LMFDB - Embedded newform 1472.4.a.bd.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1472,4,Mod(1,1472)] 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("1472.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1472.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,1,0,20,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$86.8508115285$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.167313.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 16x^{2} + 4x + 24 $ x^4 - x^3 - 16x^2 + 4x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.42018$ of defining polynomial
Character	$\chi$	$=$	1472.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.911599 q^{3} +2.21395 q^{5} +0.592081 q^{7} -26.1690 q^{9} -44.0622 q^{11} -39.4794 q^{13} +2.01823 q^{15} -89.6532 q^{17} +15.7985 q^{19} +0.539741 q^{21} +23.0000 q^{23} -120.098 q^{25} -48.4688 q^{27} +227.961 q^{29} -83.0988 q^{31} -40.1670 q^{33} +1.31084 q^{35} +201.333 q^{37} -35.9894 q^{39} +364.105 q^{41} +79.8025 q^{43} -57.9368 q^{45} +162.306 q^{47} -342.649 q^{49} -81.7277 q^{51} +637.999 q^{53} -97.5513 q^{55} +14.4019 q^{57} +830.645 q^{59} -23.1201 q^{61} -15.4942 q^{63} -87.4054 q^{65} +689.266 q^{67} +20.9668 q^{69} -701.876 q^{71} +332.442 q^{73} -109.482 q^{75} -26.0884 q^{77} +208.245 q^{79} +662.379 q^{81} -93.9343 q^{83} -198.487 q^{85} +207.809 q^{87} +290.773 q^{89} -23.3750 q^{91} -75.7528 q^{93} +34.9770 q^{95} +721.288 q^{97} +1153.06 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} + 20 q^{5} + 10 q^{7} + 23 q^{9} - 30 q^{11} + 153 q^{13} + 136 q^{15} - 68 q^{17} - 120 q^{19} + 426 q^{21} + 92 q^{23} - 76 q^{25} + 43 q^{27} + 315 q^{29} - 249 q^{31} - 504 q^{33} + 224 q^{35}+ \cdots - 1470 q^{99}+O(q^{100}) $ 4 * q + q^3 + 20 * q^5 + 10 * q^7 + 23 * q^9 - 30 * q^11 + 153 * q^13 + 136 * q^15 - 68 * q^17 - 120 * q^19 + 426 * q^21 + 92 * q^23 - 76 * q^25 + 43 * q^27 + 315 * q^29 - 249 * q^31 - 504 * q^33 + 224 * q^35 + 348 * q^37 + 61 * q^39 - 929 * q^41 + 50 * q^43 + 1028 * q^45 - 205 * q^47 + 456 * q^49 + 890 * q^51 + 578 * q^53 - 1128 * q^55 + 366 * q^57 - 664 * q^59 + 182 * q^61 - 624 * q^63 + 848 * q^65 + 1078 * q^67 + 23 * q^69 + 739 * q^71 + 921 * q^73 + 2883 * q^75 - 528 * q^77 - 1754 * q^79 + 788 * q^81 + 1020 * q^83 + 1072 * q^85 - 2121 * q^87 - 430 * q^89 + 1334 * q^91 - 1035 * q^93 - 744 * q^95 + 48 * q^97 - 1470 * 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
$3$	0.911599	0.175437	0.0877186	−	0.996145i	$-0.472042\pi$
$3$	0.911599	0.175437	0.0877186	+	0.996145i	$0.472042\pi$
$4$	0	0
$5$	2.21395	0.198021	0.0990107	−	0.995086i	$-0.468432\pi$
$5$	2.21395	0.198021	0.0990107	+	0.995086i	$0.468432\pi$
$6$	0	0
$7$	0.592081	0.0319694	0.0159847	−	0.999872i	$-0.494912\pi$
$7$	0.592081	0.0319694	0.0159847	+	0.999872i	$0.494912\pi$
$8$	0	0
$9$	−26.1690	−0.969222
$10$	0	0
$11$	−44.0622	−1.20775	−0.603875	−	0.797079i	$-0.706377\pi$
$11$	−44.0622	−1.20775	−0.603875	+	0.797079i	$0.706377\pi$
$12$	0	0
$13$	−39.4794	−0.842279	−0.421140	−	0.906996i	$-0.638370\pi$
$13$	−39.4794	−0.842279	−0.421140	+	0.906996i	$0.638370\pi$
$14$	0	0
$15$	2.01823	0.0347403
$16$	0	0
$17$	−89.6532	−1.27906	−0.639532	−	0.768764i	$-0.720872\pi$
$17$	−89.6532	−1.27906	−0.639532	+	0.768764i	$0.720872\pi$
$18$	0	0
$19$	15.7985	0.190759	0.0953794	−	0.995441i	$-0.469594\pi$
$19$	15.7985	0.190759	0.0953794	+	0.995441i	$0.469594\pi$
$20$	0	0
$21$	0.539741	0.00560862
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	−120.098	−0.960787
$26$	0	0
$27$	−48.4688	−0.345475
$28$	0	0
$29$	227.961	1.45970	0.729850	−	0.683607i	$-0.239590\pi$
$29$	227.961	1.45970	0.729850	+	0.683607i	$0.239590\pi$
$30$	0	0
$31$	−83.0988	−0.481451	−0.240726	−	0.970593i	$-0.577385\pi$
$31$	−83.0988	−0.481451	−0.240726	+	0.970593i	$0.577385\pi$
$32$	0	0
$33$	−40.1670	−0.211884
$34$	0	0
$35$	1.31084	0.00633063
$36$	0	0
$37$	201.333	0.894564	0.447282	−	0.894393i	$-0.352392\pi$
$37$	201.333	0.894564	0.447282	+	0.894393i	$0.352392\pi$
$38$	0	0
$39$	−35.9894	−0.147767
$40$	0	0
$41$	364.105	1.38692	0.693459	−	0.720496i	$-0.256086\pi$
$41$	364.105	1.38692	0.693459	+	0.720496i	$0.256086\pi$
$42$	0	0
$43$	79.8025	0.283018	0.141509	−	0.989937i	$-0.454805\pi$
$43$	79.8025	0.283018	0.141509	+	0.989937i	$0.454805\pi$
$44$	0	0
$45$	−57.9368	−0.191927
$46$	0	0
$47$	162.306	0.503719	0.251859	−	0.967764i	$-0.418958\pi$
$47$	162.306	0.503719	0.251859	+	0.967764i	$0.418958\pi$
$48$	0	0
$49$	−342.649	−0.998978
$50$	0	0
$51$	−81.7277	−0.224396
$52$	0	0
$53$	637.999	1.65351	0.826754	−	0.562563i	$-0.190185\pi$
$53$	637.999	1.65351	0.826754	+	0.562563i	$0.190185\pi$
$54$	0	0
$55$	−97.5513	−0.239160
$56$	0	0
$57$	14.4019	0.0334662
$58$	0	0
$59$	830.645	1.83289	0.916447	−	0.400156i	$-0.131044\pi$
