LMFDB - Embedded newform 8024.2.a.y.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.40425 q^{3} +3.37446 q^{5} -3.92495 q^{7} -1.02809 q^{9} +O(q^{10})$	$q-1.40425 q^{3} +3.37446 q^{5} -3.92495 q^{7} -1.02809 q^{9} +1.19459 q^{11} +1.76914 q^{13} -4.73858 q^{15} -1.00000 q^{17} -5.97168 q^{19} +5.51160 q^{21} -1.06460 q^{23} +6.38698 q^{25} +5.65643 q^{27} +5.79547 q^{29} +0.924101 q^{31} -1.67751 q^{33} -13.2446 q^{35} -6.65188 q^{37} -2.48431 q^{39} +8.89664 q^{41} +8.28892 q^{43} -3.46925 q^{45} -1.68499 q^{47} +8.40520 q^{49} +1.40425 q^{51} +3.41618 q^{53} +4.03111 q^{55} +8.38571 q^{57} -1.00000 q^{59} -6.47999 q^{61} +4.03519 q^{63} +5.96988 q^{65} -1.63189 q^{67} +1.49497 q^{69} -11.1438 q^{71} +12.4653 q^{73} -8.96890 q^{75} -4.68872 q^{77} -11.1434 q^{79} -4.85877 q^{81} +9.00811 q^{83} -3.37446 q^{85} -8.13827 q^{87} -11.8666 q^{89} -6.94377 q^{91} -1.29767 q^{93} -20.1512 q^{95} -17.1397 q^{97} -1.22815 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 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
$3$	−1.40425	−0.810743	−0.405371	−	0.914152i	$-0.632858\pi$
$3$	−1.40425	−0.810743	−0.405371	+	0.914152i	$0.632858\pi$
$4$	0	0
$5$	3.37446	1.50910	0.754552	−	0.656240i	$-0.227854\pi$
$5$	3.37446	1.50910	0.754552	+	0.656240i	$0.227854\pi$
$6$	0	0
$7$	−3.92495	−1.48349	−0.741745	−	0.670682i	$-0.766002\pi$
$7$	−3.92495	−1.48349	−0.741745	+	0.670682i	$0.766002\pi$
$8$	0	0
$9$	−1.02809	−0.342696
$10$	0	0
$11$	1.19459	0.360184	0.180092	−	0.983650i	$-0.442360\pi$
$11$	1.19459	0.360184	0.180092	+	0.983650i	$0.442360\pi$
$12$	0	0
$13$	1.76914	0.490670	0.245335	−	0.969438i	$-0.421102\pi$
$13$	1.76914	0.490670	0.245335	+	0.969438i	$0.421102\pi$
$14$	0	0
$15$	−4.73858	−1.22350
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−5.97168	−1.37000	−0.684998	−	0.728545i	$-0.740197\pi$
$19$	−5.97168	−1.37000	−0.684998	+	0.728545i	$0.740197\pi$
$20$	0	0
$21$	5.51160	1.20273
$22$	0	0
$23$	−1.06460	−0.221985	−0.110993	−	0.993821i	$-0.535403\pi$
$23$	−1.06460	−0.221985	−0.110993	+	0.993821i	$0.535403\pi$
$24$	0	0
$25$	6.38698	1.27740
$26$	0	0
$27$	5.65643	1.08858
$28$	0	0
$29$	5.79547	1.07619	0.538096	−	0.842884i	$-0.319144\pi$
$29$	5.79547	1.07619	0.538096	+	0.842884i	$0.319144\pi$
$30$	0	0
$31$	0.924101	0.165973	0.0829867	−	0.996551i	$-0.473554\pi$
$31$	0.924101	0.165973	0.0829867	+	0.996551i	$0.473554\pi$
$32$	0	0
$33$	−1.67751	−0.292016
$34$	0	0
$35$	−13.2446	−2.23874
$36$	0	0
$37$	−6.65188	−1.09356	−0.546781	−	0.837275i	$-0.684147\pi$
$37$	−6.65188	−1.09356	−0.546781	+	0.837275i	$0.684147\pi$
$38$	0	0
$39$	−2.48431	−0.397807
$40$	0	0
$41$	8.89664	1.38942	0.694711	−	0.719289i	$-0.255532\pi$
$41$	8.89664	1.38942	0.694711	+	0.719289i	$0.255532\pi$
$42$	0	0
$43$	8.28892	1.26405	0.632024	−	0.774949i	$-0.282224\pi$
$43$	8.28892	1.26405	0.632024	+	0.774949i	$0.282224\pi$
$44$	0	0
$45$	−3.46925	−0.517165
$46$	0	0
$47$	−1.68499	−0.245781	−0.122891	−	0.992420i	$-0.539216\pi$
$47$	−1.68499	−0.245781	−0.122891	+	0.992420i	$0.539216\pi$
$48$	0	0
$49$	8.40520	1.20074
$50$	0	0
$51$	1.40425	0.196634
$52$	0	0
$53$	3.41618	0.469249	0.234624	−	0.972086i	$-0.424614\pi$
$53$	3.41618	0.469249	0.234624	+	0.972086i	$0.424614\pi$
$54$	0	0
$55$	4.03111	0.543555
$56$	0	0
$57$	8.38571	1.11071
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−6.47999	−0.829677	−0.414839	−	0.909895i	$-0.636162\pi$
$61$	−6.47999	−0.829677	−0.414839	+	0.909895i	$0.636162\pi$
$62$	0	0
$63$	4.03519	0.508387
$64$	0	0
$65$	5.96988	0.740473
$66$	0	0
$67$	−1.63189	−0.199368	−0.0996838	−	0.995019i	$-0.531783\pi$
$67$	−1.63189	−0.199368	−0.0996838	+	0.995019i	$0.531783\pi$
$68$	0	0
$69$	1.49497	0.179973
$70$	0	0
$71$	−11.1438	−1.32253	−0.661265	−	0.750152i	$-0.729980\pi$
$71$	−11.1438	−1.32253	−0.661265	+	0.750152i	$0.729980\pi$
$72$	0	0
$73$	12.4653	1.45896	0.729478	−	0.684005i	$-0.239763\pi$
$73$	12.4653	1.45896	0.729478	+	0.684005i	$0.239763\pi$
$74$	0	0
$75$	−8.96890	−1.03564
$76$	0	0
$77$	−4.68872	−0.534329
$78$	0	0
$79$	−11.1434	−1.25373	−0.626863	−	0.779129i	$-0.715662\pi$
$79$	−11.1434	−1.25373	−0.626863	+	0.779129i	$0.715662\pi$
$80$	0	0
$81$	−4.85877	−0.539863
$82$	0	0
$83$	9.00811	0.988768	0.494384	−	0.869244i	$-0.335394\pi$
$83$	9.00811	0.988768	0.494384	+	0.869244i	$0.335394\pi$
$84$	0	0
$85$	−3.37446	−0.366012
$86$	0	0
$87$	−8.13827	−0.872515
$88$	0	0
$89$	−11.8666	−1.25786	−0.628928	−	0.777464i	$-0.716506\pi$
