LMFDB - Embedded newform 459.3.b.d.188.2

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			188.2
Root			$-3.15039i$ of defining polynomial
Character	$\chi$	$=$	459.188
Dual form			459.3.b.d.188.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.15039i q^{2} -5.92497 q^{4} +0.888739i q^{5} -10.5667 q^{7} +6.06443i q^{8} +2.79988 q^{10} +9.89652i q^{11} +12.5343 q^{13} +33.2892i q^{14} -4.59457 q^{16} +4.12311i q^{17} -10.5006 q^{19} -5.26576i q^{20} +31.1779 q^{22} +18.6257i q^{23} +24.2101 q^{25} -39.4881i q^{26} +62.6073 q^{28} +29.4407i q^{29} +48.9768 q^{31} +38.7324i q^{32} +12.9894 q^{34} -9.39102i q^{35} -10.0653 q^{37} +33.0812i q^{38} -5.38969 q^{40} +20.6721i q^{41} -61.9861 q^{43} -58.6366i q^{44} +58.6781 q^{46} +2.82082i q^{47} +62.6547 q^{49} -76.2715i q^{50} -74.2656 q^{52} +59.1829i q^{53} -8.79543 q^{55} -64.0809i q^{56} +92.7498 q^{58} +38.6083i q^{59} +14.2459 q^{61} -154.296i q^{62} +103.644 q^{64} +11.1397i q^{65} -126.421 q^{67} -24.4293i q^{68} -29.5854 q^{70} -25.5335i q^{71} +16.3979 q^{73} +31.7096i q^{74} +62.2161 q^{76} -104.573i q^{77} -57.9343 q^{79} -4.08338i q^{80} +65.1253 q^{82} -123.734i q^{83} -3.66437 q^{85} +195.280i q^{86} -60.0167 q^{88} +101.810i q^{89} -132.446 q^{91} -110.357i q^{92} +8.88668 q^{94} -9.33233i q^{95} -58.6859 q^{97} -197.387i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 3.15039i	− 1.57520i	−0.616189	−	0.787598i	$-0.711324\pi$
$2$	− 3.15039i	− 1.57520i	0.616189	−	0.787598i	$-0.288676\pi$
$3$	0	0
$3$	0	0
$4$	−5.92497	−1.48124
$5$	0.888739i	0.177748i	0.996043	+	0.0888739i	$0.0283268\pi$
$5$	0.888739i	0.177748i	−0.996043	+	0.0888739i	$0.971673\pi$
$6$	0	0
$7$	−10.5667	−1.50953	−0.754763	−	0.655998i	$-0.772248\pi$
$7$	−10.5667	−1.50953	−0.754763	+	0.655998i	$0.772248\pi$
$8$	6.06443i	0.758053i
$9$	0	0
$10$	2.79988	0.279988
$11$	9.89652i	0.899684i	0.893108	+	0.449842i	$0.148520\pi$
$11$	9.89652i	0.899684i	−0.893108	+	0.449842i	$0.851480\pi$
$12$	0	0
$13$	12.5343	0.964179	0.482090	−	0.876122i	$-0.339878\pi$
$13$	12.5343	0.964179	0.482090	+	0.876122i	$0.339878\pi$
$14$	33.2892i	2.37780i
$15$	0	0
$16$	−4.59457	−0.287161
$17$	4.12311i	0.242536i
$17$	4.12311i	0.242536i
$18$	0	0
$19$	−10.5006	−0.552666	−0.276333	−	0.961062i	$-0.589119\pi$
$19$	−10.5006	−0.552666	−0.276333	+	0.961062i	$0.589119\pi$
$20$	− 5.26576i	− 0.263288i
$21$	0	0
$22$	31.1779	1.41718
$23$	18.6257i	0.809811i	0.914359	+	0.404906i	$0.132696\pi$
$23$	18.6257i	0.809811i	−0.914359	+	0.404906i	$0.867304\pi$
$24$	0	0
$25$	24.2101	0.968406
$26$	− 39.4881i	− 1.51877i
$27$	0	0
$28$	62.6073	2.23598
$29$	29.4407i	1.01520i	0.861594	+	0.507599i	$0.169467\pi$
$29$	29.4407i	1.01520i	−0.861594	+	0.507599i	$0.830533\pi$
$30$	0	0
$31$	48.9768	1.57990	0.789948	−	0.613174i	$-0.210108\pi$
$31$	48.9768	1.57990	0.789948	+	0.613174i	$0.210108\pi$
$32$	38.7324i	1.21039i
$33$	0	0
$34$	12.9894	0.382041
$35$	− 9.39102i	− 0.268315i
$36$	0	0
$37$	−10.0653	−0.272035	−0.136017	−	0.990706i	$-0.543430\pi$
$37$	−10.0653	−0.272035	−0.136017	+	0.990706i	$0.543430\pi$
$38$	33.0812i	0.870557i
$39$	0	0
$40$	−5.38969	−0.134742
$41$	20.6721i	0.504198i	0.967701	+	0.252099i	$0.0811209\pi$
$41$	20.6721i	0.504198i	−0.967701	+	0.252099i	$0.918879\pi$
$42$	0	0
$43$	−61.9861	−1.44154	−0.720768	−	0.693176i	$-0.756211\pi$
$43$	−61.9861	−1.44154	−0.720768	+	0.693176i	$0.756211\pi$
$44$	− 58.6366i	− 1.33265i
$45$	0	0
$46$	58.6781	1.27561
$47$	2.82082i	0.0600174i	0.999550	+	0.0300087i	$0.00955350\pi$
$47$	2.82082i	0.0600174i	−0.999550	+	0.0300087i	$0.990447\pi$
$48$	0	0
$49$	62.6547	1.27867
$50$	− 76.2715i	− 1.52543i
$51$	0	0
$52$	−74.2656	−1.42818
$53$	59.1829i	1.11666i	0.829620	+	0.558329i	$0.188557\pi$
$53$	59.1829i	1.11666i	−0.829620	+	0.558329i	$0.811443\pi$
$54$	0	0
$55$	−8.79543	−0.159917
$56$	− 64.0809i	− 1.14430i
$57$	0	0
$58$	92.7498	1.59913
$59$	38.6083i	0.654378i	0.944959	+	0.327189i	$0.106101\pi$
$59$	38.6083i	0.654378i	−0.944959	+	0.327189i	$0.893899\pi$
$60$	0	0
$61$	14.2459	0.233539	0.116769	−	0.993159i	$-0.462746\pi$
$61$	14.2459	0.233539	0.116769	+	0.993159i	$0.462746\pi$
$62$	− 154.296i	− 2.48865i
$63$	0	0
$64$	103.644	1.61944
$65$	11.1397i	0.171381i
$66$	0	0
$67$	−126.421	−1.88689	−0.943444	−	0.331532i	$-0.892434\pi$
$67$	−126.421	−1.88689	−0.943444	+	0.331532i	$0.892434\pi$
$68$	− 24.4293i	− 0.359254i
$69$	0	0
$70$	−29.5854	−0.422649
$71$	− 25.5335i	− 0.359627i	−0.983701	−	0.179814i	$-0.942451\pi$
$71$	− 25.5335i	− 0.359627i	0.983701	−	0.179814i	$-0.0575495\pi$
$72$	0	0
$73$	16.3979	0.224628	0.112314	−	0.993673i	$-0.464174\pi$
$73$	16.3979	0.224628	0.112314	+	0.993673i	$0.464174\pi$
$74$	31.7096i	0.428508i
$75$	0	0
$76$	62.2161	0.818632
$77$	− 104.573i	− 1.35810i
$78$	0	0
$79$	−57.9343	−0.733345	−0.366672	−	0.930350i	$-0.619503\pi$
$79$	−57.9343	−0.733345	−0.366672	+	0.930350i	$0.619503\pi$
$80$	− 4.08338i	− 0.0510422i
$81$	0	0
$82$	65.1253	0.794211
$83$	− 123.734i	− 1.49077i	−0.666634	−	0.745385i	$-0.732266\pi$
