LMFDB - Embedded newform 460.4.a.b.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [460,4,Mod(1,460)] 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("460.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(460, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 460 = 2^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	460.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$27.1408786026$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 80x^{3} + 121x^{2} + 1212x + 1044 $ x^5 - 2x^4 - 80x^3 + 121x^2 + 1212x + 1044
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$6.72467$ of defining polynomial
Character	$\chi$	$=$	460.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.72467 q^{3} -5.00000 q^{5} -12.6137 q^{7} +5.77185 q^{9} -21.9786 q^{11} +37.2893 q^{13} -28.6233 q^{15} +62.5817 q^{17} -137.633 q^{19} -72.2093 q^{21} -23.0000 q^{23} +25.0000 q^{25} -121.524 q^{27} -71.8975 q^{29} -219.731 q^{31} -125.820 q^{33} +63.0685 q^{35} -296.223 q^{37} +213.469 q^{39} -305.282 q^{41} +140.234 q^{43} -28.8592 q^{45} -185.509 q^{47} -183.895 q^{49} +358.260 q^{51} +312.176 q^{53} +109.893 q^{55} -787.903 q^{57} -161.771 q^{59} +835.780 q^{61} -72.8043 q^{63} -186.447 q^{65} -24.5018 q^{67} -131.667 q^{69} -973.973 q^{71} -343.447 q^{73} +143.117 q^{75} +277.231 q^{77} +1214.62 q^{79} -851.526 q^{81} +749.055 q^{83} -312.908 q^{85} -411.590 q^{87} +811.662 q^{89} -470.356 q^{91} -1257.89 q^{93} +688.165 q^{95} +1026.90 q^{97} -126.857 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{3} - 25 q^{5} + 8 q^{7} + 30 q^{9} + 7 q^{11} + 5 q^{13} + 15 q^{15} - 24 q^{17} - 13 q^{19} - 115 q^{23} + 125 q^{25} - 204 q^{27} - 253 q^{29} - 98 q^{31} - 473 q^{33} - 40 q^{35} - 435 q^{37}+ \cdots + 111 q^{99}+O(q^{100}) $ 5 * q - 3 * q^3 - 25 * q^5 + 8 * q^7 + 30 * q^9 + 7 * q^11 + 5 * q^13 + 15 * q^15 - 24 * q^17 - 13 * q^19 - 115 * q^23 + 125 * q^25 - 204 * q^27 - 253 * q^29 - 98 * q^31 - 473 * q^33 - 40 * q^35 - 435 * q^37 - 410 * q^39 - 774 * q^41 - 498 * q^43 - 150 * q^45 - 572 * q^47 - 683 * q^49 - 657 * q^51 - 665 * q^53 - 35 * q^55 - 932 * q^57 - 763 * q^59 + 337 * q^61 - 1527 * q^63 - 25 * q^65 - 305 * q^67 + 69 * q^69 - 1504 * q^71 - 1304 * q^73 - 75 * q^75 + 182 * q^77 + 626 * q^79 - 959 * q^81 - 1703 * q^83 + 120 * q^85 - 1354 * q^87 + 646 * q^89 - 767 * q^91 - 452 * q^93 + 65 * q^95 - 233 * q^97 + 111 * q^99

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	0
$2$	0	0
$3$	5.72467	1.10171	0.550857	−	0.834600i	$-0.314301\pi$
$3$	5.72467	1.10171	0.550857	+	0.834600i	$0.314301\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−12.6137	−0.681076	−0.340538	−	0.940231i	$-0.610609\pi$
$7$	−12.6137	−0.681076	−0.340538	+	0.940231i	$0.610609\pi$
$8$	0	0
$9$	5.77185	0.213772
$10$	0	0
$11$	−21.9786	−0.602435	−0.301217	−	0.953555i	$-0.597393\pi$
$11$	−21.9786	−0.602435	−0.301217	+	0.953555i	$0.597393\pi$
$12$	0	0
$13$	37.2893	0.795554	0.397777	−	0.917482i	$-0.369782\pi$
$13$	37.2893	0.795554	0.397777	+	0.917482i	$0.369782\pi$
$14$	0	0
$15$	−28.6233	−0.492701
$16$	0	0
$17$	62.5817	0.892841	0.446420	−	0.894823i	$-0.352699\pi$
$17$	62.5817	0.892841	0.446420	+	0.894823i	$0.352699\pi$
$18$	0	0
$19$	−137.633	−1.66185	−0.830925	−	0.556384i	$-0.812188\pi$
$19$	−137.633	−1.66185	−0.830925	+	0.556384i	$0.812188\pi$
$20$	0	0
$21$	−72.2093	−0.750350
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−121.524	−0.866198
$28$	0	0
$29$	−71.8975	−0.460380	−0.230190	−	0.973146i	$-0.573935\pi$
$29$	−71.8975	−0.460380	−0.230190	+	0.973146i	$0.573935\pi$
$30$	0	0
$31$	−219.731	−1.27306	−0.636531	−	0.771251i	$-0.719631\pi$
$31$	−219.731	−1.27306	−0.636531	+	0.771251i	$0.719631\pi$
$32$	0	0
$33$	−125.820	−0.663710
$34$	0	0
$35$	63.0685	0.304586
$36$	0	0
$37$	−296.223	−1.31618	−0.658091	−	0.752939i	$-0.728636\pi$
$37$	−296.223	−1.31618	−0.658091	+	0.752939i	$0.728636\pi$
$38$	0	0
$39$	213.469	0.876473
$40$	0	0
$41$	−305.282	−1.16285	−0.581427	−	0.813599i	$-0.697505\pi$
$41$	−305.282	−1.16285	−0.581427	+	0.813599i	$0.697505\pi$
$42$	0	0
$43$	140.234	0.497338	0.248669	−	0.968588i	$-0.420007\pi$
$43$	140.234	0.497338	0.248669	+	0.968588i	$0.420007\pi$
$44$	0	0
$45$	−28.8592	−0.0956018
$46$	0	0
$47$	−185.509	−0.575730	−0.287865	−	0.957671i	$-0.592945\pi$
$47$	−185.509	−0.575730	−0.287865	+	0.957671i	$0.592945\pi$
$48$	0	0
$49$	−183.895	−0.536136
$50$	0	0
$51$	358.260	0.983654
$52$	0	0
$53$	312.176	0.809069	0.404534	−	0.914523i	$-0.367434\pi$
$53$	312.176	0.809069	0.404534	+	0.914523i	$0.367434\pi$
$54$	0	0
$55$	109.893	0.269417
$56$	0	0
$57$	−787.903	−1.83088
$58$	0	0
$59$	−161.771	−0.356964	−0.178482	−	0.983943i	$-0.557119\pi$
$59$	−161.771	−0.356964	−0.178482	+	0.983943i	$0.557119\pi$
$60$	0	0
