LMFDB - Embedded newform 2400.4.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,4,Mod(1,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2400.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$141.604584014$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.28724.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 19x + 9 $ x^3 - x^2 - 19*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.66165$ of defining polynomial
Character	$\chi$	$=$	2400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +6.18460 q^{7} +9.00000 q^{9} -1.46201 q^{11} -30.9398 q^{13} -116.603 q^{17} -149.266 q^{19} -18.5538 q^{21} +99.6477 q^{23} -27.0000 q^{27} +150.818 q^{29} -217.792 q^{31} +4.38604 q^{33} +321.542 q^{37} +92.8195 q^{39} -192.388 q^{41} -168.930 q^{43} +115.968 q^{47} -304.751 q^{49} +349.810 q^{51} -162.724 q^{53} +447.797 q^{57} +493.021 q^{59} +239.785 q^{61} +55.6614 q^{63} +720.968 q^{67} -298.943 q^{69} +179.198 q^{71} -512.622 q^{73} -9.04197 q^{77} +240.014 q^{79} +81.0000 q^{81} +631.000 q^{83} -452.455 q^{87} +501.502 q^{89} -191.350 q^{91} +653.376 q^{93} -512.059 q^{97} -13.1581 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} + 19 q^{7} + 27 q^{9} + 48 q^{11} + 63 q^{13} + 16 q^{17} + 21 q^{19} - 57 q^{21} + 140 q^{23} - 81 q^{27} + 92 q^{29} + 179 q^{31} - 144 q^{33} + 550 q^{37} - 189 q^{39} - 206 q^{41} + 279 q^{43}+ \cdots + 432 q^{99}+O(q^{100}) $ 3 * q - 9 * q^3 + 19 * q^7 + 27 * q^9 + 48 * q^11 + 63 * q^13 + 16 * q^17 + 21 * q^19 - 57 * q^21 + 140 * q^23 - 81 * q^27 + 92 * q^29 + 179 * q^31 - 144 * q^33 + 550 * q^37 - 189 * q^39 - 206 * q^41 + 279 * q^43 + 450 * q^47 + 446 * q^49 - 48 * q^51 - 434 * q^53 - 63 * q^57 + 854 * q^59 - 1103 * q^61 + 171 * q^63 + 555 * q^67 - 420 * q^69 - 970 * q^71 - 722 * q^73 + 1184 * q^77 - 856 * q^79 + 243 * q^81 + 222 * q^83 - 276 * q^87 - 1216 * q^89 - 1025 * q^91 - 537 * q^93 - 89 * q^97 + 432 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	6.18460	0.333937	0.166968	−	0.985962i	$-0.446602\pi$
$7$	6.18460	0.333937	0.166968	+	0.985962i	$0.446602\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−1.46201	−0.0400740	−0.0200370	−	0.999799i	$-0.506378\pi$
$11$	−1.46201	−0.0400740	−0.0200370	+	0.999799i	$0.506378\pi$
$12$	0	0
$13$	−30.9398	−0.660090	−0.330045	−	0.943965i	$-0.607064\pi$
$13$	−30.9398	−0.660090	−0.330045	+	0.943965i	$0.607064\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−116.603	−1.66356	−0.831778	−	0.555108i	$-0.812677\pi$
$17$	−116.603	−1.66356	−0.831778	+	0.555108i	$0.812677\pi$
$18$	0	0
$19$	−149.266	−1.80231	−0.901155	−	0.433496i	$-0.857280\pi$
$19$	−149.266	−1.80231	−0.901155	+	0.433496i	$0.857280\pi$
$20$	0	0
$21$	−18.5538	−0.192799
$22$	0	0
$23$	99.6477	0.903390	0.451695	−	0.892172i	$-0.350820\pi$
$23$	99.6477	0.903390	0.451695	+	0.892172i	$0.350820\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	150.818	0.965734	0.482867	−	0.875694i	$-0.339595\pi$
$29$	150.818	0.965734	0.482867	+	0.875694i	$0.339595\pi$
$30$	0	0
$31$	−217.792	−1.26183	−0.630913	−	0.775854i	$-0.717319\pi$
$31$	−217.792	−1.26183	−0.630913	+	0.775854i	$0.717319\pi$
$32$	0	0
$33$	4.38604	0.0231367
$34$	0	0
$35$	0	0
$36$	0	0
$37$	321.542	1.42868	0.714340	−	0.699798i	$-0.246727\pi$
$37$	321.542	1.42868	0.714340	+	0.699798i	$0.246727\pi$
$38$	0	0
$39$	92.8195	0.381103
$40$	0	0
$41$	−192.388	−0.732829	−0.366414	−	0.930452i	$-0.619415\pi$
$41$	−192.388	−0.732829	−0.366414	+	0.930452i	$0.619415\pi$
$42$	0	0
$43$	−168.930	−0.599107	−0.299554	−	0.954079i	$-0.596838\pi$
$43$	−168.930	−0.599107	−0.299554	+	0.954079i	$0.596838\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	115.968	0.359909	0.179955	−	0.983675i	$-0.442405\pi$
$47$	115.968	0.359909	0.179955	+	0.983675i	$0.442405\pi$
$48$	0	0
$49$	−304.751	−0.888486
$50$	0	0
$51$	349.810	0.960455
$52$	0	0
$53$	−162.724	−0.421732	−0.210866	−	0.977515i	$-0.567628\pi$
$53$	−162.724	−0.421732	−0.210866	+	0.977515i	$0.567628\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	447.797	1.04056
$58$	0	0
$59$	493.021	1.08790	0.543948	−	0.839119i	$-0.316929\pi$
$59$	493.021	1.08790	0.543948	+	0.839119i	$0.316929\pi$
$60$	0	0
$61$	239.785	0.503301	0.251651	−	0.967818i	$-0.419027\pi$
$61$	239.785	0.503301	0.251651	+	0.967818i	$0.419027\pi$
$62$	0	0
$63$	55.6614	0.111312
$64$	0	0
$65$	0	0
$66$	0	0
$67$	720.968	1.31463	0.657315	−	0.753616i	$-0.271692\pi$
$67$	720.968	1.31463	0.657315	+	0.753616i	$0.271692\pi$
$68$	0	0
$69$	−298.943	−0.521573
$70$	0	0
$71$	179.198	0.299534	0.149767	−	0.988721i	$-0.452148\pi$
$71$	179.198	0.299534	0.149767	+	0.988721i	$0.452148\pi$
