LMFDB - Embedded newform 6864.2.a.cb.1.4

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:	4.4.90996.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 11x^{2} - 3x + 2 $ x^4 - x^3 - 11x^2 - 3x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1716)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.63415$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.22388 q^{5} +0.874887 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.22388 q^{5} +0.874887 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +3.22388 q^{15} -1.16953 q^{17} +0.874887 q^{19} +0.874887 q^{21} -8.84117 q^{23} +5.39341 q^{25} +1.00000 q^{27} +1.65101 q^{29} +1.22388 q^{31} -1.00000 q^{33} +2.82053 q^{35} -0.820532 q^{37} +1.00000 q^{39} +11.6617 q^{41} -4.91930 q^{43} +3.22388 q^{45} +12.4140 q^{47} -6.23457 q^{49} -1.16953 q^{51} +4.44776 q^{53} -3.22388 q^{55} +0.874887 q^{57} +6.44776 q^{59} +11.2683 q^{61} +0.874887 q^{63} +3.22388 q^{65} +10.0107 q^{67} -8.84117 q^{69} +7.71605 q^{71} +5.57287 q^{73} +5.39341 q^{75} -0.874887 q^{77} -2.47154 q^{79} +1.00000 q^{81} -16.9843 q^{83} -3.77041 q^{85} +1.65101 q^{87} +14.0444 q^{89} +0.874887 q^{91} +1.22388 q^{93} +2.82053 q^{95} +10.2165 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 2 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 2 * q^5 - 2 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 2 q^{5} - 2 q^{7} + 4 q^{9} - 4 q^{11} + 4 q^{13} + 2 q^{15} + 2 q^{17} - 2 q^{19} - 2 q^{21} + 4 q^{23} + 4 q^{25} + 4 q^{27} + 12 q^{29} - 6 q^{31} - 4 q^{33} + 10 q^{35} - 2 q^{37} + 4 q^{39} + 6 q^{41} - 2 q^{43} + 2 q^{45} - 6 q^{47} + 32 q^{49} + 2 q^{51} - 4 q^{53} - 2 q^{55} - 2 q^{57} + 4 q^{59} + 22 q^{61} - 2 q^{63} + 2 q^{65} - 6 q^{67} + 4 q^{69} - 14 q^{71} + 6 q^{73} + 4 q^{75} + 2 q^{77} - 14 q^{79} + 4 q^{81} + 34 q^{85} + 12 q^{87} + 44 q^{89} - 2 q^{91} - 6 q^{93} + 10 q^{95} + 18 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 2 * q^5 - 2 * q^7 + 4 * q^9 - 4 * q^11 + 4 * q^13 + 2 * q^15 + 2 * q^17 - 2 * q^19 - 2 * q^21 + 4 * q^23 + 4 * q^25 + 4 * q^27 + 12 * q^29 - 6 * q^31 - 4 * q^33 + 10 * q^35 - 2 * q^37 + 4 * q^39 + 6 * q^41 - 2 * q^43 + 2 * q^45 - 6 * q^47 + 32 * q^49 + 2 * q^51 - 4 * q^53 - 2 * q^55 - 2 * q^57 + 4 * q^59 + 22 * q^61 - 2 * q^63 + 2 * q^65 - 6 * q^67 + 4 * q^69 - 14 * q^71 + 6 * q^73 + 4 * q^75 + 2 * q^77 - 14 * q^79 + 4 * q^81 + 34 * q^85 + 12 * q^87 + 44 * q^89 - 2 * q^91 - 6 * q^93 + 10 * q^95 + 18 * 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
$5$	3.22388	1.44176	0.720882	−	0.693058i	$-0.243737\pi$
$5$	3.22388	1.44176	0.720882	+	0.693058i	$0.243737\pi$
$6$	0	0
$7$	0.874887	0.330676	0.165338	−	0.986237i	$-0.447128\pi$
$7$	0.874887	0.330676	0.165338	+	0.986237i	$0.447128\pi$
$8$	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
$15$	3.22388	0.832402
$16$	0	0
$17$	−1.16953	−0.283652	−0.141826	−	0.989892i	$-0.545297\pi$
$17$	−1.16953	−0.283652	−0.141826	+	0.989892i	$0.545297\pi$
$18$	0	0
$19$	0.874887	0.200713	0.100356	−	0.994952i	$-0.468002\pi$
$19$	0.874887	0.200713	0.100356	+	0.994952i	$0.468002\pi$
$20$	0	0
$21$	0.874887	0.190916
$22$	0	0
$23$	−8.84117	−1.84351	−0.921755	−	0.387772i	$-0.873245\pi$
$23$	−8.84117	−1.84351	−0.921755	+	0.387772i	$0.873245\pi$
$24$	0	0
$25$	5.39341	1.07868
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.65101	0.306584	0.153292	−	0.988181i	$-0.451012\pi$
$29$	1.65101	0.306584	0.153292	+	0.988181i	$0.451012\pi$
$30$	0	0
$31$	1.22388	0.219815	0.109908	−	0.993942i	$-0.464944\pi$
$31$	1.22388	0.219815	0.109908	+	0.993942i	$0.464944\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	2.82053	0.476757
$36$	0	0
$37$	−0.820532	−0.134895	−0.0674473	−	0.997723i	$-0.521485\pi$
$37$	−0.820532	−0.134895	−0.0674473	+	0.997723i	$0.521485\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	11.6617	1.82125	0.910626	−	0.413232i	$-0.135600\pi$
$41$	11.6617	1.82125	0.910626	+	0.413232i	$0.135600\pi$
$42$	0	0
$43$	−4.91930	−0.750186	−0.375093	−	0.926987i	$-0.622389\pi$
$43$	−4.91930	−0.750186	−0.375093	+	0.926987i	$0.622389\pi$
$44$	0	0
$45$	3.22388	0.480588
$46$	0	0
$47$	12.4140	1.81077	0.905387	−	0.424587i	$-0.139581\pi$
$47$	12.4140	1.81077	0.905387	+	0.424587i	$0.139581\pi$
$48$	0	0
$49$	−6.23457	−0.890653
$50$	0	0
$51$	−1.16953	−0.163766
$52$	0	0
$53$	4.44776	0.610947	0.305473	−	0.952201i	$-0.401185\pi$
$53$	4.44776	0.610947	0.305473	+	0.952201i	$0.401185\pi$
$54$	0	0
$55$	−3.22388	−0.434708
$56$	0	0
$57$	0.874887	0.115882
$58$	0	0
$59$	6.44776	0.839427	0.419713	−	0.907657i	$-0.362131\pi$
$59$	6.44776	0.839427	0.419713	+	0.907657i	$0.362131\pi$
$60$	0	0
$61$	11.2683	1.44276	0.721379	−	0.692541i	$-0.243509\pi$
$61$	11.2683	1.44276	0.721379	+	0.692541i	$0.243509\pi$
