LMFDB - Embedded newform 8024.2.a.y.1.15

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			1.15
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)

$f(q)$	$=$	$q+0.561991 q^{3} -2.34032 q^{5} -3.24647 q^{7} -2.68417 q^{9} +O(q^{10})$	$q+0.561991 q^{3} -2.34032 q^{5} -3.24647 q^{7} -2.68417 q^{9} +0.117214 q^{11} +0.0710091 q^{13} -1.31524 q^{15} -1.00000 q^{17} +7.74546 q^{19} -1.82449 q^{21} +3.00596 q^{23} +0.477109 q^{25} -3.19445 q^{27} +7.74335 q^{29} +0.790508 q^{31} +0.0658734 q^{33} +7.59779 q^{35} +5.26024 q^{37} +0.0399065 q^{39} -4.64131 q^{41} +5.78183 q^{43} +6.28181 q^{45} -2.92091 q^{47} +3.53958 q^{49} -0.561991 q^{51} -4.98500 q^{53} -0.274319 q^{55} +4.35288 q^{57} -1.00000 q^{59} +3.39452 q^{61} +8.71407 q^{63} -0.166184 q^{65} +4.95731 q^{67} +1.68932 q^{69} +4.72617 q^{71} -11.2207 q^{73} +0.268131 q^{75} -0.380533 q^{77} -6.93936 q^{79} +6.25725 q^{81} +10.2866 q^{83} +2.34032 q^{85} +4.35169 q^{87} -15.0663 q^{89} -0.230529 q^{91} +0.444258 q^{93} -18.1269 q^{95} +1.43255 q^{97} -0.314623 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.561991	0.324466	0.162233	−	0.986753i	$-0.448130\pi$
$3$	0.561991	0.324466	0.162233	+	0.986753i	$0.448130\pi$
$4$	0	0
$5$	−2.34032	−1.04662	−0.523312	−	0.852141i	$-0.675304\pi$
$5$	−2.34032	−1.04662	−0.523312	+	0.852141i	$0.675304\pi$
$6$	0	0
$7$	−3.24647	−1.22705	−0.613525	−	0.789675i	$-0.710249\pi$
$7$	−3.24647	−1.22705	−0.613525	+	0.789675i	$0.710249\pi$
$8$	0	0
$9$	−2.68417	−0.894722
$10$	0	0
$11$	0.117214	0.0353415	0.0176707	−	0.999844i	$-0.494375\pi$
$11$	0.117214	0.0353415	0.0176707	+	0.999844i	$0.494375\pi$
$12$	0	0
$13$	0.0710091	0.0196944	0.00984719	−	0.999952i	$-0.496865\pi$
$13$	0.0710091	0.0196944	0.00984719	+	0.999952i	$0.496865\pi$
$14$	0	0
$15$	−1.31524	−0.339593
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	7.74546	1.77693	0.888465	−	0.458944i	$-0.151772\pi$
$19$	7.74546	1.77693	0.888465	+	0.458944i	$0.151772\pi$
$20$	0	0
$21$	−1.82449	−0.398136
$22$	0	0
$23$	3.00596	0.626785	0.313393	−	0.949624i	$-0.398534\pi$
$23$	3.00596	0.626785	0.313393	+	0.949624i	$0.398534\pi$
$24$	0	0
$25$	0.477109	0.0954217
$26$	0	0
$27$	−3.19445	−0.614772
$28$	0	0
$29$	7.74335	1.43790	0.718952	−	0.695060i	$-0.244622\pi$
$29$	7.74335	1.43790	0.718952	+	0.695060i	$0.244622\pi$
$30$	0	0
$31$	0.790508	0.141979	0.0709897	−	0.997477i	$-0.477384\pi$
$31$	0.790508	0.141979	0.0709897	+	0.997477i	$0.477384\pi$
$32$	0	0
$33$	0.0658734	0.0114671
$34$	0	0
$35$	7.59779	1.28426
$36$	0	0
$37$	5.26024	0.864777	0.432389	−	0.901687i	$-0.357671\pi$
$37$	5.26024	0.864777	0.432389	+	0.901687i	$0.357671\pi$
$38$	0	0
$39$	0.0399065	0.00639015
$40$	0	0
$41$	−4.64131	−0.724850	−0.362425	−	0.932013i	$-0.618051\pi$
$41$	−4.64131	−0.724850	−0.362425	+	0.932013i	$0.618051\pi$
$42$	0	0
$43$	5.78183	0.881721	0.440861	−	0.897576i	$-0.354673\pi$
$43$	5.78183	0.881721	0.440861	+	0.897576i	$0.354673\pi$
$44$	0	0
$45$	6.28181	0.936438
$46$	0	0
$47$	−2.92091	−0.426059	−0.213029	−	0.977046i	$-0.568333\pi$
$47$	−2.92091	−0.426059	−0.213029	+	0.977046i	$0.568333\pi$
$48$	0	0
$49$	3.53958	0.505654
$50$	0	0
$51$	−0.561991	−0.0786945
$52$	0	0
$53$	−4.98500	−0.684743	−0.342371	−	0.939565i	$-0.611230\pi$
$53$	−4.98500	−0.684743	−0.342371	+	0.939565i	$0.611230\pi$
$54$	0	0
$55$	−0.274319	−0.0369892
$56$	0	0
$57$	4.35288	0.576553
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	3.39452	0.434623	0.217312	−	0.976102i	$-0.430271\pi$
$61$	3.39452	0.434623	0.217312	+	0.976102i	$0.430271\pi$
$62$	0	0
$63$	8.71407	1.09787
$64$	0	0
$65$	−0.166184	−0.0206126
$66$	0	0
$67$	4.95731	0.605632	0.302816	−	0.953049i	$-0.402073\pi$
$67$	4.95731	0.605632	0.302816	+	0.953049i	$0.402073\pi$
$68$	0	0
$69$	1.68932	0.203370
$70$	0	0
$71$	4.72617	0.560893	0.280447	−	0.959870i	$-0.409517\pi$
$71$	4.72617	0.560893	0.280447	+	0.959870i	$0.409517\pi$
$72$	0	0
$73$	−11.2207	−1.31329	−0.656644	−	0.754201i	$-0.728025\pi$
$73$	−11.2207	−1.31329	−0.656644	+	0.754201i	$0.728025\pi$
$74$	0	0
$75$	0.268131	0.0309611
$76$	0	0
$77$	−0.380533	−0.0433658
$78$	0	0
$79$	−6.93936	−0.780739	−0.390369	−	0.920658i	$-0.627653\pi$
$79$	−6.93936	−0.780739	−0.390369	+	0.920658i	$0.627653\pi$
$80$	0	0
$81$	6.25725	0.695250
$82$	0	0
$83$	10.2866	1.12910	0.564548	−	0.825400i	$-0.309050\pi$
$83$	10.2866	1.12910	0.564548	+	0.825400i	$0.309050\pi$
$84$	0	0
$85$	2.34032	0.253844
$86$	0	0
$87$	4.35169	0.466550
$88$	0	0
$89$	−15.0663	−1.59703	−0.798515	−	0.601975i	$-0.794381\pi$
