LMFDB - Embedded newform 2304.4.a.bn.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2304,4,Mod(1,2304)] 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("2304.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2304, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2304.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,28,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$135.940400653$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	2304.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+10.5830 q^{5} +8.00000 q^{7} +15.8745 q^{11} -52.9150 q^{13} +14.0000 q^{17} +37.0405 q^{19} -152.000 q^{23} -13.0000 q^{25} -158.745 q^{29} +224.000 q^{31} +84.6640 q^{35} +243.409 q^{37} -70.0000 q^{41} -439.195 q^{43} -336.000 q^{47} -279.000 q^{49} -31.7490 q^{53} +168.000 q^{55} -534.442 q^{59} -95.2470 q^{61} -560.000 q^{65} -174.620 q^{67} -72.0000 q^{71} +294.000 q^{73} +126.996 q^{77} -464.000 q^{79} -545.025 q^{83} +148.162 q^{85} +266.000 q^{89} -423.320 q^{91} +392.000 q^{95} +994.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 16 q^{7} + 28 q^{17} - 304 q^{23} - 26 q^{25} + 448 q^{31} - 140 q^{41} - 672 q^{47} - 558 q^{49} + 336 q^{55} - 1120 q^{65} - 144 q^{71} + 588 q^{73} - 928 q^{79} + 532 q^{89} + 784 q^{95} + 1988 q^{97}+O(q^{100}) $ 2 * q + 16 * q^7 + 28 * q^17 - 304 * q^23 - 26 * q^25 + 448 * q^31 - 140 * q^41 - 672 * q^47 - 558 * q^49 + 336 * q^55 - 1120 * q^65 - 144 * q^71 + 588 * q^73 - 928 * q^79 + 532 * q^89 + 784 * q^95 + 1988 * 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
$5$	10.5830	0.946573	0.473286	−	0.880909i	$-0.343068\pi$
$5$	10.5830	0.946573	0.473286	+	0.880909i	$0.343068\pi$
$6$	0	0
$7$	8.00000	0.431959	0.215980	−	0.976398i	$-0.430705\pi$
$7$	8.00000	0.431959	0.215980	+	0.976398i	$0.430705\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	15.8745	0.435122	0.217561	−	0.976047i	$-0.430190\pi$
$11$	15.8745	0.435122	0.217561	+	0.976047i	$0.430190\pi$
$12$	0	0
$13$	−52.9150	−1.12892	−0.564461	−	0.825460i	$-0.690916\pi$
$13$	−52.9150	−1.12892	−0.564461	+	0.825460i	$0.690916\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	14.0000	0.199735	0.0998676	−	0.995001i	$-0.468158\pi$
$17$	14.0000	0.199735	0.0998676	+	0.995001i	$0.468158\pi$
$18$	0	0
$19$	37.0405	0.447246	0.223623	−	0.974676i	$-0.428212\pi$
$19$	37.0405	0.447246	0.223623	+	0.974676i	$0.428212\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−152.000	−1.37801	−0.689004	−	0.724757i	$-0.741952\pi$
$23$	−152.000	−1.37801	−0.689004	+	0.724757i	$0.741952\pi$
$24$	0	0
$25$	−13.0000	−0.104000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−158.745	−1.01649	−0.508245	−	0.861212i	$-0.669706\pi$
$29$	−158.745	−1.01649	−0.508245	+	0.861212i	$0.669706\pi$
$30$	0	0
$31$	224.000	1.29779	0.648897	−	0.760877i	$-0.275231\pi$
$31$	224.000	1.29779	0.648897	+	0.760877i	$0.275231\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	84.6640	0.408881
$36$	0	0
$37$	243.409	1.08152	0.540760	−	0.841177i	$-0.318137\pi$
$37$	243.409	1.08152	0.540760	+	0.841177i	$0.318137\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−70.0000	−0.266638	−0.133319	−	0.991073i	$-0.542564\pi$
$41$	−70.0000	−0.266638	−0.133319	+	0.991073i	$0.542564\pi$
$42$	0	0
$43$	−439.195	−1.55759	−0.778797	−	0.627276i	$-0.784170\pi$
$43$	−439.195	−1.55759	−0.778797	+	0.627276i	$0.784170\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−336.000	−1.04278	−0.521390	−	0.853319i	$-0.674586\pi$
$47$	−336.000	−1.04278	−0.521390	+	0.853319i	$0.674586\pi$
$48$	0	0
$49$	−279.000	−0.813411
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−31.7490	−0.0822842	−0.0411421	−	0.999153i	$-0.513100\pi$
$53$	−31.7490	−0.0822842	−0.0411421	+	0.999153i	$0.513100\pi$
$54$	0	0
$55$	168.000	0.411875
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−534.442	−1.17929	−0.589647	−	0.807661i	$-0.700733\pi$
$59$	−534.442	−1.17929	−0.589647	+	0.807661i	$0.700733\pi$
$60$	0	0
$61$	−95.2470	−0.199920	−0.0999601	−	0.994991i	$-0.531872\pi$
$61$	−95.2470	−0.199920	−0.0999601	+	0.994991i	$0.531872\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−560.000	−1.06861
$66$	0	0
$67$	−174.620	−0.318406	−0.159203	−	0.987246i	$-0.550892\pi$
$67$	−174.620	−0.318406	−0.159203	+	0.987246i	$0.550892\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−72.0000	−0.120350	−0.0601748	−	0.998188i	$-0.519166\pi$
$71$	−72.0000	−0.120350	−0.0601748	+	0.998188i	$0.519166\pi$
$72$	0	0
$73$	294.000	0.471371	0.235686	−	0.971829i	$-0.424266\pi$
$73$	294.000	0.471371	0.235686	+	0.971829i	$0.424266\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	126.996	0.187955
$78$	0	0
$79$	−464.000	−0.660811	−0.330406	−	0.943839i	$-0.607186\pi$