$89$	−11.8666	−1.25786	−0.628928	+	0.777464i	$0.716506\pi$
$90$	0	0
$91$	−6.94377	−0.727905
$92$	0	0
$93$	−1.29767	−0.134562
$94$	0	0
$95$	−20.1512	−2.06747
$96$	0	0
$97$	−17.1397	−1.74028	−0.870138	−	0.492808i	$-0.835970\pi$
$97$	−17.1397	−1.74028	−0.870138	+	0.492808i	$0.835970\pi$
$98$	0	0
$99$	−1.22815	−0.123434
$100$	0	0
$101$	6.12710	0.609670	0.304835	−	0.952405i	$-0.401399\pi$
$101$	6.12710	0.609670	0.304835	+	0.952405i	$0.401399\pi$
$102$	0	0
$103$	−18.7949	−1.85191	−0.925957	−	0.377629i	$-0.876740\pi$
$103$	−18.7949	−1.85191	−0.925957	+	0.377629i	$0.876740\pi$
$104$	0	0
$105$	18.5987	1.81504
$106$	0	0
$107$	7.62309	0.736952	0.368476	−	0.929637i	$-0.379880\pi$
$107$	7.62309	0.736952	0.368476	+	0.929637i	$0.379880\pi$
$108$	0	0
$109$	17.8941	1.71394	0.856970	−	0.515367i	$-0.172344\pi$
$109$	17.8941	1.71394	0.856970	+	0.515367i	$0.172344\pi$
$110$	0	0
$111$	9.34089	0.886598
$112$	0	0
$113$	−0.957182	−0.0900441	−0.0450220	−	0.998986i	$-0.514336\pi$
$113$	−0.957182	−0.0900441	−0.0450220	+	0.998986i	$0.514336\pi$
$114$	0	0
$115$	−3.59246	−0.334999
$116$	0	0
$117$	−1.81883	−0.168151
$118$	0	0
$119$	3.92495	0.359799
$120$	0	0
$121$	−9.57294	−0.870268
$122$	0	0
$123$	−12.4931	−1.12646
$124$	0	0
$125$	4.68031	0.418619
$126$	0	0
$127$	16.1541	1.43345	0.716723	−	0.697358i	$-0.245641\pi$
$127$	16.1541	1.43345	0.716723	+	0.697358i	$0.245641\pi$
$128$	0	0
$129$	−11.6397	−1.02482
$130$	0	0
$131$	−4.95328	−0.432770	−0.216385	−	0.976308i	$-0.569427\pi$
$131$	−4.95328	−0.432770	−0.216385	+	0.976308i	$0.569427\pi$
$132$	0	0
$133$	23.4385	2.03238
$134$	0	0
$135$	19.0874	1.64278
$136$	0	0
$137$	−3.57758	−0.305653	−0.152827	−	0.988253i	$-0.548838\pi$
$137$	−3.57758	−0.305653	−0.152827	+	0.988253i	$0.548838\pi$
$138$	0	0
$139$	7.16757	0.607946	0.303973	−	0.952681i	$-0.401687\pi$
$139$	7.16757	0.607946	0.303973	+	0.952681i	$0.401687\pi$
$140$	0	0
$141$	2.36615	0.199265
$142$	0	0
$143$	2.11340	0.176732
$144$	0	0
$145$	19.5566	1.62409
$146$	0	0
$147$	−11.8030	−0.973494
$148$	0	0
$149$	−13.3050	−1.08998	−0.544992	−	0.838441i	$-0.683467\pi$
$149$	−13.3050	−1.08998	−0.544992	+	0.838441i	$0.683467\pi$
$150$	0	0
$151$	−4.88886	−0.397850	−0.198925	−	0.980015i	$-0.563745\pi$
$151$	−4.88886	−0.397850	−0.198925	+	0.980015i	$0.563745\pi$
$152$	0	0
$153$	1.02809	0.0831161
$154$	0	0
$155$	3.11834	0.250471
$156$	0	0
$157$	2.47404	0.197450	0.0987251	−	0.995115i	$-0.468524\pi$
$157$	2.47404	0.197450	0.0987251	+	0.995115i	$0.468524\pi$
$158$	0	0
$159$	−4.79716	−0.380440
$160$	0	0
$161$	4.17851	0.329313
$162$	0	0
$163$	−5.17630	−0.405439	−0.202719	−	0.979237i	$-0.564978\pi$
$163$	−5.17630	−0.405439	−0.202719	+	0.979237i	$0.564978\pi$
$164$	0	0
$165$	−5.66068	−0.440683
$166$	0	0
$167$	−21.3190	−1.64971	−0.824857	−	0.565342i	$-0.808744\pi$
$167$	−21.3190	−1.64971	−0.824857	+	0.565342i	$0.808744\pi$
$168$	0	0
$169$	−9.87015	−0.759243
$170$	0	0
$171$	6.13941	0.469493
$172$	0	0
$173$	−23.9668	−1.82217	−0.911083	−	0.412223i	$-0.864752\pi$
$173$	−23.9668	−1.82217	−0.911083	+	0.412223i	$0.864752\pi$
$174$	0	0
$175$	−25.0686	−1.89500
$176$	0	0
$177$	1.40425	0.105550
$178$	0	0
$179$	9.00636	0.673167	0.336583	−	0.941654i	$-0.390729\pi$
$179$	9.00636	0.673167	0.336583	+	0.941654i	$0.390729\pi$
$180$	0	0
$181$	6.12584	0.455330	0.227665	−	0.973740i	$-0.426891\pi$
$181$	6.12584	0.455330	0.227665	+	0.973740i	$0.426891\pi$
$182$	0	0
$183$	9.09950	0.672655
$184$	0	0
$185$	−22.4465	−1.65030
$186$	0	0
$187$	−1.19459	−0.0873574
$188$	0	0
$189$	−22.2012	−1.61490
$190$	0	0
$191$	9.98130	0.722222	0.361111	−	0.932523i	$-0.382398\pi$
$191$	9.98130	0.722222	0.361111	+	0.932523i	$0.382398\pi$
$192$	0	0
$193$	−24.5420	−1.76657	−0.883286	−	0.468835i	$-0.844674\pi$
$193$	−24.5420	−1.76657	−0.883286	+	0.468835i	$0.844674\pi$
$194$	0	0
$195$	−8.38319	−0.600333
$196$	0	0
$197$	0.312390	0.0222569	0.0111284	−	0.999938i	$-0.496458\pi$
$197$	0.312390	0.0222569	0.0111284	+	0.999938i	$0.496458\pi$
$198$	0	0
$199$	−21.0450	−1.49184	−0.745919	−	0.666037i	$-0.767989\pi$
$199$	−21.0450	−1.49184	−0.745919	+	0.666037i	$0.767989\pi$
$200$	0	0
$201$	2.29158	0.161636
$202$	0	0
$203$	−22.7469	−1.59652
$204$	0	0
$205$	30.0213	2.09678
$206$	0	0
$207$	1.09451	0.0760736
$208$	0	0
$209$	−7.13373	−0.493451
$210$	0	0
$211$	−19.7315	−1.35837	−0.679187	−	0.733965i	$-0.737668\pi$
$211$	−19.7315	−1.35837	−0.679187	+	0.733965i	$0.737668\pi$
$212$	0	0