$89$	−15.0663	−1.59703	−0.798515	+	0.601975i	$0.794381\pi$
$90$	0	0
$91$	−0.230529	−0.0241660
$92$	0	0
$93$	0.444258	0.0460674
$94$	0	0
$95$	−18.1269	−1.85978
$96$	0	0
$97$	1.43255	0.145454	0.0727270	−	0.997352i	$-0.476830\pi$
$97$	1.43255	0.145454	0.0727270	+	0.997352i	$0.476830\pi$
$98$	0	0
$99$	−0.314623	−0.0316208
$100$	0	0
$101$	2.58501	0.257218	0.128609	−	0.991695i	$-0.458949\pi$
$101$	2.58501	0.257218	0.128609	+	0.991695i	$0.458949\pi$
$102$	0	0
$103$	−14.7383	−1.45220	−0.726102	−	0.687586i	$-0.758670\pi$
$103$	−14.7383	−1.45220	−0.726102	+	0.687586i	$0.758670\pi$
$104$	0	0
$105$	4.26989	0.416698
$106$	0	0
$107$	4.93693	0.477271	0.238636	−	0.971109i	$-0.423300\pi$
$107$	4.93693	0.477271	0.238636	+	0.971109i	$0.423300\pi$
$108$	0	0
$109$	−11.1205	−1.06515	−0.532576	−	0.846382i	$-0.678776\pi$
$109$	−11.1205	−1.06515	−0.532576	+	0.846382i	$0.678776\pi$
$110$	0	0
$111$	2.95620	0.280590
$112$	0	0
$113$	18.2007	1.71218	0.856088	−	0.516829i	$-0.172888\pi$
$113$	18.2007	1.71218	0.856088	+	0.516829i	$0.172888\pi$
$114$	0	0
$115$	−7.03491	−0.656008
$116$	0	0
$117$	−0.190600	−0.0176210
$118$	0	0
$119$	3.24647	0.297604
$120$	0	0
$121$	−10.9863	−0.998751
$122$	0	0
$123$	−2.60837	−0.235189
$124$	0	0
$125$	10.5850	0.946753
$126$	0	0
$127$	−7.96314	−0.706614	−0.353307	−	0.935507i	$-0.614943\pi$
$127$	−7.96314	−0.706614	−0.353307	+	0.935507i	$0.614943\pi$
$128$	0	0
$129$	3.24934	0.286088
$130$	0	0
$131$	0.555572	0.0485405	0.0242703	−	0.999705i	$-0.492274\pi$
$131$	0.555572	0.0485405	0.0242703	+	0.999705i	$0.492274\pi$
$132$	0	0
$133$	−25.1454	−2.18038
$134$	0	0
$135$	7.47604	0.643435
$136$	0	0
$137$	10.0742	0.860696	0.430348	−	0.902663i	$-0.358391\pi$
$137$	10.0742	0.860696	0.430348	+	0.902663i	$0.358391\pi$
$138$	0	0
$139$	−7.69079	−0.652324	−0.326162	−	0.945314i	$-0.605756\pi$
$139$	−7.69079	−0.652324	−0.326162	+	0.945314i	$0.605756\pi$
$140$	0	0
$141$	−1.64153	−0.138241
$142$	0	0
$143$	0.00832329	0.000696028 0
$144$	0	0
$145$	−18.1219	−1.50494
$146$	0	0
$147$	1.98921	0.164067
$148$	0	0
$149$	0.675664	0.0553526	0.0276763	−	0.999617i	$-0.491189\pi$
$149$	0.675664	0.0553526	0.0276763	+	0.999617i	$0.491189\pi$
$150$	0	0
$151$	10.6608	0.867560	0.433780	−	0.901019i	$-0.357180\pi$
$151$	10.6608	0.867560	0.433780	+	0.901019i	$0.357180\pi$
$152$	0	0
$153$	2.68417	0.217002
$154$	0	0
$155$	−1.85004	−0.148599
$156$	0	0
$157$	−18.3514	−1.46460	−0.732302	−	0.680980i	$-0.761554\pi$
$157$	−18.3514	−1.46460	−0.732302	+	0.680980i	$0.761554\pi$
$158$	0	0
$159$	−2.80153	−0.222176
$160$	0	0
$161$	−9.75875	−0.769097
$162$	0	0
$163$	−0.526646	−0.0412501	−0.0206250	−	0.999787i	$-0.506566\pi$
$163$	−0.526646	−0.0412501	−0.0206250	+	0.999787i	$0.506566\pi$
$164$	0	0
$165$	−0.154165	−0.0120017
$166$	0	0
$167$	−23.7882	−1.84079	−0.920395	−	0.390991i	$-0.872133\pi$
$167$	−23.7882	−1.84079	−0.920395	+	0.390991i	$0.872133\pi$
$168$	0	0
$169$	−12.9950	−0.999612
$170$	0	0
$171$	−20.7901	−1.58986
$172$	0	0
$173$	3.99732	0.303910	0.151955	−	0.988387i	$-0.451443\pi$
$173$	3.99732	0.303910	0.151955	+	0.988387i	$0.451443\pi$
$174$	0	0
$175$	−1.54892	−0.117087
$176$	0	0
$177$	−0.561991	−0.0422418
$178$	0	0
$179$	19.1128	1.42856	0.714281	−	0.699859i	$-0.246754\pi$
$179$	19.1128	1.42856	0.714281	+	0.699859i	$0.246754\pi$
$180$	0	0
$181$	17.9075	1.33106	0.665529	−	0.746372i	$-0.268206\pi$
$181$	17.9075	1.33106	0.665529	+	0.746372i	$0.268206\pi$
$182$	0	0
$183$	1.90769	0.141020
$184$	0	0
$185$	−12.3106	−0.905097
$186$	0	0
$187$	−0.117214	−0.00857157
$188$	0	0
$189$	10.3707	0.754357
$190$	0	0
$191$	−15.4930	−1.12103	−0.560516	−	0.828143i	$-0.689397\pi$
$191$	−15.4930	−1.12103	−0.560516	+	0.828143i	$0.689397\pi$
$192$	0	0
$193$	−11.2077	−0.806747	−0.403374	−	0.915035i	$-0.632163\pi$
$193$	−11.2077	−0.806747	−0.403374	+	0.915035i	$0.632163\pi$
$194$	0	0
$195$	−0.0933940	−0.00668808
$196$	0	0
$197$	−0.265499	−0.0189160	−0.00945801	−	0.999955i	$-0.503011\pi$
$197$	−0.265499	−0.0189160	−0.00945801	+	0.999955i	$0.503011\pi$
$198$	0	0
$199$	−26.7943	−1.89940	−0.949700	−	0.313162i	$-0.898612\pi$
$199$	−26.7943	−1.89940	−0.949700	+	0.313162i	$0.898612\pi$
$200$	0	0
$201$	2.78596	0.196507
$202$	0	0
$203$	−25.1386	−1.76438
$204$	0	0
$205$	10.8622	0.758646
$206$	0	0
$207$	−8.06849	−0.560798
$208$	0	0
$209$	0.907880	0.0627993
$210$	0	0
$211$	7.90000	0.543859	0.271929	−	0.962317i	$-0.412338\pi$
$211$	7.90000	0.543859	0.271929	+	0.962317i	$0.412338\pi$
