LMFDB - Embedded newform 459.3.b.d.188.10

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)

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);

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")

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			188.10
Root			$2.15341i$ of defining polynomial
Character	$\chi$	$=$	459.188
Dual form			459.3.b.d.188.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.15341i q^{2} -0.637159 q^{4} -8.83083i q^{5} -10.7410 q^{7} +7.24156i q^{8} +19.0164 q^{10} +16.2351i q^{11} +2.08942 q^{13} -23.1298i q^{14} -18.1427 q^{16} -4.12311i q^{17} +7.95171 q^{19} +5.62664i q^{20} -34.9607 q^{22} +43.0080i q^{23} -52.9835 q^{25} +4.49936i q^{26} +6.84376 q^{28} +31.0930i q^{29} -43.3950 q^{31} -10.1023i q^{32} +8.87872 q^{34} +94.8524i q^{35} -31.2688 q^{37} +17.1233i q^{38} +63.9490 q^{40} +32.0376i q^{41} -33.9170 q^{43} -10.3443i q^{44} -92.6136 q^{46} -50.5628i q^{47} +66.3701 q^{49} -114.095i q^{50} -1.33129 q^{52} -89.1797i q^{53} +143.369 q^{55} -77.7820i q^{56} -66.9559 q^{58} +47.8328i q^{59} -4.33392 q^{61} -93.4471i q^{62} -50.8163 q^{64} -18.4513i q^{65} +52.2521 q^{67} +2.62707i q^{68} -204.256 q^{70} +64.1659i q^{71} -18.4693 q^{73} -67.3345i q^{74} -5.06650 q^{76} -174.381i q^{77} +43.7991 q^{79} +160.215i q^{80} -68.9900 q^{82} +92.3664i q^{83} -36.4104 q^{85} -73.0371i q^{86} -117.567 q^{88} -3.17265i q^{89} -22.4425 q^{91} -27.4029i q^{92} +108.882 q^{94} -70.2202i q^{95} +24.8273 q^{97} +142.922i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.15341i	1.07670i	0.842720	+	0.538352i	$0.180953\pi$
$2$	2.15341i	1.07670i	−0.842720	+	0.538352i	$0.819047\pi$
$3$	0	0
$3$	0	0
$4$	−0.637159	−0.159290
$5$	− 8.83083i	− 1.76617i	−0.469217	−	0.883083i	$-0.655464\pi$
$5$	− 8.83083i	− 1.76617i	0.469217	−	0.883083i	$-0.344536\pi$
$6$	0	0
$7$	−10.7410	−1.53444	−0.767218	−	0.641387i	$-0.778359\pi$
$7$	−10.7410	−1.53444	−0.767218	+	0.641387i	$0.778359\pi$
$8$	7.24156i	0.905195i
$9$	0	0
$10$	19.0164	1.90164
$11$	16.2351i	1.47591i	0.674848	+	0.737957i	$0.264209\pi$
$11$	16.2351i	1.47591i	−0.674848	+	0.737957i	$0.735791\pi$
$12$	0	0
$13$	2.08942	0.160724	0.0803622	−	0.996766i	$-0.474392\pi$
$13$	2.08942	0.160724	0.0803622	+	0.996766i	$0.474392\pi$
$14$	− 23.1298i	− 1.65213i
$15$	0	0
$16$	−18.1427	−1.13392
$17$	− 4.12311i	− 0.242536i
$17$	− 4.12311i	− 0.242536i
$18$	0	0
$19$	7.95171	0.418511	0.209255	−	0.977861i	$-0.432896\pi$
$19$	7.95171	0.418511	0.209255	+	0.977861i	$0.432896\pi$
$20$	5.62664i	0.281332i
$21$	0	0
$22$	−34.9607	−1.58912
$23$	43.0080i	1.86991i	0.354764	+	0.934956i	$0.384561\pi$
$23$	43.0080i	1.86991i	−0.354764	+	0.934956i	$0.615439\pi$
$24$	0	0
$25$	−52.9835	−2.11934
$26$	4.49936i	0.173052i
$27$	0	0
$28$	6.84376	0.244420
$29$	31.0930i	1.07217i	0.844163	+	0.536086i	$0.180098\pi$
$29$	31.0930i	1.07217i	−0.844163	+	0.536086i	$0.819902\pi$
$30$	0	0
$31$	−43.3950	−1.39984	−0.699920	−	0.714222i	$-0.746781\pi$
$31$	−43.3950	−1.39984	−0.699920	+	0.714222i	$0.746781\pi$
$32$	− 10.1023i	− 0.315696i
$33$	0	0
$34$	8.87872	0.261139
$35$	94.8524i	2.71007i
$36$	0	0
$37$	−31.2688	−0.845103	−0.422551	−	0.906339i	$-0.638865\pi$
$37$	−31.2688	−0.845103	−0.422551	+	0.906339i	$0.638865\pi$
$38$	17.1233i	0.450612i
$39$	0	0
$40$	63.9490	1.59873
$41$	32.0376i	0.781406i	0.920517	+	0.390703i	$0.127768\pi$
$41$	32.0376i	0.781406i	−0.920517	+	0.390703i	$0.872232\pi$
$42$	0	0
$43$	−33.9170	−0.788767	−0.394384	−	0.918946i	$-0.629042\pi$
$43$	−33.9170	−0.788767	−0.394384	+	0.918946i	$0.629042\pi$
$44$	− 10.3443i	− 0.235098i
$45$	0	0
$46$	−92.6136	−2.01334
$47$	− 50.5628i	− 1.07581i	−0.843007	−	0.537903i	$-0.819217\pi$
$47$	− 50.5628i	− 1.07581i	0.843007	−	0.537903i	$-0.180783\pi$
$48$	0	0
$49$	66.3701	1.35449
$50$	− 114.095i	− 2.28190i
$51$	0	0
$52$	−1.33129	−0.0256017
$53$	− 89.1797i	− 1.68264i	−0.540541	−	0.841318i	$-0.681780\pi$
$53$	− 89.1797i	− 1.68264i	0.540541	−	0.841318i	$-0.318220\pi$
$54$	0	0
$55$	143.369	2.60671
$56$	− 77.7820i	− 1.38896i
$57$	0	0
$58$	−66.9559	−1.15441
$59$	47.8328i	0.810726i	0.914156	+	0.405363i	$0.132855\pi$
$59$	47.8328i	0.810726i	−0.914156	+	0.405363i	$0.867145\pi$
$60$	0	0
$61$	−4.33392	−0.0710478	−0.0355239	−	0.999369i	$-0.511310\pi$
$61$	−4.33392	−0.0710478	−0.0355239	+	0.999369i	$0.511310\pi$
$62$	− 93.4471i	− 1.50721i
$63$	0	0
$64$	−50.8163	−0.794005
$65$	− 18.4513i	− 0.283866i
$66$	0	0
$67$	52.2521	0.779882	0.389941	−	0.920840i	$-0.372495\pi$
$67$	52.2521	0.779882	0.389941	+	0.920840i	$0.372495\pi$
$68$	2.62707i	0.0386335i
$69$	0	0
$70$	−204.256	−2.91794
$71$	64.1659i	0.903745i	0.892082	+	0.451873i	$0.149244\pi$
$71$	64.1659i	0.903745i	−0.892082	+	0.451873i	$0.850756\pi$
$72$	0	0
$73$	−18.4693	−0.253004	−0.126502	−	0.991966i	$-0.540375\pi$
$73$	−18.4693	−0.253004	−0.126502	+	0.991966i	$0.540375\pi$
$74$	− 67.3345i	− 0.909925i
$75$	0	0
$76$	−5.06650	−0.0666645
$77$	− 174.381i	− 2.26469i
$78$	0	0
$79$	43.7991	0.554419	0.277210	−	0.960809i	$-0.410590\pi$
