LMFDB - Embedded newform 8464.2.a.bn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8464,2,Mod(1,8464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8464, 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("8464.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.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:	$67.5853802708$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4232)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.93185$ of defining polynomial
Character	$\chi$	$=$	8464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} +0.0681483 q^{5} -2.73205 q^{7} -1.00000 q^{9} +2.73205 q^{11} +0.0352762 q^{13} -0.0963763 q^{15} -1.55051 q^{17} +2.09638 q^{19} +3.86370 q^{21} -4.99536 q^{25} +5.65685 q^{27} -1.93890 q^{29} +9.32780 q^{31} -3.86370 q^{33} -0.186185 q^{35} -9.08516 q^{37} -0.0498881 q^{39} -4.46410 q^{41} +2.02922 q^{43} -0.0681483 q^{45} +0.944060 q^{47} +0.464102 q^{49} +2.19275 q^{51} +12.4670 q^{53} +0.186185 q^{55} -2.96472 q^{57} +9.47067 q^{59} +6.81017 q^{61} +2.73205 q^{63} +0.00240401 q^{65} -3.19151 q^{67} -10.4130 q^{71} +12.9236 q^{73} +7.06450 q^{75} -7.46410 q^{77} -10.1928 q^{79} -5.00000 q^{81} +7.16228 q^{83} -0.105665 q^{85} +2.74202 q^{87} -12.2037 q^{89} -0.0963763 q^{91} -13.1915 q^{93} +0.142865 q^{95} -11.1732 q^{97} -2.73205 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5} - 4 q^{7} - 4 q^{9} + 4 q^{11} - 4 q^{13} + 4 q^{15} - 16 q^{17} + 4 q^{19} + 4 q^{25} - 8 q^{29} + 8 q^{31} - 8 q^{35} - 24 q^{37} + 8 q^{39} - 4 q^{41} - 8 q^{45} + 8 q^{47} - 12 q^{49}+ \cdots - 4 q^{99}+O(q^{100}) $ 4 * q + 8 * q^5 - 4 * q^7 - 4 * q^9 + 4 * q^11 - 4 * q^13 + 4 * q^15 - 16 * q^17 + 4 * q^19 + 4 * q^25 - 8 * q^29 + 8 * q^31 - 8 * q^35 - 24 * q^37 + 8 * q^39 - 4 * q^41 - 8 * q^45 + 8 * q^47 - 12 * q^49 + 8 * q^55 - 16 * q^57 + 8 * q^59 + 4 * q^63 - 16 * q^65 + 32 * q^67 + 16 * q^75 - 16 * q^77 - 32 * q^79 - 20 * q^81 - 8 * q^83 - 44 * q^85 - 24 * q^87 - 16 * q^89 + 4 * q^91 - 8 * q^93 + 16 * q^97 - 4 * q^99

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
$5$	0.0681483	0.0304769	0.0152384	−	0.999884i	$-0.495149\pi$
$5$	0.0681483	0.0304769	0.0152384	+	0.999884i	$0.495149\pi$
$6$	0	0
$7$	−2.73205	−1.03262	−0.516309	−	0.856402i	$-0.672694\pi$
$7$	−2.73205	−1.03262	−0.516309	+	0.856402i	$0.672694\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.73205	0.823744	0.411872	−	0.911242i	$-0.364875\pi$
$11$	2.73205	0.823744	0.411872	+	0.911242i	$0.364875\pi$
$12$	0	0
$13$	0.0352762	0.00978385	0.00489193	−	0.999988i	$-0.498443\pi$
$13$	0.0352762	0.00978385	0.00489193	+	0.999988i	$0.498443\pi$
$14$	0	0
$15$	−0.0963763	−0.0248843
$16$	0	0
$17$	−1.55051	−0.376054	−0.188027	−	0.982164i	$-0.560209\pi$
$17$	−1.55051	−0.376054	−0.188027	+	0.982164i	$0.560209\pi$
$18$	0	0
$19$	2.09638	0.480942	0.240471	−	0.970656i	$-0.422698\pi$
$19$	2.09638	0.480942	0.240471	+	0.970656i	$0.422698\pi$
$20$	0	0
$21$	3.86370	0.843129
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−4.99536	−0.999071
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	−1.93890	−0.360045	−0.180022	−	0.983663i	$-0.557617\pi$
$29$	−1.93890	−0.360045	−0.180022	+	0.983663i	$0.557617\pi$
$30$	0	0
$31$	9.32780	1.67532	0.837662	−	0.546190i	$-0.183922\pi$
$31$	9.32780	1.67532	0.837662	+	0.546190i	$0.183922\pi$
$32$	0	0
$33$	−3.86370	−0.672584
$34$	0	0
$35$	−0.186185	−0.0314710
$36$	0	0
$37$	−9.08516	−1.49359	−0.746796	−	0.665053i	$-0.768409\pi$
$37$	−9.08516	−1.49359	−0.746796	+	0.665053i	$0.768409\pi$
$38$	0	0
$39$	−0.0498881	−0.00798848
$40$	0	0
$41$	−4.46410	−0.697176	−0.348588	−	0.937276i	$-0.613339\pi$
$41$	−4.46410	−0.697176	−0.348588	+	0.937276i	$0.613339\pi$
$42$	0	0
$43$	2.02922	0.309454	0.154727	−	0.987957i	$-0.450550\pi$
$43$	2.02922	0.309454	0.154727	+	0.987957i	$0.450550\pi$
$44$	0	0
$45$	−0.0681483	−0.0101590
$46$	0	0
$47$	0.944060	0.137705	0.0688526	−	0.997627i	$-0.478066\pi$
$47$	0.944060	0.137705	0.0688526	+	0.997627i	$0.478066\pi$
$48$	0	0
$49$	0.464102	0.0663002
$50$	0	0
$51$	2.19275	0.307047
$52$	0	0
$53$	12.4670	1.71248	0.856239	−	0.516581i	$-0.172795\pi$
$53$	12.4670	1.71248	0.856239	+	0.516581i	$0.172795\pi$
$54$	0	0
$55$	0.186185	0.0251051
$56$	0	0
$57$	−2.96472	−0.392687
$58$	0	0
$59$	9.47067	1.23298	0.616488	−	0.787364i	$-0.288555\pi$
$59$	9.47067	1.23298	0.616488	+	0.787364i	$0.288555\pi$
$60$	0	0
$61$	6.81017	0.871953	0.435976	−	0.899958i	$-0.356403\pi$
$61$	6.81017	0.871953	0.435976	+	0.899958i	$0.356403\pi$
$62$	0	0
$63$	2.73205	0.344206
$64$	0	0
$65$	0.00240401	0.000298181 0
$66$	0	0
$67$	−3.19151	−0.389905	−0.194952	−	0.980813i	$-0.562455\pi$