$59$	830.645	1.83289	0.916447	+	0.400156i	$0.131044\pi$
$60$	0	0
$61$	−23.1201	−0.0485283	−0.0242641	−	0.999706i	$-0.507724\pi$
$61$	−23.1201	−0.0485283	−0.0242641	+	0.999706i	$0.507724\pi$
$62$	0	0
$63$	−15.4942	−0.0309854
$64$	0	0
$65$	−87.4054	−0.166789
$66$	0	0
$67$	689.266	1.25683	0.628413	−	0.777880i	$-0.283705\pi$
$67$	689.266	1.25683	0.628413	+	0.777880i	$0.283705\pi$
$68$	0	0
$69$	20.9668	0.0365812
$70$	0	0
$71$	−701.876	−1.17320	−0.586601	−	0.809876i	$-0.699534\pi$
$71$	−701.876	−1.17320	−0.586601	+	0.809876i	$0.699534\pi$
$72$	0	0
$73$	332.442	0.533005	0.266502	−	0.963834i	$-0.414132\pi$
$73$	332.442	0.533005	0.266502	+	0.963834i	$0.414132\pi$
$74$	0	0
$75$	−109.482	−0.168558
$76$	0	0
$77$	−26.0884	−0.0386110
$78$	0	0
$79$	208.245	0.296575	0.148287	−	0.988944i	$-0.452624\pi$
$79$	208.245	0.296575	0.148287	+	0.988944i	$0.452624\pi$
$80$	0	0
$81$	662.379	0.908613
$82$	0	0
$83$	−93.9343	−0.124224	−0.0621122	−	0.998069i	$-0.519784\pi$
$83$	−93.9343	−0.124224	−0.0621122	+	0.998069i	$0.519784\pi$
$84$	0	0
$85$	−198.487	−0.253282
$86$	0	0
$87$	207.809	0.256086
$88$	0	0
$89$	290.773	0.346313	0.173157	−	0.984894i	$-0.444603\pi$
$89$	290.773	0.346313	0.173157	+	0.984894i	$0.444603\pi$
$90$	0	0
$91$	−23.3750	−0.0269271
$92$	0	0
$93$	−75.7528	−0.0844645
$94$	0	0
$95$	34.9770	0.0377743
$96$	0	0
$97$	721.288	0.755007	0.377504	−	0.926008i	$-0.376783\pi$
$97$	721.288	0.755007	0.377504	+	0.926008i	$0.376783\pi$
$98$	0	0
$99$	1153.06	1.17058
$100$	0	0
$101$	190.272	0.187453	0.0937267	−	0.995598i	$-0.470122\pi$
$101$	190.272	0.187453	0.0937267	+	0.995598i	$0.470122\pi$
$102$	0	0
$103$	−1528.28	−1.46200	−0.731001	−	0.682377i	$-0.760946\pi$
$103$	−1528.28	−1.46200	−0.731001	+	0.682377i	$0.760946\pi$
$104$	0	0
$105$	1.19496	0.00111063
$106$	0	0
$107$	−1762.11	−1.59205	−0.796026	−	0.605262i	$-0.793068\pi$
$107$	−1762.11	−1.59205	−0.796026	+	0.605262i	$0.793068\pi$
$108$	0	0
$109$	178.049	0.156459	0.0782296	−	0.996935i	$-0.475073\pi$
$109$	178.049	0.156459	0.0782296	+	0.996935i	$0.475073\pi$
$110$	0	0
$111$	183.534	0.156940
$112$	0	0
$113$	−542.027	−0.451236	−0.225618	−	0.974216i	$-0.572440\pi$
$113$	−542.027	−0.451236	−0.225618	+	0.974216i	$0.572440\pi$
$114$	0	0
$115$	50.9208	0.0412903
$116$	0	0
$117$	1033.14	0.816355
$118$	0	0
$119$	−53.0820	−0.0408909
$120$	0	0
$121$	610.475	0.458659
$122$	0	0
$123$	331.917	0.243317
$124$	0	0
$125$	−542.635	−0.388278
$126$	0	0
$127$	276.131	0.192935	0.0964673	−	0.995336i	$-0.469246\pi$
$127$	276.131	0.192935	0.0964673	+	0.995336i	$0.469246\pi$
$128$	0	0
$129$	72.7479	0.0496519
$130$	0	0
$131$	−2475.57	−1.65108	−0.825542	−	0.564341i	$-0.809130\pi$
$131$	−2475.57	−1.65108	−0.825542	+	0.564341i	$0.809130\pi$
$132$	0	0
$133$	9.35398	0.00609844
$134$	0	0
$135$	−107.307	−0.0684114
$136$	0	0
$137$	37.7868	0.0235646	0.0117823	−	0.999931i	$-0.496249\pi$
$137$	37.7868	0.0235646	0.0117823	+	0.999931i	$0.496249\pi$
$138$	0	0
$139$	975.089	0.595007	0.297503	−	0.954721i	$-0.403846\pi$
$139$	975.089	0.595007	0.297503	+	0.954721i	$0.403846\pi$
$140$	0	0
$141$	147.958	0.0883711
$142$	0	0
$143$	1739.55	1.01726
$144$	0	0
$145$	504.694	0.289052
$146$	0	0
$147$	−312.359	−0.175258
$148$	0	0
$149$	204.870	0.112642	0.0563209	−	0.998413i	$-0.482063\pi$
$149$	204.870	0.112642	0.0563209	+	0.998413i	$0.482063\pi$
$150$	0	0
$151$	2905.20	1.56571	0.782853	−	0.622207i	$-0.213764\pi$
$151$	2905.20	1.56571	0.782853	+	0.622207i	$0.213764\pi$
$152$	0	0
$153$	2346.13	1.23970
$154$	0	0
$155$	−183.976	−0.0953377
$156$	0	0
$157$	1749.50	0.889334	0.444667	−	0.895696i	$-0.353322\pi$
$157$	1749.50	0.889334	0.444667	+	0.895696i	$0.353322\pi$
$158$	0	0
$159$	581.599	0.290087
$160$	0	0
$161$	13.6179	0.00666608
$162$	0	0
$163$	−2494.40	−1.19863	−0.599315	−	0.800513i	$-0.704560\pi$
$163$	−2494.40	−1.19863	−0.599315	+	0.800513i	$0.704560\pi$
$164$	0	0
$165$	−88.9277	−0.0419576
$166$	0	0
$167$	−620.316	−0.287434	−0.143717	−	0.989619i	$-0.545906\pi$
$167$	−620.316	−0.287434	−0.143717	+	0.989619i	$0.545906\pi$
$168$	0	0
$169$	−638.374	−0.290566
$170$	0	0
$171$	−413.430	−0.184888
$172$	0	0
$173$	504.568	0.221743	0.110872	−	0.993835i	$-0.464636\pi$
$173$	504.568	0.221743	0.110872	+	0.993835i	$0.464636\pi$
$174$	0	0
$175$	−71.1080	−0.0307158
$176$	0	0
$177$	757.215	0.321558
$178$	0	0
$179$	−3033.16	−1.26653	−0.633265	−	0.773935i	$-0.718286\pi$
$179$	−3033.16	−1.26653	−0.633265	+	0.773935i	$0.718286\pi$
$180$	0	0
$181$	455.262	0.186958	0.0934790	−	0.995621i	$-0.470201\pi$
$181$	455.262	0.186958	0.0934790	+	0.995621i	$0.470201\pi$
$182$	0	0
$183$	−21.0763	−0.00851367
$184$	0	0
$185$	445.740	0.177143
$186$	0	0
$187$	3950.31	1.54479
$188$	0	0
$189$	−28.6975	−0.0110446
$190$	0	0
$191$	3231.26	1.22412	0.612058	−	0.790813i	$-0.290342\pi$