$79$	43.7991	0.554419	0.277210	+	0.960809i	$0.410590\pi$
$80$	160.215i	2.00268i
$81$	0	0
$82$	−68.9900	−0.841342
$83$	92.3664i	1.11285i	0.830898	+	0.556424i	$0.187827\pi$
$83$	92.3664i	1.11285i	−0.830898	+	0.556424i	$0.812173\pi$
$84$	0	0
$85$	−36.4104	−0.428358
$86$	− 73.0371i	− 0.849268i
$87$	0	0
$88$	−117.567	−1.33599
$89$	− 3.17265i	− 0.0356477i	−0.999841	−	0.0178239i	$-0.994326\pi$
$89$	− 3.17265i	− 0.0356477i	0.999841	−	0.0178239i	$-0.00567381\pi$
$90$	0	0
$91$	−22.4425	−0.246621
$92$	− 27.4029i	− 0.297858i
$93$	0	0
$94$	108.882	1.15832
$95$	− 70.2202i	− 0.739160i
$96$	0	0
$97$	24.8273	0.255951	0.127976	−	0.991777i	$-0.459152\pi$
$97$	24.8273	0.255951	0.127976	+	0.991777i	$0.459152\pi$
$98$	142.922i	1.45839i
$99$	0	0
$100$	33.7590	0.337590
$101$	32.3169i	0.319970i	0.987119	+	0.159985i	$0.0511445\pi$
$101$	32.3169i	0.319970i	−0.987119	+	0.159985i	$0.948855\pi$
$102$	0	0
$103$	120.835	1.17315	0.586575	−	0.809895i	$-0.300476\pi$
$103$	120.835	1.17315	0.586575	+	0.809895i	$0.300476\pi$
$104$	15.1306i	0.145487i
$105$	0	0
$106$	192.040	1.81170
$107$	− 128.021i	− 1.19646i	−0.801325	−	0.598229i	$-0.795871\pi$
$107$	− 128.021i	− 1.19646i	0.801325	−	0.598229i	$-0.204129\pi$
$108$	0	0
$109$	−11.0503	−0.101379	−0.0506896	−	0.998714i	$-0.516142\pi$
$109$	−11.0503	−0.101379	−0.0506896	+	0.998714i	$0.516142\pi$
$110$	308.732i	2.80665i
$111$	0	0
$112$	194.871	1.73992
$113$	− 133.693i	− 1.18312i	−0.806260	−	0.591562i	$-0.798512\pi$
$113$	− 133.693i	− 1.18312i	0.806260	−	0.591562i	$-0.201488\pi$
$114$	0	0
$115$	379.796	3.30257
$116$	− 19.8112i	− 0.170786i
$117$	0	0
$118$	−103.003	−0.872911
$119$	44.2865i	0.372155i
$120$	0	0
$121$	−142.577	−1.17832
$122$	− 9.33269i	− 0.0764974i
$123$	0	0
$124$	27.6495	0.222980
$125$	247.118i	1.97694i
$126$	0	0
$127$	18.1596	0.142989	0.0714944	−	0.997441i	$-0.477223\pi$
$127$	18.1596	0.142989	0.0714944	+	0.997441i	$0.477223\pi$
$128$	− 149.837i	− 1.17060i
$129$	0	0
$130$	39.7331	0.305639
$131$	− 11.3244i	− 0.0864459i	−0.999065	−	0.0432230i	$-0.986237\pi$
$131$	− 11.3244i	− 0.0864459i	0.999065	−	0.0432230i	$-0.0137626\pi$
$132$	0	0
$133$	−85.4097	−0.642178
$134$	112.520i	0.839701i
$135$	0	0
$136$	29.8577	0.219542
$137$	82.7619i	0.604101i	0.953292	+	0.302051i	$0.0976712\pi$
$137$	82.7619i	0.604101i	−0.953292	+	0.302051i	$0.902329\pi$
$138$	0	0
$139$	−222.320	−1.59943	−0.799713	−	0.600382i	$-0.795015\pi$
$139$	−222.320	−1.59943	−0.799713	+	0.600382i	$0.795015\pi$
$140$	− 60.4361i	− 0.431686i
$141$	0	0
$142$	−138.175	−0.973066
$143$	33.9218i	0.237215i
$144$	0	0
$145$	274.577	1.89363
$146$	− 39.7719i	− 0.272410i
$147$	0	0
$148$	19.9232	0.134616
$149$	− 32.1525i	− 0.215789i	−0.994162	−	0.107894i	$-0.965589\pi$
$149$	− 32.1525i	− 0.215789i	0.994162	−	0.107894i	$-0.0344108\pi$
$150$	0	0
$151$	254.580	1.68596	0.842982	−	0.537942i	$-0.180798\pi$
$151$	254.580	1.68596	0.842982	+	0.537942i	$0.180798\pi$
$152$	57.5828i	0.378834i
$153$	0	0
$154$	375.514	2.43840
$155$	383.214i	2.47235i
$156$	0	0
$157$	−189.235	−1.20532	−0.602659	−	0.797999i	$-0.705892\pi$
$157$	−189.235	−1.20532	−0.602659	+	0.797999i	$0.705892\pi$
$158$	94.3173i	0.596945i
$159$	0	0
$160$	−89.2115	−0.557572
$161$	− 461.951i	− 2.86926i
$162$	0	0
$163$	−276.990	−1.69933	−0.849663	−	0.527327i	$-0.823194\pi$
$163$	−276.990	−1.69933	−0.849663	+	0.527327i	$0.823194\pi$
$164$	− 20.4131i	− 0.124470i
$165$	0	0
$166$	−198.902	−1.19821
$167$	0.178520i	0.00106898i	1.00000		0.000534492i	$0.000170134\pi$
$167$	0.178520i	0.00106898i	−1.00000		0.000534492i	$0.999830\pi$
$168$	0	0
$169$	−164.634	−0.974168
$170$	− 78.4065i	− 0.461215i
$171$	0	0
$172$	21.6105	0.125643
$173$	− 135.509i	− 0.783287i	−0.920117	−	0.391643i	$-0.871907\pi$
$173$	− 135.509i	− 0.783287i	0.920117	−	0.391643i	$-0.128093\pi$
$174$	0	0
$175$	569.099	3.25199
$176$	− 294.547i	− 1.67356i
$177$	0	0
$178$	6.83200	0.0383820
$179$	− 25.8984i	− 0.144684i	−0.997380	−	0.0723419i	$-0.976953\pi$
$179$	− 25.8984i	− 0.144684i	0.997380	−	0.0723419i	$-0.0230473\pi$
$180$	0	0
$181$	−27.6495	−0.152760	−0.0763798	−	0.997079i	$-0.524336\pi$
$181$	−27.6495	−0.152760	−0.0763798	+	0.997079i	$0.524336\pi$
$182$	− 48.3279i	− 0.265538i
$183$	0	0
$184$	−311.445	−1.69264
$185$	276.129i	1.49259i
$186$	0	0
$187$	66.9388	0.357962
$188$	32.2166i	0.171365i
$189$	0	0
$190$	151.213	0.795856
$191$	250.341i	1.31069i	0.755331	+	0.655343i	$0.227476\pi$
$191$	250.341i	1.31069i	−0.755331	+	0.655343i	$0.772524\pi$
$192$	0	0
$193$	205.896	1.06682	0.533411	−	0.845856i	$-0.320910\pi$
$193$	205.896	1.06682	0.533411	+	0.845856i	$0.320910\pi$
$194$	53.4632i	0.275584i
$195$	0	0
$196$	−42.2883	−0.215757
$197$	188.394i	0.956316i	0.878274	+	0.478158i	$0.158695\pi$
$197$	188.394i	0.956316i	−0.878274	+	0.478158i	$0.841305\pi$
$198$	0	0
$199$	−17.3590	−0.0872310	−0.0436155	−	0.999048i	$-0.513888\pi$
$199$	−17.3590	−0.0872310	−0.0436155	+	0.999048i	$0.513888\pi$
$200$	− 383.684i	− 1.91842i
$201$	0	0
$202$	−69.5915	−0.344512
$203$	− 333.971i	− 1.64518i
$204$	0	0
$205$	282.919	1.38009
$206$	260.206i	1.26314i
