LMFDB - Embedded newform 2008.2.a.d.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2008,2,Mod(1,2008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2008, 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("2008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2008 = 2^{3} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2008.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:	$16.0339607259$
Analytic rank:	$0$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	2008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.13165 q^{3} +0.257423 q^{5} +0.578263 q^{7} +1.54395 q^{9} +O(q^{10})$	$q-2.13165 q^{3} +0.257423 q^{5} +0.578263 q^{7} +1.54395 q^{9} -1.44278 q^{11} +1.23315 q^{13} -0.548737 q^{15} +3.22998 q^{17} -2.16232 q^{19} -1.23266 q^{21} -3.59035 q^{23} -4.93373 q^{25} +3.10379 q^{27} +1.34015 q^{29} +2.91300 q^{31} +3.07551 q^{33} +0.148858 q^{35} +7.43825 q^{37} -2.62866 q^{39} +9.61132 q^{41} -4.00610 q^{43} +0.397448 q^{45} -7.55848 q^{47} -6.66561 q^{49} -6.88521 q^{51} -5.08800 q^{53} -0.371405 q^{55} +4.60933 q^{57} -4.79877 q^{59} -0.119891 q^{61} +0.892809 q^{63} +0.317442 q^{65} -8.76927 q^{67} +7.65338 q^{69} +14.2487 q^{71} +10.5417 q^{73} +10.5170 q^{75} -0.834306 q^{77} +14.7249 q^{79} -11.2481 q^{81} +9.70080 q^{83} +0.831472 q^{85} -2.85674 q^{87} +15.5478 q^{89} +0.713087 q^{91} -6.20952 q^{93} -0.556632 q^{95} +17.4458 q^{97} -2.22758 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9}+O(q^{10}) $ 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9	$ 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9} + 8 q^{11} + 8 q^{13} + 7 q^{15} + 19 q^{17} - 9 q^{19} + 9 q^{21} + 21 q^{23} + 65 q^{25} + 5 q^{27} + 10 q^{29} - 9 q^{31} + 34 q^{33} + 12 q^{35} + 11 q^{37} - 9 q^{39} + 35 q^{41} - 9 q^{43} + 29 q^{45} + 37 q^{47} + 77 q^{49} - 17 q^{51} + 38 q^{53} - 20 q^{55} + 51 q^{57} + 17 q^{59} + 22 q^{63} + 41 q^{65} + 9 q^{67} + 8 q^{69} + 13 q^{71} + 41 q^{73} + 25 q^{75} + 36 q^{77} - 36 q^{79} + 127 q^{81} + 29 q^{83} + 34 q^{85} + 10 q^{87} + 36 q^{89} - 6 q^{91} + 36 q^{93} + 25 q^{95} + 40 q^{97} + 19 q^{99}+O(q^{100}) $ 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9 + 8 * q^11 + 8 * q^13 + 7 * q^15 + 19 * q^17 - 9 * q^19 + 9 * q^21 + 21 * q^23 + 65 * q^25 + 5 * q^27 + 10 * q^29 - 9 * q^31 + 34 * q^33 + 12 * q^35 + 11 * q^37 - 9 * q^39 + 35 * q^41 - 9 * q^43 + 29 * q^45 + 37 * q^47 + 77 * q^49 - 17 * q^51 + 38 * q^53 - 20 * q^55 + 51 * q^57 + 17 * q^59 + 22 * q^63 + 41 * q^65 + 9 * q^67 + 8 * q^69 + 13 * q^71 + 41 * q^73 + 25 * q^75 + 36 * q^77 - 36 * q^79 + 127 * q^81 + 29 * q^83 + 34 * q^85 + 10 * q^87 + 36 * q^89 - 6 * q^91 + 36 * q^93 + 25 * q^95 + 40 * q^97 + 19 * 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
$3$	−2.13165	−1.23071	−0.615356	−	0.788250i	$-0.710988\pi$
$3$	−2.13165	−1.23071	−0.615356	+	0.788250i	$0.710988\pi$
$4$	0	0
$5$	0.257423	0.115123	0.0575615	−	0.998342i	$-0.481667\pi$
$5$	0.257423	0.115123	0.0575615	+	0.998342i	$0.481667\pi$
$6$	0	0
$7$	0.578263	0.218563	0.109281	−	0.994011i	$-0.465145\pi$
$7$	0.578263	0.218563	0.109281	+	0.994011i	$0.465145\pi$
$8$	0	0
$9$	1.54395	0.514650
$10$	0	0
$11$	−1.44278	−0.435015	−0.217507	−	0.976059i	$-0.569793\pi$
$11$	−1.44278	−0.435015	−0.217507	+	0.976059i	$0.569793\pi$
$12$	0	0
$13$	1.23315	0.342015	0.171008	−	0.985270i	$-0.445298\pi$
$13$	1.23315	0.342015	0.171008	+	0.985270i	$0.445298\pi$
$14$	0	0
$15$	−0.548737	−0.141683
$16$	0	0
$17$	3.22998	0.783386	0.391693	−	0.920096i	$-0.371890\pi$
$17$	3.22998	0.783386	0.391693	+	0.920096i	$0.371890\pi$
$18$	0	0
$19$	−2.16232	−0.496071	−0.248036	−	0.968751i	$-0.579785\pi$
$19$	−2.16232	−0.496071	−0.248036	+	0.968751i	$0.579785\pi$
$20$	0	0
$21$	−1.23266	−0.268988
$22$	0	0
$23$	−3.59035	−0.748639	−0.374320	−	0.927300i	$-0.622124\pi$
$23$	−3.59035	−0.748639	−0.374320	+	0.927300i	$0.622124\pi$
$24$	0	0
$25$	−4.93373	−0.986747
$26$	0	0
$27$	3.10379	0.597325
$28$	0	0
$29$	1.34015	0.248860	0.124430	−	0.992228i	$-0.460290\pi$
$29$	1.34015	0.248860	0.124430	+	0.992228i	$0.460290\pi$
$30$	0	0
$31$	2.91300	0.523191	0.261595	−	0.965178i	$-0.415751\pi$
$31$	2.91300	0.523191	0.261595	+	0.965178i	$0.415751\pi$
$32$	0	0
$33$	3.07551	0.535377
$34$	0	0
$35$	0.148858	0.0251616
$36$	0	0
$37$	7.43825	1.22284	0.611421	−	0.791306i	$-0.290598\pi$
$37$	7.43825	1.22284	0.611421	+	0.791306i	$0.290598\pi$
$38$	0	0
$39$	−2.62866	−0.420922
$40$	0	0
$41$	9.61132	1.50104	0.750518	−	0.660850i	$-0.229804\pi$
$41$	9.61132	1.50104	0.750518	+	0.660850i	$0.229804\pi$
$42$	0	0
$43$	−4.00610	−0.610925	−0.305462	−	0.952204i	$-0.598811\pi$
$43$	−4.00610	−0.610925	−0.305462	+	0.952204i	$0.598811\pi$
$44$	0	0
$45$	0.397448	0.0592481
$46$	0	0
$47$	−7.55848	−1.10252	−0.551259	−	0.834334i	$-0.685852\pi$
$47$	−7.55848	−1.10252	−0.551259	+	0.834334i	$0.685852\pi$
