LMFDB - Embedded newform 135.3.c.c.26.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,3,Mod(26,135)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(135, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("135.26");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	135.c (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:	$3.67848356886$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.3
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	135.26
Dual form			135.3.c.c.26.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.381966i q^{2} +3.85410 q^{4} -2.23607i q^{5} +3.70820 q^{7} +3.00000i q^{8} +O(q^{10})$	\(q+0.381966i q^{2} +3.85410 q^{4} -2.23607i q^{5} +3.70820 q^{7} +3.00000i q^{8} +0.854102 q^{10} -8.18034i q^{11} +7.70820 q^{13} +1.41641i q^{14} +14.2705 q^{16} +11.9443i q^{17} +5.58359 q^{19} -8.61803i q^{20} +3.12461 q^{22} +28.4164i q^{23} -5.00000 q^{25} +2.94427i q^{26} +14.2918 q^{28} -56.0689i q^{29} -31.2492 q^{31} +17.4508i q^{32} -4.56231 q^{34} -8.29180i q^{35} -53.4164 q^{37} +2.13274i q^{38} +6.70820 q^{40} +60.7639i q^{41} -71.3738 q^{43} -31.5279i q^{44} -10.8541 q^{46} -46.1378i q^{47} -35.2492 q^{49} -1.90983i q^{50} +29.7082 q^{52} +73.3607i q^{53} -18.2918 q^{55} +11.1246i q^{56} +21.4164 q^{58} -90.7639i q^{59} +19.0000 q^{61} -11.9361i q^{62} +50.4164 q^{64} -17.2361i q^{65} +0.334369 q^{67} +46.0344i q^{68} +3.16718 q^{70} +81.2624i q^{71} +50.7902 q^{73} -20.4033i q^{74} +21.5197 q^{76} -30.3344i q^{77} -48.1672 q^{79} -31.9098i q^{80} -23.2098 q^{82} +62.6656i q^{83} +26.7082 q^{85} -27.2624i q^{86} +24.5410 q^{88} -69.7082i q^{89} +28.5836 q^{91} +109.520i q^{92} +17.6231 q^{94} -12.4853i q^{95} +159.416 q^{97} -13.4640i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 12 q^{7}+O(q^{10}) $ 4 * q + 2 * q^4 - 12 * q^7	$ 4 q + 2 q^{4} - 12 q^{7} - 10 q^{10} + 4 q^{13} - 10 q^{16} + 76 q^{19} - 68 q^{22} - 20 q^{25} + 84 q^{28} + 36 q^{31} + 22 q^{34} - 160 q^{37} - 44 q^{43} - 30 q^{46} + 20 q^{49} + 92 q^{52} - 100 q^{55} + 32 q^{58} + 76 q^{61} + 148 q^{64} + 216 q^{67} + 120 q^{70} - 92 q^{73} - 142 q^{76} - 300 q^{79} - 388 q^{82} + 80 q^{85} - 36 q^{88} + 168 q^{91} - 332 q^{94} + 584 q^{97}+O(q^{100}) $ 4 * q + 2 * q^4 - 12 * q^7 - 10 * q^10 + 4 * q^13 - 10 * q^16 + 76 * q^19 - 68 * q^22 - 20 * q^25 + 84 * q^28 + 36 * q^31 + 22 * q^34 - 160 * q^37 - 44 * q^43 - 30 * q^46 + 20 * q^49 + 92 * q^52 - 100 * q^55 + 32 * q^58 + 76 * q^61 + 148 * q^64 + 216 * q^67 + 120 * q^70 - 92 * q^73 - 142 * q^76 - 300 * q^79 - 388 * q^82 + 80 * q^85 - 36 * q^88 + 168 * q^91 - 332 * q^94 + 584 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/135\mathbb{Z}\right)^\times$.

$n$	$56$	$82$
$\chi(n)$	$-1$	$1$

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 $

$2$	0.381966i	0.190983i	0.995430	+	0.0954915i	$0.0304423\pi$
$2$	0.381966i	0.190983i	−0.995430	+	0.0954915i	$0.969558\pi$
$3$	0	0
$3$	0	0
$4$	3.85410	0.963525
$5$	− 2.23607i	− 0.447214i
$5$	− 2.23607i	− 0.447214i
$6$	0	0
$7$	3.70820	0.529743	0.264872	−	0.964284i	$-0.414670\pi$
$7$	3.70820	0.529743	0.264872	+	0.964284i	$0.414670\pi$
$8$	3.00000i	0.375000i
$9$	0	0
$10$	0.854102	0.0854102
$11$	− 8.18034i	− 0.743667i	−0.928299	−	0.371834i	$-0.878729\pi$
$11$	− 8.18034i	− 0.743667i	0.928299	−	0.371834i	$-0.121271\pi$
$12$	0	0
$13$	7.70820	0.592939	0.296469	−	0.955042i	$-0.404191\pi$
$13$	7.70820	0.592939	0.296469	+	0.955042i	$0.404191\pi$
$14$	1.41641i	0.101172i
$15$	0	0
$16$	14.2705	0.891907
$17$	11.9443i	0.702604i	0.936262	+	0.351302i	$0.114261\pi$
$17$	11.9443i	0.702604i	−0.936262	+	0.351302i	$0.885739\pi$
$18$	0	0
$19$	5.58359	0.293873	0.146937	−	0.989146i	$-0.453059\pi$
$19$	5.58359	0.293873	0.146937	+	0.989146i	$0.453059\pi$
$20$	− 8.61803i	− 0.430902i
$21$	0	0
$22$	3.12461	0.142028
$23$	28.4164i	1.23550i	0.786376	+	0.617748i	$0.211955\pi$
$23$	28.4164i	1.23550i	−0.786376	+	0.617748i	$0.788045\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	2.94427i	0.113241i
$27$	0	0
$28$	14.2918	0.510421
$29$	− 56.0689i	− 1.93341i	−0.255894	−	0.966705i	$-0.582370\pi$
$29$	− 56.0689i	− 1.93341i	0.255894	−	0.966705i	$-0.417630\pi$
$30$	0	0
$31$	−31.2492	−1.00804	−0.504020	−	0.863692i	$-0.668146\pi$
$31$	−31.2492	−1.00804	−0.504020	+	0.863692i	$0.668146\pi$
$32$	17.4508i	0.545339i
$33$	0	0
$34$	−4.56231	−0.134185
$35$	− 8.29180i	− 0.236908i
$36$	0	0
$37$	−53.4164	−1.44369	−0.721843	−	0.692056i	$-0.756705\pi$
$37$	−53.4164	−1.44369	−0.721843	+	0.692056i	$0.756705\pi$
$38$	2.13274i	0.0561248i
$39$	0	0
$40$	6.70820	0.167705
$41$	60.7639i	1.48205i	0.671479	+	0.741024i	$0.265659\pi$
$41$	60.7639i	1.48205i	−0.671479	+	0.741024i	$0.734341\pi$
$42$	0	0
$43$	−71.3738	−1.65986	−0.829928	−	0.557870i	$-0.811619\pi$
$43$	−71.3738	−1.65986	−0.829928	+	0.557870i	$0.811619\pi$
$44$	− 31.5279i	− 0.716542i
$45$	0	0
$46$	−10.8541	−0.235959
$47$	− 46.1378i	− 0.981655i	−0.871257	−	0.490827i	$-0.836695\pi$
$47$	− 46.1378i	− 0.981655i	0.871257	−	0.490827i	$-0.163305\pi$
$48$	0	0
$49$	−35.2492	−0.719372
$50$	− 1.90983i	− 0.0381966i
$51$	0	0
$52$	29.7082	0.571312
$53$	73.3607i	1.38416i	0.721819	+	0.692082i	$0.243306\pi$
$53$	73.3607i	1.38416i	−0.721819	+	0.692082i	$0.756694\pi$
$54$	0	0
$55$	−18.2918	−0.332578
$56$	11.1246i	0.198654i
$57$	0	0
$58$	21.4164	0.369248
$59$	− 90.7639i	− 1.53837i	−0.639025	−	0.769186i	$-0.720662\pi$
