LMFDB - Embedded newform 8100.2.a.bd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8100,2,Mod(1,8100)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8100.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8100.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:	$64.6788256372$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.1207701504.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 14x^{4} + 43x^{2} - 36 $ x^6 - 14x^4 + 43x^2 - 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 180)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.20590$ of defining polynomial
Character	$\chi$	$=$	8100.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.76963 q^{7} +O(q^{10})$	$q-3.76963 q^{7} +3.54580 q^{11} +1.00351 q^{13} -1.56023 q^{17} +7.21013 q^{19} -6.18143 q^{23} +3.00000 q^{29} -5.78285 q^{31} +0.851576 q^{37} +3.11852 q^{41} -2.70666 q^{43} -9.74867 q^{47} +7.21013 q^{49} +11.2494 q^{53} +9.66433 q^{59} -8.21013 q^{61} -2.91806 q^{67} -7.21013 q^{71} +16.7817 q^{73} -13.3664 q^{77} +10.9930 q^{79} -10.1955 q^{83} +5.33567 q^{89} -3.78285 q^{91} -3.86209 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q + 2 q^{11} + 18 q^{29} - 6 q^{31} + 14 q^{41} + 34 q^{59} - 6 q^{61} - 6 q^{79} + 56 q^{89} + 6 q^{91}+O(q^{100}) $ 6 * q + 2 * q^11 + 18 * q^29 - 6 * q^31 + 14 * q^41 + 34 * q^59 - 6 * q^61 - 6 * q^79 + 56 * q^89 + 6 * q^91

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.76963	−1.42479	−0.712394	−	0.701780i	$-0.752389\pi$
$7$	−3.76963	−1.42479	−0.712394	+	0.701780i	$0.752389\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.54580	1.06910	0.534550	−	0.845137i	$-0.320481\pi$
$11$	3.54580	1.06910	0.534550	+	0.845137i	$0.320481\pi$
$12$	0	0
$13$	1.00351	0.278322	0.139161	−	0.990270i	$-0.455559\pi$
$13$	1.00351	0.278322	0.139161	+	0.990270i	$0.455559\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.56023	−0.378410	−0.189205	−	0.981938i	$-0.560591\pi$
$17$	−1.56023	−0.378410	−0.189205	+	0.981938i	$0.560591\pi$
$18$	0	0
$19$	7.21013	1.65412	0.827058	−	0.562116i	$-0.190013\pi$
$19$	7.21013	1.65412	0.827058	+	0.562116i	$0.190013\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.18143	−1.28892	−0.644459	−	0.764639i	$-0.722917\pi$
$23$	−6.18143	−1.28892	−0.644459	+	0.764639i	$0.722917\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−5.78285	−1.03863	−0.519315	−	0.854583i	$-0.673813\pi$
$31$	−5.78285	−1.03863	−0.519315	+	0.854583i	$0.673813\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.851576	0.139998	0.0699991	−	0.997547i	$-0.477700\pi$
$37$	0.851576	0.139998	0.0699991	+	0.997547i	$0.477700\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.11852	0.487031	0.243516	−	0.969897i	$-0.421699\pi$
$41$	3.11852	0.487031	0.243516	+	0.969897i	$0.421699\pi$
$42$	0	0
$43$	−2.70666	−0.412761	−0.206381	−	0.978472i	$-0.566169\pi$
$43$	−2.70666	−0.412761	−0.206381	+	0.978472i	$0.566169\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.74867	−1.42199	−0.710995	−	0.703197i	$-0.751755\pi$
$47$	−9.74867	−1.42199	−0.710995	+	0.703197i	$0.751755\pi$
$48$	0	0
$49$	7.21013	1.03002
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.2494	1.54523	0.772614	−	0.634876i	$-0.218949\pi$
$53$	11.2494	1.54523	0.772614	+	0.634876i	$0.218949\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.66433	1.25819	0.629094	−	0.777329i	$-0.283426\pi$
$59$	9.66433	1.25819	0.629094	+	0.777329i	$0.283426\pi$
$60$	0	0
$61$	−8.21013	−1.05120	−0.525600	−	0.850732i	$-0.676159\pi$
$61$	−8.21013	−1.05120	−0.525600	+	0.850732i	$0.676159\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.91806	−0.356497	−0.178249	−	0.983985i	$-0.557043\pi$
$67$	−2.91806	−0.356497	−0.178249	+	0.983985i	$0.557043\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.21013	−0.855685	−0.427842	−	0.903853i	$-0.640726\pi$
$71$	−7.21013	−0.855685	−0.427842	+	0.903853i	$0.640726\pi$
$72$	0	0
$73$	16.7817	1.96415	0.982074	−	0.188498i	$-0.0603618\pi$
$73$	16.7817	1.96415	0.982074	+	0.188498i	$0.0603618\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−13.3664	−1.52324
$78$	0	0
$79$	10.9930	1.23681	0.618403	−	0.785861i	$-0.287780\pi$
$79$	10.9930	1.23681	0.618403	+	0.785861i	$0.287780\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.1955	−1.11910	−0.559548	−	0.828798i	$-0.689025\pi$