$207$	0	0
$208$	−37.9076	−0.182248
$209$	129.096i	0.617686i
$210$	0	0
$211$	−188.111	−0.891519	−0.445760	−	0.895153i	$-0.647066\pi$
$211$	−188.111	−0.891519	−0.445760	+	0.895153i	$0.647066\pi$
$212$	56.8217i	0.268027i
$213$	0	0
$214$	275.681	1.28823
$215$	299.515i	1.39309i
$216$	0	0
$217$	466.108	2.14796
$218$	− 23.7958i	− 0.109155i
$219$	0	0
$220$	−91.3489	−0.415222
$221$	− 8.61488i	− 0.0389814i
$222$	0	0
$223$	−80.8240	−0.362439	−0.181220	−	0.983443i	$-0.558005\pi$
$223$	−80.8240	−0.362439	−0.181220	+	0.983443i	$0.558005\pi$
$224$	108.509i	0.484415i
$225$	0	0
$226$	287.895	1.27387
$227$	− 14.8085i	− 0.0652359i	−0.999468	−	0.0326179i	$-0.989616\pi$
$227$	− 14.8085i	− 0.0652359i	0.999468	−	0.0326179i	$-0.0103845\pi$
$228$	0	0
$229$	419.255	1.83081	0.915404	−	0.402536i	$-0.131871\pi$
$229$	419.255	1.83081	0.915404	+	0.402536i	$0.131871\pi$
$230$	817.855i	3.55589i
$231$	0	0
$232$	−225.162	−0.970526
$233$	273.497i	1.17380i	0.809658	+	0.586902i	$0.199653\pi$
$233$	273.497i	1.17380i	−0.809658	+	0.586902i	$0.800347\pi$
$234$	0	0
$235$	−446.512	−1.90005
$236$	− 30.4771i	− 0.129140i
$237$	0	0
$238$	−95.3668	−0.400701
$239$	− 212.504i	− 0.889136i	−0.895745	−	0.444568i	$-0.853357\pi$
$239$	− 212.504i	− 0.889136i	0.895745	−	0.444568i	$-0.146643\pi$
$240$	0	0
$241$	195.819	0.812526	0.406263	−	0.913756i	$-0.366832\pi$
$241$	195.819	0.812526	0.406263	+	0.913756i	$0.366832\pi$
$242$	− 307.026i	− 1.26870i
$243$	0	0
$244$	2.76140	0.0113172
$245$	− 586.103i	− 2.39226i
$246$	0	0
$247$	16.6144	0.0672649
$248$	− 314.248i	− 1.26713i
$249$	0	0
$250$	−532.145	−2.12858
$251$	393.530i	1.56785i	0.620856	+	0.783925i	$0.286785\pi$
$251$	393.530i	1.56785i	−0.620856	+	0.783925i	$0.713215\pi$
$252$	0	0
$253$	−698.237	−2.75983
$254$	39.1050i	0.153957i
$255$	0	0
$256$	119.395	0.466388
$257$	33.9160i	0.131969i	0.997821	+	0.0659845i	$0.0210188\pi$
$257$	33.9160i	0.131969i	−0.997821	+	0.0659845i	$0.978981\pi$
$258$	0	0
$259$	335.860	1.29676
$260$	11.7564i	0.0452169i
$261$	0	0
$262$	24.3861	0.0930766
$263$	− 394.277i	− 1.49915i	−0.661919	−	0.749575i	$-0.730258\pi$
$263$	− 394.277i	− 1.49915i	0.661919	−	0.749575i	$-0.269742\pi$
$264$	0	0
$265$	−787.530	−2.97181
$266$	− 183.922i	− 0.691435i
$267$	0	0
$268$	−33.2929	−0.124227
$269$	− 299.051i	− 1.11171i	−0.831279	−	0.555856i	$-0.812391\pi$
$269$	− 299.051i	− 1.11171i	0.831279	−	0.555856i	$-0.187609\pi$
$270$	0	0
$271$	−102.506	−0.378252	−0.189126	−	0.981953i	$-0.560565\pi$
$271$	−102.506	−0.378252	−0.189126	+	0.981953i	$0.560565\pi$
$272$	74.8041i	0.275015i
$273$	0	0
$274$	−178.220	−0.650438
$275$	− 860.191i	− 3.12797i
$276$	0	0
$277$	−10.0143	−0.0361528	−0.0180764	−	0.999837i	$-0.505754\pi$
$277$	−10.0143	−0.0361528	−0.0180764	+	0.999837i	$0.505754\pi$
$278$	− 478.746i	− 1.72211i
$279$	0	0
$280$	−686.879	−2.45314
$281$	147.806i	0.525999i	0.964796	+	0.262999i	$0.0847117\pi$
$281$	147.806i	0.525999i	−0.964796	+	0.262999i	$0.915288\pi$
$282$	0	0
$283$	156.649	0.553529	0.276764	−	0.960938i	$-0.410738\pi$
$283$	156.649	0.553529	0.276764	+	0.960938i	$0.410738\pi$
$284$	− 40.8839i	− 0.143957i
$285$	0	0
$286$	−73.0474	−0.255410
$287$	− 344.118i	− 1.19902i
$288$	0	0
$289$	−17.0000	−0.0588235
$290$	591.276i	2.03888i
$291$	0	0
$292$	11.7679	0.0403010
$293$	− 223.189i	− 0.761737i	−0.924629	−	0.380868i	$-0.875625\pi$
$293$	− 223.189i	− 0.761737i	0.924629	−	0.380868i	$-0.124375\pi$
$294$	0	0
$295$	422.403	1.43188
$296$	− 226.435i	− 0.764983i
$297$	0	0
$298$	69.2374	0.232340
$299$	89.8615i	0.300540i
$300$	0	0
$301$	364.304	1.21031
$302$	548.215i	1.81528i
$303$	0	0
$304$	−144.265	−0.474556
$305$	38.2721i	0.125482i
$306$	0	0
$307$	147.629	0.480876	0.240438	−	0.970664i	$-0.422709\pi$
$307$	147.629	0.480876	0.240438	+	0.970664i	$0.422709\pi$
$308$	111.109i	0.360743i
$309$	0	0
$310$	−825.215	−2.66198
$311$	158.564i	0.509852i	0.966961	+	0.254926i	$0.0820511\pi$
$311$	158.564i	0.509852i	−0.966961	+	0.254926i	$0.917949\pi$
$312$	0	0
$313$	−100.108	−0.319835	−0.159917	−	0.987130i	$-0.551123\pi$
$313$	−100.108	−0.319835	−0.159917	+	0.987130i	$0.551123\pi$
$314$	− 407.500i	− 1.29777i
$315$	0	0
$316$	−27.9070	−0.0883134
$317$	232.336i	0.732922i	0.930433	+	0.366461i	$0.119431\pi$
$317$	232.336i	0.732922i	−0.930433	+	0.366461i	$0.880569\pi$
$318$	0	0
$319$	−504.797	−1.58243
$320$	448.750i	1.40235i
$321$	0	0
$322$	994.767	3.08934
$323$	− 32.7857i	− 0.101504i
$324$	0	0
$325$	−110.705	−0.340630
$326$	− 596.472i	− 1.82967i
$327$	0	0
$328$	−232.002	−0.707325
$329$	543.098i	1.65075i
$330$	0	0
$331$	−198.488	−0.599661	−0.299830	−	0.953992i	$-0.596930\pi$
$331$	−198.488	−0.599661	−0.299830	+	0.953992i	$0.596930\pi$
$332$	− 58.8521i	− 0.177265i
$333$	0	0
$334$	−0.384427	−0.00115098
$335$	− 461.429i	− 1.37740i
$336$	0	0
$337$	10.5780	0.0313887	0.0156943	−	0.999877i	$-0.495004\pi$
$337$	10.5780	0.0313887	0.0156943	+	0.999877i	$0.495004\pi$
$338$	− 354.525i	− 1.04889i
$339$	0	0
$340$	23.1992	0.0682331
$341$	− 704.520i	− 2.06604i
$342$	0	0
$343$	−186.573	−0.543945