$62$	0	0
$63$	0.874887	0.110225
$64$	0	0
$65$	3.22388	0.399873
$66$	0	0
$67$	10.0107	1.22300	0.611500	−	0.791244i	$-0.290566\pi$
$67$	10.0107	1.22300	0.611500	+	0.791244i	$0.290566\pi$
$68$	0	0
$69$	−8.84117	−1.06435
$70$	0	0
$71$	7.71605	0.915727	0.457864	−	0.889022i	$-0.348615\pi$
$71$	7.71605	0.915727	0.457864	+	0.889022i	$0.348615\pi$
$72$	0	0
$73$	5.57287	0.652256	0.326128	−	0.945326i	$-0.394256\pi$
$73$	5.57287	0.652256	0.326128	+	0.945326i	$0.394256\pi$
$74$	0	0
$75$	5.39341	0.622777
$76$	0	0
$77$	−0.874887	−0.0997026
$78$	0	0
$79$	−2.47154	−0.278070	−0.139035	−	0.990287i	$-0.544400\pi$
$79$	−2.47154	−0.278070	−0.139035	+	0.990287i	$0.544400\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−16.9843	−1.86427	−0.932137	−	0.362105i	$-0.882058\pi$
$83$	−16.9843	−1.86427	−0.932137	+	0.362105i	$0.882058\pi$
$84$	0	0
$85$	−3.77041	−0.408958
$86$	0	0
$87$	1.65101	0.177006
$88$	0	0
$89$	14.0444	1.48870	0.744352	−	0.667787i	$-0.232758\pi$
$89$	14.0444	1.48870	0.744352	+	0.667787i	$0.232758\pi$
$90$	0	0
$91$	0.874887	0.0917131
$92$	0	0
$93$	1.22388	0.126910
$94$	0	0
$95$	2.82053	0.289380
$96$	0	0
$97$	10.2165	1.03733	0.518664	−	0.854978i	$-0.326429\pi$
$97$	10.2165	1.03733	0.518664	+	0.854978i	$0.326429\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	10.9193	1.08651	0.543255	−	0.839567i	$-0.317192\pi$
$101$	10.9193	1.08651	0.543255	+	0.839567i	$0.317192\pi$
$102$	0	0
$103$	7.14575	0.704091	0.352046	−	0.935983i	$-0.385486\pi$
$103$	7.14575	0.704091	0.352046	+	0.935983i	$0.385486\pi$
$104$	0	0
$105$	2.82053	0.275256
$106$	0	0
$107$	2.44776	0.236634	0.118317	−	0.992976i	$-0.462250\pi$
$107$	2.44776	0.236634	0.118317	+	0.992976i	$0.462250\pi$
$108$	0	0
$109$	11.3226	1.08451	0.542257	−	0.840213i	$-0.317570\pi$
$109$	11.3226	1.08451	0.542257	+	0.840213i	$0.317570\pi$
$110$	0	0
$111$	−0.820532	−0.0778814
$112$	0	0
$113$	−6.19754	−0.583015	−0.291508	−	0.956569i	$-0.594157\pi$
$113$	−6.19754	−0.583015	−0.291508	+	0.956569i	$0.594157\pi$
$114$	0	0
$115$	−28.5029	−2.65791
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−1.02320	−0.0937968
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	11.6617	1.05150
$124$	0	0
$125$	1.26829	0.113440
$126$	0	0
$127$	−16.5465	−1.46827	−0.734134	−	0.679005i	$-0.762411\pi$
$127$	−16.5465	−1.46827	−0.734134	+	0.679005i	$0.762411\pi$
$128$	0	0
$129$	−4.91930	−0.433120
$130$	0	0
$131$	16.7868	1.46667	0.733335	−	0.679867i	$-0.237963\pi$
$131$	16.7868	1.46667	0.733335	+	0.679867i	$0.237963\pi$
$132$	0	0
$133$	0.765428	0.0663710
$134$	0	0
$135$	3.22388	0.277467
$136$	0	0
$137$	−1.45514	−0.124321	−0.0621603	−	0.998066i	$-0.519799\pi$
$137$	−1.45514	−0.124321	−0.0621603	+	0.998066i	$0.519799\pi$
$138$	0	0
$139$	−6.79675	−0.576493	−0.288247	−	0.957556i	$-0.593072\pi$
$139$	−6.79675	−0.576493	−0.288247	+	0.957556i	$0.593072\pi$
$140$	0	0
$141$	12.4140	1.04545
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	5.32265	0.442022
$146$	0	0
$147$	−6.23457	−0.514219
$148$	0	0
$149$	−3.66170	−0.299978	−0.149989	−	0.988688i	$-0.547924\pi$
$149$	−3.66170	−0.299978	−0.149989	+	0.988688i	$0.547924\pi$
$150$	0	0
$151$	2.51595	0.204745	0.102373	−	0.994746i	$-0.467357\pi$
$151$	2.51595	0.204745	0.102373	+	0.994746i	$0.467357\pi$
$152$	0	0
$153$	−1.16953	−0.0945505
$154$	0	0
$155$	3.94564	0.316922
$156$	0	0
$157$	−1.35637	−0.108250	−0.0541250	−	0.998534i	$-0.517237\pi$
$157$	−1.35637	−0.108250	−0.0541250	+	0.998534i	$0.517237\pi$
$158$	0	0
$159$	4.44776	0.352730
$160$	0	0
$161$	−7.73502	−0.609605
$162$	0	0
$163$	13.6379	1.06820	0.534102	−	0.845420i	$-0.320650\pi$
$163$	13.6379	1.06820	0.534102	+	0.845420i	$0.320650\pi$
$164$	0	0
$165$	−3.22388	−0.250979
$166$	0	0
$167$	−6.25023	−0.483657	−0.241828	−	0.970319i	$-0.577747\pi$
$167$	−6.25023	−0.483657	−0.241828	+	0.970319i	$0.577747\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.874887	0.0669043
$172$	0	0
$173$	−14.7630	−1.12241	−0.561206	−	0.827676i	$-0.689663\pi$
$173$	−14.7630	−1.12241	−0.561206	+	0.827676i	$0.689663\pi$
$174$	0	0
$175$	4.71862	0.356694
$176$	0	0
$177$	6.44776	0.484643
$178$	0	0
$179$	−21.8255	−1.63132	−0.815658	−	0.578535i	$-0.803625\pi$
$179$	−21.8255	−1.63132	−0.815658	+	0.578535i	$0.803625\pi$
$180$	0	0
$181$	5.35637	0.398136	0.199068	−	0.979986i	$-0.436209\pi$
$181$	5.35637	0.398136	0.199068	+	0.979986i	$0.436209\pi$
$182$	0	0
$183$	11.2683	0.832976
$184$	0	0
$185$	−2.64530	−0.194486
$186$	0	0
$187$	1.16953	0.0855241
$188$	0	0
$189$	0.874887	0.0636387
$190$	0	0
$191$	22.4478	1.62426	0.812131	−	0.583474i	$-0.198307\pi$
$191$	22.4478	1.62426	0.812131	+	0.583474i	$0.198307\pi$
$192$	0	0
$193$	9.68158	0.696896	0.348448	−	0.937328i	$-0.386709\pi$
$193$	9.68158	0.696896	0.348448	+	0.937328i	$0.386709\pi$
$194$	0	0