$83$	− 123.734i	− 1.49077i	0.666634	−	0.745385i	$-0.267734\pi$
$84$	0	0
$85$	−3.66437	−0.0431102
$86$	195.280i	2.27070i
$87$	0	0
$88$	−60.0167	−0.682008
$89$	101.810i	1.14393i	0.820278	+	0.571965i	$0.193819\pi$
$89$	101.810i	1.14393i	−0.820278	+	0.571965i	$0.806181\pi$
$90$	0	0
$91$	−132.446	−1.45545
$92$	− 110.357i	− 1.19953i
$93$	0	0
$94$	8.88668	0.0945392
$95$	− 9.33233i	− 0.0982351i
$96$	0	0
$97$	−58.6859	−0.605010	−0.302505	−	0.953148i	$-0.597823\pi$
$97$	−58.6859	−0.605010	−0.302505	+	0.953148i	$0.597823\pi$
$98$	− 197.387i	− 2.01415i
$99$	0	0
$100$	−143.444	−1.43444
$101$	− 78.6743i	− 0.778954i	−0.921036	−	0.389477i	$-0.872656\pi$
$101$	− 78.6743i	− 0.778954i	0.921036	−	0.389477i	$-0.127344\pi$
$102$	0	0
$103$	−83.9849	−0.815388	−0.407694	−	0.913119i	$-0.633667\pi$
$103$	−83.9849	−0.815388	−0.407694	+	0.913119i	$0.633667\pi$
$104$	76.0135i	0.730899i
$105$	0	0
$106$	186.449	1.75896
$107$	169.045i	1.57986i	0.613199	+	0.789928i	$0.289882\pi$
$107$	169.045i	1.57986i	−0.613199	+	0.789928i	$0.710118\pi$
$108$	0	0
$109$	−61.1657	−0.561153	−0.280576	−	0.959832i	$-0.590526\pi$
$109$	−61.1657	−0.561153	−0.280576	+	0.959832i	$0.590526\pi$
$110$	27.7090i	0.251900i
$111$	0	0
$112$	48.5494	0.433476
$113$	3.37271i	0.0298470i	0.999889	+	0.0149235i	$0.00475047\pi$
$113$	3.37271i	0.0298470i	−0.999889	+	0.0149235i	$0.995250\pi$
$114$	0	0
$115$	−16.5534	−0.143942
$116$	− 174.436i	− 1.50375i
$117$	0	0
$118$	121.631	1.03077
$119$	− 43.5675i	− 0.366114i
$120$	0	0
$121$	23.0589	0.190569
$122$	− 44.8801i	− 0.367870i
$123$	0	0
$124$	−290.186	−2.34021
$125$	43.7350i	0.349880i
$126$	0	0
$127$	−154.265	−1.21469	−0.607343	−	0.794440i	$-0.707764\pi$
$127$	−154.265	−1.21469	−0.607343	+	0.794440i	$0.707764\pi$
$128$	− 171.590i	− 1.34055i
$129$	0	0
$130$	35.0946	0.269958
$131$	194.713i	1.48636i	0.669090	+	0.743181i	$0.266684\pi$
$131$	194.713i	1.48636i	−0.669090	+	0.743181i	$0.733316\pi$
$132$	0	0
$133$	110.957	0.834263
$134$	398.277i	2.97222i
$135$	0	0
$136$	−25.0043	−0.183855
$137$	168.707i	1.23144i	0.787965	+	0.615720i	$0.211135\pi$
$137$	168.707i	1.23144i	−0.787965	+	0.615720i	$0.788865\pi$
$138$	0	0
$139$	154.598	1.11221	0.556107	−	0.831111i	$-0.312295\pi$
$139$	154.598	1.11221	0.556107	+	0.831111i	$0.312295\pi$
$140$	55.6416i	0.397440i
$141$	0	0
$142$	−80.4407	−0.566484
$143$	124.046i	0.867456i
$144$	0	0
$145$	−26.1651	−0.180449
$146$	− 51.6597i	− 0.353834i
$147$	0	0
$148$	59.6366	0.402950
$149$	224.338i	1.50562i	0.658237	+	0.752811i	$0.271302\pi$
$149$	224.338i	1.50562i	−0.658237	+	0.752811i	$0.728698\pi$
$150$	0	0
$151$	155.240	1.02808	0.514040	−	0.857766i	$-0.328148\pi$
$151$	155.240	1.02808	0.514040	+	0.857766i	$0.328148\pi$
$152$	− 63.6804i	− 0.418950i
$153$	0	0
$154$	−329.447	−2.13927
$155$	43.5276i	0.280823i
$156$	0	0
$157$	−69.5880	−0.443236	−0.221618	−	0.975134i	$-0.571134\pi$
$157$	−69.5880	−0.443236	−0.221618	+	0.975134i	$0.571134\pi$
$158$	182.516i	1.15516i
$159$	0	0
$160$	−34.4230	−0.215144
$161$	− 196.811i	− 1.22243i
$162$	0	0
$163$	295.962	1.81572	0.907858	−	0.419277i	$-0.137716\pi$
$163$	295.962	1.81572	0.907858	+	0.419277i	$0.137716\pi$
$164$	− 122.482i	− 0.746840i
$165$	0	0
$166$	−389.811	−2.34826
$167$	247.319i	1.48095i	0.672084	+	0.740475i	$0.265399\pi$
$167$	247.319i	1.48095i	−0.672084	+	0.740475i	$0.734601\pi$
$168$	0	0
$169$	−11.8906	−0.0703588
$170$	11.5442i	0.0679070i
$171$	0	0
$172$	367.266	2.13527
$173$	− 269.938i	− 1.56034i	−0.625570	−	0.780168i	$-0.715133\pi$
$173$	− 269.938i	− 1.56034i	0.625570	−	0.780168i	$-0.284867\pi$
$174$	0	0
$175$	−255.821	−1.46183
$176$	− 45.4703i	− 0.258354i
$177$	0	0
$178$	320.741	1.80192
$179$	− 317.609i	− 1.77435i	−0.461432	−	0.887176i	$-0.652664\pi$
$179$	− 317.609i	− 1.77435i	0.461432	−	0.887176i	$-0.347336\pi$
$180$	0	0
$181$	51.4419	0.284209	0.142105	−	0.989852i	$-0.454613\pi$
$181$	51.4419	0.284209	0.142105	+	0.989852i	$0.454613\pi$
$182$	417.258i	2.29262i
$183$	0	0
$184$	−112.954	−0.613880
$185$	− 8.94542i	− 0.0483536i
$186$	0	0
$187$	−40.8044	−0.218205
$188$	− 16.7133i	− 0.0889004i
$189$	0	0
$190$	−29.4005	−0.154740
$191$	122.709i	0.642456i	0.947002	+	0.321228i	$0.104096\pi$
$191$	122.709i	0.642456i	−0.947002	+	0.321228i	$0.895904\pi$
$192$	0	0
$193$	−77.8242	−0.403234	−0.201617	−	0.979464i	$-0.564620\pi$
$193$	−77.8242	−0.403234	−0.201617	+	0.979464i	$0.564620\pi$
$194$	184.884i	0.953009i
$195$	0	0
$196$	−371.227	−1.89402
$197$	95.2181i	0.483341i	0.970358	+	0.241670i	$0.0776952\pi$
$197$	95.2181i	0.483341i	−0.970358	+	0.241670i	$0.922305\pi$
$198$	0	0
$199$	−285.742	−1.43589	−0.717946	−	0.696099i	$-0.754918\pi$
$199$	−285.742	−1.43589	−0.717946	+	0.696099i	$0.754918\pi$
$200$	146.821i	0.734103i
$201$	0	0
$202$	−247.855	−1.22701
$203$	− 311.091i	− 1.53247i
$204$	0	0
$205$	−18.3721	−0.0896201
$206$	264.585i	1.28440i
$207$	0	0
$208$	−57.5899	−0.276874
$209$	− 103.920i	− 0.497224i
$210$	0	0
$211$	−281.344	−1.33338	−0.666691	−	0.745334i	$-0.732290\pi$
$211$	−281.344	−1.33338	−0.666691	+	0.745334i	$0.732290\pi$
