LMFDB - Embedded newform 6864.2.a.bl.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:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 858)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.37228$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -2.37228 q^{5} -2.37228 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.37228 q^{5} -2.37228 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -2.37228 q^{15} -4.00000 q^{17} -4.74456 q^{19} -2.37228 q^{21} -2.37228 q^{23} +0.627719 q^{25} +1.00000 q^{27} +2.37228 q^{29} +2.74456 q^{31} +1.00000 q^{33} +5.62772 q^{35} -2.00000 q^{37} -1.00000 q^{39} -0.372281 q^{41} +8.37228 q^{43} -2.37228 q^{45} +4.00000 q^{47} -1.37228 q^{49} -4.00000 q^{51} -10.0000 q^{53} -2.37228 q^{55} -4.74456 q^{57} +5.62772 q^{59} -9.11684 q^{61} -2.37228 q^{63} +2.37228 q^{65} +5.11684 q^{67} -2.37228 q^{69} +3.25544 q^{71} +9.11684 q^{73} +0.627719 q^{75} -2.37228 q^{77} +10.0000 q^{79} +1.00000 q^{81} +8.74456 q^{83} +9.48913 q^{85} +2.37228 q^{87} +8.00000 q^{89} +2.37228 q^{91} +2.74456 q^{93} +11.2554 q^{95} -10.0000 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + q^{5} + q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + q^5 + q^7 + 2 * q^9	$ 2 q + 2 q^{3} + q^{5} + q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{13} + q^{15} - 8 q^{17} + 2 q^{19} + q^{21} + q^{23} + 7 q^{25} + 2 q^{27} - q^{29} - 6 q^{31} + 2 q^{33} + 17 q^{35} - 4 q^{37} - 2 q^{39} + 5 q^{41} + 11 q^{43} + q^{45} + 8 q^{47} + 3 q^{49} - 8 q^{51} - 20 q^{53} + q^{55} + 2 q^{57} + 17 q^{59} - q^{61} + q^{63} - q^{65} - 7 q^{67} + q^{69} + 18 q^{71} + q^{73} + 7 q^{75} + q^{77} + 20 q^{79} + 2 q^{81} + 6 q^{83} - 4 q^{85} - q^{87} + 16 q^{89} - q^{91} - 6 q^{93} + 34 q^{95} - 20 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + q^5 + q^7 + 2 * q^9 + 2 * q^11 - 2 * q^13 + q^15 - 8 * q^17 + 2 * q^19 + q^21 + q^23 + 7 * q^25 + 2 * q^27 - q^29 - 6 * q^31 + 2 * q^33 + 17 * q^35 - 4 * q^37 - 2 * q^39 + 5 * q^41 + 11 * q^43 + q^45 + 8 * q^47 + 3 * q^49 - 8 * q^51 - 20 * q^53 + q^55 + 2 * q^57 + 17 * q^59 - q^61 + q^63 - q^65 - 7 * q^67 + q^69 + 18 * q^71 + q^73 + 7 * q^75 + q^77 + 20 * q^79 + 2 * q^81 + 6 * q^83 - 4 * q^85 - q^87 + 16 * q^89 - q^91 - 6 * q^93 + 34 * q^95 - 20 * q^97 + 2 * 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
$5$	−2.37228	−1.06092	−0.530458	−	0.847711i	$-0.677980\pi$
$5$	−2.37228	−1.06092	−0.530458	+	0.847711i	$0.677980\pi$
$6$	0	0
$7$	−2.37228	−0.896638	−0.448319	−	0.893874i	$-0.647977\pi$
$7$	−2.37228	−0.896638	−0.448319	+	0.893874i	$0.647977\pi$
$8$	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
$15$	−2.37228	−0.612520
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	−4.74456	−1.08848	−0.544239	−	0.838930i	$-0.683181\pi$
$19$	−4.74456	−1.08848	−0.544239	+	0.838930i	$0.683181\pi$
$20$	0	0
$21$	−2.37228	−0.517674
$22$	0	0
$23$	−2.37228	−0.494655	−0.247327	−	0.968932i	$-0.579552\pi$
$23$	−2.37228	−0.494655	−0.247327	+	0.968932i	$0.579552\pi$
$24$	0	0
$25$	0.627719	0.125544
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.37228	0.440522	0.220261	−	0.975441i	$-0.429309\pi$
$29$	2.37228	0.440522	0.220261	+	0.975441i	$0.429309\pi$
$30$	0	0
$31$	2.74456	0.492938	0.246469	−	0.969151i	$-0.420730\pi$
$31$	2.74456	0.492938	0.246469	+	0.969151i	$0.420730\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	5.62772	0.951258
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−0.372281	−0.0581406	−0.0290703	−	0.999577i	$-0.509255\pi$
$41$	−0.372281	−0.0581406	−0.0290703	+	0.999577i	$0.509255\pi$
$42$	0	0
$43$	8.37228	1.27676	0.638380	−	0.769721i	$-0.279605\pi$
$43$	8.37228	1.27676	0.638380	+	0.769721i	$0.279605\pi$
$44$	0	0
$45$	−2.37228	−0.353639
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	−1.37228	−0.196040
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	−2.37228	−0.319878
$56$	0	0
$57$	−4.74456	−0.628433
$58$	0	0
$59$	5.62772	0.732667	0.366333	−	0.930484i	$-0.380613\pi$
$59$	5.62772	0.732667	0.366333	+	0.930484i	$0.380613\pi$
$60$	0	0
$61$	−9.11684	−1.16729	−0.583646	−	0.812008i	$-0.698374\pi$
$61$	−9.11684	−1.16729	−0.583646	+	0.812008i	$0.698374\pi$
$62$	0	0
$63$	−2.37228	−0.298879
$64$	0	0
$65$	2.37228	0.294245
$66$	0	0
$67$	5.11684	0.625122	0.312561	−	0.949898i	$-0.398813\pi$
$67$	5.11684	0.625122	0.312561	+	0.949898i	$0.398813\pi$
$68$	0	0
$69$	−2.37228	−0.285589
$70$	0	0
$71$	3.25544	0.386349	0.193175	−	0.981164i	$-0.438122\pi$
