LMFDB - Embedded newform 6864.2.a.cc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6864,2,Mod(1,6864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6864, 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("6864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6864.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:	$54.8093159474$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.23252.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 2 $ x^4 - x^3 - 6*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.565882$ of defining polynomial
Character	$\chi$	$=$	6864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.24566 q^{5} +2.40254 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.24566 q^{5} +2.40254 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} -2.24566 q^{15} +0.711354 q^{17} +2.40254 q^{19} +2.40254 q^{21} +5.53431 q^{23} +0.0429869 q^{25} +1.00000 q^{27} -4.64820 q^{29} -0.559425 q^{31} -1.00000 q^{33} -5.39529 q^{35} -4.20038 q^{37} -1.00000 q^{39} +6.66607 q^{41} -2.71135 q^{43} -2.24566 q^{45} +5.35956 q^{47} -1.22779 q^{49} +0.711354 q^{51} +9.38238 q^{53} +2.24566 q^{55} +2.40254 q^{57} +9.64590 q^{59} +3.35956 q^{61} +2.40254 q^{63} +2.24566 q^{65} -7.62804 q^{67} +5.53431 q^{69} +14.2004 q^{71} -7.15739 q^{73} +0.0429869 q^{75} -2.40254 q^{77} +12.0435 q^{79} +1.00000 q^{81} -15.5599 q^{83} -1.59746 q^{85} -4.64820 q^{87} +7.11390 q^{89} -2.40254 q^{91} -0.559425 q^{93} -5.39529 q^{95} -3.04579 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^5 - 2 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{5} - 2 q^{7} + 4 q^{9} - 4 q^{11} - 4 q^{13} + 4 q^{15} + 4 q^{17} - 2 q^{19} - 2 q^{21} + 8 q^{23} + 12 q^{25} + 4 q^{27} + 6 q^{29} + 8 q^{31} - 4 q^{33} - 14 q^{35} + 14 q^{37} - 4 q^{39} + 10 q^{41} - 12 q^{43} + 4 q^{45} - 2 q^{47} + 16 q^{49} + 4 q^{51} + 12 q^{53} - 4 q^{55} - 2 q^{57} + 8 q^{59} - 10 q^{61} - 2 q^{63} - 4 q^{65} + 8 q^{67} + 8 q^{69} + 26 q^{71} + 14 q^{73} + 12 q^{75} + 2 q^{77} + 16 q^{79} + 4 q^{81} - 8 q^{83} - 18 q^{85} + 6 q^{87} + 18 q^{89} + 2 q^{91} + 8 q^{93} - 14 q^{95} + 14 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^5 - 2 * q^7 + 4 * q^9 - 4 * q^11 - 4 * q^13 + 4 * q^15 + 4 * q^17 - 2 * q^19 - 2 * q^21 + 8 * q^23 + 12 * q^25 + 4 * q^27 + 6 * q^29 + 8 * q^31 - 4 * q^33 - 14 * q^35 + 14 * q^37 - 4 * q^39 + 10 * q^41 - 12 * q^43 + 4 * q^45 - 2 * q^47 + 16 * q^49 + 4 * q^51 + 12 * q^53 - 4 * q^55 - 2 * q^57 + 8 * q^59 - 10 * q^61 - 2 * q^63 - 4 * q^65 + 8 * q^67 + 8 * q^69 + 26 * q^71 + 14 * q^73 + 12 * q^75 + 2 * q^77 + 16 * q^79 + 4 * q^81 - 8 * q^83 - 18 * q^85 + 6 * q^87 + 18 * q^89 + 2 * q^91 + 8 * q^93 - 14 * q^95 + 14 * 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
$5$	−2.24566	−1.00429	−0.502145	−	0.864784i	$-0.667456\pi$
$5$	−2.24566	−1.00429	−0.502145	+	0.864784i	$0.667456\pi$
$6$	0	0
$7$	2.40254	0.908076	0.454038	−	0.890982i	$-0.349983\pi$
$7$	2.40254	0.908076	0.454038	+	0.890982i	$0.349983\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−2.24566	−0.579827
$16$	0	0
$17$	0.711354	0.172529	0.0862643	−	0.996272i	$-0.472507\pi$
$17$	0.711354	0.172529	0.0862643	+	0.996272i	$0.472507\pi$
$18$	0	0
$19$	2.40254	0.551181	0.275590	−	0.961275i	$-0.411127\pi$
$19$	2.40254	0.551181	0.275590	+	0.961275i	$0.411127\pi$
$20$	0	0
$21$	2.40254	0.524278
$22$	0	0
$23$	5.53431	1.15398	0.576991	−	0.816750i	$-0.304227\pi$
$23$	5.53431	1.15398	0.576991	+	0.816750i	$0.304227\pi$
$24$	0	0
$25$	0.0429869	0.00859738
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.64820	−0.863149	−0.431575	−	0.902077i	$-0.642042\pi$
$29$	−4.64820	−0.863149	−0.431575	+	0.902077i	$0.642042\pi$
$30$	0	0
$31$	−0.559425	−0.100476	−0.0502378	−	0.998737i	$-0.515998\pi$
$31$	−0.559425	−0.100476	−0.0502378	+	0.998737i	$0.515998\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−5.39529	−0.911971
$36$	0	0
$37$	−4.20038	−0.690538	−0.345269	−	0.938504i	$-0.612212\pi$
$37$	−4.20038	−0.690538	−0.345269	+	0.938504i	$0.612212\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	6.66607	1.04107	0.520533	−	0.853842i	$-0.325733\pi$
$41$	6.66607	1.04107	0.520533	+	0.853842i	$0.325733\pi$
$42$	0	0
$43$	−2.71135	−0.413478	−0.206739	−	0.978396i	$-0.566285\pi$
$43$	−2.71135	−0.413478	−0.206739	+	0.978396i	$0.566285\pi$
$44$	0	0
$45$	−2.24566	−0.334763
$46$	0	0
$47$	5.35956	0.781771	0.390886	−	0.920439i	$-0.372169\pi$
$47$	5.35956	0.781771	0.390886	+	0.920439i	$0.372169\pi$
$48$	0	0
$49$	−1.22779	−0.175399
$50$	0	0
$51$	0.711354	0.0996094
$52$	0	0
$53$	9.38238	1.28877	0.644384	−	0.764702i	$-0.277114\pi$
$53$	9.38238	1.28877	0.644384	+	0.764702i	$0.277114\pi$
$54$	0	0
$55$	2.24566	0.302805
$56$	0	0
$57$	2.40254	0.318224
$58$	0	0
$59$	9.64590	1.25579	0.627895	−	0.778298i	$-0.283917\pi$