$191$	3231.26	1.22412	0.612058	+	0.790813i	$0.290342\pi$
$192$	0	0
$193$	2085.89	0.777956	0.388978	−	0.921247i	$-0.372828\pi$
$193$	2085.89	0.777956	0.388978	+	0.921247i	$0.372828\pi$
$194$	0	0
$195$	−79.6787	−0.0292611
$196$	0	0
$197$	1227.57	0.443964	0.221982	−	0.975051i	$-0.428747\pi$
$197$	1227.57	0.443964	0.221982	+	0.975051i	$0.428747\pi$
$198$	0	0
$199$	−160.852	−0.0572990	−0.0286495	−	0.999590i	$-0.509121\pi$
$199$	−160.852	−0.0572990	−0.0286495	+	0.999590i	$0.509121\pi$
$200$	0	0
$201$	628.334	0.220494
$202$	0	0
$203$	134.972	0.0466657
$204$	0	0
$205$	806.109	0.274639
$206$	0	0
$207$	−601.887	−0.202097
$208$	0	0
$209$	−696.115	−0.230389
$210$	0	0
$211$	2066.49	0.674231	0.337116	−	0.941463i	$-0.390549\pi$
$211$	2066.49	0.674231	0.337116	+	0.941463i	$0.390549\pi$
$212$	0	0
$213$	−639.829	−0.205823
$214$	0	0
$215$	176.679	0.0560436
$216$	0	0
$217$	−49.2013	−0.0153917
$218$	0	0
$219$	303.053	0.0935089
$220$	0	0
$221$	3539.46	1.07733
$222$	0	0
$223$	−2918.95	−0.876536	−0.438268	−	0.898844i	$-0.644408\pi$
$223$	−2918.95	−0.876536	−0.438268	+	0.898844i	$0.644408\pi$
$224$	0	0
$225$	3142.85	0.931216
$226$	0	0
$227$	−3072.13	−0.898256	−0.449128	−	0.893467i	$-0.648265\pi$
$227$	−3072.13	−0.898256	−0.449128	+	0.893467i	$0.648265\pi$
$228$	0	0
$229$	3007.05	0.867737	0.433868	−	0.900976i	$-0.357148\pi$
$229$	3007.05	0.867737	0.433868	+	0.900976i	$0.357148\pi$
$230$	0	0
$231$	−23.7821	−0.00677381
$232$	0	0
$233$	5625.92	1.58183	0.790914	−	0.611927i	$-0.209605\pi$
$233$	5625.92	1.58183	0.790914	+	0.611927i	$0.209605\pi$
$234$	0	0
$235$	359.337	0.0997472
$236$	0	0
$237$	189.836	0.0520302
$238$	0	0
$239$	1533.72	0.415097	0.207548	−	0.978225i	$-0.433452\pi$
$239$	1533.72	0.415097	0.207548	+	0.978225i	$0.433452\pi$
$240$	0	0
$241$	1965.38	0.525316	0.262658	−	0.964889i	$-0.415401\pi$
$241$	1965.38	0.525316	0.262658	+	0.964889i	$0.415401\pi$
$242$	0	0
$243$	1912.48	0.504879
$244$	0	0
$245$	−758.608	−0.197819
$246$	0	0
$247$	−623.715	−0.160672
$248$	0	0
$249$	−85.6304	−0.0217936
$250$	0	0
$251$	−1342.89	−0.337700	−0.168850	−	0.985642i	$-0.554005\pi$
$251$	−1342.89	−0.337700	−0.168850	+	0.985642i	$0.554005\pi$
$252$	0	0
$253$	−1013.43	−0.251833
$254$	0	0
$255$	−180.941	−0.0444351
$256$	0	0
$257$	4699.24	1.14059	0.570293	−	0.821441i	$-0.306830\pi$
$257$	4699.24	1.14059	0.570293	+	0.821441i	$0.306830\pi$
$258$	0	0
$259$	119.205	0.0285987
$260$	0	0
$261$	−5965.51	−1.41477
$262$	0	0
$263$	−5578.08	−1.30783	−0.653914	−	0.756569i	$-0.726874\pi$
$263$	−5578.08	−1.30783	−0.653914	+	0.756569i	$0.726874\pi$
$264$	0	0
$265$	1412.50	0.327430
$266$	0	0
$267$	265.068	0.0607562
$268$	0	0
$269$	4775.55	1.08242	0.541209	−	0.840888i	$-0.317967\pi$
$269$	4775.55	1.08242	0.541209	+	0.840888i	$0.317967\pi$
$270$	0	0
$271$	−6426.28	−1.44047	−0.720237	−	0.693728i	$-0.755967\pi$
$271$	−6426.28	−1.44047	−0.720237	+	0.693728i	$0.755967\pi$
$272$	0	0
$273$	−21.3087	−0.00472402
$274$	0	0
$275$	5291.80	1.16039
$276$	0	0
$277$	2287.81	0.496249	0.248124	−	0.968728i	$-0.420186\pi$
$277$	2287.81	0.496249	0.248124	+	0.968728i	$0.420186\pi$
$278$	0	0
$279$	2174.61	0.466633
$280$	0	0
$281$	−7904.63	−1.67812	−0.839058	−	0.544041i	$-0.816894\pi$
$281$	−7904.63	−1.67812	−0.839058	+	0.544041i	$0.816894\pi$
$282$	0	0
$283$	−3873.86	−0.813699	−0.406850	−	0.913495i	$-0.633373\pi$
$283$	−3873.86	−0.813699	−0.406850	+	0.913495i	$0.633373\pi$
$284$	0	0
$285$	31.8850	0.00662702
$286$	0	0
$287$	215.580	0.0443389
$288$	0	0
$289$	3124.69	0.636005
$290$	0	0
$291$	657.525	0.132456
$292$	0	0
$293$	8432.22	1.68128	0.840640	−	0.541594i	$-0.182179\pi$
$293$	8432.22	1.68128	0.840640	+	0.541594i	$0.182179\pi$
$294$	0	0
$295$	1839.00	0.362952
$296$	0	0
$297$	2135.64	0.417247
$298$	0	0
$299$	−908.027	−0.175627
$300$	0	0
$301$	47.2496	0.00904791
$302$	0	0
$303$	173.452	0.0328863
$304$	0	0
$305$	−51.1867	−0.00960964
$306$	0	0
$307$	9341.35	1.73661	0.868305	−	0.496031i	$-0.165210\pi$
$307$	9341.35	1.73661	0.868305	+	0.496031i	$0.165210\pi$
$308$	0	0
$309$	−1393.18	−0.256489
$310$	0	0
$311$	−4888.06	−0.891242	−0.445621	−	0.895222i	$-0.647017\pi$
$311$	−4888.06	−0.891242	−0.445621	+	0.895222i	$0.647017\pi$
$312$	0	0
$313$	6333.38	1.14372	0.571859	−	0.820352i	$-0.306222\pi$
$313$	6333.38	1.14372	0.571859	+	0.820352i	$0.306222\pi$
$314$	0	0
$315$	−34.3033	−0.00613578
$316$	0	0
$317$	5279.80	0.935467	0.467734	−	0.883870i	$-0.345071\pi$
$317$	5279.80	0.935467	0.467734	+	0.883870i	$0.345071\pi$
$318$	0	0
$319$	−10044.5	−1.76295
$320$	0	0
$321$	−1606.34	−0.279305
$322$	0	0
$323$	−1416.38	−0.243993
$324$	0	0
$325$	4741.42	0.809251
$326$	0	0
$327$	162.310	0.0274488
$328$	0	0
$329$	96.0985	0.0161036
$330$	0	0
$331$	521.432	0.0865877	0.0432938	−	0.999062i	$-0.486215\pi$
