LMFDB - Embedded newform 4368.2.a.bh.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4368,2,Mod(1,4368)] 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("4368.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,-1,0,-2,0,2,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$34.8786556029$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.27492$ of defining polynomial
Character	$\chi$	$=$	4368.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.27492 q^{5} -1.00000 q^{7} +1.00000 q^{9} -5.27492 q^{11} -1.00000 q^{13} +3.27492 q^{15} -7.27492 q^{17} -5.27492 q^{19} -1.00000 q^{21} +5.27492 q^{23} +5.72508 q^{25} +1.00000 q^{27} +0.725083 q^{29} -8.00000 q^{31} -5.27492 q^{33} -3.27492 q^{35} +3.27492 q^{37} -1.00000 q^{39} -8.54983 q^{41} -5.27492 q^{43} +3.27492 q^{45} +1.00000 q^{49} -7.27492 q^{51} +10.0000 q^{53} -17.2749 q^{55} -5.27492 q^{57} -8.00000 q^{59} -4.72508 q^{61} -1.00000 q^{63} -3.27492 q^{65} +2.54983 q^{67} +5.27492 q^{69} +9.82475 q^{73} +5.72508 q^{75} +5.27492 q^{77} +2.54983 q^{79} +1.00000 q^{81} -10.5498 q^{83} -23.8248 q^{85} +0.725083 q^{87} -14.0000 q^{89} +1.00000 q^{91} -8.00000 q^{93} -17.2749 q^{95} -15.0997 q^{97} -5.27492 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - q^{5} - 2 q^{7} + 2 q^{9} - 3 q^{11} - 2 q^{13} - q^{15} - 7 q^{17} - 3 q^{19} - 2 q^{21} + 3 q^{23} + 19 q^{25} + 2 q^{27} + 9 q^{29} - 16 q^{31} - 3 q^{33} + q^{35} - q^{37} - 2 q^{39}+ \cdots - 3 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - q^5 - 2 * q^7 + 2 * q^9 - 3 * q^11 - 2 * q^13 - q^15 - 7 * q^17 - 3 * q^19 - 2 * q^21 + 3 * q^23 + 19 * q^25 + 2 * q^27 + 9 * q^29 - 16 * q^31 - 3 * q^33 + q^35 - q^37 - 2 * q^39 - 2 * q^41 - 3 * q^43 - q^45 + 2 * q^49 - 7 * q^51 + 20 * q^53 - 27 * q^55 - 3 * q^57 - 16 * q^59 - 17 * q^61 - 2 * q^63 + q^65 - 10 * q^67 + 3 * q^69 - 3 * q^73 + 19 * q^75 + 3 * q^77 - 10 * q^79 + 2 * q^81 - 6 * q^83 - 25 * q^85 + 9 * q^87 - 28 * q^89 + 2 * q^91 - 16 * q^93 - 27 * q^95 - 3 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.27492	1.46459	0.732294	−	0.680989i	$-0.238450\pi$
$5$	3.27492	1.46459	0.732294	+	0.680989i	$0.238450\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.27492	−1.59045	−0.795224	−	0.606316i	$-0.792647\pi$
$11$	−5.27492	−1.59045	−0.795224	+	0.606316i	$0.792647\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	3.27492	0.845580
$16$	0	0
$17$	−7.27492	−1.76443	−0.882213	−	0.470850i	$-0.843947\pi$
$17$	−7.27492	−1.76443	−0.882213	+	0.470850i	$0.843947\pi$
$18$	0	0
$19$	−5.27492	−1.21015	−0.605075	−	0.796169i	$-0.706857\pi$
$19$	−5.27492	−1.21015	−0.605075	+	0.796169i	$0.706857\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	5.27492	1.09990	0.549948	−	0.835199i	$-0.314647\pi$
$23$	5.27492	1.09990	0.549948	+	0.835199i	$0.314647\pi$
$24$	0	0
$25$	5.72508	1.14502
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.725083	0.134644	0.0673222	−	0.997731i	$-0.478554\pi$
$29$	0.725083	0.134644	0.0673222	+	0.997731i	$0.478554\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	−5.27492	−0.918245
$34$	0	0
$35$	−3.27492	−0.553562
$36$	0	0
$37$	3.27492	0.538393	0.269197	−	0.963085i	$-0.413242\pi$
$37$	3.27492	0.538393	0.269197	+	0.963085i	$0.413242\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−8.54983	−1.33526	−0.667630	−	0.744493i	$-0.732691\pi$
$41$	−8.54983	−1.33526	−0.667630	+	0.744493i	$0.732691\pi$
$42$	0	0
$43$	−5.27492	−0.804417	−0.402209	−	0.915548i	$-0.631757\pi$
$43$	−5.27492	−0.804417	−0.402209	+	0.915548i	$0.631757\pi$
$44$	0	0
$45$	3.27492	0.488196
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.27492	−1.01869
$52$	0	0
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	−17.2749	−2.32935
$56$	0	0
$57$	−5.27492	−0.698680
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−4.72508	−0.604985	−0.302492	−	0.953152i	$-0.597819\pi$
$61$	−4.72508	−0.604985	−0.302492	+	0.953152i	$0.597819\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−3.27492	−0.406203
$66$	0	0
$67$	2.54983	0.311512	0.155756	−	0.987796i	$-0.450219\pi$
$67$	2.54983	0.311512	0.155756	+	0.987796i	$0.450219\pi$
$68$	0	0
$69$	5.27492	0.635025
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	9.82475	1.14990	0.574950	−	0.818188i	$-0.305021\pi$
$73$	9.82475	1.14990	0.574950	+	0.818188i	$0.305021\pi$
$74$	0	0
$75$	5.72508	0.661076