$71$	3.25544	0.386349	0.193175	+	0.981164i	$0.438122\pi$
$72$	0	0
$73$	9.11684	1.06705	0.533523	−	0.845786i	$-0.320868\pi$
$73$	9.11684	1.06705	0.533523	+	0.845786i	$0.320868\pi$
$74$	0	0
$75$	0.627719	0.0724827
$76$	0	0
$77$	−2.37228	−0.270347
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.74456	0.959840	0.479920	−	0.877312i	$-0.340666\pi$
$83$	8.74456	0.959840	0.479920	+	0.877312i	$0.340666\pi$
$84$	0	0
$85$	9.48913	1.02924
$86$	0	0
$87$	2.37228	0.254335
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	2.37228	0.248683
$92$	0	0
$93$	2.74456	0.284598
$94$	0	0
$95$	11.2554	1.15478
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−3.25544	−0.323928	−0.161964	−	0.986797i	$-0.551783\pi$
$101$	−3.25544	−0.323928	−0.161964	+	0.986797i	$0.551783\pi$
$102$	0	0
$103$	−3.11684	−0.307112	−0.153556	−	0.988140i	$-0.549073\pi$
$103$	−3.11684	−0.307112	−0.153556	+	0.988140i	$0.549073\pi$
$104$	0	0
$105$	5.62772	0.549209
$106$	0	0
$107$	−6.37228	−0.616032	−0.308016	−	0.951381i	$-0.599665\pi$
$107$	−6.37228	−0.616032	−0.308016	+	0.951381i	$0.599665\pi$
$108$	0	0
$109$	1.25544	0.120249	0.0601245	−	0.998191i	$-0.480850\pi$
$109$	1.25544	0.120249	0.0601245	+	0.998191i	$0.480850\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−3.62772	−0.341267	−0.170634	−	0.985335i	$-0.554581\pi$
$113$	−3.62772	−0.341267	−0.170634	+	0.985335i	$0.554581\pi$
$114$	0	0
$115$	5.62772	0.524787
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	9.48913	0.869867
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.372281	−0.0335675
$124$	0	0
$125$	10.3723	0.927725
$126$	0	0
$127$	7.48913	0.664552	0.332276	−	0.943182i	$-0.392183\pi$
$127$	7.48913	0.664552	0.332276	+	0.943182i	$0.392183\pi$
$128$	0	0
$129$	8.37228	0.737138
$130$	0	0
$131$	10.3723	0.906230	0.453115	−	0.891452i	$-0.350313\pi$
$131$	10.3723	0.906230	0.453115	+	0.891452i	$0.350313\pi$
$132$	0	0
$133$	11.2554	0.975970
$134$	0	0
$135$	−2.37228	−0.204173
$136$	0	0
$137$	12.7446	1.08884	0.544421	−	0.838812i	$-0.316750\pi$
$137$	12.7446	1.08884	0.544421	+	0.838812i	$0.316750\pi$
$138$	0	0
$139$	20.2337	1.71620	0.858100	−	0.513483i	$-0.171645\pi$
$139$	20.2337	1.71620	0.858100	+	0.513483i	$0.171645\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−5.62772	−0.467357
$146$	0	0
$147$	−1.37228	−0.113184
$148$	0	0
$149$	5.25544	0.430542	0.215271	−	0.976554i	$-0.430936\pi$
$149$	5.25544	0.430542	0.215271	+	0.976554i	$0.430936\pi$
$150$	0	0
$151$	−21.4891	−1.74876	−0.874380	−	0.485242i	$-0.838732\pi$
$151$	−21.4891	−1.74876	−0.874380	+	0.485242i	$0.838732\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	−6.51087	−0.522966
$156$	0	0
$157$	−15.4891	−1.23617	−0.618083	−	0.786113i	$-0.712091\pi$
$157$	−15.4891	−1.23617	−0.618083	+	0.786113i	$0.712091\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	5.62772	0.443526
$162$	0	0
$163$	−3.62772	−0.284145	−0.142072	−	0.989856i	$-0.545377\pi$
$163$	−3.62772	−0.284145	−0.142072	+	0.989856i	$0.545377\pi$
$164$	0	0
$165$	−2.37228	−0.184682
$166$	0	0
$167$	15.1168	1.16978	0.584888	−	0.811114i	$-0.301138\pi$
$167$	15.1168	1.16978	0.584888	+	0.811114i	$0.301138\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.74456	−0.362826
$172$	0	0
$173$	−11.8614	−0.901806	−0.450903	−	0.892573i	$-0.648898\pi$
$173$	−11.8614	−0.901806	−0.450903	+	0.892573i	$0.648898\pi$
$174$	0	0
$175$	−1.48913	−0.112567
$176$	0	0
$177$	5.62772	0.423005
$178$	0	0
$179$	−5.48913	−0.410276	−0.205138	−	0.978733i	$-0.565764\pi$
$179$	−5.48913	−0.410276	−0.205138	+	0.978733i	$0.565764\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	−9.11684	−0.673936
$184$	0	0
$185$	4.74456	0.348827
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	−2.37228	−0.172558
$190$	0	0
$191$	−13.6277	−0.986067	−0.493034	−	0.870010i	$-0.664112\pi$
$191$	−13.6277	−0.986067	−0.493034	+	0.870010i	$0.664112\pi$
$192$	0	0
$193$	11.4891	0.827005	0.413503	−	0.910503i	$-0.364305\pi$
$193$	11.4891	0.827005	0.413503	+	0.910503i	$0.364305\pi$
$194$	0	0
$195$	2.37228	0.169883
$196$	0	0
$197$	−1.25544	−0.0894462	−0.0447231	−	0.998999i	$-0.514241\pi$
$197$	−1.25544	−0.0894462	−0.0447231	+	0.998999i	$0.514241\pi$
$198$	0	0
$199$	−23.8614	−1.69149	−0.845745	−	0.533587i	$-0.820844\pi$
$199$	−23.8614	−1.69149	−0.845745	+	0.533587i	$0.820844\pi$
$200$	0	0
$201$	5.11684	0.360914
$202$	0	0
$203$	−5.62772	−0.394988
$204$	0	0
$205$	0.883156	0.0616823
$206$	0	0
$207$	−2.37228	−0.164885
$208$	0	0
$209$	−4.74456	−0.328188
$210$	0	0
