LMFDB - Embedded newform 1638.2.a.y.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1638,2,Mod(1,1638)] 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("1638.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1638, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,0,2,1,0,2,2,0,1,-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:	$13.0794958511$
Analytic rank:	$0$
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			1.1
Root			$-3.27492$ of defining polynomial
Character	$\chi$	$=$	1638.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -3.27492 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.27492 q^{10} -5.27492 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +7.27492 q^{17} +5.27492 q^{19} -3.27492 q^{20} -5.27492 q^{22} +5.27492 q^{23} +5.72508 q^{25} -1.00000 q^{26} +1.00000 q^{28} -0.725083 q^{29} +8.00000 q^{31} +1.00000 q^{32} +7.27492 q^{34} -3.27492 q^{35} +3.27492 q^{37} +5.27492 q^{38} -3.27492 q^{40} +8.54983 q^{41} +5.27492 q^{43} -5.27492 q^{44} +5.27492 q^{46} +1.00000 q^{49} +5.72508 q^{50} -1.00000 q^{52} -10.0000 q^{53} +17.2749 q^{55} +1.00000 q^{56} -0.725083 q^{58} -8.00000 q^{59} -4.72508 q^{61} +8.00000 q^{62} +1.00000 q^{64} +3.27492 q^{65} -2.54983 q^{67} +7.27492 q^{68} -3.27492 q^{70} +9.82475 q^{73} +3.27492 q^{74} +5.27492 q^{76} -5.27492 q^{77} -2.54983 q^{79} -3.27492 q^{80} +8.54983 q^{82} -10.5498 q^{83} -23.8248 q^{85} +5.27492 q^{86} -5.27492 q^{88} +14.0000 q^{89} -1.00000 q^{91} +5.27492 q^{92} -17.2749 q^{95} -15.0997 q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} + 2 q^{8} + q^{10} - 3 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} + 7 q^{17} + 3 q^{19} + q^{20} - 3 q^{22} + 3 q^{23} + 19 q^{25} - 2 q^{26} + 2 q^{28} - 9 q^{29}+ \cdots + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + q^5 + 2 * q^7 + 2 * q^8 + q^10 - 3 * q^11 - 2 * q^13 + 2 * q^14 + 2 * q^16 + 7 * q^17 + 3 * q^19 + q^20 - 3 * q^22 + 3 * q^23 + 19 * q^25 - 2 * q^26 + 2 * q^28 - 9 * q^29 + 16 * q^31 + 2 * q^32 + 7 * q^34 + q^35 - q^37 + 3 * q^38 + q^40 + 2 * q^41 + 3 * q^43 - 3 * q^44 + 3 * q^46 + 2 * q^49 + 19 * q^50 - 2 * q^52 - 20 * q^53 + 27 * q^55 + 2 * q^56 - 9 * q^58 - 16 * q^59 - 17 * q^61 + 16 * q^62 + 2 * q^64 - q^65 + 10 * q^67 + 7 * q^68 + q^70 - 3 * q^73 - q^74 + 3 * q^76 - 3 * q^77 + 10 * q^79 + q^80 + 2 * q^82 - 6 * q^83 - 25 * q^85 + 3 * q^86 - 3 * q^88 + 28 * q^89 - 2 * q^91 + 3 * q^92 - 27 * q^95 + 2 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.27492	−1.46459	−0.732294	−	0.680989i	$-0.761550\pi$
$5$	−3.27492	−1.46459	−0.732294	+	0.680989i	$0.761550\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.27492	−1.03562
$11$	−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
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	7.27492	1.76443	0.882213	−	0.470850i	$-0.156053\pi$
$17$	7.27492	1.76443	0.882213	+	0.470850i	$0.156053\pi$
$18$	0	0
$19$	5.27492	1.21015	0.605075	−	0.796169i	$-0.293143\pi$
$19$	5.27492	1.21015	0.605075	+	0.796169i	$0.293143\pi$
$20$	−3.27492	−0.732294
$21$	0	0
$22$	−5.27492	−1.12462
$23$	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
$26$	−1.00000	−0.196116
$27$	0	0
$28$	1.00000	0.188982
$29$	−0.725083	−0.134644	−0.0673222	−	0.997731i	$-0.521446\pi$
$29$	−0.725083	−0.134644	−0.0673222	+	0.997731i	$0.521446\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	7.27492	1.24764
$35$	−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	$0.413242\pi$
$38$	5.27492	0.855705
$39$	0	0
$40$	−3.27492	−0.517810
$41$	8.54983	1.33526	0.667630	−	0.744493i	$-0.267309\pi$
$41$	8.54983	1.33526	0.667630	+	0.744493i	$0.267309\pi$
$42$	0	0
$43$	5.27492	0.804417	0.402209	−	0.915548i	$-0.368243\pi$
$43$	5.27492	0.804417	0.402209	+	0.915548i	$0.368243\pi$
$44$	−5.27492	−0.795224
$45$	0	0
$46$	5.27492	0.777744
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	5.72508	0.809649
$51$	0	0
$52$	−1.00000	−0.138675
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	17.2749	2.32935