$79$	−464.000	−0.660811	−0.330406	+	0.943839i	$0.607186\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−545.025	−0.720774	−0.360387	−	0.932803i	$-0.617355\pi$
$83$	−545.025	−0.720774	−0.360387	+	0.932803i	$0.617355\pi$
$84$	0	0
$85$	148.162	0.189064
$86$	0	0
$87$	0	0
$88$	0	0
$89$	266.000	0.316808	0.158404	−	0.987374i	$-0.449365\pi$
$89$	266.000	0.316808	0.158404	+	0.987374i	$0.449365\pi$
$90$	0	0
$91$	−423.320	−0.487649
$92$	0	0
$93$	0	0
$94$	0	0
$95$	392.000	0.423351
$96$	0	0
$97$	994.000	1.04047	0.520234	−	0.854024i	$-0.325845\pi$
$97$	994.000	1.04047	0.520234	+	0.854024i	$0.325845\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−751.393	−0.740262	−0.370131	−	0.928980i	$-0.620687\pi$
$101$	−751.393	−0.740262	−0.370131	+	0.928980i	$0.620687\pi$
$102$	0	0
$103$	−1176.00	−1.12500	−0.562499	−	0.826798i	$-0.690160\pi$
$103$	−1176.00	−1.12500	−0.562499	+	0.826798i	$0.690160\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	269.867	0.243822	0.121911	−	0.992541i	$-0.461098\pi$
$107$	269.867	0.243822	0.121911	+	0.992541i	$0.461098\pi$
$108$	0	0
$109$	1894.36	1.66465	0.832324	−	0.554290i	$-0.187010\pi$
$109$	1894.36	1.66465	0.832324	+	0.554290i	$0.187010\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1710.00	1.42357	0.711784	−	0.702398i	$-0.247887\pi$
$113$	1710.00	1.42357	0.711784	+	0.702398i	$0.247887\pi$
$114$	0	0
$115$	−1608.62	−1.30439
$116$	0	0
$117$	0	0
$118$	0	0
$119$	112.000	0.0862775
$120$	0	0
$121$	−1079.00	−0.810669
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1460.45	−1.04502
$126$	0	0
$127$	−1664.00	−1.16265	−0.581323	−	0.813673i	$-0.697465\pi$
$127$	−1664.00	−1.16265	−0.581323	+	0.813673i	$0.697465\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−672.021	−0.448204	−0.224102	−	0.974566i	$-0.571945\pi$
$131$	−672.021	−0.448204	−0.224102	+	0.974566i	$0.571945\pi$
$132$	0	0
$133$	296.324	0.193192
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1062.00	−0.662283	−0.331142	−	0.943581i	$-0.607434\pi$
$137$	−1062.00	−0.662283	−0.331142	+	0.943581i	$0.607434\pi$
$138$	0	0
$139$	2693.37	1.64352	0.821759	−	0.569835i	$-0.192993\pi$
$139$	2693.37	1.64352	0.821759	+	0.569835i	$0.192993\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−840.000	−0.491219
$144$	0	0
$145$	−1680.00	−0.962182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−793.725	−0.436406	−0.218203	−	0.975903i	$-0.570020\pi$
$149$	−793.725	−0.436406	−0.218203	+	0.975903i	$0.570020\pi$
$150$	0	0
$151$	−744.000	−0.400966	−0.200483	−	0.979697i	$-0.564251\pi$
$151$	−744.000	−0.400966	−0.200483	+	0.979697i	$0.564251\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2370.59	1.22846
$156$	0	0
$157$	−179.911	−0.0914552	−0.0457276	−	0.998954i	$-0.514561\pi$
$157$	−179.911	−0.0914552	−0.0457276	+	0.998954i	$0.514561\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1216.00	−0.595244
$162$	0	0
$163$	1772.65	0.851809	0.425905	−	0.904768i	$-0.359956\pi$
$163$	1772.65	0.851809	0.425905	+	0.904768i	$0.359956\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1960.00	−0.908200	−0.454100	−	0.890951i	$-0.650039\pi$
$167$	−1960.00	−0.908200	−0.454100	+	0.890951i	$0.650039\pi$
$168$	0	0
$169$	603.000	0.274465
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2000.19	0.879026	0.439513	−	0.898236i	$-0.355151\pi$
$173$	2000.19	0.879026	0.439513	+	0.898236i	$0.355151\pi$
$174$	0	0
$175$	−104.000	−0.0449238
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3264.86	1.36328	0.681639	−	0.731688i	$-0.261267\pi$
$179$	3264.86	1.36328	0.681639	+	0.731688i	$0.261267\pi$
$180$	0	0
$181$	−2338.84	−0.960469	−0.480235	−	0.877140i	$-0.659448\pi$
$181$	−2338.84	−0.960469	−0.480235	+	0.877140i	$0.659448\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2576.00	1.02374
$186$	0	0
$187$	222.243	0.0869092
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3904.00	−1.47897	−0.739486	−	0.673172i	$-0.764931\pi$
$191$	−3904.00	−1.47897	−0.739486	+	0.673172i	$0.764931\pi$
$192$	0	0
$193$	3330.00	1.24196	0.620981	−	0.783826i	$-0.286734\pi$
$193$	3330.00	1.24196	0.620981	+	0.783826i	$0.286734\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1195.88	0.432502	0.216251	−	0.976338i	$-0.430617\pi$
$197$	1195.88	0.432502	0.216251	+	0.976338i	$0.430617\pi$
$198$	0	0
$199$	1736.00	0.618401	0.309200	−	0.950997i	$-0.399939\pi$
$199$	1736.00	0.618401	0.309200	+	0.950997i	$0.399939\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1269.96	−0.439083
$204$	0	0
$205$	−740.810	−0.252392
$206$	0	0
$207$	0	0
$208$	0	0
$209$	588.000	0.194607
$210$	0	0
$211$	2915.62	0.951277	0.475638	−	0.879641i	$-0.342217\pi$
$211$	2915.62	0.951277	0.475638	+	0.879641i	$0.342217\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4648.00	−1.47438
