LMFDB - Embedded newform 864.4.a.q.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [864,4,Mod(1,864)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(864, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("864.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	864.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,6,0,-9,0,0,0,-18] 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:	$50.9776502450$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	864.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+15.4193 q^{5} -29.5718 q^{7} +6.88586 q^{11} -25.0384 q^{13} +98.3347 q^{17} -63.7439 q^{19} -15.5139 q^{23} +112.754 q^{25} +77.6771 q^{29} -209.031 q^{31} -455.976 q^{35} -295.347 q^{37} -213.831 q^{41} -150.974 q^{43} +478.210 q^{47} +531.494 q^{49} -427.378 q^{53} +106.175 q^{55} -798.377 q^{59} -223.346 q^{61} -386.074 q^{65} +1026.21 q^{67} -925.890 q^{71} -1070.06 q^{73} -203.628 q^{77} +860.902 q^{79} -459.709 q^{83} +1516.25 q^{85} +787.829 q^{89} +740.432 q^{91} -982.885 q^{95} +118.831 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{5} - 9 q^{7} - 18 q^{11} + 3 q^{13} + 18 q^{17} - 27 q^{19} - 90 q^{23} + 21 q^{25} + 72 q^{29} - 144 q^{31} - 450 q^{35} - 159 q^{37} - 168 q^{41} + 180 q^{43} - 522 q^{47} + 150 q^{49} - 84 q^{53}+ \cdots - 117 q^{97}+O(q^{100}) $ 3 * q + 6 * q^5 - 9 * q^7 - 18 * q^11 + 3 * q^13 + 18 * q^17 - 27 * q^19 - 90 * q^23 + 21 * q^25 + 72 * q^29 - 144 * q^31 - 450 * q^35 - 159 * q^37 - 168 * q^41 + 180 * q^43 - 522 * q^47 + 150 * q^49 - 84 * q^53 - 324 * q^55 - 1458 * q^59 + 33 * q^61 + 246 * q^65 + 837 * q^67 - 1908 * q^71 - 165 * q^73 - 954 * q^77 - 531 * q^79 - 2268 * q^83 + 708 * q^85 + 618 * q^89 + 1719 * q^91 - 2934 * q^95 - 117 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	15.4193	1.37914	0.689571	−	0.724218i	$-0.257799\pi$
$5$	15.4193	1.37914	0.689571	+	0.724218i	$0.257799\pi$
$6$	0	0
$7$	−29.5718	−1.59673	−0.798365	−	0.602174i	$-0.794301\pi$
$7$	−29.5718	−1.59673	−0.798365	+	0.602174i	$0.794301\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.88586	0.188742	0.0943711	−	0.995537i	$-0.469916\pi$
$11$	6.88586	0.188742	0.0943711	+	0.995537i	$0.469916\pi$
$12$	0	0
$13$	−25.0384	−0.534185	−0.267093	−	0.963671i	$-0.586063\pi$
$13$	−25.0384	−0.534185	−0.267093	+	0.963671i	$0.586063\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	98.3347	1.40292	0.701461	−	0.712708i	$-0.252532\pi$
$17$	98.3347	1.40292	0.701461	+	0.712708i	$0.252532\pi$
$18$	0	0
$19$	−63.7439	−0.769677	−0.384838	−	0.922984i	$-0.625743\pi$
$19$	−63.7439	−0.769677	−0.384838	+	0.922984i	$0.625743\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−15.5139	−0.140647	−0.0703233	−	0.997524i	$-0.522403\pi$
$23$	−15.5139	−0.140647	−0.0703233	+	0.997524i	$0.522403\pi$
$24$	0	0
$25$	112.754	0.902031
$26$	0	0
$27$	0	0
$28$	0	0
$29$	77.6771	0.497389	0.248694	−	0.968582i	$-0.419999\pi$
$29$	77.6771	0.497389	0.248694	+	0.968582i	$0.419999\pi$
$30$	0	0
$31$	−209.031	−1.21107	−0.605534	−	0.795819i	$-0.707041\pi$
$31$	−209.031	−1.21107	−0.605534	+	0.795819i	$0.707041\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−455.976	−2.20212
$36$	0	0
$37$	−295.347	−1.31229	−0.656145	−	0.754635i	$-0.727814\pi$
$37$	−295.347	−1.31229	−0.656145	+	0.754635i	$0.727814\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−213.831	−0.814506	−0.407253	−	0.913315i	$-0.633513\pi$
$41$	−213.831	−0.814506	−0.407253	+	0.913315i	$0.633513\pi$
$42$	0	0
$43$	−150.974	−0.535428	−0.267714	−	0.963498i	$-0.586268\pi$
$43$	−150.974	−0.535428	−0.267714	+	0.963498i	$0.586268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	478.210	1.48413	0.742065	−	0.670328i	$-0.233846\pi$
$47$	478.210	1.48413	0.742065	+	0.670328i	$0.233846\pi$
$48$	0	0
$49$	531.494	1.54954
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−427.378	−1.10764	−0.553819	−	0.832637i	$-0.686830\pi$
$53$	−427.378	−1.10764	−0.553819	+	0.832637i	$0.686830\pi$
$54$	0	0
$55$	106.175	0.260302
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−798.377	−1.76169	−0.880846	−	0.473403i	$-0.843026\pi$
$59$	−798.377	−1.76169	−0.880846	+	0.473403i	$0.843026\pi$
$60$	0	0
$61$	−223.346	−0.468795	−0.234398	−	0.972141i	$-0.575312\pi$
$61$	−223.346	−0.468795	−0.234398	+	0.972141i	$0.575312\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−386.074	−0.736717
$66$	0	0
$67$	1026.21	1.87122	0.935610	−	0.353035i	$-0.114850\pi$
$67$	1026.21	1.87122	0.935610	+	0.353035i	$0.114850\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−925.890	−1.54765	−0.773823	−	0.633402i	$-0.781658\pi$