$213$	15.6487	1.07223
$214$	0	0
$215$	27.9706	1.90758
$216$	0	0
$217$	−3.62705	−0.246220
$218$	0	0
$219$	−17.5044	−1.18284
$220$	0	0
$221$	−1.76914	−0.119005
$222$	0	0
$223$	−3.67333	−0.245984	−0.122992	−	0.992408i	$-0.539249\pi$
$223$	−3.67333	−0.245984	−0.122992	+	0.992408i	$0.539249\pi$
$224$	0	0
$225$	−6.56638	−0.437759
$226$	0	0
$227$	−10.6693	−0.708147	−0.354073	−	0.935218i	$-0.615204\pi$
$227$	−10.6693	−0.708147	−0.354073	+	0.935218i	$0.615204\pi$
$228$	0	0
$229$	25.7563	1.70203	0.851014	−	0.525144i	$-0.175988\pi$
$229$	25.7563	1.70203	0.851014	+	0.525144i	$0.175988\pi$
$230$	0	0
$231$	6.58412	0.433204
$232$	0	0
$233$	14.6812	0.961800	0.480900	−	0.876776i	$-0.340310\pi$
$233$	14.6812	0.961800	0.480900	+	0.876776i	$0.340310\pi$
$234$	0	0
$235$	−5.68594	−0.370910
$236$	0	0
$237$	15.6480	1.01645
$238$	0	0
$239$	−1.22425	−0.0791900	−0.0395950	−	0.999216i	$-0.512607\pi$
$239$	−1.22425	−0.0791900	−0.0395950	+	0.999216i	$0.512607\pi$
$240$	0	0
$241$	−8.75573	−0.564006	−0.282003	−	0.959414i	$-0.590999\pi$
$241$	−8.75573	−0.564006	−0.282003	+	0.959414i	$0.590999\pi$
$242$	0	0
$243$	−10.1464	−0.650891
$244$	0	0
$245$	28.3630	1.81205
$246$	0	0
$247$	−10.5647	−0.672217
$248$	0	0
$249$	−12.6496	−0.801637
$250$	0	0
$251$	0.410423	0.0259057	0.0129528	−	0.999916i	$-0.495877\pi$
$251$	0.410423	0.0259057	0.0129528	+	0.999916i	$0.495877\pi$
$252$	0	0
$253$	−1.27177	−0.0799555
$254$	0	0
$255$	4.73858	0.296741
$256$	0	0
$257$	−16.6060	−1.03585	−0.517927	−	0.855425i	$-0.673296\pi$
$257$	−16.6060	−1.03585	−0.517927	+	0.855425i	$0.673296\pi$
$258$	0	0
$259$	26.1083	1.62229
$260$	0	0
$261$	−5.95826	−0.368807
$262$	0	0
$263$	−26.8406	−1.65506	−0.827531	−	0.561419i	$-0.810255\pi$
$263$	−26.8406	−1.65506	−0.827531	+	0.561419i	$0.810255\pi$
$264$	0	0
$265$	11.5278	0.708145
$266$	0	0
$267$	16.6636	1.01980
$268$	0	0
$269$	−26.6572	−1.62532	−0.812659	−	0.582739i	$-0.801981\pi$
$269$	−26.6572	−1.62532	−0.812659	+	0.582739i	$0.801981\pi$
$270$	0	0
$271$	11.6422	0.707216	0.353608	−	0.935394i	$-0.384955\pi$
$271$	11.6422	0.707216	0.353608	+	0.935394i	$0.384955\pi$
$272$	0	0
$273$	9.75077	0.590143
$274$	0	0
$275$	7.62985	0.460097
$276$	0	0
$277$	−4.23304	−0.254339	−0.127169	−	0.991881i	$-0.540589\pi$
$277$	−4.23304	−0.254339	−0.127169	+	0.991881i	$0.540589\pi$
$278$	0	0
$279$	−0.950058	−0.0568785
$280$	0	0
$281$	−5.87365	−0.350392	−0.175196	−	0.984534i	$-0.556056\pi$
$281$	−5.87365	−0.350392	−0.175196	+	0.984534i	$0.556056\pi$
$282$	0	0
$283$	28.2391	1.67864	0.839321	−	0.543637i	$-0.182953\pi$
$283$	28.2391	1.67864	0.839321	+	0.543637i	$0.182953\pi$
$284$	0	0
$285$	28.2972	1.67618
$286$	0	0
$287$	−34.9188	−2.06119
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	24.0684	1.41092
$292$	0	0
$293$	20.2227	1.18142	0.590712	−	0.806882i	$-0.298847\pi$
$293$	20.2227	1.18142	0.590712	+	0.806882i	$0.298847\pi$
$294$	0	0
$295$	−3.37446	−0.196469
$296$	0	0
$297$	6.75715	0.392089
$298$	0	0
$299$	−1.88343	−0.108922
$300$	0	0
$301$	−32.5336	−1.87520
$302$	0	0
$303$	−8.60397	−0.494285
$304$	0	0
$305$	−21.8665	−1.25207
$306$	0	0
$307$	−14.1471	−0.807418	−0.403709	−	0.914887i	$-0.632279\pi$
$307$	−14.1471	−0.807418	−0.403709	+	0.914887i	$0.632279\pi$
$308$	0	0
$309$	26.3927	1.50143
$310$	0	0
$311$	8.64731	0.490344	0.245172	−	0.969480i	$-0.421156\pi$
$311$	8.64731	0.490344	0.245172	+	0.969480i	$0.421156\pi$
$312$	0	0
$313$	30.9936	1.75186	0.875932	−	0.482435i	$-0.160247\pi$
$313$	30.9936	1.75186	0.875932	+	0.482435i	$0.160247\pi$
$314$	0	0
$315$	13.6166	0.767209
$316$	0	0
$317$	33.6027	1.88732	0.943659	−	0.330919i	$-0.107359\pi$
$317$	33.6027	1.88732	0.943659	+	0.330919i	$0.107359\pi$
$318$	0	0
$319$	6.92324	0.387627
$320$	0	0
$321$	−10.7047	−0.597479
$322$	0	0
$323$	5.97168	0.332273
$324$	0	0
$325$	11.2994	0.626780
$326$	0	0
$327$	−25.1277	−1.38956
$328$	0	0
$329$	6.61351	0.364614
$330$	0	0
$331$	−4.02899	−0.221453	−0.110727	−	0.993851i	$-0.535318\pi$
$331$	−4.02899	−0.221453	−0.110727	+	0.993851i	$0.535318\pi$
$332$	0	0
$333$	6.83873	0.374760
$334$	0	0
$335$	−5.50676	−0.300866
$336$	0	0
$337$	−21.0390	−1.14607	−0.573033	−	0.819533i	$-0.694233\pi$
$337$	−21.0390	−1.14607	−0.573033	+	0.819533i	$0.694233\pi$
$338$	0	0
$339$	1.34412	0.0730026
$340$	0	0
$341$	1.10393	0.0597810
$342$	0	0
$343$	−5.51535	−0.297801
$344$	0	0
$345$	5.04471	0.271598
$346$	0	0
$347$	−9.33028	−0.500876	−0.250438	−	0.968133i	$-0.580575\pi$
