LMFDB - Embedded newform 6864.2.a.be.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3432)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.41421 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.41421 q^{5} +2.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +1.41421 q^{15} -0.585786 q^{17} -0.828427 q^{19} -2.00000 q^{21} -4.82843 q^{23} -3.00000 q^{25} -1.00000 q^{27} +5.07107 q^{29} +2.24264 q^{31} -1.00000 q^{33} -2.82843 q^{35} -0.343146 q^{37} -1.00000 q^{39} -6.82843 q^{41} -8.24264 q^{43} -1.41421 q^{45} +12.4853 q^{47} -3.00000 q^{49} +0.585786 q^{51} +2.00000 q^{53} -1.41421 q^{55} +0.828427 q^{57} +12.4853 q^{59} -10.0000 q^{61} +2.00000 q^{63} -1.41421 q^{65} +7.89949 q^{67} +4.82843 q^{69} -11.3137 q^{71} -16.4853 q^{73} +3.00000 q^{75} +2.00000 q^{77} +0.242641 q^{79} +1.00000 q^{81} -5.65685 q^{83} +0.828427 q^{85} -5.07107 q^{87} -8.24264 q^{89} +2.00000 q^{91} -2.24264 q^{93} +1.17157 q^{95} -7.65685 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 4 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 4 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{17} + 4 q^{19} - 4 q^{21} - 4 q^{23} - 6 q^{25} - 2 q^{27} - 4 q^{29} - 4 q^{31} - 2 q^{33} - 12 q^{37} - 2 q^{39} - 8 q^{41} - 8 q^{43} + 8 q^{47} - 6 q^{49} + 4 q^{51} + 4 q^{53} - 4 q^{57} + 8 q^{59} - 20 q^{61} + 4 q^{63} - 4 q^{67} + 4 q^{69} - 16 q^{73} + 6 q^{75} + 4 q^{77} - 8 q^{79} + 2 q^{81} - 4 q^{85} + 4 q^{87} - 8 q^{89} + 4 q^{91} + 4 q^{93} + 8 q^{95} - 4 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 4 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 - 4 * q^17 + 4 * q^19 - 4 * q^21 - 4 * q^23 - 6 * q^25 - 2 * q^27 - 4 * q^29 - 4 * q^31 - 2 * q^33 - 12 * q^37 - 2 * q^39 - 8 * q^41 - 8 * q^43 + 8 * q^47 - 6 * q^49 + 4 * q^51 + 4 * q^53 - 4 * q^57 + 8 * q^59 - 20 * q^61 + 4 * q^63 - 4 * q^67 + 4 * q^69 - 16 * q^73 + 6 * q^75 + 4 * q^77 - 8 * q^79 + 2 * q^81 - 4 * q^85 + 4 * q^87 - 8 * q^89 + 4 * q^91 + 4 * q^93 + 8 * q^95 - 4 * 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
$5$	−1.41421	−0.632456	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	−1.41421	−0.632456	−0.316228	+	0.948683i	$0.602416\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	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
$15$	1.41421	0.365148
$16$	0	0
$17$	−0.585786	−0.142074	−0.0710370	−	0.997474i	$-0.522631\pi$
$17$	−0.585786	−0.142074	−0.0710370	+	0.997474i	$0.522631\pi$
$18$	0	0
$19$	−0.828427	−0.190054	−0.0950271	−	0.995475i	$-0.530294\pi$
$19$	−0.828427	−0.190054	−0.0950271	+	0.995475i	$0.530294\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	−4.82843	−1.00680	−0.503398	−	0.864054i	$-0.667917\pi$
$23$	−4.82843	−1.00680	−0.503398	+	0.864054i	$0.667917\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.07107	0.941674	0.470837	−	0.882220i	$-0.343952\pi$
$29$	5.07107	0.941674	0.470837	+	0.882220i	$0.343952\pi$
$30$	0	0
$31$	2.24264	0.402790	0.201395	−	0.979510i	$-0.435452\pi$
$31$	2.24264	0.402790	0.201395	+	0.979510i	$0.435452\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−2.82843	−0.478091
$36$	0	0
$37$	−0.343146	−0.0564128	−0.0282064	−	0.999602i	$-0.508980\pi$
$37$	−0.343146	−0.0564128	−0.0282064	+	0.999602i	$0.508980\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−6.82843	−1.06642	−0.533211	−	0.845983i	$-0.679015\pi$
$41$	−6.82843	−1.06642	−0.533211	+	0.845983i	$0.679015\pi$
$42$	0	0
$43$	−8.24264	−1.25699	−0.628495	−	0.777813i	$-0.716329\pi$
$43$	−8.24264	−1.25699	−0.628495	+	0.777813i	$0.716329\pi$
$44$	0	0
$45$	−1.41421	−0.210819
$46$	0	0
$47$	12.4853	1.82117	0.910583	−	0.413327i	$-0.135633\pi$
$47$	12.4853	1.82117	0.910583	+	0.413327i	$0.135633\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0.585786	0.0820265
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	−1.41421	−0.190693
$56$	0	0
$57$	0.828427	0.109728
$58$	0	0
$59$	12.4853	1.62545	0.812723	−	0.582651i	$-0.197984\pi$
$59$	12.4853	1.62545	0.812723	+	0.582651i	$0.197984\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	−1.41421	−0.175412
$66$	0	0
$67$	7.89949	0.965077	0.482538	−	0.875875i	$-0.339715\pi$
$67$	7.89949	0.965077	0.482538	+	0.875875i	$0.339715\pi$
$68$	0	0
$69$	4.82843	0.581274
$70$	0	0
$71$	−11.3137	−1.34269	−0.671345	−	0.741145i	$-0.734283\pi$
$71$	−11.3137	−1.34269	−0.671345	+	0.741145i	$0.734283\pi$
$72$	0	0
$73$	−16.4853	−1.92946	−0.964728	−	0.263248i	$-0.915206\pi$
$73$	−16.4853	−1.92946	−0.964728	+	0.263248i	$0.915206\pi$
$74$	0	0
$75$	3.00000	0.346410
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	0.242641	0.0272992	0.0136496	−	0.999907i	$-0.495655\pi$
