LMFDB - Embedded newform 6864.2.a.cf.1.5

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:	$5$
Coefficient field:	5.5.2172244.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 8x^{3} + 16x^{2} + 5x - 8 $ x^5 - 2x^4 - 8x^3 + 16x^2 + 5x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.60211$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.52882 q^{5} +3.24835 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.52882 q^{5} +3.24835 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +3.52882 q^{15} -5.48469 q^{17} +2.44934 q^{19} +3.24835 q^{21} -2.29362 q^{23} +7.45258 q^{25} +1.00000 q^{27} +9.32897 q^{29} -2.32783 q^{31} +1.00000 q^{33} +11.4629 q^{35} -11.4537 q^{37} +1.00000 q^{39} -10.0588 q^{41} +7.09186 q^{43} +3.52882 q^{45} +2.48433 q^{47} +3.55179 q^{49} -5.48469 q^{51} +13.6449 q^{53} +3.52882 q^{55} +2.44934 q^{57} +5.35995 q^{59} -7.71007 q^{61} +3.24835 q^{63} +3.52882 q^{65} +12.2750 q^{67} -2.29362 q^{69} -1.76516 q^{71} +1.80929 q^{73} +7.45258 q^{75} +3.24835 q^{77} -3.26150 q^{79} +1.00000 q^{81} +13.1482 q^{83} -19.3545 q^{85} +9.32897 q^{87} -2.06824 q^{89} +3.24835 q^{91} -2.32783 q^{93} +8.64329 q^{95} -5.04526 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 + q^5 + 5 * q^7 + 5 * q^9	$ 5 q + 5 q^{3} + q^{5} + 5 q^{7} + 5 q^{9} + 5 q^{11} + 5 q^{13} + q^{15} + 4 q^{19} + 5 q^{21} + 5 q^{23} + 4 q^{25} + 5 q^{27} + 11 q^{29} + 8 q^{31} + 5 q^{33} + 5 q^{35} - 8 q^{37} + 5 q^{39} - q^{41} - q^{43} + q^{45} + 18 q^{47} + 10 q^{49} + 2 q^{53} + q^{55} + 4 q^{57} + 13 q^{59} + 9 q^{61} + 5 q^{63} + q^{65} - 5 q^{67} + 5 q^{69} + 24 q^{71} - 13 q^{73} + 4 q^{75} + 5 q^{77} + 6 q^{79} + 5 q^{81} + 22 q^{83} - 22 q^{85} + 11 q^{87} - 14 q^{89} + 5 q^{91} + 8 q^{93} + 32 q^{95} - 20 q^{97} + 5 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + q^5 + 5 * q^7 + 5 * q^9 + 5 * q^11 + 5 * q^13 + q^15 + 4 * q^19 + 5 * q^21 + 5 * q^23 + 4 * q^25 + 5 * q^27 + 11 * q^29 + 8 * q^31 + 5 * q^33 + 5 * q^35 - 8 * q^37 + 5 * q^39 - q^41 - q^43 + q^45 + 18 * q^47 + 10 * q^49 + 2 * q^53 + q^55 + 4 * q^57 + 13 * q^59 + 9 * q^61 + 5 * q^63 + q^65 - 5 * q^67 + 5 * q^69 + 24 * q^71 - 13 * q^73 + 4 * q^75 + 5 * q^77 + 6 * q^79 + 5 * q^81 + 22 * q^83 - 22 * q^85 + 11 * q^87 - 14 * q^89 + 5 * q^91 + 8 * q^93 + 32 * q^95 - 20 * q^97 + 5 * 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
$5$	3.52882	1.57814	0.789068	−	0.614305i	$-0.210564\pi$
$5$	3.52882	1.57814	0.789068	+	0.614305i	$0.210564\pi$
$6$	0	0
$7$	3.24835	1.22776	0.613881	−	0.789399i	$-0.289607\pi$
$7$	3.24835	1.22776	0.613881	+	0.789399i	$0.289607\pi$
$8$	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
$15$	3.52882	0.911138
$16$	0	0
$17$	−5.48469	−1.33023	−0.665117	−	0.746740i	$-0.731618\pi$
$17$	−5.48469	−1.33023	−0.665117	+	0.746740i	$0.731618\pi$
$18$	0	0
$19$	2.44934	0.561917	0.280959	−	0.959720i	$-0.409348\pi$
$19$	2.44934	0.561917	0.280959	+	0.959720i	$0.409348\pi$
$20$	0	0
$21$	3.24835	0.708849
$22$	0	0
$23$	−2.29362	−0.478252	−0.239126	−	0.970989i	$-0.576861\pi$
$23$	−2.29362	−0.478252	−0.239126	+	0.970989i	$0.576861\pi$
$24$	0	0
$25$	7.45258	1.49052
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	9.32897	1.73235	0.866173	−	0.499745i	$-0.166573\pi$
$29$	9.32897	1.73235	0.866173	+	0.499745i	$0.166573\pi$
$30$	0	0
$31$	−2.32783	−0.418091	−0.209045	−	0.977906i	$-0.567036\pi$
$31$	−2.32783	−0.418091	−0.209045	+	0.977906i	$0.567036\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	11.4629	1.93758
$36$	0	0
$37$	−11.4537	−1.88298	−0.941489	−	0.337043i	$-0.890573\pi$
$37$	−11.4537	−1.88298	−0.941489	+	0.337043i	$0.890573\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−10.0588	−1.57092	−0.785459	−	0.618914i	$-0.787573\pi$
$41$	−10.0588	−1.57092	−0.785459	+	0.618914i	$0.787573\pi$
$42$	0	0
$43$	7.09186	1.08150	0.540749	−	0.841184i	$-0.318141\pi$
$43$	7.09186	1.08150	0.540749	+	0.841184i	$0.318141\pi$
$44$	0	0
$45$	3.52882	0.526046
$46$	0	0
$47$	2.48433	0.362376	0.181188	−	0.983448i	$-0.442006\pi$
$47$	2.48433	0.362376	0.181188	+	0.983448i	$0.442006\pi$
$48$	0	0
$49$	3.55179	0.507399
$50$	0	0
$51$	−5.48469	−0.768010
$52$	0	0
$53$	13.6449	1.87427	0.937134	−	0.348970i	$-0.113468\pi$
$53$	13.6449	1.87427	0.937134	+	0.348970i	$0.113468\pi$
$54$	0	0
$55$	3.52882	0.475826
$56$	0	0
$57$	2.44934	0.324423
$58$	0	0
$59$	5.35995	0.697806	0.348903	−	0.937159i	$-0.386554\pi$
$59$	5.35995	0.697806	0.348903	+	0.937159i	$0.386554\pi$
$60$	0	0
$61$	−7.71007	−0.987173	−0.493587	−	0.869697i	$-0.664314\pi$
$61$	−7.71007	−0.987173	−0.493587	+	0.869697i	$0.664314\pi$
$62$	0	0
$63$	3.24835	0.409254
$64$	0	0
$65$	3.52882	0.437696
$66$	0	0
$67$	12.2750	1.49963	0.749816	−	0.661647i	$-0.230142\pi$