$347$	−9.33028	−0.500876	−0.250438	+	0.968133i	$0.580575\pi$
$348$	0	0
$349$	12.8849	0.689710	0.344855	−	0.938656i	$-0.387928\pi$
$349$	12.8849	0.689710	0.344855	+	0.938656i	$0.387928\pi$
$350$	0	0
$351$	10.0070	0.534135
$352$	0	0
$353$	−15.0103	−0.798916	−0.399458	−	0.916752i	$-0.630802\pi$
$353$	−15.0103	−0.798916	−0.399458	+	0.916752i	$0.630802\pi$
$354$	0	0
$355$	−37.6044	−1.99584
$356$	0	0
$357$	−5.51160	−0.291705
$358$	0	0
$359$	−34.9933	−1.84688	−0.923438	−	0.383748i	$-0.874633\pi$
$359$	−34.9933	−1.84688	−0.923438	+	0.383748i	$0.874633\pi$
$360$	0	0
$361$	16.6609	0.876890
$362$	0	0
$363$	13.4428	0.705563
$364$	0	0
$365$	42.0637	2.20172
$366$	0	0
$367$	11.5208	0.601381	0.300690	−	0.953722i	$-0.402783\pi$
$367$	11.5208	0.601381	0.300690	+	0.953722i	$0.402783\pi$
$368$	0	0
$369$	−9.14653	−0.476150
$370$	0	0
$371$	−13.4083	−0.696126
$372$	0	0
$373$	28.4863	1.47497	0.737483	−	0.675366i	$-0.236014\pi$
$373$	28.4863	1.47497	0.737483	+	0.675366i	$0.236014\pi$
$374$	0	0
$375$	−6.57231	−0.339393
$376$	0	0
$377$	10.2530	0.528055
$378$	0	0
$379$	27.4443	1.40972	0.704859	−	0.709348i	$-0.251010\pi$
$379$	27.4443	1.40972	0.704859	+	0.709348i	$0.251010\pi$
$380$	0	0
$381$	−22.6844	−1.16216
$382$	0	0
$383$	1.47669	0.0754552	0.0377276	−	0.999288i	$-0.487988\pi$
$383$	1.47669	0.0754552	0.0377276	+	0.999288i	$0.487988\pi$
$384$	0	0
$385$	−15.8219	−0.806359
$386$	0	0
$387$	−8.52174	−0.433185
$388$	0	0
$389$	−28.5353	−1.44680	−0.723399	−	0.690430i	$-0.757421\pi$
$389$	−28.5353	−1.44680	−0.723399	+	0.690430i	$0.757421\pi$
$390$	0	0
$391$	1.06460	0.0538393
$392$	0	0
$393$	6.95563	0.350865
$394$	0	0
$395$	−37.6028	−1.89200
$396$	0	0
$397$	−15.7144	−0.788681	−0.394341	−	0.918964i	$-0.629027\pi$
$397$	−15.7144	−0.788681	−0.394341	+	0.918964i	$0.629027\pi$
$398$	0	0
$399$	−32.9135	−1.64773
$400$	0	0
$401$	−10.3684	−0.517772	−0.258886	−	0.965908i	$-0.583355\pi$
$401$	−10.3684	−0.517772	−0.258886	+	0.965908i	$0.583355\pi$
$402$	0	0
$403$	1.63486	0.0814383
$404$	0	0
$405$	−16.3957	−0.814709
$406$	0	0
$407$	−7.94630	−0.393884
$408$	0	0
$409$	−26.6310	−1.31682	−0.658408	−	0.752661i	$-0.728770\pi$
$409$	−26.6310	−1.31682	−0.658408	+	0.752661i	$0.728770\pi$
$410$	0	0
$411$	5.02381	0.247806
$412$	0	0
$413$	3.92495	0.193134
$414$	0	0
$415$	30.3975	1.49215
$416$	0	0
$417$	−10.0650	−0.492887
$418$	0	0
$419$	31.6397	1.54570	0.772850	−	0.634589i	$-0.218831\pi$
$419$	31.6397	1.54570	0.772850	+	0.634589i	$0.218831\pi$
$420$	0	0
$421$	17.3788	0.846990	0.423495	−	0.905898i	$-0.360803\pi$
$421$	17.3788	0.846990	0.423495	+	0.905898i	$0.360803\pi$
$422$	0	0
$423$	1.73232	0.0842284
$424$	0	0
$425$	−6.38698	−0.309814
$426$	0	0
$427$	25.4336	1.23082
$428$	0	0
$429$	−2.96774	−0.143284
$430$	0	0
$431$	−25.2679	−1.21711	−0.608557	−	0.793510i	$-0.708251\pi$
$431$	−25.2679	−1.21711	−0.608557	+	0.793510i	$0.708251\pi$
$432$	0	0
$433$	7.42428	0.356788	0.178394	−	0.983959i	$-0.442910\pi$
$433$	7.42428	0.356788	0.178394	+	0.983959i	$0.442910\pi$
$434$	0	0
$435$	−27.4623	−1.31672
$436$	0	0
$437$	6.35747	0.304119
$438$	0	0
$439$	−21.8827	−1.04440	−0.522201	−	0.852822i	$-0.674889\pi$
$439$	−21.8827	−1.04440	−0.522201	+	0.852822i	$0.674889\pi$
$440$	0	0
$441$	−8.64130	−0.411490
$442$	0	0
$443$	10.7859	0.512455	0.256227	−	0.966617i	$-0.417520\pi$
$443$	10.7859	0.512455	0.256227	+	0.966617i	$0.417520\pi$
$444$	0	0
$445$	−40.0433	−1.89824
$446$	0	0
$447$	18.6835	0.883697
$448$	0	0
$449$	8.13970	0.384136	0.192068	−	0.981382i	$-0.438481\pi$
$449$	8.13970	0.384136	0.192068	+	0.981382i	$0.438481\pi$
$450$	0	0
$451$	10.6279	0.500447
$452$	0	0
$453$	6.86517	0.322554
$454$	0	0
$455$	−23.4315	−1.09848
$456$	0	0
$457$	−0.343927	−0.0160882	−0.00804411	−	0.999968i	$-0.502561\pi$
$457$	−0.343927	−0.0160882	−0.00804411	+	0.999968i	$0.502561\pi$
$458$	0	0
$459$	−5.65643	−0.264020
$460$	0	0
$461$	5.70741	0.265821	0.132910	−	0.991128i	$-0.457568\pi$
$461$	5.70741	0.265821	0.132910	+	0.991128i	$0.457568\pi$
$462$	0	0
$463$	16.0862	0.747589	0.373795	−	0.927512i	$-0.378057\pi$
$463$	16.0862	0.747589	0.373795	+	0.927512i	$0.378057\pi$
$464$	0	0
$465$	−4.37892	−0.203068
$466$	0	0
$467$	−9.27233	−0.429072	−0.214536	−	0.976716i	$-0.568824\pi$
$467$	−9.27233	−0.429072	−0.214536	+	0.976716i	$0.568824\pi$
$468$	0	0
$469$	6.40510	0.295760
$470$	0	0
$471$	−3.47417	−0.160081
$472$	0	0
$473$	9.90190	0.455290
$474$	0	0
$475$	−38.1410	−1.75003
$476$	0	0