$59$	9.64590	1.25579	0.627895	+	0.778298i	$0.283917\pi$
$60$	0	0
$61$	3.35956	0.430147	0.215073	−	0.976598i	$-0.431001\pi$
$61$	3.35956	0.430147	0.215073	+	0.976598i	$0.431001\pi$
$62$	0	0
$63$	2.40254	0.302692
$64$	0	0
$65$	2.24566	0.278540
$66$	0	0
$67$	−7.62804	−0.931913	−0.465957	−	0.884808i	$-0.654290\pi$
$67$	−7.62804	−0.931913	−0.465957	+	0.884808i	$0.654290\pi$
$68$	0	0
$69$	5.53431	0.666252
$70$	0	0
$71$	14.2004	1.68527	0.842637	−	0.538482i	$-0.181002\pi$
$71$	14.2004	1.68527	0.842637	+	0.538482i	$0.181002\pi$
$72$	0	0
$73$	−7.15739	−0.837709	−0.418855	−	0.908053i	$-0.637568\pi$
$73$	−7.15739	−0.837709	−0.418855	+	0.908053i	$0.637568\pi$
$74$	0	0
$75$	0.0429869	0.00496370
$76$	0	0
$77$	−2.40254	−0.273795
$78$	0	0
$79$	12.0435	1.35500	0.677499	−	0.735523i	$-0.263064\pi$
$79$	12.0435	1.35500	0.677499	+	0.735523i	$0.263064\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.5599	−1.70792	−0.853962	−	0.520335i	$-0.825807\pi$
$83$	−15.5599	−1.70792	−0.853962	+	0.520335i	$0.825807\pi$
$84$	0	0
$85$	−1.59746	−0.173269
$86$	0	0
$87$	−4.64820	−0.498340
$88$	0	0
$89$	7.11390	0.754071	0.377036	−	0.926199i	$-0.376943\pi$
$89$	7.11390	0.754071	0.377036	+	0.926199i	$0.376943\pi$
$90$	0	0
$91$	−2.40254	−0.251855
$92$	0	0
$93$	−0.559425	−0.0580096
$94$	0	0
$95$	−5.39529	−0.553545
$96$	0	0
$97$	−3.04579	−0.309253	−0.154627	−	0.987973i	$-0.549417\pi$
$97$	−3.04579	−0.309253	−0.154627	+	0.987973i	$0.549417\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	13.2027	1.31372	0.656858	−	0.754015i	$-0.271885\pi$
$101$	13.2027	1.31372	0.656858	+	0.754015i	$0.271885\pi$
$102$	0	0
$103$	−18.9423	−1.86644	−0.933221	−	0.359304i	$-0.883014\pi$
$103$	−18.9423	−1.86644	−0.933221	+	0.359304i	$0.883014\pi$
$104$	0	0
$105$	−5.39529	−0.526527
$106$	0	0
$107$	−10.7548	−1.03971	−0.519855	−	0.854254i	$-0.674014\pi$
$107$	−10.7548	−1.03971	−0.519855	+	0.854254i	$0.674014\pi$
$108$	0	0
$109$	3.28369	0.314521	0.157260	−	0.987557i	$-0.449734\pi$
$109$	3.28369	0.314521	0.157260	+	0.987557i	$0.449734\pi$
$110$	0	0
$111$	−4.20038	−0.398682
$112$	0	0
$113$	1.15918	0.109046	0.0545232	−	0.998513i	$-0.482636\pi$
$113$	1.15918	0.109046	0.0545232	+	0.998513i	$0.482636\pi$
$114$	0	0
$115$	−12.4282	−1.15893
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	1.70906	0.156669
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.66607	0.601059
$124$	0	0
$125$	11.1318	0.995655
$126$	0	0
$127$	−0.106645	−0.00946324	−0.00473162	−	0.999989i	$-0.501506\pi$
$127$	−0.106645	−0.00946324	−0.00473162	+	0.999989i	$0.501506\pi$
$128$	0	0
$129$	−2.71135	−0.238721
$130$	0	0
$131$	5.33214	0.465871	0.232936	−	0.972492i	$-0.425167\pi$
$131$	5.33214	0.465871	0.232936	+	0.972492i	$0.425167\pi$
$132$	0	0
$133$	5.77221	0.500514
$134$	0	0
$135$	−2.24566	−0.193276
$136$	0	0
$137$	22.3601	1.91035	0.955174	−	0.296043i	$-0.0956673\pi$
$137$	22.3601	1.91035	0.955174	+	0.296043i	$0.0956673\pi$
$138$	0	0
$139$	7.48902	0.635211	0.317605	−	0.948223i	$-0.397121\pi$
$139$	7.48902	0.635211	0.317605	+	0.948223i	$0.397121\pi$
$140$	0	0
$141$	5.35956	0.451356
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	10.4383	0.866852
$146$	0	0
$147$	−1.22779	−0.101267
$148$	0	0
$149$	13.7347	1.12519	0.562594	−	0.826733i	$-0.309803\pi$
$149$	13.7347	1.12519	0.562594	+	0.826733i	$0.309803\pi$
$150$	0	0
$151$	0.792374	0.0644825	0.0322412	−	0.999480i	$-0.489736\pi$
$151$	0.792374	0.0644825	0.0322412	+	0.999480i	$0.489736\pi$
$152$	0	0
$153$	0.711354	0.0575095
$154$	0	0
$155$	1.25628	0.100907
$156$	0	0
$157$	3.22054	0.257027	0.128514	−	0.991708i	$-0.458979\pi$
$157$	3.22054	0.257027	0.128514	+	0.991708i	$0.458979\pi$
$158$	0	0
$159$	9.38238	0.744071
$160$	0	0
$161$	13.2964	1.04790
$162$	0	0
$163$	−7.86874	−0.616327	−0.308164	−	0.951333i	$-0.599714\pi$
$163$	−7.86874	−0.616327	−0.308164	+	0.951333i	$0.599714\pi$
$164$	0	0
$165$	2.24566	0.174824
$166$	0	0
$167$	4.17755	0.323269	0.161634	−	0.986851i	$-0.448323\pi$
$167$	4.17755	0.323269	0.161634	+	0.986851i	$0.448323\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.40254	0.183727
$172$	0	0
$173$	3.87053	0.294271	0.147136	−	0.989116i	$-0.452995\pi$
$173$	3.87053	0.294271	0.147136	+	0.989116i	$0.452995\pi$
$174$	0	0
$175$	0.103278	0.00780707
$176$	0	0
$177$	9.64590	0.725031
$178$	0	0
$179$	−3.30651	−0.247141	−0.123570	−	0.992336i	$-0.539434\pi$
$179$	−3.30651	−0.247141	−0.123570	+	0.992336i	$0.539434\pi$
$180$	0	0
$181$	0.779459	0.0579367	0.0289684	−	0.999580i	$-0.490778\pi$
$181$	0.779459	0.0579367	0.0289684	+	0.999580i	$0.490778\pi$
$182$	0	0
$183$	3.35956	0.248345
$184$	0	0
$185$	9.43261	0.693500
$186$	0	0
$187$	−0.711354	−0.0520193
$188$	0	0
$189$	2.40254	0.174759
$190$	0	0
$191$	−12.7145	−0.919990	−0.459995	−	0.887921i	$-0.652149\pi$