$56$	1.00000	0.133631
$57$	0	0
$58$	−0.725083	−0.0952080
$59$	−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	$0.597819\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	3.27492	0.406203
$66$	0	0
$67$	−2.54983	−0.311512	−0.155756	−	0.987796i	$-0.549781\pi$
$67$	−2.54983	−0.311512	−0.155756	+	0.987796i	$0.549781\pi$
$68$	7.27492	0.882213
$69$	0	0
$70$	−3.27492	−0.391427
$71$	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	$0.305021\pi$
$74$	3.27492	0.380701
$75$	0	0
$76$	5.27492	0.605075
$77$	−5.27492	−0.601133
$78$	0	0
$79$	−2.54983	−0.286879	−0.143439	−	0.989659i	$-0.545816\pi$
$79$	−2.54983	−0.286879	−0.143439	+	0.989659i	$0.545816\pi$
$80$	−3.27492	−0.366147
$81$	0	0
$82$	8.54983	0.944171
$83$	−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
$86$	5.27492	0.568809
$87$	0	0
$88$	−5.27492	−0.562308
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	5.27492	0.549948
$93$	0	0
$94$	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	$0.778038\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	5.72508	0.572508
$101$	12.5498	1.24876	0.624378	−	0.781123i	$-0.285353\pi$
$101$	12.5498	1.24876	0.624378	+	0.781123i	$0.285353\pi$
$102$	0	0
$103$	2.72508	0.268510	0.134255	−	0.990947i	$-0.457136\pi$
$103$	2.72508	0.268510	0.134255	+	0.990947i	$0.457136\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−10.0000	−0.971286
$107$	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	$0.613277\pi$
$110$	17.2749	1.64710
$111$	0	0
$112$	1.00000	0.0944911
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−17.2749	−1.61089
$116$	−0.725083	−0.0673222
$117$	0	0
$118$	−8.00000	−0.736460
$119$	7.27492	0.666891
$120$	0	0
$121$	16.8248	1.52952
$122$	−4.72508	−0.427789
$123$	0	0
$124$	8.00000	0.718421
$125$	−2.37459	−0.212389
$126$	0	0
$127$	−5.45017	−0.483624	−0.241812	−	0.970323i	$-0.577742\pi$
$127$	−5.45017	−0.483624	−0.241812	+	0.970323i	$0.577742\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	3.27492	0.287229
$131$	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
$134$	−2.54983	−0.220272
$135$	0	0
$136$	7.27492	0.623819
$137$	−20.3746	−1.74072	−0.870359	−	0.492417i	$-0.836113\pi$
$137$	−20.3746	−1.74072	−0.870359	+	0.492417i	$0.836113\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	−3.27492	−0.276781
$141$	0	0
$142$	0	0
$143$	5.27492	0.441111
$144$	0	0
$145$	2.37459	0.197199
$146$	9.82475	0.813102
$147$	0	0
$148$	3.27492	0.269197
$149$	−7.45017	−0.610341	−0.305171	−	0.952298i	$-0.598714\pi$
$149$	−7.45017	−0.610341	−0.305171	+	0.952298i	$0.598714\pi$
$150$	0	0
$151$	−1.27492	−0.103751	−0.0518756	−	0.998654i	$-0.516520\pi$
$151$	−1.27492	−0.103751	−0.0518756	+	0.998654i	$0.516520\pi$
$152$	5.27492	0.427852
$153$	0	0
$154$	−5.27492	−0.425065
$155$	−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	$0.0746146\pi$
$158$	−2.54983	−0.202854
$159$	0	0
$160$	−3.27492	−0.258905
$161$	5.27492	0.415722
$162$	0	0
$163$	−2.54983	−0.199718	−0.0998592	−	0.995002i	$-0.531839\pi$
$163$	−2.54983	−0.199718	−0.0998592	+	0.995002i	$0.531839\pi$
$164$	8.54983	0.667630
$165$	0	0
$166$	−10.5498	−0.818826
$167$	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
$170$	−23.8248	−1.82728
$171$	0	0
$172$	5.27492	0.402209
$173$	23.0997	1.75624	0.878118	−	0.478445i	$-0.158799\pi$
$173$	23.0997	1.75624	0.878118	+	0.478445i	$0.158799\pi$
$174$	0	0
$175$	5.72508	0.432776
$176$	−5.27492	−0.397612
$177$	0	0
$178$	14.0000	1.04934
$179$	−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	$0.635350\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	5.27492	0.388872
$185$	−10.7251	−0.788524
$186$	0	0
$187$	−38.3746	−2.80623
$188$	0	0
$189$	0	0
$190$	−17.2749	−1.25325
$191$	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	$0.539628\pi$
$194$	−15.0997	−1.08409
$195$	0	0
$196$	1.00000	0.0714286
$197$	−12.5498	−0.894139	−0.447069	−	0.894499i	$-0.647532\pi$
$197$	−12.5498	−0.894139	−0.447069	+	0.894499i	$0.647532\pi$
$198$	0	0