$331$	521.432	0.0865877	0.0432938	+	0.999062i	$0.486215\pi$
$332$	0	0
$333$	−5268.67	−0.867031
$334$	0	0
$335$	1526.00	0.248878
$336$	0	0
$337$	−7697.66	−1.24427	−0.622134	−	0.782911i	$-0.713734\pi$
$337$	−7697.66	−1.24427	−0.622134	+	0.782911i	$0.713734\pi$
$338$	0	0
$339$	−494.111	−0.0791635
$340$	0	0
$341$	3661.52	0.581473
$342$	0	0
$343$	−405.960	−0.0639061
$344$	0	0
$345$	46.4193	0.00724386
$346$	0	0
$347$	−6001.92	−0.928530	−0.464265	−	0.885696i	$-0.653682\pi$
$347$	−6001.92	−0.928530	−0.464265	+	0.885696i	$0.653682\pi$
$348$	0	0
$349$	−3955.05	−0.606617	−0.303308	−	0.952892i	$-0.598091\pi$
$349$	−3955.05	−0.606617	−0.303308	+	0.952892i	$0.598091\pi$
$350$	0	0
$351$	1913.52	0.290986
$352$	0	0
$353$	765.163	0.115370	0.0576848	−	0.998335i	$-0.481628\pi$
$353$	765.163	0.115370	0.0576848	+	0.998335i	$0.481628\pi$
$354$	0	0
$355$	−1553.92	−0.232319
$356$	0	0
$357$	−48.3895	−0.00717379
$358$	0	0
$359$	8630.55	1.26881	0.634406	−	0.773000i	$-0.281245\pi$
$359$	8630.55	1.26881	0.634406	+	0.773000i	$0.281245\pi$
$360$	0	0
$361$	−6609.41	−0.963611
$362$	0	0
$363$	556.508	0.0804658
$364$	0	0
$365$	736.008	0.105546
$366$	0	0
$367$	9040.53	1.28586	0.642932	−	0.765924i	$-0.277718\pi$
$367$	9040.53	1.28586	0.642932	+	0.765924i	$0.277718\pi$
$368$	0	0
$369$	−9528.25	−1.34423
$370$	0	0
$371$	377.748	0.0528617
$372$	0	0
$373$	2798.43	0.388464	0.194232	−	0.980956i	$-0.437778\pi$
$373$	2798.43	0.388464	0.194232	+	0.980956i	$0.437778\pi$
$374$	0	0
$375$	−494.665	−0.0681184
$376$	0	0
$377$	−8999.78	−1.22948
$378$	0	0
$379$	11763.4	1.59432	0.797158	−	0.603771i	$-0.206336\pi$
$379$	11763.4	1.59432	0.797158	+	0.603771i	$0.206336\pi$
$380$	0	0
$381$	251.721	0.0338479
$382$	0	0
$383$	7350.38	0.980645	0.490323	−	0.871541i	$-0.336879\pi$
$383$	7350.38	0.980645	0.490323	+	0.871541i	$0.336879\pi$
$384$	0	0
$385$	−57.7583	−0.00764581
$386$	0	0
$387$	−2088.35	−0.274307
$388$	0	0
$389$	−13475.8	−1.75643	−0.878217	−	0.478263i	$-0.841267\pi$
$389$	−13475.8	−1.75643	−0.878217	+	0.478263i	$0.841267\pi$
$390$	0	0
$391$	−2062.02	−0.266703
$392$	0	0
$393$	−2256.73	−0.289662
$394$	0	0
$395$	461.043	0.0587281
$396$	0	0
$397$	12509.3	1.58141	0.790707	−	0.612195i	$-0.209713\pi$
$397$	12509.3	1.58141	0.790707	+	0.612195i	$0.209713\pi$
$398$	0	0
$399$	8.52707	0.00106989
$400$	0	0
$401$	3129.19	0.389687	0.194843	−	0.980834i	$-0.437580\pi$
$401$	3129.19	0.389687	0.194843	+	0.980834i	$0.437580\pi$
$402$	0	0
$403$	3280.70	0.405516
$404$	0	0
$405$	1466.47	0.179925
$406$	0	0
$407$	−8871.15	−1.08041
$408$	0	0
$409$	−961.985	−0.116301	−0.0581505	−	0.998308i	$-0.518520\pi$
$409$	−961.985	−0.116301	−0.0581505	+	0.998308i	$0.518520\pi$
$410$	0	0
$411$	34.4464	0.00413411
$412$	0	0
$413$	491.809	0.0585965
$414$	0	0
$415$	−207.966	−0.0245991
$416$	0	0
$417$	888.889	0.104386
$418$	0	0
$419$	−9209.39	−1.07377	−0.536883	−	0.843657i	$-0.680398\pi$
$419$	−9209.39	−1.07377	−0.536883	+	0.843657i	$0.680398\pi$
$420$	0	0
$421$	7913.21	0.916071	0.458036	−	0.888934i	$-0.348553\pi$
$421$	7913.21	0.916071	0.458036	+	0.888934i	$0.348553\pi$
$422$	0	0
$423$	−4247.39	−0.488215
$424$	0	0
$425$	10767.2	1.22891
$426$	0	0
$427$	−13.6890	−0.00155142
$428$	0	0
$429$	1585.77	0.178466
$430$	0	0
$431$	14530.7	1.62394	0.811971	−	0.583698i	$-0.198395\pi$
$431$	14530.7	1.62394	0.811971	+	0.583698i	$0.198395\pi$
$432$	0	0
$433$	−9133.35	−1.01367	−0.506837	−	0.862042i	$-0.669185\pi$
$433$	−9133.35	−1.01367	−0.506837	+	0.862042i	$0.669185\pi$
$434$	0	0
$435$	460.078	0.0507105
$436$	0	0
$437$	363.365	0.0397759
$438$	0	0
$439$	6418.48	0.697808	0.348904	−	0.937159i	$-0.386554\pi$
$439$	6418.48	0.697808	0.348904	+	0.937159i	$0.386554\pi$
$440$	0	0
$441$	8966.79	0.968231
$442$	0	0
$443$	9244.72	0.991489	0.495745	−	0.868468i	$-0.334895\pi$
$443$	9244.72	0.991489	0.495745	+	0.868468i	$0.334895\pi$
$444$	0	0
$445$	643.756	0.0685774
$446$	0	0
$447$	186.759	0.0197616
$448$	0	0
$449$	11826.4	1.24303	0.621515	−	0.783402i	$-0.286518\pi$
$449$	11826.4	1.24303	0.621515	+	0.783402i	$0.286518\pi$
$450$	0	0
$451$	−16043.2	−1.67505
$452$	0	0
$453$	2648.37	0.274683
$454$	0	0
$455$	−51.7511	−0.00533215
$456$	0	0
$457$	7907.62	0.809416	0.404708	−	0.914446i	$-0.367373\pi$
$457$	7907.62	0.809416	0.404708	+	0.914446i	$0.367373\pi$
$458$	0	0
$459$	4345.38	0.441885
$460$	0	0
$461$	−4955.68	−0.500670	−0.250335	−	0.968159i	$-0.580541\pi$
$461$	−4955.68	−0.500670	−0.250335	+	0.968159i	$0.580541\pi$
$462$	0	0
$463$	−13842.4	−1.38944	−0.694719	−	0.719282i	$-0.744471\pi$
$463$	−13842.4	−1.38944	−0.694719	+	0.719282i	$0.744471\pi$
$464$	0	0
$465$	−167.713	−0.0167258
$466$	0	0
$467$	−7500.26	−0.743192	−0.371596	−	0.928395i	$-0.621189\pi$
$467$	−7500.26	−0.743192	−0.371596	+	0.928395i	$0.621189\pi$