$48$	0	0
$49$	−6.66561	−0.952230
$50$	0	0
$51$	−6.88521	−0.964122
$52$	0	0
$53$	−5.08800	−0.698890	−0.349445	−	0.936957i	$-0.613630\pi$
$53$	−5.08800	−0.698890	−0.349445	+	0.936957i	$0.613630\pi$
$54$	0	0
$55$	−0.371405	−0.0500802
$56$	0	0
$57$	4.60933	0.610520
$58$	0	0
$59$	−4.79877	−0.624747	−0.312374	−	0.949959i	$-0.601124\pi$
$59$	−4.79877	−0.624747	−0.312374	+	0.949959i	$0.601124\pi$
$60$	0	0
$61$	−0.119891	−0.0153504	−0.00767522	−	0.999971i	$-0.502443\pi$
$61$	−0.119891	−0.0153504	−0.00767522	+	0.999971i	$0.502443\pi$
$62$	0	0
$63$	0.892809	0.112483
$64$	0	0
$65$	0.317442	0.0393738
$66$	0	0
$67$	−8.76927	−1.07134	−0.535668	−	0.844429i	$-0.679940\pi$
$67$	−8.76927	−1.07134	−0.535668	+	0.844429i	$0.679940\pi$
$68$	0	0
$69$	7.65338	0.921359
$70$	0	0
$71$	14.2487	1.69101	0.845504	−	0.533969i	$-0.179300\pi$
$71$	14.2487	1.69101	0.845504	+	0.533969i	$0.179300\pi$
$72$	0	0
$73$	10.5417	1.23381	0.616905	−	0.787038i	$-0.288386\pi$
$73$	10.5417	1.23381	0.616905	+	0.787038i	$0.288386\pi$
$74$	0	0
$75$	10.5170	1.21440
$76$	0	0
$77$	−0.834306	−0.0950780
$78$	0	0
$79$	14.7249	1.65669	0.828343	−	0.560222i	$-0.189284\pi$
$79$	14.7249	1.65669	0.828343	+	0.560222i	$0.189284\pi$
$80$	0	0
$81$	−11.2481	−1.24979
$82$	0	0
$83$	9.70080	1.06480	0.532401	−	0.846492i	$-0.321290\pi$
$83$	9.70080	1.06480	0.532401	+	0.846492i	$0.321290\pi$
$84$	0	0
$85$	0.831472	0.0901858
$86$	0	0
$87$	−2.85674	−0.306275
$88$	0	0
$89$	15.5478	1.64807	0.824034	−	0.566540i	$-0.191718\pi$
$89$	15.5478	1.64807	0.824034	+	0.566540i	$0.191718\pi$
$90$	0	0
$91$	0.713087	0.0747518
$92$	0	0
$93$	−6.20952	−0.643897
$94$	0	0
$95$	−0.556632	−0.0571092
$96$	0	0
$97$	17.4458	1.77135	0.885676	−	0.464303i	$-0.153695\pi$
$97$	17.4458	1.77135	0.885676	+	0.464303i	$0.153695\pi$
$98$	0	0
$99$	−2.22758	−0.223880
$100$	0	0
$101$	11.4484	1.13916	0.569580	−	0.821936i	$-0.307106\pi$
$101$	11.4484	1.13916	0.569580	+	0.821936i	$0.307106\pi$
$102$	0	0
$103$	0.147959	0.0145789	0.00728943	−	0.999973i	$-0.497680\pi$
$103$	0.147959	0.0145789	0.00728943	+	0.999973i	$0.497680\pi$
$104$	0	0
$105$	−0.317314	−0.0309667
$106$	0	0
$107$	−0.777070	−0.0751221	−0.0375611	−	0.999294i	$-0.511959\pi$
$107$	−0.777070	−0.0751221	−0.0375611	+	0.999294i	$0.511959\pi$
$108$	0	0
$109$	−2.23762	−0.214325	−0.107162	−	0.994242i	$-0.534176\pi$
$109$	−2.23762	−0.214325	−0.107162	+	0.994242i	$0.534176\pi$
$110$	0	0
$111$	−15.8558	−1.50496
$112$	0	0
$113$	10.2350	0.962831	0.481416	−	0.876492i	$-0.340123\pi$
$113$	10.2350	0.962831	0.481416	+	0.876492i	$0.340123\pi$
$114$	0	0
$115$	−0.924238	−0.0861857
$116$	0	0
$117$	1.90393	0.176018
$118$	0	0
$119$	1.86778	0.171219
$120$	0	0
$121$	−8.91839	−0.810762
$122$	0	0
$123$	−20.4880	−1.84734
$124$	0	0
$125$	−2.55717	−0.228720
$126$	0	0
$127$	18.6170	1.65200	0.825998	−	0.563674i	$-0.190612\pi$
$127$	18.6170	1.65200	0.825998	+	0.563674i	$0.190612\pi$
$128$	0	0
$129$	8.53962	0.751872
$130$	0	0
$131$	−6.50800	−0.568607	−0.284303	−	0.958734i	$-0.591762\pi$
$131$	−6.50800	−0.568607	−0.284303	+	0.958734i	$0.591762\pi$
$132$	0	0
$133$	−1.25039	−0.108423
$134$	0	0
$135$	0.798988	0.0687659
$136$	0	0
$137$	18.2925	1.56283	0.781415	−	0.624012i	$-0.214498\pi$
$137$	18.2925	1.56283	0.781415	+	0.624012i	$0.214498\pi$
$138$	0	0
$139$	−12.7384	−1.08046	−0.540231	−	0.841517i	$-0.681663\pi$
$139$	−12.7384	−1.08046	−0.540231	+	0.841517i	$0.681663\pi$
$140$	0	0
$141$	16.1121	1.35688
$142$	0	0
$143$	−1.77917	−0.148782
$144$	0	0
$145$	0.344986	0.0286495
$146$	0	0
$147$	14.2088	1.17192
$148$	0	0
$149$	3.89636	0.319203	0.159601	−	0.987182i	$-0.448979\pi$
$149$	3.89636	0.319203	0.159601	+	0.987182i	$0.448979\pi$
$150$	0	0
$151$	14.6934	1.19573	0.597865	−	0.801597i	$-0.296016\pi$
$151$	14.6934	1.19573	0.597865	+	0.801597i	$0.296016\pi$
$152$	0	0
$153$	4.98694	0.403170
$154$	0	0
$155$	0.749874	0.0602313
$156$	0	0
$157$	9.49373	0.757682	0.378841	−	0.925462i	$-0.376323\pi$
$157$	9.49373	0.757682	0.378841	+	0.925462i	$0.376323\pi$
$158$	0	0
$159$	10.8458	0.860132
$160$	0	0
$161$	−2.07617	−0.163625
$162$	0	0
$163$	−16.6323	−1.30274	−0.651369	−	0.758761i	$-0.725805\pi$
$163$	−16.6323	−1.30274	−0.651369	+	0.758761i	$0.725805\pi$
$164$	0	0
$165$	0.791707	0.0616343
$166$	0	0
$167$	5.05080	0.390843	0.195421	−	0.980719i	$-0.437393\pi$
$167$	5.05080	0.390843	0.195421	+	0.980719i	$0.437393\pi$
$168$	0	0
$169$	−11.4793	−0.883026
$170$	0	0
$171$	−3.33852	−0.255303
$172$	0	0
$173$	5.19947	0.395308	0.197654	−	0.980272i	$-0.436668\pi$
$173$	5.19947	0.395308	0.197654	+	0.980272i	$0.436668\pi$
$174$	0	0
$175$	−2.85299	−0.215666
$176$	0	0
$177$	10.2293	0.768883
$178$	0	0
$179$	4.38408	0.327681	0.163841	−	0.986487i	$-0.447612\pi$
$179$	4.38408	0.327681	0.163841	+	0.986487i	$0.447612\pi$
$180$	0	0
$181$	12.9053	0.959241	0.479621	−	0.877476i	$-0.340774\pi$