$199$	−15.8248	−1.12179	−0.560893	−	0.827888i	$-0.689542\pi$
$199$	−15.8248	−1.12179	−0.560893	+	0.827888i	$0.689542\pi$
$200$	5.72508	0.404824
$201$	0	0
$202$	12.5498	0.883003
$203$	−0.725083	−0.0508908
$204$	0	0
$205$	−28.0000	−1.95560
$206$	2.72508	0.189866
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	−27.8248	−1.92468
$210$	0	0
$211$	23.8248	1.64016	0.820082	−	0.572246i	$-0.193928\pi$
$211$	23.8248	1.64016	0.820082	+	0.572246i	$0.193928\pi$
$212$	−10.0000	−0.686803
$213$	0	0
$214$	14.5498	0.994606
$215$	−17.2749	−1.17814
$216$	0	0
$217$	8.00000	0.543075
$218$	−7.27492	−0.492720
$219$	0	0
$220$	17.2749	1.16467
$221$	−7.27492	−0.489364
$222$	0	0
$223$	−23.6495	−1.58369	−0.791844	−	0.610723i	$-0.790879\pi$
$223$	−23.6495	−1.58369	−0.791844	+	0.610723i	$0.790879\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	6.00000	0.399114
$227$	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	$0.619527\pi$
$230$	−17.2749	−1.13907
$231$	0	0
$232$	−0.725083	−0.0476040
$233$	−20.5498	−1.34626	−0.673132	−	0.739522i	$-0.735052\pi$
$233$	−20.5498	−1.34626	−0.673132	+	0.739522i	$0.735052\pi$
$234$	0	0
$235$	0	0
$236$	−8.00000	−0.520756
$237$	0	0
$238$	7.27492	0.471563
$239$	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	$0.661664\pi$
$242$	16.8248	1.08154
$243$	0	0
$244$	−4.72508	−0.302492
$245$	−3.27492	−0.209227
$246$	0	0
$247$	−5.27492	−0.335635
$248$	8.00000	0.508001
$249$	0	0
$250$	−2.37459	−0.150182
$251$	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
$254$	−5.45017	−0.341974
$255$	0	0
$256$	1.00000	0.0625000
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	3.27492	0.203493
$260$	3.27492	0.203102
$261$	0	0
$262$	15.8248	0.977656
$263$	−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
$266$	5.27492	0.323426
$267$	0	0
$268$	−2.54983	−0.155756
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	13.4502	0.817039	0.408520	−	0.912750i	$-0.366045\pi$
$271$	13.4502	0.817039	0.408520	+	0.912750i	$0.366045\pi$
$272$	7.27492	0.441107
$273$	0	0
$274$	−20.3746	−1.23087
$275$	−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	$0.597127\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	−3.27492	−0.195714
$281$	15.0997	0.900771	0.450385	−	0.892834i	$-0.351287\pi$
$281$	15.0997	0.900771	0.450385	+	0.892834i	$0.351287\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	0	0
$286$	5.27492	0.311912
$287$	8.54983	0.504681
$288$	0	0
$289$	35.9244	2.11320
$290$	2.37459	0.139440
$291$	0	0
$292$	9.82475	0.574950
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	26.1993	1.52538
$296$	3.27492	0.190351
$297$	0	0
$298$	−7.45017	−0.431577
$299$	−5.27492	−0.305056
$300$	0	0
$301$	5.27492	0.304041
$302$	−1.27492	−0.0733632
$303$	0	0
$304$	5.27492	0.302537
$305$	15.4743	0.886053
$306$	0	0
$307$	−9.09967	−0.519346	−0.259673	−	0.965697i	$-0.583615\pi$
$307$	−9.09967	−0.519346	−0.259673	+	0.965697i	$0.583615\pi$
$308$	−5.27492	−0.300566
$309$	0	0
$310$	−26.1993	−1.48802
$311$	−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	$0.333773\pi$
$314$	24.3746	1.37554
$315$	0	0
$316$	−2.54983	−0.143439
$317$	16.5498	0.929531	0.464766	−	0.885434i	$-0.346139\pi$
$317$	16.5498	0.929531	0.464766	+	0.885434i	$0.346139\pi$
$318$	0	0
$319$	3.82475	0.214145
$320$	−3.27492	−0.183073
$321$	0	0
$322$	5.27492	0.293960
$323$	38.3746	2.13522
$324$	0	0
$325$	−5.72508	−0.317570
$326$	−2.54983	−0.141222
$327$	0	0
$328$	8.54983	0.472086
$329$	0	0
$330$	0	0
$331$	15.6495	0.860174	0.430087	−	0.902787i	$-0.358483\pi$
$331$	15.6495	0.860174	0.430087	+	0.902787i	$0.358483\pi$
$332$	−10.5498	−0.578997
$333$	0	0
$334$	9.27492	0.507501
$335$	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	$0.781160\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	−23.8248	−1.29208
$341$	−42.1993	−2.28522
$342$	0	0
$343$	1.00000	0.0539949
$344$	5.27492	0.284404
$345$	0	0
$346$	23.0997	1.24185
$347$	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	$0.482953\pi$
$350$	5.72508	0.306019
$351$	0	0
