LMFDB - Embedded newform 1984.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1984,2,Mod(1,1984)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1984, 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("1984.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1984 = 2^{6} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1984.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:	$15.8423197610$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 992)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1984.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} +1.00000 q^{5} -2.41421 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.41421 q^{3} +1.00000 q^{5} -2.41421 q^{7} -1.00000 q^{9} +0.828427 q^{11} +3.41421 q^{13} -1.41421 q^{15} +2.24264 q^{17} -1.58579 q^{19} +3.41421 q^{21} -2.58579 q^{23} -4.00000 q^{25} +5.65685 q^{27} +7.07107 q^{29} -1.00000 q^{31} -1.17157 q^{33} -2.41421 q^{35} +0.343146 q^{37} -4.82843 q^{39} -6.65685 q^{41} -7.89949 q^{43} -1.00000 q^{45} -1.17157 q^{47} -1.17157 q^{49} -3.17157 q^{51} +4.00000 q^{53} +0.828427 q^{55} +2.24264 q^{57} -10.0711 q^{59} +1.75736 q^{61} +2.41421 q^{63} +3.41421 q^{65} -6.00000 q^{67} +3.65685 q^{69} -3.58579 q^{71} -11.6569 q^{73} +5.65685 q^{75} -2.00000 q^{77} +7.41421 q^{79} -5.00000 q^{81} -4.82843 q^{83} +2.24264 q^{85} -10.0000 q^{87} +3.75736 q^{89} -8.24264 q^{91} +1.41421 q^{93} -1.58579 q^{95} -9.48528 q^{97} -0.828427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^5 - 2 * q^7 - 2 * q^9	$ 2 q + 2 q^{5} - 2 q^{7} - 2 q^{9} - 4 q^{11} + 4 q^{13} - 4 q^{17} - 6 q^{19} + 4 q^{21} - 8 q^{23} - 8 q^{25} - 2 q^{31} - 8 q^{33} - 2 q^{35} + 12 q^{37} - 4 q^{39} - 2 q^{41} + 4 q^{43} - 2 q^{45} - 8 q^{47} - 8 q^{49} - 12 q^{51} + 8 q^{53} - 4 q^{55} - 4 q^{57} - 6 q^{59} + 12 q^{61} + 2 q^{63} + 4 q^{65} - 12 q^{67} - 4 q^{69} - 10 q^{71} - 12 q^{73} - 4 q^{77} + 12 q^{79} - 10 q^{81} - 4 q^{83} - 4 q^{85} - 20 q^{87} + 16 q^{89} - 8 q^{91} - 6 q^{95} - 2 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^7 - 2 * q^9 - 4 * q^11 + 4 * q^13 - 4 * q^17 - 6 * q^19 + 4 * q^21 - 8 * q^23 - 8 * q^25 - 2 * q^31 - 8 * q^33 - 2 * q^35 + 12 * q^37 - 4 * q^39 - 2 * q^41 + 4 * q^43 - 2 * q^45 - 8 * q^47 - 8 * q^49 - 12 * q^51 + 8 * q^53 - 4 * q^55 - 4 * q^57 - 6 * q^59 + 12 * q^61 + 2 * q^63 + 4 * q^65 - 12 * q^67 - 4 * q^69 - 10 * q^71 - 12 * q^73 - 4 * q^77 + 12 * q^79 - 10 * q^81 - 4 * q^83 - 4 * q^85 - 20 * q^87 + 16 * q^89 - 8 * q^91 - 6 * q^95 - 2 * q^97 + 4 * 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
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	−2.41421	−0.912487	−0.456243	−	0.889855i	$-0.650805\pi$
$7$	−2.41421	−0.912487	−0.456243	+	0.889855i	$0.650805\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0.828427	0.249780	0.124890	−	0.992171i	$-0.460142\pi$
$11$	0.828427	0.249780	0.124890	+	0.992171i	$0.460142\pi$
$12$	0	0
$13$	3.41421	0.946932	0.473466	−	0.880812i	$-0.343003\pi$
$13$	3.41421	0.946932	0.473466	+	0.880812i	$0.343003\pi$
$14$	0	0
$15$	−1.41421	−0.365148
$16$	0	0
$17$	2.24264	0.543920	0.271960	−	0.962309i	$-0.412328\pi$
$17$	2.24264	0.543920	0.271960	+	0.962309i	$0.412328\pi$
$18$	0	0
$19$	−1.58579	−0.363804	−0.181902	−	0.983317i	$-0.558225\pi$
$19$	−1.58579	−0.363804	−0.181902	+	0.983317i	$0.558225\pi$
$20$	0	0
$21$	3.41421	0.745042
$22$	0	0
$23$	−2.58579	−0.539174	−0.269587	−	0.962976i	$-0.586887\pi$
$23$	−2.58579	−0.539174	−0.269587	+	0.962976i	$0.586887\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	7.07107	1.31306	0.656532	−	0.754298i	$-0.272023\pi$
$29$	7.07107	1.31306	0.656532	+	0.754298i	$0.272023\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−1.17157	−0.203945
$34$	0	0
$35$	−2.41421	−0.408077
$36$	0	0
$37$	0.343146	0.0564128	0.0282064	−	0.999602i	$-0.491020\pi$
$37$	0.343146	0.0564128	0.0282064	+	0.999602i	$0.491020\pi$
$38$	0	0
$39$	−4.82843	−0.773167
$40$	0	0
$41$	−6.65685	−1.03963	−0.519813	−	0.854280i	$-0.673998\pi$
$41$	−6.65685	−1.03963	−0.519813	+	0.854280i	$0.673998\pi$
$42$	0	0
$43$	−7.89949	−1.20466	−0.602331	−	0.798247i	$-0.705761\pi$
$43$	−7.89949	−1.20466	−0.602331	+	0.798247i	$0.705761\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−1.17157	−0.170891	−0.0854457	−	0.996343i	$-0.527231\pi$
$47$	−1.17157	−0.170891	−0.0854457	+	0.996343i	$0.527231\pi$
$48$	0	0
$49$	−1.17157	−0.167368
$50$	0	0
$51$	−3.17157	−0.444109
$52$	0	0
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	0.828427	0.111705
$56$	0	0
$57$	2.24264	0.297045
$58$	0	0
$59$	−10.0711	−1.31114	−0.655571	−	0.755134i	$-0.727572\pi$
$59$	−10.0711	−1.31114	−0.655571	+	0.755134i	$0.727572\pi$
$60$	0	0
$61$	1.75736	0.225007	0.112503	−	0.993651i	$-0.464113\pi$