$212$	− 350.657i	− 1.65404i
$213$	0	0
$214$	532.557	2.48858
$215$	− 55.0894i	− 0.256230i
$216$	0	0
$217$	−517.522	−2.38489
$218$	192.696i	0.883926i
$219$	0	0
$220$	52.1127	0.236876
$221$	51.6804i	0.233848i
$222$	0	0
$223$	78.3787	0.351474	0.175737	−	0.984437i	$-0.443769\pi$
$223$	78.3787	0.351474	0.175737	+	0.984437i	$0.443769\pi$
$224$	− 409.273i	− 1.82711i
$225$	0	0
$226$	10.6254	0.0470148
$227$	151.503i	0.667413i	0.942677	+	0.333707i	$0.108300\pi$
$227$	151.503i	0.667413i	−0.942677	+	0.333707i	$0.891700\pi$
$228$	0	0
$229$	231.801	1.01223	0.506116	−	0.862465i	$-0.331081\pi$
$229$	231.801	1.01223	0.506116	+	0.862465i	$0.331081\pi$
$230$	52.1496i	0.226737i
$231$	0	0
$232$	−178.541	−0.769574
$233$	− 272.899i	− 1.17124i	−0.810585	−	0.585621i	$-0.800851\pi$
$233$	− 272.899i	− 1.17124i	0.810585	−	0.585621i	$-0.199149\pi$
$234$	0	0
$235$	−2.50697	−0.0106680
$236$	− 228.753i	− 0.969293i
$237$	0	0
$238$	−137.255	−0.576701
$239$	− 116.935i	− 0.489267i	−0.969616	−	0.244634i	$-0.921332\pi$
$239$	− 116.935i	− 0.489267i	0.969616	−	0.244634i	$-0.0786677\pi$
$240$	0	0
$241$	425.733	1.76653	0.883263	−	0.468877i	$-0.155341\pi$
$241$	425.733	1.76653	0.883263	+	0.468877i	$0.155341\pi$
$242$	− 72.6445i	− 0.300184i
$243$	0	0
$244$	−84.4064	−0.345928
$245$	55.6837i	0.227280i
$246$	0	0
$247$	−131.619	−0.532869
$248$	297.016i	1.19765i
$249$	0	0
$250$	137.782	0.551129
$251$	− 249.281i	− 0.993150i	−0.867994	−	0.496575i	$-0.834591\pi$
$251$	− 249.281i	− 0.993150i	0.867994	−	0.496575i	$-0.165409\pi$
$252$	0	0
$253$	−184.329	−0.728574
$254$	485.995i	1.91337i
$255$	0	0
$256$	−125.999	−0.492184
$257$	− 35.5290i	− 0.138245i	−0.997608	−	0.0691227i	$-0.977980\pi$
$257$	− 35.5290i	− 0.138245i	0.997608	−	0.0691227i	$-0.0220200\pi$
$258$	0	0
$259$	106.357	0.410644
$260$	− 66.0027i	− 0.253857i
$261$	0	0
$262$	613.424	2.34131
$263$	− 379.455i	− 1.44279i	−0.692522	−	0.721397i	$-0.743500\pi$
$263$	− 379.455i	− 1.44279i	0.692522	−	0.721397i	$-0.256500\pi$
$264$	0	0
$265$	−52.5981	−0.198483
$266$	− 349.558i	− 1.31413i
$267$	0	0
$268$	749.044	2.79494
$269$	− 15.1816i	− 0.0564370i	−0.999602	−	0.0282185i	$-0.991017\pi$
$269$	− 15.1816i	− 0.0564370i	0.999602	−	0.0282185i	$-0.00898343\pi$
$270$	0	0
$271$	−376.207	−1.38822	−0.694109	−	0.719870i	$-0.744201\pi$
$271$	−376.207	−1.38822	−0.694109	+	0.719870i	$0.744201\pi$
$272$	− 18.9439i	− 0.0696467i
$273$	0	0
$274$	531.494	1.93976
$275$	239.596i	0.871259i
$276$	0	0
$277$	−10.9140	−0.0394007	−0.0197003	−	0.999806i	$-0.506271\pi$
$277$	−10.9140	−0.0394007	−0.0197003	+	0.999806i	$0.506271\pi$
$278$	− 487.043i	− 1.75195i
$279$	0	0
$280$	56.9512	0.203397
$281$	− 317.993i	− 1.13165i	−0.824526	−	0.565824i	$-0.808558\pi$
$281$	− 317.993i	− 1.13165i	0.824526	−	0.565824i	$-0.191442\pi$
$282$	0	0
$283$	496.762	1.75534	0.877671	−	0.479264i	$-0.159096\pi$
$283$	496.762	1.75534	0.877671	+	0.479264i	$0.159096\pi$
$284$	151.286i	0.532696i
$285$	0	0
$286$	390.794	1.36641
$287$	− 218.436i	− 0.761100i
$288$	0	0
$289$	−17.0000	−0.0588235
$290$	82.4304i	0.284243i
$291$	0	0
$292$	−97.1569	−0.332729
$293$	− 461.796i	− 1.57609i	−0.615615	−	0.788047i	$-0.711092\pi$
$293$	− 461.796i	− 1.57609i	0.615615	−	0.788047i	$-0.288908\pi$
$294$	0	0
$295$	−34.3127	−0.116314
$296$	− 61.0402i	− 0.206217i
$297$	0	0
$298$	706.751	2.37165
$299$	233.460i	0.780803i
$300$	0	0
$301$	654.987	2.17604
$302$	− 489.067i	− 1.61943i
$303$	0	0
$304$	48.2460	0.158704
$305$	12.6609i	0.0415110i
$306$	0	0
$307$	−386.037	−1.25745	−0.628725	−	0.777628i	$-0.716423\pi$
$307$	−386.037	−1.25745	−0.628725	+	0.777628i	$0.716423\pi$
$308$	619.594i	2.01167i
$309$	0	0
$310$	137.129	0.442351
$311$	295.253i	0.949366i	0.880157	+	0.474683i	$0.157437\pi$
$311$	295.253i	0.949366i	−0.880157	+	0.474683i	$0.842563\pi$
$312$	0	0
$313$	−28.9955	−0.0926373	−0.0463187	−	0.998927i	$-0.514749\pi$
$313$	−28.9955	−0.0926373	−0.0463187	+	0.998927i	$0.514749\pi$
$314$	219.229i	0.698183i
$315$	0	0
$316$	343.259	1.08626
$317$	− 126.503i	− 0.399062i	−0.979891	−	0.199531i	$-0.936058\pi$
$317$	− 126.503i	− 0.399062i	0.979891	−	0.199531i	$-0.0639419\pi$
$318$	0	0
$319$	−291.361	−0.913356
$320$	92.1125i	0.287852i
$321$	0	0
$322$	−620.033	−1.92557
$323$	− 43.2953i	− 0.134041i
$324$	0	0
$325$	303.458	0.933716
$326$	− 932.396i	− 2.86011i
$327$	0	0
$328$	−125.365	−0.382209
$329$	− 29.8067i	− 0.0905978i
$330$	0	0
$331$	392.966	1.18721	0.593605	−	0.804757i	$-0.297704\pi$
$331$	392.966	1.18721	0.593605	+	0.804757i	$0.297704\pi$
$332$	733.121i	2.20819i
$333$	0	0
$334$	779.151	2.33279
$335$	− 112.356i	− 0.335390i
$336$	0	0
$337$	223.852	0.664250	0.332125	−	0.943235i	$-0.392234\pi$
$337$	223.852	0.664250	0.332125	+	0.943235i	$0.392234\pi$
$338$	37.4602i	0.110829i
$339$	0	0
$340$	21.7113	0.0638567
$341$	484.700i	1.42141i
$342$	0	0
$343$	−144.285	−0.420655
$344$	− 375.910i	− 1.09276i
$345$	0	0
$346$	−850.412	−2.45784
$347$	− 595.431i	− 1.71594i	−0.513701	−	0.857969i	$-0.671726\pi$