$67$	−3.19151	−0.389905	−0.194952	+	0.980813i	$0.562455\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.4130	−1.23579	−0.617896	−	0.786260i	$-0.712015\pi$
$71$	−10.4130	−1.23579	−0.617896	+	0.786260i	$0.712015\pi$
$72$	0	0
$73$	12.9236	1.51259	0.756294	−	0.654232i	$-0.227008\pi$
$73$	12.9236	1.51259	0.756294	+	0.654232i	$0.227008\pi$
$74$	0	0
$75$	7.06450	0.815738
$76$	0	0
$77$	−7.46410	−0.850613
$78$	0	0
$79$	−10.1928	−1.14677	−0.573387	−	0.819285i	$-0.694371\pi$
$79$	−10.1928	−1.14677	−0.573387	+	0.819285i	$0.694371\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	7.16228	0.786163	0.393081	−	0.919504i	$-0.371409\pi$
$83$	7.16228	0.786163	0.393081	+	0.919504i	$0.371409\pi$
$84$	0	0
$85$	−0.105665	−0.0114609
$86$	0	0
$87$	2.74202	0.293975
$88$	0	0
$89$	−12.2037	−1.29359	−0.646796	−	0.762663i	$-0.723891\pi$
$89$	−12.2037	−1.29359	−0.646796	+	0.762663i	$0.723891\pi$
$90$	0	0
$91$	−0.0963763	−0.0101030
$92$	0	0
$93$	−13.1915	−1.36790
$94$	0	0
$95$	0.142865	0.0146576
$96$	0	0
$97$	−11.1732	−1.13447	−0.567236	−	0.823555i	$-0.691987\pi$
$97$	−11.1732	−1.13447	−0.567236	+	0.823555i	$0.691987\pi$
$98$	0	0
$99$	−2.73205	−0.274581
$100$	0	0
$101$	−12.2280	−1.21673	−0.608367	−	0.793656i	$-0.708175\pi$
$101$	−12.2280	−1.21673	−0.608367	+	0.793656i	$0.708175\pi$
$102$	0	0
$103$	12.0905	1.19131	0.595656	−	0.803240i	$-0.296892\pi$
$103$	12.0905	1.19131	0.595656	+	0.803240i	$0.296892\pi$
$104$	0	0
$105$	0.263305	0.0256959
$106$	0	0
$107$	−5.68268	−0.549365	−0.274683	−	0.961535i	$-0.588573\pi$
$107$	−5.68268	−0.549365	−0.274683	+	0.961535i	$0.588573\pi$
$108$	0	0
$109$	8.83083	0.845840	0.422920	−	0.906167i	$-0.361005\pi$
$109$	8.83083	0.845840	0.422920	+	0.906167i	$0.361005\pi$
$110$	0	0
$111$	12.8484	1.21951
$112$	0	0
$113$	17.6593	1.66124	0.830622	−	0.556837i	$-0.187985\pi$
$113$	17.6593	1.66124	0.830622	+	0.556837i	$0.187985\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.0352762	−0.00326128
$118$	0	0
$119$	4.23607	0.388320
$120$	0	0
$121$	−3.53590	−0.321445
$122$	0	0
$123$	6.31319	0.569241
$124$	0	0
$125$	−0.681167	−0.0609254
$126$	0	0
$127$	−0.292529	−0.0259577	−0.0129789	−	0.999916i	$-0.504131\pi$
$127$	−0.292529	−0.0259577	−0.0129789	+	0.999916i	$0.504131\pi$
$128$	0	0
$129$	−2.86976	−0.252668
$130$	0	0
$131$	6.91964	0.604572	0.302286	−	0.953217i	$-0.402250\pi$
$131$	6.91964	0.604572	0.302286	+	0.953217i	$0.402250\pi$
$132$	0	0
$133$	−5.72741	−0.496629
$134$	0	0
$135$	0.385505	0.0331790
$136$	0	0
$137$	10.5103	0.897959	0.448980	−	0.893542i	$-0.351788\pi$
$137$	10.5103	0.897959	0.448980	+	0.893542i	$0.351788\pi$
$138$	0	0
$139$	16.3558	1.38728	0.693640	−	0.720322i	$-0.256006\pi$
$139$	16.3558	1.38728	0.693640	+	0.720322i	$0.256006\pi$
$140$	0	0
$141$	−1.33510	−0.112436
$142$	0	0
$143$	0.0963763	0.00805939
$144$	0	0
$145$	−0.132133	−0.0109730
$146$	0	0
$147$	−0.656339	−0.0541339
$148$	0	0
$149$	15.9817	1.30928	0.654638	−	0.755943i	$-0.272821\pi$
$149$	15.9817	1.30928	0.654638	+	0.755943i	$0.272821\pi$
$150$	0	0
$151$	11.0559	0.899720	0.449860	−	0.893099i	$-0.351474\pi$
$151$	11.0559	0.899720	0.449860	+	0.893099i	$0.351474\pi$
$152$	0	0
$153$	1.55051	0.125351
$154$	0	0
$155$	0.635674	0.0510586
$156$	0	0
$157$	12.0730	0.963527	0.481763	−	0.876301i	$-0.339996\pi$
$157$	12.0730	0.963527	0.481763	+	0.876301i	$0.339996\pi$
$158$	0	0
$159$	−17.6310	−1.39823
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−23.3271	−1.82712	−0.913559	−	0.406707i	$-0.866677\pi$
$163$	−23.3271	−1.82712	−0.913559	+	0.406707i	$0.866677\pi$
$164$	0	0
$165$	−0.263305	−0.0204983
$166$	0	0
$167$	17.5405	1.35732	0.678662	−	0.734451i	$-0.262560\pi$
$167$	17.5405	1.35732	0.678662	+	0.734451i	$0.262560\pi$
$168$	0	0
$169$	−12.9988	−0.999904
$170$	0	0
$171$	−2.09638	−0.160314
$172$	0	0
$173$	−13.4546	−1.02294	−0.511469	−	0.859302i	$-0.670899\pi$
$173$	−13.4546	−1.02294	−0.511469	+	0.859302i	$0.670899\pi$
$174$	0	0
$175$	13.6476	1.03166
$176$	0	0
$177$	−13.3935	−1.00672
$178$	0	0
$179$	−24.1562	−1.80552	−0.902761	−	0.430143i	$-0.858463\pi$
$179$	−24.1562	−1.80552	−0.902761	+	0.430143i	$0.858463\pi$
$180$	0	0
$181$	0.285961	0.0212553	0.0106277	−	0.999944i	$-0.496617\pi$
$181$	0.285961	0.0212553	0.0106277	+	0.999944i	$0.496617\pi$
$182$	0	0
$183$	−9.63103	−0.711946
$184$	0	0
$185$	−0.619139	−0.0455200
$186$	0	0
$187$	−4.23607	−0.309772
$188$	0	0
$189$	−15.4548	−1.12417
$190$	0	0
$191$	−14.5558	−1.05322	−0.526612	−	0.850106i	$-0.676538\pi$
$191$	−14.5558	−1.05322	−0.526612	+	0.850106i	$0.676538\pi$
$192$	0	0
$193$	−23.3373	−1.67985	−0.839926	−	0.542701i	$-0.817402\pi$
$193$	−23.3373	−1.67985	−0.839926	+	0.542701i	$0.817402\pi$
$194$	0	0
$195$	−0.00339979	−0.000243464 0
$196$	0	0
