LMFDB - Embedded newform 459.3.b.d.188.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [459,3,Mod(188,459)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(459, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("459.188"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 459 = 3^{3} \cdot 17 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	459.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,-20,0,0,-54] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$12.5068441341$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 34x^{10} + 395x^{8} + 1888x^{6} + 3523x^{4} + 1566x^{2} + 81 $ x^12 + 34x^10 + 395x^8 + 1888x^6 + 3523x^4 + 1566*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			188.5
Root			$-0.736967i$ of defining polynomial
Character	$\chi$	$=$	459.188
Dual form			459.3.b.d.188.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.736967i q^{2} +3.45688 q^{4} -5.75031i q^{5} -8.26227 q^{7} -5.49547i q^{8} -4.23779 q^{10} -4.53034i q^{11} -22.6247 q^{13} +6.08902i q^{14} +9.77754 q^{16} +4.12311i q^{17} -27.4537 q^{19} -19.8781i q^{20} -3.33871 q^{22} +37.3340i q^{23} -8.06606 q^{25} +16.6737i q^{26} -28.5617 q^{28} -38.4420i q^{29} +34.1630 q^{31} -29.1876i q^{32} +3.03859 q^{34} +47.5106i q^{35} -0.534811 q^{37} +20.2325i q^{38} -31.6007 q^{40} +9.07193i q^{41} -18.1671 q^{43} -15.6608i q^{44} +27.5139 q^{46} -54.7848i q^{47} +19.2651 q^{49} +5.94441i q^{50} -78.2109 q^{52} -69.1129i q^{53} -26.0508 q^{55} +45.4051i q^{56} -28.3305 q^{58} +35.9578i q^{59} -4.75066 q^{61} -25.1770i q^{62} +17.5999 q^{64} +130.099i q^{65} +53.7119 q^{67} +14.2531i q^{68} +35.0137 q^{70} -84.5435i q^{71} -109.143 q^{73} +0.394138i q^{74} -94.9043 q^{76} +37.4309i q^{77} -112.916 q^{79} -56.2239i q^{80} +6.68571 q^{82} -162.036i q^{83} +23.7091 q^{85} +13.3886i q^{86} -24.8963 q^{88} -44.0055i q^{89} +186.932 q^{91} +129.059i q^{92} -40.3746 q^{94} +157.867i q^{95} -96.2125 q^{97} -14.1978i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 20 q^{4} - 54 q^{7} - 12 q^{10} - 50 q^{13} + 108 q^{16} - 50 q^{19} - 106 q^{22} - 48 q^{25} + 382 q^{28} + 2 q^{31} - 244 q^{37} + 208 q^{40} - 78 q^{43} - 18 q^{46} + 470 q^{49} + 42 q^{52} + 290 q^{55}+ \cdots + 66 q^{97}+O(q^{100}) $ 12 * q - 20 * q^4 - 54 * q^7 - 12 * q^10 - 50 * q^13 + 108 * q^16 - 50 * q^19 - 106 * q^22 - 48 * q^25 + 382 * q^28 + 2 * q^31 - 244 * q^37 + 208 * q^40 - 78 * q^43 - 18 * q^46 + 470 * q^49 + 42 * q^52 + 290 * q^55 + 374 * q^58 - 72 * q^61 - 476 * q^64 - 602 * q^67 - 82 * q^70 - 302 * q^73 - 918 * q^76 - 28 * q^79 - 262 * q^82 - 68 * q^85 - 38 * q^88 + 428 * q^91 + 756 * q^94 + 66 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/459\mathbb{Z}\right)^\times$.

$n$	$137$	$190$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 0.736967i	− 0.368483i	−0.982881	−	0.184242i	$-0.941017\pi$
$2$	− 0.736967i	− 0.368483i	0.982881	−	0.184242i	$-0.0589829\pi$
$3$	0	0
$3$	0	0
$4$	3.45688	0.864220
$5$	− 5.75031i	− 1.15006i	−0.818132	−	0.575031i	$-0.804990\pi$
$5$	− 5.75031i	− 1.15006i	0.818132	−	0.575031i	$-0.195010\pi$
$6$	0	0
$7$	−8.26227	−1.18032	−0.590162	−	0.807285i	$-0.700936\pi$
$7$	−8.26227	−1.18032	−0.590162	+	0.807285i	$0.700936\pi$
$8$	− 5.49547i	− 0.686934i
$9$	0	0
$10$	−4.23779	−0.423779
$11$	− 4.53034i	− 0.411849i	−0.978568	−	0.205924i	$-0.933980\pi$
$11$	− 4.53034i	− 0.411849i	0.978568	−	0.205924i	$-0.0660201\pi$
$12$	0	0
$13$	−22.6247	−1.74036	−0.870181	−	0.492731i	$-0.835998\pi$
$13$	−22.6247	−1.74036	−0.870181	+	0.492731i	$0.835998\pi$
$14$	6.08902i	0.434930i
$15$	0	0
$16$	9.77754	0.611096
$17$	4.12311i	0.242536i
$17$	4.12311i	0.242536i
$18$	0	0
$19$	−27.4537	−1.44493	−0.722467	−	0.691406i	$-0.756992\pi$
$19$	−27.4537	−1.44493	−0.722467	+	0.691406i	$0.756992\pi$
$20$	− 19.8781i	− 0.993907i
$21$	0	0
$22$	−3.33871	−0.151759
$23$	37.3340i	1.62322i	0.584200	+	0.811610i	$0.301408\pi$
$23$	37.3340i	1.62322i	−0.584200	+	0.811610i	$0.698592\pi$
$24$	0	0
$25$	−8.06606	−0.322642
$26$	16.6737i	0.641295i
$27$	0	0
$28$	−28.5617	−1.02006
$29$	− 38.4420i	− 1.32559i	−0.748802	−	0.662794i	$-0.769371\pi$
$29$	− 38.4420i	− 1.32559i	0.748802	−	0.662794i	$-0.230629\pi$
$30$	0	0
$31$	34.1630	1.10203	0.551017	−	0.834494i	$-0.314240\pi$
$31$	34.1630	1.10203	0.551017	+	0.834494i	$0.314240\pi$
$32$	− 29.1876i	− 0.912113i
$33$	0	0
$34$	3.03859	0.0893703
$35$	47.5106i	1.35745i
$36$	0	0
$37$	−0.534811	−0.0144544	−0.00722718	−	0.999974i	$-0.502301\pi$
$37$	−0.534811	−0.0144544	−0.00722718	+	0.999974i	$0.502301\pi$
$38$	20.2325i	0.532434i
$39$	0	0
$40$	−31.6007	−0.790016
$41$	9.07193i	0.221267i	0.993861	+	0.110633i	$0.0352879\pi$
$41$	9.07193i	0.221267i	−0.993861	+	0.110633i	$0.964712\pi$
$42$	0	0
$43$	−18.1671	−0.422492	−0.211246	−	0.977433i	$-0.567752\pi$
$43$	−18.1671	−0.422492	−0.211246	+	0.977433i	$0.567752\pi$
$44$	− 15.6608i	− 0.355928i
$45$	0	0
$46$	27.5139	0.598129
$47$	− 54.7848i	− 1.16563i	−0.812604	−	0.582817i	$-0.801950\pi$
$47$	− 54.7848i	− 1.16563i	0.812604	−	0.582817i	$-0.198050\pi$
$48$	0	0
$49$	19.2651	0.393166
$50$	5.94441i	0.118888i
$51$	0	0
$52$	−78.2109	−1.50406
$53$	− 69.1129i	− 1.30402i	−0.758212	−	0.652008i	$-0.773927\pi$
$53$	− 69.1129i	− 1.30402i	0.758212	−	0.652008i	$-0.226073\pi$
$54$	0	0