$181$	12.9053	0.959241	0.479621	+	0.877476i	$0.340774\pi$
$182$	0	0
$183$	0.255566	0.0188920
$184$	0	0
$185$	1.91478	0.140777
$186$	0	0
$187$	−4.66016	−0.340784
$188$	0	0
$189$	1.79481	0.130553
$190$	0	0
$191$	14.6014	1.05652	0.528261	−	0.849082i	$-0.322844\pi$
$191$	14.6014	1.05652	0.528261	+	0.849082i	$0.322844\pi$
$192$	0	0
$193$	0.612232	0.0440694	0.0220347	−	0.999757i	$-0.492986\pi$
$193$	0.612232	0.0440694	0.0220347	+	0.999757i	$0.492986\pi$
$194$	0	0
$195$	−0.676677	−0.0484578
$196$	0	0
$197$	−11.7607	−0.837914	−0.418957	−	0.908006i	$-0.637604\pi$
$197$	−11.7607	−0.837914	−0.418957	+	0.908006i	$0.637604\pi$
$198$	0	0
$199$	−5.95001	−0.421785	−0.210892	−	0.977509i	$-0.567637\pi$
$199$	−5.95001	−0.421785	−0.210892	+	0.977509i	$0.567637\pi$
$200$	0	0
$201$	18.6930	1.31851
$202$	0	0
$203$	0.774961	0.0543916
$204$	0	0
$205$	2.47417	0.172804
$206$	0	0
$207$	−5.54332	−0.385287
$208$	0	0
$209$	3.11976	0.215798
$210$	0	0
$211$	23.4565	1.61481	0.807407	−	0.589995i	$-0.200870\pi$
$211$	23.4565	1.61481	0.807407	+	0.589995i	$0.200870\pi$
$212$	0	0
$213$	−30.3733	−2.08114
$214$	0	0
$215$	−1.03126	−0.0703315
$216$	0	0
$217$	1.68448	0.114350
$218$	0	0
$219$	−22.4712	−1.51846
$220$	0	0
$221$	3.98306	0.267930
$222$	0	0
$223$	−13.7654	−0.921800	−0.460900	−	0.887452i	$-0.652473\pi$
$223$	−13.7654	−0.921800	−0.460900	+	0.887452i	$0.652473\pi$
$224$	0	0
$225$	−7.61744	−0.507829
$226$	0	0
$227$	25.3420	1.68201	0.841005	−	0.541028i	$-0.181965\pi$
$227$	25.3420	1.68201	0.841005	+	0.541028i	$0.181965\pi$
$228$	0	0
$229$	−8.16364	−0.539468	−0.269734	−	0.962935i	$-0.586936\pi$
$229$	−8.16364	−0.539468	−0.269734	+	0.962935i	$0.586936\pi$
$230$	0	0
$231$	1.77845	0.117014
$232$	0	0
$233$	4.62470	0.302974	0.151487	−	0.988459i	$-0.451594\pi$
$233$	4.62470	0.302974	0.151487	+	0.988459i	$0.451594\pi$
$234$	0	0
$235$	−1.94573	−0.126925
$236$	0	0
$237$	−31.3885	−2.03890
$238$	0	0
$239$	−4.88042	−0.315688	−0.157844	−	0.987464i	$-0.550454\pi$
$239$	−4.88042	−0.315688	−0.157844	+	0.987464i	$0.550454\pi$
$240$	0	0
$241$	0.652332	0.0420204	0.0210102	−	0.999779i	$-0.493312\pi$
$241$	0.652332	0.0420204	0.0210102	+	0.999779i	$0.493312\pi$
$242$	0	0
$243$	14.6656	0.940800
$244$	0	0
$245$	−1.71588	−0.109624
$246$	0	0
$247$	−2.66648	−0.169664
$248$	0	0
$249$	−20.6788	−1.31046
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	5.18008	0.325669
$254$	0	0
$255$	−1.77241	−0.110993
$256$	0	0
$257$	−21.6636	−1.35134	−0.675670	−	0.737205i	$-0.736145\pi$
$257$	−21.6636	−1.35134	−0.675670	+	0.737205i	$0.736145\pi$
$258$	0	0
$259$	4.30127	0.267268
$260$	0	0
$261$	2.06913	0.128076
$262$	0	0
$263$	24.1926	1.49178	0.745889	−	0.666070i	$-0.232025\pi$
$263$	24.1926	1.49178	0.745889	+	0.666070i	$0.232025\pi$
$264$	0	0
$265$	−1.30977	−0.0804584
$266$	0	0
$267$	−33.1426	−2.02830
$268$	0	0
$269$	−16.0347	−0.977652	−0.488826	−	0.872381i	$-0.662575\pi$
$269$	−16.0347	−0.977652	−0.488826	+	0.872381i	$0.662575\pi$
$270$	0	0
$271$	−28.6525	−1.74051	−0.870257	−	0.492598i	$-0.836047\pi$
$271$	−28.6525	−1.74051	−0.870257	+	0.492598i	$0.836047\pi$
$272$	0	0
$273$	−1.52005	−0.0919979
$274$	0	0
$275$	7.11829	0.429249
$276$	0	0
$277$	19.8294	1.19143	0.595717	−	0.803194i	$-0.296868\pi$
$277$	19.8294	1.19143	0.595717	+	0.803194i	$0.296868\pi$
$278$	0	0
$279$	4.49753	0.269260
$280$	0	0
$281$	7.04601	0.420330	0.210165	−	0.977666i	$-0.432600\pi$
$281$	7.04601	0.420330	0.210165	+	0.977666i	$0.432600\pi$
$282$	0	0
$283$	16.2260	0.964534	0.482267	−	0.876024i	$-0.339814\pi$
$283$	16.2260	0.964534	0.482267	+	0.876024i	$0.339814\pi$
$284$	0	0
$285$	1.18655	0.0702850
$286$	0	0
$287$	5.55787	0.328070
$288$	0	0
$289$	−6.56721	−0.386306
$290$	0	0
$291$	−37.1884	−2.18002
$292$	0	0
$293$	9.06228	0.529424	0.264712	−	0.964327i	$-0.414723\pi$
$293$	9.06228	0.529424	0.264712	+	0.964327i	$0.414723\pi$
$294$	0	0
$295$	−1.23531	−0.0719228
$296$	0	0
$297$	−4.47809	−0.259845
$298$	0	0
$299$	−4.42745	−0.256046
$300$	0	0
$301$	−2.31658	−0.133525
$302$	0	0
$303$	−24.4041	−1.40198
$304$	0	0
$305$	−0.0308626	−0.00176719
$306$	0	0
$307$	−23.0105	−1.31328	−0.656638	−	0.754206i	$-0.728022\pi$
$307$	−23.0105	−1.31328	−0.656638	+	0.754206i	$0.728022\pi$
$308$	0	0
$309$	−0.315398	−0.0179424
$310$	0	0
$311$	−9.22635	−0.523179	−0.261589	−	0.965179i	$-0.584247\pi$
$311$	−9.22635	−0.523179	−0.261589	+	0.965179i	$0.584247\pi$
$312$	0	0
$313$	11.6420	0.658045	0.329022	−	0.944322i	$-0.393281\pi$
$313$	11.6420	0.658045	0.329022	+	0.944322i	$0.393281\pi$
$314$	0	0
$315$	0.229830	0.0129494
$316$	0	0
$317$	−11.7025	−0.657276	−0.328638	−	0.944456i	$-0.606590\pi$
$317$	−11.7025	−0.657276	−0.328638	+	0.944456i	$0.606590\pi$
$318$	0	0
$319$	−1.93355	−0.108258
$320$	0	0
$321$	1.65644	0.0924537
$322$	0	0
$323$	−6.98427	−0.388615
$324$	0	0
$325$	−6.08405	−0.337482
$326$	0	0
$327$	4.76983	0.263772
$328$	0	0
