LMFDB - Embedded newform 4788.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4788,2,Mod(1,4788)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4788, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4788.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4788.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$38.2323724878$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1596)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4788.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.23607 q^{5} +1.00000 q^{7} +O(q^{10})$	$q+3.23607 q^{5} +1.00000 q^{7} +2.00000 q^{11} +4.47214 q^{13} -3.23607 q^{17} -1.00000 q^{19} +2.00000 q^{23} +5.47214 q^{25} +2.76393 q^{29} +4.00000 q^{31} +3.23607 q^{35} -4.47214 q^{37} +2.47214 q^{41} -1.52786 q^{43} +4.76393 q^{47} +1.00000 q^{49} +10.1803 q^{53} +6.47214 q^{55} -4.94427 q^{59} -8.47214 q^{61} +14.4721 q^{65} -12.9443 q^{67} +2.76393 q^{71} +6.00000 q^{73} +2.00000 q^{77} +0.944272 q^{79} +11.2361 q^{83} -10.4721 q^{85} +10.4721 q^{89} +4.47214 q^{91} -3.23607 q^{95} +7.52786 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 + 2 * q^7	$ 2 q + 2 q^{5} + 2 q^{7} + 4 q^{11} - 2 q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 10 q^{29} + 8 q^{31} + 2 q^{35} - 4 q^{41} - 12 q^{43} + 14 q^{47} + 2 q^{49} - 2 q^{53} + 4 q^{55} + 8 q^{59} - 8 q^{61} + 20 q^{65} - 8 q^{67} + 10 q^{71} + 12 q^{73} + 4 q^{77} - 16 q^{79} + 18 q^{83} - 12 q^{85} + 12 q^{89} - 2 q^{95} + 24 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 2 * q^7 + 4 * q^11 - 2 * q^17 - 2 * q^19 + 4 * q^23 + 2 * q^25 + 10 * q^29 + 8 * q^31 + 2 * q^35 - 4 * q^41 - 12 * q^43 + 14 * q^47 + 2 * q^49 - 2 * q^53 + 4 * q^55 + 8 * q^59 - 8 * q^61 + 20 * q^65 - 8 * q^67 + 10 * q^71 + 12 * q^73 + 4 * q^77 - 16 * q^79 + 18 * q^83 - 12 * q^85 + 12 * q^89 - 2 * q^95 + 24 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.23607	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	4.47214	1.24035	0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214	1.24035	0.620174	+	0.784465i	$0.287062\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.23607	−0.784862	−0.392431	−	0.919781i	$-0.628366\pi$
$17$	−3.23607	−0.784862	−0.392431	+	0.919781i	$0.628366\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.76393	0.513249	0.256625	−	0.966511i	$-0.417390\pi$
$29$	2.76393	0.513249	0.256625	+	0.966511i	$0.417390\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.23607	0.546995
$36$	0	0
$37$	−4.47214	−0.735215	−0.367607	−	0.929981i	$-0.619823\pi$
$37$	−4.47214	−0.735215	−0.367607	+	0.929981i	$0.619823\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.47214	0.386083	0.193041	−	0.981191i	$-0.438165\pi$
$41$	2.47214	0.386083	0.193041	+	0.981191i	$0.438165\pi$
$42$	0	0
$43$	−1.52786	−0.232997	−0.116499	−	0.993191i	$-0.537167\pi$
$43$	−1.52786	−0.232997	−0.116499	+	0.993191i	$0.537167\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.76393	0.694891	0.347445	−	0.937700i	$-0.387049\pi$
$47$	4.76393	0.694891	0.347445	+	0.937700i	$0.387049\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.1803	1.39838	0.699189	−	0.714937i	$-0.253545\pi$
$53$	10.1803	1.39838	0.699189	+	0.714937i	$0.253545\pi$
$54$	0	0
$55$	6.47214	0.872703
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.94427	−0.643689	−0.321845	−	0.946792i	$-0.604303\pi$
$59$	−4.94427	−0.643689	−0.321845	+	0.946792i	$0.604303\pi$
$60$	0	0
$61$	−8.47214	−1.08475	−0.542373	−	0.840138i	$-0.682474\pi$
$61$	−8.47214	−1.08475	−0.542373	+	0.840138i	$0.682474\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	14.4721	1.79505
$66$	0	0
$67$	−12.9443	−1.58139	−0.790697	−	0.612207i	$-0.790282\pi$
$67$	−12.9443	−1.58139	−0.790697	+	0.612207i	$0.790282\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.76393	0.328018	0.164009	−	0.986459i	$-0.447557\pi$
$71$	2.76393	0.328018	0.164009	+	0.986459i	$0.447557\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	0.944272	0.106239	0.0531194	−	0.998588i	$-0.483084\pi$
$79$	0.944272	0.106239	0.0531194	+	0.998588i	$0.483084\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.2361	1.23332	0.616659	−	0.787230i	$-0.288486\pi$
$83$	11.2361	1.23332	0.616659	+	0.787230i	$0.288486\pi$
$84$	0	0