$76$	0	0
$77$	5.27492	0.601133
$78$	0	0
$79$	2.54983	0.286879	0.143439	−	0.989659i	$-0.454184\pi$
$79$	2.54983	0.286879	0.143439	+	0.989659i	$0.454184\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.5498	−1.15799	−0.578997	−	0.815329i	$-0.696556\pi$
$83$	−10.5498	−1.15799	−0.578997	+	0.815329i	$0.696556\pi$
$84$	0	0
$85$	−23.8248	−2.58416
$86$	0	0
$87$	0.725083	0.0777370
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	−17.2749	−1.77237
$96$	0	0
$97$	−15.0997	−1.53314	−0.766570	−	0.642161i	$-0.778038\pi$
$97$	−15.0997	−1.53314	−0.766570	+	0.642161i	$0.778038\pi$
$98$	0	0
$99$	−5.27492	−0.530149
$100$	0	0
$101$	−12.5498	−1.24876	−0.624378	−	0.781123i	$-0.714647\pi$
$101$	−12.5498	−1.24876	−0.624378	+	0.781123i	$0.714647\pi$
$102$	0	0
$103$	−2.72508	−0.268510	−0.134255	−	0.990947i	$-0.542864\pi$
$103$	−2.72508	−0.268510	−0.134255	+	0.990947i	$0.542864\pi$
$104$	0	0
$105$	−3.27492	−0.319599
$106$	0	0
$107$	14.5498	1.40659	0.703293	−	0.710900i	$-0.251712\pi$
$107$	14.5498	1.40659	0.703293	+	0.710900i	$0.251712\pi$
$108$	0	0
$109$	−7.27492	−0.696811	−0.348405	−	0.937344i	$-0.613277\pi$
$109$	−7.27492	−0.696811	−0.348405	+	0.937344i	$0.613277\pi$
$110$	0	0
$111$	3.27492	0.310841
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	17.2749	1.61089
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	7.27492	0.666891
$120$	0	0
$121$	16.8248	1.52952
$122$	0	0
$123$	−8.54983	−0.770913
$124$	0	0
$125$	2.37459	0.212389
$126$	0	0
$127$	5.45017	0.483624	0.241812	−	0.970323i	$-0.422258\pi$
$127$	5.45017	0.483624	0.241812	+	0.970323i	$0.422258\pi$
$128$	0	0
$129$	−5.27492	−0.464431
$130$	0	0
$131$	15.8248	1.38261	0.691307	−	0.722561i	$-0.257035\pi$
$131$	15.8248	1.38261	0.691307	+	0.722561i	$0.257035\pi$
$132$	0	0
$133$	5.27492	0.457393
$134$	0	0
$135$	3.27492	0.281860
$136$	0	0
$137$	20.3746	1.74072	0.870359	−	0.492417i	$-0.163887\pi$
$137$	20.3746	1.74072	0.870359	+	0.492417i	$0.163887\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.27492	0.441111
$144$	0	0
$145$	2.37459	0.197199
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	7.45017	0.610341	0.305171	−	0.952298i	$-0.401286\pi$
$149$	7.45017	0.610341	0.305171	+	0.952298i	$0.401286\pi$
$150$	0	0
$151$	1.27492	0.103751	0.0518756	−	0.998654i	$-0.483480\pi$
$151$	1.27492	0.103751	0.0518756	+	0.998654i	$0.483480\pi$
$152$	0	0
$153$	−7.27492	−0.588142
$154$	0	0
$155$	−26.1993	−2.10438
$156$	0	0
$157$	24.3746	1.94530	0.972652	−	0.232268i	$-0.0746146\pi$
$157$	24.3746	1.94530	0.972652	+	0.232268i	$0.0746146\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	−5.27492	−0.415722
$162$	0	0
$163$	2.54983	0.199718	0.0998592	−	0.995002i	$-0.468161\pi$
$163$	2.54983	0.199718	0.0998592	+	0.995002i	$0.468161\pi$
$164$	0	0
$165$	−17.2749	−1.34485
$166$	0	0
$167$	9.27492	0.717715	0.358857	−	0.933392i	$-0.383166\pi$
$167$	9.27492	0.717715	0.358857	+	0.933392i	$0.383166\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−5.27492	−0.403383
$172$	0	0
$173$	−23.0997	−1.75624	−0.878118	−	0.478445i	$-0.841201\pi$
$173$	−23.0997	−1.75624	−0.878118	+	0.478445i	$0.841201\pi$
$174$	0	0
$175$	−5.72508	−0.432776
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−11.0997	−0.825032	−0.412516	−	0.910950i	$-0.635350\pi$
$181$	−11.0997	−0.825032	−0.412516	+	0.910950i	$0.635350\pi$
$182$	0	0
$183$	−4.72508	−0.349288
$184$	0	0
$185$	10.7251	0.788524
$186$	0	0
$187$	38.3746	2.80623
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	5.27492	0.381680	0.190840	−	0.981621i	$-0.438879\pi$
$191$	5.27492	0.381680	0.190840	+	0.981621i	$0.438879\pi$
$192$	0	0
$193$	−3.45017	−0.248348	−0.124174	−	0.992260i	$-0.539628\pi$
$193$	−3.45017	−0.248348	−0.124174	+	0.992260i	$0.539628\pi$
$194$	0	0
$195$	−3.27492	−0.234522
$196$	0	0
$197$	12.5498	0.894139	0.447069	−	0.894499i	$-0.352468\pi$
$197$	12.5498	0.894139	0.447069	+	0.894499i	$0.352468\pi$
$198$	0	0
$199$	15.8248	1.12179	0.560893	−	0.827888i	$-0.310458\pi$
$199$	15.8248	1.12179	0.560893	+	0.827888i	$0.310458\pi$
$200$	0	0
$201$	2.54983	0.179851
$202$	0	0
$203$	−0.725083	−0.0508908
$204$	0	0
$205$	−28.0000	−1.95560
$206$	0	0
$207$	5.27492	0.366632
$208$	0	0
$209$	27.8248	1.92468
$210$	0	0
$211$	−23.8248	−1.64016	−0.820082	−	0.572246i	$-0.806072\pi$
$211$	−23.8248	−1.64016	−0.820082	+	0.572246i	$0.806072\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−17.2749	−1.17814