$212$	0	0
$213$	2.65606	0.181990
$214$	0	0
$215$	−13.5314	−0.922830
$216$	0	0
$217$	−2.56636	−0.174216
$218$	0	0
$219$	−6.30595	−0.426117
$220$	0	0
$221$	−0.0710091	−0.00477659
$222$	0	0
$223$	1.54628	0.103546	0.0517732	−	0.998659i	$-0.483513\pi$
$223$	1.54628	0.103546	0.0517732	+	0.998659i	$0.483513\pi$
$224$	0	0
$225$	−1.28064	−0.0853759
$226$	0	0
$227$	−14.0213	−0.930628	−0.465314	−	0.885146i	$-0.654059\pi$
$227$	−14.0213	−0.930628	−0.465314	+	0.885146i	$0.654059\pi$
$228$	0	0
$229$	−23.0153	−1.52090	−0.760448	−	0.649399i	$-0.775020\pi$
$229$	−23.0153	−1.52090	−0.760448	+	0.649399i	$0.775020\pi$
$230$	0	0
$231$	−0.213856	−0.0140707
$232$	0	0
$233$	0.251787	0.0164951	0.00824757	−	0.999966i	$-0.497375\pi$
$233$	0.251787	0.0164951	0.00824757	+	0.999966i	$0.497375\pi$
$234$	0	0
$235$	6.83588	0.445923
$236$	0	0
$237$	−3.89985	−0.253323
$238$	0	0
$239$	0.594481	0.0384538	0.0192269	−	0.999815i	$-0.493880\pi$
$239$	0.594481	0.0384538	0.0192269	+	0.999815i	$0.493880\pi$
$240$	0	0
$241$	27.3229	1.76002	0.880011	−	0.474954i	$-0.157535\pi$
$241$	27.3229	1.76002	0.880011	+	0.474954i	$0.157535\pi$
$242$	0	0
$243$	13.0999	0.840357
$244$	0	0
$245$	−8.28375	−0.529229
$246$	0	0
$247$	0.549998	0.0349956
$248$	0	0
$249$	5.78095	0.366353
$250$	0	0
$251$	−13.7857	−0.870145	−0.435073	−	0.900395i	$-0.643277\pi$
$251$	−13.7857	−0.870145	−0.435073	+	0.900395i	$0.643277\pi$
$252$	0	0
$253$	0.352341	0.0221515
$254$	0	0
$255$	1.31524	0.0823635
$256$	0	0
$257$	19.5944	1.22226	0.611132	−	0.791529i	$-0.290714\pi$
$257$	19.5944	1.22226	0.611132	+	0.791529i	$0.290714\pi$
$258$	0	0
$259$	−17.0772	−1.06113
$260$	0	0
$261$	−20.7844	−1.28652
$262$	0	0
$263$	−16.0703	−0.990939	−0.495470	−	0.868625i	$-0.665004\pi$
$263$	−16.0703	−0.990939	−0.495470	+	0.868625i	$0.665004\pi$
$264$	0	0
$265$	11.6665	0.716668
$266$	0	0
$267$	−8.46715	−0.518181
$268$	0	0
$269$	−19.6317	−1.19697	−0.598483	−	0.801135i	$-0.704230\pi$
$269$	−19.6317	−1.19697	−0.598483	+	0.801135i	$0.704230\pi$
$270$	0	0
$271$	−10.9271	−0.663771	−0.331886	−	0.943320i	$-0.607685\pi$
$271$	−10.9271	−0.663771	−0.331886	+	0.943320i	$0.607685\pi$
$272$	0	0
$273$	−0.129555	−0.00784104
$274$	0	0
$275$	0.0559240	0.00337234
$276$	0	0
$277$	31.0410	1.86507	0.932536	−	0.361077i	$-0.117591\pi$
$277$	31.0410	1.86507	0.932536	+	0.361077i	$0.117591\pi$
$278$	0	0
$279$	−2.12185	−0.127032
$280$	0	0
$281$	−11.5841	−0.691051	−0.345526	−	0.938409i	$-0.612299\pi$
$281$	−11.5841	−0.691051	−0.345526	+	0.938409i	$0.612299\pi$
$282$	0	0
$283$	12.9460	0.769558	0.384779	−	0.923009i	$-0.374278\pi$
$283$	12.9460	0.769558	0.384779	+	0.923009i	$0.374278\pi$
$284$	0	0
$285$	−10.1871	−0.603434
$286$	0	0
$287$	15.0679	0.889428
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0.805083	0.0471948
$292$	0	0
$293$	−6.28488	−0.367167	−0.183583	−	0.983004i	$-0.558770\pi$
$293$	−6.28488	−0.367167	−0.183583	+	0.983004i	$0.558770\pi$
$294$	0	0
$295$	2.34032	0.136259
$296$	0	0
$297$	−0.374435	−0.0217269
$298$	0	0
$299$	0.213450	0.0123441
$300$	0	0
$301$	−18.7706	−1.08192
$302$	0	0
$303$	1.45275	0.0834584
$304$	0	0
$305$	−7.94426	−0.454887
$306$	0	0
$307$	27.0456	1.54357	0.771787	−	0.635881i	$-0.219363\pi$
$307$	27.0456	1.54357	0.771787	+	0.635881i	$0.219363\pi$
$308$	0	0
$309$	−8.28277	−0.471191
$310$	0	0
$311$	−30.2147	−1.71332	−0.856660	−	0.515882i	$-0.827464\pi$
$311$	−30.2147	−1.71332	−0.856660	+	0.515882i	$0.827464\pi$
$312$	0	0
$313$	−18.4813	−1.04462	−0.522312	−	0.852755i	$-0.674930\pi$
$313$	−18.4813	−1.04462	−0.522312	+	0.852755i	$0.674930\pi$
$314$	0	0
$315$	−20.3937	−1.14906
$316$	0	0
$317$	−29.6956	−1.66787	−0.833935	−	0.551863i	$-0.813917\pi$
$317$	−29.6956	−1.66787	−0.833935	+	0.551863i	$0.813917\pi$
$318$	0	0
$319$	0.907632	0.0508176
$320$	0	0
$321$	2.77451	0.154858
$322$	0	0
$323$	−7.74546	−0.430969
$324$	0	0
$325$	0.0338791	0.00187927
$326$	0	0
$327$	−6.24963	−0.345605
$328$	0	0
$329$	9.48266	0.522796
$330$	0	0
$331$	12.1582	0.668276	0.334138	−	0.942524i	$-0.391555\pi$
$331$	12.1582	0.668276	0.334138	+	0.942524i	$0.391555\pi$
$332$	0	0
$333$	−14.1193	−0.773735
$334$	0	0
$335$	−11.6017	−0.633869
$336$	0	0
$337$	−14.5042	−0.790094	−0.395047	−	0.918661i	$-0.629272\pi$
$337$	−14.5042	−0.790094	−0.395047	+	0.918661i	$0.629272\pi$
$338$	0	0
$339$	10.2286	0.555542
$340$	0	0
$341$	0.0926589	0.00501776
$342$	0	0
$343$	11.2342	0.606588
$344$	0	0
$345$	−3.95355	−0.212852
$346$	0	0
$347$	−6.81828	−0.366025	−0.183012	−	0.983111i	$-0.558585\pi$