$85$	−10.4721	−1.13586
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.4721	1.11004	0.555022	−	0.831836i	$-0.312710\pi$
$89$	10.4721	1.11004	0.555022	+	0.831836i	$0.312710\pi$
$90$	0	0
$91$	4.47214	0.468807
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.23607	−0.332014
$96$	0	0
$97$	7.52786	0.764339	0.382169	−	0.924092i	$-0.375177\pi$
$97$	7.52786	0.764339	0.382169	+	0.924092i	$0.375177\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−5.70820	−0.567988	−0.283994	−	0.958826i	$-0.591660\pi$
$101$	−5.70820	−0.567988	−0.283994	+	0.958826i	$0.591660\pi$
$102$	0	0
$103$	−17.8885	−1.76261	−0.881305	−	0.472547i	$-0.843335\pi$
$103$	−17.8885	−1.76261	−0.881305	+	0.472547i	$0.843335\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.76393	0.653894	0.326947	−	0.945043i	$-0.393980\pi$
$107$	6.76393	0.653894	0.326947	+	0.945043i	$0.393980\pi$
$108$	0	0
$109$	9.41641	0.901928	0.450964	−	0.892542i	$-0.351080\pi$
$109$	9.41641	0.901928	0.450964	+	0.892542i	$0.351080\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.23607	−0.116279	−0.0581397	−	0.998308i	$-0.518517\pi$
$113$	−1.23607	−0.116279	−0.0581397	+	0.998308i	$0.518517\pi$
$114$	0	0
$115$	6.47214	0.603530
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.23607	−0.296650
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.52786	0.136656
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.18034	−0.714720	−0.357360	−	0.933967i	$-0.616323\pi$
$131$	−8.18034	−0.714720	−0.357360	+	0.933967i	$0.616323\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	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
$143$	8.94427	0.747958
$144$	0	0
$145$	8.94427	0.742781
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.4721	1.02176	0.510879	−	0.859653i	$-0.329320\pi$
$149$	12.4721	1.02176	0.510879	+	0.859653i	$0.329320\pi$
$150$	0	0
$151$	−11.4164	−0.929054	−0.464527	−	0.885559i	$-0.653776\pi$
$151$	−11.4164	−0.929054	−0.464527	+	0.885559i	$0.653776\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	12.9443	1.03971
$156$	0	0
$157$	4.47214	0.356915	0.178458	−	0.983948i	$-0.442889\pi$
$157$	4.47214	0.356915	0.178458	+	0.983948i	$0.442889\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	−10.4721	−0.820241	−0.410120	−	0.912031i	$-0.634513\pi$
$163$	−10.4721	−0.820241	−0.410120	+	0.912031i	$0.634513\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.4164	−1.19296	−0.596479	−	0.802629i	$-0.703434\pi$
$167$	−15.4164	−1.19296	−0.596479	+	0.802629i	$0.703434\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−13.5279	−1.02850	−0.514252	−	0.857639i	$-0.671930\pi$
$173$	−13.5279	−1.02850	−0.514252	+	0.857639i	$0.671930\pi$
$174$	0	0
$175$	5.47214	0.413655
$176$	0	0
$177$	0	0
$178$	0	0
$179$	19.1246	1.42944	0.714720	−	0.699410i	$-0.246554\pi$
$179$	19.1246	1.42944	0.714720	+	0.699410i	$0.246554\pi$
$180$	0	0
$181$	3.52786	0.262224	0.131112	−	0.991368i	$-0.458145\pi$
$181$	3.52786	0.262224	0.131112	+	0.991368i	$0.458145\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−14.4721	−1.06401
$186$	0	0
$187$	−6.47214	−0.473289
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.9443	−1.08133	−0.540665	−	0.841238i	$-0.681827\pi$
$191$	−14.9443	−1.08133	−0.540665	+	0.841238i	$0.681827\pi$
$192$	0	0
$193$	20.4721	1.47362	0.736808	−	0.676102i	$-0.236332\pi$
$193$	20.4721	1.47362	0.736808	+	0.676102i	$0.236332\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	−14.4721	−1.02590	−0.512951	−	0.858418i	$-0.671448\pi$
$199$	−14.4721	−1.02590	−0.512951	+	0.858418i	$0.671448\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.76393	0.193990
$204$	0	0
$205$	8.00000	0.558744
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−7.41641	−0.510567	−0.255283	−	0.966866i	$-0.582169\pi$
$211$	−7.41641	−0.510567	−0.255283	+	0.966866i	$0.582169\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4.94427	−0.337197
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−14.4721	−0.973501
$222$	0	0
$223$	12.0000	0.803579	0.401790	−	0.915732i	$-0.368388\pi$
$223$	12.0000	0.803579	0.401790	+	0.915732i	$0.368388\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.5279	1.16337	0.581683	−	0.813416i	$-0.302395\pi$
$227$	17.5279	1.16337	0.581683	+	0.813416i	$0.302395\pi$