$211$	−2.74456	−0.188943	−0.0944717	−	0.995528i	$-0.530116\pi$
$211$	−2.74456	−0.188943	−0.0944717	+	0.995528i	$0.530116\pi$
$212$	0	0
$213$	3.25544	0.223059
$214$	0	0
$215$	−19.8614	−1.35454
$216$	0	0
$217$	−6.51087	−0.441987
$218$	0	0
$219$	9.11684	0.616059
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	−10.7446	−0.719509	−0.359755	−	0.933047i	$-0.617140\pi$
$223$	−10.7446	−0.719509	−0.359755	+	0.933047i	$0.617140\pi$
$224$	0	0
$225$	0.627719	0.0418479
$226$	0	0
$227$	22.9783	1.52512	0.762560	−	0.646917i	$-0.223942\pi$
$227$	22.9783	1.52512	0.762560	+	0.646917i	$0.223942\pi$
$228$	0	0
$229$	18.6060	1.22952	0.614759	−	0.788715i	$-0.289253\pi$
$229$	18.6060	1.22952	0.614759	+	0.788715i	$0.289253\pi$
$230$	0	0
$231$	−2.37228	−0.156085
$232$	0	0
$233$	16.7446	1.09697	0.548486	−	0.836159i	$-0.315204\pi$
$233$	16.7446	1.09697	0.548486	+	0.836159i	$0.315204\pi$
$234$	0	0
$235$	−9.48913	−0.619002
$236$	0	0
$237$	10.0000	0.649570
$238$	0	0
$239$	3.86141	0.249774	0.124887	−	0.992171i	$-0.460143\pi$
$239$	3.86141	0.249774	0.124887	+	0.992171i	$0.460143\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.25544	0.207982
$246$	0	0
$247$	4.74456	0.301889
$248$	0	0
$249$	8.74456	0.554164
$250$	0	0
$251$	26.2337	1.65586	0.827928	−	0.560835i	$-0.189520\pi$
$251$	26.2337	1.65586	0.827928	+	0.560835i	$0.189520\pi$
$252$	0	0
$253$	−2.37228	−0.149144
$254$	0	0
$255$	9.48913	0.594232
$256$	0	0
$257$	13.1168	0.818206	0.409103	−	0.912488i	$-0.365842\pi$
$257$	13.1168	0.818206	0.409103	+	0.912488i	$0.365842\pi$
$258$	0	0
$259$	4.74456	0.294813
$260$	0	0
$261$	2.37228	0.146841
$262$	0	0
$263$	−5.48913	−0.338474	−0.169237	−	0.985575i	$-0.554130\pi$
$263$	−5.48913	−0.338474	−0.169237	+	0.985575i	$0.554130\pi$
$264$	0	0
$265$	23.7228	1.45728
$266$	0	0
$267$	8.00000	0.489592
$268$	0	0
$269$	8.23369	0.502017	0.251008	−	0.967985i	$-0.419238\pi$
$269$	8.23369	0.502017	0.251008	+	0.967985i	$0.419238\pi$
$270$	0	0
$271$	13.4891	0.819406	0.409703	−	0.912219i	$-0.365632\pi$
$271$	13.4891	0.819406	0.409703	+	0.912219i	$0.365632\pi$
$272$	0	0
$273$	2.37228	0.143577
$274$	0	0
$275$	0.627719	0.0378529
$276$	0	0
$277$	−11.6277	−0.698642	−0.349321	−	0.937003i	$-0.613588\pi$
$277$	−11.6277	−0.698642	−0.349321	+	0.937003i	$0.613588\pi$
$278$	0	0
$279$	2.74456	0.164313
$280$	0	0
$281$	28.3723	1.69255	0.846274	−	0.532748i	$-0.178840\pi$
$281$	28.3723	1.69255	0.846274	+	0.532748i	$0.178840\pi$
$282$	0	0
$283$	28.3723	1.68656	0.843279	−	0.537477i	$-0.180622\pi$
$283$	28.3723	1.68656	0.843279	+	0.537477i	$0.180622\pi$
$284$	0	0
$285$	11.2554	0.666715
$286$	0	0
$287$	0.883156	0.0521311
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	−4.51087	−0.263528	−0.131764	−	0.991281i	$-0.542064\pi$
$293$	−4.51087	−0.263528	−0.131764	+	0.991281i	$0.542064\pi$
$294$	0	0
$295$	−13.3505	−0.777298
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	2.37228	0.137193
$300$	0	0
$301$	−19.8614	−1.14479
$302$	0	0
$303$	−3.25544	−0.187020
$304$	0	0
$305$	21.6277	1.23840
$306$	0	0
$307$	31.7228	1.81052	0.905258	−	0.424862i	$-0.139677\pi$
$307$	31.7228	1.81052	0.905258	+	0.424862i	$0.139677\pi$
$308$	0	0
$309$	−3.11684	−0.177311
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	−10.8832	−0.615152	−0.307576	−	0.951523i	$-0.599518\pi$
$313$	−10.8832	−0.615152	−0.307576	+	0.951523i	$0.599518\pi$
$314$	0	0
$315$	5.62772	0.317086
$316$	0	0
$317$	11.8614	0.666203	0.333101	−	0.942891i	$-0.391905\pi$
$317$	11.8614	0.666203	0.333101	+	0.942891i	$0.391905\pi$
$318$	0	0
$319$	2.37228	0.132822
$320$	0	0
$321$	−6.37228	−0.355666
$322$	0	0
$323$	18.9783	1.05598
$324$	0	0
$325$	−0.627719	−0.0348196
$326$	0	0
$327$	1.25544	0.0694258
$328$	0	0
$329$	−9.48913	−0.523152
$330$	0	0
$331$	−32.3723	−1.77934	−0.889671	−	0.456603i	$-0.849066\pi$
$331$	−32.3723	−1.77934	−0.889671	+	0.456603i	$0.849066\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	−12.1386	−0.663202
$336$	0	0
$337$	−12.9783	−0.706970	−0.353485	−	0.935440i	$-0.615004\pi$
$337$	−12.9783	−0.706970	−0.353485	+	0.935440i	$0.615004\pi$
$338$	0	0
$339$	−3.62772	−0.197031
$340$	0	0
$341$	2.74456	0.148626
$342$	0	0
$343$	19.8614	1.07242
$344$	0	0
$345$	5.62772	0.302986
$346$	0	0
$347$	6.51087	0.349522	0.174761	−	0.984611i	$-0.444085\pi$
$347$	6.51087	0.349522	0.174761	+	0.984611i	$0.444085\pi$
$348$	0	0
$349$	−10.7446	−0.575143	−0.287572	−	0.957759i	$-0.592848\pi$
$349$	−10.7446	−0.575143	−0.287572	+	0.957759i	$0.592848\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−8.74456	−0.465426	−0.232713	−	0.972545i	$-0.574760\pi$
