LMFDB - Embedded newform 27.3.b.b.26.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [27,3,Mod(26,27)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(27, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("27.26");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	27.b (of order $2$, degree $1$, 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:	$0.735696713773$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	27.26
Dual form			27.3.b.b.26.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.00000i q^{2} -5.00000 q^{4} +3.00000i q^{5} +5.00000 q^{7} +3.00000i q^{8} +9.00000 q^{10} +15.0000i q^{11} -10.0000 q^{13} -15.0000i q^{14} -11.0000 q^{16} -18.0000i q^{17} -16.0000 q^{19} -15.0000i q^{20} +45.0000 q^{22} +12.0000i q^{23} +16.0000 q^{25} +30.0000i q^{26} -25.0000 q^{28} -30.0000i q^{29} -1.00000 q^{31} +45.0000i q^{32} -54.0000 q^{34} +15.0000i q^{35} +20.0000 q^{37} +48.0000i q^{38} -9.00000 q^{40} -60.0000i q^{41} +50.0000 q^{43} -75.0000i q^{44} +36.0000 q^{46} +6.00000i q^{47} -24.0000 q^{49} -48.0000i q^{50} +50.0000 q^{52} +27.0000i q^{53} -45.0000 q^{55} +15.0000i q^{56} -90.0000 q^{58} +30.0000i q^{59} -76.0000 q^{61} +3.00000i q^{62} +91.0000 q^{64} -30.0000i q^{65} -10.0000 q^{67} +90.0000i q^{68} +45.0000 q^{70} +90.0000i q^{71} +65.0000 q^{73} -60.0000i q^{74} +80.0000 q^{76} +75.0000i q^{77} +14.0000 q^{79} -33.0000i q^{80} -180.000 q^{82} -3.00000i q^{83} +54.0000 q^{85} -150.000i q^{86} -45.0000 q^{88} -90.0000i q^{89} -50.0000 q^{91} -60.0000i q^{92} +18.0000 q^{94} -48.0000i q^{95} -85.0000 q^{97} +72.0000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 10 q^{4} + 10 q^{7} + 18 q^{10} - 20 q^{13} - 22 q^{16} - 32 q^{19} + 90 q^{22} + 32 q^{25} - 50 q^{28} - 2 q^{31} - 108 q^{34} + 40 q^{37} - 18 q^{40} + 100 q^{43} + 72 q^{46} - 48 q^{49} + 100 q^{52}+ \cdots - 170 q^{97}+O(q^{100}) $ 2 * q - 10 * q^4 + 10 * q^7 + 18 * q^10 - 20 * q^13 - 22 * q^16 - 32 * q^19 + 90 * q^22 + 32 * q^25 - 50 * q^28 - 2 * q^31 - 108 * q^34 + 40 * q^37 - 18 * q^40 + 100 * q^43 + 72 * q^46 - 48 * q^49 + 100 * q^52 - 90 * q^55 - 180 * q^58 - 152 * q^61 + 182 * q^64 - 20 * q^67 + 90 * q^70 + 130 * q^73 + 160 * q^76 + 28 * q^79 - 360 * q^82 + 108 * q^85 - 90 * q^88 - 100 * q^91 + 36 * q^94 - 170 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/27\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	− 3.00000i	− 1.50000i	−0.661438	−	0.750000i	$-0.730053\pi$
$2$	− 3.00000i	− 1.50000i	0.661438	−	0.750000i	$-0.269947\pi$
$3$	0	0
$3$	0	0
$4$	−5.00000	−1.25000
$5$	3.00000i	0.600000i	0.953939	+	0.300000i	$0.0969867\pi$
$5$	3.00000i	0.600000i	−0.953939	+	0.300000i	$0.903013\pi$
$6$	0	0
$7$	5.00000	0.714286	0.357143	−	0.934050i	$-0.383751\pi$
$7$	5.00000	0.714286	0.357143	+	0.934050i	$0.383751\pi$
$8$	3.00000i	0.375000i
$9$	0	0
$10$	9.00000	0.900000
$11$	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	$0.625666\pi$
$14$	− 15.0000i	− 1.07143i
$15$	0	0
$16$	−11.0000	−0.687500
$17$	− 18.0000i	− 1.05882i	−0.848365	−	0.529412i	$-0.822413\pi$
$17$	− 18.0000i	− 1.05882i	0.848365	−	0.529412i	$-0.177587\pi$
$18$	0	0
$19$	−16.0000	−0.842105	−0.421053	−	0.907036i	$-0.638339\pi$
$19$	−16.0000	−0.842105	−0.421053	+	0.907036i	$0.638339\pi$
$20$	− 15.0000i	− 0.750000i
$21$	0	0
$22$	45.0000	2.04545
$23$	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
$26$	30.0000i	1.15385i
$27$	0	0
$28$	−25.0000	−0.892857
$29$	− 30.0000i	− 1.03448i	−0.855840	−	0.517241i	$-0.826959\pi$
$29$	− 30.0000i	− 1.03448i	0.855840	−	0.517241i	$-0.173041\pi$
$30$	0	0
$31$	−1.00000	−0.0322581	−0.0161290	−	0.999870i	$-0.505134\pi$
$31$	−1.00000	−0.0322581	−0.0161290	+	0.999870i	$0.505134\pi$
$32$	45.0000i	1.40625i
$33$	0	0
$34$	−54.0000	−1.58824
$35$	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	$0.412887\pi$
$38$	48.0000i	1.26316i
$39$	0	0
$40$	−9.00000	−0.225000
$41$	− 60.0000i	− 1.46341i	−0.681619	−	0.731707i	$-0.738724\pi$
$41$	− 60.0000i	− 1.46341i	0.681619	−	0.731707i	$-0.261276\pi$
$42$	0	0
$43$	50.0000	1.16279	0.581395	−	0.813621i	$-0.302507\pi$
$43$	50.0000	1.16279	0.581395	+	0.813621i	$0.302507\pi$
$44$	− 75.0000i	− 1.70455i
$45$	0	0
$46$	36.0000	0.782609
$47$	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
$50$	− 48.0000i	− 0.960000i
$51$	0	0
$52$	50.0000	0.961538
$53$	27.0000i	0.509434i	0.967016	+	0.254717i	$0.0819823\pi$