$347$	− 595.431i	− 1.71594i	0.513701	−	0.857969i	$-0.328274\pi$
$348$	0	0
$349$	−558.124	−1.59921	−0.799605	−	0.600526i	$-0.794958\pi$
$349$	−558.124	−1.59921	−0.799605	+	0.600526i	$0.794958\pi$
$350$	805.936i	2.30267i
$351$	0	0
$352$	−383.316	−1.08897
$353$	294.388i	0.833960i	0.908916	+	0.416980i	$0.136911\pi$
$353$	294.388i	0.833960i	−0.908916	+	0.416980i	$0.863089\pi$
$354$	0	0
$355$	22.6927	0.0639230
$356$	− 603.221i	− 1.69444i
$357$	0	0
$358$	−1000.59	−2.79495
$359$	425.659i	1.18568i	0.805320	+	0.592840i	$0.201993\pi$
$359$	425.659i	1.18568i	−0.805320	+	0.592840i	$0.798007\pi$
$360$	0	0
$361$	−250.736	−0.694561
$362$	− 162.062i	− 0.447685i
$363$	0	0
$364$	784.740	2.15588
$365$	14.5734i	0.0399272i
$366$	0	0
$367$	−237.639	−0.647518	−0.323759	−	0.946140i	$-0.604947\pi$
$367$	−237.639	−0.647518	−0.323759	+	0.946140i	$0.604947\pi$
$368$	− 85.5769i	− 0.232546i
$369$	0	0
$370$	−28.1816	−0.0761664
$371$	− 625.366i	− 1.68562i
$372$	0	0
$373$	−44.4936	−0.119286	−0.0596429	−	0.998220i	$-0.518996\pi$
$373$	−44.4936	−0.119286	−0.0596429	+	0.998220i	$0.518996\pi$
$374$	128.550i	0.343716i
$375$	0	0
$376$	−17.1066	−0.0454964
$377$	369.020i	0.978832i
$378$	0	0
$379$	350.367	0.924452	0.462226	−	0.886762i	$-0.347051\pi$
$379$	350.367	0.924452	0.462226	+	0.886762i	$0.347051\pi$
$380$	55.2939i	0.145510i
$381$	0	0
$382$	386.582	1.01199
$383$	388.918i	1.01545i	0.861518	+	0.507726i	$0.169514\pi$
$383$	388.918i	1.01545i	−0.861518	+	0.507726i	$0.830486\pi$
$384$	0	0
$385$	92.9384	0.241399
$386$	245.177i	0.635173i
$387$	0	0
$388$	347.713	0.896167
$389$	− 145.514i	− 0.374071i	−0.982353	−	0.187036i	$-0.940112\pi$
$389$	− 145.514i	− 0.374071i	0.982353	−	0.187036i	$-0.0598879\pi$
$390$	0	0
$391$	−76.7955	−0.196408
$392$	379.965i	0.969298i
$393$	0	0
$394$	299.974	0.761356
$395$	− 51.4884i	− 0.130350i
$396$	0	0
$397$	−583.455	−1.46966	−0.734831	−	0.678251i	$-0.762738\pi$
$397$	−583.455	−1.46966	−0.734831	+	0.678251i	$0.762738\pi$
$398$	900.201i	2.26181i
$399$	0	0
$400$	−111.235	−0.278088
$401$	103.586i	0.258319i	0.991624	+	0.129159i	$0.0412279\pi$
$401$	103.586i	0.258319i	−0.991624	+	0.129159i	$0.958772\pi$
$402$	0	0
$403$	613.891	1.52330
$404$	466.144i	1.15382i
$405$	0	0
$406$	−980.058	−2.41393
$407$	− 99.6114i	− 0.244745i
$408$	0	0
$409$	−120.594	−0.294851	−0.147426	−	0.989073i	$-0.547099\pi$
$409$	−120.594	−0.294851	−0.147426	+	0.989073i	$0.547099\pi$
$410$	57.8794i	0.141169i
$411$	0	0
$412$	497.609	1.20779
$413$	− 407.962i	− 0.987800i
$414$	0	0
$415$	109.967	0.264981
$416$	485.485i	1.16703i
$417$	0	0
$418$	−327.388	−0.783226
$419$	585.973i	1.39850i	0.714875	+	0.699252i	$0.246484\pi$
$419$	585.973i	1.39850i	−0.714875	+	0.699252i	$0.753516\pi$
$420$	0	0
$421$	−16.5774	−0.0393761	−0.0196881	−	0.999806i	$-0.506267\pi$
$421$	−16.5774	−0.0393761	−0.0196881	+	0.999806i	$0.506267\pi$
$422$	886.343i	2.10034i
$423$	0	0
$424$	−358.910	−0.846486
$425$	99.8210i	0.234873i
$426$	0	0
$427$	−150.532	−0.352533
$428$	− 1001.59i	− 2.34015i
$429$	0	0
$430$	−173.553	−0.403613
$431$	722.881i	1.67722i	0.544733	+	0.838609i	$0.316631\pi$
$431$	722.881i	1.67722i	−0.544733	+	0.838609i	$0.683369\pi$
$432$	0	0
$433$	559.664	1.29253	0.646264	−	0.763114i	$-0.276331\pi$
$433$	559.664	1.29253	0.646264	+	0.763114i	$0.276331\pi$
$434$	1630.40i	3.75667i
$435$	0	0
$436$	362.405	0.831204
$437$	− 195.581i	− 0.447555i
$438$	0	0
$439$	−297.111	−0.676791	−0.338395	−	0.941004i	$-0.609884\pi$
$439$	−297.111	−0.676791	−0.338395	+	0.941004i	$0.609884\pi$
$440$	− 53.3392i	− 0.121225i
$441$	0	0
$442$	162.813	0.368356
$443$	− 601.063i	− 1.35680i	−0.734692	−	0.678401i	$-0.762673\pi$
$443$	− 601.063i	− 1.35680i	0.734692	−	0.678401i	$-0.237327\pi$
$444$	0	0
$445$	−90.4824	−0.203331
$446$	− 246.924i	− 0.553641i
$447$	0	0
$448$	−1095.17	−2.44458
$449$	48.5849i	0.108207i	0.998535	+	0.0541034i	$0.0172301\pi$
$449$	48.5849i	0.108207i	−0.998535	+	0.0541034i	$0.982770\pi$
$450$	0	0
$451$	−204.582	−0.453619
$452$	− 19.9832i	− 0.0442106i
$453$	0	0
$454$	477.294	1.05131
$455$	− 117.710i	− 0.258704i
$456$	0	0
$457$	604.556	1.32288	0.661440	−	0.749998i	$-0.269946\pi$
$457$	604.556	1.32288	0.661440	+	0.749998i	$0.269946\pi$
$458$	− 730.265i	− 1.59447i
$459$	0	0
$460$	98.0782	0.213213
$461$	2.46881i	0.00535533i	0.999996	+	0.00267766i	$0.000852328\pi$
$461$	2.46881i	0.00535533i	−0.999996	+	0.00267766i	$0.999148\pi$
$462$	0	0
$463$	22.1762	0.0478967	0.0239483	−	0.999713i	$-0.492376\pi$
$463$	22.1762	0.0478967	0.0239483	+	0.999713i	$0.492376\pi$
$464$	− 135.267i	− 0.291525i
$465$	0	0
$466$	−859.740	−1.84493
$467$	− 579.900i	− 1.24176i	−0.783907	−	0.620878i	$-0.786776\pi$
$467$	− 579.900i	− 1.24176i	0.783907	−	0.620878i	$-0.213224\pi$
$468$	0	0
$469$	1335.86	2.84831
$470$	7.89794i	0.0168041i
$471$	0	0
$472$	−234.137	−0.496053
$473$	− 613.446i	− 1.29693i
$474$	0	0
$475$	−254.222	−0.535204
$476$	258.137i	0.542304i
$477$	0	0
$478$	−368.391	−0.770692
$479$	− 750.497i	− 1.56680i	−0.621519	−	0.783399i	$-0.713484\pi$
