LMFDB - Embedded newform 432.3.e.c.161.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,3,Mod(161,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("432.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	432.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$11.7711474204$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	432.161
Dual form			432.3.e.c.161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{5} -5.00000 q^{7} +O(q^{10})$	$q-3.00000i q^{5} -5.00000 q^{7} +15.0000i q^{11} -10.0000 q^{13} +18.0000i q^{17} +16.0000 q^{19} +12.0000i q^{23} +16.0000 q^{25} +30.0000i q^{29} +1.00000 q^{31} +15.0000i q^{35} +20.0000 q^{37} +60.0000i q^{41} -50.0000 q^{43} +6.00000i q^{47} -24.0000 q^{49} -27.0000i q^{53} +45.0000 q^{55} +30.0000i q^{59} -76.0000 q^{61} +30.0000i q^{65} +10.0000 q^{67} +90.0000i q^{71} +65.0000 q^{73} -75.0000i q^{77} -14.0000 q^{79} -3.00000i q^{83} +54.0000 q^{85} +90.0000i q^{89} +50.0000 q^{91} -48.0000i q^{95} -85.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 10 q^{7}+O(q^{10}) $ 2 * q - 10 * q^7	$ 2 q - 10 q^{7} - 20 q^{13} + 32 q^{19} + 32 q^{25} + 2 q^{31} + 40 q^{37} - 100 q^{43} - 48 q^{49} + 90 q^{55} - 152 q^{61} + 20 q^{67} + 130 q^{73} - 28 q^{79} + 108 q^{85} + 100 q^{91} - 170 q^{97}+O(q^{100}) $ 2 * q - 10 * q^7 - 20 * q^13 + 32 * q^19 + 32 * q^25 + 2 * q^31 + 40 * q^37 - 100 * q^43 - 48 * q^49 + 90 * q^55 - 152 * q^61 + 20 * q^67 + 130 * q^73 - 28 * q^79 + 108 * q^85 + 100 * q^91 - 170 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	− 3.00000i	− 0.600000i	−0.953939	−	0.300000i	$-0.903013\pi$
$5$	− 3.00000i	− 0.600000i	0.953939	−	0.300000i	$-0.0969867\pi$
$6$	0	0
$7$	−5.00000	−0.714286	−0.357143	−	0.934050i	$-0.616249\pi$
$7$	−5.00000	−0.714286	−0.357143	+	0.934050i	$0.616249\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	15.0000i	1.36364i	0.731522	+	0.681818i	$0.238810\pi$
$11$	15.0000i	1.36364i	−0.731522	+	0.681818i	$0.761190\pi$
$12$	0	0
$13$	−10.0000	−0.769231	−0.384615	−	0.923077i	$-0.625666\pi$
$13$	−10.0000	−0.769231	−0.384615	+	0.923077i	$0.625666\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	18.0000i	1.05882i	0.848365	+	0.529412i	$0.177587\pi$
$17$	18.0000i	1.05882i	−0.848365	+	0.529412i	$0.822413\pi$
$18$	0	0
$19$	16.0000	0.842105	0.421053	−	0.907036i	$-0.361661\pi$
$19$	16.0000	0.842105	0.421053	+	0.907036i	$0.361661\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	12.0000i	0.521739i	0.965374	+	0.260870i	$0.0840093\pi$
$23$	12.0000i	0.521739i	−0.965374	+	0.260870i	$0.915991\pi$
$24$	0	0
$25$	16.0000	0.640000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	30.0000i	1.03448i	0.855840	+	0.517241i	$0.173041\pi$
$29$	30.0000i	1.03448i	−0.855840	+	0.517241i	$0.826959\pi$
$30$	0	0
$31$	1.00000	0.0322581	0.0161290	−	0.999870i	$-0.494866\pi$
$31$	1.00000	0.0322581	0.0161290	+	0.999870i	$0.494866\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	15.0000i	0.428571i
$36$	0	0
$37$	20.0000	0.540541	0.270270	−	0.962784i	$-0.412887\pi$
$37$	20.0000	0.540541	0.270270	+	0.962784i	$0.412887\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	60.0000i	1.46341i	0.681619	+	0.731707i	$0.261276\pi$
$41$	60.0000i	1.46341i	−0.681619	+	0.731707i	$0.738724\pi$
$42$	0	0
$43$	−50.0000	−1.16279	−0.581395	−	0.813621i	$-0.697493\pi$
$43$	−50.0000	−1.16279	−0.581395	+	0.813621i	$0.697493\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.00000i	0.127660i	0.997961	+	0.0638298i	$0.0203315\pi$
$47$	6.00000i	0.127660i	−0.997961	+	0.0638298i	$0.979669\pi$
$48$	0	0
$49$	−24.0000	−0.489796
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 27.0000i	− 0.509434i	−0.967016	−	0.254717i	$-0.918018\pi$
$53$	− 27.0000i	− 0.509434i	0.967016	−	0.254717i	$-0.0819823\pi$
$54$	0	0
$55$	45.0000	0.818182
$56$	0	0
$57$	0	0
$58$	0	0
$59$	30.0000i	0.508475i	0.967142	+	0.254237i	$0.0818244\pi$
$59$	30.0000i	0.508475i	−0.967142	+	0.254237i	$0.918176\pi$
$60$	0	0
$61$	−76.0000	−1.24590	−0.622951	−	0.782261i	$-0.714066\pi$
$61$	−76.0000	−1.24590	−0.622951	+	0.782261i	$0.714066\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	30.0000i	0.461538i
$66$	0	0
$67$	10.0000	0.149254	0.0746269	−	0.997212i	$-0.476223\pi$
$67$	10.0000	0.149254	0.0746269	+	0.997212i	$0.476223\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	90.0000i	1.26761i	0.773495	+	0.633803i	$0.218507\pi$
$71$	90.0000i	1.26761i	−0.773495	+	0.633803i	$0.781493\pi$
$72$	0	0
$73$	65.0000	0.890411	0.445205	−	0.895428i	$-0.353131\pi$
$73$	65.0000	0.890411	0.445205	+	0.895428i	$0.353131\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 75.0000i	− 0.974026i
$78$	0	0
$79$	−14.0000	−0.177215	−0.0886076	−	0.996067i	$-0.528242\pi$
