
$881$	−30319.1	−1.15945	−0.579726	−	0.814811i	$-0.696841\pi$
$881$	−30319.1	−1.15945	−0.579726	+	0.814811i	$0.696841\pi$
$882$	0	0
$883$	12071.7	0.460073	0.230036	−	0.973182i	$-0.426115\pi$
$883$	12071.7	0.460073	0.230036	+	0.973182i	$0.426115\pi$
$884$	0	0
$885$	12345.7	0.468921
$886$	0	0
$887$	17077.4	0.646451	0.323225	−	0.946322i	$-0.395233\pi$
$887$	17077.4	0.646451	0.323225	+	0.946322i	$0.395233\pi$
$888$	0	0
$889$	−7765.62	−0.292970
$890$	0	0
$891$	−17368.3	−0.653041
$892$	0	0
$893$	−68220.8	−2.55646
$894$	0	0
$895$	−12122.4	−0.452745
$896$	0	0
$897$	1369.94	0.0509934
$898$	0	0
$899$	6362.49	0.236041
$900$	0	0
$901$	7059.83	0.261040
$902$	0	0
$903$	−16284.9	−0.600140
$904$	0	0
$905$	−14208.8	−0.521896
$906$	0	0
$907$	−3395.69	−0.124313	−0.0621565	−	0.998066i	$-0.519798\pi$
$907$	−3395.69	−0.124313	−0.0621565	+	0.998066i	$0.519798\pi$
$908$	0	0
$909$	2794.85	0.101980
$910$	0	0
$911$	−26847.3	−0.976390	−0.488195	−	0.872735i	$-0.662344\pi$
$911$	−26847.3	−0.976390	−0.488195	+	0.872735i	$0.662344\pi$
$912$	0	0
$913$	−3384.77	−0.122694
$914$	0	0
$915$	−3065.44	−0.110754
$916$	0	0
$917$	3632.80	0.130824
$918$	0	0
$919$	−4992.30	−0.179196	−0.0895978	−	0.995978i	$-0.528558\pi$
$919$	−4992.30	−0.179196	−0.0895978	+	0.995978i	$0.528558\pi$
$920$	0	0
$921$	31968.6	1.14376
$922$	0	0
$923$	−2593.50	−0.0924878
$924$	0	0
$925$	9164.21	0.325749
$926$	0	0
$927$	−44084.9	−1.56196
$928$	0	0
$929$	19718.4	0.696383	0.348191	−	0.937423i	$-0.386796\pi$
$929$	19718.4	0.696383	0.348191	+	0.937423i	$0.386796\pi$
$930$	0	0
$931$	36541.5	1.28636
$932$	0	0
$933$	32118.5	1.12702
$934$	0	0
$935$	13898.9	0.486141
$936$	0	0
$937$	−2954.11	−0.102995	−0.0514976	−	0.998673i	$-0.516399\pi$
$937$	−2954.11	−0.102995	−0.0514976	+	0.998673i	$0.516399\pi$
$938$	0	0
$939$	−13712.4	−0.476556
$940$	0	0
$941$	−17447.2	−0.604424	−0.302212	−	0.953241i	$-0.597725\pi$
$941$	−17447.2	−0.604424	−0.302212	+	0.953241i	$0.597725\pi$
$942$	0	0
$943$	7808.57	0.269652
$944$	0	0
$945$	−1611.70	−0.0554800
$946$	0	0
$947$	2058.23	0.0706266	0.0353133	−	0.999376i	$-0.488757\pi$
$947$	2058.23	0.0706266	0.0353133	+	0.999376i	$0.488757\pi$
$948$	0	0
$949$	738.287	0.0252538
$950$	0	0
$951$	−53838.9	−1.83580
$952$	0	0
$953$	−19875.7	−0.675590	−0.337795	−	0.941220i	$-0.609681\pi$
$953$	−19875.7	−0.675590	−0.337795	+	0.941220i	$0.609681\pi$
$954$	0	0
$955$	6832.12	0.231500
$956$	0	0
$957$	19413.2	0.655735
$958$	0	0
$959$	3853.54	0.129757
$960$	0	0
$961$	−14349.6	−0.481676
$962$	0	0
$963$	27150.1	0.908516
$964$	0	0
$965$	23487.2	0.783503
$966$	0	0
$967$	−47947.1	−1.59449	−0.797246	−	0.603655i	$-0.793711\pi$
$967$	−47947.1	−1.59449	−0.797246	+	0.603655i	$0.793711\pi$
$968$	0	0
$969$	−52435.2	−1.73835
$970$	0	0
$971$	21547.0	0.712127	0.356064	−	0.934462i	$-0.384119\pi$
$971$	21547.0	0.712127	0.356064	+	0.934462i	$0.384119\pi$
$972$	0	0
$973$	−6274.55	−0.206735
$974$	0	0
$975$	−1489.07	−0.0489111
$976$	0	0
$977$	−54581.5	−1.78733	−0.893663	−	0.448738i	$-0.851874\pi$
$977$	−54581.5	−1.78733	−0.893663	+	0.448738i	$0.851874\pi$
$978$	0	0
$979$	−43415.5	−1.41733
$980$	0	0
$981$	−5644.86	−0.183717
$982$	0	0
$983$	−44404.8	−1.44079	−0.720394	−	0.693565i	$-0.756039\pi$
$983$	−44404.8	−1.44079	−0.720394	+	0.693565i	$0.756039\pi$
$984$	0	0
$985$	3105.69	0.100462
$986$	0	0
$987$	−19931.3	−0.642778
$988$	0	0
$989$	−11445.3	−0.367986
$990$	0	0
$991$	42750.0	1.37033	0.685166	−	0.728387i	$-0.259730\pi$
$991$	42750.0	1.37033	0.685166	+	0.728387i	$0.259730\pi$
$992$	0	0
$993$	−59880.2	−1.91364
$994$	0	0
$995$	13524.0	0.430894
$996$	0	0
$997$	−5790.43	−0.183937	−0.0919683	−	0.995762i	$-0.529316\pi$
$997$	−5790.43	−0.183937	−0.0919683	+	0.995762i	$0.529316\pi$
$998$	0	0
$999$	28851.1	0.913723