$329$	−4.37079	−0.240969
$330$	0	0
$331$	−2.26578	−0.124539	−0.0622694	−	0.998059i	$-0.519834\pi$
$331$	−2.26578	−0.124539	−0.0622694	+	0.998059i	$0.519834\pi$
$332$	0	0
$333$	11.4843	0.629336
$334$	0	0
$335$	−2.25741	−0.123336
$336$	0	0
$337$	−8.58164	−0.467472	−0.233736	−	0.972300i	$-0.575095\pi$
$337$	−8.58164	−0.467472	−0.233736	+	0.972300i	$0.575095\pi$
$338$	0	0
$339$	−21.8176	−1.18497
$340$	0	0
$341$	−4.20282	−0.227596
$342$	0	0
$343$	−7.90232	−0.426685
$344$	0	0
$345$	1.97016	0.106070
$346$	0	0
$347$	17.8709	0.959361	0.479681	−	0.877443i	$-0.340752\pi$
$347$	17.8709	0.959361	0.479681	+	0.877443i	$0.340752\pi$
$348$	0	0
$349$	−22.8884	−1.22519	−0.612594	−	0.790398i	$-0.709874\pi$
$349$	−22.8884	−1.22519	−0.612594	+	0.790398i	$0.709874\pi$
$350$	0	0
$351$	3.82745	0.204294
$352$	0	0
$353$	−12.4512	−0.662710	−0.331355	−	0.943506i	$-0.607506\pi$
$353$	−12.4512	−0.662710	−0.331355	+	0.943506i	$0.607506\pi$
$354$	0	0
$355$	3.66794	0.194674
$356$	0	0
$357$	−3.98146	−0.210721
$358$	0	0
$359$	25.4272	1.34200	0.670998	−	0.741459i	$-0.265866\pi$
$359$	25.4272	1.34200	0.670998	+	0.741459i	$0.265866\pi$
$360$	0	0
$361$	−14.3244	−0.753913
$362$	0	0
$363$	19.0109	0.997814
$364$	0	0
$365$	2.71367	0.142040
$366$	0	0
$367$	34.2480	1.78773	0.893866	−	0.448335i	$-0.147983\pi$
$367$	34.2480	1.78773	0.893866	+	0.448335i	$0.147983\pi$
$368$	0	0
$369$	14.8394	0.772508
$370$	0	0
$371$	−2.94220	−0.152751
$372$	0	0
$373$	25.8031	1.33603	0.668017	−	0.744146i	$-0.267143\pi$
$373$	25.8031	1.33603	0.668017	+	0.744146i	$0.267143\pi$
$374$	0	0
$375$	5.45101	0.281489
$376$	0	0
$377$	1.65261	0.0851139
$378$	0	0
$379$	−4.65289	−0.239003	−0.119502	−	0.992834i	$-0.538130\pi$
$379$	−4.65289	−0.239003	−0.119502	+	0.992834i	$0.538130\pi$
$380$	0	0
$381$	−39.6851	−2.03313
$382$	0	0
$383$	−10.8385	−0.553823	−0.276912	−	0.960895i	$-0.589311\pi$
$383$	−10.8385	−0.553823	−0.276912	+	0.960895i	$0.589311\pi$
$384$	0	0
$385$	−0.214770	−0.0109457
$386$	0	0
$387$	−6.18522	−0.314413
$388$	0	0
$389$	−31.2682	−1.58536	−0.792679	−	0.609639i	$-0.791314\pi$
$389$	−31.2682	−1.58536	−0.792679	+	0.609639i	$0.791314\pi$
$390$	0	0
$391$	−11.5968	−0.586474
$392$	0	0
$393$	13.8728	0.699791
$394$	0	0
$395$	3.79054	0.190723
$396$	0	0
$397$	−18.4742	−0.927195	−0.463597	−	0.886046i	$-0.653442\pi$
$397$	−18.4742	−0.927195	−0.463597	+	0.886046i	$0.653442\pi$
$398$	0	0
$399$	2.66540	0.133437
$400$	0	0
$401$	10.3634	0.517525	0.258762	−	0.965941i	$-0.416685\pi$
$401$	10.3634	0.517525	0.258762	+	0.965941i	$0.416685\pi$
$402$	0	0
$403$	3.59218	0.178939
$404$	0	0
$405$	−2.89551	−0.143879
$406$	0	0
$407$	−10.7318	−0.531954
$408$	0	0
$409$	−33.5152	−1.65722	−0.828611	−	0.559825i	$-0.810868\pi$
$409$	−33.5152	−1.65722	−0.828611	+	0.559825i	$0.810868\pi$
$410$	0	0
$411$	−38.9932	−1.92339
$412$	0	0
$413$	−2.77495	−0.136546
$414$	0	0
$415$	2.49721	0.122583
$416$	0	0
$417$	27.1540	1.32974
$418$	0	0
$419$	8.01559	0.391587	0.195794	−	0.980645i	$-0.437272\pi$
$419$	8.01559	0.391587	0.195794	+	0.980645i	$0.437272\pi$
$420$	0	0
$421$	−13.1527	−0.641025	−0.320512	−	0.947244i	$-0.603855\pi$
$421$	−13.1527	−0.641025	−0.320512	+	0.947244i	$0.603855\pi$
$422$	0	0
$423$	−11.6699	−0.567411
$424$	0	0
$425$	−15.9359	−0.773004
$426$	0	0
$427$	−0.0693284	−0.00335504
$428$	0	0
$429$	3.79257	0.183107
$430$	0	0
$431$	2.41216	0.116190	0.0580948	−	0.998311i	$-0.481497\pi$
$431$	2.41216	0.116190	0.0580948	+	0.998311i	$0.481497\pi$
$432$	0	0
$433$	−15.1889	−0.729930	−0.364965	−	0.931021i	$-0.618919\pi$
$433$	−15.1889	−0.729930	−0.364965	+	0.931021i	$0.618919\pi$
$434$	0	0
$435$	−0.735391	−0.0352593
$436$	0	0
$437$	7.76350	0.371378
$438$	0	0
$439$	−5.98083	−0.285449	−0.142725	−	0.989762i	$-0.545586\pi$
$439$	−5.98083	−0.285449	−0.142725	+	0.989762i	$0.545586\pi$
$440$	0	0
$441$	−10.2914	−0.490066
$442$	0	0
$443$	−22.3495	−1.06186	−0.530928	−	0.847417i	$-0.678157\pi$
$443$	−22.3495	−1.06186	−0.530928	+	0.847417i	$0.678157\pi$
$444$	0	0
$445$	4.00237	0.189731
$446$	0	0
$447$	−8.30570	−0.392846
$448$	0	0
$449$	−13.5528	−0.639598	−0.319799	−	0.947485i	$-0.603615\pi$
$449$	−13.5528	−0.639598	−0.319799	+	0.947485i	$0.603615\pi$
$450$	0	0
$451$	−13.8670	−0.652972
$452$	0	0
$453$	−31.3212	−1.47160
$454$	0	0
$455$	0.183565	0.00860565
$456$	0	0
$457$	8.19148	0.383181	0.191591	−	0.981475i	$-0.438635\pi$
$457$	8.19148	0.383181	0.191591	+	0.981475i	$0.438635\pi$
$458$	0	0
$459$	10.0252	0.467936
$460$	0	0
$461$	13.1494	0.612430	0.306215	−	0.951962i	$-0.400937\pi$
$461$	13.1494	0.612430	0.306215	+	0.951962i	$0.400937\pi$
$462$	0	0
$463$	9.03105	0.419708	0.209854	−	0.977733i	$-0.432701\pi$
$463$	9.03105	0.419708	0.209854	+	0.977733i	$0.432701\pi$
$464$	0	0
$465$	−1.59847	−0.0741274
$466$	0	0
$467$	11.3433	0.524908	0.262454	−	0.964945i	$-0.415468\pi$
$467$	11.3433	0.524908	0.262454	+	0.964945i	$0.415468\pi$