$216$	0	0
$217$	1792.00	0.560594
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−740.810	−0.225486
$222$	0	0
$223$	1568.00	0.470857	0.235428	−	0.971892i	$-0.424351\pi$
$223$	1568.00	0.470857	0.235428	+	0.971892i	$0.424351\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1264.67	−0.369775	−0.184888	−	0.982760i	$-0.559192\pi$
$227$	−1264.67	−0.369775	−0.184888	+	0.982760i	$0.559192\pi$
$228$	0	0
$229$	5153.92	1.48725	0.743626	−	0.668595i	$-0.233104\pi$
$229$	5153.92	1.48725	0.743626	+	0.668595i	$0.233104\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−838.000	−0.235619	−0.117809	−	0.993036i	$-0.537587\pi$
$233$	−838.000	−0.235619	−0.117809	+	0.993036i	$0.537587\pi$
$234$	0	0
$235$	−3555.89	−0.987067
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6288.00	−1.70183	−0.850914	−	0.525305i	$-0.823951\pi$
$239$	−6288.00	−1.70183	−0.850914	+	0.525305i	$0.823951\pi$
$240$	0	0
$241$	−2926.00	−0.782076	−0.391038	−	0.920375i	$-0.627884\pi$
$241$	−2926.00	−0.782076	−0.391038	+	0.920375i	$0.627884\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2952.66	−0.769953
$246$	0	0
$247$	−1960.00	−0.504906
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5444.96	−1.36925	−0.684627	−	0.728894i	$-0.740035\pi$
$251$	−5444.96	−1.36925	−0.684627	+	0.728894i	$0.740035\pi$
$252$	0	0
$253$	−2412.93	−0.599602
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2562.00	−0.621841	−0.310921	−	0.950436i	$-0.600637\pi$
$257$	−2562.00	−0.621841	−0.310921	+	0.950436i	$0.600637\pi$
$258$	0	0
$259$	1947.27	0.467172
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5896.00	−1.38237	−0.691184	−	0.722679i	$-0.742911\pi$
$263$	−5896.00	−1.38237	−0.691184	+	0.722679i	$0.742911\pi$
$264$	0	0
$265$	−336.000	−0.0778880
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5365.58	−1.21615	−0.608077	−	0.793878i	$-0.708059\pi$
$269$	−5365.58	−1.21615	−0.608077	+	0.793878i	$0.708059\pi$
$270$	0	0
$271$	−1680.00	−0.376578	−0.188289	−	0.982114i	$-0.560294\pi$
$271$	−1680.00	−0.376578	−0.188289	+	0.982114i	$0.560294\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−206.369	−0.0452527
$276$	0	0
$277$	−1576.87	−0.342039	−0.171019	−	0.985268i	$-0.554706\pi$
$277$	−1576.87	−0.342039	−0.171019	+	0.985268i	$0.554706\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2742.00	−0.582114	−0.291057	−	0.956706i	$-0.594007\pi$
$281$	−2742.00	−0.582114	−0.291057	+	0.956706i	$0.594007\pi$
$282$	0	0
$283$	2989.70	0.627983	0.313991	−	0.949426i	$-0.398334\pi$
$283$	2989.70	0.627983	0.313991	+	0.949426i	$0.398334\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−560.000	−0.115177
$288$	0	0
$289$	−4717.00	−0.960106
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9238.96	1.84214	0.921068	−	0.389401i	$-0.127318\pi$
$293$	9238.96	1.84214	0.921068	+	0.389401i	$0.127318\pi$
$294$	0	0
$295$	−5656.00	−1.11629
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8043.08	1.55566
$300$	0	0
$301$	−3513.56	−0.672818
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1008.00	−0.189239
$306$	0	0
$307$	−2587.54	−0.481039	−0.240520	−	0.970644i	$-0.577318\pi$
$307$	−2587.54	−0.481039	−0.240520	+	0.970644i	$0.577318\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2744.00	−0.500315	−0.250157	−	0.968205i	$-0.580482\pi$
$311$	−2744.00	−0.500315	−0.250157	+	0.968205i	$0.580482\pi$
$312$	0	0
$313$	−2282.00	−0.412097	−0.206048	−	0.978542i	$-0.566060\pi$
$313$	−2282.00	−0.412097	−0.206048	+	0.978542i	$0.566060\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9577.62	1.69695	0.848474	−	0.529237i	$-0.177522\pi$
$317$	9577.62	1.69695	0.848474	+	0.529237i	$0.177522\pi$
$318$	0	0
$319$	−2520.00	−0.442298
$320$	0	0
$321$	0	0
$322$	0	0
$323$	518.567	0.0893308
$324$	0	0
$325$	687.895	0.117408
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2688.00	−0.450438
$330$	0	0
$331$	−4249.08	−0.705590	−0.352795	−	0.935701i	$-0.614769\pi$
$331$	−4249.08	−0.705590	−0.352795	+	0.935701i	$0.614769\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1848.00	−0.301394
$336$	0	0
$337$	6130.00	0.990868	0.495434	−	0.868646i	$-0.335009\pi$
$337$	6130.00	0.990868	0.495434	+	0.868646i	$0.335009\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3555.89	0.564699
$342$	0	0
$343$	−4976.00	−0.783320
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2481.71	−0.383935	−0.191967	−	0.981401i	$-0.561487\pi$
$347$	−2481.71	−0.383935	−0.191967	+	0.981401i	$0.561487\pi$
$348$	0	0
$349$	328.073	0.0503191	0.0251595	−	0.999683i	$-0.491991\pi$
$349$	328.073	0.0503191	0.0251595	+	0.999683i	$0.491991\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10206.0	1.53884	0.769420	−	0.638743i	$-0.220545\pi$