$195$	3.22388	0.230867
$196$	0	0
$197$	−11.5729	−0.824533	−0.412267	−	0.911063i	$-0.635263\pi$
$197$	−11.5729	−0.824533	−0.412267	+	0.911063i	$0.635263\pi$
$198$	0	0
$199$	−20.2864	−1.43806	−0.719031	−	0.694978i	$-0.755414\pi$
$199$	−20.2864	−1.43806	−0.719031	+	0.694978i	$0.755414\pi$
$200$	0	0
$201$	10.0107	0.706100
$202$	0	0
$203$	1.44444	0.101380
$204$	0	0
$205$	37.5959	2.62581
$206$	0	0
$207$	−8.84117	−0.614504
$208$	0	0
$209$	−0.874887	−0.0605172
$210$	0	0
$211$	−24.1539	−1.66282	−0.831411	−	0.555659i	$-0.812466\pi$
$211$	−24.1539	−1.66282	−0.831411	+	0.555659i	$0.812466\pi$
$212$	0	0
$213$	7.71605	0.528695
$214$	0	0
$215$	−15.8592	−1.08159
$216$	0	0
$217$	1.07076	0.0726877
$218$	0	0
$219$	5.57287	0.376580
$220$	0	0
$221$	−1.16953	−0.0786708
$222$	0	0
$223$	−0.723429	−0.0484444	−0.0242222	−	0.999707i	$-0.507711\pi$
$223$	−0.723429	−0.0484444	−0.0242222	+	0.999707i	$0.507711\pi$
$224$	0	0
$225$	5.39341	0.359560
$226$	0	0
$227$	−17.3440	−1.15116	−0.575582	−	0.817744i	$-0.695224\pi$
$227$	−17.3440	−1.15116	−0.575582	+	0.817744i	$0.695224\pi$
$228$	0	0
$229$	12.0551	0.796624	0.398312	−	0.917250i	$-0.369596\pi$
$229$	12.0551	0.796624	0.398312	+	0.917250i	$0.369596\pi$
$230$	0	0
$231$	−0.874887	−0.0575634
$232$	0	0
$233$	−11.3333	−0.742472	−0.371236	−	0.928539i	$-0.621066\pi$
$233$	−11.3333	−0.742472	−0.371236	+	0.928539i	$0.621066\pi$
$234$	0	0
$235$	40.0214	2.61071
$236$	0	0
$237$	−2.47154	−0.160544
$238$	0	0
$239$	16.4485	1.06397	0.531983	−	0.846755i	$-0.321447\pi$
$239$	16.4485	1.06397	0.531983	+	0.846755i	$0.321447\pi$
$240$	0	0
$241$	−6.94308	−0.447243	−0.223621	−	0.974676i	$-0.571788\pi$
$241$	−6.94308	−0.447243	−0.223621	+	0.974676i	$0.571788\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−20.0995	−1.28411
$246$	0	0
$247$	0.874887	0.0556677
$248$	0	0
$249$	−16.9843	−1.07634
$250$	0	0
$251$	6.94821	0.438567	0.219284	−	0.975661i	$-0.429628\pi$
$251$	6.94821	0.438567	0.219284	+	0.975661i	$0.429628\pi$
$252$	0	0
$253$	8.84117	0.555839
$254$	0	0
$255$	−3.77041	−0.236112
$256$	0	0
$257$	−20.5366	−1.28104	−0.640519	−	0.767943i	$-0.721281\pi$
$257$	−20.5366	−1.28104	−0.640519	+	0.767943i	$0.721281\pi$
$258$	0	0
$259$	−0.717873	−0.0446064
$260$	0	0
$261$	1.65101	0.102195
$262$	0	0
$263$	8.80670	0.543044	0.271522	−	0.962432i	$-0.412473\pi$
$263$	8.80670	0.543044	0.271522	+	0.962432i	$0.412473\pi$
$264$	0	0
$265$	14.3391	0.880841
$266$	0	0
$267$	14.0444	0.859504
$268$	0	0
$269$	−23.0931	−1.40801	−0.704004	−	0.710196i	$-0.748606\pi$
$269$	−23.0931	−1.40801	−0.704004	+	0.710196i	$0.748606\pi$
$270$	0	0
$271$	11.2552	0.683705	0.341853	−	0.939754i	$-0.388946\pi$
$271$	11.2552	0.683705	0.341853	+	0.939754i	$0.388946\pi$
$272$	0	0
$273$	0.874887	0.0529506
$274$	0	0
$275$	−5.39341	−0.325235
$276$	0	0
$277$	−9.14575	−0.549515	−0.274757	−	0.961514i	$-0.588598\pi$
$277$	−9.14575	−0.549515	−0.274757	+	0.961514i	$0.588598\pi$
$278$	0	0
$279$	1.22388	0.0732718
$280$	0	0
$281$	−7.57287	−0.451760	−0.225880	−	0.974155i	$-0.572526\pi$
$281$	−7.57287	−0.451760	−0.225880	+	0.974155i	$0.572526\pi$
$282$	0	0
$283$	−8.34899	−0.496296	−0.248148	−	0.968722i	$-0.579822\pi$
$283$	−8.34899	−0.496296	−0.248148	+	0.968722i	$0.579822\pi$
$284$	0	0
$285$	2.82053	0.167074
$286$	0	0
$287$	10.2027	0.602245
$288$	0	0
$289$	−15.6322	−0.919542
$290$	0	0
$291$	10.2165	0.598902
$292$	0	0
$293$	−23.5729	−1.37714	−0.688571	−	0.725169i	$-0.741762\pi$
$293$	−23.5729	−1.37714	−0.688571	+	0.725169i	$0.741762\pi$
$294$	0	0
$295$	20.7868	1.21025
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−8.84117	−0.511298
$300$	0	0
$301$	−4.30383	−0.248069
$302$	0	0
$303$	10.9193	0.627297
$304$	0	0
$305$	36.3276	2.08011
$306$	0	0
$307$	6.62466	0.378089	0.189045	−	0.981968i	$-0.439461\pi$
$307$	6.62466	0.378089	0.189045	+	0.981968i	$0.439461\pi$
$308$	0	0
$309$	7.14575	0.406507
$310$	0	0
$311$	−1.69542	−0.0961384	−0.0480692	−	0.998844i	$-0.515307\pi$
$311$	−1.69542	−0.0961384	−0.0480692	+	0.998844i	$0.515307\pi$
$312$	0	0
$313$	−12.5909	−0.711682	−0.355841	−	0.934546i	$-0.615806\pi$
$313$	−12.5909	−0.711682	−0.355841	+	0.934546i	$0.615806\pi$
$314$	0	0
$315$	2.82053	0.158919
$316$	0	0
$317$	−15.5967	−0.875995	−0.437998	−	0.898976i	$-0.644312\pi$
$317$	−15.5967	−0.875995	−0.437998	+	0.898976i	$0.644312\pi$
$318$	0	0
$319$	−1.65101	−0.0924386
$320$	0	0
$321$	2.44776	0.136621
$322$	0	0
$323$	−1.02320	−0.0569325
$324$	0	0
$325$	5.39341	0.299172
$326$	0	0
$327$	11.3226	0.626144
$328$	0	0
$329$	10.8609	0.598780
$330$	0	0
$331$	−5.44039	−0.299031	−0.149515	−	0.988759i	$-0.547771\pi$
$331$	−5.44039	−0.299031	−0.149515	+	0.988759i	$0.547771\pi$
$332$	0	0
$333$	−0.820532	−0.0449649
$334$	0	0
$335$	32.2733	1.76328
$336$	0	0
$337$	−12.4478	−0.678073	−0.339036	−	0.940773i	$-0.610101\pi$