$197$	7.78851	0.554908	0.277454	−	0.960739i	$-0.410509\pi$
$197$	7.78851	0.554908	0.277454	+	0.960739i	$0.410509\pi$
$198$	0	0
$199$	−14.6109	−1.03574	−0.517869	−	0.855460i	$-0.673275\pi$
$199$	−14.6109	−1.03574	−0.517869	+	0.855460i	$0.673275\pi$
$200$	0	0
$201$	4.51347	0.318356
$202$	0	0
$203$	5.29717	0.371789
$204$	0	0
$205$	−0.304221	−0.0212477
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.72741	0.396173
$210$	0	0
$211$	−1.27311	−0.0876444	−0.0438222	−	0.999039i	$-0.513954\pi$
$211$	−1.27311	−0.0876444	−0.0438222	+	0.999039i	$0.513954\pi$
$212$	0	0
$213$	14.7262	1.00902
$214$	0	0
$215$	0.138288	0.00943118
$216$	0	0
$217$	−25.4840	−1.72997
$218$	0	0
$219$	−18.2767	−1.23502
$220$	0	0
$221$	−0.0546961	−0.00367926
$222$	0	0
$223$	−3.87707	−0.259628	−0.129814	−	0.991538i	$-0.541438\pi$
$223$	−3.87707	−0.259628	−0.129814	+	0.991538i	$0.541438\pi$
$224$	0	0
$225$	4.99536	0.333024
$226$	0	0
$227$	−22.7967	−1.51307	−0.756536	−	0.653953i	$-0.773110\pi$
$227$	−22.7967	−1.51307	−0.756536	+	0.653953i	$0.773110\pi$
$228$	0	0
$229$	−20.8710	−1.37920	−0.689598	−	0.724193i	$-0.742213\pi$
$229$	−20.8710	−1.37920	−0.689598	+	0.724193i	$0.742213\pi$
$230$	0	0
$231$	10.5558	0.694523
$232$	0	0
$233$	24.5040	1.60531	0.802654	−	0.596445i	$-0.203420\pi$
$233$	24.5040	1.60531	0.802654	+	0.596445i	$0.203420\pi$
$234$	0	0
$235$	0.0643361	0.00419682
$236$	0	0
$237$	14.4147	0.936337
$238$	0	0
$239$	−6.06574	−0.392360	−0.196180	−	0.980568i	$-0.562854\pi$
$239$	−6.06574	−0.392360	−0.196180	+	0.980568i	$0.562854\pi$
$240$	0	0
$241$	−20.3514	−1.31095	−0.655474	−	0.755218i	$-0.727531\pi$
$241$	−20.3514	−1.31095	−0.655474	+	0.755218i	$0.727531\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	0.0316278	0.00202062
$246$	0	0
$247$	0.0739521	0.00470546
$248$	0	0
$249$	−10.1290	−0.641899
$250$	0	0
$251$	−5.28788	−0.333768	−0.166884	−	0.985977i	$-0.553371\pi$
$251$	−5.28788	−0.333768	−0.166884	+	0.985977i	$0.553371\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0.149432	0.00935782
$256$	0	0
$257$	6.05040	0.377414	0.188707	−	0.982033i	$-0.439570\pi$
$257$	6.05040	0.377414	0.188707	+	0.982033i	$0.439570\pi$
$258$	0	0
$259$	24.8211	1.54231
$260$	0	0
$261$	1.93890	0.120015
$262$	0	0
$263$	5.68165	0.350345	0.175173	−	0.984538i	$-0.443952\pi$
$263$	5.68165	0.350345	0.175173	+	0.984538i	$0.443952\pi$
$264$	0	0
$265$	0.849607	0.0521909
$266$	0	0
$267$	17.2587	1.05621
$268$	0	0
$269$	−21.5959	−1.31673	−0.658363	−	0.752700i	$-0.728751\pi$
$269$	−21.5959	−1.31673	−0.658363	+	0.752700i	$0.728751\pi$
$270$	0	0
$271$	−3.95011	−0.239952	−0.119976	−	0.992777i	$-0.538282\pi$
$271$	−3.95011	−0.239952	−0.119976	+	0.992777i	$0.538282\pi$
$272$	0	0
$273$	0.136297	0.00824905
$274$	0	0
$275$	−13.6476	−0.822979
$276$	0	0
$277$	−13.1210	−0.788362	−0.394181	−	0.919033i	$-0.628972\pi$
$277$	−13.1210	−0.788362	−0.394181	+	0.919033i	$0.628972\pi$
$278$	0	0
$279$	−9.32780	−0.558441
$280$	0	0
$281$	−30.7035	−1.83162	−0.915809	−	0.401615i	$-0.868449\pi$
$281$	−30.7035	−1.83162	−0.915809	+	0.401615i	$0.868449\pi$
$282$	0	0
$283$	23.4840	1.39598	0.697991	−	0.716107i	$-0.254078\pi$
$283$	23.4840	1.39598	0.697991	+	0.716107i	$0.254078\pi$
$284$	0	0
$285$	−0.202041	−0.0119679
$286$	0	0
$287$	12.1962	0.719916
$288$	0	0
$289$	−14.5959	−0.858583
$290$	0	0
$291$	15.8014	0.926292
$292$	0	0
$293$	16.4952	0.963661	0.481830	−	0.876265i	$-0.339972\pi$
$293$	16.4952	0.963661	0.481830	+	0.876265i	$0.339972\pi$
$294$	0	0
$295$	0.645410	0.0375773
$296$	0	0
$297$	15.4548	0.896779
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−5.54394	−0.319548
$302$	0	0
$303$	17.2930	0.993459
$304$	0	0
$305$	0.464102	0.0265744
$306$	0	0
$307$	−2.96296	−0.169105	−0.0845526	−	0.996419i	$-0.526946\pi$
$307$	−2.96296	−0.169105	−0.0845526	+	0.996419i	$0.526946\pi$
$308$	0	0
$309$	−17.0985	−0.972701
$310$	0	0
$311$	−10.7937	−0.612053	−0.306026	−	0.952023i	$-0.599000\pi$
$311$	−10.7937	−0.612053	−0.306026	+	0.952023i	$0.599000\pi$
$312$	0	0
$313$	3.76809	0.212985	0.106493	−	0.994314i	$-0.466038\pi$
$313$	3.76809	0.212985	0.106493	+	0.994314i	$0.466038\pi$
$314$	0	0
$315$	0.186185	0.0104903
$316$	0	0
$317$	−10.8671	−0.610357	−0.305179	−	0.952295i	$-0.598716\pi$
$317$	−10.8671	−0.610357	−0.305179	+	0.952295i	$0.598716\pi$
$318$	0	0
$319$	−5.29717	−0.296585
$320$	0	0
$321$	8.03652	0.448555
$322$	0	0
$323$	−3.25045	−0.180860
$324$	0	0
$325$	−0.176217	−0.00977476
$326$	0	0
$327$	−12.4887	−0.690626
$328$	0	0
$329$	−2.57922	−0.142197
$330$	0	0
$331$	2.59435	0.142598	0.0712991	−	0.997455i	$-0.477286\pi$
$331$	2.59435	0.142598	0.0712991	+	0.997455i	$0.477286\pi$
$332$	0	0
$333$	9.08516	0.497864
$334$	0	0