$347$	−6.81828	−0.366025	−0.183012	+	0.983111i	$0.558585\pi$
$348$	0	0
$349$	−18.6342	−0.997466	−0.498733	−	0.866756i	$-0.666201\pi$
$349$	−18.6342	−0.997466	−0.498733	+	0.866756i	$0.666201\pi$
$350$	0	0
$351$	−0.226835	−0.0121076
$352$	0	0
$353$	−9.44366	−0.502635	−0.251318	−	0.967905i	$-0.580864\pi$
$353$	−9.44366	−0.502635	−0.251318	+	0.967905i	$0.580864\pi$
$354$	0	0
$355$	−11.0608	−0.587044
$356$	0	0
$357$	1.82449	0.0965621
$358$	0	0
$359$	−9.72585	−0.513311	−0.256655	−	0.966503i	$-0.582621\pi$
$359$	−9.72585	−0.513311	−0.256655	+	0.966503i	$0.582621\pi$
$360$	0	0
$361$	40.9922	2.15748
$362$	0	0
$363$	−6.17418	−0.324060
$364$	0	0
$365$	26.2601	1.37452
$366$	0	0
$367$	5.40684	0.282235	0.141117	−	0.989993i	$-0.454931\pi$
$367$	5.40684	0.282235	0.141117	+	0.989993i	$0.454931\pi$
$368$	0	0
$369$	12.4580	0.648540
$370$	0	0
$371$	16.1837	0.840214
$372$	0	0
$373$	−20.6921	−1.07140	−0.535699	−	0.844409i	$-0.679952\pi$
$373$	−20.6921	−1.07140	−0.535699	+	0.844409i	$0.679952\pi$
$374$	0	0
$375$	5.94869	0.307189
$376$	0	0
$377$	0.549848	0.0283186
$378$	0	0
$379$	−21.6116	−1.11011	−0.555056	−	0.831813i	$-0.687303\pi$
$379$	−21.6116	−1.11011	−0.555056	+	0.831813i	$0.687303\pi$
$380$	0	0
$381$	−4.47521	−0.229272
$382$	0	0
$383$	10.8765	0.555762	0.277881	−	0.960615i	$-0.410368\pi$
$383$	10.8765	0.555762	0.277881	+	0.960615i	$0.410368\pi$
$384$	0	0
$385$	0.890570	0.0453877
$386$	0	0
$387$	−15.5194	−0.788895
$388$	0	0
$389$	−34.7027	−1.75950	−0.879749	−	0.475439i	$-0.842289\pi$
$389$	−34.7027	−1.75950	−0.879749	+	0.475439i	$0.842289\pi$
$390$	0	0
$391$	−3.00596	−0.152018
$392$	0	0
$393$	0.312226	0.0157497
$394$	0	0
$395$	16.2403	0.817140
$396$	0	0
$397$	16.2346	0.814790	0.407395	−	0.913252i	$-0.366437\pi$
$397$	16.2346	0.814790	0.407395	+	0.913252i	$0.366437\pi$
$398$	0	0
$399$	−14.1315	−0.707460
$400$	0	0
$401$	35.8549	1.79051	0.895255	−	0.445554i	$-0.146993\pi$
$401$	35.8549	1.79051	0.895255	+	0.445554i	$0.146993\pi$
$402$	0	0
$403$	0.0561332	0.00279620
$404$	0	0
$405$	−14.6440	−0.727665
$406$	0	0
$407$	0.616575	0.0305625
$408$	0	0
$409$	−26.4945	−1.31007	−0.655035	−	0.755599i	$-0.727346\pi$
$409$	−26.4945	−1.31007	−0.655035	+	0.755599i	$0.727346\pi$
$410$	0	0
$411$	5.66160	0.279266
$412$	0	0
$413$	3.24647	0.159748
$414$	0	0
$415$	−24.0739	−1.18174
$416$	0	0
$417$	−4.32215	−0.211657
$418$	0	0
$419$	32.8109	1.60292	0.801459	−	0.598049i	$-0.204057\pi$
$419$	32.8109	1.60292	0.801459	+	0.598049i	$0.204057\pi$
$420$	0	0
$421$	2.51760	0.122700	0.0613502	−	0.998116i	$-0.480459\pi$
$421$	2.51760	0.122700	0.0613502	+	0.998116i	$0.480459\pi$
$422$	0	0
$423$	7.84022	0.381204
$424$	0	0
$425$	−0.477109	−0.0231432
$426$	0	0
$427$	−11.0202	−0.533305
$428$	0	0
$429$	0.00467761	0.000225837 0
$430$	0	0
$431$	−9.22755	−0.444476	−0.222238	−	0.974992i	$-0.571336\pi$
$431$	−9.22755	−0.444476	−0.222238	+	0.974992i	$0.571336\pi$
$432$	0	0
$433$	6.40588	0.307847	0.153924	−	0.988083i	$-0.450809\pi$
$433$	6.40588	0.307847	0.153924	+	0.988083i	$0.450809\pi$
$434$	0	0
$435$	−10.1844	−0.488303
$436$	0	0
$437$	23.2825	1.11375
$438$	0	0
$439$	−7.39755	−0.353066	−0.176533	−	0.984295i	$-0.556488\pi$
$439$	−7.39755	−0.353066	−0.176533	+	0.984295i	$0.556488\pi$
$440$	0	0
$441$	−9.50081	−0.452420
$442$	0	0
$443$	31.8225	1.51193	0.755965	−	0.654612i	$-0.227168\pi$
$443$	31.8225	1.51193	0.755965	+	0.654612i	$0.227168\pi$
$444$	0	0
$445$	35.2601	1.67149
$446$	0	0
$447$	0.379717	0.0179600
$448$	0	0
$449$	16.0309	0.756544	0.378272	−	0.925694i	$-0.376518\pi$
$449$	16.0309	0.756544	0.378272	+	0.925694i	$0.376518\pi$
$450$	0	0
$451$	−0.544028	−0.0256173
$452$	0	0
$453$	5.99125	0.281493
$454$	0	0
$455$	0.539512	0.0252927
$456$	0	0
$457$	16.4067	0.767472	0.383736	−	0.923443i	$-0.374637\pi$
$457$	16.4067	0.767472	0.383736	+	0.923443i	$0.374637\pi$
$458$	0	0
$459$	3.19445	0.149104
$460$	0	0
$461$	22.2102	1.03443	0.517217	−	0.855854i	$-0.326968\pi$
$461$	22.2102	1.03443	0.517217	+	0.855854i	$0.326968\pi$
$462$	0	0
$463$	−36.4573	−1.69431	−0.847157	−	0.531343i	$-0.821688\pi$
$463$	−36.4573	−1.69431	−0.847157	+	0.531343i	$0.821688\pi$
$464$	0	0
$465$	−1.03971	−0.0482153
$466$	0	0
$467$	−13.6909	−0.633540	−0.316770	−	0.948502i	$-0.602598\pi$
$467$	−13.6909	−0.633540	−0.316770	+	0.948502i	$0.602598\pi$
$468$	0	0
$469$	−16.0938	−0.743141
$470$	0	0
$471$	−10.3133	−0.475214
$472$	0	0
$473$	0.677714	0.0311613
$474$	0	0
$475$	3.69543	0.169558
$476$	0	0
$477$	13.3806	0.612655
$478$	0	0