$79$	−14.0000	−0.177215	−0.0886076	+	0.996067i	$0.528242\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 3.00000i	− 0.0361446i	−0.999837	−	0.0180723i	$-0.994247\pi$
$83$	− 3.00000i	− 0.0361446i	0.999837	−	0.0180723i	$-0.00575290\pi$
$84$	0	0
$85$	54.0000	0.635294
$86$	0	0
$87$	0	0
$88$	0	0
$89$	90.0000i	1.01124i	0.862757	+	0.505618i	$0.168735\pi$
$89$	90.0000i	1.01124i	−0.862757	+	0.505618i	$0.831265\pi$
$90$	0	0
$91$	50.0000	0.549451
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 48.0000i	− 0.505263i
$96$	0	0
$97$	−85.0000	−0.876289	−0.438144	−	0.898905i	$-0.644364\pi$
$97$	−85.0000	−0.876289	−0.438144	+	0.898905i	$0.644364\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 195.000i	− 1.93069i	−0.260971	−	0.965347i	$-0.584043\pi$
$101$	− 195.000i	− 1.93069i	0.260971	−	0.965347i	$-0.415957\pi$
$102$	0	0
$103$	−170.000	−1.65049	−0.825243	−	0.564778i	$-0.808962\pi$
$103$	−170.000	−1.65049	−0.825243	+	0.564778i	$0.808962\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 189.000i	− 1.76636i	−0.469039	−	0.883178i	$-0.655400\pi$
$107$	− 189.000i	− 1.76636i	0.469039	−	0.883178i	$-0.344600\pi$
$108$	0	0
$109$	164.000	1.50459	0.752294	−	0.658828i	$-0.228948\pi$
$109$	164.000	1.50459	0.752294	+	0.658828i	$0.228948\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	24.0000i	0.212389i	0.994345	+	0.106195i	$0.0338667\pi$
$113$	24.0000i	0.212389i	−0.994345	+	0.106195i	$0.966133\pi$
$114$	0	0
$115$	36.0000	0.313043
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 90.0000i	− 0.756303i
$120$	0	0
$121$	−104.000	−0.859504
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 123.000i	− 0.984000i
$126$	0	0
$127$	205.000	1.61417	0.807087	−	0.590433i	$-0.201043\pi$
$127$	205.000	1.61417	0.807087	+	0.590433i	$0.201043\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 15.0000i	− 0.114504i	−0.998360	−	0.0572519i	$-0.981766\pi$
$131$	− 15.0000i	− 0.114504i	0.998360	−	0.0572519i	$-0.0182338\pi$
$132$	0	0
$133$	−80.0000	−0.601504
$134$	0	0
$135$	0	0
$136$	0	0
$137$	138.000i	1.00730i	0.863908	+	0.503650i	$0.168010\pi$
$137$	138.000i	1.00730i	−0.863908	+	0.503650i	$0.831990\pi$
$138$	0	0
$139$	28.0000	0.201439	0.100719	−	0.994915i	$-0.467886\pi$
$139$	28.0000	0.201439	0.100719	+	0.994915i	$0.467886\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 150.000i	− 1.04895i
$144$	0	0
$145$	90.0000	0.620690
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 75.0000i	− 0.503356i	−0.967811	−	0.251678i	$-0.919018\pi$
$149$	− 75.0000i	− 0.503356i	0.967811	−	0.251678i	$-0.0809823\pi$
$150$	0	0
$151$	−77.0000	−0.509934	−0.254967	−	0.966950i	$-0.582065\pi$
$151$	−77.0000	−0.509934	−0.254967	+	0.966950i	$0.582065\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 3.00000i	− 0.0193548i
$156$	0	0
$157$	−100.000	−0.636943	−0.318471	−	0.947932i	$-0.603169\pi$
$157$	−100.000	−0.636943	−0.318471	+	0.947932i	$0.603169\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 60.0000i	− 0.372671i
$162$	0	0
$163$	−110.000	−0.674847	−0.337423	−	0.941353i	$-0.609555\pi$
$163$	−110.000	−0.674847	−0.337423	+	0.941353i	$0.609555\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 78.0000i	− 0.467066i	−0.972349	−	0.233533i	$-0.924971\pi$
$167$	− 78.0000i	− 0.467066i	0.972349	−	0.233533i	$-0.0750287\pi$
$168$	0	0
$169$	−69.0000	−0.408284
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 177.000i	− 1.02312i	−0.859247	−	0.511561i	$-0.829068\pi$
$173$	− 177.000i	− 1.02312i	0.859247	−	0.511561i	$-0.170932\pi$
$174$	0	0
$175$	−80.0000	−0.457143
$176$	0	0
$177$	0	0
$178$	0	0
$179$	225.000i	1.25698i	0.777816	+	0.628492i	$0.216327\pi$
$179$	225.000i	1.25698i	−0.777816	+	0.628492i	$0.783673\pi$
$180$	0	0
$181$	−16.0000	−0.0883978	−0.0441989	−	0.999023i	$-0.514074\pi$
$181$	−16.0000	−0.0883978	−0.0441989	+	0.999023i	$0.514074\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 60.0000i	− 0.324324i
$186$	0	0
$187$	−270.000	−1.44385
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 30.0000i	− 0.157068i	−0.996911	−	0.0785340i	$-0.974976\pi$
$191$	− 30.0000i	− 0.157068i	0.996911	−	0.0785340i	$-0.0250239\pi$
$192$	0	0
$193$	215.000	1.11399	0.556995	−	0.830516i	$-0.311954\pi$
$193$	215.000	1.11399	0.556995	+	0.830516i	$0.311954\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 207.000i	− 1.05076i	−0.850867	−	0.525381i	$-0.823923\pi$
$197$	− 207.000i	− 1.05076i	0.850867	−	0.525381i	$-0.176077\pi$
$198$	0	0
$199$	223.000	1.12060	0.560302	−	0.828289i	$-0.310685\pi$
$199$	223.000	1.12060	0.560302	+	0.828289i	$0.310685\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 150.000i	− 0.738916i
$204$	0	0
$205$	180.000	0.878049
$206$	0	0
$207$	0	0
$208$	0	0
$209$	240.000i	1.14833i