$353$	−8.74456	−0.465426	−0.232713	+	0.972545i	$0.574760\pi$
$354$	0	0
$355$	−7.72281	−0.409884
$356$	0	0
$357$	9.48913	0.502218
$358$	0	0
$359$	15.1168	0.797837	0.398918	−	0.916986i	$-0.369386\pi$
$359$	15.1168	0.797837	0.398918	+	0.916986i	$0.369386\pi$
$360$	0	0
$361$	3.51087	0.184783
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−21.6277	−1.13205
$366$	0	0
$367$	10.9783	0.573060	0.286530	−	0.958071i	$-0.407498\pi$
$367$	10.9783	0.573060	0.286530	+	0.958071i	$0.407498\pi$
$368$	0	0
$369$	−0.372281	−0.0193802
$370$	0	0
$371$	23.7228	1.23163
$372$	0	0
$373$	−17.1168	−0.886277	−0.443138	−	0.896453i	$-0.646135\pi$
$373$	−17.1168	−0.886277	−0.443138	+	0.896453i	$0.646135\pi$
$374$	0	0
$375$	10.3723	0.535622
$376$	0	0
$377$	−2.37228	−0.122179
$378$	0	0
$379$	28.2337	1.45027	0.725134	−	0.688608i	$-0.241778\pi$
$379$	28.2337	1.45027	0.725134	+	0.688608i	$0.241778\pi$
$380$	0	0
$381$	7.48913	0.383680
$382$	0	0
$383$	23.7228	1.21218	0.606090	−	0.795396i	$-0.292737\pi$
$383$	23.7228	1.21218	0.606090	+	0.795396i	$0.292737\pi$
$384$	0	0
$385$	5.62772	0.286815
$386$	0	0
$387$	8.37228	0.425587
$388$	0	0
$389$	28.9783	1.46926	0.734628	−	0.678470i	$-0.237357\pi$
$389$	28.9783	1.46926	0.734628	+	0.678470i	$0.237357\pi$
$390$	0	0
$391$	9.48913	0.479886
$392$	0	0
$393$	10.3723	0.523212
$394$	0	0
$395$	−23.7228	−1.19362
$396$	0	0
$397$	19.6277	0.985087	0.492543	−	0.870288i	$-0.336067\pi$
$397$	19.6277	0.985087	0.492543	+	0.870288i	$0.336067\pi$
$398$	0	0
$399$	11.2554	0.563477
$400$	0	0
$401$	29.4891	1.47262	0.736308	−	0.676646i	$-0.236567\pi$
$401$	29.4891	1.47262	0.736308	+	0.676646i	$0.236567\pi$
$402$	0	0
$403$	−2.74456	−0.136716
$404$	0	0
$405$	−2.37228	−0.117880
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	−30.6060	−1.51337	−0.756684	−	0.653781i	$-0.773182\pi$
$409$	−30.6060	−1.51337	−0.756684	+	0.653781i	$0.773182\pi$
$410$	0	0
$411$	12.7446	0.628643
$412$	0	0
$413$	−13.3505	−0.656937
$414$	0	0
$415$	−20.7446	−1.01831
$416$	0	0
$417$	20.2337	0.990848
$418$	0	0
$419$	10.5109	0.513490	0.256745	−	0.966479i	$-0.417350\pi$
$419$	10.5109	0.513490	0.256745	+	0.966479i	$0.417350\pi$
$420$	0	0
$421$	−14.8832	−0.725361	−0.362680	−	0.931914i	$-0.618138\pi$
$421$	−14.8832	−0.725361	−0.362680	+	0.931914i	$0.618138\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	−2.51087	−0.121795
$426$	0	0
$427$	21.6277	1.04664
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−18.9783	−0.914150	−0.457075	−	0.889428i	$-0.651103\pi$
$431$	−18.9783	−0.914150	−0.457075	+	0.889428i	$0.651103\pi$
$432$	0	0
$433$	−13.1168	−0.630355	−0.315178	−	0.949033i	$-0.602064\pi$
$433$	−13.1168	−0.630355	−0.315178	+	0.949033i	$0.602064\pi$
$434$	0	0
$435$	−5.62772	−0.269828
$436$	0	0
$437$	11.2554	0.538421
$438$	0	0
$439$	26.7446	1.27645	0.638224	−	0.769851i	$-0.279669\pi$
$439$	26.7446	1.27645	0.638224	+	0.769851i	$0.279669\pi$
$440$	0	0
$441$	−1.37228	−0.0653467
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	−18.9783	−0.899655
$446$	0	0
$447$	5.25544	0.248574
$448$	0	0
$449$	−34.9783	−1.65073	−0.825363	−	0.564603i	$-0.809029\pi$
$449$	−34.9783	−1.65073	−0.825363	+	0.564603i	$0.809029\pi$
$450$	0	0
$451$	−0.372281	−0.0175300
$452$	0	0
$453$	−21.4891	−1.00965
$454$	0	0
$455$	−5.62772	−0.263832
$456$	0	0
$457$	0.372281	0.0174146	0.00870729	−	0.999962i	$-0.497228\pi$
$457$	0.372281	0.0174146	0.00870729	+	0.999962i	$0.497228\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	36.2337	1.68757	0.843786	−	0.536680i	$-0.180322\pi$
$461$	36.2337	1.68757	0.843786	+	0.536680i	$0.180322\pi$
$462$	0	0
$463$	−12.5109	−0.581430	−0.290715	−	0.956810i	$-0.593893\pi$
$463$	−12.5109	−0.581430	−0.290715	+	0.956810i	$0.593893\pi$
$464$	0	0
$465$	−6.51087	−0.301935
$466$	0	0
$467$	−38.9783	−1.80370	−0.901849	−	0.432051i	$-0.857790\pi$
$467$	−38.9783	−1.80370	−0.901849	+	0.432051i	$0.857790\pi$
$468$	0	0
$469$	−12.1386	−0.560508
$470$	0	0
$471$	−15.4891	−0.713701
$472$	0	0
$473$	8.37228	0.384958
$474$	0	0
$475$	−2.97825	−0.136652
$476$	0	0
$477$	−10.0000	−0.457869
$478$	0	0
$479$	−21.3505	−0.975531	−0.487765	−	0.872975i	$-0.662188\pi$
$479$	−21.3505	−0.975531	−0.487765	+	0.872975i	$0.662188\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	0	0
$483$	5.62772	0.256070
$484$	0	0
$485$	23.7228	1.07720
$486$	0	0
$487$	−38.4674	−1.74312	−0.871562	−	0.490286i	$-0.836892\pi$
$487$	−38.4674	−1.74312	−0.871562	+	0.490286i	$0.836892\pi$
$488$	0	0
