LMFDB - Embedded newform 6384.2.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6384,2,Mod(1,6384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6384.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6384.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:	$50.9764966504$
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			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	6384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.23607 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.23607 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} +4.47214 q^{13} -3.23607 q^{15} +3.23607 q^{17} +1.00000 q^{19} -1.00000 q^{21} +2.00000 q^{23} +5.47214 q^{25} +1.00000 q^{27} -2.76393 q^{29} -4.00000 q^{31} +2.00000 q^{33} +3.23607 q^{35} -4.47214 q^{37} +4.47214 q^{39} -2.47214 q^{41} +1.52786 q^{43} -3.23607 q^{45} +4.76393 q^{47} +1.00000 q^{49} +3.23607 q^{51} -10.1803 q^{53} -6.47214 q^{55} +1.00000 q^{57} -4.94427 q^{59} -8.47214 q^{61} -1.00000 q^{63} -14.4721 q^{65} +12.9443 q^{67} +2.00000 q^{69} +2.76393 q^{71} +6.00000 q^{73} +5.47214 q^{75} -2.00000 q^{77} -0.944272 q^{79} +1.00000 q^{81} +11.2361 q^{83} -10.4721 q^{85} -2.76393 q^{87} -10.4721 q^{89} -4.47214 q^{91} -4.00000 q^{93} -3.23607 q^{95} +7.52786 q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{15} + 2 q^{17} + 2 q^{19} - 2 q^{21} + 4 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} - 8 q^{31} + 4 q^{33} + 2 q^{35} + 4 q^{41} + 12 q^{43} - 2 q^{45} + 14 q^{47} + 2 q^{49} + 2 q^{51} + 2 q^{53} - 4 q^{55} + 2 q^{57} + 8 q^{59} - 8 q^{61} - 2 q^{63} - 20 q^{65} + 8 q^{67} + 4 q^{69} + 10 q^{71} + 12 q^{73} + 2 q^{75} - 4 q^{77} + 16 q^{79} + 2 q^{81} + 18 q^{83} - 12 q^{85} - 10 q^{87} - 12 q^{89} - 8 q^{93} - 2 q^{95} + 24 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 4 * q^11 - 2 * q^15 + 2 * q^17 + 2 * q^19 - 2 * q^21 + 4 * q^23 + 2 * q^25 + 2 * q^27 - 10 * q^29 - 8 * q^31 + 4 * q^33 + 2 * q^35 + 4 * q^41 + 12 * q^43 - 2 * q^45 + 14 * q^47 + 2 * q^49 + 2 * q^51 + 2 * q^53 - 4 * q^55 + 2 * q^57 + 8 * q^59 - 8 * q^61 - 2 * q^63 - 20 * q^65 + 8 * q^67 + 4 * q^69 + 10 * q^71 + 12 * q^73 + 2 * q^75 - 4 * q^77 + 16 * q^79 + 2 * q^81 + 18 * q^83 - 12 * q^85 - 10 * q^87 - 12 * q^89 - 8 * q^93 - 2 * q^95 + 24 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	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
$15$	−3.23607	−0.835549
$16$	0	0
$17$	3.23607	0.784862	0.392431	−	0.919781i	$-0.371634\pi$
$17$	3.23607	0.784862	0.392431	+	0.919781i	$0.371634\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	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
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.76393	−0.513249	−0.256625	−	0.966511i	$-0.582610\pi$
$29$	−2.76393	−0.513249	−0.256625	+	0.966511i	$0.582610\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	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
$39$	4.47214	0.716115
$40$	0	0
$41$	−2.47214	−0.386083	−0.193041	−	0.981191i	$-0.561835\pi$
$41$	−2.47214	−0.386083	−0.193041	+	0.981191i	$0.561835\pi$
$42$	0	0
$43$	1.52786	0.232997	0.116499	−	0.993191i	$-0.462833\pi$
$43$	1.52786	0.232997	0.116499	+	0.993191i	$0.462833\pi$
$44$	0	0
$45$	−3.23607	−0.482405
$46$	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
$51$	3.23607	0.453140
$52$	0	0
$53$	−10.1803	−1.39838	−0.699189	−	0.714937i	$-0.746455\pi$
$53$	−10.1803	−1.39838	−0.699189	+	0.714937i	$0.746455\pi$
$54$	0	0
$55$	−6.47214	−0.872703
$56$	0	0