$61$	835.780	1.75427	0.877137	−	0.480240i	$-0.159450\pi$
$61$	835.780	1.75427	0.877137	+	0.480240i	$0.159450\pi$
$62$	0	0
$63$	−72.8043	−0.145595
$64$	0	0
$65$	−186.447	−0.355783
$66$	0	0
$67$	−24.5018	−0.0446772	−0.0223386	−	0.999750i	$-0.507111\pi$
$67$	−24.5018	−0.0446772	−0.0223386	+	0.999750i	$0.507111\pi$
$68$	0	0
$69$	−131.667	−0.229723
$70$	0	0
$71$	−973.973	−1.62802	−0.814009	−	0.580852i	$-0.802719\pi$
$71$	−973.973	−1.62802	−0.814009	+	0.580852i	$0.802719\pi$
$72$	0	0
$73$	−343.447	−0.550650	−0.275325	−	0.961351i	$-0.588786\pi$
$73$	−343.447	−0.550650	−0.275325	+	0.961351i	$0.588786\pi$
$74$	0	0
$75$	143.117	0.220343
$76$	0	0
$77$	277.231	0.410304
$78$	0	0
$79$	1214.62	1.72981	0.864905	−	0.501935i	$-0.167378\pi$
$79$	1214.62	1.72981	0.864905	+	0.501935i	$0.167378\pi$
$80$	0	0
$81$	−851.526	−1.16807
$82$	0	0
$83$	749.055	0.990597	0.495298	−	0.868723i	$-0.335059\pi$
$83$	749.055	0.990597	0.495298	+	0.868723i	$0.335059\pi$
$84$	0	0
$85$	−312.908	−0.399290
$86$	0	0
$87$	−411.590	−0.507207
$88$	0	0
$89$	811.662	0.966696	0.483348	−	0.875428i	$-0.339421\pi$
$89$	811.662	0.966696	0.483348	+	0.875428i	$0.339421\pi$
$90$	0	0
$91$	−470.356	−0.541833
$92$	0	0
$93$	−1257.89	−1.40255
$94$	0	0
$95$	688.165	0.743202
$96$	0	0
$97$	1026.90	1.07491	0.537455	−	0.843293i	$-0.319386\pi$
$97$	1026.90	1.07491	0.537455	+	0.843293i	$0.319386\pi$
$98$	0	0
$99$	−126.857	−0.128784
$100$	0	0
$101$	−788.330	−0.776651	−0.388326	−	0.921522i	$-0.626946\pi$
$101$	−788.330	−0.776651	−0.388326	+	0.921522i	$0.626946\pi$
$102$	0	0
$103$	471.764	0.451304	0.225652	−	0.974208i	$-0.427549\pi$
$103$	471.764	0.451304	0.225652	+	0.974208i	$0.427549\pi$
$104$	0	0
$105$	361.046	0.335567
$106$	0	0
$107$	−1402.85	−1.26746	−0.633732	−	0.773553i	$-0.718478\pi$
$107$	−1402.85	−1.26746	−0.633732	+	0.773553i	$0.718478\pi$
$108$	0	0
$109$	−2151.31	−1.89044	−0.945221	−	0.326432i	$-0.894154\pi$
$109$	−2151.31	−1.89044	−0.945221	+	0.326432i	$0.894154\pi$
$110$	0	0
$111$	−1695.78	−1.45005
$112$	0	0
$113$	754.833	0.628396	0.314198	−	0.949358i	$-0.398264\pi$
$113$	754.833	0.628396	0.314198	+	0.949358i	$0.398264\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	215.228	0.170067
$118$	0	0
$119$	−789.387	−0.608092
$120$	0	0
$121$	−847.943	−0.637072
$122$	0	0
$123$	−1747.64	−1.28113
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	2161.28	1.51010	0.755050	−	0.655667i	$-0.227612\pi$
$127$	2161.28	1.51010	0.755050	+	0.655667i	$0.227612\pi$
$128$	0	0
$129$	802.796	0.547924
$130$	0	0
$131$	397.736	0.265270	0.132635	−	0.991165i	$-0.457656\pi$
$131$	397.736	0.265270	0.132635	+	0.991165i	$0.457656\pi$
$132$	0	0
$133$	1736.06	1.13185
$134$	0	0
$135$	607.621	0.387375
$136$	0	0
$137$	−1660.73	−1.03566	−0.517831	−	0.855483i	$-0.673260\pi$
$137$	−1660.73	−1.03566	−0.517831	+	0.855483i	$0.673260\pi$
$138$	0	0
$139$	−997.853	−0.608897	−0.304449	−	0.952529i	$-0.598472\pi$
$139$	−997.853	−0.608897	−0.304449	+	0.952529i	$0.598472\pi$
$140$	0	0
$141$	−1061.98	−0.634289
$142$	0	0
$143$	−819.566	−0.479270
$144$	0	0
$145$	359.488	0.205888
$146$	0	0
$147$	−1052.74	−0.590668
$148$	0	0
$149$	3014.50	1.65743	0.828715	−	0.559671i	$-0.189072\pi$
$149$	3014.50	1.65743	0.828715	+	0.559671i	$0.189072\pi$
$150$	0	0
$151$	−684.908	−0.369120	−0.184560	−	0.982821i	$-0.559086\pi$
$151$	−684.908	−0.369120	−0.184560	+	0.982821i	$0.559086\pi$
$152$	0	0
$153$	361.212	0.190864
$154$	0	0
$155$	1098.66	0.569330
$156$	0	0
$157$	727.598	0.369864	0.184932	−	0.982751i	$-0.440794\pi$
$157$	727.598	0.369864	0.184932	+	0.982751i	$0.440794\pi$
$158$	0	0
$159$	1787.10	0.891362
$160$	0	0
$161$	290.115	0.142014
$162$	0	0
$163$	−768.718	−0.369390	−0.184695	−	0.982796i	$-0.559130\pi$
$163$	−768.718	−0.369390	−0.184695	+	0.982796i	$0.559130\pi$
$164$	0	0
$165$	629.100	0.296820
$166$	0	0
$167$	−243.397	−0.112782	−0.0563910	−	0.998409i	$-0.517959\pi$
$167$	−243.397	−0.112782	−0.0563910	+	0.998409i	$0.517959\pi$
$168$	0	0
$169$	−806.505	−0.367094
$170$	0	0
$171$	−794.396	−0.355257
$172$	0	0
$173$	−201.538	−0.0885702	−0.0442851	−	0.999019i	$-0.514101\pi$
$173$	−201.538	−0.0885702	−0.0442851	+	0.999019i	$0.514101\pi$
$174$	0	0
$175$	−315.342	−0.136215
$176$	0	0
$177$	−926.088	−0.393272
$178$	0	0
$179$	−722.564	−0.301715	−0.150857	−	0.988556i	$-0.548203\pi$
$179$	−722.564	−0.301715	−0.150857	+	0.988556i	$0.548203\pi$
$180$	0	0
$181$	2117.76	0.869678	0.434839	−	0.900508i	$-0.356805\pi$
$181$	2117.76	0.869678	0.434839	+	0.900508i	$0.356805\pi$
$182$	0	0
$183$	4784.57	1.93271
$184$	0	0
$185$	1481.11	0.588614
$186$	0	0
$187$	−1375.46	−0.537878
$188$	0	0
$189$	1532.87	0.589946
$190$	0	0
$191$	82.9624	0.0314290	0.0157145	−	0.999877i	$-0.494998\pi$
$191$	82.9624	0.0314290	0.0157145	+	0.999877i	$0.494998\pi$