$83$	−10.1955	−1.11910	−0.559548	+	0.828798i	$0.689025\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.33567	0.565580	0.282790	−	0.959182i	$-0.408740\pi$
$89$	5.33567	0.565580	0.282790	+	0.959182i	$0.408740\pi$
$90$	0	0
$91$	−3.78285	−0.396550
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.86209	−0.392136	−0.196068	−	0.980590i	$-0.562817\pi$
$97$	−3.86209	−0.392136	−0.196068	+	0.980590i	$0.562817\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.75593	−0.473233	−0.236616	−	0.971603i	$-0.576038\pi$
$101$	−4.75593	−0.473233	−0.236616	+	0.971603i	$0.576038\pi$
$102$	0	0
$103$	18.0890	1.78237	0.891183	−	0.453643i	$-0.149876\pi$
$103$	18.0890	1.78237	0.891183	+	0.453643i	$0.149876\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.61770	0.349737	0.174868	−	0.984592i	$-0.444050\pi$
$107$	3.61770	0.349737	0.174868	+	0.984592i	$0.444050\pi$
$108$	0	0
$109$	−9.63741	−0.923096	−0.461548	−	0.887115i	$-0.652706\pi$
$109$	−9.63741	−0.923096	−0.461548	+	0.887115i	$0.652706\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.55823	−0.334730	−0.167365	−	0.985895i	$-0.553526\pi$
$113$	−3.55823	−0.334730	−0.167365	+	0.985895i	$0.553526\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.88148	0.539154
$120$	0	0
$121$	1.57272	0.142974
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−6.78015	−0.601641	−0.300820	−	0.953681i	$-0.597261\pi$
$127$	−6.78015	−0.601641	−0.300820	+	0.953681i	$0.597261\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.42728	0.299443	0.149721	−	0.988728i	$-0.452162\pi$
$131$	3.42728	0.299443	0.149721	+	0.988728i	$0.452162\pi$
$132$	0	0
$133$	−27.1795	−2.35676
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.9266	1.27527	0.637633	−	0.770340i	$-0.279914\pi$
$137$	14.9266	1.27527	0.637633	+	0.770340i	$0.279914\pi$
$138$	0	0
$139$	−5.42728	−0.460336	−0.230168	−	0.973151i	$-0.573928\pi$
$139$	−5.42728	−0.460336	−0.230168	+	0.973151i	$0.573928\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.55823	0.297554
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.53878	−0.453754	−0.226877	−	0.973923i	$-0.572852\pi$
$149$	−5.53878	−0.453754	−0.226877	+	0.973923i	$0.572852\pi$
$150$	0	0
$151$	0.217152	0.0176716	0.00883580	−	0.999961i	$-0.497187\pi$
$151$	0.217152	0.0176716	0.00883580	+	0.999961i	$0.497187\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.15894	0.172302	0.0861511	−	0.996282i	$-0.472543\pi$
$157$	2.15894	0.172302	0.0861511	+	0.996282i	$0.472543\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	23.3017	1.83643
$162$	0	0
$163$	8.39084	0.657221	0.328611	−	0.944465i	$-0.393420\pi$
$163$	8.39084	0.657221	0.328611	+	0.944465i	$0.393420\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.62121	0.357600	0.178800	−	0.983885i	$-0.442779\pi$
$167$	4.62121	0.357600	0.178800	+	0.983885i	$0.442779\pi$
$168$	0	0
$169$	−11.9930	−0.922537
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.45029	−0.110264	−0.0551318	−	0.998479i	$-0.517558\pi$
$173$	−1.45029	−0.110264	−0.0551318	+	0.998479i	$0.517558\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	19.4472	1.45355	0.726775	−	0.686876i	$-0.241018\pi$
$179$	19.4472	1.45355	0.726775	+	0.686876i	$0.241018\pi$
$180$	0	0
$181$	−10.4273	−0.775054	−0.387527	−	0.921858i	$-0.626671\pi$
$181$	−10.4273	−0.775054	−0.387527	+	0.921858i	$0.626671\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5.53225	−0.404558
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.78285	0.273717	0.136859	−	0.990591i	$-0.456299\pi$
$191$	3.78285	0.273717	0.136859	+	0.990591i	$0.456299\pi$
$192$	0	0
$193$	−14.0750	−1.01314	−0.506571	−	0.862198i	$-0.669087\pi$
$193$	−14.0750	−1.01314	−0.506571	+	0.862198i	$0.669087\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.5044	1.53212	0.766061	−	0.642768i	$-0.222214\pi$
$197$	21.5044	1.53212	0.766061	+	0.642768i	$0.222214\pi$
$198$	0	0
$199$	7.21013	0.511112	0.255556	−	0.966794i	$-0.417741\pi$
$199$	7.21013	0.511112	0.255556	+	0.966794i	$0.417741\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−11.3089	−0.793729
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	25.5657	1.76842
$210$	0	0
$211$	3.78285	0.260422	0.130211	−	0.991486i	$-0.458435\pi$
$211$	3.78285	0.260422	0.130211	+	0.991486i	$0.458435\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	21.7992	1.47983
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.56570	−0.105320