$468$	0	0
$469$	−5.07094	−0.234154
$470$	0	0
$471$	−20.2373	−0.932488
$472$	0	0
$473$	5.77992	0.265761
$474$	0	0
$475$	10.6683	0.489497
$476$	0	0
$477$	−7.85562	−0.359684
$478$	0	0
$479$	−5.01937	−0.229341	−0.114670	−	0.993404i	$-0.536581\pi$
$479$	−5.01937	−0.229341	−0.114670	+	0.993404i	$0.536581\pi$
$480$	0	0
$481$	9.17251	0.418230
$482$	0	0
$483$	4.42567	0.201375
$484$	0	0
$485$	4.49095	0.203924
$486$	0	0
$487$	−2.51042	−0.113758	−0.0568789	−	0.998381i	$-0.518115\pi$
$487$	−2.51042	−0.113758	−0.0568789	+	0.998381i	$0.518115\pi$
$488$	0	0
$489$	35.4542	1.60330
$490$	0	0
$491$	−2.31426	−0.104441	−0.0522205	−	0.998636i	$-0.516630\pi$
$491$	−2.31426	−0.104441	−0.0522205	+	0.998636i	$0.516630\pi$
$492$	0	0
$493$	4.32867	0.194954
$494$	0	0
$495$	−0.573431	−0.0257738
$496$	0	0
$497$	8.23948	0.369591
$498$	0	0
$499$	−8.76208	−0.392244	−0.196122	−	0.980579i	$-0.562835\pi$
$499$	−8.76208	−0.392244	−0.196122	+	0.980579i	$0.562835\pi$
$500$	0	0
$501$	−10.7666	−0.481015
$502$	0	0
$503$	−4.97063	−0.221630	−0.110815	−	0.993841i	$-0.535346\pi$
$503$	−4.97063	−0.221630	−0.110815	+	0.993841i	$0.535346\pi$
$504$	0	0
$505$	2.94709	0.131144
$506$	0	0
$507$	24.4700	1.08675
$508$	0	0
$509$	−11.5632	−0.512529	−0.256265	−	0.966607i	$-0.582492\pi$
$509$	−11.5632	−0.512529	−0.256265	+	0.966607i	$0.582492\pi$
$510$	0	0
$511$	6.09586	0.269665
$512$	0	0
$513$	−6.71141	−0.296316
$514$	0	0
$515$	0.0380881	0.00167836
$516$	0	0
$517$	10.9052	0.479611
$518$	0	0
$519$	−11.0835	−0.486511
$520$	0	0
$521$	22.1108	0.968691	0.484346	−	0.874877i	$-0.339058\pi$
$521$	22.1108	0.968691	0.484346	+	0.874877i	$0.339058\pi$
$522$	0	0
$523$	18.6154	0.813993	0.406997	−	0.913430i	$-0.366576\pi$
$523$	18.6154	0.813993	0.406997	+	0.913430i	$0.366576\pi$
$524$	0	0
$525$	6.08160	0.265423
$526$	0	0
$527$	9.40895	0.409860
$528$	0	0
$529$	−10.1094	−0.439539
$530$	0	0
$531$	−7.40907	−0.321526
$532$	0	0
$533$	11.8522	0.513377
$534$	0	0
$535$	−0.200036	−0.00864829
$536$	0	0
$537$	−9.34533	−0.403281
$538$	0	0
$539$	9.61701	0.414234
$540$	0	0
$541$	−36.3161	−1.56135	−0.780675	−	0.624937i	$-0.785124\pi$
$541$	−36.3161	−1.56135	−0.780675	+	0.624937i	$0.785124\pi$
$542$	0	0
$543$	−27.5096	−1.18055
$544$	0	0
$545$	−0.576014	−0.0246737
$546$	0	0
$547$	−2.47047	−0.105630	−0.0528148	−	0.998604i	$-0.516819\pi$
$547$	−2.47047	−0.105630	−0.0528148	+	0.998604i	$0.516819\pi$
$548$	0	0
$549$	−0.185105	−0.00790011
$550$	0	0
$551$	−2.89784	−0.123452
$552$	0	0
$553$	8.51489	0.362090
$554$	0	0
$555$	−4.08164	−0.173256
$556$	0	0
$557$	−39.3494	−1.66729	−0.833644	−	0.552302i	$-0.813749\pi$
$557$	−39.3494	−1.66729	−0.833644	+	0.552302i	$0.813749\pi$
$558$	0	0
$559$	−4.94014	−0.208945
$560$	0	0
$561$	9.93384	0.419407
$562$	0	0
$563$	−12.6798	−0.534389	−0.267195	−	0.963643i	$-0.586097\pi$
$563$	−12.6798	−0.534389	−0.267195	+	0.963643i	$0.586097\pi$
$564$	0	0
$565$	2.63473	0.110844
$566$	0	0
$567$	−6.50434	−0.273157
$568$	0	0
$569$	39.6216	1.66102	0.830512	−	0.557000i	$-0.188048\pi$
$569$	39.6216	1.66102	0.830512	+	0.557000i	$0.188048\pi$
$570$	0	0
$571$	−10.9909	−0.459954	−0.229977	−	0.973196i	$-0.573865\pi$
$571$	−10.9909	−0.459954	−0.229977	+	0.973196i	$0.573865\pi$
$572$	0	0
$573$	−31.1252	−1.30027
$574$	0	0
$575$	17.7138	0.738717
$576$	0	0
$577$	22.5319	0.938017	0.469008	−	0.883194i	$-0.344611\pi$
$577$	22.5319	0.938017	0.469008	+	0.883194i	$0.344611\pi$
$578$	0	0
$579$	−1.30507	−0.0542367
$580$	0	0
$581$	5.60961	0.232726
$582$	0	0
$583$	7.34086	0.304027
$584$	0	0
$585$	0.490115	0.0202638
$586$	0	0
$587$	39.4321	1.62754	0.813769	−	0.581189i	$-0.197412\pi$
$587$	39.4321	1.62754	0.813769	+	0.581189i	$0.197412\pi$
$588$	0	0
$589$	−6.29886	−0.259540
$590$	0	0
$591$	25.0697	1.03123
$592$	0	0
$593$	−41.9828	−1.72403	−0.862013	−	0.506887i	$-0.830796\pi$
$593$	−41.9828	−1.72403	−0.862013	+	0.506887i	$0.830796\pi$
$594$	0	0
$595$	0.480809	0.0197113
$596$	0	0
$597$	12.6834	0.519096
$598$	0	0
$599$	−15.8875	−0.649146	−0.324573	−	0.945861i	$-0.605221\pi$
$599$	−15.8875	−0.649146	−0.324573	+	0.945861i	$0.605221\pi$
$600$	0	0
$601$	39.4266	1.60825	0.804123	−	0.594463i	$-0.202635\pi$
$601$	39.4266	1.60825	0.804123	+	0.594463i	$0.202635\pi$
$602$	0	0
$603$	−13.5393	−0.551364
$604$	0	0
$605$	−2.29580	−0.0933374
$606$	0	0
$607$	5.29224	0.214805	0.107403	−	0.994216i	$-0.465747\pi$
$607$	5.29224	0.214805	0.107403	+	0.994216i	$0.465747\pi$
$608$	0	0
$609$	−1.65195	−0.0669403
$610$	0	0
$611$	−9.32077	−0.377078
$612$	0	0
$613$	34.1195	1.37808	0.689038	−	0.724725i	$-0.258033\pi$
$613$	34.1195	1.37808	0.689038	+	0.724725i	$0.258033\pi$
$614$	0	0
$615$	−5.27408	−0.212672
$616$	0	0
$617$	29.4738	1.18657	0.593285	−	0.804992i	$-0.297831\pi$
$617$	29.4738	1.18657	0.593285	+	0.804992i	$0.297831\pi$
$618$	0	0
$619$	−32.8210	−1.31919	−0.659593	−	0.751623i	$-0.729271\pi$