$353$	10206.0	1.53884	0.769420	+	0.638743i	$0.220545\pi$
$354$	0	0
$355$	−761.976	−0.113920
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3176.00	−0.466916	−0.233458	−	0.972367i	$-0.575004\pi$
$359$	−3176.00	−0.466916	−0.233458	+	0.972367i	$0.575004\pi$
$360$	0	0
$361$	−5487.00	−0.799971
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3111.40	0.446187
$366$	0	0
$367$	−11760.0	−1.67266	−0.836331	−	0.548225i	$-0.815304\pi$
$367$	−11760.0	−1.67266	−0.836331	+	0.548225i	$0.815304\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−253.992	−0.0355434
$372$	0	0
$373$	−10974.6	−1.52344	−0.761719	−	0.647908i	$-0.775644\pi$
$373$	−10974.6	−1.52344	−0.761719	+	0.647908i	$0.775644\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8400.00	1.14754
$378$	0	0
$379$	3074.36	0.416674	0.208337	−	0.978057i	$-0.433195\pi$
$379$	3074.36	0.416674	0.208337	+	0.978057i	$0.433195\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2688.00	−0.358617	−0.179309	−	0.983793i	$-0.557386\pi$
$383$	−2688.00	−0.358617	−0.179309	+	0.983793i	$0.557386\pi$
$384$	0	0
$385$	1344.00	0.177913
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10487.8	−1.36697	−0.683484	−	0.729966i	$-0.739536\pi$
$389$	−10487.8	−1.36697	−0.683484	+	0.729966i	$0.739536\pi$
$390$	0	0
$391$	−2128.00	−0.275237
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4910.51	−0.625506
$396$	0	0
$397$	5704.24	0.721127	0.360564	−	0.932735i	$-0.382584\pi$
$397$	5704.24	0.721127	0.360564	+	0.932735i	$0.382584\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12402.0	−1.54445	−0.772227	−	0.635346i	$-0.780857\pi$
$401$	−12402.0	−1.54445	−0.772227	+	0.635346i	$0.780857\pi$
$402$	0	0
$403$	−11853.0	−1.46511
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3864.00	0.470593
$408$	0	0
$409$	12278.0	1.48437	0.742186	−	0.670194i	$-0.233789\pi$
$409$	12278.0	1.48437	0.742186	+	0.670194i	$0.233789\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4275.53	−0.509407
$414$	0	0
$415$	−5768.00	−0.682265
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−8207.12	−0.956907	−0.478454	−	0.878113i	$-0.658802\pi$
$419$	−8207.12	−0.956907	−0.478454	+	0.878113i	$0.658802\pi$
$420$	0	0
$421$	−1449.87	−0.167844	−0.0839221	−	0.996472i	$-0.526745\pi$
$421$	−1449.87	−0.167844	−0.0839221	+	0.996472i	$0.526745\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−182.000	−0.0207725
$426$	0	0
$427$	−761.976	−0.0863574
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7632.00	−0.852948	−0.426474	−	0.904500i	$-0.640244\pi$
$431$	−7632.00	−0.852948	−0.426474	+	0.904500i	$0.640244\pi$
$432$	0	0
$433$	3794.00	0.421081	0.210540	−	0.977585i	$-0.432478\pi$
$433$	3794.00	0.421081	0.210540	+	0.977585i	$0.432478\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5630.16	−0.616309
$438$	0	0
$439$	1848.00	0.200912	0.100456	−	0.994942i	$-0.467970\pi$
$439$	1848.00	0.200912	0.100456	+	0.994942i	$0.467970\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12334.5	1.32287	0.661433	−	0.750004i	$-0.269949\pi$
$443$	12334.5	1.32287	0.661433	+	0.750004i	$0.269949\pi$
$444$	0	0
$445$	2815.08	0.299882
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3582.00	0.376492	0.188246	−	0.982122i	$-0.439720\pi$
$449$	3582.00	0.376492	0.188246	+	0.982122i	$0.439720\pi$
$450$	0	0
$451$	−1111.22	−0.116020
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4480.00	−0.461595
$456$	0	0
$457$	−2714.00	−0.277802	−0.138901	−	0.990306i	$-0.544357\pi$
$457$	−2714.00	−0.277802	−0.138901	+	0.990306i	$0.544357\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8349.99	0.843596	0.421798	−	0.906690i	$-0.361399\pi$
$461$	8349.99	0.843596	0.421798	+	0.906690i	$0.361399\pi$
$462$	0	0
$463$	2224.00	0.223236	0.111618	−	0.993751i	$-0.464397\pi$
$463$	2224.00	0.223236	0.111618	+	0.993751i	$0.464397\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10292.0	1.01982	0.509910	−	0.860228i	$-0.329679\pi$
$467$	10292.0	1.01982	0.509910	+	0.860228i	$0.329679\pi$
$468$	0	0
$469$	−1396.96	−0.137538
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6972.00	−0.677744
$474$	0	0
$475$	−481.527	−0.0465136
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17696.0	−1.68800	−0.843999	−	0.536345i	$-0.819805\pi$
$479$	−17696.0	−1.68800	−0.843999	+	0.536345i	$0.819805\pi$
$480$	0	0
$481$	−12880.0	−1.22095
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10519.5	0.984879
$486$	0	0
$487$	−1304.00	−0.121334	−0.0606672	−	0.998158i	$-0.519323\pi$
$487$	−1304.00	−0.121334	−0.0606672	+	0.998158i	$0.519323\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−16662.9	−1.53154	−0.765772	−	0.643112i	$-0.777643\pi$
$491$	−16662.9	−1.53154	−0.765772	+	0.643112i	$0.777643\pi$
$492$	0	0