$72$	0	0
$73$	−512.622	−0.821889	−0.410944	−	0.911660i	$-0.634801\pi$
$73$	−512.622	−0.821889	−0.410944	+	0.911660i	$0.634801\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−9.04197	−0.0133822
$78$	0	0
$79$	240.014	0.341819	0.170910	−	0.985287i	$-0.445329\pi$
$79$	240.014	0.341819	0.170910	+	0.985287i	$0.445329\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	631.000	0.834472	0.417236	−	0.908798i	$-0.362999\pi$
$83$	631.000	0.834472	0.417236	+	0.908798i	$0.362999\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−452.455	−0.557567
$88$	0	0
$89$	501.502	0.597293	0.298646	−	0.954364i	$-0.403465\pi$
$89$	501.502	0.597293	0.298646	+	0.954364i	$0.403465\pi$
$90$	0	0
$91$	−191.350	−0.220428
$92$	0	0
$93$	653.376	0.728515
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−512.059	−0.535997	−0.267998	−	0.963419i	$-0.586362\pi$
$97$	−512.059	−0.535997	−0.267998	+	0.963419i	$0.586362\pi$
$98$	0	0
$99$	−13.1581	−0.0133580
$100$	0	0
$101$	−1558.82	−1.53573	−0.767863	−	0.640614i	$-0.778680\pi$
$101$	−1558.82	−1.53573	−0.767863	+	0.640614i	$0.778680\pi$
$102$	0	0
$103$	636.521	0.608915	0.304458	−	0.952526i	$-0.401525\pi$
$103$	636.521	0.608915	0.304458	+	0.952526i	$0.401525\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1214.36	−1.09716	−0.548580	−	0.836098i	$-0.684832\pi$
$107$	−1214.36	−1.09716	−0.548580	+	0.836098i	$0.684832\pi$
$108$	0	0
$109$	284.921	0.250371	0.125186	−	0.992133i	$-0.460047\pi$
$109$	284.921	0.250371	0.125186	+	0.992133i	$0.460047\pi$
$110$	0	0
$111$	−964.626	−0.824849
$112$	0	0
$113$	−634.021	−0.527820	−0.263910	−	0.964547i	$-0.585012\pi$
$113$	−634.021	−0.527820	−0.263910	+	0.964547i	$0.585012\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−278.459	−0.220030
$118$	0	0
$119$	−721.144	−0.555523
$120$	0	0
$121$	−1328.86	−0.998394
$122$	0	0
$123$	577.164	0.423099
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−190.261	−0.132937	−0.0664684	−	0.997789i	$-0.521173\pi$
$127$	−190.261	−0.132937	−0.0664684	+	0.997789i	$0.521173\pi$
$128$	0	0
$129$	506.791	0.345895
$130$	0	0
$131$	−1538.27	−1.02595	−0.512975	−	0.858404i	$-0.671457\pi$
$131$	−1538.27	−1.02595	−0.512975	+	0.858404i	$0.671457\pi$
$132$	0	0
$133$	−923.148	−0.601858
$134$	0	0
$135$	0	0
$136$	0	0
$137$	586.299	0.365627	0.182814	−	0.983148i	$-0.441480\pi$
$137$	586.299	0.365627	0.182814	+	0.983148i	$0.441480\pi$
$138$	0	0
$139$	1226.37	0.748343	0.374171	−	0.927360i	$-0.377927\pi$
$139$	1226.37	0.748343	0.374171	+	0.927360i	$0.377927\pi$
$140$	0	0
$141$	−347.905	−0.207794
$142$	0	0
$143$	45.2345	0.0264524
$144$	0	0
$145$	0	0
$146$	0	0
$147$	914.252	0.512968
$148$	0	0
$149$	−3164.73	−1.74003	−0.870016	−	0.493023i	$-0.835892\pi$
$149$	−3164.73	−1.74003	−0.870016	+	0.493023i	$0.835892\pi$
$150$	0	0
$151$	108.000	0.0582048	0.0291024	−	0.999576i	$-0.490735\pi$
$151$	108.000	0.0582048	0.0291024	+	0.999576i	$0.490735\pi$
$152$	0	0
$153$	−1049.43	−0.554519
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2092.95	1.06392	0.531959	−	0.846770i	$-0.321456\pi$
$157$	2092.95	1.06392	0.531959	+	0.846770i	$0.321456\pi$
$158$	0	0
$159$	488.171	0.243487
$160$	0	0
$161$	616.281	0.301675
$162$	0	0
$163$	3803.83	1.82785	0.913924	−	0.405886i	$-0.133037\pi$
$163$	3803.83	1.82785	0.913924	+	0.405886i	$0.133037\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3534.64	1.63784	0.818919	−	0.573909i	$-0.194574\pi$
$167$	3534.64	1.63784	0.818919	+	0.573909i	$0.194574\pi$
$168$	0	0
$169$	−1239.73	−0.564282
$170$	0	0
$171$	−1343.39	−0.600770
$172$	0	0
$173$	1636.70	0.719281	0.359641	−	0.933091i	$-0.382899\pi$
$173$	1636.70	0.719281	0.359641	+	0.933091i	$0.382899\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1479.06	−0.628097
$178$	0	0
$179$	1896.64	0.791966	0.395983	−	0.918258i	$-0.370404\pi$
$179$	1896.64	0.791966	0.395983	+	0.918258i	$0.370404\pi$
$180$	0	0
$181$	848.277	0.348353	0.174177	−	0.984714i	$-0.444274\pi$
$181$	848.277	0.348353	0.174177	+	0.984714i	$0.444274\pi$
$182$	0	0
$183$	−719.356	−0.290581
$184$	0	0
$185$	0	0
$186$	0	0
$187$	170.476	0.0666653
$188$	0	0
$189$	−166.984	−0.0642662
$190$	0	0
$191$	−3265.20	−1.23697	−0.618487	−	0.785795i	$-0.712254\pi$
$191$	−3265.20	−1.23697	−0.618487	+	0.785795i	$0.712254\pi$
$192$	0	0
$193$	778.836	0.290476	0.145238	−	0.989397i	$-0.453605\pi$
$193$	778.836	0.290476	0.145238	+	0.989397i	$0.453605\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3262.24	1.17982	0.589912	−	0.807468i	$-0.299162\pi$
$197$	3262.24	1.17982	0.589912	+	0.807468i	$0.299162\pi$
$198$	0	0
$199$	1005.79	0.358284	0.179142	−	0.983823i	$-0.442668\pi$