$61$	1.75736	0.225007	0.112503	+	0.993651i	$0.464113\pi$
$62$	0	0
$63$	2.41421	0.304162
$64$	0	0
$65$	3.41421	0.423481
$66$	0	0
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	0	0
$69$	3.65685	0.440234
$70$	0	0
$71$	−3.58579	−0.425555	−0.212777	−	0.977101i	$-0.568251\pi$
$71$	−3.58579	−0.425555	−0.212777	+	0.977101i	$0.568251\pi$
$72$	0	0
$73$	−11.6569	−1.36433	−0.682166	−	0.731198i	$-0.738962\pi$
$73$	−11.6569	−1.36433	−0.682166	+	0.731198i	$0.738962\pi$
$74$	0	0
$75$	5.65685	0.653197
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	7.41421	0.834164	0.417082	−	0.908869i	$-0.363053\pi$
$79$	7.41421	0.834164	0.417082	+	0.908869i	$0.363053\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−4.82843	−0.529989	−0.264994	−	0.964250i	$-0.585370\pi$
$83$	−4.82843	−0.529989	−0.264994	+	0.964250i	$0.585370\pi$
$84$	0	0
$85$	2.24264	0.243249
$86$	0	0
$87$	−10.0000	−1.07211
$88$	0	0
$89$	3.75736	0.398279	0.199140	−	0.979971i	$-0.436185\pi$
$89$	3.75736	0.398279	0.199140	+	0.979971i	$0.436185\pi$
$90$	0	0
$91$	−8.24264	−0.864064
$92$	0	0
$93$	1.41421	0.146647
$94$	0	0
$95$	−1.58579	−0.162698
$96$	0	0
$97$	−9.48528	−0.963084	−0.481542	−	0.876423i	$-0.659923\pi$
$97$	−9.48528	−0.963084	−0.481542	+	0.876423i	$0.659923\pi$
$98$	0	0
$99$	−0.828427	−0.0832601
$100$	0	0
$101$	−15.1421	−1.50670	−0.753349	−	0.657620i	$-0.771563\pi$
$101$	−15.1421	−1.50670	−0.753349	+	0.657620i	$0.771563\pi$
$102$	0	0
$103$	3.24264	0.319507	0.159753	−	0.987157i	$-0.448930\pi$
$103$	3.24264	0.319507	0.159753	+	0.987157i	$0.448930\pi$
$104$	0	0
$105$	3.41421	0.333193
$106$	0	0
$107$	−3.58579	−0.346651	−0.173326	−	0.984865i	$-0.555451\pi$
$107$	−3.58579	−0.346651	−0.173326	+	0.984865i	$0.555451\pi$
$108$	0	0
$109$	−4.31371	−0.413178	−0.206589	−	0.978428i	$-0.566236\pi$
$109$	−4.31371	−0.413178	−0.206589	+	0.978428i	$0.566236\pi$
$110$	0	0
$111$	−0.485281	−0.0460609
$112$	0	0
$113$	9.48528	0.892300	0.446150	−	0.894958i	$-0.352795\pi$
$113$	9.48528	0.892300	0.446150	+	0.894958i	$0.352795\pi$
$114$	0	0
$115$	−2.58579	−0.241126
$116$	0	0
$117$	−3.41421	−0.315644
$118$	0	0
$119$	−5.41421	−0.496320
$120$	0	0
$121$	−10.3137	−0.937610
$122$	0	0
$123$	9.41421	0.848851
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−4.82843	−0.428454	−0.214227	−	0.976784i	$-0.568723\pi$
$127$	−4.82843	−0.428454	−0.214227	+	0.976784i	$0.568723\pi$
$128$	0	0
$129$	11.1716	0.983602
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	3.82843	0.331967
$134$	0	0
$135$	5.65685	0.486864
$136$	0	0
$137$	3.41421	0.291696	0.145848	−	0.989307i	$-0.453409\pi$
$137$	3.41421	0.291696	0.145848	+	0.989307i	$0.453409\pi$
$138$	0	0
$139$	12.1421	1.02988	0.514941	−	0.857225i	$-0.327814\pi$
$139$	12.1421	1.02988	0.514941	+	0.857225i	$0.327814\pi$
$140$	0	0
$141$	1.65685	0.139532
$142$	0	0
$143$	2.82843	0.236525
$144$	0	0
$145$	7.07107	0.587220
$146$	0	0
$147$	1.65685	0.136655
$148$	0	0
$149$	−8.48528	−0.695141	−0.347571	−	0.937654i	$-0.612993\pi$
$149$	−8.48528	−0.695141	−0.347571	+	0.937654i	$0.612993\pi$
$150$	0	0
$151$	−13.7574	−1.11956	−0.559779	−	0.828642i	$-0.689114\pi$
$151$	−13.7574	−1.11956	−0.559779	+	0.828642i	$0.689114\pi$
$152$	0	0
$153$	−2.24264	−0.181307
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	15.1421	1.20847	0.604237	−	0.796805i	$-0.293478\pi$
$157$	15.1421	1.20847	0.604237	+	0.796805i	$0.293478\pi$
$158$	0	0
$159$	−5.65685	−0.448618
$160$	0	0
$161$	6.24264	0.491989
$162$	0	0
$163$	−1.58579	−0.124208	−0.0621042	−	0.998070i	$-0.519781\pi$
$163$	−1.58579	−0.124208	−0.0621042	+	0.998070i	$0.519781\pi$
$164$	0	0
$165$	−1.17157	−0.0912068
$166$	0	0
$167$	−24.9706	−1.93228	−0.966140	−	0.258018i	$-0.916931\pi$
$167$	−24.9706	−1.93228	−0.966140	+	0.258018i	$0.916931\pi$
$168$	0	0
$169$	−1.34315	−0.103319
$170$	0	0
$171$	1.58579	0.121268
$172$	0	0
$173$	8.48528	0.645124	0.322562	−	0.946548i	$-0.395456\pi$
$173$	8.48528	0.645124	0.322562	+	0.946548i	$0.395456\pi$
$174$	0	0
$175$	9.65685	0.729990
$176$	0	0
$177$	14.2426	1.07054
$178$	0	0
$179$	−15.7574	−1.17776	−0.588880	−	0.808220i	$-0.700431\pi$
$179$	−15.7574	−1.17776	−0.588880	+	0.808220i	$0.700431\pi$
$180$	0	0
$181$	−11.8995	−0.884482	−0.442241	−	0.896896i	$-0.645816\pi$
$181$	−11.8995	−0.884482	−0.442241	+	0.896896i	$0.645816\pi$
$182$	0	0
$183$	−2.48528	−0.183717
$184$	0	0
$185$	0.343146	0.0252286
$186$	0	0
$187$	1.85786	0.135860
$188$	0	0
$189$	−13.6569	−0.993390
$190$	0	0
$191$	4.75736	0.344230	0.172115	−	0.985077i	$-0.444940\pi$
$191$	4.75736	0.344230	0.172115	+	0.985077i	$0.444940\pi$
$192$	0	0
$193$	−21.9706	−1.58148	−0.790738	−	0.612155i	$-0.790303\pi$
$193$	−21.9706	−1.58148	−0.790738	+	0.612155i	$0.790303\pi$
$194$	0	0
$195$	−4.82843	−0.345771