$71$	−925.890	−1.54765	−0.773823	+	0.633402i	$0.781658\pi$
$72$	0	0
$73$	−1070.06	−1.71564	−0.857819	−	0.513953i	$-0.828181\pi$
$73$	−1070.06	−1.71564	−0.857819	+	0.513953i	$0.828181\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−203.628	−0.301370
$78$	0	0
$79$	860.902	1.22606	0.613032	−	0.790058i	$-0.289950\pi$
$79$	860.902	1.22606	0.613032	+	0.790058i	$0.289950\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−459.709	−0.607947	−0.303974	−	0.952680i	$-0.598314\pi$
$83$	−459.709	−0.607947	−0.303974	+	0.952680i	$0.598314\pi$
$84$	0	0
$85$	1516.25	1.93483
$86$	0	0
$87$	0	0
$88$	0	0
$89$	787.829	0.938312	0.469156	−	0.883115i	$-0.344558\pi$
$89$	787.829	0.938312	0.469156	+	0.883115i	$0.344558\pi$
$90$	0	0
$91$	740.432	0.852949
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−982.885	−1.06149
$96$	0	0
$97$	118.831	0.124386	0.0621930	−	0.998064i	$-0.480191\pi$
$97$	118.831	0.124386	0.0621930	+	0.998064i	$0.480191\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1973.05	1.94382	0.971910	−	0.235354i	$-0.0756249\pi$
$101$	1973.05	1.94382	0.971910	+	0.235354i	$0.0756249\pi$
$102$	0	0
$103$	−1655.41	−1.58362	−0.791809	−	0.610769i	$-0.790861\pi$
$103$	−1655.41	−1.58362	−0.791809	+	0.610769i	$0.790861\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1258.09	−1.13667	−0.568335	−	0.822797i	$-0.692412\pi$
$107$	−1258.09	−1.13667	−0.568335	+	0.822797i	$0.692412\pi$
$108$	0	0
$109$	−775.418	−0.681390	−0.340695	−	0.940174i	$-0.610662\pi$
$109$	−775.418	−0.681390	−0.340695	+	0.940174i	$0.610662\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	543.228	0.452235	0.226118	−	0.974100i	$-0.427397\pi$
$113$	543.228	0.452235	0.226118	+	0.974100i	$0.427397\pi$
$114$	0	0
$115$	−239.213	−0.193972
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2907.94	−2.24008
$120$	0	0
$121$	−1283.58	−0.964376
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−188.825	−0.135112
$126$	0	0
$127$	−1597.27	−1.11602	−0.558011	−	0.829834i	$-0.688435\pi$
$127$	−1597.27	−1.11602	−0.558011	+	0.829834i	$0.688435\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	362.313	0.241644	0.120822	−	0.992674i	$-0.461447\pi$
$131$	362.313	0.241644	0.120822	+	0.992674i	$0.461447\pi$
$132$	0	0
$133$	1885.02	1.22897
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−245.324	−0.152989	−0.0764944	−	0.997070i	$-0.524373\pi$
$137$	−245.324	−0.152989	−0.0764944	+	0.997070i	$0.524373\pi$
$138$	0	0
$139$	1709.35	1.04306	0.521528	−	0.853234i	$-0.325362\pi$
$139$	1709.35	1.04306	0.521528	+	0.853234i	$0.325362\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−172.411	−0.100823
$144$	0	0
$145$	1197.72	0.685969
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3358.98	−1.84683	−0.923417	−	0.383799i	$-0.874615\pi$
$149$	−3358.98	−1.84683	−0.923417	+	0.383799i	$0.874615\pi$
$150$	0	0
$151$	−2191.12	−1.18087	−0.590434	−	0.807086i	$-0.701043\pi$
$151$	−2191.12	−1.18087	−0.590434	+	0.807086i	$0.701043\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3223.11	−1.67023
$156$	0	0
$157$	−1514.37	−0.769809	−0.384904	−	0.922956i	$-0.625766\pi$
$157$	−1514.37	−0.769809	−0.384904	+	0.922956i	$0.625766\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	458.775	0.224575
$162$	0	0
$163$	−432.967	−0.208053	−0.104026	−	0.994575i	$-0.533173\pi$
$163$	−432.967	−0.208053	−0.104026	+	0.994575i	$0.533173\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3084.91	−1.42945	−0.714723	−	0.699407i	$-0.753447\pi$
$167$	−3084.91	−1.42945	−0.714723	+	0.699407i	$0.753447\pi$
$168$	0	0
$169$	−1570.08	−0.714646
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2186.79	0.961032	0.480516	−	0.876986i	$-0.340449\pi$
$173$	2186.79	0.961032	0.480516	+	0.876986i	$0.340449\pi$
$174$	0	0
$175$	−3334.34	−1.44030
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3485.04	1.45522	0.727609	−	0.685992i	$-0.240632\pi$
$179$	3485.04	1.45522	0.727609	+	0.685992i	$0.240632\pi$
$180$	0	0
$181$	575.709	0.236420	0.118210	−	0.992989i	$-0.462284\pi$
$181$	575.709	0.236420	0.118210	+	0.992989i	$0.462284\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4554.04	−1.80983
$186$	0	0
$187$	677.119	0.264790
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3913.44	−1.48255	−0.741273	−	0.671203i	$-0.765778\pi$
$191$	−3913.44	−1.48255	−0.741273	+	0.671203i	$0.765778\pi$
$192$	0	0
$193$	1103.37	0.411513	0.205757	−	0.978603i	$-0.434034\pi$
$193$	1103.37	0.411513	0.205757	+	0.978603i	$0.434034\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1564.65	−0.565872	−0.282936	−	0.959139i	$-0.591308\pi$
$197$	−1564.65	−0.565872	−0.282936	+	0.959139i	$0.591308\pi$
