LMFDB - Embedded newform 6864.2.a.u.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:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
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
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)

$f(q)$

$=$

$q+1.00000 q^{3} -1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} -1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} -1.00000 q^{15} -4.00000 q^{17} +2.00000 q^{19} +3.00000 q^{21} +1.00000 q^{23} -4.00000 q^{25} +1.00000 q^{27} -9.00000 q^{29} +4.00000 q^{31} -1.00000 q^{33} -3.00000 q^{35} -6.00000 q^{37} -1.00000 q^{39} +1.00000 q^{41} -11.0000 q^{43} -1.00000 q^{45} +2.00000 q^{49} -4.00000 q^{51} -10.0000 q^{53} +1.00000 q^{55} +2.00000 q^{57} +3.00000 q^{59} +5.00000 q^{61} +3.00000 q^{63} +1.00000 q^{65} -3.00000 q^{67} +1.00000 q^{69} -10.0000 q^{71} +9.00000 q^{73} -4.00000 q^{75} -3.00000 q^{77} -10.0000 q^{79} +1.00000 q^{81} +6.00000 q^{83} +4.00000 q^{85} -9.00000 q^{87} -8.00000 q^{89} -3.00000 q^{91} +4.00000 q^{93} -2.00000 q^{95} +2.00000 q^{97} -1.00000 q^{99} +O(q^{100})$

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
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	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
$15$	−1.00000	−0.258199
$16$	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
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−3.00000	−0.507093
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	1.00000	0.156174	0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000	0.156174	0.0780869	+	0.996947i	$0.475119\pi$
$42$	0	0
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	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
$55$	1.00000	0.134840
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	0	0
$63$	3.00000	0.377964
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	−3.00000	−0.341882
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	0	0
$87$	−9.00000	−0.964901
$88$	0	0
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	11.0000	1.08386	0.541931	−	0.840423i	$-0.317693\pi$
$103$	11.0000	1.08386	0.541931	+	0.840423i	$0.317693\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	0	0
$107$	−7.00000	−0.676716	−0.338358	−	0.941018i	$-0.609871\pi$
$107$	−7.00000	−0.676716	−0.338358	+	0.941018i	$0.609871\pi$
$108$	0	0
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	5.00000	0.470360	0.235180	−	0.971952i	$-0.424432\pi$
$113$	5.00000	0.470360	0.235180	+	0.971952i	$0.424432\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−12.0000	−1.10004
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.00000	0.0901670
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	0	0
$129$	−11.0000	−0.968496
$130$	0	0
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	9.00000	0.747409
$146$	0	0
$147$	2.00000	0.164957
$148$	0	0
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	0	0
$151$	−24.0000	−1.95309	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000	−1.95309	−0.976546	+	0.215308i	$0.930924\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	3.00000	0.236433
$162$	0	0
$163$	−9.00000	−0.704934	−0.352467	−	0.935824i	$-0.614657\pi$
$163$	−9.00000	−0.704934	−0.352467	+	0.935824i	$0.614657\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	0	0
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	−19.0000	−1.44454	−0.722272	−	0.691609i	$-0.756902\pi$
$173$	−19.0000	−1.44454	−0.722272	+	0.691609i	$0.756902\pi$