$493$	−2222.43	−0.203029
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−576.000	−0.0519862
$498$	0	0
$499$	−3095.53	−0.277705	−0.138853	−	0.990313i	$-0.544341\pi$
$499$	−3095.53	−0.277705	−0.138853	+	0.990313i	$0.544341\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−19320.0	−1.71260	−0.856298	−	0.516481i	$-0.827242\pi$
$503$	−19320.0	−1.71260	−0.856298	+	0.516481i	$0.827242\pi$
$504$	0	0
$505$	−7952.00	−0.700712
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−4476.61	−0.389828	−0.194914	−	0.980820i	$-0.562443\pi$
$509$	−4476.61	−0.389828	−0.194914	+	0.980820i	$0.562443\pi$
$510$	0	0
$511$	2352.00	0.203613
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−12445.6	−1.06489
$516$	0	0
$517$	−5333.83	−0.453737
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2982.00	−0.250756	−0.125378	−	0.992109i	$-0.540014\pi$
$521$	−2982.00	−0.250756	−0.125378	+	0.992109i	$0.540014\pi$
$522$	0	0
$523$	2016.06	0.168559	0.0842794	−	0.996442i	$-0.473141\pi$
$523$	2016.06	0.168559	0.0842794	+	0.996442i	$0.473141\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3136.00	0.259215
$528$	0	0
$529$	10937.0	0.898907
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3704.05	0.301014
$534$	0	0
$535$	2856.00	0.230796
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4428.99	−0.353933
$540$	0	0
$541$	−15419.4	−1.22539	−0.612693	−	0.790321i	$-0.709914\pi$
$541$	−15419.4	−1.22539	−0.612693	+	0.790321i	$0.709914\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	20048.0	1.57571
$546$	0	0
$547$	12609.7	0.985649	0.492824	−	0.870129i	$-0.335965\pi$
$547$	12609.7	0.985649	0.492824	+	0.870129i	$0.335965\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5880.00	−0.454621
$552$	0	0
$553$	−3712.00	−0.285444
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7143.53	−0.543413	−0.271706	−	0.962380i	$-0.587588\pi$
$557$	−7143.53	−0.543413	−0.271706	+	0.962380i	$0.587588\pi$
$558$	0	0
$559$	23240.0	1.75840
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7572.14	−0.566834	−0.283417	−	0.958997i	$-0.591468\pi$
$563$	−7572.14	−0.566834	−0.283417	+	0.958997i	$0.591468\pi$
$564$	0	0
$565$	18096.9	1.34751
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15594.0	1.14892	0.574459	−	0.818533i	$-0.305212\pi$
$569$	15594.0	1.14892	0.574459	+	0.818533i	$0.305212\pi$
$570$	0	0
$571$	−16737.0	−1.22666	−0.613330	−	0.789827i	$-0.710170\pi$
$571$	−16737.0	−1.22666	−0.613330	+	0.789827i	$0.710170\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1976.00	0.143313
$576$	0	0
$577$	6594.00	0.475757	0.237879	−	0.971295i	$-0.423548\pi$
$577$	6594.00	0.475757	0.237879	+	0.971295i	$0.423548\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4360.20	−0.311345
$582$	0	0
$583$	−504.000	−0.0358037
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23213.8	1.63226	0.816130	−	0.577868i	$-0.196115\pi$
$587$	23213.8	1.63226	0.816130	+	0.577868i	$0.196115\pi$
$588$	0	0
$589$	8297.08	0.580433
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14322.0	−0.991794	−0.495897	−	0.868381i	$-0.665161\pi$
$593$	−14322.0	−0.991794	−0.495897	+	0.868381i	$0.665161\pi$
$594$	0	0
$595$	1185.30	0.0816679
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−16088.0	−1.09739	−0.548696	−	0.836022i	$-0.684876\pi$
$599$	−16088.0	−1.09739	−0.548696	+	0.836022i	$0.684876\pi$
$600$	0	0
$601$	21238.0	1.44146	0.720729	−	0.693217i	$-0.243807\pi$
$601$	21238.0	1.44146	0.720729	+	0.693217i	$0.243807\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−11419.1	−0.767357
$606$	0	0
$607$	−13664.0	−0.913681	−0.456841	−	0.889549i	$-0.651019\pi$
$607$	−13664.0	−0.913681	−0.456841	+	0.889549i	$0.651019\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	17779.4	1.17722
$612$	0	0
$613$	20393.5	1.34369	0.671846	−	0.740690i	$-0.265501\pi$
$613$	20393.5	1.34369	0.671846	+	0.740690i	$0.265501\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3782.00	−0.246771	−0.123385	−	0.992359i	$-0.539375\pi$
$617$	−3782.00	−0.246771	−0.123385	+	0.992359i	$0.539375\pi$
$618$	0	0
$619$	5825.94	0.378295	0.189147	−	0.981949i	$-0.439428\pi$
$619$	5825.94	0.378295	0.189147	+	0.981949i	$0.439428\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2128.00	0.136848
$624$	0	0
$625$	−13831.0	−0.885184
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3407.73	0.216017
$630$	0	0
$631$	−2056.00	−0.129712	−0.0648558	−	0.997895i	$-0.520659\pi$
$631$	−2056.00	−0.129712	−0.0648558	+	0.997895i	$0.520659\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−17610.1	−1.10053
$636$	0	0
$637$	14763.3	0.918278
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−11842.0	−0.729689	−0.364845	−	0.931068i	$-0.618878\pi$
$641$	−11842.0	−0.729689	−0.364845	+	0.931068i	$0.618878\pi$