$59$	− 90.7639i	− 1.53837i	0.639025	−	0.769186i	$-0.279338\pi$
$60$	0	0
$61$	19.0000	0.311475	0.155738	−	0.987798i	$-0.450225\pi$
$61$	19.0000	0.311475	0.155738	+	0.987798i	$0.450225\pi$
$62$	− 11.9361i	− 0.192518i
$63$	0	0
$64$	50.4164	0.787756
$65$	− 17.2361i	− 0.265170i
$66$	0	0
$67$	0.334369	0.00499058	0.00249529	−	0.999997i	$-0.499206\pi$
$67$	0.334369	0.00499058	0.00249529	+	0.999997i	$0.499206\pi$
$68$	46.0344i	0.676977i
$69$	0	0
$70$	3.16718	0.0452455
$71$	81.2624i	1.14454i	0.820065	+	0.572270i	$0.193937\pi$
$71$	81.2624i	1.14454i	−0.820065	+	0.572270i	$0.806063\pi$
$72$	0	0
$73$	50.7902	0.695757	0.347878	−	0.937540i	$-0.386902\pi$
$73$	50.7902	0.695757	0.347878	+	0.937540i	$0.386902\pi$
$74$	− 20.4033i	− 0.275720i
$75$	0	0
$76$	21.5197	0.283154
$77$	− 30.3344i	− 0.393953i
$78$	0	0
$79$	−48.1672	−0.609711	−0.304856	−	0.952399i	$-0.598608\pi$
$79$	−48.1672	−0.609711	−0.304856	+	0.952399i	$0.598608\pi$
$80$	− 31.9098i	− 0.398873i
$81$	0	0
$82$	−23.2098	−0.283046
$83$	62.6656i	0.755008i	0.926008	+	0.377504i	$0.123217\pi$
$83$	62.6656i	0.755008i	−0.926008	+	0.377504i	$0.876783\pi$
$84$	0	0
$85$	26.7082	0.314214
$86$	− 27.2624i	− 0.317004i
$87$	0	0
$88$	24.5410	0.278875
$89$	− 69.7082i	− 0.783238i	−0.920127	−	0.391619i	$-0.871915\pi$
$89$	− 69.7082i	− 0.783238i	0.920127	−	0.391619i	$-0.128085\pi$
$90$	0	0
$91$	28.5836	0.314105
$92$	109.520i	1.19043i
$93$	0	0
$94$	17.6231	0.187479
$95$	− 12.4853i	− 0.131424i
$96$	0	0
$97$	159.416	1.64347	0.821734	−	0.569871i	$-0.193007\pi$
$97$	159.416	1.64347	0.821734	+	0.569871i	$0.193007\pi$
$98$	− 13.4640i	− 0.137388i
$99$	0	0
$100$	−19.2705	−0.192705
$101$	24.0000i	0.237624i	0.992917	+	0.118812i	$0.0379085\pi$
$101$	24.0000i	0.237624i	−0.992917	+	0.118812i	$0.962091\pi$
$102$	0	0
$103$	−25.1672	−0.244342	−0.122171	−	0.992509i	$-0.538986\pi$
$103$	−25.1672	−0.244342	−0.122171	+	0.992509i	$0.538986\pi$
$104$	23.1246i	0.222352i
$105$	0	0
$106$	−28.0213	−0.264352
$107$	47.6656i	0.445473i	0.974879	+	0.222737i	$0.0714990\pi$
$107$	47.6656i	0.445473i	−0.974879	+	0.222737i	$0.928501\pi$
$108$	0	0
$109$	164.915	1.51298	0.756490	−	0.654005i	$-0.226913\pi$
$109$	164.915	1.51298	0.756490	+	0.654005i	$0.226913\pi$
$110$	− 6.98684i	− 0.0635168i
$111$	0	0
$112$	52.9180	0.472482
$113$	− 68.3870i	− 0.605195i	−0.953118	−	0.302597i	$-0.902146\pi$
$113$	− 68.3870i	− 0.605195i	0.953118	−	0.302597i	$-0.0978537\pi$
$114$	0	0
$115$	63.5410	0.552531
$116$	− 216.095i	− 1.86289i
$117$	0	0
$118$	34.6687	0.293803
$119$	44.2918i	0.372200i
$120$	0	0
$121$	54.0820	0.446959
$122$	7.25735i	0.0594865i
$123$	0	0
$124$	−120.438	−0.971272
$125$	11.1803i	0.0894427i
$126$	0	0
$127$	−200.748	−1.58069	−0.790345	−	0.612662i	$-0.790099\pi$
$127$	−200.748	−1.58069	−0.790345	+	0.612662i	$0.790099\pi$
$128$	89.0608i	0.695787i
$129$	0	0
$130$	6.58359	0.0506430
$131$	− 8.16408i	− 0.0623212i	−0.999514	−	0.0311606i	$-0.990080\pi$
$131$	− 8.16408i	− 0.0623212i	0.999514	−	0.0311606i	$-0.00992033\pi$
$132$	0	0
$133$	20.7051	0.155677
$134$	0.127717i	0 0.000953115i
$135$	0	0
$136$	−35.8328	−0.263477
$137$	50.3313i	0.367381i	0.982984	+	0.183691i	$0.0588045\pi$
$137$	50.3313i	0.367381i	−0.982984	+	0.183691i	$0.941196\pi$
$138$	0	0
$139$	−126.498	−0.910061	−0.455030	−	0.890476i	$-0.650372\pi$
$139$	−126.498	−0.910061	−0.455030	+	0.890476i	$0.650372\pi$
$140$	− 31.9574i	− 0.228267i
$141$	0	0
$142$	−31.0395	−0.218588
$143$	− 63.0557i	− 0.440949i
$144$	0	0
$145$	−125.374	−0.864647
$146$	19.4001i	0.132878i
$147$	0	0
$148$	−205.872	−1.39103
$149$	− 15.8197i	− 0.106172i	−0.998590	−	0.0530861i	$-0.983094\pi$
$149$	− 15.8197i	− 0.106172i	0.998590	−	0.0530861i	$-0.0169058\pi$
$150$	0	0
$151$	−138.997	−0.920509	−0.460255	−	0.887787i	$-0.652242\pi$
$151$	−138.997	−0.920509	−0.460255	+	0.887787i	$0.652242\pi$
$152$	16.7508i	0.110202i
$153$	0	0
$154$	11.5867	0.0752383
$155$	69.8754i	0.450809i
$156$	0	0
$157$	190.705	1.21468	0.607341	−	0.794441i	$-0.292236\pi$
$157$	190.705	1.21468	0.607341	+	0.794441i	$0.292236\pi$
$158$	− 18.3982i	− 0.116444i
$159$	0	0
$160$	39.0213	0.243883
$161$	105.374i	0.654496i
$162$	0	0
$163$	−36.8328	−0.225968	−0.112984	−	0.993597i	$-0.536041\pi$
$163$	−36.8328	−0.225968	−0.112984	+	0.993597i	$0.536041\pi$
$164$	234.190i	1.42799i
$165$	0	0
$166$	−23.9361	−0.144194
$167$	− 258.580i	− 1.54839i	−0.632950	−	0.774193i	$-0.718156\pi$
$167$	− 258.580i	− 1.54839i	0.632950	−	0.774193i	$-0.281844\pi$
$168$	0	0
$169$	−109.584	−0.648424
$170$	10.2016i	0.0600096i
$171$	0	0
$172$	−275.082	−1.59931
$173$	− 182.331i	− 1.05394i	−0.849885	−	0.526969i	$-0.823328\pi$
$173$	− 182.331i	− 1.05394i	0.849885	−	0.526969i	$-0.176672\pi$
$174$	0	0
$175$	−18.5410	−0.105949
$176$	− 116.738i	− 0.663282i
$177$	0	0
$178$	26.6262	0.149585
$179$	256.774i	1.43449i	0.696820	+	0.717246i	$0.254598\pi$
$179$	256.774i	1.43449i	−0.696820	+	0.717246i	$0.745402\pi$
$180$	0	0
$181$	180.082	0.994928	0.497464	−	0.867485i	$-0.334265\pi$
$181$	180.082	0.994928	0.497464	+	0.867485i	$0.334265\pi$
$182$	10.9180i	0.0599888i
$183$	0	0
$184$	−85.2492	−0.463311
$185$	119.443i	0.645636i
$186$	0	0
$187$	97.7082	0.522504
$188$	− 177.820i	− 0.945849i
$189$	0	0
$190$	4.76896	0.0250998
$191$	128.387i	0.672183i	0.941829	+	0.336092i	$0.109105\pi$
$191$	128.387i	0.672183i	−0.941829	+	0.336092i	$0.890895\pi$
$192$	0	0
$193$	64.2918	0.333118	0.166559	−	0.986031i	$-0.446734\pi$