$191$	−12.7145	−0.919990	−0.459995	+	0.887921i	$0.652149\pi$
$192$	0	0
$193$	10.0484	0.723303	0.361652	−	0.932313i	$-0.382213\pi$
$193$	10.0484	0.723303	0.361652	+	0.932313i	$0.382213\pi$
$194$	0	0
$195$	2.24566	0.160815
$196$	0	0
$197$	−7.03109	−0.500944	−0.250472	−	0.968124i	$-0.580586\pi$
$197$	−7.03109	−0.500944	−0.250472	+	0.968124i	$0.580586\pi$
$198$	0	0
$199$	3.06861	0.217528	0.108764	−	0.994068i	$-0.465311\pi$
$199$	3.06861	0.217528	0.108764	+	0.994068i	$0.465311\pi$
$200$	0	0
$201$	−7.62804	−0.538040
$202$	0	0
$203$	−11.1675	−0.783805
$204$	0	0
$205$	−14.9697	−1.04553
$206$	0	0
$207$	5.53431	0.384661
$208$	0	0
$209$	−2.40254	−0.166187
$210$	0	0
$211$	21.3756	1.47156	0.735780	−	0.677221i	$-0.236816\pi$
$211$	21.3756	1.47156	0.735780	+	0.677221i	$0.236816\pi$
$212$	0	0
$213$	14.2004	0.972994
$214$	0	0
$215$	6.08878	0.415251
$216$	0	0
$217$	−1.34404	−0.0912395
$218$	0	0
$219$	−7.15739	−0.483652
$220$	0	0
$221$	−0.711354	−0.0478508
$222$	0	0
$223$	20.3326	1.36157	0.680787	−	0.732481i	$-0.261638\pi$
$223$	20.3326	1.36157	0.680787	+	0.732481i	$0.261638\pi$
$224$	0	0
$225$	0.0429869	0.00286579
$226$	0	0
$227$	−0.488516	−0.0324239	−0.0162120	−	0.999869i	$-0.505161\pi$
$227$	−0.488516	−0.0324239	−0.0162120	+	0.999869i	$0.505161\pi$
$228$	0	0
$229$	−18.1243	−1.19769	−0.598844	−	0.800866i	$-0.704373\pi$
$229$	−18.1243	−1.19769	−0.598844	+	0.800866i	$0.704373\pi$
$230$	0	0
$231$	−2.40254	−0.158076
$232$	0	0
$233$	26.5622	1.74015	0.870075	−	0.492920i	$-0.164070\pi$
$233$	26.5622	1.74015	0.870075	+	0.492920i	$0.164070\pi$
$234$	0	0
$235$	−12.0357	−0.785125
$236$	0	0
$237$	12.0435	0.782309
$238$	0	0
$239$	17.9982	1.16421	0.582104	−	0.813115i	$-0.302230\pi$
$239$	17.9982	1.16421	0.582104	+	0.813115i	$0.302230\pi$
$240$	0	0
$241$	5.33673	0.343769	0.171885	−	0.985117i	$-0.445014\pi$
$241$	5.33673	0.343769	0.171885	+	0.985117i	$0.445014\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.75720	0.176151
$246$	0	0
$247$	−2.40254	−0.152870
$248$	0	0
$249$	−15.5599	−0.986071
$250$	0	0
$251$	−21.8235	−1.37748	−0.688742	−	0.725006i	$-0.741837\pi$
$251$	−21.8235	−1.37748	−0.688742	+	0.725006i	$0.741837\pi$
$252$	0	0
$253$	−5.53431	−0.347939
$254$	0	0
$255$	−1.59746	−0.100037
$256$	0	0
$257$	9.87370	0.615904	0.307952	−	0.951402i	$-0.400356\pi$
$257$	9.87370	0.615904	0.307952	+	0.951402i	$0.400356\pi$
$258$	0	0
$259$	−10.0916	−0.627060
$260$	0	0
$261$	−4.64820	−0.287716
$262$	0	0
$263$	−32.3650	−1.99571	−0.997856	−	0.0654451i	$-0.979153\pi$
$263$	−32.3650	−1.99571	−0.997856	+	0.0654451i	$0.979153\pi$
$264$	0	0
$265$	−21.0696	−1.29430
$266$	0	0
$267$	7.11390	0.435363
$268$	0	0
$269$	20.9324	1.27627	0.638136	−	0.769924i	$-0.279706\pi$
$269$	20.9324	1.27627	0.638136	+	0.769924i	$0.279706\pi$
$270$	0	0
$271$	18.7676	1.14005	0.570024	−	0.821628i	$-0.306934\pi$
$271$	18.7676	1.14005	0.570024	+	0.821628i	$0.306934\pi$
$272$	0	0
$273$	−2.40254	−0.145408
$274$	0	0
$275$	−0.0429869	−0.00259221
$276$	0	0
$277$	16.4153	0.986297	0.493148	−	0.869945i	$-0.335846\pi$
$277$	16.4153	0.986297	0.493148	+	0.869945i	$0.335846\pi$
$278$	0	0
$279$	−0.559425	−0.0334919
$280$	0	0
$281$	−4.41245	−0.263225	−0.131612	−	0.991301i	$-0.542015\pi$
$281$	−4.41245	−0.263225	−0.131612	+	0.991301i	$0.542015\pi$
$282$	0	0
$283$	0.734175	0.0436422	0.0218211	−	0.999762i	$-0.493054\pi$
$283$	0.734175	0.0436422	0.0218211	+	0.999762i	$0.493054\pi$
$284$	0	0
$285$	−5.39529	−0.319589
$286$	0	0
$287$	16.0155	0.945366
$288$	0	0
$289$	−16.4940	−0.970234
$290$	0	0
$291$	−3.04579	−0.178547
$292$	0	0
$293$	−0.893861	−0.0522199	−0.0261100	−	0.999659i	$-0.508312\pi$
$293$	−0.893861	−0.0522199	−0.0261100	+	0.999659i	$0.508312\pi$
$294$	0	0
$295$	−21.6614	−1.26118
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−5.53431	−0.320057
$300$	0	0
$301$	−6.51414	−0.375469
$302$	0	0
$303$	13.2027	0.758474
$304$	0	0
$305$	−7.54442	−0.431992
$306$	0	0
$307$	−7.55713	−0.431308	−0.215654	−	0.976470i	$-0.569188\pi$
$307$	−7.55713	−0.431308	−0.215654	+	0.976470i	$0.569188\pi$
$308$	0	0
$309$	−18.9423	−1.07759
$310$	0	0
$311$	−4.42536	−0.250939	−0.125470	−	0.992097i	$-0.540044\pi$
$311$	−4.42536	−0.250939	−0.125470	+	0.992097i	$0.540044\pi$
$312$	0	0
$313$	27.0585	1.52944	0.764718	−	0.644365i	$-0.222878\pi$
$313$	27.0585	1.52944	0.764718	+	0.644365i	$0.222878\pi$
$314$	0	0
$315$	−5.39529	−0.303990
$316$	0	0
$317$	11.6410	0.653821	0.326910	−	0.945055i	$-0.393992\pi$
$317$	11.6410	0.653821	0.326910	+	0.945055i	$0.393992\pi$
$318$	0	0
$319$	4.64820	0.260249
$320$	0	0
$321$	−10.7548	−0.600277
$322$	0	0
$323$	1.70906	0.0950945
$324$	0	0
$325$	−0.0429869	−0.00238448
$326$	0	0
$327$	3.28369	0.181589
$328$	0	0
$329$	12.8766	0.709908
$330$	0	0
$331$	13.2914	0.730564	0.365282	−	0.930897i	$-0.380973\pi$