$228$	0	0
$229$	16.4721	1.08851	0.544255	−	0.838920i	$-0.316813\pi$
$229$	16.4721	1.08851	0.544255	+	0.838920i	$0.316813\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.41641	−0.0927920	−0.0463960	−	0.998923i	$-0.514774\pi$
$233$	−1.41641	−0.0927920	−0.0463960	+	0.998923i	$0.514774\pi$
$234$	0	0
$235$	15.4164	1.00566
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.52786	0.228199	0.114099	−	0.993469i	$-0.463602\pi$
$239$	3.52786	0.228199	0.114099	+	0.993469i	$0.463602\pi$
$240$	0	0
$241$	7.52786	0.484912	0.242456	−	0.970162i	$-0.422047\pi$
$241$	7.52786	0.484912	0.242456	+	0.970162i	$0.422047\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.23607	0.206745
$246$	0	0
$247$	−4.47214	−0.284555
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.29180	−0.144657	−0.0723284	−	0.997381i	$-0.523043\pi$
$251$	−2.29180	−0.144657	−0.0723284	+	0.997381i	$0.523043\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−21.8885	−1.36537	−0.682685	−	0.730713i	$-0.739188\pi$
$257$	−21.8885	−1.36537	−0.682685	+	0.730713i	$0.739188\pi$
$258$	0	0
$259$	−4.47214	−0.277885
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.3607	1.13217	0.566084	−	0.824348i	$-0.308458\pi$
$263$	18.3607	1.13217	0.566084	+	0.824348i	$0.308458\pi$
$264$	0	0
$265$	32.9443	2.02375
$266$	0	0
$267$	0	0
$268$	0	0
$269$	17.8885	1.09068	0.545342	−	0.838214i	$-0.316400\pi$
$269$	17.8885	1.09068	0.545342	+	0.838214i	$0.316400\pi$
$270$	0	0
$271$	−24.3607	−1.47981	−0.739903	−	0.672714i	$-0.765129\pi$
$271$	−24.3607	−1.47981	−0.739903	+	0.672714i	$0.765129\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10.9443	0.659964
$276$	0	0
$277$	7.52786	0.452306	0.226153	−	0.974092i	$-0.427385\pi$
$277$	7.52786	0.452306	0.226153	+	0.974092i	$0.427385\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7.70820	0.459833	0.229916	−	0.973210i	$-0.426155\pi$
$281$	7.70820	0.459833	0.229916	+	0.973210i	$0.426155\pi$
$282$	0	0
$283$	23.4164	1.39196	0.695980	−	0.718061i	$-0.254970\pi$
$283$	23.4164	1.39196	0.695980	+	0.718061i	$0.254970\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.47214	0.145926
$288$	0	0
$289$	−6.52786	−0.383992
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.52786	0.322941	0.161471	−	0.986878i	$-0.448376\pi$
$293$	5.52786	0.322941	0.161471	+	0.986878i	$0.448376\pi$
$294$	0	0
$295$	−16.0000	−0.931556
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.94427	0.517261
$300$	0	0
$301$	−1.52786	−0.0880646
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−27.4164	−1.56986
$306$	0	0
$307$	−4.94427	−0.282185	−0.141092	−	0.989996i	$-0.545061\pi$
$307$	−4.94427	−0.282185	−0.141092	+	0.989996i	$0.545061\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	27.5967	1.56487	0.782434	−	0.622734i	$-0.213978\pi$
$311$	27.5967	1.56487	0.782434	+	0.622734i	$0.213978\pi$
$312$	0	0
$313$	−25.4164	−1.43662	−0.718310	−	0.695723i	$-0.755084\pi$
$313$	−25.4164	−1.43662	−0.718310	+	0.695723i	$0.755084\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.7639	1.27855	0.639275	−	0.768978i	$-0.279235\pi$
$317$	22.7639	1.27855	0.639275	+	0.768978i	$0.279235\pi$
$318$	0	0
$319$	5.52786	0.309501
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.23607	0.180060
$324$	0	0
$325$	24.4721	1.35747
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.76393	0.262644
$330$	0	0
$331$	12.3607	0.679404	0.339702	−	0.940533i	$-0.389674\pi$
$331$	12.3607	0.679404	0.339702	+	0.940533i	$0.389674\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−41.8885	−2.28862
$336$	0	0
$337$	11.5279	0.627963	0.313981	−	0.949429i	$-0.398337\pi$
$337$	11.5279	0.627963	0.313981	+	0.949429i	$0.398337\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−29.4164	−1.57916	−0.789578	−	0.613651i	$-0.789700\pi$
$347$	−29.4164	−1.57916	−0.789578	+	0.613651i	$0.789700\pi$
$348$	0	0
$349$	−11.8885	−0.636379	−0.318190	−	0.948027i	$-0.603075\pi$
$349$	−11.8885	−0.636379	−0.318190	+	0.948027i	$0.603075\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−37.1246	−1.97594	−0.987972	−	0.154634i	$-0.950580\pi$
$353$	−37.1246	−1.97594	−0.987972	+	0.154634i	$0.950580\pi$
$354$	0	0
$355$	8.94427	0.474713
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	19.4164	1.01630