$642$	0	0
$643$	16250.2	0.996649	0.498325	−	0.866991i	$-0.333949\pi$
$643$	16250.2	0.996649	0.498325	+	0.866991i	$0.333949\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19320.0	1.17395	0.586976	−	0.809604i	$-0.300318\pi$
$647$	19320.0	1.17395	0.586976	+	0.809604i	$0.300318\pi$
$648$	0	0
$649$	−8484.00	−0.513137
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2317.68	−0.138894	−0.0694470	−	0.997586i	$-0.522123\pi$
$653$	−2317.68	−0.138894	−0.0694470	+	0.997586i	$0.522123\pi$
$654$	0	0
$655$	−7112.00	−0.424258
$656$	0	0
$657$	0	0
$658$	0	0
$659$	27732.8	1.63932	0.819662	−	0.572847i	$-0.194161\pi$
$659$	27732.8	1.63932	0.819662	+	0.572847i	$0.194161\pi$
$660$	0	0
$661$	22467.7	1.32208	0.661039	−	0.750352i	$-0.270116\pi$
$661$	22467.7	1.32208	0.661039	+	0.750352i	$0.270116\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3136.00	0.182870
$666$	0	0
$667$	24129.3	1.40073
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1512.00	−0.0869897
$672$	0	0
$673$	−10078.0	−0.577234	−0.288617	−	0.957445i	$-0.593195\pi$
$673$	−10078.0	−0.577234	−0.288617	+	0.957445i	$0.593195\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−16160.2	−0.917413	−0.458707	−	0.888588i	$-0.651687\pi$
$677$	−16160.2	−0.917413	−0.458707	+	0.888588i	$0.651687\pi$
$678$	0	0
$679$	7952.00	0.449440
$680$	0	0
$681$	0	0
$682$	0	0
$683$	16356.0	0.916320	0.458160	−	0.888870i	$-0.348509\pi$
$683$	16356.0	0.916320	0.458160	+	0.888870i	$0.348509\pi$
$684$	0	0
$685$	−11239.2	−0.626899
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1680.00	0.0928925
$690$	0	0
$691$	29246.1	1.61009	0.805047	−	0.593211i	$-0.202140\pi$
$691$	29246.1	1.61009	0.805047	+	0.593211i	$0.202140\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	28504.0	1.55571
$696$	0	0
$697$	−980.000	−0.0532570
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2465.84	0.132858	0.0664290	−	0.997791i	$-0.478839\pi$
$701$	2465.84	0.132858	0.0664290	+	0.997791i	$0.478839\pi$
$702$	0	0
$703$	9016.00	0.483705
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6011.15	−0.319763
$708$	0	0
$709$	−31674.9	−1.67782	−0.838912	−	0.544267i	$-0.816808\pi$
$709$	−31674.9	−1.67782	−0.838912	+	0.544267i	$0.816808\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−34048.0	−1.78837
$714$	0	0
$715$	−8889.72	−0.464975
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9296.00	0.482173	0.241086	−	0.970504i	$-0.422496\pi$
$719$	9296.00	0.482173	0.241086	+	0.970504i	$0.422496\pi$
$720$	0	0
$721$	−9408.00	−0.485953
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2063.69	0.105715
$726$	0	0
$727$	−21672.0	−1.10560	−0.552799	−	0.833315i	$-0.686440\pi$
$727$	−21672.0	−1.10560	−0.552799	+	0.833315i	$0.686440\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−6148.73	−0.311106
$732$	0	0
$733$	9471.79	0.477283	0.238642	−	0.971108i	$-0.423298\pi$
$733$	9471.79	0.477283	0.238642	+	0.971108i	$0.423298\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2772.00	−0.138545
$738$	0	0
$739$	−6863.08	−0.341627	−0.170814	−	0.985303i	$-0.554640\pi$
$739$	−6863.08	−0.341627	−0.170814	+	0.985303i	$0.554640\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17432.0	0.860724	0.430362	−	0.902656i	$-0.358386\pi$
$743$	17432.0	0.860724	0.430362	+	0.902656i	$0.358386\pi$
$744$	0	0
$745$	−8400.00	−0.413090
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2158.93	0.105321
$750$	0	0
$751$	−11632.0	−0.565190	−0.282595	−	0.959239i	$-0.591195\pi$
$751$	−11632.0	−0.565190	−0.282595	+	0.959239i	$0.591195\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−7873.76	−0.379543
$756$	0	0
$757$	−16731.7	−0.803336	−0.401668	−	0.915785i	$-0.631569\pi$
$757$	−16731.7	−0.803336	−0.401668	+	0.915785i	$0.631569\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	39466.0	1.87995	0.939975	−	0.341244i	$-0.110848\pi$
$761$	39466.0	1.87995	0.939975	+	0.341244i	$0.110848\pi$
$762$	0	0
$763$	15154.9	0.719060
$764$	0	0
$765$	0	0
$766$	0	0
$767$	28280.0	1.33133
$768$	0	0
$769$	35266.0	1.65374	0.826869	−	0.562395i	$-0.190120\pi$
$769$	35266.0	1.65374	0.826869	+	0.562395i	$0.190120\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−16244.9	−0.755872	−0.377936	−	0.925832i	$-0.623366\pi$
$773$	−16244.9	−0.755872	−0.377936	+	0.925832i	$0.623366\pi$
$774$	0	0
$775$	−2912.00	−0.134970
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2592.84	−0.119253
$780$	0	0
$781$	−1142.96	−0.0523668
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1904.00	−0.0865690
$786$	0	0
$787$	−34844.5	−1.57824	−0.789119	−	0.614240i	$-0.789463\pi$
$787$	−34844.5	−1.57824	−0.789119	+	0.614240i	$0.789463\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	13680.0	0.614924