$331$	13.2914	0.730564	0.365282	+	0.930897i	$0.380973\pi$
$332$	0	0
$333$	−4.20038	−0.230179
$334$	0	0
$335$	17.1300	0.935910
$336$	0	0
$337$	−8.35410	−0.455077	−0.227538	−	0.973769i	$-0.573068\pi$
$337$	−8.35410	−0.455077	−0.227538	+	0.973769i	$0.573068\pi$
$338$	0	0
$339$	1.15918	0.0629580
$340$	0	0
$341$	0.559425	0.0302945
$342$	0	0
$343$	−19.7676	−1.06735
$344$	0	0
$345$	−12.4282	−0.669110
$346$	0	0
$347$	27.7101	1.48755	0.743777	−	0.668428i	$-0.233032\pi$
$347$	27.7101	1.48755	0.743777	+	0.668428i	$0.233032\pi$
$348$	0	0
$349$	4.17296	0.223373	0.111687	−	0.993743i	$-0.464375\pi$
$349$	4.17296	0.223373	0.111687	+	0.993743i	$0.464375\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	5.26302	0.280122	0.140061	−	0.990143i	$-0.455270\pi$
$353$	5.26302	0.280122	0.140061	+	0.990143i	$0.455270\pi$
$354$	0	0
$355$	−31.8892	−1.69250
$356$	0	0
$357$	1.70906	0.0904529
$358$	0	0
$359$	32.4135	1.71072	0.855359	−	0.518036i	$-0.173337\pi$
$359$	32.4135	1.71072	0.855359	+	0.518036i	$0.173337\pi$
$360$	0	0
$361$	−13.2278	−0.696200
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	16.0731	0.841302
$366$	0	0
$367$	13.5196	0.705717	0.352859	−	0.935677i	$-0.385210\pi$
$367$	13.5196	0.705717	0.352859	+	0.935677i	$0.385210\pi$
$368$	0	0
$369$	6.66607	0.347022
$370$	0	0
$371$	22.5416	1.17030
$372$	0	0
$373$	−7.24515	−0.375140	−0.187570	−	0.982251i	$-0.560061\pi$
$373$	−7.24515	−0.375140	−0.187570	+	0.982251i	$0.560061\pi$
$374$	0	0
$375$	11.1318	0.574842
$376$	0	0
$377$	4.64820	0.239395
$378$	0	0
$379$	−10.7955	−0.554529	−0.277265	−	0.960794i	$-0.589428\pi$
$379$	−10.7955	−0.554529	−0.277265	+	0.960794i	$0.589428\pi$
$380$	0	0
$381$	−0.106645	−0.00546360
$382$	0	0
$383$	25.1701	1.28613	0.643066	−	0.765811i	$-0.277662\pi$
$383$	25.1701	1.28613	0.643066	+	0.765811i	$0.277662\pi$
$384$	0	0
$385$	5.39529	0.274970
$386$	0	0
$387$	−2.71135	−0.137826
$388$	0	0
$389$	−12.1273	−0.614879	−0.307440	−	0.951568i	$-0.599472\pi$
$389$	−12.1273	−0.614879	−0.307440	+	0.951568i	$0.599472\pi$
$390$	0	0
$391$	3.93685	0.199095
$392$	0	0
$393$	5.33214	0.268971
$394$	0	0
$395$	−27.0456	−1.36081
$396$	0	0
$397$	17.9368	0.900225	0.450112	−	0.892972i	$-0.351384\pi$
$397$	17.9368	0.900225	0.450112	+	0.892972i	$0.351384\pi$
$398$	0	0
$399$	5.77221	0.288972
$400$	0	0
$401$	−17.2237	−0.860111	−0.430055	−	0.902803i	$-0.641506\pi$
$401$	−17.2237	−0.860111	−0.430055	+	0.902803i	$0.641506\pi$
$402$	0	0
$403$	0.559425	0.0278669
$404$	0	0
$405$	−2.24566	−0.111588
$406$	0	0
$407$	4.20038	0.208205
$408$	0	0
$409$	7.96247	0.393719	0.196859	−	0.980432i	$-0.436926\pi$
$409$	7.96247	0.393719	0.196859	+	0.980432i	$0.436926\pi$
$410$	0	0
$411$	22.3601	1.10294
$412$	0	0
$413$	23.1747	1.14035
$414$	0	0
$415$	34.9423	1.71525
$416$	0	0
$417$	7.48902	0.366739
$418$	0	0
$419$	−28.1628	−1.37585	−0.687923	−	0.725784i	$-0.741477\pi$
$419$	−28.1628	−1.37585	−0.687923	+	0.725784i	$0.741477\pi$
$420$	0	0
$421$	23.0795	1.12483	0.562414	−	0.826856i	$-0.309873\pi$
$421$	23.0795	1.12483	0.562414	+	0.826856i	$0.309873\pi$
$422$	0	0
$423$	5.35956	0.260590
$424$	0	0
$425$	0.0305789	0.00148329
$426$	0	0
$427$	8.07147	0.390606
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	−26.3148	−1.26754	−0.633769	−	0.773522i	$-0.718493\pi$
$431$	−26.3148	−1.26754	−0.633769	+	0.773522i	$0.718493\pi$
$432$	0	0
$433$	14.5271	0.698126	0.349063	−	0.937099i	$-0.386500\pi$
$433$	14.5271	0.698126	0.349063	+	0.937099i	$0.386500\pi$
$434$	0	0
$435$	10.4383	0.500477
$436$	0	0
$437$	13.2964	0.636053
$438$	0	0
$439$	−6.53941	−0.312109	−0.156054	−	0.987748i	$-0.549878\pi$
$439$	−6.53941	−0.312109	−0.156054	+	0.987748i	$0.549878\pi$
$440$	0	0
$441$	−1.22779	−0.0584663
$442$	0	0
$443$	30.9479	1.47038	0.735190	−	0.677861i	$-0.237093\pi$
$443$	30.9479	1.47038	0.735190	+	0.677861i	$0.237093\pi$
$444$	0	0
$445$	−15.9754	−0.757306
$446$	0	0
$447$	13.7347	0.649628
$448$	0	0
$449$	9.80559	0.462754	0.231377	−	0.972864i	$-0.425677\pi$
$449$	9.80559	0.462754	0.231377	+	0.972864i	$0.425677\pi$
$450$	0	0
$451$	−6.66607	−0.313893
$452$	0	0
$453$	0.792374	0.0372290
$454$	0	0
$455$	5.39529	0.252935
$456$	0	0
$457$	−13.4694	−0.630070	−0.315035	−	0.949080i	$-0.602016\pi$
$457$	−13.4694	−0.630070	−0.315035	+	0.949080i	$0.602016\pi$
$458$	0	0
$459$	0.711354	0.0332031
$460$	0	0
$461$	−20.4538	−0.952628	−0.476314	−	0.879275i	$-0.658027\pi$
$461$	−20.4538	−0.952628	−0.476314	+	0.879275i	$0.658027\pi$
$462$	0	0
$463$	7.07357	0.328736	0.164368	−	0.986399i	$-0.447441\pi$
$463$	7.07357	0.328736	0.164368	+	0.986399i	$0.447441\pi$
$464$	0	0
$465$	1.25628	0.0582585
$466$	0	0
$467$	−26.6945	−1.23527	−0.617637	−	0.786463i	$-0.711910\pi$
$467$	−26.6945	−1.23527	−0.617637	+	0.786463i	$0.711910\pi$
$468$	0	0
$469$	−18.3267	−0.846247
$470$	0	0