$479$	− 750.497i	− 1.56680i	0.621519	−	0.783399i	$-0.286516\pi$
$480$	0	0
$481$	−126.162	−0.262290
$482$	− 1341.23i	− 2.78263i
$483$	0	0
$484$	−136.623	−0.282280
$485$	− 52.1565i	− 0.107539i
$486$	0	0
$487$	515.880	1.05930	0.529651	−	0.848216i	$-0.322323\pi$
$487$	515.880	1.05930	0.529651	+	0.848216i	$0.322323\pi$
$488$	86.3931i	0.177035i
$489$	0	0
$490$	175.425	0.358011
$491$	542.940i	1.10578i	0.833253	+	0.552892i	$0.186476\pi$
$491$	542.940i	1.10578i	−0.833253	+	0.552892i	$0.813524\pi$
$492$	0	0
$493$	−121.387	−0.246221
$494$	414.650i	0.839373i
$495$	0	0
$496$	−225.027	−0.453684
$497$	269.805i	0.542867i
$498$	0	0
$499$	863.555	1.73057	0.865286	−	0.501279i	$-0.167137\pi$
$499$	863.555	1.73057	0.865286	+	0.501279i	$0.167137\pi$
$500$	− 259.129i	− 0.518257i
$501$	0	0
$502$	−785.332	−1.56441
$503$	9.54565i	0.0189774i	0.999955	+	0.00948872i	$0.00302040\pi$
$503$	9.54565i	0.0189774i	−0.999955	+	0.00948872i	$0.996980\pi$
$504$	0	0
$505$	69.9210	0.138457
$506$	580.709i	1.14765i
$507$	0	0
$508$	914.016	1.79925
$509$	509.699i	1.00137i	0.865629	+	0.500687i	$0.166919\pi$
$509$	509.699i	1.00137i	−0.865629	+	0.500687i	$0.833081\pi$
$510$	0	0
$511$	−173.271	−0.339082
$512$	− 289.413i	− 0.565259i
$513$	0	0
$514$	−111.930	−0.217764
$515$	− 74.6407i	− 0.144933i
$516$	0	0
$517$	−27.9163	−0.0539967
$518$	− 335.065i	− 0.646844i
$519$	0	0
$520$	−67.5562	−0.129916
$521$	542.030i	1.04036i	0.854055	+	0.520182i	$0.174136\pi$
$521$	542.030i	1.04036i	−0.854055	+	0.520182i	$0.825864\pi$
$522$	0	0
$523$	−369.678	−0.706842	−0.353421	−	0.935464i	$-0.614982\pi$
$523$	−369.678	−0.706842	−0.353421	+	0.935464i	$0.614982\pi$
$524$	− 1153.67i	− 2.20166i
$525$	0	0
$526$	−1195.43	−2.27268
$527$	201.936i	0.383181i
$528$	0	0
$529$	182.085	0.344206
$530$	165.705i	0.312650i
$531$	0	0
$532$	−657.417	−1.23575
$533$	259.111i	0.486137i
$534$	0	0
$535$	−150.237	−0.280816
$536$	− 766.674i	− 1.43036i
$537$	0	0
$538$	−47.8279	−0.0888994
$539$	620.063i	1.15040i
$540$	0	0
$541$	725.661	1.34133	0.670666	−	0.741759i	$-0.266008\pi$
$541$	725.661	1.34133	0.670666	+	0.741759i	$0.266008\pi$
$542$	1185.20i	2.18672i
$543$	0	0
$544$	−159.698	−0.293562
$545$	− 54.3603i	− 0.0997437i
$546$	0	0
$547$	−822.618	−1.50387	−0.751936	−	0.659237i	$-0.770880\pi$
$547$	−822.618	−1.50387	−0.751936	+	0.659237i	$0.770880\pi$
$548$	− 999.586i	− 1.82406i
$549$	0	0
$550$	754.822	1.37240
$551$	− 309.147i	− 0.561064i
$552$	0	0
$553$	612.173	1.10700
$554$	34.3833i	0.0620638i
$555$	0	0
$556$	−915.987	−1.64746
$557$	− 62.6284i	− 0.112439i	−0.998418	−	0.0562194i	$-0.982095\pi$
$557$	− 62.6284i	− 0.112439i	0.998418	−	0.0562194i	$-0.0179046\pi$
$558$	0	0
$559$	−776.954	−1.38990
$560$	43.1477i	0.0770495i
$561$	0	0
$562$	−1001.80	−1.78257
$563$	208.133i	0.369686i	0.982768	+	0.184843i	$0.0591777\pi$
$563$	208.133i	0.369686i	−0.982768	+	0.184843i	$0.940822\pi$
$564$	0	0
$565$	−2.99746	−0.00530523
$566$	− 1564.99i	− 2.76501i
$567$	0	0
$568$	154.846	0.272617
$569$	746.533i	1.31201i	0.754757	+	0.656004i	$0.227755\pi$
$569$	746.533i	1.31201i	−0.754757	+	0.656004i	$0.772245\pi$
$570$	0	0
$571$	241.951	0.423732	0.211866	−	0.977299i	$-0.432046\pi$
$571$	241.951	0.423732	0.211866	+	0.977299i	$0.432046\pi$
$572$	− 734.971i	− 1.28491i
$573$	0	0
$574$	−688.158	−1.19888
$575$	450.930i	0.784226i
$576$	0	0
$577$	1140.42	1.97646	0.988228	−	0.152986i	$-0.0488890\pi$
$577$	1140.42	1.97646	0.988228	+	0.152986i	$0.0488890\pi$
$578$	53.5567i	0.0926586i
$579$	0	0
$580$	155.028	0.267289
$581$	1307.46i	2.25036i
$582$	0	0
$583$	−585.704	−1.00464
$584$	99.4436i	0.170280i
$585$	0	0
$586$	−1454.84	−2.48266
$587$	34.1484i	0.0581744i	0.999577	+	0.0290872i	$0.00926004\pi$
$587$	34.1484i	0.0581744i	−0.999577	+	0.0290872i	$0.990740\pi$
$588$	0	0
$589$	−514.288	−0.873154
$590$	108.099i	0.183218i
$591$	0	0
$592$	46.2457	0.0781177
$593$	− 33.4618i	− 0.0564279i	−0.999602	−	0.0282140i	$-0.991018\pi$
$593$	− 33.4618i	− 0.0564279i	0.999602	−	0.0282140i	$-0.00898198\pi$
$594$	0	0
$595$	38.7202	0.0650759
$596$	− 1329.19i	− 2.23019i
$597$	0	0
$598$	735.491	1.22992
$599$	− 840.770i	− 1.40362i	−0.712363	−	0.701811i	$-0.752375\pi$
$599$	− 840.770i	− 1.40362i	0.712363	−	0.701811i	$-0.247625\pi$
$600$	0	0
$601$	830.903	1.38253	0.691267	−	0.722599i	$-0.257053\pi$
$601$	830.903	1.38253	0.691267	+	0.722599i	$0.257053\pi$
$602$	− 2063.47i	− 3.42768i
$603$	0	0
$604$	−919.794	−1.52284
$605$	20.4933i	0.0338733i
$606$	0	0
$607$	−633.043	−1.04290	−0.521452	−	0.853280i	$-0.674610\pi$
$607$	−633.043	−1.04290	−0.521452	+	0.853280i	$0.674610\pi$
$608$	− 406.715i	− 0.668940i
$609$	0	0
$610$	39.8867	0.0653880
$611$	35.3570i	0.0578675i
$612$	0	0
$613$	−822.606	−1.34193	−0.670967	−	0.741487i	$-0.734121\pi$
$613$	−822.606	−1.34193	−0.670967	+	0.741487i	$0.734121\pi$
$614$	1216.17i	1.98073i
$615$	0	0
$616$	634.177	1.02951
$617$	− 1197.99i	− 1.94164i	−0.239810	−	0.970820i	$-0.577085\pi$
$617$	− 1197.99i	− 1.94164i	0.239810	−	0.970820i	$-0.422915\pi$
$618$	0	0
$619$	−510.165	−0.824176	−0.412088	−	0.911144i	$-0.635200\pi$