$57$	1.00000	0.132453
$58$	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
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−14.4721	−1.79505
$66$	0	0
$67$	12.9443	1.58139	0.790697	−	0.612207i	$-0.209718\pi$
$67$	12.9443	1.58139	0.790697	+	0.612207i	$0.209718\pi$
$68$	0	0
$69$	2.00000	0.240772
$70$	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
$75$	5.47214	0.631868
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	−0.944272	−0.106239	−0.0531194	−	0.998588i	$-0.516916\pi$
$79$	−0.944272	−0.106239	−0.0531194	+	0.998588i	$0.516916\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	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
$87$	−2.76393	−0.296325
$88$	0	0
$89$	−10.4721	−1.11004	−0.555022	−	0.831836i	$-0.687290\pi$
$89$	−10.4721	−1.11004	−0.555022	+	0.831836i	$0.687290\pi$
$90$	0	0
$91$	−4.47214	−0.468807
$92$	0	0
$93$	−4.00000	−0.414781
$94$	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
$99$	2.00000	0.201008
$100$	0	0
$101$	5.70820	0.567988	0.283994	−	0.958826i	$-0.408340\pi$
$101$	5.70820	0.567988	0.283994	+	0.958826i	$0.408340\pi$
$102$	0	0
$103$	17.8885	1.76261	0.881305	−	0.472547i	$-0.156665\pi$
$103$	17.8885	1.76261	0.881305	+	0.472547i	$0.156665\pi$
$104$	0	0
$105$	3.23607	0.315808
$106$	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
$111$	−4.47214	−0.424476
$112$	0	0
$113$	1.23607	0.116279	0.0581397	−	0.998308i	$-0.481483\pi$
$113$	1.23607	0.116279	0.0581397	+	0.998308i	$0.481483\pi$
$114$	0	0
$115$	−6.47214	−0.603530
$116$	0	0
$117$	4.47214	0.413449
$118$	0	0
$119$	−3.23607	−0.296650
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−2.47214	−0.222905
$124$	0	0
$125$	−1.52786	−0.136656
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	0	0
$129$	1.52786	0.134521
$130$	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
$135$	−3.23607	−0.278516
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	4.76393	0.401195
$142$	0	0
$143$	8.94427	0.747958
$144$	0	0
$145$	8.94427	0.742781
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−12.4721	−1.02176	−0.510879	−	0.859653i	$-0.670680\pi$
$149$	−12.4721	−1.02176	−0.510879	+	0.859653i	$0.670680\pi$
$150$	0	0
$151$	11.4164	0.929054	0.464527	−	0.885559i	$-0.346224\pi$
$151$	11.4164	0.929054	0.464527	+	0.885559i	$0.346224\pi$
$152$	0	0
$153$	3.23607	0.261621
$154$	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
$159$	−10.1803	−0.807353
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	10.4721	0.820241	0.410120	−	0.912031i	$-0.365487\pi$
$163$	10.4721	0.820241	0.410120	+	0.912031i	$0.365487\pi$
$164$	0	0
$165$	−6.47214	−0.503855
$166$	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
$171$	1.00000	0.0764719
$172$	0	0
$173$	13.5279	1.02850	0.514252	−	0.857639i	$-0.328070\pi$
$173$	13.5279	1.02850	0.514252	+	0.857639i	$0.328070\pi$
$174$	0	0
$175$	−5.47214	−0.413655
$176$	0	0
$177$	−4.94427	−0.371634
$178$	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
$183$	−8.47214	−0.626278
$184$	0	0
$185$	14.4721	1.06401
$186$	0	0
$187$	6.47214	0.473289
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	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
$195$	−14.4721	−1.03637
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	14.4721	1.02590	0.512951	−	0.858418i	$-0.328552\pi$
$199$	14.4721	1.02590	0.512951	+	0.858418i	$0.328552\pi$
$200$	0	0
$201$	12.9443	0.913019
$202$	0	0
$203$	2.76393	0.193990
$204$	0	0
$205$	8.00000	0.558744
$206$	0	0
$207$	2.00000	0.139010