$174$	0	0
$175$	−12.0000	−0.907115
$176$	0	0
$177$	3.00000	0.225494
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	5.00000	0.369611
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	4.00000	0.292509
$188$	0	0
$189$	3.00000	0.218218
$190$	0	0
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	−3.00000	−0.212664	−0.106332	−	0.994331i	$-0.533911\pi$
$199$	−3.00000	−0.212664	−0.106332	+	0.994331i	$0.533911\pi$
$200$	0	0
$201$	−3.00000	−0.211604
$202$	0	0
$203$	−27.0000	−1.89503
$204$	0	0
$205$	−1.00000	−0.0698430
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	0	0
$213$	−10.0000	−0.685189
$214$	0	0
$215$	11.0000	0.750194
$216$	0	0
$217$	12.0000	0.814613
$218$	0	0
$219$	9.00000	0.608164
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	20.0000	1.33930	0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000	1.33930	0.669650	+	0.742677i	$0.266444\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	0	0
$229$	25.0000	1.65205	0.826023	−	0.563636i	$-0.190598\pi$
$229$	25.0000	1.65205	0.826023	+	0.563636i	$0.190598\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−10.0000	−0.649570
$238$	0	0
$239$	5.00000	0.323423	0.161712	−	0.986838i	$-0.448299\pi$
$239$	5.00000	0.323423	0.161712	+	0.986838i	$0.448299\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.00000	−0.127775
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	0	0
$255$	4.00000	0.250490
$256$	0	0
$257$	−25.0000	−1.55946	−0.779729	−	0.626118i	$-0.784643\pi$
$257$	−25.0000	−1.55946	−0.779729	+	0.626118i	$0.784643\pi$
$258$	0	0
$259$	−18.0000	−1.11847
$260$	0	0
$261$	−9.00000	−0.557086
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	0	0
$267$	−8.00000	−0.489592
$268$	0	0
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	−3.00000	−0.181568
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	5.00000	0.300421	0.150210	−	0.988654i	$-0.452005\pi$
$277$	5.00000	0.300421	0.150210	+	0.988654i	$0.452005\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	−29.0000	−1.72999	−0.864997	−	0.501776i	$-0.832680\pi$
$281$	−29.0000	−1.72999	−0.864997	+	0.501776i	$0.832680\pi$
$282$	0	0
$283$	9.00000	0.534994	0.267497	−	0.963559i	$-0.413803\pi$
$283$	9.00000	0.534994	0.267497	+	0.963559i	$0.413803\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	3.00000	0.177084
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	−3.00000	−0.174667
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−1.00000	−0.0578315
$300$	0	0
$301$	−33.0000	−1.90209
$302$	0	0
$303$	−10.0000	−0.574485
$304$	0	0
$305$	−5.00000	−0.286299
$306$	0	0
$307$	10.0000	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$308$	0	0
$309$	11.0000	0.625768
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	19.0000	1.07394	0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000	1.07394	0.536972	+	0.843600i	$0.319568\pi$
$314$	0	0
$315$	−3.00000	−0.169031
$316$	0	0
$317$	−3.00000	−0.168497	−0.0842484	−	0.996445i	$-0.526849\pi$
$317$	−3.00000	−0.168497	−0.0842484	+	0.996445i	$0.526849\pi$
$318$	0	0
$319$	9.00000	0.503903
$320$	0	0
$321$	−7.00000	−0.390702
$322$	0	0
$323$	−8.00000	−0.445132
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	16.0000	0.884802
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−19.0000	−1.04433	−0.522167	−	0.852843i	$-0.674876\pi$
$331$	−19.0000	−1.04433	−0.522167	+	0.852843i	$0.674876\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	3.00000	0.163908
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	5.00000	0.271563
