LMFDB - Embedded newform 1728.4.a.bk.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1728,4,Mod(1,1728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1728.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-22,0,0,0,0] 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:	$101.955300490$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 27)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1728.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+16.9706 q^{5} -11.0000 q^{7} +16.9706 q^{11} -29.0000 q^{13} -50.9117 q^{17} +29.0000 q^{19} -84.8528 q^{23} +163.000 q^{25} -271.529 q^{29} +268.000 q^{31} -186.676 q^{35} -83.0000 q^{37} -271.529 q^{41} -232.000 q^{43} +390.323 q^{47} -222.000 q^{49} -305.470 q^{53} +288.000 q^{55} +288.500 q^{59} -767.000 q^{61} -492.146 q^{65} -511.000 q^{67} -712.764 q^{71} +137.000 q^{73} -186.676 q^{77} +475.000 q^{79} +576.999 q^{83} -864.000 q^{85} -254.558 q^{89} +319.000 q^{91} +492.146 q^{95} +821.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 22 q^{7} - 58 q^{13} + 58 q^{19} + 326 q^{25} + 536 q^{31} - 166 q^{37} - 464 q^{43} - 444 q^{49} + 576 q^{55} - 1534 q^{61} - 1022 q^{67} + 274 q^{73} + 950 q^{79} - 1728 q^{85} + 638 q^{91} + 1642 q^{97}+O(q^{100}) $ 2 * q - 22 * q^7 - 58 * q^13 + 58 * q^19 + 326 * q^25 + 536 * q^31 - 166 * q^37 - 464 * q^43 - 444 * q^49 + 576 * q^55 - 1534 * q^61 - 1022 * q^67 + 274 * q^73 + 950 * q^79 - 1728 * q^85 + 638 * q^91 + 1642 * 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
$5$	16.9706	1.51789	0.758947	−	0.651153i	$-0.225714\pi$
$5$	16.9706	1.51789	0.758947	+	0.651153i	$0.225714\pi$
$6$	0	0
$7$	−11.0000	−0.593944	−0.296972	−	0.954886i	$-0.595977\pi$
$7$	−11.0000	−0.593944	−0.296972	+	0.954886i	$0.595977\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	16.9706	0.465165	0.232583	−	0.972577i	$-0.425282\pi$
$11$	16.9706	0.465165	0.232583	+	0.972577i	$0.425282\pi$
$12$	0	0
$13$	−29.0000	−0.618704	−0.309352	−	0.950948i	$-0.600112\pi$
$13$	−29.0000	−0.618704	−0.309352	+	0.950948i	$0.600112\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−50.9117	−0.726347	−0.363173	−	0.931722i	$-0.618307\pi$
$17$	−50.9117	−0.726347	−0.363173	+	0.931722i	$0.618307\pi$
$18$	0	0
$19$	29.0000	0.350161	0.175080	−	0.984554i	$-0.443981\pi$
$19$	29.0000	0.350161	0.175080	+	0.984554i	$0.443981\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−84.8528	−0.769262	−0.384631	−	0.923070i	$-0.625671\pi$
$23$	−84.8528	−0.769262	−0.384631	+	0.923070i	$0.625671\pi$
$24$	0	0
$25$	163.000	1.30400
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−271.529	−1.73868	−0.869339	−	0.494216i	$-0.835455\pi$
$29$	−271.529	−1.73868	−0.869339	+	0.494216i	$0.835455\pi$
$30$	0	0
$31$	268.000	1.55272	0.776358	−	0.630292i	$-0.217065\pi$
$31$	268.000	1.55272	0.776358	+	0.630292i	$0.217065\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−186.676	−0.901544
$36$	0	0
$37$	−83.0000	−0.368787	−0.184393	−	0.982853i	$-0.559032\pi$
$37$	−83.0000	−0.368787	−0.184393	+	0.982853i	$0.559032\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−271.529	−1.03429	−0.517143	−	0.855899i	$-0.673004\pi$
$41$	−271.529	−1.03429	−0.517143	+	0.855899i	$0.673004\pi$
$42$	0	0
$43$	−232.000	−0.822783	−0.411391	−	0.911459i	$-0.634957\pi$
$43$	−232.000	−0.822783	−0.411391	+	0.911459i	$0.634957\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	390.323	1.21137	0.605686	−	0.795704i	$-0.292899\pi$
$47$	390.323	1.21137	0.605686	+	0.795704i	$0.292899\pi$
$48$	0	0
$49$	−222.000	−0.647230
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−305.470	−0.791690	−0.395845	−	0.918317i	$-0.629548\pi$
$53$	−305.470	−0.791690	−0.395845	+	0.918317i	$0.629548\pi$
$54$	0	0
$55$	288.000	0.706071
$56$	0	0
$57$	0	0
$58$	0	0
$59$	288.500	0.636601	0.318300	−	0.947990i	$-0.396888\pi$
$59$	288.500	0.636601	0.318300	+	0.947990i	$0.396888\pi$
$60$	0	0
$61$	−767.000	−1.60991	−0.804953	−	0.593338i	$-0.797810\pi$
$61$	−767.000	−1.60991	−0.804953	+	0.593338i	$0.797810\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−492.146	−0.939127
$66$	0	0
$67$	−511.000	−0.931770	−0.465885	−	0.884845i	$-0.654264\pi$
$67$	−511.000	−0.931770	−0.465885	+	0.884845i	$0.654264\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−712.764	−1.19140	−0.595701	−	0.803207i	$-0.703126\pi$
$71$	−712.764	−1.19140	−0.595701	+	0.803207i	$0.703126\pi$
$72$	0	0
$73$	137.000	0.219653	0.109826	−	0.993951i	$-0.464971\pi$
$73$	137.000	0.219653	0.109826	+	0.993951i	$0.464971\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−186.676	−0.276282
$78$	0	0
