LMFDB - Embedded newform 8000.2.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8000,2,Mod(1,8000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8000, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8000 = 2^{6} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8000.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:	$63.8803216170$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.12400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 12x^{2} + 31 $ x^4 - 12*x^2 + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 500)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.86986$ of defining polynomial
Character	$\chi$	$=$	8000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.77367 q^{3} +4.64352 q^{7} +0.145898 q^{9} +O(q^{10})$	$q-1.77367 q^{3} +4.64352 q^{7} +0.145898 q^{9} -4.47214 q^{11} +5.73971 q^{13} -3.54734 q^{17} +1.23607 q^{19} -8.23607 q^{21} +1.09619 q^{23} +5.06223 q^{27} -4.61803 q^{29} +6.00000 q^{31} +7.93208 q^{33} -2.19237 q^{37} -10.1803 q^{39} -2.61803 q^{41} -3.96604 q^{43} -7.51338 q^{47} +14.5623 q^{49} +6.29180 q^{51} +11.4794 q^{53} -2.19237 q^{57} +12.4721 q^{59} -1.14590 q^{61} +0.677481 q^{63} -2.19237 q^{67} -1.94427 q^{69} +5.23607 q^{71} -5.73971 q^{73} -20.7665 q^{77} +9.70820 q^{79} -9.41641 q^{81} +6.83590 q^{83} +8.19086 q^{87} -11.3262 q^{89} +26.6525 q^{91} -10.6420 q^{93} -5.73971 q^{97} -0.652476 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 14 q^{9}+O(q^{10}) $ 4 * q + 14 * q^9	$ 4 q + 14 q^{9} - 4 q^{19} - 24 q^{21} - 14 q^{29} + 24 q^{31} + 4 q^{39} - 6 q^{41} + 18 q^{49} + 52 q^{51} + 32 q^{59} - 18 q^{61} + 28 q^{69} + 12 q^{71} + 12 q^{79} + 16 q^{81} - 14 q^{89} + 44 q^{91} + 60 q^{99}+O(q^{100}) $ 4 * q + 14 * q^9 - 4 * q^19 - 24 * q^21 - 14 * q^29 + 24 * q^31 + 4 * q^39 - 6 * q^41 + 18 * q^49 + 52 * q^51 + 32 * q^59 - 18 * q^61 + 28 * q^69 + 12 * q^71 + 12 * q^79 + 16 * q^81 - 14 * q^89 + 44 * q^91 + 60 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.77367	−1.02403	−0.512014	−	0.858977i	$-0.671100\pi$
$3$	−1.77367	−1.02403	−0.512014	+	0.858977i	$0.671100\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.64352	1.75509	0.877543	−	0.479497i	$-0.159181\pi$
$7$	4.64352	1.75509	0.877543	+	0.479497i	$0.159181\pi$
$8$	0	0
$9$	0.145898	0.0486327
$10$	0	0
$11$	−4.47214	−1.34840	−0.674200	−	0.738549i	$-0.735511\pi$
$11$	−4.47214	−1.34840	−0.674200	+	0.738549i	$0.735511\pi$
$12$	0	0
$13$	5.73971	1.59191	0.795955	−	0.605356i	$-0.206969\pi$
$13$	5.73971	1.59191	0.795955	+	0.605356i	$0.206969\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.54734	−0.860355	−0.430178	−	0.902744i	$-0.641549\pi$
$17$	−3.54734	−0.860355	−0.430178	+	0.902744i	$0.641549\pi$
$18$	0	0
$19$	1.23607	0.283573	0.141787	−	0.989897i	$-0.454715\pi$
$19$	1.23607	0.283573	0.141787	+	0.989897i	$0.454715\pi$
$20$	0	0
$21$	−8.23607	−1.79726
$22$	0	0
$23$	1.09619	0.228571	0.114285	−	0.993448i	$-0.463542\pi$
$23$	1.09619	0.228571	0.114285	+	0.993448i	$0.463542\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.06223	0.974226
$28$	0	0
$29$	−4.61803	−0.857547	−0.428774	−	0.903412i	$-0.641054\pi$
$29$	−4.61803	−0.857547	−0.428774	+	0.903412i	$0.641054\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	0	0
$33$	7.93208	1.38080
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.19237	−0.360424	−0.180212	−	0.983628i	$-0.557678\pi$
$37$	−2.19237	−0.360424	−0.180212	+	0.983628i	$0.557678\pi$
$38$	0	0
$39$	−10.1803	−1.63016
$40$	0	0
$41$	−2.61803	−0.408868	−0.204434	−	0.978880i	$-0.565535\pi$
$41$	−2.61803	−0.408868	−0.204434	+	0.978880i	$0.565535\pi$
$42$	0	0
$43$	−3.96604	−0.604816	−0.302408	−	0.953179i	$-0.597790\pi$
$43$	−3.96604	−0.604816	−0.302408	+	0.953179i	$0.597790\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.51338	−1.09594	−0.547969	−	0.836498i	$-0.684599\pi$
$47$	−7.51338	−1.09594	−0.547969	+	0.836498i	$0.684599\pi$
$48$	0	0
$49$	14.5623	2.08033
$50$	0	0
$51$	6.29180	0.881028
$52$	0	0
$53$	11.4794	1.57682	0.788410	−	0.615150i	$-0.210905\pi$
$53$	11.4794	1.57682	0.788410	+	0.615150i	$0.210905\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.19237	−0.290387
$58$	0	0
$59$	12.4721	1.62373	0.811867	−	0.583843i	$-0.198451\pi$
$59$	12.4721	1.62373	0.811867	+	0.583843i	$0.198451\pi$
$60$	0	0
$61$	−1.14590	−0.146717	−0.0733586	−	0.997306i	$-0.523372\pi$
$61$	−1.14590	−0.146717	−0.0733586	+	0.997306i	$0.523372\pi$
$62$	0	0
$63$	0.677481	0.0853546
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.19237	−0.267841	−0.133921	−	0.990992i	$-0.542757\pi$
$67$	−2.19237	−0.267841	−0.133921	+	0.990992i	$0.542757\pi$
$68$	0	0
$69$	−1.94427	−0.234063