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	7.41641	0.510567	0.255283	−	0.966866i	$-0.417831\pi$
$211$	7.41641	0.510567	0.255283	+	0.966866i	$0.417831\pi$
$212$	0	0
$213$	2.76393	0.189382
$214$	0	0
$215$	−4.94427	−0.337197
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	6.00000	0.405442
$220$	0	0
$221$	14.4721	0.973501
$222$	0	0
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	0	0
$225$	5.47214	0.364809
$226$	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
$231$	−2.00000	−0.131590
$232$	0	0
$233$	1.41641	0.0927920	0.0463960	−	0.998923i	$-0.485226\pi$
$233$	1.41641	0.0927920	0.0463960	+	0.998923i	$0.485226\pi$
$234$	0	0
$235$	−15.4164	−1.00566
$236$	0	0
$237$	−0.944272	−0.0613371
$238$	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
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.23607	−0.206745
$246$	0	0
$247$	4.47214	0.284555
$248$	0	0
$249$	11.2361	0.712057
$250$	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
$255$	−10.4721	−0.655791
$256$	0	0
$257$	21.8885	1.36537	0.682685	−	0.730713i	$-0.260812\pi$
$257$	21.8885	1.36537	0.682685	+	0.730713i	$0.260812\pi$
$258$	0	0
$259$	4.47214	0.277885
$260$	0	0
$261$	−2.76393	−0.171083
$262$	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
$267$	−10.4721	−0.640884
$268$	0	0
$269$	−17.8885	−1.09068	−0.545342	−	0.838214i	$-0.683600\pi$
$269$	−17.8885	−1.09068	−0.545342	+	0.838214i	$0.683600\pi$
$270$	0	0
$271$	24.3607	1.47981	0.739903	−	0.672714i	$-0.234871\pi$
$271$	24.3607	1.47981	0.739903	+	0.672714i	$0.234871\pi$
$272$	0	0
$273$	−4.47214	−0.270666
$274$	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
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−7.70820	−0.459833	−0.229916	−	0.973210i	$-0.573845\pi$
$281$	−7.70820	−0.459833	−0.229916	+	0.973210i	$0.573845\pi$
$282$	0	0
$283$	−23.4164	−1.39196	−0.695980	−	0.718061i	$-0.745030\pi$
$283$	−23.4164	−1.39196	−0.695980	+	0.718061i	$0.745030\pi$
$284$	0	0
$285$	−3.23607	−0.191688
$286$	0	0
$287$	2.47214	0.145926
$288$	0	0
$289$	−6.52786	−0.383992
$290$	0	0
$291$	7.52786	0.441291
$292$	0	0
$293$	−5.52786	−0.322941	−0.161471	−	0.986878i	$-0.551624\pi$
$293$	−5.52786	−0.322941	−0.161471	+	0.986878i	$0.551624\pi$
$294$	0	0
$295$	16.0000	0.931556
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	8.94427	0.517261
$300$	0	0
$301$	−1.52786	−0.0880646
$302$	0	0
$303$	5.70820	0.327928
$304$	0	0
$305$	27.4164	1.56986
$306$	0	0
$307$	4.94427	0.282185	0.141092	−	0.989996i	$-0.454939\pi$
$307$	4.94427	0.282185	0.141092	+	0.989996i	$0.454939\pi$
$308$	0	0
$309$	17.8885	1.01764
$310$	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
$315$	3.23607	0.182332
$316$	0	0
$317$	−22.7639	−1.27855	−0.639275	−	0.768978i	$-0.720765\pi$
$317$	−22.7639	−1.27855	−0.639275	+	0.768978i	$0.720765\pi$
$318$	0	0
$319$	−5.52786	−0.309501
$320$	0	0
$321$	6.76393	0.377526
$322$	0	0
$323$	3.23607	0.180060
$324$	0	0
$325$	24.4721	1.35747
$326$	0	0
$327$	9.41641	0.520729
$328$	0	0
$329$	−4.76393	−0.262644
$330$	0	0
$331$	−12.3607	−0.679404	−0.339702	−	0.940533i	$-0.610326\pi$
$331$	−12.3607	−0.679404	−0.339702	+	0.940533i	$0.610326\pi$
$332$	0	0
$333$	−4.47214	−0.245072
$334$	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
$339$	1.23607	0.0671340
$340$	0	0
$341$	−8.00000	−0.433224
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−6.47214	−0.348448
$346$	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
$351$	4.47214	0.238705
$352$	0	0