$352$	−5.27492	−0.281154
$353$	3.09967	0.164979	0.0824894	−	0.996592i	$-0.473713\pi$
$353$	3.09967	0.164979	0.0824894	+	0.996592i	$0.473713\pi$
$354$	0	0
$355$	0	0
$356$	14.0000	0.741999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	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
$362$	−11.0997	−0.583386
$363$	0	0
$364$	−1.00000	−0.0524142
$365$	−32.1752	−1.68413
$366$	0	0
$367$	−33.0997	−1.72779	−0.863894	−	0.503673i	$-0.831982\pi$
$367$	−33.0997	−1.72779	−0.863894	+	0.503673i	$0.831982\pi$
$368$	5.27492	0.274974
$369$	0	0
$370$	−10.7251	−0.557571
$371$	−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	$0.404203\pi$
$374$	−38.3746	−1.98430
$375$	0	0
$376$	0	0
$377$	0.725083	0.0373437
$378$	0	0
$379$	−2.90033	−0.148980	−0.0744900	−	0.997222i	$-0.523733\pi$
$379$	−2.90033	−0.148980	−0.0744900	+	0.997222i	$0.523733\pi$
$380$	−17.2749	−0.886185
$381$	0	0
$382$	5.27492	0.269888
$383$	−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
$386$	−3.45017	−0.175609
$387$	0	0
$388$	−15.0997	−0.766570
$389$	32.1993	1.63257	0.816286	−	0.577649i	$-0.196030\pi$
$389$	32.1993	1.63257	0.816286	+	0.577649i	$0.196030\pi$
$390$	0	0
$391$	38.3746	1.94069
$392$	1.00000	0.0505076
$393$	0	0
$394$	−12.5498	−0.632252
$395$	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	$0.614267\pi$
$398$	−15.8248	−0.793223
$399$	0	0
$400$	5.72508	0.286254
$401$	−27.0997	−1.35329	−0.676646	−	0.736308i	$-0.736567\pi$
$401$	−27.0997	−1.35329	−0.676646	+	0.736308i	$0.736567\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	12.5498	0.624378
$405$	0	0
$406$	−0.725083	−0.0359853
$407$	−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	$0.763932\pi$
$410$	−28.0000	−1.38282
$411$	0	0
$412$	2.72508	0.134255
$413$	−8.00000	−0.393654
$414$	0	0
$415$	34.5498	1.69598
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	−27.8248	−1.36095
$419$	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	$0.729604\pi$
$422$	23.8248	1.15977
$423$	0	0
$424$	−10.0000	−0.485643
$425$	41.6495	2.02030
$426$	0	0
$427$	−4.72508	−0.228663
$428$	14.5498	0.703293
$429$	0	0
$430$	−17.2749	−0.833070
$431$	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	$0.674144\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	−7.27492	−0.348405
$437$	27.8248	1.33104
$438$	0	0
$439$	7.82475	0.373455	0.186728	−	0.982412i	$-0.440212\pi$
$439$	7.82475	0.373455	0.186728	+	0.982412i	$0.440212\pi$
$440$	17.2749	0.823549
$441$	0	0
$442$	−7.27492	−0.346033
$443$	−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
$446$	−23.6495	−1.11984
$447$	0	0
$448$	1.00000	0.0472456
$449$	−7.27492	−0.343325	−0.171662	−	0.985156i	$-0.554914\pi$
$449$	−7.27492	−0.343325	−0.171662	+	0.985156i	$0.554914\pi$
$450$	0	0
$451$	−45.0997	−2.12366
$452$	6.00000	0.282216
$453$	0	0
$454$	5.45017	0.255789
$455$	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	$0.525714\pi$
$458$	−11.0997	−0.518653
$459$	0	0
$460$	−17.2749	−0.805447
$461$	−34.9244	−1.62659	−0.813296	−	0.581850i	$-0.802329\pi$
$461$	−34.9244	−1.62659	−0.813296	+	0.581850i	$0.802329\pi$
$462$	0	0
$463$	22.7251	1.05612	0.528062	−	0.849206i	$-0.322919\pi$
$463$	22.7251	1.05612	0.528062	+	0.849206i	$0.322919\pi$
$464$	−0.725083	−0.0336611
$465$	0	0
$466$	−20.5498	−0.951953
$467$	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
$471$	0	0
$472$	−8.00000	−0.368230
$473$	−27.8248	−1.27938
$474$	0	0
$475$	30.1993	1.38564
$476$	7.27492	0.333445
$477$	0	0
$478$	5.09967	0.233253
$479$	−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
$482$	−15.0997	−0.687771
$483$	0	0
$484$	16.8248	0.764761
$485$	49.4502	2.24542
$486$	0	0
$487$	42.1993	1.91223	0.956117	−	0.292984i	$-0.0946484\pi$
$487$	42.1993	1.91223	0.956117	+	0.292984i	$0.0946484\pi$
$488$	−4.72508	−0.213894
$489$	0	0
$490$	−3.27492	−0.147946
$491$	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
$494$	−5.27492	−0.237330
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	0	0
$499$	−5.09967	−0.228293	−0.114146	−	0.993464i	$-0.536413\pi$