$192$	0	0
$193$	1350.43	0.503657	0.251829	−	0.967772i	$-0.418968\pi$
$193$	1350.43	0.503657	0.251829	+	0.967772i	$0.418968\pi$
$194$	0	0
$195$	−1067.35	−0.391970
$196$	0	0
$197$	4317.69	1.56154	0.780769	−	0.624820i	$-0.214828\pi$
$197$	4317.69	1.56154	0.780769	+	0.624820i	$0.214828\pi$
$198$	0	0
$199$	3026.27	1.07802	0.539011	−	0.842299i	$-0.318798\pi$
$199$	3026.27	1.07802	0.539011	+	0.842299i	$0.318798\pi$
$200$	0	0
$201$	−140.265	−0.0492215
$202$	0	0
$203$	906.893	0.313554
$204$	0	0
$205$	1526.41	0.520044
$206$	0	0
$207$	−132.752	−0.0445746
$208$	0	0
$209$	3024.97	1.00116
$210$	0	0
$211$	2292.01	0.747812	0.373906	−	0.927467i	$-0.378018\pi$
$211$	2292.01	0.747812	0.373906	+	0.927467i	$0.378018\pi$
$212$	0	0
$213$	−5575.67	−1.79361
$214$	0	0
$215$	−701.172	−0.222417
$216$	0	0
$217$	2771.62	0.867051
$218$	0	0
$219$	−1966.12	−0.606659
$220$	0	0
$221$	2333.63	0.710303
$222$	0	0
$223$	−2997.29	−0.900061	−0.450030	−	0.893013i	$-0.648587\pi$
$223$	−2997.29	−0.900061	−0.450030	+	0.893013i	$0.648587\pi$
$224$	0	0
$225$	144.296	0.0427544
$226$	0	0
$227$	−5786.16	−1.69181	−0.845905	−	0.533333i	$-0.820939\pi$
$227$	−5786.16	−1.69181	−0.845905	+	0.533333i	$0.820939\pi$
$228$	0	0
$229$	4516.35	1.30327	0.651635	−	0.758532i	$-0.274083\pi$
$229$	4516.35	1.30327	0.651635	+	0.758532i	$0.274083\pi$
$230$	0	0
$231$	1587.05	0.452037
$232$	0	0
$233$	2990.75	0.840903	0.420452	−	0.907315i	$-0.361872\pi$
$233$	2990.75	0.840903	0.420452	+	0.907315i	$0.361872\pi$
$234$	0	0
$235$	927.546	0.257474
$236$	0	0
$237$	6953.28	1.90575
$238$	0	0
$239$	4043.74	1.09443	0.547213	−	0.836994i	$-0.315689\pi$
$239$	4043.74	1.09443	0.547213	+	0.836994i	$0.315689\pi$
$240$	0	0
$241$	3653.64	0.976564	0.488282	−	0.872686i	$-0.337624\pi$
$241$	3653.64	0.976564	0.488282	+	0.872686i	$0.337624\pi$
$242$	0	0
$243$	−1593.55	−0.420684
$244$	0	0
$245$	919.473	0.239767
$246$	0	0
$247$	−5132.24	−1.32209
$248$	0	0
$249$	4288.09	1.09135
$250$	0	0
$251$	−6799.45	−1.70987	−0.854935	−	0.518735i	$-0.826403\pi$
$251$	−6799.45	−1.70987	−0.854935	+	0.518735i	$0.826403\pi$
$252$	0	0
$253$	505.507	0.125616
$254$	0	0
$255$	−1791.30	−0.439904
$256$	0	0
$257$	179.925	0.0436710	0.0218355	−	0.999762i	$-0.493049\pi$
$257$	179.925	0.0436710	0.0218355	+	0.999762i	$0.493049\pi$
$258$	0	0
$259$	3736.46	0.896419
$260$	0	0
$261$	−414.981	−0.0984165
$262$	0	0
$263$	−1148.48	−0.269271	−0.134635	−	0.990895i	$-0.542986\pi$
$263$	−1148.48	−0.269271	−0.134635	+	0.990895i	$0.542986\pi$
$264$	0	0
$265$	−1560.88	−0.361827
$266$	0	0
$267$	4646.50	1.06502
$268$	0	0
$269$	2999.90	0.679953	0.339976	−	0.940434i	$-0.389581\pi$
$269$	2999.90	0.679953	0.339976	+	0.940434i	$0.389581\pi$
$270$	0	0
$271$	5587.83	1.25253	0.626266	−	0.779609i	$-0.284582\pi$
$271$	5587.83	1.25253	0.626266	+	0.779609i	$0.284582\pi$
$272$	0	0
$273$	−2692.64	−0.596944
$274$	0	0
$275$	−549.464	−0.120487
$276$	0	0
$277$	6219.06	1.34898	0.674489	−	0.738285i	$-0.264364\pi$
$277$	6219.06	1.34898	0.674489	+	0.738285i	$0.264364\pi$
$278$	0	0
$279$	−1268.26	−0.272145
$280$	0	0
$281$	−6770.91	−1.43743	−0.718717	−	0.695303i	$-0.755270\pi$
$281$	−6770.91	−1.43743	−0.718717	+	0.695303i	$0.755270\pi$
$282$	0	0
$283$	−7811.42	−1.64078	−0.820390	−	0.571804i	$-0.806244\pi$
$283$	−7811.42	−1.64078	−0.820390	+	0.571804i	$0.806244\pi$
$284$	0	0
$285$	3939.52	0.818796
$286$	0	0
$287$	3850.73	0.791991
$288$	0	0
$289$	−996.531	−0.202836
$290$	0	0
$291$	5878.68	1.18424
$292$	0	0
$293$	−8415.02	−1.67785	−0.838926	−	0.544246i	$-0.816816\pi$
$293$	−8415.02	−1.67785	−0.838926	+	0.544246i	$0.816816\pi$
$294$	0	0
$295$	808.857	0.159639
$296$	0	0
$297$	2670.93	0.521828
$298$	0	0
$299$	−857.655	−0.165884
$300$	0	0
$301$	−1768.87	−0.338725
$302$	0	0
$303$	−4512.93	−0.855647
$304$	0	0
$305$	−4178.90	−0.784535
$306$	0	0
$307$	−9633.23	−1.79087	−0.895435	−	0.445192i	$-0.853135\pi$
$307$	−9633.23	−1.79087	−0.895435	+	0.445192i	$0.853135\pi$
$308$	0	0
$309$	2700.69	0.497207
$310$	0	0
$311$	−7000.64	−1.27643	−0.638216	−	0.769858i	$-0.720327\pi$
$311$	−7000.64	−1.27643	−0.638216	+	0.769858i	$0.720327\pi$
$312$	0	0
$313$	−2954.79	−0.533593	−0.266796	−	0.963753i	$-0.585965\pi$
$313$	−2954.79	−0.533593	−0.266796	+	0.963753i	$0.585965\pi$
$314$	0	0
$315$	364.022	0.0651120
$316$	0	0
$317$	−4015.68	−0.711492	−0.355746	−	0.934583i	$-0.615773\pi$
$317$	−4015.68	−0.711492	−0.355746	+	0.934583i	$0.615773\pi$
$318$	0	0
$319$	1580.20	0.277349
$320$	0	0
$321$	−8030.85	−1.39638
$322$	0	0
$323$	−8613.30	−1.48377
$324$	0	0
$325$	932.234	0.159111
$326$	0	0
$327$	−12315.5	−2.08272
$328$	0	0
$329$	2339.96	0.392116
$330$	0	0
$331$	596.567	0.0990644	0.0495322	−	0.998773i	$-0.484227\pi$
$331$	596.567	0.0990644	0.0495322	+	0.998773i	$0.484227\pi$
$332$	0	0