$471$	3.22054	0.148395
$472$	0	0
$473$	2.71135	0.124668
$474$	0	0
$475$	0.103278	0.00473871
$476$	0	0
$477$	9.38238	0.429590
$478$	0	0
$479$	9.90132	0.452403	0.226201	−	0.974081i	$-0.427369\pi$
$479$	9.90132	0.452403	0.226201	+	0.974081i	$0.427369\pi$
$480$	0	0
$481$	4.20038	0.191521
$482$	0	0
$483$	13.2964	0.605007
$484$	0	0
$485$	6.83981	0.310580
$486$	0	0
$487$	34.1323	1.54668	0.773340	−	0.633991i	$-0.218584\pi$
$487$	34.1323	1.54668	0.773340	+	0.633991i	$0.218584\pi$
$488$	0	0
$489$	−7.86874	−0.355837
$490$	0	0
$491$	−22.4236	−1.01196	−0.505981	−	0.862545i	$-0.668869\pi$
$491$	−22.4236	−1.01196	−0.505981	+	0.862545i	$0.668869\pi$
$492$	0	0
$493$	−3.30651	−0.148918
$494$	0	0
$495$	2.24566	0.100935
$496$	0	0
$497$	34.1170	1.53036
$498$	0	0
$499$	−0.731372	−0.0327407	−0.0163704	−	0.999866i	$-0.505211\pi$
$499$	−0.731372	−0.0327407	−0.0163704	+	0.999866i	$0.505211\pi$
$500$	0	0
$501$	4.17755	0.186639
$502$	0	0
$503$	−1.47396	−0.0657205	−0.0328603	−	0.999460i	$-0.510462\pi$
$503$	−1.47396	−0.0657205	−0.0328603	+	0.999460i	$0.510462\pi$
$504$	0	0
$505$	−29.6487	−1.31935
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−9.85884	−0.436985	−0.218493	−	0.975839i	$-0.570114\pi$
$509$	−9.85884	−0.436985	−0.218493	+	0.975839i	$0.570114\pi$
$510$	0	0
$511$	−17.1959	−0.760703
$512$	0	0
$513$	2.40254	0.106075
$514$	0	0
$515$	42.5380	1.87445
$516$	0	0
$517$	−5.35956	−0.235713
$518$	0	0
$519$	3.87053	0.169898
$520$	0	0
$521$	8.76475	0.383991	0.191995	−	0.981396i	$-0.438504\pi$
$521$	8.76475	0.383991	0.191995	+	0.981396i	$0.438504\pi$
$522$	0	0
$523$	−26.7983	−1.17181	−0.585905	−	0.810380i	$-0.699261\pi$
$523$	−26.7983	−1.17181	−0.585905	+	0.810380i	$0.699261\pi$
$524$	0	0
$525$	0.103278	0.00450741
$526$	0	0
$527$	−0.397949	−0.0173349
$528$	0	0
$529$	7.62854	0.331676
$530$	0	0
$531$	9.64590	0.418597
$532$	0	0
$533$	−6.66607	−0.288740
$534$	0	0
$535$	24.1517	1.04417
$536$	0	0
$537$	−3.30651	−0.142687
$538$	0	0
$539$	1.22779	0.0528847
$540$	0	0
$541$	27.3377	1.17534	0.587671	−	0.809100i	$-0.300045\pi$
$541$	27.3377	1.17534	0.587671	+	0.809100i	$0.300045\pi$
$542$	0	0
$543$	0.779459	0.0334498
$544$	0	0
$545$	−7.37406	−0.315870
$546$	0	0
$547$	6.02067	0.257425	0.128713	−	0.991682i	$-0.458916\pi$
$547$	6.02067	0.257425	0.128713	+	0.991682i	$0.458916\pi$
$548$	0	0
$549$	3.35956	0.143382
$550$	0	0
$551$	−11.1675	−0.475752
$552$	0	0
$553$	28.9350	1.23044
$554$	0	0
$555$	9.43261	0.400392
$556$	0	0
$557$	7.95257	0.336961	0.168481	−	0.985705i	$-0.446114\pi$
$557$	7.95257	0.336961	0.168481	+	0.985705i	$0.446114\pi$
$558$	0	0
$559$	2.71135	0.114678
$560$	0	0
$561$	−0.711354	−0.0300334
$562$	0	0
$563$	−10.0403	−0.423149	−0.211575	−	0.977362i	$-0.567859\pi$
$563$	−10.0403	−0.423149	−0.211575	+	0.977362i	$0.567859\pi$
$564$	0	0
$565$	−2.60312	−0.109514
$566$	0	0
$567$	2.40254	0.100897
$568$	0	0
$569$	−3.60241	−0.151021	−0.0755105	−	0.997145i	$-0.524059\pi$
$569$	−3.60241	−0.151021	−0.0755105	+	0.997145i	$0.524059\pi$
$570$	0	0
$571$	11.5621	0.483858	0.241929	−	0.970294i	$-0.422220\pi$
$571$	11.5621	0.483858	0.241929	+	0.970294i	$0.422220\pi$
$572$	0	0
$573$	−12.7145	−0.531157
$574$	0	0
$575$	0.237903	0.00992123
$576$	0	0
$577$	12.4959	0.520212	0.260106	−	0.965580i	$-0.416243\pi$
$577$	12.4959	0.520212	0.260106	+	0.965580i	$0.416243\pi$
$578$	0	0
$579$	10.0484	0.417599
$580$	0	0
$581$	−37.3834	−1.55092
$582$	0	0
$583$	−9.38238	−0.388578
$584$	0	0
$585$	2.24566	0.0928466
$586$	0	0
$587$	−21.6003	−0.891538	−0.445769	−	0.895148i	$-0.647070\pi$
$587$	−21.6003	−0.891538	−0.445769	+	0.895148i	$0.647070\pi$
$588$	0	0
$589$	−1.34404	−0.0553803
$590$	0	0
$591$	−7.03109	−0.289220
$592$	0	0
$593$	0.134421	0.00552001	0.00276000	−	0.999996i	$-0.499121\pi$
$593$	0.134421	0.00552001	0.00276000	+	0.999996i	$0.499121\pi$
$594$	0	0
$595$	−3.83796	−0.157341
$596$	0	0
$597$	3.06861	0.125590
$598$	0	0
$599$	−29.0696	−1.18775	−0.593876	−	0.804556i	$-0.702403\pi$
$599$	−29.0696	−1.18775	−0.593876	+	0.804556i	$0.702403\pi$
$600$	0	0
$601$	−10.9370	−0.446129	−0.223065	−	0.974804i	$-0.571606\pi$
$601$	−10.9370	−0.446129	−0.223065	+	0.974804i	$0.571606\pi$
$602$	0	0
$603$	−7.62804	−0.310638
$604$	0	0
$605$	−2.24566	−0.0912990
$606$	0	0
$607$	12.4204	0.504129	0.252064	−	0.967710i	$-0.418891\pi$
$607$	12.4204	0.504129	0.252064	+	0.967710i	$0.418891\pi$
$608$	0	0
$609$	−11.1675	−0.452530
$610$	0	0
$611$	−5.35956	−0.216824
$612$	0	0
$613$	−39.0311	−1.57645	−0.788225	−	0.615387i	$-0.789000\pi$
$613$	−39.0311	−1.57645	−0.788225	+	0.615387i	$0.789000\pi$
$614$	0	0
$615$	−14.9697	−0.603638
$616$	0	0
$617$	−40.5346	−1.63186	−0.815931	−	0.578149i	$-0.803775\pi$
$617$	−40.5346	−1.63186	−0.815931	+	0.578149i	$0.803775\pi$
$618$	0	0
$619$	−21.6664	−0.870845	−0.435423	−	0.900226i	$-0.643401\pi$