$55$	−26.0508	−0.473652
$56$	45.4051i	0.810805i
$57$	0	0
$58$	−28.3305	−0.488457
$59$	35.9578i	0.609455i	0.952440	+	0.304727i	$0.0985654\pi$
$59$	35.9578i	0.609455i	−0.952440	+	0.304727i	$0.901435\pi$
$60$	0	0
$61$	−4.75066	−0.0778796	−0.0389398	−	0.999242i	$-0.512398\pi$
$61$	−4.75066	−0.0778796	−0.0389398	+	0.999242i	$0.512398\pi$
$62$	− 25.1770i	− 0.406081i
$63$	0	0
$64$	17.5999	0.274998
$65$	130.099i	2.00152i
$66$	0	0
$67$	53.7119	0.801670	0.400835	−	0.916150i	$-0.368720\pi$
$67$	53.7119	0.801670	0.400835	+	0.916150i	$0.368720\pi$
$68$	14.2531i	0.209604i
$69$	0	0
$70$	35.0137	0.500196
$71$	− 84.5435i	− 1.19075i	−0.803447	−	0.595376i	$-0.797003\pi$
$71$	− 84.5435i	− 1.19075i	0.803447	−	0.595376i	$-0.202997\pi$
$72$	0	0
$73$	−109.143	−1.49511	−0.747554	−	0.664201i	$-0.768772\pi$
$73$	−109.143	−1.49511	−0.747554	+	0.664201i	$0.768772\pi$
$74$	0.394138i	0.00532619i
$75$	0	0
$76$	−94.9043	−1.24874
$77$	37.4309i	0.486115i
$78$	0	0
$79$	−112.916	−1.42931	−0.714657	−	0.699475i	$-0.753417\pi$
$79$	−112.916	−1.42931	−0.714657	+	0.699475i	$0.753417\pi$
$80$	− 56.2239i	− 0.702799i
$81$	0	0
$82$	6.68571	0.0815330
$83$	− 162.036i	− 1.95224i	−0.217230	−	0.976120i	$-0.569702\pi$
$83$	− 162.036i	− 1.95224i	0.217230	−	0.976120i	$-0.430298\pi$
$84$	0	0
$85$	23.7091	0.278931
$86$	13.3886i	0.155681i
$87$	0	0
$88$	−24.8963	−0.282913
$89$	− 44.0055i	− 0.494443i	−0.968959	−	0.247222i	$-0.920482\pi$
$89$	− 44.0055i	− 0.494443i	0.968959	−	0.247222i	$-0.0795176\pi$
$90$	0	0
$91$	186.932	2.05419
$92$	129.059i	1.40282i
$93$	0	0
$94$	−40.3746	−0.429517
$95$	157.867i	1.66176i
$96$	0	0
$97$	−96.2125	−0.991882	−0.495941	−	0.868356i	$-0.665177\pi$
$97$	−96.2125	−0.991882	−0.495941	+	0.868356i	$0.665177\pi$
$98$	− 14.1978i	− 0.144875i
$99$	0	0
$100$	−27.8834	−0.278834
$101$	− 42.8093i	− 0.423854i	−0.977285	−	0.211927i	$-0.932026\pi$
$101$	− 42.8093i	− 0.423854i	0.977285	−	0.211927i	$-0.0679739\pi$
$102$	0	0
$103$	48.2637	0.468579	0.234290	−	0.972167i	$-0.424724\pi$
$103$	48.2637	0.468579	0.234290	+	0.972167i	$0.424724\pi$
$104$	124.333i	1.19551i
$105$	0	0
$106$	−50.9339	−0.480508
$107$	137.538i	1.28540i	0.766117	+	0.642701i	$0.222186\pi$
$107$	137.538i	1.28540i	−0.766117	+	0.642701i	$0.777814\pi$
$108$	0	0
$109$	197.392	1.81094	0.905469	−	0.424413i	$-0.139520\pi$
$109$	197.392	1.81094	0.905469	+	0.424413i	$0.139520\pi$
$110$	19.1986i	0.174533i
$111$	0	0
$112$	−80.7847	−0.721292
$113$	52.2099i	0.462034i	0.972950	+	0.231017i	$0.0742054\pi$
$113$	52.2099i	0.462034i	−0.972950	+	0.231017i	$0.925795\pi$
$114$	0	0
$115$	214.682	1.86680
$116$	− 132.889i	− 1.14560i
$117$	0	0
$118$	26.4997	0.224574
$119$	− 34.0662i	− 0.286271i
$120$	0	0
$121$	100.476	0.830380
$122$	3.50108i	0.0286973i
$123$	0	0
$124$	118.098	0.952399
$125$	− 97.3754i	− 0.779003i
$126$	0	0
$127$	140.810	1.10874	0.554369	−	0.832271i	$-0.312960\pi$
$127$	140.810	1.10874	0.554369	+	0.832271i	$0.312960\pi$
$128$	− 129.721i	− 1.01344i
$129$	0	0
$130$	95.8787	0.737528
$131$	− 0.900209i	− 0.00687183i	−0.999994	−	0.00343591i	$-0.998906\pi$
$131$	− 0.900209i	− 0.00687183i	0.999994	−	0.00343591i	$-0.00109369\pi$
$132$	0	0
$133$	226.830	1.70549
$134$	− 39.5839i	− 0.295402i
$135$	0	0
$136$	22.6584	0.166606
$137$	− 110.159i	− 0.804079i	−0.915622	−	0.402039i	$-0.868302\pi$
$137$	− 110.159i	− 0.804079i	0.915622	−	0.402039i	$-0.131698\pi$
$138$	0	0
$139$	182.122	1.31023	0.655115	−	0.755529i	$-0.272620\pi$
$139$	182.122	1.31023	0.655115	+	0.755529i	$0.272620\pi$
$140$	164.239i	1.17313i
$141$	0	0
$142$	−62.3057	−0.438772
$143$	102.498i	0.716767i
$144$	0	0
$145$	−221.054	−1.52451
$146$	80.4347i	0.550923i
$147$	0	0
$148$	−1.84878	−0.0124917
$149$	− 164.181i	− 1.10189i	−0.834543	−	0.550943i	$-0.814268\pi$
$149$	− 164.181i	− 1.10189i	0.834543	−	0.550943i	$-0.185732\pi$
$150$	0	0
$151$	36.5332	0.241941	0.120971	−	0.992656i	$-0.461399\pi$
$151$	36.5332	0.241941	0.120971	+	0.992656i	$0.461399\pi$
$152$	150.871i	0.992574i
$153$	0	0
$154$	27.5853	0.179125
$155$	− 196.448i	− 1.26741i
$156$	0	0
$157$	−91.2220	−0.581032	−0.290516	−	0.956870i	$-0.593827\pi$
$157$	−91.2220	−0.581032	−0.290516	+	0.956870i	$0.593827\pi$
$158$	83.2152i	0.526679i
$159$	0	0
$160$	−167.838	−1.04899
$161$	− 308.464i	− 1.91593i
$162$	0	0
$163$	−107.845	−0.661623	−0.330812	−	0.943697i	$-0.607322\pi$
$163$	−107.845	−0.661623	−0.330812	+	0.943697i	$0.607322\pi$
$164$	31.3606i	0.191223i
$165$	0	0
$166$	−119.415	−0.719368
$167$	203.473i	1.21840i	0.793017	+	0.609200i	$0.208509\pi$
$167$	203.473i	1.21840i	−0.793017	+	0.609200i	$0.791491\pi$
$168$	0	0
$169$	342.878	2.02886
$170$	− 17.4728i	− 0.102781i
$171$	0	0
$172$	−62.8016	−0.365126
$173$	− 7.57072i	− 0.0437614i	−0.999761	−	0.0218807i	$-0.993035\pi$
$173$	− 7.57072i	− 0.0437614i	0.999761	−	0.0218807i	$-0.00696540\pi$
$174$	0	0
$175$	66.6439	0.380822
$176$	− 44.2956i	− 0.251679i
$177$	0	0
$178$	−32.4305	−0.182194
$179$	− 331.700i	− 1.85307i	−0.376207	−	0.926536i	$-0.622772\pi$
$179$	− 331.700i	− 1.85307i	0.376207	−	0.926536i	$-0.377228\pi$
$180$	0	0
$181$	−93.5512	−0.516857	−0.258429	−	0.966030i	$-0.583205\pi$
$181$	−93.5512	−0.516857	−0.258429	+	0.966030i	$0.583205\pi$
$182$	− 137.762i	− 0.756936i
$183$	0	0
$184$	205.168	1.11504
$185$	3.07533i	0.0166234i
$186$	0	0