$199$	1005.79	0.358284	0.179142	+	0.983823i	$0.442668\pi$
$200$	0	0
$201$	−2162.90	−0.759002
$202$	0	0
$203$	932.751	0.322494
$204$	0	0
$205$	0	0
$206$	0	0
$207$	896.829	0.301130
$208$	0	0
$209$	218.229	0.0722258
$210$	0	0
$211$	1903.03	0.620900	0.310450	−	0.950590i	$-0.399520\pi$
$211$	1903.03	0.620900	0.310450	+	0.950590i	$0.399520\pi$
$212$	0	0
$213$	−537.594	−0.172936
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1346.96	−0.421370
$218$	0	0
$219$	1537.87	0.474518
$220$	0	0
$221$	3607.69	1.09810
$222$	0	0
$223$	4060.30	1.21927	0.609636	−	0.792682i	$-0.291316\pi$
$223$	4060.30	1.21927	0.609636	+	0.792682i	$0.291316\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	341.242	0.0997756	0.0498878	−	0.998755i	$-0.484114\pi$
$227$	341.242	0.0997756	0.0498878	+	0.998755i	$0.484114\pi$
$228$	0	0
$229$	4503.67	1.29961	0.649806	−	0.760100i	$-0.274850\pi$
$229$	4503.67	1.29961	0.649806	+	0.760100i	$0.274850\pi$
$230$	0	0
$231$	27.1259	0.00772621
$232$	0	0
$233$	1200.67	0.337589	0.168795	−	0.985651i	$-0.446013\pi$
$233$	1200.67	0.337589	0.168795	+	0.985651i	$0.446013\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−720.043	−0.197350
$238$	0	0
$239$	6422.87	1.73833	0.869165	−	0.494522i	$-0.164657\pi$
$239$	6422.87	1.73833	0.869165	+	0.494522i	$0.164657\pi$
$240$	0	0
$241$	−2105.29	−0.562711	−0.281356	−	0.959604i	$-0.590784\pi$
$241$	−2105.29	−0.562711	−0.281356	+	0.959604i	$0.590784\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4618.26	1.18969
$248$	0	0
$249$	−1893.00	−0.481783
$250$	0	0
$251$	1521.25	0.382551	0.191275	−	0.981536i	$-0.438738\pi$
$251$	1521.25	0.382551	0.191275	+	0.981536i	$0.438738\pi$
$252$	0	0
$253$	−145.686	−0.0362024
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4392.25	−1.06607	−0.533037	−	0.846092i	$-0.678949\pi$
$257$	−4392.25	−1.06607	−0.533037	+	0.846092i	$0.678949\pi$
$258$	0	0
$259$	1988.61	0.477089
$260$	0	0
$261$	1357.37	0.321911
$262$	0	0
$263$	7995.63	1.87464	0.937322	−	0.348464i	$-0.113297\pi$
$263$	7995.63	1.87464	0.937322	+	0.348464i	$0.113297\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1504.50	−0.344847
$268$	0	0
$269$	−262.524	−0.0595031	−0.0297516	−	0.999557i	$-0.509472\pi$
$269$	−262.524	−0.0595031	−0.0297516	+	0.999557i	$0.509472\pi$
$270$	0	0
$271$	−8782.03	−1.96853	−0.984263	−	0.176710i	$-0.943455\pi$
$271$	−8782.03	−1.96853	−0.984263	+	0.176710i	$0.943455\pi$
$272$	0	0
$273$	574.051	0.127264
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1946.87	−0.422297	−0.211148	−	0.977454i	$-0.567720\pi$
$277$	−1946.87	−0.422297	−0.211148	+	0.977454i	$0.567720\pi$
$278$	0	0
$279$	−1960.13	−0.420609
$280$	0	0
$281$	−939.459	−0.199443	−0.0997214	−	0.995015i	$-0.531795\pi$
$281$	−939.459	−0.199443	−0.0997214	+	0.995015i	$0.531795\pi$
$282$	0	0
$283$	4353.15	0.914373	0.457187	−	0.889371i	$-0.348857\pi$
$283$	4353.15	0.914373	0.457187	+	0.889371i	$0.348857\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1189.84	−0.244719
$288$	0	0
$289$	8683.33	1.76742
$290$	0	0
$291$	1536.18	0.309458
$292$	0	0
$293$	−4418.82	−0.881059	−0.440530	−	0.897738i	$-0.645209\pi$
$293$	−4418.82	−0.881059	−0.440530	+	0.897738i	$0.645209\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	39.4744	0.00771224
$298$	0	0
$299$	−3083.08	−0.596318
$300$	0	0
$301$	−1044.77	−0.200064
$302$	0	0
$303$	4676.46	0.886652
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1822.92	−0.338890	−0.169445	−	0.985540i	$-0.554198\pi$
$307$	−1822.92	−0.338890	−0.169445	+	0.985540i	$0.554198\pi$
$308$	0	0
$309$	−1909.56	−0.351557
$310$	0	0
$311$	−6883.95	−1.25515	−0.627577	−	0.778554i	$-0.715953\pi$
$311$	−6883.95	−1.25515	−0.627577	+	0.778554i	$0.715953\pi$
$312$	0	0
$313$	−3831.83	−0.691973	−0.345987	−	0.938239i	$-0.612456\pi$
$313$	−3831.83	−0.691973	−0.345987	+	0.938239i	$0.612456\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2202.07	−0.390160	−0.195080	−	0.980787i	$-0.562497\pi$
$317$	−2202.07	−0.390160	−0.195080	+	0.980787i	$0.562497\pi$
$318$	0	0
$319$	−220.499	−0.0387008
$320$	0	0
$321$	3643.07	0.633446
$322$	0	0
$323$	17404.9	2.99824
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−854.762	−0.144552
$328$	0	0
$329$	717.218	0.120187
$330$	0	0
$331$	−4628.98	−0.768675	−0.384338	−	0.923193i	$-0.625570\pi$
$331$	−4628.98	−0.768675	−0.384338	+	0.923193i	$0.625570\pi$
$332$	0	0
$333$	2893.88	0.476227
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8122.20	1.31289	0.656446	−	0.754373i	$-0.272059\pi$
$337$	8122.20	1.31289	0.656446	+	0.754373i	$0.272059\pi$
$338$	0	0
$339$	1902.06	0.304737