$198$	0	0
$199$	−721.657	−0.257070	−0.128535	−	0.991705i	$-0.541027\pi$
$199$	−721.657	−0.257070	−0.128535	+	0.991705i	$0.541027\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2297.05	−0.794195
$204$	0	0
$205$	−3297.11	−1.12332
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−438.932	−0.145270
$210$	0	0
$211$	2478.16	0.808546	0.404273	−	0.914638i	$-0.367525\pi$
$211$	2478.16	0.808546	0.404273	+	0.914638i	$0.367525\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2327.92	−0.738431
$216$	0	0
$217$	6181.44	1.93375
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2462.15	−0.749420
$222$	0	0
$223$	418.660	0.125720	0.0628600	−	0.998022i	$-0.479978\pi$
$223$	418.660	0.125720	0.0628600	+	0.998022i	$0.479978\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2316.20	0.677232	0.338616	−	0.940925i	$-0.390041\pi$
$227$	2316.20	0.677232	0.338616	+	0.940925i	$0.390041\pi$
$228$	0	0
$229$	−2242.01	−0.646971	−0.323486	−	0.946233i	$-0.604855\pi$
$229$	−2242.01	−0.646971	−0.323486	+	0.946233i	$0.604855\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6174.06	1.73595	0.867974	−	0.496610i	$-0.165422\pi$
$233$	6174.06	1.73595	0.867974	+	0.496610i	$0.165422\pi$
$234$	0	0
$235$	7373.65	2.04683
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5495.44	1.48732	0.743662	−	0.668555i	$-0.233087\pi$
$239$	5495.44	1.48732	0.743662	+	0.668555i	$0.233087\pi$
$240$	0	0
$241$	1219.96	0.326076	0.163038	−	0.986620i	$-0.447871\pi$
$241$	1219.96	0.326076	0.163038	+	0.986620i	$0.447871\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	8195.24	2.13704
$246$	0	0
$247$	1596.05	0.411150
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3586.35	−0.901867	−0.450933	−	0.892558i	$-0.648909\pi$
$251$	−3586.35	−0.901867	−0.450933	+	0.892558i	$0.648909\pi$
$252$	0	0
$253$	−106.827	−0.0265460
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1122.37	−0.272418	−0.136209	−	0.990680i	$-0.543492\pi$
$257$	−1122.37	−0.272418	−0.136209	+	0.990680i	$0.543492\pi$
$258$	0	0
$259$	8733.95	2.09537
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5943.77	1.39357	0.696784	−	0.717281i	$-0.254614\pi$
$263$	5943.77	1.39357	0.696784	+	0.717281i	$0.254614\pi$
$264$	0	0
$265$	−6589.85	−1.52759
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−171.733	−0.0389247	−0.0194624	−	0.999811i	$-0.506195\pi$
$269$	−171.733	−0.0389247	−0.0194624	+	0.999811i	$0.506195\pi$
$270$	0	0
$271$	−3144.04	−0.704749	−0.352375	−	0.935859i	$-0.614626\pi$
$271$	−3144.04	−0.704749	−0.352375	+	0.935859i	$0.614626\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	776.408	0.170251
$276$	0	0
$277$	249.879	0.0542012	0.0271006	−	0.999633i	$-0.491373\pi$
$277$	249.879	0.0542012	0.0271006	+	0.999633i	$0.491373\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6972.42	1.48021	0.740106	−	0.672490i	$-0.234775\pi$
$281$	6972.42	1.48021	0.740106	+	0.672490i	$0.234775\pi$
$282$	0	0
$283$	2886.27	0.606257	0.303128	−	0.952950i	$-0.401969\pi$
$283$	2886.27	0.606257	0.303128	+	0.952950i	$0.401969\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6323.37	1.30055
$288$	0	0
$289$	4756.71	0.968188
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1729.41	−0.344824	−0.172412	−	0.985025i	$-0.555156\pi$
$293$	−1729.41	−0.344824	−0.172412	+	0.985025i	$0.555156\pi$
$294$	0	0
$295$	−12310.4	−2.42962
$296$	0	0
$297$	0	0
$298$	0	0
$299$	388.444	0.0751314
$300$	0	0
$301$	4464.59	0.854933
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3443.83	−0.646535
$306$	0	0
$307$	−1977.30	−0.367592	−0.183796	−	0.982964i	$-0.558839\pi$
$307$	−1977.30	−0.367592	−0.183796	+	0.982964i	$0.558839\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−238.817	−0.0435436	−0.0217718	−	0.999763i	$-0.506931\pi$
$311$	−238.817	−0.0435436	−0.0217718	+	0.999763i	$0.506931\pi$
$312$	0	0
$313$	−762.610	−0.137716	−0.0688582	−	0.997626i	$-0.521936\pi$
$313$	−762.610	−0.137716	−0.0688582	+	0.997626i	$0.521936\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2026.48	0.359048	0.179524	−	0.983754i	$-0.442544\pi$
$317$	2026.48	0.359048	0.179524	+	0.983754i	$0.442544\pi$
$318$	0	0
$319$	534.873	0.0938783
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6268.24	−1.07980
$324$	0	0
$325$	−2823.18	−0.481852
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−14141.6	−2.36975
$330$	0	0
$331$	3556.18	0.590530	0.295265	−	0.955415i	$-0.404592\pi$
$331$	3556.18	0.590530	0.295265	+	0.955415i	$0.404592\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	15823.4	2.58068
$336$	0	0
$337$	−11124.3	−1.79815	−0.899075	−	0.437794i	$-0.855760\pi$