$619$	−21.6664	−0.870845	−0.435423	+	0.900226i	$0.643401\pi$
$620$	0	0
$621$	5.53431	0.222084
$622$	0	0
$623$	17.0914	0.684754
$624$	0	0
$625$	−25.2131	−1.00852
$626$	0	0
$627$	−2.40254	−0.0959483
$628$	0	0
$629$	−2.98795	−0.119137
$630$	0	0
$631$	49.5401	1.97216	0.986080	−	0.166275i	$-0.0531738\pi$
$631$	49.5401	1.97216	0.986080	+	0.166275i	$0.0531738\pi$
$632$	0	0
$633$	21.3756	0.849605
$634$	0	0
$635$	0.239489	0.00950383
$636$	0	0
$637$	1.22779	0.0486469
$638$	0	0
$639$	14.2004	0.561758
$640$	0	0
$641$	−39.0795	−1.54355	−0.771774	−	0.635896i	$-0.780631\pi$
$641$	−39.0795	−1.54355	−0.771774	+	0.635896i	$0.780631\pi$
$642$	0	0
$643$	35.4013	1.39609	0.698045	−	0.716054i	$-0.254054\pi$
$643$	35.4013	1.39609	0.698045	+	0.716054i	$0.254054\pi$
$644$	0	0
$645$	6.08878	0.239745
$646$	0	0
$647$	−27.8344	−1.09428	−0.547141	−	0.837040i	$-0.684284\pi$
$647$	−27.8344	−1.09428	−0.547141	+	0.837040i	$0.684284\pi$
$648$	0	0
$649$	−9.64590	−0.378635
$650$	0	0
$651$	−1.34404	−0.0526771
$652$	0	0
$653$	32.1372	1.25763	0.628813	−	0.777556i	$-0.283541\pi$
$653$	32.1372	1.25763	0.628813	+	0.777556i	$0.283541\pi$
$654$	0	0
$655$	−11.9742	−0.467870
$656$	0	0
$657$	−7.15739	−0.279236
$658$	0	0
$659$	−19.6231	−0.764407	−0.382203	−	0.924078i	$-0.624835\pi$
$659$	−19.6231	−0.764407	−0.382203	+	0.924078i	$0.624835\pi$
$660$	0	0
$661$	14.7145	0.572328	0.286164	−	0.958181i	$-0.407620\pi$
$661$	14.7145	0.572328	0.286164	+	0.958181i	$0.407620\pi$
$662$	0	0
$663$	−0.711354	−0.0276267
$664$	0	0
$665$	−12.9624	−0.502661
$666$	0	0
$667$	−25.7246	−0.996059
$668$	0	0
$669$	20.3326	0.786106
$670$	0	0
$671$	−3.35956	−0.129694
$672$	0	0
$673$	−19.6185	−0.756237	−0.378119	−	0.925757i	$-0.623429\pi$
$673$	−19.6185	−0.756237	−0.378119	+	0.925757i	$0.623429\pi$
$674$	0	0
$675$	0.0429869	0.00165457
$676$	0	0
$677$	−25.9446	−0.997132	−0.498566	−	0.866852i	$-0.666140\pi$
$677$	−25.9446	−0.997132	−0.498566	+	0.866852i	$0.666140\pi$
$678$	0	0
$679$	−7.31764	−0.280825
$680$	0	0
$681$	−0.488516	−0.0187200
$682$	0	0
$683$	35.3550	1.35282	0.676410	−	0.736525i	$-0.263535\pi$
$683$	35.3550	1.35282	0.676410	+	0.736525i	$0.263535\pi$
$684$	0	0
$685$	−50.2131	−1.91854
$686$	0	0
$687$	−18.1243	−0.691486
$688$	0	0
$689$	−9.38238	−0.357440
$690$	0	0
$691$	−31.2118	−1.18735	−0.593676	−	0.804704i	$-0.702324\pi$
$691$	−31.2118	−1.18735	−0.593676	+	0.804704i	$0.702324\pi$
$692$	0	0
$693$	−2.40254	−0.0912650
$694$	0	0
$695$	−16.8178	−0.637935
$696$	0	0
$697$	4.74193	0.179613
$698$	0	0
$699$	26.5622	1.00468
$700$	0	0
$701$	46.5054	1.75648	0.878242	−	0.478216i	$-0.158716\pi$
$701$	46.5054	1.75648	0.878242	+	0.478216i	$0.158716\pi$
$702$	0	0
$703$	−10.0916	−0.380611
$704$	0	0
$705$	−12.0357	−0.453292
$706$	0	0
$707$	31.7200	1.19295
$708$	0	0
$709$	33.1929	1.24659	0.623293	−	0.781988i	$-0.285794\pi$
$709$	33.1929	1.24659	0.623293	+	0.781988i	$0.285794\pi$
$710$	0	0
$711$	12.0435	0.451666
$712$	0	0
$713$	−3.09603	−0.115947
$714$	0	0
$715$	−2.24566	−0.0839829
$716$	0	0
$717$	17.9982	0.672155
$718$	0	0
$719$	−3.24617	−0.121062	−0.0605308	−	0.998166i	$-0.519279\pi$
$719$	−3.24617	−0.121062	−0.0605308	+	0.998166i	$0.519279\pi$
$720$	0	0
$721$	−45.5097	−1.69487
$722$	0	0
$723$	5.33673	0.198475
$724$	0	0
$725$	−0.199812	−0.00742082
$726$	0	0
$727$	−32.9433	−1.22180	−0.610900	−	0.791708i	$-0.709192\pi$
$727$	−32.9433	−1.22180	−0.610900	+	0.791708i	$0.709192\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.92873	−0.0713367
$732$	0	0
$733$	−19.2579	−0.711305	−0.355653	−	0.934618i	$-0.615741\pi$
$733$	−19.2579	−0.711305	−0.355653	+	0.934618i	$0.615741\pi$
$734$	0	0
$735$	2.75720	0.101701
$736$	0	0
$737$	7.62804	0.280982
$738$	0	0
$739$	16.7980	0.617924	0.308962	−	0.951074i	$-0.400018\pi$
$739$	16.7980	0.617924	0.308962	+	0.951074i	$0.400018\pi$
$740$	0	0
$741$	−2.40254	−0.0882596
$742$	0	0
$743$	−30.6285	−1.12365	−0.561826	−	0.827255i	$-0.689901\pi$
$743$	−30.6285	−1.12365	−0.561826	+	0.827255i	$0.689901\pi$
$744$	0	0
$745$	−30.8434	−1.13002
$746$	0	0
$747$	−15.5599	−0.569308
$748$	0	0
$749$	−25.8390	−0.944136
$750$	0	0
$751$	27.9643	1.02043	0.510215	−	0.860047i	$-0.329566\pi$
$751$	27.9643	1.02043	0.510215	+	0.860047i	$0.329566\pi$
$752$	0	0
$753$	−21.8235	−0.795291
$754$	0	0
$755$	−1.77940	−0.0647590
$756$	0	0
$757$	7.62129	0.277001	0.138500	−	0.990362i	$-0.455772\pi$
$757$	7.62129	0.277001	0.138500	+	0.990362i	$0.455772\pi$
$758$	0	0
$759$	−5.53431	−0.200883
$760$	0	0
$761$	19.6844	0.713561	0.356780	−	0.934188i	$-0.383874\pi$
$761$	19.6844	0.713561	0.356780	+	0.934188i	$0.383874\pi$
$762$	0	0
$763$	7.88921	0.285609
$764$	0	0
$765$	−1.59746	−0.0577562
$766$	0	0
$767$	−9.64590	−0.348293
$768$	0	0
$769$	−20.8196	−0.750773	−0.375387	−	0.926868i	$-0.622490\pi$
$769$	−20.8196	−0.750773	−0.375387	+	0.926868i	$0.622490\pi$