$353$	37.1246	1.97594	0.987972	−	0.154634i	$-0.0494198\pi$
$353$	37.1246	1.97594	0.987972	+	0.154634i	$0.0494198\pi$
$354$	0	0
$355$	−8.94427	−0.474713
$356$	0	0
$357$	−3.23607	−0.171271
$358$	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
$363$	−7.00000	−0.367405
$364$	0	0
$365$	−19.4164	−1.01630
$366$	0	0
$367$	32.3607	1.68921	0.844607	−	0.535387i	$-0.179834\pi$
$367$	32.3607	1.68921	0.844607	+	0.535387i	$0.179834\pi$
$368$	0	0
$369$	−2.47214	−0.128694
$370$	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
$375$	−1.52786	−0.0788986
$376$	0	0
$377$	−12.3607	−0.636607
$378$	0	0
$379$	18.4721	0.948850	0.474425	−	0.880296i	$-0.342656\pi$
$379$	18.4721	0.948850	0.474425	+	0.880296i	$0.342656\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	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
$387$	1.52786	0.0776657
$388$	0	0
$389$	14.3607	0.728115	0.364058	−	0.931376i	$-0.381391\pi$
$389$	14.3607	0.728115	0.364058	+	0.931376i	$0.381391\pi$
$390$	0	0
$391$	6.47214	0.327310
$392$	0	0
$393$	−8.18034	−0.412644
$394$	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
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	14.7639	0.737276	0.368638	−	0.929573i	$-0.379824\pi$
$401$	14.7639	0.737276	0.368638	+	0.929573i	$0.379824\pi$
$402$	0	0
$403$	−17.8885	−0.891092
$404$	0	0
$405$	−3.23607	−0.160802
$406$	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
$411$	−2.00000	−0.0986527
$412$	0	0
$413$	4.94427	0.243292
$414$	0	0
$415$	−36.3607	−1.78488
$416$	0	0
$417$	4.00000	0.195881
$418$	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
$423$	4.76393	0.231630
$424$	0	0
$425$	17.7082	0.858974
$426$	0	0
$427$	8.47214	0.409995
$428$	0	0
$429$	8.94427	0.431834
$430$	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
$435$	8.94427	0.428845
$436$	0	0
$437$	2.00000	0.0956730
$438$	0	0
$439$	32.9443	1.57234	0.786172	−	0.618008i	$-0.212060\pi$
$439$	32.9443	1.57234	0.786172	+	0.618008i	$0.212060\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	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
$447$	−12.4721	−0.589912
$448$	0	0
$449$	−8.29180	−0.391314	−0.195657	−	0.980672i	$-0.562684\pi$
$449$	−8.29180	−0.391314	−0.195657	+	0.980672i	$0.562684\pi$
$450$	0	0
$451$	−4.94427	−0.232817
$452$	0	0
$453$	11.4164	0.536390
$454$	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
$459$	3.23607	0.151047
$460$	0	0
$461$	−28.1803	−1.31249	−0.656245	−	0.754548i	$-0.727856\pi$
$461$	−28.1803	−1.31249	−0.656245	+	0.754548i	$0.727856\pi$
$462$	0	0
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	0	0
$465$	12.9443	0.600276
$466$	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
$471$	4.47214	0.206065
$472$	0	0
$473$	3.05573	0.140503
$474$	0	0
$475$	5.47214	0.251079
$476$	0	0
$477$	−10.1803	−0.466126
$478$	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
$483$	−2.00000	−0.0910032
$484$	0	0
$485$	−24.3607	−1.10616
$486$	0	0
$487$	−30.4721	−1.38082	−0.690412	−	0.723416i	$-0.742571\pi$
$487$	−30.4721	−1.38082	−0.690412	+	0.723416i	$0.742571\pi$
$488$	0	0
$489$	10.4721	0.473566
$490$	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
$495$	−6.47214	−0.290901
$496$	0	0
$497$	−2.76393	−0.123979
$498$	0	0
$499$	2.11146	0.0945218	0.0472609	−	0.998883i	$-0.484951\pi$
$499$	2.11146	0.0945218	0.0472609	+	0.998883i	$0.484951\pi$
$500$	0	0
$501$	−15.4164	−0.688754
$502$	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
$507$	7.00000	0.310881
$508$	0	0
$509$	−28.9443	−1.28293	−0.641466	−	0.767151i	$-0.721674\pi$