$344$	− 245.612i	− 0.713989i
$345$	0	0
$346$	291.805	0.843367
$347$	322.802i	0.930265i	0.885241	+	0.465132i	$0.153993\pi$
$347$	322.802i	0.930265i	−0.885241	+	0.465132i	$0.846007\pi$
$348$	0	0
$349$	176.530	0.505816	0.252908	−	0.967490i	$-0.418613\pi$
$349$	176.530	0.505816	0.252908	+	0.967490i	$0.418613\pi$
$350$	1225.50i	3.50143i
$351$	0	0
$352$	164.011	0.465940
$353$	17.8128i	0.0504611i	0.999682	+	0.0252305i	$0.00803198\pi$
$353$	17.8128i	0.0504611i	−0.999682	+	0.0252305i	$0.991968\pi$
$354$	0	0
$355$	566.638	1.59616
$356$	2.02148i	0.00567832i
$357$	0	0
$358$	55.7698	0.155782
$359$	299.351i	0.833846i	0.908942	+	0.416923i	$0.136892\pi$
$359$	299.351i	0.833846i	−0.908942	+	0.416923i	$0.863108\pi$
$360$	0	0
$361$	−297.770	−0.824849
$362$	− 59.5406i	− 0.164477i
$363$	0	0
$364$	14.2995	0.0392842
$365$	163.099i	0.446847i
$366$	0	0
$367$	−580.018	−1.58043	−0.790215	−	0.612830i	$-0.790031\pi$
$367$	−580.018	−1.58043	−0.790215	+	0.612830i	$0.790031\pi$
$368$	− 780.279i	− 2.12032i
$369$	0	0
$370$	−594.619	−1.60708
$371$	957.883i	2.58189i
$372$	0	0
$373$	206.015	0.552320	0.276160	−	0.961112i	$-0.410938\pi$
$373$	206.015	0.552320	0.276160	+	0.961112i	$0.410938\pi$
$374$	144.147i	0.385419i
$375$	0	0
$376$	366.154	0.973814
$377$	64.9662i	0.172324i
$378$	0	0
$379$	−140.930	−0.371848	−0.185924	−	0.982564i	$-0.559528\pi$
$379$	−140.930	−0.371848	−0.185924	+	0.982564i	$0.559528\pi$
$380$	44.7414i	0.117741i
$381$	0	0
$382$	−539.086	−1.41122
$383$	− 387.640i	− 1.01212i	−0.862500	−	0.506058i	$-0.831102\pi$
$383$	− 387.640i	− 1.01212i	0.862500	−	0.506058i	$-0.168898\pi$
$384$	0	0
$385$	−1539.93	−3.99983
$386$	443.379i	1.14865i
$387$	0	0
$388$	−15.8189	−0.0407704
$389$	− 3.28650i	− 0.00844860i	−0.999991	−	0.00422430i	$-0.998655\pi$
$389$	− 3.28650i	− 0.00844860i	0.999991	−	0.00422430i	$-0.00134464\pi$
$390$	0	0
$391$	177.326	0.453520
$392$	480.623i	1.22608i
$393$	0	0
$394$	−405.689	−1.02967
$395$	− 386.783i	− 0.979197i
$396$	0	0
$397$	12.7657	0.0321553	0.0160777	−	0.999871i	$-0.494882\pi$
$397$	12.7657	0.0321553	0.0160777	+	0.999871i	$0.494882\pi$
$398$	− 37.3809i	− 0.0939219i
$399$	0	0
$400$	961.263	2.40316
$401$	666.354i	1.66173i	0.556473	+	0.830866i	$0.312154\pi$
$401$	666.354i	1.66173i	−0.556473	+	0.830866i	$0.687846\pi$
$402$	0	0
$403$	−90.6702	−0.224988
$404$	− 20.5910i	− 0.0509679i
$405$	0	0
$406$	719.176	1.77137
$407$	− 507.651i	− 1.24730i
$408$	0	0
$409$	553.417	1.35310	0.676549	−	0.736398i	$-0.263475\pi$
$409$	553.417	1.35310	0.676549	+	0.736398i	$0.263475\pi$
$410$	609.239i	1.48595i
$411$	0	0
$412$	−76.9908	−0.186871
$413$	− 513.775i	− 1.24401i
$414$	0	0
$415$	815.672	1.96547
$416$	− 21.1079i	− 0.0507401i
$417$	0	0
$418$	−277.997	−0.665065
$419$	339.048i	0.809184i	0.914497	+	0.404592i	$0.132587\pi$
$419$	339.048i	0.809184i	−0.914497	+	0.404592i	$0.867413\pi$
$420$	0	0
$421$	269.994	0.641317	0.320658	−	0.947195i	$-0.396096\pi$
$421$	269.994	0.641317	0.320658	+	0.947195i	$0.396096\pi$
$422$	− 405.079i	− 0.959902i
$423$	0	0
$424$	645.800	1.52311
$425$	218.457i	0.514016i
$426$	0	0
$427$	46.5508	0.109018
$428$	81.5698i	0.190584i
$429$	0	0
$430$	−644.978	−1.49995
$431$	− 810.847i	− 1.88131i	−0.339358	−	0.940657i	$-0.610210\pi$
$431$	− 810.847i	− 1.88131i	0.339358	−	0.940657i	$-0.389790\pi$
$432$	0	0
$433$	−334.556	−0.772648	−0.386324	−	0.922363i	$-0.626255\pi$
$433$	−334.556	−0.772648	−0.386324	+	0.922363i	$0.626255\pi$
$434$	1003.72i	2.31272i
$435$	0	0
$436$	7.04082	0.0161487
$437$	341.987i	0.782579i
$438$	0	0
$439$	322.793	0.735292	0.367646	−	0.929966i	$-0.380164\pi$
$439$	322.793	0.735292	0.367646	+	0.929966i	$0.380164\pi$
$440$	1038.22i	2.35958i
$441$	0	0
$442$	18.5513	0.0419714
$443$	487.141i	1.09964i	0.835283	+	0.549820i	$0.185304\pi$
$443$	487.141i	1.09964i	−0.835283	+	0.549820i	$0.814696\pi$
$444$	0	0
$445$	−28.0171	−0.0629598
$446$	− 174.047i	− 0.390240i
$447$	0	0
$448$	545.821	1.21835
$449$	887.941i	1.97760i	0.149258	+	0.988798i	$0.452312\pi$
$449$	887.941i	1.97760i	−0.149258	+	0.988798i	$0.547688\pi$
$450$	0	0
$451$	−520.133	−1.15329
$452$	85.1837i	0.188460i
$453$	0	0
$454$	31.8888	0.0702397
$455$	198.186i	0.435574i
$456$	0	0
$457$	11.7191	0.0256436	0.0128218	−	0.999918i	$-0.495919\pi$
$457$	11.7191	0.0256436	0.0128218	+	0.999918i	$0.495919\pi$
$458$	902.827i	1.97124i
$459$	0	0
$460$	−241.991	−0.526066
$461$	− 79.0941i	− 0.171571i	−0.996314	−	0.0857854i	$-0.972660\pi$
$461$	− 79.0941i	− 0.171571i	0.996314	−	0.0857854i	$-0.0273399\pi$
$462$	0	0
$463$	192.265	0.415259	0.207630	−	0.978208i	$-0.433425\pi$
$463$	192.265	0.415259	0.207630	+	0.978208i	$0.433425\pi$
$464$	− 564.110i	− 1.21575i
$465$	0	0
$466$	−588.949	−1.26384
$467$	− 632.971i	− 1.35540i	−0.735339	−	0.677700i	$-0.762977\pi$
$467$	− 632.971i	− 1.35540i	0.735339	−	0.677700i	$-0.237023\pi$
$468$	0	0
$469$	−561.242	−1.19668
$470$	− 961.521i	− 2.04579i
$471$	0	0
$472$	−346.384	−0.733865
$473$	− 550.644i	− 1.16415i
$474$	0	0
$475$	−421.310	−0.886968
$476$	− 28.2175i	− 0.0592805i
$477$	0	0
