LMFDB - Embedded newform 1368.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,4,Mod(1,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1368.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1368.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:	$80.7146128879$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.27492$ of defining polynomial
Character	$\chi$	$=$	1368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.27492 q^{5} +12.0997 q^{7} +O(q^{10})$	$q-1.27492 q^{5} +12.0997 q^{7} -18.9244 q^{11} -55.9244 q^{13} +89.7492 q^{17} -19.0000 q^{19} +135.072 q^{23} -123.375 q^{25} +102.474 q^{29} -103.698 q^{31} -15.4261 q^{35} +29.6977 q^{37} -234.743 q^{41} -53.3231 q^{43} +33.3713 q^{47} -196.598 q^{49} -93.1752 q^{53} +24.1271 q^{55} -637.320 q^{59} -125.571 q^{61} +71.2990 q^{65} +119.375 q^{67} -18.5083 q^{71} -394.794 q^{73} -228.979 q^{77} +303.341 q^{79} +394.337 q^{83} -114.423 q^{85} +1021.88 q^{89} -676.667 q^{91} +24.2234 q^{95} +1322.82 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5 q^{5} - 6 q^{7}+O(q^{10}) $ 2 * q + 5 * q^5 - 6 * q^7	$ 2 q + 5 q^{5} - 6 q^{7} + 15 q^{11} - 59 q^{13} + 104 q^{17} - 38 q^{19} + 21 q^{23} - 209 q^{25} + 137 q^{29} + 4 q^{31} - 129 q^{35} - 152 q^{37} + 210 q^{41} + 67 q^{43} - 273 q^{47} - 212 q^{49} - 209 q^{53} + 237 q^{55} - 799 q^{59} + 149 q^{61} + 52 q^{65} + 201 q^{67} - 792 q^{71} - 246 q^{73} - 843 q^{77} - 254 q^{79} - 374 q^{83} - 25 q^{85} + 564 q^{89} - 621 q^{91} - 95 q^{95} - 178 q^{97}+O(q^{100}) $ 2 * q + 5 * q^5 - 6 * q^7 + 15 * q^11 - 59 * q^13 + 104 * q^17 - 38 * q^19 + 21 * q^23 - 209 * q^25 + 137 * q^29 + 4 * q^31 - 129 * q^35 - 152 * q^37 + 210 * q^41 + 67 * q^43 - 273 * q^47 - 212 * q^49 - 209 * q^53 + 237 * q^55 - 799 * q^59 + 149 * q^61 + 52 * q^65 + 201 * q^67 - 792 * q^71 - 246 * q^73 - 843 * q^77 - 254 * q^79 - 374 * q^83 - 25 * q^85 + 564 * q^89 - 621 * q^91 - 95 * q^95 - 178 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.27492	−0.114032	−0.0570160	−	0.998373i	$-0.518159\pi$
$5$	−1.27492	−0.114032	−0.0570160	+	0.998373i	$0.518159\pi$
$6$	0	0
$7$	12.0997	0.653321	0.326660	−	0.945142i	$-0.394077\pi$
$7$	12.0997	0.653321	0.326660	+	0.945142i	$0.394077\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−18.9244	−0.518721	−0.259360	−	0.965781i	$-0.583512\pi$
$11$	−18.9244	−0.518721	−0.259360	+	0.965781i	$0.583512\pi$
$12$	0	0
$13$	−55.9244	−1.19313	−0.596563	−	0.802566i	$-0.703468\pi$
$13$	−55.9244	−1.19313	−0.596563	+	0.802566i	$0.703468\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	89.7492	1.28043	0.640217	−	0.768194i	$-0.278845\pi$
$17$	89.7492	1.28043	0.640217	+	0.768194i	$0.278845\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	135.072	1.22454	0.612272	−	0.790647i	$-0.290256\pi$
$23$	135.072	1.22454	0.612272	+	0.790647i	$0.290256\pi$
$24$	0	0
$25$	−123.375	−0.986997
$26$	0	0
$27$	0	0
$28$	0	0
$29$	102.474	0.656172	0.328086	−	0.944648i	$-0.393596\pi$
$29$	102.474	0.656172	0.328086	+	0.944648i	$0.393596\pi$
$30$	0	0
$31$	−103.698	−0.600795	−0.300398	−	0.953814i	$-0.597119\pi$
$31$	−103.698	−0.600795	−0.300398	+	0.953814i	$0.597119\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−15.4261	−0.0744995
$36$	0	0
$37$	29.6977	0.131953	0.0659766	−	0.997821i	$-0.478984\pi$
$37$	29.6977	0.131953	0.0659766	+	0.997821i	$0.478984\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−234.743	−0.894162	−0.447081	−	0.894494i	$-0.647536\pi$
$41$	−234.743	−0.894162	−0.447081	+	0.894494i	$0.647536\pi$
$42$	0	0
$43$	−53.3231	−0.189109	−0.0945546	−	0.995520i	$-0.530143\pi$