$67$	12.2750	1.49963	0.749816	+	0.661647i	$0.230142\pi$
$68$	0	0
$69$	−2.29362	−0.276119
$70$	0	0
$71$	−1.76516	−0.209486	−0.104743	−	0.994499i	$-0.533402\pi$
$71$	−1.76516	−0.209486	−0.104743	+	0.994499i	$0.533402\pi$
$72$	0	0
$73$	1.80929	0.211761	0.105881	−	0.994379i	$-0.466234\pi$
$73$	1.80929	0.211761	0.105881	+	0.994379i	$0.466234\pi$
$74$	0	0
$75$	7.45258	0.860549
$76$	0	0
$77$	3.24835	0.370184
$78$	0	0
$79$	−3.26150	−0.366948	−0.183474	−	0.983025i	$-0.558734\pi$
$79$	−3.26150	−0.366948	−0.183474	+	0.983025i	$0.558734\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.1482	1.44320	0.721600	−	0.692311i	$-0.243407\pi$
$83$	13.1482	1.44320	0.721600	+	0.692311i	$0.243407\pi$
$84$	0	0
$85$	−19.3545	−2.09929
$86$	0	0
$87$	9.32897	1.00017
$88$	0	0
$89$	−2.06824	−0.219233	−0.109616	−	0.993974i	$-0.534962\pi$
$89$	−2.06824	−0.219233	−0.109616	+	0.993974i	$0.534962\pi$
$90$	0	0
$91$	3.24835	0.340520
$92$	0	0
$93$	−2.32783	−0.241385
$94$	0	0
$95$	8.64329	0.886783
$96$	0	0
$97$	−5.04526	−0.512269	−0.256135	−	0.966641i	$-0.582449\pi$
$97$	−5.04526	−0.512269	−0.256135	+	0.966641i	$0.582449\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	0.897457	0.0893003	0.0446502	−	0.999003i	$-0.485783\pi$
$101$	0.897457	0.0893003	0.0446502	+	0.999003i	$0.485783\pi$
$102$	0	0
$103$	−1.69769	−0.167279	−0.0836394	−	0.996496i	$-0.526654\pi$
$103$	−1.69769	−0.167279	−0.0836394	+	0.996496i	$0.526654\pi$
$104$	0	0
$105$	11.4629	1.11866
$106$	0	0
$107$	−8.44389	−0.816301	−0.408151	−	0.912915i	$-0.633826\pi$
$107$	−8.44389	−0.816301	−0.408151	+	0.912915i	$0.633826\pi$
$108$	0	0
$109$	5.30344	0.507978	0.253989	−	0.967207i	$-0.418257\pi$
$109$	5.30344	0.507978	0.253989	+	0.967207i	$0.418257\pi$
$110$	0	0
$111$	−11.4537	−1.08714
$112$	0	0
$113$	13.4569	1.26592	0.632961	−	0.774183i	$-0.281839\pi$
$113$	13.4569	1.26592	0.632961	+	0.774183i	$0.281839\pi$
$114$	0	0
$115$	−8.09376	−0.754747
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−17.8162	−1.63321
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−10.0588	−0.906969
$124$	0	0
$125$	8.65470	0.774100
$126$	0	0
$127$	9.11492	0.808818	0.404409	−	0.914578i	$-0.367477\pi$
$127$	9.11492	0.808818	0.404409	+	0.914578i	$0.367477\pi$
$128$	0	0
$129$	7.09186	0.624403
$130$	0	0
$131$	6.88954	0.601942	0.300971	−	0.953633i	$-0.402689\pi$
$131$	6.88954	0.601942	0.300971	+	0.953633i	$0.402689\pi$
$132$	0	0
$133$	7.95632	0.689901
$134$	0	0
$135$	3.52882	0.303713
$136$	0	0
$137$	−9.28257	−0.793063	−0.396532	−	0.918021i	$-0.629786\pi$
$137$	−9.28257	−0.793063	−0.396532	+	0.918021i	$0.629786\pi$
$138$	0	0
$139$	16.3512	1.38689	0.693444	−	0.720511i	$-0.256092\pi$
$139$	16.3512	1.38689	0.693444	+	0.720511i	$0.256092\pi$
$140$	0	0
$141$	2.48433	0.209218
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	32.9203	2.73388
$146$	0	0
$147$	3.55179	0.292947
$148$	0	0
$149$	6.85779	0.561812	0.280906	−	0.959735i	$-0.409365\pi$
$149$	6.85779	0.561812	0.280906	+	0.959735i	$0.409365\pi$
$150$	0	0
$151$	−2.36108	−0.192142	−0.0960711	−	0.995374i	$-0.530628\pi$
$151$	−2.36108	−0.192142	−0.0960711	+	0.995374i	$0.530628\pi$
$152$	0	0
$153$	−5.48469	−0.443411
$154$	0	0
$155$	−8.21450	−0.659805
$156$	0	0
$157$	−17.2539	−1.37701	−0.688504	−	0.725233i	$-0.741732\pi$
$157$	−17.2539	−1.37701	−0.688504	+	0.725233i	$0.741732\pi$
$158$	0	0
$159$	13.6449	1.08211
$160$	0	0
$161$	−7.45048	−0.587180
$162$	0	0
$163$	−8.66881	−0.678994	−0.339497	−	0.940607i	$-0.610257\pi$
$163$	−8.66881	−0.678994	−0.339497	+	0.940607i	$0.610257\pi$
$164$	0	0
$165$	3.52882	0.274718
$166$	0	0
$167$	13.6094	1.05313	0.526565	−	0.850135i	$-0.323480\pi$
$167$	13.6094	1.05313	0.526565	+	0.850135i	$0.323480\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.44934	0.187306
$172$	0	0
$173$	−12.2378	−0.930419	−0.465210	−	0.885201i	$-0.654021\pi$
$173$	−12.2378	−0.930419	−0.465210	+	0.885201i	$0.654021\pi$
$174$	0	0
$175$	24.2086	1.83000
$176$	0	0
$177$	5.35995	0.402878
$178$	0	0
$179$	10.7549	0.803858	0.401929	−	0.915671i	$-0.368340\pi$
$179$	10.7549	0.803858	0.401929	+	0.915671i	$0.368340\pi$
$180$	0	0
$181$	−8.68491	−0.645544	−0.322772	−	0.946477i	$-0.604615\pi$
$181$	−8.68491	−0.645544	−0.322772	+	0.946477i	$0.604615\pi$
$182$	0	0
$183$	−7.71007	−0.569945
$184$	0	0
$185$	−40.4181	−2.97160
$186$	0	0
$187$	−5.48469	−0.401080
$188$	0	0
$189$	3.24835	0.236283
$190$	0	0
$191$	−14.8523	−1.07468	−0.537339	−	0.843367i	$-0.680570\pi$
$191$	−14.8523	−1.07468	−0.537339	+	0.843367i	$0.680570\pi$
$192$	0	0
$193$	−7.50698	−0.540364	−0.270182	−	0.962809i	$-0.587084\pi$
$193$	−7.50698	−0.540364	−0.270182	+	0.962809i	$0.587084\pi$
$194$	0	0
$195$	3.52882	0.252704
$196$	0	0