$509$	−28.9443	−1.28293	−0.641466	+	0.767151i	$0.721674\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−57.8885	−2.55087
$516$	0	0
$517$	9.52786	0.419035
$518$	0	0
$519$	13.5279	0.593807
$520$	0	0
$521$	−22.8328	−1.00032	−0.500162	−	0.865932i	$-0.666726\pi$
$521$	−22.8328	−1.00032	−0.500162	+	0.865932i	$0.666726\pi$
$522$	0	0
$523$	12.9443	0.566013	0.283007	−	0.959118i	$-0.408668\pi$
$523$	12.9443	0.566013	0.283007	+	0.959118i	$0.408668\pi$
$524$	0	0
$525$	−5.47214	−0.238824
$526$	0	0
$527$	−12.9443	−0.563861
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−4.94427	−0.214563
$532$	0	0
$533$	−11.0557	−0.478877
$534$	0	0
$535$	−21.8885	−0.946324
$536$	0	0
$537$	19.1246	0.825288
$538$	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
$543$	3.52786	0.151395
$544$	0	0
$545$	−30.4721	−1.30528
$546$	0	0
$547$	−38.8328	−1.66037	−0.830186	−	0.557487i	$-0.811766\pi$
$547$	−38.8328	−1.66037	−0.830186	+	0.557487i	$0.811766\pi$
$548$	0	0
$549$	−8.47214	−0.361582
$550$	0	0
$551$	−2.76393	−0.117747
$552$	0	0
$553$	0.944272	0.0401545
$554$	0	0
$555$	14.4721	0.614308
$556$	0	0
$557$	−1.41641	−0.0600151	−0.0300076	−	0.999550i	$-0.509553\pi$
$557$	−1.41641	−0.0600151	−0.0300076	+	0.999550i	$0.509553\pi$
$558$	0	0
$559$	6.83282	0.288997
$560$	0	0
$561$	6.47214	0.273254
$562$	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
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−2.76393	−0.115870	−0.0579350	−	0.998320i	$-0.518452\pi$
$569$	−2.76393	−0.115870	−0.0579350	+	0.998320i	$0.518452\pi$
$570$	0	0
$571$	−25.8885	−1.08340	−0.541701	−	0.840571i	$-0.682219\pi$
$571$	−25.8885	−1.08340	−0.541701	+	0.840571i	$0.682219\pi$
$572$	0	0
$573$	−14.9443	−0.624306
$574$	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
$579$	20.4721	0.850793
$580$	0	0
$581$	−11.2361	−0.466151
$582$	0	0
$583$	−20.3607	−0.843253
$584$	0	0
$585$	−14.4721	−0.598349
$586$	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
$591$	−6.00000	−0.246807
$592$	0	0
$593$	−21.1246	−0.867484	−0.433742	−	0.901037i	$-0.642807\pi$
$593$	−21.1246	−0.867484	−0.433742	+	0.901037i	$0.642807\pi$
$594$	0	0
$595$	10.4721	0.429316
$596$	0	0
$597$	14.4721	0.592305
$598$	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
$603$	12.9443	0.527132
$604$	0	0
$605$	22.6525	0.920954
$606$	0	0
$607$	−14.8328	−0.602045	−0.301023	−	0.953617i	$-0.597328\pi$
$607$	−14.8328	−0.602045	−0.301023	+	0.953617i	$0.597328\pi$
$608$	0	0
$609$	2.76393	0.112000
$610$	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
$615$	8.00000	0.322591
$616$	0	0
$617$	−14.9443	−0.601634	−0.300817	−	0.953682i	$-0.597259\pi$
$617$	−14.9443	−0.601634	−0.300817	+	0.953682i	$0.597259\pi$
$618$	0	0
$619$	12.0000	0.482321	0.241160	−	0.970485i	$-0.422472\pi$
$619$	12.0000	0.482321	0.241160	+	0.970485i	$0.422472\pi$
$620$	0	0
$621$	2.00000	0.0802572
$622$	0	0
$623$	10.4721	0.419557
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	2.00000	0.0798723
$628$	0	0
$629$	−14.4721	−0.577042
$630$	0	0
$631$	24.3607	0.969783	0.484892	−	0.874574i	$-0.338859\pi$
$631$	24.3607	0.969783	0.484892	+	0.874574i	$0.338859\pi$
$632$	0	0
$633$	7.41641	0.294776
$634$	0	0
$635$	−38.8328	−1.54103
$636$	0	0
$637$	4.47214	0.177192
$638$	0	0
$639$	2.76393	0.109339
$640$	0	0
$641$	22.7639	0.899121	0.449561	−	0.893250i	$-0.351581\pi$
$641$	22.7639	0.899121	0.449561	+	0.893250i	$0.351581\pi$
$642$	0	0
$643$	20.3607	0.802947	0.401473	−	0.915871i	$-0.368498\pi$