$337$	−11124.3	−1.79815	−0.899075	+	0.437794i	$0.855760\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1439.36	−0.228580
$342$	0	0
$343$	−5574.10	−0.877473
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5442.81	0.842033	0.421016	−	0.907053i	$-0.361674\pi$
$347$	5442.81	0.842033	0.421016	+	0.907053i	$0.361674\pi$
$348$	0	0
$349$	8082.25	1.23964	0.619818	−	0.784745i	$-0.287206\pi$
$349$	8082.25	1.23964	0.619818	+	0.784745i	$0.287206\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9229.59	1.39162	0.695809	−	0.718227i	$-0.255046\pi$
$353$	9229.59	1.39162	0.695809	+	0.718227i	$0.255046\pi$
$354$	0	0
$355$	−14276.5	−2.13442
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1042.32	0.153235	0.0766175	−	0.997061i	$-0.475588\pi$
$359$	1042.32	0.153235	0.0766175	+	0.997061i	$0.475588\pi$
$360$	0	0
$361$	−2795.71	−0.407598
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−16499.6	−2.36611
$366$	0	0
$367$	−537.465	−0.0764453	−0.0382226	−	0.999269i	$-0.512170\pi$
$367$	−537.465	−0.0764453	−0.0382226	+	0.999269i	$0.512170\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	12638.3	1.76860
$372$	0	0
$373$	9364.17	1.29989	0.649944	−	0.759982i	$-0.274792\pi$
$373$	9364.17	1.29989	0.649944	+	0.759982i	$0.274792\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1944.91	−0.265698
$378$	0	0
$379$	5153.18	0.698419	0.349210	−	0.937045i	$-0.386450\pi$
$379$	5153.18	0.698419	0.349210	+	0.937045i	$0.386450\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4524.09	0.603577	0.301789	−	0.953375i	$-0.402416\pi$
$383$	4524.09	0.603577	0.301789	+	0.953375i	$0.402416\pi$
$384$	0	0
$385$	−3139.79	−0.415632
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−6873.06	−0.895831	−0.447915	−	0.894076i	$-0.647833\pi$
$389$	−6873.06	−0.895831	−0.447915	+	0.894076i	$0.647833\pi$
$390$	0	0
$391$	−1525.55	−0.197316
$392$	0	0
$393$	0	0
$394$	0	0
$395$	13274.5	1.69092
$396$	0	0
$397$	−2235.66	−0.282631	−0.141316	−	0.989965i	$-0.545133\pi$
$397$	−2235.66	−0.282631	−0.141316	+	0.989965i	$0.545133\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5635.28	−0.701777	−0.350889	−	0.936417i	$-0.614120\pi$
$401$	−5635.28	−0.701777	−0.350889	+	0.936417i	$0.614120\pi$
$402$	0	0
$403$	5233.81	0.646935
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2033.72	−0.247685
$408$	0	0
$409$	12300.2	1.48705	0.743525	−	0.668708i	$-0.233152\pi$
$409$	12300.2	1.48705	0.743525	+	0.668708i	$0.233152\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	23609.5	2.81295
$414$	0	0
$415$	−7088.38	−0.838446
$416$	0	0
$417$	0	0
$418$	0	0
$419$	935.396	0.109062	0.0545311	−	0.998512i	$-0.482634\pi$
$419$	935.396	0.109062	0.0545311	+	0.998512i	$0.482634\pi$
$420$	0	0
$421$	−8532.08	−0.987716	−0.493858	−	0.869543i	$-0.664414\pi$
$421$	−8532.08	−0.987716	−0.493858	+	0.869543i	$0.664414\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	11087.6	1.26548
$426$	0	0
$427$	6604.75	0.748539
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4312.88	−0.482005	−0.241003	−	0.970524i	$-0.577476\pi$
$431$	−4312.88	−0.482005	−0.241003	+	0.970524i	$0.577476\pi$
$432$	0	0
$433$	−11728.0	−1.30164	−0.650820	−	0.759232i	$-0.725575\pi$
$433$	−11728.0	−1.30164	−0.650820	+	0.759232i	$0.725575\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	988.917	0.108252
$438$	0	0
$439$	−12399.1	−1.34801	−0.674004	−	0.738727i	$-0.735427\pi$
$439$	−12399.1	−1.34801	−0.674004	+	0.738727i	$0.735427\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7551.97	0.809943	0.404972	−	0.914329i	$-0.367281\pi$
$443$	7551.97	0.809943	0.404972	+	0.914329i	$0.367281\pi$
$444$	0	0
$445$	12147.8	1.29406
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−9921.19	−1.04278	−0.521392	−	0.853317i	$-0.674587\pi$
$449$	−9921.19	−1.04278	−0.521392	+	0.853317i	$0.674587\pi$
$450$	0	0
$451$	−1472.41	−0.153732
$452$	0	0
$453$	0	0
$454$	0	0
$455$	11416.9	1.17634
$456$	0	0
$457$	4668.89	0.477903	0.238951	−	0.971032i	$-0.423196\pi$
$457$	4668.89	0.477903	0.238951	+	0.971032i	$0.423196\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9571.43	0.966997	0.483499	−	0.875345i	$-0.339366\pi$
$461$	9571.43	0.966997	0.483499	+	0.875345i	$0.339366\pi$
$462$	0	0
$463$	13977.7	1.40302	0.701509	−	0.712661i	$-0.252510\pi$
$463$	13977.7	1.40302	0.701509	+	0.712661i	$0.252510\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14140.3	1.40115	0.700573	−	0.713581i	$-0.252928\pi$
$467$	14140.3	1.40115	0.700573	+	0.713581i	$0.252928\pi$
$468$	0	0
$469$	−30347.0	−2.98783
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1039.59	−0.101058