$43$	−53.3231	−0.189109	−0.0945546	+	0.995520i	$0.530143\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	33.3713	0.103568	0.0517841	−	0.998658i	$-0.483509\pi$
$47$	33.3713	0.103568	0.0517841	+	0.998658i	$0.483509\pi$
$48$	0	0
$49$	−196.598	−0.573172
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−93.1752	−0.241483	−0.120742	−	0.992684i	$-0.538527\pi$
$53$	−93.1752	−0.241483	−0.120742	+	0.992684i	$0.538527\pi$
$54$	0	0
$55$	24.1271	0.0591508
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−637.320	−1.40630	−0.703152	−	0.711039i	$-0.748225\pi$
$59$	−637.320	−1.40630	−0.703152	+	0.711039i	$0.748225\pi$
$60$	0	0
$61$	−125.571	−0.263568	−0.131784	−	0.991278i	$-0.542071\pi$
$61$	−125.571	−0.263568	−0.131784	+	0.991278i	$0.542071\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	71.2990	0.136055
$66$	0	0
$67$	119.375	0.217671	0.108835	−	0.994060i	$-0.465288\pi$
$67$	119.375	0.217671	0.108835	+	0.994060i	$0.465288\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−18.5083	−0.0309370	−0.0154685	−	0.999880i	$-0.504924\pi$
$71$	−18.5083	−0.0309370	−0.0154685	+	0.999880i	$0.504924\pi$
$72$	0	0
$73$	−394.794	−0.632975	−0.316487	−	0.948597i	$-0.602503\pi$
$73$	−394.794	−0.632975	−0.316487	+	0.948597i	$0.602503\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−228.979	−0.338891
$78$	0	0
$79$	303.341	0.432006	0.216003	−	0.976393i	$-0.430698\pi$
$79$	303.341	0.432006	0.216003	+	0.976393i	$0.430698\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	394.337	0.521496	0.260748	−	0.965407i	$-0.416031\pi$
$83$	394.337	0.521496	0.260748	+	0.965407i	$0.416031\pi$
$84$	0	0
$85$	−114.423	−0.146010
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1021.88	1.21707	0.608536	−	0.793526i	$-0.291757\pi$
$89$	1021.88	1.21707	0.608536	+	0.793526i	$0.291757\pi$
$90$	0	0
$91$	−676.667	−0.779494
$92$	0	0
$93$	0	0
$94$	0	0
$95$	24.2234	0.0261607
$96$	0	0
$97$	1322.82	1.38466	0.692330	−	0.721581i	$-0.256584\pi$
$97$	1322.82	1.38466	0.692330	+	0.721581i	$0.256584\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1923.06	−1.89457	−0.947285	−	0.320391i	$-0.896186\pi$
$101$	−1923.06	−1.89457	−0.947285	+	0.320391i	$0.896186\pi$
$102$	0	0
$103$	467.774	0.447487	0.223743	−	0.974648i	$-0.428172\pi$
$103$	467.774	0.447487	0.223743	+	0.974648i	$0.428172\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−260.468	−0.235330	−0.117665	−	0.993053i	$-0.537541\pi$
$107$	−260.468	−0.235330	−0.117665	+	0.993053i	$0.537541\pi$
$108$	0	0
$109$	−511.856	−0.449789	−0.224894	−	0.974383i	$-0.572204\pi$
$109$	−511.856	−0.449789	−0.224894	+	0.974383i	$0.572204\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1453.48	1.21002	0.605008	−	0.796220i	$-0.293170\pi$
$113$	1453.48	1.21002	0.605008	+	0.796220i	$0.293170\pi$
$114$	0	0
$115$	−172.206	−0.139637
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1085.94	0.836534
$120$	0	0
$121$	−972.866	−0.730929
$122$	0	0
$123$	0	0
$124$	0	0
$125$	316.657	0.226581
$126$	0	0
$127$	−2166.29	−1.51360	−0.756799	−	0.653647i	$-0.773238\pi$
$127$	−2166.29	−1.51360	−0.756799	+	0.653647i	$0.773238\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−329.310	−0.219633	−0.109817	−	0.993952i	$-0.535026\pi$
$131$	−329.310	−0.219633	−0.109817	+	0.993952i	$0.535026\pi$
$132$	0	0
$133$	−229.894	−0.149882
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−736.919	−0.459556	−0.229778	−	0.973243i	$-0.573800\pi$
$137$	−736.919	−0.459556	−0.229778	+	0.973243i	$0.573800\pi$
$138$	0	0
$139$	−3041.10	−1.85571	−0.927853	−	0.372947i	$-0.878347\pi$
$139$	−3041.10	−1.85571	−0.927853	+	0.372947i	$0.878347\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1058.34	0.618899
$144$	0	0