$619$	−32.8210	−1.31919	−0.659593	+	0.751623i	$0.729271\pi$
$620$	0	0
$621$	−11.1437	−0.447181
$622$	0	0
$623$	8.99074	0.360206
$624$	0	0
$625$	24.0104	0.960416
$626$	0	0
$627$	−6.65025	−0.265585
$628$	0	0
$629$	24.0254	0.957957
$630$	0	0
$631$	−13.2532	−0.527601	−0.263801	−	0.964577i	$-0.584976\pi$
$631$	−13.2532	−0.527601	−0.263801	+	0.964577i	$0.584976\pi$
$632$	0	0
$633$	−50.0012	−1.98737
$634$	0	0
$635$	4.79245	0.190183
$636$	0	0
$637$	−8.21972	−0.325677
$638$	0	0
$639$	21.9993	0.870278
$640$	0	0
$641$	1.34546	0.0531426	0.0265713	−	0.999647i	$-0.491541\pi$
$641$	1.34546	0.0531426	0.0265713	+	0.999647i	$0.491541\pi$
$642$	0	0
$643$	−12.2114	−0.481569	−0.240785	−	0.970579i	$-0.577405\pi$
$643$	−12.2114	−0.481569	−0.240785	+	0.970579i	$0.577405\pi$
$644$	0	0
$645$	2.19830	0.0865578
$646$	0	0
$647$	28.5704	1.12322	0.561608	−	0.827403i	$-0.310183\pi$
$647$	28.5704	1.12322	0.561608	+	0.827403i	$0.310183\pi$
$648$	0	0
$649$	6.92358	0.271774
$650$	0	0
$651$	−3.59073	−0.140732
$652$	0	0
$653$	15.3953	0.602463	0.301231	−	0.953551i	$-0.402602\pi$
$653$	15.3953	0.602463	0.301231	+	0.953551i	$0.402602\pi$
$654$	0	0
$655$	−1.67531	−0.0654597
$656$	0	0
$657$	16.2758	0.634981
$658$	0	0
$659$	24.7538	0.964272	0.482136	−	0.876096i	$-0.339861\pi$
$659$	24.7538	0.964272	0.482136	+	0.876096i	$0.339861\pi$
$660$	0	0
$661$	−19.0121	−0.739487	−0.369743	−	0.929134i	$-0.620554\pi$
$661$	−19.0121	−0.739487	−0.369743	+	0.929134i	$0.620554\pi$
$662$	0	0
$663$	−8.49052	−0.329744
$664$	0	0
$665$	−0.321879	−0.0124820
$666$	0	0
$667$	−4.81162	−0.186306
$668$	0	0
$669$	29.3431	1.13447
$670$	0	0
$671$	0.172976	0.00667767
$672$	0	0
$673$	16.1863	0.623935	0.311967	−	0.950093i	$-0.399012\pi$
$673$	16.1863	0.623935	0.311967	+	0.950093i	$0.399012\pi$
$674$	0	0
$675$	−15.3133	−0.589409
$676$	0	0
$677$	34.9382	1.34278	0.671392	−	0.741102i	$-0.265697\pi$
$677$	34.9382	1.34278	0.671392	+	0.741102i	$0.265697\pi$
$678$	0	0
$679$	10.0883	0.387152
$680$	0	0
$681$	−54.0204	−2.07007
$682$	0	0
$683$	39.9757	1.52963	0.764813	−	0.644252i	$-0.222831\pi$
$683$	39.9757	1.52963	0.764813	+	0.644252i	$0.222831\pi$
$684$	0	0
$685$	4.70890	0.179918
$686$	0	0
$687$	17.4021	0.663930
$688$	0	0
$689$	−6.27428	−0.239031
$690$	0	0
$691$	−3.27679	−0.124655	−0.0623275	−	0.998056i	$-0.519852\pi$
$691$	−3.27679	−0.124655	−0.0623275	+	0.998056i	$0.519852\pi$
$692$	0	0
$693$	−1.28813	−0.0489319
$694$	0	0
$695$	−3.27917	−0.124386
$696$	0	0
$697$	31.0444	1.17589
$698$	0	0
$699$	−9.85826	−0.372873
$700$	0	0
$701$	−22.9428	−0.866538	−0.433269	−	0.901265i	$-0.642640\pi$
$701$	−22.9428	−0.866538	−0.433269	+	0.901265i	$0.642640\pi$
$702$	0	0
$703$	−16.0839	−0.606616
$704$	0	0
$705$	4.14762	0.156208
$706$	0	0
$707$	6.62020	0.248978
$708$	0	0
$709$	−51.3035	−1.92674	−0.963371	−	0.268173i	$-0.913580\pi$
$709$	−51.3035	−1.92674	−0.963371	+	0.268173i	$0.913580\pi$
$710$	0	0
$711$	22.7346	0.852614
$712$	0	0
$713$	−10.4587	−0.391681
$714$	0	0
$715$	−0.457999	−0.0171282
$716$	0	0
$717$	10.4034	0.388521
$718$	0	0
$719$	−4.63057	−0.172691	−0.0863456	−	0.996265i	$-0.527519\pi$
$719$	−4.63057	−0.172691	−0.0863456	+	0.996265i	$0.527519\pi$
$720$	0	0
$721$	0.0855594	0.00318640
$722$	0	0
$723$	−1.39055	−0.0517149
$724$	0	0
$725$	−6.61196	−0.245562
$726$	0	0
$727$	−0.697832	−0.0258812	−0.0129406	−	0.999916i	$-0.504119\pi$
$727$	−0.697832	−0.0258812	−0.0129406	+	0.999916i	$0.504119\pi$
$728$	0	0
$729$	2.48218	0.0919327
$730$	0	0
$731$	−12.9396	−0.478590
$732$	0	0
$733$	−13.5235	−0.499502	−0.249751	−	0.968310i	$-0.580349\pi$
$733$	−13.5235	−0.499502	−0.249751	+	0.968310i	$0.580349\pi$
$734$	0	0
$735$	3.65767	0.134915
$736$	0	0
$737$	12.6521	0.466047
$738$	0	0
$739$	49.4982	1.82082	0.910410	−	0.413708i	$-0.135767\pi$
$739$	49.4982	1.82082	0.910410	+	0.413708i	$0.135767\pi$
$740$	0	0
$741$	5.68401	0.208807
$742$	0	0
$743$	28.6466	1.05094	0.525472	−	0.850811i	$-0.323889\pi$
$743$	28.6466	1.05094	0.525472	+	0.850811i	$0.323889\pi$
$744$	0	0
$745$	1.00301	0.0367476
$746$	0	0
$747$	14.9776	0.548000
$748$	0	0
$749$	−0.449350	−0.0164189
$750$	0	0
$751$	−12.6351	−0.461061	−0.230530	−	0.973065i	$-0.574046\pi$
$751$	−12.6351	−0.461061	−0.230530	+	0.973065i	$0.574046\pi$
$752$	0	0
$753$	2.13165	0.0776818
$754$	0	0
$755$	3.78241	0.137656
$756$	0	0
$757$	27.6027	1.00324	0.501619	−	0.865089i	$-0.332738\pi$
$757$	27.6027	1.00324	0.501619	+	0.865089i	$0.332738\pi$
$758$	0	0
$759$	−11.0421	−0.400805
$760$	0	0
$761$	15.9504	0.578201	0.289100	−	0.957299i	$-0.406644\pi$
$761$	15.9504	0.578201	0.289100	+	0.957299i	$0.406644\pi$
$762$	0	0
$763$	−1.29393	−0.0468434
$764$	0	0
$765$	1.28375	0.0464141
$766$	0	0
$767$	−5.91762	−0.213673
$768$	0	0
$769$	42.2253	1.52268	0.761342	−	0.648350i	$-0.224541\pi$
$769$	42.2253	1.52268	0.761342	+	0.648350i	$0.224541\pi$
$770$	0	0
$771$	46.1793	1.66311
$772$	0	0