$478$	457.606	0.957336
$479$	622.366i	1.29930i	0.760232	+	0.649652i	$0.225085\pi$
$479$	622.366i	1.29930i	−0.760232	+	0.649652i	$0.774915\pi$
$480$	0	0
$481$	−65.3336	−0.135829
$482$	421.677i	0.874849i
$483$	0	0
$484$	90.8442	0.187695
$485$	− 219.245i	− 0.452052i
$486$	0	0
$487$	756.159	1.55269	0.776344	−	0.630309i	$-0.217072\pi$
$487$	756.159	1.55269	0.776344	+	0.630309i	$0.217072\pi$
$488$	− 31.3843i	− 0.0643122i
$489$	0	0
$490$	1262.12	2.57575
$491$	− 434.325i	− 0.884571i	−0.896874	−	0.442286i	$-0.854168\pi$
$491$	− 434.325i	− 0.884571i	0.896874	−	0.442286i	$-0.145832\pi$
$492$	0	0
$493$	128.200	0.260040
$494$	35.7776i	0.0724243i
$495$	0	0
$496$	787.301	1.58730
$497$	− 689.209i	− 1.38674i
$498$	0	0
$499$	−274.131	−0.549360	−0.274680	−	0.961536i	$-0.588572\pi$
$499$	−274.131	−0.549360	−0.274680	+	0.961536i	$0.588572\pi$
$500$	− 157.453i	− 0.314907i
$501$	0	0
$502$	−847.430	−1.68811
$503$	− 356.757i	− 0.709259i	−0.935007	−	0.354630i	$-0.884607\pi$
$503$	− 356.757i	− 0.709259i	0.935007	−	0.354630i	$-0.115393\pi$
$504$	0	0
$505$	285.385	0.565119
$506$	− 1503.59i	− 2.97152i
$507$	0	0
$508$	−11.5705	−0.0227767
$509$	− 189.234i	− 0.371775i	−0.982571	−	0.185888i	$-0.940484\pi$
$509$	− 189.234i	− 0.371775i	0.982571	−	0.185888i	$-0.0595161\pi$
$510$	0	0
$511$	198.380	0.388218
$512$	− 342.243i	− 0.668443i
$513$	0	0
$514$	−73.0350	−0.142091
$515$	− 1067.07i	− 2.07198i
$516$	0	0
$517$	820.890	1.58780
$518$	723.243i	1.39622i
$519$	0	0
$520$	133.616	0.256954
$521$	395.986i	0.760049i	0.924976	+	0.380025i	$0.124085\pi$
$521$	395.986i	0.760049i	−0.924976	+	0.380025i	$0.875915\pi$
$522$	0	0
$523$	257.549	0.492445	0.246223	−	0.969213i	$-0.420811\pi$
$523$	257.549	0.492445	0.246223	+	0.969213i	$0.420811\pi$
$524$	7.21545i	0.0137700i
$525$	0	0
$526$	849.038	1.61414
$527$	178.922i	0.339511i
$528$	0	0
$529$	−1320.69	−2.49657
$530$	− 1695.87i	− 3.19976i
$531$	0	0
$532$	54.4196	0.102292
$533$	66.9399i	0.125591i
$534$	0	0
$535$	−1130.53	−2.11314
$536$	378.387i	0.705946i
$537$	0	0
$538$	643.977	1.19698
$539$	1077.52i	1.99911i
$540$	0	0
$541$	609.909	1.12737	0.563687	−	0.825989i	$-0.309383\pi$
$541$	609.909	1.12737	0.563687	+	0.825989i	$0.309383\pi$
$542$	− 220.738i	− 0.407265i
$543$	0	0
$544$	−41.6528	−0.0765676
$545$	97.5835i	0.179052i
$546$	0	0
$547$	945.075	1.72774	0.863871	−	0.503713i	$-0.168033\pi$
$547$	945.075	1.72774	0.863871	+	0.503713i	$0.168033\pi$
$548$	− 52.7325i	− 0.0962272i
$549$	0	0
$550$	1852.34	3.36789
$551$	247.243i	0.448716i
$552$	0	0
$553$	−470.449	−0.850721
$554$	− 21.5649i	− 0.0389259i
$555$	0	0
$556$	141.653	0.254772
$557$	− 137.874i	− 0.247529i	−0.992312	−	0.123764i	$-0.960503\pi$
$557$	− 137.874i	− 0.247529i	0.992312	−	0.123764i	$-0.0394967\pi$
$558$	0	0
$559$	−70.8667	−0.126774
$560$	− 1720.87i	− 3.07299i
$561$	0	0
$562$	−318.286	−0.566344
$563$	930.409i	1.65259i	0.563236	+	0.826296i	$0.309556\pi$
$563$	930.409i	1.65259i	−0.563236	+	0.826296i	$0.690444\pi$
$564$	0	0
$565$	−1180.62	−2.08959
$566$	337.328i	0.595986i
$567$	0	0
$568$	−464.662	−0.818066
$569$	1044.38i	1.83547i	0.397194	+	0.917735i	$0.369984\pi$
$569$	1044.38i	1.83547i	−0.397194	+	0.917735i	$0.630016\pi$
$570$	0	0
$571$	−478.476	−0.837961	−0.418981	−	0.907995i	$-0.637613\pi$
$571$	−478.476	−0.837961	−0.418981	+	0.907995i	$0.637613\pi$
$572$	− 21.6136i	− 0.0377860i
$573$	0	0
$574$	741.025	1.29098
$575$	− 2278.71i	− 3.96298i
$576$	0	0
$577$	497.918	0.862943	0.431472	−	0.902126i	$-0.357994\pi$
$577$	497.918	0.862943	0.431472	+	0.902126i	$0.357994\pi$
$578$	− 36.6079i	− 0.0633355i
$579$	0	0
$580$	−174.949	−0.301637
$581$	− 992.112i	− 1.70759i
$582$	0	0
$583$	1447.84	2.48342
$584$	− 133.747i	− 0.229018i
$585$	0	0
$586$	480.616	0.820165
$587$	− 72.4672i	− 0.123454i	−0.998093	−	0.0617268i	$-0.980339\pi$
$587$	− 72.4672i	− 0.123454i	0.998093	−	0.0617268i	$-0.0196607\pi$
$588$	0	0
$589$	−345.064	−0.585848
$590$	909.606i	1.54171i
$591$	0	0
$592$	567.300	0.958276
$593$	600.699i	1.01298i	0.862245	+	0.506491i	$0.169058\pi$
$593$	600.699i	1.01298i	−0.862245	+	0.506491i	$0.830942\pi$
$594$	0	0
$595$	391.086	0.657288
$596$	20.4863i	0.0343729i
$597$	0	0
$598$	−193.508	−0.323593
$599$	102.906i	0.171797i	0.996304	+	0.0858983i	$0.0273760\pi$
$599$	102.906i	0.171797i	−0.996304	+	0.0858983i	$0.972624\pi$
$600$	0	0
$601$	−631.616	−1.05094	−0.525470	−	0.850812i	$-0.676111\pi$
$601$	−631.616	−1.05094	−0.525470	+	0.850812i	$0.676111\pi$
$602$	784.495i	1.30315i
$603$	0	0
$604$	−162.208	−0.268557
$605$	1259.07i	2.08111i
$606$	0	0
$607$	366.006	0.602975	0.301487	−	0.953470i	$-0.402517\pi$
$607$	366.006	0.602975	0.301487	+	0.953470i	$0.402517\pi$
$608$	− 80.3304i	− 0.132122i
$609$	0	0
$610$	−82.4154	−0.135107
$611$	− 105.647i	− 0.172908i
$612$	0	0
$613$	−711.746	−1.16109	−0.580543	−	0.814229i	$-0.697160\pi$
$613$	−711.746	−1.16109	−0.580543	+	0.814229i	$0.697160\pi$
$614$	317.905i	0.517761i
$615$	0	0
$616$	1262.79	2.04999
$617$	248.749i	0.403159i	0.979472	+	0.201579i	$0.0646074\pi$
$617$	248.749i	0.403159i	−0.979472	+	0.201579i	$0.935393\pi$