$366$	0	0
$367$	−32.3607	−1.68921	−0.844607	−	0.535387i	$-0.820166\pi$
$367$	−32.3607	−1.68921	−0.844607	+	0.535387i	$0.820166\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10.1803	0.528537
$372$	0	0
$373$	−16.8328	−0.871570	−0.435785	−	0.900051i	$-0.643529\pi$
$373$	−16.8328	−0.871570	−0.435785	+	0.900051i	$0.643529\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.3607	0.636607
$378$	0	0
$379$	−18.4721	−0.948850	−0.474425	−	0.880296i	$-0.657344\pi$
$379$	−18.4721	−0.948850	−0.474425	+	0.880296i	$0.657344\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−19.4164	−0.992132	−0.496066	−	0.868285i	$-0.665223\pi$
$383$	−19.4164	−0.992132	−0.496066	+	0.868285i	$0.665223\pi$
$384$	0	0
$385$	6.47214	0.329851
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−14.3607	−0.728115	−0.364058	−	0.931376i	$-0.618609\pi$
$389$	−14.3607	−0.728115	−0.364058	+	0.931376i	$0.618609\pi$
$390$	0	0
$391$	−6.47214	−0.327310
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.05573	0.153750
$396$	0	0
$397$	10.9443	0.549277	0.274639	−	0.961548i	$-0.411442\pi$
$397$	10.9443	0.549277	0.274639	+	0.961548i	$0.411442\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.7639	−0.737276	−0.368638	−	0.929573i	$-0.620176\pi$
$401$	−14.7639	−0.737276	−0.368638	+	0.929573i	$0.620176\pi$
$402$	0	0
$403$	17.8885	0.891092
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.94427	−0.443351
$408$	0	0
$409$	32.4721	1.60564	0.802822	−	0.596219i	$-0.203331\pi$
$409$	32.4721	1.60564	0.802822	+	0.596219i	$0.203331\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.94427	−0.243292
$414$	0	0
$415$	36.3607	1.78488
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−11.8197	−0.577428	−0.288714	−	0.957415i	$-0.593228\pi$
$419$	−11.8197	−0.577428	−0.288714	+	0.957415i	$0.593228\pi$
$420$	0	0
$421$	−9.05573	−0.441349	−0.220675	−	0.975347i	$-0.570826\pi$
$421$	−9.05573	−0.441349	−0.220675	+	0.975347i	$0.570826\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−17.7082	−0.858974
$426$	0	0
$427$	−8.47214	−0.409995
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−36.0689	−1.73738	−0.868688	−	0.495359i	$-0.835037\pi$
$431$	−36.0689	−1.73738	−0.868688	+	0.495359i	$0.835037\pi$
$432$	0	0
$433$	34.3607	1.65127	0.825634	−	0.564205i	$-0.190817\pi$
$433$	34.3607	1.65127	0.825634	+	0.564205i	$0.190817\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.00000	−0.0956730
$438$	0	0
$439$	−32.9443	−1.57234	−0.786172	−	0.618008i	$-0.787940\pi$
$439$	−32.9443	−1.57234	−0.786172	+	0.618008i	$0.787940\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	22.3607	1.06239	0.531194	−	0.847250i	$-0.321744\pi$
$443$	22.3607	1.06239	0.531194	+	0.847250i	$0.321744\pi$
$444$	0	0
$445$	33.8885	1.60647
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8.29180	0.391314	0.195657	−	0.980672i	$-0.437316\pi$
$449$	8.29180	0.391314	0.195657	+	0.980672i	$0.437316\pi$
$450$	0	0
$451$	4.94427	0.232817
$452$	0	0
$453$	0	0
$454$	0	0
$455$	14.4721	0.678464
$456$	0	0
$457$	41.4164	1.93738	0.968689	−	0.248278i	$-0.0798645\pi$
$457$	41.4164	1.93738	0.968689	+	0.248278i	$0.0798645\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	28.1803	1.31249	0.656245	−	0.754548i	$-0.272144\pi$
$461$	28.1803	1.31249	0.656245	+	0.754548i	$0.272144\pi$
$462$	0	0
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.5967	0.721731	0.360866	−	0.932618i	$-0.382481\pi$
$467$	15.5967	0.721731	0.360866	+	0.932618i	$0.382481\pi$
$468$	0	0
$469$	−12.9443	−0.597711
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.05573	−0.140503
$474$	0	0
$475$	−5.47214	−0.251079
$476$	0	0
$477$	0	0
$478$	0	0
$479$	27.2361	1.24445	0.622224	−	0.782839i	$-0.286229\pi$
$479$	27.2361	1.24445	0.622224	+	0.782839i	$0.286229\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	24.3607	1.10616
$486$	0	0
$487$	30.4721	1.38082	0.690412	−	0.723416i	$-0.257429\pi$
$487$	30.4721	1.38082	0.690412	+	0.723416i	$0.257429\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−23.3050	−1.05174	−0.525869	−	0.850566i	$-0.676260\pi$
$491$	−23.3050	−1.05174	−0.525869	+	0.850566i	$0.676260\pi$
$492$	0	0
$493$	−8.94427	−0.402830
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.76393	0.123979
$498$	0	0