$53$	27.0000i	0.509434i	−0.967016	+	0.254717i	$0.918018\pi$
$54$	0	0
$55$	−45.0000	−0.818182
$56$	15.0000i	0.267857i
$57$	0	0
$58$	−90.0000	−1.55172
$59$	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	$0.714066\pi$
$62$	3.00000i	0.0483871i
$63$	0	0
$64$	91.0000	1.42188
$65$	− 30.0000i	− 0.461538i
$66$	0	0
$67$	−10.0000	−0.149254	−0.0746269	−	0.997212i	$-0.523777\pi$
$67$	−10.0000	−0.149254	−0.0746269	+	0.997212i	$0.523777\pi$
$68$	90.0000i	1.32353i
$69$	0	0
$70$	45.0000	0.642857
$71$	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	$0.353131\pi$
$74$	− 60.0000i	− 0.810811i
$75$	0	0
$76$	80.0000	1.05263
$77$	75.0000i	0.974026i
$78$	0	0
$79$	14.0000	0.177215	0.0886076	−	0.996067i	$-0.471758\pi$
$79$	14.0000	0.177215	0.0886076	+	0.996067i	$0.471758\pi$
$80$	− 33.0000i	− 0.412500i
$81$	0	0
$82$	−180.000	−2.19512
$83$	− 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
$86$	− 150.000i	− 1.74419i
$87$	0	0
$88$	−45.0000	−0.511364
$89$	− 90.0000i	− 1.01124i	−0.862757	−	0.505618i	$-0.831265\pi$
$89$	− 90.0000i	− 1.01124i	0.862757	−	0.505618i	$-0.168735\pi$
$90$	0	0
$91$	−50.0000	−0.549451
$92$	− 60.0000i	− 0.652174i
$93$	0	0
$94$	18.0000	0.191489
$95$	− 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	$0.644364\pi$
$98$	72.0000i	0.734694i
$99$	0	0
$100$	−80.0000	−0.800000
$101$	195.000i	1.93069i	0.260971	+	0.965347i	$0.415957\pi$
$101$	195.000i	1.93069i	−0.260971	+	0.965347i	$0.584043\pi$
$102$	0	0
$103$	170.000	1.65049	0.825243	−	0.564778i	$-0.191038\pi$
$103$	170.000	1.65049	0.825243	+	0.564778i	$0.191038\pi$
$104$	− 30.0000i	− 0.288462i
$105$	0	0
$106$	81.0000	0.764151
$107$	− 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	$0.228948\pi$
$110$	135.000i	1.22727i
$111$	0	0
$112$	−55.0000	−0.491071
$113$	− 24.0000i	− 0.212389i	−0.994345	−	0.106195i	$-0.966133\pi$
$113$	− 24.0000i	− 0.212389i	0.994345	−	0.106195i	$-0.0338667\pi$
$114$	0	0
$115$	−36.0000	−0.313043
$116$	150.000i	1.29310i
$117$	0	0
$118$	90.0000	0.762712
$119$	− 90.0000i	− 0.756303i
$120$	0	0
$121$	−104.000	−0.859504
$122$	228.000i	1.86885i
$123$	0	0
$124$	5.00000	0.0403226
$125$	123.000i	0.984000i
$126$	0	0
$127$	−205.000	−1.61417	−0.807087	−	0.590433i	$-0.798957\pi$
$127$	−205.000	−1.61417	−0.807087	+	0.590433i	$0.798957\pi$
$128$	− 93.0000i	− 0.726562i
$129$	0	0
$130$	−90.0000	−0.692308
$131$	− 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
$134$	30.0000i	0.223881i
$135$	0	0
$136$	54.0000	0.397059
$137$	− 138.000i	− 1.00730i	−0.863908	−	0.503650i	$-0.831990\pi$
$137$	− 138.000i	− 1.00730i	0.863908	−	0.503650i	$-0.168010\pi$
$138$	0	0
$139$	−28.0000	−0.201439	−0.100719	−	0.994915i	$-0.532114\pi$
$139$	−28.0000	−0.201439	−0.100719	+	0.994915i	$0.532114\pi$
$140$	− 75.0000i	− 0.535714i
$141$	0	0
$142$	270.000	1.90141
$143$	− 150.000i	− 1.04895i
$144$	0	0
$145$	90.0000	0.620690
$146$	− 195.000i	− 1.33562i
$147$	0	0
$148$	−100.000	−0.675676
$149$	75.0000i	0.503356i	0.967811	+	0.251678i	$0.0809823\pi$
$149$	75.0000i	0.503356i	−0.967811	+	0.251678i	$0.919018\pi$
$150$	0	0
$151$	77.0000	0.509934	0.254967	−	0.966950i	$-0.417935\pi$
$151$	77.0000	0.509934	0.254967	+	0.966950i	$0.417935\pi$
$152$	− 48.0000i	− 0.315789i
$153$	0	0
$154$	225.000	1.46104
$155$	− 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	$0.603169\pi$
$158$	− 42.0000i	− 0.265823i
$159$	0	0
$160$	−135.000	−0.843750
$161$	60.0000i	0.372671i
$162$	0	0
$163$	110.000	0.674847	0.337423	−	0.941353i	$-0.390445\pi$
$163$	110.000	0.674847	0.337423	+	0.941353i	$0.390445\pi$
$164$	300.000i	1.82927i
$165$	0	0
$166$	−9.00000	−0.0542169
$167$	− 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
$170$	− 162.000i	− 0.952941i
$171$	0	0
$172$	−250.000	−1.45349
$173$	177.000i	1.02312i	0.859247	+	0.511561i	$0.170932\pi$
$173$	177.000i	1.02312i	−0.859247	+	0.511561i	$0.829068\pi$
$174$	0	0
$175$	80.0000	0.457143
$176$	− 165.000i	− 0.937500i
$177$	0	0
$178$	−270.000	−1.51685
$179$	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	$0.514074\pi$
$182$	150.000i	0.824176i
$183$	0	0
$184$	−36.0000	−0.195652
$185$	60.0000i	0.324324i
$186$	0	0
$187$	270.000	1.44385
$188$	− 30.0000i	− 0.159574i
$189$	0	0
$190$	−144.000	−0.757895
$191$	− 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	$0.311954\pi$
$194$	255.000i	1.31443i
$195$	0	0
$196$	120.000	0.612245