$337$	−12.4478	−0.678073	−0.339036	+	0.940773i	$0.610101\pi$
$338$	0	0
$339$	−6.19754	−0.336604
$340$	0	0
$341$	−1.22388	−0.0662768
$342$	0	0
$343$	−11.5788	−0.625194
$344$	0	0
$345$	−28.5029	−1.53454
$346$	0	0
$347$	29.9663	1.60867	0.804337	−	0.594173i	$-0.202521\pi$
$347$	29.9663	1.60867	0.804337	+	0.594173i	$0.202521\pi$
$348$	0	0
$349$	4.40649	0.235874	0.117937	−	0.993021i	$-0.462372\pi$
$349$	4.40649	0.235874	0.117937	+	0.993021i	$0.462372\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−9.76047	−0.519497	−0.259749	−	0.965676i	$-0.583640\pi$
$353$	−9.76047	−0.519497	−0.259749	+	0.965676i	$0.583640\pi$
$354$	0	0
$355$	24.8756	1.32026
$356$	0	0
$357$	−1.02320	−0.0541536
$358$	0	0
$359$	22.9162	1.20947	0.604734	−	0.796427i	$-0.293279\pi$
$359$	22.9162	1.20947	0.604734	+	0.796427i	$0.293279\pi$
$360$	0	0
$361$	−18.2346	−0.959714
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	17.9663	0.940398
$366$	0	0
$367$	−9.48480	−0.495102	−0.247551	−	0.968875i	$-0.579626\pi$
$367$	−9.48480	−0.495102	−0.247551	+	0.968875i	$0.579626\pi$
$368$	0	0
$369$	11.6617	0.607084
$370$	0	0
$371$	3.89129	0.202026
$372$	0	0
$373$	−22.0888	−1.14372	−0.571858	−	0.820353i	$-0.693777\pi$
$373$	−22.0888	−1.14372	−0.571858	+	0.820353i	$0.693777\pi$
$374$	0	0
$375$	1.26829	0.0654944
$376$	0	0
$377$	1.65101	0.0850312
$378$	0	0
$379$	−17.1375	−0.880293	−0.440146	−	0.897926i	$-0.645073\pi$
$379$	−17.1375	−0.880293	−0.440146	+	0.897926i	$0.645073\pi$
$380$	0	0
$381$	−16.5465	−0.847704
$382$	0	0
$383$	−2.33905	−0.119520	−0.0597599	−	0.998213i	$-0.519034\pi$
$383$	−2.33905	−0.119520	−0.0597599	+	0.998213i	$0.519034\pi$
$384$	0	0
$385$	−2.82053	−0.143748
$386$	0	0
$387$	−4.91930	−0.250062
$388$	0	0
$389$	1.12586	0.0570834	0.0285417	−	0.999593i	$-0.490914\pi$
$389$	1.12586	0.0570834	0.0285417	+	0.999593i	$0.490914\pi$
$390$	0	0
$391$	10.3400	0.522915
$392$	0	0
$393$	16.7868	0.846783
$394$	0	0
$395$	−7.96794	−0.400911
$396$	0	0
$397$	−17.2156	−0.864026	−0.432013	−	0.901867i	$-0.642197\pi$
$397$	−17.2156	−0.864026	−0.432013	+	0.901867i	$0.642197\pi$
$398$	0	0
$399$	0.765428	0.0383193
$400$	0	0
$401$	25.9580	1.29628	0.648140	−	0.761521i	$-0.275547\pi$
$401$	25.9580	1.29628	0.648140	+	0.761521i	$0.275547\pi$
$402$	0	0
$403$	1.22388	0.0609658
$404$	0	0
$405$	3.22388	0.160196
$406$	0	0
$407$	0.820532	0.0406723
$408$	0	0
$409$	−34.6873	−1.71518	−0.857589	−	0.514336i	$-0.828038\pi$
$409$	−34.6873	−1.71518	−0.857589	+	0.514336i	$0.828038\pi$
$410$	0	0
$411$	−1.45514	−0.0717766
$412$	0	0
$413$	5.64106	0.277579
$414$	0	0
$415$	−54.7555	−2.68784
$416$	0	0
$417$	−6.79675	−0.332838
$418$	0	0
$419$	1.78424	0.0871660	0.0435830	−	0.999050i	$-0.486123\pi$
$419$	1.78424	0.0871660	0.0435830	+	0.999050i	$0.486123\pi$
$420$	0	0
$421$	−18.2864	−0.891223	−0.445611	−	0.895227i	$-0.647014\pi$
$421$	−18.2864	−0.891223	−0.445611	+	0.895227i	$0.647014\pi$
$422$	0	0
$423$	12.4140	0.603591
$424$	0	0
$425$	−6.30772	−0.305970
$426$	0	0
$427$	9.85849	0.477086
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−24.0413	−1.15803	−0.579014	−	0.815318i	$-0.696562\pi$
$431$	−24.0413	−1.15803	−0.579014	+	0.815318i	$0.696562\pi$
$432$	0	0
$433$	−16.2915	−0.782919	−0.391460	−	0.920195i	$-0.628030\pi$
$433$	−16.2915	−0.782919	−0.391460	+	0.920195i	$0.628030\pi$
$434$	0	0
$435$	5.32265	0.255201
$436$	0	0
$437$	−7.73502	−0.370016
$438$	0	0
$439$	−21.4198	−1.02231	−0.511154	−	0.859489i	$-0.670782\pi$
$439$	−21.4198	−1.02231	−0.511154	+	0.859489i	$0.670782\pi$
$440$	0	0
$441$	−6.23457	−0.296884
$442$	0	0
$443$	8.50045	0.403869	0.201934	−	0.979399i	$-0.435277\pi$
$443$	8.50045	0.403869	0.201934	+	0.979399i	$0.435277\pi$
$444$	0	0
$445$	45.2775	2.14636
$446$	0	0
$447$	−3.66170	−0.173192
$448$	0	0
$449$	32.4058	1.52932	0.764661	−	0.644432i	$-0.222906\pi$
$449$	32.4058	1.52932	0.764661	+	0.644432i	$0.222906\pi$
$450$	0	0
$451$	−11.6617	−0.549128
$452$	0	0
$453$	2.51595	0.118210
$454$	0	0
$455$	2.82053	0.132229
$456$	0	0
$457$	−35.6823	−1.66915	−0.834575	−	0.550895i	$-0.814286\pi$
$457$	−35.6823	−1.66915	−0.834575	+	0.550895i	$0.814286\pi$
$458$	0	0
$459$	−1.16953	−0.0545888
$460$	0	0
$461$	−30.9851	−1.44312	−0.721560	−	0.692352i	$-0.756575\pi$
$461$	−30.9851	−1.44312	−0.721560	+	0.692352i	$0.756575\pi$
$462$	0	0
$463$	−5.50783	−0.255970	−0.127985	−	0.991776i	$-0.540851\pi$
$463$	−5.50783	−0.255970	−0.127985	+	0.991776i	$0.540851\pi$
$464$	0	0
$465$	3.94564	0.182975
$466$	0	0
$467$	33.8056	1.56434	0.782169	−	0.623066i	$-0.214113\pi$
$467$	33.8056	1.56434	0.782169	+	0.623066i	$0.214113\pi$
$468$	0	0
$469$	8.75823	0.404417
$470$	0	0
$471$	−1.35637	−0.0624982
$472$	0	0
$473$	4.91930	0.226190
$474$	0	0
$475$	4.71862	0.216505
$476$	0	0
$477$	4.44776	0.203649