$333$	−1709.75	−0.281363
$334$	0	0
$335$	122.509	0.0199802
$336$	0	0
$337$	2162.05	0.349479	0.174739	−	0.984615i	$-0.444092\pi$
$337$	2162.05	0.349479	0.174739	+	0.984615i	$0.444092\pi$
$338$	0	0
$339$	4321.17	0.692312
$340$	0	0
$341$	4829.38	0.766937
$342$	0	0
$343$	6646.09	1.04622
$344$	0	0
$345$	658.337	0.102735
$346$	0	0
$347$	−8014.61	−1.23990	−0.619952	−	0.784639i	$-0.712848\pi$
$347$	−8014.61	−1.23990	−0.619952	+	0.784639i	$0.712848\pi$
$348$	0	0
$349$	3264.77	0.500743	0.250371	−	0.968150i	$-0.419447\pi$
$349$	3264.77	0.500743	0.250371	+	0.968150i	$0.419447\pi$
$350$	0	0
$351$	−4531.56	−0.689107
$352$	0	0
$353$	−6659.27	−1.00407	−0.502036	−	0.864847i	$-0.667416\pi$
$353$	−6659.27	−1.00407	−0.502036	+	0.864847i	$0.667416\pi$
$354$	0	0
$355$	4869.86	0.728072
$356$	0	0
$357$	−4518.98	−0.669943
$358$	0	0
$359$	670.397	0.0985577	0.0492789	−	0.998785i	$-0.484308\pi$
$359$	670.397	0.0985577	0.0492789	+	0.998785i	$0.484308\pi$
$360$	0	0
$361$	12083.8	1.76175
$362$	0	0
$363$	−4854.19	−0.701871
$364$	0	0
$365$	1717.24	0.246258
$366$	0	0
$367$	3528.04	0.501804	0.250902	−	0.968013i	$-0.419273\pi$
$367$	3528.04	0.501804	0.250902	+	0.968013i	$0.419273\pi$
$368$	0	0
$369$	−1762.04	−0.248586
$370$	0	0
$371$	−3937.69	−0.551037
$372$	0	0
$373$	−2718.66	−0.377391	−0.188695	−	0.982036i	$-0.560426\pi$
$373$	−2718.66	−0.377391	−0.188695	+	0.982036i	$0.560426\pi$
$374$	0	0
$375$	−715.584	−0.0985402
$376$	0	0
$377$	−2681.01	−0.366258
$378$	0	0
$379$	10773.8	1.46019	0.730096	−	0.683345i	$-0.239475\pi$
$379$	10773.8	1.46019	0.730096	+	0.683345i	$0.239475\pi$
$380$	0	0
$381$	12372.6	1.66370
$382$	0	0
$383$	−6104.18	−0.814384	−0.407192	−	0.913343i	$-0.633492\pi$
$383$	−6104.18	−0.814384	−0.407192	+	0.913343i	$0.633492\pi$
$384$	0	0
$385$	−1386.15	−0.183493
$386$	0	0
$387$	809.412	0.106317
$388$	0	0
$389$	−5056.94	−0.659118	−0.329559	−	0.944135i	$-0.606900\pi$
$389$	−5056.94	−0.659118	−0.329559	+	0.944135i	$0.606900\pi$
$390$	0	0
$391$	−1439.38	−0.186170
$392$	0	0
$393$	2276.91	0.292251
$394$	0	0
$395$	−6073.08	−0.773595
$396$	0	0
$397$	8320.52	1.05188	0.525938	−	0.850523i	$-0.323714\pi$
$397$	8320.52	1.05188	0.525938	+	0.850523i	$0.323714\pi$
$398$	0	0
$399$	9938.37	1.24697
$400$	0	0
$401$	1253.78	0.156137	0.0780685	−	0.996948i	$-0.475125\pi$
$401$	1253.78	0.156137	0.0780685	+	0.996948i	$0.475125\pi$
$402$	0	0
$403$	−8193.64	−1.01279
$404$	0	0
$405$	4257.63	0.522378
$406$	0	0
$407$	6510.55	0.792914
$408$	0	0
$409$	−11283.0	−1.36408	−0.682041	−	0.731314i	$-0.738907\pi$
$409$	−11283.0	−1.36408	−0.682041	+	0.731314i	$0.738907\pi$
$410$	0	0
$411$	−9507.13	−1.14100
$412$	0	0
$413$	2040.54	0.243119
$414$	0	0
$415$	−3745.28	−0.443008
$416$	0	0
$417$	−5712.38	−0.670830
$418$	0	0
$419$	4178.10	0.487144	0.243572	−	0.969883i	$-0.421681\pi$
$419$	4178.10	0.487144	0.243572	+	0.969883i	$0.421681\pi$
$420$	0	0
$421$	−11693.0	−1.35364	−0.676821	−	0.736147i	$-0.736643\pi$
$421$	−11693.0	−1.35364	−0.676821	+	0.736147i	$0.736643\pi$
$422$	0	0
$423$	−1070.73	−0.123075
$424$	0	0
$425$	1564.54	0.178568
$426$	0	0
$427$	−10542.3	−1.19479
$428$	0	0
$429$	−4691.74	−0.528018
$430$	0	0
$431$	−11158.2	−1.24703	−0.623515	−	0.781812i	$-0.714296\pi$
$431$	−11158.2	−1.24703	−0.623515	+	0.781812i	$0.714296\pi$
$432$	0	0
$433$	−13192.6	−1.46419	−0.732096	−	0.681201i	$-0.761458\pi$
$433$	−13192.6	−1.46419	−0.732096	+	0.681201i	$0.761458\pi$
$434$	0	0
$435$	2057.95	0.226830
$436$	0	0
$437$	3165.56	0.346520
$438$	0	0
$439$	−1659.40	−0.180407	−0.0902037	−	0.995923i	$-0.528752\pi$
$439$	−1659.40	−0.180407	−0.0902037	+	0.995923i	$0.528752\pi$
$440$	0	0
$441$	−1061.41	−0.114611
$442$	0	0
$443$	5144.57	0.551752	0.275876	−	0.961193i	$-0.411032\pi$
$443$	5144.57	0.551752	0.275876	+	0.961193i	$0.411032\pi$
$444$	0	0
$445$	−4058.31	−0.432320
$446$	0	0
$447$	17257.0	1.82601
$448$	0	0
$449$	5469.59	0.574891	0.287446	−	0.957797i	$-0.407194\pi$
$449$	5469.59	0.574891	0.287446	+	0.957797i	$0.407194\pi$
$450$	0	0
$451$	6709.65	0.700544
$452$	0	0
$453$	−3920.87	−0.406664
$454$	0	0
$455$	2351.78	0.242315
$456$	0	0
$457$	6063.04	0.620606	0.310303	−	0.950638i	$-0.399569\pi$
$457$	6063.04	0.620606	0.310303	+	0.950638i	$0.399569\pi$
$458$	0	0
$459$	−7605.19	−0.773377
$460$	0	0
$461$	4025.39	0.406684	0.203342	−	0.979108i	$-0.434820\pi$
$461$	4025.39	0.406684	0.203342	+	0.979108i	$0.434820\pi$
$462$	0	0
$463$	−4617.49	−0.463483	−0.231742	−	0.972777i	$-0.574442\pi$
$463$	−4617.49	−0.463483	−0.231742	+	0.972777i	$0.574442\pi$
$464$	0	0
$465$	6289.45	0.627239
$466$	0	0
$467$	4012.00	0.397544	0.198772	−	0.980046i	$-0.436305\pi$
$467$	4012.00	0.397544	0.198772	+	0.980046i	$0.436305\pi$
$468$	0	0
$469$	309.058	0.0304285
$470$	0	0
$471$	4165.26	0.407484
$472$	0	0
$473$	−3082.15	−0.299614