$222$	0	0
$223$	2.91806	0.195408	0.0977038	−	0.995216i	$-0.468850\pi$
$223$	2.91806	0.195408	0.0977038	+	0.995216i	$0.468850\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.11846	−0.339724	−0.169862	−	0.985468i	$-0.554332\pi$
$227$	−5.11846	−0.339724	−0.169862	+	0.985468i	$0.554332\pi$
$228$	0	0
$229$	−9.42026	−0.622508	−0.311254	−	0.950327i	$-0.600749\pi$
$229$	−9.42026	−0.622508	−0.311254	+	0.950327i	$0.600749\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	26.1850	1.71544	0.857719	−	0.514118i	$-0.171881\pi$
$233$	26.1850	1.71544	0.857719	+	0.514118i	$0.171881\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.7559	1.47196	0.735979	−	0.677004i	$-0.236722\pi$
$239$	22.7559	1.47196	0.735979	+	0.677004i	$0.236722\pi$
$240$	0	0
$241$	16.6304	1.07126	0.535629	−	0.844454i	$-0.320075\pi$
$241$	16.6304	1.07126	0.535629	+	0.844454i	$0.320075\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	7.23541	0.460378
$248$	0	0
$249$	0	0
$250$	0	0
$251$	20.7758	1.31136	0.655679	−	0.755040i	$-0.272382\pi$
$251$	20.7758	1.31136	0.655679	+	0.755040i	$0.272382\pi$
$252$	0	0
$253$	−21.9181	−1.37798
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.0596	1.18890	0.594451	−	0.804132i	$-0.297369\pi$
$257$	19.0596	1.18890	0.594451	+	0.804132i	$0.297369\pi$
$258$	0	0
$259$	−3.21013	−0.199468
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.13798	0.501809	0.250905	−	0.968012i	$-0.419272\pi$
$263$	8.13798	0.501809	0.250905	+	0.968012i	$0.419272\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	25.6374	1.56314	0.781570	−	0.623817i	$-0.214419\pi$
$269$	25.6374	1.56314	0.781570	+	0.623817i	$0.214419\pi$
$270$	0	0
$271$	−3.21013	−0.195001	−0.0975007	−	0.995235i	$-0.531085\pi$
$271$	−3.21013	−0.195001	−0.0975007	+	0.995235i	$0.531085\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−0.699646	−0.0420377	−0.0210188	−	0.999779i	$-0.506691\pi$
$277$	−0.699646	−0.0420377	−0.0210188	+	0.999779i	$0.506691\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	30.6304	1.82726	0.913628	−	0.406552i	$-0.133269\pi$
$281$	30.6304	1.82726	0.913628	+	0.406552i	$0.133269\pi$
$282$	0	0
$283$	−9.60575	−0.571002	−0.285501	−	0.958378i	$-0.592160\pi$
$283$	−9.60575	−0.571002	−0.285501	+	0.958378i	$0.592160\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−11.7557	−0.693916
$288$	0	0
$289$	−14.5657	−0.856806
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.01052	0.175876	0.0879381	−	0.996126i	$-0.471972\pi$
$293$	3.01052	0.175876	0.0879381	+	0.996126i	$0.471972\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.20310	−0.358735
$300$	0	0
$301$	10.2031	0.588097
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−14.8671	−0.848512	−0.424256	−	0.905542i	$-0.639464\pi$
$307$	−14.8671	−0.848512	−0.424256	+	0.905542i	$0.639464\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3.07171	−0.174181	−0.0870905	−	0.996200i	$-0.527757\pi$
$311$	−3.07171	−0.174181	−0.0870905	+	0.996200i	$0.527757\pi$
$312$	0	0
$313$	23.0136	1.30080	0.650402	−	0.759590i	$-0.274600\pi$
$313$	23.0136	1.30080	0.650402	+	0.759590i	$0.274600\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.00900354	−0.000505689 0	−0.000252844	−	1.00000i	$-0.500080\pi$
$317$	−0.00900354	−0.000505689 0	−0.000252844		1.00000i	$0.500080\pi$
$318$	0	0
$319$	10.6374	0.595581
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−11.2494	−0.625935
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	36.7489	2.02603
$330$	0	0
$331$	8.99298	0.494299	0.247149	−	0.968977i	$-0.420506\pi$
$331$	8.99298	0.494299	0.247149	+	0.968977i	$0.420506\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−5.26139	−0.286606	−0.143303	−	0.989679i	$-0.545772\pi$
$337$	−5.26139	−0.286606	−0.143303	+	0.989679i	$0.545772\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−20.5048	−1.11040
$342$	0	0
$343$	−0.792107	−0.0427698
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.4954	1.15393	0.576965	−	0.816769i	$-0.304237\pi$
$347$	21.4954	1.15393	0.576965	+	0.816769i	$0.304237\pi$
$348$	0	0
$349$	−12.5657	−0.672626	−0.336313	−	0.941750i	$-0.609180\pi$
$349$	−12.5657	−0.672626	−0.336313	+	0.941750i	$0.609180\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	25.0626	1.33395	0.666973	−	0.745081i	$-0.267589\pi$