$79$	475.000	0.676477	0.338238	−	0.941060i	$-0.390169\pi$
$79$	475.000	0.676477	0.338238	+	0.941060i	$0.390169\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	576.999	0.763059	0.381529	−	0.924357i	$-0.375397\pi$
$83$	576.999	0.763059	0.381529	+	0.924357i	$0.375397\pi$
$84$	0	0
$85$	−864.000	−1.10252
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−254.558	−0.303181	−0.151591	−	0.988443i	$-0.548440\pi$
$89$	−254.558	−0.303181	−0.151591	+	0.988443i	$0.548440\pi$
$90$	0	0
$91$	319.000	0.367476
$92$	0	0
$93$	0	0
$94$	0	0
$95$	492.146	0.531507
$96$	0	0
$97$	821.000	0.859381	0.429690	−	0.902976i	$-0.358623\pi$
$97$	821.000	0.859381	0.429690	+	0.902976i	$0.358623\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	543.058	0.535013	0.267506	−	0.963556i	$-0.413800\pi$
$101$	543.058	0.535013	0.267506	+	0.963556i	$0.413800\pi$
$102$	0	0
$103$	−839.000	−0.802613	−0.401306	−	0.915944i	$-0.631444\pi$
$103$	−839.000	−0.802613	−0.401306	+	0.915944i	$0.631444\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−763.675	−0.689975	−0.344987	−	0.938607i	$-0.612117\pi$
$107$	−763.675	−0.689975	−0.344987	+	0.938607i	$0.612117\pi$
$108$	0	0
$109$	−218.000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−218.000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1510.38	1.25739	0.628693	−	0.777654i	$-0.283590\pi$
$113$	1510.38	1.25739	0.628693	+	0.777654i	$0.283590\pi$
$114$	0	0
$115$	−1440.00	−1.16766
$116$	0	0
$117$	0	0
$118$	0	0
$119$	560.029	0.431410
$120$	0	0
$121$	−1043.00	−0.783621
$122$	0	0
$123$	0	0
$124$	0	0
$125$	644.881	0.461440
$126$	0	0
$127$	−1244.00	−0.869190	−0.434595	−	0.900626i	$-0.643109\pi$
$127$	−1244.00	−0.869190	−0.434595	+	0.900626i	$0.643109\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2511.64	−1.67514	−0.837570	−	0.546330i	$-0.816024\pi$
$131$	−2511.64	−1.67514	−0.837570	+	0.546330i	$0.816024\pi$
$132$	0	0
$133$	−319.000	−0.207976
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1340.67	0.836070	0.418035	−	0.908431i	$-0.362719\pi$
$137$	1340.67	0.836070	0.418035	+	0.908431i	$0.362719\pi$
$138$	0	0
$139$	947.000	0.577867	0.288933	−	0.957349i	$-0.406699\pi$
$139$	947.000	0.577867	0.288933	+	0.957349i	$0.406699\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−492.146	−0.287800
$144$	0	0
$145$	−4608.00	−2.63913
$146$	0	0
$147$	0	0
$148$	0	0
$149$	576.999	0.317246	0.158623	−	0.987339i	$-0.449295\pi$
$149$	576.999	0.317246	0.158623	+	0.987339i	$0.449295\pi$
$150$	0	0
$151$	2311.00	1.24547	0.622737	−	0.782431i	$-0.286021\pi$
$151$	2311.00	1.24547	0.622737	+	0.782431i	$0.286021\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4548.11	2.35686
$156$	0	0
$157$	−1622.00	−0.824520	−0.412260	−	0.911066i	$-0.635261\pi$
$157$	−1622.00	−0.824520	−0.412260	+	0.911066i	$0.635261\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	933.381	0.456899
$162$	0	0
$163$	2243.00	1.07782	0.538912	−	0.842362i	$-0.318836\pi$
$163$	2243.00	1.07782	0.538912	+	0.842362i	$0.318836\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2732.26	−1.26604	−0.633020	−	0.774135i	$-0.718185\pi$
$167$	−2732.26	−1.26604	−0.633020	+	0.774135i	$0.718185\pi$
$168$	0	0
$169$	−1356.00	−0.617205
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1357.65	0.596646	0.298323	−	0.954465i	$-0.403573\pi$
$173$	1357.65	0.596646	0.298323	+	0.954465i	$0.403573\pi$
$174$	0	0
$175$	−1793.00	−0.774503
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2341.94	−0.977903	−0.488952	−	0.872311i	$-0.662621\pi$
$179$	−2341.94	−0.977903	−0.488952	+	0.872311i	$0.662621\pi$
$180$	0	0
$181$	1591.00	0.653360	0.326680	−	0.945135i	$-0.394070\pi$
$181$	1591.00	0.653360	0.326680	+	0.945135i	$0.394070\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1408.56	−0.559779
$186$	0	0
$187$	−864.000	−0.337871
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4768.73	1.80656	0.903280	−	0.429051i	$-0.141152\pi$
$191$	4768.73	1.80656	0.903280	+	0.429051i	$0.141152\pi$
$192$	0	0
$193$	−3481.00	−1.29828	−0.649140	−	0.760669i	$-0.724871\pi$
$193$	−3481.00	−1.29828	−0.649140	+	0.760669i	$0.724871\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2392.85	0.865398	0.432699	−	0.901538i	$-0.357561\pi$
$197$	2392.85	0.865398	0.432699	+	0.901538i	$0.357561\pi$
$198$	0	0
$199$	−2351.00	−0.837477	−0.418739	−	0.908107i	$-0.637528\pi$
$199$	−2351.00	−0.837477	−0.418739	+	0.908107i	$0.637528\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2986.82	1.03268
$204$	0	0
$205$	−4608.00	−1.56994
$206$	0	0
$207$	0	0
$208$	0	0
$209$	492.146	0.162883
$210$	0	0
$211$	1703.00	0.555637	0.277818	−	0.960634i	$-0.410389\pi$