$489$	−3.62772	−0.164051
$490$	0	0
$491$	34.3723	1.55120	0.775600	−	0.631225i	$-0.217448\pi$
$491$	34.3723	1.55120	0.775600	+	0.631225i	$0.217448\pi$
$492$	0	0
$493$	−9.48913	−0.427369
$494$	0	0
$495$	−2.37228	−0.106626
$496$	0	0
$497$	−7.72281	−0.346416
$498$	0	0
$499$	−6.13859	−0.274801	−0.137401	−	0.990516i	$-0.543875\pi$
$499$	−6.13859	−0.274801	−0.137401	+	0.990516i	$0.543875\pi$
$500$	0	0
$501$	15.1168	0.675371
$502$	0	0
$503$	−5.48913	−0.244748	−0.122374	−	0.992484i	$-0.539051\pi$
$503$	−5.48913	−0.244748	−0.122374	+	0.992484i	$0.539051\pi$
$504$	0	0
$505$	7.72281	0.343661
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	15.2554	0.676185	0.338093	−	0.941113i	$-0.390218\pi$
$509$	15.2554	0.676185	0.338093	+	0.941113i	$0.390218\pi$
$510$	0	0
$511$	−21.6277	−0.956754
$512$	0	0
$513$	−4.74456	−0.209478
$514$	0	0
$515$	7.39403	0.325820
$516$	0	0
$517$	4.00000	0.175920
$518$	0	0
$519$	−11.8614	−0.520658
$520$	0	0
$521$	33.1168	1.45088	0.725438	−	0.688288i	$-0.241637\pi$
$521$	33.1168	1.45088	0.725438	+	0.688288i	$0.241637\pi$
$522$	0	0
$523$	−18.7446	−0.819642	−0.409821	−	0.912166i	$-0.634409\pi$
$523$	−18.7446	−0.819642	−0.409821	+	0.912166i	$0.634409\pi$
$524$	0	0
$525$	−1.48913	−0.0649908
$526$	0	0
$527$	−10.9783	−0.478220
$528$	0	0
$529$	−17.3723	−0.755317
$530$	0	0
$531$	5.62772	0.244222
$532$	0	0
$533$	0.372281	0.0161253
$534$	0	0
$535$	15.1168	0.653558
$536$	0	0
$537$	−5.48913	−0.236873
$538$	0	0
$539$	−1.37228	−0.0591083
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	0	0
$543$	10.0000	0.429141
$544$	0	0
$545$	−2.97825	−0.127574
$546$	0	0
$547$	2.88316	0.123275	0.0616374	−	0.998099i	$-0.480368\pi$
$547$	2.88316	0.123275	0.0616374	+	0.998099i	$0.480368\pi$
$548$	0	0
$549$	−9.11684	−0.389097
$550$	0	0
$551$	−11.2554	−0.479498
$552$	0	0
$553$	−23.7228	−1.00880
$554$	0	0
$555$	4.74456	0.201395
$556$	0	0
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	0	0
$559$	−8.37228	−0.354110
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	−34.9783	−1.47416	−0.737079	−	0.675807i	$-0.763795\pi$
$563$	−34.9783	−1.47416	−0.737079	+	0.675807i	$0.763795\pi$
$564$	0	0
$565$	8.60597	0.362056
$566$	0	0
$567$	−2.37228	−0.0996265
$568$	0	0
$569$	30.2337	1.26746	0.633731	−	0.773553i	$-0.281523\pi$
$569$	30.2337	1.26746	0.633731	+	0.773553i	$0.281523\pi$
$570$	0	0
$571$	−5.11684	−0.214133	−0.107067	−	0.994252i	$-0.534146\pi$
$571$	−5.11684	−0.214133	−0.107067	+	0.994252i	$0.534146\pi$
$572$	0	0
$573$	−13.6277	−0.569306
$574$	0	0
$575$	−1.48913	−0.0621008
$576$	0	0
$577$	0.233688	0.00972856	0.00486428	−	0.999988i	$-0.498452\pi$
$577$	0.233688	0.00972856	0.00486428	+	0.999988i	$0.498452\pi$
$578$	0	0
$579$	11.4891	0.477472
$580$	0	0
$581$	−20.7446	−0.860629
$582$	0	0
$583$	−10.0000	−0.414158
$584$	0	0
$585$	2.37228	0.0980818
$586$	0	0
$587$	10.3723	0.428110	0.214055	−	0.976822i	$-0.431333\pi$
$587$	10.3723	0.428110	0.214055	+	0.976822i	$0.431333\pi$
$588$	0	0
$589$	−13.0217	−0.536552
$590$	0	0
$591$	−1.25544	−0.0516418
$592$	0	0
$593$	−28.9783	−1.18999	−0.594997	−	0.803728i	$-0.702847\pi$
$593$	−28.9783	−1.18999	−0.594997	+	0.803728i	$0.702847\pi$
$594$	0	0
$595$	−22.5109	−0.922856
$596$	0	0
$597$	−23.8614	−0.976582
$598$	0	0
$599$	31.1168	1.27140	0.635700	−	0.771936i	$-0.280712\pi$
$599$	31.1168	1.27140	0.635700	+	0.771936i	$0.280712\pi$
$600$	0	0
$601$	−1.25544	−0.0512104	−0.0256052	−	0.999672i	$-0.508151\pi$
$601$	−1.25544	−0.0512104	−0.0256052	+	0.999672i	$0.508151\pi$
$602$	0	0
$603$	5.11684	0.208374
$604$	0	0
$605$	−2.37228	−0.0964470
$606$	0	0
$607$	32.9783	1.33855	0.669273	−	0.743017i	$-0.266606\pi$
$607$	32.9783	1.33855	0.669273	+	0.743017i	$0.266606\pi$
$608$	0	0
$609$	−5.62772	−0.228047
$610$	0	0
$611$	−4.00000	−0.161823
$612$	0	0
$613$	−33.7228	−1.36205	−0.681026	−	0.732259i	$-0.738466\pi$
$613$	−33.7228	−1.36205	−0.681026	+	0.732259i	$0.738466\pi$
$614$	0	0
$615$	0.883156	0.0356123
$616$	0	0
$617$	−3.25544	−0.131059	−0.0655295	−	0.997851i	$-0.520874\pi$
$617$	−3.25544	−0.131059	−0.0655295	+	0.997851i	$0.520874\pi$
$618$	0	0
$619$	5.11684	0.205663	0.102832	−	0.994699i	$-0.467210\pi$
$619$	5.11684	0.205663	0.102832	+	0.994699i	$0.467210\pi$
$620$	0	0
$621$	−2.37228	−0.0951964
$622$	0	0
$623$	−18.9783	−0.760348
$624$	0	0
$625$	−27.7446	−1.10978
$626$	0	0
$627$	−4.74456	−0.189480
$628$	0	0
$629$	8.00000	0.318981
$630$	0	0
$631$	−30.0000	−1.19428	−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000	−1.19428	−0.597141	+	0.802137i	$0.703697\pi$