$187$	18.6791	0.0998881
$188$	− 189.384i	− 1.00736i
$189$	0	0
$190$	116.343	0.612332
$191$	130.458i	0.683024i	0.939877	+	0.341512i	$0.110939\pi$
$191$	130.458i	0.683024i	−0.939877	+	0.341512i	$0.889061\pi$
$192$	0	0
$193$	−41.5533	−0.215302	−0.107651	−	0.994189i	$-0.534333\pi$
$193$	−41.5533	−0.215302	−0.107651	+	0.994189i	$0.534333\pi$
$194$	70.9054i	0.365492i
$195$	0	0
$196$	66.5972	0.339782
$197$	37.7643i	0.191697i	0.995396	+	0.0958485i	$0.0305564\pi$
$197$	37.7643i	0.191697i	−0.995396	+	0.0958485i	$0.969444\pi$
$198$	0	0
$199$	301.860	1.51688	0.758442	−	0.651741i	$-0.225961\pi$
$199$	301.860	1.51688	0.758442	+	0.651741i	$0.225961\pi$
$200$	44.3268i	0.221634i
$201$	0	0
$202$	−31.5490	−0.156183
$203$	317.618i	1.56462i
$204$	0	0
$205$	52.1664	0.254470
$206$	− 35.5687i	− 0.172664i
$207$	0	0
$208$	−221.214	−1.06353
$209$	124.375i	0.595094i
$210$	0	0
$211$	108.604	0.514711	0.257356	−	0.966317i	$-0.417149\pi$
$211$	108.604	0.514711	0.257356	+	0.966317i	$0.417149\pi$
$212$	− 238.915i	− 1.12696i
$213$	0	0
$214$	101.361	0.473649
$215$	104.467i	0.485892i
$216$	0	0
$217$	−282.264	−1.30076
$218$	− 145.471i	− 0.667300i
$219$	0	0
$220$	−90.0547	−0.409339
$221$	− 93.2841i	− 0.422100i
$222$	0	0
$223$	−29.2060	−0.130969	−0.0654843	−	0.997854i	$-0.520859\pi$
$223$	−29.2060	−0.130969	−0.0654843	+	0.997854i	$0.520859\pi$
$224$	241.156i	1.07659i
$225$	0	0
$226$	38.4769	0.170252
$227$	158.247i	0.697124i	0.937286	+	0.348562i	$0.113330\pi$
$227$	158.247i	0.697124i	−0.937286	+	0.348562i	$0.886670\pi$
$228$	0	0
$229$	−243.276	−1.06234	−0.531170	−	0.847266i	$-0.678247\pi$
$229$	−243.276	−1.06234	−0.531170	+	0.847266i	$0.678247\pi$
$230$	− 158.214i	− 0.687886i
$231$	0	0
$232$	−211.257	−0.910591
$233$	378.974i	1.62650i	0.581916	+	0.813249i	$0.302303\pi$
$233$	378.974i	1.62650i	−0.581916	+	0.813249i	$0.697697\pi$
$234$	0	0
$235$	−315.029	−1.34055
$236$	124.302i	0.526703i
$237$	0	0
$238$	−25.1057	−0.105486
$239$	285.070i	1.19276i	0.802702	+	0.596381i	$0.203395\pi$
$239$	285.070i	1.19276i	−0.802702	+	0.596381i	$0.796605\pi$
$240$	0	0
$241$	−108.978	−0.452190	−0.226095	−	0.974105i	$-0.572596\pi$
$241$	−108.978	−0.452190	−0.226095	+	0.974105i	$0.572596\pi$
$242$	− 74.0475i	− 0.305981i
$243$	0	0
$244$	−16.4225	−0.0673051
$245$	− 110.780i	− 0.452165i
$246$	0	0
$247$	621.133	2.51471
$248$	− 187.742i	− 0.757024i
$249$	0	0
$250$	−71.7624	−0.287050
$251$	214.535i	0.854721i	0.904081	+	0.427361i	$0.140557\pi$
$251$	214.535i	0.854721i	−0.904081	+	0.427361i	$0.859443\pi$
$252$	0	0
$253$	169.136	0.668521
$254$	− 103.772i	− 0.408551i
$255$	0	0
$256$	−25.2005	−0.0984393
$257$	− 370.911i	− 1.44324i	−0.692292	−	0.721618i	$-0.743399\pi$
$257$	− 370.911i	− 1.44324i	0.692292	−	0.721618i	$-0.256601\pi$
$258$	0	0
$259$	4.41876	0.0170608
$260$	449.737i	1.72976i
$261$	0	0
$262$	−0.663424	−0.00253215
$263$	152.898i	0.581360i	0.956820	+	0.290680i	$0.0938815\pi$
$263$	152.898i	0.581360i	−0.956820	+	0.290680i	$0.906119\pi$
$264$	0	0
$265$	−397.420	−1.49970
$266$	− 167.166i	− 0.628445i
$267$	0	0
$268$	185.676	0.692819
$269$	130.516i	0.485191i	0.970127	+	0.242596i	$0.0779988\pi$
$269$	130.516i	0.485191i	−0.970127	+	0.242596i	$0.922001\pi$
$270$	0	0
$271$	−100.177	−0.369656	−0.184828	−	0.982771i	$-0.559173\pi$
$271$	−100.177	−0.369656	−0.184828	+	0.982771i	$0.559173\pi$
$272$	40.3138i	0.148213i
$273$	0	0
$274$	−81.1833	−0.296289
$275$	36.5420i	0.132880i
$276$	0	0
$277$	−18.5107	−0.0668256	−0.0334128	−	0.999442i	$-0.510638\pi$
$277$	−18.5107	−0.0668256	−0.0334128	+	0.999442i	$0.510638\pi$
$278$	− 134.218i	− 0.482798i
$279$	0	0
$280$	261.093	0.932476
$281$	313.029i	1.11398i	0.830518	+	0.556991i	$0.188044\pi$
$281$	313.029i	1.11398i	−0.830518	+	0.556991i	$0.811956\pi$
$282$	0	0
$283$	−261.138	−0.922751	−0.461375	−	0.887205i	$-0.652644\pi$
$283$	−261.138	−0.922751	−0.461375	+	0.887205i	$0.652644\pi$
$284$	− 292.257i	− 1.02907i
$285$	0	0
$286$	75.5373	0.264117
$287$	− 74.9547i	− 0.261166i
$288$	0	0
$289$	−17.0000	−0.0588235
$290$	162.909i	0.561755i
$291$	0	0
$292$	−377.294	−1.29210
$293$	337.949i	1.15341i	0.816952	+	0.576705i	$0.195662\pi$
$293$	337.949i	1.15341i	−0.816952	+	0.576705i	$0.804338\pi$
$294$	0	0
$295$	206.769	0.700911
$296$	2.93904i	0.00992919i
$297$	0	0
$298$	−120.996	−0.406027
$299$	− 844.672i	− 2.82499i
$300$	0	0
$301$	150.102	0.498677
$302$	− 26.9237i	− 0.0891514i
$303$	0	0
$304$	−268.430	−0.882994
$305$	27.3177i	0.0895664i
$306$	0	0
$307$	206.927	0.674030	0.337015	−	0.941499i	$-0.390583\pi$
$307$	206.927	0.674030	0.337015	+	0.941499i	$0.390583\pi$
$308$	129.394i	0.420111i
$309$	0	0
$310$	−144.776	−0.467018
$311$	− 27.3306i	− 0.0878798i	−0.999034	−	0.0439399i	$-0.986009\pi$
$311$	− 27.3306i	− 0.0878798i	0.999034	−	0.0439399i	$-0.0139910\pi$
$312$	0	0
$313$	−247.988	−0.792294	−0.396147	−	0.918187i	$-0.629653\pi$
$313$	−247.988	−0.792294	−0.396147	+	0.918187i	$0.629653\pi$
$314$	67.2275i	0.214100i
$315$	0	0
$316$	−390.337	−1.23524
$317$	− 187.835i	− 0.592538i	−0.955104	−	0.296269i	$-0.904257\pi$
$317$	− 187.835i	− 0.592538i	0.955104	−	0.296269i	$-0.0957426\pi$
$318$	0	0
$319$	−174.155	−0.545942
$320$	− 101.205i	− 0.316265i
$321$	0	0
$322$	−227.328	−0.705987
$323$	− 113.195i	− 0.350448i
$324$	0	0
$325$	182.492	0.561514