$211$	1703.00	0.555637	0.277818	+	0.960634i	$0.410389\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3937.17	−1.24890
$216$	0	0
$217$	−2948.00	−0.922227
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1476.44	0.449394
$222$	0	0
$223$	−1388.00	−0.416804	−0.208402	−	0.978043i	$-0.566826\pi$
$223$	−1388.00	−0.416804	−0.208402	+	0.978043i	$0.566826\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4717.82	−1.37944	−0.689719	−	0.724077i	$-0.742266\pi$
$227$	−4717.82	−1.37944	−0.689719	+	0.724077i	$0.742266\pi$
$228$	0	0
$229$	−434.000	−0.125238	−0.0626191	−	0.998038i	$-0.519945\pi$
$229$	−434.000	−0.125238	−0.0626191	+	0.998038i	$0.519945\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3461.99	0.973403	0.486701	−	0.873568i	$-0.338200\pi$
$233$	3461.99	0.973403	0.486701	+	0.873568i	$0.338200\pi$
$234$	0	0
$235$	6624.00	1.83873
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2817.11	0.762443	0.381222	−	0.924484i	$-0.375503\pi$
$239$	2817.11	0.762443	0.381222	+	0.924484i	$0.375503\pi$
$240$	0	0
$241$	−2095.00	−0.559962	−0.279981	−	0.960006i	$-0.590328\pi$
$241$	−2095.00	−0.559962	−0.279981	+	0.960006i	$0.590328\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3767.46	−0.982427
$246$	0	0
$247$	−841.000	−0.216646
$248$	0	0
$249$	0	0
$250$	0	0
$251$	203.647	0.0512114	0.0256057	−	0.999672i	$-0.491849\pi$
$251$	203.647	0.0512114	0.0256057	+	0.999672i	$0.491849\pi$
$252$	0	0
$253$	−1440.00	−0.357834
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1798.88	−0.436619	−0.218309	−	0.975880i	$-0.570054\pi$
$257$	−1798.88	−0.436619	−0.218309	+	0.975880i	$0.570054\pi$
$258$	0	0
$259$	913.000	0.219039
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−373.352	−0.0875357	−0.0437679	−	0.999042i	$-0.513936\pi$
$263$	−373.352	−0.0875357	−0.0437679	+	0.999042i	$0.513936\pi$
$264$	0	0
$265$	−5184.00	−1.20170
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7484.02	−1.69631	−0.848157	−	0.529744i	$-0.822288\pi$
$269$	−7484.02	−1.69631	−0.848157	+	0.529744i	$0.822288\pi$
$270$	0	0
$271$	3319.00	0.743966	0.371983	−	0.928239i	$-0.378678\pi$
$271$	3319.00	0.743966	0.371983	+	0.928239i	$0.378678\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2766.20	0.606575
$276$	0	0
$277$	−8354.00	−1.81207	−0.906035	−	0.423203i	$-0.860906\pi$
$277$	−8354.00	−1.81207	−0.906035	+	0.423203i	$0.860906\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2579.53	−0.547621	−0.273811	−	0.961784i	$-0.588284\pi$
$281$	−2579.53	−0.547621	−0.273811	+	0.961784i	$0.588284\pi$
$282$	0	0
$283$	−6208.00	−1.30398	−0.651992	−	0.758226i	$-0.726066\pi$
$283$	−6208.00	−1.30398	−0.651992	+	0.758226i	$0.726066\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2986.82	0.614308
$288$	0	0
$289$	−2321.00	−0.472420
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6194.26	−1.23506	−0.617529	−	0.786548i	$-0.711866\pi$
$293$	−6194.26	−1.23506	−0.617529	+	0.786548i	$0.711866\pi$
$294$	0	0
$295$	4896.00	0.966292
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2460.73	0.475946
$300$	0	0
$301$	2552.00	0.488687
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−13016.4	−2.44367
$306$	0	0
$307$	−2320.00	−0.431301	−0.215650	−	0.976471i	$-0.569187\pi$
$307$	−2320.00	−0.431301	−0.215650	+	0.976471i	$0.569187\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−797.616	−0.145430	−0.0727149	−	0.997353i	$-0.523166\pi$
$311$	−797.616	−0.145430	−0.0727149	+	0.997353i	$0.523166\pi$
$312$	0	0
$313$	1307.00	0.236026	0.118013	−	0.993012i	$-0.462348\pi$
$313$	1307.00	0.236026	0.118013	+	0.993012i	$0.462348\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2070.41	0.366832	0.183416	−	0.983035i	$-0.441284\pi$
$317$	2070.41	0.366832	0.183416	+	0.983035i	$0.441284\pi$
$318$	0	0
$319$	−4608.00	−0.808773
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1476.44	−0.254338
$324$	0	0
$325$	−4727.00	−0.806790
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4293.55	−0.719487
$330$	0	0
$331$	−5173.00	−0.859014	−0.429507	−	0.903063i	$-0.641313\pi$
$331$	−5173.00	−0.859014	−0.429507	+	0.903063i	$0.641313\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8671.96	−1.41433
$336$	0	0
$337$	2621.00	0.423665	0.211832	−	0.977306i	$-0.432057\pi$
$337$	2621.00	0.423665	0.211832	+	0.977306i	$0.432057\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4548.11	0.722270
$342$	0	0
$343$	6215.00	0.978363
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7704.64	−1.19195	−0.595975	−	0.803003i	$-0.703234\pi$
$347$	−7704.64	−1.19195	−0.595975	+	0.803003i	$0.703234\pi$
$348$	0	0
$349$	−1955.00	−0.299853	−0.149927	−	0.988697i	$-0.547904\pi$