$468$	0	0
$469$	408.102	0.0401799
$470$	0	0
$471$	1594.84	0.156022
$472$	0	0
$473$	−3516.27	−0.341815
$474$	0	0
$475$	−1897.37	−0.183279
$476$	0	0
$477$	−16695.8	−1.60262
$478$	0	0
$479$	20080.4	1.91545	0.957723	−	0.287691i	$-0.0928876\pi$
$479$	20080.4	1.91545	0.957723	+	0.287691i	$0.0928876\pi$
$480$	0	0
$481$	−7948.50	−0.753472
$482$	0	0
$483$	12.4140	0.00116948
$484$	0	0
$485$	1596.89	0.149508
$486$	0	0
$487$	−9758.80	−0.908035	−0.454018	−	0.890993i	$-0.650010\pi$
$487$	−9758.80	−0.908035	−0.454018	+	0.890993i	$0.650010\pi$
$488$	0	0
$489$	−2273.90	−0.210284
$490$	0	0
$491$	3286.03	0.302029	0.151015	−	0.988532i	$-0.451746\pi$
$491$	3286.03	0.302029	0.151015	+	0.988532i	$0.451746\pi$
$492$	0	0
$493$	−20437.4	−1.86705
$494$	0	0
$495$	2552.82	0.231799
$496$	0	0
$497$	−415.568	−0.0375066
$498$	0	0
$499$	−1092.31	−0.0979927	−0.0489964	−	0.998799i	$-0.515602\pi$
$499$	−1092.31	−0.0979927	−0.0489964	+	0.998799i	$0.515602\pi$
$500$	0	0
$501$	−565.479	−0.0504266
$502$	0	0
$503$	−16200.1	−1.43603	−0.718017	−	0.696026i	$-0.754950\pi$
$503$	−16200.1	−1.43603	−0.718017	+	0.696026i	$0.754950\pi$
$504$	0	0
$505$	421.253	0.0371198
$506$	0	0
$507$	−581.940	−0.0509761
$508$	0	0
$509$	−16182.5	−1.40919	−0.704593	−	0.709612i	$-0.748870\pi$
$509$	−16182.5	−1.40919	−0.704593	+	0.709612i	$0.748870\pi$
$510$	0	0
$511$	196.832	0.0170398
$512$	0	0
$513$	−765.732	−0.0659023
$514$	0	0
$515$	−3383.54	−0.289508
$516$	0	0
$517$	−7151.56	−0.608366
$518$	0	0
$519$	459.964	0.0389020
$520$	0	0
$521$	7129.58	0.599525	0.299763	−	0.954014i	$-0.403093\pi$
$521$	7129.58	0.599525	0.299763	+	0.954014i	$0.403093\pi$
$522$	0	0
$523$	12642.8	1.05704	0.528520	−	0.848921i	$-0.322747\pi$
$523$	12642.8	1.05704	0.528520	+	0.848921i	$0.322747\pi$
$524$	0	0
$525$	−64.8220	−0.00538869
$526$	0	0
$527$	7450.08	0.615807
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−21737.1	−1.77648
$532$	0	0
$533$	−14374.7	−1.16817
$534$	0	0
$535$	−3901.22	−0.315261
$536$	0	0
$537$	−2765.02	−0.222197
$538$	0	0
$539$	15097.9	1.20652
$540$	0	0
$541$	9070.56	0.720839	0.360419	−	0.932790i	$-0.382634\pi$
$541$	9070.56	0.720839	0.360419	+	0.932790i	$0.382634\pi$
$542$	0	0
$543$	415.017	0.0327994
$544$	0	0
$545$	394.192	0.0309823
$546$	0	0
$547$	−8523.38	−0.666241	−0.333120	−	0.942884i	$-0.608102\pi$
$547$	−8523.38	−0.666241	−0.333120	+	0.942884i	$0.608102\pi$
$548$	0	0
$549$	605.030	0.0470347
$550$	0	0
$551$	3601.43	0.278451
$552$	0	0
$553$	123.298	0.00948131
$554$	0	0
$555$	406.336	0.0310775
$556$	0	0
$557$	−5346.61	−0.406720	−0.203360	−	0.979104i	$-0.565186\pi$
$557$	−5346.61	−0.406720	−0.203360	+	0.979104i	$0.565186\pi$
$558$	0	0
$559$	−3150.56	−0.238380
$560$	0	0
$561$	3601.10	0.271014
$562$	0	0
$563$	6453.26	0.483077	0.241539	−	0.970391i	$-0.422348\pi$
$563$	6453.26	0.483077	0.241539	+	0.970391i	$0.422348\pi$
$564$	0	0
$565$	−1200.02	−0.0893543
$566$	0	0
$567$	392.182	0.0290478
$568$	0	0
$569$	−10772.7	−0.793702	−0.396851	−	0.917883i	$-0.629897\pi$
$569$	−10772.7	−0.793702	−0.396851	+	0.917883i	$0.629897\pi$
$570$	0	0
$571$	−7263.48	−0.532342	−0.266171	−	0.963926i	$-0.585759\pi$
$571$	−7263.48	−0.532342	−0.266171	+	0.963926i	$0.585759\pi$
$572$	0	0
$573$	2945.62	0.214755
$574$	0	0
$575$	−2762.26	−0.200338
$576$	0	0
$577$	−18974.9	−1.36904	−0.684519	−	0.728995i	$-0.739988\pi$
$577$	−18974.9	−1.36904	−0.684519	+	0.728995i	$0.739988\pi$
$578$	0	0
$579$	1901.49	0.136482
$580$	0	0
$581$	−55.6168	−0.00397138
$582$	0	0
$583$	−28111.6	−1.99702
$584$	0	0
$585$	2287.31	0.161656
$586$	0	0
$587$	15576.7	1.09527	0.547633	−	0.836719i	$-0.315529\pi$
$587$	15576.7	1.09527	0.547633	+	0.836719i	$0.315529\pi$
$588$	0	0
$589$	−1312.83	−0.0918411
$590$	0	0
$591$	1119.05	0.0778879
$592$	0	0
$593$	16186.3	1.12090	0.560450	−	0.828188i	$-0.310628\pi$
$593$	16186.3	1.12090	0.560450	+	0.828188i	$0.310628\pi$
$594$	0	0
$595$	−117.521	−0.00809728
$596$	0	0
$597$	−146.633	−0.0100524
$598$	0	0
$599$	−2758.10	−0.188135	−0.0940676	−	0.995566i	$-0.529987\pi$
$599$	−2758.10	−0.188135	−0.0940676	+	0.995566i	$0.529987\pi$
$600$	0	0
$601$	20723.3	1.40652	0.703261	−	0.710932i	$-0.251727\pi$
$601$	20723.3	1.40652	0.703261	+	0.710932i	$0.251727\pi$
$602$	0	0
$603$	−18037.4	−1.21814
$604$	0	0
$605$	1351.56	0.0908243
$606$	0	0
$607$	27245.6	1.82185	0.910927	−	0.412567i	$-0.135368\pi$
$607$	27245.6	1.82185	0.910927	+	0.412567i	$0.135368\pi$
$608$	0	0
$609$	123.040	0.00818691
$610$	0	0
$611$	−6407.76	−0.424272
$612$	0	0
$613$	−27208.2	−1.79271	−0.896353	−	0.443341i	$-0.853793\pi$
$613$	−27208.2	−1.79271	−0.896353	+	0.443341i	$0.853793\pi$
$614$	0	0
$615$	734.848	0.0481820
$616$	0	0
$617$	−13368.1	−0.872251	−0.436125	−	0.899886i	$-0.643650\pi$
$617$	−13368.1	−0.872251	−0.436125	+	0.899886i	$0.643650\pi$