$353$	25.0626	1.33395	0.666973	+	0.745081i	$0.267589\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−23.1961	−1.22424	−0.612121	−	0.790764i	$-0.709684\pi$
$359$	−23.1961	−1.22424	−0.612121	+	0.790764i	$0.709684\pi$
$360$	0	0
$361$	32.9860	1.73610
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.7852	0.928379	0.464190	−	0.885736i	$-0.346346\pi$
$367$	17.7852	0.928379	0.464190	+	0.885736i	$0.346346\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−42.4062	−2.20162
$372$	0	0
$373$	12.9196	0.668951	0.334475	−	0.942404i	$-0.391441\pi$
$373$	12.9196	0.668951	0.334475	+	0.942404i	$0.391441\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.01052	0.155050
$378$	0	0
$379$	−5.56570	−0.285891	−0.142945	−	0.989731i	$-0.545657\pi$
$379$	−5.56570	−0.285891	−0.142945	+	0.989731i	$0.545657\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	29.7433	1.51981	0.759905	−	0.650034i	$-0.225245\pi$
$383$	29.7433	1.51981	0.759905	+	0.650034i	$0.225245\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7.55283	0.382944	0.191472	−	0.981498i	$-0.438674\pi$
$389$	7.55283	0.382944	0.191472	+	0.981498i	$0.438674\pi$
$390$	0	0
$391$	9.64443	0.487740
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	30.1571	1.51354	0.756770	−	0.653682i	$-0.226776\pi$
$397$	30.1571	1.51354	0.756770	+	0.653682i	$0.226776\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	21.6643	1.08186	0.540932	−	0.841066i	$-0.318071\pi$
$401$	21.6643	1.08186	0.540932	+	0.841066i	$0.318071\pi$
$402$	0	0
$403$	−5.80312	−0.289074
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.01952	0.149672
$408$	0	0
$409$	−8.13841	−0.402419	−0.201209	−	0.979548i	$-0.564487\pi$
$409$	−8.13841	−0.402419	−0.201209	+	0.979548i	$0.564487\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−36.4310	−1.79265
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−13.2947	−0.649489	−0.324745	−	0.945802i	$-0.605278\pi$
$419$	−13.2947	−0.649489	−0.324745	+	0.945802i	$0.605278\pi$
$420$	0	0
$421$	20.1384	0.981486	0.490743	−	0.871304i	$-0.336725\pi$
$421$	20.1384	0.981486	0.490743	+	0.871304i	$0.336725\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	30.9492	1.49774
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−25.8027	−1.24287	−0.621437	−	0.783464i	$-0.713451\pi$
$431$	−25.8027	−1.24287	−0.621437	+	0.783464i	$0.713451\pi$
$432$	0	0
$433$	−6.38383	−0.306787	−0.153394	−	0.988165i	$-0.549020\pi$
$433$	−6.38383	−0.306787	−0.153394	+	0.988165i	$0.549020\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−44.5689	−2.13202
$438$	0	0
$439$	10.9930	0.524666	0.262333	−	0.964977i	$-0.415508\pi$
$439$	10.9930	0.524666	0.262333	+	0.964977i	$0.415508\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−25.2320	−1.19881	−0.599404	−	0.800447i	$-0.704596\pi$
$443$	−25.2320	−1.19881	−0.599404	+	0.800447i	$0.704596\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	19.5657	0.923362	0.461681	−	0.887046i	$-0.347246\pi$
$449$	19.5657	0.923362	0.461681	+	0.887046i	$0.347246\pi$
$450$	0	0
$451$	11.0577	0.520685
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.53576	0.305730	0.152865	−	0.988247i	$-0.451150\pi$
$457$	6.53576	0.305730	0.152865	+	0.988247i	$0.451150\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	35.4203	1.64969	0.824843	−	0.565362i	$-0.191264\pi$
$461$	35.4203	1.64969	0.824843	+	0.565362i	$0.191264\pi$
$462$	0	0
$463$	−16.2670	−0.755990	−0.377995	−	0.925808i	$-0.623386\pi$
$463$	−16.2670	−0.755990	−0.377995	+	0.925808i	$0.623386\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15.8112	−0.731654	−0.365827	−	0.930683i	$-0.619214\pi$
$467$	−15.8112	−0.731654	−0.365827	+	0.930683i	$0.619214\pi$
$468$	0	0
$469$	11.0000	0.507933
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−9.59728	−0.441283
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	32.7419	1.49601	0.748007	−	0.663690i	$-0.231011\pi$
$479$	32.7419	1.49601	0.748007	+	0.663690i	$0.231011\pi$
$480$	0	0
$481$	0.854561	0.0389646
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−36.4220	−1.65044	−0.825218	−	0.564814i	$-0.808948\pi$
$487$	−36.4220	−1.65044	−0.825218	+	0.564814i	$0.808948\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3.42728	0.154671	0.0773355	−	0.997005i	$-0.475359\pi$
$491$	3.42728	0.154671	0.0773355	+	0.997005i	$0.475359\pi$
$492$	0	0
$493$	−4.68068	−0.210807
$494$	0	0
$495$	0	0
$496$	0	0
$497$	27.1795	1.21917