$193$	64.2918	0.333118	0.166559	+	0.986031i	$0.446734\pi$
$194$	60.8916i	0.313874i
$195$	0	0
$196$	−135.854	−0.693133
$197$	− 217.997i	− 1.10658i	−0.832988	−	0.553292i	$-0.813372\pi$
$197$	− 217.997i	− 1.10658i	0.832988	−	0.553292i	$-0.186628\pi$
$198$	0	0
$199$	320.249	1.60929	0.804646	−	0.593754i	$-0.202355\pi$
$199$	320.249	1.60929	0.804646	+	0.593754i	$0.202355\pi$
$200$	− 15.0000i	− 0.0750000i
$201$	0	0
$202$	−9.16718	−0.0453821
$203$	− 207.915i	− 1.02421i
$204$	0	0
$205$	135.872	0.662792
$206$	− 9.61301i	− 0.0466651i
$207$	0	0
$208$	110.000	0.528846
$209$	− 45.6757i	− 0.218544i
$210$	0	0
$211$	−63.8328	−0.302525	−0.151263	−	0.988494i	$-0.548334\pi$
$211$	−63.8328	−0.302525	−0.151263	+	0.988494i	$0.548334\pi$
$212$	282.740i	1.33368i
$213$	0	0
$214$	−18.2067	−0.0850778
$215$	159.597i	0.742310i
$216$	0	0
$217$	−115.878	−0.534002
$218$	62.9919i	0.288954i
$219$	0	0
$220$	−70.4984	−0.320447
$221$	92.0689i	0.416601i
$222$	0	0
$223$	−8.24922	−0.0369920	−0.0184960	−	0.999829i	$-0.505888\pi$
$223$	−8.24922	−0.0369920	−0.0184960	+	0.999829i	$0.505888\pi$
$224$	64.7113i	0.288890i
$225$	0	0
$226$	26.1215	0.115582
$227$	− 339.525i	− 1.49570i	−0.663866	−	0.747852i	$-0.731085\pi$
$227$	− 339.525i	− 1.49570i	0.663866	−	0.747852i	$-0.268915\pi$
$228$	0	0
$229$	202.495	0.884259	0.442130	−	0.896951i	$-0.354223\pi$
$229$	202.495	0.884259	0.442130	+	0.896951i	$0.354223\pi$
$230$	24.2705i	0.105524i
$231$	0	0
$232$	168.207	0.725029
$233$	19.0820i	0.0818972i	0.999161	+	0.0409486i	$0.0130380\pi$
$233$	19.0820i	0.0818972i	−0.999161	+	0.0409486i	$0.986962\pi$
$234$	0	0
$235$	−103.167	−0.439009
$236$	− 349.813i	− 1.48226i
$237$	0	0
$238$	−16.9180	−0.0710839
$239$	272.610i	1.14063i	0.821427	+	0.570314i	$0.193178\pi$
$239$	272.610i	1.14063i	−0.821427	+	0.570314i	$0.806822\pi$
$240$	0	0
$241$	232.580	0.965064	0.482532	−	0.875878i	$-0.339717\pi$
$241$	232.580	0.965064	0.482532	+	0.875878i	$0.339717\pi$
$242$	20.6575i	0.0853616i
$243$	0	0
$244$	73.2279	0.300114
$245$	78.8197i	0.321713i
$246$	0	0
$247$	43.0395	0.174249
$248$	− 93.7477i	− 0.378015i
$249$	0	0
$250$	−4.27051	−0.0170820
$251$	14.4195i	0.0574483i	0.999587	+	0.0287241i	$0.00914443\pi$
$251$	14.4195i	0.0574483i	−0.999587	+	0.0287241i	$0.990856\pi$
$252$	0	0
$253$	232.456	0.918798
$254$	− 76.6788i	− 0.301885i
$255$	0	0
$256$	167.647	0.654873
$257$	165.748i	0.644933i	0.946581	+	0.322466i	$0.104512\pi$
$257$	165.748i	0.644933i	−0.946581	+	0.322466i	$0.895488\pi$
$258$	0	0
$259$	−198.079	−0.764784
$260$	− 66.4296i	− 0.255498i
$261$	0	0
$262$	3.11840	0.0119023
$263$	− 520.026i	− 1.97729i	−0.150282	−	0.988643i	$-0.548018\pi$
$263$	− 520.026i	− 1.97729i	0.150282	−	0.988643i	$-0.451982\pi$
$264$	0	0
$265$	164.039	0.619017
$266$	7.90864i	0.0297317i
$267$	0	0
$268$	1.28869	0.00480855
$269$	87.3576i	0.324749i	0.986729	+	0.162375i	$0.0519153\pi$
$269$	87.3576i	0.324749i	−0.986729	+	0.162375i	$0.948085\pi$
$270$	0	0
$271$	35.5836	0.131305	0.0656524	−	0.997843i	$-0.479087\pi$
$271$	35.5836	0.131305	0.0656524	+	0.997843i	$0.479087\pi$
$272$	170.451i	0.626658i
$273$	0	0
$274$	−19.2248	−0.0701636
$275$	40.9017i	0.148733i
$276$	0	0
$277$	147.374	0.532036	0.266018	−	0.963968i	$-0.414292\pi$
$277$	147.374	0.532036	0.266018	+	0.963968i	$0.414292\pi$
$278$	− 48.3181i	− 0.173806i
$279$	0	0
$280$	24.8754	0.0888407
$281$	− 190.869i	− 0.679250i	−0.940561	−	0.339625i	$-0.889700\pi$
$281$	− 190.869i	− 0.679250i	0.940561	−	0.339625i	$-0.110300\pi$
$282$	0	0
$283$	117.708	0.415930	0.207965	−	0.978136i	$-0.433316\pi$
$283$	117.708	0.415930	0.207965	+	0.978136i	$0.433316\pi$
$284$	313.193i	1.10279i
$285$	0	0
$286$	24.0851	0.0842138
$287$	225.325i	0.785105i
$288$	0	0
$289$	146.334	0.506347
$290$	− 47.8885i	− 0.165133i
$291$	0	0
$292$	195.751	0.670379
$293$	413.243i	1.41039i	0.709016	+	0.705193i	$0.249140\pi$
$293$	413.243i	1.41039i	−0.709016	+	0.705193i	$0.750860\pi$
$294$	0	0
$295$	−202.954	−0.687981
$296$	− 160.249i	− 0.541383i
$297$	0	0
$298$	6.04257	0.0202771
$299$	219.039i	0.732573i
$300$	0	0
$301$	−264.669	−0.879298
$302$	− 53.0921i	− 0.175802i
$303$	0	0
$304$	79.6807	0.262108
$305$	− 42.4853i	− 0.139296i
$306$	0	0
$307$	−458.158	−1.49237	−0.746185	−	0.665738i	$-0.768117\pi$
$307$	−458.158	−1.49237	−0.746185	+	0.665738i	$0.768117\pi$
$308$	− 116.912i	− 0.379584i
$309$	0	0
$310$	−26.6900	−0.0860968
$311$	− 118.043i	− 0.379558i	−0.981827	−	0.189779i	$-0.939223\pi$
$311$	− 118.043i	− 0.379558i	0.981827	−	0.189779i	$-0.0607772\pi$
$312$	0	0
$313$	−412.827	−1.31893	−0.659467	−	0.751733i	$-0.729218\pi$
$313$	−412.827	−1.31893	−0.659467	+	0.751733i	$0.729218\pi$
$314$	72.8429i	0.231984i
$315$	0	0
$316$	−185.641	−0.587472
$317$	− 269.944i	− 0.851559i	−0.904827	−	0.425780i	$-0.860000\pi$
$317$	− 269.944i	− 0.851559i	0.904827	−	0.425780i	$-0.140000\pi$
$318$	0	0
$319$	−458.663	−1.43781
$320$	− 112.735i	− 0.352295i
$321$	0	0
$322$	−40.2492	−0.124998
$323$	66.6919i	0.206477i
$324$	0	0
$325$	−38.5410	−0.118588
$326$	− 14.0689i	− 0.0431561i
$327$	0	0
$328$	−182.292	−0.555768
$329$	− 171.088i	− 0.520025i
$330$	0	0
$331$	9.91486	0.0299542	0.0149771	−	0.999888i	$-0.495232\pi$
$331$	9.91486	0.0299542	0.0149771	+	0.999888i	$0.495232\pi$
$332$	241.520i	0.727469i
$333$	0	0
$334$	98.7690	0.295715
$335$	− 0.747671i	− 0.00223185i
$336$	0	0
$337$	−436.450	−1.29510	−0.647551	−	0.762022i	$-0.724207\pi$