$197$	207.000i	1.05076i	0.850867	+	0.525381i	$0.176077\pi$
$197$	207.000i	1.05076i	−0.850867	+	0.525381i	$0.823923\pi$
$198$	0	0
$199$	−223.000	−1.12060	−0.560302	−	0.828289i	$-0.689315\pi$
$199$	−223.000	−1.12060	−0.560302	+	0.828289i	$0.689315\pi$
$200$	48.0000i	0.240000i
$201$	0	0
$202$	585.000	2.89604
$203$	− 150.000i	− 0.738916i
$204$	0	0
$205$	180.000	0.878049
$206$	− 510.000i	− 2.47573i
$207$	0	0
$208$	110.000	0.528846
$209$	− 240.000i	− 1.14833i
$210$	0	0
$211$	−316.000	−1.49763	−0.748815	−	0.662779i	$-0.769377\pi$
$211$	−316.000	−1.49763	−0.748815	+	0.662779i	$0.769377\pi$
$212$	− 135.000i	− 0.636792i
$213$	0	0
$214$	−567.000	−2.64953
$215$	150.000i	0.697674i
$216$	0	0
$217$	−5.00000	−0.0230415
$218$	− 492.000i	− 2.25688i
$219$	0	0
$220$	225.000	1.02273
$221$	180.000i	0.814480i
$222$	0	0
$223$	−130.000	−0.582960	−0.291480	−	0.956577i	$-0.594148\pi$
$223$	−130.000	−0.582960	−0.291480	+	0.956577i	$0.594148\pi$
$224$	225.000i	1.00446i
$225$	0	0
$226$	−72.0000	−0.318584
$227$	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	$0.664264\pi$
$230$	108.000i	0.469565i
$231$	0	0
$232$	90.0000	0.387931
$233$	− 234.000i	− 1.00429i	−0.864783	−	0.502146i	$-0.832544\pi$
$233$	− 234.000i	− 1.00429i	0.864783	−	0.502146i	$-0.167456\pi$
$234$	0	0
$235$	−18.0000	−0.0765957
$236$	− 150.000i	− 0.635593i
$237$	0	0
$238$	−270.000	−1.13445
$239$	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	$0.490753\pi$
$242$	312.000i	1.28926i
$243$	0	0
$244$	380.000	1.55738
$245$	− 72.0000i	− 0.293878i
$246$	0	0
$247$	160.000	0.647773
$248$	− 3.00000i	− 0.0120968i
$249$	0	0
$250$	369.000	1.47600
$251$	− 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
$254$	615.000i	2.42126i
$255$	0	0
$256$	85.0000	0.332031
$257$	− 438.000i	− 1.70428i	−0.523314	−	0.852140i	$-0.675304\pi$
$257$	− 438.000i	− 1.70428i	0.523314	−	0.852140i	$-0.324696\pi$
$258$	0	0
$259$	100.000	0.386100
$260$	150.000i	0.576923i
$261$	0	0
$262$	−45.0000	−0.171756
$263$	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
$266$	240.000i	0.902256i
$267$	0	0
$268$	50.0000	0.186567
$269$	− 270.000i	− 1.00372i	−0.864950	−	0.501859i	$-0.832650\pi$
$269$	− 270.000i	− 1.00372i	0.864950	−	0.501859i	$-0.167350\pi$
$270$	0	0
$271$	299.000	1.10332	0.551661	−	0.834069i	$-0.313994\pi$
$271$	299.000	1.10332	0.551661	+	0.834069i	$0.313994\pi$
$272$	198.000i	0.727941i
$273$	0	0
$274$	−414.000	−1.51095
$275$	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	$0.418679\pi$
$278$	84.0000i	0.302158i
$279$	0	0
$280$	−45.0000	−0.160714
$281$	150.000i	0.533808i	0.963723	+	0.266904i	$0.0860006\pi$
$281$	150.000i	0.533808i	−0.963723	+	0.266904i	$0.913999\pi$
$282$	0	0
$283$	−280.000	−0.989399	−0.494700	−	0.869064i	$-0.664722\pi$
$283$	−280.000	−0.989399	−0.494700	+	0.869064i	$0.664722\pi$
$284$	− 450.000i	− 1.58451i
$285$	0	0
$286$	−450.000	−1.57343
$287$	− 300.000i	− 1.04530i
$288$	0	0
$289$	−35.0000	−0.121107
$290$	− 270.000i	− 0.931034i
$291$	0	0
$292$	−325.000	−1.11301
$293$	− 258.000i	− 0.880546i	−0.897864	−	0.440273i	$-0.854882\pi$
$293$	− 258.000i	− 0.880546i	0.897864	−	0.440273i	$-0.145118\pi$
$294$	0	0
$295$	−90.0000	−0.305085
$296$	60.0000i	0.202703i
$297$	0	0
$298$	225.000	0.755034
$299$	− 120.000i	− 0.401338i
$300$	0	0
$301$	250.000	0.830565
$302$	− 231.000i	− 0.764901i
$303$	0	0
$304$	176.000	0.578947
$305$	− 228.000i	− 0.747541i
$306$	0	0
$307$	290.000	0.944625	0.472313	−	0.881431i	$-0.343419\pi$
$307$	290.000	0.944625	0.472313	+	0.881431i	$0.343419\pi$
$308$	− 375.000i	− 1.21753i
$309$	0	0
$310$	−9.00000	−0.0290323
$311$	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	$0.404505\pi$
$314$	300.000i	0.955414i
$315$	0	0
$316$	−70.0000	−0.221519
$317$	− 183.000i	− 0.577287i	−0.957437	−	0.288644i	$-0.906796\pi$
$317$	− 183.000i	− 0.577287i	0.957437	−	0.288644i	$-0.0932042\pi$
$318$	0	0
$319$	450.000	1.41066
$320$	273.000i	0.853125i
$321$	0	0
$322$	180.000	0.559006
$323$	288.000i	0.891641i
$324$	0	0
$325$	−160.000	−0.492308
$326$	− 330.000i	− 1.01227i
$327$	0	0
$328$	180.000	0.548780
$329$	30.0000i	0.0911854i
$330$	0	0
$331$	−238.000	−0.719033	−0.359517	−	0.933139i	$-0.617058\pi$
$331$	−238.000	−0.719033	−0.359517	+	0.933139i	$0.617058\pi$
$332$	15.0000i	0.0451807i
$333$	0	0
$334$	−234.000	−0.700599
$335$	− 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	$0.504723\pi$