$618$	0	0
$619$	255.280	0.412407	0.206204	−	0.978509i	$-0.433889\pi$
$619$	255.280	0.412407	0.206204	+	0.978509i	$0.433889\pi$
$620$	− 244.168i	− 0.393820i
$621$	0	0
$622$	−341.453	−0.548959
$623$	34.0776i	0.0546991i
$624$	0	0
$625$	857.667	1.37227
$626$	− 215.574i	− 0.344367i
$627$	0	0
$628$	120.573	0.191995
$629$	128.925i	0.204968i
$630$	0	0
$631$	127.689	0.202360	0.101180	−	0.994868i	$-0.467738\pi$
$631$	127.689	0.202360	0.101180	+	0.994868i	$0.467738\pi$
$632$	317.174i	0.501858i
$633$	0	0
$634$	−500.314	−0.789139
$635$	− 160.364i	− 0.252542i
$636$	0	0
$637$	138.675	0.217700
$638$	− 1087.03i	− 1.70381i
$639$	0	0
$640$	−1323.19	−2.06748
$641$	235.225i	0.366965i	0.983023	+	0.183483i	$0.0587371\pi$
$641$	235.225i	0.366965i	−0.983023	+	0.183483i	$0.941263\pi$
$642$	0	0
$643$	−355.072	−0.552211	−0.276105	−	0.961127i	$-0.589044\pi$
$643$	−355.072	−0.552211	−0.276105	+	0.961127i	$0.589044\pi$
$644$	294.336i	0.457044i
$645$	0	0
$646$	70.6010	0.109289
$647$	419.285i	0.648045i	0.946049	+	0.324023i	$0.105035\pi$
$647$	419.285i	0.648045i	−0.946049	+	0.324023i	$0.894965\pi$
$648$	0	0
$649$	−776.568	−1.19656
$650$	− 238.392i	− 0.366757i
$651$	0	0
$652$	176.487	0.270685
$653$	− 90.6357i	− 0.138799i	−0.997589	−	0.0693994i	$-0.977892\pi$
$653$	− 90.6357i	− 0.138799i	0.997589	−	0.0693994i	$-0.0221083\pi$
$654$	0	0
$655$	−100.004	−0.152678
$656$	− 581.248i	− 0.886049i
$657$	0	0
$658$	−1169.51	−1.77737
$659$	968.353i	1.46943i	0.678377	+	0.734714i	$0.262684\pi$
$659$	968.353i	1.46943i	−0.678377	+	0.734714i	$0.737316\pi$
$660$	0	0
$661$	−725.240	−1.09719	−0.548593	−	0.836089i	$-0.684836\pi$
$661$	−725.240	−1.09719	−0.548593	+	0.836089i	$0.684836\pi$
$662$	− 427.425i	− 0.645657i
$663$	0	0
$664$	−668.877	−1.00735
$665$	754.238i	1.13419i
$666$	0	0
$667$	−1337.25	−2.00487
$668$	− 0.113746i	0 0.000170278i
$669$	0	0
$670$	993.645	1.48305
$671$	− 70.3614i	− 0.104861i
$672$	0	0
$673$	−615.604	−0.914716	−0.457358	−	0.889283i	$-0.651204\pi$
$673$	−615.604	−0.914716	−0.457358	+	0.889283i	$0.651204\pi$
$674$	22.7787i	0.0337963i
$675$	0	0
$676$	104.898	0.155175
$677$	138.151i	0.204064i	0.994781	+	0.102032i	$0.0325344\pi$
$677$	138.151i	0.204064i	−0.994781	+	0.102032i	$0.967466\pi$
$678$	0	0
$679$	−266.671	−0.392741
$680$	− 263.668i	− 0.387748i
$681$	0	0
$682$	1517.12	2.22451
$683$	762.513i	1.11642i	0.829701	+	0.558209i	$0.188511\pi$
$683$	762.513i	1.11642i	−0.829701	+	0.558209i	$0.811489\pi$
$684$	0	0
$685$	730.856	1.06694
$686$	− 401.767i	− 0.585667i
$687$	0	0
$688$	615.345	0.894396
$689$	− 186.333i	− 0.270440i
$690$	0	0
$691$	903.484	1.30750	0.653751	−	0.756709i	$-0.273194\pi$
$691$	903.484	1.30750	0.653751	+	0.756709i	$0.273194\pi$
$692$	86.3406i	0.124770i
$693$	0	0
$694$	−695.124	−1.00162
$695$	1963.27i	2.82485i
$696$	0	0
$697$	132.095	0.189519
$698$	380.141i	0.544614i
$699$	0	0
$700$	−362.607	−0.518009
$701$	− 818.788i	− 1.16803i	−0.811743	−	0.584014i	$-0.801481\pi$
$701$	− 818.788i	− 1.16803i	0.811743	−	0.584014i	$-0.198519\pi$
$702$	0	0
$703$	−248.640	−0.353685
$704$	− 825.006i	− 1.17188i
$705$	0	0
$706$	−38.3581	−0.0543316
$707$	− 347.118i	− 0.490973i
$708$	0	0
$709$	406.551	0.573415	0.286708	−	0.958018i	$-0.407439\pi$
$709$	406.551	0.573415	0.286708	+	0.958018i	$0.407439\pi$
$710$	1220.20i	1.71860i
$711$	0	0
$712$	22.9749	0.0322682
$713$	− 1866.33i	− 2.61758i
$714$	0	0
$715$	299.557	0.418962
$716$	16.5014i	0.0230467i
$717$	0	0
$718$	−644.624	−0.897805
$719$	577.298i	0.802919i	0.915877	+	0.401459i	$0.131497\pi$
$719$	577.298i	0.802919i	−0.915877	+	0.401459i	$0.868503\pi$
$720$	0	0
$721$	−1297.89	−1.80012
$722$	− 641.221i	− 0.888117i
$723$	0	0
$724$	17.6171	0.0243330
$725$	− 1647.42i	− 2.27230i
$726$	0	0
$727$	834.418	1.14776	0.573878	−	0.818941i	$-0.305438\pi$
$727$	834.418	1.14776	0.573878	+	0.818941i	$0.305438\pi$
$728$	− 162.519i	− 0.223240i
$729$	0	0
$730$	−351.219	−0.481122
$731$	139.843i	0.191304i
$732$	0	0
$733$	20.6440	0.0281637	0.0140818	−	0.999901i	$-0.495517\pi$
$733$	20.6440	0.0281637	0.0140818	+	0.999901i	$0.495517\pi$
$734$	− 1249.01i	− 1.70165i
$735$	0	0
$736$	434.479	0.590324
$737$	848.316i	1.15104i
$738$	0	0
$739$	728.199	0.985385	0.492692	−	0.870204i	$-0.336013\pi$
$739$	728.199	0.985385	0.492692	+	0.870204i	$0.336013\pi$
$740$	− 175.938i	− 0.237755i
$741$	0	0
$742$	−2062.71	−2.77993
$743$	− 50.5350i	− 0.0680148i	−0.999422	−	0.0340074i	$-0.989173\pi$
$743$	− 50.5350i	− 0.0680148i	0.999422	−	0.0340074i	$-0.0108270\pi$
$744$	0	0
$745$	−283.933	−0.381119
$746$	443.635i	0.594684i
$747$	0	0
$748$	−42.6507	−0.0570197
$749$	1375.08i	1.83589i
$750$	0	0
$751$	797.006	1.06126	0.530630	−	0.847604i	$-0.321955\pi$
$751$	797.006	1.06126	0.530630	+	0.847604i	$0.321955\pi$
$752$	917.345i	1.21987i
$753$	0	0
$754$	−139.899	−0.185542
$755$	− 2248.16i	− 2.97769i
$756$	0	0
$757$	92.1377	0.121714	0.0608571	−	0.998146i	$-0.480617\pi$
$757$	92.1377	0.121714	0.0608571	+	0.998146i	$0.480617\pi$
$758$	− 303.481i	− 0.400370i
$759$	0	0
$760$	508.504	0.669084