$210$	0	0
$211$	316.000	1.49763	0.748815	−	0.662779i	$-0.230623\pi$
$211$	316.000	1.49763	0.748815	+	0.662779i	$0.230623\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	150.000i	0.697674i
$216$	0	0
$217$	−5.00000	−0.0230415
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 180.000i	− 0.814480i
$222$	0	0
$223$	130.000	0.582960	0.291480	−	0.956577i	$-0.405852\pi$
$223$	130.000	0.582960	0.291480	+	0.956577i	$0.405852\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	42.0000i	0.185022i	0.995712	+	0.0925110i	$0.0294893\pi$
$227$	42.0000i	0.185022i	−0.995712	+	0.0925110i	$0.970511\pi$
$228$	0	0
$229$	−226.000	−0.986900	−0.493450	−	0.869774i	$-0.664264\pi$
$229$	−226.000	−0.986900	−0.493450	+	0.869774i	$0.664264\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	234.000i	1.00429i	0.864783	+	0.502146i	$0.167456\pi$
$233$	234.000i	1.00429i	−0.864783	+	0.502146i	$0.832544\pi$
$234$	0	0
$235$	18.0000	0.0765957
$236$	0	0
$237$	0	0
$238$	0	0
$239$	120.000i	0.502092i	0.967975	+	0.251046i	$0.0807746\pi$
$239$	120.000i	0.502092i	−0.967975	+	0.251046i	$0.919225\pi$
$240$	0	0
$241$	14.0000	0.0580913	0.0290456	−	0.999578i	$-0.490753\pi$
$241$	14.0000	0.0580913	0.0290456	+	0.999578i	$0.490753\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	72.0000i	0.293878i
$246$	0	0
$247$	−160.000	−0.647773
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 90.0000i	− 0.358566i	−0.983798	−	0.179283i	$-0.942622\pi$
$251$	− 90.0000i	− 0.358566i	0.983798	−	0.179283i	$-0.0573777\pi$
$252$	0	0
$253$	−180.000	−0.711462
$254$	0	0
$255$	0	0
$256$	0	0
$257$	438.000i	1.70428i	0.523314	+	0.852140i	$0.324696\pi$
$257$	438.000i	1.70428i	−0.523314	+	0.852140i	$0.675304\pi$
$258$	0	0
$259$	−100.000	−0.386100
$260$	0	0
$261$	0	0
$262$	0	0
$263$	276.000i	1.04943i	0.851278	+	0.524715i	$0.175828\pi$
$263$	276.000i	1.04943i	−0.851278	+	0.524715i	$0.824172\pi$
$264$	0	0
$265$	−81.0000	−0.305660
$266$	0	0
$267$	0	0
$268$	0	0
$269$	270.000i	1.00372i	0.864950	+	0.501859i	$0.167350\pi$
$269$	270.000i	1.00372i	−0.864950	+	0.501859i	$0.832650\pi$
$270$	0	0
$271$	−299.000	−1.10332	−0.551661	−	0.834069i	$-0.686006\pi$
$271$	−299.000	−1.10332	−0.551661	+	0.834069i	$0.686006\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	240.000i	0.872727i
$276$	0	0
$277$	140.000	0.505415	0.252708	−	0.967543i	$-0.418679\pi$
$277$	140.000	0.505415	0.252708	+	0.967543i	$0.418679\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 150.000i	− 0.533808i	−0.963723	−	0.266904i	$-0.913999\pi$
$281$	− 150.000i	− 0.533808i	0.963723	−	0.266904i	$-0.0860006\pi$
$282$	0	0
$283$	280.000	0.989399	0.494700	−	0.869064i	$-0.335278\pi$
$283$	280.000	0.989399	0.494700	+	0.869064i	$0.335278\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 300.000i	− 1.04530i
$288$	0	0
$289$	−35.0000	−0.121107
$290$	0	0
$291$	0	0
$292$	0	0
$293$	258.000i	0.880546i	0.897864	+	0.440273i	$0.145118\pi$
$293$	258.000i	0.880546i	−0.897864	+	0.440273i	$0.854882\pi$
$294$	0	0
$295$	90.0000	0.305085
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 120.000i	− 0.401338i
$300$	0	0
$301$	250.000	0.830565
$302$	0	0
$303$	0	0
$304$	0	0
$305$	228.000i	0.747541i
$306$	0	0
$307$	−290.000	−0.944625	−0.472313	−	0.881431i	$-0.656581\pi$
$307$	−290.000	−0.944625	−0.472313	+	0.881431i	$0.656581\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	480.000i	1.54341i	0.635982	+	0.771704i	$0.280595\pi$
$311$	480.000i	1.54341i	−0.635982	+	0.771704i	$0.719405\pi$
$312$	0	0
$313$	185.000	0.591054	0.295527	−	0.955334i	$-0.404505\pi$
$313$	185.000	0.591054	0.295527	+	0.955334i	$0.404505\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	183.000i	0.577287i	0.957437	+	0.288644i	$0.0932042\pi$
$317$	183.000i	0.577287i	−0.957437	+	0.288644i	$0.906796\pi$
$318$	0	0
$319$	−450.000	−1.41066
$320$	0	0
$321$	0	0
$322$	0	0
$323$	288.000i	0.891641i
$324$	0	0
$325$	−160.000	−0.492308
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 30.0000i	− 0.0911854i
$330$	0	0
$331$	238.000	0.719033	0.359517	−	0.933139i	$-0.382942\pi$
$331$	238.000	0.719033	0.359517	+	0.933139i	$0.382942\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 30.0000i	− 0.0895522i
$336$	0	0
$337$	−10.0000	−0.0296736	−0.0148368	−	0.999890i	$-0.504723\pi$
$337$	−10.0000	−0.0296736	−0.0148368	+	0.999890i	$0.504723\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	15.0000i	0.0439883i
$342$	0	0
$343$	365.000	1.06414
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 69.0000i	− 0.198847i	−0.995045	−	0.0994236i	$-0.968300\pi$
$347$	− 69.0000i	− 0.198847i	0.995045	−	0.0994236i	$-0.0316999\pi$
$348$	0	0
$349$	−256.000	−0.733524	−0.366762	−	0.930315i	$-0.619534\pi$