$499$	−5.09967	−0.228293	−0.114146	+	0.993464i	$0.536413\pi$
$500$	−2.37459	−0.106195
$501$	0	0
$502$	28.9244	1.29096
$503$	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
$506$	−27.8248	−1.23696
$507$	0	0
$508$	−5.45017	−0.241812
$509$	−42.9244	−1.90259	−0.951296	−	0.308280i	$-0.900247\pi$
$509$	−42.9244	−1.90259	−0.951296	+	0.308280i	$0.900247\pi$
$510$	0	0
$511$	9.82475	0.434621
$512$	1.00000	0.0441942
$513$	0	0
$514$	14.0000	0.617514
$515$	−8.92442	−0.393257
$516$	0	0
$517$	0	0
$518$	3.27492	0.143892
$519$	0	0
$520$	3.27492	0.143615
$521$	−3.27492	−0.143477	−0.0717384	−	0.997423i	$-0.522855\pi$
$521$	−3.27492	−0.143477	−0.0717384	+	0.997423i	$0.522855\pi$
$522$	0	0
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	15.8248	0.691307
$525$	0	0
$526$	−28.0000	−1.22086
$527$	58.1993	2.53520
$528$	0	0
$529$	4.82475	0.209772
$530$	32.7492	1.42253
$531$	0	0
$532$	5.27492	0.228697
$533$	−8.54983	−0.370334
$534$	0	0
$535$	−47.6495	−2.06007
$536$	−2.54983	−0.110136
$537$	0	0
$538$	−6.00000	−0.258678
$539$	−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	$0.684463\pi$
$542$	13.4502	0.577734
$543$	0	0
$544$	7.27492	0.311910
$545$	23.8248	1.02054
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	−20.3746	−0.870359
$549$	0	0
$550$	−30.1993	−1.28770
$551$	−3.82475	−0.162940
$552$	0	0
$553$	−2.54983	−0.108430
$554$	−10.0000	−0.424859
$555$	0	0
$556$	4.00000	0.169638
$557$	−30.7492	−1.30288	−0.651442	−	0.758698i	$-0.725836\pi$
$557$	−30.7492	−1.30288	−0.651442	+	0.758698i	$0.725836\pi$
$558$	0	0
$559$	−5.27492	−0.223105
$560$	−3.27492	−0.138391
$561$	0	0
$562$	15.0997	0.636941
$563$	−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
$566$	−4.00000	−0.168133
$567$	0	0
$568$	0	0
$569$	27.0997	1.13608	0.568039	−	0.823002i	$-0.307702\pi$
$569$	27.0997	1.13608	0.568039	+	0.823002i	$0.307702\pi$
$570$	0	0
$571$	−6.90033	−0.288770	−0.144385	−	0.989522i	$-0.546120\pi$
$571$	−6.90033	−0.288770	−0.144385	+	0.989522i	$0.546120\pi$
$572$	5.27492	0.220555
$573$	0	0
$574$	8.54983	0.356863
$575$	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	$0.802643\pi$
$578$	35.9244	1.49426
$579$	0	0
$580$	2.37459	0.0985993
$581$	−10.5498	−0.437681
$582$	0	0
$583$	52.7492	2.18465
$584$	9.82475	0.406551
$585$	0	0
$586$	6.00000	0.247858
$587$	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
$590$	26.1993	1.07861
$591$	0	0
$592$	3.27492	0.134598
$593$	−23.0997	−0.948590	−0.474295	−	0.880366i	$-0.657297\pi$
$593$	−23.0997	−0.948590	−0.474295	+	0.880366i	$0.657297\pi$
$594$	0	0
$595$	−23.8248	−0.976720
$596$	−7.45017	−0.305171
$597$	0	0
$598$	−5.27492	−0.215707
$599$	−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	$0.645572\pi$
$602$	5.27492	0.214990
$603$	0	0
$604$	−1.27492	−0.0518756
$605$	−55.0997	−2.24012
$606$	0	0
$607$	31.8248	1.29173	0.645863	−	0.763453i	$-0.276498\pi$
$607$	31.8248	1.29173	0.645863	+	0.763453i	$0.276498\pi$
$608$	5.27492	0.213926
$609$	0	0
$610$	15.4743	0.626534
$611$	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	$0.443619\pi$
$614$	−9.09967	−0.367233
$615$	0	0
$616$	−5.27492	−0.212532
$617$	−1.82475	−0.0734617	−0.0367309	−	0.999325i	$-0.511694\pi$
$617$	−1.82475	−0.0734617	−0.0367309	+	0.999325i	$0.511694\pi$
$618$	0	0
$619$	37.2749	1.49821	0.749103	−	0.662454i	$-0.230485\pi$
$619$	37.2749	1.49821	0.749103	+	0.662454i	$0.230485\pi$
$620$	−26.1993	−1.05219
$621$	0	0
$622$	−18.5498	−0.743781
$623$	14.0000	0.560898
$624$	0	0
$625$	−20.8488	−0.833954
$626$	17.6495	0.705416
$627$	0	0
$628$	24.3746	0.972652
$629$	23.8248	0.949955
$630$	0	0
$631$	11.8248	0.470736	0.235368	−	0.971906i	$-0.424370\pi$
$631$	11.8248	0.470736	0.235368	+	0.971906i	$0.424370\pi$
$632$	−2.54983	−0.101427
$633$	0	0
$634$	16.5498	0.657278
$635$	17.8488	0.708310
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	3.82475	0.151423
$639$	0	0
$640$	−3.27492	−0.129452
$641$	−15.0997	−0.596401	−0.298201	−	0.954503i	$-0.596386\pi$