$70$	0	0
$71$	5.23607	0.621407	0.310703	−	0.950507i	$-0.399435\pi$
$71$	5.23607	0.621407	0.310703	+	0.950507i	$0.399435\pi$
$72$	0	0
$73$	−5.73971	−0.671782	−0.335891	−	0.941901i	$-0.609037\pi$
$73$	−5.73971	−0.671782	−0.335891	+	0.941901i	$0.609037\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−20.7665	−2.36656
$78$	0	0
$79$	9.70820	1.09226	0.546129	−	0.837701i	$-0.316101\pi$
$79$	9.70820	1.09226	0.546129	+	0.837701i	$0.316101\pi$
$80$	0	0
$81$	−9.41641	−1.04627
$82$	0	0
$83$	6.83590	0.750337	0.375169	−	0.926957i	$-0.377585\pi$
$83$	6.83590	0.750337	0.375169	+	0.926957i	$0.377585\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.19086	0.878152
$88$	0	0
$89$	−11.3262	−1.20058	−0.600289	−	0.799783i	$-0.704948\pi$
$89$	−11.3262	−1.20058	−0.600289	+	0.799783i	$0.704948\pi$
$90$	0	0
$91$	26.6525	2.79394
$92$	0	0
$93$	−10.6420	−1.10352
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−5.73971	−0.582779	−0.291390	−	0.956604i	$-0.594118\pi$
$97$	−5.73971	−0.582779	−0.291390	+	0.956604i	$0.594118\pi$
$98$	0	0
$99$	−0.652476	−0.0655763
$100$	0	0
$101$	19.3262	1.92303	0.961516	−	0.274748i	$-0.0885944\pi$
$101$	19.3262	1.92303	0.961516	+	0.274748i	$0.0885944\pi$
$102$	0	0
$103$	9.28705	0.915080	0.457540	−	0.889189i	$-0.348731\pi$
$103$	9.28705	0.915080	0.457540	+	0.889189i	$0.348731\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.418706	−0.0404779	−0.0202389	−	0.999795i	$-0.506443\pi$
$107$	−0.418706	−0.0404779	−0.0202389	+	0.999795i	$0.506443\pi$
$108$	0	0
$109$	−13.3820	−1.28176	−0.640880	−	0.767641i	$-0.721430\pi$
$109$	−13.3820	−1.28176	−0.640880	+	0.767641i	$0.721430\pi$
$110$	0	0
$111$	3.88854	0.369084
$112$	0	0
$113$	17.2191	1.61984	0.809920	−	0.586541i	$-0.199511\pi$
$113$	17.2191	1.61984	0.809920	+	0.586541i	$0.199511\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.837412	0.0774188
$118$	0	0
$119$	−16.4721	−1.51000
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	4.64352	0.418692
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−14.6080	−1.29625	−0.648127	−	0.761532i	$-0.724448\pi$
$127$	−14.6080	−1.29625	−0.648127	+	0.761532i	$0.724448\pi$
$128$	0	0
$129$	7.03444	0.619348
$130$	0	0
$131$	7.23607	0.632218	0.316109	−	0.948723i	$-0.397623\pi$
$131$	7.23607	0.632218	0.316109	+	0.948723i	$0.397623\pi$
$132$	0	0
$133$	5.73971	0.497696
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.0268	−1.28382	−0.641911	−	0.766779i	$-0.721858\pi$
$137$	−15.0268	−1.28382	−0.641911	+	0.766779i	$0.721858\pi$
$138$	0	0
$139$	2.29180	0.194388	0.0971938	−	0.995265i	$-0.469013\pi$
$139$	2.29180	0.194388	0.0971938	+	0.995265i	$0.469013\pi$
$140$	0	0
$141$	13.3262	1.12227
$142$	0	0
$143$	−25.6688	−2.14653
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−25.8287	−2.13031
$148$	0	0
$149$	−9.32624	−0.764035	−0.382018	−	0.924155i	$-0.624771\pi$
$149$	−9.32624	−0.764035	−0.382018	+	0.924155i	$0.624771\pi$
$150$	0	0
$151$	−1.70820	−0.139012	−0.0695058	−	0.997582i	$-0.522142\pi$
$151$	−1.70820	−0.139012	−0.0695058	+	0.997582i	$0.522142\pi$
$152$	0	0
$153$	−0.517549	−0.0418414
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.0268	1.19927	0.599633	−	0.800275i	$-0.295313\pi$
$157$	15.0268	1.19927	0.599633	+	0.800275i	$0.295313\pi$
$158$	0	0
$159$	−20.3607	−1.61471
$160$	0	0
$161$	5.09017	0.401162
$162$	0	0
$163$	15.2855	1.19726	0.598628	−	0.801027i	$-0.295713\pi$
$163$	15.2855	1.19726	0.598628	+	0.801027i	$0.295713\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.77367	0.137251	0.0686253	−	0.997643i	$-0.478139\pi$
$167$	1.77367	0.137251	0.0686253	+	0.997643i	$0.478139\pi$
$168$	0	0
$169$	19.9443	1.53417
$170$	0	0
$171$	0.180340	0.0137909
$172$	0	0
$173$	−22.1214	−1.68186	−0.840931	−	0.541143i	$-0.817992\pi$
$173$	−22.1214	−1.68186	−0.840931	+	0.541143i	$0.817992\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−22.1214	−1.66275
$178$	0	0
$179$	−3.52786	−0.263685	−0.131842	−	0.991271i	$-0.542089\pi$
$179$	−3.52786	−0.263685	−0.131842	+	0.991271i	$0.542089\pi$
$180$	0	0
$181$	−2.43769	−0.181192	−0.0905962	−	0.995888i	$-0.528877\pi$
$181$	−2.43769	−0.181192	−0.0905962	+	0.995888i	$0.528877\pi$
$182$	0	0
$183$	2.03244	0.150242
$184$	0	0
$185$	0	0
$186$	0	0
$187$	15.8642	1.16010
$188$	0	0
$189$	23.5066	1.70985
$190$	0	0
$191$	8.65248	0.626071	0.313036	−	0.949741i	$-0.398654\pi$
$191$	8.65248	0.626071	0.313036	+	0.949741i	$0.398654\pi$
$192$	0	0
$193$	−9.28705	−0.668496	−0.334248	−	0.942485i	$-0.608482\pi$
$193$	−9.28705	−0.668496	−0.334248	+	0.942485i	$0.608482\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.6420	−0.758212	−0.379106	−	0.925353i	$-0.623768\pi$