$618$	0	0
$619$	15188.6	0.986240	0.493120	−	0.869961i	$-0.335856\pi$
$619$	15188.6	0.986240	0.493120	+	0.869961i	$0.335856\pi$
$620$	0	0
$621$	−1114.78	−0.0720365
$622$	0	0
$623$	172.161	0.0110714
$624$	0	0
$625$	13810.9	0.883900
$626$	0	0
$627$	−634.577	−0.0404188
$628$	0	0
$629$	−18050.1	−1.14420
$630$	0	0
$631$	22687.4	1.43133	0.715665	−	0.698444i	$-0.246124\pi$
$631$	22687.4	1.43133	0.715665	+	0.698444i	$0.246124\pi$
$632$	0	0
$633$	1883.81	0.118285
$634$	0	0
$635$	611.341	0.0382052
$636$	0	0
$637$	13527.6	0.841418
$638$	0	0
$639$	18367.4	1.13709
$640$	0	0
$641$	−18278.3	−1.12629	−0.563143	−	0.826360i	$-0.690408\pi$
$641$	−18278.3	−1.12629	−0.563143	+	0.826360i	$0.690408\pi$
$642$	0	0
$643$	6862.31	0.420876	0.210438	−	0.977607i	$-0.432511\pi$
$643$	6862.31	0.420876	0.210438	+	0.977607i	$0.432511\pi$
$644$	0	0
$645$	161.060	0.00983214
$646$	0	0
$647$	−15451.9	−0.938913	−0.469456	−	0.882956i	$-0.655550\pi$
$647$	−15451.9	−0.938913	−0.469456	+	0.882956i	$0.655550\pi$
$648$	0	0
$649$	−36600.0	−2.21368
$650$	0	0
$651$	−44.8518	−0.00270028
$652$	0	0
$653$	−2222.53	−0.133192	−0.0665959	−	0.997780i	$-0.521214\pi$
$653$	−2222.53	−0.133192	−0.0665959	+	0.997780i	$0.521214\pi$
$654$	0	0
$655$	−5480.79	−0.326950
$656$	0	0
$657$	−8699.66	−0.516600
$658$	0	0
$659$	−21327.7	−1.26071	−0.630357	−	0.776305i	$-0.717092\pi$
$659$	−21327.7	−1.26071	−0.630357	+	0.776305i	$0.717092\pi$
$660$	0	0
$661$	33548.0	1.97408	0.987038	−	0.160486i	$-0.0513061\pi$
$661$	33548.0	1.97408	0.987038	+	0.160486i	$0.0513061\pi$
$662$	0	0
$663$	3226.57	0.189004
$664$	0	0
$665$	20.7092	0.00120762
$666$	0	0
$667$	5243.10	0.304369
$668$	0	0
$669$	−2660.91	−0.153777
$670$	0	0
$671$	1018.72	0.0586100
$672$	0	0
$673$	4007.13	0.229515	0.114758	−	0.993394i	$-0.463391\pi$
$673$	4007.13	0.229515	0.114758	+	0.993394i	$0.463391\pi$
$674$	0	0
$675$	5821.02	0.331928
$676$	0	0
$677$	−24051.8	−1.36541	−0.682707	−	0.730692i	$-0.739198\pi$
$677$	−24051.8	−1.36541	−0.682707	+	0.730692i	$0.739198\pi$
$678$	0	0
$679$	427.061	0.0241371
$680$	0	0
$681$	−2800.55	−0.157588
$682$	0	0
$683$	7112.48	0.398465	0.199233	−	0.979952i	$-0.436155\pi$
$683$	7112.48	0.398465	0.199233	+	0.979952i	$0.436155\pi$
$684$	0	0
$685$	83.6581	0.00466629
$686$	0	0
$687$	2741.23	0.152233
$688$	0	0
$689$	−25187.9	−1.39272
$690$	0	0
$691$	−11373.1	−0.626124	−0.313062	−	0.949733i	$-0.601355\pi$
$691$	−11373.1	−0.626124	−0.313062	+	0.949733i	$0.601355\pi$
$692$	0	0
$693$	682.707	0.0374226
$694$	0	0
$695$	2158.79	0.117824
$696$	0	0
$697$	−32643.2	−1.77396
$698$	0	0
$699$	5128.58	0.277512
$700$	0	0
$701$	−21075.0	−1.13551	−0.567754	−	0.823198i	$-0.692187\pi$
$701$	−21075.0	−1.13551	−0.567754	+	0.823198i	$0.692187\pi$
$702$	0	0
$703$	3180.74	0.170646
$704$	0	0
$705$	327.571	0.0174994
$706$	0	0
$707$	112.657	0.00599277
$708$	0	0
$709$	33021.6	1.74916	0.874580	−	0.484882i	$-0.161137\pi$
$709$	33021.6	1.74916	0.874580	+	0.484882i	$0.161137\pi$
$710$	0	0
$711$	−5449.56	−0.287446
$712$	0	0
$713$	−1911.27	−0.100390
$714$	0	0
$715$	3851.27	0.201440
$716$	0	0
$717$	1398.14	0.0728234
$718$	0	0
$719$	−22281.2	−1.15570	−0.577851	−	0.816143i	$-0.696108\pi$
$719$	−22281.2	−1.15570	−0.577851	+	0.816143i	$0.696108\pi$
$720$	0	0
$721$	−904.868	−0.0467393
$722$	0	0
$723$	1791.64	0.0921600
$724$	0	0
$725$	−27377.8	−1.40246
$726$	0	0
$727$	−24966.4	−1.27366	−0.636832	−	0.771003i	$-0.719756\pi$
$727$	−24966.4	−1.27366	−0.636832	+	0.771003i	$0.719756\pi$
$728$	0	0
$729$	−16140.8	−0.820038
$730$	0	0
$731$	−7154.55	−0.361998
$732$	0	0
$733$	569.397	0.0286919	0.0143460	−	0.999897i	$-0.495433\pi$
$733$	569.397	0.0286919	0.0143460	+	0.999897i	$0.495433\pi$
$734$	0	0
$735$	−691.546	−0.0347048
$736$	0	0
$737$	−30370.6	−1.51793
$738$	0	0
$739$	−6590.88	−0.328078	−0.164039	−	0.986454i	$-0.552452\pi$
$739$	−6590.88	−0.328078	−0.164039	+	0.986454i	$0.552452\pi$
$740$	0	0
$741$	−568.577	−0.0281879
$742$	0	0
$743$	31525.6	1.55661	0.778305	−	0.627887i	$-0.216080\pi$
$743$	31525.6	1.55661	0.778305	+	0.627887i	$0.216080\pi$
$744$	0	0
$745$	453.572	0.0223055
$746$	0	0
$747$	2458.17	0.120401
$748$	0	0
$749$	−1043.31	−0.0508969
$750$	0	0
$751$	28931.0	1.40573	0.702867	−	0.711321i	$-0.251903\pi$
$751$	28931.0	1.40573	0.702867	+	0.711321i	$0.251903\pi$
$752$	0	0
$753$	−1224.18	−0.0592451
$754$	0	0
$755$	6431.95	0.310043
$756$	0	0
$757$	−11377.9	−0.546284	−0.273142	−	0.961974i	$-0.588063\pi$
$757$	−11377.9	−0.546284	−0.273142	+	0.961974i	$0.588063\pi$
$758$	0	0
$759$	−923.841	−0.0441809
$760$	0	0
$761$	−8236.18	−0.392328	−0.196164	−	0.980571i	$-0.562848\pi$
$761$	−8236.18	−0.392328	−0.196164	+	0.980571i	$0.562848\pi$
$762$	0	0
$763$	105.420	0.00500190
$764$	0	0
$765$	5194.22	0.245487
$766$	0	0
$767$	−32793.4	−1.54381