$474$	0	0
$475$	−3440.82	−0.332370
$476$	0	0
$477$	1801.83	0.172956
$478$	0	0
$479$	−7317.02	−0.697961	−0.348980	−	0.937130i	$-0.613472\pi$
$479$	−7317.02	−0.697961	−0.348980	+	0.937130i	$0.613472\pi$
$480$	0	0
$481$	−11046.0	−1.04709
$482$	0	0
$483$	1660.81	0.156459
$484$	0	0
$485$	−5134.51	−0.480714
$486$	0	0
$487$	−3708.67	−0.345084	−0.172542	−	0.985002i	$-0.555198\pi$
$487$	−3708.67	−0.345084	−0.172542	+	0.985002i	$0.555198\pi$
$488$	0	0
$489$	−4400.66	−0.406962
$490$	0	0
$491$	6689.69	0.614871	0.307435	−	0.951569i	$-0.400529\pi$
$491$	6689.69	0.614871	0.307435	+	0.951569i	$0.400529\pi$
$492$	0	0
$493$	−4499.47	−0.411046
$494$	0	0
$495$	634.284	0.0575938
$496$	0	0
$497$	12285.4	1.10880
$498$	0	0
$499$	17381.9	1.55936	0.779678	−	0.626180i	$-0.215383\pi$
$499$	17381.9	1.55936	0.779678	+	0.626180i	$0.215383\pi$
$500$	0	0
$501$	−1393.37	−0.124253
$502$	0	0
$503$	9771.20	0.866156	0.433078	−	0.901357i	$-0.357428\pi$
$503$	9771.20	0.866156	0.433078	+	0.901357i	$0.357428\pi$
$504$	0	0
$505$	3941.65	0.347329
$506$	0	0
$507$	−4616.97	−0.404432
$508$	0	0
$509$	5820.46	0.506852	0.253426	−	0.967355i	$-0.418443\pi$
$509$	5820.46	0.506852	0.253426	+	0.967355i	$0.418443\pi$
$510$	0	0
$511$	4332.14	0.375035
$512$	0	0
$513$	16725.7	1.43949
$514$	0	0
$515$	−2358.82	−0.201829
$516$	0	0
$517$	4077.22	0.346840
$518$	0	0
$519$	−1153.74	−0.0975789
$520$	0	0
$521$	−6357.64	−0.534613	−0.267307	−	0.963612i	$-0.586134\pi$
$521$	−6357.64	−0.534613	−0.267307	+	0.963612i	$0.586134\pi$
$522$	0	0
$523$	−10556.2	−0.882580	−0.441290	−	0.897364i	$-0.645479\pi$
$523$	−10556.2	−0.882580	−0.441290	+	0.897364i	$0.645479\pi$
$524$	0	0
$525$	−1805.23	−0.150070
$526$	0	0
$527$	−13751.2	−1.13664
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−933.720	−0.0763089
$532$	0	0
$533$	−11383.8	−0.925113
$534$	0	0
$535$	7014.25	0.566827
$536$	0	0
$537$	−4136.44	−0.332403
$538$	0	0
$539$	4041.74	0.322987
$540$	0	0
$541$	−16883.4	−1.34172	−0.670862	−	0.741582i	$-0.734076\pi$
$541$	−16883.4	−1.34172	−0.670862	+	0.741582i	$0.734076\pi$
$542$	0	0
$543$	12123.5	0.958136
$544$	0	0
$545$	10756.5	0.845431
$546$	0	0
$547$	−14318.4	−1.11922	−0.559608	−	0.828757i	$-0.689048\pi$
$547$	−14318.4	−1.11922	−0.559608	+	0.828757i	$0.689048\pi$
$548$	0	0
$549$	4824.00	0.375015
$550$	0	0
$551$	9895.47	0.765084
$552$	0	0
$553$	−15320.8	−1.17813
$554$	0	0
$555$	8478.89	0.648484
$556$	0	0
$557$	−11764.7	−0.894946	−0.447473	−	0.894297i	$-0.647676\pi$
$557$	−11764.7	−0.894946	−0.447473	+	0.894297i	$0.647676\pi$
$558$	0	0
$559$	5229.25	0.395660
$560$	0	0
$561$	−7874.03	−0.592588
$562$	0	0
$563$	−21201.3	−1.58709	−0.793543	−	0.608514i	$-0.791766\pi$
$563$	−21201.3	−1.58709	−0.793543	+	0.608514i	$0.791766\pi$
$564$	0	0
$565$	−3774.17	−0.281027
$566$	0	0
$567$	10740.9	0.795546
$568$	0	0
$569$	−17206.5	−1.26772	−0.633861	−	0.773447i	$-0.718531\pi$
$569$	−17206.5	−1.26772	−0.633861	+	0.773447i	$0.718531\pi$
$570$	0	0
$571$	12208.3	0.894746	0.447373	−	0.894348i	$-0.352360\pi$
$571$	12208.3	0.894746	0.447373	+	0.894348i	$0.352360\pi$
$572$	0	0
$573$	474.932	0.0346258
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	12432.0	0.896967	0.448484	−	0.893791i	$-0.351964\pi$
$577$	12432.0	0.896967	0.448484	+	0.893791i	$0.351964\pi$
$578$	0	0
$579$	7730.75	0.554886
$580$	0	0
$581$	−9448.36	−0.674671
$582$	0	0
$583$	−6861.17	−0.487411
$584$	0	0
$585$	−1076.14	−0.0760564
$586$	0	0
$587$	−18732.7	−1.31718	−0.658588	−	0.752504i	$-0.728846\pi$
$587$	−18732.7	−1.31718	−0.658588	+	0.752504i	$0.728846\pi$
$588$	0	0
$589$	30242.3	2.11564
$590$	0	0
$591$	24717.4	1.72037
$592$	0	0
$593$	16320.1	1.13016	0.565081	−	0.825036i	$-0.308845\pi$
$593$	16320.1	1.13016	0.565081	+	0.825036i	$0.308845\pi$
$594$	0	0
$595$	3946.93	0.271947
$596$	0	0
$597$	17324.4	1.18767
$598$	0	0
$599$	−7015.52	−0.478542	−0.239271	−	0.970953i	$-0.576908\pi$
$599$	−7015.52	−0.478542	−0.239271	+	0.970953i	$0.576908\pi$
$600$	0	0
$601$	−4517.10	−0.306583	−0.153292	−	0.988181i	$-0.548987\pi$
$601$	−4517.10	−0.306583	−0.153292	+	0.988181i	$0.548987\pi$
$602$	0	0
$603$	−141.421	−0.00955074
$604$	0	0
$605$	4239.72	0.284907
$606$	0	0
$607$	−9861.96	−0.659448	−0.329724	−	0.944077i	$-0.606956\pi$
$607$	−9861.96	−0.659448	−0.329724	+	0.944077i	$0.606956\pi$
$608$	0	0
$609$	5191.67	0.345447
$610$	0	0
$611$	−6917.52	−0.458024
$612$	0	0
$613$	20044.7	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$613$	20044.7	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$614$	0	0
$615$	8738.19	0.572939
$616$	0	0
$617$	−3604.26	−0.235174	−0.117587	−	0.993063i	$-0.537516\pi$
$617$	−3604.26	−0.235174	−0.117587	+	0.993063i	$0.537516\pi$
$618$	0	0
$619$	14864.7	0.965204	0.482602	−	0.875840i	$-0.339692\pi$