$197$	−10.6420	−0.758212	−0.379106	+	0.925353i	$0.623768\pi$
$198$	0	0
$199$	19.7082	1.39708	0.698539	−	0.715572i	$-0.253834\pi$
$199$	19.7082	1.39708	0.698539	+	0.715572i	$0.253834\pi$
$200$	0	0
$201$	3.88854	0.274277
$202$	0	0
$203$	−21.4439	−1.50507
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.159932	0.0111160
$208$	0	0
$209$	−5.52786	−0.382370
$210$	0	0
$211$	−2.29180	−0.157774	−0.0788869	−	0.996884i	$-0.525137\pi$
$211$	−2.29180	−0.157774	−0.0788869	+	0.996884i	$0.525137\pi$
$212$	0	0
$213$	−9.28705	−0.636338
$214$	0	0
$215$	0	0
$216$	0	0
$217$	27.8611	1.89134
$218$	0	0
$219$	10.1803	0.687924
$220$	0	0
$221$	−20.3607	−1.36961
$222$	0	0
$223$	16.8004	1.12504	0.562520	−	0.826784i	$-0.309832\pi$
$223$	16.8004	1.12504	0.562520	+	0.826784i	$0.309832\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.83590	−0.453714	−0.226857	−	0.973928i	$-0.572845\pi$
$227$	−6.83590	−0.453714	−0.226857	+	0.973928i	$0.572845\pi$
$228$	0	0
$229$	−4.85410	−0.320768	−0.160384	−	0.987055i	$-0.551273\pi$
$229$	−4.85410	−0.320768	−0.160384	+	0.987055i	$0.551273\pi$
$230$	0	0
$231$	36.8328	2.42342
$232$	0	0
$233$	−19.4115	−1.27169	−0.635845	−	0.771817i	$-0.719348\pi$
$233$	−19.4115	−1.27169	−0.635845	+	0.771817i	$0.719348\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−17.2191	−1.11850
$238$	0	0
$239$	14.2918	0.924459	0.462230	−	0.886760i	$-0.347050\pi$
$239$	14.2918	0.924459	0.462230	+	0.886760i	$0.347050\pi$
$240$	0	0
$241$	20.3820	1.31292	0.656459	−	0.754362i	$-0.272054\pi$
$241$	20.3820	1.31292	0.656459	+	0.754362i	$0.272054\pi$
$242$	0	0
$243$	1.51489	0.0971805
$244$	0	0
$245$	0	0
$246$	0	0
$247$	7.09467	0.451423
$248$	0	0
$249$	−12.1246	−0.768366
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−4.90230	−0.308205
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.1893	0.885107	0.442553	−	0.896742i	$-0.354073\pi$
$257$	14.1893	0.885107	0.442553	+	0.896742i	$0.354073\pi$
$258$	0	0
$259$	−10.1803	−0.632576
$260$	0	0
$261$	−0.673762	−0.0417048
$262$	0	0
$263$	−1.77367	−0.109369	−0.0546845	−	0.998504i	$-0.517415\pi$
$263$	−1.77367	−0.109369	−0.0546845	+	0.998504i	$0.517415\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	20.0890	1.22943
$268$	0	0
$269$	10.9443	0.667284	0.333642	−	0.942700i	$-0.391722\pi$
$269$	10.9443	0.667284	0.333642	+	0.942700i	$0.391722\pi$
$270$	0	0
$271$	5.52786	0.335794	0.167897	−	0.985805i	$-0.446302\pi$
$271$	5.52786	0.335794	0.167897	+	0.985805i	$0.446302\pi$
$272$	0	0
$273$	−47.2726	−2.86107
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−19.4115	−1.16632	−0.583162	−	0.812356i	$-0.698185\pi$
$277$	−19.4115	−1.16632	−0.583162	+	0.812356i	$0.698185\pi$
$278$	0	0
$279$	0.875388	0.0524081
$280$	0	0
$281$	−11.0344	−0.658260	−0.329130	−	0.944285i	$-0.606755\pi$
$281$	−11.0344	−0.658260	−0.329130	+	0.944285i	$0.606755\pi$
$282$	0	0
$283$	−9.28705	−0.552058	−0.276029	−	0.961149i	$-0.589019\pi$
$283$	−9.28705	−0.552058	−0.276029	+	0.961149i	$0.589019\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.1569	−0.717599
$288$	0	0
$289$	−4.41641	−0.259789
$290$	0	0
$291$	10.1803	0.596782
$292$	0	0
$293$	−12.8344	−0.749793	−0.374896	−	0.927067i	$-0.622322\pi$
$293$	−12.8344	−0.749793	−0.374896	+	0.927067i	$0.622322\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−22.6390	−1.31365
$298$	0	0
$299$	6.29180	0.363864
$300$	0	0
$301$	−18.4164	−1.06150
$302$	0	0
$303$	−34.2783	−1.96924
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3.28856	0.187688	0.0938441	−	0.995587i	$-0.470084\pi$
$307$	3.28856	0.187688	0.0938441	+	0.995587i	$0.470084\pi$
$308$	0	0
$309$	−16.4721	−0.937067
$310$	0	0
$311$	25.4164	1.44123	0.720616	−	0.693334i	$-0.243859\pi$
$311$	25.4164	1.44123	0.720616	+	0.693334i	$0.243859\pi$
$312$	0	0
$313$	18.5741	1.04987	0.524935	−	0.851142i	$-0.324090\pi$
$313$	18.5741	1.04987	0.524935	+	0.851142i	$0.324090\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.3817	0.920089	0.460044	−	0.887896i	$-0.347833\pi$
$317$	16.3817	0.920089	0.460044	+	0.887896i	$0.347833\pi$
$318$	0	0
$319$	20.6525	1.15632
$320$	0	0
$321$	0.742646	0.0414504
$322$	0	0
$323$	−4.38475	−0.243974
$324$	0	0
$325$	0	0
$326$	0	0
$327$	23.7352	1.31256
$328$	0	0
$329$	−34.8885	−1.92347
$330$	0	0
$331$	10.2918	0.565688	0.282844	−	0.959166i	$-0.408722\pi$
$331$	10.2918	0.565688	0.282844	+	0.959166i	$0.408722\pi$
$332$	0	0
$333$	−0.319863	−0.0175284
$334$	0	0
$335$	0	0
$336$	0	0
$337$	27.8611	1.51769	0.758846	−	0.651270i	$-0.225763\pi$
$337$	27.8611	1.51769	0.758846	+	0.651270i	$0.225763\pi$
$338$	0	0
$339$	−30.5410	−1.65876
$340$	0	0
$341$	−26.8328	−1.45308
$342$	0	0
$343$	35.1157	1.89607