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	9.82475	0.663895
$220$	0	0
$221$	7.27492	0.489364
$222$	0	0
$223$	23.6495	1.58369	0.791844	−	0.610723i	$-0.209121\pi$
$223$	23.6495	1.58369	0.791844	+	0.610723i	$0.209121\pi$
$224$	0	0
$225$	5.72508	0.381672
$226$	0	0
$227$	5.45017	0.361740	0.180870	−	0.983507i	$-0.442109\pi$
$227$	5.45017	0.361740	0.180870	+	0.983507i	$0.442109\pi$
$228$	0	0
$229$	−11.0997	−0.733487	−0.366743	−	0.930322i	$-0.619527\pi$
$229$	−11.0997	−0.733487	−0.366743	+	0.930322i	$0.619527\pi$
$230$	0	0
$231$	5.27492	0.347064
$232$	0	0
$233$	20.5498	1.34626	0.673132	−	0.739522i	$-0.264948\pi$
$233$	20.5498	1.34626	0.673132	+	0.739522i	$0.264948\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.54983	0.165630
$238$	0	0
$239$	5.09967	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$239$	5.09967	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$240$	0	0
$241$	−15.0997	−0.972655	−0.486328	−	0.873777i	$-0.661664\pi$
$241$	−15.0997	−0.972655	−0.486328	+	0.873777i	$0.661664\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.27492	0.209227
$246$	0	0
$247$	5.27492	0.335635
$248$	0	0
$249$	−10.5498	−0.668569
$250$	0	0
$251$	28.9244	1.82569	0.912847	−	0.408303i	$-0.133879\pi$
$251$	28.9244	1.82569	0.912847	+	0.408303i	$0.133879\pi$
$252$	0	0
$253$	−27.8248	−1.74933
$254$	0	0
$255$	−23.8248	−1.49196
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	−3.27492	−0.203493
$260$	0	0
$261$	0.725083	0.0448815
$262$	0	0
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	0	0
$265$	32.7492	2.01177
$266$	0	0
$267$	−14.0000	−0.856786
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−13.4502	−0.817039	−0.408520	−	0.912750i	$-0.633955\pi$
$271$	−13.4502	−0.817039	−0.408520	+	0.912750i	$0.633955\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	−30.1993	−1.82109
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−15.0997	−0.900771	−0.450385	−	0.892834i	$-0.648713\pi$
$281$	−15.0997	−0.900771	−0.450385	+	0.892834i	$0.648713\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	−17.2749	−1.02328
$286$	0	0
$287$	8.54983	0.504681
$288$	0	0
$289$	35.9244	2.11320
$290$	0	0
$291$	−15.0997	−0.885158
$292$	0	0
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−26.1993	−1.52538
$296$	0	0
$297$	−5.27492	−0.306082
$298$	0	0
$299$	−5.27492	−0.305056
$300$	0	0
$301$	5.27492	0.304041
$302$	0	0
$303$	−12.5498	−0.720969
$304$	0	0
$305$	−15.4743	−0.886053
$306$	0	0
$307$	9.09967	0.519346	0.259673	−	0.965697i	$-0.416385\pi$
$307$	9.09967	0.519346	0.259673	+	0.965697i	$0.416385\pi$
$308$	0	0
$309$	−2.72508	−0.155025
$310$	0	0
$311$	−18.5498	−1.05186	−0.525932	−	0.850526i	$-0.676283\pi$
$311$	−18.5498	−1.05186	−0.525932	+	0.850526i	$0.676283\pi$
$312$	0	0
$313$	17.6495	0.997609	0.498804	−	0.866715i	$-0.333773\pi$
$313$	17.6495	0.997609	0.498804	+	0.866715i	$0.333773\pi$
$314$	0	0
$315$	−3.27492	−0.184521
$316$	0	0
$317$	−16.5498	−0.929531	−0.464766	−	0.885434i	$-0.653861\pi$
$317$	−16.5498	−0.929531	−0.464766	+	0.885434i	$0.653861\pi$
$318$	0	0
$319$	−3.82475	−0.214145
$320$	0	0
$321$	14.5498	0.812093
$322$	0	0
$323$	38.3746	2.13522
$324$	0	0
$325$	−5.72508	−0.317570
$326$	0	0
$327$	−7.27492	−0.402304
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−15.6495	−0.860174	−0.430087	−	0.902787i	$-0.641517\pi$
$331$	−15.6495	−0.860174	−0.430087	+	0.902787i	$0.641517\pi$
$332$	0	0
$333$	3.27492	0.179464
$334$	0	0
$335$	8.35050	0.456236
$336$	0	0
$337$	−28.3746	−1.54566	−0.772831	−	0.634612i	$-0.781160\pi$
$337$	−28.3746	−1.54566	−0.772831	+	0.634612i	$0.781160\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	42.1993	2.28522
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	17.2749	0.930050
$346$	0	0
$347$	25.0997	1.34742	0.673710	−	0.738995i	$-0.264700\pi$
$347$	25.0997	1.34742	0.673710	+	0.738995i	$0.264700\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−3.09967	−0.164979	−0.0824894	−	0.996592i	$-0.526287\pi$
$353$	−3.09967	−0.164979	−0.0824894	+	0.996592i	$0.526287\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	7.27492	0.385029
$358$	0	0
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	0	0
$361$	8.82475	0.464461
$362$	0	0
$363$	16.8248	0.883070
$364$	0	0
$365$	32.1752	1.68413
$366$	0	0
$367$	33.0997	1.72779	0.863894	−	0.503673i	$-0.168018\pi$