$340$	0	0
$341$	318.415	0.0505664
$342$	0	0
$343$	−4006.08	−0.630635
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6603.58	1.02161	0.510805	−	0.859696i	$-0.329347\pi$
$347$	6603.58	1.02161	0.510805	+	0.859696i	$0.329347\pi$
$348$	0	0
$349$	−7941.66	−1.21807	−0.609036	−	0.793142i	$-0.708444\pi$
$349$	−7941.66	−1.21807	−0.609036	+	0.793142i	$0.708444\pi$
$350$	0	0
$351$	835.376	0.127034
$352$	0	0
$353$	5576.09	0.840751	0.420376	−	0.907350i	$-0.361898\pi$
$353$	5576.09	0.840751	0.420376	+	0.907350i	$0.361898\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2163.43	0.320731
$358$	0	0
$359$	6048.44	0.889205	0.444602	−	0.895728i	$-0.353345\pi$
$359$	6048.44	0.889205	0.444602	+	0.895728i	$0.353345\pi$
$360$	0	0
$361$	15421.3	2.24832
$362$	0	0
$363$	3986.59	0.576423
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10719.0	1.52460	0.762298	−	0.647227i	$-0.224071\pi$
$367$	10719.0	1.52460	0.762298	+	0.647227i	$0.224071\pi$
$368$	0	0
$369$	−1731.49	−0.244276
$370$	0	0
$371$	−1006.38	−0.140832
$372$	0	0
$373$	10144.6	1.40823	0.704115	−	0.710086i	$-0.251344\pi$
$373$	10144.6	1.40823	0.704115	+	0.710086i	$0.251344\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4666.30	−0.637471
$378$	0	0
$379$	14272.9	1.93443	0.967216	−	0.253957i	$-0.0817322\pi$
$379$	14272.9	1.93443	0.967216	+	0.253957i	$0.0817322\pi$
$380$	0	0
$381$	570.784	0.0767511
$382$	0	0
$383$	−11654.0	−1.55480	−0.777401	−	0.629005i	$-0.783462\pi$
$383$	−11654.0	−1.55480	−0.777401	+	0.629005i	$0.783462\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1520.37	−0.199702
$388$	0	0
$389$	2532.78	0.330121	0.165061	−	0.986283i	$-0.447218\pi$
$389$	2532.78	0.330121	0.165061	+	0.986283i	$0.447218\pi$
$390$	0	0
$391$	−11619.2	−1.50284
$392$	0	0
$393$	4614.81	0.592332
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7635.48	0.965273	0.482637	−	0.875821i	$-0.339679\pi$
$397$	7635.48	0.965273	0.482637	+	0.875821i	$0.339679\pi$
$398$	0	0
$399$	2769.44	0.347483
$400$	0	0
$401$	−9890.62	−1.23171	−0.615853	−	0.787861i	$-0.711188\pi$
$401$	−9890.62	−1.23171	−0.615853	+	0.787861i	$0.711188\pi$
$402$	0	0
$403$	6738.45	0.832918
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−470.099	−0.0572529
$408$	0	0
$409$	11704.6	1.41504	0.707522	−	0.706691i	$-0.249813\pi$
$409$	11704.6	1.41504	0.707522	+	0.706691i	$0.249813\pi$
$410$	0	0
$411$	−1758.90	−0.211095
$412$	0	0
$413$	3049.14	0.363289
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−3679.12	−0.432056
$418$	0	0
$419$	15336.9	1.78821	0.894103	−	0.447862i	$-0.147814\pi$
$419$	15336.9	1.78821	0.894103	+	0.447862i	$0.147814\pi$
$420$	0	0
$421$	14291.7	1.65448	0.827239	−	0.561850i	$-0.189910\pi$
$421$	14291.7	1.65448	0.827239	+	0.561850i	$0.189910\pi$
$422$	0	0
$423$	1043.72	0.119970
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1482.98	0.168071
$428$	0	0
$429$	−135.703	−0.0152723
$430$	0	0
$431$	−6508.14	−0.727345	−0.363673	−	0.931527i	$-0.618477\pi$
$431$	−6508.14	−0.727345	−0.363673	+	0.931527i	$0.618477\pi$
$432$	0	0
$433$	−10144.7	−1.12592	−0.562962	−	0.826482i	$-0.690338\pi$
$433$	−10144.7	−1.12592	−0.562962	+	0.826482i	$0.690338\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−14874.0	−1.62819
$438$	0	0
$439$	1137.25	0.123640	0.0618199	−	0.998087i	$-0.480310\pi$
$439$	1137.25	0.123640	0.0618199	+	0.998087i	$0.480310\pi$
$440$	0	0
$441$	−2742.76	−0.296162
$442$	0	0
$443$	7794.17	0.835919	0.417960	−	0.908466i	$-0.362745\pi$
$443$	7794.17	0.835919	0.417960	+	0.908466i	$0.362745\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	9494.19	1.00461
$448$	0	0
$449$	−5182.89	−0.544756	−0.272378	−	0.962190i	$-0.587810\pi$
$449$	−5182.89	−0.544756	−0.272378	+	0.962190i	$0.587810\pi$
$450$	0	0
$451$	281.274	0.0293674
$452$	0	0
$453$	−324.000	−0.0336046
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5982.62	−0.612374	−0.306187	−	0.951971i	$-0.599053\pi$
$457$	−5982.62	−0.612374	−0.306187	+	0.951971i	$0.599053\pi$
$458$	0	0
$459$	3148.29	0.320152
$460$	0	0
$461$	2540.96	0.256713	0.128356	−	0.991728i	$-0.459030\pi$
$461$	2540.96	0.256713	0.128356	+	0.991728i	$0.459030\pi$
$462$	0	0
$463$	15902.7	1.59625	0.798123	−	0.602494i	$-0.205826\pi$
$463$	15902.7	1.59625	0.798123	+	0.602494i	$0.205826\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2931.74	−0.290503	−0.145251	−	0.989395i	$-0.546399\pi$
$467$	−2931.74	−0.290503	−0.145251	+	0.989395i	$0.546399\pi$
$468$	0	0
$469$	4458.90	0.439004
$470$	0	0
$471$	−6278.84	−0.614254
$472$	0	0
$473$	246.978	0.0240086
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1464.51	−0.140577