$335$	−0.217496	−0.0118831
$336$	0	0
$337$	0.738352	0.0402206	0.0201103	−	0.999798i	$-0.493598\pi$
$337$	0.738352	0.0402206	0.0201103	+	0.999798i	$0.493598\pi$
$338$	0	0
$339$	−24.9740	−1.35640
$340$	0	0
$341$	25.4840	1.38004
$342$	0	0
$343$	17.8564	0.964155
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.5785	−0.889980	−0.444990	−	0.895536i	$-0.646793\pi$
$347$	−16.5785	−0.889980	−0.444990	+	0.895536i	$0.646793\pi$
$348$	0	0
$349$	4.27010	0.228573	0.114287	−	0.993448i	$-0.463542\pi$
$349$	4.27010	0.228573	0.114287	+	0.993448i	$0.463542\pi$
$350$	0	0
$351$	0.199552	0.0106513
$352$	0	0
$353$	−2.33850	−0.124466	−0.0622329	−	0.998062i	$-0.519822\pi$
$353$	−2.33850	−0.124466	−0.0622329	+	0.998062i	$0.519822\pi$
$354$	0	0
$355$	−0.709627	−0.0376631
$356$	0	0
$357$	−5.99071	−0.317062
$358$	0	0
$359$	25.1030	1.32488	0.662442	−	0.749113i	$-0.269520\pi$
$359$	25.1030	1.32488	0.662442	+	0.749113i	$0.269520\pi$
$360$	0	0
$361$	−14.6052	−0.768695
$362$	0	0
$363$	5.00052	0.262459
$364$	0	0
$365$	0.880719	0.0460989
$366$	0	0
$367$	0.0165356	0.000863149 0	0.000431575	−	1.00000i	$-0.499863\pi$
$367$	0.0165356	0.000863149 0	0.000431575		1.00000i	$0.499863\pi$
$368$	0	0
$369$	4.46410	0.232392
$370$	0	0
$371$	−34.0605	−1.76833
$372$	0	0
$373$	−17.3392	−0.897789	−0.448894	−	0.893585i	$-0.648182\pi$
$373$	−17.3392	−0.897789	−0.448894	+	0.893585i	$0.648182\pi$
$374$	0	0
$375$	0.963316	0.0497454
$376$	0	0
$377$	−0.0683970	−0.00352262
$378$	0	0
$379$	17.1657	0.881742	0.440871	−	0.897570i	$-0.354670\pi$
$379$	17.1657	0.881742	0.440871	+	0.897570i	$0.354670\pi$
$380$	0	0
$381$	0.413698	0.0211944
$382$	0	0
$383$	−9.14089	−0.467078	−0.233539	−	0.972347i	$-0.575031\pi$
$383$	−9.14089	−0.467078	−0.233539	+	0.972347i	$0.575031\pi$
$384$	0	0
$385$	−0.508666	−0.0259240
$386$	0	0
$387$	−2.02922	−0.103151
$388$	0	0
$389$	−16.9441	−0.859098	−0.429549	−	0.903044i	$-0.641327\pi$
$389$	−16.9441	−0.859098	−0.429549	+	0.903044i	$0.641327\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−9.78585	−0.493631
$394$	0	0
$395$	−0.694619	−0.0349501
$396$	0	0
$397$	−24.7546	−1.24240	−0.621200	−	0.783652i	$-0.713354\pi$
$397$	−24.7546	−1.24240	−0.621200	+	0.783652i	$0.713354\pi$
$398$	0	0
$399$	8.09978	0.405496
$400$	0	0
$401$	−6.27427	−0.313322	−0.156661	−	0.987652i	$-0.550073\pi$
$401$	−6.27427	−0.313322	−0.156661	+	0.987652i	$0.550073\pi$
$402$	0	0
$403$	0.329049	0.0163911
$404$	0	0
$405$	−0.340742	−0.0169316
$406$	0	0
$407$	−24.8211	−1.23034
$408$	0	0
$409$	16.7967	0.830544	0.415272	−	0.909697i	$-0.363686\pi$
$409$	16.7967	0.830544	0.415272	+	0.909697i	$0.363686\pi$
$410$	0	0
$411$	−14.8639	−0.733180
$412$	0	0
$413$	−25.8744	−1.27319
$414$	0	0
$415$	0.488098	0.0239598
$416$	0	0
$417$	−23.1306	−1.13271
$418$	0	0
$419$	0.716546	0.0350056	0.0175028	−	0.999847i	$-0.494428\pi$
$419$	0.716546	0.0350056	0.0175028	+	0.999847i	$0.494428\pi$
$420$	0	0
$421$	2.07911	0.101330	0.0506649	−	0.998716i	$-0.483866\pi$
$421$	2.07911	0.101330	0.0506649	+	0.998716i	$0.483866\pi$
$422$	0	0
$423$	−0.944060	−0.0459017
$424$	0	0
$425$	7.74535	0.375705
$426$	0	0
$427$	−18.6057	−0.900394
$428$	0	0
$429$	−0.136297	−0.00658047
$430$	0	0
$431$	21.4899	1.03513	0.517567	−	0.855643i	$-0.326838\pi$
$431$	21.4899	1.03513	0.517567	+	0.855643i	$0.326838\pi$
$432$	0	0
$433$	24.1654	1.16132	0.580658	−	0.814147i	$-0.302795\pi$
$433$	24.1654	1.16132	0.580658	+	0.814147i	$0.302795\pi$
$434$	0	0
$435$	0.186864	0.00895944
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−13.9081	−0.663795	−0.331898	−	0.943315i	$-0.607689\pi$
$439$	−13.9081	−0.663795	−0.331898	+	0.943315i	$0.607689\pi$
$440$	0	0
$441$	−0.464102	−0.0221001
$442$	0	0
$443$	−15.6476	−0.743438	−0.371719	−	0.928345i	$-0.621232\pi$
$443$	−15.6476	−0.743438	−0.371719	+	0.928345i	$0.621232\pi$
$444$	0	0
$445$	−0.831663	−0.0394246
$446$	0	0
$447$	−22.6016	−1.06902
$448$	0	0
$449$	3.78710	0.178724	0.0893621	−	0.995999i	$-0.471517\pi$
$449$	3.78710	0.178724	0.0893621	+	0.995999i	$0.471517\pi$
$450$	0	0
$451$	−12.1962	−0.574294
$452$	0	0
$453$	−15.6355	−0.734618
$454$	0	0
$455$	−0.00656789	−0.000307907 0
$456$	0	0
$457$	8.61917	0.403188	0.201594	−	0.979469i	$-0.435388\pi$
$457$	8.61917	0.403188	0.201594	+	0.979469i	$0.435388\pi$
$458$	0	0
$459$	−8.77101	−0.409396
$460$	0	0
$461$	−19.6474	−0.915071	−0.457535	−	0.889191i	$-0.651268\pi$
$461$	−19.6474	−0.915071	−0.457535	+	0.889191i	$0.651268\pi$
$462$	0	0
$463$	10.8201	0.502853	0.251426	−	0.967876i	$-0.419100\pi$
$463$	10.8201	0.502853	0.251426	+	0.967876i	$0.419100\pi$
$464$	0	0
$465$	−0.898979	−0.0416892
$466$	0	0
$467$	8.92140	0.412833	0.206417	−	0.978464i	$-0.433820\pi$
$467$	8.92140	0.412833	0.206417	+	0.978464i	$0.433820\pi$