$474$	0	0
$475$	−7187.38	−0.694273
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11658.2	−1.11206	−0.556030	−	0.831162i	$-0.687676\pi$
$479$	−11658.2	−1.11206	−0.556030	+	0.831162i	$0.687676\pi$
$480$	0	0
$481$	7395.02	0.701006
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1832.28	0.171546
$486$	0	0
$487$	13681.7	1.27305	0.636525	−	0.771256i	$-0.280371\pi$
$487$	13681.7	1.27305	0.636525	+	0.771256i	$0.280371\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9540.92	0.876936	0.438468	−	0.898747i	$-0.355521\pi$
$491$	9540.92	0.876936	0.438468	+	0.898747i	$0.355521\pi$
$492$	0	0
$493$	7638.35	0.697797
$494$	0	0
$495$	0	0
$496$	0	0
$497$	27380.3	2.47117
$498$	0	0
$499$	4986.46	0.447344	0.223672	−	0.974665i	$-0.428196\pi$
$499$	4986.46	0.447344	0.223672	+	0.974665i	$0.428196\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18568.5	1.64598	0.822989	−	0.568058i	$-0.192305\pi$
$503$	18568.5	1.64598	0.822989	+	0.568058i	$0.192305\pi$
$504$	0	0
$505$	30423.0	2.68080
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−20855.8	−1.81614	−0.908072	−	0.418815i	$-0.862446\pi$
$509$	−20855.8	−1.81614	−0.908072	+	0.418815i	$0.862446\pi$
$510$	0	0
$511$	31643.8	2.73941
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−25525.3	−2.18403
$516$	0	0
$517$	3292.89	0.280118
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4457.44	0.374826	0.187413	−	0.982281i	$-0.439990\pi$
$521$	4457.44	0.374826	0.187413	+	0.982281i	$0.439990\pi$
$522$	0	0
$523$	16688.1	1.39526	0.697629	−	0.716459i	$-0.254238\pi$
$523$	16688.1	1.39526	0.697629	+	0.716459i	$0.254238\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−20555.0	−1.69903
$528$	0	0
$529$	−11926.3	−0.980219
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5353.99	0.435097
$534$	0	0
$535$	−19398.8	−1.56763
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3659.79	0.292464
$540$	0	0
$541$	−24636.0	−1.95783	−0.978914	−	0.204272i	$-0.934517\pi$
$541$	−24636.0	−1.95783	−0.978914	+	0.204272i	$0.934517\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−11956.4	−0.939734
$546$	0	0
$547$	2762.55	0.215938	0.107969	−	0.994154i	$-0.465565\pi$
$547$	2762.55	0.215938	0.107969	+	0.994154i	$0.465565\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4951.44	−0.382828
$552$	0	0
$553$	−25458.4	−1.95769
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5054.03	0.384464	0.192232	−	0.981350i	$-0.438427\pi$
$557$	5054.03	0.384464	0.192232	+	0.981350i	$0.438427\pi$
$558$	0	0
$559$	3780.16	0.286018
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18442.2	1.38054	0.690272	−	0.723550i	$-0.257491\pi$
$563$	18442.2	1.38054	0.690272	+	0.723550i	$0.257491\pi$
$564$	0	0
$565$	8376.18	0.623697
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11382.1	−0.838600	−0.419300	−	0.907848i	$-0.637724\pi$
$569$	−11382.1	−0.838600	−0.419300	+	0.907848i	$0.637724\pi$
$570$	0	0
$571$	10276.8	0.753192	0.376596	−	0.926378i	$-0.377094\pi$
$571$	10276.8	0.753192	0.376596	+	0.926378i	$0.377094\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1749.25	−0.126868
$576$	0	0
$577$	17124.8	1.23555	0.617776	−	0.786354i	$-0.288034\pi$
$577$	17124.8	1.23555	0.617776	+	0.786354i	$0.288034\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	13594.4	0.970727
$582$	0	0
$583$	−2942.86	−0.209058
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9957.79	0.700174	0.350087	−	0.936717i	$-0.386152\pi$
$587$	9957.79	0.700174	0.350087	+	0.936717i	$0.386152\pi$
$588$	0	0
$589$	13324.5	0.932131
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−17146.9	−1.18742	−0.593708	−	0.804681i	$-0.702336\pi$
$593$	−17146.9	−1.18742	−0.593708	+	0.804681i	$0.702336\pi$
$594$	0	0
$595$	−44838.3	−3.08939
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5219.71	0.356046	0.178023	−	0.984026i	$-0.443030\pi$
$599$	5219.71	0.356046	0.178023	+	0.984026i	$0.443030\pi$
$600$	0	0
$601$	19507.5	1.32401	0.662004	−	0.749500i	$-0.269706\pi$
$601$	19507.5	1.32401	0.662004	+	0.749500i	$0.269706\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−19791.9	−1.33001
$606$	0	0
$607$	−1566.17	−0.104726	−0.0523632	−	0.998628i	$-0.516675\pi$
$607$	−1566.17	−0.104726	−0.0523632	+	0.998628i	$0.516675\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−11973.6	−0.792801
$612$	0	0
$613$	13199.4	0.869690	0.434845	−	0.900505i	$-0.356803\pi$
$613$	13199.4	0.869690	0.434845	+	0.900505i	$0.356803\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−17339.0	−1.13135	−0.565675	−	0.824628i	$-0.691384\pi$
$617$	−17339.0	−1.13135	−0.565675	+	0.824628i	$0.691384\pi$