$499$	−2.11146	−0.0945218	−0.0472609	−	0.998883i	$-0.515049\pi$
$499$	−2.11146	−0.0945218	−0.0472609	+	0.998883i	$0.515049\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−17.1246	−0.763549	−0.381774	−	0.924256i	$-0.624687\pi$
$503$	−17.1246	−0.763549	−0.381774	+	0.924256i	$0.624687\pi$
$504$	0	0
$505$	−18.4721	−0.821999
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.9443	1.28293	0.641466	−	0.767151i	$-0.278326\pi$
$509$	28.9443	1.28293	0.641466	+	0.767151i	$0.278326\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−57.8885	−2.55087
$516$	0	0
$517$	9.52786	0.419035
$518$	0	0
$519$	0	0
$520$	0	0
$521$	22.8328	1.00032	0.500162	−	0.865932i	$-0.333274\pi$
$521$	22.8328	1.00032	0.500162	+	0.865932i	$0.333274\pi$
$522$	0	0
$523$	−12.9443	−0.566013	−0.283007	−	0.959118i	$-0.591332\pi$
$523$	−12.9443	−0.566013	−0.283007	+	0.959118i	$0.591332\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.9443	−0.563861
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	0	0
$532$	0	0
$533$	11.0557	0.478877
$534$	0	0
$535$	21.8885	0.946324
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	30.4721	1.30528
$546$	0	0
$547$	38.8328	1.66037	0.830186	−	0.557487i	$-0.188234\pi$
$547$	38.8328	1.66037	0.830186	+	0.557487i	$0.188234\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.76393	−0.117747
$552$	0	0
$553$	0.944272	0.0401545
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.41641	0.0600151	0.0300076	−	0.999550i	$-0.490447\pi$
$557$	1.41641	0.0600151	0.0300076	+	0.999550i	$0.490447\pi$
$558$	0	0
$559$	−6.83282	−0.288997
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.36068	0.352361	0.176180	−	0.984358i	$-0.443626\pi$
$563$	8.36068	0.352361	0.176180	+	0.984358i	$0.443626\pi$
$564$	0	0
$565$	−4.00000	−0.168281
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.76393	0.115870	0.0579350	−	0.998320i	$-0.481548\pi$
$569$	2.76393	0.115870	0.0579350	+	0.998320i	$0.481548\pi$
$570$	0	0
$571$	25.8885	1.08340	0.541701	−	0.840571i	$-0.317781\pi$
$571$	25.8885	1.08340	0.541701	+	0.840571i	$0.317781\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	10.9443	0.456408
$576$	0	0
$577$	13.4164	0.558532	0.279266	−	0.960214i	$-0.409909\pi$
$577$	13.4164	0.558532	0.279266	+	0.960214i	$0.409909\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	11.2361	0.466151
$582$	0	0
$583$	20.3607	0.843253
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−26.2918	−1.08518	−0.542589	−	0.839998i	$-0.682556\pi$
$587$	−26.2918	−1.08518	−0.542589	+	0.839998i	$0.682556\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21.1246	0.867484	0.433742	−	0.901037i	$-0.357193\pi$
$593$	21.1246	0.867484	0.433742	+	0.901037i	$0.357193\pi$
$594$	0	0
$595$	−10.4721	−0.429316
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.23607	0.0505044	0.0252522	−	0.999681i	$-0.491961\pi$
$599$	1.23607	0.0505044	0.0252522	+	0.999681i	$0.491961\pi$
$600$	0	0
$601$	3.52786	0.143905	0.0719523	−	0.997408i	$-0.477077\pi$
$601$	3.52786	0.143905	0.0719523	+	0.997408i	$0.477077\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−22.6525	−0.920954
$606$	0	0
$607$	14.8328	0.602045	0.301023	−	0.953617i	$-0.402672\pi$
$607$	14.8328	0.602045	0.301023	+	0.953617i	$0.402672\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	21.3050	0.861906
$612$	0	0
$613$	−44.4721	−1.79621	−0.898106	−	0.439778i	$-0.855057\pi$
$613$	−44.4721	−1.79621	−0.898106	+	0.439778i	$0.855057\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.9443	0.601634	0.300817	−	0.953682i	$-0.402741\pi$
$617$	14.9443	0.601634	0.300817	+	0.953682i	$0.402741\pi$
$618$	0	0
$619$	−12.0000	−0.482321	−0.241160	−	0.970485i	$-0.577528\pi$
$619$	−12.0000	−0.482321	−0.241160	+	0.970485i	$0.577528\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.4721	0.419557
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14.4721	0.577042
$630$	0	0
$631$	−24.3607	−0.969783	−0.484892	−	0.874574i	$-0.661141\pi$
$631$	−24.3607	−0.969783	−0.484892	+	0.874574i	$0.661141\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−38.8328	−1.54103
$636$	0	0
$637$	4.47214	0.177192
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−22.7639	−0.899121	−0.449561	−	0.893250i	$-0.648419\pi$