$468$	0	0
$469$	8.71936	0.402623
$470$	0	0
$471$	−17.0737	−0.786716
$472$	0	0
$473$	5.54394	0.254911
$474$	0	0
$475$	−10.4721	−0.480495
$476$	0	0
$477$	−12.4670	−0.570826
$478$	0	0
$479$	40.8166	1.86496	0.932480	−	0.361221i	$-0.117640\pi$
$479$	40.8166	1.86496	0.932480	+	0.361221i	$0.117640\pi$
$480$	0	0
$481$	−0.320490	−0.0146131
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.761438	−0.0345751
$486$	0	0
$487$	−23.2520	−1.05365	−0.526825	−	0.849974i	$-0.676618\pi$
$487$	−23.2520	−1.05365	−0.526825	+	0.849974i	$0.676618\pi$
$488$	0	0
$489$	32.9895	1.49184
$490$	0	0
$491$	−17.1502	−0.773977	−0.386988	−	0.922085i	$-0.626485\pi$
$491$	−17.1502	−0.773977	−0.386988	+	0.922085i	$0.626485\pi$
$492$	0	0
$493$	3.00628	0.135396
$494$	0	0
$495$	−0.186185	−0.00836838
$496$	0	0
$497$	28.4488	1.27610
$498$	0	0
$499$	−8.37718	−0.375014	−0.187507	−	0.982263i	$-0.560041\pi$
$499$	−8.37718	−0.375014	−0.187507	+	0.982263i	$0.560041\pi$
$500$	0	0
$501$	−24.8060	−1.10825
$502$	0	0
$503$	38.8931	1.73416	0.867079	−	0.498171i	$-0.165995\pi$
$503$	38.8931	1.73416	0.867079	+	0.498171i	$0.165995\pi$
$504$	0	0
$505$	−0.833320	−0.0370823
$506$	0	0
$507$	18.3830	0.816418
$508$	0	0
$509$	−8.98930	−0.398444	−0.199222	−	0.979954i	$-0.563841\pi$
$509$	−8.98930	−0.398444	−0.199222	+	0.979954i	$0.563841\pi$
$510$	0	0
$511$	−35.3078	−1.56193
$512$	0	0
$513$	11.8589	0.523583
$514$	0	0
$515$	0.823947	0.0363074
$516$	0	0
$517$	2.57922	0.113434
$518$	0	0
$519$	19.0277	0.835225
$520$	0	0
$521$	−16.8229	−0.737024	−0.368512	−	0.929623i	$-0.620133\pi$
$521$	−16.8229	−0.737024	−0.368512	+	0.929623i	$0.620133\pi$
$522$	0	0
$523$	−35.7552	−1.56346	−0.781732	−	0.623614i	$-0.785664\pi$
$523$	−35.7552	−1.56346	−0.781732	+	0.623614i	$0.785664\pi$
$524$	0	0
$525$	−19.3006	−0.842346
$526$	0	0
$527$	−14.4629	−0.630012
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−9.47067	−0.410992
$532$	0	0
$533$	−0.157476	−0.00682106
$534$	0	0
$535$	−0.387265	−0.0167429
$536$	0	0
$537$	34.1621	1.47420
$538$	0	0
$539$	1.26795	0.0546144
$540$	0	0
$541$	−28.5663	−1.22816	−0.614081	−	0.789243i	$-0.710473\pi$
$541$	−28.5663	−1.22816	−0.614081	+	0.789243i	$0.710473\pi$
$542$	0	0
$543$	−0.404410	−0.0173549
$544$	0	0
$545$	0.601807	0.0257786
$546$	0	0
$547$	−32.6050	−1.39409	−0.697044	−	0.717028i	$-0.745502\pi$
$547$	−32.6050	−1.39409	−0.697044	+	0.717028i	$0.745502\pi$
$548$	0	0
$549$	−6.81017	−0.290651
$550$	0	0
$551$	−4.06466	−0.173160
$552$	0	0
$553$	27.8471	1.18418
$554$	0	0
$555$	0.875595	0.0371669
$556$	0	0
$557$	15.4950	0.656544	0.328272	−	0.944583i	$-0.393534\pi$
$557$	15.4950	0.656544	0.328272	+	0.944583i	$0.393534\pi$
$558$	0	0
$559$	0.0715833	0.00302765
$560$	0	0
$561$	5.99071	0.252928
$562$	0	0
$563$	−35.2328	−1.48489	−0.742444	−	0.669909i	$-0.766333\pi$
$563$	−35.2328	−1.48489	−0.742444	+	0.669909i	$0.766333\pi$
$564$	0	0
$565$	1.20345	0.0506295
$566$	0	0
$567$	13.6603	0.573677
$568$	0	0
$569$	−40.4512	−1.69580	−0.847901	−	0.530155i	$-0.822134\pi$
$569$	−40.4512	−1.69580	−0.847901	+	0.530155i	$0.822134\pi$
$570$	0	0
$571$	10.9316	0.457473	0.228737	−	0.973488i	$-0.426541\pi$
$571$	10.9316	0.457473	0.228737	+	0.973488i	$0.426541\pi$
$572$	0	0
$573$	20.5851	0.859953
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−24.0584	−1.00157	−0.500783	−	0.865573i	$-0.666955\pi$
$577$	−24.0584	−1.00157	−0.500783	+	0.865573i	$0.666955\pi$
$578$	0	0
$579$	33.0039	1.37159
$580$	0	0
$581$	−19.5677	−0.811806
$582$	0	0
$583$	34.0605	1.41064
$584$	0	0
$585$	−0.00240401	−9.93937e−5 0
$586$	0	0
$587$	−15.2786	−0.630617	−0.315309	−	0.948989i	$-0.602108\pi$
$587$	−15.2786	−0.630617	−0.315309	+	0.948989i	$0.602108\pi$
$588$	0	0
$589$	19.5546	0.805733
$590$	0	0
$591$	−11.0146	−0.453081
$592$	0	0
$593$	−7.20080	−0.295701	−0.147851	−	0.989010i	$-0.547235\pi$
$593$	−7.20080	−0.295701	−0.147851	+	0.989010i	$0.547235\pi$
$594$	0	0
$595$	0.288681	0.0118348
$596$	0	0
$597$	20.6629	0.845676
$598$	0	0
$599$	−48.1145	−1.96591	−0.982953	−	0.183859i	$-0.941141\pi$
$599$	−48.1145	−1.96591	−0.982953	+	0.183859i	$0.941141\pi$
$600$	0	0
$601$	−40.0748	−1.63468	−0.817342	−	0.576153i	$-0.804553\pi$
$601$	−40.0748	−1.63468	−0.817342	+	0.576153i	$0.804553\pi$
$602$	0	0
$603$	3.19151	0.129968
$604$	0	0
$605$	−0.240966	−0.00979665
$606$	0	0
$607$	14.0990	0.572263	0.286131	−	0.958190i	$-0.407631\pi$
$607$	14.0990	0.572263	0.286131	+	0.958190i	$0.407631\pi$
$608$	0	0
$609$	−7.49133	−0.303564
$610$	0	0
$611$	0.0333028	0.00134729
$612$	0	0
$613$	−7.95355	−0.321241	−0.160620	−	0.987016i	$-0.551349\pi$
$613$	−7.95355	−0.321241	−0.160620	+	0.987016i	$0.551349\pi$
$614$	0	0
$615$	0.430234	0.0173487