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	−15.0000	−0.809924
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	10.0000	0.530745
$356$	0	0
$357$	−12.0000	−0.635107
$358$	0	0
$359$	−13.0000	−0.686114	−0.343057	−	0.939315i	$-0.611462\pi$
$359$	−13.0000	−0.686114	−0.343057	+	0.939315i	$0.611462\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−9.00000	−0.471082
$366$	0	0
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	0	0
$369$	1.00000	0.0520579
$370$	0	0
$371$	−30.0000	−1.55752
$372$	0	0
$373$	−11.0000	−0.569558	−0.284779	−	0.958593i	$-0.591920\pi$
$373$	−11.0000	−0.569558	−0.284779	+	0.958593i	$0.591920\pi$
$374$	0	0
$375$	9.00000	0.464758
$376$	0	0
$377$	9.00000	0.463524
$378$	0	0
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	0	0
$381$	14.0000	0.717242
$382$	0	0
$383$	−14.0000	−0.715367	−0.357683	−	0.933843i	$-0.616433\pi$
$383$	−14.0000	−0.715367	−0.357683	+	0.933843i	$0.616433\pi$
$384$	0	0
$385$	3.00000	0.152894
$386$	0	0
$387$	−11.0000	−0.559161
$388$	0	0
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	3.00000	0.151330
$394$	0	0
$395$	10.0000	0.503155
$396$	0	0
$397$	−19.0000	−0.953583	−0.476791	−	0.879017i	$-0.658200\pi$
$397$	−19.0000	−0.953583	−0.476791	+	0.879017i	$0.658200\pi$
$398$	0	0
$399$	6.00000	0.300376
$400$	0	0
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	7.00000	0.346128	0.173064	−	0.984911i	$-0.444633\pi$
$409$	7.00000	0.346128	0.173064	+	0.984911i	$0.444633\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	0	0
$413$	9.00000	0.442861
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	5.00000	0.243685	0.121843	−	0.992549i	$-0.461120\pi$
$421$	5.00000	0.243685	0.121843	+	0.992549i	$0.461120\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	16.0000	0.776114
$426$	0	0
$427$	15.0000	0.725901
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	0	0
$433$	5.00000	0.240285	0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000	0.240285	0.120142	+	0.992757i	$0.461665\pi$
$434$	0	0
$435$	9.00000	0.431517
$436$	0	0
$437$	2.00000	0.0956730
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	8.00000	0.379236
$446$	0	0
$447$	4.00000	0.189194
$448$	0	0
$449$	−36.0000	−1.69895	−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000	−1.69895	−0.849473	+	0.527633i	$0.823080\pi$
$450$	0	0
$451$	−1.00000	−0.0470882
$452$	0	0
$453$	−24.0000	−1.12762
$454$	0	0
$455$	3.00000	0.140642
$456$	0	0
$457$	−37.0000	−1.73079	−0.865393	−	0.501093i	$-0.832931\pi$
$457$	−37.0000	−1.73079	−0.865393	+	0.501093i	$0.832931\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	0	0
$463$	34.0000	1.58011	0.790057	−	0.613033i	$-0.210051\pi$
$463$	34.0000	1.58011	0.790057	+	0.613033i	$0.210051\pi$
$464$	0	0
$465$	−4.00000	−0.185496
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	−9.00000	−0.415581
$470$	0	0
$471$	−22.0000	−1.01371
$472$	0	0
$473$	11.0000	0.505781
$474$	0	0
$475$	−8.00000	−0.367065
$476$	0	0
$477$	−10.0000	−0.457869
$478$	0	0
$479$	31.0000	1.41643	0.708213	−	0.705999i	$-0.249502\pi$
$479$	31.0000	1.41643	0.708213	+	0.705999i	$0.249502\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	3.00000	0.136505
$484$	0	0
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	0	0
$489$	−9.00000	−0.406994
$490$	0	0
$491$	−29.0000	−1.30875	−0.654376	−	0.756169i	$-0.727069\pi$
$491$	−29.0000	−1.30875	−0.654376	+	0.756169i	$0.727069\pi$
$492$	0	0
$493$	36.0000	1.62136
$494$	0	0