$196$	0	0
$197$	−16.0000	−1.13995	−0.569976	−	0.821661i	$-0.693048\pi$
$197$	−16.0000	−1.13995	−0.569976	+	0.821661i	$0.693048\pi$
$198$	0	0
$199$	0.828427	0.0587256	0.0293628	−	0.999569i	$-0.490652\pi$
$199$	0.828427	0.0587256	0.0293628	+	0.999569i	$0.490652\pi$
$200$	0	0
$201$	8.48528	0.598506
$202$	0	0
$203$	−17.0711	−1.19815
$204$	0	0
$205$	−6.65685	−0.464935
$206$	0	0
$207$	2.58579	0.179725
$208$	0	0
$209$	−1.31371	−0.0908711
$210$	0	0
$211$	1.24264	0.0855469	0.0427735	−	0.999085i	$-0.486381\pi$
$211$	1.24264	0.0855469	0.0427735	+	0.999085i	$0.486381\pi$
$212$	0	0
$213$	5.07107	0.347464
$214$	0	0
$215$	−7.89949	−0.538741
$216$	0	0
$217$	2.41421	0.163887
$218$	0	0
$219$	16.4853	1.11397
$220$	0	0
$221$	7.65685	0.515056
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	0.485281	0.0322093	0.0161046	−	0.999870i	$-0.494874\pi$
$227$	0.485281	0.0322093	0.0161046	+	0.999870i	$0.494874\pi$
$228$	0	0
$229$	−15.6569	−1.03463	−0.517317	−	0.855794i	$-0.673069\pi$
$229$	−15.6569	−1.03463	−0.517317	+	0.855794i	$0.673069\pi$
$230$	0	0
$231$	2.82843	0.186097
$232$	0	0
$233$	21.9706	1.43934	0.719670	−	0.694317i	$-0.244293\pi$
$233$	21.9706	1.43934	0.719670	+	0.694317i	$0.244293\pi$
$234$	0	0
$235$	−1.17157	−0.0764250
$236$	0	0
$237$	−10.4853	−0.681092
$238$	0	0
$239$	2.72792	0.176455	0.0882273	−	0.996100i	$-0.471880\pi$
$239$	2.72792	0.176455	0.0882273	+	0.996100i	$0.471880\pi$
$240$	0	0
$241$	5.31371	0.342286	0.171143	−	0.985246i	$-0.445254\pi$
$241$	5.31371	0.342286	0.171143	+	0.985246i	$0.445254\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	−1.17157	−0.0748490
$246$	0	0
$247$	−5.41421	−0.344498
$248$	0	0
$249$	6.82843	0.432734
$250$	0	0
$251$	11.8995	0.751089	0.375545	−	0.926804i	$-0.377456\pi$
$251$	11.8995	0.751089	0.375545	+	0.926804i	$0.377456\pi$
$252$	0	0
$253$	−2.14214	−0.134675
$254$	0	0
$255$	−3.17157	−0.198612
$256$	0	0
$257$	−15.1421	−0.944540	−0.472270	−	0.881454i	$-0.656565\pi$
$257$	−15.1421	−0.944540	−0.472270	+	0.881454i	$0.656565\pi$
$258$	0	0
$259$	−0.828427	−0.0514760
$260$	0	0
$261$	−7.07107	−0.437688
$262$	0	0
$263$	29.6985	1.83129	0.915644	−	0.401991i	$-0.131682\pi$
$263$	29.6985	1.83129	0.915644	+	0.401991i	$0.131682\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	−5.31371	−0.325194
$268$	0	0
$269$	−18.3431	−1.11840	−0.559201	−	0.829032i	$-0.688892\pi$
$269$	−18.3431	−1.11840	−0.559201	+	0.829032i	$0.688892\pi$
$270$	0	0
$271$	1.07107	0.0650627	0.0325314	−	0.999471i	$-0.489643\pi$
$271$	1.07107	0.0650627	0.0325314	+	0.999471i	$0.489643\pi$
$272$	0	0
$273$	11.6569	0.705505
$274$	0	0
$275$	−3.31371	−0.199824
$276$	0	0
$277$	20.7279	1.24542	0.622710	−	0.782453i	$-0.286032\pi$
$277$	20.7279	1.24542	0.622710	+	0.782453i	$0.286032\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	−7.97056	−0.475484	−0.237742	−	0.971328i	$-0.576407\pi$
$281$	−7.97056	−0.475484	−0.237742	+	0.971328i	$0.576407\pi$
$282$	0	0
$283$	25.6569	1.52514	0.762571	−	0.646905i	$-0.223937\pi$
$283$	25.6569	1.52514	0.762571	+	0.646905i	$0.223937\pi$
$284$	0	0
$285$	2.24264	0.132843
$286$	0	0
$287$	16.0711	0.948645
$288$	0	0
$289$	−11.9706	−0.704151
$290$	0	0
$291$	13.4142	0.786355
$292$	0	0
$293$	−4.00000	−0.233682	−0.116841	−	0.993151i	$-0.537277\pi$
$293$	−4.00000	−0.233682	−0.116841	+	0.993151i	$0.537277\pi$
$294$	0	0
$295$	−10.0711	−0.586360
$296$	0	0
$297$	4.68629	0.271926
$298$	0	0
$299$	−8.82843	−0.510561
$300$	0	0
$301$	19.0711	1.09924
$302$	0	0
$303$	21.4142	1.23021
$304$	0	0
$305$	1.75736	0.100626
$306$	0	0
$307$	−18.4142	−1.05095	−0.525477	−	0.850808i	$-0.676113\pi$
$307$	−18.4142	−1.05095	−0.525477	+	0.850808i	$0.676113\pi$
$308$	0	0
$309$	−4.58579	−0.260876
$310$	0	0
$311$	−25.8701	−1.46696	−0.733478	−	0.679713i	$-0.762104\pi$
$311$	−25.8701	−1.46696	−0.733478	+	0.679713i	$0.762104\pi$
$312$	0	0
$313$	8.38478	0.473936	0.236968	−	0.971518i	$-0.423846\pi$
$313$	8.38478	0.473936	0.236968	+	0.971518i	$0.423846\pi$
$314$	0	0
$315$	2.41421	0.136026
$316$	0	0
$317$	13.8284	0.776682	0.388341	−	0.921516i	$-0.373048\pi$
$317$	13.8284	0.776682	0.388341	+	0.921516i	$0.373048\pi$
$318$	0	0
$319$	5.85786	0.327977
$320$	0	0
$321$	5.07107	0.283039
$322$	0	0
$323$	−3.55635	−0.197881
$324$	0	0
$325$	−13.6569	−0.757546
$326$	0	0
$327$	6.10051	0.337359
$328$	0	0
$329$	2.82843	0.155936
$330$	0	0
$331$	28.8284	1.58455	0.792277	−	0.610162i	$-0.208896\pi$
$331$	28.8284	1.58455	0.792277	+	0.610162i	$0.208896\pi$
$332$	0	0
$333$	−0.343146	−0.0188043
$334$	0	0
$335$	−6.00000	−0.327815
$336$	0	0
$337$	12.0416	0.655949	0.327975	−	0.944687i	$-0.393634\pi$
$337$	12.0416	0.655949	0.327975	+	0.944687i	$0.393634\pi$
$338$	0	0
$339$	−13.4142	−0.728560
$340$	0	0
$341$	−0.828427	−0.0448618