$770$	0	0
$771$	9.87370	0.355592
$772$	0	0
$773$	−29.8429	−1.07337	−0.536687	−	0.843781i	$-0.680325\pi$
$773$	−29.8429	−1.07337	−0.536687	+	0.843781i	$0.680325\pi$
$774$	0	0
$775$	−0.0240479	−0.000863827 0
$776$	0	0
$777$	−10.0916	−0.362033
$778$	0	0
$779$	16.0155	0.573815
$780$	0	0
$781$	−14.2004	−0.508129
$782$	0	0
$783$	−4.64820	−0.166113
$784$	0	0
$785$	−7.23224	−0.258130
$786$	0	0
$787$	1.76941	0.0630725	0.0315362	−	0.999503i	$-0.489960\pi$
$787$	1.76941	0.0630725	0.0315362	+	0.999503i	$0.489960\pi$
$788$	0	0
$789$	−32.3650	−1.15223
$790$	0	0
$791$	2.78498	0.0990224
$792$	0	0
$793$	−3.35956	−0.119301
$794$	0	0
$795$	−21.0696	−0.747263
$796$	0	0
$797$	5.94875	0.210716	0.105358	−	0.994434i	$-0.466401\pi$
$797$	5.94875	0.210716	0.105358	+	0.994434i	$0.466401\pi$
$798$	0	0
$799$	3.81254	0.134878
$800$	0	0
$801$	7.11390	0.251357
$802$	0	0
$803$	7.15739	0.252579
$804$	0	0
$805$	−29.8592	−1.05240
$806$	0	0
$807$	20.9324	0.736855
$808$	0	0
$809$	−40.9858	−1.44098	−0.720492	−	0.693463i	$-0.756084\pi$
$809$	−40.9858	−1.44098	−0.720492	+	0.693463i	$0.756084\pi$
$810$	0	0
$811$	10.8545	0.381155	0.190577	−	0.981672i	$-0.438964\pi$
$811$	10.8545	0.381155	0.190577	+	0.981672i	$0.438964\pi$
$812$	0	0
$813$	18.7676	0.658207
$814$	0	0
$815$	17.6705	0.618971
$816$	0	0
$817$	−6.51414	−0.227901
$818$	0	0
$819$	−2.40254	−0.0839516
$820$	0	0
$821$	18.8277	0.657091	0.328546	−	0.944488i	$-0.393442\pi$
$821$	18.8277	0.657091	0.328546	+	0.944488i	$0.393442\pi$
$822$	0	0
$823$	8.35941	0.291391	0.145695	−	0.989330i	$-0.453458\pi$
$823$	8.35941	0.291391	0.145695	+	0.989330i	$0.453458\pi$
$824$	0	0
$825$	−0.0429869	−0.00149661
$826$	0	0
$827$	34.6770	1.20584	0.602919	−	0.797803i	$-0.294004\pi$
$827$	34.6770	1.20584	0.602919	+	0.797803i	$0.294004\pi$
$828$	0	0
$829$	45.4750	1.57941	0.789706	−	0.613486i	$-0.210233\pi$
$829$	45.4750	1.57941	0.789706	+	0.613486i	$0.210233\pi$
$830$	0	0
$831$	16.4153	0.569439
$832$	0	0
$833$	−0.873394	−0.0302613
$834$	0	0
$835$	−9.38136	−0.324655
$836$	0	0
$837$	−0.559425	−0.0193365
$838$	0	0
$839$	32.1776	1.11089	0.555446	−	0.831552i	$-0.312547\pi$
$839$	32.1776	1.11089	0.555446	+	0.831552i	$0.312547\pi$
$840$	0	0
$841$	−7.39422	−0.254973
$842$	0	0
$843$	−4.41245	−0.151973
$844$	0	0
$845$	−2.24566	−0.0772530
$846$	0	0
$847$	2.40254	0.0825523
$848$	0	0
$849$	0.734175	0.0251968
$850$	0	0
$851$	−23.2462	−0.796868
$852$	0	0
$853$	26.0969	0.893541	0.446770	−	0.894649i	$-0.352574\pi$
$853$	26.0969	0.893541	0.446770	+	0.894649i	$0.352574\pi$
$854$	0	0
$855$	−5.39529	−0.184515
$856$	0	0
$857$	−51.1862	−1.74849	−0.874243	−	0.485488i	$-0.838642\pi$
$857$	−51.1862	−1.74849	−0.874243	+	0.485488i	$0.838642\pi$
$858$	0	0
$859$	−29.7520	−1.01512	−0.507562	−	0.861615i	$-0.669453\pi$
$859$	−29.7520	−1.01512	−0.507562	+	0.861615i	$0.669453\pi$
$860$	0	0
$861$	16.0155	0.545807
$862$	0	0
$863$	−24.1326	−0.821484	−0.410742	−	0.911752i	$-0.634730\pi$
$863$	−24.1326	−0.821484	−0.410742	+	0.911752i	$0.634730\pi$
$864$	0	0
$865$	−8.69190	−0.295533
$866$	0	0
$867$	−16.4940	−0.560165
$868$	0	0
$869$	−12.0435	−0.408548
$870$	0	0
$871$	7.62804	0.258466
$872$	0	0
$873$	−3.04579	−0.103084
$874$	0	0
$875$	26.7445	0.904130
$876$	0	0
$877$	−40.5829	−1.37039	−0.685194	−	0.728361i	$-0.740282\pi$
$877$	−40.5829	−1.37039	−0.685194	+	0.728361i	$0.740282\pi$
$878$	0	0
$879$	−0.893861	−0.0301492
$880$	0	0
$881$	16.6735	0.561743	0.280872	−	0.959745i	$-0.409376\pi$
$881$	16.6735	0.561743	0.280872	+	0.959745i	$0.409376\pi$
$882$	0	0
$883$	−8.30386	−0.279447	−0.139724	−	0.990191i	$-0.544621\pi$
$883$	−8.30386	−0.279447	−0.139724	+	0.990191i	$0.544621\pi$
$884$	0	0
$885$	−21.6614	−0.728141
$886$	0	0
$887$	−3.57744	−0.120119	−0.0600593	−	0.998195i	$-0.519129\pi$
$887$	−3.57744	−0.120119	−0.0600593	+	0.998195i	$0.519129\pi$
$888$	0	0
$889$	−0.256220	−0.00859333
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	12.8766	0.430898
$894$	0	0
$895$	7.42531	0.248201
$896$	0	0
$897$	−5.53431	−0.184785
$898$	0	0
$899$	2.60032	0.0867255
$900$	0	0
$901$	6.67419	0.222349
$902$	0	0
$903$	−6.51414	−0.216777
$904$	0	0
$905$	−1.75040	−0.0581853
$906$	0	0
$907$	−52.3650	−1.73875	−0.869376	−	0.494151i	$-0.835479\pi$
$907$	−52.3650	−1.73875	−0.869376	+	0.494151i	$0.835479\pi$
$908$	0	0
$909$	13.2027	0.437905
$910$	0	0
$911$	−7.70605	−0.255313	−0.127656	−	0.991818i	$-0.540745\pi$
$911$	−7.70605	−0.255313	−0.127656	+	0.991818i	$0.540745\pi$
$912$	0	0
$913$	15.5599	0.514959
$914$	0	0
$915$	−7.54442	−0.249411
$916$	0	0
$917$	12.8107	0.423046
$918$	0	0
$919$	3.91918	0.129282	0.0646410	−	0.997909i	$-0.479410\pi$
$919$	3.91918	0.129282	0.0646410	+	0.997909i	$0.479410\pi$
$920$	0	0
$921$	−7.55713	−0.249016
$922$	0	0
$923$	−14.2004	−0.467411
$924$	0	0