$79$	0.242641	0.0272992	0.0136496	+	0.999907i	$0.495655\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.65685	−0.620920	−0.310460	−	0.950586i	$-0.600483\pi$
$83$	−5.65685	−0.620920	−0.310460	+	0.950586i	$0.600483\pi$
$84$	0	0
$85$	0.828427	0.0898555
$86$	0	0
$87$	−5.07107	−0.543676
$88$	0	0
$89$	−8.24264	−0.873718	−0.436859	−	0.899530i	$-0.643909\pi$
$89$	−8.24264	−0.873718	−0.436859	+	0.899530i	$0.643909\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	−2.24264	−0.232551
$94$	0	0
$95$	1.17157	0.120201
$96$	0	0
$97$	−7.65685	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$97$	−7.65685	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	6.24264	0.621166	0.310583	−	0.950546i	$-0.399476\pi$
$101$	6.24264	0.621166	0.310583	+	0.950546i	$0.399476\pi$
$102$	0	0
$103$	−1.17157	−0.115439	−0.0577193	−	0.998333i	$-0.518383\pi$
$103$	−1.17157	−0.115439	−0.0577193	+	0.998333i	$0.518383\pi$
$104$	0	0
$105$	2.82843	0.276026
$106$	0	0
$107$	16.4853	1.59369	0.796846	−	0.604182i	$-0.206500\pi$
$107$	16.4853	1.59369	0.796846	+	0.604182i	$0.206500\pi$
$108$	0	0
$109$	−10.8284	−1.03718	−0.518588	−	0.855024i	$-0.673542\pi$
$109$	−10.8284	−1.03718	−0.518588	+	0.855024i	$0.673542\pi$
$110$	0	0
$111$	0.343146	0.0325700
$112$	0	0
$113$	7.65685	0.720296	0.360148	−	0.932895i	$-0.382726\pi$
$113$	7.65685	0.720296	0.360148	+	0.932895i	$0.382726\pi$
$114$	0	0
$115$	6.82843	0.636754
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−1.17157	−0.107398
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.82843	0.615699
$124$	0	0
$125$	11.3137	1.01193
$126$	0	0
$127$	12.2426	1.08636	0.543179	−	0.839617i	$-0.317220\pi$
$127$	12.2426	1.08636	0.543179	+	0.839617i	$0.317220\pi$
$128$	0	0
$129$	8.24264	0.725724
$130$	0	0
$131$	9.65685	0.843723	0.421862	−	0.906660i	$-0.361377\pi$
$131$	9.65685	0.843723	0.421862	+	0.906660i	$0.361377\pi$
$132$	0	0
$133$	−1.65685	−0.143667
$134$	0	0
$135$	1.41421	0.121716
$136$	0	0
$137$	12.7279	1.08742	0.543710	−	0.839273i	$-0.317019\pi$
$137$	12.7279	1.08742	0.543710	+	0.839273i	$0.317019\pi$
$138$	0	0
$139$	17.8995	1.51822	0.759108	−	0.650965i	$-0.225636\pi$
$139$	17.8995	1.51822	0.759108	+	0.650965i	$0.225636\pi$
$140$	0	0
$141$	−12.4853	−1.05145
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−7.17157	−0.595567
$146$	0	0
$147$	3.00000	0.247436
$148$	0	0
$149$	−11.3137	−0.926855	−0.463428	−	0.886135i	$-0.653381\pi$
$149$	−11.3137	−0.926855	−0.463428	+	0.886135i	$0.653381\pi$
$150$	0	0
$151$	−8.14214	−0.662598	−0.331299	−	0.943526i	$-0.607487\pi$
$151$	−8.14214	−0.662598	−0.331299	+	0.943526i	$0.607487\pi$
$152$	0	0
$153$	−0.585786	−0.0473580
$154$	0	0
$155$	−3.17157	−0.254747
$156$	0	0
$157$	−19.3137	−1.54140	−0.770701	−	0.637197i	$-0.780094\pi$
$157$	−19.3137	−1.54140	−0.770701	+	0.637197i	$0.780094\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	−9.65685	−0.761067
$162$	0	0
$163$	−24.3848	−1.90996	−0.954982	−	0.296665i	$-0.904125\pi$
$163$	−24.3848	−1.90996	−0.954982	+	0.296665i	$0.904125\pi$
$164$	0	0
$165$	1.41421	0.110096
$166$	0	0
$167$	−9.65685	−0.747270	−0.373635	−	0.927576i	$-0.621889\pi$
$167$	−9.65685	−0.747270	−0.373635	+	0.927576i	$0.621889\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.828427	−0.0633514
$172$	0	0
$173$	−2.72792	−0.207400	−0.103700	−	0.994609i	$-0.533068\pi$
$173$	−2.72792	−0.207400	−0.103700	+	0.994609i	$0.533068\pi$
$174$	0	0
$175$	−6.00000	−0.453557
$176$	0	0
$177$	−12.4853	−0.938451
$178$	0	0
$179$	−6.48528	−0.484733	−0.242366	−	0.970185i	$-0.577924\pi$
$179$	−6.48528	−0.484733	−0.242366	+	0.970185i	$0.577924\pi$
$180$	0	0
$181$	9.65685	0.717788	0.358894	−	0.933378i	$-0.383154\pi$
$181$	9.65685	0.717788	0.358894	+	0.933378i	$0.383154\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	0	0
$185$	0.485281	0.0356786
$186$	0	0
$187$	−0.585786	−0.0428369
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	3.31371	0.239772	0.119886	−	0.992788i	$-0.461747\pi$
$191$	3.31371	0.239772	0.119886	+	0.992788i	$0.461747\pi$
$192$	0	0
$193$	2.14214	0.154194	0.0770971	−	0.997024i	$-0.475435\pi$
$193$	2.14214	0.154194	0.0770971	+	0.997024i	$0.475435\pi$
$194$	0	0
$195$	1.41421	0.101274
$196$	0	0
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	0	0
$199$	18.1421	1.28606	0.643031	−	0.765840i	$-0.277677\pi$
$199$	18.1421	1.28606	0.643031	+	0.765840i	$0.277677\pi$
$200$	0	0
$201$	−7.89949	−0.557187
$202$	0	0
$203$	10.1421	0.711838
$204$	0	0
$205$	9.65685	0.674464
$206$	0	0
$207$	−4.82843	−0.335599
$208$	0	0
$209$	−0.828427	−0.0573035
$210$	0	0