$145$	−130.646	−0.0748247
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2156.31	−1.18558	−0.592790	−	0.805357i	$-0.701974\pi$
$149$	−2156.31	−1.18558	−0.592790	+	0.805357i	$0.701974\pi$
$150$	0	0
$151$	−1816.60	−0.979024	−0.489512	−	0.871997i	$-0.662825\pi$
$151$	−1816.60	−0.979024	−0.489512	+	0.871997i	$0.662825\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	132.206	0.0685099
$156$	0	0
$157$	−1118.10	−0.568368	−0.284184	−	0.958770i	$-0.591723\pi$
$157$	−1118.10	−0.568368	−0.284184	+	0.958770i	$0.591723\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1634.33	0.800020
$162$	0	0
$163$	−3304.95	−1.58812	−0.794060	−	0.607839i	$-0.792037\pi$
$163$	−3304.95	−1.58812	−0.794060	+	0.607839i	$0.792037\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1750.74	−0.811237	−0.405618	−	0.914043i	$-0.632944\pi$
$167$	−1750.74	−0.811237	−0.405618	+	0.914043i	$0.632944\pi$
$168$	0	0
$169$	930.541	0.423551
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2698.18	−1.18577	−0.592887	−	0.805286i	$-0.702012\pi$
$173$	−2698.18	−1.18577	−0.592887	+	0.805286i	$0.702012\pi$
$174$	0	0
$175$	−1492.79	−0.644825
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3867.80	−1.61505	−0.807523	−	0.589836i	$-0.799192\pi$
$179$	−3867.80	−1.61505	−0.807523	+	0.589836i	$0.799192\pi$
$180$	0	0
$181$	3858.74	1.58463	0.792315	−	0.610112i	$-0.208875\pi$
$181$	3858.74	1.58463	0.792315	+	0.610112i	$0.208875\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−37.8621	−0.0150469
$186$	0	0
$187$	−1698.45	−0.664187
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4111.06	−1.55741	−0.778706	−	0.627389i	$-0.784124\pi$
$191$	−4111.06	−1.55741	−0.778706	+	0.627389i	$0.784124\pi$
$192$	0	0
$193$	−1825.36	−0.680789	−0.340395	−	0.940283i	$-0.610561\pi$
$193$	−1825.36	−0.680789	−0.340395	+	0.940283i	$0.610561\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	767.088	0.277425	0.138713	−	0.990333i	$-0.455704\pi$
$197$	767.088	0.277425	0.138713	+	0.990333i	$0.455704\pi$
$198$	0	0
$199$	3176.43	1.13151	0.565757	−	0.824572i	$-0.308584\pi$
$199$	3176.43	1.13151	0.565757	+	0.824572i	$0.308584\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1239.90	0.428691
$204$	0	0
$205$	299.277	0.101963
$206$	0	0
$207$	0	0
$208$	0	0
$209$	359.564	0.119003
$210$	0	0
$211$	2460.54	0.802797	0.401399	−	0.915903i	$-0.368524\pi$
$211$	2460.54	0.802797	0.401399	+	0.915903i	$0.368524\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	67.9825	0.0215645
$216$	0	0
$217$	−1254.71	−0.392512
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5019.17	−1.52772
$222$	0	0
$223$	3731.73	1.12061	0.560304	−	0.828287i	$-0.310684\pi$
$223$	3731.73	1.12061	0.560304	+	0.828287i	$0.310684\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−653.188	−0.190985	−0.0954926	−	0.995430i	$-0.530443\pi$
$227$	−653.188	−0.190985	−0.0954926	+	0.995430i	$0.530443\pi$
$228$	0	0
$229$	340.511	0.0982602	0.0491301	−	0.998792i	$-0.484355\pi$
$229$	340.511	0.0982602	0.0491301	+	0.998792i	$0.484355\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3936.99	1.10696	0.553478	−	0.832864i	$-0.313300\pi$
$233$	3936.99	1.10696	0.553478	+	0.832864i	$0.313300\pi$
$234$	0	0
$235$	−42.5456	−0.0118101
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4762.18	−1.28887	−0.644434	−	0.764660i	$-0.722907\pi$
$239$	−4762.18	−1.28887	−0.644434	+	0.764660i	$0.722907\pi$
$240$	0	0
$241$	−3893.55	−1.04069	−0.520343	−	0.853957i	$-0.674196\pi$
$241$	−3893.55	−1.04069	−0.520343	+	0.853957i	$0.674196\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	250.646	0.0653600
$246$	0	0
$247$	1062.56	0.273722