$478$	0	0
$479$	−27.7704	−1.26886	−0.634431	−	0.772979i	$-0.718766\pi$
$479$	−27.7704	−1.26886	−0.634431	+	0.772979i	$0.718766\pi$
$480$	0	0
$481$	−0.820532	−0.0374130
$482$	0	0
$483$	−7.73502	−0.351956
$484$	0	0
$485$	32.9368	1.49558
$486$	0	0
$487$	1.39912	0.0634000	0.0317000	−	0.999497i	$-0.489908\pi$
$487$	1.39912	0.0634000	0.0317000	+	0.999497i	$0.489908\pi$
$488$	0	0
$489$	13.6379	0.616728
$490$	0	0
$491$	32.7892	1.47976	0.739879	−	0.672741i	$-0.234883\pi$
$491$	32.7892	1.47976	0.739879	+	0.672741i	$0.234883\pi$
$492$	0	0
$493$	−1.93089	−0.0869631
$494$	0	0
$495$	−3.22388	−0.144903
$496$	0	0
$497$	6.75068	0.302809
$498$	0	0
$499$	40.1555	1.79761	0.898804	−	0.438350i	$-0.144437\pi$
$499$	40.1555	1.79761	0.898804	+	0.438350i	$0.144437\pi$
$500$	0	0
$501$	−6.25023	−0.279239
$502$	0	0
$503$	−9.72989	−0.433834	−0.216917	−	0.976190i	$-0.569600\pi$
$503$	−9.72989	−0.433834	−0.216917	+	0.976190i	$0.569600\pi$
$504$	0	0
$505$	35.2025	1.56649
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−7.44039	−0.329789	−0.164895	−	0.986311i	$-0.552728\pi$
$509$	−7.44039	−0.329789	−0.164895	+	0.986311i	$0.552728\pi$
$510$	0	0
$511$	4.87564	0.215685
$512$	0	0
$513$	0.874887	0.0386272
$514$	0	0
$515$	23.0370	1.01513
$516$	0	0
$517$	−12.4140	−0.545969
$518$	0	0
$519$	−14.7630	−0.648025
$520$	0	0
$521$	22.3951	0.981146	0.490573	−	0.871400i	$-0.336788\pi$
$521$	22.3951	0.981146	0.490573	+	0.871400i	$0.336788\pi$
$522$	0	0
$523$	−3.32579	−0.145427	−0.0727133	−	0.997353i	$-0.523166\pi$
$523$	−3.32579	−0.145427	−0.0727133	+	0.997353i	$0.523166\pi$
$524$	0	0
$525$	4.71862	0.205938
$526$	0	0
$527$	−1.43136	−0.0623510
$528$	0	0
$529$	55.1662	2.39853
$530$	0	0
$531$	6.44776	0.279809
$532$	0	0
$533$	11.6617	0.505124
$534$	0	0
$535$	7.89129	0.341170
$536$	0	0
$537$	−21.8255	−0.941840
$538$	0	0
$539$	6.23457	0.268542
$540$	0	0
$541$	−36.9696	−1.58945	−0.794724	−	0.606972i	$-0.792384\pi$
$541$	−36.9696	−1.58945	−0.794724	+	0.606972i	$0.792384\pi$
$542$	0	0
$543$	5.35637	0.229864
$544$	0	0
$545$	36.5029	1.56361
$546$	0	0
$547$	−31.6724	−1.35421	−0.677107	−	0.735885i	$-0.736767\pi$
$547$	−31.6724	−1.35421	−0.677107	+	0.735885i	$0.736767\pi$
$548$	0	0
$549$	11.2683	0.480919
$550$	0	0
$551$	1.44444	0.0615354
$552$	0	0
$553$	−2.16232	−0.0919511
$554$	0	0
$555$	−2.64530	−0.112287
$556$	0	0
$557$	−4.88002	−0.206773	−0.103387	−	0.994641i	$-0.532968\pi$
$557$	−4.88002	−0.206773	−0.103387	+	0.994641i	$0.532968\pi$
$558$	0	0
$559$	−4.91930	−0.208064
$560$	0	0
$561$	1.16953	0.0493774
$562$	0	0
$563$	−8.60916	−0.362833	−0.181416	−	0.983406i	$-0.558068\pi$
$563$	−8.60916	−0.362833	−0.181416	+	0.983406i	$0.558068\pi$
$564$	0	0
$565$	−19.9801	−0.840570
$566$	0	0
$567$	0.874887	0.0367418
$568$	0	0
$569$	33.5646	1.40710	0.703551	−	0.710645i	$-0.251597\pi$
$569$	33.5646	1.40710	0.703551	+	0.710645i	$0.251597\pi$
$570$	0	0
$571$	−45.8362	−1.91819	−0.959093	−	0.283092i	$-0.908640\pi$
$571$	−45.8362	−1.91819	−0.959093	+	0.283092i	$0.908640\pi$
$572$	0	0
$573$	22.4478	0.937769
$574$	0	0
$575$	−47.6840	−1.98856
$576$	0	0
$577$	33.7185	1.40372	0.701859	−	0.712316i	$-0.252354\pi$
$577$	33.7185	1.40372	0.701859	+	0.712316i	$0.252354\pi$
$578$	0	0
$579$	9.68158	0.402353
$580$	0	0
$581$	−14.8594	−0.616471
$582$	0	0
$583$	−4.44776	−0.184207
$584$	0	0
$585$	3.22388	0.133291
$586$	0	0
$587$	16.8067	0.693687	0.346843	−	0.937923i	$-0.387254\pi$
$587$	16.8067	0.693687	0.346843	+	0.937923i	$0.387254\pi$
$588$	0	0
$589$	1.07076	0.0441198
$590$	0	0
$591$	−11.5729	−0.476044
$592$	0	0
$593$	−17.6090	−0.723115	−0.361558	−	0.932350i	$-0.617755\pi$
$593$	−17.6090	−0.723115	−0.361558	+	0.932350i	$0.617755\pi$
$594$	0	0
$595$	−3.29868	−0.135233
$596$	0	0
$597$	−20.2864	−0.830265
$598$	0	0
$599$	−20.5218	−0.838499	−0.419250	−	0.907871i	$-0.637707\pi$
$599$	−20.5218	−0.838499	−0.419250	+	0.907871i	$0.637707\pi$
$600$	0	0
$601$	11.3580	0.463304	0.231652	−	0.972799i	$-0.425587\pi$
$601$	11.3580	0.463304	0.231652	+	0.972799i	$0.425587\pi$
$602$	0	0
$603$	10.0107	0.407667
$604$	0	0
$605$	3.22388	0.131069
$606$	0	0
$607$	34.8181	1.41322	0.706612	−	0.707601i	$-0.250223\pi$
$607$	34.8181	1.41322	0.706612	+	0.707601i	$0.250223\pi$
$608$	0	0
$609$	1.44444	0.0585318
$610$	0	0
$611$	12.4140	0.502218
$612$	0	0
$613$	−41.3967	−1.67200	−0.835999	−	0.548731i	$-0.815111\pi$
$613$	−41.3967	−1.67200	−0.835999	+	0.548731i	$0.815111\pi$
$614$	0	0
$615$	37.5959	1.51601
$616$	0	0
$617$	46.7647	1.88268	0.941338	−	0.337465i	$-0.109570\pi$
$617$	46.7647	1.88268	0.941338	+	0.337465i	$0.109570\pi$
$618$	0	0
$619$	−17.0288	−0.684444	−0.342222	−	0.939619i	$-0.611179\pi$
$619$	−17.0288	−0.684444	−0.342222	+	0.939619i	$0.611179\pi$
$620$	0	0
$621$	−8.84117	−0.354784
$622$	0	0
$623$	12.2873	0.492279