$632$	0	0
$633$	−2.74456	−0.109087
$634$	0	0
$635$	−17.7663	−0.705035
$636$	0	0
$637$	1.37228	0.0543718
$638$	0	0
$639$	3.25544	0.128783
$640$	0	0
$641$	−44.8397	−1.77106	−0.885530	−	0.464582i	$-0.846205\pi$
$641$	−44.8397	−1.77106	−0.885530	+	0.464582i	$0.846205\pi$
$642$	0	0
$643$	13.2554	0.522743	0.261372	−	0.965238i	$-0.415825\pi$
$643$	13.2554	0.522743	0.261372	+	0.965238i	$0.415825\pi$
$644$	0	0
$645$	−19.8614	−0.782042
$646$	0	0
$647$	10.9783	0.431600	0.215800	−	0.976438i	$-0.430764\pi$
$647$	10.9783	0.431600	0.215800	+	0.976438i	$0.430764\pi$
$648$	0	0
$649$	5.62772	0.220907
$650$	0	0
$651$	−6.51087	−0.255181
$652$	0	0
$653$	−15.4891	−0.606136	−0.303068	−	0.952969i	$-0.598011\pi$
$653$	−15.4891	−0.606136	−0.303068	+	0.952969i	$0.598011\pi$
$654$	0	0
$655$	−24.6060	−0.961435
$656$	0	0
$657$	9.11684	0.355682
$658$	0	0
$659$	13.4891	0.525462	0.262731	−	0.964869i	$-0.415377\pi$
$659$	13.4891	0.525462	0.262731	+	0.964869i	$0.415377\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	4.00000	0.155347
$664$	0	0
$665$	−26.7011	−1.03542
$666$	0	0
$667$	−5.62772	−0.217906
$668$	0	0
$669$	−10.7446	−0.415409
$670$	0	0
$671$	−9.11684	−0.351952
$672$	0	0
$673$	24.2337	0.934140	0.467070	−	0.884220i	$-0.345310\pi$
$673$	24.2337	0.934140	0.467070	+	0.884220i	$0.345310\pi$
$674$	0	0
$675$	0.627719	0.0241609
$676$	0	0
$677$	−2.23369	−0.0858476	−0.0429238	−	0.999078i	$-0.513667\pi$
$677$	−2.23369	−0.0858476	−0.0429238	+	0.999078i	$0.513667\pi$
$678$	0	0
$679$	23.7228	0.910398
$680$	0	0
$681$	22.9783	0.880528
$682$	0	0
$683$	51.5842	1.97382	0.986908	−	0.161286i	$-0.0515643\pi$
$683$	51.5842	1.97382	0.986908	+	0.161286i	$0.0515643\pi$
$684$	0	0
$685$	−30.2337	−1.15517
$686$	0	0
$687$	18.6060	0.709862
$688$	0	0
$689$	10.0000	0.380970
$690$	0	0
$691$	−31.2119	−1.18736	−0.593679	−	0.804702i	$-0.702325\pi$
$691$	−31.2119	−1.18736	−0.593679	+	0.804702i	$0.702325\pi$
$692$	0	0
$693$	−2.37228	−0.0901155
$694$	0	0
$695$	−48.0000	−1.82074
$696$	0	0
$697$	1.48913	0.0564046
$698$	0	0
$699$	16.7446	0.633338
$700$	0	0
$701$	46.8397	1.76911	0.884555	−	0.466436i	$-0.154462\pi$
$701$	46.8397	1.76911	0.884555	+	0.466436i	$0.154462\pi$
$702$	0	0
$703$	9.48913	0.357889
$704$	0	0
$705$	−9.48913	−0.357381
$706$	0	0
$707$	7.72281	0.290446
$708$	0	0
$709$	23.6277	0.887358	0.443679	−	0.896186i	$-0.353673\pi$
$709$	23.6277	0.887358	0.443679	+	0.896186i	$0.353673\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	0	0
$713$	−6.51087	−0.243834
$714$	0	0
$715$	2.37228	0.0887183
$716$	0	0
$717$	3.86141	0.144207
$718$	0	0
$719$	7.39403	0.275751	0.137875	−	0.990450i	$-0.455973\pi$
$719$	7.39403	0.275751	0.137875	+	0.990450i	$0.455973\pi$
$720$	0	0
$721$	7.39403	0.275368
$722$	0	0
$723$	14.0000	0.520666
$724$	0	0
$725$	1.48913	0.0553047
$726$	0	0
$727$	−4.00000	−0.148352	−0.0741759	−	0.997245i	$-0.523633\pi$
$727$	−4.00000	−0.148352	−0.0741759	+	0.997245i	$0.523633\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−33.4891	−1.23864
$732$	0	0
$733$	0.978251	0.0361325	0.0180663	−	0.999837i	$-0.494249\pi$
$733$	0.978251	0.0361325	0.0180663	+	0.999837i	$0.494249\pi$
$734$	0	0
$735$	3.25544	0.120079
$736$	0	0
$737$	5.11684	0.188481
$738$	0	0
$739$	−32.4674	−1.19433	−0.597166	−	0.802118i	$-0.703707\pi$
$739$	−32.4674	−1.19433	−0.597166	+	0.802118i	$0.703707\pi$
$740$	0	0
$741$	4.74456	0.174296
$742$	0	0
$743$	−38.8397	−1.42489	−0.712444	−	0.701729i	$-0.752412\pi$
$743$	−38.8397	−1.42489	−0.712444	+	0.701729i	$0.752412\pi$
$744$	0	0
$745$	−12.4674	−0.456769
$746$	0	0
$747$	8.74456	0.319947
$748$	0	0
$749$	15.1168	0.552357
$750$	0	0
$751$	−34.3723	−1.25426	−0.627131	−	0.778914i	$-0.715771\pi$
$751$	−34.3723	−1.25426	−0.627131	+	0.778914i	$0.715771\pi$
$752$	0	0
$753$	26.2337	0.956009
$754$	0	0
$755$	50.9783	1.85529
$756$	0	0
$757$	28.9783	1.05323	0.526616	−	0.850103i	$-0.323461\pi$
$757$	28.9783	1.05323	0.526616	+	0.850103i	$0.323461\pi$
$758$	0	0
$759$	−2.37228	−0.0861084
$760$	0	0
$761$	−41.1168	−1.49048	−0.745242	−	0.666794i	$-0.767666\pi$
$761$	−41.1168	−1.49048	−0.745242	+	0.666794i	$0.767666\pi$
$762$	0	0
$763$	−2.97825	−0.107820
$764$	0	0
$765$	9.48913	0.343080
$766$	0	0
$767$	−5.62772	−0.203205
$768$	0	0
$769$	−28.3723	−1.02313	−0.511565	−	0.859244i	$-0.670934\pi$
$769$	−28.3723	−1.02313	−0.511565	+	0.859244i	$0.670934\pi$
$770$	0	0
$771$	13.1168	0.472392
$772$	0	0
$773$	38.2337	1.37517	0.687585	−	0.726104i	$-0.258671\pi$
$773$	38.2337	1.37517	0.687585	+	0.726104i	$0.258671\pi$