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1384.13	−0.348069	−0.174034	−	0.984740i	$-0.555680\pi$
$251$	−1384.13	−0.348069	−0.174034	+	0.984740i	$0.555680\pi$
$252$	0	0
$253$	−2556.16	−0.635196
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4645.01	1.12742	0.563711	−	0.825972i	$-0.309373\pi$
$257$	4645.01	1.12742	0.563711	+	0.825972i	$0.309373\pi$
$258$	0	0
$259$	359.332	0.0862078
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2151.41	0.504416	0.252208	−	0.967673i	$-0.418843\pi$
$263$	2151.41	0.504416	0.252208	+	0.967673i	$0.418843\pi$
$264$	0	0
$265$	118.791	0.0275368
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5768.66	1.30752	0.653758	−	0.756704i	$-0.273192\pi$
$269$	5768.66	1.30752	0.653758	+	0.756704i	$0.273192\pi$
$270$	0	0
$271$	−6859.23	−1.53752	−0.768761	−	0.639537i	$-0.779126\pi$
$271$	−6859.23	−1.53752	−0.768761	+	0.639537i	$0.779126\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2334.79	0.511976
$276$	0	0
$277$	1237.24	0.268371	0.134185	−	0.990956i	$-0.457158\pi$
$277$	1237.24	0.268371	0.134185	+	0.990956i	$0.457158\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4355.21	−0.924592	−0.462296	−	0.886726i	$-0.652974\pi$
$281$	−4355.21	−0.924592	−0.462296	+	0.886726i	$0.652974\pi$
$282$	0	0
$283$	−3651.29	−0.766949	−0.383474	−	0.923551i	$-0.625273\pi$
$283$	−3651.29	−0.766949	−0.383474	+	0.923551i	$0.625273\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2840.31	−0.584174
$288$	0	0
$289$	3141.91	0.639510
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3391.25	0.676174	0.338087	−	0.941115i	$-0.390220\pi$
$293$	3391.25	0.676174	0.338087	+	0.941115i	$0.390220\pi$
$294$	0	0
$295$	812.530	0.160364
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7553.84	−1.46104
$300$	0	0
$301$	−645.192	−0.123549
$302$	0	0
$303$	0	0
$304$	0	0
$305$	160.092	0.0300552
$306$	0	0
$307$	4343.35	0.807452	0.403726	−	0.914880i	$-0.367715\pi$
$307$	4343.35	0.807452	0.403726	+	0.914880i	$0.367715\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5671.28	−1.03405	−0.517024	−	0.855971i	$-0.672960\pi$
$311$	−5671.28	−1.03405	−0.517024	+	0.855971i	$0.672960\pi$
$312$	0	0
$313$	−9449.71	−1.70648	−0.853241	−	0.521516i	$-0.825367\pi$
$313$	−9449.71	−1.70648	−0.853241	+	0.521516i	$0.825367\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5390.75	0.955125	0.477562	−	0.878598i	$-0.341520\pi$
$317$	5390.75	0.955125	0.477562	+	0.878598i	$0.341520\pi$
$318$	0	0
$319$	−1939.27	−0.340370
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1705.23	−0.293752
$324$	0	0
$325$	6899.65	1.17761
$326$	0	0
$327$	0	0
$328$	0	0
$329$	403.781	0.0676632
$330$	0	0
$331$	−9230.14	−1.53273	−0.766366	−	0.642404i	$-0.777937\pi$
$331$	−9230.14	−1.53273	−0.766366	+	0.642404i	$0.777937\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−152.193	−0.0248214
$336$	0	0
$337$	−7815.90	−1.26338	−0.631690	−	0.775221i	$-0.717638\pi$
$337$	−7815.90	−1.26338	−0.631690	+	0.775221i	$0.717638\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1962.42	0.311645
$342$	0	0
$343$	−6528.96	−1.02779
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−680.076	−0.105211	−0.0526057	−	0.998615i	$-0.516753\pi$
$347$	−680.076	−0.105211	−0.0526057	+	0.998615i	$0.516753\pi$
$348$	0	0
$349$	3641.72	0.558558	0.279279	−	0.960210i	$-0.409905\pi$
$349$	3641.72	0.558558	0.279279	+	0.960210i	$0.409905\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7885.46	1.18895	0.594477	−	0.804113i	$-0.297359\pi$
$353$	7885.46	1.18895	0.594477	+	0.804113i	$0.297359\pi$
$354$	0	0
$355$	23.5965	0.00352781
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2862.17	0.420779	0.210389	−	0.977618i	$-0.432527\pi$