$478$	0	0
$479$	−14722.5	−1.40436	−0.702179	−	0.712001i	$-0.747789\pi$
$479$	−14722.5	−1.40436	−0.702179	+	0.712001i	$0.747789\pi$
$480$	0	0
$481$	−9948.46	−0.943058
$482$	0	0
$483$	−1848.84	−0.174172
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−9604.32	−0.893662	−0.446831	−	0.894618i	$-0.647447\pi$
$487$	−9604.32	−0.893662	−0.446831	+	0.894618i	$0.647447\pi$
$488$	0	0
$489$	−11411.5	−1.05531
$490$	0	0
$491$	12095.5	1.11174	0.555870	−	0.831269i	$-0.312385\pi$
$491$	12095.5	1.11174	0.555870	+	0.831269i	$0.312385\pi$
$492$	0	0
$493$	−17585.9	−1.60655
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1108.27	0.100025
$498$	0	0
$499$	−348.001	−0.0312198	−0.0156099	−	0.999878i	$-0.504969\pi$
$499$	−348.001	−0.0312198	−0.0156099	+	0.999878i	$0.504969\pi$
$500$	0	0
$501$	−10603.9	−0.945606
$502$	0	0
$503$	8767.40	0.777175	0.388587	−	0.921412i	$-0.372963\pi$
$503$	8767.40	0.777175	0.388587	+	0.921412i	$0.372963\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3719.18	0.325788
$508$	0	0
$509$	1405.66	0.122406	0.0612031	−	0.998125i	$-0.480506\pi$
$509$	1405.66	0.122406	0.0612031	+	0.998125i	$0.480506\pi$
$510$	0	0
$511$	−3170.36	−0.274459
$512$	0	0
$513$	4030.17	0.346855
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−169.547	−0.0144230
$518$	0	0
$519$	−4910.09	−0.415277
$520$	0	0
$521$	5431.44	0.456729	0.228364	−	0.973576i	$-0.426662\pi$
$521$	5431.44	0.456729	0.228364	+	0.973576i	$0.426662\pi$
$522$	0	0
$523$	18655.4	1.55974	0.779868	−	0.625944i	$-0.215286\pi$
$523$	18655.4	1.55974	0.779868	+	0.625944i	$0.215286\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	25395.3	2.09912
$528$	0	0
$529$	−2237.34	−0.183886
$530$	0	0
$531$	4437.19	0.362632
$532$	0	0
$533$	5952.46	0.483733
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−5689.93	−0.457242
$538$	0	0
$539$	445.550	0.0356052
$540$	0	0
$541$	−17070.1	−1.35656	−0.678281	−	0.734803i	$-0.737275\pi$
$541$	−17070.1	−1.35656	−0.678281	+	0.734803i	$0.737275\pi$
$542$	0	0
$543$	−2544.83	−0.201122
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−17569.2	−1.37331	−0.686657	−	0.726981i	$-0.740923\pi$
$547$	−17569.2	−1.37331	−0.686657	+	0.726981i	$0.740923\pi$
$548$	0	0
$549$	2158.07	0.167767
$550$	0	0
$551$	−22512.0	−1.74055
$552$	0	0
$553$	1484.39	0.114146
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12574.2	0.956529	0.478264	−	0.878216i	$-0.341266\pi$
$557$	12574.2	0.956529	0.478264	+	0.878216i	$0.341266\pi$
$558$	0	0
$559$	5226.67	0.395465
$560$	0	0
$561$	−511.427	−0.0384892
$562$	0	0
$563$	2946.14	0.220541	0.110271	−	0.993902i	$-0.464828\pi$
$563$	2946.14	0.220541	0.110271	+	0.993902i	$0.464828\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	500.952	0.0371041
$568$	0	0
$569$	−12393.3	−0.913098	−0.456549	−	0.889698i	$-0.650915\pi$
$569$	−12393.3	−0.913098	−0.456549	+	0.889698i	$0.650915\pi$
$570$	0	0
$571$	20840.7	1.52742	0.763708	−	0.645562i	$-0.223377\pi$
$571$	20840.7	1.52742	0.763708	+	0.645562i	$0.223377\pi$
$572$	0	0
$573$	9795.61	0.714167
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6808.37	0.491224	0.245612	−	0.969368i	$-0.421011\pi$
$577$	6808.37	0.491224	0.245612	+	0.969368i	$0.421011\pi$
$578$	0	0
$579$	−2336.51	−0.167706
$580$	0	0
$581$	3902.48	0.278661
$582$	0	0
$583$	237.904	0.0169005
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−21014.4	−1.47761	−0.738804	−	0.673920i	$-0.764609\pi$
$587$	−21014.4	−1.47761	−0.738804	+	0.673920i	$0.764609\pi$
$588$	0	0
$589$	32508.9	2.27420
$590$	0	0
$591$	−9786.73	−0.681172
$592$	0	0
$593$	20550.4	1.42311	0.711554	−	0.702632i	$-0.247992\pi$
$593$	20550.4	1.42311	0.711554	+	0.702632i	$0.247992\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3017.37	−0.206856
$598$	0	0
$599$	7083.42	0.483173	0.241586	−	0.970379i	$-0.422332\pi$
$599$	7083.42	0.483173	0.241586	+	0.970379i	$0.422332\pi$
$600$	0	0
$601$	19174.3	1.30139	0.650696	−	0.759338i	$-0.274477\pi$
$601$	19174.3	1.30139	0.650696	+	0.759338i	$0.274477\pi$
$602$	0	0
$603$	6488.71	0.438210
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7315.15	0.489148	0.244574	−	0.969631i	$-0.421352\pi$
$607$	7315.15	0.489148	0.244574	+	0.969631i	$0.421352\pi$
$608$	0	0
$609$	−2798.25	−0.186192
$610$	0	0
$611$	−3588.04	−0.237572
$612$	0	0
$613$	−21302.2	−1.40357	−0.701786	−	0.712388i	$-0.747614\pi$
$613$	−21302.2	−1.40357	−0.701786	+	0.712388i	$0.747614\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24210.1	1.57968	0.789839	−	0.613314i	$-0.210164\pi$
$617$	24210.1	1.57968	0.789839	+	0.613314i	$0.210164\pi$
$618$	0	0
$619$	8667.47	0.562803	0.281401	−	0.959590i	$-0.409201\pi$