$367$	33.0997	1.72779	0.863894	+	0.503673i	$0.168018\pi$
$368$	0	0
$369$	−8.54983	−0.445087
$370$	0	0
$371$	−10.0000	−0.519174
$372$	0	0
$373$	11.4502	0.592867	0.296434	−	0.955053i	$-0.404203\pi$
$373$	11.4502	0.592867	0.296434	+	0.955053i	$0.404203\pi$
$374$	0	0
$375$	2.37459	0.122623
$376$	0	0
$377$	−0.725083	−0.0373437
$378$	0	0
$379$	2.90033	0.148980	0.0744900	−	0.997222i	$-0.476267\pi$
$379$	2.90033	0.148980	0.0744900	+	0.997222i	$0.476267\pi$
$380$	0	0
$381$	5.45017	0.279220
$382$	0	0
$383$	−22.7251	−1.16120	−0.580599	−	0.814190i	$-0.697181\pi$
$383$	−22.7251	−1.16120	−0.580599	+	0.814190i	$0.697181\pi$
$384$	0	0
$385$	17.2749	0.880411
$386$	0	0
$387$	−5.27492	−0.268139
$388$	0	0
$389$	−32.1993	−1.63257	−0.816286	−	0.577649i	$-0.803970\pi$
$389$	−32.1993	−1.63257	−0.816286	+	0.577649i	$0.803970\pi$
$390$	0	0
$391$	−38.3746	−1.94069
$392$	0	0
$393$	15.8248	0.798253
$394$	0	0
$395$	8.35050	0.420159
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	0	0
$399$	5.27492	0.264076
$400$	0	0
$401$	27.0997	1.35329	0.676646	−	0.736308i	$-0.263433\pi$
$401$	27.0997	1.35329	0.676646	+	0.736308i	$0.263433\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	0	0
$405$	3.27492	0.162732
$406$	0	0
$407$	−17.2749	−0.856286
$408$	0	0
$409$	−29.8248	−1.47474	−0.737370	−	0.675490i	$-0.763932\pi$
$409$	−29.8248	−1.47474	−0.737370	+	0.675490i	$0.763932\pi$
$410$	0	0
$411$	20.3746	1.00500
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	−34.5498	−1.69598
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	5.27492	0.257697	0.128848	−	0.991664i	$-0.458872\pi$
$419$	5.27492	0.257697	0.128848	+	0.991664i	$0.458872\pi$
$420$	0	0
$421$	−27.0997	−1.32076	−0.660379	−	0.750933i	$-0.729604\pi$
$421$	−27.0997	−1.32076	−0.660379	+	0.750933i	$0.729604\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−41.6495	−2.02030
$426$	0	0
$427$	4.72508	0.228663
$428$	0	0
$429$	5.27492	0.254675
$430$	0	0
$431$	26.5498	1.27886	0.639430	−	0.768849i	$-0.279170\pi$
$431$	26.5498	1.27886	0.639430	+	0.768849i	$0.279170\pi$
$432$	0	0
$433$	−21.6495	−1.04041	−0.520204	−	0.854042i	$-0.674144\pi$
$433$	−21.6495	−1.04041	−0.520204	+	0.854042i	$0.674144\pi$
$434$	0	0
$435$	2.37459	0.113853
$436$	0	0
$437$	−27.8248	−1.33104
$438$	0	0
$439$	−7.82475	−0.373455	−0.186728	−	0.982412i	$-0.559788\pi$
$439$	−7.82475	−0.373455	−0.186728	+	0.982412i	$0.559788\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−22.5498	−1.07137	−0.535687	−	0.844416i	$-0.679947\pi$
$443$	−22.5498	−1.07137	−0.535687	+	0.844416i	$0.679947\pi$
$444$	0	0
$445$	−45.8488	−2.17344
$446$	0	0
$447$	7.45017	0.352381
$448$	0	0
$449$	7.27492	0.343325	0.171662	−	0.985156i	$-0.445086\pi$
$449$	7.27492	0.343325	0.171662	+	0.985156i	$0.445086\pi$
$450$	0	0
$451$	45.0997	2.12366
$452$	0	0
$453$	1.27492	0.0599008
$454$	0	0
$455$	3.27492	0.153530
$456$	0	0
$457$	−3.45017	−0.161392	−0.0806960	−	0.996739i	$-0.525714\pi$
$457$	−3.45017	−0.161392	−0.0806960	+	0.996739i	$0.525714\pi$
$458$	0	0
$459$	−7.27492	−0.339564
$460$	0	0
$461$	34.9244	1.62659	0.813296	−	0.581850i	$-0.197671\pi$
$461$	34.9244	1.62659	0.813296	+	0.581850i	$0.197671\pi$
$462$	0	0
$463$	−22.7251	−1.05612	−0.528062	−	0.849206i	$-0.677081\pi$
$463$	−22.7251	−1.05612	−0.528062	+	0.849206i	$0.677081\pi$
$464$	0	0
$465$	−26.1993	−1.21497
$466$	0	0
$467$	18.7251	0.866493	0.433247	−	0.901275i	$-0.357368\pi$
$467$	18.7251	0.866493	0.433247	+	0.901275i	$0.357368\pi$
$468$	0	0
$469$	−2.54983	−0.117740
$470$	0	0
$471$	24.3746	1.12312
$472$	0	0
$473$	27.8248	1.27938
$474$	0	0
$475$	−30.1993	−1.38564
$476$	0	0
$477$	10.0000	0.457869
$478$	0	0
$479$	−14.3746	−0.656792	−0.328396	−	0.944540i	$-0.606508\pi$
$479$	−14.3746	−0.656792	−0.328396	+	0.944540i	$0.606508\pi$
$480$	0	0
$481$	−3.27492	−0.149323
$482$	0	0
$483$	−5.27492	−0.240017
$484$	0	0
$485$	−49.4502	−2.24542
$486$	0	0
$487$	−42.1993	−1.91223	−0.956117	−	0.292984i	$-0.905352\pi$
$487$	−42.1993	−1.91223	−0.956117	+	0.292984i	$0.905352\pi$
$488$	0	0
$489$	2.54983	0.115307
$490$	0	0
$491$	6.54983	0.295590	0.147795	−	0.989018i	$-0.452782\pi$
$491$	6.54983	0.295590	0.147795	+	0.989018i	$0.452782\pi$
$492$	0	0
$493$	−5.27492	−0.237570
$494$	0	0
$495$	−17.2749	−0.776450
$496$	0	0
$497$	0	0
$498$	0	0
$499$	5.09967	0.228293	0.114146	−	0.993464i	$-0.463587\pi$