$616$	0	0
$617$	40.9991	1.65056	0.825280	−	0.564724i	$-0.191017\pi$
$617$	40.9991	1.65056	0.825280	+	0.564724i	$0.191017\pi$
$618$	0	0
$619$	31.9882	1.28572	0.642858	−	0.765986i	$-0.277749\pi$
$619$	31.9882	1.28572	0.642858	+	0.765986i	$0.277749\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	33.3412	1.33579
$624$	0	0
$625$	24.9304	0.997214
$626$	0	0
$627$	−8.09978	−0.323474
$628$	0	0
$629$	14.0866	0.561671
$630$	0	0
$631$	−37.6183	−1.49756	−0.748781	−	0.662817i	$-0.769361\pi$
$631$	−37.6183	−1.49756	−0.748781	+	0.662817i	$0.769361\pi$
$632$	0	0
$633$	1.80045	0.0715614
$634$	0	0
$635$	−0.0199354	−0.000791110 0
$636$	0	0
$637$	0.0163717	0.000648672 0
$638$	0	0
$639$	10.4130	0.411931
$640$	0	0
$641$	2.27749	0.0899553	0.0449776	−	0.998988i	$-0.485678\pi$
$641$	2.27749	0.0899553	0.0449776	+	0.998988i	$0.485678\pi$
$642$	0	0
$643$	12.3718	0.487896	0.243948	−	0.969788i	$-0.421557\pi$
$643$	12.3718	0.487896	0.243948	+	0.969788i	$0.421557\pi$
$644$	0	0
$645$	−0.195569	−0.00770053
$646$	0	0
$647$	20.4986	0.805885	0.402943	−	0.915225i	$-0.367987\pi$
$647$	20.4986	0.805885	0.402943	+	0.915225i	$0.367987\pi$
$648$	0	0
$649$	25.8744	1.01566
$650$	0	0
$651$	36.0399	1.41251
$652$	0	0
$653$	−12.1482	−0.475395	−0.237698	−	0.971339i	$-0.576393\pi$
$653$	−12.1482	−0.475395	−0.237698	+	0.971339i	$0.576393\pi$
$654$	0	0
$655$	0.471562	0.0184255
$656$	0	0
$657$	−12.9236	−0.504196
$658$	0	0
$659$	−28.5145	−1.11077	−0.555384	−	0.831594i	$-0.687429\pi$
$659$	−28.5145	−1.11077	−0.555384	+	0.831594i	$0.687429\pi$
$660$	0	0
$661$	−23.6946	−0.921611	−0.460806	−	0.887501i	$-0.652439\pi$
$661$	−23.6946	−0.921611	−0.460806	+	0.887501i	$0.652439\pi$
$662$	0	0
$663$	0.0773519	0.00300410
$664$	0	0
$665$	−0.390313	−0.0151357
$666$	0	0
$667$	0	0
$668$	0	0
$669$	5.48301	0.211985
$670$	0	0
$671$	18.6057	0.718266
$672$	0	0
$673$	18.9681	0.731166	0.365583	−	0.930779i	$-0.380870\pi$
$673$	18.9681	0.731166	0.365583	+	0.930779i	$0.380870\pi$
$674$	0	0
$675$	−28.2580	−1.08765
$676$	0	0
$677$	51.3952	1.97528	0.987639	−	0.156743i	$-0.0500994\pi$
$677$	51.3952	1.97528	0.987639	+	0.156743i	$0.0500994\pi$
$678$	0	0
$679$	30.5259	1.17148
$680$	0	0
$681$	32.2394	1.23542
$682$	0	0
$683$	−23.5327	−0.900452	−0.450226	−	0.892915i	$-0.648657\pi$
$683$	−23.5327	−0.900452	−0.450226	+	0.892915i	$0.648657\pi$
$684$	0	0
$685$	0.716262	0.0273670
$686$	0	0
$687$	29.5161	1.12611
$688$	0	0
$689$	0.439789	0.0167546
$690$	0	0
$691$	−32.5927	−1.23988	−0.619942	−	0.784648i	$-0.712844\pi$
$691$	−32.5927	−1.23988	−0.619942	+	0.784648i	$0.712844\pi$
$692$	0	0
$693$	7.46410	0.283538
$694$	0	0
$695$	1.11462	0.0422799
$696$	0	0
$697$	6.92164	0.262176
$698$	0	0
$699$	−34.6538	−1.31073
$700$	0	0
$701$	−4.75811	−0.179711	−0.0898556	−	0.995955i	$-0.528641\pi$
$701$	−4.75811	−0.179711	−0.0898556	+	0.995955i	$0.528641\pi$
$702$	0	0
$703$	−19.0459	−0.718331
$704$	0	0
$705$	−0.0909850	−0.00342669
$706$	0	0
$707$	33.4076	1.25642
$708$	0	0
$709$	−9.01702	−0.338641	−0.169321	−	0.985561i	$-0.554157\pi$
$709$	−9.01702	−0.338641	−0.169321	+	0.985561i	$0.554157\pi$
$710$	0	0
$711$	10.1928	0.382258
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0.00656789	0.000245625 0
$716$	0	0
$717$	8.57826	0.320361
$718$	0	0
$719$	−42.5042	−1.58514	−0.792569	−	0.609782i	$-0.791257\pi$
$719$	−42.5042	−1.58514	−0.792569	+	0.609782i	$0.791257\pi$
$720$	0	0
$721$	−33.0318	−1.23017
$722$	0	0
$723$	28.7812	1.07038
$724$	0	0
$725$	9.68549	0.359710
$726$	0	0
$727$	−3.63000	−0.134629	−0.0673146	−	0.997732i	$-0.521443\pi$
$727$	−3.63000	−0.134629	−0.0673146	+	0.997732i	$0.521443\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	−3.14633	−0.116371
$732$	0	0
$733$	−5.13643	−0.189718	−0.0948591	−	0.995491i	$-0.530240\pi$
$733$	−5.13643	−0.189718	−0.0948591	+	0.995491i	$0.530240\pi$
$734$	0	0
$735$	−0.0447284	−0.00164983
$736$	0	0
$737$	−8.71936	−0.321182
$738$	0	0
$739$	4.97079	0.182854	0.0914268	−	0.995812i	$-0.470857\pi$
$739$	4.97079	0.182854	0.0914268	+	0.995812i	$0.470857\pi$
$740$	0	0
$741$	−0.104584	−0.00384199
$742$	0	0
$743$	−15.0951	−0.553787	−0.276893	−	0.960901i	$-0.589305\pi$
$743$	−15.0951	−0.553787	−0.276893	+	0.960901i	$0.589305\pi$
$744$	0	0
$745$	1.08913	0.0399026
$746$	0	0
$747$	−7.16228	−0.262054
$748$	0	0
$749$	15.5254	0.567285
$750$	0	0
$751$	−1.38912	−0.0506897	−0.0253448	−	0.999679i	$-0.508068\pi$
$751$	−1.38912	−0.0506897	−0.0253448	+	0.999679i	$0.508068\pi$
$752$	0	0
$753$	7.47820	0.272521
$754$	0	0
$755$	0.753444	0.0274206
$756$	0	0
$757$	−13.3076	−0.483671	−0.241836	−	0.970317i	$-0.577749\pi$
$757$	−13.3076	−0.483671	−0.241836	+	0.970317i	$0.577749\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−19.0244	−0.689633	−0.344816	−	0.938670i	$-0.612059\pi$