$337$	−436.450	−1.29510	−0.647551	+	0.762022i	$0.724207\pi$
$338$	− 41.8572i	− 0.123838i
$339$	0	0
$340$	102.936	0.302753
$341$	255.629i	0.749646i
$342$	0	0
$343$	−312.413	−0.910826
$344$	− 214.122i	− 0.622446i
$345$	0	0
$346$	69.6443	0.201284
$347$	− 25.3839i	− 0.0731524i	−0.999331	−	0.0365762i	$-0.988355\pi$
$347$	− 25.3839i	− 0.0731524i	0.999331	−	0.0365762i	$-0.0116452\pi$
$348$	0	0
$349$	−72.0820	−0.206539	−0.103269	−	0.994653i	$-0.532930\pi$
$349$	−72.0820	−0.206539	−0.103269	+	0.994653i	$0.532930\pi$
$350$	− 7.08204i	− 0.0202344i
$351$	0	0
$352$	142.754	0.405551
$353$	392.164i	1.11095i	0.831534	+	0.555473i	$0.187463\pi$
$353$	392.164i	1.11095i	−0.831534	+	0.555473i	$0.812537\pi$
$354$	0	0
$355$	181.708	0.511854
$356$	− 268.663i	− 0.754670i
$357$	0	0
$358$	−98.0789	−0.273964
$359$	− 399.994i	− 1.11419i	−0.830449	−	0.557094i	$-0.811916\pi$
$359$	− 399.994i	− 1.11419i	0.830449	−	0.557094i	$-0.188084\pi$
$360$	0	0
$361$	−329.823	−0.913639
$362$	68.7852i	0.190014i
$363$	0	0
$364$	110.164	0.302649
$365$	− 113.570i	− 0.311152i
$366$	0	0
$367$	206.371	0.562318	0.281159	−	0.959661i	$-0.409281\pi$
$367$	206.371	0.562318	0.281159	+	0.959661i	$0.409281\pi$
$368$	405.517i	1.10195i
$369$	0	0
$370$	−45.6231	−0.123306
$371$	272.036i	0.733252i
$372$	0	0
$373$	354.833	0.951294	0.475647	−	0.879636i	$-0.342214\pi$
$373$	354.833	0.951294	0.475647	+	0.879636i	$0.342214\pi$
$374$	37.3212i	0.0997893i
$375$	0	0
$376$	138.413	0.368120
$377$	− 432.190i	− 1.14639i
$378$	0	0
$379$	−423.997	−1.11873	−0.559363	−	0.828923i	$-0.688954\pi$
$379$	−423.997	−1.11873	−0.559363	+	0.828923i	$0.688954\pi$
$380$	− 48.1196i	− 0.126630i
$381$	0	0
$382$	−49.0395	−0.128376
$383$	− 207.158i	− 0.540882i	−0.962737	−	0.270441i	$-0.912830\pi$
$383$	− 207.158i	− 0.540882i	0.962737	−	0.270441i	$-0.0871695\pi$
$384$	0	0
$385$	−67.8297	−0.176181
$386$	24.5573i	0.0636199i
$387$	0	0
$388$	614.407	1.58352
$389$	− 439.957i	− 1.13100i	−0.824750	−	0.565498i	$-0.808684\pi$
$389$	− 439.957i	− 1.13100i	0.824750	−	0.565498i	$-0.191316\pi$
$390$	0	0
$391$	−339.413	−0.868065
$392$	− 105.748i	− 0.269764i
$393$	0	0
$394$	83.2674	0.211339
$395$	107.705i	0.272671i
$396$	0	0
$397$	453.872	1.14326	0.571628	−	0.820513i	$-0.306312\pi$
$397$	453.872	1.14326	0.571628	+	0.820513i	$0.306312\pi$
$398$	122.324i	0.307348i
$399$	0	0
$400$	−71.3525	−0.178381
$401$	556.407i	1.38755i	0.720192	+	0.693774i	$0.244054\pi$
$401$	556.407i	1.38755i	−0.720192	+	0.693774i	$0.755946\pi$
$402$	0	0
$403$	−240.875	−0.597706
$404$	92.4984i	0.228957i
$405$	0	0
$406$	79.4164	0.195607
$407$	436.964i	1.07362i
$408$	0	0
$409$	75.6625	0.184994	0.0924970	−	0.995713i	$-0.470515\pi$
$409$	75.6625	0.184994	0.0924970	+	0.995713i	$0.470515\pi$
$410$	51.8986i	0.126582i
$411$	0	0
$412$	−96.9969	−0.235429
$413$	− 336.571i	− 0.814942i
$414$	0	0
$415$	140.125	0.337650
$416$	134.515i	0.323353i
$417$	0	0
$418$	17.4466	0.0417382
$419$	− 521.060i	− 1.24358i	−0.783184	−	0.621789i	$-0.786406\pi$
$419$	− 521.060i	− 1.24358i	0.783184	−	0.621789i	$-0.213594\pi$
$420$	0	0
$421$	114.745	0.272552	0.136276	−	0.990671i	$-0.456487\pi$
$421$	114.745	0.272552	0.136276	+	0.990671i	$0.456487\pi$
$422$	− 24.3820i	− 0.0577772i
$423$	0	0
$424$	−220.082	−0.519061
$425$	− 59.7214i	− 0.140521i
$426$	0	0
$427$	70.4559	0.165002
$428$	183.708i	0.429225i
$429$	0	0
$430$	−60.9605	−0.141769
$431$	233.204i	0.541076i	0.962709	+	0.270538i	$0.0872015\pi$
$431$	233.204i	0.541076i	−0.962709	+	0.270538i	$0.912799\pi$
$432$	0	0
$433$	−501.702	−1.15867	−0.579333	−	0.815091i	$-0.696687\pi$
$433$	−501.702	−1.15867	−0.579333	+	0.815091i	$0.696687\pi$
$434$	− 44.2616i	− 0.101985i
$435$	0	0
$436$	635.599	1.45780
$437$	158.666i	0.363079i
$438$	0	0
$439$	365.997	0.833706	0.416853	−	0.908974i	$-0.363133\pi$
$439$	365.997	0.833706	0.416853	+	0.908974i	$0.363133\pi$
$440$	− 54.8754i	− 0.124717i
$441$	0	0
$442$	−35.1672	−0.0795638
$443$	632.902i	1.42867i	0.699802	+	0.714337i	$0.253272\pi$
$443$	632.902i	1.42867i	−0.699802	+	0.714337i	$0.746728\pi$
$444$	0	0
$445$	−155.872	−0.350275
$446$	− 3.15092i	− 0.00706485i
$447$	0	0
$448$	186.954	0.417309
$449$	− 796.344i	− 1.77360i	−0.462158	−	0.886798i	$-0.652925\pi$
$449$	− 796.344i	− 1.77360i	0.462158	−	0.886798i	$-0.347075\pi$
$450$	0	0
$451$	497.070	1.10215
$452$	− 263.570i	− 0.583120i
$453$	0	0
$454$	129.687	0.285654
$455$	− 63.9149i	− 0.140472i
$456$	0	0
$457$	−202.577	−0.443277	−0.221638	−	0.975129i	$-0.571140\pi$
$457$	−202.577	−0.443277	−0.221638	+	0.975129i	$0.571140\pi$
$458$	77.3463i	0.168878i
$459$	0	0
$460$	244.894	0.532377
$461$	727.909i	1.57898i	0.613765	+	0.789489i	$0.289654\pi$
$461$	727.909i	1.57898i	−0.613765	+	0.789489i	$0.710346\pi$
$462$	0	0
$463$	493.003	1.06480	0.532401	−	0.846492i	$-0.321290\pi$
$463$	493.003	1.06480	0.532401	+	0.846492i	$0.321290\pi$
$464$	− 800.132i	− 1.72442i
$465$	0	0
$466$	−7.28869	−0.0156410
$467$	− 453.190i	− 0.970429i	−0.874395	−	0.485215i	$-0.838741\pi$
$467$	− 453.190i	− 0.970429i	0.874395	−	0.485215i	$-0.161259\pi$
$468$	0	0
$469$	1.23991	0.00264372
$470$	− 39.4064i	− 0.0838433i
$471$	0	0
$472$	272.292	0.576889
$473$	583.862i	1.23438i
$474$	0	0
$475$	−27.9180	−0.0587747
$476$	170.705i	0.358624i
$477$	0	0
$478$	−104.128	−0.217840
$479$	319.544i	0.667107i	0.942731	+	0.333553i	$0.108248\pi$
$479$	319.544i	0.667107i	−0.942731	+	0.333553i	$0.891752\pi$