$499$	5.09967	0.228293	0.114146	+	0.993464i	$0.463587\pi$
$500$	0	0
$501$	9.27492	0.414373
$502$	0	0
$503$	18.5498	0.827096	0.413548	−	0.910482i	$-0.364289\pi$
$503$	18.5498	0.827096	0.413548	+	0.910482i	$0.364289\pi$
$504$	0	0
$505$	−41.0997	−1.82891
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	42.9244	1.90259	0.951296	−	0.308280i	$-0.0997533\pi$
$509$	42.9244	1.90259	0.951296	+	0.308280i	$0.0997533\pi$
$510$	0	0
$511$	−9.82475	−0.434621
$512$	0	0
$513$	−5.27492	−0.232893
$514$	0	0
$515$	−8.92442	−0.393257
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−23.0997	−1.01396
$520$	0	0
$521$	3.27492	0.143477	0.0717384	−	0.997423i	$-0.477145\pi$
$521$	3.27492	0.143477	0.0717384	+	0.997423i	$0.477145\pi$
$522$	0	0
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	0	0
$525$	−5.72508	−0.249863
$526$	0	0
$527$	58.1993	2.53520
$528$	0	0
$529$	4.82475	0.209772
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	8.54983	0.370334
$534$	0	0
$535$	47.6495	2.06007
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	−5.27492	−0.227207
$540$	0	0
$541$	−25.4743	−1.09522	−0.547612	−	0.836732i	$-0.684463\pi$
$541$	−25.4743	−1.09522	−0.547612	+	0.836732i	$0.684463\pi$
$542$	0	0
$543$	−11.0997	−0.476332
$544$	0	0
$545$	−23.8248	−1.02054
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	0	0
$549$	−4.72508	−0.201662
$550$	0	0
$551$	−3.82475	−0.162940
$552$	0	0
$553$	−2.54983	−0.108430
$554$	0	0
$555$	10.7251	0.455254
$556$	0	0
$557$	30.7492	1.30288	0.651442	−	0.758698i	$-0.274164\pi$
$557$	30.7492	1.30288	0.651442	+	0.758698i	$0.274164\pi$
$558$	0	0
$559$	5.27492	0.223105
$560$	0	0
$561$	38.3746	1.62018
$562$	0	0
$563$	−2.37459	−0.100077	−0.0500384	−	0.998747i	$-0.515934\pi$
$563$	−2.37459	−0.100077	−0.0500384	+	0.998747i	$0.515934\pi$
$564$	0	0
$565$	−19.6495	−0.826661
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−27.0997	−1.13608	−0.568039	−	0.823002i	$-0.692298\pi$
$569$	−27.0997	−1.13608	−0.568039	+	0.823002i	$0.692298\pi$
$570$	0	0
$571$	6.90033	0.288770	0.144385	−	0.989522i	$-0.453880\pi$
$571$	6.90033	0.288770	0.144385	+	0.989522i	$0.453880\pi$
$572$	0	0
$573$	5.27492	0.220363
$574$	0	0
$575$	30.1993	1.25940
$576$	0	0
$577$	−39.0997	−1.62774	−0.813870	−	0.581047i	$-0.802643\pi$
$577$	−39.0997	−1.62774	−0.813870	+	0.581047i	$0.802643\pi$
$578$	0	0
$579$	−3.45017	−0.143384
$580$	0	0
$581$	10.5498	0.437681
$582$	0	0
$583$	−52.7492	−2.18465
$584$	0	0
$585$	−3.27492	−0.135401
$586$	0	0
$587$	28.7492	1.18661	0.593303	−	0.804979i	$-0.297824\pi$
$587$	28.7492	1.18661	0.593303	+	0.804979i	$0.297824\pi$
$588$	0	0
$589$	42.1993	1.73879
$590$	0	0
$591$	12.5498	0.516231
$592$	0	0
$593$	23.0997	0.948590	0.474295	−	0.880366i	$-0.342703\pi$
$593$	23.0997	0.948590	0.474295	+	0.880366i	$0.342703\pi$
$594$	0	0
$595$	23.8248	0.976720
$596$	0	0
$597$	15.8248	0.647664
$598$	0	0
$599$	−29.2749	−1.19614	−0.598070	−	0.801444i	$-0.704066\pi$
$599$	−29.2749	−1.19614	−0.598070	+	0.801444i	$0.704066\pi$
$600$	0	0
$601$	−21.6495	−0.883102	−0.441551	−	0.897236i	$-0.645572\pi$
$601$	−21.6495	−0.883102	−0.441551	+	0.897236i	$0.645572\pi$
$602$	0	0
$603$	2.54983	0.103837
$604$	0	0
$605$	55.0997	2.24012
$606$	0	0
$607$	−31.8248	−1.29173	−0.645863	−	0.763453i	$-0.723502\pi$
$607$	−31.8248	−1.29173	−0.645863	+	0.763453i	$0.723502\pi$
$608$	0	0
$609$	−0.725083	−0.0293818
$610$	0	0
$611$	0	0
$612$	0	0
$613$	8.72508	0.352403	0.176201	−	0.984354i	$-0.443619\pi$
$613$	8.72508	0.352403	0.176201	+	0.984354i	$0.443619\pi$
$614$	0	0
$615$	−28.0000	−1.12907
$616$	0	0
$617$	1.82475	0.0734617	0.0367309	−	0.999325i	$-0.488306\pi$
$617$	1.82475	0.0734617	0.0367309	+	0.999325i	$0.488306\pi$
$618$	0	0
$619$	−37.2749	−1.49821	−0.749103	−	0.662454i	$-0.769515\pi$
$619$	−37.2749	−1.49821	−0.749103	+	0.662454i	$0.769515\pi$
$620$	0	0
$621$	5.27492	0.211675
$622$	0	0
$623$	14.0000	0.560898
$624$	0	0
$625$	−20.8488	−0.833954
$626$	0	0
$627$	27.8248	1.11121
$628$	0	0
$629$	−23.8248	−0.949955
$630$	0	0
$631$	−11.8248	−0.470736	−0.235368	−	0.971906i	$-0.575630\pi$
$631$	−11.8248	−0.470736	−0.235368	+	0.971906i	$0.575630\pi$
$632$	0	0
$633$	−23.8248	−0.946949
$634$	0	0
$635$	17.8488	0.708310
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	0	0
$640$	0	0
$641$	15.0997	0.596401	0.298201	−	0.954503i	$-0.403614\pi$
$641$	15.0997	0.596401	0.298201	+	0.954503i	$0.403614\pi$
$642$	0	0