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.2474	1.40903	0.704517	−	0.709687i	$-0.251164\pi$
$347$	26.2474	1.40903	0.704517	+	0.709687i	$0.251164\pi$
$348$	0	0
$349$	25.9787	1.39061	0.695304	−	0.718715i	$-0.255270\pi$
$349$	25.9787	1.39061	0.695304	+	0.718715i	$0.255270\pi$
$350$	0	0
$351$	29.0557	1.55088
$352$	0	0
$353$	5.73971	0.305494	0.152747	−	0.988265i	$-0.451188\pi$
$353$	5.73971	0.305494	0.152747	+	0.988265i	$0.451188\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	29.2161	1.54628
$358$	0	0
$359$	−14.9443	−0.788729	−0.394364	−	0.918954i	$-0.629035\pi$
$359$	−14.9443	−0.788729	−0.394364	+	0.918954i	$0.629035\pi$
$360$	0	0
$361$	−17.4721	−0.919586
$362$	0	0
$363$	−15.9630	−0.837841
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.28856	0.171662	0.0858308	−	0.996310i	$-0.472646\pi$
$367$	3.28856	0.171662	0.0858308	+	0.996310i	$0.472646\pi$
$368$	0	0
$369$	−0.381966	−0.0198844
$370$	0	0
$371$	53.3050	2.76746
$372$	0	0
$373$	−6.57712	−0.340550	−0.170275	−	0.985397i	$-0.554466\pi$
$373$	−6.57712	−0.340550	−0.170275	+	0.985397i	$0.554466\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−26.5062	−1.36514
$378$	0	0
$379$	−24.3607	−1.25132	−0.625662	−	0.780094i	$-0.715171\pi$
$379$	−24.3607	−1.25132	−0.625662	+	0.780094i	$0.715171\pi$
$380$	0	0
$381$	25.9098	1.32740
$382$	0	0
$383$	23.3775	1.19454	0.597268	−	0.802041i	$-0.296253\pi$
$383$	23.3775	1.19454	0.597268	+	0.802041i	$0.296253\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.578638	−0.0294138
$388$	0	0
$389$	−12.2148	−0.619314	−0.309657	−	0.950848i	$-0.600214\pi$
$389$	−12.2148	−0.619314	−0.309657	+	0.950848i	$0.600214\pi$
$390$	0	0
$391$	−3.88854	−0.196652
$392$	0	0
$393$	−12.8344	−0.647409
$394$	0	0
$395$	0	0
$396$	0	0
$397$	21.6039	1.08427	0.542134	−	0.840292i	$-0.317617\pi$
$397$	21.6039	1.08427	0.542134	+	0.840292i	$0.317617\pi$
$398$	0	0
$399$	−10.1803	−0.509654
$400$	0	0
$401$	−20.4508	−1.02127	−0.510633	−	0.859799i	$-0.670589\pi$
$401$	−20.4508	−1.02127	−0.510633	+	0.859799i	$0.670589\pi$
$402$	0	0
$403$	34.4383	1.71549
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.80460	0.485996
$408$	0	0
$409$	19.6180	0.970049	0.485025	−	0.874500i	$-0.338811\pi$
$409$	19.6180	0.970049	0.485025	+	0.874500i	$0.338811\pi$
$410$	0	0
$411$	26.6525	1.31467
$412$	0	0
$413$	57.9147	2.84979
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4.06489	−0.199058
$418$	0	0
$419$	16.3607	0.799272	0.399636	−	0.916674i	$-0.369137\pi$
$419$	16.3607	0.799272	0.399636	+	0.916674i	$0.369137\pi$
$420$	0	0
$421$	8.79837	0.428807	0.214403	−	0.976745i	$-0.431219\pi$
$421$	8.79837	0.428807	0.214403	+	0.976745i	$0.431219\pi$
$422$	0	0
$423$	−1.09619	−0.0532984
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−5.32100	−0.257501
$428$	0	0
$429$	45.5279	2.19811
$430$	0	0
$431$	23.2361	1.11924	0.559621	−	0.828749i	$-0.310947\pi$
$431$	23.2361	1.11924	0.559621	+	0.828749i	$0.310947\pi$
$432$	0	0
$433$	11.9970	0.576538	0.288269	−	0.957550i	$-0.406920\pi$
$433$	11.9970	0.576538	0.288269	+	0.957550i	$0.406920\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.35496	0.0648166
$438$	0	0
$439$	23.1246	1.10368	0.551839	−	0.833951i	$-0.313926\pi$
$439$	23.1246	1.10368	0.551839	+	0.833951i	$0.313926\pi$
$440$	0	0
$441$	2.12461	0.101172
$442$	0	0
$443$	−9.02827	−0.428946	−0.214473	−	0.976730i	$-0.568803\pi$
$443$	−9.02827	−0.428946	−0.214473	+	0.976730i	$0.568803\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	16.5416	0.782393
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	11.7082	0.551318
$452$	0	0
$453$	3.02979	0.142352
$454$	0	0
$455$	0	0
$456$	0	0
$457$	37.1482	1.73772	0.868859	−	0.495059i	$-0.164854\pi$
$457$	37.1482	1.73772	0.868859	+	0.495059i	$0.164854\pi$
$458$	0	0
$459$	−17.9574	−0.838181
$460$	0	0
$461$	29.7984	1.38785	0.693924	−	0.720048i	$-0.255880\pi$
$461$	29.7984	1.38785	0.693924	+	0.720048i	$0.255880\pi$
$462$	0	0
$463$	−21.8627	−1.01604	−0.508022	−	0.861344i	$-0.669623\pi$
$463$	−21.8627	−1.01604	−0.508022	+	0.861344i	$0.669623\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−31.1497	−1.44144	−0.720718	−	0.693228i	$-0.756188\pi$
$467$	−31.1497	−1.44144	−0.720718	+	0.693228i	$0.756188\pi$
$468$	0	0
$469$	−10.1803	−0.470084
$470$	0	0
$471$	−26.6525	−1.22808
$472$	0	0
$473$	17.7367	0.815533
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.67482	0.0766850
$478$	0	0
$479$	14.2918	0.653009	0.326504	−	0.945196i	$-0.394129\pi$
$479$	14.2918	0.653009	0.326504	+	0.945196i	$0.394129\pi$
$480$	0	0
$481$	−12.5836	−0.573762
$482$	0	0
$483$	−9.02827	−0.410801
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−5.32100	−0.241118	−0.120559	−	0.992706i	$-0.538469\pi$