$480$	0	0
$481$	−411.745	−0.856018
$482$	88.8378i	0.184311i
$483$	0	0
$484$	208.438	0.430656
$485$	− 356.466i	− 0.734981i
$486$	0	0
$487$	792.869	1.62807	0.814034	−	0.580817i	$-0.197267\pi$
$487$	792.869	1.62807	0.814034	+	0.580817i	$0.197267\pi$
$488$	57.0000i	0.116803i
$489$	0	0
$490$	−30.1064	−0.0614417
$491$	− 369.437i	− 0.752417i	−0.926535	−	0.376208i	$-0.877228\pi$
$491$	− 369.437i	− 0.752417i	0.926535	−	0.376208i	$-0.122772\pi$
$492$	0	0
$493$	669.702	1.35842
$494$	16.4396i	0.0332786i
$495$	0	0
$496$	−445.942	−0.899077
$497$	301.337i	0.606313i
$498$	0	0
$499$	609.912	1.22227	0.611134	−	0.791527i	$-0.290714\pi$
$499$	609.912	1.22227	0.611134	+	0.791527i	$0.290714\pi$
$500$	43.0902i	0.0861803i
$501$	0	0
$502$	−5.50776	−0.0109716
$503$	305.308i	0.606974i	0.952836	+	0.303487i	$0.0981509\pi$
$503$	305.308i	0.606974i	−0.952836	+	0.303487i	$0.901849\pi$
$504$	0	0
$505$	53.6656	0.106269
$506$	88.7902i	0.175475i
$507$	0	0
$508$	−773.702	−1.52304
$509$	− 204.478i	− 0.401726i	−0.979619	−	0.200863i	$-0.935625\pi$
$509$	− 204.478i	− 0.401726i	0.979619	−	0.200863i	$-0.0643745\pi$
$510$	0	0
$511$	188.341	0.368573
$512$	420.279i	0.820857i
$513$	0	0
$514$	−63.3100	−0.123171
$515$	56.2755i	0.109273i
$516$	0	0
$517$	−377.423	−0.730024
$518$	− 75.6594i	− 0.146061i
$519$	0	0
$520$	51.7082	0.0994389
$521$	− 590.227i	− 1.13287i	−0.824105	−	0.566436i	$-0.808322\pi$
$521$	− 590.227i	− 1.13287i	0.824105	−	0.566436i	$-0.191678\pi$
$522$	0	0
$523$	−19.7933	−0.0378458	−0.0189229	−	0.999821i	$-0.506024\pi$
$523$	−19.7933	−0.0378458	−0.0189229	+	0.999821i	$0.506024\pi$
$524$	− 31.4652i	− 0.0600481i
$525$	0	0
$526$	198.632	0.377628
$527$	− 373.249i	− 0.708253i
$528$	0	0
$529$	−278.492	−0.526450
$530$	62.6575i	0.118222i
$531$	0	0
$532$	79.7996	0.149999
$533$	468.381i	0.878763i
$534$	0	0
$535$	106.584	0.199222
$536$	1.00311i	0.00187147i
$537$	0	0
$538$	−33.3676	−0.0620216
$539$	288.351i	0.534973i
$540$	0	0
$541$	428.164	0.791431	0.395715	−	0.918373i	$-0.370497\pi$
$541$	428.164	0.791431	0.395715	+	0.918373i	$0.370497\pi$
$542$	13.5917i	0.0250770i
$543$	0	0
$544$	−208.438	−0.383158
$545$	− 368.761i	− 0.676625i
$546$	0	0
$547$	−982.705	−1.79654	−0.898268	−	0.439448i	$-0.855174\pi$
$547$	−982.705	−1.79654	−0.898268	+	0.439448i	$0.855174\pi$
$548$	193.982i	0.353981i
$549$	0	0
$550$	−15.6231	−0.0284056
$551$	− 313.066i	− 0.568177i
$552$	0	0
$553$	−178.614	−0.322990
$554$	56.2918i	0.101610i
$555$	0	0
$556$	−487.538	−0.876867
$557$	310.839i	0.558059i	0.960282	+	0.279030i	$0.0900128\pi$
$557$	310.839i	0.558059i	−0.960282	+	0.279030i	$0.909987\pi$
$558$	0	0
$559$	−550.164	−0.984193
$560$	− 118.328i	− 0.211300i
$561$	0	0
$562$	72.9055	0.129725
$563$	− 850.997i	− 1.51154i	−0.654837	−	0.755770i	$-0.727263\pi$
$563$	− 850.997i	− 1.51154i	0.654837	−	0.755770i	$-0.272737\pi$
$564$	0	0
$565$	−152.918	−0.270651
$566$	44.9605i	0.0794356i
$567$	0	0
$568$	−243.787	−0.429203
$569$	− 292.377i	− 0.513843i	−0.966432	−	0.256922i	$-0.917292\pi$
$569$	− 292.377i	− 0.513843i	0.966432	−	0.256922i	$-0.0827083\pi$
$570$	0	0
$571$	−357.754	−0.626539	−0.313270	−	0.949664i	$-0.601424\pi$
$571$	−357.754	−0.626539	−0.313270	+	0.949664i	$0.601424\pi$
$572$	− 243.023i	− 0.424866i
$573$	0	0
$574$	−86.0665	−0.149942
$575$	− 142.082i	− 0.247099i
$576$	0	0
$577$	216.456	0.375140	0.187570	−	0.982251i	$-0.439939\pi$
$577$	216.456	0.375140	0.187570	+	0.982251i	$0.439939\pi$
$578$	55.8948i	0.0967037i
$579$	0	0
$580$	−483.204	−0.833110
$581$	232.377i	0.399960i
$582$	0	0
$583$	600.115	1.02936
$584$	152.371i	0.260909i
$585$	0	0
$586$	−157.845	−0.269360
$587$	578.489i	0.985501i	0.870171	+	0.492751i	$0.164008\pi$
$587$	578.489i	0.985501i	−0.870171	+	0.492751i	$0.835992\pi$
$588$	0	0
$589$	−174.483	−0.296236
$590$	− 77.5217i	− 0.131393i
$591$	0	0
$592$	−762.279	−1.28763
$593$	159.971i	0.269765i	0.990862	+	0.134882i	$0.0430657\pi$
$593$	159.971i	0.269765i	−0.990862	+	0.134882i	$0.956934\pi$
$594$	0	0
$595$	99.0395	0.166453
$596$	− 60.9706i	− 0.102300i
$597$	0	0
$598$	−83.6656	−0.139909
$599$	453.755i	0.757520i	0.925495	+	0.378760i	$0.123649\pi$
$599$	453.755i	0.757520i	−0.925495	+	0.378760i	$0.876351\pi$
$600$	0	0
$601$	−653.584	−1.08749	−0.543747	−	0.839249i	$-0.682995\pi$
$601$	−653.584	−1.08749	−0.543747	+	0.839249i	$0.682995\pi$
$602$	− 101.094i	− 0.167931i
$603$	0	0
$604$	−535.708	−0.886934
$605$	− 120.931i	− 0.199886i
$606$	0	0
$607$	−184.626	−0.304162	−0.152081	−	0.988368i	$-0.548597\pi$
$607$	−184.626	−0.304162	−0.152081	+	0.988368i	$0.548597\pi$
$608$	97.4384i	0.160261i
$609$	0	0
$610$	16.2279	0.0266032
$611$	− 355.639i	− 0.582061i
$612$	0	0
$613$	−903.416	−1.47376	−0.736881	−	0.676022i	$-0.763702\pi$
$613$	−903.416	−1.47376	−0.736881	+	0.676022i	$0.763702\pi$
$614$	− 175.001i	− 0.285017i
$615$	0	0
$616$	91.0031	0.147732
$617$	386.489i	0.626401i	0.949687	+	0.313200i	$0.101401\pi$
$617$	386.489i	0.626401i	−0.949687	+	0.313200i	$0.898599\pi$
$618$	0	0
$619$	1165.90	1.88353	0.941763	−	0.336278i	$-0.109168\pi$
$619$	1165.90	1.88353	0.941763	+	0.336278i	$0.109168\pi$
$620$	269.307i	0.434366i
$621$	0	0
$622$	45.0883	0.0724891
$623$	− 258.492i	− 0.414915i
$624$	0	0
$625$	25.0000	0.0400000
$626$	− 157.686i	− 0.251894i
$627$	0	0
$628$	734.997	1.17038
$629$	− 638.020i	− 1.01434i
$630$	0	0
$631$	50.8297	0.0805542	0.0402771	−	0.999189i	$-0.487176\pi$