$495$	1.00000	0.0449467
$496$	0	0
$497$	−30.0000	−1.34568
$498$	0	0
$499$	−17.0000	−0.761025	−0.380512	−	0.924776i	$-0.624252\pi$
$499$	−17.0000	−0.761025	−0.380512	+	0.924776i	$0.624252\pi$
$500$	0	0
$501$	3.00000	0.134030
$502$	0	0
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	0	0
$511$	27.0000	1.19441
$512$	0	0
$513$	2.00000	0.0883022
$514$	0	0
$515$	−11.0000	−0.484718
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−19.0000	−0.834007
$520$	0	0
$521$	−37.0000	−1.62100	−0.810500	−	0.585739i	$-0.800804\pi$
$521$	−37.0000	−1.62100	−0.810500	+	0.585739i	$0.800804\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	−12.0000	−0.523723
$526$	0	0
$527$	−16.0000	−0.696971
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	3.00000	0.130189
$532$	0	0
$533$	−1.00000	−0.0433148
$534$	0	0
$535$	7.00000	0.302636
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	14.0000	0.600798
$544$	0	0
$545$	−16.0000	−0.685365
$546$	0	0
$547$	5.00000	0.213785	0.106892	−	0.994271i	$-0.465910\pi$
$547$	5.00000	0.213785	0.106892	+	0.994271i	$0.465910\pi$
$548$	0	0
$549$	5.00000	0.213395
$550$	0	0
$551$	−18.0000	−0.766826
$552$	0	0
$553$	−30.0000	−1.27573
$554$	0	0
$555$	6.00000	0.254686
$556$	0	0
$557$	−14.0000	−0.593199	−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000	−0.593199	−0.296600	+	0.955002i	$0.595853\pi$
$558$	0	0
$559$	11.0000	0.465250
$560$	0	0
$561$	4.00000	0.168880
$562$	0	0
$563$	−32.0000	−1.34864	−0.674320	−	0.738440i	$-0.735563\pi$
$563$	−32.0000	−1.34864	−0.674320	+	0.738440i	$0.735563\pi$
$564$	0	0
$565$	−5.00000	−0.210352
$566$	0	0
$567$	3.00000	0.125988
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	29.0000	1.21361	0.606806	−	0.794850i	$-0.292450\pi$
$571$	29.0000	1.21361	0.606806	+	0.794850i	$0.292450\pi$
$572$	0	0
$573$	15.0000	0.626634
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	4.00000	0.166522	0.0832611	−	0.996528i	$-0.473466\pi$
$577$	4.00000	0.166522	0.0832611	+	0.996528i	$0.473466\pi$
$578$	0	0
$579$	−6.00000	−0.249351
$580$	0	0
$581$	18.0000	0.746766
$582$	0	0
$583$	10.0000	0.414158
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	37.0000	1.52715	0.763577	−	0.645717i	$-0.223441\pi$
$587$	37.0000	1.52715	0.763577	+	0.645717i	$0.223441\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	12.0000	0.493614
$592$	0	0
$593$	26.0000	1.06769	0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000	1.06769	0.533846	+	0.845582i	$0.320746\pi$
$594$	0	0
$595$	12.0000	0.491952
$596$	0	0
$597$	−3.00000	−0.122782
$598$	0	0
$599$	13.0000	0.531166	0.265583	−	0.964088i	$-0.414436\pi$
$599$	13.0000	0.531166	0.265583	+	0.964088i	$0.414436\pi$
$600$	0	0
$601$	−24.0000	−0.978980	−0.489490	−	0.872009i	$-0.662817\pi$
$601$	−24.0000	−0.978980	−0.489490	+	0.872009i	$0.662817\pi$
$602$	0	0
$603$	−3.00000	−0.122169
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	0	0
$609$	−27.0000	−1.09410
$610$	0	0
$611$	0	0
$612$	0	0
$613$	24.0000	0.969351	0.484675	−	0.874694i	$-0.338938\pi$
$613$	24.0000	0.969351	0.484675	+	0.874694i	$0.338938\pi$
$614$	0	0
$615$	−1.00000	−0.0403239
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	−11.0000	−0.442127	−0.221064	−	0.975259i	$-0.570953\pi$
$619$	−11.0000	−0.442127	−0.221064	+	0.975259i	$0.570953\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−24.0000	−0.961540
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−2.00000	−0.0798723
$628$	0	0
$629$	24.0000	0.956943
$630$	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