$487$	−5.32100	−0.241118	−0.120559	+	0.992706i	$0.538469\pi$
$488$	0	0
$489$	−27.1115	−1.22602
$490$	0	0
$491$	−6.18034	−0.278915	−0.139457	−	0.990228i	$-0.544536\pi$
$491$	−6.18034	−0.278915	−0.139457	+	0.990228i	$0.544536\pi$
$492$	0	0
$493$	16.3817	0.737795
$494$	0	0
$495$	0	0
$496$	0	0
$497$	24.3138	1.09062
$498$	0	0
$499$	39.1246	1.75146	0.875729	−	0.482803i	$-0.160381\pi$
$499$	39.1246	1.75146	0.875729	+	0.482803i	$0.160381\pi$
$500$	0	0
$501$	−3.14590	−0.140548
$502$	0	0
$503$	35.2146	1.57014	0.785070	−	0.619407i	$-0.212627\pi$
$503$	35.2146	1.57014	0.785070	+	0.619407i	$0.212627\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−35.3745	−1.57104
$508$	0	0
$509$	10.5836	0.469109	0.234555	−	0.972103i	$-0.424637\pi$
$509$	10.5836	0.469109	0.234555	+	0.972103i	$0.424637\pi$
$510$	0	0
$511$	−26.6525	−1.17904
$512$	0	0
$513$	6.25726	0.276265
$514$	0	0
$515$	0	0
$516$	0	0
$517$	33.6008	1.47776
$518$	0	0
$519$	39.2361	1.72227
$520$	0	0
$521$	−12.9098	−0.565590	−0.282795	−	0.959180i	$-0.591262\pi$
$521$	−12.9098	−0.565590	−0.282795	+	0.959180i	$0.591262\pi$
$522$	0	0
$523$	7.51338	0.328537	0.164269	−	0.986416i	$-0.447474\pi$
$523$	7.51338	0.328537	0.164269	+	0.986416i	$0.447474\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−21.2840	−0.927146
$528$	0	0
$529$	−21.7984	−0.947755
$530$	0	0
$531$	1.81966	0.0789665
$532$	0	0
$533$	−15.0268	−0.650881
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6.25726	0.270021
$538$	0	0
$539$	−65.1246	−2.80512
$540$	0	0
$541$	−33.2705	−1.43041	−0.715205	−	0.698914i	$-0.753667\pi$
$541$	−33.2705	−1.43041	−0.715205	+	0.698914i	$0.753667\pi$
$542$	0	0
$543$	4.32366	0.185546
$544$	0	0
$545$	0	0
$546$	0	0
$547$	24.5726	1.05065	0.525324	−	0.850902i	$-0.323944\pi$
$547$	24.5726	1.05065	0.525324	+	0.850902i	$0.323944\pi$
$548$	0	0
$549$	−0.167184	−0.00713525
$550$	0	0
$551$	−5.70820	−0.243178
$552$	0	0
$553$	45.0803	1.91701
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−23.7963	−1.00828	−0.504140	−	0.863622i	$-0.668190\pi$
$557$	−23.7963	−1.00828	−0.504140	+	0.863622i	$0.668190\pi$
$558$	0	0
$559$	−22.7639	−0.962812
$560$	0	0
$561$	−28.1378	−1.18798
$562$	0	0
$563$	13.6718	0.576197	0.288099	−	0.957601i	$-0.406977\pi$
$563$	13.6718	0.576197	0.288099	+	0.957601i	$0.406977\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−43.7253	−1.83629
$568$	0	0
$569$	−26.9098	−1.12812	−0.564059	−	0.825734i	$-0.690761\pi$
$569$	−26.9098	−1.12812	−0.564059	+	0.825734i	$0.690761\pi$
$570$	0	0
$571$	44.3607	1.85644	0.928218	−	0.372036i	$-0.121340\pi$
$571$	44.3607	1.85644	0.928218	+	0.372036i	$0.121340\pi$
$572$	0	0
$573$	−15.3466	−0.641114
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.70992	0.112816	0.0564078	−	0.998408i	$-0.482035\pi$
$577$	2.70992	0.112816	0.0564078	+	0.998408i	$0.482035\pi$
$578$	0	0
$579$	16.4721	0.684559
$580$	0	0
$581$	31.7426	1.31691
$582$	0	0
$583$	−51.3375	−2.12618
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−34.9558	−1.44278	−0.721390	−	0.692529i	$-0.756497\pi$
$587$	−34.9558	−1.44278	−0.721390	+	0.692529i	$0.756497\pi$
$588$	0	0
$589$	7.41641	0.305588
$590$	0	0
$591$	18.8754	0.776430
$592$	0	0
$593$	13.6718	0.561433	0.280717	−	0.959791i	$-0.409428\pi$
$593$	13.6718	0.561433	0.280717	+	0.959791i	$0.409428\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−34.9558	−1.43065
$598$	0	0
$599$	19.5967	0.800701	0.400351	−	0.916362i	$-0.368888\pi$
$599$	19.5967	0.800701	0.400351	+	0.916362i	$0.368888\pi$
$600$	0	0
$601$	42.6869	1.74124	0.870618	−	0.491960i	$-0.163719\pi$
$601$	42.6869	1.74124	0.870618	+	0.491960i	$0.163719\pi$
$602$	0	0
$603$	−0.319863	−0.0130258
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−27.8611	−1.13085	−0.565424	−	0.824800i	$-0.691288\pi$
$607$	−27.8611	−1.13085	−0.565424	+	0.824800i	$0.691288\pi$
$608$	0	0
$609$	38.0344	1.54123
$610$	0	0
$611$	−43.1246	−1.74464
$612$	0	0
$613$	2.70992	0.109453	0.0547264	−	0.998501i	$-0.482571\pi$
$613$	2.70992	0.109453	0.0547264	+	0.998501i	$0.482571\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.1214	0.890575	0.445288	−	0.895388i	$-0.353101\pi$
$617$	22.1214	0.890575	0.445288	+	0.895388i	$0.353101\pi$
$618$	0	0
$619$	2.76393	0.111092	0.0555459	−	0.998456i	$-0.482310\pi$
$619$	2.76393	0.111092	0.0555459	+	0.998456i	$0.482310\pi$
$620$	0	0
$621$	5.54915	0.222680
$622$	0	0
$623$	−52.5936	−2.10712
$624$	0	0
$625$	0	0
$626$	0	0
$627$	9.80460	0.391558
$628$	0	0
$629$	7.77709	0.310093
$630$	0	0
$631$	28.8328	1.14782	0.573908	−	0.818920i	$-0.305427\pi$
$631$	28.8328	1.14782	0.573908	+	0.818920i	$0.305427\pi$
$632$	0	0