$326$	79.4778i	0.243797i
$327$	0	0
$328$	49.8545	0.151996
$329$	452.647i	1.37583i
$330$	0	0
$331$	−341.100	−1.03051	−0.515257	−	0.857036i	$-0.672303\pi$
$331$	−341.100	−1.03051	−0.515257	+	0.857036i	$0.672303\pi$
$332$	− 560.139i	− 1.68717i
$333$	0	0
$334$	149.953	0.448960
$335$	− 308.860i	− 0.921970i
$336$	0	0
$337$	−552.709	−1.64008	−0.820042	−	0.572303i	$-0.806050\pi$
$337$	−552.709	−1.64008	−0.820042	+	0.572303i	$0.806050\pi$
$338$	− 252.689i	− 0.747602i
$339$	0	0
$340$	81.9596	0.241058
$341$	− 154.770i	− 0.453871i
$342$	0	0
$343$	245.678	0.716261
$344$	99.8370i	0.290224i
$345$	0	0
$346$	−5.57937	−0.0161253
$347$	− 233.876i	− 0.673995i	−0.941506	−	0.336997i	$-0.890589\pi$
$347$	− 233.876i	− 0.673995i	0.941506	−	0.336997i	$-0.109411\pi$
$348$	0	0
$349$	−200.132	−0.573444	−0.286722	−	0.958014i	$-0.592566\pi$
$349$	−200.132	−0.573444	−0.286722	+	0.958014i	$0.592566\pi$
$350$	− 49.1143i	− 0.140327i
$351$	0	0
$352$	−132.230	−0.375653
$353$	7.33355i	0.0207749i	0.999946	+	0.0103875i	$0.00330649\pi$
$353$	7.33355i	0.0207749i	−0.999946	+	0.0103875i	$0.996694\pi$
$354$	0	0
$355$	−486.151	−1.36944
$356$	− 152.122i	− 0.427308i
$357$	0	0
$358$	−244.452	−0.682826
$359$	− 492.659i	− 1.37231i	−0.727456	−	0.686154i	$-0.759298\pi$
$359$	− 492.659i	− 1.37231i	0.727456	−	0.686154i	$-0.240702\pi$
$360$	0	0
$361$	392.707	1.08783
$362$	68.9441i	0.190453i
$363$	0	0
$364$	646.200	1.77527
$365$	627.606i	1.71947i
$366$	0	0
$367$	497.347	1.35517	0.677585	−	0.735445i	$-0.263027\pi$
$367$	497.347	1.35517	0.677585	+	0.735445i	$0.263027\pi$
$368$	365.035i	0.991944i
$369$	0	0
$370$	2.26642	0.00612545
$371$	571.029i	1.53916i
$372$	0	0
$373$	243.368	0.652460	0.326230	−	0.945290i	$-0.394222\pi$
$373$	243.368	0.652460	0.326230	+	0.945290i	$0.394222\pi$
$374$	− 13.7658i	− 0.0368071i
$375$	0	0
$376$	−301.068	−0.800713
$377$	869.740i	2.30700i
$378$	0	0
$379$	−71.9585	−0.189864	−0.0949321	−	0.995484i	$-0.530263\pi$
$379$	−71.9585	−0.189864	−0.0949321	+	0.995484i	$0.530263\pi$
$380$	545.729i	1.43613i
$381$	0	0
$382$	96.1429	0.251683
$383$	− 199.114i	− 0.519879i	−0.965625	−	0.259940i	$-0.916297\pi$
$383$	− 199.114i	− 0.519879i	0.965625	−	0.259940i	$-0.0837027\pi$
$384$	0	0
$385$	215.239	0.559063
$386$	30.6234i	0.0793352i
$387$	0	0
$388$	−332.595	−0.857204
$389$	− 340.779i	− 0.876038i	−0.898966	−	0.438019i	$-0.855680\pi$
$389$	− 340.779i	− 0.876038i	0.898966	−	0.438019i	$-0.144320\pi$
$390$	0	0
$391$	−153.932	−0.393689
$392$	− 105.871i	− 0.270079i
$393$	0	0
$394$	27.8310	0.0706371
$395$	649.301i	1.64380i
$396$	0	0
$397$	429.345	1.08147	0.540737	−	0.841192i	$-0.318146\pi$
$397$	429.345	1.08147	0.540737	+	0.841192i	$0.318146\pi$
$398$	− 222.461i	− 0.558946i
$399$	0	0
$400$	−78.8662	−0.197166
$401$	− 719.715i	− 1.79480i	−0.441216	−	0.897401i	$-0.645453\pi$
$401$	− 719.715i	− 1.79480i	0.441216	−	0.897401i	$-0.354547\pi$
$402$	0	0
$403$	−772.929	−1.91794
$404$	− 147.986i	− 0.366303i
$405$	0	0
$406$	234.074	0.576537
$407$	2.42288i	0.00595301i
$408$	0	0
$409$	253.316	0.619355	0.309677	−	0.950842i	$-0.399779\pi$
$409$	253.316	0.619355	0.309677	+	0.950842i	$0.399779\pi$
$410$	− 38.4449i	− 0.0937680i
$411$	0	0
$412$	166.842	0.404956
$413$	− 297.093i	− 0.719355i
$414$	0	0
$415$	−931.757	−2.24520
$416$	660.361i	1.58741i
$417$	0	0
$418$	91.6600	0.219282
$419$	200.345i	0.478151i	0.971001	+	0.239076i	$0.0768444\pi$
$419$	200.345i	0.478151i	−0.971001	+	0.239076i	$0.923156\pi$
$420$	0	0
$421$	−50.3166	−0.119517	−0.0597584	−	0.998213i	$-0.519033\pi$
$421$	−50.3166	−0.119517	−0.0597584	+	0.998213i	$0.519033\pi$
$422$	− 80.0376i	− 0.189663i
$423$	0	0
$424$	−379.808	−0.895773
$425$	− 33.2572i	− 0.0782522i
$426$	0	0
$427$	39.2512	0.0919232
$428$	475.452i	1.11087i
$429$	0	0
$430$	76.9885	0.179043
$431$	− 424.583i	− 0.985111i	−0.870281	−	0.492556i	$-0.836063\pi$
$431$	− 424.583i	− 0.985111i	0.870281	−	0.492556i	$-0.163937\pi$
$432$	0	0
$433$	−799.376	−1.84613	−0.923067	−	0.384639i	$-0.874326\pi$
$433$	−799.376	−1.84613	−0.923067	+	0.384639i	$0.874326\pi$
$434$	208.019i	0.479307i
$435$	0	0
$436$	682.361	1.56505
$437$	− 1024.96i	− 2.34544i
$438$	0	0
$439$	−368.695	−0.839852	−0.419926	−	0.907558i	$-0.637944\pi$
$439$	−368.695	−0.839852	−0.419926	+	0.907558i	$0.637944\pi$
$440$	143.162i	0.325367i
$441$	0	0
$442$	−68.7473	−0.155537
$443$	− 110.882i	− 0.250298i	−0.992138	−	0.125149i	$-0.960059\pi$
$443$	− 110.882i	− 0.250298i	0.992138	−	0.125149i	$-0.0399408\pi$
$444$	0	0
$445$	−253.045	−0.568640
$446$	21.5238i	0.0482597i
$447$	0	0
$448$	−145.415	−0.324587
$449$	215.373i	0.479672i	0.970813	+	0.239836i	$0.0770937\pi$
$449$	215.373i	0.479672i	−0.970813	+	0.239836i	$0.922906\pi$
$450$	0	0
$451$	41.0989	0.0911284
$452$	180.483i	0.399299i
$453$	0	0
$454$	116.623	0.256878
$455$	− 1074.91i	− 2.36245i
$456$	0	0
$457$	459.587	1.00566	0.502830	−	0.864385i	$-0.332292\pi$
$457$	459.587	1.00566	0.502830	+	0.864385i	$0.332292\pi$
$458$	179.286i	0.391454i
$459$	0	0
$460$	742.131	1.61333
$461$	− 349.049i	− 0.757157i	−0.925569	−	0.378578i	$-0.876413\pi$
$461$	− 349.049i	− 0.757157i	0.925569	−	0.378578i	$-0.123587\pi$
$462$	0	0
$463$	266.427	0.575437	0.287719	−	0.957715i	$-0.407103\pi$
$463$	266.427	0.575437	0.287719	+	0.957715i	$0.407103\pi$
$464$	− 375.869i	− 0.810062i
$465$	0	0