$197$	−3.19553	−0.227672	−0.113836	−	0.993500i	$-0.536314\pi$
$197$	−3.19553	−0.227672	−0.113836	+	0.993500i	$0.536314\pi$
$198$	0	0
$199$	−19.3205	−1.36959	−0.684796	−	0.728735i	$-0.740109\pi$
$199$	−19.3205	−1.36959	−0.684796	+	0.728735i	$0.740109\pi$
$200$	0	0
$201$	12.2750	0.865813
$202$	0	0
$203$	30.3038	2.12691
$204$	0	0
$205$	−35.4956	−2.47912
$206$	0	0
$207$	−2.29362	−0.159417
$208$	0	0
$209$	2.44934	0.169424
$210$	0	0
$211$	−4.58374	−0.315558	−0.157779	−	0.987474i	$-0.550433\pi$
$211$	−4.58374	−0.315558	−0.157779	+	0.987474i	$0.550433\pi$
$212$	0	0
$213$	−1.76516	−0.120947
$214$	0	0
$215$	25.0259	1.70675
$216$	0	0
$217$	−7.56162	−0.513316
$218$	0	0
$219$	1.80929	0.122260
$220$	0	0
$221$	−5.48469	−0.368940
$222$	0	0
$223$	15.3212	1.02599	0.512993	−	0.858393i	$-0.328537\pi$
$223$	15.3212	1.02599	0.512993	+	0.858393i	$0.328537\pi$
$224$	0	0
$225$	7.45258	0.496838
$226$	0	0
$227$	−5.66821	−0.376212	−0.188106	−	0.982149i	$-0.560235\pi$
$227$	−5.66821	−0.376212	−0.188106	+	0.982149i	$0.560235\pi$
$228$	0	0
$229$	6.49994	0.429528	0.214764	−	0.976666i	$-0.431102\pi$
$229$	6.49994	0.429528	0.214764	+	0.976666i	$0.431102\pi$
$230$	0	0
$231$	3.24835	0.213726
$232$	0	0
$233$	−11.5430	−0.756208	−0.378104	−	0.925763i	$-0.623424\pi$
$233$	−11.5430	−0.756208	−0.378104	+	0.925763i	$0.623424\pi$
$234$	0	0
$235$	8.76675	0.571880
$236$	0	0
$237$	−3.26150	−0.211857
$238$	0	0
$239$	7.78299	0.503440	0.251720	−	0.967800i	$-0.419004\pi$
$239$	7.78299	0.503440	0.251720	+	0.967800i	$0.419004\pi$
$240$	0	0
$241$	−26.6383	−1.71592	−0.857961	−	0.513714i	$-0.828269\pi$
$241$	−26.6383	−1.71592	−0.857961	+	0.513714i	$0.828269\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	12.5336	0.800745
$246$	0	0
$247$	2.44934	0.155848
$248$	0	0
$249$	13.1482	0.833231
$250$	0	0
$251$	−18.5880	−1.17326	−0.586631	−	0.809854i	$-0.699546\pi$
$251$	−18.5880	−1.17326	−0.586631	+	0.809854i	$0.699546\pi$
$252$	0	0
$253$	−2.29362	−0.144198
$254$	0	0
$255$	−19.3545	−1.21203
$256$	0	0
$257$	−4.23233	−0.264006	−0.132003	−	0.991249i	$-0.542141\pi$
$257$	−4.23233	−0.264006	−0.132003	+	0.991249i	$0.542141\pi$
$258$	0	0
$259$	−37.2057	−2.31185
$260$	0	0
$261$	9.32897	0.577449
$262$	0	0
$263$	−27.2699	−1.68154	−0.840768	−	0.541396i	$-0.817896\pi$
$263$	−27.2699	−1.68154	−0.840768	+	0.541396i	$0.817896\pi$
$264$	0	0
$265$	48.1503	2.95785
$266$	0	0
$267$	−2.06824	−0.126574
$268$	0	0
$269$	−17.3334	−1.05684	−0.528419	−	0.848984i	$-0.677215\pi$
$269$	−17.3334	−1.05684	−0.528419	+	0.848984i	$0.677215\pi$
$270$	0	0
$271$	−19.9402	−1.21128	−0.605640	−	0.795739i	$-0.707083\pi$
$271$	−19.9402	−1.21128	−0.605640	+	0.795739i	$0.707083\pi$
$272$	0	0
$273$	3.24835	0.196599
$274$	0	0
$275$	7.45258	0.449407
$276$	0	0
$277$	8.21405	0.493534	0.246767	−	0.969075i	$-0.420632\pi$
$277$	8.21405	0.493534	0.246767	+	0.969075i	$0.420632\pi$
$278$	0	0
$279$	−2.32783	−0.139364
$280$	0	0
$281$	−15.8135	−0.943354	−0.471677	−	0.881772i	$-0.656351\pi$
$281$	−15.8135	−0.943354	−0.471677	+	0.881772i	$0.656351\pi$
$282$	0	0
$283$	−11.2407	−0.668191	−0.334095	−	0.942539i	$-0.608431\pi$
$283$	−11.2407	−0.668191	−0.334095	+	0.942539i	$0.608431\pi$
$284$	0	0
$285$	8.64329	0.511984
$286$	0	0
$287$	−32.6745	−1.92871
$288$	0	0
$289$	13.0818	0.769520
$290$	0	0
$291$	−5.04526	−0.295759
$292$	0	0
$293$	−16.4756	−0.962517	−0.481259	−	0.876579i	$-0.659820\pi$
$293$	−16.4756	−0.962517	−0.481259	+	0.876579i	$0.659820\pi$
$294$	0	0
$295$	18.9143	1.10123
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−2.29362	−0.132643
$300$	0	0
$301$	23.0368	1.32782
$302$	0	0
$303$	0.897457	0.0515576
$304$	0	0
$305$	−27.2075	−1.55789
$306$	0	0
$307$	30.0991	1.71785	0.858924	−	0.512102i	$-0.171133\pi$
$307$	30.0991	1.71785	0.858924	+	0.512102i	$0.171133\pi$
$308$	0	0
$309$	−1.69769	−0.0965784
$310$	0	0
$311$	18.6929	1.05998	0.529989	−	0.848004i	$-0.322196\pi$
$311$	18.6929	1.05998	0.529989	+	0.848004i	$0.322196\pi$
$312$	0	0
$313$	7.73114	0.436990	0.218495	−	0.975838i	$-0.429885\pi$
$313$	7.73114	0.436990	0.218495	+	0.975838i	$0.429885\pi$
$314$	0	0
$315$	11.4629	0.645859
$316$	0	0
$317$	−24.2897	−1.36424	−0.682122	−	0.731239i	$-0.738943\pi$
$317$	−24.2897	−1.36424	−0.682122	+	0.731239i	$0.738943\pi$
$318$	0	0
$319$	9.32897	0.522322
$320$	0	0
$321$	−8.44389	−0.471292
$322$	0	0
$323$	−13.4339	−0.747481
$324$	0	0
$325$	7.45258	0.413395
$326$	0	0
$327$	5.30344	0.293281
$328$	0	0
$329$	8.06997	0.444912
$330$	0	0
$331$	2.62003	0.144010	0.0720050	−	0.997404i	$-0.477060\pi$
$331$	2.62003	0.144010	0.0720050	+	0.997404i	$0.477060\pi$
$332$	0	0
$333$	−11.4537	−0.627659
$334$	0	0
$335$	43.3163	2.36662
$336$	0	0
$337$	−14.3625	−0.782375	−0.391188	−	0.920311i	$-0.627936\pi$