$479$	−16.3360	−0.746410	−0.373205	−	0.927749i	$-0.621741\pi$
$479$	−16.3360	−0.746410	−0.373205	+	0.927749i	$0.621741\pi$
$480$	0	0
$481$	0.373525	0.0170313
$482$	0	0
$483$	−5.48433	−0.249546
$484$	0	0
$485$	−3.35264	−0.152236
$486$	0	0
$487$	22.8668	1.03619	0.518096	−	0.855322i	$-0.326641\pi$
$487$	22.8668	1.03619	0.518096	+	0.855322i	$0.326641\pi$
$488$	0	0
$489$	−0.295970	−0.0133842
$490$	0	0
$491$	−4.25023	−0.191810	−0.0959051	−	0.995390i	$-0.530575\pi$
$491$	−4.25023	−0.191810	−0.0959051	+	0.995390i	$0.530575\pi$
$492$	0	0
$493$	−7.74335	−0.348743
$494$	0	0
$495$	0.736319	0.0330951
$496$	0	0
$497$	−15.3434	−0.688244
$498$	0	0
$499$	−21.2155	−0.949736	−0.474868	−	0.880057i	$-0.657504\pi$
$499$	−21.2155	−0.949736	−0.474868	+	0.880057i	$0.657504\pi$
$500$	0	0
$501$	−13.3688	−0.597273
$502$	0	0
$503$	−14.1322	−0.630123	−0.315061	−	0.949071i	$-0.602025\pi$
$503$	−14.1322	−0.630123	−0.315061	+	0.949071i	$0.602025\pi$
$504$	0	0
$505$	−6.04976	−0.269211
$506$	0	0
$507$	−7.30305	−0.324340
$508$	0	0
$509$	12.0552	0.534337	0.267169	−	0.963650i	$-0.413912\pi$
$509$	12.0552	0.534337	0.267169	+	0.963650i	$0.413912\pi$
$510$	0	0
$511$	36.4278	1.61147
$512$	0	0
$513$	−24.7425	−1.09241
$514$	0	0
$515$	34.4923	1.51991
$516$	0	0
$517$	−0.342373	−0.0150575
$518$	0	0
$519$	2.24646	0.0986085
$520$	0	0
$521$	3.95491	0.173268	0.0866338	−	0.996240i	$-0.472389\pi$
$521$	3.95491	0.173268	0.0866338	+	0.996240i	$0.472389\pi$
$522$	0	0
$523$	26.7347	1.16903	0.584514	−	0.811384i	$-0.301285\pi$
$523$	26.7347	1.16903	0.584514	+	0.811384i	$0.301285\pi$
$524$	0	0
$525$	−0.870479	−0.0379908
$526$	0	0
$527$	−0.790508	−0.0344351
$528$	0	0
$529$	−13.9642	−0.607140
$530$	0	0
$531$	2.68417	0.116483
$532$	0	0
$533$	−0.329575	−0.0142755
$534$	0	0
$535$	−11.5540	−0.499523
$536$	0	0
$537$	10.7412	0.463519
$538$	0	0
$539$	0.414889	0.0178706
$540$	0	0
$541$	28.7459	1.23588	0.617942	−	0.786224i	$-0.287967\pi$
$541$	28.7459	1.23588	0.617942	+	0.786224i	$0.287967\pi$
$542$	0	0
$543$	10.0639	0.431882
$544$	0	0
$545$	26.0256	1.11481
$546$	0	0
$547$	27.4103	1.17198	0.585990	−	0.810319i	$-0.300706\pi$
$547$	27.4103	1.17198	0.585990	+	0.810319i	$0.300706\pi$
$548$	0	0
$549$	−9.11144	−0.388867
$550$	0	0
$551$	59.9758	2.55505
$552$	0	0
$553$	22.5284	0.958006
$554$	0	0
$555$	−6.91847	−0.293673
$556$	0	0
$557$	8.56909	0.363084	0.181542	−	0.983383i	$-0.441891\pi$
$557$	8.56909	0.363084	0.181542	+	0.983383i	$0.441891\pi$
$558$	0	0
$559$	0.410563	0.0173650
$560$	0	0
$561$	−0.0658734	−0.00278118
$562$	0	0
$563$	−25.0077	−1.05395	−0.526974	−	0.849881i	$-0.676673\pi$
$563$	−25.0077	−1.05395	−0.526974	+	0.849881i	$0.676673\pi$
$564$	0	0
$565$	−42.5955	−1.79201
$566$	0	0
$567$	−20.3140	−0.853107
$568$	0	0
$569$	−19.6405	−0.823373	−0.411687	−	0.911325i	$-0.635060\pi$
$569$	−19.6405	−0.823373	−0.411687	+	0.911325i	$0.635060\pi$
$570$	0	0
$571$	−27.2253	−1.13934	−0.569672	−	0.821872i	$-0.692930\pi$
$571$	−27.2253	−1.13934	−0.569672	+	0.821872i	$0.692930\pi$
$572$	0	0
$573$	−8.70691	−0.363737
$574$	0	0
$575$	1.43417	0.0598089
$576$	0	0
$577$	18.1573	0.755900	0.377950	−	0.925826i	$-0.376629\pi$
$577$	18.1573	0.755900	0.377950	+	0.925826i	$0.376629\pi$
$578$	0	0
$579$	−6.29862	−0.261762
$580$	0	0
$581$	−33.3950	−1.38546
$582$	0	0
$583$	−0.584314	−0.0241998
$584$	0	0
$585$	0.446066	0.0184426
$586$	0	0
$587$	−5.86489	−0.242070	−0.121035	−	0.992648i	$-0.538621\pi$
$587$	−5.86489	−0.242070	−0.121035	+	0.992648i	$0.538621\pi$
$588$	0	0
$589$	6.12285	0.252288
$590$	0	0
$591$	−0.149208	−0.00613760
$592$	0	0
$593$	10.6228	0.436225	0.218113	−	0.975924i	$-0.430010\pi$
$593$	10.6228	0.436225	0.218113	+	0.975924i	$0.430010\pi$
$594$	0	0
$595$	−7.59779	−0.311479
$596$	0	0
$597$	−15.0582	−0.616290
$598$	0	0
$599$	−26.8939	−1.09885	−0.549427	−	0.835541i	$-0.685154\pi$
$599$	−26.8939	−1.09885	−0.549427	+	0.835541i	$0.685154\pi$
$600$	0	0
$601$	−4.06315	−0.165739	−0.0828697	−	0.996560i	$-0.526409\pi$
$601$	−4.06315	−0.165739	−0.0828697	+	0.996560i	$0.526409\pi$
$602$	0	0
$603$	−13.3062	−0.541872
$604$	0	0
$605$	25.7114	1.04532
$606$	0	0
$607$	15.3846	0.624441	0.312221	−	0.950010i	$-0.398927\pi$
$607$	15.3846	0.624441	0.312221	+	0.950010i	$0.398927\pi$
$608$	0	0
$609$	−14.1276	−0.572481
$610$	0	0
$611$	−0.207411	−0.00839097
$612$	0	0
$613$	40.0791	1.61878	0.809391	−	0.587271i	$-0.199798\pi$
$613$	40.0791	1.61878	0.809391	+	0.587271i	$0.199798\pi$
$614$	0	0
$615$	6.10443	0.246154
$616$	0	0