$633$	−20.0000	−0.794929
$634$	0	0
$635$	−14.0000	−0.555573
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	−10.0000	−0.395594
$640$	0	0
$641$	35.0000	1.38242	0.691208	−	0.722655i	$-0.257079\pi$
$641$	35.0000	1.38242	0.691208	+	0.722655i	$0.257079\pi$
$642$	0	0
$643$	36.0000	1.41970	0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000	1.41970	0.709851	+	0.704352i	$0.248762\pi$
$644$	0	0
$645$	11.0000	0.433125
$646$	0	0
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	0	0
$649$	−3.00000	−0.117760
$650$	0	0
$651$	12.0000	0.470317
$652$	0	0
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	−3.00000	−0.117220
$656$	0	0
$657$	9.00000	0.351123
$658$	0	0
$659$	40.0000	1.55818	0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000	1.55818	0.779089	+	0.626913i	$0.215682\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	0	0
$663$	4.00000	0.155347
$664$	0	0
$665$	−6.00000	−0.232670
$666$	0	0
$667$	−9.00000	−0.348481
$668$	0	0
$669$	20.0000	0.773245
$670$	0	0
$671$	−5.00000	−0.193023
$672$	0	0
$673$	−16.0000	−0.616755	−0.308377	−	0.951264i	$-0.599786\pi$
$673$	−16.0000	−0.616755	−0.308377	+	0.951264i	$0.599786\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	0	0
$679$	6.00000	0.230259
$680$	0	0
$681$	−8.00000	−0.306561
$682$	0	0
$683$	−29.0000	−1.10965	−0.554827	−	0.831966i	$-0.687216\pi$
$683$	−29.0000	−1.10965	−0.554827	+	0.831966i	$0.687216\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	0	0
$687$	25.0000	0.953809
$688$	0	0
$689$	10.0000	0.380970
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	−3.00000	−0.113961
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	14.0000	0.529529
$700$	0	0
$701$	35.0000	1.32193	0.660966	−	0.750416i	$-0.270147\pi$
$701$	35.0000	1.32193	0.660966	+	0.750416i	$0.270147\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−30.0000	−1.12827
$708$	0	0
$709$	−39.0000	−1.46468	−0.732338	−	0.680941i	$-0.761571\pi$
$709$	−39.0000	−1.46468	−0.732338	+	0.680941i	$0.761571\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	0	0
$713$	4.00000	0.149801
$714$	0	0
$715$	−1.00000	−0.0373979
$716$	0	0
$717$	5.00000	0.186728
$718$	0	0
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	0	0
$721$	33.0000	1.22898
$722$	0	0
$723$	2.00000	0.0743808
$724$	0	0
$725$	36.0000	1.33701
$726$	0	0
$727$	4.00000	0.148352	0.0741759	−	0.997245i	$-0.476367\pi$
$727$	4.00000	0.148352	0.0741759	+	0.997245i	$0.476367\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	44.0000	1.62740
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	3.00000	0.110506
$738$	0	0
$739$	24.0000	0.882854	0.441427	−	0.897297i	$-0.354472\pi$
$739$	24.0000	0.882854	0.441427	+	0.897297i	$0.354472\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	−29.0000	−1.06391	−0.531953	−	0.846774i	$-0.678542\pi$
$743$	−29.0000	−1.06391	−0.531953	+	0.846774i	$0.678542\pi$
$744$	0	0
$745$	−4.00000	−0.146549
$746$	0	0
$747$	6.00000	0.219529
$748$	0	0
$749$	−21.0000	−0.767323
$750$	0	0
$751$	53.0000	1.93400	0.966999	−	0.254781i	$-0.0820034\pi$
$751$	53.0000	1.93400	0.966999	+	0.254781i	$0.0820034\pi$
$752$	0	0
$753$	6.00000	0.218652
$754$	0	0
$755$	24.0000	0.873449
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	0	0
$759$	−1.00000	−0.0362977
$760$	0	0
$761$	35.0000	1.26875	0.634375	−	0.773026i	$-0.281258\pi$
$761$	35.0000	1.26875	0.634375	+	0.773026i	$0.281258\pi$
$762$	0	0
$763$	48.0000	1.73772
$764$	0	0
$765$	4.00000	0.144620