$211$	2.58579	0.178013	0.0890064	−	0.996031i	$-0.471631\pi$
$211$	2.58579	0.178013	0.0890064	+	0.996031i	$0.471631\pi$
$212$	0	0
$213$	11.3137	0.775203
$214$	0	0
$215$	11.6569	0.794991
$216$	0	0
$217$	4.48528	0.304481
$218$	0	0
$219$	16.4853	1.11397
$220$	0	0
$221$	−0.585786	−0.0394043
$222$	0	0
$223$	−25.5563	−1.71138	−0.855690	−	0.517489i	$-0.826867\pi$
$223$	−25.5563	−1.71138	−0.855690	+	0.517489i	$0.826867\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	0	0
$227$	25.3137	1.68013	0.840065	−	0.542486i	$-0.182517\pi$
$227$	25.3137	1.68013	0.840065	+	0.542486i	$0.182517\pi$
$228$	0	0
$229$	3.17157	0.209583	0.104792	−	0.994494i	$-0.466582\pi$
$229$	3.17157	0.209583	0.104792	+	0.994494i	$0.466582\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	−1.07107	−0.0701680	−0.0350840	−	0.999384i	$-0.511170\pi$
$233$	−1.07107	−0.0701680	−0.0350840	+	0.999384i	$0.511170\pi$
$234$	0	0
$235$	−17.6569	−1.15181
$236$	0	0
$237$	−0.242641	−0.0157612
$238$	0	0
$239$	−26.0000	−1.68180	−0.840900	−	0.541190i	$-0.817974\pi$
$239$	−26.0000	−1.68180	−0.840900	+	0.541190i	$0.817974\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	4.24264	0.271052
$246$	0	0
$247$	−0.828427	−0.0527116
$248$	0	0
$249$	5.65685	0.358489
$250$	0	0
$251$	−23.3137	−1.47155	−0.735774	−	0.677227i	$-0.763181\pi$
$251$	−23.3137	−1.47155	−0.735774	+	0.677227i	$0.763181\pi$
$252$	0	0
$253$	−4.82843	−0.303561
$254$	0	0
$255$	−0.828427	−0.0518781
$256$	0	0
$257$	−14.4853	−0.903567	−0.451784	−	0.892128i	$-0.649212\pi$
$257$	−14.4853	−0.903567	−0.451784	+	0.892128i	$0.649212\pi$
$258$	0	0
$259$	−0.686292	−0.0426441
$260$	0	0
$261$	5.07107	0.313891
$262$	0	0
$263$	−19.7990	−1.22086	−0.610429	−	0.792071i	$-0.709003\pi$
$263$	−19.7990	−1.22086	−0.610429	+	0.792071i	$0.709003\pi$
$264$	0	0
$265$	−2.82843	−0.173749
$266$	0	0
$267$	8.24264	0.504441
$268$	0	0
$269$	−0.142136	−0.00866616	−0.00433308	−	0.999991i	$-0.501379\pi$
$269$	−0.142136	−0.00866616	−0.00433308	+	0.999991i	$0.501379\pi$
$270$	0	0
$271$	3.17157	0.192659	0.0963297	−	0.995349i	$-0.469290\pi$
$271$	3.17157	0.192659	0.0963297	+	0.995349i	$0.469290\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	−3.00000	−0.180907
$276$	0	0
$277$	4.34315	0.260954	0.130477	−	0.991451i	$-0.458349\pi$
$277$	4.34315	0.260954	0.130477	+	0.991451i	$0.458349\pi$
$278$	0	0
$279$	2.24264	0.134263
$280$	0	0
$281$	2.82843	0.168730	0.0843649	−	0.996435i	$-0.473114\pi$
$281$	2.82843	0.168730	0.0843649	+	0.996435i	$0.473114\pi$
$282$	0	0
$283$	1.41421	0.0840663	0.0420331	−	0.999116i	$-0.486616\pi$
$283$	1.41421	0.0840663	0.0420331	+	0.999116i	$0.486616\pi$
$284$	0	0
$285$	−1.17157	−0.0693980
$286$	0	0
$287$	−13.6569	−0.806139
$288$	0	0
$289$	−16.6569	−0.979815
$290$	0	0
$291$	7.65685	0.448853
$292$	0	0
$293$	−3.31371	−0.193589	−0.0967945	−	0.995304i	$-0.530859\pi$
$293$	−3.31371	−0.193589	−0.0967945	+	0.995304i	$0.530859\pi$
$294$	0	0
$295$	−17.6569	−1.02802
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−4.82843	−0.279235
$300$	0	0
$301$	−16.4853	−0.950196
$302$	0	0
$303$	−6.24264	−0.358630
$304$	0	0
$305$	14.1421	0.809776
$306$	0	0
$307$	22.4853	1.28330	0.641651	−	0.766996i	$-0.278250\pi$
$307$	22.4853	1.28330	0.641651	+	0.766996i	$0.278250\pi$
$308$	0	0
$309$	1.17157	0.0666485
$310$	0	0
$311$	−28.1421	−1.59579	−0.797897	−	0.602794i	$-0.794054\pi$
$311$	−28.1421	−1.59579	−0.797897	+	0.602794i	$0.794054\pi$
$312$	0	0
$313$	−24.9706	−1.41142	−0.705710	−	0.708501i	$-0.749372\pi$
$313$	−24.9706	−1.41142	−0.705710	+	0.708501i	$0.749372\pi$
$314$	0	0
$315$	−2.82843	−0.159364
$316$	0	0
$317$	34.3848	1.93124	0.965621	−	0.259955i	$-0.0837077\pi$
$317$	34.3848	1.93124	0.965621	+	0.259955i	$0.0837077\pi$
$318$	0	0
$319$	5.07107	0.283925
$320$	0	0
$321$	−16.4853	−0.920119
$322$	0	0
$323$	0.485281	0.0270018
$324$	0	0
$325$	−3.00000	−0.166410
$326$	0	0
$327$	10.8284	0.598813
$328$	0	0
$329$	24.9706	1.37667
$330$	0	0
$331$	−22.0416	−1.21152	−0.605759	−	0.795648i	$-0.707130\pi$
$331$	−22.0416	−1.21152	−0.605759	+	0.795648i	$0.707130\pi$
$332$	0	0
$333$	−0.343146	−0.0188043
$334$	0	0
$335$	−11.1716	−0.610368
$336$	0	0
$337$	−14.4853	−0.789064	−0.394532	−	0.918882i	$-0.629093\pi$
$337$	−14.4853	−0.789064	−0.394532	+	0.918882i	$0.629093\pi$
$338$	0	0
$339$	−7.65685	−0.415863
$340$	0	0
$341$	2.24264	0.121446
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	−6.82843	−0.367630
$346$	0	0
$347$	−15.7990	−0.848134	−0.424067	−	0.905631i	$-0.639398\pi$
$347$	−15.7990	−0.848134	−0.424067	+	0.905631i	$0.639398\pi$
$348$	0	0
$349$	−29.3137	−1.56913	−0.784563	−	0.620049i	$-0.787113\pi$