$761$	−19.0244	−0.689633	−0.344816	+	0.938670i	$0.612059\pi$
$762$	0	0
$763$	−24.1263	−0.873430
$764$	0	0
$765$	0.105665	0.00382032
$766$	0	0
$767$	0.334089	0.0120633
$768$	0	0
$769$	−8.92267	−0.321760	−0.160880	−	0.986974i	$-0.551433\pi$
$769$	−8.92267	−0.321760	−0.160880	+	0.986974i	$0.551433\pi$
$770$	0	0
$771$	−8.55656	−0.308157
$772$	0	0
$773$	13.9476	0.501661	0.250831	−	0.968031i	$-0.419296\pi$
$773$	13.9476	0.501661	0.250831	+	0.968031i	$0.419296\pi$
$774$	0	0
$775$	−46.5957	−1.67377
$776$	0	0
$777$	−35.1024	−1.25929
$778$	0	0
$779$	−9.35844	−0.335301
$780$	0	0
$781$	−28.4488	−1.01798
$782$	0	0
$783$	−10.9681	−0.391967
$784$	0	0
$785$	0.822752	0.0293653
$786$	0	0
$787$	−46.5275	−1.65853	−0.829264	−	0.558858i	$-0.811240\pi$
$787$	−46.5275	−1.65853	−0.829264	+	0.558858i	$0.811240\pi$
$788$	0	0
$789$	−8.03506	−0.286056
$790$	0	0
$791$	−48.2460	−1.71543
$792$	0	0
$793$	0.240237	0.00853105
$794$	0	0
$795$	−1.20153	−0.0426137
$796$	0	0
$797$	−26.8136	−0.949787	−0.474893	−	0.880043i	$-0.657513\pi$
$797$	−26.8136	−0.949787	−0.474893	+	0.880043i	$0.657513\pi$
$798$	0	0
$799$	−1.46377	−0.0517846
$800$	0	0
$801$	12.2037	0.431197
$802$	0	0
$803$	35.3078	1.24599
$804$	0	0
$805$	0	0
$806$	0	0
$807$	30.5412	1.07510
$808$	0	0
$809$	−30.1971	−1.06167	−0.530836	−	0.847475i	$-0.678122\pi$
$809$	−30.1971	−1.06167	−0.530836	+	0.847475i	$0.678122\pi$
$810$	0	0
$811$	−13.0141	−0.456987	−0.228494	−	0.973545i	$-0.573380\pi$
$811$	−13.0141	−0.456987	−0.228494	+	0.973545i	$0.573380\pi$
$812$	0	0
$813$	5.58630	0.195920
$814$	0	0
$815$	−1.58970	−0.0556848
$816$	0	0
$817$	4.25402	0.148829
$818$	0	0
$819$	0.0963763	0.00336766
$820$	0	0
$821$	−9.40743	−0.328322	−0.164161	−	0.986434i	$-0.552492\pi$
$821$	−9.40743	−0.328322	−0.164161	+	0.986434i	$0.552492\pi$
$822$	0	0
$823$	36.4402	1.27023	0.635113	−	0.772419i	$-0.280954\pi$
$823$	36.4402	1.27023	0.635113	+	0.772419i	$0.280954\pi$
$824$	0	0
$825$	19.3006	0.671960
$826$	0	0
$827$	3.73081	0.129733	0.0648664	−	0.997894i	$-0.479338\pi$
$827$	3.73081	0.129733	0.0648664	+	0.997894i	$0.479338\pi$
$828$	0	0
$829$	44.1131	1.53211	0.766055	−	0.642775i	$-0.222217\pi$
$829$	44.1131	1.53211	0.766055	+	0.642775i	$0.222217\pi$
$830$	0	0
$831$	18.5558	0.643695
$832$	0	0
$833$	−0.719594	−0.0249325
$834$	0	0
$835$	1.19536	0.0413670
$836$	0	0
$837$	52.7660	1.82386
$838$	0	0
$839$	24.8304	0.857240	0.428620	−	0.903485i	$-0.359000\pi$
$839$	24.8304	0.857240	0.428620	+	0.903485i	$0.359000\pi$
$840$	0	0
$841$	−25.2407	−0.870368
$842$	0	0
$843$	43.4213	1.49551
$844$	0	0
$845$	−0.885844	−0.0304740
$846$	0	0
$847$	9.66025	0.331930
$848$	0	0
$849$	−33.2114	−1.13981
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−50.5309	−1.73015	−0.865073	−	0.501646i	$-0.832728\pi$
$853$	−50.5309	−1.73015	−0.865073	+	0.501646i	$0.832728\pi$
$854$	0	0
$855$	−0.142865	−0.00488587
$856$	0	0
$857$	12.5956	0.430257	0.215129	−	0.976586i	$-0.430983\pi$
$857$	12.5956	0.430257	0.215129	+	0.976586i	$0.430983\pi$
$858$	0	0
$859$	−33.2765	−1.13538	−0.567689	−	0.823243i	$-0.692162\pi$
$859$	−33.2765	−1.13538	−0.567689	+	0.823243i	$0.692162\pi$
$860$	0	0
$861$	−17.2480	−0.587809
$862$	0	0
$863$	−18.5024	−0.629830	−0.314915	−	0.949120i	$-0.601976\pi$
$863$	−18.5024	−0.629830	−0.314915	+	0.949120i	$0.601976\pi$
$864$	0	0
$865$	−0.916912	−0.0311759
$866$	0	0
$867$	20.6417	0.701030
$868$	0	0
$869$	−27.8471	−0.944649
$870$	0	0
$871$	−0.112584	−0.00381477
$872$	0	0
$873$	11.1732	0.378157
$874$	0	0
$875$	1.86098	0.0629127
$876$	0	0
$877$	−10.9677	−0.370354	−0.185177	−	0.982705i	$-0.559286\pi$
$877$	−10.9677	−0.370354	−0.185177	+	0.982705i	$0.559286\pi$
$878$	0	0
$879$	−23.3278	−0.786826
$880$	0	0
$881$	−0.688997	−0.0232129	−0.0116065	−	0.999933i	$-0.503695\pi$
$881$	−0.688997	−0.0232129	−0.0116065	+	0.999933i	$0.503695\pi$
$882$	0	0
$883$	41.2530	1.38827	0.694137	−	0.719843i	$-0.255786\pi$
$883$	41.2530	1.38827	0.694137	+	0.719843i	$0.255786\pi$
$884$	0	0
$885$	−0.912748	−0.0306817
$886$	0	0
$887$	21.1578	0.710410	0.355205	−	0.934789i	$-0.384411\pi$
$887$	21.1578	0.710410	0.355205	+	0.934789i	$0.384411\pi$
$888$	0	0
$889$	0.799203	0.0268044
$890$	0	0
$891$	−13.6603	−0.457636
$892$	0	0
$893$	1.97910	0.0662282
$894$	0	0
$895$	−1.64621	−0.0550266
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−18.0857	−0.603191
$900$	0	0
$901$	−19.3302	−0.643984
$902$	0	0
$903$	7.84032	0.260909
$904$	0	0
$905$	0.0194878	0.000647795 0
$906$	0	0
$907$	17.2277	0.572035	0.286017	−	0.958224i	$-0.407668\pi$
$907$	17.2277	0.572035	0.286017	+	0.958224i	$0.407668\pi$
$908$	0	0
$909$	12.2280	0.405578
$910$	0	0