$925$	−0.180561	−0.00593681
$926$	0	0
$927$	−18.9423	−0.622147
$928$	0	0
$929$	3.86415	0.126779	0.0633893	−	0.997989i	$-0.479809\pi$
$929$	3.86415	0.126779	0.0633893	+	0.997989i	$0.479809\pi$
$930$	0	0
$931$	−2.94982	−0.0966765
$932$	0	0
$933$	−4.42536	−0.144880
$934$	0	0
$935$	1.59746	0.0522425
$936$	0	0
$937$	45.4062	1.48336	0.741678	−	0.670756i	$-0.234030\pi$
$937$	45.4062	1.48336	0.741678	+	0.670756i	$0.234030\pi$
$938$	0	0
$939$	27.0585	0.883021
$940$	0	0
$941$	7.12165	0.232159	0.116080	−	0.993240i	$-0.462967\pi$
$941$	7.12165	0.232159	0.116080	+	0.993240i	$0.462967\pi$
$942$	0	0
$943$	36.8921	1.20137
$944$	0	0
$945$	−5.39529	−0.175509
$946$	0	0
$947$	−25.2297	−0.819854	−0.409927	−	0.912118i	$-0.634446\pi$
$947$	−25.2297	−0.819854	−0.409927	+	0.912118i	$0.634446\pi$
$948$	0	0
$949$	7.15739	0.232339
$950$	0	0
$951$	11.6410	0.377484
$952$	0	0
$953$	16.0564	0.520118	0.260059	−	0.965593i	$-0.416258\pi$
$953$	16.0564	0.520118	0.260059	+	0.965593i	$0.416258\pi$
$954$	0	0
$955$	28.5525	0.923936
$956$	0	0
$957$	4.64820	0.150255
$958$	0	0
$959$	53.7210	1.73474
$960$	0	0
$961$	−30.6870	−0.989905
$962$	0	0
$963$	−10.7548	−0.346570
$964$	0	0
$965$	−22.5654	−0.726406
$966$	0	0
$967$	3.82167	0.122897	0.0614483	−	0.998110i	$-0.480428\pi$
$967$	3.82167	0.122897	0.0614483	+	0.998110i	$0.480428\pi$
$968$	0	0
$969$	1.70906	0.0549028
$970$	0	0
$971$	20.4310	0.655661	0.327831	−	0.944737i	$-0.393683\pi$
$971$	20.4310	0.655661	0.327831	+	0.944737i	$0.393683\pi$
$972$	0	0
$973$	17.9927	0.576819
$974$	0	0
$975$	−0.0429869	−0.00137668
$976$	0	0
$977$	−7.38834	−0.236374	−0.118187	−	0.992991i	$-0.537708\pi$
$977$	−7.38834	−0.236374	−0.118187	+	0.992991i	$0.537708\pi$
$978$	0	0
$979$	−7.11390	−0.227361
$980$	0	0
$981$	3.28369	0.104840
$982$	0	0
$983$	−34.8418	−1.11128	−0.555641	−	0.831422i	$-0.687527\pi$
$983$	−34.8418	−1.11128	−0.555641	+	0.831422i	$0.687527\pi$
$984$	0	0
$985$	15.7894	0.503093
$986$	0	0
$987$	12.8766	0.409865
$988$	0	0
$989$	−15.0055	−0.477146
$990$	0	0
$991$	17.1188	0.543798	0.271899	−	0.962326i	$-0.412348\pi$
$991$	17.1188	0.543798	0.271899	+	0.962326i	$0.412348\pi$
$992$	0	0
$993$	13.2914	0.421791
$994$	0	0
$995$	−6.89106	−0.218461
$996$	0	0
$997$	−58.1382	−1.84126	−0.920628	−	0.390440i	$-0.872323\pi$
$997$	−58.1382	−1.84126	−0.920628	+	0.390440i	$0.872323\pi$
$998$	0	0
$999$	−4.20038	−0.132894

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.cc.1.1		4
4.3	odd	2		3432.2.a.t.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.t.1.1	✓	4	4.3	odd	2
6864.2.a.cc.1.1		4	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:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.24566 q^{5} +2.40254 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.24566 q^{5} +2.40254 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} -2.24566 q^{15} +0.711354 q^{17} +2.40254 q^{19} +2.40254 q^{21} +5.53431 q^{23} +0.0429869 q^{25} +1.00000 q^{27} -4.64820 q^{29} -0.559425 q^{31} -1.00000 q^{33} -5.39529 q^{35} -4.20038 q^{37} -1.00000 q^{39} +6.66607 q^{41} -2.71135 q^{43} -2.24566 q^{45} +5.35956 q^{47} -1.22779 q^{49} +0.711354 q^{51} +9.38238 q^{53} +2.24566 q^{55} +2.40254 q^{57} +9.64590 q^{59} +3.35956 q^{61} +2.40254 q^{63} +2.24566 q^{65} -7.62804 q^{67} +5.53431 q^{69} +14.2004 q^{71} -7.15739 q^{73} +0.0429869 q^{75} -2.40254 q^{77} +12.0435 q^{79} +1.00000 q^{81} -15.5599 q^{83} -1.59746 q^{85} -4.64820 q^{87} +7.11390 q^{89} -2.40254 q^{91} -0.559425 q^{93} -5.39529 q^{95} -3.04579 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^5 - 2 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{5} - 2 q^{7} + 4 q^{9} - 4 q^{11} - 4 q^{13} + 4 q^{15} + 4 q^{17} - 2 q^{19} - 2 q^{21} + 8 q^{23} + 12 q^{25} + 4 q^{27} + 6 q^{29} + 8 q^{31} - 4 q^{33} - 14 q^{35} + 14 q^{37} - 4 q^{39} + 10 q^{41} - 12 q^{43} + 4 q^{45} - 2 q^{47} + 16 q^{49} + 4 q^{51} + 12 q^{53} - 4 q^{55} - 2 q^{57} + 8 q^{59} - 10 q^{61} - 2 q^{63} - 4 q^{65} + 8 q^{67} + 8 q^{69} + 26 q^{71} + 14 q^{73} + 12 q^{75} + 2 q^{77} + 16 q^{79} + 4 q^{81} - 8 q^{83} - 18 q^{85} + 6 q^{87} + 18 q^{89} + 2 q^{91} + 8 q^{93} - 14 q^{95} + 14 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^5 - 2 * q^7 + 4 * q^9 - 4 * q^11 - 4 * q^13 + 4 * q^15 + 4 * q^17 - 2 * q^19 - 2 * q^21 + 8 * q^23 + 12 * q^25 + 4 * q^27 + 6 * q^29 + 8 * q^31 - 4 * q^33 - 14 * q^35 + 14 * q^37 - 4 * q^39 + 10 * q^41 - 12 * q^43 + 4 * q^45 - 2 * q^47 + 16 * q^49 + 4 * q^51 + 12 * q^53 - 4 * q^55 - 2 * q^57 + 8 * q^59 - 10 * q^61 - 2 * q^63 - 4 * q^65 + 8 * q^67 + 8 * q^69 + 26 * q^71 + 14 * q^73 + 12 * q^75 + 2 * q^77 + 16 * q^79 + 4 * q^81 - 8 * q^83 - 18 * q^85 + 6 * q^87 + 18 * q^89 + 2 * q^91 + 8 * q^93 - 14 * q^95 + 14 * q^97 - 4 * q^99

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