$631$	50.8297	0.0805542	0.0402771	+	0.999189i	$0.487176\pi$
$632$	− 144.502i	− 0.228642i
$633$	0	0
$634$	103.110	0.162633
$635$	448.885i	0.706906i
$636$	0	0
$637$	−271.708	−0.426543
$638$	− 175.193i	− 0.274598i
$639$	0	0
$640$	199.146	0.311165
$641$	− 548.754i	− 0.856090i	−0.903757	−	0.428045i	$-0.859202\pi$
$641$	− 548.754i	− 0.856090i	0.903757	−	0.428045i	$-0.140798\pi$
$642$	0	0
$643$	526.711	0.819147	0.409573	−	0.912277i	$-0.365678\pi$
$643$	526.711	0.819147	0.409573	+	0.912277i	$0.365678\pi$
$644$	406.122i	0.630623i
$645$	0	0
$646$	−25.4741	−0.0394335
$647$	694.607i	1.07358i	0.843716	+	0.536790i	$0.180363\pi$
$647$	694.607i	1.07358i	−0.843716	+	0.536790i	$0.819637\pi$
$648$	0	0
$649$	−742.480	−1.14404
$650$	− 14.7214i	− 0.0226482i
$651$	0	0
$652$	−141.957	−0.217726
$653$	1122.29i	1.71867i	0.511412	+	0.859336i	$0.329123\pi$
$653$	1122.29i	1.71867i	−0.511412	+	0.859336i	$0.670877\pi$
$654$	0	0
$655$	−18.2554	−0.0278709
$656$	867.132i	1.32185i
$657$	0	0
$658$	65.3499	0.0993160
$659$	670.488i	1.01743i	0.860934	+	0.508717i	$0.169880\pi$
$659$	670.488i	1.01743i	−0.860934	+	0.508717i	$0.830120\pi$
$660$	0	0
$661$	1088.15	1.64622	0.823110	−	0.567882i	$-0.192237\pi$
$661$	1088.15	1.64622	0.823110	+	0.567882i	$0.192237\pi$
$662$	3.78714i	0.00572075i
$663$	0	0
$664$	−187.997	−0.283128
$665$	− 46.2980i	− 0.0696211i
$666$	0	0
$667$	1593.28	2.38872
$668$	− 996.596i	− 1.49191i
$669$	0	0
$670$	0.285585	0.000426246 0
$671$	− 155.426i	− 0.231634i
$672$	0	0
$673$	−638.371	−0.948545	−0.474272	−	0.880378i	$-0.657289\pi$
$673$	−638.371	−0.948545	−0.474272	+	0.880378i	$0.657289\pi$
$674$	− 166.709i	− 0.247343i
$675$	0	0
$676$	−422.346	−0.624773
$677$	− 798.158i	− 1.17896i	−0.807782	−	0.589481i	$-0.799332\pi$
$677$	− 798.158i	− 1.17896i	0.807782	−	0.589481i	$-0.200668\pi$
$678$	0	0
$679$	591.149	0.870616
$680$	80.1246i	0.117830i
$681$	0	0
$682$	−97.6417	−0.143170
$683$	− 757.505i	− 1.10908i	−0.832156	−	0.554542i	$-0.812893\pi$
$683$	− 757.505i	− 1.10908i	0.832156	−	0.554542i	$-0.187107\pi$
$684$	0	0
$685$	112.544	0.164298
$686$	− 119.331i	− 0.173952i
$687$	0	0
$688$	−1018.54	−1.48044
$689$	565.479i	0.820724i
$690$	0	0
$691$	702.331	1.01640	0.508199	−	0.861240i	$-0.330311\pi$
$691$	702.331	1.01640	0.508199	+	0.861240i	$0.330311\pi$
$692$	− 702.723i	− 1.01550i
$693$	0	0
$694$	9.69578	0.0139709
$695$	282.859i	0.406992i
$696$	0	0
$697$	−725.781	−1.04129
$698$	− 27.5329i	− 0.0394454i
$699$	0	0
$700$	−71.4590	−0.102084
$701$	− 406.616i	− 0.580052i	−0.957019	−	0.290026i	$-0.906336\pi$
$701$	− 406.616i	− 0.580052i	0.957019	−	0.290026i	$-0.0936639\pi$
$702$	0	0
$703$	−298.255	−0.424261
$704$	− 412.423i	− 0.585829i
$705$	0	0
$706$	−149.793	−0.212172
$707$	88.9969i	0.125880i
$708$	0	0
$709$	564.322	0.795941	0.397970	−	0.917398i	$-0.369715\pi$
$709$	564.322	0.795941	0.397970	+	0.917398i	$0.369715\pi$
$710$	69.4064i	0.0977554i
$711$	0	0
$712$	209.125	0.293714
$713$	− 887.991i	− 1.24543i
$714$	0	0
$715$	−140.997	−0.197198
$716$	989.633i	1.38217i
$717$	0	0
$718$	152.784	0.212791
$719$	1340.07i	1.86379i	0.362725	+	0.931896i	$0.381846\pi$
$719$	1340.07i	1.86379i	−0.362725	+	0.931896i	$0.618154\pi$
$720$	0	0
$721$	−93.3251	−0.129438
$722$	− 125.981i	− 0.174489i
$723$	0	0
$724$	694.055	0.958639
$725$	280.344i	0.386682i
$726$	0	0
$727$	130.991	0.180180	0.0900899	−	0.995934i	$-0.471285\pi$
$727$	130.991	0.180180	0.0900899	+	0.995934i	$0.471285\pi$
$728$	85.7508i	0.117790i
$729$	0	0
$730$	43.3800	0.0594247
$731$	− 852.508i	− 1.16622i
$732$	0	0
$733$	1414.48	1.92971	0.964857	−	0.262777i	$-0.0846383\pi$
$733$	1414.48	1.92971	0.964857	+	0.262777i	$0.0846383\pi$
$734$	78.8266i	0.107393i
$735$	0	0
$736$	−495.890	−0.673764
$737$	− 2.73525i	− 0.00371133i
$738$	0	0
$739$	10.5805	0.0143173	0.00715865	−	0.999974i	$-0.497721\pi$
$739$	10.5805	0.0143173	0.00715865	+	0.999974i	$0.497721\pi$
$740$	460.344i	0.622087i
$741$	0	0
$742$	−103.909	−0.140039
$743$	− 794.053i	− 1.06871i	−0.845260	−	0.534356i	$-0.820554\pi$
$743$	− 794.053i	− 1.06871i	0.845260	−	0.534356i	$-0.179446\pi$
$744$	0	0
$745$	−35.3738	−0.0474817
$746$	135.534i	0.181681i
$747$	0	0
$748$	376.577	0.503446
$749$	176.754i	0.235986i
$750$	0	0
$751$	−53.5109	−0.0712528	−0.0356264	−	0.999365i	$-0.511343\pi$
$751$	−53.5109	−0.0712528	−0.0356264	+	0.999365i	$0.511343\pi$
$752$	− 658.409i	− 0.875544i
$753$	0	0
$754$	165.082	0.218942
$755$	310.807i	0.411664i
$756$	0	0
$757$	921.745	1.21763	0.608814	−	0.793313i	$-0.291646\pi$
$757$	921.745	1.21763	0.608814	+	0.793313i	$0.291646\pi$
$758$	− 161.952i	− 0.213658i
$759$	0	0
$760$	37.4559	0.0492840
$761$	578.115i	0.759678i	0.925053	+	0.379839i	$0.124021\pi$
$761$	578.115i	0.759678i	−0.925053	+	0.379839i	$0.875979\pi$
$762$	0	0
$763$	611.538	0.801491
$764$	494.817i	0.647666i
$765$	0	0
$766$	79.1273	0.103299
$767$	− 699.627i	− 0.912160i
$768$	0	0
$769$	−651.656	−0.847407	−0.423704	−	0.905801i	$-0.639270\pi$
$769$	−651.656	−0.847407	−0.423704	+	0.905801i	$0.639270\pi$
$770$	− 25.9086i	− 0.0336476i
$771$	0	0
$772$	247.787	0.320968
$773$	389.754i	0.504209i	0.967700	+	0.252105i	$0.0811227\pi$
$773$	389.754i	0.504209i	−0.967700	+	0.252105i	$0.918877\pi$
$774$	0	0
$775$	156.246	0.201608
$776$	478.249i	0.616301i
$777$	0	0
$778$	168.049	0.216001
$779$	339.281i	0.435534i
$780$	0	0
$781$	664.754	0.851157
$782$	− 129.644i	− 0.165786i
$783$	0	0
$784$	−503.024	−0.641613