$766$	0	0
$767$	−3.00000	−0.108324
$768$	0	0
$769$	49.0000	1.76699	0.883493	−	0.468445i	$-0.155186\pi$
$769$	49.0000	1.76699	0.883493	+	0.468445i	$0.155186\pi$
$770$	0	0
$771$	−25.0000	−0.900353
$772$	0	0
$773$	−2.00000	−0.0719350	−0.0359675	−	0.999353i	$-0.511451\pi$
$773$	−2.00000	−0.0719350	−0.0359675	+	0.999353i	$0.511451\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	0	0
$777$	−18.0000	−0.645746
$778$	0	0
$779$	2.00000	0.0716574
$780$	0	0
$781$	10.0000	0.357828
$782$	0	0
$783$	−9.00000	−0.321634
$784$	0	0
$785$	22.0000	0.785214
$786$	0	0
$787$	−44.0000	−1.56843	−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000	−1.56843	−0.784215	+	0.620489i	$0.786934\pi$
$788$	0	0
$789$	24.0000	0.854423
$790$	0	0
$791$	15.0000	0.533339
$792$	0	0
$793$	−5.00000	−0.177555
$794$	0	0
$795$	10.0000	0.354663
$796$	0	0
$797$	28.0000	0.991811	0.495905	−	0.868377i	$-0.334836\pi$
$797$	28.0000	0.991811	0.495905	+	0.868377i	$0.334836\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−8.00000	−0.282666
$802$	0	0
$803$	−9.00000	−0.317603
$804$	0	0
$805$	−3.00000	−0.105736
$806$	0	0
$807$	−4.00000	−0.140807
$808$	0	0
$809$	−50.0000	−1.75791	−0.878953	−	0.476908i	$-0.841757\pi$
$809$	−50.0000	−1.75791	−0.878953	+	0.476908i	$0.841757\pi$
$810$	0	0
$811$	−56.0000	−1.96643	−0.983213	−	0.182462i	$-0.941593\pi$
$811$	−56.0000	−1.96643	−0.983213	+	0.182462i	$0.941593\pi$
$812$	0	0
$813$	16.0000	0.561144
$814$	0	0
$815$	9.00000	0.315256
$816$	0	0
$817$	−22.0000	−0.769683
$818$	0	0
$819$	−3.00000	−0.104828
$820$	0	0
$821$	34.0000	1.18661	0.593304	−	0.804978i	$-0.297823\pi$
$821$	34.0000	1.18661	0.593304	+	0.804978i	$0.297823\pi$
$822$	0	0
$823$	−7.00000	−0.244005	−0.122002	−	0.992530i	$-0.538932\pi$
$823$	−7.00000	−0.244005	−0.122002	+	0.992530i	$0.538932\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	−28.0000	−0.973655	−0.486828	−	0.873498i	$-0.661846\pi$
$827$	−28.0000	−0.973655	−0.486828	+	0.873498i	$0.661846\pi$
$828$	0	0
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	0	0
$831$	5.00000	0.173448
$832$	0	0
$833$	−8.00000	−0.277184
$834$	0	0
$835$	−3.00000	−0.103819
$836$	0	0
$837$	4.00000	0.138260
$838$	0	0
$839$	−14.0000	−0.483334	−0.241667	−	0.970359i	$-0.577694\pi$
$839$	−14.0000	−0.483334	−0.241667	+	0.970359i	$0.577694\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	−29.0000	−0.998813
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	3.00000	0.103081
$848$	0	0
$849$	9.00000	0.308879
$850$	0	0
$851$	−6.00000	−0.205677
$852$	0	0
$853$	16.0000	0.547830	0.273915	−	0.961754i	$-0.411681\pi$
$853$	16.0000	0.547830	0.273915	+	0.961754i	$0.411681\pi$
$854$	0	0
$855$	−2.00000	−0.0683986
$856$	0	0
$857$	−12.0000	−0.409912	−0.204956	−	0.978771i	$-0.565705\pi$
$857$	−12.0000	−0.409912	−0.204956	+	0.978771i	$0.565705\pi$
$858$	0	0
$859$	−32.0000	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$860$	0	0
$861$	3.00000	0.102240
$862$	0	0
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	0	0
$865$	19.0000	0.646019
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	10.0000	0.339227
$870$	0	0
$871$	3.00000	0.101651
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	27.0000	0.912767
$876$	0	0
$877$	36.0000	1.21563	0.607817	−	0.794077i	$-0.292045\pi$
$877$	36.0000	1.21563	0.607817	+	0.794077i	$0.292045\pi$
$878$	0	0
$879$	−18.0000	−0.607125
$880$	0	0
$881$	9.00000	0.303218	0.151609	−	0.988441i	$-0.451555\pi$
$881$	9.00000	0.303218	0.151609	+	0.988441i	$0.451555\pi$