$349$	−256.000	−0.733524	−0.366762	+	0.930315i	$0.619534\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 456.000i	− 1.29178i	−0.763428	−	0.645892i	$-0.776485\pi$
$353$	− 456.000i	− 1.29178i	0.763428	−	0.645892i	$-0.223515\pi$
$354$	0	0
$355$	270.000	0.760563
$356$	0	0
$357$	0	0
$358$	0	0
$359$	450.000i	1.25348i	0.779228	+	0.626741i	$0.215612\pi$
$359$	450.000i	1.25348i	−0.779228	+	0.626741i	$0.784388\pi$
$360$	0	0
$361$	−105.000	−0.290859
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 195.000i	− 0.534247i
$366$	0	0
$367$	625.000	1.70300	0.851499	−	0.524357i	$-0.175694\pi$
$367$	625.000	1.70300	0.851499	+	0.524357i	$0.175694\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	135.000i	0.363881i
$372$	0	0
$373$	170.000	0.455764	0.227882	−	0.973689i	$-0.426820\pi$
$373$	170.000	0.455764	0.227882	+	0.973689i	$0.426820\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 300.000i	− 0.795756i
$378$	0	0
$379$	−704.000	−1.85752	−0.928760	−	0.370682i	$-0.879124\pi$
$379$	−704.000	−1.85752	−0.928760	+	0.370682i	$0.879124\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 618.000i	− 1.61358i	−0.590840	−	0.806789i	$-0.701204\pi$
$383$	− 618.000i	− 1.61358i	0.590840	−	0.806789i	$-0.298796\pi$
$384$	0	0
$385$	−225.000	−0.584416
$386$	0	0
$387$	0	0
$388$	0	0
$389$	525.000i	1.34961i	0.737994	+	0.674807i	$0.235773\pi$
$389$	525.000i	1.34961i	−0.737994	+	0.674807i	$0.764227\pi$
$390$	0	0
$391$	−216.000	−0.552430
$392$	0	0
$393$	0	0
$394$	0	0
$395$	42.0000i	0.106329i
$396$	0	0
$397$	−70.0000	−0.176322	−0.0881612	−	0.996106i	$-0.528099\pi$
$397$	−70.0000	−0.176322	−0.0881612	+	0.996106i	$0.528099\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 120.000i	− 0.299252i	−0.988743	−	0.149626i	$-0.952193\pi$
$401$	− 120.000i	− 0.299252i	0.988743	−	0.149626i	$-0.0478069\pi$
$402$	0	0
$403$	−10.0000	−0.0248139
$404$	0	0
$405$	0	0
$406$	0	0
$407$	300.000i	0.737101i
$408$	0	0
$409$	269.000	0.657702	0.328851	−	0.944382i	$-0.393339\pi$
$409$	269.000	0.657702	0.328851	+	0.944382i	$0.393339\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 150.000i	− 0.363196i
$414$	0	0
$415$	−9.00000	−0.0216867
$416$	0	0
$417$	0	0
$418$	0	0
$419$	210.000i	0.501193i	0.968092	+	0.250597i	$0.0806268\pi$
$419$	210.000i	0.501193i	−0.968092	+	0.250597i	$0.919373\pi$
$420$	0	0
$421$	644.000	1.52969	0.764846	−	0.644214i	$-0.222815\pi$
$421$	644.000	1.52969	0.764846	+	0.644214i	$0.222815\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	288.000i	0.677647i
$426$	0	0
$427$	380.000	0.889930
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 270.000i	− 0.626450i	−0.949679	−	0.313225i	$-0.898591\pi$
$431$	− 270.000i	− 0.626450i	0.949679	−	0.313225i	$-0.101409\pi$
$432$	0	0
$433$	−565.000	−1.30485	−0.652425	−	0.757853i	$-0.726248\pi$
$433$	−565.000	−1.30485	−0.652425	+	0.757853i	$0.726248\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	192.000i	0.439359i
$438$	0	0
$439$	211.000	0.480638	0.240319	−	0.970694i	$-0.422748\pi$
$439$	211.000	0.480638	0.240319	+	0.970694i	$0.422748\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 498.000i	− 1.12415i	−0.827085	−	0.562077i	$-0.810003\pi$
$443$	− 498.000i	− 1.12415i	0.827085	−	0.562077i	$-0.189997\pi$
$444$	0	0
$445$	270.000	0.606742
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 360.000i	− 0.801782i	−0.916126	−	0.400891i	$-0.868701\pi$
$449$	− 360.000i	− 0.801782i	0.916126	−	0.400891i	$-0.131299\pi$
$450$	0	0
$451$	−900.000	−1.99557
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 150.000i	− 0.329670i
$456$	0	0
$457$	365.000	0.798687	0.399344	−	0.916801i	$-0.369238\pi$
$457$	365.000	0.798687	0.399344	+	0.916801i	$0.369238\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 105.000i	− 0.227766i	−0.993494	−	0.113883i	$-0.963671\pi$
$461$	− 105.000i	− 0.227766i	0.993494	−	0.113883i	$-0.0363289\pi$
$462$	0	0
$463$	−215.000	−0.464363	−0.232181	−	0.972672i	$-0.574586\pi$
$463$	−215.000	−0.464363	−0.232181	+	0.972672i	$0.574586\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 63.0000i	− 0.134904i	−0.997723	−	0.0674518i	$-0.978513\pi$
$467$	− 63.0000i	− 0.134904i	0.997723	−	0.0674518i	$-0.0214869\pi$
$468$	0	0
$469$	−50.0000	−0.106610
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 750.000i	− 1.58562i
$474$	0	0
$475$	256.000	0.538947
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 750.000i	− 1.56576i	−0.622171	−	0.782881i	$-0.713749\pi$
$479$	− 750.000i	− 1.56576i	0.622171	−	0.782881i	$-0.286251\pi$
$480$	0	0
$481$	−200.000	−0.415800
$482$	0	0
$483$	0	0
$484$	0	0
$485$	255.000i	0.525773i
$486$	0	0
$487$	−110.000	−0.225873	−0.112936	−	0.993602i	$-0.536026\pi$