$619$	−510.165	−0.824176	−0.412088	+	0.911144i	$0.635200\pi$
$620$	− 257.900i	− 0.415967i
$621$	0	0
$622$	930.163	1.49544
$623$	− 1075.79i	− 1.72679i
$624$	0	0
$625$	566.385	0.906215
$626$	91.3472i	0.145922i
$627$	0	0
$628$	412.307	0.656540
$629$	− 41.5003i	− 0.0659782i
$630$	0	0
$631$	−6.87574	−0.0108966	−0.00544829	−	0.999985i	$-0.501734\pi$
$631$	−6.87574	−0.0108966	−0.00544829	+	0.999985i	$0.501734\pi$
$632$	− 351.338i	− 0.555915i
$633$	0	0
$634$	−398.533	−0.628601
$635$	− 137.101i	− 0.215908i
$636$	0	0
$637$	785.334	1.23286
$638$	917.901i	1.43872i
$639$	0	0
$640$	152.499	0.238279
$641$	475.717i	0.742149i	0.928603	+	0.371074i	$0.121010\pi$
$641$	475.717i	0.742149i	−0.928603	+	0.371074i	$0.878990\pi$
$642$	0	0
$643$	−52.1901	−0.0811666	−0.0405833	−	0.999176i	$-0.512922\pi$
$643$	−52.1901	−0.0811666	−0.0405833	+	0.999176i	$0.512922\pi$
$644$	1166.10i	1.81072i
$645$	0	0
$646$	−136.397	−0.211141
$647$	350.025i	0.540997i	0.962720	+	0.270499i	$0.0871885\pi$
$647$	350.025i	0.540997i	−0.962720	+	0.270499i	$0.912811\pi$
$648$	0	0
$649$	−382.088	−0.588733
$650$	− 956.011i	− 1.47079i
$651$	0	0
$652$	−1753.57	−2.68952
$653$	168.972i	0.258763i	0.991595	+	0.129382i	$0.0412992\pi$
$653$	168.972i	0.258763i	−0.991595	+	0.129382i	$0.958701\pi$
$654$	0	0
$655$	−173.049	−0.264198
$656$	− 94.9796i	− 0.144786i
$657$	0	0
$658$	−93.9027	−0.142709
$659$	1256.97i	1.90740i	0.300764	+	0.953699i	$0.402758\pi$
$659$	1256.97i	1.90740i	−0.300764	+	0.953699i	$0.597242\pi$
$660$	0	0
$661$	527.830	0.798533	0.399267	−	0.916835i	$-0.369265\pi$
$661$	527.830	0.798533	0.399267	+	0.916835i	$0.369265\pi$
$662$	− 1238.00i	− 1.87009i
$663$	0	0
$664$	750.376	1.13008
$665$	98.6118i	0.148288i
$666$	0	0
$667$	−548.353	−0.822118
$668$	− 1465.36i	− 2.19365i
$669$	0	0
$670$	−353.965	−0.528305
$671$	140.985i	0.210111i
$672$	0	0
$673$	148.226	0.220247	0.110124	−	0.993918i	$-0.464875\pi$
$673$	148.226	0.220247	0.110124	+	0.993918i	$0.464875\pi$
$674$	− 705.223i	− 1.04632i
$675$	0	0
$676$	70.4517	0.104218
$677$	− 1112.16i	− 1.64278i	−0.570366	−	0.821390i	$-0.693199\pi$
$677$	− 1112.16i	− 1.64278i	0.570366	−	0.821390i	$-0.306801\pi$
$678$	0	0
$679$	620.115	0.913277
$680$	− 22.2223i	− 0.0326798i
$681$	0	0
$682$	1526.99	2.23899
$683$	87.0626i	0.127471i	0.997967	+	0.0637354i	$0.0203014\pi$
$683$	87.0626i	0.127471i	−0.997967	+	0.0637354i	$0.979699\pi$
$684$	0	0
$685$	−149.937	−0.218886
$686$	454.554i	0.662615i
$687$	0	0
$688$	284.799	0.413953
$689$	741.817i	1.07666i
$690$	0	0
$691$	−652.515	−0.944305	−0.472152	−	0.881517i	$-0.656523\pi$
$691$	−652.515	−0.944305	−0.472152	+	0.881517i	$0.656523\pi$
$692$	1599.38i	2.31124i
$693$	0	0
$694$	−1875.84	−2.70294
$695$	137.397i	0.197694i
$696$	0	0
$697$	−85.2334	−0.122286
$698$	1758.31i	2.51907i
$699$	0	0
$700$	1515.73	2.16533
$701$	− 648.629i	− 0.925290i	−0.886544	−	0.462645i	$-0.846900\pi$
$701$	− 648.629i	− 0.925290i	0.886544	−	0.462645i	$-0.153100\pi$
$702$	0	0
$703$	105.692	0.150344
$704$	1025.72i	1.45698i
$705$	0	0
$706$	927.437	1.31365
$707$	831.327i	1.17585i
$708$	0	0
$709$	−733.868	−1.03508	−0.517538	−	0.855661i	$-0.673151\pi$
$709$	−733.868	−1.03508	−0.517538	+	0.855661i	$0.673151\pi$
$710$	− 71.4908i	− 0.100691i
$711$	0	0
$712$	−617.418	−0.867160
$713$	912.224i	1.27942i
$714$	0	0
$715$	−110.245	−0.154188
$716$	1881.82i	2.62825i
$717$	0	0
$718$	1340.99	1.86768
$719$	847.039i	1.17808i	0.808104	+	0.589039i	$0.200494\pi$
$719$	847.039i	1.17808i	−0.808104	+	0.589039i	$0.799506\pi$
$720$	0	0
$721$	887.442	1.23085
$722$	789.918i	1.09407i
$723$	0	0
$724$	−304.792	−0.420983
$725$	712.764i	0.983123i
$726$	0	0
$727$	896.272	1.23284	0.616418	−	0.787419i	$-0.288583\pi$
$727$	896.272	1.23284	0.616418	+	0.787419i	$0.288583\pi$
$728$	− 803.210i	− 1.10331i
$729$	0	0
$730$	45.9120	0.0628931
$731$	− 255.575i	− 0.349624i
$732$	0	0
$733$	−583.413	−0.795924	−0.397962	−	0.917402i	$-0.630282\pi$
$733$	−583.413	−0.795924	−0.397962	+	0.917402i	$0.630282\pi$
$734$	748.656i	1.01997i
$735$	0	0
$736$	−721.417	−0.980186
$737$	− 1251.13i	− 1.69760i
$738$	0	0
$739$	460.901	0.623682	0.311841	−	0.950134i	$-0.399054\pi$
$739$	460.901	0.623682	0.311841	+	0.950134i	$0.399054\pi$
$740$	53.0014i	0.0716235i
$741$	0	0
$742$	−1970.15	−2.65519
$743$	− 678.005i	− 0.912524i	−0.889846	−	0.456262i	$-0.849188\pi$
$743$	− 678.005i	− 0.912524i	0.889846	−	0.456262i	$-0.150812\pi$
$744$	0	0
$745$	−199.378	−0.267621
$746$	140.172i	0.187899i
$747$	0	0
$748$	241.765	0.323215
$749$	− 1786.24i	− 2.38483i
$750$	0	0
$751$	−770.775	−1.02633	−0.513166	−	0.858290i	$-0.671527\pi$
$751$	−770.775	−1.02633	−0.513166	+	0.858290i	$0.671527\pi$
$752$	− 12.9604i	− 0.0172346i
$753$	0	0
$754$	1162.56	1.54185
$755$	137.968i	0.182739i
$756$	0	0
$757$	783.588	1.03512	0.517561	−	0.855646i	$-0.326840\pi$
$757$	783.588	1.03512	0.517561	+	0.855646i	$0.326840\pi$
$758$	− 1103.79i	− 1.45619i
$759$	0	0
$760$	56.5953	0.0744675
$761$	233.162i	0.306389i	0.988196	+	0.153195i	$0.0489561\pi$