$785$	− 426.430i	− 0.543222i
$786$	0	0
$787$	−36.4559	−0.0463226	−0.0231613	−	0.999732i	$-0.507373\pi$
$787$	−36.4559	−0.0463226	−0.0231613	+	0.999732i	$0.507373\pi$
$788$	− 840.182i	− 1.06622i
$789$	0	0
$790$	−41.1397	−0.0520756
$791$	− 253.593i	− 0.320598i
$792$	0	0
$793$	146.456	0.184686
$794$	173.364i	0.218342i
$795$	0	0
$796$	1234.27	1.55059
$797$	731.466i	0.917774i	0.888495	+	0.458887i	$0.151752\pi$
$797$	731.466i	0.917774i	−0.888495	+	0.458887i	$0.848248\pi$
$798$	0	0
$799$	551.082	0.689715
$800$	− 87.2542i	− 0.109068i
$801$	0	0
$802$	−212.529	−0.264998
$803$	− 415.481i	− 0.517412i
$804$	0	0
$805$	235.623	0.292699
$806$	− 92.0062i	− 0.114152i
$807$	0	0
$808$	−72.0000	−0.0891089
$809$	595.767i	0.736424i	0.929742	+	0.368212i	$0.120030\pi$
$809$	595.767i	0.736424i	−0.929742	+	0.368212i	$0.879970\pi$
$810$	0	0
$811$	180.177	0.222166	0.111083	−	0.993811i	$-0.464568\pi$
$811$	180.177	0.222166	0.111083	+	0.993811i	$0.464568\pi$
$812$	− 801.325i	− 0.986854i
$813$	0	0
$814$	−166.906	−0.205044
$815$	82.3607i	0.101056i
$816$	0	0
$817$	−398.522	−0.487787
$818$	28.9005i	0.0353307i
$819$	0	0
$820$	523.666	0.638617
$821$	31.3375i	0.0381699i	0.999818	+	0.0190849i	$0.00607529\pi$
$821$	31.3375i	0.0381699i	−0.999818	+	0.0190849i	$0.993925\pi$
$822$	0	0
$823$	−234.803	−0.285301	−0.142650	−	0.989773i	$-0.545562\pi$
$823$	−234.803	−0.285301	−0.142650	+	0.989773i	$0.545562\pi$
$824$	− 75.5016i	− 0.0916281i
$825$	0	0
$826$	128.559	0.155640
$827$	− 47.8003i	− 0.0577996i	−0.999582	−	0.0288998i	$-0.990800\pi$
$827$	− 47.8003i	− 0.0577996i	0.999582	−	0.0288998i	$-0.00920038\pi$
$828$	0	0
$829$	−292.760	−0.353148	−0.176574	−	0.984287i	$-0.556502\pi$
$829$	−292.760	−0.353148	−0.176574	+	0.984287i	$0.556502\pi$
$830$	53.5228i	0.0644853i
$831$	0	0
$832$	388.620	0.467091
$833$	− 421.026i	− 0.505434i
$834$	0	0
$835$	−578.204	−0.692459
$836$	− 176.039i	− 0.210573i
$837$	0	0
$838$	199.027	0.237502
$839$	494.387i	0.589257i	0.955612	+	0.294629i	$0.0951960\pi$
$839$	494.387i	0.589257i	−0.955612	+	0.294629i	$0.904804\pi$
$840$	0	0
$841$	−2302.72	−2.73807
$842$	43.8285i	0.0520529i
$843$	0	0
$844$	−246.018	−0.291491
$845$	245.036i	0.289984i
$846$	0	0
$847$	200.547	0.236774
$848$	1046.89i	1.23455i
$849$	0	0
$850$	22.8115	0.0268371
$851$	− 1517.90i	− 1.78367i
$852$	0	0
$853$	−1151.65	−1.35011	−0.675057	−	0.737766i	$-0.735881\pi$
$853$	−1151.65	−1.35011	−0.675057	+	0.737766i	$0.735881\pi$
$854$	26.9117i	0.0315126i
$855$	0	0
$856$	−142.997	−0.167052
$857$	1098.02i	1.28124i	0.767858	+	0.640620i	$0.221323\pi$
$857$	1098.02i	1.28124i	−0.767858	+	0.640620i	$0.778677\pi$
$858$	0	0
$859$	−290.757	−0.338483	−0.169242	−	0.985575i	$-0.554132\pi$
$859$	−290.757	−0.338483	−0.169242	+	0.985575i	$0.554132\pi$
$860$	615.102i	0.715235i
$861$	0	0
$862$	−89.0758	−0.103336
$863$	− 1069.28i	− 1.23903i	−0.784985	−	0.619514i	$-0.787330\pi$
$863$	− 1069.28i	− 1.23903i	0.784985	−	0.619514i	$-0.212670\pi$
$864$	0	0
$865$	−407.705	−0.471335
$866$	− 191.633i	− 0.221285i
$867$	0	0
$868$	−446.608	−0.514525
$869$	394.024i	0.453422i
$870$	0	0
$871$	2.57738	0.00295911
$872$	494.745i	0.567368i
$873$	0	0
$874$	−60.6049	−0.0693420
$875$	41.4590i	0.0473817i
$876$	0	0
$877$	−160.043	−0.182489	−0.0912443	−	0.995829i	$-0.529084\pi$
$877$	−160.043	−0.182489	−0.0912443	+	0.995829i	$0.529084\pi$
$878$	139.798i	0.159224i
$879$	0	0
$880$	−261.033	−0.296629
$881$	− 712.502i	− 0.808743i	−0.914595	−	0.404371i	$-0.867490\pi$
$881$	− 712.502i	− 0.808743i	0.914595	−	0.404371i	$-0.132510\pi$
$882$	0	0
$883$	−1244.12	−1.40897	−0.704486	−	0.709718i	$-0.748822\pi$
$883$	−1244.12	−1.40897	−0.704486	+	0.709718i	$0.748822\pi$
$884$	354.843i	0.401406i
$885$	0	0
$886$	−241.747	−0.272852
$887$	− 429.827i	− 0.484585i	−0.970203	−	0.242292i	$-0.922101\pi$
$887$	− 429.827i	− 0.484585i	0.970203	−	0.242292i	$-0.0778993\pi$
$888$	0	0
$889$	−744.413	−0.837360
$890$	− 59.5379i	− 0.0668965i
$891$	0	0
$892$	−31.7933	−0.0356428
$893$	− 257.614i	− 0.288482i
$894$	0	0
$895$	574.164	0.641524
$896$	330.255i	0.368589i
$897$	0	0
$898$	304.177	0.338727
$899$	1752.11i	1.94895i
$900$	0	0
$901$	−876.240	−0.972519
$902$	189.864i	0.210492i
$903$	0	0
$904$	205.161	0.226948
$905$	− 402.676i	− 0.444946i
$906$	0	0
$907$	−553.240	−0.609967	−0.304983	−	0.952358i	$-0.598651\pi$
$907$	−553.240	−0.609967	−0.304983	+	0.952358i	$0.598651\pi$
$908$	− 1308.56i	− 1.44115i
$909$	0	0
$910$	24.4133	0.0268278
$911$	− 667.275i	− 0.732464i	−0.930524	−	0.366232i	$-0.880648\pi$
$911$	− 667.275i	− 0.732464i	0.930524	−	0.366232i	$-0.119352\pi$
$912$	0	0
$913$	512.626	0.561474
$914$	− 77.3777i	− 0.0846583i
$915$	0	0
$916$	780.438	0.852006
$917$	− 30.2741i	− 0.0330143i
$918$	0	0
$919$	−338.000	−0.367791	−0.183896	−	0.982946i	$-0.558871\pi$
$919$	−338.000	−0.367791	−0.183896	+	0.982946i	$0.558871\pi$
$920$	190.623i	0.207199i
$921$	0	0
$922$	−278.036	−0.301558
$923$	626.387i	0.678642i
$924$	0	0
$925$	267.082	0.288737
$926$	188.310i	0.203359i
$927$	0	0
$928$	978.450	1.05436
$929$	− 216.829i	− 0.233400i	−0.993167	−	0.116700i	$-0.962768\pi$
$929$	− 216.829i	− 0.233400i	0.993167	−	0.116700i	$-0.0372317\pi$
$930$	0	0
$931$	−196.817	−0.211404
$932$	73.5441i	0.0789100i
$933$	0	0
$934$	173.103	0.185335
$935$	− 218.482i	− 0.233671i
$936$	0	0
$937$	−851.647	−0.908908	−0.454454	−	0.890770i	$-0.650166\pi$
$937$	−851.647	−0.908908	−0.454454	+	0.890770i	$0.650166\pi$