$792$	0	0
$793$	5040.00	0.225694
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2550.50	0.113354	0.0566772	−	0.998393i	$-0.481949\pi$
$797$	2550.50	0.113354	0.0566772	+	0.998393i	$0.481949\pi$
$798$	0	0
$799$	−4704.00	−0.208280
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4667.11	0.205104
$804$	0	0
$805$	−12868.9	−0.563441
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−24390.0	−1.05996	−0.529979	−	0.848010i	$-0.677800\pi$
$809$	−24390.0	−1.05996	−0.529979	+	0.848010i	$0.677800\pi$
$810$	0	0
$811$	−9582.91	−0.414922	−0.207461	−	0.978243i	$-0.566520\pi$
$811$	−9582.91	−0.414922	−0.207461	+	0.978243i	$0.566520\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	18760.0	0.806300
$816$	0	0
$817$	−16268.0	−0.696628
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8773.31	0.372948	0.186474	−	0.982460i	$-0.440294\pi$
$821$	8773.31	0.372948	0.186474	+	0.982460i	$0.440294\pi$
$822$	0	0
$823$	21688.0	0.918586	0.459293	−	0.888285i	$-0.348103\pi$
$823$	21688.0	0.918586	0.459293	+	0.888285i	$0.348103\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19446.3	0.817670	0.408835	−	0.912608i	$-0.365935\pi$
$827$	19446.3	0.817670	0.408835	+	0.912608i	$0.365935\pi$
$828$	0	0
$829$	19546.8	0.818925	0.409462	−	0.912327i	$-0.365716\pi$
$829$	19546.8	0.818925	0.409462	+	0.912327i	$0.365716\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3906.00	−0.162467
$834$	0	0
$835$	−20742.7	−0.859677
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−18760.0	−0.771951	−0.385976	−	0.922509i	$-0.626135\pi$
$839$	−18760.0	−0.771951	−0.385976	+	0.922509i	$0.626135\pi$
$840$	0	0
$841$	811.000	0.0332527
$842$	0	0
$843$	0	0
$844$	0	0
$845$	6381.55	0.259801
$846$	0	0
$847$	−8632.00	−0.350176
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−36998.2	−1.49034
$852$	0	0
$853$	28732.9	1.15333	0.576667	−	0.816979i	$-0.304353\pi$
$853$	28732.9	1.15333	0.576667	+	0.816979i	$0.304353\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8778.00	0.349884	0.174942	−	0.984579i	$-0.444026\pi$
$857$	8778.00	0.349884	0.174942	+	0.984579i	$0.444026\pi$
$858$	0	0
$859$	−5646.03	−0.224261	−0.112130	−	0.993693i	$-0.535767\pi$
$859$	−5646.03	−0.224261	−0.112130	+	0.993693i	$0.535767\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9312.00	0.367305	0.183652	−	0.982991i	$-0.441208\pi$
$863$	9312.00	0.367305	0.183652	+	0.982991i	$0.441208\pi$
$864$	0	0
$865$	21168.0	0.832062
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−7365.77	−0.287534
$870$	0	0
$871$	9240.00	0.359455
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−11683.6	−0.451405
$876$	0	0
$877$	−137.579	−0.00529728	−0.00264864	−	0.999996i	$-0.500843\pi$
$877$	−137.579	−0.00529728	−0.00264864	+	0.999996i	$0.500843\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	31150.0	1.19123	0.595613	−	0.803272i	$-0.296909\pi$
$881$	31150.0	1.19123	0.595613	+	0.803272i	$0.296909\pi$
$882$	0	0
$883$	−12577.9	−0.479366	−0.239683	−	0.970851i	$-0.577043\pi$
$883$	−12577.9	−0.479366	−0.239683	+	0.970851i	$0.577043\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37128.0	1.40545	0.702726	−	0.711460i	$-0.251966\pi$
$887$	37128.0	1.40545	0.702726	+	0.711460i	$0.251966\pi$
$888$	0	0
$889$	−13312.0	−0.502216
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−12445.6	−0.466379
$894$	0	0
$895$	34552.0	1.29044
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−35558.9	−1.31919
$900$	0	0
$901$	−444.486	−0.0164351
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−24752.0	−0.909154
$906$	0	0
$907$	35204.4	1.28880	0.644400	−	0.764688i	$-0.277107\pi$
$907$	35204.4	1.28880	0.644400	+	0.764688i	$0.277107\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	10512.0	0.382303	0.191152	−	0.981561i	$-0.438778\pi$
$911$	10512.0	0.382303	0.191152	+	0.981561i	$0.438778\pi$
$912$	0	0
$913$	−8652.00	−0.313625
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5376.17	−0.193606
$918$	0	0
$919$	46104.0	1.65488	0.827438	−	0.561557i	$-0.189798\pi$
$919$	46104.0	1.65488	0.827438	+	0.561557i	$0.189798\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3809.88	0.135865
$924$	0	0
$925$	−3164.32	−0.112478
$926$	0	0
$927$	0	0
$928$	0	0
$929$	5726.00	0.202222	0.101111	−	0.994875i	$-0.467760\pi$
$929$	5726.00	0.202222	0.101111	+	0.994875i	$0.467760\pi$
$930$	0	0
$931$	−10334.3	−0.363795
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2352.00	0.0822659
$936$	0	0
$937$	−1274.00	−0.0444181	−0.0222091	−	0.999753i	$-0.507070\pi$
$937$	−1274.00	−0.0444181	−0.0222091	+	0.999753i	$0.507070\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−26446.9	−0.916201	−0.458101	−	0.888900i	$-0.651470\pi$
$941$	−26446.9	−0.916201	−0.458101	+	0.888900i	$0.651470\pi$