$349$	−1955.00	−0.299853	−0.149927	+	0.988697i	$0.547904\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1187.94	0.179115	0.0895576	−	0.995982i	$-0.471455\pi$
$353$	1187.94	0.179115	0.0895576	+	0.995982i	$0.471455\pi$
$354$	0	0
$355$	−12096.0	−1.80842
$356$	0	0
$357$	0	0
$358$	0	0
$359$	560.029	0.0823320	0.0411660	−	0.999152i	$-0.486893\pi$
$359$	560.029	0.0823320	0.0411660	+	0.999152i	$0.486893\pi$
$360$	0	0
$361$	−6018.00	−0.877387
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2324.97	0.333409
$366$	0	0
$367$	−7895.00	−1.12293	−0.561465	−	0.827500i	$-0.689762\pi$
$367$	−7895.00	−1.12293	−0.561465	+	0.827500i	$0.689762\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3360.17	0.470219
$372$	0	0
$373$	−9803.00	−1.36080	−0.680402	−	0.732839i	$-0.738195\pi$
$373$	−9803.00	−1.36080	−0.680402	+	0.732839i	$0.738195\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7874.34	1.07573
$378$	0	0
$379$	10505.0	1.42376	0.711881	−	0.702300i	$-0.247844\pi$
$379$	10505.0	1.42376	0.711881	+	0.702300i	$0.247844\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1086.12	0.144903	0.0724516	−	0.997372i	$-0.476918\pi$
$383$	1086.12	0.144903	0.0724516	+	0.997372i	$0.476918\pi$
$384$	0	0
$385$	−3168.00	−0.419367
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2155.26	−0.280915	−0.140458	−	0.990087i	$-0.544857\pi$
$389$	−2155.26	−0.280915	−0.140458	+	0.990087i	$0.544857\pi$
$390$	0	0
$391$	4320.00	0.558751
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8061.02	1.02682
$396$	0	0
$397$	−12422.0	−1.57038	−0.785192	−	0.619253i	$-0.787436\pi$
$397$	−12422.0	−1.57038	−0.785192	+	0.619253i	$0.787436\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	15511.1	1.93164	0.965819	−	0.259216i	$-0.0834642\pi$
$401$	15511.1	1.93164	0.965819	+	0.259216i	$0.0834642\pi$
$402$	0	0
$403$	−7772.00	−0.960672
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1408.56	−0.171547
$408$	0	0
$409$	7265.00	0.878316	0.439158	−	0.898410i	$-0.355277\pi$
$409$	7265.00	0.878316	0.439158	+	0.898410i	$0.355277\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3173.50	−0.378105
$414$	0	0
$415$	9792.00	1.15824
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3173.50	−0.370013	−0.185006	−	0.982737i	$-0.559231\pi$
$419$	−3173.50	−0.370013	−0.185006	+	0.982737i	$0.559231\pi$
$420$	0	0
$421$	−3413.00	−0.395106	−0.197553	−	0.980292i	$-0.563299\pi$
$421$	−3413.00	−0.395106	−0.197553	+	0.980292i	$0.563299\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−8298.61	−0.947156
$426$	0	0
$427$	8437.00	0.956194
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12677.0	1.41678	0.708388	−	0.705824i	$-0.249423\pi$
$431$	12677.0	1.41678	0.708388	+	0.705824i	$0.249423\pi$
$432$	0	0
$433$	8642.00	0.959141	0.479570	−	0.877503i	$-0.340792\pi$
$433$	8642.00	0.959141	0.479570	+	0.877503i	$0.340792\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2460.73	−0.269366
$438$	0	0
$439$	−524.000	−0.0569685	−0.0284842	−	0.999594i	$-0.509068\pi$
$439$	−524.000	−0.0569685	−0.0284842	+	0.999594i	$0.509068\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−18158.5	−1.94749	−0.973743	−	0.227649i	$-0.926896\pi$
$443$	−18158.5	−1.94749	−0.973743	+	0.227649i	$0.926896\pi$
$444$	0	0
$445$	−4320.00	−0.460197
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3309.26	0.347825	0.173913	−	0.984761i	$-0.444359\pi$
$449$	3309.26	0.347825	0.173913	+	0.984761i	$0.444359\pi$
$450$	0	0
$451$	−4608.00	−0.481114
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5413.61	0.557789
$456$	0	0
$457$	−9466.00	−0.968930	−0.484465	−	0.874811i	$-0.660986\pi$
$457$	−9466.00	−0.968930	−0.484465	+	0.874811i	$0.660986\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3241.38	0.327475	0.163738	−	0.986504i	$-0.447645\pi$
$461$	3241.38	0.327475	0.163738	+	0.986504i	$0.447645\pi$
$462$	0	0
$463$	−11315.0	−1.13575	−0.567875	−	0.823115i	$-0.692234\pi$
$463$	−11315.0	−1.13575	−0.567875	+	0.823115i	$0.692234\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17462.7	1.73036	0.865180	−	0.501462i	$-0.167204\pi$
$467$	17462.7	1.73036	0.865180	+	0.501462i	$0.167204\pi$
$468$	0	0
$469$	5621.00	0.553419
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3937.17	−0.382730
$474$	0	0
$475$	4727.00	0.456610
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8926.52	−0.851488	−0.425744	−	0.904844i	$-0.639988\pi$
$479$	−8926.52	−0.851488	−0.425744	+	0.904844i	$0.639988\pi$
$480$	0	0
$481$	2407.00	0.228170
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13932.8	1.30445
$486$	0	0
$487$	18493.0	1.72073	0.860367	−	0.509674i	$-0.170234\pi$