$342$	0	0
$343$	19.7279	1.06521
$344$	0	0
$345$	3.65685	0.196878
$346$	0	0
$347$	27.0711	1.45325	0.726626	−	0.687034i	$-0.241088\pi$
$347$	27.0711	1.45325	0.726626	+	0.687034i	$0.241088\pi$
$348$	0	0
$349$	−13.1716	−0.705058	−0.352529	−	0.935801i	$-0.614678\pi$
$349$	−13.1716	−0.705058	−0.352529	+	0.935801i	$0.614678\pi$
$350$	0	0
$351$	19.3137	1.03089
$352$	0	0
$353$	0.970563	0.0516578	0.0258289	−	0.999666i	$-0.491777\pi$
$353$	0.970563	0.0516578	0.0258289	+	0.999666i	$0.491777\pi$
$354$	0	0
$355$	−3.58579	−0.190314
$356$	0	0
$357$	7.65685	0.405244
$358$	0	0
$359$	33.8701	1.78759	0.893797	−	0.448472i	$-0.148032\pi$
$359$	33.8701	1.78759	0.893797	+	0.448472i	$0.148032\pi$
$360$	0	0
$361$	−16.4853	−0.867646
$362$	0	0
$363$	14.5858	0.765555
$364$	0	0
$365$	−11.6569	−0.610148
$366$	0	0
$367$	−8.14214	−0.425016	−0.212508	−	0.977159i	$-0.568163\pi$
$367$	−8.14214	−0.425016	−0.212508	+	0.977159i	$0.568163\pi$
$368$	0	0
$369$	6.65685	0.346542
$370$	0	0
$371$	−9.65685	−0.501359
$372$	0	0
$373$	25.4853	1.31958	0.659789	−	0.751451i	$-0.270646\pi$
$373$	25.4853	1.31958	0.659789	+	0.751451i	$0.270646\pi$
$374$	0	0
$375$	12.7279	0.657267
$376$	0	0
$377$	24.1421	1.24338
$378$	0	0
$379$	14.3431	0.736758	0.368379	−	0.929676i	$-0.379913\pi$
$379$	14.3431	0.736758	0.368379	+	0.929676i	$0.379913\pi$
$380$	0	0
$381$	6.82843	0.349831
$382$	0	0
$383$	9.51472	0.486179	0.243090	−	0.970004i	$-0.421839\pi$
$383$	9.51472	0.486179	0.243090	+	0.970004i	$0.421839\pi$
$384$	0	0
$385$	−2.00000	−0.101929
$386$	0	0
$387$	7.89949	0.401554
$388$	0	0
$389$	0.686292	0.0347964	0.0173982	−	0.999849i	$-0.494462\pi$
$389$	0.686292	0.0347964	0.0173982	+	0.999849i	$0.494462\pi$
$390$	0	0
$391$	−5.79899	−0.293268
$392$	0	0
$393$	−5.65685	−0.285351
$394$	0	0
$395$	7.41421	0.373050
$396$	0	0
$397$	28.4558	1.42816	0.714079	−	0.700065i	$-0.246846\pi$
$397$	28.4558	1.42816	0.714079	+	0.700065i	$0.246846\pi$
$398$	0	0
$399$	−5.41421	−0.271050
$400$	0	0
$401$	−26.3848	−1.31759	−0.658796	−	0.752321i	$-0.728934\pi$
$401$	−26.3848	−1.31759	−0.658796	+	0.752321i	$0.728934\pi$
$402$	0	0
$403$	−3.41421	−0.170074
$404$	0	0
$405$	−5.00000	−0.248452
$406$	0	0
$407$	0.284271	0.0140908
$408$	0	0
$409$	19.2132	0.950032	0.475016	−	0.879977i	$-0.342442\pi$
$409$	19.2132	0.950032	0.475016	+	0.879977i	$0.342442\pi$
$410$	0	0
$411$	−4.82843	−0.238169
$412$	0	0
$413$	24.3137	1.19640
$414$	0	0
$415$	−4.82843	−0.237018
$416$	0	0
$417$	−17.1716	−0.840896
$418$	0	0
$419$	27.7279	1.35460	0.677299	−	0.735708i	$-0.263150\pi$
$419$	27.7279	1.35460	0.677299	+	0.735708i	$0.263150\pi$
$420$	0	0
$421$	30.3137	1.47740	0.738700	−	0.674034i	$-0.235440\pi$
$421$	30.3137	1.47740	0.738700	+	0.674034i	$0.235440\pi$
$422$	0	0
$423$	1.17157	0.0569638
$424$	0	0
$425$	−8.97056	−0.435136
$426$	0	0
$427$	−4.24264	−0.205316
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	−10.9706	−0.528433	−0.264217	−	0.964463i	$-0.585113\pi$
$431$	−10.9706	−0.528433	−0.264217	+	0.964463i	$0.585113\pi$
$432$	0	0
$433$	10.3848	0.499061	0.249530	−	0.968367i	$-0.419724\pi$
$433$	10.3848	0.499061	0.249530	+	0.968367i	$0.419724\pi$
$434$	0	0
$435$	−10.0000	−0.479463
$436$	0	0
$437$	4.10051	0.196154
$438$	0	0
$439$	37.3848	1.78428	0.892139	−	0.451761i	$-0.149204\pi$
$439$	37.3848	1.78428	0.892139	+	0.451761i	$0.149204\pi$
$440$	0	0
$441$	1.17157	0.0557892
$442$	0	0
$443$	2.41421	0.114703	0.0573514	−	0.998354i	$-0.481734\pi$
$443$	2.41421	0.114703	0.0573514	+	0.998354i	$0.481734\pi$
$444$	0	0
$445$	3.75736	0.178116
$446$	0	0
$447$	12.0000	0.567581
$448$	0	0
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	−5.51472	−0.259678
$452$	0	0
$453$	19.4558	0.914115
$454$	0	0
$455$	−8.24264	−0.386421
$456$	0	0
$457$	−40.9706	−1.91652	−0.958261	−	0.285895i	$-0.907709\pi$
$457$	−40.9706	−1.91652	−0.958261	+	0.285895i	$0.907709\pi$
$458$	0	0
$459$	12.6863	0.592145
$460$	0	0
$461$	−1.31371	−0.0611855	−0.0305928	−	0.999532i	$-0.509739\pi$
$461$	−1.31371	−0.0611855	−0.0305928	+	0.999532i	$0.509739\pi$
$462$	0	0
$463$	27.1716	1.26277	0.631385	−	0.775469i	$-0.282487\pi$
$463$	27.1716	1.26277	0.631385	+	0.775469i	$0.282487\pi$
$464$	0	0
$465$	1.41421	0.0655826
$466$	0	0
$467$	5.72792	0.265057	0.132528	−	0.991179i	$-0.457690\pi$
$467$	5.72792	0.265057	0.132528	+	0.991179i	$0.457690\pi$
$468$	0	0
$469$	14.4853	0.668868
$470$	0	0
$471$	−21.4142	−0.986715
$472$	0	0
$473$	−6.54416	−0.300901
$474$	0	0
$475$	6.34315	0.291043
$476$	0	0
$477$	−4.00000	−0.183147
$478$	0	0
$479$	−1.10051	−0.0502834	−0.0251417	−	0.999684i	$-0.508004\pi$
$479$	−1.10051	−0.0502834	−0.0251417	+	0.999684i	$0.508004\pi$
$480$	0	0
$481$	1.17157	0.0534191
$482$	0	0
$483$	−8.82843	−0.401707
$484$	0	0
$485$	−9.48528	−0.430704