$618$	0	0
$619$	−29803.1	−1.93520	−0.967601	−	0.252486i	$-0.918752\pi$
$619$	−29803.1	−1.93520	−0.967601	+	0.252486i	$0.918752\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−23297.6	−1.49823
$624$	0	0
$625$	−17005.8	−1.08837
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−29042.8	−1.84104
$630$	0	0
$631$	8792.76	0.554729	0.277365	−	0.960765i	$-0.410539\pi$
$631$	8792.76	0.554729	0.277365	+	0.960765i	$0.410539\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−24628.7	−1.53915
$636$	0	0
$637$	−13307.8	−0.827744
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4317.68	0.266050	0.133025	−	0.991113i	$-0.457531\pi$
$641$	4317.68	0.266050	0.133025	+	0.991113i	$0.457531\pi$
$642$	0	0
$643$	−8621.03	−0.528741	−0.264370	−	0.964421i	$-0.585164\pi$
$643$	−8621.03	−0.528741	−0.264370	+	0.964421i	$0.585164\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13067.3	−0.794018	−0.397009	−	0.917815i	$-0.629952\pi$
$647$	−13067.3	−0.794018	−0.397009	+	0.917815i	$0.629952\pi$
$648$	0	0
$649$	−5497.51	−0.332506
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3553.56	−0.212958	−0.106479	−	0.994315i	$-0.533958\pi$
$653$	−3553.56	−0.212958	−0.106479	+	0.994315i	$0.533958\pi$
$654$	0	0
$655$	5586.60	0.333262
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−7006.61	−0.414171	−0.207086	−	0.978323i	$-0.566398\pi$
$659$	−7006.61	−0.414171	−0.207086	+	0.978323i	$0.566398\pi$
$660$	0	0
$661$	−8529.88	−0.501927	−0.250964	−	0.967996i	$-0.580747\pi$
$661$	−8529.88	−0.501927	−0.250964	+	0.967996i	$0.580747\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	29065.7	1.69492
$666$	0	0
$667$	−1205.07	−0.0699560
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1537.93	−0.0884814
$672$	0	0
$673$	−9810.77	−0.561928	−0.280964	−	0.959718i	$-0.590654\pi$
$673$	−9810.77	−0.561928	−0.280964	+	0.959718i	$0.590654\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7583.91	0.430537	0.215268	−	0.976555i	$-0.430937\pi$
$677$	7583.91	0.430537	0.215268	+	0.976555i	$0.430937\pi$
$678$	0	0
$679$	−3514.04	−0.198611
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−13524.5	−0.757690	−0.378845	−	0.925460i	$-0.623679\pi$
$683$	−13524.5	−0.757690	−0.378845	+	0.925460i	$0.623679\pi$
$684$	0	0
$685$	−3782.72	−0.210993
$686$	0	0
$687$	0	0
$688$	0	0
$689$	10700.9	0.591684
$690$	0	0
$691$	−27600.4	−1.51949	−0.759747	−	0.650219i	$-0.774677\pi$
$691$	−27600.4	−1.51949	−0.759747	+	0.650219i	$0.774677\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	26356.9	1.43852
$696$	0	0
$697$	−21027.0	−1.14269
$698$	0	0
$699$	0	0
$700$	0	0
$701$	19843.9	1.06918	0.534589	−	0.845112i	$-0.320466\pi$
$701$	19843.9	1.06918	0.534589	+	0.845112i	$0.320466\pi$
$702$	0	0
$703$	18826.6	1.01004
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−58346.7	−3.10375
$708$	0	0
$709$	20305.8	1.07560	0.537801	−	0.843072i	$-0.319255\pi$
$709$	20305.8	1.07560	0.537801	+	0.843072i	$0.319255\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3242.89	0.170333
$714$	0	0
$715$	−2658.45	−0.139050
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−18258.3	−0.947038	−0.473519	−	0.880783i	$-0.657017\pi$
$719$	−18258.3	−0.947038	−0.473519	+	0.880783i	$0.657017\pi$
$720$	0	0
$721$	48953.6	2.52861
$722$	0	0
$723$	0	0
$724$	0	0
$725$	8758.40	0.448660
$726$	0	0
$727$	19790.1	1.00959	0.504797	−	0.863238i	$-0.331567\pi$
$727$	19790.1	1.00959	0.504797	+	0.863238i	$0.331567\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−14846.0	−0.751163
$732$	0	0
$733$	28712.7	1.44683	0.723415	−	0.690414i	$-0.242571\pi$
$733$	28712.7	1.44683	0.723415	+	0.690414i	$0.242571\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7066.35	0.353178
$738$	0	0
$739$	−18118.2	−0.901881	−0.450940	−	0.892554i	$-0.648911\pi$
$739$	−18118.2	−0.901881	−0.450940	+	0.892554i	$0.648911\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1931.54	−0.0953717	−0.0476859	−	0.998862i	$-0.515185\pi$
$743$	−1931.54	−0.0953717	−0.0476859	+	0.998862i	$0.515185\pi$
$744$	0	0
$745$	−51793.0	−2.54704
$746$	0	0
$747$	0	0
$748$	0	0
$749$	37203.9	1.81496
$750$	0	0
$751$	7810.34	0.379499	0.189749	−	0.981833i	$-0.439232\pi$
$751$	7810.34	0.379499	0.189749	+	0.981833i	$0.439232\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−33785.5	−1.62858
$756$	0	0
$757$	8626.50	0.414181	0.207091	−	0.978322i	$-0.433600\pi$
$757$	8626.50	0.414181	0.207091	+	0.978322i	$0.433600\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−5766.05	−0.274664	−0.137332	−	0.990525i	$-0.543853\pi$
$761$	−5766.05	−0.274664	−0.137332	+	0.990525i	$0.543853\pi$
$762$	0	0