$487$	18493.0	1.72073	0.860367	+	0.509674i	$0.170234\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12812.8	1.17766	0.588831	−	0.808256i	$-0.299588\pi$
$491$	12812.8	1.17766	0.588831	+	0.808256i	$0.299588\pi$
$492$	0	0
$493$	13824.0	1.26288
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7840.40	0.707626
$498$	0	0
$499$	−12256.0	−1.09951	−0.549753	−	0.835327i	$-0.685278\pi$
$499$	−12256.0	−1.09951	−0.549753	+	0.835327i	$0.685278\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7382.19	0.654385	0.327193	−	0.944958i	$-0.393897\pi$
$503$	7382.19	0.654385	0.327193	+	0.944958i	$0.393897\pi$
$504$	0	0
$505$	9216.00	0.812092
$506$	0	0
$507$	0	0
$508$	0	0
$509$	20992.6	1.82806	0.914028	−	0.405652i	$-0.132956\pi$
$509$	20992.6	1.82806	0.914028	+	0.405652i	$0.132956\pi$
$510$	0	0
$511$	−1507.00	−0.130461
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−14238.3	−1.21828
$516$	0	0
$517$	6624.00	0.563488
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12269.7	1.03176	0.515879	−	0.856661i	$-0.327465\pi$
$521$	12269.7	1.03176	0.515879	+	0.856661i	$0.327465\pi$
$522$	0	0
$523$	6833.00	0.571293	0.285646	−	0.958335i	$-0.407792\pi$
$523$	6833.00	0.571293	0.285646	+	0.958335i	$0.407792\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−13644.3	−1.12781
$528$	0	0
$529$	−4967.00	−0.408235
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7874.34	0.639917
$534$	0	0
$535$	−12960.0	−1.04731
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3767.46	−0.301069
$540$	0	0
$541$	10555.0	0.838808	0.419404	−	0.907800i	$-0.362239\pi$
$541$	10555.0	0.838808	0.419404	+	0.907800i	$0.362239\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3699.58	−0.290776
$546$	0	0
$547$	17291.0	1.35157	0.675786	−	0.737098i	$-0.263804\pi$
$547$	17291.0	1.35157	0.675786	+	0.737098i	$0.263804\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7874.34	−0.608817
$552$	0	0
$553$	−5225.00	−0.401790
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10335.1	−0.786196	−0.393098	−	0.919497i	$-0.628597\pi$
$557$	−10335.1	−0.786196	−0.393098	+	0.919497i	$0.628597\pi$
$558$	0	0
$559$	6728.00	0.509059
$560$	0	0
$561$	0	0
$562$	0	0
$563$	16427.5	1.22973	0.614864	−	0.788633i	$-0.289211\pi$
$563$	16427.5	1.22973	0.614864	+	0.788633i	$0.289211\pi$
$564$	0	0
$565$	25632.0	1.90858
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15697.8	1.15656	0.578282	−	0.815837i	$-0.303723\pi$
$569$	15697.8	1.15656	0.578282	+	0.815837i	$0.303723\pi$
$570$	0	0
$571$	−4075.00	−0.298658	−0.149329	−	0.988788i	$-0.547711\pi$
$571$	−4075.00	−0.298658	−0.149329	+	0.988788i	$0.547711\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−13831.0	−1.00312
$576$	0	0
$577$	6995.00	0.504689	0.252345	−	0.967637i	$-0.418798\pi$
$577$	6995.00	0.504689	0.252345	+	0.967637i	$0.418798\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6346.99	−0.453214
$582$	0	0
$583$	−5184.00	−0.368266
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5583.32	−0.392586	−0.196293	−	0.980545i	$-0.562890\pi$
$587$	−5583.32	−0.392586	−0.196293	+	0.980545i	$0.562890\pi$
$588$	0	0
$589$	7772.00	0.543701
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14968.0	−1.03653	−0.518266	−	0.855219i	$-0.673422\pi$
$593$	−14968.0	−1.03653	−0.518266	+	0.855219i	$0.673422\pi$
$594$	0	0
$595$	9504.00	0.654834
$596$	0	0
$597$	0	0
$598$	0	0
$599$	18192.4	1.24094	0.620470	−	0.784230i	$-0.286942\pi$
$599$	18192.4	1.24094	0.620470	+	0.784230i	$0.286942\pi$
$600$	0	0
$601$	−6550.00	−0.444559	−0.222280	−	0.974983i	$-0.571350\pi$
$601$	−6550.00	−0.444559	−0.222280	+	0.974983i	$0.571350\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−17700.3	−1.18945
$606$	0	0
$607$	−12827.0	−0.857713	−0.428857	−	0.903373i	$-0.641083\pi$
$607$	−12827.0	−0.857713	−0.428857	+	0.903373i	$0.641083\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−11319.4	−0.749480
$612$	0	0
$613$	−18767.0	−1.23653	−0.618264	−	0.785970i	$-0.712164\pi$
$613$	−18767.0	−1.23653	−0.618264	+	0.785970i	$0.712164\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−7551.90	−0.492752	−0.246376	−	0.969174i	$-0.579240\pi$
$617$	−7551.90	−0.492752	−0.246376	+	0.969174i	$0.579240\pi$
$618$	0	0
$619$	24581.0	1.59611	0.798056	−	0.602583i	$-0.205862\pi$
$619$	24581.0	1.59611	0.798056	+	0.602583i	$0.205862\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2800.14	0.180073
$624$	0	0
$625$	−9431.00	−0.603584
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4225.67	0.267867
$630$	0	0
$631$	18223.0	1.14968	0.574838	−	0.818267i	$-0.305065\pi$