$477$	−3.51214	−0.160810
$478$	0	0
$479$	−24.6333	−1.12552	−0.562761	−	0.826619i	$-0.690261\pi$
$479$	−24.6333	−1.12552	−0.562761	+	0.826619i	$0.690261\pi$
$480$	0	0
$481$	−11.7681	−0.536579
$482$	0	0
$483$	−5.86767	−0.266988
$484$	0	0
$485$	−57.8373	−2.62626
$486$	0	0
$487$	6.32303	0.286524	0.143262	−	0.989685i	$-0.454241\pi$
$487$	6.32303	0.286524	0.143262	+	0.989685i	$0.454241\pi$
$488$	0	0
$489$	7.26880	0.328707
$490$	0	0
$491$	−19.7251	−0.890183	−0.445092	−	0.895485i	$-0.646829\pi$
$491$	−19.7251	−0.890183	−0.445092	+	0.895485i	$0.646829\pi$
$492$	0	0
$493$	−5.79547	−0.261015
$494$	0	0
$495$	−4.14434	−0.186274
$496$	0	0
$497$	43.7390	1.96196
$498$	0	0
$499$	18.0540	0.808206	0.404103	−	0.914713i	$-0.367584\pi$
$499$	18.0540	0.808206	0.404103	+	0.914713i	$0.367584\pi$
$500$	0	0
$501$	29.9371	1.33749
$502$	0	0
$503$	−3.00127	−0.133820	−0.0669099	−	0.997759i	$-0.521314\pi$
$503$	−3.00127	−0.133820	−0.0669099	+	0.997759i	$0.521314\pi$
$504$	0	0
$505$	20.6757	0.920055
$506$	0	0
$507$	13.8601	0.615550
$508$	0	0
$509$	−42.9672	−1.90449	−0.952244	−	0.305339i	$-0.901230\pi$
$509$	−42.9672	−1.90449	−0.952244	+	0.305339i	$0.901230\pi$
$510$	0	0
$511$	−48.9257	−2.16435
$512$	0	0
$513$	−33.7784	−1.49135
$514$	0	0
$515$	−63.4226	−2.79473
$516$	0	0
$517$	−2.01288	−0.0885265
$518$	0	0
$519$	33.6554	1.47731
$520$	0	0
$521$	−34.9603	−1.53164	−0.765819	−	0.643056i	$-0.777666\pi$
$521$	−34.9603	−1.53164	−0.765819	+	0.643056i	$0.777666\pi$
$522$	0	0
$523$	−3.07882	−0.134628	−0.0673138	−	0.997732i	$-0.521443\pi$
$523$	−3.07882	−0.134628	−0.0673138	+	0.997732i	$0.521443\pi$
$524$	0	0
$525$	35.2024	1.53636
$526$	0	0
$527$	−0.924101	−0.0402545
$528$	0	0
$529$	−21.8666	−0.950723
$530$	0	0
$531$	1.02809	0.0446153
$532$	0	0
$533$	15.7394	0.681748
$534$	0	0
$535$	25.7238	1.11214
$536$	0	0
$537$	−12.6472	−0.545765
$538$	0	0
$539$	10.0408	0.432488
$540$	0	0
$541$	−26.6997	−1.14791	−0.573955	−	0.818887i	$-0.694592\pi$
$541$	−26.6997	−1.14791	−0.573955	+	0.818887i	$0.694592\pi$
$542$	0	0
$543$	−8.60219	−0.369155
$544$	0	0
$545$	60.3828	2.58651
$546$	0	0
$547$	8.39937	0.359131	0.179566	−	0.983746i	$-0.442531\pi$
$547$	8.39937	0.359131	0.179566	+	0.983746i	$0.442531\pi$
$548$	0	0
$549$	6.66200	0.284327
$550$	0	0
$551$	−34.6087	−1.47438
$552$	0	0
$553$	43.7371	1.85989
$554$	0	0
$555$	31.5204	1.33797
$556$	0	0
$557$	−39.2640	−1.66367	−0.831834	−	0.555024i	$-0.812709\pi$
$557$	−39.2640	−1.66367	−0.831834	+	0.555024i	$0.812709\pi$
$558$	0	0
$559$	14.6642	0.620231
$560$	0	0
$561$	1.67751	0.0708244
$562$	0	0
$563$	−31.7135	−1.33657	−0.668283	−	0.743907i	$-0.732971\pi$
$563$	−31.7135	−1.33657	−0.668283	+	0.743907i	$0.732971\pi$
$564$	0	0
$565$	−3.22997	−0.135886
$566$	0	0
$567$	19.0704	0.800881
$568$	0	0
$569$	−3.99799	−0.167604	−0.0838021	−	0.996482i	$-0.526706\pi$
$569$	−3.99799	−0.167604	−0.0838021	+	0.996482i	$0.526706\pi$
$570$	0	0
$571$	−3.72745	−0.155989	−0.0779945	−	0.996954i	$-0.524852\pi$
$571$	−3.72745	−0.155989	−0.0779945	+	0.996954i	$0.524852\pi$
$572$	0	0
$573$	−14.0162	−0.585536
$574$	0	0
$575$	−6.79961	−0.283563
$576$	0	0
$577$	−41.5054	−1.72789	−0.863945	−	0.503586i	$-0.832014\pi$
$577$	−41.5054	−1.72789	−0.863945	+	0.503586i	$0.832014\pi$
$578$	0	0
$579$	34.4630	1.43223
$580$	0	0
$581$	−35.3563	−1.46683
$582$	0	0
$583$	4.08095	0.169016
$584$	0	0
$585$	−6.13757	−0.253757
$586$	0	0
$587$	−29.9602	−1.23659	−0.618294	−	0.785947i	$-0.712176\pi$
$587$	−29.9602	−1.23659	−0.618294	+	0.785947i	$0.712176\pi$
$588$	0	0
$589$	−5.51843	−0.227383
$590$	0	0
$591$	−0.438673	−0.0180446
$592$	0	0
$593$	23.1764	0.951741	0.475870	−	0.879515i	$-0.342133\pi$
$593$	23.1764	0.951741	0.475870	+	0.879515i	$0.342133\pi$
$594$	0	0
$595$	13.2446	0.542975
$596$	0	0
$597$	29.5523	1.20950
$598$	0	0
$599$	17.7787	0.726418	0.363209	−	0.931708i	$-0.381681\pi$
$599$	17.7787	0.726418	0.363209	+	0.931708i	$0.381681\pi$
$600$	0	0
$601$	−3.03049	−0.123616	−0.0618081	−	0.998088i	$-0.519687\pi$
$601$	−3.03049	−0.123616	−0.0618081	+	0.998088i	$0.519687\pi$
$602$	0	0
$603$	1.67773	0.0683225
$604$	0	0
$605$	−32.3035	−1.31332
$606$	0	0
$607$	−18.8912	−0.766769	−0.383385	−	0.923589i	$-0.625242\pi$
$607$	−18.8912	−0.766769	−0.383385	+	0.923589i	$0.625242\pi$
$608$	0	0
$609$	31.9423	1.29437
$610$	0	0
$611$	−2.98098	−0.120598
$612$	0	0
$613$	4.81066	0.194301	0.0971504	−	0.995270i	$-0.469027\pi$
$613$	4.81066	0.194301	0.0971504	+	0.995270i	$0.469027\pi$