$773$	14.9447	0.537523	0.268761	−	0.963207i	$-0.413386\pi$
$773$	14.9447	0.537523	0.268761	+	0.963207i	$0.413386\pi$
$774$	0	0
$775$	−14.3720	−0.516257
$776$	0	0
$777$	−9.16881	−0.328929
$778$	0	0
$779$	−20.7828	−0.744620
$780$	0	0
$781$	−20.5577	−0.735613
$782$	0	0
$783$	4.15956	0.148650
$784$	0	0
$785$	2.44390	0.0872267
$786$	0	0
$787$	−19.8262	−0.706727	−0.353364	−	0.935486i	$-0.614962\pi$
$787$	−19.8262	−0.706727	−0.353364	+	0.935486i	$0.614962\pi$
$788$	0	0
$789$	−51.5702	−1.83595
$790$	0	0
$791$	5.91854	0.210439
$792$	0	0
$793$	−0.147844	−0.00525008
$794$	0	0
$795$	2.79197	0.0990210
$796$	0	0
$797$	12.3151	0.436224	0.218112	−	0.975924i	$-0.430010\pi$
$797$	12.3151	0.436224	0.218112	+	0.975924i	$0.430010\pi$
$798$	0	0
$799$	−24.4138	−0.863697
$800$	0	0
$801$	24.0051	0.848179
$802$	0	0
$803$	−15.2093	−0.536725
$804$	0	0
$805$	−0.534453	−0.0188370
$806$	0	0
$807$	34.1804	1.20321
$808$	0	0
$809$	1.95284	0.0686581	0.0343291	−	0.999411i	$-0.489071\pi$
$809$	1.95284	0.0686581	0.0343291	+	0.999411i	$0.489071\pi$
$810$	0	0
$811$	−4.28269	−0.150386	−0.0751928	−	0.997169i	$-0.523957\pi$
$811$	−4.28269	−0.150386	−0.0751928	+	0.997169i	$0.523957\pi$
$812$	0	0
$813$	61.0772	2.14207
$814$	0	0
$815$	−4.28153	−0.149975
$816$	0	0
$817$	8.66249	0.303062
$818$	0	0
$819$	1.10097	0.0384710
$820$	0	0
$821$	−10.1730	−0.355040	−0.177520	−	0.984117i	$-0.556808\pi$
$821$	−10.1730	−0.355040	−0.177520	+	0.984117i	$0.556808\pi$
$822$	0	0
$823$	−9.65644	−0.336602	−0.168301	−	0.985736i	$-0.553828\pi$
$823$	−9.65644	−0.336602	−0.168301	+	0.985736i	$0.553828\pi$
$824$	0	0
$825$	−15.1737	−0.528282
$826$	0	0
$827$	55.4357	1.92769	0.963845	−	0.266464i	$-0.0858553\pi$
$827$	55.4357	1.92769	0.963845	+	0.266464i	$0.0858553\pi$
$828$	0	0
$829$	−20.0529	−0.696465	−0.348232	−	0.937408i	$-0.613218\pi$
$829$	−20.0529	−0.696465	−0.348232	+	0.937408i	$0.613218\pi$
$830$	0	0
$831$	−42.2695	−1.46631
$832$	0	0
$833$	−21.5298	−0.745964
$834$	0	0
$835$	1.30019	0.0449950
$836$	0	0
$837$	9.04136	0.312515
$838$	0	0
$839$	33.8398	1.16828	0.584139	−	0.811654i	$-0.301432\pi$
$839$	33.8398	1.16828	0.584139	+	0.811654i	$0.301432\pi$
$840$	0	0
$841$	−27.2040	−0.938069
$842$	0	0
$843$	−15.0197	−0.517305
$844$	0	0
$845$	−2.95504	−0.101657
$846$	0	0
$847$	−5.15717	−0.177202
$848$	0	0
$849$	−34.5882	−1.18706
$850$	0	0
$851$	−26.7059	−0.915467
$852$	0	0
$853$	−43.8531	−1.50150	−0.750751	−	0.660586i	$-0.770308\pi$
$853$	−43.8531	−1.50150	−0.750751	+	0.660586i	$0.770308\pi$
$854$	0	0
$855$	−0.859412	−0.0293913
$856$	0	0
$857$	−16.7010	−0.570496	−0.285248	−	0.958454i	$-0.592076\pi$
$857$	−16.7010	−0.570496	−0.285248	+	0.958454i	$0.592076\pi$
$858$	0	0
$859$	31.2057	1.06473	0.532363	−	0.846516i	$-0.321304\pi$
$859$	31.2057	1.06473	0.532363	+	0.846516i	$0.321304\pi$
$860$	0	0
$861$	−11.8475	−0.403760
$862$	0	0
$863$	28.2924	0.963083	0.481542	−	0.876423i	$-0.340077\pi$
$863$	28.2924	0.963083	0.481542	+	0.876423i	$0.340077\pi$
$864$	0	0
$865$	1.33846	0.0455091
$866$	0	0
$867$	13.9990	0.475431
$868$	0	0
$869$	−21.2449	−0.720682
$870$	0	0
$871$	−10.8138	−0.366413
$872$	0	0
$873$	26.9355	0.911627
$874$	0	0
$875$	−1.47872	−0.0499898
$876$	0	0
$877$	−42.6838	−1.44133	−0.720665	−	0.693284i	$-0.756163\pi$
$877$	−42.6838	−1.44133	−0.720665	+	0.693284i	$0.756163\pi$
$878$	0	0
$879$	−19.3177	−0.651568
$880$	0	0
$881$	20.3040	0.684060	0.342030	−	0.939689i	$-0.388885\pi$
$881$	20.3040	0.684060	0.342030	+	0.939689i	$0.388885\pi$
$882$	0	0
$883$	47.4337	1.59627	0.798135	−	0.602479i	$-0.205820\pi$
$883$	47.4337	1.59627	0.798135	+	0.602479i	$0.205820\pi$
$884$	0	0
$885$	2.63326	0.0885162
$886$	0	0
$887$	21.3167	0.715746	0.357873	−	0.933770i	$-0.383502\pi$
$887$	21.3167	0.715746	0.357873	+	0.933770i	$0.383502\pi$
$888$	0	0
$889$	10.7655	0.361065
$890$	0	0
$891$	16.2285	0.543675
$892$	0	0
$893$	16.3439	0.546927
$894$	0	0
$895$	1.12856	0.0377237
$896$	0	0
$897$	9.43779	0.315119
$898$	0	0
$899$	3.90387	0.130201
$900$	0	0
$901$	−16.4341	−0.547501
$902$	0	0
$903$	4.93815	0.164331
$904$	0	0
$905$	3.32211	0.110431
$906$	0	0
$907$	−17.8871	−0.593931	−0.296966	−	0.954888i	$-0.595975\pi$
$907$	−17.8871	−0.593931	−0.296966	+	0.954888i	$0.595975\pi$
$908$	0	0
$909$	17.6758	0.586269
$910$	0	0
$911$	−4.03861	−0.133805	−0.0669026	−	0.997760i	$-0.521312\pi$
$911$	−4.03861	−0.133805	−0.0669026	+	0.997760i	$0.521312\pi$
$912$	0	0
$913$	−13.9961	−0.463204
$914$	0	0
$915$	0.0657885	0.00217490
$916$	0	0
$917$	−3.76334	−0.124276
$918$	0	0
$919$	17.2681	0.569624	0.284812	−	0.958583i	$-0.408069\pi$
$919$	17.2681	0.569624	0.284812	+	0.958583i	$0.408069\pi$
$920$	0	0
$921$	49.0504	1.61626
$922$	0	0
$923$	17.5708	0.578350
$924$	0	0
$925$	−36.6984	−1.20663
$926$	0	0
$927$	0.228442	0.00750302
$928$	0	0
$929$	−32.2922	−1.05947	−0.529737	−	0.848162i	$-0.677709\pi$