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.q.1.5		5
4.3	odd	2		460.4.a.a.1.1	✓	5
20.3	even	4		2300.4.c.c.1749.1		10
20.7	even	4		2300.4.c.c.1749.10		10
20.19	odd	2		2300.4.a.d.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.4.a.a.1.1	✓	5	4.3	odd	2
1840.4.a.q.1.5		5	1.1	even	1	trivial
2300.4.a.d.1.5		5	20.19	odd	2
2300.4.c.c.1749.1		10	20.3	even	4
2300.4.c.c.1749.10		10	20.7	even	4

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)

\(f(q)\)	\(=\)	\(q+7.99061 q^{3} +5.00000 q^{5} -4.09549 q^{7} +36.8498 q^{9} +O(q^{10})\)	\(q+7.99061 q^{3} +5.00000 q^{5} -4.09549 q^{7} +36.8498 q^{9} +47.4497 q^{11} -7.45409 q^{13} +39.9530 q^{15} +58.5837 q^{17} -112.013 q^{19} -32.7255 q^{21} -23.0000 q^{23} +25.0000 q^{25} +78.7060 q^{27} +51.2016 q^{29} +124.263 q^{31} +379.152 q^{33} -20.4775 q^{35} +366.568 q^{37} -59.5627 q^{39} -339.503 q^{41} +497.620 q^{43} +184.249 q^{45} +609.046 q^{47} -326.227 q^{49} +468.119 q^{51} +120.509 q^{53} +237.248 q^{55} -895.048 q^{57} +309.004 q^{59} -76.7261 q^{61} -150.918 q^{63} -37.2704 q^{65} -502.624 q^{67} -183.784 q^{69} +347.930 q^{71} -99.0446 q^{73} +199.765 q^{75} -194.330 q^{77} +990.840 q^{79} -366.036 q^{81} -71.3340 q^{83} +292.918 q^{85} +409.132 q^{87} -914.981 q^{89} +30.5282 q^{91} +992.940 q^{93} -560.063 q^{95} -258.519 q^{97} +1748.51 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 7 q^{3} + 25 q^{5} + 20 q^{7} + 30 q^{9}+O(q^{10}) \) 5 * q + 7 * q^3 + 25 * q^5 + 20 * q^7 + 30 * q^9	\( 5 q + 7 q^{3} + 25 q^{5} + 20 q^{7} + 30 q^{9} + 63 q^{11} - 99 q^{13} + 35 q^{15} - 44 q^{17} + 199 q^{19} - 68 q^{21} - 115 q^{23} + 125 q^{25} + 28 q^{27} + 231 q^{29} + 518 q^{31} - 111 q^{33} + 100 q^{35} - 113 q^{37} + 974 q^{39} - 174 q^{41} + 298 q^{43} + 150 q^{45} + 360 q^{47} - 163 q^{49} + 1163 q^{51} + 217 q^{53} + 315 q^{55} - 684 q^{57} + 1551 q^{59} - 737 q^{61} + 1353 q^{63} - 495 q^{65} + 539 q^{67} - 161 q^{69} + 1736 q^{71} - 628 q^{73} + 175 q^{75} - 882 q^{77} + 2954 q^{79} - 495 q^{81} + 153 q^{83} - 220 q^{85} + 274 q^{87} - 1558 q^{89} + 37 q^{91} + 584 q^{93} + 995 q^{95} - 1375 q^{97} + 2487 q^{99}+O(q^{100}) \) 5 * q + 7 * q^3 + 25 * q^5 + 20 * q^7 + 30 * q^9 + 63 * q^11 - 99 * q^13 + 35 * q^15 - 44 * q^17 + 199 * q^19 - 68 * q^21 - 115 * q^23 + 125 * q^25 + 28 * q^27 + 231 * q^29 + 518 * q^31 - 111 * q^33 + 100 * q^35 - 113 * q^37 + 974 * q^39 - 174 * q^41 + 298 * q^43 + 150 * q^45 + 360 * q^47 - 163 * q^49 + 1163 * q^51 + 217 * q^53 + 315 * q^55 - 684 * q^57 + 1551 * q^59 - 737 * q^61 + 1353 * q^63 - 495 * q^65 + 539 * q^67 - 161 * q^69 + 1736 * q^71 - 628 * q^73 + 175 * q^75 - 882 * q^77 + 2954 * q^79 - 495 * q^81 + 153 * q^83 - 220 * q^85 + 274 * q^87 - 1558 * q^89 + 37 * q^91 + 584 * q^93 + 995 * q^95 - 1375 * q^97 + 2487 * q^99

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