$643$	−28.9244	−1.14067	−0.570334	−	0.821413i	$-0.693186\pi$
$643$	−28.9244	−1.14067	−0.570334	+	0.821413i	$0.693186\pi$
$644$	0	0
$645$	−17.2749	−0.680199
$646$	0	0
$647$	−42.1993	−1.65903	−0.829514	−	0.558487i	$-0.811382\pi$
$647$	−42.1993	−1.65903	−0.829514	+	0.558487i	$0.811382\pi$
$648$	0	0
$649$	42.1993	1.65647
$650$	0	0
$651$	8.00000	0.313545
$652$	0	0
$653$	32.3746	1.26692	0.633458	−	0.773777i	$-0.281635\pi$
$653$	32.3746	1.26692	0.633458	+	0.773777i	$0.281635\pi$
$654$	0	0
$655$	51.8248	2.02496
$656$	0	0
$657$	9.82475	0.383300
$658$	0	0
$659$	22.5498	0.878417	0.439208	−	0.898385i	$-0.355259\pi$
$659$	22.5498	0.878417	0.439208	+	0.898385i	$0.355259\pi$
$660$	0	0
$661$	20.1993	0.785663	0.392832	−	0.919610i	$-0.371496\pi$
$661$	20.1993	0.785663	0.392832	+	0.919610i	$0.371496\pi$
$662$	0	0
$663$	7.27492	0.282534
$664$	0	0
$665$	17.2749	0.669893
$666$	0	0
$667$	3.82475	0.148095
$668$	0	0
$669$	23.6495	0.914343
$670$	0	0
$671$	24.9244	0.962197
$672$	0	0
$673$	8.72508	0.336327	0.168164	−	0.985759i	$-0.446216\pi$
$673$	8.72508	0.336327	0.168164	+	0.985759i	$0.446216\pi$
$674$	0	0
$675$	5.72508	0.220359
$676$	0	0
$677$	−44.1993	−1.69872	−0.849359	−	0.527815i	$-0.823011\pi$
$677$	−44.1993	−1.69872	−0.849359	+	0.527815i	$0.823011\pi$
$678$	0	0
$679$	15.0997	0.579472
$680$	0	0
$681$	5.45017	0.208851
$682$	0	0
$683$	34.3746	1.31531	0.657653	−	0.753321i	$-0.271549\pi$
$683$	34.3746	1.31531	0.657653	+	0.753321i	$0.271549\pi$
$684$	0	0
$685$	66.7251	2.54943
$686$	0	0
$687$	−11.0997	−0.423479
$688$	0	0
$689$	−10.0000	−0.380970
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	0	0
$693$	5.27492	0.200378
$694$	0	0
$695$	−13.0997	−0.496899
$696$	0	0
$697$	62.1993	2.35597
$698$	0	0
$699$	20.5498	0.777266
$700$	0	0
$701$	15.0997	0.570307	0.285153	−	0.958482i	$-0.407955\pi$
$701$	15.0997	0.570307	0.285153	+	0.958482i	$0.407955\pi$
$702$	0	0
$703$	−17.2749	−0.651536
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.5498	0.471985
$708$	0	0
$709$	7.09967	0.266634	0.133317	−	0.991073i	$-0.457437\pi$
$709$	7.09967	0.266634	0.133317	+	0.991073i	$0.457437\pi$
$710$	0	0
$711$	2.54983	0.0956263
$712$	0	0
$713$	−42.1993	−1.58038
$714$	0	0
$715$	17.2749	0.646045
$716$	0	0
$717$	5.09967	0.190451
$718$	0	0
$719$	−15.6495	−0.583628	−0.291814	−	0.956475i	$-0.594259\pi$
$719$	−15.6495	−0.583628	−0.291814	+	0.956475i	$0.594259\pi$
$720$	0	0
$721$	2.72508	0.101487
$722$	0	0
$723$	−15.0997	−0.561563
$724$	0	0
$725$	4.15116	0.154170
$726$	0	0
$727$	2.37459	0.0880685	0.0440343	−	0.999030i	$-0.485979\pi$
$727$	2.37459	0.0880685	0.0440343	+	0.999030i	$0.485979\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	38.3746	1.41934
$732$	0	0
$733$	−0.549834	−0.0203086	−0.0101543	−	0.999948i	$-0.503232\pi$
$733$	−0.549834	−0.0203086	−0.0101543	+	0.999948i	$0.503232\pi$
$734$	0	0
$735$	3.27492	0.120797
$736$	0	0
$737$	−13.4502	−0.495443
$738$	0	0
$739$	7.64950	0.281392	0.140696	−	0.990053i	$-0.455066\pi$
$739$	7.64950	0.281392	0.140696	+	0.990053i	$0.455066\pi$
$740$	0	0
$741$	5.27492	0.193779
$742$	0	0
$743$	−26.5498	−0.974019	−0.487009	−	0.873397i	$-0.661912\pi$
$743$	−26.5498	−0.974019	−0.487009	+	0.873397i	$0.661912\pi$
$744$	0	0
$745$	24.3987	0.893898
$746$	0	0
$747$	−10.5498	−0.385998
$748$	0	0
$749$	−14.5498	−0.531639
$750$	0	0
$751$	50.1993	1.83180	0.915900	−	0.401407i	$-0.131479\pi$
$751$	50.1993	1.83180	0.915900	+	0.401407i	$0.131479\pi$
$752$	0	0
$753$	28.9244	1.05406
$754$	0	0
$755$	4.17525	0.151953
$756$	0	0
$757$	−4.90033	−0.178106	−0.0890528	−	0.996027i	$-0.528384\pi$
$757$	−4.90033	−0.178106	−0.0890528	+	0.996027i	$0.528384\pi$
$758$	0	0
$759$	−27.8248	−1.00997
$760$	0	0
$761$	23.4502	0.850068	0.425034	−	0.905177i	$-0.360262\pi$
$761$	23.4502	0.850068	0.425034	+	0.905177i	$0.360262\pi$
$762$	0	0
$763$	7.27492	0.263370
$764$	0	0
$765$	−23.8248	−0.861386
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	10.1752	0.366929	0.183464	−	0.983026i	$-0.441269\pi$
$769$	10.1752	0.366929	0.183464	+	0.983026i	$0.441269\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	0	0
$773$	42.9244	1.54388	0.771942	−	0.635693i	$-0.219286\pi$
$773$	42.9244	1.54388	0.771942	+	0.635693i	$0.219286\pi$
$774$	0	0
$775$	−45.8007	−1.64521
$776$	0	0
$777$	−3.27492	−0.117487
$778$	0	0
$779$	45.0997	1.61586
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0.725083	0.0259123