$466$	279.291	0.599337
$467$	306.559i	0.656443i	0.944601	+	0.328221i	$0.106449\pi$
$467$	306.559i	0.656443i	−0.944601	+	0.328221i	$0.893551\pi$
$468$	0	0
$469$	−443.782	−0.946231
$470$	232.166i	0.493971i
$471$	0	0
$472$	197.605	0.418655
$473$	82.3033i	0.174003i
$474$	0	0
$475$	221.443	0.466196
$476$	− 117.763i	− 0.247401i
$477$	0	0
$478$	210.087	0.439513
$479$	− 384.989i	− 0.803734i	−0.915698	−	0.401867i	$-0.868361\pi$
$479$	− 384.989i	− 0.803734i	0.915698	−	0.401867i	$-0.131639\pi$
$480$	0	0
$481$	12.1000	0.0251558
$482$	80.3129i	0.166624i
$483$	0	0
$484$	347.334	0.717631
$485$	553.252i	1.14073i
$486$	0	0
$487$	584.764	1.20075	0.600374	−	0.799719i	$-0.295018\pi$
$487$	584.764	1.20075	0.600374	+	0.799719i	$0.295018\pi$
$488$	26.1071i	0.0534982i
$489$	0	0
$490$	−81.6415	−0.166615
$491$	217.884i	0.443755i	0.975075	+	0.221877i	$0.0712185\pi$
$491$	217.884i	0.443755i	−0.975075	+	0.221877i	$0.928782\pi$
$492$	0	0
$493$	158.501	0.321502
$494$	− 457.754i	− 0.926628i
$495$	0	0
$496$	334.031	0.673449
$497$	698.521i	1.40547i
$498$	0	0
$499$	−151.280	−0.303166	−0.151583	−	0.988445i	$-0.548437\pi$
$499$	−151.280	−0.303166	−0.151583	+	0.988445i	$0.548437\pi$
$500$	− 336.615i	− 0.673230i
$501$	0	0
$502$	158.105	0.314951
$503$	8.00028i	0.0159051i	0.999968	+	0.00795257i	$0.00253141\pi$
$503$	8.00028i	0.0159051i	−0.999968	+	0.00795257i	$0.997469\pi$
$504$	0	0
$505$	−246.166	−0.487458
$506$	− 124.647i	− 0.246339i
$507$	0	0
$508$	486.762	0.958194
$509$	− 97.1115i	− 0.190789i	−0.995440	−	0.0953944i	$-0.969589\pi$
$509$	− 97.1115i	− 0.190789i	0.995440	−	0.0953944i	$-0.0304112\pi$
$510$	0	0
$511$	901.769	1.76471
$512$	− 500.312i	− 0.977172i
$513$	0	0
$514$	−273.349	−0.531808
$515$	− 277.531i	− 0.538895i
$516$	0	0
$517$	−248.194	−0.480065
$518$	− 3.25648i	− 0.00628663i
$519$	0	0
$520$	714.956	1.37492
$521$	424.535i	0.814847i	0.913239	+	0.407424i	$0.133573\pi$
$521$	424.535i	0.814847i	−0.913239	+	0.407424i	$0.866427\pi$
$522$	0	0
$523$	683.906	1.30766	0.653830	−	0.756641i	$-0.273161\pi$
$523$	683.906	1.30766	0.653830	+	0.756641i	$0.273161\pi$
$524$	− 3.11192i	− 0.00593877i
$525$	0	0
$526$	112.680	0.214221
$527$	140.858i	0.267282i
$528$	0	0
$529$	−864.831	−1.63484
$530$	292.885i	0.552614i
$531$	0	0
$532$	784.125	1.47392
$533$	− 205.250i	− 0.385084i
$534$	0	0
$535$	790.886	1.47829
$536$	− 295.172i	− 0.550694i
$537$	0	0
$538$	96.1863	0.178785
$539$	− 87.2775i	− 0.161925i
$540$	0	0
$541$	444.490	0.821608	0.410804	−	0.911724i	$-0.365248\pi$
$541$	444.490	0.821608	0.410804	+	0.911724i	$0.365248\pi$
$542$	73.8270i	0.136212i
$543$	0	0
$544$	120.344	0.221220
$545$	− 1135.07i	− 2.08269i
$546$	0	0
$547$	−771.036	−1.40957	−0.704786	−	0.709420i	$-0.748957\pi$
$547$	−771.036	−1.40957	−0.704786	+	0.709420i	$0.748957\pi$
$548$	− 380.806i	− 0.694901i
$549$	0	0
$550$	26.9302	0.0489640
$551$	1055.38i	1.91538i
$552$	0	0
$553$	932.941	1.68706
$554$	13.6418i	0.0246241i
$555$	0	0
$556$	629.574	1.13233
$557$	781.224i	1.40256i	0.712888	+	0.701278i	$0.247387\pi$
$557$	781.224i	1.40256i	−0.712888	+	0.701278i	$0.752613\pi$
$558$	0	0
$559$	411.026	0.735289
$560$	464.537i	0.829530i
$561$	0	0
$562$	230.692	0.410484
$563$	− 746.517i	− 1.32596i	−0.748636	−	0.662981i	$-0.769291\pi$
$563$	− 746.517i	− 1.32596i	0.748636	−	0.662981i	$-0.230709\pi$
$564$	0	0
$565$	300.223	0.531368
$566$	192.450i	0.340018i
$567$	0	0
$568$	−464.606	−0.817969
$569$	796.201i	1.39930i	0.714487	+	0.699649i	$0.246660\pi$
$569$	796.201i	1.39930i	−0.714487	+	0.699649i	$0.753340\pi$
$570$	0	0
$571$	556.430	0.974484	0.487242	−	0.873267i	$-0.338003\pi$
$571$	556.430	0.974484	0.487242	+	0.873267i	$0.338003\pi$
$572$	354.322i	0.619444i
$573$	0	0
$574$	−55.2391	−0.0962354
$575$	− 301.138i	− 0.523719i
$576$	0	0
$577$	−864.828	−1.49883	−0.749417	−	0.662098i	$-0.769666\pi$
$577$	−864.828	−1.49883	−0.749417	+	0.662098i	$0.769666\pi$
$578$	12.5284i	0.0216755i
$579$	0	0
$580$	−764.156	−1.31751
$581$	1338.79i	2.30428i
$582$	0	0
$583$	−313.105	−0.537058
$584$	599.792i	1.02704i
$585$	0	0
$586$	249.057	0.425013
$587$	− 766.397i	− 1.30562i	−0.757523	−	0.652809i	$-0.773591\pi$
$587$	− 766.397i	− 1.30562i	0.757523	−	0.652809i	$-0.226409\pi$
$588$	0	0
$589$	−937.903	−1.59236
$590$	− 152.382i	− 0.258274i
$591$	0	0
$592$	−5.22914	−0.00883301
$593$	− 21.5288i	− 0.0363049i	−0.999835	−	0.0181525i	$-0.994222\pi$
$593$	− 21.5288i	− 0.0363049i	0.999835	−	0.0181525i	$-0.00577843\pi$
$594$	0	0
$595$	−195.891	−0.329229
$596$	− 567.555i	− 0.952273i
$597$	0	0
$598$	−622.495	−1.04096
$599$	121.869i	0.203454i	0.994812	+	0.101727i	$0.0324369\pi$
$599$	121.869i	0.203454i	−0.994812	+	0.101727i	$0.967563\pi$
$600$	0	0
$601$	−30.5002	−0.0507491	−0.0253746	−	0.999678i	$-0.508078\pi$
$601$	−30.5002	−0.0507491	−0.0253746	+	0.999678i	$0.508078\pi$
$602$	− 110.620i	− 0.183754i
$603$	0	0
$604$	126.291	0.209091
$605$	− 577.768i	− 0.954989i
$606$	0	0
$607$	0.0533785	8.79383e−5 0	4.39691e−5	−	1.00000i	$-0.499986\pi$
$607$	0.0533785	8.79383e−5 0	4.39691e−5		1.00000i	$0.499986\pi$
$608$	801.309i	1.31794i
$609$	0	0
$610$	20.1323	0.0330037
$611$	1239.49i	2.02863i
$612$	0	0
$613$	−1089.81	−1.77782	−0.888912	−	0.458078i	$-0.848538\pi$
$613$	−1089.81	−1.77782	−0.888912	+	0.458078i	$0.848538\pi$
$614$	− 152.499i	− 0.248369i