$774$	0	0
$775$	1.72281	0.0618853
$776$	0	0
$777$	4.74456	0.170210
$778$	0	0
$779$	1.76631	0.0632847
$780$	0	0
$781$	3.25544	0.116489
$782$	0	0
$783$	2.37228	0.0847784
$784$	0	0
$785$	36.7446	1.31147
$786$	0	0
$787$	−46.9783	−1.67459	−0.837297	−	0.546748i	$-0.815865\pi$
$787$	−46.9783	−1.67459	−0.837297	+	0.546748i	$0.815865\pi$
$788$	0	0
$789$	−5.48913	−0.195418
$790$	0	0
$791$	8.60597	0.305993
$792$	0	0
$793$	9.11684	0.323749
$794$	0	0
$795$	23.7228	0.841361
$796$	0	0
$797$	−29.7228	−1.05284	−0.526418	−	0.850226i	$-0.676465\pi$
$797$	−29.7228	−1.05284	−0.526418	+	0.850226i	$0.676465\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	8.00000	0.282666
$802$	0	0
$803$	9.11684	0.321726
$804$	0	0
$805$	−13.3505	−0.470544
$806$	0	0
$807$	8.23369	0.289840
$808$	0	0
$809$	−27.2554	−0.958250	−0.479125	−	0.877747i	$-0.659046\pi$
$809$	−27.2554	−0.958250	−0.479125	+	0.877747i	$0.659046\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	13.4891	0.473084
$814$	0	0
$815$	8.60597	0.301454
$816$	0	0
$817$	−39.7228	−1.38973
$818$	0	0
$819$	2.37228	0.0828942
$820$	0	0
$821$	40.9783	1.43015	0.715075	−	0.699047i	$-0.246392\pi$
$821$	40.9783	1.43015	0.715075	+	0.699047i	$0.246392\pi$
$822$	0	0
$823$	−3.39403	−0.118308	−0.0591542	−	0.998249i	$-0.518840\pi$
$823$	−3.39403	−0.118308	−0.0591542	+	0.998249i	$0.518840\pi$
$824$	0	0
$825$	0.627719	0.0218544
$826$	0	0
$827$	−24.4674	−0.850814	−0.425407	−	0.905002i	$-0.639869\pi$
$827$	−24.4674	−0.850814	−0.425407	+	0.905002i	$0.639869\pi$
$828$	0	0
$829$	48.9783	1.70108	0.850542	−	0.525906i	$-0.176274\pi$
$829$	48.9783	1.70108	0.850542	+	0.525906i	$0.176274\pi$
$830$	0	0
$831$	−11.6277	−0.403361
$832$	0	0
$833$	5.48913	0.190187
$834$	0	0
$835$	−35.8614	−1.24104
$836$	0	0
$837$	2.74456	0.0948660
$838$	0	0
$839$	24.7446	0.854277	0.427139	−	0.904186i	$-0.359522\pi$
$839$	24.7446	0.854277	0.427139	+	0.904186i	$0.359522\pi$
$840$	0	0
$841$	−23.3723	−0.805941
$842$	0	0
$843$	28.3723	0.977193
$844$	0	0
$845$	−2.37228	−0.0816090
$846$	0	0
$847$	−2.37228	−0.0815126
$848$	0	0
$849$	28.3723	0.973734
$850$	0	0
$851$	4.74456	0.162642
$852$	0	0
$853$	0.233688	0.00800132	0.00400066	−	0.999992i	$-0.498727\pi$
$853$	0.233688	0.00800132	0.00400066	+	0.999992i	$0.498727\pi$
$854$	0	0
$855$	11.2554	0.384928
$856$	0	0
$857$	5.02175	0.171540	0.0857698	−	0.996315i	$-0.472665\pi$
$857$	5.02175	0.171540	0.0857698	+	0.996315i	$0.472665\pi$
$858$	0	0
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	0	0
$861$	0.883156	0.0300979
$862$	0	0
$863$	−6.51087	−0.221633	−0.110816	−	0.993841i	$-0.535347\pi$
$863$	−6.51087	−0.221633	−0.110816	+	0.993841i	$0.535347\pi$
$864$	0	0
$865$	28.1386	0.956741
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	10.0000	0.339227
$870$	0	0
$871$	−5.11684	−0.173378
$872$	0	0
$873$	−10.0000	−0.338449
$874$	0	0
$875$	−24.6060	−0.831834
$876$	0	0
$877$	−5.72281	−0.193246	−0.0966228	−	0.995321i	$-0.530804\pi$
$877$	−5.72281	−0.193246	−0.0966228	+	0.995321i	$0.530804\pi$
$878$	0	0
$879$	−4.51087	−0.152148
$880$	0	0
$881$	−2.13859	−0.0720510	−0.0360255	−	0.999351i	$-0.511470\pi$
$881$	−2.13859	−0.0720510	−0.0360255	+	0.999351i	$0.511470\pi$
$882$	0	0
$883$	−41.2119	−1.38689	−0.693446	−	0.720509i	$-0.743908\pi$
$883$	−41.2119	−1.38689	−0.693446	+	0.720509i	$0.743908\pi$
$884$	0	0
$885$	−13.3505	−0.448773
$886$	0	0
$887$	−22.9783	−0.771534	−0.385767	−	0.922596i	$-0.626063\pi$
$887$	−22.9783	−0.771534	−0.385767	+	0.922596i	$0.626063\pi$
$888$	0	0
$889$	−17.7663	−0.595863
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−18.9783	−0.635083
$894$	0	0
$895$	13.0217	0.435269
$896$	0	0
$897$	2.37228	0.0792082
$898$	0	0
$899$	6.51087	0.217150
$900$	0	0
$901$	40.0000	1.33259
$902$	0	0
$903$	−19.8614	−0.660946
$904$	0	0
$905$	−23.7228	−0.788573
$906$	0	0
$907$	−32.4674	−1.07806	−0.539031	−	0.842286i	$-0.681209\pi$
$907$	−32.4674	−1.07806	−0.539031	+	0.842286i	$0.681209\pi$
$908$	0	0
$909$	−3.25544	−0.107976
$910$	0	0
$911$	−2.97825	−0.0986738	−0.0493369	−	0.998782i	$-0.515711\pi$
$911$	−2.97825	−0.0986738	−0.0493369	+	0.998782i	$0.515711\pi$
$912$	0	0
$913$	8.74456	0.289403
$914$	0	0
$915$	21.6277	0.714990
$916$	0	0
$917$	−24.6060	−0.812561
$918$	0	0
$919$	56.7011	1.87040	0.935198	−	0.354126i	$-0.115222\pi$
$919$	56.7011	1.87040	0.935198	+	0.354126i	$0.115222\pi$
$920$	0	0
$921$	31.7228	1.04530
$922$	0	0
$923$	−3.25544	−0.107154
$924$	0	0
$925$	−1.25544	−0.0412785
$926$	0	0
$927$	−3.11684	−0.102371
$928$	0	0