$349$	−29.3137	−1.56913	−0.784563	+	0.620049i	$0.787113\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−20.2426	−1.07741	−0.538704	−	0.842495i	$-0.681086\pi$
$353$	−20.2426	−1.07741	−0.538704	+	0.842495i	$0.681086\pi$
$354$	0	0
$355$	16.0000	0.849192
$356$	0	0
$357$	1.17157	0.0620062
$358$	0	0
$359$	−3.65685	−0.193001	−0.0965007	−	0.995333i	$-0.530765\pi$
$359$	−3.65685	−0.193001	−0.0965007	+	0.995333i	$0.530765\pi$
$360$	0	0
$361$	−18.3137	−0.963879
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	23.3137	1.22030
$366$	0	0
$367$	−20.9706	−1.09465	−0.547327	−	0.836919i	$-0.684355\pi$
$367$	−20.9706	−1.09465	−0.547327	+	0.836919i	$0.684355\pi$
$368$	0	0
$369$	−6.82843	−0.355474
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	4.14214	0.214472	0.107236	−	0.994234i	$-0.465800\pi$
$373$	4.14214	0.214472	0.107236	+	0.994234i	$0.465800\pi$
$374$	0	0
$375$	−11.3137	−0.584237
$376$	0	0
$377$	5.07107	0.261173
$378$	0	0
$379$	−15.2132	−0.781450	−0.390725	−	0.920508i	$-0.627776\pi$
$379$	−15.2132	−0.781450	−0.390725	+	0.920508i	$0.627776\pi$
$380$	0	0
$381$	−12.2426	−0.627209
$382$	0	0
$383$	−9.85786	−0.503713	−0.251857	−	0.967765i	$-0.581041\pi$
$383$	−9.85786	−0.503713	−0.251857	+	0.967765i	$0.581041\pi$
$384$	0	0
$385$	−2.82843	−0.144150
$386$	0	0
$387$	−8.24264	−0.418997
$388$	0	0
$389$	−20.1421	−1.02125	−0.510623	−	0.859804i	$-0.670585\pi$
$389$	−20.1421	−1.02125	−0.510623	+	0.859804i	$0.670585\pi$
$390$	0	0
$391$	2.82843	0.143040
$392$	0	0
$393$	−9.65685	−0.487124
$394$	0	0
$395$	−0.343146	−0.0172655
$396$	0	0
$397$	−10.6863	−0.536330	−0.268165	−	0.963373i	$-0.586417\pi$
$397$	−10.6863	−0.536330	−0.268165	+	0.963373i	$0.586417\pi$
$398$	0	0
$399$	1.65685	0.0829465
$400$	0	0
$401$	13.4142	0.669874	0.334937	−	0.942241i	$-0.391285\pi$
$401$	13.4142	0.669874	0.334937	+	0.942241i	$0.391285\pi$
$402$	0	0
$403$	2.24264	0.111714
$404$	0	0
$405$	−1.41421	−0.0702728
$406$	0	0
$407$	−0.343146	−0.0170091
$408$	0	0
$409$	11.7990	0.583423	0.291711	−	0.956506i	$-0.405775\pi$
$409$	11.7990	0.583423	0.291711	+	0.956506i	$0.405775\pi$
$410$	0	0
$411$	−12.7279	−0.627822
$412$	0	0
$413$	24.9706	1.22872
$414$	0	0
$415$	8.00000	0.392705
$416$	0	0
$417$	−17.8995	−0.876542
$418$	0	0
$419$	−3.17157	−0.154941	−0.0774707	−	0.996995i	$-0.524684\pi$
$419$	−3.17157	−0.154941	−0.0774707	+	0.996995i	$0.524684\pi$
$420$	0	0
$421$	10.9706	0.534673	0.267336	−	0.963603i	$-0.413857\pi$
$421$	10.9706	0.534673	0.267336	+	0.963603i	$0.413857\pi$
$422$	0	0
$423$	12.4853	0.607055
$424$	0	0
$425$	1.75736	0.0852444
$426$	0	0
$427$	−20.0000	−0.967868
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	29.9411	1.44221	0.721107	−	0.692824i	$-0.243634\pi$
$431$	29.9411	1.44221	0.721107	+	0.692824i	$0.243634\pi$
$432$	0	0
$433$	−9.31371	−0.447588	−0.223794	−	0.974636i	$-0.571844\pi$
$433$	−9.31371	−0.447588	−0.223794	+	0.974636i	$0.571844\pi$
$434$	0	0
$435$	7.17157	0.343851
$436$	0	0
$437$	4.00000	0.191346
$438$	0	0
$439$	−40.2426	−1.92068	−0.960338	−	0.278838i	$-0.910051\pi$
$439$	−40.2426	−1.92068	−0.960338	+	0.278838i	$0.910051\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	18.6274	0.885015	0.442508	−	0.896765i	$-0.354089\pi$
$443$	18.6274	0.885015	0.442508	+	0.896765i	$0.354089\pi$
$444$	0	0
$445$	11.6569	0.552588
$446$	0	0
$447$	11.3137	0.535120
$448$	0	0
$449$	−5.41421	−0.255513	−0.127756	−	0.991806i	$-0.540778\pi$
$449$	−5.41421	−0.255513	−0.127756	+	0.991806i	$0.540778\pi$
$450$	0	0
$451$	−6.82843	−0.321538
$452$	0	0
$453$	8.14214	0.382551
$454$	0	0
$455$	−2.82843	−0.132599
$456$	0	0
$457$	−14.6863	−0.686996	−0.343498	−	0.939153i	$-0.611612\pi$
$457$	−14.6863	−0.686996	−0.343498	+	0.939153i	$0.611612\pi$
$458$	0	0
$459$	0.585786	0.0273422
$460$	0	0
$461$	−19.7990	−0.922131	−0.461065	−	0.887366i	$-0.652533\pi$
$461$	−19.7990	−0.922131	−0.461065	+	0.887366i	$0.652533\pi$
$462$	0	0
$463$	38.2426	1.77729	0.888643	−	0.458599i	$-0.151649\pi$
$463$	38.2426	1.77729	0.888643	+	0.458599i	$0.151649\pi$
$464$	0	0
$465$	3.17157	0.147078
$466$	0	0
$467$	−15.4558	−0.715211	−0.357606	−	0.933873i	$-0.616407\pi$
$467$	−15.4558	−0.715211	−0.357606	+	0.933873i	$0.616407\pi$
$468$	0	0
$469$	15.7990	0.729530
$470$	0	0
$471$	19.3137	0.889929
$472$	0	0
$473$	−8.24264	−0.378997
$474$	0	0
$475$	2.48528	0.114033
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	−30.9706	−1.41508	−0.707541	−	0.706673i	$-0.750195\pi$
$479$	−30.9706	−1.41508	−0.707541	+	0.706673i	$0.750195\pi$
$480$	0	0
$481$	−0.343146	−0.0156461
$482$	0	0
$483$	9.65685	0.439402
$484$	0	0
$485$	10.8284	0.491694
$486$	0	0