$486$	0	0
$487$	−36.5269	−1.65519	−0.827596	−	0.561324i	$-0.810292\pi$
$487$	−36.5269	−1.65519	−0.827596	+	0.561324i	$0.810292\pi$
$488$	0	0
$489$	2.24264	0.101416
$490$	0	0
$491$	−14.3431	−0.647297	−0.323649	−	0.946177i	$-0.604910\pi$
$491$	−14.3431	−0.647297	−0.323649	+	0.946177i	$0.604910\pi$
$492$	0	0
$493$	15.8579	0.714202
$494$	0	0
$495$	−0.828427	−0.0372350
$496$	0	0
$497$	8.65685	0.388313
$498$	0	0
$499$	−4.14214	−0.185427	−0.0927137	−	0.995693i	$-0.529554\pi$
$499$	−4.14214	−0.185427	−0.0927137	+	0.995693i	$0.529554\pi$
$500$	0	0
$501$	35.3137	1.57770
$502$	0	0
$503$	−34.5563	−1.54079	−0.770396	−	0.637566i	$-0.779941\pi$
$503$	−34.5563	−1.54079	−0.770396	+	0.637566i	$0.779941\pi$
$504$	0	0
$505$	−15.1421	−0.673816
$506$	0	0
$507$	1.89949	0.0843595
$508$	0	0
$509$	−15.5563	−0.689523	−0.344762	−	0.938690i	$-0.612040\pi$
$509$	−15.5563	−0.689523	−0.344762	+	0.938690i	$0.612040\pi$
$510$	0	0
$511$	28.1421	1.24493
$512$	0	0
$513$	−8.97056	−0.396060
$514$	0	0
$515$	3.24264	0.142888
$516$	0	0
$517$	−0.970563	−0.0426853
$518$	0	0
$519$	−12.0000	−0.526742
$520$	0	0
$521$	−15.7990	−0.692166	−0.346083	−	0.938204i	$-0.612488\pi$
$521$	−15.7990	−0.692166	−0.346083	+	0.938204i	$0.612488\pi$
$522$	0	0
$523$	39.5563	1.72968	0.864839	−	0.502049i	$-0.167420\pi$
$523$	39.5563	1.72968	0.864839	+	0.502049i	$0.167420\pi$
$524$	0	0
$525$	−13.6569	−0.596034
$526$	0	0
$527$	−2.24264	−0.0976910
$528$	0	0
$529$	−16.3137	−0.709292
$530$	0	0
$531$	10.0711	0.437047
$532$	0	0
$533$	−22.7279	−0.984456
$534$	0	0
$535$	−3.58579	−0.155027
$536$	0	0
$537$	22.2843	0.961637
$538$	0	0
$539$	−0.970563	−0.0418051
$540$	0	0
$541$	−38.4558	−1.65335	−0.826673	−	0.562683i	$-0.809769\pi$
$541$	−38.4558	−1.65335	−0.826673	+	0.562683i	$0.809769\pi$
$542$	0	0
$543$	16.8284	0.722177
$544$	0	0
$545$	−4.31371	−0.184779
$546$	0	0
$547$	15.3848	0.657806	0.328903	−	0.944364i	$-0.393321\pi$
$547$	15.3848	0.657806	0.328903	+	0.944364i	$0.393321\pi$
$548$	0	0
$549$	−1.75736	−0.0750023
$550$	0	0
$551$	−11.2132	−0.477699
$552$	0	0
$553$	−17.8995	−0.761164
$554$	0	0
$555$	−0.485281	−0.0205990
$556$	0	0
$557$	−11.3137	−0.479377	−0.239689	−	0.970850i	$-0.577045\pi$
$557$	−11.3137	−0.479377	−0.239689	+	0.970850i	$0.577045\pi$
$558$	0	0
$559$	−26.9706	−1.14073
$560$	0	0
$561$	−2.62742	−0.110930
$562$	0	0
$563$	−9.92893	−0.418455	−0.209227	−	0.977867i	$-0.567095\pi$
$563$	−9.92893	−0.418455	−0.209227	+	0.977867i	$0.567095\pi$
$564$	0	0
$565$	9.48528	0.399049
$566$	0	0
$567$	12.0711	0.506937
$568$	0	0
$569$	−19.3137	−0.809673	−0.404836	−	0.914389i	$-0.632672\pi$
$569$	−19.3137	−0.809673	−0.404836	+	0.914389i	$0.632672\pi$
$570$	0	0
$571$	26.8701	1.12448	0.562238	−	0.826975i	$-0.309940\pi$
$571$	26.8701	1.12448	0.562238	+	0.826975i	$0.309940\pi$
$572$	0	0
$573$	−6.72792	−0.281063
$574$	0	0
$575$	10.3431	0.431339
$576$	0	0
$577$	−34.1421	−1.42136	−0.710678	−	0.703518i	$-0.751612\pi$
$577$	−34.1421	−1.42136	−0.710678	+	0.703518i	$0.751612\pi$
$578$	0	0
$579$	31.0711	1.29127
$580$	0	0
$581$	11.6569	0.483608
$582$	0	0
$583$	3.31371	0.137240
$584$	0	0
$585$	−3.41421	−0.141160
$586$	0	0
$587$	−31.3137	−1.29246	−0.646228	−	0.763145i	$-0.723654\pi$
$587$	−31.3137	−1.29246	−0.646228	+	0.763145i	$0.723654\pi$
$588$	0	0
$589$	1.58579	0.0653412
$590$	0	0
$591$	22.6274	0.930768
$592$	0	0
$593$	−14.3137	−0.587794	−0.293897	−	0.955837i	$-0.594952\pi$
$593$	−14.3137	−0.587794	−0.293897	+	0.955837i	$0.594952\pi$
$594$	0	0
$595$	−5.41421	−0.221961
$596$	0	0
$597$	−1.17157	−0.0479493
$598$	0	0
$599$	−28.3553	−1.15857	−0.579284	−	0.815126i	$-0.696668\pi$
$599$	−28.3553	−1.15857	−0.579284	+	0.815126i	$0.696668\pi$
$600$	0	0
$601$	−7.75736	−0.316429	−0.158215	−	0.987405i	$-0.550574\pi$
$601$	−7.75736	−0.316429	−0.158215	+	0.987405i	$0.550574\pi$
$602$	0	0
$603$	6.00000	0.244339
$604$	0	0
$605$	−10.3137	−0.419312
$606$	0	0
$607$	40.4853	1.64325	0.821623	−	0.570031i	$-0.193069\pi$
$607$	40.4853	1.64325	0.821623	+	0.570031i	$0.193069\pi$
$608$	0	0
$609$	24.1421	0.978289
$610$	0	0
$611$	−4.00000	−0.161823
$612$	0	0
$613$	17.3137	0.699294	0.349647	−	0.936881i	$-0.386302\pi$
$613$	17.3137	0.699294	0.349647	+	0.936881i	$0.386302\pi$
$614$	0	0
$615$	9.41421	0.379618
$616$	0	0
$617$	27.3137	1.09961	0.549804	−	0.835294i	$-0.314702\pi$
$617$	27.3137	1.09961	0.549804	+	0.835294i	$0.314702\pi$
$618$	0	0
$619$	9.07107	0.364597	0.182298	−	0.983243i	$-0.441646\pi$
$619$	9.07107	0.364597	0.182298	+	0.983243i	$0.441646\pi$
$620$	0	0
$621$	−14.6274	−0.586978
$622$	0	0
$623$	−9.07107	−0.363425
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	1.85786	0.0741960
$628$	0	0
$629$	0.769553	0.0306841
$630$	0	0
$631$	29.7990	1.18628	0.593140	−	0.805100i	$-0.297888\pi$