$763$	22930.5	1.08800
$764$	0	0
$765$	0	0
$766$	0	0
$767$	19990.1	0.941070
$768$	0	0
$769$	−21691.7	−1.01720	−0.508598	−	0.861004i	$-0.669836\pi$
$769$	−21691.7	−1.01720	−0.508598	+	0.861004i	$0.669836\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−3779.52	−0.175860	−0.0879300	−	0.996127i	$-0.528025\pi$
$773$	−3779.52	−0.175860	−0.0879300	+	0.996127i	$0.528025\pi$
$774$	0	0
$775$	−23569.1	−1.09242
$776$	0	0
$777$	0	0
$778$	0	0
$779$	13630.4	0.626906
$780$	0	0
$781$	−6375.55	−0.292106
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−23350.5	−1.06168
$786$	0	0
$787$	−28812.2	−1.30501	−0.652505	−	0.757784i	$-0.726282\pi$
$787$	−28812.2	−1.30501	−0.652505	+	0.757784i	$0.726282\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−16064.3	−0.722098
$792$	0	0
$793$	5592.23	0.250423
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1226.06	−0.0544908	−0.0272454	−	0.999629i	$-0.508674\pi$
$797$	−1226.06	−0.0544908	−0.0272454	+	0.999629i	$0.508674\pi$
$798$	0	0
$799$	47024.6	2.08212
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−7368.31	−0.323813
$804$	0	0
$805$	7073.97	0.309720
$806$	0	0
$807$	0	0
$808$	0	0
$809$	4199.27	0.182495	0.0912476	−	0.995828i	$-0.470915\pi$
$809$	4199.27	0.182495	0.0912476	+	0.995828i	$0.470915\pi$
$810$	0	0
$811$	24531.4	1.06216	0.531082	−	0.847320i	$-0.321786\pi$
$811$	24531.4	1.06216	0.531082	+	0.847320i	$0.321786\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−6676.04	−0.286934
$816$	0	0
$817$	9623.70	0.412106
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10092.8	−0.429040	−0.214520	−	0.976720i	$-0.568819\pi$
$821$	−10092.8	−0.429040	−0.214520	+	0.976720i	$0.568819\pi$
$822$	0	0
$823$	14657.0	0.620791	0.310395	−	0.950608i	$-0.399539\pi$
$823$	14657.0	0.620791	0.310395	+	0.950608i	$0.399539\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−19762.8	−0.830978	−0.415489	−	0.909598i	$-0.636390\pi$
$827$	−19762.8	−0.830978	−0.415489	+	0.909598i	$0.636390\pi$
$828$	0	0
$829$	−8418.19	−0.352685	−0.176342	−	0.984329i	$-0.556427\pi$
$829$	−8418.19	−0.352685	−0.176342	+	0.984329i	$0.556427\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	52264.2	2.17389
$834$	0	0
$835$	−47567.1	−1.97141
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−14894.9	−0.612908	−0.306454	−	0.951886i	$-0.599142\pi$
$839$	−14894.9	−0.612908	−0.306454	+	0.951886i	$0.599142\pi$
$840$	0	0
$841$	−18355.3	−0.752604
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−24209.4	−0.985598
$846$	0	0
$847$	37958.0	1.53985
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4581.99	0.184569
$852$	0	0
$853$	9678.25	0.388484	0.194242	−	0.980954i	$-0.437775\pi$
$853$	9678.25	0.388484	0.194242	+	0.980954i	$0.437775\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−39511.8	−1.57491	−0.787455	−	0.616373i	$-0.788601\pi$
$857$	−39511.8	−1.57491	−0.787455	+	0.616373i	$0.788601\pi$
$858$	0	0
$859$	37536.2	1.49094	0.745471	−	0.666539i	$-0.232225\pi$
$859$	37536.2	1.49094	0.745471	+	0.666539i	$0.232225\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−17185.4	−0.677866	−0.338933	−	0.940811i	$-0.610066\pi$
$863$	−17185.4	−0.677866	−0.338933	+	0.940811i	$0.610066\pi$
$864$	0	0
$865$	33718.7	1.32540
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5928.05	0.231410
$870$	0	0
$871$	−25694.7	−0.999578
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5583.91	0.215738
$876$	0	0
$877$	−19722.5	−0.759385	−0.379693	−	0.925113i	$-0.623970\pi$
$877$	−19722.5	−0.759385	−0.379693	+	0.925113i	$0.623970\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−17073.8	−0.652931	−0.326465	−	0.945209i	$-0.605858\pi$
$881$	−17073.8	−0.652931	−0.326465	+	0.945209i	$0.605858\pi$
$882$	0	0
$883$	15877.2	0.605108	0.302554	−	0.953132i	$-0.402161\pi$
$883$	15877.2	0.605108	0.302554	+	0.953132i	$0.402161\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−38270.1	−1.44869	−0.724343	−	0.689440i	$-0.757857\pi$
$887$	−38270.1	−1.44869	−0.724343	+	0.689440i	$0.757857\pi$
$888$	0	0
$889$	47234.2	1.78198
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−30483.0	−1.14230
$894$	0	0
$895$	53736.8	2.00695
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−16236.9	−0.602372
$900$	0	0
$901$	−42026.0	−1.55393
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8877.01	0.326057
$906$	0	0
$907$	27813.0	1.01821	0.509104	−	0.860705i	$-0.329977\pi$
$907$	27813.0	1.01821	0.509104	+	0.860705i	$0.329977\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−27203.6	−0.989347	−0.494674	−	0.869079i	$-0.664712\pi$
$911$	−27203.6	−0.989347	−0.494674	+	0.869079i	$0.664712\pi$