$338$	207.000i	0.612426i
$339$	0	0
$340$	−270.000	−0.794118
$341$	− 15.0000i	− 0.0439883i
$342$	0	0
$343$	−365.000	−1.06414
$344$	150.000i	0.436047i
$345$	0	0
$346$	531.000	1.53468
$347$	− 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	$0.619534\pi$
$350$	− 240.000i	− 0.685714i
$351$	0	0
$352$	−675.000	−1.91761
$353$	456.000i	1.29178i	0.763428	+	0.645892i	$0.223515\pi$
$353$	456.000i	1.29178i	−0.763428	+	0.645892i	$0.776485\pi$
$354$	0	0
$355$	−270.000	−0.760563
$356$	450.000i	1.26404i
$357$	0	0
$358$	675.000	1.88547
$359$	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
$362$	48.0000i	0.132597i
$363$	0	0
$364$	250.000	0.686813
$365$	195.000i	0.534247i
$366$	0	0
$367$	−625.000	−1.70300	−0.851499	−	0.524357i	$-0.824306\pi$
$367$	−625.000	−1.70300	−0.851499	+	0.524357i	$0.824306\pi$
$368$	− 132.000i	− 0.358696i
$369$	0	0
$370$	180.000	0.486486
$371$	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	$0.426820\pi$
$374$	− 810.000i	− 2.16578i
$375$	0	0
$376$	−18.0000	−0.0478723
$377$	300.000i	0.795756i
$378$	0	0
$379$	704.000	1.85752	0.928760	−	0.370682i	$-0.120876\pi$
$379$	704.000	1.85752	0.928760	+	0.370682i	$0.120876\pi$
$380$	240.000i	0.631579i
$381$	0	0
$382$	−90.0000	−0.235602
$383$	− 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
$386$	− 645.000i	− 1.67098i
$387$	0	0
$388$	425.000	1.09536
$389$	− 525.000i	− 1.34961i	−0.737994	−	0.674807i	$-0.764227\pi$
$389$	− 525.000i	− 1.34961i	0.737994	−	0.674807i	$-0.235773\pi$
$390$	0	0
$391$	216.000	0.552430
$392$	− 72.0000i	− 0.183673i
$393$	0	0
$394$	621.000	1.57614
$395$	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	$0.528099\pi$
$398$	669.000i	1.68090i
$399$	0	0
$400$	−176.000	−0.440000
$401$	120.000i	0.299252i	0.988743	+	0.149626i	$0.0478069\pi$
$401$	120.000i	0.299252i	−0.988743	+	0.149626i	$0.952193\pi$
$402$	0	0
$403$	10.0000	0.0248139
$404$	− 975.000i	− 2.41337i
$405$	0	0
$406$	−450.000	−1.10837
$407$	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	$0.393339\pi$
$410$	− 540.000i	− 1.31707i
$411$	0	0
$412$	−850.000	−2.06311
$413$	150.000i	0.363196i
$414$	0	0
$415$	9.00000	0.0216867
$416$	− 450.000i	− 1.08173i
$417$	0	0
$418$	−720.000	−1.72249
$419$	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	$0.222815\pi$
$422$	948.000i	2.24645i
$423$	0	0
$424$	−81.0000	−0.191038
$425$	− 288.000i	− 0.677647i
$426$	0	0
$427$	−380.000	−0.889930
$428$	945.000i	2.20794i
$429$	0	0
$430$	450.000	1.04651
$431$	− 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	$0.726248\pi$
$434$	15.0000i	0.0345622i
$435$	0	0
$436$	−820.000	−1.88073
$437$	− 192.000i	− 0.439359i
$438$	0	0
$439$	−211.000	−0.480638	−0.240319	−	0.970694i	$-0.577252\pi$
$439$	−211.000	−0.480638	−0.240319	+	0.970694i	$0.577252\pi$
$440$	− 135.000i	− 0.306818i
$441$	0	0
$442$	540.000	1.22172
$443$	− 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
$446$	390.000i	0.874439i
$447$	0	0
$448$	455.000	1.01562
$449$	360.000i	0.801782i	0.916126	+	0.400891i	$0.131299\pi$
$449$	360.000i	0.801782i	−0.916126	+	0.400891i	$0.868701\pi$
$450$	0	0
$451$	900.000	1.99557
$452$	120.000i	0.265487i
$453$	0	0
$454$	126.000	0.277533
$455$	− 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	$0.369238\pi$
$458$	678.000i	1.48035i
$459$	0	0
$460$	180.000	0.391304
$461$	105.000i	0.227766i	0.993494	+	0.113883i	$0.0363289\pi$
$461$	105.000i	0.227766i	−0.993494	+	0.113883i	$0.963671\pi$
$462$	0	0
$463$	215.000	0.464363	0.232181	−	0.972672i	$-0.425414\pi$
$463$	215.000	0.464363	0.232181	+	0.972672i	$0.425414\pi$
$464$	330.000i	0.711207i
$465$	0	0
$466$	−702.000	−1.50644
$467$	− 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
$470$	54.0000i	0.114894i
$471$	0	0
$472$	−90.0000	−0.190678
$473$	750.000i	1.58562i
$474$	0	0
$475$	−256.000	−0.538947
$476$	450.000i	0.945378i
$477$	0	0
$478$	360.000	0.753138
$479$	− 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
$482$	− 42.0000i	− 0.0871369i
$483$	0	0
$484$	520.000	1.07438
$485$	− 255.000i	− 0.525773i
$486$	0	0
$487$	110.000	0.225873	0.112936	−	0.993602i	$-0.463974\pi$
$487$	110.000	0.225873	0.112936	+	0.993602i	$0.463974\pi$
$488$	− 228.000i	− 0.467213i
$489$	0	0
$490$	−216.000	−0.440816
$491$	− 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
$494$	− 480.000i	− 0.971660i
$495$	0	0
$496$	11.0000	0.0221774
$497$	450.000i	0.905433i