$761$	345.415i	0.453896i	0.973907	+	0.226948i	$0.0728747\pi$
$761$	345.415i	0.453896i	−0.973907	+	0.226948i	$0.927125\pi$
$762$	0	0
$763$	118.692	0.155560
$764$	− 159.507i	− 0.208779i
$765$	0	0
$766$	834.747	1.08975
$767$	99.9427i	0.130303i
$768$	0	0
$769$	925.147	1.20305	0.601526	−	0.798853i	$-0.294559\pi$
$769$	925.147	1.20305	0.601526	+	0.798853i	$0.294559\pi$
$770$	− 3316.10i	− 4.30663i
$771$	0	0
$772$	−131.189	−0.169934
$773$	394.040i	0.509754i	0.966973	+	0.254877i	$0.0820350\pi$
$773$	394.040i	0.509754i	−0.966973	+	0.254877i	$0.917965\pi$
$774$	0	0
$775$	2299.22	2.96674
$776$	179.788i	0.231686i
$777$	0	0
$778$	7.07718	0.00909663
$779$	254.754i	0.327027i
$780$	0	0
$781$	−1041.74	−1.33385
$782$	381.856i	0.488307i
$783$	0	0
$784$	−1204.13	−1.53588
$785$	1671.10i	2.12879i
$786$	0	0
$787$	−1198.52	−1.52290	−0.761448	−	0.648226i	$-0.775511\pi$
$787$	−1198.52	−1.52290	−0.761448	+	0.648226i	$0.775511\pi$
$788$	− 120.037i	− 0.152331i
$789$	0	0
$790$	832.900	1.05430
$791$	1436.00i	1.81543i
$792$	0	0
$793$	−9.05536	−0.0114191
$794$	27.4897i	0.0346217i
$795$	0	0
$796$	11.0604	0.0138950
$797$	489.381i	0.614028i	0.951705	+	0.307014i	$0.0993299\pi$
$797$	489.381i	0.614028i	−0.951705	+	0.307014i	$0.900670\pi$
$798$	0	0
$799$	−208.476	−0.260921
$800$	535.255i	0.669068i
$801$	0	0
$802$	−1434.93	−1.78919
$803$	− 299.850i	− 0.373412i
$804$	0	0
$805$	−4079.41	−5.06759
$806$	− 195.250i	− 0.242245i
$807$	0	0
$808$	−234.025	−0.289635
$809$	− 294.463i	− 0.363984i	−0.983300	−	0.181992i	$-0.941745\pi$
$809$	− 294.463i	− 0.363984i	0.983300	−	0.181992i	$-0.0582545\pi$
$810$	0	0
$811$	−639.238	−0.788209	−0.394105	−	0.919066i	$-0.628945\pi$
$811$	−639.238	−0.788209	−0.394105	+	0.919066i	$0.628945\pi$
$812$	212.793i	0.262060i
$813$	0	0
$814$	1093.18	1.34297
$815$	2446.05i	3.00129i
$816$	0	0
$817$	−269.698	−0.330108
$818$	1191.73i	1.45688i
$819$	0	0
$820$	−180.264	−0.219835
$821$	− 158.726i	− 0.193332i	−0.995317	−	0.0966660i	$-0.969182\pi$
$821$	− 158.726i	− 0.193332i	0.995317	−	0.0966660i	$-0.0308179\pi$
$822$	0	0
$823$	−1222.65	−1.48560	−0.742800	−	0.669514i	$-0.766503\pi$
$823$	−1222.65	−1.48560	−0.742800	+	0.669514i	$0.766503\pi$
$824$	875.031i	1.06193i
$825$	0	0
$826$	1106.37	1.33943
$827$	− 1268.36i	− 1.53369i	−0.641834	−	0.766843i	$-0.721826\pi$
$827$	− 1268.36i	− 1.53369i	0.641834	−	0.766843i	$-0.278174\pi$
$828$	0	0
$829$	−510.662	−0.615998	−0.307999	−	0.951387i	$-0.599659\pi$
$829$	−510.662	−0.615998	−0.307999	+	0.951387i	$0.599659\pi$
$830$	1756.47i	2.11623i
$831$	0	0
$832$	−106.176	−0.127616
$833$	− 273.651i	− 0.328512i
$834$	0	0
$835$	1.57648	0.00188800
$836$	− 82.2550i	− 0.0983911i
$837$	0	0
$838$	−730.109	−0.871251
$839$	− 468.220i	− 0.558069i	−0.960281	−	0.279034i	$-0.909986\pi$
$839$	− 468.220i	− 0.558069i	0.960281	−	0.279034i	$-0.0900143\pi$
$840$	0	0
$841$	−125.775	−0.149554
$842$	581.408i	0.690508i
$843$	0	0
$844$	119.856	0.142010
$845$	1453.86i	1.72054i
$846$	0	0
$847$	1531.43	1.80806
$848$	1617.96i	1.90797i
$849$	0	0
$850$	−470.426	−0.553443
$851$	− 1344.81i	− 1.58027i
$852$	0	0
$853$	−466.860	−0.547315	−0.273657	−	0.961827i	$-0.588233\pi$
$853$	−466.860	−0.547315	−0.273657	+	0.961827i	$0.588233\pi$
$854$	100.243i	0.117380i
$855$	0	0
$856$	927.073	1.08303
$857$	− 1165.53i	− 1.36001i	−0.733205	−	0.680007i	$-0.761977\pi$
$857$	− 1165.53i	− 1.36001i	0.733205	−	0.680007i	$-0.238023\pi$
$858$	0	0
$859$	1287.48	1.49881	0.749406	−	0.662111i	$-0.230339\pi$
$859$	1287.48	1.49881	0.749406	+	0.662111i	$0.230339\pi$
$860$	− 190.839i	− 0.221906i
$861$	0	0
$862$	1746.08	2.02562
$863$	1512.53i	1.75264i	0.481731	+	0.876319i	$0.340008\pi$
$863$	1512.53i	1.75264i	−0.481731	+	0.876319i	$0.659992\pi$
$864$	0	0
$865$	−1196.65	−1.38341
$866$	− 720.436i	− 0.831912i
$867$	0	0
$868$	−296.985	−0.342149
$869$	711.081i	0.818275i
$870$	0	0
$871$	109.176	0.125346
$872$	− 80.0216i	− 0.0917679i
$873$	0	0
$874$	−736.437	−0.842605
$875$	− 2654.30i	− 3.03349i
$876$	0	0
$877$	−1072.27	−1.22266	−0.611331	−	0.791375i	$-0.709365\pi$
$877$	−1072.27	−1.22266	−0.611331	+	0.791375i	$0.709365\pi$
$878$	695.104i	0.791691i
$879$	0	0
$880$	−2601.10	−2.95579
$881$	1031.56i	1.17090i	0.810710	+	0.585448i	$0.199081\pi$
$881$	1031.56i	1.17090i	−0.810710	+	0.585448i	$0.800919\pi$
$882$	0	0
$883$	−1761.12	−1.99448	−0.997239	−	0.0742568i	$-0.976342\pi$
$883$	−1761.12	−1.99448	−0.997239	+	0.0742568i	$0.976342\pi$
$884$	5.48905i	0.00620934i
$885$	0	0
$886$	−1049.01	−1.18399
$887$	− 1377.74i	− 1.55326i	−0.629955	−	0.776631i	$-0.716927\pi$
$887$	− 1377.74i	− 1.55326i	0.629955	−	0.776631i	$-0.283073\pi$
$888$	0	0
$889$	−195.053	−0.219407
$890$	− 60.3322i	− 0.0677890i
$891$	0	0
$892$	51.4978	0.0577329
$893$	− 402.061i	− 0.450236i
$894$	0	0
$895$	−228.704	−0.255536
$896$	1609.41i	1.79622i
$897$	0	0
$898$	−1912.10	−2.12928
$899$	− 1349.28i	− 1.50087i
$900$	0	0
$901$	−367.697	−0.408099
$902$	− 1120.06i	− 1.24175i
$903$	0	0
$904$	968.146	1.07096
$905$	244.168i	0.269799i
$906$	0	0