$624$	0	0
$625$	−22.8782	−0.915128
$626$	0	0
$627$	−0.874887	−0.0349396
$628$	0	0
$629$	0.959633	0.0382631
$630$	0	0
$631$	−22.5810	−0.898935	−0.449468	−	0.893297i	$-0.648386\pi$
$631$	−22.5810	−0.898935	−0.449468	+	0.893297i	$0.648386\pi$
$632$	0	0
$633$	−24.1539	−0.960030
$634$	0	0
$635$	−53.3440	−2.11689
$636$	0	0
$637$	−6.23457	−0.247023
$638$	0	0
$639$	7.71605	0.305242
$640$	0	0
$641$	31.6149	1.24871	0.624357	−	0.781139i	$-0.285361\pi$
$641$	31.6149	1.24871	0.624357	+	0.781139i	$0.285361\pi$
$642$	0	0
$643$	8.06338	0.317989	0.158994	−	0.987279i	$-0.449175\pi$
$643$	8.06338	0.317989	0.158994	+	0.987279i	$0.449175\pi$
$644$	0	0
$645$	−15.8592	−0.624457
$646$	0	0
$647$	8.15627	0.320656	0.160328	−	0.987064i	$-0.448745\pi$
$647$	8.15627	0.320656	0.160328	+	0.987064i	$0.448745\pi$
$648$	0	0
$649$	−6.44776	−0.253097
$650$	0	0
$651$	1.07076	0.0419663
$652$	0	0
$653$	−24.9317	−0.975651	−0.487826	−	0.872941i	$-0.662210\pi$
$653$	−24.9317	−0.975651	−0.487826	+	0.872941i	$0.662210\pi$
$654$	0	0
$655$	54.1187	2.11459
$656$	0	0
$657$	5.57287	0.217419
$658$	0	0
$659$	36.9558	1.43959	0.719796	−	0.694186i	$-0.244235\pi$
$659$	36.9558	1.43959	0.719796	+	0.694186i	$0.244235\pi$
$660$	0	0
$661$	0.906942	0.0352760	0.0176380	−	0.999844i	$-0.494385\pi$
$661$	0.906942	0.0352760	0.0176380	+	0.999844i	$0.494385\pi$
$662$	0	0
$663$	−1.16953	−0.0454206
$664$	0	0
$665$	2.46765	0.0956913
$666$	0	0
$667$	−14.5968	−0.565191
$668$	0	0
$669$	−0.723429	−0.0279694
$670$	0	0
$671$	−11.2683	−0.435008
$672$	0	0
$673$	25.7573	0.992872	0.496436	−	0.868073i	$-0.334642\pi$
$673$	25.7573	0.992872	0.496436	+	0.868073i	$0.334642\pi$
$674$	0	0
$675$	5.39341	0.207592
$676$	0	0
$677$	−15.6871	−0.602906	−0.301453	−	0.953481i	$-0.597472\pi$
$677$	−15.6871	−0.602906	−0.301453	+	0.953481i	$0.597472\pi$
$678$	0	0
$679$	8.93829	0.343020
$680$	0	0
$681$	−17.3440	−0.664625
$682$	0	0
$683$	32.6116	1.24785	0.623924	−	0.781485i	$-0.285538\pi$
$683$	32.6116	1.24785	0.623924	+	0.781485i	$0.285538\pi$
$684$	0	0
$685$	−4.69119	−0.179241
$686$	0	0
$687$	12.0551	0.459931
$688$	0	0
$689$	4.44776	0.169446
$690$	0	0
$691$	27.2100	1.03512	0.517559	−	0.855647i	$-0.326841\pi$
$691$	27.2100	1.03512	0.517559	+	0.855647i	$0.326841\pi$
$692$	0	0
$693$	−0.874887	−0.0332342
$694$	0	0
$695$	−21.9119	−0.831167
$696$	0	0
$697$	−13.6386	−0.516601
$698$	0	0
$699$	−11.3333	−0.428666
$700$	0	0
$701$	32.3153	1.22053	0.610266	−	0.792197i	$-0.291063\pi$
$701$	32.3153	1.22053	0.610266	+	0.792197i	$0.291063\pi$
$702$	0	0
$703$	−0.717873	−0.0270751
$704$	0	0
$705$	40.0214	1.50729
$706$	0	0
$707$	9.55316	0.359283
$708$	0	0
$709$	49.4923	1.85872	0.929362	−	0.369170i	$-0.120358\pi$
$709$	49.4923	1.85872	0.929362	+	0.369170i	$0.120358\pi$
$710$	0	0
$711$	−2.47154	−0.0926899
$712$	0	0
$713$	−10.8205	−0.405232
$714$	0	0
$715$	−3.22388	−0.120566
$716$	0	0
$717$	16.4485	0.614281
$718$	0	0
$719$	−40.0825	−1.49483	−0.747413	−	0.664359i	$-0.768704\pi$
$719$	−40.0825	−1.49483	−0.747413	+	0.664359i	$0.768704\pi$
$720$	0	0
$721$	6.25172	0.232826
$722$	0	0
$723$	−6.94308	−0.258216
$724$	0	0
$725$	8.90455	0.330707
$726$	0	0
$727$	2.98948	0.110874	0.0554369	−	0.998462i	$-0.482345\pi$
$727$	2.98948	0.110874	0.0554369	+	0.998462i	$0.482345\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.75324	0.212791
$732$	0	0
$733$	−9.28651	−0.343005	−0.171503	−	0.985184i	$-0.554862\pi$
$733$	−9.28651	−0.343005	−0.171503	+	0.985184i	$0.554862\pi$
$734$	0	0
$735$	−20.0995	−0.741382
$736$	0	0
$737$	−10.0107	−0.368749
$738$	0	0
$739$	31.6304	1.16354	0.581771	−	0.813352i	$-0.302360\pi$
$739$	31.6304	1.16354	0.581771	+	0.813352i	$0.302360\pi$
$740$	0	0
$741$	0.874887	0.0321398
$742$	0	0
$743$	20.3752	0.747493	0.373747	−	0.927531i	$-0.378073\pi$
$743$	20.3752	0.747493	0.373747	+	0.927531i	$0.378073\pi$
$744$	0	0
$745$	−11.8049	−0.432497
$746$	0	0
$747$	−16.9843	−0.621425
$748$	0	0
$749$	2.14151	0.0782492
$750$	0	0
$751$	14.6254	0.533689	0.266844	−	0.963740i	$-0.414019\pi$
$751$	14.6254	0.533689	0.266844	+	0.963740i	$0.414019\pi$
$752$	0	0
$753$	6.94821	0.253207
$754$	0	0
$755$	8.11112	0.295194
$756$	0	0
$757$	34.5762	1.25669	0.628347	−	0.777934i	$-0.283732\pi$
$757$	34.5762	1.25669	0.628347	+	0.777934i	$0.283732\pi$
$758$	0	0
$759$	8.84117	0.320914
$760$	0	0
$761$	13.4314	0.486886	0.243443	−	0.969915i	$-0.421723\pi$
$761$	13.4314	0.486886	0.243443	+	0.969915i	$0.421723\pi$
$762$	0	0
$763$	9.90604	0.358623
$764$	0	0
$765$	−3.77041	−0.136319
$766$	0	0
$767$	6.44776	0.232815
$768$	0	0
$769$	−6.21742	−0.224206	−0.112103	−	0.993697i	$-0.535759\pi$
$769$	−6.21742	−0.224206	−0.112103	+	0.993697i	$0.535759\pi$
$770$	0	0
$771$	−20.5366	−0.739607
$772$	0	0
$773$	48.0805	1.72934	0.864669	−	0.502343i	$-0.167528\pi$