$359$	2862.17	0.420779	0.210389	+	0.977618i	$0.432527\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	503.330	0.0721794
$366$	0	0
$367$	−9783.29	−1.39151	−0.695754	−	0.718280i	$-0.744930\pi$
$367$	−9783.29	−1.39151	−0.695754	+	0.718280i	$0.744930\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1127.39	−0.157766
$372$	0	0
$373$	6551.84	0.909495	0.454747	−	0.890621i	$-0.349730\pi$
$373$	6551.84	0.909495	0.454747	+	0.890621i	$0.349730\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5730.81	−0.782896
$378$	0	0
$379$	−2228.78	−0.302071	−0.151035	−	0.988528i	$-0.548261\pi$
$379$	−2228.78	−0.302071	−0.151035	+	0.988528i	$0.548261\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5223.85	0.696935	0.348468	−	0.937321i	$-0.386702\pi$
$383$	5223.85	0.696935	0.348468	+	0.937321i	$0.386702\pi$
$384$	0	0
$385$	291.930	0.0386444
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7672.26	0.999998	0.499999	−	0.866026i	$-0.333334\pi$
$389$	7672.26	0.999998	0.499999	+	0.866026i	$0.333334\pi$
$390$	0	0
$391$	12122.6	1.56795
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−386.734	−0.0492625
$396$	0	0
$397$	−13564.0	−1.71475	−0.857375	−	0.514692i	$-0.827906\pi$
$397$	−13564.0	−1.71475	−0.857375	+	0.514692i	$0.827906\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6488.39	0.808017	0.404009	−	0.914755i	$-0.367617\pi$
$401$	6488.39	0.808017	0.404009	+	0.914755i	$0.367617\pi$
$402$	0	0
$403$	5799.23	0.716825
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−562.011	−0.0684469
$408$	0	0
$409$	14696.2	1.77672	0.888360	−	0.459147i	$-0.151845\pi$
$409$	14696.2	1.77672	0.888360	+	0.459147i	$0.151845\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−7711.36	−0.918768
$414$	0	0
$415$	−502.747	−0.0594672
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9492.08	−1.10673	−0.553363	−	0.832940i	$-0.686656\pi$
$419$	−9492.08	−1.10673	−0.553363	+	0.832940i	$0.686656\pi$
$420$	0	0
$421$	−13028.4	−1.50823	−0.754116	−	0.656741i	$-0.771935\pi$
$421$	−13028.4	−1.50823	−0.754116	+	0.656741i	$0.771935\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−11072.8	−1.26378
$426$	0	0
$427$	−1519.36	−0.172195
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5404.99	0.604058	0.302029	−	0.953299i	$-0.402336\pi$
$431$	5404.99	0.604058	0.302029	+	0.953299i	$0.402336\pi$
$432$	0	0
$433$	16745.4	1.85850	0.929252	−	0.369448i	$-0.120453\pi$
$433$	16745.4	1.85850	0.929252	+	0.369448i	$0.120453\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2566.37	−0.280930
$438$	0	0
$439$	13422.6	1.45928	0.729641	−	0.683831i	$-0.239687\pi$
$439$	13422.6	1.45928	0.729641	+	0.683831i	$0.239687\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14048.7	1.50672	0.753358	−	0.657611i	$-0.228433\pi$
$443$	14048.7	1.50672	0.753358	+	0.657611i	$0.228433\pi$
$444$	0	0
$445$	−1302.82	−0.138785
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−9841.11	−1.03437	−0.517183	−	0.855875i	$-0.673020\pi$
$449$	−9841.11	−1.03437	−0.517183	+	0.855875i	$0.673020\pi$
$450$	0	0
$451$	4442.37	0.463820
$452$	0	0
$453$	0	0
$454$	0	0
$455$	862.694	0.0888873
$456$	0	0
$457$	4443.15	0.454796	0.227398	−	0.973802i	$-0.426978\pi$
$457$	4443.15	0.454796	0.227398	+	0.973802i	$0.426978\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12288.1	1.24146	0.620728	−	0.784026i	$-0.286837\pi$
$461$	12288.1	1.24146	0.620728	+	0.784026i	$0.286837\pi$
$462$	0	0
$463$	4814.38	0.483246	0.241623	−	0.970370i	$-0.422320\pi$
$463$	4814.38	0.483246	0.241623	+	0.970370i	$0.422320\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9863.03	−0.977316	−0.488658	−	0.872475i	$-0.662513\pi$