$619$	14864.7	0.965204	0.482602	+	0.875840i	$0.339692\pi$
$620$	0	0
$621$	2795.06	0.180615
$622$	0	0
$623$	−10238.1	−0.658393
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	17317.0	1.10299
$628$	0	0
$629$	−18538.1	−1.17514
$630$	0	0
$631$	12677.7	0.799828	0.399914	−	0.916553i	$-0.369040\pi$
$631$	12677.7	0.799828	0.399914	+	0.916553i	$0.369040\pi$
$632$	0	0
$633$	13121.0	0.823875
$634$	0	0
$635$	−10806.4	−0.675337
$636$	0	0
$637$	−6857.31	−0.426525
$638$	0	0
$639$	−5621.62	−0.348025
$640$	0	0
$641$	−19168.5	−1.18114	−0.590571	−	0.806986i	$-0.701097\pi$
$641$	−19168.5	−1.18114	−0.590571	+	0.806986i	$0.701097\pi$
$642$	0	0
$643$	31298.6	1.91959	0.959793	−	0.280707i	$-0.0905690\pi$
$643$	31298.6	1.91959	0.959793	+	0.280707i	$0.0905690\pi$
$644$	0	0
$645$	−4013.98	−0.245039
$646$	0	0
$647$	−26340.3	−1.60053	−0.800267	−	0.599643i	$-0.795309\pi$
$647$	−26340.3	−1.60053	−0.800267	+	0.599643i	$0.795309\pi$
$648$	0	0
$649$	3555.50	0.215047
$650$	0	0
$651$	15866.6	0.955242
$652$	0	0
$653$	27113.8	1.62488	0.812439	−	0.583046i	$-0.198139\pi$
$653$	27113.8	1.62488	0.812439	+	0.583046i	$0.198139\pi$
$654$	0	0
$655$	−1988.68	−0.118632
$656$	0	0
$657$	−1982.33	−0.117714
$658$	0	0
$659$	−24125.5	−1.42609	−0.713046	−	0.701117i	$-0.752685\pi$
$659$	−24125.5	−1.42609	−0.713046	+	0.701117i	$0.752685\pi$
$660$	0	0
$661$	1776.69	0.104546	0.0522732	−	0.998633i	$-0.483353\pi$
$661$	1776.69	0.104546	0.0522732	+	0.998633i	$0.483353\pi$
$662$	0	0
$663$	13359.3	0.782550
$664$	0	0
$665$	−8680.30	−0.506177
$666$	0	0
$667$	1653.64	0.0959960
$668$	0	0
$669$	−17158.5	−0.991609
$670$	0	0
$671$	−18369.2	−1.05684
$672$	0	0
$673$	−14137.6	−0.809756	−0.404878	−	0.914371i	$-0.632686\pi$
$673$	−14137.6	−0.809756	−0.404878	+	0.914371i	$0.632686\pi$
$674$	0	0
$675$	−3038.10	−0.173240
$676$	0	0
$677$	−9057.23	−0.514177	−0.257088	−	0.966388i	$-0.582763\pi$
$677$	−9057.23	−0.514177	−0.257088	+	0.966388i	$0.582763\pi$
$678$	0	0
$679$	−12953.0	−0.732094
$680$	0	0
$681$	−33123.9	−1.86389
$682$	0	0
$683$	23206.6	1.30011	0.650054	−	0.759888i	$-0.274746\pi$
$683$	23206.6	1.30011	0.650054	+	0.759888i	$0.274746\pi$
$684$	0	0
$685$	8303.65	0.463162
$686$	0	0
$687$	25854.6	1.43583
$688$	0	0
$689$	11640.8	0.643658
$690$	0	0
$691$	2060.79	0.113453	0.0567265	−	0.998390i	$-0.481934\pi$
$691$	2060.79	0.113453	0.0567265	+	0.998390i	$0.481934\pi$
$692$	0	0
$693$	1600.13	0.0877115
$694$	0	0
$695$	4989.26	0.272307
$696$	0	0
$697$	−19105.0	−1.03824
$698$	0	0
$699$	17121.1	0.926434
$700$	0	0
$701$	4057.66	0.218624	0.109312	−	0.994007i	$-0.465135\pi$
$701$	4057.66	0.218624	0.109312	+	0.994007i	$0.465135\pi$
$702$	0	0
$703$	40770.0	2.18730
$704$	0	0
$705$	5309.90	0.283663
$706$	0	0
$707$	9943.76	0.528958
$708$	0	0
$709$	18777.0	0.994622	0.497311	−	0.867572i	$-0.334321\pi$
$709$	18777.0	0.994622	0.497311	+	0.867572i	$0.334321\pi$
$710$	0	0
$711$	7010.58	0.369785
$712$	0	0
$713$	5053.82	0.265452
$714$	0	0
$715$	4097.83	0.214336
$716$	0	0
$717$	23149.1	1.20574
$718$	0	0
$719$	−2217.63	−0.115026	−0.0575129	−	0.998345i	$-0.518317\pi$
$719$	−2217.63	−0.115026	−0.0575129	+	0.998345i	$0.518317\pi$
$720$	0	0
$721$	−5950.69	−0.307372
$722$	0	0
$723$	20915.9	1.07589
$724$	0	0
$725$	−1797.44	−0.0920761
$726$	0	0
$727$	−23162.5	−1.18164	−0.590818	−	0.806805i	$-0.701195\pi$
$727$	−23162.5	−1.18164	−0.590818	+	0.806805i	$0.701195\pi$
$728$	0	0
$729$	13868.6	0.704600
$730$	0	0
$731$	8776.11	0.444044
$732$	0	0
$733$	11686.4	0.588875	0.294438	−	0.955671i	$-0.404868\pi$
$733$	11686.4	0.588875	0.294438	+	0.955671i	$0.404868\pi$
$734$	0	0
$735$	5263.68	0.264155
$736$	0	0
$737$	538.514	0.0269151
$738$	0	0
$739$	−7408.28	−0.368766	−0.184383	−	0.982854i	$-0.559029\pi$
$739$	−7408.28	−0.368766	−0.184383	+	0.982854i	$0.559029\pi$
$740$	0	0
$741$	−29380.4	−1.45657
$742$	0	0
$743$	−7008.68	−0.346061	−0.173031	−	0.984916i	$-0.555356\pi$
$743$	−7008.68	−0.346061	−0.173031	+	0.984916i	$0.555356\pi$
$744$	0	0
$745$	−15072.5	−0.741225
$746$	0	0
$747$	4323.43	0.211762
$748$	0	0
$749$	17695.1	0.863238
$750$	0	0
$751$	10916.9	0.530443	0.265222	−	0.964187i	$-0.414555\pi$
$751$	10916.9	0.530443	0.265222	+	0.964187i	$0.414555\pi$
$752$	0	0
$753$	−38924.6	−1.88379
$754$	0	0
$755$	3424.54	0.165075
$756$	0	0
$757$	14124.7	0.678165	0.339083	−	0.940757i	$-0.389883\pi$
$757$	14124.7	0.678165	0.339083	+	0.940757i	$0.389883\pi$
$758$	0	0
$759$	2893.86	0.138393
$760$	0	0
$761$	6947.27	0.330931	0.165465	−	0.986216i	$-0.447087\pi$
$761$	6947.27	0.330931	0.165465	+	0.986216i	$0.447087\pi$
$762$	0	0
$763$	27136.0	1.28753
$764$	0	0
$765$	−1806.06	−0.0853572
$766$	0	0
$767$	−6032.35	−0.283984
$768$	0	0
$769$	19409.1	0.910158	0.455079	−	0.890451i	$-0.349611\pi$
$769$	19409.1	0.910158	0.455079	+	0.890451i	$0.349611\pi$