$617$	21.6832	0.872932	0.436466	−	0.899721i	$-0.356230\pi$
$617$	21.6832	0.872932	0.436466	+	0.899721i	$0.356230\pi$
$618$	0	0
$619$	−29.5712	−1.18857	−0.594283	−	0.804256i	$-0.702564\pi$
$619$	−29.5712	−1.18857	−0.594283	+	0.804256i	$0.702564\pi$
$620$	0	0
$621$	−9.60237	−0.385330
$622$	0	0
$623$	48.9125	1.95964
$624$	0	0
$625$	−27.1579	−1.08632
$626$	0	0
$627$	0.510220	0.0203762
$628$	0	0
$629$	−5.26024	−0.209739
$630$	0	0
$631$	−45.2129	−1.79990	−0.899948	−	0.435998i	$-0.856396\pi$
$631$	−45.2129	−1.79990	−0.899948	+	0.435998i	$0.856396\pi$
$632$	0	0
$633$	4.43973	0.176463
$634$	0	0
$635$	18.6363	0.739559
$636$	0	0
$637$	0.251342	0.00995854
$638$	0	0
$639$	−12.6858	−0.501843
$640$	0	0
$641$	12.7187	0.502360	0.251180	−	0.967940i	$-0.419181\pi$
$641$	12.7187	0.502360	0.251180	+	0.967940i	$0.419181\pi$
$642$	0	0
$643$	−18.2174	−0.718423	−0.359212	−	0.933256i	$-0.616954\pi$
$643$	−18.2174	−0.718423	−0.359212	+	0.933256i	$0.616954\pi$
$644$	0	0
$645$	−7.60449	−0.299427
$646$	0	0
$647$	−36.2887	−1.42666	−0.713329	−	0.700830i	$-0.752813\pi$
$647$	−36.2887	−1.42666	−0.713329	+	0.700830i	$0.752813\pi$
$648$	0	0
$649$	−0.117214	−0.00460107
$650$	0	0
$651$	−1.44227	−0.0565271
$652$	0	0
$653$	39.7221	1.55445	0.777224	−	0.629224i	$-0.216627\pi$
$653$	39.7221	1.55445	0.777224	+	0.629224i	$0.216627\pi$
$654$	0	0
$655$	−1.30022	−0.0508037
$656$	0	0
$657$	30.1183	1.17503
$658$	0	0
$659$	28.3533	1.10449	0.552244	−	0.833682i	$-0.313772\pi$
$659$	28.3533	1.10449	0.552244	+	0.833682i	$0.313772\pi$
$660$	0	0
$661$	−5.05785	−0.196727	−0.0983637	−	0.995151i	$-0.531361\pi$
$661$	−5.05785	−0.196727	−0.0983637	+	0.995151i	$0.531361\pi$
$662$	0	0
$663$	−0.0399065	−0.00154984
$664$	0	0
$665$	58.8484	2.28204
$666$	0	0
$667$	23.2762	0.901256
$668$	0	0
$669$	0.868993	0.0335972
$670$	0	0
$671$	0.397886	0.0153602
$672$	0	0
$673$	−25.7206	−0.991456	−0.495728	−	0.868478i	$-0.665099\pi$
$673$	−25.7206	−0.991456	−0.495728	+	0.868478i	$0.665099\pi$
$674$	0	0
$675$	−1.52410	−0.0586626
$676$	0	0
$677$	18.6800	0.717932	0.358966	−	0.933351i	$-0.383129\pi$
$677$	18.6800	0.717932	0.358966	+	0.933351i	$0.383129\pi$
$678$	0	0
$679$	−4.65075	−0.178479
$680$	0	0
$681$	−7.87986	−0.301957
$682$	0	0
$683$	−1.59557	−0.0610527	−0.0305263	−	0.999534i	$-0.509718\pi$
$683$	−1.59557	−0.0610527	−0.0305263	+	0.999534i	$0.509718\pi$
$684$	0	0
$685$	−23.5768	−0.900825
$686$	0	0
$687$	−12.9344	−0.493478
$688$	0	0
$689$	−0.353981	−0.0134856
$690$	0	0
$691$	−25.4625	−0.968639	−0.484319	−	0.874891i	$-0.660933\pi$
$691$	−25.4625	−0.968639	−0.484319	+	0.874891i	$0.660933\pi$
$692$	0	0
$693$	1.02141	0.0388003
$694$	0	0
$695$	17.9989	0.682738
$696$	0	0
$697$	4.64131	0.175802
$698$	0	0
$699$	0.141502	0.00535210
$700$	0	0
$701$	−46.7845	−1.76702	−0.883512	−	0.468408i	$-0.844828\pi$
$701$	−46.7845	−1.76702	−0.883512	+	0.468408i	$0.844828\pi$
$702$	0	0
$703$	40.7430	1.53665
$704$	0	0
$705$	3.84170	0.144687
$706$	0	0
$707$	−8.39216	−0.315620
$708$	0	0
$709$	−3.43353	−0.128949	−0.0644745	−	0.997919i	$-0.520537\pi$
$709$	−3.43353	−0.128949	−0.0644745	+	0.997919i	$0.520537\pi$
$710$	0	0
$711$	18.6264	0.698544
$712$	0	0
$713$	2.37623	0.0889906
$714$	0	0
$715$	−0.0194792	−0.000728480 0
$716$	0	0
$717$	0.334093	0.0124769
$718$	0	0
$719$	−4.30519	−0.160556	−0.0802782	−	0.996772i	$-0.525581\pi$
$719$	−4.30519	−0.160556	−0.0802782	+	0.996772i	$0.525581\pi$
$720$	0	0
$721$	47.8474	1.78193
$722$	0	0
$723$	15.3552	0.571066
$724$	0	0
$725$	3.69442	0.137207
$726$	0	0
$727$	−35.1387	−1.30322	−0.651611	−	0.758553i	$-0.725906\pi$
$727$	−35.1387	−1.30322	−0.651611	+	0.758553i	$0.725906\pi$
$728$	0	0
$729$	−11.4097	−0.422583
$730$	0	0
$731$	−5.78183	−0.213849
$732$	0	0
$733$	13.1437	0.485473	0.242736	−	0.970092i	$-0.421955\pi$
$733$	13.1437	0.485473	0.242736	+	0.970092i	$0.421955\pi$
$734$	0	0
$735$	−4.65539	−0.171717
$736$	0	0
$737$	0.581068	0.0214039
$738$	0	0
$739$	−7.70317	−0.283366	−0.141683	−	0.989912i	$-0.545251\pi$
$739$	−7.70317	−0.283366	−0.141683	+	0.989912i	$0.545251\pi$
$740$	0	0
$741$	0.309094	0.0113549
$742$	0	0
$743$	12.0613	0.442487	0.221244	−	0.975219i	$-0.428988\pi$
$743$	12.0613	0.442487	0.221244	+	0.975219i	$0.428988\pi$
$744$	0	0
$745$	−1.58127	−0.0579333
$746$	0	0
$747$	−27.6108	−1.01023
$748$	0	0
$749$	−16.0276	−0.585636
$750$	0	0
$751$	−0.932935	−0.0340433	−0.0170216	−	0.999855i	$-0.505418\pi$
$751$	−0.932935	−0.0340433	−0.0170216	+	0.999855i	$0.505418\pi$
$752$	0	0
$753$	−7.74743	−0.282332
$754$	0	0