$641$	−15.0997	−0.596401	−0.298201	+	0.954503i	$0.596386\pi$
$642$	0	0
$643$	28.9244	1.14067	0.570334	−	0.821413i	$-0.306814\pi$
$643$	28.9244	1.14067	0.570334	+	0.821413i	$0.306814\pi$
$644$	5.27492	0.207861
$645$	0	0
$646$	38.3746	1.50983
$647$	−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
$650$	−5.72508	−0.224556
$651$	0	0
$652$	−2.54983	−0.0998592
$653$	−32.3746	−1.26692	−0.633458	−	0.773777i	$-0.718365\pi$
$653$	−32.3746	−1.26692	−0.633458	+	0.773777i	$0.718365\pi$
$654$	0	0
$655$	−51.8248	−2.02496
$656$	8.54983	0.333815
$657$	0	0
$658$	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	$0.371496\pi$
$662$	15.6495	0.608235
$663$	0	0
$664$	−10.5498	−0.409413
$665$	−17.2749	−0.669893
$666$	0	0
$667$	−3.82475	−0.148095
$668$	9.27492	0.358857
$669$	0	0
$670$	8.35050	0.322608
$671$	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	$0.446216\pi$
$674$	−28.3746	−1.09295
$675$	0	0
$676$	1.00000	0.0384615
$677$	44.1993	1.69872	0.849359	−	0.527815i	$-0.176989\pi$
$677$	44.1993	1.69872	0.849359	+	0.527815i	$0.176989\pi$
$678$	0	0
$679$	−15.0997	−0.579472
$680$	−23.8248	−0.913638
$681$	0	0
$682$	−42.1993	−1.61590
$683$	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
$686$	1.00000	0.0381802
$687$	0	0
$688$	5.27492	0.201104
$689$	10.0000	0.380970
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	23.0997	0.878118
$693$	0	0
$694$	25.0997	0.952770
$695$	−13.0997	−0.496899
$696$	0	0
$697$	62.1993	2.35597
$698$	2.00000	0.0757011
$699$	0	0
$700$	5.72508	0.216388
$701$	−15.0997	−0.570307	−0.285153	−	0.958482i	$-0.592045\pi$
$701$	−15.0997	−0.570307	−0.285153	+	0.958482i	$0.592045\pi$
$702$	0	0
$703$	17.2749	0.651536
$704$	−5.27492	−0.198806
$705$	0	0
$706$	3.09967	0.116658
$707$	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
$711$	0	0
$712$	14.0000	0.524672
$713$	42.1993	1.58038
$714$	0	0
$715$	−17.2749	−0.646045
$716$	−12.0000	−0.448461
$717$	0	0
$718$	32.0000	1.19423
$719$	−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
$722$	8.82475	0.328423
$723$	0	0
$724$	−11.0997	−0.412516
$725$	−4.15116	−0.154170
$726$	0	0
$727$	−2.37459	−0.0880685	−0.0440343	−	0.999030i	$-0.514021\pi$
$727$	−2.37459	−0.0880685	−0.0440343	+	0.999030i	$0.514021\pi$
$728$	−1.00000	−0.0370625
$729$	0	0
$730$	−32.1752	−1.19086
$731$	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	$0.503232\pi$
$734$	−33.0997	−1.22173
$735$	0	0
$736$	5.27492	0.194436
$737$	13.4502	0.495443
$738$	0	0
$739$	−7.64950	−0.281392	−0.140696	−	0.990053i	$-0.544934\pi$
$739$	−7.64950	−0.281392	−0.140696	+	0.990053i	$0.544934\pi$
$740$	−10.7251	−0.394262
$741$	0	0
$742$	−10.0000	−0.367112
$743$	−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
$746$	11.4502	0.419220
$747$	0	0
$748$	−38.3746	−1.40311
$749$	14.5498	0.531639
$750$	0	0
$751$	−50.1993	−1.83180	−0.915900	−	0.401407i	$-0.868521\pi$
$751$	−50.1993	−1.83180	−0.915900	+	0.401407i	$0.868521\pi$
$752$	0	0
$753$	0	0
$754$	0.725083	0.0264060
$755$	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	$0.528384\pi$
$758$	−2.90033	−0.105345
$759$	0	0
$760$	−17.2749	−0.626627
$761$	−23.4502	−0.850068	−0.425034	−	0.905177i	$-0.639738\pi$
$761$	−23.4502	−0.850068	−0.425034	+	0.905177i	$0.639738\pi$
$762$	0	0
$763$	−7.27492	−0.263370
$764$	5.27492	0.190840
$765$	0	0
$766$	−22.7251	−0.821091
$767$	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	$0.441269\pi$
$770$	17.2749	0.622545
$771$	0	0
$772$	−3.45017	−0.124174
$773$	−42.9244	−1.54388	−0.771942	−	0.635693i	$-0.780714\pi$
$773$	−42.9244	−1.54388	−0.771942	+	0.635693i	$0.780714\pi$
$774$	0	0
$775$	45.8007	1.64521
$776$	−15.0997	−0.542047
$777$	0	0
$778$	32.1993	1.15440
$779$	45.0997	1.61586
$780$	0	0
$781$	0	0
$782$	38.3746	1.37227
$783$	0	0
$784$	1.00000	0.0357143
$785$	−79.8248	−2.84907
$786$	0	0
$787$	−2.72508	−0.0971387	−0.0485694	−	0.998820i	$-0.515466\pi$
$787$	−2.72508	−0.0971387	−0.0485694	+	0.998820i	$0.515466\pi$
$788$	−12.5498	−0.447069
$789$	0	0
$790$	8.35050	0.297097
$791$	6.00000	0.213335