$337$	−14.3625	−0.782375	−0.391188	+	0.920311i	$0.627936\pi$
$338$	0	0
$339$	13.4569	0.730881
$340$	0	0
$341$	−2.32783	−0.126059
$342$	0	0
$343$	−11.2010	−0.604797
$344$	0	0
$345$	−8.09376	−0.435754
$346$	0	0
$347$	34.3771	1.84546	0.922731	−	0.385445i	$-0.125952\pi$
$347$	34.3771	1.84546	0.922731	+	0.385445i	$0.125952\pi$
$348$	0	0
$349$	14.4553	0.773774	0.386887	−	0.922127i	$-0.373550\pi$
$349$	14.4553	0.773774	0.386887	+	0.922127i	$0.373550\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	14.3564	0.764114	0.382057	−	0.924139i	$-0.375216\pi$
$353$	14.3564	0.764114	0.382057	+	0.924139i	$0.375216\pi$
$354$	0	0
$355$	−6.22893	−0.330597
$356$	0	0
$357$	−17.8162	−0.942934
$358$	0	0
$359$	9.33888	0.492887	0.246444	−	0.969157i	$-0.420738\pi$
$359$	9.33888	0.492887	0.246444	+	0.969157i	$0.420738\pi$
$360$	0	0
$361$	−13.0007	−0.684249
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	6.38466	0.334188
$366$	0	0
$367$	−27.8507	−1.45379	−0.726897	−	0.686746i	$-0.759038\pi$
$367$	−27.8507	−1.45379	−0.726897	+	0.686746i	$0.759038\pi$
$368$	0	0
$369$	−10.0588	−0.523639
$370$	0	0
$371$	44.3234	2.30115
$372$	0	0
$373$	25.1455	1.30198	0.650992	−	0.759084i	$-0.274353\pi$
$373$	25.1455	1.30198	0.650992	+	0.759084i	$0.274353\pi$
$374$	0	0
$375$	8.65470	0.446927
$376$	0	0
$377$	9.32897	0.480466
$378$	0	0
$379$	−33.4424	−1.71782	−0.858911	−	0.512125i	$-0.828859\pi$
$379$	−33.4424	−1.71782	−0.858911	+	0.512125i	$0.828859\pi$
$380$	0	0
$381$	9.11492	0.466971
$382$	0	0
$383$	28.2495	1.44348	0.721741	−	0.692164i	$-0.243342\pi$
$383$	28.2495	1.44348	0.721741	+	0.692164i	$0.243342\pi$
$384$	0	0
$385$	11.4629	0.584201
$386$	0	0
$387$	7.09186	0.360499
$388$	0	0
$389$	−15.8301	−0.802620	−0.401310	−	0.915942i	$-0.631445\pi$
$389$	−15.8301	−0.802620	−0.401310	+	0.915942i	$0.631445\pi$
$390$	0	0
$391$	12.5798	0.636187
$392$	0	0
$393$	6.88954	0.347531
$394$	0	0
$395$	−11.5093	−0.579093
$396$	0	0
$397$	−4.96388	−0.249130	−0.124565	−	0.992211i	$-0.539754\pi$
$397$	−4.96388	−0.249130	−0.124565	+	0.992211i	$0.539754\pi$
$398$	0	0
$399$	7.95632	0.398314
$400$	0	0
$401$	−28.8698	−1.44169	−0.720846	−	0.693096i	$-0.756246\pi$
$401$	−28.8698	−1.44169	−0.720846	+	0.693096i	$0.756246\pi$
$402$	0	0
$403$	−2.32783	−0.115958
$404$	0	0
$405$	3.52882	0.175349
$406$	0	0
$407$	−11.4537	−0.567739
$408$	0	0
$409$	26.6445	1.31748	0.658742	−	0.752369i	$-0.271089\pi$
$409$	26.6445	1.31748	0.658742	+	0.752369i	$0.271089\pi$
$410$	0	0
$411$	−9.28257	−0.457875
$412$	0	0
$413$	17.4110	0.856739
$414$	0	0
$415$	46.3975	2.27757
$416$	0	0
$417$	16.3512	0.800720
$418$	0	0
$419$	5.83882	0.285245	0.142623	−	0.989777i	$-0.454447\pi$
$419$	5.83882	0.285245	0.142623	+	0.989777i	$0.454447\pi$
$420$	0	0
$421$	37.3214	1.81893	0.909466	−	0.415778i	$-0.136491\pi$
$421$	37.3214	1.81893	0.909466	+	0.415778i	$0.136491\pi$
$422$	0	0
$423$	2.48433	0.120792
$424$	0	0
$425$	−40.8751	−1.98273
$426$	0	0
$427$	−25.0450	−1.21201
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	−20.7222	−0.998152	−0.499076	−	0.866558i	$-0.666327\pi$
$431$	−20.7222	−0.998152	−0.499076	+	0.866558i	$0.666327\pi$
$432$	0	0
$433$	−24.7204	−1.18799	−0.593993	−	0.804470i	$-0.702449\pi$
$433$	−24.7204	−1.18799	−0.593993	+	0.804470i	$0.702449\pi$
$434$	0	0
$435$	32.9203	1.57841
$436$	0	0
$437$	−5.61785	−0.268738
$438$	0	0
$439$	−1.77211	−0.0845783	−0.0422892	−	0.999105i	$-0.513465\pi$
$439$	−1.77211	−0.0845783	−0.0422892	+	0.999105i	$0.513465\pi$
$440$	0	0
$441$	3.55179	0.169133
$442$	0	0
$443$	−6.43320	−0.305651	−0.152825	−	0.988253i	$-0.548837\pi$
$443$	−6.43320	−0.305651	−0.152825	+	0.988253i	$0.548837\pi$
$444$	0	0
$445$	−7.29844	−0.345979
$446$	0	0
$447$	6.85779	0.324362
$448$	0	0
$449$	29.4607	1.39034	0.695169	−	0.718847i	$-0.255330\pi$
$449$	29.4607	1.39034	0.695169	+	0.718847i	$0.255330\pi$
$450$	0	0
$451$	−10.0588	−0.473649
$452$	0	0
$453$	−2.36108	−0.110933
$454$	0	0
$455$	11.4629	0.537387
$456$	0	0
$457$	−40.3142	−1.88582	−0.942910	−	0.333048i	$-0.891923\pi$
$457$	−40.3142	−1.88582	−0.942910	+	0.333048i	$0.891923\pi$
$458$	0	0
$459$	−5.48469	−0.256003
$460$	0	0
$461$	18.1182	0.843851	0.421925	−	0.906631i	$-0.361354\pi$
$461$	18.1182	0.843851	0.421925	+	0.906631i	$0.361354\pi$
$462$	0	0
$463$	−24.9917	−1.16146	−0.580731	−	0.814096i	$-0.697233\pi$
$463$	−24.9917	−1.16146	−0.580731	+	0.814096i	$0.697233\pi$
$464$	0	0
$465$	−8.21450	−0.380938
$466$	0	0
$467$	−20.7835	−0.961743	−0.480872	−	0.876791i	$-0.659680\pi$
$467$	−20.7835	−0.961743	−0.480872	+	0.876791i	$0.659680\pi$
$468$	0	0
$469$	39.8736	1.84119
$470$	0	0
$471$	−17.2539	−0.795016
$472$	0	0
$473$	7.09186	0.326084
$474$	0	0
$475$	18.2539	0.837546
$476$	0	0
$477$	13.6449	0.624756