$615$	0	0
$616$	205.700	0.333929
$617$	778.150i	1.26118i	0.776115	+	0.630592i	$0.217188\pi$
$617$	778.150i	1.26118i	−0.776115	+	0.630592i	$0.782812\pi$
$618$	0	0
$619$	937.604	1.51471	0.757354	−	0.653004i	$-0.226492\pi$
$619$	937.604	1.51471	0.757354	+	0.653004i	$0.226492\pi$
$620$	− 679.097i	− 1.09532i
$621$	0	0
$622$	−20.1417	−0.0323822
$623$	363.585i	0.583604i
$624$	0	0
$625$	−761.590	−1.21854
$626$	182.759i	0.291947i
$627$	0	0
$628$	−315.343	−0.502139
$629$	− 2.20508i	− 0.00350570i
$630$	0	0
$631$	856.821	1.35788	0.678939	−	0.734195i	$-0.262440\pi$
$631$	856.821	1.35788	0.678939	+	0.734195i	$0.262440\pi$
$632$	620.526i	0.981845i
$633$	0	0
$634$	−138.428	−0.218340
$635$	− 809.700i	− 1.27512i
$636$	0	0
$637$	−435.868	−0.684251
$638$	128.347i	0.201170i
$639$	0	0
$640$	−745.936	−1.16552
$641$	− 1042.26i	− 1.62599i	−0.582268	−	0.812997i	$-0.697835\pi$
$641$	− 1042.26i	− 1.62599i	0.582268	−	0.812997i	$-0.302165\pi$
$642$	0	0
$643$	−14.0719	−0.0218848	−0.0109424	−	0.999940i	$-0.503483\pi$
$643$	−14.0719	−0.0218848	−0.0109424	+	0.999940i	$0.503483\pi$
$644$	− 1066.32i	− 1.65578i
$645$	0	0
$646$	−83.4207	−0.129134
$647$	626.827i	0.968821i	0.874841	+	0.484411i	$0.160966\pi$
$647$	626.827i	0.968821i	−0.874841	+	0.484411i	$0.839034\pi$
$648$	0	0
$649$	162.901	0.251003
$650$	− 134.491i	− 0.206909i
$651$	0	0
$652$	−372.806	−0.571788
$653$	208.665i	0.319549i	0.987154	+	0.159774i	$0.0510767\pi$
$653$	208.665i	0.319549i	−0.987154	+	0.159774i	$0.948923\pi$
$654$	0	0
$655$	−5.17648	−0.00790303
$656$	88.7012i	0.135215i
$657$	0	0
$658$	333.585	0.506969
$659$	− 760.139i	− 1.15347i	−0.816930	−	0.576737i	$-0.804326\pi$
$659$	− 760.139i	− 1.15347i	0.816930	−	0.576737i	$-0.195674\pi$
$660$	0	0
$661$	221.854	0.335634	0.167817	−	0.985818i	$-0.446328\pi$
$661$	221.854	0.335634	0.167817	+	0.985818i	$0.446328\pi$
$662$	251.379i	0.379727i
$663$	0	0
$664$	−890.464	−1.34106
$665$	− 1304.34i	− 1.96142i
$666$	0	0
$667$	1435.20	2.15172
$668$	703.381i	1.05297i
$669$	0	0
$670$	−227.619	−0.339730
$671$	21.5221i	0.0320746i
$672$	0	0
$673$	−55.1837	−0.0819965	−0.0409983	−	0.999159i	$-0.513054\pi$
$673$	−55.1837	−0.0819965	−0.0409983	+	0.999159i	$0.513054\pi$
$674$	407.328i	0.604344i
$675$	0	0
$676$	1185.29	1.75338
$677$	− 714.307i	− 1.05511i	−0.849522	−	0.527553i	$-0.823109\pi$
$677$	− 714.307i	− 1.05511i	0.849522	−	0.527553i	$-0.176891\pi$
$678$	0	0
$679$	794.934	1.17074
$680$	− 130.293i	− 0.191607i
$681$	0	0
$682$	−114.060	−0.167244
$683$	1102.08i	1.61359i	0.590835	+	0.806793i	$0.298799\pi$
$683$	1102.08i	1.61359i	−0.590835	+	0.806793i	$0.701201\pi$
$684$	0	0
$685$	−633.447	−0.924740
$686$	− 181.056i	− 0.263930i
$687$	0	0
$688$	−177.630	−0.258183
$689$	1563.66i	2.26946i
$690$	0	0
$691$	−8.42852	−0.0121976	−0.00609879	−	0.999981i	$-0.501941\pi$
$691$	−8.42852	−0.0121976	−0.00609879	+	0.999981i	$0.501941\pi$
$692$	− 26.1711i	− 0.0378195i
$693$	0	0
$694$	−172.359	−0.248356
$695$	− 1047.26i	− 1.50684i
$696$	0	0
$697$	−37.4045	−0.0536650
$698$	147.491i	0.211305i
$699$	0	0
$700$	230.380	0.329114
$701$	− 823.596i	− 1.17489i	−0.809265	−	0.587444i	$-0.800134\pi$
$701$	− 823.596i	− 1.17489i	0.809265	−	0.587444i	$-0.199866\pi$
$702$	0	0
$703$	14.6826	0.0208856
$704$	− 79.7334i	− 0.113258i
$705$	0	0
$706$	5.40458	0.00765522
$707$	353.702i	0.500285i
$708$	0	0
$709$	−598.044	−0.843504	−0.421752	−	0.906711i	$-0.638585\pi$
$709$	−598.044	−0.843504	−0.421752	+	0.906711i	$0.638585\pi$
$710$	358.277i	0.504615i
$711$	0	0
$712$	−241.831	−0.339650
$713$	1275.44i	1.78884i
$714$	0	0
$715$	589.393	0.824326
$716$	− 1146.65i	− 1.60146i
$717$	0	0
$718$	−363.073	−0.505673
$719$	1087.21i	1.51211i	0.654508	+	0.756055i	$0.272876\pi$
$719$	1087.21i	1.51211i	−0.654508	+	0.756055i	$0.727124\pi$
$720$	0	0
$721$	−398.768	−0.553076
$722$	− 289.412i	− 0.400848i
$723$	0	0
$724$	−323.395	−0.446679
$725$	310.075i	0.427690i
$726$	0	0
$727$	483.010	0.664388	0.332194	−	0.943211i	$-0.392211\pi$
$727$	483.010	0.664388	0.332194	+	0.943211i	$0.392211\pi$
$728$	− 1027.28i	− 1.41109i
$729$	0	0
$730$	462.524	0.633595
$731$	− 74.9051i	− 0.102469i
$732$	0	0
$733$	−1382.66	−1.88631	−0.943153	−	0.332359i	$-0.892156\pi$
$733$	−1382.66	−1.88631	−0.943153	+	0.332359i	$0.892156\pi$
$734$	− 366.528i	− 0.499357i
$735$	0	0
$736$	1089.69	1.48056
$737$	− 243.333i	− 0.330167i
$738$	0	0
$739$	203.552	0.275443	0.137721	−	0.990471i	$-0.456022\pi$
$739$	203.552	0.275443	0.137721	+	0.990471i	$0.456022\pi$
$740$	10.6310i	0.0143663i
$741$	0	0
$742$	420.829	0.567155
$743$	− 49.0580i	− 0.0660269i	−0.999455	−	0.0330135i	$-0.989490\pi$
$743$	− 49.0580i	− 0.0660269i	0.999455	−	0.0330135i	$-0.0105104\pi$
$744$	0	0
$745$	−944.092	−1.26724
$746$	− 179.354i	− 0.240421i
$747$	0	0
$748$	64.5713	0.0863253
$749$	− 1136.38i	− 1.51719i
$750$	0	0
$751$	280.297	0.373231	0.186616	−	0.982433i	$-0.440248\pi$
$751$	280.297	0.373231	0.186616	+	0.982433i	$0.440248\pi$
$752$	− 535.661i	− 0.712315i
$753$	0	0
$754$	640.969	0.850092
$755$	− 210.077i	− 0.278248i
$756$	0	0
$757$	−1036.08	−1.36867	−0.684333	−	0.729170i	$-0.739906\pi$
$757$	−1036.08	−1.36867	−0.684333	+	0.729170i	$0.739906\pi$
$758$	53.0310i	0.0699618i
$759$	0	0
$760$	867.556	1.14152
$761$	− 41.5943i	− 0.0546575i	−0.999627	−	0.0273287i	$-0.991300\pi$