$938$	0.473602i	0 0.000504906i
$939$	0	0
$940$	−397.617	−0.422997
$941$	− 1073.42i	− 1.14073i	−0.821392	−	0.570363i	$-0.806802\pi$
$941$	− 1073.42i	− 1.14073i	0.821392	−	0.570363i	$-0.193198\pi$
$942$	0	0
$943$	−1726.69	−1.83106
$944$	− 1295.25i	− 1.37208i
$945$	0	0
$946$	−223.016	−0.235746
$947$	1645.82i	1.73793i	0.494873	+	0.868965i	$0.335215\pi$
$947$	1645.82i	1.73793i	−0.494873	+	0.868965i	$0.664785\pi$
$948$	0	0
$949$	391.502	0.412541
$950$	− 10.6637i	− 0.0112250i
$951$	0	0
$952$	−132.875	−0.139575
$953$	1139.03i	1.19520i	0.801793	+	0.597602i	$0.203880\pi$
$953$	1139.03i	1.19520i	−0.801793	+	0.597602i	$0.796120\pi$
$954$	0	0
$955$	287.082	0.300609
$956$	1050.67i	1.09902i
$957$	0	0
$958$	−122.055	−0.127406
$959$	186.639i	0.194618i
$960$	0	0
$961$	15.5140	0.0161436
$962$	− 157.272i	− 0.163485i
$963$	0	0
$964$	896.389	0.929864
$965$	− 143.761i	− 0.148975i
$966$	0	0
$967$	−357.489	−0.369689	−0.184844	−	0.982768i	$-0.559178\pi$
$967$	−357.489	−0.369689	−0.184844	+	0.982768i	$0.559178\pi$
$968$	162.246i	0.167610i
$969$	0	0
$970$	136.158	0.140369
$971$	− 46.5836i	− 0.0479749i	−0.999712	−	0.0239874i	$-0.992364\pi$
$971$	− 46.5836i	− 0.0479749i	0.999712	−	0.0239874i	$-0.00763617\pi$
$972$	0	0
$973$	−469.082	−0.482099
$974$	302.849i	0.310933i
$975$	0	0
$976$	271.140	0.277807
$977$	346.616i	0.354776i	0.984141	+	0.177388i	$0.0567647\pi$
$977$	346.616i	0.354776i	−0.984141	+	0.177388i	$0.943235\pi$
$978$	0	0
$979$	−570.237	−0.582469
$980$	303.779i	0.309979i
$981$	0	0
$982$	141.112	0.143699
$983$	− 846.975i	− 0.861623i	−0.902442	−	0.430811i	$-0.858227\pi$
$983$	− 846.975i	− 0.861623i	0.902442	−	0.430811i	$-0.141773\pi$
$984$	0	0
$985$	−487.456	−0.494879
$986$	255.803i	0.259435i
$987$	0	0
$988$	165.878	0.167893
$989$	− 2028.19i	− 2.05075i
$990$	0	0
$991$	1304.33	1.31617	0.658085	−	0.752943i	$-0.271367\pi$
$991$	1304.33	1.31617	0.658085	+	0.752943i	$0.271367\pi$
$992$	− 545.326i	− 0.549723i
$993$	0	0
$994$	−115.101	−0.115795
$995$	− 716.099i	− 0.719698i
$996$	0	0
$997$	1835.77	1.84130	0.920649	−	0.390391i	$-0.127660\pi$
$997$	1835.77	1.84130	0.920649	+	0.390391i	$0.127660\pi$
$998$	232.966i	0.233432i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.3.c.c.26.3	yes	4
3.2	odd	2	inner	135.3.c.c.26.2	✓	4
4.3	odd	2		2160.3.l.g.161.1		4
5.2	odd	4		675.3.d.e.674.4		4
5.3	odd	4		675.3.d.i.674.1		4
5.4	even	2		675.3.c.p.26.2		4
9.2	odd	6		405.3.i.c.296.2		8
9.4	even	3		405.3.i.c.26.2		8
9.5	odd	6		405.3.i.c.26.3		8
9.7	even	3		405.3.i.c.296.3		8
12.11	even	2		2160.3.l.g.161.3		4
15.2	even	4		675.3.d.i.674.2		4
15.8	even	4		675.3.d.e.674.3		4
15.14	odd	2		675.3.c.p.26.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.3.c.c.26.2	✓	4	3.2	odd	2	inner
135.3.c.c.26.3	yes	4	1.1	even	1	trivial
405.3.i.c.26.2		8	9.4	even	3
405.3.i.c.26.3		8	9.5	odd	6
405.3.i.c.296.2		8	9.2	odd	6
405.3.i.c.296.3		8	9.7	even	3
675.3.c.p.26.2		4	5.4	even	2
675.3.c.p.26.3		4	15.14	odd	2
675.3.d.e.674.3		4	15.8	even	4
675.3.d.e.674.4		4	5.2	odd	4
675.3.d.i.674.1		4	5.3	odd	4
675.3.d.i.674.2		4	15.2	even	4
2160.3.l.g.161.1		4	4.3	odd	2
2160.3.l.g.161.3		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	135.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.381966i q^{2} +3.85410 q^{4} -2.23607i q^{5} +3.70820 q^{7} +3.00000i q^{8} +O(q^{10})\)	\(q+0.381966i q^{2} +3.85410 q^{4} -2.23607i q^{5} +3.70820 q^{7} +3.00000i q^{8} +0.854102 q^{10} -8.18034i q^{11} +7.70820 q^{13} +1.41641i q^{14} +14.2705 q^{16} +11.9443i q^{17} +5.58359 q^{19} -8.61803i q^{20} +3.12461 q^{22} +28.4164i q^{23} -5.00000 q^{25} +2.94427i q^{26} +14.2918 q^{28} -56.0689i q^{29} -31.2492 q^{31} +17.4508i q^{32} -4.56231 q^{34} -8.29180i q^{35} -53.4164 q^{37} +2.13274i q^{38} +6.70820 q^{40} +60.7639i q^{41} -71.3738 q^{43} -31.5279i q^{44} -10.8541 q^{46} -46.1378i q^{47} -35.2492 q^{49} -1.90983i q^{50} +29.7082 q^{52} +73.3607i q^{53} -18.2918 q^{55} +11.1246i q^{56} +21.4164 q^{58} -90.7639i q^{59} +19.0000 q^{61} -11.9361i q^{62} +50.4164 q^{64} -17.2361i q^{65} +0.334369 q^{67} +46.0344i q^{68} +3.16718 q^{70} +81.2624i q^{71} +50.7902 q^{73} -20.4033i q^{74} +21.5197 q^{76} -30.3344i q^{77} -48.1672 q^{79} -31.9098i q^{80} -23.2098 q^{82} +62.6656i q^{83} +26.7082 q^{85} -27.2624i q^{86} +24.5410 q^{88} -69.7082i q^{89} +28.5836 q^{91} +109.520i q^{92} +17.6231 q^{94} -12.4853i q^{95} +159.416 q^{97} -13.4640i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 12 q^{7}+O(q^{10}) \) 4 * q + 2 * q^4 - 12 * q^7	\( 4 q + 2 q^{4} - 12 q^{7} - 10 q^{10} + 4 q^{13} - 10 q^{16} + 76 q^{19} - 68 q^{22} - 20 q^{25} + 84 q^{28} + 36 q^{31} + 22 q^{34} - 160 q^{37} - 44 q^{43} - 30 q^{46} + 20 q^{49} + 92 q^{52} - 100 q^{55} + 32 q^{58} + 76 q^{61} + 148 q^{64} + 216 q^{67} + 120 q^{70} - 92 q^{73} - 142 q^{76} - 300 q^{79} - 388 q^{82} + 80 q^{85} - 36 q^{88} + 168 q^{91} - 332 q^{94} + 584 q^{97}+O(q^{100}) \) 4 * q + 2 * q^4 - 12 * q^7 - 10 * q^10 + 4 * q^13 - 10 * q^16 + 76 * q^19 - 68 * q^22 - 20 * q^25 + 84 * q^28 + 36 * q^31 + 22 * q^34 - 160 * q^37 - 44 * q^43 - 30 * q^46 + 20 * q^49 + 92 * q^52 - 100 * q^55 + 32 * q^58 + 76 * q^61 + 148 * q^64 + 216 * q^67 + 120 * q^70 - 92 * q^73 - 142 * q^76 - 300 * q^79 - 388 * q^82 + 80 * q^85 - 36 * q^88 + 168 * q^91 - 332 * q^94 + 584 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more