$478$	0	0
$479$	36.5844	1.67159	0.835793	−	0.549045i	$-0.185008\pi$
$479$	36.5844	1.67159	0.835793	+	0.549045i	$0.185008\pi$
$480$	0	0
$481$	−11.4537	−0.522244
$482$	0	0
$483$	−7.45048	−0.339008
$484$	0	0
$485$	−17.8038	−0.808430
$486$	0	0
$487$	−21.6052	−0.979025	−0.489513	−	0.871996i	$-0.662825\pi$
$487$	−21.6052	−0.979025	−0.489513	+	0.871996i	$0.662825\pi$
$488$	0	0
$489$	−8.66881	−0.392017
$490$	0	0
$491$	17.8732	0.806607	0.403304	−	0.915066i	$-0.367862\pi$
$491$	17.8732	0.806607	0.403304	+	0.915066i	$0.367862\pi$
$492$	0	0
$493$	−51.1665	−2.30442
$494$	0	0
$495$	3.52882	0.158609
$496$	0	0
$497$	−5.73386	−0.257199
$498$	0	0
$499$	−26.1343	−1.16993	−0.584967	−	0.811057i	$-0.698892\pi$
$499$	−26.1343	−1.16993	−0.584967	+	0.811057i	$0.698892\pi$
$500$	0	0
$501$	13.6094	0.608025
$502$	0	0
$503$	44.3670	1.97823	0.989114	−	0.147149i	$-0.0470098\pi$
$503$	44.3670	1.97823	0.989114	+	0.147149i	$0.0470098\pi$
$504$	0	0
$505$	3.16697	0.140928
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−0.450386	−0.0199630	−0.00998151	−	0.999950i	$-0.503177\pi$
$509$	−0.450386	−0.0199630	−0.00998151	+	0.999950i	$0.503177\pi$
$510$	0	0
$511$	5.87721	0.259992
$512$	0	0
$513$	2.44934	0.108141
$514$	0	0
$515$	−5.99086	−0.263989
$516$	0	0
$517$	2.48433	0.109261
$518$	0	0
$519$	−12.2378	−0.537178
$520$	0	0
$521$	38.0188	1.66563	0.832817	−	0.553549i	$-0.186727\pi$
$521$	38.0188	1.66563	0.832817	+	0.553549i	$0.186727\pi$
$522$	0	0
$523$	−19.7961	−0.865625	−0.432812	−	0.901484i	$-0.642479\pi$
$523$	−19.7961	−0.865625	−0.432812	+	0.901484i	$0.642479\pi$
$524$	0	0
$525$	24.2086	1.05655
$526$	0	0
$527$	12.7674	0.556158
$528$	0	0
$529$	−17.7393	−0.771275
$530$	0	0
$531$	5.35995	0.232602
$532$	0	0
$533$	−10.0588	−0.435694
$534$	0	0
$535$	−29.7970	−1.28824
$536$	0	0
$537$	10.7549	0.464107
$538$	0	0
$539$	3.55179	0.152987
$540$	0	0
$541$	3.16993	0.136286	0.0681429	−	0.997676i	$-0.478293\pi$
$541$	3.16993	0.136286	0.0681429	+	0.997676i	$0.478293\pi$
$542$	0	0
$543$	−8.68491	−0.372705
$544$	0	0
$545$	18.7149	0.801658
$546$	0	0
$547$	−16.1394	−0.690071	−0.345035	−	0.938590i	$-0.612133\pi$
$547$	−16.1394	−0.690071	−0.345035	+	0.938590i	$0.612133\pi$
$548$	0	0
$549$	−7.71007	−0.329058
$550$	0	0
$551$	22.8498	0.973435
$552$	0	0
$553$	−10.5945	−0.450524
$554$	0	0
$555$	−40.4181	−1.71565
$556$	0	0
$557$	39.5561	1.67605	0.838023	−	0.545635i	$-0.183711\pi$
$557$	39.5561	1.67605	0.838023	+	0.545635i	$0.183711\pi$
$558$	0	0
$559$	7.09186	0.299954
$560$	0	0
$561$	−5.48469	−0.231564
$562$	0	0
$563$	7.62505	0.321357	0.160679	−	0.987007i	$-0.448632\pi$
$563$	7.62505	0.321357	0.160679	+	0.987007i	$0.448632\pi$
$564$	0	0
$565$	47.4871	1.99780
$566$	0	0
$567$	3.24835	0.136418
$568$	0	0
$569$	−14.2286	−0.596495	−0.298247	−	0.954489i	$-0.596402\pi$
$569$	−14.2286	−0.596495	−0.298247	+	0.954489i	$0.596402\pi$
$570$	0	0
$571$	−0.707256	−0.0295978	−0.0147989	−	0.999890i	$-0.504711\pi$
$571$	−0.707256	−0.0295978	−0.0147989	+	0.999890i	$0.504711\pi$
$572$	0	0
$573$	−14.8523	−0.620465
$574$	0	0
$575$	−17.0934	−0.712842
$576$	0	0
$577$	29.3254	1.22083	0.610416	−	0.792081i	$-0.291002\pi$
$577$	29.3254	1.22083	0.610416	+	0.792081i	$0.291002\pi$
$578$	0	0
$579$	−7.50698	−0.311980
$580$	0	0
$581$	42.7099	1.77190
$582$	0	0
$583$	13.6449	0.565113
$584$	0	0
$585$	3.52882	0.145899
$586$	0	0
$587$	−30.0849	−1.24174	−0.620869	−	0.783914i	$-0.713220\pi$
$587$	−30.0849	−1.24174	−0.620869	+	0.783914i	$0.713220\pi$
$588$	0	0
$589$	−5.70165	−0.234933
$590$	0	0
$591$	−3.19553	−0.131447
$592$	0	0
$593$	−21.5857	−0.886418	−0.443209	−	0.896418i	$-0.646160\pi$
$593$	−21.5857	−0.886418	−0.443209	+	0.896418i	$0.646160\pi$
$594$	0	0
$595$	−62.8702	−2.57743
$596$	0	0
$597$	−19.3205	−0.790734
$598$	0	0
$599$	−31.7749	−1.29829	−0.649143	−	0.760667i	$-0.724872\pi$
$599$	−31.7749	−1.29829	−0.649143	+	0.760667i	$0.724872\pi$
$600$	0	0
$601$	−8.09700	−0.330283	−0.165142	−	0.986270i	$-0.552808\pi$
$601$	−8.09700	−0.330283	−0.165142	+	0.986270i	$0.552808\pi$
$602$	0	0
$603$	12.2750	0.499877
$604$	0	0
$605$	3.52882	0.143467
$606$	0	0
$607$	9.11492	0.369963	0.184982	−	0.982742i	$-0.440777\pi$
$607$	9.11492	0.369963	0.184982	+	0.982742i	$0.440777\pi$
$608$	0	0
$609$	30.3038	1.22797
$610$	0	0
$611$	2.48433	0.100505
$612$	0	0
$613$	−2.60183	−0.105087	−0.0525435	−	0.998619i	$-0.516733\pi$
$613$	−2.60183	−0.105087	−0.0525435	+	0.998619i	$0.516733\pi$
$614$	0	0
$615$	−35.4956	−1.43132
$616$	0	0
$617$	−20.8466	−0.839254	−0.419627	−	0.907697i	$-0.637839\pi$
$617$	−20.8466	−0.839254	−0.419627	+	0.907697i	$0.637839\pi$
$618$	0	0
$619$	31.8886	1.28171	0.640855	−	0.767662i	$-0.278580\pi$
$619$	31.8886	1.28171	0.640855	+	0.767662i	$0.278580\pi$
$620$	0	0