$761$	233.162i	0.306389i	−0.988196	+	0.153195i	$0.951044\pi$
$762$	0	0
$763$	646.318	0.847075
$764$	− 727.049i	− 0.951634i
$765$	0	0
$766$	1225.25	1.59954
$767$	483.929i	0.630938i
$768$	0	0
$769$	602.730	0.783784	0.391892	−	0.920011i	$-0.371821\pi$
$769$	602.730	0.783784	0.391892	+	0.920011i	$0.371821\pi$
$770$	− 292.793i	− 0.380250i
$771$	0	0
$772$	461.107	0.597288
$773$	807.361i	1.04445i	0.852807	+	0.522226i	$0.174898\pi$
$773$	807.361i	1.04445i	−0.852807	+	0.522226i	$0.825102\pi$
$774$	0	0
$775$	1185.73	1.52998
$776$	− 355.897i	− 0.458630i
$777$	0	0
$778$	−458.425	−0.589235
$779$	− 217.071i	− 0.278653i
$780$	0	0
$781$	252.693	0.323551
$782$	241.936i	0.309381i
$783$	0	0
$784$	−287.871	−0.367183
$785$	− 61.8456i	− 0.0787842i
$786$	0	0
$787$	−223.724	−0.284275	−0.142137	−	0.989847i	$-0.545397\pi$
$787$	−223.724	−0.284275	−0.142137	+	0.989847i	$0.545397\pi$
$788$	− 564.165i	− 0.715945i
$789$	0	0
$790$	−162.209	−0.205328
$791$	− 35.6383i	− 0.0450548i
$792$	0	0
$793$	178.562	0.225173
$794$	1838.11i	2.31500i
$795$	0	0
$796$	1693.02	2.12691
$797$	1159.10i	1.45433i	0.686463	+	0.727165i	$0.259163\pi$
$797$	1159.10i	1.45433i	−0.686463	+	0.727165i	$0.740837\pi$
$798$	0	0
$799$	−11.6305	−0.0145564
$800$	937.717i	1.17215i
$801$	0	0
$802$	326.336	0.406903
$803$	162.282i	0.202094i
$804$	0	0
$805$	174.914	0.217284
$806$	− 1934.00i	− 2.39950i
$807$	0	0
$808$	477.115	0.590489
$809$	− 274.817i	− 0.339700i	−0.985470	−	0.169850i	$-0.945672\pi$
$809$	− 274.817i	− 0.339700i	0.985470	−	0.169850i	$-0.0543283\pi$
$810$	0	0
$811$	−264.647	−0.326322	−0.163161	−	0.986599i	$-0.552169\pi$
$811$	−264.647	−0.326322	−0.163161	+	0.986599i	$0.552169\pi$
$812$	1843.20i	2.26996i
$813$	0	0
$814$	−313.815	−0.385522
$815$	263.033i	0.322740i
$816$	0	0
$817$	650.894	0.796687
$818$	379.919i	0.464449i
$819$	0	0
$820$	108.854	0.132749
$821$	− 129.094i	− 0.157239i	−0.996905	−	0.0786197i	$-0.974949\pi$
$821$	− 129.094i	− 0.157239i	0.996905	−	0.0786197i	$-0.0250513\pi$
$822$	0	0
$823$	−476.112	−0.578507	−0.289254	−	0.957252i	$-0.593407\pi$
$823$	−476.112	−0.578507	−0.289254	+	0.957252i	$0.593407\pi$
$824$	− 509.320i	− 0.618107i
$825$	0	0
$826$	−1285.24	−1.55598
$827$	576.036i	0.696537i	0.937395	+	0.348268i	$0.113230\pi$
$827$	576.036i	0.696537i	−0.937395	+	0.348268i	$0.886770\pi$
$828$	0	0
$829$	931.225	1.12331	0.561655	−	0.827371i	$-0.310165\pi$
$829$	931.225	1.12331	0.561655	+	0.827371i	$0.310165\pi$
$830$	− 346.440i	− 0.417398i
$831$	0	0
$832$	1299.11	1.56143
$833$	258.332i	0.310122i
$834$	0	0
$835$	−219.802	−0.263236
$836$	615.723i	0.736510i
$837$	0	0
$838$	1846.05	2.20292
$839$	324.575i	0.386859i	0.981114	+	0.193430i	$0.0619611\pi$
$839$	324.575i	0.386859i	−0.981114	+	0.193430i	$0.938039\pi$
$840$	0	0
$841$	−25.7558	−0.0306252
$842$	52.2252i	0.0620252i
$843$	0	0
$844$	1666.95	1.97506
$845$	− 10.5677i	− 0.0125061i
$846$	0	0
$847$	−243.656	−0.287669
$848$	− 271.920i	− 0.320660i
$849$	0	0
$850$	314.475	0.369971
$851$	− 187.473i	− 0.220297i
$852$	0	0
$853$	14.1695	0.0166114	0.00830569	−	0.999966i	$-0.497356\pi$
$853$	14.1695	0.0166114	0.00830569	+	0.999966i	$0.497356\pi$
$854$	474.234i	0.555309i
$855$	0	0
$856$	−1025.16	−1.19762
$857$	915.266i	1.06799i	0.845488	+	0.533994i	$0.179310\pi$
$857$	915.266i	1.06799i	−0.845488	+	0.533994i	$0.820690\pi$
$858$	0	0
$859$	−1414.02	−1.64612	−0.823060	−	0.567954i	$-0.807735\pi$
$859$	−1414.02	−1.64612	−0.823060	+	0.567954i	$0.807735\pi$
$860$	326.404i	0.379539i
$861$	0	0
$862$	2277.36	2.64195
$863$	900.463i	1.04341i	0.853126	+	0.521705i	$0.174704\pi$
$863$	900.463i	1.04341i	−0.853126	+	0.521705i	$0.825296\pi$
$864$	0	0
$865$	239.905	0.277346
$866$	− 1763.16i	− 2.03598i
$867$	0	0
$868$	3066.30	3.53261
$869$	− 573.347i	− 0.659778i
$870$	0	0
$871$	−1584.61	−1.81930
$872$	− 370.935i	− 0.425384i
$873$	0	0
$874$	−616.158	−0.704987
$875$	− 462.133i	− 0.528153i
$876$	0	0
$877$	−1192.09	−1.35928	−0.679639	−	0.733547i	$-0.737863\pi$
$877$	−1192.09	−1.35928	−0.679639	+	0.733547i	$0.737863\pi$
$878$	936.017i	1.06608i
$879$	0	0
$880$	40.4112	0.0459218
$881$	− 1088.01i	− 1.23497i	−0.786582	−	0.617486i	$-0.788151\pi$
$881$	− 1088.01i	− 1.23497i	0.786582	−	0.617486i	$-0.211849\pi$
$882$	0	0
$883$	354.975	0.402011	0.201005	−	0.979590i	$-0.435579\pi$
$883$	354.975	0.402011	0.201005	+	0.979590i	$0.435579\pi$
$884$	− 306.205i	− 0.346386i
$885$	0	0
$886$	−1893.59	−2.13723
$887$	648.965i	0.731640i	0.930686	+	0.365820i	$0.119211\pi$
$887$	648.965i	0.731640i	−0.930686	+	0.365820i	$0.880789\pi$
$888$	0	0
$889$	1630.07	1.83360
$890$	285.055i	0.320287i
$891$	0	0
$892$	−464.392	−0.520619
$893$	− 29.6204i	− 0.0331695i
$894$	0	0
$895$	282.271	0.315387
$896$	1813.13i	2.02359i
$897$	0	0
$898$	153.061	0.170447
$899$	1441.91i	1.60391i
$900$	0	0
$901$	−244.017	−0.270829
$902$	644.514i	0.714539i
$903$	0	0
$904$	−20.4535	−0.0226256
$905$	45.7184i	0.0505176i
$906$	0	0
$907$	1386.46	1.52862	0.764308	−	0.644851i	$-0.223081\pi$
$907$	1386.46	1.52862	0.764308	+	0.644851i	$0.223081\pi$