$467$	−9863.03	−0.977316	−0.488658	+	0.872475i	$0.662513\pi$
$468$	0	0
$469$	1444.39	0.142209
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1009.11	0.0980949
$474$	0	0
$475$	2344.12	0.226433
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−12743.3	−1.21557	−0.607785	−	0.794102i	$-0.707942\pi$
$479$	−12743.3	−1.21557	−0.607785	+	0.794102i	$0.707942\pi$
$480$	0	0
$481$	−1660.83	−0.157437
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1686.48	−0.157896
$486$	0	0
$487$	−4200.07	−0.390807	−0.195404	−	0.980723i	$-0.562602\pi$
$487$	−4200.07	−0.390807	−0.195404	+	0.980723i	$0.562602\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	11292.3	1.03791	0.518955	−	0.854802i	$-0.326321\pi$
$491$	11292.3	1.03791	0.518955	+	0.854802i	$0.326321\pi$
$492$	0	0
$493$	9196.98	0.840185
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−223.944	−0.0202118
$498$	0	0
$499$	−12126.6	−1.08790	−0.543948	−	0.839119i	$-0.683071\pi$
$499$	−12126.6	−1.08790	−0.543948	+	0.839119i	$0.683071\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2033.61	0.180267	0.0901336	−	0.995930i	$-0.471271\pi$
$503$	2033.61	0.180267	0.0901336	+	0.995930i	$0.471271\pi$
$504$	0	0
$505$	2451.74	0.216042
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1367.41	0.119076	0.0595379	−	0.998226i	$-0.481037\pi$
$509$	1367.41	0.119076	0.0595379	+	0.998226i	$0.481037\pi$
$510$	0	0
$511$	−4776.88	−0.413535
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−596.373	−0.0510279
$516$	0	0
$517$	−631.532	−0.0537229
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9613.98	0.808438	0.404219	−	0.914662i	$-0.367543\pi$
$521$	9613.98	0.808438	0.404219	+	0.914662i	$0.367543\pi$
$522$	0	0
$523$	−8185.79	−0.684397	−0.342199	−	0.939628i	$-0.611172\pi$
$523$	−8185.79	−0.684397	−0.342199	+	0.939628i	$0.611172\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9306.78	−0.769278
$528$	0	0
$529$	6077.52	0.499508
$530$	0	0
$531$	0	0
$532$	0	0
$533$	13127.8	1.06685
$534$	0	0
$535$	332.075	0.0268352
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3720.50	0.297316
$540$	0	0
$541$	−1869.73	−0.148587	−0.0742937	−	0.997236i	$-0.523670\pi$
$541$	−1869.73	−0.148587	−0.0742937	+	0.997236i	$0.523670\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	652.575	0.0512903
$546$	0	0
$547$	4024.62	0.314589	0.157295	−	0.987552i	$-0.449723\pi$
$547$	4024.62	0.314589	0.157295	+	0.987552i	$0.449723\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1947.01	−0.150536
$552$	0	0
$553$	3670.32	0.282239
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15750.7	1.19816	0.599082	−	0.800688i	$-0.295532\pi$
$557$	15750.7	1.19816	0.599082	+	0.800688i	$0.295532\pi$
$558$	0	0
$559$	2982.06	0.225631
$560$	0	0
$561$	0	0
$562$	0	0
$563$	641.091	0.0479907	0.0239954	−	0.999712i	$-0.492361\pi$
$563$	641.091	0.0479907	0.0239954	+	0.999712i	$0.492361\pi$
$564$	0	0
$565$	−1853.06	−0.137981
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18392.3	1.35509	0.677544	−	0.735482i	$-0.263044\pi$
$569$	18392.3	1.35509	0.677544	+	0.735482i	$0.263044\pi$
$570$	0	0
$571$	1400.26	0.102625	0.0513126	−	0.998683i	$-0.483660\pi$
$571$	1400.26	0.102625	0.0513126	+	0.998683i	$0.483660\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−16664.5	−1.20862
$576$	0	0
$577$	23464.7	1.69298	0.846489	−	0.532406i	$-0.178712\pi$
$577$	23464.7	1.69298	0.846489	+	0.532406i	$0.178712\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4771.35	0.340704
$582$	0	0
$583$	1763.29	0.125262
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1336.76	0.0939934	0.0469967	−	0.998895i	$-0.485035\pi$
$587$	1336.76	0.0939934	0.0469967	+	0.998895i	$0.485035\pi$