$614$	0	0
$615$	−42.1574	−1.69995
$616$	0	0
$617$	29.9632	1.20627	0.603137	−	0.797638i	$-0.293917\pi$
$617$	29.9632	1.20627	0.603137	+	0.797638i	$0.293917\pi$
$618$	0	0
$619$	32.7428	1.31605	0.658023	−	0.752998i	$-0.271393\pi$
$619$	32.7428	1.31605	0.658023	+	0.752998i	$0.271393\pi$
$620$	0	0
$621$	−6.02186	−0.241649
$622$	0	0
$623$	46.5757	1.86602
$624$	0	0
$625$	−16.1414	−0.645656
$626$	0	0
$627$	10.0175	0.400062
$628$	0	0
$629$	6.65188	0.265228
$630$	0	0
$631$	33.5903	1.33721	0.668605	−	0.743618i	$-0.266892\pi$
$631$	33.5903	1.33721	0.668605	+	0.743618i	$0.266892\pi$
$632$	0	0
$633$	27.7080	1.10129
$634$	0	0
$635$	54.5114	2.16322
$636$	0	0
$637$	14.8700	0.589169
$638$	0	0
$639$	11.4569	0.453226
$640$	0	0
$641$	−13.3976	−0.529175	−0.264587	−	0.964362i	$-0.585236\pi$
$641$	−13.3976	−0.529175	−0.264587	+	0.964362i	$0.585236\pi$
$642$	0	0
$643$	10.2416	0.403890	0.201945	−	0.979397i	$-0.435274\pi$
$643$	10.2416	0.403890	0.201945	+	0.979397i	$0.435274\pi$
$644$	0	0
$645$	−39.2777	−1.54656
$646$	0	0
$647$	34.3309	1.34969	0.674843	−	0.737961i	$-0.264211\pi$
$647$	34.3309	1.34969	0.674843	+	0.737961i	$0.264211\pi$
$648$	0	0
$649$	−1.19459	−0.0468920
$650$	0	0
$651$	5.09327	0.199621
$652$	0	0
$653$	−17.8490	−0.698487	−0.349243	−	0.937032i	$-0.613561\pi$
$653$	−17.8490	−0.698487	−0.349243	+	0.937032i	$0.613561\pi$
$654$	0	0
$655$	−16.7147	−0.653095
$656$	0	0
$657$	−12.8155	−0.499979
$658$	0	0
$659$	−9.67092	−0.376726	−0.188363	−	0.982100i	$-0.560318\pi$
$659$	−9.67092	−0.376726	−0.188363	+	0.982100i	$0.560318\pi$
$660$	0	0
$661$	−14.4194	−0.560848	−0.280424	−	0.959876i	$-0.590475\pi$
$661$	−14.4194	−0.560848	−0.280424	+	0.959876i	$0.590475\pi$
$662$	0	0
$663$	2.48431	0.0964825
$664$	0	0
$665$	79.0923	3.06707
$666$	0	0
$667$	−6.16988	−0.238899
$668$	0	0
$669$	5.15826	0.199430
$670$	0	0
$671$	−7.74096	−0.298836
$672$	0	0
$673$	−21.4315	−0.826125	−0.413063	−	0.910703i	$-0.635541\pi$
$673$	−21.4315	−0.826125	−0.413063	+	0.910703i	$0.635541\pi$
$674$	0	0
$675$	36.1275	1.39055
$676$	0	0
$677$	48.9977	1.88313	0.941567	−	0.336827i	$-0.109354\pi$
$677$	48.9977	1.88313	0.941567	+	0.336827i	$0.109354\pi$
$678$	0	0
$679$	67.2725	2.58168
$680$	0	0
$681$	14.9824	0.574125
$682$	0	0
$683$	5.38815	0.206172	0.103086	−	0.994672i	$-0.467128\pi$
$683$	5.38815	0.206172	0.103086	+	0.994672i	$0.467128\pi$
$684$	0	0
$685$	−12.0724	−0.461262
$686$	0	0
$687$	−36.1683	−1.37991
$688$	0	0
$689$	6.04369	0.230246
$690$	0	0
$691$	−16.8920	−0.642602	−0.321301	−	0.946977i	$-0.604120\pi$
$691$	−16.8920	−0.642602	−0.321301	+	0.946977i	$0.604120\pi$
$692$	0	0
$693$	4.82042	0.183113
$694$	0	0
$695$	24.1867	0.917453
$696$	0	0
$697$	−8.89664	−0.336984
$698$	0	0
$699$	−20.6161	−0.779772
$700$	0	0
$701$	−47.6074	−1.79811	−0.899053	−	0.437841i	$-0.855743\pi$
$701$	−47.6074	−1.79811	−0.899053	+	0.437841i	$0.855743\pi$
$702$	0	0
$703$	39.7229	1.49818
$704$	0	0
$705$	7.98447	0.300712
$706$	0	0
$707$	−24.0486	−0.904439
$708$	0	0
$709$	−37.3586	−1.40303	−0.701516	−	0.712654i	$-0.747493\pi$
$709$	−37.3586	−1.40303	−0.701516	+	0.712654i	$0.747493\pi$
$710$	0	0
$711$	11.4564	0.429648
$712$	0	0
$713$	−0.983802	−0.0368437
$714$	0	0
$715$	7.13159	0.266706
$716$	0	0
$717$	1.71915	0.0642027
$718$	0	0
$719$	1.01318	0.0377852	0.0188926	−	0.999822i	$-0.493986\pi$
$719$	1.01318	0.0377852	0.0188926	+	0.999822i	$0.493986\pi$
$720$	0	0
$721$	73.7689	2.74730
$722$	0	0
$723$	12.2952	0.457264
$724$	0	0
$725$	37.0155	1.37472
$726$	0	0
$727$	−41.6767	−1.54570	−0.772851	−	0.634587i	$-0.781170\pi$
$727$	−41.6767	−1.54570	−0.772851	+	0.634587i	$0.781170\pi$
$728$	0	0
$729$	28.8243	1.06757
$730$	0	0
$731$	−8.28892	−0.306577
$732$	0	0
$733$	45.0442	1.66375	0.831873	−	0.554967i	$-0.187269\pi$
$733$	45.0442	1.66375	0.831873	+	0.554967i	$0.187269\pi$
$734$	0	0
$735$	−39.8287	−1.46910
$736$	0	0
$737$	−1.94945	−0.0718090
$738$	0	0
$739$	14.9186	0.548789	0.274394	−	0.961617i	$-0.411523\pi$
$739$	14.9186	0.548789	0.274394	+	0.961617i	$0.411523\pi$
$740$	0	0
$741$	14.8355	0.544995
$742$	0	0
$743$	19.0492	0.698847	0.349424	−	0.936965i	$-0.386377\pi$
$743$	19.0492	0.698847	0.349424	+	0.936965i	$0.386377\pi$
$744$	0	0
$745$	−44.8970	−1.64490
$746$	0	0
$747$	−9.26114	−0.338847
$748$	0	0
$749$	−29.9202	−1.09326
$750$	0	0
$751$	−42.6468	−1.55621	−0.778103	−	0.628137i	$-0.783818\pi$
$751$	−42.6468	−1.55621	−0.778103	+	0.628137i	$0.783818\pi$
$752$	0	0