$768$	0	0
$769$	11253.6	0.527719	0.263860	−	0.964561i	$-0.415004\pi$
$769$	11253.6	0.527719	0.263860	+	0.964561i	$0.415004\pi$
$770$	0	0
$771$	4283.82	0.200101
$772$	0	0
$773$	10650.2	0.495553	0.247776	−	0.968817i	$-0.420300\pi$
$773$	10650.2	0.495553	0.247776	+	0.968817i	$0.420300\pi$
$774$	0	0
$775$	9980.04	0.462572
$776$	0	0
$777$	108.667	0.00501727
$778$	0	0
$779$	5752.30	0.264567
$780$	0	0
$781$	30926.2	1.41693
$782$	0	0
$783$	−11049.0	−0.504290
$784$	0	0
$785$	3873.31	0.176107
$786$	0	0
$787$	−23276.3	−1.05427	−0.527136	−	0.849781i	$-0.676734\pi$
$787$	−23276.3	−1.05427	−0.527136	+	0.849781i	$0.676734\pi$
$788$	0	0
$789$	−5084.97	−0.229442
$790$	0	0
$791$	−320.924	−0.0144257
$792$	0	0
$793$	912.769	0.0408744
$794$	0	0
$795$	1287.63	0.0574435
$796$	0	0
$797$	2322.42	0.103217	0.0516087	−	0.998667i	$-0.483565\pi$
$797$	2322.42	0.103217	0.0516087	+	0.998667i	$0.483565\pi$
$798$	0	0
$799$	−14551.3	−0.644289
$800$	0	0
$801$	−7609.23	−0.335654
$802$	0	0
$803$	−14648.1	−0.643736
$804$	0	0
$805$	30.1493	0.00132003
$806$	0	0
$807$	4353.39	0.189897
$808$	0	0
$809$	10930.5	0.475024	0.237512	−	0.971385i	$-0.423668\pi$
$809$	10930.5	0.475024	0.237512	+	0.971385i	$0.423668\pi$
$810$	0	0
$811$	−3850.82	−0.166733	−0.0833667	−	0.996519i	$-0.526567\pi$
$811$	−3850.82	−0.166733	−0.0833667	+	0.996519i	$0.526567\pi$
$812$	0	0
$813$	−5858.19	−0.252713
$814$	0	0
$815$	−5522.48	−0.237355
$816$	0	0
$817$	1260.76	0.0539882
$818$	0	0
$819$	611.701	0.0260984
$820$	0	0
$821$	−6360.26	−0.270371	−0.135186	−	0.990820i	$-0.543163\pi$
$821$	−6360.26	−0.270371	−0.135186	+	0.990820i	$0.543163\pi$
$822$	0	0
$823$	−22339.4	−0.946178	−0.473089	−	0.881015i	$-0.656861\pi$
$823$	−22339.4	−0.946178	−0.473089	+	0.881015i	$0.656861\pi$
$824$	0	0
$825$	4824.00	0.203576
$826$	0	0
$827$	40000.1	1.68191	0.840955	−	0.541106i	$-0.181994\pi$
$827$	40000.1	1.68191	0.840955	+	0.541106i	$0.181994\pi$
$828$	0	0
$829$	27206.2	1.13982	0.569910	−	0.821707i	$-0.306978\pi$
$829$	27206.2	1.13982	0.569910	+	0.821707i	$0.306978\pi$
$830$	0	0
$831$	2085.56	0.0870605
$832$	0	0
$833$	30719.6	1.27776
$834$	0	0
$835$	−1373.35	−0.0569181
$836$	0	0
$837$	4027.70	0.166329
$838$	0	0
$839$	−30155.3	−1.24086	−0.620428	−	0.784264i	$-0.713041\pi$
$839$	−30155.3	−1.24086	−0.620428	+	0.784264i	$0.713041\pi$
$840$	0	0
$841$	27577.3	1.13073
$842$	0	0
$843$	−7205.85	−0.294404
$844$	0	0
$845$	−1413.33	−0.0575383
$846$	0	0
$847$	361.451	0.0146630
$848$	0	0
$849$	−3531.40	−0.142753
$850$	0	0
$851$	4630.65	0.186529
$852$	0	0
$853$	18469.2	0.741354	0.370677	−	0.928762i	$-0.379126\pi$
$853$	18469.2	0.741354	0.370677	+	0.928762i	$0.379126\pi$
$854$	0	0
$855$	−915.312	−0.0366117
$856$	0	0
$857$	−32360.5	−1.28986	−0.644932	−	0.764240i	$-0.723114\pi$
$857$	−32360.5	−1.28986	−0.644932	+	0.764240i	$0.723114\pi$
$858$	0	0
$859$	36501.0	1.44982	0.724911	−	0.688843i	$-0.241881\pi$
$859$	36501.0	1.44982	0.724911	+	0.688843i	$0.241881\pi$
$860$	0	0
$861$	196.522	0.00777870
$862$	0	0
$863$	708.078	0.0279296	0.0139648	−	0.999902i	$-0.495555\pi$
$863$	708.078	0.0279296	0.0139648	+	0.999902i	$0.495555\pi$
$864$	0	0
$865$	1117.09	0.0439099
$866$	0	0
$867$	2848.47	0.111579
$868$	0	0
$869$	−9175.72	−0.358188
$870$	0	0
$871$	−27211.9	−1.05860
$872$	0	0
$873$	−18875.4	−0.731769
$874$	0	0
$875$	−321.284	−0.0124130
$876$	0	0
$877$	−12368.7	−0.476237	−0.238119	−	0.971236i	$-0.576531\pi$
$877$	−12368.7	−0.476237	−0.238119	+	0.971236i	$0.576531\pi$
$878$	0	0
$879$	7686.80	0.294959
$880$	0	0
$881$	27775.3	1.06217	0.531086	−	0.847318i	$-0.321784\pi$
$881$	27775.3	1.06217	0.531086	+	0.847318i	$0.321784\pi$
$882$	0	0
$883$	16798.0	0.640202	0.320101	−	0.947383i	$-0.396283\pi$
$883$	16798.0	0.640202	0.320101	+	0.947383i	$0.396283\pi$
$884$	0	0
$885$	1676.43	0.0636754
$886$	0	0
$887$	−20731.8	−0.784786	−0.392393	−	0.919798i	$-0.628353\pi$
$887$	−20731.8	−0.784786	−0.392393	+	0.919798i	$0.628353\pi$
$888$	0	0
$889$	163.492	0.00616800
$890$	0	0
$891$	−29185.8	−1.09738
$892$	0	0
$893$	2564.19	0.0960888
$894$	0	0
$895$	−6715.25	−0.250800
$896$	0	0
$897$	−827.756	−0.0308116
$898$	0	0
$899$	−18943.3	−0.702775
$900$	0	0
$901$	−57198.7	−2.11494
$902$	0	0
$903$	43.0727	0.00158734
$904$	0	0
$905$	1007.93	0.0370217
$906$	0	0
$907$	3295.84	0.120658	0.0603289	−	0.998179i	$-0.480785\pi$
$907$	3295.84	0.120658	0.0603289	+	0.998179i	$0.480785\pi$
$908$	0	0
$909$	−4979.23	−0.181684
$910$	0	0
$911$	43398.7	1.57834	0.789168	−	0.614178i	$-0.210512\pi$
$911$	43398.7	1.57834	0.789168	+	0.614178i	$0.210512\pi$
$912$	0	0
$913$	4138.95	0.150032
$914$	0	0
$915$	−46.6617	−0.00168589
$916$	0	0
$917$	−1465.74	−0.0527841
$918$	0	0
$919$	−6085.18	−0.218424	−0.109212	−	0.994018i	$-0.534833\pi$
$919$	−6085.18	−0.218424	−0.109212	+	0.994018i	$0.534833\pi$