$498$	0	0
$499$	−766.000	−1.53507	−0.767535	−	0.641007i	$-0.778517\pi$
$499$	−766.000	−1.53507	−0.767535	+	0.641007i	$0.778517\pi$
$500$	− 615.000i	− 1.23000i
$501$	0	0
$502$	−270.000	−0.537849
$503$	− 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
$506$	540.000i	1.06719i
$507$	0	0
$508$	1025.00	2.01772
$509$	− 555.000i	− 1.09037i	−0.838315	−	0.545187i	$-0.816459\pi$
$509$	− 555.000i	− 1.09037i	0.838315	−	0.545187i	$-0.183541\pi$
$510$	0	0
$511$	325.000	0.636008
$512$	− 627.000i	− 1.22461i
$513$	0	0
$514$	−1314.00	−2.55642
$515$	510.000i	0.990291i
$516$	0	0
$517$	−90.0000	−0.174081
$518$	− 300.000i	− 0.579151i
$519$	0	0
$520$	90.0000	0.173077
$521$	450.000i	0.863724i	0.901940	+	0.431862i	$0.142143\pi$
$521$	450.000i	0.863724i	−0.901940	+	0.431862i	$0.857857\pi$
$522$	0	0
$523$	−250.000	−0.478011	−0.239006	−	0.971018i	$-0.576821\pi$
$523$	−250.000	−0.478011	−0.239006	+	0.971018i	$0.576821\pi$
$524$	75.0000i	0.143130i
$525$	0	0
$526$	828.000	1.57414
$527$	18.0000i	0.0341556i
$528$	0	0
$529$	385.000	0.727788
$530$	243.000i	0.458491i
$531$	0	0
$532$	400.000	0.751880
$533$	600.000i	1.12570i
$534$	0	0
$535$	567.000	1.05981
$536$	− 30.0000i	− 0.0559701i
$537$	0	0
$538$	−810.000	−1.50558
$539$	− 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	$0.579671\pi$
$542$	− 897.000i	− 1.65498i
$543$	0	0
$544$	810.000	1.48897
$545$	492.000i	0.902752i
$546$	0	0
$547$	410.000	0.749543	0.374771	−	0.927117i	$-0.377721\pi$
$547$	410.000	0.749543	0.374771	+	0.927117i	$0.377721\pi$
$548$	690.000i	1.25912i
$549$	0	0
$550$	720.000	1.30909
$551$	480.000i	0.871143i
$552$	0	0
$553$	70.0000	0.126582
$554$	− 420.000i	− 0.758123i
$555$	0	0
$556$	140.000	0.251799
$557$	− 639.000i	− 1.14722i	−0.819130	−	0.573609i	$-0.805543\pi$
$557$	− 639.000i	− 1.14722i	0.819130	−	0.573609i	$-0.194457\pi$
$558$	0	0
$559$	−500.000	−0.894454
$560$	− 165.000i	− 0.294643i
$561$	0	0
$562$	450.000	0.800712
$563$	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
$566$	840.000i	1.48410i
$567$	0	0
$568$	−270.000	−0.475352
$569$	240.000i	0.421793i	0.977508	+	0.210896i	$0.0676382\pi$
$569$	240.000i	0.421793i	−0.977508	+	0.210896i	$0.932362\pi$
$570$	0	0
$571$	−946.000	−1.65674	−0.828371	−	0.560179i	$-0.810732\pi$
$571$	−946.000	−1.65674	−0.828371	+	0.560179i	$0.810732\pi$
$572$	750.000i	1.31119i
$573$	0	0
$574$	−900.000	−1.56794
$575$	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	$0.244491\pi$
$578$	105.000i	0.181661i
$579$	0	0
$580$	−450.000	−0.775862
$581$	− 15.0000i	− 0.0258176i
$582$	0	0
$583$	−405.000	−0.694683
$584$	195.000i	0.333904i
$585$	0	0
$586$	−774.000	−1.32082
$587$	− 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
$590$	270.000i	0.457627i
$591$	0	0
$592$	−220.000	−0.371622
$593$	702.000i	1.18381i	0.806007	+	0.591906i	$0.201624\pi$
$593$	702.000i	1.18381i	−0.806007	+	0.591906i	$0.798376\pi$
$594$	0	0
$595$	270.000	0.453782
$596$	− 375.000i	− 0.629195i
$597$	0	0
$598$	−360.000	−0.602007
$599$	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	$0.242780\pi$
$602$	− 750.000i	− 1.24585i
$603$	0	0
$604$	−385.000	−0.637417
$605$	− 312.000i	− 0.515702i
$606$	0	0
$607$	530.000	0.873147	0.436573	−	0.899669i	$-0.356192\pi$
$607$	530.000	0.873147	0.436573	+	0.899669i	$0.356192\pi$
$608$	− 720.000i	− 1.18421i
$609$	0	0
$610$	−684.000	−1.12131
$611$	− 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	$0.518184\pi$
$614$	− 870.000i	− 1.41694i
$615$	0	0
$616$	−225.000	−0.365260
$617$	552.000i	0.894652i	0.894371	+	0.447326i	$0.147624\pi$
$617$	552.000i	0.894652i	−0.894371	+	0.447326i	$0.852376\pi$
$618$	0	0
$619$	662.000	1.06947	0.534733	−	0.845021i	$-0.320412\pi$
$619$	662.000	1.06947	0.534733	+	0.845021i	$0.320412\pi$
$620$	15.0000i	0.0241935i
$621$	0	0
$622$	1440.00	2.31511
$623$	− 450.000i	− 0.722311i
$624$	0	0
$625$	31.0000	0.0496000
$626$	− 555.000i	− 0.886581i
$627$	0	0
$628$	500.000	0.796178
$629$	− 360.000i	− 0.572337i
$630$	0	0
$631$	−331.000	−0.524564	−0.262282	−	0.964991i	$-0.584475\pi$
$631$	−331.000	−0.524564	−0.262282	+	0.964991i	$0.584475\pi$
$632$	42.0000i	0.0664557i
$633$	0	0
$634$	−549.000	−0.865931
$635$	− 615.000i	− 0.968504i
$636$	0	0
$637$	240.000	0.376766
$638$	− 1350.00i	− 2.11599i
$639$	0	0
$640$	279.000	0.435937
$641$	60.0000i	0.0936037i	0.998904	+	0.0468019i	$0.0149029\pi$
$641$	60.0000i	0.0936037i	−0.998904	+	0.0468019i	$0.985097\pi$