$631$	29.7990	1.18628	0.593140	+	0.805100i	$0.297888\pi$
$632$	0	0
$633$	−1.75736	−0.0698488
$634$	0	0
$635$	−4.82843	−0.191610
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	3.58579	0.141852
$640$	0	0
$641$	−37.6569	−1.48736	−0.743678	−	0.668538i	$-0.766920\pi$
$641$	−37.6569	−1.48736	−0.743678	+	0.668538i	$0.766920\pi$
$642$	0	0
$643$	43.1716	1.70252	0.851260	−	0.524744i	$-0.175839\pi$
$643$	43.1716	1.70252	0.851260	+	0.524744i	$0.175839\pi$
$644$	0	0
$645$	11.1716	0.439880
$646$	0	0
$647$	−24.6863	−0.970518	−0.485259	−	0.874370i	$-0.661275\pi$
$647$	−24.6863	−0.970518	−0.485259	+	0.874370i	$0.661275\pi$
$648$	0	0
$649$	−8.34315	−0.327497
$650$	0	0
$651$	−3.41421	−0.133814
$652$	0	0
$653$	10.3431	0.404759	0.202379	−	0.979307i	$-0.435133\pi$
$653$	10.3431	0.404759	0.202379	+	0.979307i	$0.435133\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	0	0
$657$	11.6569	0.454777
$658$	0	0
$659$	−37.2426	−1.45077	−0.725384	−	0.688345i	$-0.758338\pi$
$659$	−37.2426	−1.45077	−0.725384	+	0.688345i	$0.758338\pi$
$660$	0	0
$661$	−11.1421	−0.433379	−0.216689	−	0.976241i	$-0.569526\pi$
$661$	−11.1421	−0.433379	−0.216689	+	0.976241i	$0.569526\pi$
$662$	0	0
$663$	−10.8284	−0.420541
$664$	0	0
$665$	3.82843	0.148460
$666$	0	0
$667$	−18.2843	−0.707970
$668$	0	0
$669$	−5.65685	−0.218707
$670$	0	0
$671$	1.45584	0.0562022
$672$	0	0
$673$	−28.5269	−1.09963	−0.549816	−	0.835286i	$-0.685302\pi$
$673$	−28.5269	−1.09963	−0.549816	+	0.835286i	$0.685302\pi$
$674$	0	0
$675$	−22.6274	−0.870930
$676$	0	0
$677$	31.9411	1.22760	0.613799	−	0.789463i	$-0.289641\pi$
$677$	31.9411	1.22760	0.613799	+	0.789463i	$0.289641\pi$
$678$	0	0
$679$	22.8995	0.878802
$680$	0	0
$681$	−0.686292	−0.0262987
$682$	0	0
$683$	−35.7279	−1.36709	−0.683545	−	0.729908i	$-0.739563\pi$
$683$	−35.7279	−1.36709	−0.683545	+	0.729908i	$0.739563\pi$
$684$	0	0
$685$	3.41421	0.130450
$686$	0	0
$687$	22.1421	0.844775
$688$	0	0
$689$	13.6569	0.520285
$690$	0	0
$691$	−39.0416	−1.48521	−0.742607	−	0.669728i	$-0.766411\pi$
$691$	−39.0416	−1.48521	−0.742607	+	0.669728i	$0.766411\pi$
$692$	0	0
$693$	2.00000	0.0759737
$694$	0	0
$695$	12.1421	0.460577
$696$	0	0
$697$	−14.9289	−0.565474
$698$	0	0
$699$	−31.0711	−1.17522
$700$	0	0
$701$	−35.4853	−1.34026	−0.670130	−	0.742243i	$-0.733762\pi$
$701$	−35.4853	−1.34026	−0.670130	+	0.742243i	$0.733762\pi$
$702$	0	0
$703$	−0.544156	−0.0205232
$704$	0	0
$705$	1.65685	0.0624007
$706$	0	0
$707$	36.5563	1.37484
$708$	0	0
$709$	−5.65685	−0.212448	−0.106224	−	0.994342i	$-0.533876\pi$
$709$	−5.65685	−0.212448	−0.106224	+	0.994342i	$0.533876\pi$
$710$	0	0
$711$	−7.41421	−0.278055
$712$	0	0
$713$	2.58579	0.0968385
$714$	0	0
$715$	2.82843	0.105777
$716$	0	0
$717$	−3.85786	−0.144075
$718$	0	0
$719$	−33.0711	−1.23334	−0.616671	−	0.787221i	$-0.711519\pi$
$719$	−33.0711	−1.23334	−0.616671	+	0.787221i	$0.711519\pi$
$720$	0	0
$721$	−7.82843	−0.291546
$722$	0	0
$723$	−7.51472	−0.279475
$724$	0	0
$725$	−28.2843	−1.05045
$726$	0	0
$727$	9.10051	0.337519	0.168760	−	0.985657i	$-0.446024\pi$
$727$	9.10051	0.337519	0.168760	+	0.985657i	$0.446024\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	−17.7157	−0.655240
$732$	0	0
$733$	22.1716	0.818926	0.409463	−	0.912327i	$-0.365716\pi$
$733$	22.1716	0.818926	0.409463	+	0.912327i	$0.365716\pi$
$734$	0	0
$735$	1.65685	0.0611140
$736$	0	0
$737$	−4.97056	−0.183093
$738$	0	0
$739$	−4.87006	−0.179148	−0.0895740	−	0.995980i	$-0.528551\pi$
$739$	−4.87006	−0.179148	−0.0895740	+	0.995980i	$0.528551\pi$
$740$	0	0
$741$	7.65685	0.281282
$742$	0	0
$743$	−2.44365	−0.0896489	−0.0448244	−	0.998995i	$-0.514273\pi$
$743$	−2.44365	−0.0896489	−0.0448244	+	0.998995i	$0.514273\pi$
$744$	0	0
$745$	−8.48528	−0.310877
$746$	0	0
$747$	4.82843	0.176663
$748$	0	0
$749$	8.65685	0.316315
$750$	0	0
$751$	−32.0711	−1.17029	−0.585145	−	0.810929i	$-0.698962\pi$
$751$	−32.0711	−1.17029	−0.585145	+	0.810929i	$0.698962\pi$
$752$	0	0
$753$	−16.8284	−0.613262
$754$	0	0
$755$	−13.7574	−0.500682
$756$	0	0
$757$	3.75736	0.136564	0.0682818	−	0.997666i	$-0.478248\pi$
$757$	3.75736	0.136564	0.0682818	+	0.997666i	$0.478248\pi$
$758$	0	0
$759$	3.02944	0.109962
$760$	0	0
$761$	−2.68629	−0.0973780	−0.0486890	−	0.998814i	$-0.515504\pi$
$761$	−2.68629	−0.0973780	−0.0486890	+	0.998814i	$0.515504\pi$
$762$	0	0
$763$	10.4142	0.377020
$764$	0	0
$765$	−2.24264	−0.0810828
$766$	0	0
$767$	−34.3848	−1.24156
$768$	0	0
$769$	45.4853	1.64024	0.820121	−	0.572191i	$-0.193906\pi$
$769$	45.4853	1.64024	0.820121	+	0.572191i	$0.193906\pi$
$770$	0	0
$771$	21.4142	0.771214
$772$	0	0
$773$	−27.0711	−0.973679	−0.486839	−	0.873492i	$-0.661850\pi$
$773$	−27.0711	−0.973679	−0.486839	+	0.873492i	$0.661850\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	1.17157	0.0420299