$631$	18223.0	1.14968	0.574838	+	0.818267i	$0.305065\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−21111.4	−1.31934
$636$	0	0
$637$	6438.00	0.400444
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1595.23	0.0982963	0.0491481	−	0.998792i	$-0.484349\pi$
$641$	1595.23	0.0982963	0.0491481	+	0.998792i	$0.484349\pi$
$642$	0	0
$643$	−26296.0	−1.61277	−0.806386	−	0.591389i	$-0.798580\pi$
$643$	−26296.0	−1.61277	−0.806386	+	0.591389i	$0.798580\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	25659.5	1.55916	0.779582	−	0.626301i	$-0.215432\pi$
$647$	25659.5	1.55916	0.779582	+	0.626301i	$0.215432\pi$
$648$	0	0
$649$	4896.00	0.296125
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−441.235	−0.0264423	−0.0132212	−	0.999913i	$-0.504209\pi$
$653$	−441.235	−0.0264423	−0.0132212	+	0.999913i	$0.504209\pi$
$654$	0	0
$655$	−42624.0	−2.54268
$656$	0	0
$657$	0	0
$658$	0	0
$659$	27051.1	1.59903	0.799515	−	0.600647i	$-0.205090\pi$
$659$	27051.1	1.59903	0.799515	+	0.600647i	$0.205090\pi$
$660$	0	0
$661$	−623.000	−0.0366594	−0.0183297	−	0.999832i	$-0.505835\pi$
$661$	−623.000	−0.0366594	−0.0183297	+	0.999832i	$0.505835\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5413.61	−0.315685
$666$	0	0
$667$	23040.0	1.33750
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−13016.4	−0.748872
$672$	0	0
$673$	27875.0	1.59659	0.798293	−	0.602269i	$-0.205737\pi$
$673$	27875.0	1.59659	0.798293	+	0.602269i	$0.205737\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6601.55	0.374768	0.187384	−	0.982287i	$-0.439999\pi$
$677$	6601.55	0.374768	0.187384	+	0.982287i	$0.439999\pi$
$678$	0	0
$679$	−9031.00	−0.510424
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−15680.8	−0.878491	−0.439245	−	0.898367i	$-0.644754\pi$
$683$	−15680.8	−0.878491	−0.439245	+	0.898367i	$0.644754\pi$
$684$	0	0
$685$	22752.0	1.26906
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8858.63	0.489822
$690$	0	0
$691$	−10600.0	−0.583564	−0.291782	−	0.956485i	$-0.594248\pi$
$691$	−10600.0	−0.583564	−0.291782	+	0.956485i	$0.594248\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	16071.1	0.877140
$696$	0	0
$697$	13824.0	0.751250
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−13593.4	−0.732406	−0.366203	−	0.930535i	$-0.619342\pi$
$701$	−13593.4	−0.732406	−0.366203	+	0.930535i	$0.619342\pi$
$702$	0	0
$703$	−2407.00	−0.129135
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5973.64	−0.317768
$708$	0	0
$709$	33523.0	1.77572	0.887858	−	0.460117i	$-0.152193\pi$
$709$	33523.0	1.77572	0.887858	+	0.460117i	$0.152193\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−22740.6	−1.19445
$714$	0	0
$715$	−8352.00	−0.436849
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−31870.7	−1.65310	−0.826549	−	0.562865i	$-0.809699\pi$
$719$	−31870.7	−1.65310	−0.826549	+	0.562865i	$0.809699\pi$
$720$	0	0
$721$	9229.00	0.476707
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−44259.2	−2.26724
$726$	0	0
$727$	13084.0	0.667481	0.333741	−	0.942665i	$-0.391689\pi$
$727$	13084.0	0.667481	0.333741	+	0.942665i	$0.391689\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	11811.5	0.597626
$732$	0	0
$733$	19222.0	0.968596	0.484298	−	0.874903i	$-0.339075\pi$
$733$	19222.0	0.968596	0.484298	+	0.874903i	$0.339075\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8671.96	−0.433427
$738$	0	0
$739$	6320.00	0.314594	0.157297	−	0.987551i	$-0.449722\pi$
$739$	6320.00	0.314594	0.157297	+	0.987551i	$0.449722\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−4157.79	−0.205295	−0.102648	−	0.994718i	$-0.532731\pi$
$743$	−4157.79	−0.205295	−0.102648	+	0.994718i	$0.532731\pi$
$744$	0	0
$745$	9792.00	0.481545
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8400.43	0.409806
$750$	0	0
$751$	−20333.0	−0.987965	−0.493982	−	0.869472i	$-0.664459\pi$
$751$	−20333.0	−0.987965	−0.493982	+	0.869472i	$0.664459\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	39219.0	1.89050
$756$	0	0
$757$	14011.0	0.672706	0.336353	−	0.941736i	$-0.390806\pi$
$757$	14011.0	0.672706	0.336353	+	0.941736i	$0.390806\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−25981.9	−1.23764	−0.618820	−	0.785533i	$-0.712389\pi$
$761$	−25981.9	−1.23764	−0.618820	+	0.785533i	$0.712389\pi$
$762$	0	0
$763$	2398.00	0.113779
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8366.49	−0.393867
$768$	0	0
$769$	−6289.00	−0.294912	−0.147456	−	0.989069i	$-0.547108\pi$
$769$	−6289.00	−0.294912	−0.147456	+	0.989069i	$0.547108\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7229.46	0.336385	0.168192	−	0.985754i	$-0.446207\pi$
$773$	7229.46	0.336385	0.168192	+	0.985754i	$0.446207\pi$