$642$	0	0
$643$	440.000	0.684292	0.342146	−	0.939647i	$-0.388846\pi$
$643$	440.000	0.684292	0.342146	+	0.939647i	$0.388846\pi$
$644$	− 300.000i	− 0.465839i
$645$	0	0
$646$	864.000	1.33746
$647$	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
$650$	480.000i	0.738462i
$651$	0	0
$652$	−550.000	−0.843558
$653$	− 483.000i	− 0.739663i	−0.929099	−	0.369832i	$-0.879415\pi$
$653$	− 483.000i	− 0.739663i	0.929099	−	0.369832i	$-0.120585\pi$
$654$	0	0
$655$	45.0000	0.0687023
$656$	660.000i	1.00610i
$657$	0	0
$658$	90.0000	0.136778
$659$	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	$0.747694\pi$
$662$	714.000i	1.07855i
$663$	0	0
$664$	9.00000	0.0135542
$665$	− 240.000i	− 0.360902i
$666$	0	0
$667$	360.000	0.539730
$668$	390.000i	0.583832i
$669$	0	0
$670$	−90.0000	−0.134328
$671$	− 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	$0.761319\pi$
$674$	30.0000i	0.0445104i
$675$	0	0
$676$	345.000	0.510355
$677$	− 354.000i	− 0.522895i	−0.965218	−	0.261448i	$-0.915800\pi$
$677$	− 354.000i	− 0.522895i	0.965218	−	0.261448i	$-0.0841998\pi$
$678$	0	0
$679$	−425.000	−0.625920
$680$	162.000i	0.238235i
$681$	0	0
$682$	−45.0000	−0.0659824
$683$	− 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
$686$	1095.00i	1.59621i
$687$	0	0
$688$	−550.000	−0.799419
$689$	− 270.000i	− 0.391872i
$690$	0	0
$691$	−436.000	−0.630970	−0.315485	−	0.948931i	$-0.602167\pi$
$691$	−436.000	−0.630970	−0.315485	+	0.948931i	$0.602167\pi$
$692$	− 885.000i	− 1.27890i
$693$	0	0
$694$	−207.000	−0.298271
$695$	− 84.0000i	− 0.120863i
$696$	0	0
$697$	−1080.00	−1.54950
$698$	768.000i	1.10029i
$699$	0	0
$700$	−400.000	−0.571429
$701$	− 135.000i	− 0.192582i	−0.995353	−	0.0962910i	$-0.969302\pi$
$701$	− 135.000i	− 0.192582i	0.995353	−	0.0962910i	$-0.0306979\pi$
$702$	0	0
$703$	−320.000	−0.455192
$704$	1365.00i	1.93892i
$705$	0	0
$706$	1368.00	1.93768
$707$	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	$0.492816\pi$
$710$	810.000i	1.14085i
$711$	0	0
$712$	270.000	0.379213
$713$	− 12.0000i	− 0.0168303i
$714$	0	0
$715$	450.000	0.629371
$716$	− 1125.00i	− 1.57123i
$717$	0	0
$718$	1350.00	1.88022
$719$	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
$722$	315.000i	0.436288i
$723$	0	0
$724$	80.0000	0.110497
$725$	− 480.000i	− 0.662069i
$726$	0	0
$727$	−175.000	−0.240715	−0.120358	−	0.992731i	$-0.538404\pi$
$727$	−175.000	−0.240715	−0.120358	+	0.992731i	$0.538404\pi$
$728$	− 150.000i	− 0.206044i
$729$	0	0
$730$	585.000	0.801370
$731$	− 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	$0.209421\pi$
$734$	1875.00i	2.55450i
$735$	0	0
$736$	−540.000	−0.733696
$737$	− 150.000i	− 0.203528i
$738$	0	0
$739$	−1006.00	−1.36130	−0.680650	−	0.732609i	$-0.738302\pi$
$739$	−1006.00	−1.36130	−0.680650	+	0.732609i	$0.738302\pi$
$740$	− 300.000i	− 0.405405i
$741$	0	0
$742$	405.000	0.545822
$743$	− 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
$746$	− 510.000i	− 0.683646i
$747$	0	0
$748$	−1350.00	−1.80481
$749$	− 945.000i	− 1.26168i
$750$	0	0
$751$	359.000	0.478029	0.239015	−	0.971016i	$-0.423176\pi$
$751$	359.000	0.478029	0.239015	+	0.971016i	$0.423176\pi$
$752$	− 66.0000i	− 0.0877660i
$753$	0	0
$754$	900.000	1.19363
$755$	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	$0.591667\pi$
$758$	− 2112.00i	− 2.78628i
$759$	0	0
$760$	144.000	0.189474
$761$	− 1320.00i	− 1.73456i	−0.497821	−	0.867280i	$-0.665866\pi$
$761$	− 1320.00i	− 1.73456i	0.497821	−	0.867280i	$-0.334134\pi$
$762$	0	0
$763$	820.000	1.07471
$764$	150.000i	0.196335i
$765$	0	0
$766$	−1854.00	−2.42037
$767$	− 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	$0.194753\pi$
$770$	675.000i	0.876623i
$771$	0	0
$772$	−1075.00	−1.39249
$773$	522.000i	0.675291i	0.941273	+	0.337646i	$0.109631\pi$
$773$	522.000i	0.675291i	−0.941273	+	0.337646i	$0.890369\pi$
$774$	0	0
$775$	−16.0000	−0.0206452
$776$	− 255.000i	− 0.328608i
$777$	0	0
$778$	−1575.00	−2.02442
$779$	960.000i	1.23235i
$780$	0	0
$781$	−1350.00	−1.72855
$782$	− 648.000i	− 0.828645i
$783$	0	0
$784$	264.000	0.336735
$785$	− 300.000i	− 0.382166i
$786$	0	0
$787$	−460.000	−0.584498	−0.292249	−	0.956342i	$-0.594404\pi$
$787$	−460.000	−0.584498	−0.292249	+	0.956342i	$0.594404\pi$
$788$	− 1035.00i	− 1.31345i
$789$	0	0
$790$	126.000	0.159494
$791$	− 120.000i	− 0.151707i
$792$	0	0
$793$	760.000	0.958386
$794$	210.000i	0.264484i