$619$	8667.47	0.562803	0.281401	+	0.959590i	$0.409201\pi$
$620$	0	0
$621$	−2690.49	−0.173858
$622$	0	0
$623$	3101.58	0.199458
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−654.686	−0.0416996
$628$	0	0
$629$	−37492.9	−2.37669
$630$	0	0
$631$	−4831.15	−0.304794	−0.152397	−	0.988319i	$-0.548699\pi$
$631$	−4831.15	−0.304794	−0.152397	+	0.988319i	$0.548699\pi$
$632$	0	0
$633$	−5709.08	−0.358477
$634$	0	0
$635$	0	0
$636$	0	0
$637$	9428.94	0.586481
$638$	0	0
$639$	1612.78	0.0998445
$640$	0	0
$641$	−205.527	−0.0126643	−0.00633216	−	0.999980i	$-0.502016\pi$
$641$	−205.527	−0.0126643	−0.00633216	+	0.999980i	$0.502016\pi$
$642$	0	0
$643$	−5324.60	−0.326566	−0.163283	−	0.986579i	$-0.552208\pi$
$643$	−5324.60	−0.326566	−0.163283	+	0.986579i	$0.552208\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20996.4	1.27582	0.637908	−	0.770113i	$-0.279800\pi$
$647$	20996.4	1.27582	0.637908	+	0.770113i	$0.279800\pi$
$648$	0	0
$649$	−720.804	−0.0435963
$650$	0	0
$651$	4040.87	0.243278
$652$	0	0
$653$	3441.98	0.206271	0.103136	−	0.994667i	$-0.467112\pi$
$653$	3441.98	0.206271	0.103136	+	0.994667i	$0.467112\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−4613.60	−0.273963
$658$	0	0
$659$	57.3161	0.00338804	0.00169402	−	0.999999i	$-0.499461\pi$
$659$	57.3161	0.00338804	0.00169402	+	0.999999i	$0.499461\pi$
$660$	0	0
$661$	−196.228	−0.0115467	−0.00577335	−	0.999983i	$-0.501838\pi$
$661$	−196.228	−0.0115467	−0.00577335	+	0.999983i	$0.501838\pi$
$662$	0	0
$663$	−10823.1	−0.633986
$664$	0	0
$665$	0	0
$666$	0	0
$667$	15028.7	0.872434
$668$	0	0
$669$	−12180.9	−0.703947
$670$	0	0
$671$	−350.570	−0.0201693
$672$	0	0
$673$	−21773.5	−1.24711	−0.623556	−	0.781779i	$-0.714312\pi$
$673$	−21773.5	−1.24711	−0.623556	+	0.781779i	$0.714312\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14527.9	−0.824748	−0.412374	−	0.911015i	$-0.635300\pi$
$677$	−14527.9	−0.824748	−0.412374	+	0.911015i	$0.635300\pi$
$678$	0	0
$679$	−3166.88	−0.178989
$680$	0	0
$681$	−1023.73	−0.0576054
$682$	0	0
$683$	−2333.00	−0.130703	−0.0653513	−	0.997862i	$-0.520817\pi$
$683$	−2333.00	−0.130703	−0.0653513	+	0.997862i	$0.520817\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−13511.0	−0.750331
$688$	0	0
$689$	5034.64	0.278381
$690$	0	0
$691$	10962.4	0.603517	0.301759	−	0.953384i	$-0.402426\pi$
$691$	10962.4	0.603517	0.301759	+	0.953384i	$0.402426\pi$
$692$	0	0
$693$	−81.3777	−0.00446073
$694$	0	0
$695$	0	0
$696$	0	0
$697$	22433.1	1.21910
$698$	0	0
$699$	−3602.00	−0.194907
$700$	0	0
$701$	−15041.6	−0.810433	−0.405216	−	0.914221i	$-0.632804\pi$
$701$	−15041.6	−0.810433	−0.405216	+	0.914221i	$0.632804\pi$
$702$	0	0
$703$	−47995.2	−2.57493
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9640.67	−0.512836
$708$	0	0
$709$	35670.3	1.88946	0.944729	−	0.327852i	$-0.106325\pi$
$709$	35670.3	1.88946	0.944729	+	0.327852i	$0.106325\pi$
$710$	0	0
$711$	2160.13	0.113940
$712$	0	0
$713$	−21702.5	−1.13992
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−19268.6	−1.00363
$718$	0	0
$719$	−6897.57	−0.357769	−0.178884	−	0.983870i	$-0.557249\pi$
$719$	−6897.57	−0.357769	−0.178884	+	0.983870i	$0.557249\pi$
$720$	0	0
$721$	3936.63	0.203339
$722$	0	0
$723$	6315.86	0.324882
$724$	0	0
$725$	0	0
$726$	0	0
$727$	19188.3	0.978894	0.489447	−	0.872033i	$-0.337199\pi$
$727$	19188.3	0.978894	0.489447	+	0.872033i	$0.337199\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	19697.8	0.996649
$732$	0	0
$733$	18810.8	0.947876	0.473938	−	0.880558i	$-0.342832\pi$
$733$	18810.8	0.947876	0.473938	+	0.880558i	$0.342832\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1054.07	−0.0526825
$738$	0	0
$739$	−7662.10	−0.381400	−0.190700	−	0.981648i	$-0.561076\pi$
$739$	−7662.10	−0.381400	−0.190700	+	0.981648i	$0.561076\pi$
$740$	0	0
$741$	−13854.8	−0.686866
$742$	0	0
$743$	−5156.24	−0.254595	−0.127297	−	0.991865i	$-0.540630\pi$
$743$	−5156.24	−0.254595	−0.127297	+	0.991865i	$0.540630\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	5679.00	0.278157
$748$	0	0
$749$	−7510.30	−0.366382
$750$	0	0
$751$	−28137.3	−1.36717	−0.683586	−	0.729870i	$-0.739581\pi$
$751$	−28137.3	−1.36717	−0.683586	+	0.729870i	$0.739581\pi$
$752$	0	0
$753$	−4563.74	−0.220866
$754$	0	0
$755$	0	0
$756$	0	0
$757$	17662.8	0.848039	0.424020	−	0.905653i	$-0.360619\pi$
$757$	17662.8	0.848039	0.424020	+	0.905653i	$0.360619\pi$
$758$	0	0
$759$	437.059	0.0209015
$760$	0	0
$761$	−41190.7	−1.96211	−0.981053	−	0.193742i	$-0.937938\pi$
$761$	−41190.7	−1.96211	−0.981053	+	0.193742i	$0.937938\pi$
$762$	0	0
$763$	1762.12	0.0836081