$774$	0	0
$775$	43684.0	2.02474
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7874.34	−0.362166
$780$	0	0
$781$	−12096.0	−0.554198
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−27526.3	−1.25153
$786$	0	0
$787$	−25675.0	−1.16292	−0.581458	−	0.813576i	$-0.697518\pi$
$787$	−25675.0	−1.16292	−0.581458	+	0.813576i	$0.697518\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−16614.2	−0.746817
$792$	0	0
$793$	22243.0	0.996056
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3326.23	0.147831	0.0739154	−	0.997265i	$-0.476451\pi$
$797$	3326.23	0.147831	0.0739154	+	0.997265i	$0.476451\pi$
$798$	0	0
$799$	−19872.0	−0.879876
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2324.97	0.102175
$804$	0	0
$805$	15840.0	0.693524
$806$	0	0
$807$	0	0
$808$	0	0
$809$	10080.5	0.438087	0.219043	−	0.975715i	$-0.429706\pi$
$809$	10080.5	0.438087	0.219043	+	0.975715i	$0.429706\pi$
$810$	0	0
$811$	14312.0	0.619682	0.309841	−	0.950788i	$-0.399724\pi$
$811$	14312.0	0.619682	0.309841	+	0.950788i	$0.399724\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	38065.0	1.63602
$816$	0	0
$817$	−6728.00	−0.288106
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2766.20	−0.117590	−0.0587948	−	0.998270i	$-0.518726\pi$
$821$	−2766.20	−0.117590	−0.0587948	+	0.998270i	$0.518726\pi$
$822$	0	0
$823$	33343.0	1.41223	0.706114	−	0.708098i	$-0.250447\pi$
$823$	33343.0	1.41223	0.706114	+	0.708098i	$0.250447\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−18379.1	−0.772799	−0.386399	−	0.922332i	$-0.626281\pi$
$827$	−18379.1	−0.772799	−0.386399	+	0.922332i	$0.626281\pi$
$828$	0	0
$829$	−3593.00	−0.150531	−0.0752654	−	0.997164i	$-0.523980\pi$
$829$	−3593.00	−0.150531	−0.0752654	+	0.997164i	$0.523980\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	11302.4	0.470114
$834$	0	0
$835$	−46368.0	−1.92171
$836$	0	0
$837$	0	0
$838$	0	0
$839$	17140.3	0.705301	0.352651	−	0.935755i	$-0.385280\pi$
$839$	17140.3	0.705301	0.352651	+	0.935755i	$0.385280\pi$
$840$	0	0
$841$	49339.0	2.02300
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−23012.1	−0.936852
$846$	0	0
$847$	11473.0	0.465427
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7042.78	0.283694
$852$	0	0
$853$	4741.00	0.190303	0.0951517	−	0.995463i	$-0.469666\pi$
$853$	4741.00	0.190303	0.0951517	+	0.995463i	$0.469666\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11981.2	0.477562	0.238781	−	0.971073i	$-0.423252\pi$
$857$	11981.2	0.477562	0.238781	+	0.971073i	$0.423252\pi$
$858$	0	0
$859$	6887.00	0.273552	0.136776	−	0.990602i	$-0.456326\pi$
$859$	6887.00	0.273552	0.136776	+	0.990602i	$0.456326\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	8400.43	0.331349	0.165674	−	0.986181i	$-0.447020\pi$
$863$	8400.43	0.331349	0.165674	+	0.986181i	$0.447020\pi$
$864$	0	0
$865$	23040.0	0.905646
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8061.02	0.314674
$870$	0	0
$871$	14819.0	0.576490
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−7093.70	−0.274069
$876$	0	0
$877$	−13475.0	−0.518835	−0.259418	−	0.965765i	$-0.583531\pi$
$877$	−13475.0	−0.518835	−0.259418	+	0.965765i	$0.583531\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−5243.90	−0.200535	−0.100268	−	0.994961i	$-0.531970\pi$
$881$	−5243.90	−0.200535	−0.100268	+	0.994961i	$0.531970\pi$
$882$	0	0
$883$	−7909.00	−0.301426	−0.150713	−	0.988578i	$-0.548157\pi$
$883$	−7909.00	−0.301426	−0.150713	+	0.988578i	$0.548157\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−35672.1	−1.35034	−0.675171	−	0.737662i	$-0.735930\pi$
$887$	−35672.1	−1.35034	−0.675171	+	0.737662i	$0.735930\pi$
$888$	0	0
$889$	13684.0	0.516250
$890$	0	0
$891$	0	0
$892$	0	0
$893$	11319.4	0.424175
$894$	0	0
$895$	−39744.0	−1.48435
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−72769.8	−2.69967
$900$	0	0
$901$	15552.0	0.575041
$902$	0	0
$903$	0	0
$904$	0	0
$905$	27000.2	0.991730
$906$	0	0
$907$	−16999.0	−0.622318	−0.311159	−	0.950358i	$-0.600717\pi$
$907$	−16999.0	−0.622318	−0.311159	+	0.950358i	$0.600717\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	39032.3	1.41954	0.709768	−	0.704435i	$-0.248800\pi$
$911$	39032.3	1.41954	0.709768	+	0.704435i	$0.248800\pi$
$912$	0	0
$913$	9792.00	0.354948
$914$	0	0
$915$	0	0
$916$	0	0
$917$	27628.1	0.994939
$918$	0	0
$919$	28348.0	1.01753	0.508767	−	0.860904i	$-0.330101\pi$
$919$	28348.0	1.01753	0.508767	+	0.860904i	$0.330101\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	20670.1	0.737125
$924$	0	0
$925$	−13529.0	−0.480898
$926$	0	0
$927$	0	0
$928$	0	0
$929$	33160.5	1.17111	0.585554	−	0.810633i	$-0.300877\pi$