$487$	43.0122	1.94907	0.974534	−	0.224239i	$-0.0719895\pi$
$487$	43.0122	1.94907	0.974534	+	0.224239i	$0.0719895\pi$
$488$	0	0
$489$	24.3848	1.10272
$490$	0	0
$491$	18.1421	0.818743	0.409372	−	0.912368i	$-0.365748\pi$
$491$	18.1421	0.818743	0.409372	+	0.912368i	$0.365748\pi$
$492$	0	0
$493$	−2.97056	−0.133787
$494$	0	0
$495$	−1.41421	−0.0635642
$496$	0	0
$497$	−22.6274	−1.01498
$498$	0	0
$499$	−14.7279	−0.659312	−0.329656	−	0.944101i	$-0.606933\pi$
$499$	−14.7279	−0.659312	−0.329656	+	0.944101i	$0.606933\pi$
$500$	0	0
$501$	9.65685	0.431436
$502$	0	0
$503$	−16.4853	−0.735042	−0.367521	−	0.930015i	$-0.619793\pi$
$503$	−16.4853	−0.735042	−0.367521	+	0.930015i	$0.619793\pi$
$504$	0	0
$505$	−8.82843	−0.392860
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−12.7279	−0.564155	−0.282078	−	0.959392i	$-0.591024\pi$
$509$	−12.7279	−0.564155	−0.282078	+	0.959392i	$0.591024\pi$
$510$	0	0
$511$	−32.9706	−1.45853
$512$	0	0
$513$	0.828427	0.0365760
$514$	0	0
$515$	1.65685	0.0730097
$516$	0	0
$517$	12.4853	0.549102
$518$	0	0
$519$	2.72792	0.119742
$520$	0	0
$521$	23.6569	1.03643	0.518213	−	0.855252i	$-0.326598\pi$
$521$	23.6569	1.03643	0.518213	+	0.855252i	$0.326598\pi$
$522$	0	0
$523$	37.2132	1.62722	0.813610	−	0.581411i	$-0.197499\pi$
$523$	37.2132	1.62722	0.813610	+	0.581411i	$0.197499\pi$
$524$	0	0
$525$	6.00000	0.261861
$526$	0	0
$527$	−1.31371	−0.0572260
$528$	0	0
$529$	0.313708	0.0136395
$530$	0	0
$531$	12.4853	0.541815
$532$	0	0
$533$	−6.82843	−0.295772
$534$	0	0
$535$	−23.3137	−1.00794
$536$	0	0
$537$	6.48528	0.279861
$538$	0	0
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−23.6569	−1.01709	−0.508544	−	0.861036i	$-0.669816\pi$
$541$	−23.6569	−1.01709	−0.508544	+	0.861036i	$0.669816\pi$
$542$	0	0
$543$	−9.65685	−0.414415
$544$	0	0
$545$	15.3137	0.655967
$546$	0	0
$547$	−0.928932	−0.0397183	−0.0198591	−	0.999803i	$-0.506322\pi$
$547$	−0.928932	−0.0397183	−0.0198591	+	0.999803i	$0.506322\pi$
$548$	0	0
$549$	−10.0000	−0.426790
$550$	0	0
$551$	−4.20101	−0.178969
$552$	0	0
$553$	0.485281	0.0206363
$554$	0	0
$555$	−0.485281	−0.0205990
$556$	0	0
$557$	−30.6274	−1.29773	−0.648863	−	0.760906i	$-0.724755\pi$
$557$	−30.6274	−1.29773	−0.648863	+	0.760906i	$0.724755\pi$
$558$	0	0
$559$	−8.24264	−0.348627
$560$	0	0
$561$	0.585786	0.0247319
$562$	0	0
$563$	5.65685	0.238408	0.119204	−	0.992870i	$-0.461966\pi$
$563$	5.65685	0.238408	0.119204	+	0.992870i	$0.461966\pi$
$564$	0	0
$565$	−10.8284	−0.455555
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	6.24264	0.261705	0.130853	−	0.991402i	$-0.458229\pi$
$569$	6.24264	0.261705	0.130853	+	0.991402i	$0.458229\pi$
$570$	0	0
$571$	−44.7279	−1.87181	−0.935903	−	0.352259i	$-0.885414\pi$
$571$	−44.7279	−1.87181	−0.935903	+	0.352259i	$0.885414\pi$
$572$	0	0
$573$	−3.31371	−0.138432
$574$	0	0
$575$	14.4853	0.604078
$576$	0	0
$577$	34.9706	1.45584	0.727922	−	0.685660i	$-0.240486\pi$
$577$	34.9706	1.45584	0.727922	+	0.685660i	$0.240486\pi$
$578$	0	0
$579$	−2.14214	−0.0890241
$580$	0	0
$581$	−11.3137	−0.469372
$582$	0	0
$583$	2.00000	0.0828315
$584$	0	0
$585$	−1.41421	−0.0584705
$586$	0	0
$587$	−25.4558	−1.05068	−0.525338	−	0.850894i	$-0.676061\pi$
$587$	−25.4558	−1.05068	−0.525338	+	0.850894i	$0.676061\pi$
$588$	0	0
$589$	−1.85786	−0.0765520
$590$	0	0
$591$	8.00000	0.329076
$592$	0	0
$593$	26.6274	1.09346	0.546728	−	0.837310i	$-0.315873\pi$
$593$	26.6274	1.09346	0.546728	+	0.837310i	$0.315873\pi$
$594$	0	0
$595$	1.65685	0.0679244
$596$	0	0
$597$	−18.1421	−0.742508
$598$	0	0
$599$	−4.68629	−0.191477	−0.0957383	−	0.995407i	$-0.530521\pi$
$599$	−4.68629	−0.191477	−0.0957383	+	0.995407i	$0.530521\pi$
$600$	0	0
$601$	16.3431	0.666651	0.333325	−	0.942812i	$-0.391829\pi$
$601$	16.3431	0.666651	0.333325	+	0.942812i	$0.391829\pi$
$602$	0	0
$603$	7.89949	0.321692
$604$	0	0
$605$	−1.41421	−0.0574960
$606$	0	0
$607$	−1.41421	−0.0574012	−0.0287006	−	0.999588i	$-0.509137\pi$
$607$	−1.41421	−0.0574012	−0.0287006	+	0.999588i	$0.509137\pi$
$608$	0	0
$609$	−10.1421	−0.410980
$610$	0	0
$611$	12.4853	0.505100
$612$	0	0
$613$	1.17157	0.0473194	0.0236597	−	0.999720i	$-0.492468\pi$
$613$	1.17157	0.0473194	0.0236597	+	0.999720i	$0.492468\pi$
$614$	0	0
$615$	−9.65685	−0.389402
$616$	0	0
$617$	−32.0416	−1.28995	−0.644974	−	0.764205i	$-0.723132\pi$
$617$	−32.0416	−1.28995	−0.644974	+	0.764205i	$0.723132\pi$
$618$	0	0
$619$	24.3848	0.980107	0.490053	−	0.871692i	$-0.336977\pi$
$619$	24.3848	0.980107	0.490053	+	0.871692i	$0.336977\pi$
$620$	0	0
$621$	4.82843	0.193758
$622$	0	0
$623$	−16.4853	−0.660469
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0.828427	0.0330842