$795$	0	0
$796$	1115.00	1.40075
$797$	237.000i	0.297365i	0.988885	+	0.148683i	$0.0475033\pi$
$797$	237.000i	0.297365i	−0.988885	+	0.148683i	$0.952497\pi$
$798$	0	0
$799$	108.000	0.135169
$800$	720.000i	0.900000i
$801$	0	0
$802$	360.000	0.448878
$803$	975.000i	1.21420i
$804$	0	0
$805$	−180.000	−0.223602
$806$	− 30.0000i	− 0.0372208i
$807$	0	0
$808$	−585.000	−0.724010
$809$	− 810.000i	− 1.00124i	−0.865668	−	0.500618i	$-0.833106\pi$
$809$	− 810.000i	− 1.00124i	0.865668	−	0.500618i	$-0.166894\pi$
$810$	0	0
$811$	272.000	0.335388	0.167694	−	0.985839i	$-0.446368\pi$
$811$	272.000	0.335388	0.167694	+	0.985839i	$0.446368\pi$
$812$	750.000i	0.923645i
$813$	0	0
$814$	900.000	1.10565
$815$	330.000i	0.404908i
$816$	0	0
$817$	−800.000	−0.979192
$818$	− 807.000i	− 0.986553i
$819$	0	0
$820$	−900.000	−1.09756
$821$	− 390.000i	− 0.475030i	−0.971384	−	0.237515i	$-0.923667\pi$
$821$	− 390.000i	− 0.475030i	0.971384	−	0.237515i	$-0.0763330\pi$
$822$	0	0
$823$	1205.00	1.46416	0.732078	−	0.681221i	$-0.238551\pi$
$823$	1205.00	1.46416	0.732078	+	0.681221i	$0.238551\pi$
$824$	510.000i	0.618932i
$825$	0	0
$826$	450.000	0.544794
$827$	− 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	$0.164298\pi$
$830$	− 27.0000i	− 0.0325301i
$831$	0	0
$832$	−910.000	−1.09375
$833$	432.000i	0.518607i
$834$	0	0
$835$	234.000	0.280240
$836$	1200.00i	1.43541i
$837$	0	0
$838$	630.000	0.751790
$839$	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
$842$	− 1932.00i	− 2.29454i
$843$	0	0
$844$	1580.00	1.87204
$845$	− 207.000i	− 0.244970i
$846$	0	0
$847$	−520.000	−0.613932
$848$	− 297.000i	− 0.350236i
$849$	0	0
$850$	−864.000	−1.01647
$851$	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	$0.387595\pi$
$854$	1140.00i	1.33489i
$855$	0	0
$856$	567.000	0.662383
$857$	1302.00i	1.51925i	0.650359	+	0.759627i	$0.274618\pi$
$857$	1302.00i	1.51925i	−0.650359	+	0.759627i	$0.725382\pi$
$858$	0	0
$859$	−316.000	−0.367870	−0.183935	−	0.982938i	$-0.558884\pi$
$859$	−316.000	−0.367870	−0.183935	+	0.982938i	$0.558884\pi$
$860$	− 750.000i	− 0.872093i
$861$	0	0
$862$	−810.000	−0.939675
$863$	− 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
$866$	1695.00i	1.95727i
$867$	0	0
$868$	25.0000	0.0288018
$869$	210.000i	0.241657i
$870$	0	0
$871$	100.000	0.114811
$872$	492.000i	0.564220i
$873$	0	0
$874$	−576.000	−0.659039
$875$	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	$0.601525\pi$
$878$	633.000i	0.720957i
$879$	0	0
$880$	495.000	0.562500
$881$	90.0000i	0.102157i	0.998695	+	0.0510783i	$0.0162658\pi$
$881$	90.0000i	0.102157i	−0.998695	+	0.0510783i	$0.983734\pi$
$882$	0	0
$883$	−880.000	−0.996602	−0.498301	−	0.867004i	$-0.666043\pi$
$883$	−880.000	−0.996602	−0.498301	+	0.867004i	$0.666043\pi$
$884$	− 900.000i	− 1.01810i
$885$	0	0
$886$	−1494.00	−1.68623
$887$	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
$890$	− 810.000i	− 0.910112i
$891$	0	0
$892$	650.000	0.728700
$893$	− 96.0000i	− 0.107503i
$894$	0	0
$895$	−675.000	−0.754190
$896$	− 465.000i	− 0.518973i
$897$	0	0
$898$	1080.00	1.20267
$899$	30.0000i	0.0333704i
$900$	0	0
$901$	486.000	0.539401
$902$	− 2700.00i	− 2.99335i
$903$	0	0
$904$	72.0000	0.0796460
$905$	− 48.0000i	− 0.0530387i
$906$	0	0
$907$	−1300.00	−1.43330	−0.716648	−	0.697435i	$-0.754325\pi$
$907$	−1300.00	−1.43330	−0.716648	+	0.697435i	$0.754325\pi$
$908$	− 210.000i	− 0.231278i
$909$	0	0
$910$	−450.000	−0.494505
$911$	− 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
$914$	− 1095.00i	− 1.19803i
$915$	0	0
$916$	1130.00	1.23362
$917$	− 75.0000i	− 0.0817884i
$918$	0	0
$919$	137.000	0.149075	0.0745375	−	0.997218i	$-0.476252\pi$
$919$	137.000	0.149075	0.0745375	+	0.997218i	$0.476252\pi$
$920$	− 108.000i	− 0.117391i
$921$	0	0
$922$	315.000	0.341649
$923$	− 900.000i	− 0.975081i
$924$	0	0
$925$	320.000	0.345946
$926$	− 645.000i	− 0.696544i
$927$	0	0
$928$	1350.00	1.45474
$929$	− 660.000i	− 0.710441i	−0.934782	−	0.355221i	$-0.884406\pi$
$929$	− 660.000i	− 0.710441i	0.934782	−	0.355221i	$-0.115594\pi$
$930$	0	0
$931$	384.000	0.412460
$932$	1170.00i	1.25536i
$933$	0	0
$934$	−189.000	−0.202355
$935$	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	$0.395363\pi$
$938$	150.000i	0.159915i
$939$	0	0
$940$	90.0000	0.0957447
$941$	1605.00i	1.70563i	0.522211	+	0.852816i	$0.325107\pi$
$941$	1605.00i	1.70563i	−0.522211	+	0.852816i	$0.674893\pi$