$929$	−32.2922	−1.05947	−0.529737	+	0.848162i	$0.677709\pi$
$930$	0	0
$931$	14.4132	0.472374
$932$	0	0
$933$	19.6674	0.643882
$934$	0	0
$935$	−1.19963	−0.0392321
$936$	0	0
$937$	−11.5818	−0.378362	−0.189181	−	0.981942i	$-0.560583\pi$
$937$	−11.5818	−0.378362	−0.189181	+	0.981942i	$0.560583\pi$
$938$	0	0
$939$	−24.8167	−0.809863
$940$	0	0
$941$	−32.0018	−1.04323	−0.521614	−	0.853182i	$-0.674670\pi$
$941$	−32.0018	−1.04323	−0.521614	+	0.853182i	$0.674670\pi$
$942$	0	0
$943$	−34.5080	−1.12373
$944$	0	0
$945$	0.462025	0.0150297
$946$	0	0
$947$	28.0987	0.913085	0.456543	−	0.889702i	$-0.349088\pi$
$947$	28.0987	0.913085	0.456543	+	0.889702i	$0.349088\pi$
$948$	0	0
$949$	12.9995	0.421982
$950$	0	0
$951$	24.9456	0.808917
$952$	0	0
$953$	11.9394	0.386755	0.193377	−	0.981124i	$-0.438056\pi$
$953$	11.9394	0.386755	0.193377	+	0.981124i	$0.438056\pi$
$954$	0	0
$955$	3.75874	0.121630
$956$	0	0
$957$	4.12165	0.133234
$958$	0	0
$959$	10.5778	0.341576
$960$	0	0
$961$	−22.5144	−0.726271
$962$	0	0
$963$	−1.19976	−0.0386616
$964$	0	0
$965$	0.157603	0.00507341
$966$	0	0
$967$	−37.0856	−1.19259	−0.596296	−	0.802764i	$-0.703362\pi$
$967$	−37.0856	−1.19259	−0.596296	+	0.802764i	$0.703362\pi$
$968$	0	0
$969$	14.8881	0.478273
$970$	0	0
$971$	3.36029	0.107837	0.0539184	−	0.998545i	$-0.482829\pi$
$971$	3.36029	0.107837	0.0539184	+	0.998545i	$0.482829\pi$
$972$	0	0
$973$	−7.36617	−0.236149
$974$	0	0
$975$	12.9691	0.415343
$976$	0	0
$977$	−24.6162	−0.787543	−0.393772	−	0.919208i	$-0.628830\pi$
$977$	−24.6162	−0.787543	−0.393772	+	0.919208i	$0.628830\pi$
$978$	0	0
$979$	−22.4321	−0.716934
$980$	0	0
$981$	−3.45477	−0.110302
$982$	0	0
$983$	−38.0516	−1.21366	−0.606829	−	0.794832i	$-0.707559\pi$
$983$	−38.0516	−1.21366	−0.606829	+	0.794832i	$0.707559\pi$
$984$	0	0
$985$	−3.02747	−0.0964632
$986$	0	0
$987$	9.31701	0.296564
$988$	0	0
$989$	14.3833	0.457362
$990$	0	0
$991$	−41.3513	−1.31357	−0.656783	−	0.754080i	$-0.728083\pi$
$991$	−41.3513	−1.31357	−0.656783	+	0.754080i	$0.728083\pi$
$992$	0	0
$993$	4.82987	0.153271
$994$	0	0
$995$	−1.53167	−0.0485572
$996$	0	0
$997$	−5.97938	−0.189369	−0.0946845	−	0.995507i	$-0.530184\pi$
$997$	−5.97938	−0.189369	−0.0946845	+	0.995507i	$0.530184\pi$
$998$	0	0
$999$	23.0868	0.730434

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2008.2.a.d.1.6	✓	23
4.3	odd	2		4016.2.a.m.1.18		23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.d.1.6	✓	23	1.1	even	1	trivial
4016.2.a.m.1.18		23	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2008 = 2^{3} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2008.a (trivial)

\(f(q)\)	\(=\)	\(q-2.13165 q^{3} +0.257423 q^{5} +0.578263 q^{7} +1.54395 q^{9} +O(q^{10})\)	\(q-2.13165 q^{3} +0.257423 q^{5} +0.578263 q^{7} +1.54395 q^{9} -1.44278 q^{11} +1.23315 q^{13} -0.548737 q^{15} +3.22998 q^{17} -2.16232 q^{19} -1.23266 q^{21} -3.59035 q^{23} -4.93373 q^{25} +3.10379 q^{27} +1.34015 q^{29} +2.91300 q^{31} +3.07551 q^{33} +0.148858 q^{35} +7.43825 q^{37} -2.62866 q^{39} +9.61132 q^{41} -4.00610 q^{43} +0.397448 q^{45} -7.55848 q^{47} -6.66561 q^{49} -6.88521 q^{51} -5.08800 q^{53} -0.371405 q^{55} +4.60933 q^{57} -4.79877 q^{59} -0.119891 q^{61} +0.892809 q^{63} +0.317442 q^{65} -8.76927 q^{67} +7.65338 q^{69} +14.2487 q^{71} +10.5417 q^{73} +10.5170 q^{75} -0.834306 q^{77} +14.7249 q^{79} -11.2481 q^{81} +9.70080 q^{83} +0.831472 q^{85} -2.85674 q^{87} +15.5478 q^{89} +0.713087 q^{91} -6.20952 q^{93} -0.556632 q^{95} +17.4458 q^{97} -2.22758 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9}+O(q^{10}) \) 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9	\( 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9} + 8 q^{11} + 8 q^{13} + 7 q^{15} + 19 q^{17} - 9 q^{19} + 9 q^{21} + 21 q^{23} + 65 q^{25} + 5 q^{27} + 10 q^{29} - 9 q^{31} + 34 q^{33} + 12 q^{35} + 11 q^{37} - 9 q^{39} + 35 q^{41} - 9 q^{43} + 29 q^{45} + 37 q^{47} + 77 q^{49} - 17 q^{51} + 38 q^{53} - 20 q^{55} + 51 q^{57} + 17 q^{59} + 22 q^{63} + 41 q^{65} + 9 q^{67} + 8 q^{69} + 13 q^{71} + 41 q^{73} + 25 q^{75} + 36 q^{77} - 36 q^{79} + 127 q^{81} + 29 q^{83} + 34 q^{85} + 10 q^{87} + 36 q^{89} - 6 q^{91} + 36 q^{93} + 25 q^{95} + 40 q^{97} + 19 q^{99}+O(q^{100}) \) 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9 + 8 * q^11 + 8 * q^13 + 7 * q^15 + 19 * q^17 - 9 * q^19 + 9 * q^21 + 21 * q^23 + 65 * q^25 + 5 * q^27 + 10 * q^29 - 9 * q^31 + 34 * q^33 + 12 * q^35 + 11 * q^37 - 9 * q^39 + 35 * q^41 - 9 * q^43 + 29 * q^45 + 37 * q^47 + 77 * q^49 - 17 * q^51 + 38 * q^53 - 20 * q^55 + 51 * q^57 + 17 * q^59 + 22 * q^63 + 41 * q^65 + 9 * q^67 + 8 * q^69 + 13 * q^71 + 41 * q^73 + 25 * q^75 + 36 * q^77 - 36 * q^79 + 127 * q^81 + 29 * q^83 + 34 * q^85 + 10 * q^87 + 36 * q^89 - 6 * q^91 + 36 * q^93 + 25 * q^95 + 40 * q^97 + 19 * q^99

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