$773$	48.0805	1.72934	0.864669	+	0.502343i	$0.167528\pi$
$774$	0	0
$775$	6.60088	0.237111
$776$	0	0
$777$	−0.717873	−0.0257535
$778$	0	0
$779$	10.2027	0.365549
$780$	0	0
$781$	−7.71605	−0.276102
$782$	0	0
$783$	1.65101	0.0590022
$784$	0	0
$785$	−4.37277	−0.156071
$786$	0	0
$787$	8.76618	0.312480	0.156240	−	0.987719i	$-0.450063\pi$
$787$	8.76618	0.312480	0.156240	+	0.987719i	$0.450063\pi$
$788$	0	0
$789$	8.80670	0.313527
$790$	0	0
$791$	−5.42214	−0.192789
$792$	0	0
$793$	11.2683	0.400149
$794$	0	0
$795$	14.3391	0.508554
$796$	0	0
$797$	−25.7185	−0.910995	−0.455497	−	0.890237i	$-0.650539\pi$
$797$	−25.7185	−0.910995	−0.455497	+	0.890237i	$0.650539\pi$
$798$	0	0
$799$	−14.5185	−0.513629
$800$	0	0
$801$	14.0444	0.496235
$802$	0	0
$803$	−5.57287	−0.196662
$804$	0	0
$805$	−24.9368	−0.878906
$806$	0	0
$807$	−23.0931	−0.812914
$808$	0	0
$809$	−41.6272	−1.46354	−0.731768	−	0.681554i	$-0.761305\pi$
$809$	−41.6272	−1.46354	−0.731768	+	0.681554i	$0.761305\pi$
$810$	0	0
$811$	32.1983	1.13063	0.565317	−	0.824874i	$-0.308754\pi$
$811$	32.1983	1.13063	0.565317	+	0.824874i	$0.308754\pi$
$812$	0	0
$813$	11.2552	0.394737
$814$	0	0
$815$	43.9670	1.54010
$816$	0	0
$817$	−4.30383	−0.150572
$818$	0	0
$819$	0.874887	0.0305710
$820$	0	0
$821$	−26.8948	−0.938634	−0.469317	−	0.883030i	$-0.655500\pi$
$821$	−26.8948	−0.938634	−0.469317	+	0.883030i	$0.655500\pi$
$822$	0	0
$823$	−0.442626	−0.0154290	−0.00771448	−	0.999970i	$-0.502456\pi$
$823$	−0.442626	−0.0154290	−0.00771448	+	0.999970i	$0.502456\pi$
$824$	0	0
$825$	−5.39341	−0.187774
$826$	0	0
$827$	−21.5891	−0.750727	−0.375364	−	0.926878i	$-0.622482\pi$
$827$	−21.5891	−0.750727	−0.375364	+	0.926878i	$0.622482\pi$
$828$	0	0
$829$	−20.4691	−0.710923	−0.355461	−	0.934691i	$-0.615676\pi$
$829$	−20.4691	−0.710923	−0.355461	+	0.934691i	$0.615676\pi$
$830$	0	0
$831$	−9.14575	−0.317262
$832$	0	0
$833$	7.29149	0.252635
$834$	0	0
$835$	−20.1500	−0.697319
$836$	0	0
$837$	1.22388	0.0423035
$838$	0	0
$839$	20.0015	0.690528	0.345264	−	0.938506i	$-0.387789\pi$
$839$	20.0015	0.690528	0.345264	+	0.938506i	$0.387789\pi$
$840$	0	0
$841$	−26.2742	−0.906006
$842$	0	0
$843$	−7.57287	−0.260824
$844$	0	0
$845$	3.22388	0.110905
$846$	0	0
$847$	0.874887	0.0300615
$848$	0	0
$849$	−8.34899	−0.286537
$850$	0	0
$851$	7.25446	0.248680
$852$	0	0
$853$	26.7194	0.914854	0.457427	−	0.889247i	$-0.348771\pi$
$853$	26.7194	0.914854	0.457427	+	0.889247i	$0.348771\pi$
$854$	0	0
$855$	2.82053	0.0964602
$856$	0	0
$857$	−4.45257	−0.152097	−0.0760484	−	0.997104i	$-0.524230\pi$
$857$	−4.45257	−0.152097	−0.0760484	+	0.997104i	$0.524230\pi$
$858$	0	0
$859$	−1.59351	−0.0543698	−0.0271849	−	0.999630i	$-0.508654\pi$
$859$	−1.59351	−0.0543698	−0.0271849	+	0.999630i	$0.508654\pi$
$860$	0	0
$861$	10.2027	0.347706
$862$	0	0
$863$	12.9154	0.439646	0.219823	−	0.975540i	$-0.429452\pi$
$863$	12.9154	0.439646	0.219823	+	0.975540i	$0.429452\pi$
$864$	0	0
$865$	−47.5943	−1.61825
$866$	0	0
$867$	−15.6322	−0.530898
$868$	0	0
$869$	2.47154	0.0838412
$870$	0	0
$871$	10.0107	0.339199
$872$	0	0
$873$	10.2165	0.345776
$874$	0	0
$875$	1.10961	0.0375118
$876$	0	0
$877$	−4.37882	−0.147862	−0.0739312	−	0.997263i	$-0.523555\pi$
$877$	−4.37882	−0.147862	−0.0739312	+	0.997263i	$0.523555\pi$
$878$	0	0
$879$	−23.5729	−0.795093
$880$	0	0
$881$	4.42638	0.149128	0.0745642	−	0.997216i	$-0.476243\pi$
$881$	4.42638	0.149128	0.0745642	+	0.997216i	$0.476243\pi$
$882$	0	0
$883$	−33.8998	−1.14082	−0.570409	−	0.821361i	$-0.693215\pi$
$883$	−33.8998	−1.14082	−0.570409	+	0.821361i	$0.693215\pi$
$884$	0	0
$885$	20.7868	0.698741
$886$	0	0
$887$	−47.6995	−1.60159	−0.800796	−	0.598937i	$-0.795590\pi$
$887$	−47.6995	−1.60159	−0.800796	+	0.598937i	$0.795590\pi$
$888$	0	0
$889$	−14.4763	−0.485521
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	10.8609	0.363446
$894$	0	0
$895$	−70.3628	−2.35197
$896$	0	0
$897$	−8.84117	−0.295198
$898$	0	0
$899$	2.02063	0.0673919
$900$	0	0
$901$	−5.20177	−0.173296
$902$	0	0
$903$	−4.30383	−0.143223
$904$	0	0
$905$	17.2683	0.574017
$906$	0	0
$907$	−26.1629	−0.868725	−0.434362	−	0.900738i	$-0.643026\pi$
$907$	−26.1629	−0.868725	−0.434362	+	0.900738i	$0.643026\pi$
$908$	0	0
$909$	10.9193	0.362170
$910$	0	0
$911$	−10.6929	−0.354270	−0.177135	−	0.984187i	$-0.556683\pi$
$911$	−10.6929	−0.354270	−0.177135	+	0.984187i	$0.556683\pi$
$912$	0	0
$913$	16.9843	0.562100
$914$	0	0
$915$	36.3276	1.20095
$916$	0	0
$917$	14.6866	0.484993
$918$	0	0
$919$	−4.72418	−0.155836	−0.0779181	−	0.996960i	$-0.524827\pi$
$919$	−4.72418	−0.155836	−0.0779181	+	0.996960i	$0.524827\pi$
$920$	0	0
$921$	6.62466	0.218290
$922$	0	0
$923$	7.71605	0.253977
$924$	0	0
$925$	−4.42546	−0.145508
$926$	0	0
$927$	7.14575	0.234697
$928$	0	0
$929$	8.75715	0.287313	0.143656	−	0.989628i	$-0.454114\pi$