$633$	4.06489	0.161565
$634$	0	0
$635$	0	0
$636$	0	0
$637$	83.5834	3.31170
$638$	0	0
$639$	0.763932	0.0302207
$640$	0	0
$641$	25.7426	1.01677	0.508387	−	0.861129i	$-0.330242\pi$
$641$	25.7426	1.01677	0.508387	+	0.861129i	$0.330242\pi$
$642$	0	0
$643$	39.5993	1.56165	0.780823	−	0.624753i	$-0.214800\pi$
$643$	39.5993	1.56165	0.780823	+	0.624753i	$0.214800\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−23.4764	−0.922952	−0.461476	−	0.887153i	$-0.652680\pi$
$647$	−23.4764	−0.922952	−0.461476	+	0.887153i	$0.652680\pi$
$648$	0	0
$649$	−55.7771	−2.18944
$650$	0	0
$651$	−49.4164	−1.93678
$652$	0	0
$653$	39.3406	1.53952	0.769758	−	0.638336i	$-0.220377\pi$
$653$	39.3406	1.53952	0.769758	+	0.638336i	$0.220377\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−0.837412	−0.0326706
$658$	0	0
$659$	−20.0689	−0.781773	−0.390886	−	0.920439i	$-0.627831\pi$
$659$	−20.0689	−0.781773	−0.390886	+	0.920439i	$0.627831\pi$
$660$	0	0
$661$	−6.61803	−0.257412	−0.128706	−	0.991683i	$-0.541082\pi$
$661$	−6.61803	−0.257412	−0.128706	+	0.991683i	$0.541082\pi$
$662$	0	0
$663$	36.1131	1.40252
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.06223	−0.196010
$668$	0	0
$669$	−29.7984	−1.15207
$670$	0	0
$671$	5.12461	0.197833
$672$	0	0
$673$	27.0237	1.04169	0.520844	−	0.853652i	$-0.325617\pi$
$673$	27.0237	1.04169	0.520844	+	0.853652i	$0.325617\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.09467	0.272670	0.136335	−	0.990663i	$-0.456468\pi$
$677$	7.09467	0.272670	0.136335	+	0.990663i	$0.456468\pi$
$678$	0	0
$679$	−26.6525	−1.02283
$680$	0	0
$681$	12.1246	0.464616
$682$	0	0
$683$	−1.77367	−0.0678675	−0.0339338	−	0.999424i	$-0.510804\pi$
$683$	−1.77367	−0.0678675	−0.0339338	+	0.999424i	$0.510804\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	8.60957	0.328475
$688$	0	0
$689$	65.8885	2.51015
$690$	0	0
$691$	33.7082	1.28232	0.641160	−	0.767407i	$-0.278453\pi$
$691$	33.7082	1.28232	0.641160	+	0.767407i	$0.278453\pi$
$692$	0	0
$693$	−3.02979	−0.115092
$694$	0	0
$695$	0	0
$696$	0	0
$697$	9.28705	0.351772
$698$	0	0
$699$	34.4296	1.30225
$700$	0	0
$701$	7.88854	0.297946	0.148973	−	0.988841i	$-0.452403\pi$
$701$	7.88854	0.297946	0.148973	+	0.988841i	$0.452403\pi$
$702$	0	0
$703$	−2.70992	−0.102207
$704$	0	0
$705$	0	0
$706$	0	0
$707$	89.7418	3.37509
$708$	0	0
$709$	−4.49342	−0.168754	−0.0843770	−	0.996434i	$-0.526890\pi$
$709$	−4.49342	−0.168754	−0.0843770	+	0.996434i	$0.526890\pi$
$710$	0	0
$711$	1.41641	0.0531194
$712$	0	0
$713$	6.57712	0.246315
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−25.3489	−0.946672
$718$	0	0
$719$	−47.0132	−1.75329	−0.876647	−	0.481133i	$-0.840225\pi$
$719$	−47.0132	−1.75329	−0.876647	+	0.481133i	$0.840225\pi$
$720$	0	0
$721$	43.1246	1.60604
$722$	0	0
$723$	−36.1508	−1.34446
$724$	0	0
$725$	0	0
$726$	0	0
$727$	29.1173	1.07990	0.539950	−	0.841697i	$-0.318443\pi$
$727$	29.1173	1.07990	0.539950	+	0.841697i	$0.318443\pi$
$728$	0	0
$729$	25.5623	0.946752
$730$	0	0
$731$	14.0689	0.520356
$732$	0	0
$733$	−15.5443	−0.574142	−0.287071	−	0.957909i	$-0.592682\pi$
$733$	−15.5443	−0.574142	−0.287071	+	0.957909i	$0.592682\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.80460	0.361157
$738$	0	0
$739$	18.2918	0.672875	0.336437	−	0.941706i	$-0.390778\pi$
$739$	18.2918	0.672875	0.336437	+	0.941706i	$0.390778\pi$
$740$	0	0
$741$	−12.5836	−0.462270
$742$	0	0
$743$	16.3817	0.600987	0.300493	−	0.953784i	$-0.402849\pi$
$743$	16.3817	0.600987	0.300493	+	0.953784i	$0.402849\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.997344	0.0364909
$748$	0	0
$749$	−1.94427	−0.0710421
$750$	0	0
$751$	−2.29180	−0.0836288	−0.0418144	−	0.999125i	$-0.513314\pi$
$751$	−2.29180	−0.0836288	−0.0418144	+	0.999125i	$0.513314\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	4.06489	0.147741	0.0738704	−	0.997268i	$-0.476465\pi$
$757$	4.06489	0.147741	0.0738704	+	0.997268i	$0.476465\pi$
$758$	0	0
$759$	8.69505	0.315610
$760$	0	0
$761$	36.2705	1.31480	0.657402	−	0.753540i	$-0.271655\pi$
$761$	36.2705	1.31480	0.657402	+	0.753540i	$0.271655\pi$
$762$	0	0
$763$	−62.1395	−2.24960
$764$	0	0
$765$	0	0
$766$	0	0
$767$	71.5864	2.58484
$768$	0	0
$769$	14.4377	0.520637	0.260318	−	0.965523i	$-0.416173\pi$
$769$	14.4377	0.520637	0.260318	+	0.965523i	$0.416173\pi$
$770$	0	0
$771$	−25.1672	−0.906374
$772$	0	0
$773$	9.80460	0.352647	0.176323	−	0.984332i	$-0.443580\pi$
$773$	9.80460	0.352647	0.176323	+	0.984332i	$0.443580\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	18.0565	0.647775
$778$	0	0
$779$	−3.23607	−0.115944
$780$	0	0
$781$	−23.4164	−0.837905
$782$	0	0
$783$	−23.3775	−0.835445