$784$	0	0
$785$	79.8248	2.84907
$786$	0	0
$787$	2.72508	0.0971387	0.0485694	−	0.998820i	$-0.484534\pi$
$787$	2.72508	0.0971387	0.0485694	+	0.998820i	$0.484534\pi$
$788$	0	0
$789$	−28.0000	−0.996826
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	4.72508	0.167793
$794$	0	0
$795$	32.7492	1.16149
$796$	0	0
$797$	29.6495	1.05024	0.525120	−	0.851028i	$-0.324021\pi$
$797$	29.6495	1.05024	0.525120	+	0.851028i	$0.324021\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−14.0000	−0.494666
$802$	0	0
$803$	−51.8248	−1.82886
$804$	0	0
$805$	−17.2749	−0.608861
$806$	0	0
$807$	6.00000	0.211210
$808$	0	0
$809$	4.19934	0.147641	0.0738204	−	0.997272i	$-0.476481\pi$
$809$	4.19934	0.147641	0.0738204	+	0.997272i	$0.476481\pi$
$810$	0	0
$811$	−26.3746	−0.926137	−0.463068	−	0.886322i	$-0.653252\pi$
$811$	−26.3746	−0.926137	−0.463068	+	0.886322i	$0.653252\pi$
$812$	0	0
$813$	−13.4502	−0.471718
$814$	0	0
$815$	8.35050	0.292505
$816$	0	0
$817$	27.8248	0.973465
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	−13.6495	−0.476371	−0.238185	−	0.971220i	$-0.576553\pi$
$821$	−13.6495	−0.476371	−0.238185	+	0.971220i	$0.576553\pi$
$822$	0	0
$823$	−5.09967	−0.177763	−0.0888816	−	0.996042i	$-0.528329\pi$
$823$	−5.09967	−0.177763	−0.0888816	+	0.996042i	$0.528329\pi$
$824$	0	0
$825$	−30.1993	−1.05141
$826$	0	0
$827$	18.7251	0.651135	0.325567	−	0.945519i	$-0.394445\pi$
$827$	18.7251	0.651135	0.325567	+	0.945519i	$0.394445\pi$
$828$	0	0
$829$	19.2749	0.669446	0.334723	−	0.942317i	$-0.391357\pi$
$829$	19.2749	0.669446	0.334723	+	0.942317i	$0.391357\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	0	0
$833$	−7.27492	−0.252061
$834$	0	0
$835$	30.3746	1.05116
$836$	0	0
$837$	−8.00000	−0.276520
$838$	0	0
$839$	−53.0997	−1.83320	−0.916602	−	0.399801i	$-0.869079\pi$
$839$	−53.0997	−1.83320	−0.916602	+	0.399801i	$0.869079\pi$
$840$	0	0
$841$	−28.4743	−0.981871
$842$	0	0
$843$	−15.0997	−0.520060
$844$	0	0
$845$	3.27492	0.112661
$846$	0	0
$847$	−16.8248	−0.578105
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	17.2749	0.592177
$852$	0	0
$853$	17.6495	0.604307	0.302154	−	0.953259i	$-0.402294\pi$
$853$	17.6495	0.604307	0.302154	+	0.953259i	$0.402294\pi$
$854$	0	0
$855$	−17.2749	−0.590790
$856$	0	0
$857$	−11.0997	−0.379157	−0.189579	−	0.981866i	$-0.560712\pi$
$857$	−11.0997	−0.379157	−0.189579	+	0.981866i	$0.560712\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	8.54983	0.291378
$862$	0	0
$863$	15.6495	0.532715	0.266358	−	0.963874i	$-0.414180\pi$
$863$	15.6495	0.532715	0.266358	+	0.963874i	$0.414180\pi$
$864$	0	0
$865$	−75.6495	−2.57216
$866$	0	0
$867$	35.9244	1.22006
$868$	0	0
$869$	−13.4502	−0.456266
$870$	0	0
$871$	−2.54983	−0.0863978
$872$	0	0
$873$	−15.0997	−0.511046
$874$	0	0
$875$	−2.37459	−0.0802757
$876$	0	0
$877$	20.9003	0.705754	0.352877	−	0.935670i	$-0.385203\pi$
$877$	20.9003	0.705754	0.352877	+	0.935670i	$0.385203\pi$
$878$	0	0
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−39.2749	−1.32321	−0.661603	−	0.749854i	$-0.730123\pi$
$881$	−39.2749	−1.32321	−0.661603	+	0.749854i	$0.730123\pi$
$882$	0	0
$883$	20.9244	0.704163	0.352081	−	0.935969i	$-0.385474\pi$
$883$	20.9244	0.704163	0.352081	+	0.935969i	$0.385474\pi$
$884$	0	0
$885$	−26.1993	−0.880681
$886$	0	0
$887$	2.90033	0.0973836	0.0486918	−	0.998814i	$-0.484495\pi$
$887$	2.90033	0.0973836	0.0486918	+	0.998814i	$0.484495\pi$
$888$	0	0
$889$	−5.45017	−0.182793
$890$	0	0
$891$	−5.27492	−0.176716
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−39.2990	−1.31362
$896$	0	0
$897$	−5.27492	−0.176124
$898$	0	0
$899$	−5.80066	−0.193463
$900$	0	0
$901$	−72.7492	−2.42363
$902$	0	0
$903$	5.27492	0.175538
$904$	0	0
$905$	−36.3505	−1.20833
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	−12.5498	−0.416252
$910$	0	0
$911$	−15.4743	−0.512685	−0.256342	−	0.966586i	$-0.582517\pi$
$911$	−15.4743	−0.512685	−0.256342	+	0.966586i	$0.582517\pi$
$912$	0	0
$913$	55.6495	1.84173
$914$	0	0
$915$	−15.4743	−0.511563
$916$	0	0
$917$	−15.8248	−0.522579
$918$	0	0
$919$	−2.54983	−0.0841113	−0.0420556	−	0.999115i	$-0.513391\pi$
$919$	−2.54983	−0.0841113	−0.0420556	+	0.999115i	$0.513391\pi$
$920$	0	0
$921$	9.09967	0.299844
$922$	0	0
$923$	0	0
$924$	0	0
$925$	18.7492	0.616469
$926$	0	0
$927$	−2.72508	−0.0895035
$928$	0	0
$929$	−0.549834	−0.0180395	−0.00901974	−	0.999959i	$-0.502871\pi$