$778$	0	0
$779$	10.5563	0.378220
$780$	0	0
$781$	−2.97056	−0.106295
$782$	0	0
$783$	40.0000	1.42948
$784$	0	0
$785$	15.1421	0.540446
$786$	0	0
$787$	−30.7279	−1.09533	−0.547666	−	0.836697i	$-0.684484\pi$
$787$	−30.7279	−1.09533	−0.547666	+	0.836697i	$0.684484\pi$
$788$	0	0
$789$	−42.0000	−1.49524
$790$	0	0
$791$	−22.8995	−0.814212
$792$	0	0
$793$	6.00000	0.213066
$794$	0	0
$795$	−5.65685	−0.200628
$796$	0	0
$797$	34.6274	1.22657	0.613283	−	0.789863i	$-0.289849\pi$
$797$	34.6274	1.22657	0.613283	+	0.789863i	$0.289849\pi$
$798$	0	0
$799$	−2.62742	−0.0929513
$800$	0	0
$801$	−3.75736	−0.132760
$802$	0	0
$803$	−9.65685	−0.340783
$804$	0	0
$805$	6.24264	0.220024
$806$	0	0
$807$	25.9411	0.913171
$808$	0	0
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	0	0
$811$	−3.02944	−0.106378	−0.0531890	−	0.998584i	$-0.516939\pi$
$811$	−3.02944	−0.106378	−0.0531890	+	0.998584i	$0.516939\pi$
$812$	0	0
$813$	−1.51472	−0.0531235
$814$	0	0
$815$	−1.58579	−0.0555477
$816$	0	0
$817$	12.5269	0.438261
$818$	0	0
$819$	8.24264	0.288021
$820$	0	0
$821$	−17.4142	−0.607760	−0.303880	−	0.952710i	$-0.598282\pi$
$821$	−17.4142	−0.607760	−0.303880	+	0.952710i	$0.598282\pi$
$822$	0	0
$823$	24.2426	0.845045	0.422523	−	0.906352i	$-0.361145\pi$
$823$	24.2426	0.845045	0.422523	+	0.906352i	$0.361145\pi$
$824$	0	0
$825$	4.68629	0.163156
$826$	0	0
$827$	2.58579	0.0899166	0.0449583	−	0.998989i	$-0.485685\pi$
$827$	2.58579	0.0899166	0.0449583	+	0.998989i	$0.485685\pi$
$828$	0	0
$829$	11.9411	0.414732	0.207366	−	0.978263i	$-0.433511\pi$
$829$	11.9411	0.414732	0.207366	+	0.978263i	$0.433511\pi$
$830$	0	0
$831$	−29.3137	−1.01688
$832$	0	0
$833$	−2.62742	−0.0910346
$834$	0	0
$835$	−24.9706	−0.864142
$836$	0	0
$837$	−5.65685	−0.195529
$838$	0	0
$839$	−24.9706	−0.862080	−0.431040	−	0.902333i	$-0.641853\pi$
$839$	−24.9706	−0.862080	−0.431040	+	0.902333i	$0.641853\pi$
$840$	0	0
$841$	21.0000	0.724138
$842$	0	0
$843$	11.2721	0.388231
$844$	0	0
$845$	−1.34315	−0.0462056
$846$	0	0
$847$	24.8995	0.855557
$848$	0	0
$849$	−36.2843	−1.24527
$850$	0	0
$851$	−0.887302	−0.0304163
$852$	0	0
$853$	2.34315	0.0802278	0.0401139	−	0.999195i	$-0.487228\pi$
$853$	2.34315	0.0802278	0.0401139	+	0.999195i	$0.487228\pi$
$854$	0	0
$855$	1.58579	0.0542328
$856$	0	0
$857$	27.5147	0.939885	0.469942	−	0.882697i	$-0.344275\pi$
$857$	27.5147	0.939885	0.469942	+	0.882697i	$0.344275\pi$
$858$	0	0
$859$	10.3431	0.352904	0.176452	−	0.984309i	$-0.443538\pi$
$859$	10.3431	0.352904	0.176452	+	0.984309i	$0.443538\pi$
$860$	0	0
$861$	−22.7279	−0.774566
$862$	0	0
$863$	−32.3848	−1.10239	−0.551195	−	0.834376i	$-0.685828\pi$
$863$	−32.3848	−1.10239	−0.551195	+	0.834376i	$0.685828\pi$
$864$	0	0
$865$	8.48528	0.288508
$866$	0	0
$867$	16.9289	0.574937
$868$	0	0
$869$	6.14214	0.208358
$870$	0	0
$871$	−20.4853	−0.694117
$872$	0	0
$873$	9.48528	0.321028
$874$	0	0
$875$	21.7279	0.734538
$876$	0	0
$877$	−4.65685	−0.157251	−0.0786254	−	0.996904i	$-0.525053\pi$
$877$	−4.65685	−0.157251	−0.0786254	+	0.996904i	$0.525053\pi$
$878$	0	0
$879$	5.65685	0.190801
$880$	0	0
$881$	50.6274	1.70568	0.852841	−	0.522171i	$-0.174878\pi$
$881$	50.6274	1.70568	0.852841	+	0.522171i	$0.174878\pi$
$882$	0	0
$883$	27.5980	0.928746	0.464373	−	0.885640i	$-0.346280\pi$
$883$	27.5980	0.928746	0.464373	+	0.885640i	$0.346280\pi$
$884$	0	0
$885$	14.2426	0.478761
$886$	0	0
$887$	38.8406	1.30414	0.652070	−	0.758159i	$-0.273901\pi$
$887$	38.8406	1.30414	0.652070	+	0.758159i	$0.273901\pi$
$888$	0	0
$889$	11.6569	0.390958
$890$	0	0
$891$	−4.14214	−0.138767
$892$	0	0
$893$	1.85786	0.0621711
$894$	0	0
$895$	−15.7574	−0.526710
$896$	0	0
$897$	12.4853	0.416871
$898$	0	0
$899$	−7.07107	−0.235833
$900$	0	0
$901$	8.97056	0.298853
$902$	0	0
$903$	−26.9706	−0.897524
$904$	0	0
$905$	−11.8995	−0.395553
$906$	0	0
$907$	22.6985	0.753691	0.376845	−	0.926276i	$-0.377009\pi$
$907$	22.6985	0.753691	0.376845	+	0.926276i	$0.377009\pi$
$908$	0	0
$909$	15.1421	0.502233
$910$	0	0
$911$	−56.8701	−1.88419	−0.942095	−	0.335347i	$-0.891146\pi$
$911$	−56.8701	−1.88419	−0.942095	+	0.335347i	$0.891146\pi$
$912$	0	0
$913$	−4.00000	−0.132381
$914$	0	0
$915$	−2.48528	−0.0821609
$916$	0	0
$917$	−9.65685	−0.318897
$918$	0	0
$919$	−25.4558	−0.839711	−0.419855	−	0.907591i	$-0.637919\pi$
$919$	−25.4558	−0.839711	−0.419855	+	0.907591i	$0.637919\pi$
$920$	0	0
$921$	26.0416	0.858101
$922$	0	0
$923$	−12.2426	−0.402971
$924$	0	0
$925$	−1.37258	−0.0451303
$926$	0	0
$927$	−3.24264	−0.106502
$928$	0	0
$929$	35.3137	1.15861	0.579303	−	0.815113i	$-0.303325\pi$
$929$	35.3137	1.15861	0.579303	+	0.815113i	$0.303325\pi$
$930$	0	0
$931$	1.85786	0.0608890
$932$	0	0
$933$	36.5858	1.19776
$934$	0	0