$621$	−2.29362	−0.0920397
$622$	0	0
$623$	−6.71837	−0.269166
$624$	0	0
$625$	−6.72199	−0.268880
$626$	0	0
$627$	2.44934	0.0978173
$628$	0	0
$629$	62.8201	2.50480
$630$	0	0
$631$	−5.38401	−0.214334	−0.107167	−	0.994241i	$-0.534178\pi$
$631$	−5.38401	−0.214334	−0.107167	+	0.994241i	$0.534178\pi$
$632$	0	0
$633$	−4.58374	−0.182187
$634$	0	0
$635$	32.1649	1.27643
$636$	0	0
$637$	3.55179	0.140727
$638$	0	0
$639$	−1.76516	−0.0698286
$640$	0	0
$641$	16.3920	0.647446	0.323723	−	0.946152i	$-0.395065\pi$
$641$	16.3920	0.647446	0.323723	+	0.946152i	$0.395065\pi$
$642$	0	0
$643$	−45.9975	−1.81396	−0.906981	−	0.421171i	$-0.861619\pi$
$643$	−45.9975	−1.81396	−0.906981	+	0.421171i	$0.861619\pi$
$644$	0	0
$645$	25.0259	0.985393
$646$	0	0
$647$	−15.2406	−0.599171	−0.299585	−	0.954069i	$-0.596848\pi$
$647$	−15.2406	−0.599171	−0.299585	+	0.954069i	$0.596848\pi$
$648$	0	0
$649$	5.35995	0.210396
$650$	0	0
$651$	−7.56162	−0.296363
$652$	0	0
$653$	−3.21660	−0.125875	−0.0629376	−	0.998017i	$-0.520047\pi$
$653$	−3.21660	−0.125875	−0.0629376	+	0.998017i	$0.520047\pi$
$654$	0	0
$655$	24.3120	0.949947
$656$	0	0
$657$	1.80929	0.0705871
$658$	0	0
$659$	1.42094	0.0553521	0.0276761	−	0.999617i	$-0.491189\pi$
$659$	1.42094	0.0553521	0.0276761	+	0.999617i	$0.491189\pi$
$660$	0	0
$661$	20.5478	0.799215	0.399608	−	0.916686i	$-0.369146\pi$
$661$	20.5478	0.799215	0.399608	+	0.916686i	$0.369146\pi$
$662$	0	0
$663$	−5.48469	−0.213008
$664$	0	0
$665$	28.0764	1.08876
$666$	0	0
$667$	−21.3971	−0.828498
$668$	0	0
$669$	15.3212	0.592353
$670$	0	0
$671$	−7.71007	−0.297644
$672$	0	0
$673$	9.13806	0.352246	0.176123	−	0.984368i	$-0.443644\pi$
$673$	9.13806	0.352246	0.176123	+	0.984368i	$0.443644\pi$
$674$	0	0
$675$	7.45258	0.286850
$676$	0	0
$677$	26.6437	1.02400	0.512000	−	0.858985i	$-0.328905\pi$
$677$	26.6437	1.02400	0.512000	+	0.858985i	$0.328905\pi$
$678$	0	0
$679$	−16.3888	−0.628944
$680$	0	0
$681$	−5.66821	−0.217206
$682$	0	0
$683$	22.9749	0.879110	0.439555	−	0.898216i	$-0.355136\pi$
$683$	22.9749	0.879110	0.439555	+	0.898216i	$0.355136\pi$
$684$	0	0
$685$	−32.7565	−1.25156
$686$	0	0
$687$	6.49994	0.247988
$688$	0	0
$689$	13.6449	0.519828
$690$	0	0
$691$	24.0127	0.913488	0.456744	−	0.889598i	$-0.349016\pi$
$691$	24.0127	0.913488	0.456744	+	0.889598i	$0.349016\pi$
$692$	0	0
$693$	3.24835	0.123395
$694$	0	0
$695$	57.7003	2.18870
$696$	0	0
$697$	55.1693	2.08969
$698$	0	0
$699$	−11.5430	−0.436597
$700$	0	0
$701$	4.66740	0.176285	0.0881425	−	0.996108i	$-0.471907\pi$
$701$	4.66740	0.176285	0.0881425	+	0.996108i	$0.471907\pi$
$702$	0	0
$703$	−28.0540	−1.05808
$704$	0	0
$705$	8.76675	0.330175
$706$	0	0
$707$	2.91526	0.109640
$708$	0	0
$709$	−13.9375	−0.523432	−0.261716	−	0.965145i	$-0.584288\pi$
$709$	−13.9375	−0.523432	−0.261716	+	0.965145i	$0.584288\pi$
$710$	0	0
$711$	−3.26150	−0.122316
$712$	0	0
$713$	5.33915	0.199953
$714$	0	0
$715$	3.52882	0.131970
$716$	0	0
$717$	7.78299	0.290661
$718$	0	0
$719$	25.7087	0.958773	0.479386	−	0.877604i	$-0.340859\pi$
$719$	25.7087	0.958773	0.479386	+	0.877604i	$0.340859\pi$
$720$	0	0
$721$	−5.51471	−0.205378
$722$	0	0
$723$	−26.6383	−0.990688
$724$	0	0
$725$	69.5248	2.58209
$726$	0	0
$727$	7.75698	0.287690	0.143845	−	0.989600i	$-0.454053\pi$
$727$	7.75698	0.287690	0.143845	+	0.989600i	$0.454053\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−38.8966	−1.43864
$732$	0	0
$733$	51.9270	1.91797	0.958984	−	0.283461i	$-0.0914826\pi$
$733$	51.9270	1.91797	0.958984	+	0.283461i	$0.0914826\pi$
$734$	0	0
$735$	12.5336	0.462310
$736$	0	0
$737$	12.2750	0.452156
$738$	0	0
$739$	−3.00728	−0.110625	−0.0553123	−	0.998469i	$-0.517615\pi$
$739$	−3.00728	−0.110625	−0.0553123	+	0.998469i	$0.517615\pi$
$740$	0	0
$741$	2.44934	0.0899788
$742$	0	0
$743$	−42.4774	−1.55834	−0.779172	−	0.626810i	$-0.784360\pi$
$743$	−42.4774	−1.55834	−0.779172	+	0.626810i	$0.784360\pi$
$744$	0	0
$745$	24.1999	0.886616
$746$	0	0
$747$	13.1482	0.481066
$748$	0	0
$749$	−27.4287	−1.00222
$750$	0	0
$751$	−4.12506	−0.150526	−0.0752628	−	0.997164i	$-0.523980\pi$
$751$	−4.12506	−0.150526	−0.0752628	+	0.997164i	$0.523980\pi$
$752$	0	0
$753$	−18.5880	−0.677383
$754$	0	0
$755$	−8.33184	−0.303227
$756$	0	0
$757$	−24.6428	−0.895657	−0.447828	−	0.894119i	$-0.647802\pi$
$757$	−24.6428	−0.895657	−0.447828	+	0.894119i	$0.647802\pi$
$758$	0	0
$759$	−2.29362	−0.0832530
$760$	0	0
$761$	51.6672	1.87293	0.936467	−	0.350756i	$-0.114075\pi$
$761$	51.6672	1.87293	0.936467	+	0.350756i	$0.114075\pi$
$762$	0	0
$763$	17.2274	0.623675
$764$	0	0
$765$	−19.3545	−0.699763
$766$	0	0
$767$	5.35995	0.193536
$768$	0	0
$769$	−1.60716	−0.0579558	−0.0289779	−	0.999580i	$-0.509225\pi$
$769$	−1.60716	−0.0579558	−0.0289779	+	0.999580i	$0.509225\pi$