$770$	0	0
$771$	1030.01	0.0481129
$772$	0	0
$773$	−32794.2	−1.52591	−0.762954	−	0.646453i	$-0.776252\pi$
$773$	−32794.2	−1.52591	−0.762954	+	0.646453i	$0.776252\pi$
$774$	0	0
$775$	−5493.28	−0.254612
$776$	0	0
$777$	21390.0	0.987597
$778$	0	0
$779$	42016.8	1.93249
$780$	0	0
$781$	21406.5	0.980775
$782$	0	0
$783$	8737.29	0.398781
$784$	0	0
$785$	−3637.99	−0.165408
$786$	0	0
$787$	1234.92	0.0559341	0.0279670	−	0.999609i	$-0.491097\pi$
$787$	1234.92	0.0559341	0.0279670	+	0.999609i	$0.491097\pi$
$788$	0	0
$789$	−6574.66	−0.296659
$790$	0	0
$791$	−9521.24	−0.427985
$792$	0	0
$793$	31165.7	1.39562
$794$	0	0
$795$	−8935.52	−0.398629
$796$	0	0
$797$	−19464.3	−0.865070	−0.432535	−	0.901617i	$-0.642381\pi$
$797$	−19464.3	−0.865070	−0.432535	+	0.901617i	$0.642381\pi$
$798$	0	0
$799$	−11609.5	−0.514035
$800$	0	0
$801$	4684.79	0.206653
$802$	0	0
$803$	7548.48	0.331731
$804$	0	0
$805$	−1450.58	−0.0635106
$806$	0	0
$807$	17173.4	0.749113
$808$	0	0
$809$	−33587.0	−1.45965	−0.729824	−	0.683635i	$-0.760398\pi$
$809$	−33587.0	−1.45965	−0.729824	+	0.683635i	$0.760398\pi$
$810$	0	0
$811$	−2612.03	−0.113096	−0.0565479	−	0.998400i	$-0.518009\pi$
$811$	−2612.03	−0.113096	−0.0565479	+	0.998400i	$0.518009\pi$
$812$	0	0
$813$	31988.5	1.37993
$814$	0	0
$815$	3843.59	0.165196
$816$	0	0
$817$	−19300.9	−0.826502
$818$	0	0
$819$	−2714.83	−0.115829
$820$	0	0
$821$	−17811.2	−0.757146	−0.378573	−	0.925572i	$-0.623585\pi$
$821$	−17811.2	−0.757146	−0.378573	+	0.925572i	$0.623585\pi$
$822$	0	0
$823$	30535.0	1.29330	0.646649	−	0.762788i	$-0.276170\pi$
$823$	30535.0	1.29330	0.646649	+	0.762788i	$0.276170\pi$
$824$	0	0
$825$	−3145.50	−0.132742
$826$	0	0
$827$	−3344.05	−0.140609	−0.0703047	−	0.997526i	$-0.522397\pi$
$827$	−3344.05	−0.140609	−0.0703047	+	0.997526i	$0.522397\pi$
$828$	0	0
$829$	−24905.6	−1.04344	−0.521718	−	0.853118i	$-0.674709\pi$
$829$	−24905.6	−1.04344	−0.521718	+	0.853118i	$0.674709\pi$
$830$	0	0
$831$	35602.1	1.48619
$832$	0	0
$833$	−11508.4	−0.478684
$834$	0	0
$835$	1216.98	0.0504377
$836$	0	0
$837$	26702.7	1.10272
$838$	0	0
$839$	−38233.9	−1.57328	−0.786639	−	0.617413i	$-0.788181\pi$
$839$	−38233.9	−1.57328	−0.786639	+	0.617413i	$0.788181\pi$
$840$	0	0
$841$	−19219.7	−0.788050
$842$	0	0
$843$	−38761.2	−1.58364
$844$	0	0
$845$	4032.52	0.164169
$846$	0	0
$847$	10695.7	0.433894
$848$	0	0
$849$	−44717.8	−1.80767
$850$	0	0
$851$	6813.12	0.274443
$852$	0	0
$853$	21297.4	0.854875	0.427438	−	0.904045i	$-0.359416\pi$
$853$	21297.4	0.854875	0.427438	+	0.904045i	$0.359416\pi$
$854$	0	0
$855$	3971.98	0.158876
$856$	0	0
$857$	−2100.08	−0.0837074	−0.0418537	−	0.999124i	$-0.513326\pi$
$857$	−2100.08	−0.0837074	−0.0418537	+	0.999124i	$0.513326\pi$
$858$	0	0
$859$	−16555.4	−0.657582	−0.328791	−	0.944403i	$-0.606641\pi$
$859$	−16555.4	−0.657582	−0.328791	+	0.944403i	$0.606641\pi$
$860$	0	0
$861$	22044.2	0.872547
$862$	0	0
$863$	−40225.0	−1.58664	−0.793322	−	0.608802i	$-0.791650\pi$
$863$	−40225.0	−1.58664	−0.793322	+	0.608802i	$0.791650\pi$
$864$	0	0
$865$	1007.69	0.0396098
$866$	0	0
$867$	−5704.81	−0.223467
$868$	0	0
$869$	−26695.5	−1.04210
$870$	0	0
$871$	−913.656	−0.0355431
$872$	0	0
$873$	5927.13	0.229786
$874$	0	0
$875$	1576.71	0.0609173
$876$	0	0
$877$	35335.8	1.36055	0.680277	−	0.732955i	$-0.261859\pi$
$877$	35335.8	1.36055	0.680277	+	0.732955i	$0.261859\pi$
$878$	0	0
$879$	−48173.2	−1.84851
$880$	0	0
$881$	28460.9	1.08839	0.544196	−	0.838958i	$-0.316835\pi$
$881$	28460.9	1.08839	0.544196	+	0.838958i	$0.316835\pi$
$882$	0	0
$883$	−35713.3	−1.36110	−0.680548	−	0.732703i	$-0.738258\pi$
$883$	−35713.3	−1.36110	−0.680548	+	0.732703i	$0.738258\pi$
$884$	0	0
$885$	4630.44	0.175876
$886$	0	0
$887$	−354.759	−0.0134292	−0.00671458	−	0.999977i	$-0.502137\pi$
$887$	−354.759	−0.0134292	−0.00671458	+	0.999977i	$0.502137\pi$
$888$	0	0
$889$	−27261.8	−1.02849
$890$	0	0
$891$	18715.3	0.703688
$892$	0	0
$893$	25532.2	0.956777
$894$	0	0
$895$	3612.82	0.134931
$896$	0	0
$897$	−4909.79	−0.182757
$898$	0	0
$899$	15798.1	0.586093
$900$	0	0
$901$	19536.5	0.722370
$902$	0	0
$903$	−10126.2	−0.373178
$904$	0	0
$905$	−10588.8	−0.388932
$906$	0	0
$907$	−41414.8	−1.51616	−0.758080	−	0.652162i	$-0.773862\pi$
$907$	−41414.8	−1.51616	−0.758080	+	0.652162i	$0.773862\pi$
$908$	0	0
$909$	−4550.12	−0.166026
$910$	0	0
$911$	4521.49	0.164439	0.0822194	−	0.996614i	$-0.473799\pi$
$911$	4521.49	0.164439	0.0822194	+	0.996614i	$0.473799\pi$
$912$	0	0
$913$	−16463.2	−0.596770
$914$	0	0
$915$	−23922.8	−0.864333
$916$	0	0
$917$	−5016.92	−0.180669
$918$	0	0
$919$	−31097.0	−1.11621	−0.558104	−	0.829771i	$-0.688471\pi$
$919$	−31097.0	−1.11621	−0.558104	+	0.829771i	$0.688471\pi$
$920$	0	0
$921$	−55147.0	−1.97303
$922$	0	0
$923$	−36318.8	−1.29518
$924$	0	0