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−15254.0	−0.718109
$768$	0	0
$769$	19122.6	0.896720	0.448360	−	0.893853i	$-0.352008\pi$
$769$	19122.6	0.896720	0.448360	+	0.893853i	$0.352008\pi$
$770$	0	0
$771$	13176.8	0.615498
$772$	0	0
$773$	−27745.0	−1.29097	−0.645484	−	0.763774i	$-0.723344\pi$
$773$	−27745.0	−1.29097	−0.645484	+	0.763774i	$0.723344\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−5965.82	−0.275448
$778$	0	0
$779$	28717.0	1.32079
$780$	0	0
$781$	−261.990	−0.0120035
$782$	0	0
$783$	−4072.10	−0.185856
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−12025.5	−0.544680	−0.272340	−	0.962201i	$-0.587798\pi$
$787$	−12025.5	−0.544680	−0.272340	+	0.962201i	$0.587798\pi$
$788$	0	0
$789$	−23986.9	−1.08233
$790$	0	0
$791$	−3921.17	−0.176259
$792$	0	0
$793$	−7418.92	−0.332224
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−28403.9	−1.26238	−0.631190	−	0.775628i	$-0.717433\pi$
$797$	−28403.9	−1.26238	−0.631190	+	0.775628i	$0.717433\pi$
$798$	0	0
$799$	−13522.3	−0.598729
$800$	0	0
$801$	4513.51	0.199098
$802$	0	0
$803$	749.461	0.0329364
$804$	0	0
$805$	0	0
$806$	0	0
$807$	787.571	0.0343541
$808$	0	0
$809$	411.382	0.0178782	0.00893908	−	0.999960i	$-0.497155\pi$
$809$	411.382	0.0178782	0.00893908	+	0.999960i	$0.497155\pi$
$810$	0	0
$811$	469.428	0.0203254	0.0101627	−	0.999948i	$-0.496765\pi$
$811$	469.428	0.0203254	0.0101627	+	0.999948i	$0.496765\pi$
$812$	0	0
$813$	26346.1	1.13653
$814$	0	0
$815$	0	0
$816$	0	0
$817$	25215.5	1.07978
$818$	0	0
$819$	−1722.15	−0.0734761
$820$	0	0
$821$	−10910.5	−0.463799	−0.231899	−	0.972740i	$-0.574494\pi$
$821$	−10910.5	−0.463799	−0.231899	+	0.972740i	$0.574494\pi$
$822$	0	0
$823$	−14117.9	−0.597958	−0.298979	−	0.954260i	$-0.596646\pi$
$823$	−14117.9	−0.597958	−0.298979	+	0.954260i	$0.596646\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−19048.5	−0.800943	−0.400472	−	0.916309i	$-0.631154\pi$
$827$	−19048.5	−0.800943	−0.400472	+	0.916309i	$0.631154\pi$
$828$	0	0
$829$	22284.8	0.933635	0.466818	−	0.884354i	$-0.345400\pi$
$829$	22284.8	0.933635	0.466818	+	0.884354i	$0.345400\pi$
$830$	0	0
$831$	5840.62	0.243813
$832$	0	0
$833$	35534.9	1.47805
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5880.38	0.242838
$838$	0	0
$839$	−39588.2	−1.62901	−0.814503	−	0.580159i	$-0.802990\pi$
$839$	−39588.2	−1.62901	−0.814503	+	0.580159i	$0.802990\pi$
$840$	0	0
$841$	−1642.79	−0.0673579
$842$	0	0
$843$	2818.38	0.115148
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−8218.48	−0.333401
$848$	0	0
$849$	−13059.4	−0.527914
$850$	0	0
$851$	32040.9	1.29066
$852$	0	0
$853$	−36096.1	−1.44889	−0.724447	−	0.689330i	$-0.757905\pi$
$853$	−36096.1	−1.44889	−0.724447	+	0.689330i	$0.757905\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22517.1	−0.897515	−0.448757	−	0.893654i	$-0.648133\pi$
$857$	−22517.1	−0.897515	−0.448757	+	0.893654i	$0.648133\pi$
$858$	0	0
$859$	−20070.2	−0.797190	−0.398595	−	0.917127i	$-0.630502\pi$
$859$	−20070.2	−0.797190	−0.398595	+	0.917127i	$0.630502\pi$
$860$	0	0
$861$	3569.53	0.141288
$862$	0	0
$863$	−45334.0	−1.78817	−0.894083	−	0.447902i	$-0.852172\pi$
$863$	−45334.0	−1.78817	−0.894083	+	0.447902i	$0.852172\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−26050.0	−1.02042
$868$	0	0
$869$	−350.905	−0.0136981
$870$	0	0
$871$	−22306.6	−0.867774
$872$	0	0
$873$	−4608.53	−0.178666
$874$	0	0
$875$	0	0
$876$	0	0
$877$	25479.8	0.981061	0.490531	−	0.871424i	$-0.336803\pi$
$877$	25479.8	0.981061	0.490531	+	0.871424i	$0.336803\pi$
$878$	0	0
$879$	13256.5	0.508680
$880$	0	0
$881$	26865.2	1.02737	0.513683	−	0.857980i	$-0.328281\pi$
$881$	26865.2	1.02737	0.513683	+	0.857980i	$0.328281\pi$
$882$	0	0
$883$	31954.4	1.21784	0.608920	−	0.793232i	$-0.291603\pi$
$883$	31954.4	1.21784	0.608920	+	0.793232i	$0.291603\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−27421.2	−1.03801	−0.519005	−	0.854771i	$-0.673697\pi$
$887$	−27421.2	−1.03801	−0.519005	+	0.854771i	$0.673697\pi$
$888$	0	0
$889$	−1176.69	−0.0443925
$890$	0	0
$891$	−118.423	−0.00445267
$892$	0	0
$893$	−17310.1	−0.648668
$894$	0	0
$895$	0	0
$896$	0	0
$897$	9249.25	0.344285
$898$	0	0
$899$	−32847.1	−1.21859
$900$	0	0
$901$	18974.1	0.701575
$902$	0	0
$903$	3134.30	0.115507
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−27087.1	−0.991634	−0.495817	−	0.868427i	$-0.665131\pi$
$907$	−27087.1	−0.991634	−0.495817	+	0.868427i	$0.665131\pi$
$908$	0	0
$909$	−14029.4	−0.511909
$910$	0	0
$911$	11780.7	0.428444	0.214222	−	0.976785i	$-0.431278\pi$
$911$	11780.7	0.428444	0.214222	+	0.976785i	$0.431278\pi$
$912$	0	0
$913$	−922.530	−0.0334406