$487$	−110.000	−0.225873	−0.112936	+	0.993602i	$0.536026\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 645.000i	− 1.31365i	−0.754045	−	0.656823i	$-0.771900\pi$
$491$	− 645.000i	− 1.31365i	0.754045	−	0.656823i	$-0.228100\pi$
$492$	0	0
$493$	−540.000	−1.09533
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 450.000i	− 0.905433i
$498$	0	0
$499$	766.000	1.53507	0.767535	−	0.641007i	$-0.221483\pi$
$499$	766.000	1.53507	0.767535	+	0.641007i	$0.221483\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 828.000i	− 1.64612i	−0.567952	−	0.823062i	$-0.692264\pi$
$503$	− 828.000i	− 1.64612i	0.567952	−	0.823062i	$-0.307736\pi$
$504$	0	0
$505$	−585.000	−1.15842
$506$	0	0
$507$	0	0
$508$	0	0
$509$	555.000i	1.09037i	0.838315	+	0.545187i	$0.183541\pi$
$509$	555.000i	1.09037i	−0.838315	+	0.545187i	$0.816459\pi$
$510$	0	0
$511$	−325.000	−0.636008
$512$	0	0
$513$	0	0
$514$	0	0
$515$	510.000i	0.990291i
$516$	0	0
$517$	−90.0000	−0.174081
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 450.000i	− 0.863724i	−0.901940	−	0.431862i	$-0.857857\pi$
$521$	− 450.000i	− 0.863724i	0.901940	−	0.431862i	$-0.142143\pi$
$522$	0	0
$523$	250.000	0.478011	0.239006	−	0.971018i	$-0.423179\pi$
$523$	250.000	0.478011	0.239006	+	0.971018i	$0.423179\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	18.0000i	0.0341556i
$528$	0	0
$529$	385.000	0.727788
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 600.000i	− 1.12570i
$534$	0	0
$535$	−567.000	−1.05981
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 360.000i	− 0.667904i
$540$	0	0
$541$	−268.000	−0.495379	−0.247689	−	0.968839i	$-0.579671\pi$
$541$	−268.000	−0.495379	−0.247689	+	0.968839i	$0.579671\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 492.000i	− 0.902752i
$546$	0	0
$547$	−410.000	−0.749543	−0.374771	−	0.927117i	$-0.622279\pi$
$547$	−410.000	−0.749543	−0.374771	+	0.927117i	$0.622279\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	480.000i	0.871143i
$552$	0	0
$553$	70.0000	0.126582
$554$	0	0
$555$	0	0
$556$	0	0
$557$	639.000i	1.14722i	0.819130	+	0.573609i	$0.194457\pi$
$557$	639.000i	1.14722i	−0.819130	+	0.573609i	$0.805543\pi$
$558$	0	0
$559$	500.000	0.894454
$560$	0	0
$561$	0	0
$562$	0	0
$563$	201.000i	0.357016i	0.983938	+	0.178508i	$0.0571270\pi$
$563$	201.000i	0.357016i	−0.983938	+	0.178508i	$0.942873\pi$
$564$	0	0
$565$	72.0000	0.127434
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 240.000i	− 0.421793i	−0.977508	−	0.210896i	$-0.932362\pi$
$569$	− 240.000i	− 0.421793i	0.977508	−	0.210896i	$-0.0676382\pi$
$570$	0	0
$571$	946.000	1.65674	0.828371	−	0.560179i	$-0.189268\pi$
$571$	946.000	1.65674	0.828371	+	0.560179i	$0.189268\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	192.000i	0.333913i
$576$	0	0
$577$	830.000	1.43847	0.719237	−	0.694764i	$-0.244491\pi$
$577$	830.000	1.43847	0.719237	+	0.694764i	$0.244491\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	15.0000i	0.0258176i
$582$	0	0
$583$	405.000	0.694683
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 453.000i	− 0.771721i	−0.922557	−	0.385860i	$-0.873905\pi$
$587$	− 453.000i	− 0.771721i	0.922557	−	0.385860i	$-0.126095\pi$
$588$	0	0
$589$	16.0000	0.0271647
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 702.000i	− 1.18381i	−0.806007	−	0.591906i	$-0.798376\pi$
$593$	− 702.000i	− 1.18381i	0.806007	−	0.591906i	$-0.201624\pi$
$594$	0	0
$595$	−270.000	−0.453782
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1110.00i	1.85309i	0.376186	+	0.926544i	$0.377235\pi$
$599$	1110.00i	1.85309i	−0.376186	+	0.926544i	$0.622765\pi$
$600$	0	0
$601$	869.000	1.44592	0.722962	−	0.690888i	$-0.242780\pi$
$601$	869.000	1.44592	0.722962	+	0.690888i	$0.242780\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	312.000i	0.515702i
$606$	0	0
$607$	−530.000	−0.873147	−0.436573	−	0.899669i	$-0.643808\pi$
$607$	−530.000	−0.873147	−0.436573	+	0.899669i	$0.643808\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 60.0000i	− 0.0981997i
$612$	0	0
$613$	−70.0000	−0.114192	−0.0570962	−	0.998369i	$-0.518184\pi$
$613$	−70.0000	−0.114192	−0.0570962	+	0.998369i	$0.518184\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 552.000i	− 0.894652i	−0.894371	−	0.447326i	$-0.852376\pi$
$617$	− 552.000i	− 0.894652i	0.894371	−	0.447326i	$-0.147624\pi$
$618$	0	0
$619$	−662.000	−1.06947	−0.534733	−	0.845021i	$-0.679588\pi$
$619$	−662.000	−1.06947	−0.534733	+	0.845021i	$0.679588\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 450.000i	− 0.722311i
$624$	0	0
$625$	31.0000	0.0496000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	360.000i	0.572337i
$630$	0	0
$631$	331.000	0.524564	0.262282	−	0.964991i	$-0.415525\pi$
$631$	331.000	0.524564	0.262282	+	0.964991i	$0.415525\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 615.000i	− 0.968504i