$942$	0	0
$943$	720.000	0.763521
$944$	− 330.000i	− 0.349576i
$945$	0	0
$946$	2250.00	2.37844
$947$	− 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
$950$	768.000i	0.808421i
$951$	0	0
$952$	270.000	0.283613
$953$	− 144.000i	− 0.151102i	−0.997142	−	0.0755509i	$-0.975928\pi$
$953$	− 144.000i	− 0.151102i	0.997142	−	0.0755509i	$-0.0240715\pi$
$954$	0	0
$955$	90.0000	0.0942408
$956$	− 600.000i	− 0.627615i
$957$	0	0
$958$	−2250.00	−2.34864
$959$	− 690.000i	− 0.719499i
$960$	0	0
$961$	−960.000	−0.998959
$962$	600.000i	0.623701i
$963$	0	0
$964$	−70.0000	−0.0726141
$965$	645.000i	0.668394i
$966$	0	0
$967$	845.000	0.873837	0.436918	−	0.899501i	$-0.356070\pi$
$967$	845.000	0.873837	0.436918	+	0.899501i	$0.356070\pi$
$968$	− 312.000i	− 0.322314i
$969$	0	0
$970$	−765.000	−0.788660
$971$	− 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
$974$	− 330.000i	− 0.338809i
$975$	0	0
$976$	836.000	0.856557
$977$	246.000i	0.251791i	0.992043	+	0.125896i	$0.0401804\pi$
$977$	246.000i	0.251791i	−0.992043	+	0.125896i	$0.959820\pi$
$978$	0	0
$979$	1350.00	1.37896
$980$	360.000i	0.367347i
$981$	0	0
$982$	−1935.00	−1.97047
$983$	− 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
$986$	1620.00i	1.64300i
$987$	0	0
$988$	−800.000	−0.809717
$989$	600.000i	0.606673i
$990$	0	0
$991$	−1501.00	−1.51463	−0.757316	−	0.653049i	$-0.773490\pi$
$991$	−1501.00	−1.51463	−0.757316	+	0.653049i	$0.773490\pi$
$992$	− 45.0000i	− 0.0453629i
$993$	0	0
$994$	1350.00	1.35815
$995$	− 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	$0.373802\pi$
$998$	2298.00i	2.30261i
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.b.b.26.1	✓	2	1.1	even	1	trivial
27.3.b.b.26.2	yes	2	3.2	odd	2	inner
81.3.d.b.26.1		4	9.5	odd	6
81.3.d.b.26.2		4	9.4	even	3
81.3.d.b.53.1		4	9.7	even	3
81.3.d.b.53.2		4	9.2	odd	6
432.3.e.c.161.1		2	12.11	even	2
432.3.e.c.161.2		2	4.3	odd	2
675.3.c.h.26.1		2	15.14	odd	2
675.3.c.h.26.2		2	5.4	even	2
675.3.d.a.674.1		2	5.3	odd	4
675.3.d.a.674.2		2	15.2	even	4
675.3.d.d.674.1		2	15.8	even	4
675.3.d.d.674.2		2	5.2	odd	4
1296.3.q.j.593.1		4	36.23	even	6
1296.3.q.j.593.2		4	36.31	odd	6
1296.3.q.j.1025.1		4	36.7	odd	6
1296.3.q.j.1025.2		4	36.11	even	6
1728.3.e.g.1025.1		2	8.3	odd	2
1728.3.e.g.1025.2		2	24.11	even	2
1728.3.e.m.1025.1		2	8.5	even	2
1728.3.e.m.1025.2		2	24.5	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:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	27.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{2} -5.00000 q^{4} +3.00000i q^{5} +5.00000 q^{7} +3.00000i q^{8} +9.00000 q^{10} +15.0000i q^{11} -10.0000 q^{13} -15.0000i q^{14} -11.0000 q^{16} -18.0000i q^{17} -16.0000 q^{19} -15.0000i q^{20} +45.0000 q^{22} +12.0000i q^{23} +16.0000 q^{25} +30.0000i q^{26} -25.0000 q^{28} -30.0000i q^{29} -1.00000 q^{31} +45.0000i q^{32} -54.0000 q^{34} +15.0000i q^{35} +20.0000 q^{37} +48.0000i q^{38} -9.00000 q^{40} -60.0000i q^{41} +50.0000 q^{43} -75.0000i q^{44} +36.0000 q^{46} +6.00000i q^{47} -24.0000 q^{49} -48.0000i q^{50} +50.0000 q^{52} +27.0000i q^{53} -45.0000 q^{55} +15.0000i q^{56} -90.0000 q^{58} +30.0000i q^{59} -76.0000 q^{61} +3.00000i q^{62} +91.0000 q^{64} -30.0000i q^{65} -10.0000 q^{67} +90.0000i q^{68} +45.0000 q^{70} +90.0000i q^{71} +65.0000 q^{73} -60.0000i q^{74} +80.0000 q^{76} +75.0000i q^{77} +14.0000 q^{79} -33.0000i q^{80} -180.000 q^{82} -3.00000i q^{83} +54.0000 q^{85} -150.000i q^{86} -45.0000 q^{88} -90.0000i q^{89} -50.0000 q^{91} -60.0000i q^{92} +18.0000 q^{94} -48.0000i q^{95} -85.0000 q^{97} +72.0000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{4} + 10 q^{7} + 18 q^{10} - 20 q^{13} - 22 q^{16} - 32 q^{19} + 90 q^{22} + 32 q^{25} - 50 q^{28} - 2 q^{31} - 108 q^{34} + 40 q^{37} - 18 q^{40} + 100 q^{43} + 72 q^{46} - 48 q^{49} + 100 q^{52}+ \cdots - 170 q^{97}+O(q^{100}) \) 2 * q - 10 * q^4 + 10 * q^7 + 18 * q^10 - 20 * q^13 - 22 * q^16 - 32 * q^19 + 90 * q^22 + 32 * q^25 - 50 * q^28 - 2 * q^31 - 108 * q^34 + 40 * q^37 - 18 * q^40 + 100 * q^43 + 72 * q^46 - 48 * q^49 + 100 * q^52 - 90 * q^55 - 180 * q^58 - 152 * q^61 + 182 * q^64 - 20 * q^67 + 90 * q^70 + 130 * q^73 + 160 * q^76 + 28 * q^79 - 360 * q^82 + 108 * q^85 - 90 * q^88 - 100 * q^91 + 36 * q^94 - 170 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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