$929$	28.4674	0.933984	0.466992	−	0.884261i	$-0.345338\pi$
$929$	28.4674	0.933984	0.466992	+	0.884261i	$0.345338\pi$
$930$	0	0
$931$	6.51087	0.213385
$932$	0	0
$933$	−8.00000	−0.261908
$934$	0	0
$935$	9.48913	0.310328
$936$	0	0
$937$	43.2119	1.41167	0.705836	−	0.708375i	$-0.250571\pi$
$937$	43.2119	1.41167	0.705836	+	0.708375i	$0.250571\pi$
$938$	0	0
$939$	−10.8832	−0.355158
$940$	0	0
$941$	−21.2554	−0.692907	−0.346454	−	0.938067i	$-0.612614\pi$
$941$	−21.2554	−0.692907	−0.346454	+	0.938067i	$0.612614\pi$
$942$	0	0
$943$	0.883156	0.0287595
$944$	0	0
$945$	5.62772	0.183070
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	−9.11684	−0.295945
$950$	0	0
$951$	11.8614	0.384632
$952$	0	0
$953$	−34.9783	−1.13306	−0.566528	−	0.824042i	$-0.691714\pi$
$953$	−34.9783	−1.13306	−0.566528	+	0.824042i	$0.691714\pi$
$954$	0	0
$955$	32.3288	1.04613
$956$	0	0
$957$	2.37228	0.0766850
$958$	0	0
$959$	−30.2337	−0.976297
$960$	0	0
$961$	−23.4674	−0.757012
$962$	0	0
$963$	−6.37228	−0.205344
$964$	0	0
$965$	−27.2554	−0.877384
$966$	0	0
$967$	42.3723	1.36260	0.681300	−	0.732004i	$-0.261415\pi$
$967$	42.3723	1.36260	0.681300	+	0.732004i	$0.261415\pi$
$968$	0	0
$969$	18.9783	0.609669
$970$	0	0
$971$	7.25544	0.232838	0.116419	−	0.993200i	$-0.462858\pi$
$971$	7.25544	0.232838	0.116419	+	0.993200i	$0.462858\pi$
$972$	0	0
$973$	−48.0000	−1.53881
$974$	0	0
$975$	−0.627719	−0.0201031
$976$	0	0
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	0	0
$979$	8.00000	0.255681
$980$	0	0
$981$	1.25544	0.0400830
$982$	0	0
$983$	7.25544	0.231413	0.115706	−	0.993283i	$-0.463087\pi$
$983$	7.25544	0.231413	0.115706	+	0.993283i	$0.463087\pi$
$984$	0	0
$985$	2.97825	0.0948950
$986$	0	0
$987$	−9.48913	−0.302042
$988$	0	0
$989$	−19.8614	−0.631556
$990$	0	0
$991$	−33.6277	−1.06822	−0.534110	−	0.845415i	$-0.679353\pi$
$991$	−33.6277	−1.06822	−0.534110	+	0.845415i	$0.679353\pi$
$992$	0	0
$993$	−32.3723	−1.02730
$994$	0	0
$995$	56.6060	1.79453
$996$	0	0
$997$	6.60597	0.209213	0.104607	−	0.994514i	$-0.466642\pi$
$997$	6.60597	0.209213	0.104607	+	0.994514i	$0.466642\pi$
$998$	0	0
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bl.1.1		2
4.3	odd	2		858.2.a.o.1.1	✓	2
12.11	even	2		2574.2.a.bc.1.2		2
44.43	even	2		9438.2.a.cb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
858.2.a.o.1.1	✓	2	4.3	odd	2
2574.2.a.bc.1.2		2	12.11	even	2
6864.2.a.bl.1.1		2	1.1	even	1	trivial
9438.2.a.cb.1.1		2	44.43	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.37228 q^{5} -2.37228 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.37228 q^{5} -2.37228 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -2.37228 q^{15} -4.00000 q^{17} -4.74456 q^{19} -2.37228 q^{21} -2.37228 q^{23} +0.627719 q^{25} +1.00000 q^{27} +2.37228 q^{29} +2.74456 q^{31} +1.00000 q^{33} +5.62772 q^{35} -2.00000 q^{37} -1.00000 q^{39} -0.372281 q^{41} +8.37228 q^{43} -2.37228 q^{45} +4.00000 q^{47} -1.37228 q^{49} -4.00000 q^{51} -10.0000 q^{53} -2.37228 q^{55} -4.74456 q^{57} +5.62772 q^{59} -9.11684 q^{61} -2.37228 q^{63} +2.37228 q^{65} +5.11684 q^{67} -2.37228 q^{69} +3.25544 q^{71} +9.11684 q^{73} +0.627719 q^{75} -2.37228 q^{77} +10.0000 q^{79} +1.00000 q^{81} +8.74456 q^{83} +9.48913 q^{85} +2.37228 q^{87} +8.00000 q^{89} +2.37228 q^{91} +2.74456 q^{93} +11.2554 q^{95} -10.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + q^5 + q^7 + 2 * q^9	\( 2 q + 2 q^{3} + q^{5} + q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{13} + q^{15} - 8 q^{17} + 2 q^{19} + q^{21} + q^{23} + 7 q^{25} + 2 q^{27} - q^{29} - 6 q^{31} + 2 q^{33} + 17 q^{35} - 4 q^{37} - 2 q^{39} + 5 q^{41} + 11 q^{43} + q^{45} + 8 q^{47} + 3 q^{49} - 8 q^{51} - 20 q^{53} + q^{55} + 2 q^{57} + 17 q^{59} - q^{61} + q^{63} - q^{65} - 7 q^{67} + q^{69} + 18 q^{71} + q^{73} + 7 q^{75} + q^{77} + 20 q^{79} + 2 q^{81} + 6 q^{83} - 4 q^{85} - q^{87} + 16 q^{89} - q^{91} - 6 q^{93} + 34 q^{95} - 20 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + q^5 + q^7 + 2 * q^9 + 2 * q^11 - 2 * q^13 + q^15 - 8 * q^17 + 2 * q^19 + q^21 + q^23 + 7 * q^25 + 2 * q^27 - q^29 - 6 * q^31 + 2 * q^33 + 17 * q^35 - 4 * q^37 - 2 * q^39 + 5 * q^41 + 11 * q^43 + q^45 + 8 * q^47 + 3 * q^49 - 8 * q^51 - 20 * q^53 + q^55 + 2 * q^57 + 17 * q^59 - q^61 + q^63 - q^65 - 7 * q^67 + q^69 + 18 * q^71 + q^73 + 7 * q^75 + q^77 + 20 * q^79 + 2 * q^81 + 6 * q^83 - 4 * q^85 - q^87 + 16 * q^89 - q^91 - 6 * q^93 + 34 * q^95 - 20 * q^97 + 2 * q^99

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