$636$	0	0
$637$	240.000	0.376766
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 60.0000i	− 0.0936037i	−0.998904	−	0.0468019i	$-0.985097\pi$
$641$	− 60.0000i	− 0.0936037i	0.998904	−	0.0468019i	$-0.0149029\pi$
$642$	0	0
$643$	−440.000	−0.684292	−0.342146	−	0.939647i	$-0.611154\pi$
$643$	−440.000	−0.684292	−0.342146	+	0.939647i	$0.611154\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	972.000i	1.50232i	0.660121	+	0.751159i	$0.270505\pi$
$647$	972.000i	1.50232i	−0.660121	+	0.751159i	$0.729495\pi$
$648$	0	0
$649$	−450.000	−0.693374
$650$	0	0
$651$	0	0
$652$	0	0
$653$	483.000i	0.739663i	0.929099	+	0.369832i	$0.120585\pi$
$653$	483.000i	0.739663i	−0.929099	+	0.369832i	$0.879415\pi$
$654$	0	0
$655$	−45.0000	−0.0687023
$656$	0	0
$657$	0	0
$658$	0	0
$659$	825.000i	1.25190i	0.779864	+	0.625948i	$0.215288\pi$
$659$	825.000i	1.25190i	−0.779864	+	0.625948i	$0.784712\pi$
$660$	0	0
$661$	−928.000	−1.40393	−0.701967	−	0.712210i	$-0.747694\pi$
$661$	−928.000	−1.40393	−0.701967	+	0.712210i	$0.747694\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	240.000i	0.360902i
$666$	0	0
$667$	−360.000	−0.539730
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1140.00i	− 1.69896i
$672$	0	0
$673$	−985.000	−1.46360	−0.731798	−	0.681522i	$-0.761319\pi$
$673$	−985.000	−1.46360	−0.731798	+	0.681522i	$0.761319\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	354.000i	0.522895i	0.965218	+	0.261448i	$0.0841998\pi$
$677$	354.000i	0.522895i	−0.965218	+	0.261448i	$0.915800\pi$
$678$	0	0
$679$	425.000	0.625920
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 198.000i	− 0.289898i	−0.989439	−	0.144949i	$-0.953698\pi$
$683$	− 198.000i	− 0.289898i	0.989439	−	0.144949i	$-0.0463017\pi$
$684$	0	0
$685$	414.000	0.604380
$686$	0	0
$687$	0	0
$688$	0	0
$689$	270.000i	0.391872i
$690$	0	0
$691$	436.000	0.630970	0.315485	−	0.948931i	$-0.397833\pi$
$691$	436.000	0.630970	0.315485	+	0.948931i	$0.397833\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 84.0000i	− 0.120863i
$696$	0	0
$697$	−1080.00	−1.54950
$698$	0	0
$699$	0	0
$700$	0	0
$701$	135.000i	0.192582i	0.995353	+	0.0962910i	$0.0306979\pi$
$701$	135.000i	0.192582i	−0.995353	+	0.0962910i	$0.969302\pi$
$702$	0	0
$703$	320.000	0.455192
$704$	0	0
$705$	0	0
$706$	0	0
$707$	975.000i	1.37907i
$708$	0	0
$709$	32.0000	0.0451340	0.0225670	−	0.999745i	$-0.492816\pi$
$709$	32.0000	0.0451340	0.0225670	+	0.999745i	$0.492816\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	12.0000i	0.0168303i
$714$	0	0
$715$	−450.000	−0.629371
$716$	0	0
$717$	0	0
$718$	0	0
$719$	900.000i	1.25174i	0.779928	+	0.625869i	$0.215256\pi$
$719$	900.000i	1.25174i	−0.779928	+	0.625869i	$0.784744\pi$
$720$	0	0
$721$	850.000	1.17892
$722$	0	0
$723$	0	0
$724$	0	0
$725$	480.000i	0.662069i
$726$	0	0
$727$	175.000	0.240715	0.120358	−	0.992731i	$-0.461596\pi$
$727$	175.000	0.240715	0.120358	+	0.992731i	$0.461596\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 900.000i	− 1.23119i
$732$	0	0
$733$	1160.00	1.58254	0.791269	−	0.611469i	$-0.209421\pi$
$733$	1160.00	1.58254	0.791269	+	0.611469i	$0.209421\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	150.000i	0.203528i
$738$	0	0
$739$	1006.00	1.36130	0.680650	−	0.732609i	$-0.261698\pi$
$739$	1006.00	1.36130	0.680650	+	0.732609i	$0.261698\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 114.000i	− 0.153432i	−0.997053	−	0.0767160i	$-0.975557\pi$
$743$	− 114.000i	− 0.153432i	0.997053	−	0.0767160i	$-0.0244435\pi$
$744$	0	0
$745$	−225.000	−0.302013
$746$	0	0
$747$	0	0
$748$	0	0
$749$	945.000i	1.26168i
$750$	0	0
$751$	−359.000	−0.478029	−0.239015	−	0.971016i	$-0.576824\pi$
$751$	−359.000	−0.478029	−0.239015	+	0.971016i	$0.576824\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	231.000i	0.305960i
$756$	0	0
$757$	−430.000	−0.568032	−0.284016	−	0.958820i	$-0.591667\pi$
$757$	−430.000	−0.568032	−0.284016	+	0.958820i	$0.591667\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1320.00i	1.73456i	0.497821	+	0.867280i	$0.334134\pi$
$761$	1320.00i	1.73456i	−0.497821	+	0.867280i	$0.665866\pi$
$762$	0	0
$763$	−820.000	−1.07471
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 300.000i	− 0.391134i
$768$	0	0
$769$	1259.00	1.63719	0.818596	−	0.574370i	$-0.194753\pi$
$769$	1259.00	1.63719	0.818596	+	0.574370i	$0.194753\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 522.000i	− 0.675291i	−0.941273	−	0.337646i	$-0.890369\pi$
$773$	− 522.000i	− 0.675291i	0.941273	−	0.337646i	$-0.109631\pi$
$774$	0	0
$775$	16.0000	0.0206452
$776$	0	0
$777$	0	0
$778$	0	0
$779$	960.000i	1.23235i
$780$	0	0
$781$	−1350.00	−1.72855
$782$	0	0
$783$	0	0
$784$	0	0
$785$	300.000i	0.382166i