$628$	0	0
$629$	0.201010	0.00801480
$630$	0	0
$631$	16.8701	0.671586	0.335793	−	0.941936i	$-0.390996\pi$
$631$	16.8701	0.671586	0.335793	+	0.941936i	$0.390996\pi$
$632$	0	0
$633$	−2.58579	−0.102776
$634$	0	0
$635$	−17.3137	−0.687074
$636$	0	0
$637$	−3.00000	−0.118864
$638$	0	0
$639$	−11.3137	−0.447563
$640$	0	0
$641$	28.6274	1.13071	0.565357	−	0.824846i	$-0.308738\pi$
$641$	28.6274	1.13071	0.565357	+	0.824846i	$0.308738\pi$
$642$	0	0
$643$	34.7279	1.36954	0.684768	−	0.728761i	$-0.259904\pi$
$643$	34.7279	1.36954	0.684768	+	0.728761i	$0.259904\pi$
$644$	0	0
$645$	−11.6569	−0.458988
$646$	0	0
$647$	−16.9706	−0.667182	−0.333591	−	0.942718i	$-0.608260\pi$
$647$	−16.9706	−0.667182	−0.333591	+	0.942718i	$0.608260\pi$
$648$	0	0
$649$	12.4853	0.490090
$650$	0	0
$651$	−4.48528	−0.175792
$652$	0	0
$653$	13.5147	0.528872	0.264436	−	0.964403i	$-0.414814\pi$
$653$	13.5147	0.528872	0.264436	+	0.964403i	$0.414814\pi$
$654$	0	0
$655$	−13.6569	−0.533617
$656$	0	0
$657$	−16.4853	−0.643152
$658$	0	0
$659$	4.68629	0.182552	0.0912760	−	0.995826i	$-0.470905\pi$
$659$	4.68629	0.182552	0.0912760	+	0.995826i	$0.470905\pi$
$660$	0	0
$661$	−40.1421	−1.56135	−0.780674	−	0.624938i	$-0.785124\pi$
$661$	−40.1421	−1.56135	−0.780674	+	0.624938i	$0.785124\pi$
$662$	0	0
$663$	0.585786	0.0227501
$664$	0	0
$665$	2.34315	0.0908633
$666$	0	0
$667$	−24.4853	−0.948074
$668$	0	0
$669$	25.5563	0.988066
$670$	0	0
$671$	−10.0000	−0.386046
$672$	0	0
$673$	17.5147	0.675143	0.337571	−	0.941300i	$-0.390395\pi$
$673$	17.5147	0.675143	0.337571	+	0.941300i	$0.390395\pi$
$674$	0	0
$675$	3.00000	0.115470
$676$	0	0
$677$	−7.41421	−0.284951	−0.142476	−	0.989798i	$-0.545506\pi$
$677$	−7.41421	−0.284951	−0.142476	+	0.989798i	$0.545506\pi$
$678$	0	0
$679$	−15.3137	−0.587686
$680$	0	0
$681$	−25.3137	−0.970023
$682$	0	0
$683$	−8.00000	−0.306111	−0.153056	−	0.988218i	$-0.548911\pi$
$683$	−8.00000	−0.306111	−0.153056	+	0.988218i	$0.548911\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	0	0
$687$	−3.17157	−0.121003
$688$	0	0
$689$	2.00000	0.0761939
$690$	0	0
$691$	12.8701	0.489600	0.244800	−	0.969574i	$-0.421278\pi$
$691$	12.8701	0.489600	0.244800	+	0.969574i	$0.421278\pi$
$692$	0	0
$693$	2.00000	0.0759737
$694$	0	0
$695$	−25.3137	−0.960204
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	1.07107	0.0405115
$700$	0	0
$701$	−52.3848	−1.97855	−0.989273	−	0.146080i	$-0.953334\pi$
$701$	−52.3848	−1.97855	−0.989273	+	0.146080i	$0.953334\pi$
$702$	0	0
$703$	0.284271	0.0107215
$704$	0	0
$705$	17.6569	0.664996
$706$	0	0
$707$	12.4853	0.469557
$708$	0	0
$709$	21.5147	0.808002	0.404001	−	0.914758i	$-0.367619\pi$
$709$	21.5147	0.808002	0.404001	+	0.914758i	$0.367619\pi$
$710$	0	0
$711$	0.242641	0.00909974
$712$	0	0
$713$	−10.8284	−0.405528
$714$	0	0
$715$	−1.41421	−0.0528886
$716$	0	0
$717$	26.0000	0.970988
$718$	0	0
$719$	−27.3137	−1.01863	−0.509315	−	0.860580i	$-0.670101\pi$
$719$	−27.3137	−1.01863	−0.509315	+	0.860580i	$0.670101\pi$
$720$	0	0
$721$	−2.34315	−0.0872633
$722$	0	0
$723$	14.0000	0.520666
$724$	0	0
$725$	−15.2132	−0.565004
$726$	0	0
$727$	−41.4558	−1.53751	−0.768756	−	0.639542i	$-0.779124\pi$
$727$	−41.4558	−1.53751	−0.768756	+	0.639542i	$0.779124\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.82843	0.178586
$732$	0	0
$733$	−19.5147	−0.720793	−0.360396	−	0.932799i	$-0.617359\pi$
$733$	−19.5147	−0.720793	−0.360396	+	0.932799i	$0.617359\pi$
$734$	0	0
$735$	−4.24264	−0.156492
$736$	0	0
$737$	7.89949	0.290982
$738$	0	0
$739$	−8.34315	−0.306908	−0.153454	−	0.988156i	$-0.549040\pi$
$739$	−8.34315	−0.306908	−0.153454	+	0.988156i	$0.549040\pi$
$740$	0	0
$741$	0.828427	0.0304330
$742$	0	0
$743$	51.3137	1.88252	0.941259	−	0.337686i	$-0.109644\pi$
$743$	51.3137	1.88252	0.941259	+	0.337686i	$0.109644\pi$
$744$	0	0
$745$	16.0000	0.586195
$746$	0	0
$747$	−5.65685	−0.206973
$748$	0	0
$749$	32.9706	1.20472
$750$	0	0
$751$	−18.6274	−0.679724	−0.339862	−	0.940475i	$-0.610380\pi$
$751$	−18.6274	−0.679724	−0.339862	+	0.940475i	$0.610380\pi$
$752$	0	0
$753$	23.3137	0.849599
$754$	0	0
$755$	11.5147	0.419064
$756$	0	0
$757$	11.3137	0.411204	0.205602	−	0.978636i	$-0.434085\pi$
$757$	11.3137	0.411204	0.205602	+	0.978636i	$0.434085\pi$
$758$	0	0
$759$	4.82843	0.175261
$760$	0	0
$761$	39.5980	1.43543	0.717713	−	0.696339i	$-0.245189\pi$
$761$	39.5980	1.43543	0.717713	+	0.696339i	$0.245189\pi$
$762$	0	0
$763$	−21.6569	−0.784031
$764$	0	0
$765$	0.828427	0.0299518
$766$	0	0
$767$	12.4853	0.450817
$768$	0	0
$769$	−37.5980	−1.35582	−0.677909	−	0.735146i	$-0.737114\pi$