$792$	0	0
$793$	4.72508	0.167793
$794$	−14.0000	−0.496841
$795$	0	0
$796$	−15.8248	−0.560893
$797$	−29.6495	−1.05024	−0.525120	−	0.851028i	$-0.675979\pi$
$797$	−29.6495	−1.05024	−0.525120	+	0.851028i	$0.675979\pi$
$798$	0	0
$799$	0	0
$800$	5.72508	0.202412
$801$	0	0
$802$	−27.0997	−0.956923
$803$	−51.8248	−1.82886
$804$	0	0
$805$	−17.2749	−0.608861
$806$	−8.00000	−0.281788
$807$	0	0
$808$	12.5498	0.441502
$809$	−4.19934	−0.147641	−0.0738204	−	0.997272i	$-0.523519\pi$
$809$	−4.19934	−0.147641	−0.0738204	+	0.997272i	$0.523519\pi$
$810$	0	0
$811$	26.3746	0.926137	0.463068	−	0.886322i	$-0.346748\pi$
$811$	26.3746	0.926137	0.463068	+	0.886322i	$0.346748\pi$
$812$	−0.725083	−0.0254454
$813$	0	0
$814$	−17.2749	−0.605486
$815$	8.35050	0.292505
$816$	0	0
$817$	27.8248	0.973465
$818$	−29.8248	−1.04280
$819$	0	0
$820$	−28.0000	−0.977802
$821$	13.6495	0.476371	0.238185	−	0.971220i	$-0.423447\pi$
$821$	13.6495	0.476371	0.238185	+	0.971220i	$0.423447\pi$
$822$	0	0
$823$	5.09967	0.177763	0.0888816	−	0.996042i	$-0.471671\pi$
$823$	5.09967	0.177763	0.0888816	+	0.996042i	$0.471671\pi$
$824$	2.72508	0.0949328
$825$	0	0
$826$	−8.00000	−0.278356
$827$	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	$0.391357\pi$
$830$	34.5498	1.19924
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	7.27492	0.252061
$834$	0	0
$835$	−30.3746	−1.05116
$836$	−27.8248	−0.962339
$837$	0	0
$838$	5.27492	0.182219
$839$	−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
$842$	−27.0997	−0.933916
$843$	0	0
$844$	23.8248	0.820082
$845$	−3.27492	−0.112661
$846$	0	0
$847$	16.8248	0.578105
$848$	−10.0000	−0.343401
$849$	0	0
$850$	41.6495	1.42857
$851$	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	$0.402294\pi$
$854$	−4.72508	−0.161689
$855$	0	0
$856$	14.5498	0.497303
$857$	11.0997	0.379157	0.189579	−	0.981866i	$-0.439288\pi$
$857$	11.0997	0.379157	0.189579	+	0.981866i	$0.439288\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	−17.2749	−0.589070
$861$	0	0
$862$	26.5498	0.904291
$863$	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
$866$	−21.6495	−0.735680
$867$	0	0
$868$	8.00000	0.271538
$869$	13.4502	0.456266
$870$	0	0
$871$	2.54983	0.0863978
$872$	−7.27492	−0.246360
$873$	0	0
$874$	27.8248	0.941186
$875$	−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	$0.385203\pi$
$878$	7.82475	0.264073
$879$	0	0
$880$	17.2749	0.582337
$881$	39.2749	1.32321	0.661603	−	0.749854i	$-0.269877\pi$
$881$	39.2749	1.32321	0.661603	+	0.749854i	$0.269877\pi$
$882$	0	0
$883$	−20.9244	−0.704163	−0.352081	−	0.935969i	$-0.614526\pi$
$883$	−20.9244	−0.704163	−0.352081	+	0.935969i	$0.614526\pi$
$884$	−7.27492	−0.244682
$885$	0	0
$886$	−22.5498	−0.757577
$887$	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
$890$	−45.8488	−1.53686
$891$	0	0
$892$	−23.6495	−0.791844
$893$	0	0
$894$	0	0
$895$	39.2990	1.31362
$896$	1.00000	0.0334077
$897$	0	0
$898$	−7.27492	−0.242767
$899$	−5.80066	−0.193463
$900$	0	0
$901$	−72.7492	−2.42363
$902$	−45.0997	−1.50165
$903$	0	0
$904$	6.00000	0.199557
$905$	36.3505	1.20833
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	5.45017	0.180870
$909$	0	0
$910$	3.27492	0.108562
$911$	−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
$914$	−3.45017	−0.114121
$915$	0	0
$916$	−11.0997	−0.366743
$917$	15.8248	0.522579
$918$	0	0
$919$	2.54983	0.0841113	0.0420556	−	0.999115i	$-0.486609\pi$
$919$	2.54983	0.0841113	0.0420556	+	0.999115i	$0.486609\pi$
$920$	−17.2749	−0.569537
$921$	0	0
$922$	−34.9244	−1.15017
$923$	0	0
$924$	0	0
$925$	18.7492	0.616469
$926$	22.7251	0.746793
$927$	0	0
$928$	−0.725083	−0.0238020
$929$	0.549834	0.0180395	0.00901974	−	0.999959i	$-0.497129\pi$
$929$	0.549834	0.0180395	0.00901974	+	0.999959i	$0.497129\pi$
$930$	0	0
$931$	5.27492	0.172878
$932$	−20.5498	−0.673132
$933$	0	0
$934$	18.7251	0.612703
$935$	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	$0.220585\pi$
$938$	−2.54983	−0.0832550
$939$	0	0
$940$	0	0
$941$	−20.9003	−0.681331	−0.340666	−	0.940185i	$-0.610652\pi$