$643$	20.3607	0.802947	0.401473	+	0.915871i	$0.368498\pi$
$644$	0	0
$645$	−4.94427	−0.194681
$646$	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
$651$	4.00000	0.156772
$652$	0	0
$653$	−2.00000	−0.0782660	−0.0391330	−	0.999234i	$-0.512460\pi$
$653$	−2.00000	−0.0782660	−0.0391330	+	0.999234i	$0.512460\pi$
$654$	0	0
$655$	26.4721	1.03435
$656$	0	0
$657$	6.00000	0.234082
$658$	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
$663$	14.4721	0.562051
$664$	0	0
$665$	3.23607	0.125489
$666$	0	0
$667$	−5.52786	−0.214040
$668$	0	0
$669$	−12.0000	−0.463947
$670$	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
$675$	5.47214	0.210623
$676$	0	0
$677$	47.7771	1.83622	0.918111	−	0.396323i	$-0.129714\pi$
$677$	47.7771	1.83622	0.918111	+	0.396323i	$0.129714\pi$
$678$	0	0
$679$	−7.52786	−0.288893
$680$	0	0
$681$	17.5279	0.671669
$682$	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
$687$	16.4721	0.628451
$688$	0	0
$689$	−45.5279	−1.73447
$690$	0	0
$691$	−17.3050	−0.658311	−0.329156	−	0.944276i	$-0.606764\pi$
$691$	−17.3050	−0.658311	−0.329156	+	0.944276i	$0.606764\pi$
$692$	0	0
$693$	−2.00000	−0.0759737
$694$	0	0
$695$	−12.9443	−0.491004
$696$	0	0
$697$	−8.00000	−0.303022
$698$	0	0
$699$	1.41641	0.0535735
$700$	0	0
$701$	−13.0557	−0.493108	−0.246554	−	0.969129i	$-0.579298\pi$
$701$	−13.0557	−0.493108	−0.246554	+	0.969129i	$0.579298\pi$
$702$	0	0
$703$	−4.47214	−0.168670
$704$	0	0
$705$	−15.4164	−0.580616
$706$	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
$711$	−0.944272	−0.0354130
$712$	0	0
$713$	−8.00000	−0.299602
$714$	0	0
$715$	−28.9443	−1.08245
$716$	0	0
$717$	3.52786	0.131750
$718$	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
$723$	7.52786	0.279964
$724$	0	0
$725$	−15.1246	−0.561714
$726$	0	0
$727$	40.3607	1.49689	0.748447	−	0.663194i	$-0.230800\pi$
$727$	40.3607	1.49689	0.748447	+	0.663194i	$0.230800\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	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
$735$	−3.23607	−0.119364
$736$	0	0
$737$	25.8885	0.953617
$738$	0	0
$739$	13.5279	0.497631	0.248815	−	0.968551i	$-0.419959\pi$
$739$	13.5279	0.497631	0.248815	+	0.968551i	$0.419959\pi$
$740$	0	0
$741$	4.47214	0.164288
$742$	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
$747$	11.2361	0.411106
$748$	0	0
$749$	−6.76393	−0.247149
$750$	0	0
$751$	6.47214	0.236172	0.118086	−	0.993003i	$-0.462324\pi$
$751$	6.47214	0.236172	0.118086	+	0.993003i	$0.462324\pi$
$752$	0	0
$753$	−2.29180	−0.0835177
$754$	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
$759$	4.00000	0.145191
$760$	0	0
$761$	0.763932	0.0276925	0.0138463	−	0.999904i	$-0.495592\pi$
$761$	0.763932	0.0276925	0.0138463	+	0.999904i	$0.495592\pi$
$762$	0	0
$763$	−9.41641	−0.340897
$764$	0	0
$765$	−10.4721	−0.378621
$766$	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
$771$	21.8885	0.788297
$772$	0	0
$773$	−13.5279	−0.486563	−0.243282	−	0.969956i	$-0.578224\pi$
$773$	−13.5279	−0.486563	−0.243282	+	0.969956i	$0.578224\pi$
$774$	0	0
$775$	−21.8885	−0.786260
$776$	0	0
$777$	4.47214	0.160437
$778$	0	0
$779$	−2.47214	−0.0885735
$780$	0	0
$781$	5.52786	0.197803
$782$	0	0
$783$	−2.76393	−0.0987749
$784$	0	0
$785$	−14.4721	−0.516533
$786$	0	0
$787$	14.8328	0.528733	0.264366	−	0.964422i	$-0.414837\pi$
$787$	14.8328	0.528733	0.264366	+	0.964422i	$0.414837\pi$
$788$	0	0
$789$	18.3607	0.653658
$790$	0	0
$791$	−1.23607	−0.0439495
$792$	0	0
$793$	−37.8885	−1.34546
$794$	0	0
$795$	32.9443	1.16841
$796$	0	0