$769$	−37.5980	−1.35582	−0.677909	+	0.735146i	$0.737114\pi$
$770$	0	0
$771$	14.4853	0.521675
$772$	0	0
$773$	44.0416	1.58407	0.792034	−	0.610477i	$-0.209022\pi$
$773$	44.0416	1.58407	0.792034	+	0.610477i	$0.209022\pi$
$774$	0	0
$775$	−6.72792	−0.241674
$776$	0	0
$777$	0.686292	0.0246206
$778$	0	0
$779$	5.65685	0.202678
$780$	0	0
$781$	−11.3137	−0.404836
$782$	0	0
$783$	−5.07107	−0.181225
$784$	0	0
$785$	27.3137	0.974868
$786$	0	0
$787$	9.02944	0.321865	0.160932	−	0.986965i	$-0.448550\pi$
$787$	9.02944	0.321865	0.160932	+	0.986965i	$0.448550\pi$
$788$	0	0
$789$	19.7990	0.704863
$790$	0	0
$791$	15.3137	0.544493
$792$	0	0
$793$	−10.0000	−0.355110
$794$	0	0
$795$	2.82843	0.100314
$796$	0	0
$797$	39.1716	1.38753	0.693764	−	0.720202i	$-0.255951\pi$
$797$	39.1716	1.38753	0.693764	+	0.720202i	$0.255951\pi$
$798$	0	0
$799$	−7.31371	−0.258740
$800$	0	0
$801$	−8.24264	−0.291239
$802$	0	0
$803$	−16.4853	−0.581753
$804$	0	0
$805$	13.6569	0.481341
$806$	0	0
$807$	0.142136	0.00500341
$808$	0	0
$809$	5.27208	0.185356	0.0926782	−	0.995696i	$-0.470457\pi$
$809$	5.27208	0.185356	0.0926782	+	0.995696i	$0.470457\pi$
$810$	0	0
$811$	−3.85786	−0.135468	−0.0677340	−	0.997703i	$-0.521577\pi$
$811$	−3.85786	−0.135468	−0.0677340	+	0.997703i	$0.521577\pi$
$812$	0	0
$813$	−3.17157	−0.111232
$814$	0	0
$815$	34.4853	1.20797
$816$	0	0
$817$	6.82843	0.238896
$818$	0	0
$819$	2.00000	0.0698857
$820$	0	0
$821$	43.1127	1.50464	0.752322	−	0.658796i	$-0.228934\pi$
$821$	43.1127	1.50464	0.752322	+	0.658796i	$0.228934\pi$
$822$	0	0
$823$	0.485281	0.0169158	0.00845792	−	0.999964i	$-0.497308\pi$
$823$	0.485281	0.0169158	0.00845792	+	0.999964i	$0.497308\pi$
$824$	0	0
$825$	3.00000	0.104447
$826$	0	0
$827$	−4.34315	−0.151026	−0.0755130	−	0.997145i	$-0.524059\pi$
$827$	−4.34315	−0.151026	−0.0755130	+	0.997145i	$0.524059\pi$
$828$	0	0
$829$	31.9411	1.10936	0.554681	−	0.832063i	$-0.312840\pi$
$829$	31.9411	1.10936	0.554681	+	0.832063i	$0.312840\pi$
$830$	0	0
$831$	−4.34315	−0.150662
$832$	0	0
$833$	1.75736	0.0608889
$834$	0	0
$835$	13.6569	0.472615
$836$	0	0
$837$	−2.24264	−0.0775170
$838$	0	0
$839$	−39.5980	−1.36707	−0.683537	−	0.729916i	$-0.739559\pi$
$839$	−39.5980	−1.36707	−0.683537	+	0.729916i	$0.739559\pi$
$840$	0	0
$841$	−3.28427	−0.113251
$842$	0	0
$843$	−2.82843	−0.0974162
$844$	0	0
$845$	−1.41421	−0.0486504
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	−1.41421	−0.0485357
$850$	0	0
$851$	1.65685	0.0567962
$852$	0	0
$853$	−15.9411	−0.545814	−0.272907	−	0.962040i	$-0.587985\pi$
$853$	−15.9411	−0.545814	−0.272907	+	0.962040i	$0.587985\pi$
$854$	0	0
$855$	1.17157	0.0400669
$856$	0	0
$857$	−54.2426	−1.85289	−0.926447	−	0.376426i	$-0.877153\pi$
$857$	−54.2426	−1.85289	−0.926447	+	0.376426i	$0.877153\pi$
$858$	0	0
$859$	37.4558	1.27798	0.638988	−	0.769216i	$-0.279353\pi$
$859$	37.4558	1.27798	0.638988	+	0.769216i	$0.279353\pi$
$860$	0	0
$861$	13.6569	0.465424
$862$	0	0
$863$	−50.8284	−1.73022	−0.865110	−	0.501582i	$-0.832751\pi$
$863$	−50.8284	−1.73022	−0.865110	+	0.501582i	$0.832751\pi$
$864$	0	0
$865$	3.85786	0.131171
$866$	0	0
$867$	16.6569	0.565696
$868$	0	0
$869$	0.242641	0.00823102
$870$	0	0
$871$	7.89949	0.267664
$872$	0	0
$873$	−7.65685	−0.259145
$874$	0	0
$875$	22.6274	0.764946
$876$	0	0
$877$	1.31371	0.0443608	0.0221804	−	0.999754i	$-0.492939\pi$
$877$	1.31371	0.0443608	0.0221804	+	0.999754i	$0.492939\pi$
$878$	0	0
$879$	3.31371	0.111769
$880$	0	0
$881$	45.3137	1.52666	0.763329	−	0.646010i	$-0.223564\pi$
$881$	45.3137	1.52666	0.763329	+	0.646010i	$0.223564\pi$
$882$	0	0
$883$	−10.8284	−0.364406	−0.182203	−	0.983261i	$-0.558323\pi$
$883$	−10.8284	−0.364406	−0.182203	+	0.983261i	$0.558323\pi$
$884$	0	0
$885$	17.6569	0.593529
$886$	0	0
$887$	50.6274	1.69990	0.849951	−	0.526862i	$-0.176631\pi$
$887$	50.6274	1.69990	0.849951	+	0.526862i	$0.176631\pi$
$888$	0	0
$889$	24.4853	0.821210
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−10.3431	−0.346120
$894$	0	0
$895$	9.17157	0.306572
$896$	0	0
$897$	4.82843	0.161216
$898$	0	0
$899$	11.3726	0.379297
$900$	0	0
$901$	−1.17157	−0.0390308
$902$	0	0
$903$	16.4853	0.548596
$904$	0	0
$905$	−13.6569	−0.453969
$906$	0	0
$907$	−12.6863	−0.421241	−0.210621	−	0.977568i	$-0.567548\pi$
$907$	−12.6863	−0.421241	−0.210621	+	0.977568i	$0.567548\pi$
$908$	0	0
$909$	6.24264	0.207055
$910$	0	0
$911$	−29.6569	−0.982575	−0.491288	−	0.870997i	$-0.663474\pi$
$911$	−29.6569	−0.982575	−0.491288	+	0.870997i	$0.663474\pi$
$912$	0	0
$913$	−5.65685	−0.187215
$914$	0	0
$915$	−14.1421	−0.467525
$916$	0	0
$917$	19.3137	0.637795
$918$	0	0
$919$	24.2426	0.799691	0.399845	−	0.916583i	$-0.369064\pi$