$755$	−24.9496	−0.908009
$756$	0	0
$757$	−53.0408	−1.92780	−0.963900	−	0.266264i	$-0.914211\pi$
$757$	−53.0408	−1.92780	−0.963900	+	0.266264i	$0.914211\pi$
$758$	0	0
$759$	0.198013	0.00718740
$760$	0	0
$761$	3.05757	0.110837	0.0554184	−	0.998463i	$-0.482351\pi$
$761$	3.05757	0.110837	0.0554184	+	0.998463i	$0.482351\pi$
$762$	0	0
$763$	36.1025	1.30700
$764$	0	0
$765$	−6.28181	−0.227119
$766$	0	0
$767$	−0.0710091	−0.00256399
$768$	0	0
$769$	53.7907	1.93974	0.969870	−	0.243622i	$-0.0783355\pi$
$769$	53.7907	1.93974	0.969870	+	0.243622i	$0.0783355\pi$
$770$	0	0
$771$	11.0119	0.396583
$772$	0	0
$773$	16.8997	0.607838	0.303919	−	0.952698i	$-0.401705\pi$
$773$	16.8997	0.607838	0.303919	+	0.952698i	$0.401705\pi$
$774$	0	0
$775$	0.377158	0.0135479
$776$	0	0
$777$	−9.59723	−0.344299
$778$	0	0
$779$	−35.9491	−1.28801
$780$	0	0
$781$	0.553975	0.0198228
$782$	0	0
$783$	−24.7357	−0.883983
$784$	0	0
$785$	42.9483	1.53289
$786$	0	0
$787$	−34.6168	−1.23395	−0.616977	−	0.786981i	$-0.711643\pi$
$787$	−34.6168	−1.23395	−0.616977	+	0.786981i	$0.711643\pi$
$788$	0	0
$789$	−9.03138	−0.321526
$790$	0	0
$791$	−59.0880	−2.10093
$792$	0	0
$793$	0.241042	0.00855964
$794$	0	0
$795$	6.55648	0.232534
$796$	0	0
$797$	43.4475	1.53899	0.769494	−	0.638654i	$-0.220508\pi$
$797$	43.4475	1.53899	0.769494	+	0.638654i	$0.220508\pi$
$798$	0	0
$799$	2.92091	0.103334
$800$	0	0
$801$	40.4406	1.42890
$802$	0	0
$803$	−1.31523	−0.0464135
$804$	0	0
$805$	22.8386	0.804956
$806$	0	0
$807$	−11.0328	−0.388374
$808$	0	0
$809$	27.7921	0.977118	0.488559	−	0.872531i	$-0.337523\pi$
$809$	27.7921	0.977118	0.488559	+	0.872531i	$0.337523\pi$
$810$	0	0
$811$	−36.2383	−1.27250	−0.636249	−	0.771484i	$-0.719515\pi$
$811$	−36.2383	−1.27250	−0.636249	+	0.771484i	$0.719515\pi$
$812$	0	0
$813$	−6.14091	−0.215371
$814$	0	0
$815$	1.23252	0.0431733
$816$	0	0
$817$	44.7830	1.56676
$818$	0	0
$819$	0.618778	0.0216219
$820$	0	0
$821$	0.953989	0.0332945	0.0166472	−	0.999861i	$-0.494701\pi$
$821$	0.953989	0.0332945	0.0166472	+	0.999861i	$0.494701\pi$
$822$	0	0
$823$	51.7352	1.80338	0.901689	−	0.432386i	$-0.142328\pi$
$823$	51.7352	1.80338	0.901689	+	0.432386i	$0.142328\pi$
$824$	0	0
$825$	0.0314288	0.00109421
$826$	0	0
$827$	−15.3047	−0.532198	−0.266099	−	0.963946i	$-0.585735\pi$
$827$	−15.3047	−0.532198	−0.266099	+	0.963946i	$0.585735\pi$
$828$	0	0
$829$	−39.2601	−1.36356	−0.681779	−	0.731558i	$-0.738794\pi$
$829$	−39.2601	−1.36356	−0.681779	+	0.731558i	$0.738794\pi$
$830$	0	0
$831$	17.4447	0.605152
$832$	0	0
$833$	−3.53958	−0.122639
$834$	0	0
$835$	55.6722	1.92661
$836$	0	0
$837$	−2.52524	−0.0872849
$838$	0	0
$839$	−50.3490	−1.73824	−0.869119	−	0.494602i	$-0.835314\pi$
$839$	−50.3490	−1.73824	−0.869119	+	0.494602i	$0.835314\pi$
$840$	0	0
$841$	30.9594	1.06757
$842$	0	0
$843$	−6.51017	−0.224222
$844$	0	0
$845$	30.4124	1.04622
$846$	0	0
$847$	35.6666	1.22552
$848$	0	0
$849$	7.27552	0.249695
$850$	0	0
$851$	15.8120	0.542030
$852$	0	0
$853$	−13.5918	−0.465374	−0.232687	−	0.972552i	$-0.574752\pi$
$853$	−13.5918	−0.465374	−0.232687	+	0.972552i	$0.574752\pi$
$854$	0	0
$855$	48.6556	1.66398
$856$	0	0
$857$	26.9831	0.921727	0.460863	−	0.887471i	$-0.347540\pi$
$857$	26.9831	0.921727	0.460863	+	0.887471i	$0.347540\pi$
$858$	0	0
$859$	19.3515	0.660266	0.330133	−	0.943934i	$-0.392906\pi$
$859$	19.3515	0.660266	0.330133	+	0.943934i	$0.392906\pi$
$860$	0	0
$861$	8.46801	0.288589
$862$	0	0
$863$	−13.9246	−0.474000	−0.237000	−	0.971510i	$-0.576164\pi$
$863$	−13.9246	−0.474000	−0.237000	+	0.971510i	$0.576164\pi$
$864$	0	0
$865$	−9.35501	−0.318080
$866$	0	0
$867$	0.561991	0.0190862
$868$	0	0
$869$	−0.813392	−0.0275925
$870$	0	0
$871$	0.352014	0.0119275
$872$	0	0
$873$	−3.84522	−0.130141
$874$	0	0
$875$	−34.3640	−1.16171
$876$	0	0
$877$	53.1789	1.79572	0.897861	−	0.440279i	$-0.145120\pi$
$877$	53.1789	1.79572	0.897861	+	0.440279i	$0.145120\pi$
$878$	0	0
$879$	−3.53204	−0.119133
$880$	0	0
$881$	18.4648	0.622095	0.311047	−	0.950394i	$-0.399320\pi$
$881$	18.4648	0.622095	0.311047	+	0.950394i	$0.399320\pi$
$882$	0	0
$883$	−6.12038	−0.205967	−0.102984	−	0.994683i	$-0.532839\pi$
$883$	−6.12038	−0.205967	−0.102984	+	0.994683i	$0.532839\pi$
$884$	0	0
$885$	1.31524	0.0442113
$886$	0	0
$887$	−22.9700	−0.771258	−0.385629	−	0.922654i	$-0.626016\pi$
$887$	−22.9700	−0.771258	−0.385629	+	0.922654i	$0.626016\pi$
$888$	0	0
$889$	25.8521	0.867052
$890$	0	0
$891$	0.733439	0.0245711
$892$	0	0
$893$	−22.6238	−0.757077
$894$	0	0