$925$	−7405.57	−0.263236
$926$	0	0
$927$	2722.95	0.0964761
$928$	0	0
$929$	11243.2	0.397070	0.198535	−	0.980094i	$-0.436382\pi$
$929$	11243.2	0.397070	0.198535	+	0.980094i	$0.436382\pi$
$930$	0	0
$931$	25310.0	0.890978
$932$	0	0
$933$	−40076.4	−1.40626
$934$	0	0
$935$	6877.28	0.240547
$936$	0	0
$937$	−31275.3	−1.09042	−0.545209	−	0.838300i	$-0.683550\pi$
$937$	−31275.3	−1.09042	−0.545209	+	0.838300i	$0.683550\pi$
$938$	0	0
$939$	−16915.2	−0.587866
$940$	0	0
$941$	−19726.6	−0.683387	−0.341694	−	0.939811i	$-0.611001\pi$
$941$	−19726.6	−0.683387	−0.341694	+	0.939811i	$0.611001\pi$
$942$	0	0
$943$	7021.48	0.242472
$944$	0	0
$945$	−7664.35	−0.263832
$946$	0	0
$947$	−9073.85	−0.311363	−0.155681	−	0.987807i	$-0.549757\pi$
$947$	−9073.85	−0.311363	−0.155681	+	0.987807i	$0.549757\pi$
$948$	0	0
$949$	−12806.9	−0.438072
$950$	0	0
$951$	−22988.4	−0.783860
$952$	0	0
$953$	20332.8	0.691125	0.345563	−	0.938396i	$-0.387688\pi$
$953$	20332.8	0.691125	0.345563	+	0.938396i	$0.387688\pi$
$954$	0	0
$955$	−414.812	−0.0140555
$956$	0	0
$957$	9046.14	0.305559
$958$	0	0
$959$	20947.9	0.705364
$960$	0	0
$961$	18490.8	0.620686
$962$	0	0
$963$	−8097.03	−0.270948
$964$	0	0
$965$	−6752.14	−0.225242
$966$	0	0
$967$	−10303.5	−0.342646	−0.171323	−	0.985215i	$-0.554804\pi$
$967$	−10303.5	−0.342646	−0.171323	+	0.985215i	$0.554804\pi$
$968$	0	0
$969$	−49308.3	−1.63469
$970$	0	0
$971$	−937.151	−0.0309728	−0.0154864	−	0.999880i	$-0.504930\pi$
$971$	−937.151	−0.0309728	−0.0154864	+	0.999880i	$0.504930\pi$
$972$	0	0
$973$	12586.6	0.414705
$974$	0	0
$975$	5336.73	0.175295
$976$	0	0
$977$	−23339.0	−0.764258	−0.382129	−	0.924109i	$-0.624809\pi$
$977$	−23339.0	−0.764258	−0.382129	+	0.924109i	$0.624809\pi$
$978$	0	0
$979$	−17839.2	−0.582372
$980$	0	0
$981$	−12417.0	−0.404124
$982$	0	0
$983$	26638.6	0.864334	0.432167	−	0.901794i	$-0.357749\pi$
$983$	26638.6	0.864334	0.432167	+	0.901794i	$0.357749\pi$
$984$	0	0
$985$	−21588.5	−0.698341
$986$	0	0
$987$	13395.5	0.431999
$988$	0	0
$989$	−3225.39	−0.103702
$990$	0	0
$991$	−24631.4	−0.789548	−0.394774	−	0.918778i	$-0.629177\pi$
$991$	−24631.4	−0.789548	−0.394774	+	0.918778i	$0.629177\pi$
$992$	0	0
$993$	3415.15	0.109141
$994$	0	0
$995$	−15131.3	−0.482106
$996$	0	0
$997$	−31438.3	−0.998656	−0.499328	−	0.866413i	$-0.666420\pi$
$997$	−31438.3	−0.998656	−0.499328	+	0.866413i	$0.666420\pi$
$998$	0	0
$999$	35998.2	1.14007

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.4.a.b.1.4	✓	5
4.3	odd	2		1840.4.a.o.1.2		5
5.2	odd	4		2300.4.c.d.1749.3		10
5.3	odd	4		2300.4.c.d.1749.8		10
5.4	even	2		2300.4.a.c.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.4.a.b.1.4	✓	5	1.1	even	1	trivial
1840.4.a.o.1.2		5	4.3	odd	2
2300.4.a.c.1.2		5	5.4	even	2
2300.4.c.d.1749.3		10	5.2	odd	4
2300.4.c.d.1749.8		10	5.3	odd	4

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:	\( N \)	\(=\)	\( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	460.a (trivial)

\(f(q)\)	\(=\)	\(q+5.72467 q^{3} -5.00000 q^{5} -12.6137 q^{7} +5.77185 q^{9} -21.9786 q^{11} +37.2893 q^{13} -28.6233 q^{15} +62.5817 q^{17} -137.633 q^{19} -72.2093 q^{21} -23.0000 q^{23} +25.0000 q^{25} -121.524 q^{27} -71.8975 q^{29} -219.731 q^{31} -125.820 q^{33} +63.0685 q^{35} -296.223 q^{37} +213.469 q^{39} -305.282 q^{41} +140.234 q^{43} -28.8592 q^{45} -185.509 q^{47} -183.895 q^{49} +358.260 q^{51} +312.176 q^{53} +109.893 q^{55} -787.903 q^{57} -161.771 q^{59} +835.780 q^{61} -72.8043 q^{63} -186.447 q^{65} -24.5018 q^{67} -131.667 q^{69} -973.973 q^{71} -343.447 q^{73} +143.117 q^{75} +277.231 q^{77} +1214.62 q^{79} -851.526 q^{81} +749.055 q^{83} -312.908 q^{85} -411.590 q^{87} +811.662 q^{89} -470.356 q^{91} -1257.89 q^{93} +688.165 q^{95} +1026.90 q^{97} -126.857 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{3} - 25 q^{5} + 8 q^{7} + 30 q^{9} + 7 q^{11} + 5 q^{13} + 15 q^{15} - 24 q^{17} - 13 q^{19} - 115 q^{23} + 125 q^{25} - 204 q^{27} - 253 q^{29} - 98 q^{31} - 473 q^{33} - 40 q^{35} - 435 q^{37}+ \cdots + 111 q^{99}+O(q^{100}) \) 5 * q - 3 * q^3 - 25 * q^5 + 8 * q^7 + 30 * q^9 + 7 * q^11 + 5 * q^13 + 15 * q^15 - 24 * q^17 - 13 * q^19 - 115 * q^23 + 125 * q^25 - 204 * q^27 - 253 * q^29 - 98 * q^31 - 473 * q^33 - 40 * q^35 - 435 * q^37 - 410 * q^39 - 774 * q^41 - 498 * q^43 - 150 * q^45 - 572 * q^47 - 683 * q^49 - 657 * q^51 - 665 * q^53 - 35 * q^55 - 932 * q^57 - 763 * q^59 + 337 * q^61 - 1527 * q^63 - 25 * q^65 - 305 * q^67 + 69 * q^69 - 1504 * q^71 - 1304 * q^73 - 75 * q^75 + 182 * q^77 + 626 * q^79 - 959 * q^81 - 1703 * q^83 + 120 * q^85 - 1354 * q^87 + 646 * q^89 - 767 * q^91 - 452 * q^93 + 65 * q^95 - 233 * q^97 + 111 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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