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−44.1440	−1.57356	−0.786782	−	0.617231i	$-0.788254\pi$
$787$	−44.1440	−1.57356	−0.786782	+	0.617231i	$0.788254\pi$
$788$	0	0
$789$	3.14590	0.111997
$790$	0	0
$791$	79.9574	2.84296
$792$	0	0
$793$	−6.57712	−0.233560
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−10.1245	−0.358627	−0.179313	−	0.983792i	$-0.557388\pi$
$797$	−10.1245	−0.358627	−0.179313	+	0.983792i	$0.557388\pi$
$798$	0	0
$799$	26.6525	0.942897
$800$	0	0
$801$	−1.65248	−0.0583874
$802$	0	0
$803$	25.6688	0.905831
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−19.4115	−0.683317
$808$	0	0
$809$	−7.90983	−0.278095	−0.139047	−	0.990286i	$-0.544404\pi$
$809$	−7.90983	−0.278095	−0.139047	+	0.990286i	$0.544404\pi$
$810$	0	0
$811$	−0.832816	−0.0292441	−0.0146221	−	0.999893i	$-0.504655\pi$
$811$	−0.832816	−0.0292441	−0.0146221	+	0.999893i	$0.504655\pi$
$812$	0	0
$813$	−9.80460	−0.343862
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−4.90230	−0.171510
$818$	0	0
$819$	3.88854	0.135877
$820$	0	0
$821$	27.2148	0.949802	0.474901	−	0.880039i	$-0.342484\pi$
$821$	27.2148	0.949802	0.474901	+	0.880039i	$0.342484\pi$
$822$	0	0
$823$	−18.0565	−0.629412	−0.314706	−	0.949189i	$-0.601906\pi$
$823$	−18.0565	−0.629412	−0.314706	+	0.949189i	$0.601906\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−20.7665	−0.722121	−0.361060	−	0.932542i	$-0.617585\pi$
$827$	−20.7665	−0.722121	−0.361060	+	0.932542i	$0.617585\pi$
$828$	0	0
$829$	23.2016	0.805826	0.402913	−	0.915238i	$-0.367998\pi$
$829$	23.2016	0.805826	0.402913	+	0.915238i	$0.367998\pi$
$830$	0	0
$831$	34.4296	1.19435
$832$	0	0
$833$	−51.6574	−1.78982
$834$	0	0
$835$	0	0
$836$	0	0
$837$	30.3734	1.04986
$838$	0	0
$839$	−15.2361	−0.526007	−0.263004	−	0.964795i	$-0.584713\pi$
$839$	−15.2361	−0.526007	−0.263004	+	0.964795i	$0.584713\pi$
$840$	0	0
$841$	−7.67376	−0.264612
$842$	0	0
$843$	19.5714	0.674076
$844$	0	0
$845$	0	0
$846$	0	0
$847$	41.7917	1.43598
$848$	0	0
$849$	16.4721	0.565322
$850$	0	0
$851$	−2.40325	−0.0823824
$852$	0	0
$853$	25.6688	0.878882	0.439441	−	0.898272i	$-0.355177\pi$
$853$	25.6688	0.878882	0.439441	+	0.898272i	$0.355177\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−45.5978	−1.55759	−0.778796	−	0.627277i	$-0.784169\pi$
$857$	−45.5978	−1.55759	−0.778796	+	0.627277i	$0.784169\pi$
$858$	0	0
$859$	30.8328	1.05200	0.526001	−	0.850484i	$-0.323691\pi$
$859$	30.8328	1.05200	0.526001	+	0.850484i	$0.323691\pi$
$860$	0	0
$861$	21.5623	0.734841
$862$	0	0
$863$	26.7650	0.911090	0.455545	−	0.890213i	$-0.349444\pi$
$863$	26.7650	0.911090	0.455545	+	0.890213i	$0.349444\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	7.83324	0.266031
$868$	0	0
$869$	−43.4164	−1.47280
$870$	0	0
$871$	−12.5836	−0.426379
$872$	0	0
$873$	−0.837412	−0.0283421
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−6.57712	−0.222094	−0.111047	−	0.993815i	$-0.535420\pi$
$877$	−6.57712	−0.222094	−0.111047	+	0.993815i	$0.535420\pi$
$878$	0	0
$879$	22.7639	0.767808
$880$	0	0
$881$	−3.97871	−0.134046	−0.0670231	−	0.997751i	$-0.521350\pi$
$881$	−3.97871	−0.134046	−0.0670231	+	0.997751i	$0.521350\pi$
$882$	0	0
$883$	13.9306	0.468801	0.234400	−	0.972140i	$-0.424687\pi$
$883$	13.9306	0.468801	0.234400	+	0.972140i	$0.424687\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.28856	0.110419	0.0552095	−	0.998475i	$-0.482417\pi$
$887$	3.28856	0.110419	0.0552095	+	0.998475i	$0.482417\pi$
$888$	0	0
$889$	−67.8328	−2.27504
$890$	0	0
$891$	42.1115	1.41079
$892$	0	0
$893$	−9.28705	−0.310779
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−11.1596	−0.372607
$898$	0	0
$899$	−27.7082	−0.924120
$900$	0	0
$901$	−40.7214	−1.35663
$902$	0	0
$903$	32.6646	1.08701
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−58.1734	−1.93162	−0.965808	−	0.259257i	$-0.916522\pi$
$907$	−58.1734	−1.93162	−0.965808	+	0.259257i	$0.916522\pi$
$908$	0	0
$909$	2.81966	0.0935222
$910$	0	0
$911$	−44.8328	−1.48538	−0.742689	−	0.669637i	$-0.766450\pi$
$911$	−44.8328	−1.48538	−0.742689	+	0.669637i	$0.766450\pi$
$912$	0	0
$913$	−30.5711	−1.01175
$914$	0	0
$915$	0	0
$916$	0	0
$917$	33.6008	1.10960
$918$	0	0
$919$	−13.2361	−0.436618	−0.218309	−	0.975880i	$-0.570054\pi$
$919$	−13.2361	−0.436618	−0.218309	+	0.975880i	$0.570054\pi$
$920$	0	0
$921$	−5.83282	−0.192198
$922$	0	0
$923$	30.0535	0.989223
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.35496	0.0445028
$928$	0	0
$929$	29.5066	0.968079	0.484040	−	0.875046i	$-0.339169\pi$
$929$	29.5066	0.968079	0.484040	+	0.875046i	$0.339169\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	0	0
$933$	−45.0803	−1.47586
$934$	0	0
$935$	0	0
$936$	0	0
$937$	18.5741	0.606789	0.303395	−	0.952865i	$-0.401880\pi$