$786$	0	0
$787$	460.000	0.584498	0.292249	−	0.956342i	$-0.405596\pi$
$787$	460.000	0.584498	0.292249	+	0.956342i	$0.405596\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 120.000i	− 0.151707i
$792$	0	0
$793$	760.000	0.958386
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 237.000i	− 0.297365i	−0.988885	−	0.148683i	$-0.952497\pi$
$797$	− 237.000i	− 0.297365i	0.988885	−	0.148683i	$-0.0475033\pi$
$798$	0	0
$799$	−108.000	−0.135169
$800$	0	0
$801$	0	0
$802$	0	0
$803$	975.000i	1.21420i
$804$	0	0
$805$	−180.000	−0.223602
$806$	0	0
$807$	0	0
$808$	0	0
$809$	810.000i	1.00124i	0.865668	+	0.500618i	$0.166894\pi$
$809$	810.000i	1.00124i	−0.865668	+	0.500618i	$0.833106\pi$
$810$	0	0
$811$	−272.000	−0.335388	−0.167694	−	0.985839i	$-0.553632\pi$
$811$	−272.000	−0.335388	−0.167694	+	0.985839i	$0.553632\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	330.000i	0.404908i
$816$	0	0
$817$	−800.000	−0.979192
$818$	0	0
$819$	0	0
$820$	0	0
$821$	390.000i	0.475030i	0.971384	+	0.237515i	$0.0763330\pi$
$821$	390.000i	0.475030i	−0.971384	+	0.237515i	$0.923667\pi$
$822$	0	0
$823$	−1205.00	−1.46416	−0.732078	−	0.681221i	$-0.761449\pi$
$823$	−1205.00	−1.46416	−0.732078	+	0.681221i	$0.761449\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 18.0000i	− 0.0217654i	−0.999941	−	0.0108827i	$-0.996536\pi$
$827$	− 18.0000i	− 0.0217654i	0.999941	−	0.0108827i	$-0.00346414\pi$
$828$	0	0
$829$	1442.00	1.73945	0.869723	−	0.493541i	$-0.164298\pi$
$829$	1442.00	1.73945	0.869723	+	0.493541i	$0.164298\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 432.000i	− 0.518607i
$834$	0	0
$835$	−234.000	−0.280240
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1590.00i	1.89511i	0.319588	+	0.947557i	$0.396456\pi$
$839$	1590.00i	1.89511i	−0.319588	+	0.947557i	$0.603544\pi$
$840$	0	0
$841$	−59.0000	−0.0701546
$842$	0	0
$843$	0	0
$844$	0	0
$845$	207.000i	0.244970i
$846$	0	0
$847$	520.000	0.613932
$848$	0	0
$849$	0	0
$850$	0	0
$851$	240.000i	0.282021i
$852$	0	0
$853$	590.000	0.691676	0.345838	−	0.938294i	$-0.387595\pi$
$853$	590.000	0.691676	0.345838	+	0.938294i	$0.387595\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1302.00i	− 1.51925i	−0.650359	−	0.759627i	$-0.725382\pi$
$857$	− 1302.00i	− 1.51925i	0.650359	−	0.759627i	$-0.274618\pi$
$858$	0	0
$859$	316.000	0.367870	0.183935	−	0.982938i	$-0.441116\pi$
$859$	316.000	0.367870	0.183935	+	0.982938i	$0.441116\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1188.00i	− 1.37659i	−0.725429	−	0.688297i	$-0.758359\pi$
$863$	− 1188.00i	− 1.37659i	0.725429	−	0.688297i	$-0.241641\pi$
$864$	0	0
$865$	−531.000	−0.613873
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 210.000i	− 0.241657i
$870$	0	0
$871$	−100.000	−0.114811
$872$	0	0
$873$	0	0
$874$	0	0
$875$	615.000i	0.702857i
$876$	0	0
$877$	−550.000	−0.627138	−0.313569	−	0.949565i	$-0.601525\pi$
$877$	−550.000	−0.627138	−0.313569	+	0.949565i	$0.601525\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 90.0000i	− 0.102157i	−0.998695	−	0.0510783i	$-0.983734\pi$
$881$	− 90.0000i	− 0.102157i	0.998695	−	0.0510783i	$-0.0162658\pi$
$882$	0	0
$883$	880.000	0.996602	0.498301	−	0.867004i	$-0.333957\pi$
$883$	880.000	0.996602	0.498301	+	0.867004i	$0.333957\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	282.000i	0.317926i	0.987285	+	0.158963i	$0.0508150\pi$
$887$	282.000i	0.317926i	−0.987285	+	0.158963i	$0.949185\pi$
$888$	0	0
$889$	−1025.00	−1.15298
$890$	0	0
$891$	0	0
$892$	0	0
$893$	96.0000i	0.107503i
$894$	0	0
$895$	675.000	0.754190
$896$	0	0
$897$	0	0
$898$	0	0
$899$	30.0000i	0.0333704i
$900$	0	0
$901$	486.000	0.539401
$902$	0	0
$903$	0	0
$904$	0	0
$905$	48.0000i	0.0530387i
$906$	0	0
$907$	1300.00	1.43330	0.716648	−	0.697435i	$-0.245675\pi$
$907$	1300.00	1.43330	0.716648	+	0.697435i	$0.245675\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 210.000i	− 0.230516i	−0.993336	−	0.115258i	$-0.963231\pi$
$911$	− 210.000i	− 0.230516i	0.993336	−	0.115258i	$-0.0367695\pi$
$912$	0	0
$913$	45.0000	0.0492881
$914$	0	0
$915$	0	0
$916$	0	0
$917$	75.0000i	0.0817884i
$918$	0	0
$919$	−137.000	−0.149075	−0.0745375	−	0.997218i	$-0.523748\pi$
$919$	−137.000	−0.149075	−0.0745375	+	0.997218i	$0.523748\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 900.000i	− 0.975081i
$924$	0	0
$925$	320.000	0.345946
$926$	0	0
$927$	0	0
$928$	0	0
$929$	660.000i	0.710441i	0.934782	+	0.355221i	$0.115594\pi$
$929$	660.000i	0.710441i	−0.934782	+	0.355221i	$0.884406\pi$
$930$	0	0
$931$	−384.000	−0.412460
$932$	0	0
$933$	0	0
$934$	0	0
$935$	810.000i	0.866310i
$936$	0	0
$937$	605.000	0.645678	0.322839	−	0.946454i	$-0.395363\pi$