$498$	0	0
$499$	3.42728	0.153426	0.0767131	−	0.997053i	$-0.475557\pi$
$499$	3.42728	0.153426	0.0767131	+	0.997053i	$0.475557\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−17.4219	−0.776802	−0.388401	−	0.921490i	$-0.626973\pi$
$503$	−17.4219	−0.776802	−0.388401	+	0.921490i	$0.626973\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.1454	0.626986	0.313493	−	0.949591i	$-0.398501\pi$
$509$	14.1454	0.626986	0.313493	+	0.949591i	$0.398501\pi$
$510$	0	0
$511$	−63.2608	−2.79849
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−34.5669	−1.52025
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20.2300	0.886293	0.443147	−	0.896449i	$-0.353862\pi$
$521$	20.2300	0.886293	0.443147	+	0.896449i	$0.353862\pi$
$522$	0	0
$523$	−36.3295	−1.58858	−0.794289	−	0.607540i	$-0.792156\pi$
$523$	−36.3295	−1.58858	−0.794289	+	0.607540i	$0.792156\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.02255	0.393028
$528$	0	0
$529$	15.2101	0.661310
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.12946	0.135552
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	25.5657	1.10119
$540$	0	0
$541$	24.8475	1.06828	0.534140	−	0.845396i	$-0.320636\pi$
$541$	24.8475	1.06828	0.534140	+	0.845396i	$0.320636\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	32.1046	1.37269	0.686347	−	0.727274i	$-0.259213\pi$
$547$	32.1046	1.37269	0.686347	+	0.727274i	$0.259213\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	21.6304	0.921485
$552$	0	0
$553$	−41.4395	−1.76219
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−17.3894	−0.736813	−0.368406	−	0.929665i	$-0.620096\pi$
$557$	−17.3894	−0.736813	−0.368406	+	0.929665i	$0.620096\pi$
$558$	0	0
$559$	−2.71615	−0.114881
$560$	0	0
$561$	0	0
$562$	0	0
$563$	27.3399	1.15224	0.576120	−	0.817365i	$-0.304566\pi$
$563$	27.3399	1.15224	0.576120	+	0.817365i	$0.304566\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15.1902	0.636808	0.318404	−	0.947955i	$-0.396853\pi$
$569$	15.1902	0.636808	0.318404	+	0.947955i	$0.396853\pi$
$570$	0	0
$571$	−38.7021	−1.61963	−0.809816	−	0.586684i	$-0.800433\pi$
$571$	−38.7021	−1.61963	−0.809816	+	0.586684i	$0.800433\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3.82910	−0.159408	−0.0797038	−	0.996819i	$-0.525397\pi$
$577$	−3.82910	−0.159408	−0.0797038	+	0.996819i	$0.525397\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	38.4331	1.59447
$582$	0	0
$583$	39.8883	1.65200
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.6843	−1.01883	−0.509415	−	0.860521i	$-0.670138\pi$
$587$	−24.6843	−1.01883	−0.509415	+	0.860521i	$0.670138\pi$
$588$	0	0
$589$	−41.6951	−1.71802
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−38.4290	−1.57809	−0.789044	−	0.614336i	$-0.789424\pi$
$593$	−38.4290	−1.57809	−0.789044	+	0.614336i	$0.789424\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−15.4273	−0.630342	−0.315171	−	0.949035i	$-0.602062\pi$
$599$	−15.4273	−0.630342	−0.315171	+	0.949035i	$0.602062\pi$
$600$	0	0
$601$	−0.572719	−0.0233617	−0.0116809	−	0.999932i	$-0.503718\pi$
$601$	−0.572719	−0.0233617	−0.0116809	+	0.999932i	$0.503718\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−9.05803	−0.367654	−0.183827	−	0.982959i	$-0.558849\pi$
$607$	−9.05803	−0.367654	−0.183827	+	0.982959i	$0.558849\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9.78285	−0.395772
$612$	0	0
$613$	6.99155	0.282386	0.141193	−	0.989982i	$-0.454906\pi$
$613$	6.99155	0.282386	0.141193	+	0.989982i	$0.454906\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−19.8342	−0.798495	−0.399247	−	0.916843i	$-0.630729\pi$
$617$	−19.8342	−0.798495	−0.399247	+	0.916843i	$0.630729\pi$
$618$	0	0
$619$	−28.2031	−1.13358	−0.566789	−	0.823863i	$-0.691815\pi$
$619$	−28.2031	−1.13358	−0.566789	+	0.823863i	$0.691815\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−20.1135	−0.805832
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.32865	−0.0529768
$630$	0	0
$631$	−7.27482	−0.289606	−0.144803	−	0.989461i	$-0.546255\pi$
$631$	−7.27482	−0.289606	−0.144803	+	0.989461i	$0.546255\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	7.23541	0.286677
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−12.8674	−0.508233	−0.254116	−	0.967174i	$-0.581785\pi$
$641$	−12.8674	−0.508233	−0.254116	+	0.967174i	$0.581785\pi$
$642$	0	0
$643$	25.2320	0.995053	0.497526	−	0.867449i	$-0.334242\pi$