$908$	− 897.651i	− 0.988602i
$909$	0	0
$910$	−370.833	−0.407509
$911$	− 1187.33i	− 1.30333i	−0.758507	−	0.651665i	$-0.774071\pi$
$911$	− 1187.33i	− 1.30333i	0.758507	−	0.651665i	$-0.225929\pi$
$912$	0	0
$913$	1224.54	1.34122
$914$	− 1904.59i	− 2.08380i
$915$	0	0
$916$	−1373.42	−1.49936
$917$	− 2057.47i	− 2.24370i
$918$	0	0
$919$	−1607.55	−1.74924	−0.874619	−	0.484812i	$-0.838888\pi$
$919$	−1607.55	−1.74924	−0.874619	+	0.484812i	$0.838888\pi$
$920$	− 100.387i	− 0.109116i
$921$	0	0
$922$	7.77771	0.00843569
$923$	− 320.046i	− 0.346745i
$924$	0	0
$925$	−243.682	−0.263440
$926$	− 69.8636i	− 0.0754467i
$927$	0	0
$928$	−1140.31	−1.22878
$929$	407.142i	0.438258i	0.975696	+	0.219129i	$0.0703216\pi$
$929$	407.142i	0.438258i	−0.975696	+	0.219129i	$0.929678\pi$
$930$	0	0
$931$	−657.915	−0.706675
$932$	1616.92i	1.73489i
$933$	0	0
$934$	−1826.91	−1.95601
$935$	− 36.2645i	− 0.0387855i
$936$	0	0
$937$	459.581	0.490482	0.245241	−	0.969462i	$-0.421133\pi$
$937$	459.581	0.490482	0.245241	+	0.969462i	$0.421133\pi$
$938$	− 4208.47i	− 4.48664i
$939$	0	0
$940$	14.8537	0.0158018
$941$	− 163.667i	− 0.173929i	−0.996211	−	0.0869645i	$-0.972283\pi$
$941$	− 163.667i	− 0.173929i	0.996211	−	0.0869645i	$-0.0277167\pi$
$942$	0	0
$943$	−385.032	−0.408305
$944$	− 177.389i	− 0.187912i
$945$	0	0
$946$	−1932.60	−2.04291
$947$	125.473i	0.132495i	0.997803	+	0.0662476i	$0.0211027\pi$
$947$	125.473i	0.132495i	−0.997803	+	0.0662476i	$0.978897\pi$
$948$	0	0
$949$	205.536	0.216582
$950$	800.900i	0.843052i
$951$	0	0
$952$	264.212	0.277534
$953$	295.864i	0.310456i	0.987879	+	0.155228i	$0.0496112\pi$
$953$	295.864i	0.310456i	−0.987879	+	0.155228i	$0.950389\pi$
$954$	0	0
$955$	−109.056	−0.114195
$956$	692.836i	0.724724i
$957$	0	0
$958$	−2364.36	−2.46802
$959$	− 1782.67i	− 1.85889i
$960$	0	0
$961$	1437.72	1.49607
$962$	397.459i	0.413159i
$963$	0	0
$964$	−2522.46	−2.61666
$965$	− 69.1654i	− 0.0716740i
$966$	0	0
$967$	671.822	0.694748	0.347374	−	0.937727i	$-0.387073\pi$
$967$	671.822	0.694748	0.347374	+	0.937727i	$0.387073\pi$
$968$	139.839i	0.144462i
$969$	0	0
$970$	−164.313	−0.169395
$971$	375.020i	0.386221i	0.981177	+	0.193110i	$0.0618575\pi$
$971$	375.020i	0.386221i	−0.981177	+	0.193110i	$0.938142\pi$
$972$	0	0
$973$	−1633.58	−1.67891
$974$	− 1625.22i	− 1.66861i
$975$	0	0
$976$	−65.4537	−0.0670632
$977$	− 653.608i	− 0.668995i	−0.942397	−	0.334497i	$-0.891434\pi$
$977$	− 653.608i	− 0.668995i	0.942397	−	0.334497i	$-0.108566\pi$
$978$	0	0
$979$	−1007.56	−1.02918
$980$	− 329.924i	− 0.336658i
$981$	0	0
$982$	1710.48	1.74183
$983$	820.263i	0.834448i	0.908804	+	0.417224i	$0.136997\pi$
$983$	820.263i	0.834448i	−0.908804	+	0.417224i	$0.863003\pi$
$984$	0	0
$985$	−84.6240	−0.0859127
$986$	382.417i	0.387847i
$987$	0	0
$988$	779.836	0.789308
$989$	− 1154.53i	− 1.16737i
$990$	0	0
$991$	1531.08	1.54498	0.772491	−	0.635026i	$-0.219011\pi$
$991$	1531.08	1.54498	0.772491	+	0.635026i	$0.219011\pi$
$992$	1896.99i	1.91229i
$993$	0	0
$994$	849.991	0.855121
$995$	− 253.951i	− 0.255227i
$996$	0	0
$997$	−178.236	−0.178772	−0.0893860	−	0.995997i	$-0.528490\pi$
$997$	−178.236	−0.178772	−0.0893860	+	0.995997i	$0.528490\pi$
$998$	− 2720.54i	− 2.72599i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	459.3.b.d.188.2	✓	12
3.2	odd	2	inner	459.3.b.d.188.11	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
459.3.b.d.188.2	✓	12	1.1	even	1	trivial
459.3.b.d.188.11	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)

\(f(q)\)	\(=\)	\(q-3.15039i q^{2} -5.92497 q^{4} +0.888739i q^{5} -10.5667 q^{7} +6.06443i q^{8} +2.79988 q^{10} +9.89652i q^{11} +12.5343 q^{13} +33.2892i q^{14} -4.59457 q^{16} +4.12311i q^{17} -10.5006 q^{19} -5.26576i q^{20} +31.1779 q^{22} +18.6257i q^{23} +24.2101 q^{25} -39.4881i q^{26} +62.6073 q^{28} +29.4407i q^{29} +48.9768 q^{31} +38.7324i q^{32} +12.9894 q^{34} -9.39102i q^{35} -10.0653 q^{37} +33.0812i q^{38} -5.38969 q^{40} +20.6721i q^{41} -61.9861 q^{43} -58.6366i q^{44} +58.6781 q^{46} +2.82082i q^{47} +62.6547 q^{49} -76.2715i q^{50} -74.2656 q^{52} +59.1829i q^{53} -8.79543 q^{55} -64.0809i q^{56} +92.7498 q^{58} +38.6083i q^{59} +14.2459 q^{61} -154.296i q^{62} +103.644 q^{64} +11.1397i q^{65} -126.421 q^{67} -24.4293i q^{68} -29.5854 q^{70} -25.5335i q^{71} +16.3979 q^{73} +31.7096i q^{74} +62.2161 q^{76} -104.573i q^{77} -57.9343 q^{79} -4.08338i q^{80} +65.1253 q^{82} -123.734i q^{83} -3.66437 q^{85} +195.280i q^{86} -60.0167 q^{88} +101.810i q^{89} -132.446 q^{91} -110.357i q^{92} +8.88668 q^{94} -9.33233i q^{95} -58.6859 q^{97} -197.387i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 20 q^{4} - 54 q^{7} - 12 q^{10} - 50 q^{13} + 108 q^{16} - 50 q^{19} - 106 q^{22} - 48 q^{25} + 382 q^{28} + 2 q^{31} - 244 q^{37} + 208 q^{40} - 78 q^{43} - 18 q^{46} + 470 q^{49} + 42 q^{52} + 290 q^{55}+ \cdots + 66 q^{97}+O(q^{100}) \) 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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