$937$	18.5741	0.606789	0.303395	+	0.952865i	$0.401880\pi$
$938$	0	0
$939$	−32.9443	−1.07510
$940$	0	0
$941$	1.41641	0.0461736	0.0230868	−	0.999733i	$-0.492651\pi$
$941$	1.41641	0.0461736	0.0230868	+	0.999733i	$0.492651\pi$
$942$	0	0
$943$	−2.86986	−0.0934553
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.9630	0.518728	0.259364	−	0.965780i	$-0.416487\pi$
$947$	15.9630	0.518728	0.259364	+	0.965780i	$0.416487\pi$
$948$	0	0
$949$	−32.9443	−1.06942
$950$	0	0
$951$	−29.0557	−0.942197
$952$	0	0
$953$	30.0535	0.973529	0.486764	−	0.873533i	$-0.338177\pi$
$953$	30.0535	0.973529	0.486764	+	0.873533i	$0.338177\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−36.6306	−1.18410
$958$	0	0
$959$	−69.7771	−2.25322
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	−0.0610884	−0.00196855
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−29.9547	−0.963277	−0.481639	−	0.876370i	$-0.659958\pi$
$967$	−29.9547	−0.963277	−0.481639	+	0.876370i	$0.659958\pi$
$968$	0	0
$969$	7.77709	0.249836
$970$	0	0
$971$	−43.2361	−1.38751	−0.693756	−	0.720210i	$-0.744045\pi$
$971$	−43.2361	−1.38751	−0.693756	+	0.720210i	$0.744045\pi$
$972$	0	0
$973$	10.6420	0.341167
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30.8909	−0.988288	−0.494144	−	0.869380i	$-0.664518\pi$
$977$	−30.8909	−0.988288	−0.494144	+	0.869380i	$0.664518\pi$
$978$	0	0
$979$	50.6525	1.61886
$980$	0	0
$981$	−1.95240	−0.0623354
$982$	0	0
$983$	27.8611	0.888632	0.444316	−	0.895870i	$-0.353447\pi$
$983$	27.8611	0.888632	0.444316	+	0.895870i	$0.353447\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	61.8807	1.96968
$988$	0	0
$989$	−4.34752	−0.138243
$990$	0	0
$991$	−39.7771	−1.26356	−0.631780	−	0.775147i	$-0.717676\pi$
$991$	−39.7771	−1.26356	−0.631780	+	0.775147i	$0.717676\pi$
$992$	0	0
$993$	−18.2542	−0.579280
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−29.2161	−0.925283	−0.462642	−	0.886545i	$-0.653098\pi$
$997$	−29.2161	−0.925283	−0.462642	+	0.886545i	$0.653098\pi$
$998$	0	0
$999$	−11.0983	−0.351135

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8000.2.a.bl.1.2		4
4.3	odd	2		8000.2.a.bm.1.3		4
5.4	even	2	inner	8000.2.a.bl.1.3		4
8.3	odd	2		2000.2.a.p.1.2		4
8.5	even	2		500.2.a.c.1.3	yes	4
20.19	odd	2		8000.2.a.bm.1.2		4
24.5	odd	2		4500.2.a.q.1.4		4
40.3	even	4		2000.2.c.a.1249.2		4
40.13	odd	4		500.2.c.a.249.3		4
40.19	odd	2		2000.2.a.p.1.3		4
40.27	even	4		2000.2.c.a.1249.3		4
40.29	even	2		500.2.a.c.1.2	✓	4
40.37	odd	4		500.2.c.a.249.2		4
120.29	odd	2		4500.2.a.q.1.1		4
120.53	even	4		4500.2.d.e.4249.1		4
120.77	even	4		4500.2.d.e.4249.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
500.2.a.c.1.2	✓	4	40.29	even	2
500.2.a.c.1.3	yes	4	8.5	even	2
500.2.c.a.249.2		4	40.37	odd	4
500.2.c.a.249.3		4	40.13	odd	4
2000.2.a.p.1.2		4	8.3	odd	2
2000.2.a.p.1.3		4	40.19	odd	2
2000.2.c.a.1249.2		4	40.3	even	4
2000.2.c.a.1249.3		4	40.27	even	4
4500.2.a.q.1.1		4	120.29	odd	2
4500.2.a.q.1.4		4	24.5	odd	2
4500.2.d.e.4249.1		4	120.53	even	4
4500.2.d.e.4249.4		4	120.77	even	4
8000.2.a.bl.1.2		4	1.1	even	1	trivial
8000.2.a.bl.1.3		4	5.4	even	2	inner
8000.2.a.bm.1.2		4	20.19	odd	2
8000.2.a.bm.1.3		4	4.3	odd	2

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8000 = 2^{6} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8000.a (trivial)

\(f(q)\)	\(=\)	\(q-1.77367 q^{3} +4.64352 q^{7} +0.145898 q^{9} +O(q^{10})\)	\(q-1.77367 q^{3} +4.64352 q^{7} +0.145898 q^{9} -4.47214 q^{11} +5.73971 q^{13} -3.54734 q^{17} +1.23607 q^{19} -8.23607 q^{21} +1.09619 q^{23} +5.06223 q^{27} -4.61803 q^{29} +6.00000 q^{31} +7.93208 q^{33} -2.19237 q^{37} -10.1803 q^{39} -2.61803 q^{41} -3.96604 q^{43} -7.51338 q^{47} +14.5623 q^{49} +6.29180 q^{51} +11.4794 q^{53} -2.19237 q^{57} +12.4721 q^{59} -1.14590 q^{61} +0.677481 q^{63} -2.19237 q^{67} -1.94427 q^{69} +5.23607 q^{71} -5.73971 q^{73} -20.7665 q^{77} +9.70820 q^{79} -9.41641 q^{81} +6.83590 q^{83} +8.19086 q^{87} -11.3262 q^{89} +26.6525 q^{91} -10.6420 q^{93} -5.73971 q^{97} -0.652476 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 14 q^{9}+O(q^{10}) \) 4 * q + 14 * q^9	\( 4 q + 14 q^{9} - 4 q^{19} - 24 q^{21} - 14 q^{29} + 24 q^{31} + 4 q^{39} - 6 q^{41} + 18 q^{49} + 52 q^{51} + 32 q^{59} - 18 q^{61} + 28 q^{69} + 12 q^{71} + 12 q^{79} + 16 q^{81} - 14 q^{89} + 44 q^{91} + 60 q^{99}+O(q^{100}) \) 4 * q + 14 * q^9 - 4 * q^19 - 24 * q^21 - 14 * q^29 + 24 * q^31 + 4 * q^39 - 6 * q^41 + 18 * q^49 + 52 * q^51 + 32 * q^59 - 18 * q^61 + 28 * q^69 + 12 * q^71 + 12 * q^79 + 16 * q^81 - 14 * q^89 + 44 * q^91 + 60 * q^99

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