$942$	0	0
$943$	10640.0	0.367430
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23922.9	0.820897	0.410448	−	0.911884i	$-0.365372\pi$
$947$	23922.9	0.820897	0.410448	+	0.911884i	$0.365372\pi$
$948$	0	0
$949$	−15557.0	−0.532141
$950$	0	0
$951$	0	0
$952$	0	0
$953$	38250.0	1.30015	0.650073	−	0.759872i	$-0.274738\pi$
$953$	38250.0	1.30015	0.650073	+	0.759872i	$0.274738\pi$
$954$	0	0
$955$	−41316.1	−1.39995
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8496.00	−0.286079
$960$	0	0
$961$	20385.0	0.684267
$962$	0	0
$963$	0	0
$964$	0	0
$965$	35241.4	1.17561
$966$	0	0
$967$	−4664.00	−0.155103	−0.0775513	−	0.996988i	$-0.524710\pi$
$967$	−4664.00	−0.155103	−0.0775513	+	0.996988i	$0.524710\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−30971.2	−1.02360	−0.511798	−	0.859106i	$-0.671020\pi$
$971$	−30971.2	−1.02360	−0.511798	+	0.859106i	$0.671020\pi$
$972$	0	0
$973$	21547.0	0.709933
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4814.00	0.157639	0.0788196	−	0.996889i	$-0.474885\pi$
$977$	4814.00	0.157639	0.0788196	+	0.996889i	$0.474885\pi$
$978$	0	0
$979$	4222.62	0.137850
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−12376.0	−0.401560	−0.200780	−	0.979636i	$-0.564348\pi$
$983$	−12376.0	−0.401560	−0.200780	+	0.979636i	$0.564348\pi$
$984$	0	0
$985$	12656.0	0.409395
$986$	0	0
$987$	0	0
$988$	0	0
$989$	66757.6	2.14638
$990$	0	0
$991$	45344.0	1.45348	0.726740	−	0.686912i	$-0.241034\pi$
$991$	45344.0	1.45348	0.726740	+	0.686912i	$0.241034\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	18372.1	0.585361
$996$	0	0
$997$	−26002.4	−0.825984	−0.412992	−	0.910735i	$-0.635516\pi$
$997$	−26002.4	−0.825984	−0.412992	+	0.910735i	$0.635516\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.4.a.bn.1.2		2
3.2	odd	2		256.4.a.l.1.1		2
4.3	odd	2		2304.4.a.v.1.2		2
8.3	odd	2		2304.4.a.v.1.1		2
8.5	even	2	inner	2304.4.a.bn.1.1		2
12.11	even	2		256.4.a.j.1.2		2
16.3	odd	4		288.4.d.a.145.1		2
16.5	even	4		72.4.d.b.37.2		2
16.11	odd	4		288.4.d.a.145.2		2
16.13	even	4		72.4.d.b.37.1		2
24.5	odd	2		256.4.a.l.1.2		2
24.11	even	2		256.4.a.j.1.1		2
48.5	odd	4		8.4.b.a.5.1	✓	2
48.11	even	4		32.4.b.a.17.1		2
48.29	odd	4		8.4.b.a.5.2	yes	2
48.35	even	4		32.4.b.a.17.2		2
240.29	odd	4		200.4.d.a.101.1		2
240.53	even	4		200.4.f.a.149.2		4
240.59	even	4		800.4.d.a.401.2		2
240.77	even	4		200.4.f.a.149.1		4
240.83	odd	4		800.4.f.a.49.1		4
240.107	odd	4		800.4.f.a.49.2		4
240.149	odd	4		200.4.d.a.101.2		2
240.173	even	4		200.4.f.a.149.4		4
240.179	even	4		800.4.d.a.401.1		2
240.197	even	4		200.4.f.a.149.3		4
240.203	odd	4		800.4.f.a.49.3		4
240.227	odd	4		800.4.f.a.49.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.4.b.a.5.1	✓	2	48.5	odd	4
8.4.b.a.5.2	yes	2	48.29	odd	4
32.4.b.a.17.1		2	48.11	even	4
32.4.b.a.17.2		2	48.35	even	4
72.4.d.b.37.1		2	16.13	even	4
72.4.d.b.37.2		2	16.5	even	4
200.4.d.a.101.1		2	240.29	odd	4
200.4.d.a.101.2		2	240.149	odd	4
200.4.f.a.149.1		4	240.77	even	4
200.4.f.a.149.2		4	240.53	even	4
200.4.f.a.149.3		4	240.197	even	4
200.4.f.a.149.4		4	240.173	even	4
256.4.a.j.1.1		2	24.11	even	2
256.4.a.j.1.2		2	12.11	even	2
256.4.a.l.1.1		2	3.2	odd	2
256.4.a.l.1.2		2	24.5	odd	2
288.4.d.a.145.1		2	16.3	odd	4
288.4.d.a.145.2		2	16.11	odd	4
800.4.d.a.401.1		2	240.179	even	4
800.4.d.a.401.2		2	240.59	even	4
800.4.f.a.49.1		4	240.83	odd	4
800.4.f.a.49.2		4	240.107	odd	4
800.4.f.a.49.3		4	240.203	odd	4
800.4.f.a.49.4		4	240.227	odd	4
2304.4.a.v.1.1		2	8.3	odd	2
2304.4.a.v.1.2		2	4.3	odd	2
2304.4.a.bn.1.1		2	8.5	even	2	inner
2304.4.a.bn.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2304.a (trivial)

\(f(q)\)	\(=\)	\(q+10.5830 q^{5} +8.00000 q^{7} +15.8745 q^{11} -52.9150 q^{13} +14.0000 q^{17} +37.0405 q^{19} -152.000 q^{23} -13.0000 q^{25} -158.745 q^{29} +224.000 q^{31} +84.6640 q^{35} +243.409 q^{37} -70.0000 q^{41} -439.195 q^{43} -336.000 q^{47} -279.000 q^{49} -31.7490 q^{53} +168.000 q^{55} -534.442 q^{59} -95.2470 q^{61} -560.000 q^{65} -174.620 q^{67} -72.0000 q^{71} +294.000 q^{73} +126.996 q^{77} -464.000 q^{79} -545.025 q^{83} +148.162 q^{85} +266.000 q^{89} -423.320 q^{91} +392.000 q^{95} +994.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{7} + 28 q^{17} - 304 q^{23} - 26 q^{25} + 448 q^{31} - 140 q^{41} - 672 q^{47} - 558 q^{49} + 336 q^{55} - 1120 q^{65} - 144 q^{71} + 588 q^{73} - 928 q^{79} + 532 q^{89} + 784 q^{95} + 1988 q^{97}+O(q^{100}) \) 2 * q + 16 * q^7 + 28 * q^17 - 304 * q^23 - 26 * q^25 + 448 * q^31 - 140 * q^41 - 672 * q^47 - 558 * q^49 + 336 * q^55 - 1120 * q^65 - 144 * q^71 + 588 * q^73 - 928 * q^79 + 532 * q^89 + 784 * q^95 + 1988 * q^97

Introduction

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