$895$	−44.7302	−1.49517
$896$	0	0
$897$	0.119957	0.00400525
$898$	0	0
$899$	6.12117	0.204153
$900$	0	0
$901$	4.98500	0.166075
$902$	0	0
$903$	−10.5489	−0.351045
$904$	0	0
$905$	−41.9094	−1.39312
$906$	0	0
$907$	−34.5731	−1.14798	−0.573990	−	0.818862i	$-0.694605\pi$
$907$	−34.5731	−1.14798	−0.573990	+	0.818862i	$0.694605\pi$
$908$	0	0
$909$	−6.93860	−0.230139
$910$	0	0
$911$	16.7005	0.553312	0.276656	−	0.960969i	$-0.410774\pi$
$911$	16.7005	0.553312	0.276656	+	0.960969i	$0.410774\pi$
$912$	0	0
$913$	1.20573	0.0399039
$914$	0	0
$915$	−4.46460	−0.147595
$916$	0	0
$917$	−1.80365	−0.0595617
$918$	0	0
$919$	−19.5469	−0.644793	−0.322397	−	0.946605i	$-0.604488\pi$
$919$	−19.5469	−0.644793	−0.322397	+	0.946605i	$0.604488\pi$
$920$	0	0
$921$	15.1994	0.500837
$922$	0	0
$923$	0.335601	0.0110464
$924$	0	0
$925$	2.50970	0.0825185
$926$	0	0
$927$	39.5600	1.29932
$928$	0	0
$929$	43.8673	1.43924	0.719620	−	0.694368i	$-0.244316\pi$
$929$	43.8673	1.43924	0.719620	+	0.694368i	$0.244316\pi$
$930$	0	0
$931$	27.4157	0.898512
$932$	0	0
$933$	−16.9804	−0.555913
$934$	0	0
$935$	0.274319	0.00897121
$936$	0	0
$937$	−27.1172	−0.885880	−0.442940	−	0.896551i	$-0.646065\pi$
$937$	−27.1172	−0.885880	−0.442940	+	0.896551i	$0.646065\pi$
$938$	0	0
$939$	−10.3863	−0.338944
$940$	0	0
$941$	8.17539	0.266510	0.133255	−	0.991082i	$-0.457457\pi$
$941$	8.17539	0.266510	0.133255	+	0.991082i	$0.457457\pi$
$942$	0	0
$943$	−13.9516	−0.454325
$944$	0	0
$945$	−24.2708	−0.789528
$946$	0	0
$947$	−0.425925	−0.0138407	−0.00692036	−	0.999976i	$-0.502203\pi$
$947$	−0.425925	−0.0138407	−0.00692036	+	0.999976i	$0.502203\pi$
$948$	0	0
$949$	−0.796774	−0.0258644
$950$	0	0
$951$	−16.6886	−0.541166
$952$	0	0
$953$	3.14977	0.102031	0.0510155	−	0.998698i	$-0.483754\pi$
$953$	3.14977	0.102031	0.0510155	+	0.998698i	$0.483754\pi$
$954$	0	0
$955$	36.2586	1.17330
$956$	0	0
$957$	0.510081	0.0164886
$958$	0	0
$959$	−32.7055	−1.05612
$960$	0	0
$961$	−30.3751	−0.979842
$962$	0	0
$963$	−13.2515	−0.427025
$964$	0	0
$965$	26.2296	0.844361
$966$	0	0
$967$	−27.7630	−0.892798	−0.446399	−	0.894834i	$-0.647294\pi$
$967$	−27.7630	−0.892798	−0.446399	+	0.894834i	$0.647294\pi$
$968$	0	0
$969$	−4.35288	−0.139835
$970$	0	0
$971$	−18.8431	−0.604705	−0.302353	−	0.953196i	$-0.597772\pi$
$971$	−18.8431	−0.604705	−0.302353	+	0.953196i	$0.597772\pi$
$972$	0	0
$973$	24.9679	0.800435
$974$	0	0
$975$	0.0190397	0.000609759 0
$976$	0	0
$977$	−21.6818	−0.693663	−0.346832	−	0.937927i	$-0.612742\pi$
$977$	−21.6818	−0.693663	−0.346832	+	0.937927i	$0.612742\pi$
$978$	0	0
$979$	−1.76599	−0.0564414
$980$	0	0
$981$	29.8493	0.953016
$982$	0	0
$983$	−48.2565	−1.53914	−0.769571	−	0.638561i	$-0.779530\pi$
$983$	−48.2565	−1.53914	−0.769571	+	0.638561i	$0.779530\pi$
$984$	0	0
$985$	0.621353	0.0197980
$986$	0	0
$987$	5.32917	0.169629
$988$	0	0
$989$	17.3799	0.552650
$990$	0	0
$991$	−20.3350	−0.645963	−0.322981	−	0.946405i	$-0.604685\pi$
$991$	−20.3350	−0.645963	−0.322981	+	0.946405i	$0.604685\pi$
$992$	0	0
$993$	6.83281	0.216833
$994$	0	0
$995$	62.7074	1.98796
$996$	0	0
$997$	−13.3323	−0.422237	−0.211119	−	0.977460i	$-0.567711\pi$
$997$	−13.3323	−0.422237	−0.211119	+	0.977460i	$0.567711\pi$
$998$	0	0
$999$	−16.8036	−0.531641

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.y.1.15	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.y.1.15	✓	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)

\(f(q)\)	\(=\)	\(q+0.561991 q^{3} -2.34032 q^{5} -3.24647 q^{7} -2.68417 q^{9} +O(q^{10})\)	\(q+0.561991 q^{3} -2.34032 q^{5} -3.24647 q^{7} -2.68417 q^{9} +0.117214 q^{11} +0.0710091 q^{13} -1.31524 q^{15} -1.00000 q^{17} +7.74546 q^{19} -1.82449 q^{21} +3.00596 q^{23} +0.477109 q^{25} -3.19445 q^{27} +7.74335 q^{29} +0.790508 q^{31} +0.0658734 q^{33} +7.59779 q^{35} +5.26024 q^{37} +0.0399065 q^{39} -4.64131 q^{41} +5.78183 q^{43} +6.28181 q^{45} -2.92091 q^{47} +3.53958 q^{49} -0.561991 q^{51} -4.98500 q^{53} -0.274319 q^{55} +4.35288 q^{57} -1.00000 q^{59} +3.39452 q^{61} +8.71407 q^{63} -0.166184 q^{65} +4.95731 q^{67} +1.68932 q^{69} +4.72617 q^{71} -11.2207 q^{73} +0.268131 q^{75} -0.380533 q^{77} -6.93936 q^{79} +6.25725 q^{81} +10.2866 q^{83} +2.34032 q^{85} +4.35169 q^{87} -15.0663 q^{89} -0.230529 q^{91} +0.444258 q^{93} -18.1269 q^{95} +1.43255 q^{97} -0.314623 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) \) 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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