$643$	25.2320	0.995053	0.497526	+	0.867449i	$0.334242\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−23.5618	−0.926311	−0.463156	−	0.886277i	$-0.653283\pi$
$647$	−23.5618	−0.926311	−0.463156	+	0.886277i	$0.653283\pi$
$648$	0	0
$649$	34.2678	1.34513
$650$	0	0
$651$	0	0
$652$	0	0
$653$	17.3804	0.680147	0.340074	−	0.940399i	$-0.389548\pi$
$653$	17.3804	0.680147	0.340074	+	0.940399i	$0.389548\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15.6643	0.610195	0.305098	−	0.952321i	$-0.401311\pi$
$659$	15.6643	0.610195	0.305098	+	0.952321i	$0.401311\pi$
$660$	0	0
$661$	23.4273	0.911216	0.455608	−	0.890181i	$-0.349422\pi$
$661$	23.4273	0.911216	0.455608	+	0.890181i	$0.349422\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−18.5443	−0.718038
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−29.1115	−1.12384
$672$	0	0
$673$	28.7308	1.10749	0.553745	−	0.832687i	$-0.313198\pi$
$673$	28.7308	1.10749	0.553745	+	0.832687i	$0.313198\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−50.3540	−1.93526	−0.967632	−	0.252367i	$-0.918791\pi$
$677$	−50.3540	−1.93526	−0.967632	+	0.252367i	$0.918791\pi$
$678$	0	0
$679$	14.5587	0.558711
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−11.5953	−0.443681	−0.221841	−	0.975083i	$-0.571206\pi$
$683$	−11.5953	−0.443681	−0.221841	+	0.975083i	$0.571206\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	11.2889	0.430072
$690$	0	0
$691$	45.8475	1.74412	0.872061	−	0.489397i	$-0.162783\pi$
$691$	45.8475	1.74412	0.872061	+	0.489397i	$0.162783\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−4.86560	−0.184298
$698$	0	0
$699$	0	0
$700$	0	0
$701$	38.1115	1.43945	0.719726	−	0.694259i	$-0.244268\pi$
$701$	38.1115	1.43945	0.719726	+	0.694259i	$0.244268\pi$
$702$	0	0
$703$	6.13997	0.231573
$704$	0	0
$705$	0	0
$706$	0	0
$707$	17.9281	0.674256
$708$	0	0
$709$	−2.28887	−0.0859602	−0.0429801	−	0.999076i	$-0.513685\pi$
$709$	−2.28887	−0.0859602	−0.0429801	+	0.999076i	$0.513685\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	35.7463	1.33871
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	16.6035	0.619205	0.309602	−	0.950866i	$-0.399804\pi$
$719$	16.6035	0.619205	0.309602	+	0.950866i	$0.399804\pi$
$720$	0	0
$721$	−68.1891	−2.53949
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−38.6074	−1.43187	−0.715934	−	0.698168i	$-0.753999\pi$
$727$	−38.6074	−1.43187	−0.715934	+	0.698168i	$0.753999\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.22300	0.156193
$732$	0	0
$733$	33.9002	1.25213	0.626066	−	0.779770i	$-0.284664\pi$
$733$	33.9002	1.25213	0.626066	+	0.779770i	$0.284664\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.3469	−0.381131
$738$	0	0
$739$	−36.8405	−1.35520	−0.677600	−	0.735431i	$-0.736980\pi$
$739$	−36.8405	−1.35520	−0.677600	+	0.735431i	$0.736980\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−11.4278	−0.419247	−0.209623	−	0.977782i	$-0.567224\pi$
$743$	−11.4278	−0.419247	−0.209623	+	0.977782i	$0.567224\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−13.6374	−0.498300
$750$	0	0
$751$	−16.2031	−0.591260	−0.295630	−	0.955303i	$-0.595529\pi$
$751$	−16.2031	−0.591260	−0.295630	+	0.955303i	$0.595529\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.08698	0.293926	0.146963	−	0.989142i	$-0.453050\pi$
$757$	8.08698	0.293926	0.146963	+	0.989142i	$0.453050\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−21.4062	−0.775974	−0.387987	−	0.921665i	$-0.626830\pi$
$761$	−21.4062	−0.775974	−0.387987	+	0.921665i	$0.626830\pi$
$762$	0	0
$763$	36.3295	1.31522
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.69821	0.350182
$768$	0	0
$769$	51.4062	1.85376	0.926878	−	0.375364i	$-0.122482\pi$
$769$	51.4062	1.85376	0.926878	+	0.375364i	$0.122482\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	22.4849	0.805607
$780$	0	0
$781$	−25.5657	−0.914813
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−4.71367	−0.168024	−0.0840121	−	0.996465i	$-0.526773\pi$
$787$	−4.71367	−0.168024	−0.0840121	+	0.996465i	$0.526773\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	13.4132	0.476920
$792$	0	0
$793$	−8.23891	−0.292572
$794$	0	0
$795$	0	0
$796$	0	0
$797$	10.2639	0.363567	0.181784	−	0.983339i	$-0.441813\pi$
$797$	10.2639	0.363567	0.181784	+	0.983339i	$0.441813\pi$
$798$	0	0
$799$	15.2101	0.538096
$800$	0	0
$801$	0	0
$802$	0	0
$803$	59.5045	2.09987