$770$	0	0
$771$	−4.23233	−0.152424
$772$	0	0
$773$	8.38325	0.301524	0.150762	−	0.988570i	$-0.451827\pi$
$773$	8.38325	0.301524	0.150762	+	0.988570i	$0.451827\pi$
$774$	0	0
$775$	−17.3483	−0.623171
$776$	0	0
$777$	−37.2057	−1.33475
$778$	0	0
$779$	−24.6374	−0.882726
$780$	0	0
$781$	−1.76516	−0.0631624
$782$	0	0
$783$	9.32897	0.333390
$784$	0	0
$785$	−60.8858	−2.17311
$786$	0	0
$787$	−37.3531	−1.33150	−0.665748	−	0.746177i	$-0.731887\pi$
$787$	−37.3531	−1.33150	−0.665748	+	0.746177i	$0.731887\pi$
$788$	0	0
$789$	−27.2699	−0.970835
$790$	0	0
$791$	43.7129	1.55425
$792$	0	0
$793$	−7.71007	−0.273793
$794$	0	0
$795$	48.1503	1.70772
$796$	0	0
$797$	−4.12175	−0.146000	−0.0730000	−	0.997332i	$-0.523257\pi$
$797$	−4.12175	−0.146000	−0.0730000	+	0.997332i	$0.523257\pi$
$798$	0	0
$799$	−13.6258	−0.482045
$800$	0	0
$801$	−2.06824	−0.0730776
$802$	0	0
$803$	1.80929	0.0638484
$804$	0	0
$805$	−26.2914	−0.926650
$806$	0	0
$807$	−17.3334	−0.610166
$808$	0	0
$809$	43.6064	1.53312	0.766560	−	0.642172i	$-0.221967\pi$
$809$	43.6064	1.53312	0.766560	+	0.642172i	$0.221967\pi$
$810$	0	0
$811$	0.946046	0.0332202	0.0166101	−	0.999862i	$-0.494713\pi$
$811$	0.946046	0.0332202	0.0166101	+	0.999862i	$0.494713\pi$
$812$	0	0
$813$	−19.9402	−0.699333
$814$	0	0
$815$	−30.5907	−1.07154
$816$	0	0
$817$	17.3704	0.607713
$818$	0	0
$819$	3.24835	0.113507
$820$	0	0
$821$	−13.2226	−0.461470	−0.230735	−	0.973017i	$-0.574113\pi$
$821$	−13.2226	−0.461470	−0.230735	+	0.973017i	$0.574113\pi$
$822$	0	0
$823$	4.96024	0.172903	0.0864516	−	0.996256i	$-0.472447\pi$
$823$	4.96024	0.172903	0.0864516	+	0.996256i	$0.472447\pi$
$824$	0	0
$825$	7.45258	0.259465
$826$	0	0
$827$	−36.0721	−1.25435	−0.627175	−	0.778878i	$-0.715789\pi$
$827$	−36.0721	−1.25435	−0.627175	+	0.778878i	$0.715789\pi$
$828$	0	0
$829$	−11.8500	−0.411566	−0.205783	−	0.978598i	$-0.565974\pi$
$829$	−11.8500	−0.411566	−0.205783	+	0.978598i	$0.565974\pi$
$830$	0	0
$831$	8.21405	0.284942
$832$	0	0
$833$	−19.4805	−0.674959
$834$	0	0
$835$	48.0253	1.66198
$836$	0	0
$837$	−2.32783	−0.0804616
$838$	0	0
$839$	−43.2541	−1.49330	−0.746648	−	0.665219i	$-0.768338\pi$
$839$	−43.2541	−1.49330	−0.746648	+	0.665219i	$0.768338\pi$
$840$	0	0
$841$	58.0296	2.00102
$842$	0	0
$843$	−15.8135	−0.544645
$844$	0	0
$845$	3.52882	0.121395
$846$	0	0
$847$	3.24835	0.111615
$848$	0	0
$849$	−11.2407	−0.385780
$850$	0	0
$851$	26.2704	0.900539
$852$	0	0
$853$	−0.307251	−0.0105201	−0.00526003	−	0.999986i	$-0.501674\pi$
$853$	−0.307251	−0.0105201	−0.00526003	+	0.999986i	$0.501674\pi$
$854$	0	0
$855$	8.64329	0.295594
$856$	0	0
$857$	49.7992	1.70111	0.850553	−	0.525889i	$-0.176267\pi$
$857$	49.7992	1.70111	0.850553	+	0.525889i	$0.176267\pi$
$858$	0	0
$859$	−49.1125	−1.67570	−0.837849	−	0.545903i	$-0.816187\pi$
$859$	−49.1125	−1.67570	−0.837849	+	0.545903i	$0.816187\pi$
$860$	0	0
$861$	−32.6745	−1.11354
$862$	0	0
$863$	−27.7615	−0.945013	−0.472507	−	0.881327i	$-0.656651\pi$
$863$	−27.7615	−0.945013	−0.472507	+	0.881327i	$0.656651\pi$
$864$	0	0
$865$	−43.1848	−1.46833
$866$	0	0
$867$	13.0818	0.444283
$868$	0	0
$869$	−3.26150	−0.110639
$870$	0	0
$871$	12.2750	0.415923
$872$	0	0
$873$	−5.04526	−0.170756
$874$	0	0
$875$	28.1135	0.950410
$876$	0	0
$877$	−1.79857	−0.0607333	−0.0303666	−	0.999539i	$-0.509667\pi$
$877$	−1.79857	−0.0607333	−0.0303666	+	0.999539i	$0.509667\pi$
$878$	0	0
$879$	−16.4756	−0.555710
$880$	0	0
$881$	16.1505	0.544126	0.272063	−	0.962279i	$-0.412294\pi$
$881$	16.1505	0.544126	0.272063	+	0.962279i	$0.412294\pi$
$882$	0	0
$883$	−27.0031	−0.908725	−0.454363	−	0.890817i	$-0.650133\pi$
$883$	−27.0031	−0.908725	−0.454363	+	0.890817i	$0.650133\pi$
$884$	0	0
$885$	18.9143	0.635797
$886$	0	0
$887$	49.9740	1.67796	0.838981	−	0.544161i	$-0.183152\pi$
$887$	49.9740	1.67796	0.838981	+	0.544161i	$0.183152\pi$
$888$	0	0
$889$	29.6085	0.993036
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	6.08497	0.203626
$894$	0	0
$895$	37.9520	1.26860
$896$	0	0
$897$	−2.29362	−0.0765816
$898$	0	0
$899$	−21.7163	−0.724278
$900$	0	0
$901$	−74.8379	−2.49321
$902$	0	0
$903$	23.0368	0.766618
$904$	0	0
$905$	−30.6475	−1.01876
$906$	0	0
$907$	50.6063	1.68036	0.840178	−	0.542311i	$-0.182450\pi$
$907$	50.6063	1.68036	0.840178	+	0.542311i	$0.182450\pi$
$908$	0	0
$909$	0.897457	0.0297668
$910$	0	0
$911$	46.7755	1.54974	0.774871	−	0.632120i	$-0.217815\pi$
$911$	46.7755	1.54974	0.774871	+	0.632120i	$0.217815\pi$
$912$	0	0
$913$	13.1482	0.435141
$914$	0	0
$915$	−27.2075	−0.899451
$916$	0	0
$917$	22.3797	0.739041
$918$	0	0
$919$	−34.6524	−1.14308	−0.571539	−	0.820575i	$-0.693653\pi$
$919$	−34.6524	−1.14308	−0.571539	+	0.820575i	$0.693653\pi$
$920$	0	0
$921$	30.0991	0.991800
$922$	0	0