$761$	− 41.5943i	− 0.0546575i	0.999627	−	0.0273287i	$-0.00870009\pi$
$762$	0	0
$763$	−1630.91	−2.13749
$764$	450.977i	0.590283i
$765$	0	0
$766$	−146.740	−0.191567
$767$	− 813.536i	− 1.06067i
$768$	0	0
$769$	−482.877	−0.627928	−0.313964	−	0.949435i	$-0.601657\pi$
$769$	−482.877	−0.627928	−0.313964	+	0.949435i	$0.601657\pi$
$770$	− 158.624i	− 0.206005i
$771$	0	0
$772$	−143.645	−0.186068
$773$	− 1448.02i	− 1.87324i	−0.350341	−	0.936622i	$-0.613934\pi$
$773$	− 1448.02i	− 1.87324i	0.350341	−	0.936622i	$-0.386066\pi$
$774$	0	0
$775$	−275.561	−0.355562
$776$	528.733i	0.681357i
$777$	0	0
$778$	−251.143	−0.322805
$779$	− 249.058i	− 0.319715i
$780$	0	0
$781$	−383.010	−0.490410
$782$	113.443i	0.145068i
$783$	0	0
$784$	188.366	0.240262
$785$	524.555i	0.668222i
$786$	0	0
$787$	423.469	0.538080	0.269040	−	0.963129i	$-0.413294\pi$
$787$	423.469	0.538080	0.269040	+	0.963129i	$0.413294\pi$
$788$	130.547i	0.165668i
$789$	0	0
$790$	478.513	0.605713
$791$	− 431.372i	− 0.545351i
$792$	0	0
$793$	107.482	0.135539
$794$	− 316.413i	− 0.398505i
$795$	0	0
$796$	1043.49	1.31092
$797$	952.166i	1.19469i	0.801985	+	0.597344i	$0.203777\pi$
$797$	952.166i	1.19469i	−0.801985	+	0.597344i	$0.796223\pi$
$798$	0	0
$799$	225.883	0.282708
$800$	235.429i	0.294286i
$801$	0	0
$802$	−530.406	−0.661354
$803$	494.455i	0.615759i
$804$	0	0
$805$	−1773.76	−2.20343
$806$	569.623i	0.706728i
$807$	0	0
$808$	−235.257	−0.291160
$809$	− 522.791i	− 0.646219i	−0.946362	−	0.323110i	$-0.895272\pi$
$809$	− 522.791i	− 0.646219i	0.946362	−	0.323110i	$-0.104728\pi$
$810$	0	0
$811$	−1074.27	−1.32462	−0.662310	−	0.749230i	$-0.730424\pi$
$811$	−1074.27	−1.32462	−0.662310	+	0.749230i	$0.730424\pi$
$812$	1097.97i	1.35218i
$813$	0	0
$814$	1.78558	0.00219359
$815$	620.140i	0.760908i
$816$	0	0
$817$	498.756	0.610472
$818$	− 186.685i	− 0.228222i
$819$	0	0
$820$	180.333	0.219918
$821$	− 475.241i	− 0.578856i	−0.957200	−	0.289428i	$-0.906535\pi$
$821$	− 475.241i	− 0.578856i	0.957200	−	0.289428i	$-0.0934651\pi$
$822$	0	0
$823$	−121.758	−0.147944	−0.0739721	−	0.997260i	$-0.523568\pi$
$823$	−121.758	−0.147944	−0.0739721	+	0.997260i	$0.523568\pi$
$824$	− 265.232i	− 0.321883i
$825$	0	0
$826$	−218.948	−0.265070
$827$	− 339.395i	− 0.410393i	−0.978721	−	0.205197i	$-0.934217\pi$
$827$	− 339.395i	− 0.410393i	0.978721	−	0.205197i	$-0.0657834\pi$
$828$	0	0
$829$	−351.251	−0.423704	−0.211852	−	0.977302i	$-0.567949\pi$
$829$	−351.251	−0.423704	−0.211852	+	0.977302i	$0.567949\pi$
$830$	686.674i	0.827318i
$831$	0	0
$832$	−398.192	−0.478597
$833$	79.4321i	0.0953567i
$834$	0	0
$835$	1170.03	1.40123
$836$	429.948i	0.514292i
$837$	0	0
$838$	147.648	0.176191
$839$	1008.84i	1.20243i	0.799087	+	0.601215i	$0.205317\pi$
$839$	1008.84i	1.20243i	−0.799087	+	0.601215i	$0.794683\pi$
$840$	0	0
$841$	−636.789	−0.757181
$842$	37.0816i	0.0440399i
$843$	0	0
$844$	375.431	0.444824
$845$	− 1971.65i	− 2.33332i
$846$	0	0
$847$	−830.160	−0.980118
$848$	− 675.754i	− 0.796880i
$849$	0	0
$850$	−24.5094	−0.0288346
$851$	− 19.9667i	− 0.0234626i
$852$	0	0
$853$	1032.31	1.21021	0.605107	−	0.796144i	$-0.293130\pi$
$853$	1032.31	1.21021	0.605107	+	0.796144i	$0.293130\pi$
$854$	− 28.9268i	− 0.0338722i
$855$	0	0
$856$	755.836	0.882986
$857$	− 1107.61i	− 1.29243i	−0.763155	−	0.646216i	$-0.776351\pi$
$857$	− 1107.61i	− 1.29243i	0.763155	−	0.646216i	$-0.223649\pi$
$858$	0	0
$859$	−438.057	−0.509961	−0.254981	−	0.966946i	$-0.582069\pi$
$859$	−438.057	−0.509961	−0.254981	+	0.966946i	$0.582069\pi$
$860$	361.129i	0.419917i
$861$	0	0
$862$	−312.903	−0.362997
$863$	− 1513.43i	− 1.75369i	−0.480775	−	0.876844i	$-0.659645\pi$
$863$	− 1513.43i	− 1.75369i	0.480775	−	0.876844i	$-0.340355\pi$
$864$	0	0
$865$	−43.5340	−0.0503283
$866$	589.113i	0.680270i
$867$	0	0
$868$	−975.754	−1.12414
$869$	511.547i	0.588662i
$870$	0	0
$871$	−1215.22	−1.39520
$872$	− 1084.76i	− 1.24399i
$873$	0	0
$874$	−755.360	−0.864257
$875$	804.542i	0.919477i
$876$	0	0
$877$	1343.35	1.53175	0.765876	−	0.642988i	$-0.222306\pi$
$877$	1343.35	1.53175	0.765876	+	0.642988i	$0.222306\pi$
$878$	271.716i	0.309471i
$879$	0	0
$880$	−254.713	−0.289447
$881$	1307.71i	1.48434i	0.670211	+	0.742171i	$0.266204\pi$
$881$	1307.71i	1.48434i	−0.670211	+	0.742171i	$0.733796\pi$
$882$	0	0
$883$	−13.3044	−0.0150673	−0.00753364	−	0.999972i	$-0.502398\pi$
$883$	−13.3044	−0.0150673	−0.00753364	+	0.999972i	$0.502398\pi$
$884$	− 322.472i	− 0.364787i
$885$	0	0
$886$	−81.7162	−0.0922305
$887$	− 240.899i	− 0.271588i	−0.990737	−	0.135794i	$-0.956641\pi$
$887$	− 240.899i	− 0.271588i	0.990737	−	0.135794i	$-0.0433585\pi$
$888$	0	0
$889$	−1163.41	−1.30867
$890$	186.486i	0.209534i
$891$	0	0
$892$	−100.962	−0.113186
$893$	1504.05i	1.68426i
$894$	0	0
$895$	−1907.38	−2.13115
$896$	1071.79i	1.19619i
$897$	0	0
$898$	158.722	0.176751
$899$	− 1313.30i	− 1.46084i
$900$	0	0
$901$	284.960	0.316270
$902$	− 30.2885i	− 0.0335793i
$903$	0	0
$904$	286.918	0.317387
$905$	537.948i	0.594418i
$906$	0	0
$907$	891.448	0.982853	0.491426	−	0.870919i	$-0.336476\pi$
$907$	891.448	0.982853	0.491426	+	0.870919i	$0.336476\pi$
$908$	547.041i	0.602468i
$909$	0	0
$910$	−792.176	−0.870523
$911$	150.667i	0.165387i	0.996575	+	0.0826933i	$0.0263522\pi$
$911$	150.667i	0.165387i	−0.996575	+	0.0826933i	$0.973648\pi$
$912$	0	0