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−21.9860	−0.772985	−0.386492	−	0.922293i	$-0.626313\pi$
$809$	−21.9860	−0.772985	−0.386492	+	0.922293i	$0.626313\pi$
$810$	0	0
$811$	−25.6951	−0.902276	−0.451138	−	0.892454i	$-0.648982\pi$
$811$	−25.6951	−0.902276	−0.451138	+	0.892454i	$0.648982\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−19.5153	−0.682756
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−24.3933	−0.851333	−0.425667	−	0.904880i	$-0.639960\pi$
$821$	−24.3933	−0.851333	−0.425667	+	0.904880i	$0.639960\pi$
$822$	0	0
$823$	−2.61420	−0.0911252	−0.0455626	−	0.998961i	$-0.514508\pi$
$823$	−2.61420	−0.0911252	−0.0455626	+	0.998961i	$0.514508\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−0.0684725	−0.00238102	−0.00119051	−	0.999999i	$-0.500379\pi$
$827$	−0.0684725	−0.00238102	−0.00119051	+	0.999999i	$0.500379\pi$
$828$	0	0
$829$	−24.7828	−0.860744	−0.430372	−	0.902652i	$-0.641618\pi$
$829$	−24.7828	−0.860744	−0.430372	+	0.902652i	$0.641618\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−11.2494	−0.389770
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−45.2409	−1.56189	−0.780944	−	0.624601i	$-0.785262\pi$
$839$	−45.2409	−1.56189	−0.780944	+	0.624601i	$0.785262\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−5.92857	−0.203708
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−5.26396	−0.180446
$852$	0	0
$853$	20.2150	0.692148	0.346074	−	0.938207i	$-0.387515\pi$
$853$	20.2150	0.692148	0.346074	+	0.938207i	$0.387515\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	26.8427	0.916929	0.458464	−	0.888713i	$-0.348400\pi$
$857$	26.8427	0.916929	0.458464	+	0.888713i	$0.348400\pi$
$858$	0	0
$859$	3.36259	0.114730	0.0573651	−	0.998353i	$-0.481730\pi$
$859$	3.36259	0.114730	0.0573651	+	0.998353i	$0.481730\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.4524	0.628126	0.314063	−	0.949402i	$-0.398310\pi$
$863$	18.4524	0.628126	0.314063	+	0.949402i	$0.398310\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	38.9789	1.32227
$870$	0	0
$871$	−2.92829	−0.0992212
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	23.1985	0.783358	0.391679	−	0.920102i	$-0.371894\pi$
$877$	23.1985	0.783358	0.391679	+	0.920102i	$0.371894\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.34854	−0.0791244	−0.0395622	−	0.999217i	$-0.512596\pi$
$881$	−2.34854	−0.0791244	−0.0395622	+	0.999217i	$0.512596\pi$
$882$	0	0
$883$	−16.3264	−0.549428	−0.274714	−	0.961526i	$-0.588583\pi$
$883$	−16.3264	−0.549428	−0.274714	+	0.961526i	$0.588583\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26.8427	0.901289	0.450645	−	0.892703i	$-0.351194\pi$
$887$	26.8427	0.901289	0.450645	+	0.892703i	$0.351194\pi$
$888$	0	0
$889$	25.5587	0.857210
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−70.2892	−2.35214
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−17.3485	−0.578606
$900$	0	0
$901$	−17.5516	−0.584730
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−7.84259	−0.260409	−0.130204	−	0.991487i	$-0.541563\pi$
$907$	−7.84259	−0.260409	−0.130204	+	0.991487i	$0.541563\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.57272	−0.0852380	−0.0426190	−	0.999091i	$-0.513570\pi$
$911$	−2.57272	−0.0852380	−0.0426190	+	0.999091i	$0.513570\pi$
$912$	0	0
$913$	−36.1511	−1.19643
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−12.9196	−0.426642
$918$	0	0
$919$	42.4062	1.39885	0.699426	−	0.714705i	$-0.253439\pi$
$919$	42.4062	1.39885	0.699426	+	0.714705i	$0.253439\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−7.23541	−0.238156
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−51.8335	−1.70060	−0.850301	−	0.526297i	$-0.823580\pi$
$929$	−51.8335	−1.70060	−0.850301	+	0.526297i	$0.823580\pi$
$930$	0	0
$931$	51.9860	1.70377
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	16.9666	0.554275	0.277137	−	0.960830i	$-0.410614\pi$
$937$	16.9666	0.554275	0.277137	+	0.960830i	$0.410614\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−5.65730	−0.184423	−0.0922114	−	0.995739i	$-0.529394\pi$
$941$	−5.65730	−0.184423	−0.0922114	+	0.995739i	$0.529394\pi$
$942$	0	0
$943$	−19.2769	−0.627744
$944$	0	0
$945$	0	0
$946$	0	0
$947$	35.3860	1.14989	0.574945	−	0.818192i	$-0.305023\pi$
$947$	35.3860	1.14989	0.574945	+	0.818192i	$0.305023\pi$
$948$	0	0
$949$	16.8405	0.546666