$797$	46.2492	1.63823	0.819116	−	0.573628i	$-0.194465\pi$
$797$	46.2492	1.63823	0.819116	+	0.573628i	$0.194465\pi$
$798$	0	0
$799$	15.4164	0.545393
$800$	0	0
$801$	−10.4721	−0.370015
$802$	0	0
$803$	12.0000	0.423471
$804$	0	0
$805$	6.47214	0.228113
$806$	0	0
$807$	−17.8885	−0.629707
$808$	0	0
$809$	40.8328	1.43561	0.717803	−	0.696247i	$-0.245148\pi$
$809$	40.8328	1.43561	0.717803	+	0.696247i	$0.245148\pi$
$810$	0	0
$811$	−26.8328	−0.942228	−0.471114	−	0.882072i	$-0.656148\pi$
$811$	−26.8328	−0.942228	−0.471114	+	0.882072i	$0.656148\pi$
$812$	0	0
$813$	24.3607	0.854366
$814$	0	0
$815$	−33.8885	−1.18706
$816$	0	0
$817$	1.52786	0.0534532
$818$	0	0
$819$	−4.47214	−0.156269
$820$	0	0
$821$	53.4164	1.86425	0.932123	−	0.362142i	$-0.117955\pi$
$821$	53.4164	1.86425	0.932123	+	0.362142i	$0.117955\pi$
$822$	0	0
$823$	48.3607	1.68575	0.842874	−	0.538112i	$-0.180862\pi$
$823$	48.3607	1.68575	0.842874	+	0.538112i	$0.180862\pi$
$824$	0	0
$825$	10.9443	0.381031
$826$	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
$831$	7.52786	0.261139
$832$	0	0
$833$	3.23607	0.112123
$834$	0	0
$835$	49.8885	1.72646
$836$	0	0
$837$	−4.00000	−0.138260
$838$	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
$843$	−7.70820	−0.265485
$844$	0	0
$845$	−22.6525	−0.779269
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	−23.4164	−0.803649
$850$	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
$855$	−3.23607	−0.110671
$856$	0	0
$857$	13.5279	0.462103	0.231052	−	0.972942i	$-0.425783\pi$
$857$	13.5279	0.462103	0.231052	+	0.972942i	$0.425783\pi$
$858$	0	0
$859$	15.0557	0.513695	0.256847	−	0.966452i	$-0.417316\pi$
$859$	15.0557	0.513695	0.256847	+	0.966452i	$0.417316\pi$
$860$	0	0
$861$	2.47214	0.0842502
$862$	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
$867$	−6.52786	−0.221698
$868$	0	0
$869$	−1.88854	−0.0640645
$870$	0	0
$871$	57.8885	1.96148
$872$	0	0
$873$	7.52786	0.254780
$874$	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
$879$	−5.52786	−0.186450
$880$	0	0
$881$	−19.5967	−0.660231	−0.330116	−	0.943941i	$-0.607088\pi$
$881$	−19.5967	−0.660231	−0.330116	+	0.943941i	$0.607088\pi$
$882$	0	0
$883$	−0.944272	−0.0317773	−0.0158886	−	0.999874i	$-0.505058\pi$
$883$	−0.944272	−0.0317773	−0.0158886	+	0.999874i	$0.505058\pi$
$884$	0	0
$885$	16.0000	0.537834
$886$	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
$891$	2.00000	0.0670025
$892$	0	0
$893$	4.76393	0.159419
$894$	0	0
$895$	−61.8885	−2.06871
$896$	0	0
$897$	8.94427	0.298641
$898$	0	0
$899$	11.0557	0.368729
$900$	0	0
$901$	−32.9443	−1.09753
$902$	0	0
$903$	−1.52786	−0.0508441
$904$	0	0
$905$	−11.4164	−0.379494
$906$	0	0
$907$	15.4164	0.511893	0.255947	−	0.966691i	$-0.417613\pi$
$907$	15.4164	0.511893	0.255947	+	0.966691i	$0.417613\pi$
$908$	0	0
$909$	5.70820	0.189329
$910$	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
$915$	27.4164	0.906358
$916$	0	0
$917$	8.18034	0.270139
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	4.94427	0.162919
$922$	0	0
$923$	12.3607	0.406857
$924$	0	0
$925$	−24.4721	−0.804639
$926$	0	0
$927$	17.8885	0.587537
$928$	0	0
$929$	−20.5410	−0.673929	−0.336964	−	0.941517i	$-0.609400\pi$
$929$	−20.5410	−0.673929	−0.336964	+	0.941517i	$0.609400\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	27.5967	0.903477
$934$	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
$939$	−25.4164	−0.829433
$940$	0	0
$941$	17.8885	0.583150	0.291575	−	0.956548i	$-0.405821\pi$
$941$	17.8885	0.583150	0.291575	+	0.956548i	$0.405821\pi$