$929$	8.75715	0.287313	0.143656	+	0.989628i	$0.454114\pi$
$930$	0	0
$931$	−5.45455	−0.178766
$932$	0	0
$933$	−1.69542	−0.0555055
$934$	0	0
$935$	3.77041	0.123306
$936$	0	0
$937$	−4.64288	−0.151676	−0.0758382	−	0.997120i	$-0.524163\pi$
$937$	−4.64288	−0.151676	−0.0758382	+	0.997120i	$0.524163\pi$
$938$	0	0
$939$	−12.5909	−0.410890
$940$	0	0
$941$	14.4536	0.471175	0.235588	−	0.971853i	$-0.424299\pi$
$941$	14.4536	0.471175	0.235588	+	0.971853i	$0.424299\pi$
$942$	0	0
$943$	−103.103	−3.35750
$944$	0	0
$945$	2.82053	0.0917519
$946$	0	0
$947$	30.1638	0.980192	0.490096	−	0.871668i	$-0.336962\pi$
$947$	30.1638	0.980192	0.490096	+	0.871668i	$0.336962\pi$
$948$	0	0
$949$	5.57287	0.180903
$950$	0	0
$951$	−15.5967	−0.505756
$952$	0	0
$953$	−0.0987677	−0.00319940	−0.00159970	−	0.999999i	$-0.500509\pi$
$953$	−0.0987677	−0.00319940	−0.00159970	+	0.999999i	$0.500509\pi$
$954$	0	0
$955$	72.3689	2.34180
$956$	0	0
$957$	−1.65101	−0.0533695
$958$	0	0
$959$	−1.27308	−0.0411099
$960$	0	0
$961$	−29.5021	−0.951681
$962$	0	0
$963$	2.44776	0.0788780
$964$	0	0
$965$	31.2123	1.00476
$966$	0	0
$967$	−34.4684	−1.10843	−0.554214	−	0.832374i	$-0.686981\pi$
$967$	−34.4684	−1.10843	−0.554214	+	0.832374i	$0.686981\pi$
$968$	0	0
$969$	−1.02320	−0.0328700
$970$	0	0
$971$	21.9818	0.705429	0.352714	−	0.935731i	$-0.385259\pi$
$971$	21.9818	0.705429	0.352714	+	0.935731i	$0.385259\pi$
$972$	0	0
$973$	−5.94639	−0.190633
$974$	0	0
$975$	5.39341	0.172727
$976$	0	0
$977$	−46.8725	−1.49958	−0.749792	−	0.661674i	$-0.769846\pi$
$977$	−46.8725	−1.49958	−0.749792	+	0.661674i	$0.769846\pi$
$978$	0	0
$979$	−14.0444	−0.448861
$980$	0	0
$981$	11.3226	0.361504
$982$	0	0
$983$	15.1060	0.481806	0.240903	−	0.970549i	$-0.422556\pi$
$983$	15.1060	0.481806	0.240903	+	0.970549i	$0.422556\pi$
$984$	0	0
$985$	−37.3096	−1.18878
$986$	0	0
$987$	10.8609	0.345706
$988$	0	0
$989$	43.4923	1.38298
$990$	0	0
$991$	−37.6348	−1.19551	−0.597754	−	0.801680i	$-0.703940\pi$
$991$	−37.6348	−1.19551	−0.597754	+	0.801680i	$0.703940\pi$
$992$	0	0
$993$	−5.44039	−0.172645
$994$	0	0
$995$	−65.4008	−2.07334
$996$	0	0
$997$	−6.52183	−0.206549	−0.103274	−	0.994653i	$-0.532932\pi$
$997$	−6.52183	−0.206549	−0.103274	+	0.994653i	$0.532932\pi$
$998$	0	0
$999$	−0.820532	−0.0259605

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.cb.1.4		4
4.3	odd	2		1716.2.a.i.1.4	✓	4
12.11	even	2		5148.2.a.q.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1716.2.a.i.1.4	✓	4	4.3	odd	2
5148.2.a.q.1.1		4	12.11	even	2
6864.2.a.cb.1.4		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.22388 q^{5} +0.874887 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.22388 q^{5} +0.874887 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +3.22388 q^{15} -1.16953 q^{17} +0.874887 q^{19} +0.874887 q^{21} -8.84117 q^{23} +5.39341 q^{25} +1.00000 q^{27} +1.65101 q^{29} +1.22388 q^{31} -1.00000 q^{33} +2.82053 q^{35} -0.820532 q^{37} +1.00000 q^{39} +11.6617 q^{41} -4.91930 q^{43} +3.22388 q^{45} +12.4140 q^{47} -6.23457 q^{49} -1.16953 q^{51} +4.44776 q^{53} -3.22388 q^{55} +0.874887 q^{57} +6.44776 q^{59} +11.2683 q^{61} +0.874887 q^{63} +3.22388 q^{65} +10.0107 q^{67} -8.84117 q^{69} +7.71605 q^{71} +5.57287 q^{73} +5.39341 q^{75} -0.874887 q^{77} -2.47154 q^{79} +1.00000 q^{81} -16.9843 q^{83} -3.77041 q^{85} +1.65101 q^{87} +14.0444 q^{89} +0.874887 q^{91} +1.22388 q^{93} +2.82053 q^{95} +10.2165 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 2 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 2 * q^5 - 2 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 2 q^{5} - 2 q^{7} + 4 q^{9} - 4 q^{11} + 4 q^{13} + 2 q^{15} + 2 q^{17} - 2 q^{19} - 2 q^{21} + 4 q^{23} + 4 q^{25} + 4 q^{27} + 12 q^{29} - 6 q^{31} - 4 q^{33} + 10 q^{35} - 2 q^{37} + 4 q^{39} + 6 q^{41} - 2 q^{43} + 2 q^{45} - 6 q^{47} + 32 q^{49} + 2 q^{51} - 4 q^{53} - 2 q^{55} - 2 q^{57} + 4 q^{59} + 22 q^{61} - 2 q^{63} + 2 q^{65} - 6 q^{67} + 4 q^{69} - 14 q^{71} + 6 q^{73} + 4 q^{75} + 2 q^{77} - 14 q^{79} + 4 q^{81} + 34 q^{85} + 12 q^{87} + 44 q^{89} - 2 q^{91} - 6 q^{93} + 10 q^{95} + 18 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 2 * q^5 - 2 * q^7 + 4 * q^9 - 4 * q^11 + 4 * q^13 + 2 * q^15 + 2 * q^17 - 2 * q^19 - 2 * q^21 + 4 * q^23 + 4 * q^25 + 4 * q^27 + 12 * q^29 - 6 * q^31 - 4 * q^33 + 10 * q^35 - 2 * q^37 + 4 * q^39 + 6 * q^41 - 2 * q^43 + 2 * q^45 - 6 * q^47 + 32 * q^49 + 2 * q^51 - 4 * q^53 - 2 * q^55 - 2 * q^57 + 4 * q^59 + 22 * q^61 - 2 * q^63 + 2 * q^65 - 6 * q^67 + 4 * q^69 - 14 * q^71 + 6 * q^73 + 4 * q^75 + 2 * q^77 - 14 * q^79 + 4 * q^81 + 34 * q^85 + 12 * q^87 + 44 * q^89 - 2 * q^91 - 6 * q^93 + 10 * q^95 + 18 * q^97 - 4 * q^99

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