$753$	−0.576336	−0.0210028
$754$	0	0
$755$	−16.4973	−0.600397
$756$	0	0
$757$	−29.5878	−1.07539	−0.537693	−	0.843141i	$-0.680704\pi$
$757$	−29.5878	−1.07539	−0.537693	+	0.843141i	$0.680704\pi$
$758$	0	0
$759$	1.78588	0.0648234
$760$	0	0
$761$	−8.80505	−0.319183	−0.159591	−	0.987183i	$-0.551018\pi$
$761$	−8.80505	−0.319183	−0.159591	+	0.987183i	$0.551018\pi$
$762$	0	0
$763$	−70.2332	−2.54261
$764$	0	0
$765$	3.46925	0.125431
$766$	0	0
$767$	−1.76914	−0.0638798
$768$	0	0
$769$	−23.6237	−0.851892	−0.425946	−	0.904749i	$-0.640059\pi$
$769$	−23.6237	−0.851892	−0.425946	+	0.904749i	$0.640059\pi$
$770$	0	0
$771$	23.3190	0.839812
$772$	0	0
$773$	6.83478	0.245830	0.122915	−	0.992417i	$-0.460776\pi$
$773$	6.83478	0.245830	0.122915	+	0.992417i	$0.460776\pi$
$774$	0	0
$775$	5.90221	0.212014
$776$	0	0
$777$	−36.6625	−1.31526
$778$	0	0
$779$	−53.1278	−1.90350
$780$	0	0
$781$	−13.3124	−0.476354
$782$	0	0
$783$	32.7817	1.17152
$784$	0	0
$785$	8.34856	0.297973
$786$	0	0
$787$	−1.88497	−0.0671918	−0.0335959	−	0.999435i	$-0.510696\pi$
$787$	−1.88497	−0.0671918	−0.0335959	+	0.999435i	$0.510696\pi$
$788$	0	0
$789$	37.6909	1.34183
$790$	0	0
$791$	3.75689	0.133580
$792$	0	0
$793$	−11.4640	−0.407098
$794$	0	0
$795$	−16.1878	−0.574123
$796$	0	0
$797$	−21.5434	−0.763108	−0.381554	−	0.924347i	$-0.624611\pi$
$797$	−21.5434	−0.763108	−0.381554	+	0.924347i	$0.624611\pi$
$798$	0	0
$799$	1.68499	0.0596108
$800$	0	0
$801$	12.1999	0.431063
$802$	0	0
$803$	14.8910	0.525492
$804$	0	0
$805$	14.1002	0.496968
$806$	0	0
$807$	37.4333	1.31771
$808$	0	0
$809$	6.69530	0.235394	0.117697	−	0.993050i	$-0.462449\pi$
$809$	6.69530	0.235394	0.117697	+	0.993050i	$0.462449\pi$
$810$	0	0
$811$	−25.8625	−0.908156	−0.454078	−	0.890962i	$-0.650031\pi$
$811$	−25.8625	−0.908156	−0.454078	+	0.890962i	$0.650031\pi$
$812$	0	0
$813$	−16.3486	−0.573370
$814$	0	0
$815$	−17.4672	−0.611849
$816$	0	0
$817$	−49.4987	−1.73174
$818$	0	0
$819$	7.13881	0.249450
$820$	0	0
$821$	−39.2418	−1.36955	−0.684773	−	0.728756i	$-0.740099\pi$
$821$	−39.2418	−1.36955	−0.684773	+	0.728756i	$0.740099\pi$
$822$	0	0
$823$	−6.96178	−0.242672	−0.121336	−	0.992611i	$-0.538718\pi$
$823$	−6.96178	−0.242672	−0.121336	+	0.992611i	$0.538718\pi$
$824$	0	0
$825$	−10.7142	−0.373021
$826$	0	0
$827$	−6.82120	−0.237196	−0.118598	−	0.992942i	$-0.537840\pi$
$827$	−6.82120	−0.237196	−0.118598	+	0.992942i	$0.537840\pi$
$828$	0	0
$829$	1.33307	0.0462994	0.0231497	−	0.999732i	$-0.492631\pi$
$829$	1.33307	0.0462994	0.0231497	+	0.999732i	$0.492631\pi$
$830$	0	0
$831$	5.94424	0.206203
$832$	0	0
$833$	−8.40520	−0.291223
$834$	0	0
$835$	−71.9401	−2.48959
$836$	0	0
$837$	5.22712	0.180676
$838$	0	0
$839$	−29.1530	−1.00647	−0.503237	−	0.864148i	$-0.667858\pi$
$839$	−29.1530	−1.00647	−0.503237	+	0.864148i	$0.667858\pi$
$840$	0	0
$841$	4.58747	0.158189
$842$	0	0
$843$	8.24805	0.284078
$844$	0	0
$845$	−33.3064	−1.14578
$846$	0	0
$847$	37.5733	1.29103
$848$	0	0
$849$	−39.6547	−1.36095
$850$	0	0
$851$	7.08162	0.242755
$852$	0	0
$853$	−16.0424	−0.549281	−0.274641	−	0.961547i	$-0.588559\pi$
$853$	−16.0424	−0.549281	−0.274641	+	0.961547i	$0.588559\pi$
$854$	0	0
$855$	20.7172	0.708514
$856$	0	0
$857$	−7.48834	−0.255797	−0.127898	−	0.991787i	$-0.540823\pi$
$857$	−7.48834	−0.255797	−0.127898	+	0.991787i	$0.540823\pi$
$858$	0	0
$859$	−25.4552	−0.868519	−0.434259	−	0.900788i	$-0.642990\pi$
$859$	−25.4552	−0.868519	−0.434259	+	0.900788i	$0.642990\pi$
$860$	0	0
$861$	49.0347	1.67110
$862$	0	0
$863$	31.5577	1.07424	0.537118	−	0.843507i	$-0.319513\pi$
$863$	31.5577	1.07424	0.537118	+	0.843507i	$0.319513\pi$
$864$	0	0
$865$	−80.8752	−2.74984
$866$	0	0
$867$	−1.40425	−0.0476907
$868$	0	0
$869$	−13.3118	−0.451572
$870$	0	0
$871$	−2.88704	−0.0978238
$872$	0	0
$873$	17.6212	0.596386
$874$	0	0
$875$	−18.3700	−0.621018
$876$	0	0
$877$	12.6969	0.428744	0.214372	−	0.976752i	$-0.431230\pi$
$877$	12.6969	0.428744	0.214372	+	0.976752i	$0.431230\pi$
$878$	0	0
$879$	−28.3977	−0.957832
$880$	0	0
$881$	−30.8661	−1.03991	−0.519953	−	0.854195i	$-0.674051\pi$
$881$	−30.8661	−1.03991	−0.519953	+	0.854195i	$0.674051\pi$
$882$	0	0
$883$	39.9064	1.34296	0.671479	−	0.741024i	$-0.265660\pi$
$883$	39.9064	1.34296	0.671479	+	0.741024i	$0.265660\pi$
$884$	0	0
$885$	4.73858	0.159286
$886$	0	0
$887$	36.6940	1.23207	0.616033	−	0.787721i	$-0.288739\pi$
$887$	36.6940	1.23207	0.616033	+	0.787721i	$0.288739\pi$
$888$	0	0
$889$	−63.4040	−2.12650