$920$	0	0
$921$	8515.56	0.304666
$922$	0	0
$923$	27709.7	0.988164
$924$	0	0
$925$	−24179.7	−0.859486
$926$	0	0
$927$	39993.6	1.41700
$928$	0	0
$929$	−2445.73	−0.0863745	−0.0431872	−	0.999067i	$-0.513751\pi$
$929$	−2445.73	−0.0863745	−0.0431872	+	0.999067i	$0.513751\pi$
$930$	0	0
$931$	−5413.33	−0.190564
$932$	0	0
$933$	−4455.95	−0.156357
$934$	0	0
$935$	8745.79	0.305901
$936$	0	0
$937$	−31333.5	−1.09244	−0.546222	−	0.837641i	$-0.683934\pi$
$937$	−31333.5	−1.09244	−0.546222	+	0.837641i	$0.683934\pi$
$938$	0	0
$939$	5773.50	0.200651
$940$	0	0
$941$	−6917.40	−0.239640	−0.119820	−	0.992796i	$-0.538232\pi$
$941$	−6917.40	−0.239640	−0.119820	+	0.992796i	$0.538232\pi$
$942$	0	0
$943$	8374.41	0.289192
$944$	0	0
$945$	−63.5347	−0.00218707
$946$	0	0
$947$	14798.9	0.507814	0.253907	−	0.967229i	$-0.418284\pi$
$947$	14798.9	0.507814	0.253907	+	0.967229i	$0.418284\pi$
$948$	0	0
$949$	−13124.6	−0.448939
$950$	0	0
$951$	4813.06	0.164116
$952$	0	0
$953$	27008.8	0.918049	0.459024	−	0.888424i	$-0.348199\pi$
$953$	27008.8	0.918049	0.459024	+	0.888424i	$0.348199\pi$
$954$	0	0
$955$	7153.85	0.242401
$956$	0	0
$957$	−9156.52	−0.309288
$958$	0	0
$959$	22.3729	0.000753345 0
$960$	0	0
$961$	−22885.6	−0.768205
$962$	0	0
$963$	46112.6	1.54305
$964$	0	0
$965$	4618.05	0.154052
$966$	0	0
$967$	−30982.1	−1.03032	−0.515158	−	0.857095i	$-0.672267\pi$
$967$	−30982.1	−1.03032	−0.515158	+	0.857095i	$0.672267\pi$
$968$	0	0
$969$	−1291.17	−0.0428054
$970$	0	0
$971$	−11197.3	−0.370070	−0.185035	−	0.982732i	$-0.559240\pi$
$971$	−11197.3	−0.370070	−0.185035	+	0.982732i	$0.559240\pi$
$972$	0	0
$973$	577.332	0.0190220
$974$	0	0
$975$	4322.27	0.141973
$976$	0	0
$977$	35631.1	1.16678	0.583388	−	0.812194i	$-0.301727\pi$
$977$	35631.1	1.16678	0.583388	+	0.812194i	$0.301727\pi$
$978$	0	0
$979$	−12812.1	−0.418259
$980$	0	0
$981$	−4659.38	−0.151644
$982$	0	0
$983$	−59201.0	−1.92087	−0.960437	−	0.278497i	$-0.910164\pi$
$983$	−59201.0	−1.92087	−0.960437	+	0.278497i	$0.910164\pi$
$984$	0	0
$985$	2717.78	0.0879145
$986$	0	0
$987$	87.6032	0.00282517
$988$	0	0
$989$	1835.46	0.0590133
$990$	0	0
$991$	49053.2	1.57238	0.786189	−	0.617986i	$-0.212051\pi$
$991$	49053.2	1.57238	0.786189	+	0.617986i	$0.212051\pi$
$992$	0	0
$993$	475.337	0.0151907
$994$	0	0
$995$	−356.118	−0.0113464
$996$	0	0
$997$	58201.2	1.84880	0.924399	−	0.381428i	$-0.124567\pi$
$997$	58201.2	1.84880	0.924399	+	0.381428i	$0.124567\pi$
$998$	0	0
$999$	−9758.34	−0.309049

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.4.a.bd.1.3		4
4.3	odd	2		1472.4.a.ba.1.2		4
8.3	odd	2		184.4.a.e.1.3	✓	4
8.5	even	2		368.4.a.n.1.2		4
24.11	even	2		1656.4.a.n.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.4.a.e.1.3	✓	4	8.3	odd	2
368.4.a.n.1.2		4	8.5	even	2
1472.4.a.ba.1.2		4	4.3	odd	2
1472.4.a.bd.1.3		4	1.1	even	1	trivial
1656.4.a.n.1.2		4	24.11	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

\(f(q)\)	\(=\)	\(q+0.911599 q^{3} +2.21395 q^{5} +0.592081 q^{7} -26.1690 q^{9} -44.0622 q^{11} -39.4794 q^{13} +2.01823 q^{15} -89.6532 q^{17} +15.7985 q^{19} +0.539741 q^{21} +23.0000 q^{23} -120.098 q^{25} -48.4688 q^{27} +227.961 q^{29} -83.0988 q^{31} -40.1670 q^{33} +1.31084 q^{35} +201.333 q^{37} -35.9894 q^{39} +364.105 q^{41} +79.8025 q^{43} -57.9368 q^{45} +162.306 q^{47} -342.649 q^{49} -81.7277 q^{51} +637.999 q^{53} -97.5513 q^{55} +14.4019 q^{57} +830.645 q^{59} -23.1201 q^{61} -15.4942 q^{63} -87.4054 q^{65} +689.266 q^{67} +20.9668 q^{69} -701.876 q^{71} +332.442 q^{73} -109.482 q^{75} -26.0884 q^{77} +208.245 q^{79} +662.379 q^{81} -93.9343 q^{83} -198.487 q^{85} +207.809 q^{87} +290.773 q^{89} -23.3750 q^{91} -75.7528 q^{93} +34.9770 q^{95} +721.288 q^{97} +1153.06 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} + 20 q^{5} + 10 q^{7} + 23 q^{9} - 30 q^{11} + 153 q^{13} + 136 q^{15} - 68 q^{17} - 120 q^{19} + 426 q^{21} + 92 q^{23} - 76 q^{25} + 43 q^{27} + 315 q^{29} - 249 q^{31} - 504 q^{33} + 224 q^{35}+ \cdots - 1470 q^{99}+O(q^{100}) \) 4 * q + q^3 + 20 * q^5 + 10 * q^7 + 23 * q^9 - 30 * q^11 + 153 * q^13 + 136 * q^15 - 68 * q^17 - 120 * q^19 + 426 * q^21 + 92 * q^23 - 76 * q^25 + 43 * q^27 + 315 * q^29 - 249 * q^31 - 504 * q^33 + 224 * q^35 + 348 * q^37 + 61 * q^39 - 929 * q^41 + 50 * q^43 + 1028 * q^45 - 205 * q^47 + 456 * q^49 + 890 * q^51 + 578 * q^53 - 1128 * q^55 + 366 * q^57 - 664 * q^59 + 182 * q^61 - 624 * q^63 + 848 * q^65 + 1078 * q^67 + 23 * q^69 + 739 * q^71 + 921 * q^73 + 2883 * q^75 - 528 * q^77 - 1754 * q^79 + 788 * q^81 + 1020 * q^83 + 1072 * q^85 - 2121 * q^87 - 430 * q^89 + 1334 * q^91 - 1035 * q^93 - 744 * q^95 + 48 * q^97 - 1470 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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