$919$	24.2426	0.799691	0.399845	+	0.916583i	$0.369064\pi$
$920$	0	0
$921$	−22.4853	−0.740915
$922$	0	0
$923$	−11.3137	−0.372395
$924$	0	0
$925$	1.02944	0.0338477
$926$	0	0
$927$	−1.17157	−0.0384795
$928$	0	0
$929$	38.1005	1.25004	0.625019	−	0.780610i	$-0.285091\pi$
$929$	38.1005	1.25004	0.625019	+	0.780610i	$0.285091\pi$
$930$	0	0
$931$	2.48528	0.0814518
$932$	0	0
$933$	28.1421	0.921332
$934$	0	0
$935$	0.828427	0.0270925
$936$	0	0
$937$	59.4558	1.94234	0.971169	−	0.238393i	$-0.0766205\pi$
$937$	59.4558	1.94234	0.971169	+	0.238393i	$0.0766205\pi$
$938$	0	0
$939$	24.9706	0.814884
$940$	0	0
$941$	−17.4558	−0.569044	−0.284522	−	0.958669i	$-0.591835\pi$
$941$	−17.4558	−0.569044	−0.284522	+	0.958669i	$0.591835\pi$
$942$	0	0
$943$	32.9706	1.07367
$944$	0	0
$945$	2.82843	0.0920087
$946$	0	0
$947$	42.1421	1.36944	0.684718	−	0.728808i	$-0.259926\pi$
$947$	42.1421	1.36944	0.684718	+	0.728808i	$0.259926\pi$
$948$	0	0
$949$	−16.4853	−0.535135
$950$	0	0
$951$	−34.3848	−1.11500
$952$	0	0
$953$	12.1005	0.391974	0.195987	−	0.980607i	$-0.437209\pi$
$953$	12.1005	0.391974	0.195987	+	0.980607i	$0.437209\pi$
$954$	0	0
$955$	−4.68629	−0.151645
$956$	0	0
$957$	−5.07107	−0.163924
$958$	0	0
$959$	25.4558	0.822012
$960$	0	0
$961$	−25.9706	−0.837760
$962$	0	0
$963$	16.4853	0.531231
$964$	0	0
$965$	−3.02944	−0.0975210
$966$	0	0
$967$	−36.6274	−1.17786	−0.588929	−	0.808185i	$-0.700450\pi$
$967$	−36.6274	−1.17786	−0.588929	+	0.808185i	$0.700450\pi$
$968$	0	0
$969$	−0.485281	−0.0155895
$970$	0	0
$971$	−16.1421	−0.518026	−0.259013	−	0.965874i	$-0.583397\pi$
$971$	−16.1421	−0.518026	−0.259013	+	0.965874i	$0.583397\pi$
$972$	0	0
$973$	35.7990	1.14766
$974$	0	0
$975$	3.00000	0.0960769
$976$	0	0
$977$	−59.0711	−1.88985	−0.944925	−	0.327286i	$-0.893866\pi$
$977$	−59.0711	−1.88985	−0.944925	+	0.327286i	$0.893866\pi$
$978$	0	0
$979$	−8.24264	−0.263436
$980$	0	0
$981$	−10.8284	−0.345725
$982$	0	0
$983$	−32.0000	−1.02064	−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000	−1.02064	−0.510321	+	0.859984i	$0.670473\pi$
$984$	0	0
$985$	11.3137	0.360485
$986$	0	0
$987$	−24.9706	−0.794822
$988$	0	0
$989$	39.7990	1.26553
$990$	0	0
$991$	−7.79899	−0.247743	−0.123872	−	0.992298i	$-0.539531\pi$
$991$	−7.79899	−0.247743	−0.123872	+	0.992298i	$0.539531\pi$
$992$	0	0
$993$	22.0416	0.699470
$994$	0	0
$995$	−25.6569	−0.813377
$996$	0	0
$997$	0.142136	0.00450148	0.00225074	−	0.999997i	$-0.499284\pi$
$997$	0.142136	0.00450148	0.00225074	+	0.999997i	$0.499284\pi$
$998$	0	0
$999$	0.343146	0.0108567

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.m.1.1	✓	2	4.3	odd	2
6864.2.a.be.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.41421 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.41421 q^{5} +2.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +1.41421 q^{15} -0.585786 q^{17} -0.828427 q^{19} -2.00000 q^{21} -4.82843 q^{23} -3.00000 q^{25} -1.00000 q^{27} +5.07107 q^{29} +2.24264 q^{31} -1.00000 q^{33} -2.82843 q^{35} -0.343146 q^{37} -1.00000 q^{39} -6.82843 q^{41} -8.24264 q^{43} -1.41421 q^{45} +12.4853 q^{47} -3.00000 q^{49} +0.585786 q^{51} +2.00000 q^{53} -1.41421 q^{55} +0.828427 q^{57} +12.4853 q^{59} -10.0000 q^{61} +2.00000 q^{63} -1.41421 q^{65} +7.89949 q^{67} +4.82843 q^{69} -11.3137 q^{71} -16.4853 q^{73} +3.00000 q^{75} +2.00000 q^{77} +0.242641 q^{79} +1.00000 q^{81} -5.65685 q^{83} +0.828427 q^{85} -5.07107 q^{87} -8.24264 q^{89} +2.00000 q^{91} -2.24264 q^{93} +1.17157 q^{95} -7.65685 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 4 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 4 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{17} + 4 q^{19} - 4 q^{21} - 4 q^{23} - 6 q^{25} - 2 q^{27} - 4 q^{29} - 4 q^{31} - 2 q^{33} - 12 q^{37} - 2 q^{39} - 8 q^{41} - 8 q^{43} + 8 q^{47} - 6 q^{49} + 4 q^{51} + 4 q^{53} - 4 q^{57} + 8 q^{59} - 20 q^{61} + 4 q^{63} - 4 q^{67} + 4 q^{69} - 16 q^{73} + 6 q^{75} + 4 q^{77} - 8 q^{79} + 2 q^{81} - 4 q^{85} + 4 q^{87} - 8 q^{89} + 4 q^{91} + 4 q^{93} + 8 q^{95} - 4 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 4 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 - 4 * q^17 + 4 * q^19 - 4 * q^21 - 4 * q^23 - 6 * q^25 - 2 * q^27 - 4 * q^29 - 4 * q^31 - 2 * q^33 - 12 * q^37 - 2 * q^39 - 8 * q^41 - 8 * q^43 + 8 * q^47 - 6 * q^49 + 4 * q^51 + 4 * q^53 - 4 * q^57 + 8 * q^59 - 20 * q^61 + 4 * q^63 - 4 * q^67 + 4 * q^69 - 16 * q^73 + 6 * q^75 + 4 * q^77 - 8 * q^79 + 2 * q^81 - 4 * q^85 + 4 * q^87 - 8 * q^89 + 4 * q^91 + 4 * q^93 + 8 * q^95 - 4 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more