$882$	0	0
$883$	−26.0000	−0.874970	−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000	−0.874970	−0.437485	+	0.899226i	$0.644131\pi$
$884$	0	0
$885$	−3.00000	−0.100844
$886$	0	0
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	0	0
$889$	42.0000	1.40863
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	0	0
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	−1.00000	−0.0333890
$898$	0	0
$899$	−36.0000	−1.20067
$900$	0	0
$901$	40.0000	1.33259
$902$	0	0
$903$	−33.0000	−1.09817
$904$	0	0
$905$	−14.0000	−0.465376
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	0	0
$909$	−10.0000	−0.331679
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	−6.00000	−0.198571
$914$	0	0
$915$	−5.00000	−0.165295
$916$	0	0
$917$	9.00000	0.297206
$918$	0	0
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	0	0
$921$	10.0000	0.329511
$922$	0	0
$923$	10.0000	0.329154
$924$	0	0
$925$	24.0000	0.789115
$926$	0	0
$927$	11.0000	0.361287
$928$	0	0
$929$	−12.0000	−0.393707	−0.196854	−	0.980433i	$-0.563072\pi$
$929$	−12.0000	−0.393707	−0.196854	+	0.980433i	$0.563072\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	0	0
$933$	8.00000	0.261908
$934$	0	0
$935$	−4.00000	−0.130814
$936$	0	0
$937$	20.0000	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$938$	0	0
$939$	19.0000	0.620042
$940$	0	0
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	0	0
$943$	1.00000	0.0325645
$944$	0	0
$945$	−3.00000	−0.0975900
$946$	0	0
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	0	0
$949$	−9.00000	−0.292152
$950$	0	0
$951$	−3.00000	−0.0972817
$952$	0	0
$953$	44.0000	1.42530	0.712650	−	0.701520i	$-0.247495\pi$
$953$	44.0000	1.42530	0.712650	+	0.701520i	$0.247495\pi$
$954$	0	0
$955$	−15.0000	−0.485389
$956$	0	0
$957$	9.00000	0.290929
$958$	0	0
$959$	−54.0000	−1.74375
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−7.00000	−0.225572
$964$	0	0
$965$	6.00000	0.193147
$966$	0	0
$967$	13.0000	0.418052	0.209026	−	0.977910i	$-0.432971\pi$
$967$	13.0000	0.418052	0.209026	+	0.977910i	$0.432971\pi$
$968$	0	0
$969$	−8.00000	−0.256997
$970$	0	0
$971$	−34.0000	−1.09111	−0.545556	−	0.838074i	$-0.683681\pi$
$971$	−34.0000	−1.09111	−0.545556	+	0.838074i	$0.683681\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	−48.0000	−1.53566	−0.767828	−	0.640656i	$-0.778662\pi$
$977$	−48.0000	−1.53566	−0.767828	+	0.640656i	$0.778662\pi$
$978$	0	0
$979$	8.00000	0.255681
$980$	0	0
$981$	16.0000	0.510841
$982$	0	0
$983$	−38.0000	−1.21201	−0.606006	−	0.795460i	$-0.707229\pi$
$983$	−38.0000	−1.21201	−0.606006	+	0.795460i	$0.707229\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11.0000	−0.349780
$990$	0	0
$991$	47.0000	1.49300	0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000	1.49300	0.746502	+	0.665383i	$0.231732\pi$
$992$	0	0
$993$	−19.0000	−0.602947
$994$	0	0
$995$	3.00000	0.0951064
$996$	0	0
$997$	−37.0000	−1.17180	−0.585901	−	0.810383i	$-0.699259\pi$
$997$	−37.0000	−1.17180	−0.585901	+	0.810383i	$0.699259\pi$
$998$	0	0
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.u.1.1		1
4.3	odd	2		858.2.a.g.1.1	✓	1
12.11	even	2		2574.2.a.i.1.1		1
44.43	even	2		9438.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
858.2.a.g.1.1	✓	1	4.3	odd	2
2574.2.a.i.1.1		1	12.11	even	2
6864.2.a.u.1.1		1	1.1	even	1	trivial
9438.2.a.d.1.1		1	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)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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