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9513.59	−0.342602
$918$	0	0
$919$	−54875.4	−1.96972	−0.984860	−	0.173351i	$-0.944541\pi$
$919$	−54875.4	−1.96972	−0.984860	+	0.173351i	$0.944541\pi$
$920$	0	0
$921$	5468.75	0.195658
$922$	0	0
$923$	−5544.35	−0.197719
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5728.69	0.202972
$928$	0	0
$929$	−14186.1	−0.501002	−0.250501	−	0.968116i	$-0.580595\pi$
$929$	−14186.1	−0.501002	−0.250501	+	0.968116i	$0.580595\pi$
$930$	0	0
$931$	45488.8	1.60133
$932$	0	0
$933$	20651.9	0.724664
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5461.34	0.190410	0.0952049	−	0.995458i	$-0.469649\pi$
$937$	5461.34	0.190410	0.0952049	+	0.995458i	$0.469649\pi$
$938$	0	0
$939$	11495.5	0.399511
$940$	0	0
$941$	13692.3	0.474341	0.237170	−	0.971468i	$-0.423780\pi$
$941$	13692.3	0.474341	0.237170	+	0.971468i	$0.423780\pi$
$942$	0	0
$943$	−19171.0	−0.662030
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8167.92	0.280276	0.140138	−	0.990132i	$-0.455245\pi$
$947$	8167.92	0.280276	0.140138	+	0.990132i	$0.455245\pi$
$948$	0	0
$949$	15860.4	0.542520
$950$	0	0
$951$	6606.22	0.225259
$952$	0	0
$953$	46746.0	1.58893	0.794465	−	0.607309i	$-0.207751\pi$
$953$	46746.0	1.58893	0.794465	+	0.607309i	$0.207751\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	661.496	0.0223439
$958$	0	0
$959$	3626.02	0.122096
$960$	0	0
$961$	17642.4	0.592204
$962$	0	0
$963$	−10929.2	−0.365720
$964$	0	0
$965$	0	0
$966$	0	0
$967$	23433.9	0.779299	0.389649	−	0.920963i	$-0.372596\pi$
$967$	23433.9	0.779299	0.389649	+	0.920963i	$0.372596\pi$
$968$	0	0
$969$	−52214.6	−1.73104
$970$	0	0
$971$	20510.1	0.677857	0.338929	−	0.940812i	$-0.389935\pi$
$971$	20510.1	0.677857	0.338929	+	0.940812i	$0.389935\pi$
$972$	0	0
$973$	7584.63	0.249899
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−14429.5	−0.472508	−0.236254	−	0.971691i	$-0.575920\pi$
$977$	−14429.5	−0.472508	−0.236254	+	0.971691i	$0.575920\pi$
$978$	0	0
$979$	−733.202	−0.0239359
$980$	0	0
$981$	2564.29	0.0834570
$982$	0	0
$983$	−55279.5	−1.79363	−0.896817	−	0.442402i	$-0.854127\pi$
$983$	−55279.5	−1.79363	−0.896817	+	0.442402i	$0.854127\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2151.65	−0.0693899
$988$	0	0
$989$	−16833.5	−0.541228
$990$	0	0
$991$	46458.6	1.48921	0.744604	−	0.667506i	$-0.232638\pi$
$991$	46458.6	1.48921	0.744604	+	0.667506i	$0.232638\pi$
$992$	0	0
$993$	13886.9	0.443795
$994$	0	0
$995$	0	0
$996$	0	0
$997$	21047.6	0.668591	0.334295	−	0.942468i	$-0.391502\pi$
$997$	21047.6	0.668591	0.334295	+	0.942468i	$0.391502\pi$
$998$	0	0
$999$	−8681.64	−0.274950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.4.a.bn.1.2	yes	3
4.3	odd	2		2400.4.a.bo.1.2	yes	3
5.4	even	2		2400.4.a.bp.1.2	yes	3
20.19	odd	2		2400.4.a.bm.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2400.4.a.bm.1.2	✓	3	20.19	odd	2
2400.4.a.bn.1.2	yes	3	1.1	even	1	trivial
2400.4.a.bo.1.2	yes	3	4.3	odd	2
2400.4.a.bp.1.2	yes	3	5.4	even	2

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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2400.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +6.18460 q^{7} +9.00000 q^{9} -1.46201 q^{11} -30.9398 q^{13} -116.603 q^{17} -149.266 q^{19} -18.5538 q^{21} +99.6477 q^{23} -27.0000 q^{27} +150.818 q^{29} -217.792 q^{31} +4.38604 q^{33} +321.542 q^{37} +92.8195 q^{39} -192.388 q^{41} -168.930 q^{43} +115.968 q^{47} -304.751 q^{49} +349.810 q^{51} -162.724 q^{53} +447.797 q^{57} +493.021 q^{59} +239.785 q^{61} +55.6614 q^{63} +720.968 q^{67} -298.943 q^{69} +179.198 q^{71} -512.622 q^{73} -9.04197 q^{77} +240.014 q^{79} +81.0000 q^{81} +631.000 q^{83} -452.455 q^{87} +501.502 q^{89} -191.350 q^{91} +653.376 q^{93} -512.059 q^{97} -13.1581 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} + 19 q^{7} + 27 q^{9} + 48 q^{11} + 63 q^{13} + 16 q^{17} + 21 q^{19} - 57 q^{21} + 140 q^{23} - 81 q^{27} + 92 q^{29} + 179 q^{31} - 144 q^{33} + 550 q^{37} - 189 q^{39} - 206 q^{41} + 279 q^{43}+ \cdots + 432 q^{99}+O(q^{100}) \) 3 * q - 9 * q^3 + 19 * q^7 + 27 * q^9 + 48 * q^11 + 63 * q^13 + 16 * q^17 + 21 * q^19 - 57 * q^21 + 140 * q^23 - 81 * q^27 + 92 * q^29 + 179 * q^31 - 144 * q^33 + 550 * q^37 - 189 * q^39 - 206 * q^41 + 279 * q^43 + 450 * q^47 + 446 * q^49 - 48 * q^51 - 434 * q^53 - 63 * q^57 + 854 * q^59 - 1103 * q^61 + 171 * q^63 + 555 * q^67 - 420 * q^69 - 970 * q^71 - 722 * q^73 + 1184 * q^77 - 856 * q^79 + 243 * q^81 + 222 * q^83 - 276 * q^87 - 1216 * q^89 - 1025 * q^91 - 537 * q^93 - 89 * q^97 + 432 * q^99

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