$641$	−22.7639	−0.899121	−0.449561	+	0.893250i	$0.648419\pi$
$642$	0	0
$643$	−20.3607	−0.802947	−0.401473	−	0.915871i	$-0.631502\pi$
$643$	−20.3607	−0.802947	−0.401473	+	0.915871i	$0.631502\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	40.5410	1.59383	0.796916	−	0.604090i	$-0.206463\pi$
$647$	40.5410	1.59383	0.796916	+	0.604090i	$0.206463\pi$
$648$	0	0
$649$	−9.88854	−0.388159
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.00000	0.0782660	0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000	0.0782660	0.0391330	+	0.999234i	$0.487540\pi$
$654$	0	0
$655$	−26.4721	−1.03435
$656$	0	0
$657$	0	0
$658$	0	0
$659$	42.1803	1.64311	0.821556	−	0.570127i	$-0.193106\pi$
$659$	42.1803	1.64311	0.821556	+	0.570127i	$0.193106\pi$
$660$	0	0
$661$	3.52786	0.137218	0.0686090	−	0.997644i	$-0.478144\pi$
$661$	3.52786	0.137218	0.0686090	+	0.997644i	$0.478144\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.23607	−0.125489
$666$	0	0
$667$	5.52786	0.214040
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−16.9443	−0.654126
$672$	0	0
$673$	−3.52786	−0.135989	−0.0679946	−	0.997686i	$-0.521660\pi$
$673$	−3.52786	−0.135989	−0.0679946	+	0.997686i	$0.521660\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−47.7771	−1.83622	−0.918111	−	0.396323i	$-0.870286\pi$
$677$	−47.7771	−1.83622	−0.918111	+	0.396323i	$0.870286\pi$
$678$	0	0
$679$	7.52786	0.288893
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−6.76393	−0.258815	−0.129407	−	0.991592i	$-0.541307\pi$
$683$	−6.76393	−0.258815	−0.129407	+	0.991592i	$0.541307\pi$
$684$	0	0
$685$	6.47214	0.247288
$686$	0	0
$687$	0	0
$688$	0	0
$689$	45.5279	1.73447
$690$	0	0
$691$	17.3050	0.658311	0.329156	−	0.944276i	$-0.393236\pi$
$691$	17.3050	0.658311	0.329156	+	0.944276i	$0.393236\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12.9443	−0.491004
$696$	0	0
$697$	−8.00000	−0.303022
$698$	0	0
$699$	0	0
$700$	0	0
$701$	13.0557	0.493108	0.246554	−	0.969129i	$-0.420702\pi$
$701$	13.0557	0.493108	0.246554	+	0.969129i	$0.420702\pi$
$702$	0	0
$703$	4.47214	0.168670
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.70820	−0.214679
$708$	0	0
$709$	−49.4164	−1.85587	−0.927936	−	0.372739i	$-0.878419\pi$
$709$	−49.4164	−1.85587	−0.927936	+	0.372739i	$0.878419\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	28.9443	1.08245
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.29180	0.0854696	0.0427348	−	0.999086i	$-0.486393\pi$
$719$	2.29180	0.0854696	0.0427348	+	0.999086i	$0.486393\pi$
$720$	0	0
$721$	−17.8885	−0.666204
$722$	0	0
$723$	0	0
$724$	0	0
$725$	15.1246	0.561714
$726$	0	0
$727$	−40.3607	−1.49689	−0.748447	−	0.663194i	$-0.769200\pi$
$727$	−40.3607	−1.49689	−0.748447	+	0.663194i	$0.769200\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.94427	0.182871
$732$	0	0
$733$	−26.9443	−0.995209	−0.497605	−	0.867404i	$-0.665787\pi$
$733$	−26.9443	−0.995209	−0.497605	+	0.867404i	$0.665787\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−25.8885	−0.953617
$738$	0	0
$739$	−13.5279	−0.497631	−0.248815	−	0.968551i	$-0.580041\pi$
$739$	−13.5279	−0.497631	−0.248815	+	0.968551i	$0.580041\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−42.1803	−1.54745	−0.773723	−	0.633524i	$-0.781608\pi$
$743$	−42.1803	−1.54745	−0.773723	+	0.633524i	$0.781608\pi$
$744$	0	0
$745$	40.3607	1.47870
$746$	0	0
$747$	0	0
$748$	0	0
$749$	6.76393	0.247149
$750$	0	0
$751$	−6.47214	−0.236172	−0.118086	−	0.993003i	$-0.537676\pi$
$751$	−6.47214	−0.236172	−0.118086	+	0.993003i	$0.537676\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−36.9443	−1.34454
$756$	0	0
$757$	−42.7214	−1.55273	−0.776367	−	0.630281i	$-0.782940\pi$
$757$	−42.7214	−1.55273	−0.776367	+	0.630281i	$0.782940\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−0.763932	−0.0276925	−0.0138463	−	0.999904i	$-0.504408\pi$
$761$	−0.763932	−0.0276925	−0.0138463	+	0.999904i	$0.504408\pi$
$762$	0	0
$763$	9.41641	0.340897
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−22.1115	−0.798398
$768$	0	0
$769$	19.5279	0.704193	0.352096	−	0.935964i	$-0.385469\pi$
$769$	19.5279	0.704193	0.352096	+	0.935964i	$0.385469\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	13.5279	0.486563	0.243282	−	0.969956i	$-0.421776\pi$