$913$	−734.078	−0.804028
$914$	− 338.700i	− 0.370569i
$915$	0	0
$916$	−840.975	−0.918095
$917$	7.43777i	0.00811099i
$918$	0	0
$919$	−746.572	−0.812374	−0.406187	−	0.913790i	$-0.633142\pi$
$919$	−746.572	−0.812374	−0.406187	+	0.913790i	$0.633142\pi$
$920$	− 1179.78i	− 1.28237i
$921$	0	0
$922$	−257.238	−0.279000
$923$	1912.77i	2.07234i
$924$	0	0
$925$	4.31382	0.00466359
$926$	− 196.348i	− 0.212039i
$927$	0	0
$928$	−1122.03	−1.20908
$929$	1353.32i	1.45675i	0.685179	+	0.728375i	$0.259724\pi$
$929$	1353.32i	1.45675i	−0.685179	+	0.728375i	$0.740276\pi$
$930$	0	0
$931$	−528.900	−0.568098
$932$	1310.07i	1.40565i
$933$	0	0
$934$	225.924	0.241888
$935$	− 107.410i	− 0.114877i
$936$	0	0
$937$	1037.51	1.10727	0.553635	−	0.832760i	$-0.313240\pi$
$937$	1037.51	1.10727	0.553635	+	0.832760i	$0.313240\pi$
$938$	327.053i	0.348670i
$939$	0	0
$940$	−1089.02	−1.15853
$941$	− 434.238i	− 0.461465i	−0.973017	−	0.230732i	$-0.925888\pi$
$941$	− 434.238i	− 0.461465i	0.973017	−	0.230732i	$-0.0741122\pi$
$942$	0	0
$943$	−338.692	−0.359164
$944$	351.579i	0.372436i
$945$	0	0
$946$	60.6548	0.0641171
$947$	− 943.690i	− 0.996505i	−0.867032	−	0.498253i	$-0.833975\pi$
$947$	− 943.690i	− 0.996505i	0.867032	−	0.498253i	$-0.166025\pi$
$948$	0	0
$949$	2469.33	2.60203
$950$	− 163.196i	− 0.171786i
$951$	0	0
$952$	−187.210	−0.196649
$953$	− 1654.02i	− 1.73559i	−0.496922	−	0.867795i	$-0.665537\pi$
$953$	− 1654.02i	− 1.73559i	0.496922	−	0.867795i	$-0.334463\pi$
$954$	0	0
$955$	750.172	0.785520
$956$	985.453i	1.03081i
$957$	0	0
$958$	−283.724	−0.296163
$959$	910.162i	0.949074i
$960$	0	0
$961$	206.113	0.214478
$962$	− 8.91726i	− 0.00926950i
$963$	0	0
$964$	−376.723	−0.390791
$965$	238.944i	0.247611i
$966$	0	0
$967$	1854.64	1.91793	0.958965	−	0.283525i	$-0.0915040\pi$
$967$	1854.64	1.91793	0.958965	+	0.283525i	$0.0915040\pi$
$968$	− 552.163i	− 0.570416i
$969$	0	0
$970$	407.728	0.420338
$971$	231.019i	0.237918i	0.992899	+	0.118959i	$0.0379558\pi$
$971$	231.019i	0.237918i	−0.992899	+	0.118959i	$0.962044\pi$
$972$	0	0
$973$	−1504.74	−1.54650
$974$	− 430.952i	− 0.442456i
$975$	0	0
$976$	−46.4498	−0.0475920
$977$	1391.81i	1.42458i	0.701887	+	0.712289i	$0.252341\pi$
$977$	1391.81i	1.42458i	−0.701887	+	0.712289i	$0.747659\pi$
$978$	0	0
$979$	−199.360	−0.203636
$980$	− 382.955i	− 0.390770i
$981$	0	0
$982$	160.573	0.163516
$983$	− 307.299i	− 0.312613i	−0.987709	−	0.156307i	$-0.950041\pi$
$983$	− 307.299i	− 0.312613i	0.987709	−	0.156307i	$-0.0499588\pi$
$984$	0	0
$985$	217.156	0.220463
$986$	− 116.810i	− 0.118468i
$987$	0	0
$988$	2147.18	2.17326
$989$	− 678.253i	− 0.685797i
$990$	0	0
$991$	−935.751	−0.944249	−0.472125	−	0.881532i	$-0.656513\pi$
$991$	−935.751	−0.944249	−0.472125	+	0.881532i	$0.656513\pi$
$992$	− 997.137i	− 1.00518i
$993$	0	0
$994$	514.787	0.517894
$995$	− 1735.79i	− 1.74451i
$996$	0	0
$997$	1046.91	1.05006	0.525030	−	0.851084i	$-0.324054\pi$
$997$	1046.91	1.05006	0.525030	+	0.851084i	$0.324054\pi$
$998$	111.488i	0.111712i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	459.3.b.d.188.5	✓	12
3.2	odd	2	inner	459.3.b.d.188.8	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
459.3.b.d.188.5	✓	12	1.1	even	1	trivial
459.3.b.d.188.8	yes	12	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 459 = 3^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	459.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.736967i q^{2} +3.45688 q^{4} -5.75031i q^{5} -8.26227 q^{7} -5.49547i q^{8} -4.23779 q^{10} -4.53034i q^{11} -22.6247 q^{13} +6.08902i q^{14} +9.77754 q^{16} +4.12311i q^{17} -27.4537 q^{19} -19.8781i q^{20} -3.33871 q^{22} +37.3340i q^{23} -8.06606 q^{25} +16.6737i q^{26} -28.5617 q^{28} -38.4420i q^{29} +34.1630 q^{31} -29.1876i q^{32} +3.03859 q^{34} +47.5106i q^{35} -0.534811 q^{37} +20.2325i q^{38} -31.6007 q^{40} +9.07193i q^{41} -18.1671 q^{43} -15.6608i q^{44} +27.5139 q^{46} -54.7848i q^{47} +19.2651 q^{49} +5.94441i q^{50} -78.2109 q^{52} -69.1129i q^{53} -26.0508 q^{55} +45.4051i q^{56} -28.3305 q^{58} +35.9578i q^{59} -4.75066 q^{61} -25.1770i q^{62} +17.5999 q^{64} +130.099i q^{65} +53.7119 q^{67} +14.2531i q^{68} +35.0137 q^{70} -84.5435i q^{71} -109.143 q^{73} +0.394138i q^{74} -94.9043 q^{76} +37.4309i q^{77} -112.916 q^{79} -56.2239i q^{80} +6.68571 q^{82} -162.036i q^{83} +23.7091 q^{85} +13.3886i q^{86} -24.8963 q^{88} -44.0055i q^{89} +186.932 q^{91} +129.059i q^{92} -40.3746 q^{94} +157.867i q^{95} -96.2125 q^{97} -14.1978i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 20 q^{4} - 54 q^{7} - 12 q^{10} - 50 q^{13} + 108 q^{16} - 50 q^{19} - 106 q^{22} - 48 q^{25} + 382 q^{28} + 2 q^{31} - 244 q^{37} + 208 q^{40} - 78 q^{43} - 18 q^{46} + 470 q^{49} + 42 q^{52} + 290 q^{55}+ \cdots + 66 q^{97}+O(q^{100}) \) 12 * q - 20 * q^4 - 54 * q^7 - 12 * q^10 - 50 * q^13 + 108 * q^16 - 50 * q^19 - 106 * q^22 - 48 * q^25 + 382 * q^28 + 2 * q^31 - 244 * q^37 + 208 * q^40 - 78 * q^43 - 18 * q^46 + 470 * q^49 + 42 * q^52 + 290 * q^55 + 374 * q^58 - 72 * q^61 - 476 * q^64 - 602 * q^67 - 82 * q^70 - 302 * q^73 - 918 * q^76 - 28 * q^79 - 262 * q^82 - 68 * q^85 - 38 * q^88 + 428 * q^91 + 756 * q^94 + 66 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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