$941$	−20.9003	−0.681331	−0.340666	+	0.940185i	$0.610652\pi$
$942$	0	0
$943$	45.0997	1.46865
$944$	−8.00000	−0.260378
$945$	0	0
$946$	−27.8248	−0.904661
$947$	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
$950$	30.1993	0.979796
$951$	0	0
$952$	7.27492	0.235781
$953$	13.6495	0.442151	0.221075	−	0.975257i	$-0.429043\pi$
$953$	13.6495	0.442151	0.221075	+	0.975257i	$0.429043\pi$
$954$	0	0
$955$	−17.2749	−0.559003
$956$	5.09967	0.164935
$957$	0	0
$958$	−14.3746	−0.464422
$959$	−20.3746	−0.657930
$960$	0	0
$961$	33.0000	1.06452
$962$	−3.27492	−0.105588
$963$	0	0
$964$	−15.0997	−0.486328
$965$	11.2990	0.363728
$966$	0	0
$967$	20.1752	0.648792	0.324396	−	0.945921i	$-0.394839\pi$
$967$	20.1752	0.648792	0.324396	+	0.945921i	$0.394839\pi$
$968$	16.8248	0.540768
$969$	0	0
$970$	49.4502	1.58775
$971$	−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
$974$	42.1993	1.35215
$975$	0	0
$976$	−4.72508	−0.151246
$977$	−38.9244	−1.24530	−0.622651	−	0.782499i	$-0.713944\pi$
$977$	−38.9244	−1.24530	−0.622651	+	0.782499i	$0.713944\pi$
$978$	0	0
$979$	−73.8488	−2.36022
$980$	−3.27492	−0.104613
$981$	0	0
$982$	6.54983	0.209014
$983$	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
$986$	−5.27492	−0.167988
$987$	0	0
$988$	−5.27492	−0.167817
$989$	27.8248	0.884776
$990$	0	0
$991$	28.7492	0.913248	0.456624	−	0.889660i	$-0.349059\pi$
$991$	28.7492	0.913248	0.456624	+	0.889660i	$0.349059\pi$
$992$	8.00000	0.254000
$993$	0	0
$994$	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	$0.739103\pi$
$998$	−5.09967	−0.161427
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.a.y.1.1		2
3.2	odd	2		546.2.a.h.1.2	✓	2
12.11	even	2		4368.2.a.bh.1.2		2
21.20	even	2		3822.2.a.bm.1.1		2
39.38	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	3.2	odd	2
1638.2.a.y.1.1		2	1.1	even	1	trivial
3822.2.a.bm.1.1		2	21.20	even	2
4368.2.a.bh.1.2		2	12.11	even	2
7098.2.a.bu.1.1		2	39.38	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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.27492 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.27492 q^{10} -5.27492 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +7.27492 q^{17} +5.27492 q^{19} -3.27492 q^{20} -5.27492 q^{22} +5.27492 q^{23} +5.72508 q^{25} -1.00000 q^{26} +1.00000 q^{28} -0.725083 q^{29} +8.00000 q^{31} +1.00000 q^{32} +7.27492 q^{34} -3.27492 q^{35} +3.27492 q^{37} +5.27492 q^{38} -3.27492 q^{40} +8.54983 q^{41} +5.27492 q^{43} -5.27492 q^{44} +5.27492 q^{46} +1.00000 q^{49} +5.72508 q^{50} -1.00000 q^{52} -10.0000 q^{53} +17.2749 q^{55} +1.00000 q^{56} -0.725083 q^{58} -8.00000 q^{59} -4.72508 q^{61} +8.00000 q^{62} +1.00000 q^{64} +3.27492 q^{65} -2.54983 q^{67} +7.27492 q^{68} -3.27492 q^{70} +9.82475 q^{73} +3.27492 q^{74} +5.27492 q^{76} -5.27492 q^{77} -2.54983 q^{79} -3.27492 q^{80} +8.54983 q^{82} -10.5498 q^{83} -23.8248 q^{85} +5.27492 q^{86} -5.27492 q^{88} +14.0000 q^{89} -1.00000 q^{91} +5.27492 q^{92} -17.2749 q^{95} -15.0997 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} + 2 q^{8} + q^{10} - 3 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} + 7 q^{17} + 3 q^{19} + q^{20} - 3 q^{22} + 3 q^{23} + 19 q^{25} - 2 q^{26} + 2 q^{28} - 9 q^{29}+ \cdots + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + q^5 + 2 * q^7 + 2 * q^8 + q^10 - 3 * q^11 - 2 * q^13 + 2 * q^14 + 2 * q^16 + 7 * q^17 + 3 * q^19 + q^20 - 3 * q^22 + 3 * q^23 + 19 * q^25 - 2 * q^26 + 2 * q^28 - 9 * q^29 + 16 * q^31 + 2 * q^32 + 7 * q^34 + q^35 - q^37 + 3 * q^38 + q^40 + 2 * q^41 + 3 * q^43 - 3 * q^44 + 3 * q^46 + 2 * q^49 + 19 * q^50 - 2 * q^52 - 20 * q^53 + 27 * q^55 + 2 * q^56 - 9 * q^58 - 16 * q^59 - 17 * q^61 + 16 * q^62 + 2 * q^64 - q^65 + 10 * q^67 + 7 * q^68 + q^70 - 3 * q^73 - q^74 + 3 * q^76 - 3 * q^77 + 10 * q^79 + q^80 + 2 * q^82 - 6 * q^83 - 25 * q^85 + 3 * q^86 - 3 * q^88 + 28 * q^89 - 2 * q^91 + 3 * q^92 - 27 * q^95 + 2 * q^98

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