$773$	13.5279	0.486563	0.243282	+	0.969956i	$0.421776\pi$
$774$	0	0
$775$	21.8885	0.786260
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.47214	−0.0885735
$780$	0	0
$781$	5.52786	0.197803
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.4721	0.516533
$786$	0	0
$787$	−14.8328	−0.528733	−0.264366	−	0.964422i	$-0.585163\pi$
$787$	−14.8328	−0.528733	−0.264366	+	0.964422i	$0.585163\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.23607	−0.0439495
$792$	0	0
$793$	−37.8885	−1.34546
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−46.2492	−1.63823	−0.819116	−	0.573628i	$-0.805535\pi$
$797$	−46.2492	−1.63823	−0.819116	+	0.573628i	$0.805535\pi$
$798$	0	0
$799$	−15.4164	−0.545393
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12.0000	0.423471
$804$	0	0
$805$	6.47214	0.228113
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−40.8328	−1.43561	−0.717803	−	0.696247i	$-0.754852\pi$
$809$	−40.8328	−1.43561	−0.717803	+	0.696247i	$0.754852\pi$
$810$	0	0
$811$	26.8328	0.942228	0.471114	−	0.882072i	$-0.343852\pi$
$811$	26.8328	0.942228	0.471114	+	0.882072i	$0.343852\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−33.8885	−1.18706
$816$	0	0
$817$	1.52786	0.0534532
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−53.4164	−1.86425	−0.932123	−	0.362142i	$-0.882045\pi$
$821$	−53.4164	−1.86425	−0.932123	+	0.362142i	$0.882045\pi$
$822$	0	0
$823$	−48.3607	−1.68575	−0.842874	−	0.538112i	$-0.819138\pi$
$823$	−48.3607	−1.68575	−0.842874	+	0.538112i	$0.819138\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−22.1803	−0.771286	−0.385643	−	0.922648i	$-0.626020\pi$
$827$	−22.1803	−0.771286	−0.385643	+	0.922648i	$0.626020\pi$
$828$	0	0
$829$	−30.3607	−1.05447	−0.527235	−	0.849720i	$-0.676771\pi$
$829$	−30.3607	−1.05447	−0.527235	+	0.849720i	$0.676771\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.23607	−0.112123
$834$	0	0
$835$	−49.8885	−1.72646
$836$	0	0
$837$	0	0
$838$	0	0
$839$	15.0557	0.519781	0.259891	−	0.965638i	$-0.416313\pi$
$839$	15.0557	0.519781	0.259891	+	0.965638i	$0.416313\pi$
$840$	0	0
$841$	−21.3607	−0.736575
$842$	0	0
$843$	0	0
$844$	0	0
$845$	22.6525	0.779269
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−8.94427	−0.306606
$852$	0	0
$853$	−3.16718	−0.108442	−0.0542212	−	0.998529i	$-0.517268\pi$
$853$	−3.16718	−0.108442	−0.0542212	+	0.998529i	$0.517268\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−13.5279	−0.462103	−0.231052	−	0.972942i	$-0.574217\pi$
$857$	−13.5279	−0.462103	−0.231052	+	0.972942i	$0.574217\pi$
$858$	0	0
$859$	−15.0557	−0.513695	−0.256847	−	0.966452i	$-0.582684\pi$
$859$	−15.0557	−0.513695	−0.256847	+	0.966452i	$0.582684\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−9.59675	−0.326677	−0.163339	−	0.986570i	$-0.552226\pi$
$863$	−9.59675	−0.326677	−0.163339	+	0.986570i	$0.552226\pi$
$864$	0	0
$865$	−43.7771	−1.48847
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.88854	0.0640645
$870$	0	0
$871$	−57.8885	−1.96148
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.52786	0.0516512
$876$	0	0
$877$	20.8328	0.703474	0.351737	−	0.936099i	$-0.385591\pi$
$877$	20.8328	0.703474	0.351737	+	0.936099i	$0.385591\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	19.5967	0.660231	0.330116	−	0.943941i	$-0.392912\pi$
$881$	19.5967	0.660231	0.330116	+	0.943941i	$0.392912\pi$
$882$	0	0
$883$	0.944272	0.0317773	0.0158886	−	0.999874i	$-0.494942\pi$
$883$	0.944272	0.0317773	0.0158886	+	0.999874i	$0.494942\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.3050	0.849657	0.424829	−	0.905274i	$-0.360334\pi$
$887$	25.3050	0.849657	0.424829	+	0.905274i	$0.360334\pi$
$888$	0	0
$889$	−12.0000	−0.402467
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4.76393	−0.159419
$894$	0	0
$895$	61.8885	2.06871
$896$	0	0
$897$	0	0
$898$	0	0
$899$	11.0557	0.368729
$900$	0	0
$901$	−32.9443	−1.09753
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11.4164	0.379494
$906$	0	0
$907$	−15.4164	−0.511893	−0.255947	−	0.966691i	$-0.582387\pi$
$907$	−15.4164	−0.511893	−0.255947	+	0.966691i	$0.582387\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	41.2361	1.36621	0.683106	−	0.730319i	$-0.260629\pi$
$911$	41.2361	1.36621	0.683106	+	0.730319i	$0.260629\pi$
$912$	0	0