$588$	0	0
$589$	1970.26	0.137832
$590$	0	0
$591$	0	0
$592$	0	0
$593$	14890.5	1.03116	0.515581	−	0.856841i	$-0.327576\pi$
$593$	14890.5	1.03116	0.515581	+	0.856841i	$0.327576\pi$
$594$	0	0
$595$	−1384.48	−0.0953917
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−22648.7	−1.54491	−0.772456	−	0.635068i	$-0.780972\pi$
$599$	−22648.7	−1.54491	−0.772456	+	0.635068i	$0.780972\pi$
$600$	0	0
$601$	8930.64	0.606137	0.303069	−	0.952969i	$-0.401989\pi$
$601$	8930.64	0.606137	0.303069	+	0.952969i	$0.401989\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1240.32	0.0833493
$606$	0	0
$607$	−5611.75	−0.375245	−0.187623	−	0.982241i	$-0.560078\pi$
$607$	−5611.75	−0.375245	−0.187623	+	0.982241i	$0.560078\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1866.27	−0.123570
$612$	0	0
$613$	−16212.8	−1.06823	−0.534117	−	0.845410i	$-0.679356\pi$
$613$	−16212.8	−1.06823	−0.534117	+	0.845410i	$0.679356\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26562.7	1.73318	0.866592	−	0.499017i	$-0.166306\pi$
$617$	26562.7	1.73318	0.866592	+	0.499017i	$0.166306\pi$
$618$	0	0
$619$	816.376	0.0530096	0.0265048	−	0.999649i	$-0.491562\pi$
$619$	816.376	0.0530096	0.0265048	+	0.999649i	$0.491562\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12364.5	0.795139
$624$	0	0
$625$	15018.1	0.961159
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2665.34	0.168957
$630$	0	0
$631$	29218.6	1.84338	0.921690	−	0.387928i	$-0.126809\pi$
$631$	29218.6	1.84338	0.921690	+	0.387928i	$0.126809\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2761.84	0.172599
$636$	0	0
$637$	10994.6	0.683867
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4831.98	0.297740	0.148870	−	0.988857i	$-0.452436\pi$
$641$	4831.98	0.297740	0.148870	+	0.988857i	$0.452436\pi$
$642$	0	0
$643$	25259.5	1.54920	0.774602	−	0.632449i	$-0.217950\pi$
$643$	25259.5	1.54920	0.774602	+	0.632449i	$0.217950\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−11004.6	−0.668676	−0.334338	−	0.942453i	$-0.608513\pi$
$647$	−11004.6	−0.668676	−0.334338	+	0.942453i	$0.608513\pi$
$648$	0	0
$649$	12060.9	0.729479
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5034.69	−0.301720	−0.150860	−	0.988555i	$-0.548204\pi$
$653$	−5034.69	−0.301720	−0.150860	+	0.988555i	$0.548204\pi$
$654$	0	0
$655$	419.843	0.0250452
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5027.20	0.297165	0.148583	−	0.988900i	$-0.452529\pi$
$659$	5027.20	0.297165	0.148583	+	0.988900i	$0.452529\pi$
$660$	0	0
$661$	−28126.0	−1.65503	−0.827515	−	0.561444i	$-0.810246\pi$
$661$	−28126.0	−1.65503	−0.827515	+	0.561444i	$0.810246\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	293.095	0.0170914
$666$	0	0
$667$	13841.4	0.803512
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2376.35	0.136718
$672$	0	0
$673$	15864.9	0.908689	0.454344	−	0.890826i	$-0.349874\pi$
$673$	15864.9	0.908689	0.454344	+	0.890826i	$0.349874\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−15911.8	−0.903308	−0.451654	−	0.892193i	$-0.649166\pi$
$677$	−15911.8	−0.903308	−0.451654	+	0.892193i	$0.649166\pi$
$678$	0	0
$679$	16005.7	0.904626
$680$	0	0
$681$	0	0
$682$	0	0
$683$	30172.0	1.69033	0.845167	−	0.534502i	$-0.179501\pi$
$683$	30172.0	1.69033	0.845167	+	0.534502i	$0.179501\pi$
$684$	0	0
$685$	939.510	0.0524042
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5210.77	0.288120
$690$	0	0
$691$	−14213.9	−0.782522	−0.391261	−	0.920280i	$-0.627961\pi$
$691$	−14213.9	−0.782522	−0.391261	+	0.920280i	$0.627961\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3877.16	0.211610
$696$	0	0
$697$	−21067.9	−1.14491