$929$	33160.5	1.17111	0.585554	+	0.810633i	$0.300877\pi$
$930$	0	0
$931$	−6438.00	−0.226635
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−14662.6	−0.512853
$936$	0	0
$937$	−133.000	−0.00463706	−0.00231853	−	0.999997i	$-0.500738\pi$
$937$	−133.000	−0.00463706	−0.00231853	+	0.999997i	$0.500738\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	48790.4	1.69024	0.845122	−	0.534573i	$-0.179527\pi$
$941$	48790.4	1.69024	0.845122	+	0.534573i	$0.179527\pi$
$942$	0	0
$943$	23040.0	0.795637
$944$	0	0
$945$	0	0
$946$	0	0
$947$	45328.4	1.55541	0.777705	−	0.628629i	$-0.216383\pi$
$947$	45328.4	1.55541	0.777705	+	0.628629i	$0.216383\pi$
$948$	0	0
$949$	−3973.00	−0.135900
$950$	0	0
$951$	0	0
$952$	0	0
$953$	43122.2	1.46576	0.732878	−	0.680360i	$-0.238177\pi$
$953$	43122.2	1.46576	0.732878	+	0.680360i	$0.238177\pi$
$954$	0	0
$955$	80928.0	2.74217
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−14747.4	−0.496579
$960$	0	0
$961$	42033.0	1.41093
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−59074.5	−1.97065
$966$	0	0
$967$	−22061.0	−0.733644	−0.366822	−	0.930291i	$-0.619554\pi$
$967$	−22061.0	−0.733644	−0.366822	+	0.930291i	$0.619554\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	33449.0	1.10549	0.552744	−	0.833351i	$-0.313581\pi$
$971$	33449.0	1.10549	0.552744	+	0.833351i	$0.313581\pi$
$972$	0	0
$973$	−10417.0	−0.343221
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20602.3	0.674642	0.337321	−	0.941390i	$-0.390479\pi$
$977$	20602.3	0.674642	0.337321	+	0.941390i	$0.390479\pi$
$978$	0	0
$979$	−4320.00	−0.141029
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−11930.3	−0.387098	−0.193549	−	0.981091i	$-0.562000\pi$
$983$	−11930.3	−0.387098	−0.193549	+	0.981091i	$0.562000\pi$
$984$	0	0
$985$	40608.0	1.31358
$986$	0	0
$987$	0	0
$988$	0	0
$989$	19685.9	0.632936
$990$	0	0
$991$	35017.0	1.12245	0.561227	−	0.827662i	$-0.310330\pi$
$991$	35017.0	1.12245	0.561227	+	0.827662i	$0.310330\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−39897.8	−1.27120
$996$	0	0
$997$	−13646.0	−0.433474	−0.216737	−	0.976230i	$-0.569541\pi$
$997$	−13646.0	−0.433474	−0.216737	+	0.976230i	$0.569541\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.a.bk.1.2		2
3.2	odd	2	inner	1728.4.a.bk.1.1		2
4.3	odd	2		1728.4.a.bp.1.2		2
8.3	odd	2		27.4.a.c.1.2	yes	2
8.5	even	2		432.4.a.q.1.1		2
12.11	even	2		1728.4.a.bp.1.1		2
24.5	odd	2		432.4.a.q.1.2		2
24.11	even	2		27.4.a.c.1.1	✓	2
40.3	even	4		675.4.b.i.649.1		4
40.19	odd	2		675.4.a.n.1.1		2
40.27	even	4		675.4.b.i.649.4		4
56.27	even	2		1323.4.a.t.1.2		2
72.11	even	6		81.4.c.e.28.2		4
72.43	odd	6		81.4.c.e.28.1		4
72.59	even	6		81.4.c.e.55.2		4
72.67	odd	6		81.4.c.e.55.1		4
120.59	even	2		675.4.a.n.1.2		2
120.83	odd	4		675.4.b.i.649.3		4
120.107	odd	4		675.4.b.i.649.2		4
168.83	odd	2		1323.4.a.t.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.a.c.1.1	✓	2	24.11	even	2
27.4.a.c.1.2	yes	2	8.3	odd	2
81.4.c.e.28.1		4	72.43	odd	6
81.4.c.e.28.2		4	72.11	even	6
81.4.c.e.55.1		4	72.67	odd	6
81.4.c.e.55.2		4	72.59	even	6
432.4.a.q.1.1		2	8.5	even	2
432.4.a.q.1.2		2	24.5	odd	2
675.4.a.n.1.1		2	40.19	odd	2
675.4.a.n.1.2		2	120.59	even	2
675.4.b.i.649.1		4	40.3	even	4
675.4.b.i.649.2		4	120.107	odd	4
675.4.b.i.649.3		4	120.83	odd	4
675.4.b.i.649.4		4	40.27	even	4
1323.4.a.t.1.1		2	168.83	odd	2
1323.4.a.t.1.2		2	56.27	even	2
1728.4.a.bk.1.1		2	3.2	odd	2	inner
1728.4.a.bk.1.2		2	1.1	even	1	trivial
1728.4.a.bp.1.1		2	12.11	even	2
1728.4.a.bp.1.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.a (trivial)

\(f(q)\)	\(=\)	\(q+16.9706 q^{5} -11.0000 q^{7} +16.9706 q^{11} -29.0000 q^{13} -50.9117 q^{17} +29.0000 q^{19} -84.8528 q^{23} +163.000 q^{25} -271.529 q^{29} +268.000 q^{31} -186.676 q^{35} -83.0000 q^{37} -271.529 q^{41} -232.000 q^{43} +390.323 q^{47} -222.000 q^{49} -305.470 q^{53} +288.000 q^{55} +288.500 q^{59} -767.000 q^{61} -492.146 q^{65} -511.000 q^{67} -712.764 q^{71} +137.000 q^{73} -186.676 q^{77} +475.000 q^{79} +576.999 q^{83} -864.000 q^{85} -254.558 q^{89} +319.000 q^{91} +492.146 q^{95} +821.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 22 q^{7} - 58 q^{13} + 58 q^{19} + 326 q^{25} + 536 q^{31} - 166 q^{37} - 464 q^{43} - 444 q^{49} + 576 q^{55} - 1534 q^{61} - 1022 q^{67} + 274 q^{73} + 950 q^{79} - 1728 q^{85} + 638 q^{91} + 1642 q^{97}+O(q^{100}) \) 2 * q - 22 * q^7 - 58 * q^13 + 58 * q^19 + 326 * q^25 + 536 * q^31 - 166 * q^37 - 464 * q^43 - 444 * q^49 + 576 * q^55 - 1534 * q^61 - 1022 * q^67 + 274 * q^73 + 950 * q^79 - 1728 * q^85 + 638 * q^91 + 1642 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more