$912$	0	0
$913$	−3165.49	−0.114745
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−10714.2	−0.385840
$918$	0	0
$919$	5542.67	0.198951	0.0994754	−	0.995040i	$-0.468284\pi$
$919$	5542.67	0.198951	0.0994754	+	0.995040i	$0.468284\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	23182.8	0.826730
$924$	0	0
$925$	−33301.5	−1.18373
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−48638.7	−1.71774	−0.858872	−	0.512191i	$-0.828834\pi$
$929$	−48638.7	−1.71774	−0.858872	+	0.512191i	$0.828834\pi$
$930$	0	0
$931$	−33879.5	−1.19265
$932$	0	0
$933$	0	0
$934$	0	0
$935$	10440.7	0.365184
$936$	0	0
$937$	15364.9	0.535698	0.267849	−	0.963461i	$-0.413687\pi$
$937$	15364.9	0.535698	0.267849	+	0.963461i	$0.413687\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−13853.8	−0.479936	−0.239968	−	0.970781i	$-0.577137\pi$
$941$	−13853.8	−0.479936	−0.239968	+	0.970781i	$0.577137\pi$
$942$	0	0
$943$	3317.35	0.114558
$944$	0	0
$945$	0	0
$946$	0	0
$947$	43336.3	1.48705	0.743527	−	0.668706i	$-0.233151\pi$
$947$	43336.3	1.48705	0.743527	+	0.668706i	$0.233151\pi$
$948$	0	0
$949$	26792.7	0.916468
$950$	0	0
$951$	0	0
$952$	0	0
$953$	14064.4	0.478060	0.239030	−	0.971012i	$-0.423171\pi$
$953$	14064.4	0.478060	0.239030	+	0.971012i	$0.423171\pi$
$954$	0	0
$955$	−60342.4	−2.04464
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7254.68	0.244282
$960$	0	0
$961$	13903.1	0.466687
$962$	0	0
$963$	0	0
$964$	0	0
$965$	17013.1	0.567535
$966$	0	0
$967$	−2110.43	−0.0701828	−0.0350914	−	0.999384i	$-0.511172\pi$
$967$	−2110.43	−0.0701828	−0.0350914	+	0.999384i	$0.511172\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	31181.7	1.03055	0.515277	−	0.857024i	$-0.327689\pi$
$971$	31181.7	1.03055	0.515277	+	0.857024i	$0.327689\pi$
$972$	0	0
$973$	−50548.5	−1.66548
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13460.4	0.440775	0.220388	−	0.975412i	$-0.429268\pi$
$977$	13460.4	0.440775	0.220388	+	0.975412i	$0.429268\pi$
$978$	0	0
$979$	5424.88	0.177099
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−36001.9	−1.16814	−0.584071	−	0.811703i	$-0.698541\pi$
$983$	−36001.9	−1.16814	−0.584071	+	0.811703i	$0.698541\pi$
$984$	0	0
$985$	−24125.8	−0.780417
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2342.20	0.0753061
$990$	0	0
$991$	16783.4	0.537986	0.268993	−	0.963142i	$-0.413309\pi$
$991$	16783.4	0.537986	0.268993	+	0.963142i	$0.413309\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−11127.4	−0.354536
$996$	0	0
$997$	26441.8	0.839940	0.419970	−	0.907538i	$-0.362041\pi$
$997$	26441.8	0.839940	0.419970	+	0.907538i	$0.362041\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.4.a.q.1.3	yes	3
3.2	odd	2		864.4.a.k.1.1	✓	3
4.3	odd	2		864.4.a.r.1.3	yes	3
8.3	odd	2		1728.4.a.bx.1.1		3
8.5	even	2		1728.4.a.bw.1.1		3
12.11	even	2		864.4.a.l.1.1	yes	3
24.5	odd	2		1728.4.a.cc.1.3		3
24.11	even	2		1728.4.a.cd.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.4.a.k.1.1	✓	3	3.2	odd	2
864.4.a.l.1.1	yes	3	12.11	even	2
864.4.a.q.1.3	yes	3	1.1	even	1	trivial
864.4.a.r.1.3	yes	3	4.3	odd	2
1728.4.a.bw.1.1		3	8.5	even	2
1728.4.a.bx.1.1		3	8.3	odd	2
1728.4.a.cc.1.3		3	24.5	odd	2
1728.4.a.cd.1.3		3	24.11	even	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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	864.a (trivial)

\(f(q)\)	\(=\)	\(q+15.4193 q^{5} -29.5718 q^{7} +6.88586 q^{11} -25.0384 q^{13} +98.3347 q^{17} -63.7439 q^{19} -15.5139 q^{23} +112.754 q^{25} +77.6771 q^{29} -209.031 q^{31} -455.976 q^{35} -295.347 q^{37} -213.831 q^{41} -150.974 q^{43} +478.210 q^{47} +531.494 q^{49} -427.378 q^{53} +106.175 q^{55} -798.377 q^{59} -223.346 q^{61} -386.074 q^{65} +1026.21 q^{67} -925.890 q^{71} -1070.06 q^{73} -203.628 q^{77} +860.902 q^{79} -459.709 q^{83} +1516.25 q^{85} +787.829 q^{89} +740.432 q^{91} -982.885 q^{95} +118.831 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{5} - 9 q^{7} - 18 q^{11} + 3 q^{13} + 18 q^{17} - 27 q^{19} - 90 q^{23} + 21 q^{25} + 72 q^{29} - 144 q^{31} - 450 q^{35} - 159 q^{37} - 168 q^{41} + 180 q^{43} - 522 q^{47} + 150 q^{49} - 84 q^{53}+ \cdots - 117 q^{97}+O(q^{100}) \) 3 * q + 6 * q^5 - 9 * q^7 - 18 * q^11 + 3 * q^13 + 18 * q^17 - 27 * q^19 - 90 * q^23 + 21 * q^25 + 72 * q^29 - 144 * q^31 - 450 * q^35 - 159 * q^37 - 168 * q^41 + 180 * q^43 - 522 * q^47 + 150 * q^49 - 84 * q^53 - 324 * q^55 - 1458 * q^59 + 33 * q^61 + 246 * q^65 + 837 * q^67 - 1908 * q^71 - 165 * q^73 - 954 * q^77 - 531 * q^79 - 2268 * q^83 + 708 * q^85 + 618 * q^89 + 1719 * q^91 - 2934 * q^95 - 117 * q^97

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