$937$	605.000	0.645678	0.322839	+	0.946454i	$0.395363\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1605.00i	− 1.70563i	−0.522211	−	0.852816i	$-0.674893\pi$
$941$	− 1605.00i	− 1.70563i	0.522211	−	0.852816i	$-0.325107\pi$
$942$	0	0
$943$	−720.000	−0.763521
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 543.000i	− 0.573390i	−0.958022	−	0.286695i	$-0.907443\pi$
$947$	− 543.000i	− 0.573390i	0.958022	−	0.286695i	$-0.0925566\pi$
$948$	0	0
$949$	−650.000	−0.684932
$950$	0	0
$951$	0	0
$952$	0	0
$953$	144.000i	0.151102i	0.997142	+	0.0755509i	$0.0240715\pi$
$953$	144.000i	0.151102i	−0.997142	+	0.0755509i	$0.975928\pi$
$954$	0	0
$955$	−90.0000	−0.0942408
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 690.000i	− 0.719499i
$960$	0	0
$961$	−960.000	−0.998959
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 645.000i	− 0.668394i
$966$	0	0
$967$	−845.000	−0.873837	−0.436918	−	0.899501i	$-0.643930\pi$
$967$	−845.000	−0.873837	−0.436918	+	0.899501i	$0.643930\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 405.000i	− 0.417096i	−0.978012	−	0.208548i	$-0.933126\pi$
$971$	− 405.000i	− 0.417096i	0.978012	−	0.208548i	$-0.0668737\pi$
$972$	0	0
$973$	−140.000	−0.143885
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 246.000i	− 0.251791i	−0.992043	−	0.125896i	$-0.959820\pi$
$977$	− 246.000i	− 0.251791i	0.992043	−	0.125896i	$-0.0401804\pi$
$978$	0	0
$979$	−1350.00	−1.37896
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1038.00i	− 1.05595i	−0.849260	−	0.527976i	$-0.822951\pi$
$983$	− 1038.00i	− 1.05595i	0.849260	−	0.527976i	$-0.177049\pi$
$984$	0	0
$985$	−621.000	−0.630457
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 600.000i	− 0.606673i
$990$	0	0
$991$	1501.00	1.51463	0.757316	−	0.653049i	$-0.226510\pi$
$991$	1501.00	1.51463	0.757316	+	0.653049i	$0.226510\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 669.000i	− 0.672362i
$996$	0	0
$997$	770.000	0.772317	0.386158	−	0.922432i	$-0.373802\pi$
$997$	770.000	0.772317	0.386158	+	0.922432i	$0.373802\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.3.e.c.161.1		2
3.2	odd	2	inner	432.3.e.c.161.2		2
4.3	odd	2		27.3.b.b.26.2	yes	2
8.3	odd	2		1728.3.e.m.1025.2		2
8.5	even	2		1728.3.e.g.1025.2		2
9.2	odd	6		1296.3.q.j.1025.1		4
9.4	even	3		1296.3.q.j.593.1		4
9.5	odd	6		1296.3.q.j.593.2		4
9.7	even	3		1296.3.q.j.1025.2		4
12.11	even	2		27.3.b.b.26.1	✓	2
20.3	even	4		675.3.d.d.674.1		2
20.7	even	4		675.3.d.a.674.2		2
20.19	odd	2		675.3.c.h.26.1		2
24.5	odd	2		1728.3.e.g.1025.1		2
24.11	even	2		1728.3.e.m.1025.1		2
36.7	odd	6		81.3.d.b.53.2		4
36.11	even	6		81.3.d.b.53.1		4
36.23	even	6		81.3.d.b.26.2		4
36.31	odd	6		81.3.d.b.26.1		4
60.23	odd	4		675.3.d.a.674.1		2
60.47	odd	4		675.3.d.d.674.2		2
60.59	even	2		675.3.c.h.26.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.b.b.26.1	✓	2	12.11	even	2
27.3.b.b.26.2	yes	2	4.3	odd	2
81.3.d.b.26.1		4	36.31	odd	6
81.3.d.b.26.2		4	36.23	even	6
81.3.d.b.53.1		4	36.11	even	6
81.3.d.b.53.2		4	36.7	odd	6
432.3.e.c.161.1		2	1.1	even	1	trivial
432.3.e.c.161.2		2	3.2	odd	2	inner
675.3.c.h.26.1		2	20.19	odd	2
675.3.c.h.26.2		2	60.59	even	2
675.3.d.a.674.1		2	60.23	odd	4
675.3.d.a.674.2		2	20.7	even	4
675.3.d.d.674.1		2	20.3	even	4
675.3.d.d.674.2		2	60.47	odd	4
1296.3.q.j.593.1		4	9.4	even	3
1296.3.q.j.593.2		4	9.5	odd	6
1296.3.q.j.1025.1		4	9.2	odd	6
1296.3.q.j.1025.2		4	9.7	even	3
1728.3.e.g.1025.1		2	24.5	odd	2
1728.3.e.g.1025.2		2	8.5	even	2
1728.3.e.m.1025.1		2	24.11	even	2
1728.3.e.m.1025.2		2	8.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	432.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{5} -5.00000 q^{7} +O(q^{10})\)	\(q-3.00000i q^{5} -5.00000 q^{7} +15.0000i q^{11} -10.0000 q^{13} +18.0000i q^{17} +16.0000 q^{19} +12.0000i q^{23} +16.0000 q^{25} +30.0000i q^{29} +1.00000 q^{31} +15.0000i q^{35} +20.0000 q^{37} +60.0000i q^{41} -50.0000 q^{43} +6.00000i q^{47} -24.0000 q^{49} -27.0000i q^{53} +45.0000 q^{55} +30.0000i q^{59} -76.0000 q^{61} +30.0000i q^{65} +10.0000 q^{67} +90.0000i q^{71} +65.0000 q^{73} -75.0000i q^{77} -14.0000 q^{79} -3.00000i q^{83} +54.0000 q^{85} +90.0000i q^{89} +50.0000 q^{91} -48.0000i q^{95} -85.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{7}+O(q^{10}) \) 2 * q - 10 * q^7	\( 2 q - 10 q^{7} - 20 q^{13} + 32 q^{19} + 32 q^{25} + 2 q^{31} + 40 q^{37} - 100 q^{43} - 48 q^{49} + 90 q^{55} - 152 q^{61} + 20 q^{67} + 130 q^{73} - 28 q^{79} + 108 q^{85} + 100 q^{91} - 170 q^{97}+O(q^{100}) \) 2 * q - 10 * q^7 - 20 * q^13 + 32 * q^19 + 32 * q^25 + 2 * q^31 + 40 * q^37 - 100 * q^43 - 48 * q^49 + 90 * q^55 - 152 * q^61 + 20 * q^67 + 130 * q^73 - 28 * q^79 + 108 * q^85 + 100 * q^91 - 170 * q^97

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