$929$	−0.549834	−0.0180395	−0.00901974	+	0.999959i	$0.502871\pi$
$930$	0	0
$931$	−5.27492	−0.172878
$932$	0	0
$933$	−18.5498	−0.607294
$934$	0	0
$935$	125.674	4.10997
$936$	0	0
$937$	47.0997	1.53868	0.769340	−	0.638840i	$-0.220585\pi$
$937$	47.0997	1.53868	0.769340	+	0.638840i	$0.220585\pi$
$938$	0	0
$939$	17.6495	0.575970
$940$	0	0
$941$	20.9003	0.681331	0.340666	−	0.940185i	$-0.389348\pi$
$941$	20.9003	0.681331	0.340666	+	0.940185i	$0.389348\pi$
$942$	0	0
$943$	−45.0997	−1.46865
$944$	0	0
$945$	−3.27492	−0.106533
$946$	0	0
$947$	39.8248	1.29413	0.647065	−	0.762435i	$-0.275996\pi$
$947$	39.8248	1.29413	0.647065	+	0.762435i	$0.275996\pi$
$948$	0	0
$949$	−9.82475	−0.318925
$950$	0	0
$951$	−16.5498	−0.536665
$952$	0	0
$953$	−13.6495	−0.442151	−0.221075	−	0.975257i	$-0.570957\pi$
$953$	−13.6495	−0.442151	−0.221075	+	0.975257i	$0.570957\pi$
$954$	0	0
$955$	17.2749	0.559003
$956$	0	0
$957$	−3.82475	−0.123637
$958$	0	0
$959$	−20.3746	−0.657930
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	14.5498	0.468862
$964$	0	0
$965$	−11.2990	−0.363728
$966$	0	0
$967$	−20.1752	−0.648792	−0.324396	−	0.945921i	$-0.605161\pi$
$967$	−20.1752	−0.648792	−0.324396	+	0.945921i	$0.605161\pi$
$968$	0	0
$969$	38.3746	1.23277
$970$	0	0
$971$	−1.09967	−0.0352901	−0.0176450	−	0.999844i	$-0.505617\pi$
$971$	−1.09967	−0.0352901	−0.0176450	+	0.999844i	$0.505617\pi$
$972$	0	0
$973$	4.00000	0.128234
$974$	0	0
$975$	−5.72508	−0.183349
$976$	0	0
$977$	38.9244	1.24530	0.622651	−	0.782499i	$-0.286056\pi$
$977$	38.9244	1.24530	0.622651	+	0.782499i	$0.286056\pi$
$978$	0	0
$979$	73.8488	2.36022
$980$	0	0
$981$	−7.27492	−0.232270
$982$	0	0
$983$	22.0241	0.702459	0.351230	−	0.936289i	$-0.385764\pi$
$983$	22.0241	0.702459	0.351230	+	0.936289i	$0.385764\pi$
$984$	0	0
$985$	41.0997	1.30954
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−27.8248	−0.884776
$990$	0	0
$991$	−28.7492	−0.913248	−0.456624	−	0.889660i	$-0.650941\pi$
$991$	−28.7492	−0.913248	−0.456624	+	0.889660i	$0.650941\pi$
$992$	0	0
$993$	−15.6495	−0.496622
$994$	0	0
$995$	51.8248	1.64296
$996$	0	0
$997$	−43.0997	−1.36498	−0.682490	−	0.730895i	$-0.739103\pi$
$997$	−43.0997	−1.36498	−0.682490	+	0.730895i	$0.739103\pi$
$998$	0	0
$999$	3.27492	0.103614

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.bh.1.2		2
4.3	odd	2		546.2.a.h.1.2	✓	2
12.11	even	2		1638.2.a.y.1.1		2
28.27	even	2		3822.2.a.bm.1.1		2
52.51	odd	2		7098.2.a.bu.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.a.h.1.2	✓	2	4.3	odd	2
1638.2.a.y.1.1		2	12.11	even	2
3822.2.a.bm.1.1		2	28.27	even	2
4368.2.a.bh.1.2		2	1.1	even	1	trivial
7098.2.a.bu.1.1		2	52.51	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.27492 q^{5} -1.00000 q^{7} +1.00000 q^{9} -5.27492 q^{11} -1.00000 q^{13} +3.27492 q^{15} -7.27492 q^{17} -5.27492 q^{19} -1.00000 q^{21} +5.27492 q^{23} +5.72508 q^{25} +1.00000 q^{27} +0.725083 q^{29} -8.00000 q^{31} -5.27492 q^{33} -3.27492 q^{35} +3.27492 q^{37} -1.00000 q^{39} -8.54983 q^{41} -5.27492 q^{43} +3.27492 q^{45} +1.00000 q^{49} -7.27492 q^{51} +10.0000 q^{53} -17.2749 q^{55} -5.27492 q^{57} -8.00000 q^{59} -4.72508 q^{61} -1.00000 q^{63} -3.27492 q^{65} +2.54983 q^{67} +5.27492 q^{69} +9.82475 q^{73} +5.72508 q^{75} +5.27492 q^{77} +2.54983 q^{79} +1.00000 q^{81} -10.5498 q^{83} -23.8248 q^{85} +0.725083 q^{87} -14.0000 q^{89} +1.00000 q^{91} -8.00000 q^{93} -17.2749 q^{95} -15.0997 q^{97} -5.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - q^{5} - 2 q^{7} + 2 q^{9} - 3 q^{11} - 2 q^{13} - q^{15} - 7 q^{17} - 3 q^{19} - 2 q^{21} + 3 q^{23} + 19 q^{25} + 2 q^{27} + 9 q^{29} - 16 q^{31} - 3 q^{33} + q^{35} - q^{37} - 2 q^{39}+ \cdots - 3 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - q^5 - 2 * q^7 + 2 * q^9 - 3 * q^11 - 2 * q^13 - q^15 - 7 * q^17 - 3 * q^19 - 2 * q^21 + 3 * q^23 + 19 * q^25 + 2 * q^27 + 9 * q^29 - 16 * q^31 - 3 * q^33 + q^35 - q^37 - 2 * q^39 - 2 * q^41 - 3 * q^43 - q^45 + 2 * q^49 - 7 * q^51 + 20 * q^53 - 27 * q^55 - 3 * q^57 - 16 * q^59 - 17 * q^61 - 2 * q^63 + q^65 - 10 * q^67 + 3 * q^69 - 3 * q^73 + 19 * q^75 + 3 * q^77 - 10 * q^79 + 2 * q^81 - 6 * q^83 - 25 * q^85 + 9 * q^87 - 28 * q^89 + 2 * q^91 - 16 * q^93 - 27 * q^95 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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