$911$	−17.5735	−0.582236	−0.291118	−	0.956687i	$-0.594027\pi$
$911$	−17.5735	−0.582236	−0.291118	+	0.956687i	$0.594027\pi$
$912$	0	0
$913$	19.5677	0.647597
$914$	0	0
$915$	−0.656339	−0.0216979
$916$	0	0
$917$	−18.9048	−0.624292
$918$	0	0
$919$	39.8155	1.31339	0.656696	−	0.754155i	$-0.271953\pi$
$919$	39.8155	1.31339	0.656696	+	0.754155i	$0.271953\pi$
$920$	0	0
$921$	4.19026	0.138074
$922$	0	0
$923$	−0.367330	−0.0120908
$924$	0	0
$925$	45.3836	1.49220
$926$	0	0
$927$	−12.0905	−0.397104
$928$	0	0
$929$	24.3057	0.797443	0.398721	−	0.917072i	$-0.369454\pi$
$929$	24.3057	0.797443	0.398721	+	0.917072i	$0.369454\pi$
$930$	0	0
$931$	0.972932	0.0318865
$932$	0	0
$933$	15.2645	0.499739
$934$	0	0
$935$	−0.288681	−0.00944089
$936$	0	0
$937$	−1.23805	−0.0404452	−0.0202226	−	0.999796i	$-0.506437\pi$
$937$	−1.23805	−0.0404452	−0.0202226	+	0.999796i	$0.506437\pi$
$938$	0	0
$939$	−5.32889	−0.173902
$940$	0	0
$941$	−59.6443	−1.94435	−0.972175	−	0.234255i	$-0.924735\pi$
$941$	−59.6443	−1.94435	−0.972175	+	0.234255i	$0.924735\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−1.05322	−0.0342612
$946$	0	0
$947$	−27.6324	−0.897934	−0.448967	−	0.893548i	$-0.648208\pi$
$947$	−27.6324	−0.897934	−0.448967	+	0.893548i	$0.648208\pi$
$948$	0	0
$949$	0.455894	0.0147989
$950$	0	0
$951$	15.3684	0.498355
$952$	0	0
$953$	20.8999	0.677015	0.338507	−	0.940964i	$-0.390078\pi$
$953$	20.8999	0.677015	0.338507	+	0.940964i	$0.390078\pi$
$954$	0	0
$955$	−0.991956	−0.0320989
$956$	0	0
$957$	7.49133	0.242160
$958$	0	0
$959$	−28.7148	−0.927249
$960$	0	0
$961$	56.0079	1.80671
$962$	0	0
$963$	5.68268	0.183122
$964$	0	0
$965$	−1.59040	−0.0511966
$966$	0	0
$967$	−44.0684	−1.41714	−0.708572	−	0.705639i	$-0.750660\pi$
$967$	−44.0684	−1.41714	−0.708572	+	0.705639i	$0.750660\pi$
$968$	0	0
$969$	4.59683	0.147672
$970$	0	0
$971$	46.0272	1.47708	0.738541	−	0.674208i	$-0.235515\pi$
$971$	46.0272	1.47708	0.738541	+	0.674208i	$0.235515\pi$
$972$	0	0
$973$	−44.6848	−1.43253
$974$	0	0
$975$	0.249209	0.00798106
$976$	0	0
$977$	22.9525	0.734317	0.367158	−	0.930158i	$-0.380331\pi$
$977$	22.9525	0.734317	0.367158	+	0.930158i	$0.380331\pi$
$978$	0	0
$979$	−33.3412	−1.06559
$980$	0	0
$981$	−8.83083	−0.281947
$982$	0	0
$983$	33.9119	1.08162	0.540811	−	0.841144i	$-0.318117\pi$
$983$	33.9119	1.08162	0.540811	+	0.841144i	$0.318117\pi$
$984$	0	0
$985$	0.530774	0.0169119
$986$	0	0
$987$	3.64757	0.116103
$988$	0	0
$989$	0	0
$990$	0	0
$991$	51.2991	1.62957	0.814785	−	0.579763i	$-0.196855\pi$
$991$	51.2991	1.62957	0.814785	+	0.579763i	$0.196855\pi$
$992$	0	0
$993$	−3.66896	−0.116431
$994$	0	0
$995$	−0.995707	−0.0315660
$996$	0	0
$997$	38.2421	1.21114	0.605569	−	0.795793i	$-0.292945\pi$
$997$	38.2421	1.21114	0.605569	+	0.795793i	$0.292945\pi$
$998$	0	0
$999$	−51.3934	−1.62602

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.bn.1.1		4
4.3	odd	2		4232.2.a.u.1.3	yes	4
23.22	odd	2		8464.2.a.bl.1.2		4
92.91	even	2		4232.2.a.s.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4232.2.a.s.1.4	✓	4	92.91	even	2
4232.2.a.u.1.3	yes	4	4.3	odd	2
8464.2.a.bl.1.2		4	23.22	odd	2
8464.2.a.bn.1.1		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} +0.0681483 q^{5} -2.73205 q^{7} -1.00000 q^{9} +2.73205 q^{11} +0.0352762 q^{13} -0.0963763 q^{15} -1.55051 q^{17} +2.09638 q^{19} +3.86370 q^{21} -4.99536 q^{25} +5.65685 q^{27} -1.93890 q^{29} +9.32780 q^{31} -3.86370 q^{33} -0.186185 q^{35} -9.08516 q^{37} -0.0498881 q^{39} -4.46410 q^{41} +2.02922 q^{43} -0.0681483 q^{45} +0.944060 q^{47} +0.464102 q^{49} +2.19275 q^{51} +12.4670 q^{53} +0.186185 q^{55} -2.96472 q^{57} +9.47067 q^{59} +6.81017 q^{61} +2.73205 q^{63} +0.00240401 q^{65} -3.19151 q^{67} -10.4130 q^{71} +12.9236 q^{73} +7.06450 q^{75} -7.46410 q^{77} -10.1928 q^{79} -5.00000 q^{81} +7.16228 q^{83} -0.105665 q^{85} +2.74202 q^{87} -12.2037 q^{89} -0.0963763 q^{91} -13.1915 q^{93} +0.142865 q^{95} -11.1732 q^{97} -2.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5} - 4 q^{7} - 4 q^{9} + 4 q^{11} - 4 q^{13} + 4 q^{15} - 16 q^{17} + 4 q^{19} + 4 q^{25} - 8 q^{29} + 8 q^{31} - 8 q^{35} - 24 q^{37} + 8 q^{39} - 4 q^{41} - 8 q^{45} + 8 q^{47} - 12 q^{49}+ \cdots - 4 q^{99}+O(q^{100}) \) 4 * q + 8 * q^5 - 4 * q^7 - 4 * q^9 + 4 * q^11 - 4 * q^13 + 4 * q^15 - 16 * q^17 + 4 * q^19 + 4 * q^25 - 8 * q^29 + 8 * q^31 - 8 * q^35 - 24 * q^37 + 8 * q^39 - 4 * q^41 - 8 * q^45 + 8 * q^47 - 12 * q^49 + 8 * q^55 - 16 * q^57 + 8 * q^59 + 4 * q^63 - 16 * q^65 + 32 * q^67 + 16 * q^75 - 16 * q^77 - 32 * q^79 - 20 * q^81 - 8 * q^83 - 44 * q^85 - 24 * q^87 - 16 * q^89 + 4 * q^91 - 8 * q^93 + 16 * q^97 - 4 * q^99

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