$942$	0	0
$943$	−4.94427	−0.161008
$944$	0	0
$945$	3.23607	0.105269
$946$	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
$951$	−22.7639	−0.738171
$952$	0	0
$953$	38.5410	1.24847	0.624233	−	0.781238i	$-0.285412\pi$
$953$	38.5410	1.24847	0.624233	+	0.781238i	$0.285412\pi$
$954$	0	0
$955$	48.3607	1.56491
$956$	0	0
$957$	−5.52786	−0.178690
$958$	0	0
$959$	2.00000	0.0645834
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	6.76393	0.217965
$964$	0	0
$965$	−66.2492	−2.13264
$966$	0	0
$967$	19.4164	0.624390	0.312195	−	0.950018i	$-0.398936\pi$
$967$	19.4164	0.624390	0.312195	+	0.950018i	$0.398936\pi$
$968$	0	0
$969$	3.23607	0.103957
$970$	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
$975$	24.4721	0.783736
$976$	0	0
$977$	−4.29180	−0.137307	−0.0686534	−	0.997641i	$-0.521870\pi$
$977$	−4.29180	−0.137307	−0.0686534	+	0.997641i	$0.521870\pi$
$978$	0	0
$979$	−20.9443	−0.669382
$980$	0	0
$981$	9.41641	0.300643
$982$	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
$987$	−4.76393	−0.151638
$988$	0	0
$989$	3.05573	0.0971665
$990$	0	0
$991$	12.5836	0.399731	0.199865	−	0.979823i	$-0.435949\pi$
$991$	12.5836	0.399731	0.199865	+	0.979823i	$0.435949\pi$
$992$	0	0
$993$	−12.3607	−0.392254
$994$	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
$999$	−4.47214	−0.141492

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1596.2.a.g.1.1	✓	2	4.3	odd	2
4788.2.a.m.1.2		2	12.11	even	2
6384.2.a.bm.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6384.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.23607 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.23607 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} +4.47214 q^{13} -3.23607 q^{15} +3.23607 q^{17} +1.00000 q^{19} -1.00000 q^{21} +2.00000 q^{23} +5.47214 q^{25} +1.00000 q^{27} -2.76393 q^{29} -4.00000 q^{31} +2.00000 q^{33} +3.23607 q^{35} -4.47214 q^{37} +4.47214 q^{39} -2.47214 q^{41} +1.52786 q^{43} -3.23607 q^{45} +4.76393 q^{47} +1.00000 q^{49} +3.23607 q^{51} -10.1803 q^{53} -6.47214 q^{55} +1.00000 q^{57} -4.94427 q^{59} -8.47214 q^{61} -1.00000 q^{63} -14.4721 q^{65} +12.9443 q^{67} +2.00000 q^{69} +2.76393 q^{71} +6.00000 q^{73} +5.47214 q^{75} -2.00000 q^{77} -0.944272 q^{79} +1.00000 q^{81} +11.2361 q^{83} -10.4721 q^{85} -2.76393 q^{87} -10.4721 q^{89} -4.47214 q^{91} -4.00000 q^{93} -3.23607 q^{95} +7.52786 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{15} + 2 q^{17} + 2 q^{19} - 2 q^{21} + 4 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} - 8 q^{31} + 4 q^{33} + 2 q^{35} + 4 q^{41} + 12 q^{43} - 2 q^{45} + 14 q^{47} + 2 q^{49} + 2 q^{51} + 2 q^{53} - 4 q^{55} + 2 q^{57} + 8 q^{59} - 8 q^{61} - 2 q^{63} - 20 q^{65} + 8 q^{67} + 4 q^{69} + 10 q^{71} + 12 q^{73} + 2 q^{75} - 4 q^{77} + 16 q^{79} + 2 q^{81} + 18 q^{83} - 12 q^{85} - 10 q^{87} - 12 q^{89} - 8 q^{93} - 2 q^{95} + 24 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 4 * q^11 - 2 * q^15 + 2 * q^17 + 2 * q^19 - 2 * q^21 + 4 * q^23 + 2 * q^25 + 2 * q^27 - 10 * q^29 - 8 * q^31 + 4 * q^33 + 2 * q^35 + 4 * q^41 + 12 * q^43 - 2 * q^45 + 14 * q^47 + 2 * q^49 + 2 * q^51 + 2 * q^53 - 4 * q^55 + 2 * q^57 + 8 * q^59 - 8 * q^61 - 2 * q^63 - 20 * q^65 + 8 * q^67 + 4 * q^69 + 10 * q^71 + 12 * q^73 + 2 * q^75 - 4 * q^77 + 16 * q^79 + 2 * q^81 + 18 * q^83 - 12 * q^85 - 10 * q^87 - 12 * q^89 - 8 * q^93 - 2 * q^95 + 24 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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