$890$	0	0
$891$	−5.80426	−0.194450
$892$	0	0
$893$	10.0622	0.336720
$894$	0	0
$895$	30.3916	1.01588
$896$	0	0
$897$	2.64480	0.0883074
$898$	0	0
$899$	5.35560	0.178619
$900$	0	0
$901$	−3.41618	−0.113809
$902$	0	0
$903$	45.6852	1.52031
$904$	0	0
$905$	20.6714	0.687140
$906$	0	0
$907$	32.5219	1.07987	0.539936	−	0.841706i	$-0.318448\pi$
$907$	32.5219	1.07987	0.539936	+	0.841706i	$0.318448\pi$
$908$	0	0
$909$	−6.29921	−0.208932
$910$	0	0
$911$	−42.9944	−1.42447	−0.712233	−	0.701943i	$-0.752316\pi$
$911$	−42.9944	−1.42447	−0.712233	+	0.701943i	$0.752316\pi$
$912$	0	0
$913$	10.7610	0.356138
$914$	0	0
$915$	30.7059	1.01511
$916$	0	0
$917$	19.4414	0.642010
$918$	0	0
$919$	45.3225	1.49505	0.747526	−	0.664233i	$-0.231242\pi$
$919$	45.3225	1.49505	0.747526	+	0.664233i	$0.231242\pi$
$920$	0	0
$921$	19.8660	0.654608
$922$	0	0
$923$	−19.7150	−0.648927
$924$	0	0
$925$	−42.4854	−1.39691
$926$	0	0
$927$	19.3228	0.634644
$928$	0	0
$929$	−5.76911	−0.189278	−0.0946392	−	0.995512i	$-0.530170\pi$
$929$	−5.76911	−0.189278	−0.0946392	+	0.995512i	$0.530170\pi$
$930$	0	0
$931$	−50.1932	−1.64501
$932$	0	0
$933$	−12.1430	−0.397543
$934$	0	0
$935$	−4.03111	−0.131831
$936$	0	0
$937$	14.7630	0.482286	0.241143	−	0.970490i	$-0.422478\pi$
$937$	14.7630	0.482286	0.241143	+	0.970490i	$0.422478\pi$
$938$	0	0
$939$	−43.5227	−1.42031
$940$	0	0
$941$	44.1338	1.43872	0.719361	−	0.694636i	$-0.244435\pi$
$941$	44.1338	1.43872	0.719361	+	0.694636i	$0.244435\pi$
$942$	0	0
$943$	−9.47140	−0.308431
$944$	0	0
$945$	−74.9171	−2.43705
$946$	0	0
$947$	−54.8411	−1.78210	−0.891048	−	0.453909i	$-0.850029\pi$
$947$	−54.8411	−1.78210	−0.891048	+	0.453909i	$0.850029\pi$
$948$	0	0
$949$	22.0529	0.715866
$950$	0	0
$951$	−47.1866	−1.53013
$952$	0	0
$953$	−7.64403	−0.247615	−0.123807	−	0.992306i	$-0.539510\pi$
$953$	−7.64403	−0.247615	−0.123807	+	0.992306i	$0.539510\pi$
$954$	0	0
$955$	33.6815	1.08991
$956$	0	0
$957$	−9.72194	−0.314266
$958$	0	0
$959$	14.0418	0.453433
$960$	0	0
$961$	−30.1460	−0.972453
$962$	0	0
$963$	−7.83722	−0.252551
$964$	0	0
$965$	−82.8160	−2.66594
$966$	0	0
$967$	26.0353	0.837240	0.418620	−	0.908161i	$-0.362514\pi$
$967$	26.0353	0.837240	0.418620	+	0.908161i	$0.362514\pi$
$968$	0	0
$969$	−8.38571	−0.269388
$970$	0	0
$971$	55.6088	1.78457	0.892285	−	0.451472i	$-0.149101\pi$
$971$	55.6088	1.78457	0.892285	+	0.451472i	$0.149101\pi$
$972$	0	0
$973$	−28.1323	−0.901881
$974$	0	0
$975$	−15.8672	−0.508158
$976$	0	0
$977$	33.1632	1.06098	0.530492	−	0.847690i	$-0.322007\pi$
$977$	33.1632	1.06098	0.530492	+	0.847690i	$0.322007\pi$
$978$	0	0
$979$	−14.1758	−0.453060
$980$	0	0
$981$	−18.3967	−0.587361
$982$	0	0
$983$	−31.0265	−0.989592	−0.494796	−	0.869009i	$-0.664757\pi$
$983$	−31.0265	−0.989592	−0.494796	+	0.869009i	$0.664757\pi$
$984$	0	0
$985$	1.05415	0.0335879
$986$	0	0
$987$	−9.28700	−0.295608
$988$	0	0
$989$	−8.82442	−0.280600
$990$	0	0
$991$	−5.28206	−0.167790	−0.0838950	−	0.996475i	$-0.526736\pi$
$991$	−5.28206	−0.167790	−0.0838950	+	0.996475i	$0.526736\pi$
$992$	0	0
$993$	5.65770	0.179542
$994$	0	0
$995$	−71.0154	−2.25134
$996$	0	0
$997$	−1.97901	−0.0626759	−0.0313379	−	0.999509i	$-0.509977\pi$
$997$	−1.97901	−0.0626759	−0.0313379	+	0.999509i	$0.509977\pi$
$998$	0	0
$999$	−37.6259	−1.19043

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.y.1.9	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.y.1.9	✓	23	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.40425 q^{3} +3.37446 q^{5} -3.92495 q^{7} -1.02809 q^{9} +O(q^{10})\)	\(q-1.40425 q^{3} +3.37446 q^{5} -3.92495 q^{7} -1.02809 q^{9} +1.19459 q^{11} +1.76914 q^{13} -4.73858 q^{15} -1.00000 q^{17} -5.97168 q^{19} +5.51160 q^{21} -1.06460 q^{23} +6.38698 q^{25} +5.65643 q^{27} +5.79547 q^{29} +0.924101 q^{31} -1.67751 q^{33} -13.2446 q^{35} -6.65188 q^{37} -2.48431 q^{39} +8.89664 q^{41} +8.28892 q^{43} -3.46925 q^{45} -1.68499 q^{47} +8.40520 q^{49} +1.40425 q^{51} +3.41618 q^{53} +4.03111 q^{55} +8.38571 q^{57} -1.00000 q^{59} -6.47999 q^{61} +4.03519 q^{63} +5.96988 q^{65} -1.63189 q^{67} +1.49497 q^{69} -11.1438 q^{71} +12.4653 q^{73} -8.96890 q^{75} -4.68872 q^{77} -11.1434 q^{79} -4.85877 q^{81} +9.00811 q^{83} -3.37446 q^{85} -8.13827 q^{87} -11.8666 q^{89} -6.94377 q^{91} -1.29767 q^{93} -20.1512 q^{95} -17.1397 q^{97} -1.22815 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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