$935$	1.85786	0.0607587
$936$	0	0
$937$	47.7990	1.56152	0.780762	−	0.624828i	$-0.214831\pi$
$937$	47.7990	1.56152	0.780762	+	0.624828i	$0.214831\pi$
$938$	0	0
$939$	−11.8579	−0.386967
$940$	0	0
$941$	−16.3431	−0.532771	−0.266386	−	0.963867i	$-0.585829\pi$
$941$	−16.3431	−0.532771	−0.266386	+	0.963867i	$0.585829\pi$
$942$	0	0
$943$	17.2132	0.560539
$944$	0	0
$945$	−13.6569	−0.444258
$946$	0	0
$947$	−53.7990	−1.74823	−0.874116	−	0.485717i	$-0.838559\pi$
$947$	−53.7990	−1.74823	−0.874116	+	0.485717i	$0.838559\pi$
$948$	0	0
$949$	−39.7990	−1.29193
$950$	0	0
$951$	−19.5563	−0.634158
$952$	0	0
$953$	−22.4437	−0.727021	−0.363511	−	0.931590i	$-0.618422\pi$
$953$	−22.4437	−0.727021	−0.363511	+	0.931590i	$0.618422\pi$
$954$	0	0
$955$	4.75736	0.153945
$956$	0	0
$957$	−8.28427	−0.267792
$958$	0	0
$959$	−8.24264	−0.266169
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	3.58579	0.115550
$964$	0	0
$965$	−21.9706	−0.707257
$966$	0	0
$967$	−26.3431	−0.847138	−0.423569	−	0.905864i	$-0.639223\pi$
$967$	−26.3431	−0.847138	−0.423569	+	0.905864i	$0.639223\pi$
$968$	0	0
$969$	5.02944	0.161569
$970$	0	0
$971$	−59.5980	−1.91259	−0.956295	−	0.292403i	$-0.905545\pi$
$971$	−59.5980	−1.91259	−0.956295	+	0.292403i	$0.905545\pi$
$972$	0	0
$973$	−29.3137	−0.939754
$974$	0	0
$975$	19.3137	0.618534
$976$	0	0
$977$	5.97056	0.191015	0.0955076	−	0.995429i	$-0.469553\pi$
$977$	5.97056	0.191015	0.0955076	+	0.995429i	$0.469553\pi$
$978$	0	0
$979$	3.11270	0.0994823
$980$	0	0
$981$	4.31371	0.137726
$982$	0	0
$983$	18.4853	0.589589	0.294794	−	0.955561i	$-0.404749\pi$
$983$	18.4853	0.589589	0.294794	+	0.955561i	$0.404749\pi$
$984$	0	0
$985$	−16.0000	−0.509802
$986$	0	0
$987$	−4.00000	−0.127321
$988$	0	0
$989$	20.4264	0.649522
$990$	0	0
$991$	−29.8995	−0.949789	−0.474894	−	0.880043i	$-0.657514\pi$
$991$	−29.8995	−0.949789	−0.474894	+	0.880043i	$0.657514\pi$
$992$	0	0
$993$	−40.7696	−1.29378
$994$	0	0
$995$	0.828427	0.0262629
$996$	0	0
$997$	13.0000	0.411714	0.205857	−	0.978582i	$-0.434002\pi$
$997$	13.0000	0.411714	0.205857	+	0.978582i	$0.434002\pi$
$998$	0	0
$999$	1.94113	0.0614145

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1984.2.a.p.1.1		2
4.3	odd	2		1984.2.a.q.1.2		2
8.3	odd	2		992.2.a.b.1.1	yes	2
8.5	even	2		992.2.a.a.1.2	✓	2
24.5	odd	2		8928.2.a.s.1.1		2
24.11	even	2		8928.2.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
992.2.a.a.1.2	✓	2	8.5	even	2
992.2.a.b.1.1	yes	2	8.3	odd	2
1984.2.a.p.1.1		2	1.1	even	1	trivial
1984.2.a.q.1.2		2	4.3	odd	2
8928.2.a.s.1.1		2	24.5	odd	2
8928.2.a.v.1.2		2	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1984 = 2^{6} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1984.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} +1.00000 q^{5} -2.41421 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{3} +1.00000 q^{5} -2.41421 q^{7} -1.00000 q^{9} +0.828427 q^{11} +3.41421 q^{13} -1.41421 q^{15} +2.24264 q^{17} -1.58579 q^{19} +3.41421 q^{21} -2.58579 q^{23} -4.00000 q^{25} +5.65685 q^{27} +7.07107 q^{29} -1.00000 q^{31} -1.17157 q^{33} -2.41421 q^{35} +0.343146 q^{37} -4.82843 q^{39} -6.65685 q^{41} -7.89949 q^{43} -1.00000 q^{45} -1.17157 q^{47} -1.17157 q^{49} -3.17157 q^{51} +4.00000 q^{53} +0.828427 q^{55} +2.24264 q^{57} -10.0711 q^{59} +1.75736 q^{61} +2.41421 q^{63} +3.41421 q^{65} -6.00000 q^{67} +3.65685 q^{69} -3.58579 q^{71} -11.6569 q^{73} +5.65685 q^{75} -2.00000 q^{77} +7.41421 q^{79} -5.00000 q^{81} -4.82843 q^{83} +2.24264 q^{85} -10.0000 q^{87} +3.75736 q^{89} -8.24264 q^{91} +1.41421 q^{93} -1.58579 q^{95} -9.48528 q^{97} -0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^5 - 2 * q^7 - 2 * q^9	\( 2 q + 2 q^{5} - 2 q^{7} - 2 q^{9} - 4 q^{11} + 4 q^{13} - 4 q^{17} - 6 q^{19} + 4 q^{21} - 8 q^{23} - 8 q^{25} - 2 q^{31} - 8 q^{33} - 2 q^{35} + 12 q^{37} - 4 q^{39} - 2 q^{41} + 4 q^{43} - 2 q^{45} - 8 q^{47} - 8 q^{49} - 12 q^{51} + 8 q^{53} - 4 q^{55} - 4 q^{57} - 6 q^{59} + 12 q^{61} + 2 q^{63} + 4 q^{65} - 12 q^{67} - 4 q^{69} - 10 q^{71} - 12 q^{73} - 4 q^{77} + 12 q^{79} - 10 q^{81} - 4 q^{83} - 4 q^{85} - 20 q^{87} + 16 q^{89} - 8 q^{91} - 6 q^{95} - 2 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^7 - 2 * q^9 - 4 * q^11 + 4 * q^13 - 4 * q^17 - 6 * q^19 + 4 * q^21 - 8 * q^23 - 8 * q^25 - 2 * q^31 - 8 * q^33 - 2 * q^35 + 12 * q^37 - 4 * q^39 - 2 * q^41 + 4 * q^43 - 2 * q^45 - 8 * q^47 - 8 * q^49 - 12 * q^51 + 8 * q^53 - 4 * q^55 - 4 * q^57 - 6 * q^59 + 12 * q^61 + 2 * q^63 + 4 * q^65 - 12 * q^67 - 4 * q^69 - 10 * q^71 - 12 * q^73 - 4 * q^77 + 12 * q^79 - 10 * q^81 - 4 * q^83 - 4 * q^85 - 20 * q^87 + 16 * q^89 - 8 * q^91 - 6 * q^95 - 2 * q^97 + 4 * q^99

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