$923$	−1.76516	−0.0581009
$924$	0	0
$925$	−85.3596	−2.80661
$926$	0	0
$927$	−1.69769	−0.0557596
$928$	0	0
$929$	−48.8020	−1.60114	−0.800570	−	0.599239i	$-0.795470\pi$
$929$	−48.8020	−1.60114	−0.800570	+	0.599239i	$0.795470\pi$
$930$	0	0
$931$	8.69956	0.285116
$932$	0	0
$933$	18.6929	0.611979
$934$	0	0
$935$	−19.3545	−0.632960
$936$	0	0
$937$	−7.99427	−0.261161	−0.130581	−	0.991438i	$-0.541684\pi$
$937$	−7.99427	−0.261161	−0.130581	+	0.991438i	$0.541684\pi$
$938$	0	0
$939$	7.73114	0.252296
$940$	0	0
$941$	58.2642	1.89936	0.949679	−	0.313225i	$-0.101409\pi$
$941$	58.2642	1.89936	0.949679	+	0.313225i	$0.101409\pi$
$942$	0	0
$943$	23.0710	0.751295
$944$	0	0
$945$	11.4629	0.372887
$946$	0	0
$947$	−31.8587	−1.03527	−0.517634	−	0.855602i	$-0.673187\pi$
$947$	−31.8587	−1.03527	−0.517634	+	0.855602i	$0.673187\pi$
$948$	0	0
$949$	1.80929	0.0587320
$950$	0	0
$951$	−24.2897	−0.787646
$952$	0	0
$953$	15.4081	0.499117	0.249558	−	0.968360i	$-0.419715\pi$
$953$	15.4081	0.499117	0.249558	+	0.968360i	$0.419715\pi$
$954$	0	0
$955$	−52.4112	−1.69599
$956$	0	0
$957$	9.32897	0.301563
$958$	0	0
$959$	−30.1531	−0.973693
$960$	0	0
$961$	−25.5812	−0.825200
$962$	0	0
$963$	−8.44389	−0.272100
$964$	0	0
$965$	−26.4908	−0.852769
$966$	0	0
$967$	37.6657	1.21125	0.605624	−	0.795751i	$-0.292924\pi$
$967$	37.6657	1.21125	0.605624	+	0.795751i	$0.292924\pi$
$968$	0	0
$969$	−13.4339	−0.431558
$970$	0	0
$971$	21.2714	0.682632	0.341316	−	0.939949i	$-0.389127\pi$
$971$	21.2714	0.682632	0.341316	+	0.939949i	$0.389127\pi$
$972$	0	0
$973$	53.1144	1.70277
$974$	0	0
$975$	7.45258	0.238673
$976$	0	0
$977$	−40.9806	−1.31109	−0.655543	−	0.755158i	$-0.727560\pi$
$977$	−40.9806	−1.31109	−0.655543	+	0.755158i	$0.727560\pi$
$978$	0	0
$979$	−2.06824	−0.0661012
$980$	0	0
$981$	5.30344	0.169326
$982$	0	0
$983$	−21.0862	−0.672546	−0.336273	−	0.941765i	$-0.609166\pi$
$983$	−21.0862	−0.672546	−0.336273	+	0.941765i	$0.609166\pi$
$984$	0	0
$985$	−11.2765	−0.359298
$986$	0	0
$987$	8.06997	0.256870
$988$	0	0
$989$	−16.2660	−0.517229
$990$	0	0
$991$	−33.8837	−1.07635	−0.538175	−	0.842833i	$-0.680886\pi$
$991$	−33.8837	−1.07635	−0.538175	+	0.842833i	$0.680886\pi$
$992$	0	0
$993$	2.62003	0.0831442
$994$	0	0
$995$	−68.1785	−2.16140
$996$	0	0
$997$	30.9741	0.980959	0.490480	−	0.871453i	$-0.336822\pi$
$997$	30.9741	0.980959	0.490480	+	0.871453i	$0.336822\pi$
$998$	0	0
$999$	−11.4537	−0.362379

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.cf.1.5		5
4.3	odd	2		3432.2.a.w.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.w.1.5	✓	5	4.3	odd	2
6864.2.a.cf.1.5		5	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.52882 q^{5} +3.24835 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.52882 q^{5} +3.24835 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +3.52882 q^{15} -5.48469 q^{17} +2.44934 q^{19} +3.24835 q^{21} -2.29362 q^{23} +7.45258 q^{25} +1.00000 q^{27} +9.32897 q^{29} -2.32783 q^{31} +1.00000 q^{33} +11.4629 q^{35} -11.4537 q^{37} +1.00000 q^{39} -10.0588 q^{41} +7.09186 q^{43} +3.52882 q^{45} +2.48433 q^{47} +3.55179 q^{49} -5.48469 q^{51} +13.6449 q^{53} +3.52882 q^{55} +2.44934 q^{57} +5.35995 q^{59} -7.71007 q^{61} +3.24835 q^{63} +3.52882 q^{65} +12.2750 q^{67} -2.29362 q^{69} -1.76516 q^{71} +1.80929 q^{73} +7.45258 q^{75} +3.24835 q^{77} -3.26150 q^{79} +1.00000 q^{81} +13.1482 q^{83} -19.3545 q^{85} +9.32897 q^{87} -2.06824 q^{89} +3.24835 q^{91} -2.32783 q^{93} +8.64329 q^{95} -5.04526 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 + q^5 + 5 * q^7 + 5 * q^9	\( 5 q + 5 q^{3} + q^{5} + 5 q^{7} + 5 q^{9} + 5 q^{11} + 5 q^{13} + q^{15} + 4 q^{19} + 5 q^{21} + 5 q^{23} + 4 q^{25} + 5 q^{27} + 11 q^{29} + 8 q^{31} + 5 q^{33} + 5 q^{35} - 8 q^{37} + 5 q^{39} - q^{41} - q^{43} + q^{45} + 18 q^{47} + 10 q^{49} + 2 q^{53} + q^{55} + 4 q^{57} + 13 q^{59} + 9 q^{61} + 5 q^{63} + q^{65} - 5 q^{67} + 5 q^{69} + 24 q^{71} - 13 q^{73} + 4 q^{75} + 5 q^{77} + 6 q^{79} + 5 q^{81} + 22 q^{83} - 22 q^{85} + 11 q^{87} - 14 q^{89} + 5 q^{91} + 8 q^{93} + 32 q^{95} - 20 q^{97} + 5 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + q^5 + 5 * q^7 + 5 * q^9 + 5 * q^11 + 5 * q^13 + q^15 + 4 * q^19 + 5 * q^21 + 5 * q^23 + 4 * q^25 + 5 * q^27 + 11 * q^29 + 8 * q^31 + 5 * q^33 + 5 * q^35 - 8 * q^37 + 5 * q^39 - q^41 - q^43 + q^45 + 18 * q^47 + 10 * q^49 + 2 * q^53 + q^55 + 4 * q^57 + 13 * q^59 + 9 * q^61 + 5 * q^63 + q^65 - 5 * q^67 + 5 * q^69 + 24 * q^71 - 13 * q^73 + 4 * q^75 + 5 * q^77 + 6 * q^79 + 5 * q^81 + 22 * q^83 - 22 * q^85 + 11 * q^87 - 14 * q^89 + 5 * q^91 + 8 * q^93 + 32 * q^95 - 20 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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