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.66116	−0.0538101	−0.0269051	−	0.999638i	$-0.508565\pi$
$953$	−1.66116	−0.0538101	−0.0269051	+	0.999638i	$0.508565\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−56.2678	−1.81698
$960$	0	0
$961$	2.44133	0.0787525
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−20.7362	−0.666832	−0.333416	−	0.942780i	$-0.608201\pi$
$967$	−20.7362	−0.666832	−0.333416	+	0.942780i	$0.608201\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−20.9193	−0.671331	−0.335665	−	0.941981i	$-0.608961\pi$
$971$	−20.9193	−0.671331	−0.335665	+	0.941981i	$0.608961\pi$
$972$	0	0
$973$	20.4589	0.655881
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.89158	0.0925098	0.0462549	−	0.998930i	$-0.485271\pi$
$977$	2.89158	0.0925098	0.0462549	+	0.998930i	$0.485271\pi$
$978$	0	0
$979$	18.9193	0.604662
$980$	0	0
$981$	0	0
$982$	0	0
$983$	6.30037	0.200951	0.100475	−	0.994940i	$-0.467964\pi$
$983$	6.30037	0.200951	0.100475	+	0.994940i	$0.467964\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.7310	0.532016
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−9.57927	−0.303378	−0.151689	−	0.988428i	$-0.548471\pi$
$997$	−9.57927	−0.303378	−0.151689	+	0.988428i	$0.548471\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8100.2.a.bd.1.1		6
3.2	odd	2		8100.2.a.bc.1.1		6
5.2	odd	4		1620.2.d.d.649.2		6
5.3	odd	4		1620.2.d.d.649.1		6
5.4	even	2	inner	8100.2.a.bd.1.6		6
9.2	odd	6		900.2.i.f.301.4		12
9.4	even	3		2700.2.i.f.1801.6		12
9.5	odd	6		900.2.i.f.601.4		12
9.7	even	3		2700.2.i.f.901.6		12
15.2	even	4		1620.2.d.c.649.5		6
15.8	even	4		1620.2.d.c.649.6		6
15.14	odd	2		8100.2.a.bc.1.6		6
45.2	even	12		180.2.r.a.49.1	✓	12
45.4	even	6		2700.2.i.f.1801.1		12
45.7	odd	12		540.2.r.a.469.4		12
45.13	odd	12		540.2.r.a.289.4		12
45.14	odd	6		900.2.i.f.601.3		12
45.22	odd	12		540.2.r.a.289.6		12
45.23	even	12		180.2.r.a.169.1	yes	12
45.29	odd	6		900.2.i.f.301.3		12
45.32	even	12		180.2.r.a.169.6	yes	12
45.34	even	6		2700.2.i.f.901.1		12
45.38	even	12		180.2.r.a.49.6	yes	12
45.43	odd	12		540.2.r.a.469.6		12
180.7	even	12		2160.2.by.e.1009.4		12
180.23	odd	12		720.2.by.e.529.6		12
180.43	even	12		2160.2.by.e.1009.6		12
180.47	odd	12		720.2.by.e.49.6		12
180.67	even	12		2160.2.by.e.289.6		12
180.83	odd	12		720.2.by.e.49.1		12
180.103	even	12		2160.2.by.e.289.4		12
180.167	odd	12		720.2.by.e.529.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.r.a.49.1	✓	12	45.2	even	12
180.2.r.a.49.6	yes	12	45.38	even	12
180.2.r.a.169.1	yes	12	45.23	even	12
180.2.r.a.169.6	yes	12	45.32	even	12
540.2.r.a.289.4		12	45.13	odd	12
540.2.r.a.289.6		12	45.22	odd	12
540.2.r.a.469.4		12	45.7	odd	12
540.2.r.a.469.6		12	45.43	odd	12
720.2.by.e.49.1		12	180.83	odd	12
720.2.by.e.49.6		12	180.47	odd	12
720.2.by.e.529.1		12	180.167	odd	12
720.2.by.e.529.6		12	180.23	odd	12
900.2.i.f.301.3		12	45.29	odd	6
900.2.i.f.301.4		12	9.2	odd	6
900.2.i.f.601.3		12	45.14	odd	6
900.2.i.f.601.4		12	9.5	odd	6
1620.2.d.c.649.5		6	15.2	even	4
1620.2.d.c.649.6		6	15.8	even	4
1620.2.d.d.649.1		6	5.3	odd	4
1620.2.d.d.649.2		6	5.2	odd	4
2160.2.by.e.289.4		12	180.103	even	12
2160.2.by.e.289.6		12	180.67	even	12
2160.2.by.e.1009.4		12	180.7	even	12
2160.2.by.e.1009.6		12	180.43	even	12
2700.2.i.f.901.1		12	45.34	even	6
2700.2.i.f.901.6		12	9.7	even	3
2700.2.i.f.1801.1		12	45.4	even	6
2700.2.i.f.1801.6		12	9.4	even	3
8100.2.a.bc.1.1		6	3.2	odd	2
8100.2.a.bc.1.6		6	15.14	odd	2
8100.2.a.bd.1.1		6	1.1	even	1	trivial
8100.2.a.bd.1.6		6	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.a (trivial)

\(f(q)\)	\(=\)	\(q-3.76963 q^{7} +O(q^{10})\)	\(q-3.76963 q^{7} +3.54580 q^{11} +1.00351 q^{13} -1.56023 q^{17} +7.21013 q^{19} -6.18143 q^{23} +3.00000 q^{29} -5.78285 q^{31} +0.851576 q^{37} +3.11852 q^{41} -2.70666 q^{43} -9.74867 q^{47} +7.21013 q^{49} +11.2494 q^{53} +9.66433 q^{59} -8.21013 q^{61} -2.91806 q^{67} -7.21013 q^{71} +16.7817 q^{73} -13.3664 q^{77} +10.9930 q^{79} -10.1955 q^{83} +5.33567 q^{89} -3.78285 q^{91} -3.86209 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q + 2 q^{11} + 18 q^{29} - 6 q^{31} + 14 q^{41} + 34 q^{59} - 6 q^{61} - 6 q^{79} + 56 q^{89} + 6 q^{91}+O(q^{100}) \) 6 * q + 2 * q^11 + 18 * q^29 - 6 * q^31 + 14 * q^41 + 34 * q^59 - 6 * q^61 - 6 * q^79 + 56 * q^89 + 6 * q^91

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