$913$	22.4721	0.743719
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.18034	−0.270139
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	12.3607	0.406857
$924$	0	0
$925$	−24.4721	−0.804639
$926$	0	0
$927$	0	0
$928$	0	0
$929$	20.5410	0.673929	0.336964	−	0.941517i	$-0.390600\pi$
$929$	20.5410	0.673929	0.336964	+	0.941517i	$0.390600\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−20.9443	−0.684951
$936$	0	0
$937$	37.7771	1.23412	0.617062	−	0.786915i	$-0.288323\pi$
$937$	37.7771	1.23412	0.617062	+	0.786915i	$0.288323\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−17.8885	−0.583150	−0.291575	−	0.956548i	$-0.594179\pi$
$941$	−17.8885	−0.583150	−0.291575	+	0.956548i	$0.594179\pi$
$942$	0	0
$943$	4.94427	0.161008
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−60.8328	−1.97680	−0.988400	−	0.151870i	$-0.951470\pi$
$947$	−60.8328	−1.97680	−0.988400	+	0.151870i	$0.951470\pi$
$948$	0	0
$949$	26.8328	0.871030
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−38.5410	−1.24847	−0.624233	−	0.781238i	$-0.714588\pi$
$953$	−38.5410	−1.24847	−0.624233	+	0.781238i	$0.714588\pi$
$954$	0	0
$955$	−48.3607	−1.56491
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.00000	0.0645834
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	0	0
$964$	0	0
$965$	66.2492	2.13264
$966$	0	0
$967$	−19.4164	−0.624390	−0.312195	−	0.950018i	$-0.601064\pi$
$967$	−19.4164	−0.624390	−0.312195	+	0.950018i	$0.601064\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−15.0557	−0.483161	−0.241581	−	0.970381i	$-0.577666\pi$
$971$	−15.0557	−0.483161	−0.241581	+	0.970381i	$0.577666\pi$
$972$	0	0
$973$	−4.00000	−0.128234
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.29180	0.137307	0.0686534	−	0.997641i	$-0.478130\pi$
$977$	4.29180	0.137307	0.0686534	+	0.997641i	$0.478130\pi$
$978$	0	0
$979$	20.9443	0.669382
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−12.9443	−0.412858	−0.206429	−	0.978462i	$-0.566184\pi$
$983$	−12.9443	−0.412858	−0.206429	+	0.978462i	$0.566184\pi$
$984$	0	0
$985$	19.4164	0.618658
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.05573	−0.0971665
$990$	0	0
$991$	−12.5836	−0.399731	−0.199865	−	0.979823i	$-0.564051\pi$
$991$	−12.5836	−0.399731	−0.199865	+	0.979823i	$0.564051\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−46.8328	−1.48470
$996$	0	0
$997$	54.3607	1.72162	0.860810	−	0.508926i	$-0.169957\pi$
$997$	54.3607	1.72162	0.860810	+	0.508926i	$0.169957\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4788.2.a.m.1.2		2
3.2	odd	2		1596.2.a.g.1.1	✓	2
12.11	even	2		6384.2.a.bm.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1596.2.a.g.1.1	✓	2	3.2	odd	2
4788.2.a.m.1.2		2	1.1	even	1	trivial
6384.2.a.bm.1.1		2	12.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4788.a (trivial)

\(f(q)\)	\(=\)	\(q+3.23607 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+3.23607 q^{5} +1.00000 q^{7} +2.00000 q^{11} +4.47214 q^{13} -3.23607 q^{17} -1.00000 q^{19} +2.00000 q^{23} +5.47214 q^{25} +2.76393 q^{29} +4.00000 q^{31} +3.23607 q^{35} -4.47214 q^{37} +2.47214 q^{41} -1.52786 q^{43} +4.76393 q^{47} +1.00000 q^{49} +10.1803 q^{53} +6.47214 q^{55} -4.94427 q^{59} -8.47214 q^{61} +14.4721 q^{65} -12.9443 q^{67} +2.76393 q^{71} +6.00000 q^{73} +2.00000 q^{77} +0.944272 q^{79} +11.2361 q^{83} -10.4721 q^{85} +10.4721 q^{89} +4.47214 q^{91} -3.23607 q^{95} +7.52786 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 + 2 * q^7	\( 2 q + 2 q^{5} + 2 q^{7} + 4 q^{11} - 2 q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 10 q^{29} + 8 q^{31} + 2 q^{35} - 4 q^{41} - 12 q^{43} + 14 q^{47} + 2 q^{49} - 2 q^{53} + 4 q^{55} + 8 q^{59} - 8 q^{61} + 20 q^{65} - 8 q^{67} + 10 q^{71} + 12 q^{73} + 4 q^{77} - 16 q^{79} + 18 q^{83} - 12 q^{85} + 12 q^{89} - 2 q^{95} + 24 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 2 * q^7 + 4 * q^11 - 2 * q^17 - 2 * q^19 + 4 * q^23 + 2 * q^25 + 10 * q^29 + 8 * q^31 + 2 * q^35 - 4 * q^41 - 12 * q^43 + 14 * q^47 + 2 * q^49 - 2 * q^53 + 4 * q^55 + 8 * q^59 - 8 * q^61 + 20 * q^65 - 8 * q^67 + 10 * q^71 + 12 * q^73 + 4 * q^77 - 16 * q^79 + 18 * q^83 - 12 * q^85 + 12 * q^89 - 2 * q^95 + 24 * q^97

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