$907$	1380.88	1.52246	0.761232	−	0.648479i	$-0.224595\pi$
$907$	1380.88	1.52246	0.761232	+	0.648479i	$0.224595\pi$
$908$	9.43540i	0.0103914i
$909$	0	0
$910$	−426.775	−0.468984
$911$	693.024i	0.760729i	0.924837	+	0.380365i	$0.124201\pi$
$911$	693.024i	0.760729i	−0.924837	+	0.380365i	$0.875799\pi$
$912$	0	0
$913$	−1499.57	−1.64247
$914$	25.2360i	0.0276105i
$915$	0	0
$916$	−267.132	−0.291629
$917$	121.636i	0.132646i
$918$	0	0
$919$	378.985	0.412388	0.206194	−	0.978511i	$-0.433892\pi$
$919$	378.985	0.412388	0.206194	+	0.978511i	$0.433892\pi$
$920$	2750.32i	2.98947i
$921$	0	0
$922$	170.322	0.184731
$923$	134.069i	0.145254i
$924$	0	0
$925$	1656.73	1.79106
$926$	414.025i	0.447111i
$927$	0	0
$928$	314.110	0.338481
$929$	− 913.872i	− 0.983715i	−0.870676	−	0.491858i	$-0.836318\pi$
$929$	− 913.872i	− 0.983715i	0.870676	−	0.491858i	$-0.163682\pi$
$930$	0	0
$931$	527.756	0.566870
$932$	− 174.261i	− 0.186975i
$933$	0	0
$934$	1363.04	1.45936
$935$	− 591.125i	− 0.632220i
$936$	0	0
$937$	1574.11	1.67994	0.839972	−	0.542629i	$-0.182571\pi$
$937$	1574.11	1.67994	0.839972	+	0.542629i	$0.182571\pi$
$938$	− 1208.58i	− 1.28847i
$939$	0	0
$940$	284.499	0.302659
$941$	− 575.242i	− 0.611309i	−0.952142	−	0.305655i	$-0.901125\pi$
$941$	− 575.242i	− 0.611309i	0.952142	−	0.305655i	$-0.0988753\pi$
$942$	0	0
$943$	−1377.87	−1.46116
$944$	− 867.815i	− 0.919295i
$945$	0	0
$946$	1185.76	1.25345
$947$	200.497i	0.211718i	0.994381	+	0.105859i	$0.0337592\pi$
$947$	200.497i	0.211718i	−0.994381	+	0.105859i	$0.966241\pi$
$948$	0	0
$949$	−38.5901	−0.0406639
$950$	− 907.251i	− 0.955001i
$951$	0	0
$952$	−320.703	−0.336873
$953$	61.9288i	0.0649830i	0.999472	+	0.0324915i	$0.0103442\pi$
$953$	61.9288i	0.0649830i	−0.999472	+	0.0324915i	$0.989656\pi$
$954$	0	0
$955$	2210.72	2.31489
$956$	135.399i	0.141630i
$957$	0	0
$958$	−1340.21	−1.39896
$959$	− 888.949i	− 0.926954i
$960$	0	0
$961$	922.127	0.959549
$962$	− 140.690i	− 0.146247i
$963$	0	0
$964$	−124.768	−0.129427
$965$	− 1818.24i	− 1.88418i
$966$	0	0
$967$	−942.391	−0.974551	−0.487276	−	0.873248i	$-0.662009\pi$
$967$	−942.391	−0.974551	−0.487276	+	0.873248i	$0.662009\pi$
$968$	− 1032.48i	− 1.06661i
$969$	0	0
$970$	472.125	0.486726
$971$	591.900i	0.609578i	0.952420	+	0.304789i	$0.0985859\pi$
$971$	591.900i	0.609578i	−0.952420	+	0.304789i	$0.901414\pi$
$972$	0	0
$973$	2387.95	2.45422
$974$	1628.32i	1.67178i
$975$	0	0
$976$	78.6288	0.0805623
$977$	− 1024.42i	− 1.04854i	−0.851552	−	0.524271i	$-0.824338\pi$
$977$	− 1024.42i	− 1.04854i	0.851552	−	0.524271i	$-0.175662\pi$
$978$	0	0
$979$	51.5081	0.0526130
$980$	373.441i	0.381062i
$981$	0	0
$982$	935.277	0.952421
$983$	− 31.1053i	− 0.0316432i	−0.999875	−	0.0158216i	$-0.994964\pi$
$983$	− 31.1053i	− 0.0316432i	0.999875	−	0.0158216i	$-0.00503639\pi$
$984$	0	0
$985$	1663.68	1.68901
$986$	276.066i	0.279986i
$987$	0	0
$988$	−10.5860	−0.0107146
$989$	− 1458.70i	− 1.47493i
$990$	0	0
$991$	938.169	0.946689	0.473344	−	0.880877i	$-0.343047\pi$
$991$	938.169	0.946689	0.473344	+	0.880877i	$0.343047\pi$
$992$	438.389i	0.441924i
$993$	0	0
$994$	1484.15	1.49311
$995$	153.294i	0.154064i
$996$	0	0
$997$	738.410	0.740632	0.370316	−	0.928906i	$-0.379249\pi$
$997$	738.410	0.740632	0.370316	+	0.928906i	$0.379249\pi$
$998$	− 590.315i	− 0.591498i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	459.3.b.d.188.10	yes	12
3.2	odd	2	inner	459.3.b.d.188.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
459.3.b.d.188.3	✓	12	3.2	odd	2	inner
459.3.b.d.188.10	yes	12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.15341i q^{2} -0.637159 q^{4} -8.83083i q^{5} -10.7410 q^{7} +7.24156i q^{8} +19.0164 q^{10} +16.2351i q^{11} +2.08942 q^{13} -23.1298i q^{14} -18.1427 q^{16} -4.12311i q^{17} +7.95171 q^{19} +5.62664i q^{20} -34.9607 q^{22} +43.0080i q^{23} -52.9835 q^{25} +4.49936i q^{26} +6.84376 q^{28} +31.0930i q^{29} -43.3950 q^{31} -10.1023i q^{32} +8.87872 q^{34} +94.8524i q^{35} -31.2688 q^{37} +17.1233i q^{38} +63.9490 q^{40} +32.0376i q^{41} -33.9170 q^{43} -10.3443i q^{44} -92.6136 q^{46} -50.5628i q^{47} +66.3701 q^{49} -114.095i q^{50} -1.33129 q^{52} -89.1797i q^{53} +143.369 q^{55} -77.7820i q^{56} -66.9559 q^{58} +47.8328i q^{59} -4.33392 q^{61} -93.4471i q^{62} -50.8163 q^{64} -18.4513i q^{65} +52.2521 q^{67} +2.62707i q^{68} -204.256 q^{70} +64.1659i q^{71} -18.4693 q^{73} -67.3345i q^{74} -5.06650 q^{76} -174.381i q^{77} +43.7991 q^{79} +160.215i q^{80} -68.9900 q^{82} +92.3664i q^{83} -36.4104 q^{85} -73.0371i q^{86} -117.567 q^{88} -3.17265i q^{89} -22.4425 q^{91} -27.4029i q^{92} +108.882 q^{94} -70.2202i q^{95} +24.8273 q^{97} +142.922i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 20 q^{4} - 54 q^{7} - 12 q^{10} - 50 q^{13} + 108 q^{16} - 50 q^{19} - 106 q^{22} - 48 q^{25} + 382 q^{28} + 2 q^{31} - 244 q^{37} + 208 q^{40} - 78 q^{43} - 18 q^{46} + 470 q^{49} + 42 q^{52} + 290 q^{55}+ \cdots + 66 q^{97}+O(q^{100}) \) 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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