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−25215.2	−1.35858	−0.679292	−	0.733869i	$-0.737713\pi$
$701$	−25215.2	−1.35858	−0.679292	+	0.733869i	$0.737713\pi$
$702$	0	0
$703$	−564.256	−0.0302721
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−23268.4	−1.23776
$708$	0	0
$709$	34384.4	1.82134	0.910672	−	0.413130i	$-0.135564\pi$
$709$	34384.4	1.82134	0.910672	+	0.413130i	$0.135564\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−14006.7	−0.735700
$714$	0	0
$715$	−1349.29	−0.0705744
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4418.46	−0.229180	−0.114590	−	0.993413i	$-0.536555\pi$
$719$	−4418.46	−0.229180	−0.114590	+	0.993413i	$0.536555\pi$
$720$	0	0
$721$	5659.91	0.292353
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−12642.7	−0.647640
$726$	0	0
$727$	805.044	0.0410694	0.0205347	−	0.999789i	$-0.493463\pi$
$727$	805.044	0.0410694	0.0205347	+	0.999789i	$0.493463\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4785.70	−0.242142
$732$	0	0
$733$	−1662.37	−0.0837668	−0.0418834	−	0.999123i	$-0.513336\pi$
$733$	−1662.37	−0.0837668	−0.0418834	+	0.999123i	$0.513336\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2259.09	−0.112910
$738$	0	0
$739$	3145.59	0.156580	0.0782899	−	0.996931i	$-0.475054\pi$
$739$	3145.59	0.156580	0.0782899	+	0.996931i	$0.475054\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−35469.5	−1.75134	−0.875672	−	0.482906i	$-0.839581\pi$
$743$	−35469.5	−1.75134	−0.875672	+	0.482906i	$0.839581\pi$
$744$	0	0
$745$	2749.11	0.135194
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3151.57	−0.153746
$750$	0	0
$751$	−29978.8	−1.45665	−0.728324	−	0.685233i	$-0.759700\pi$
$751$	−29978.8	−1.45665	−0.728324	+	0.685233i	$0.759700\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2316.01	0.111640
$756$	0	0
$757$	−33882.6	−1.62680	−0.813398	−	0.581707i	$-0.802385\pi$
$757$	−33882.6	−1.62680	−0.813398	+	0.581707i	$0.802385\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−9023.05	−0.429810	−0.214905	−	0.976635i	$-0.568944\pi$
$761$	−9023.05	−0.429810	−0.214905	+	0.976635i	$0.568944\pi$
$762$	0	0
$763$	−6193.29	−0.293856
$764$	0	0
$765$	0	0
$766$	0	0
$767$	35641.7	1.67790
$768$	0	0
$769$	−773.311	−0.0362631	−0.0181315	−	0.999836i	$-0.505772\pi$
$769$	−773.311	−0.0362631	−0.0181315	+	0.999836i	$0.505772\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−19778.7	−0.920299	−0.460150	−	0.887841i	$-0.652204\pi$
$773$	−19778.7	−0.920299	−0.460150	+	0.887841i	$0.652204\pi$
$774$	0	0
$775$	12793.7	0.592983
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4460.11	0.205135
$780$	0	0
$781$	350.258	0.0160477
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1425.48	0.0648122
$786$	0	0
$787$	32308.3	1.46336	0.731681	−	0.681648i	$-0.238736\pi$
$787$	32308.3	1.46336	0.731681	+	0.681648i	$0.238736\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	17586.6	0.790528
$792$	0	0
$793$	7022.46	0.314470
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−40290.5	−1.79067	−0.895335	−	0.445394i	$-0.853064\pi$
$797$	−40290.5	−1.79067	−0.895335	+	0.445394i	$0.853064\pi$
$798$	0	0
$799$	2995.04	0.132612
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7471.25	0.328337
$804$	0	0
$805$	−2083.64	−0.0912279
$806$	0	0
$807$	0	0
$808$	0	0
$809$	16600.1	0.721417	0.360709	−	0.932679i	$-0.382535\pi$
$809$	16600.1	0.721417	0.360709	+	0.932679i	$0.382535\pi$
$810$	0	0
$811$	9586.41	0.415073	0.207537	−	0.978227i	$-0.433455\pi$
$811$	9586.41	0.415073	0.207537	+	0.978227i	$0.433455\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4213.54	0.181097
$816$	0	0
$817$	1013.14	0.0433846
$818$	0	0
$819$	0	0
$820$	0	0
$821$	33381.4	1.41902	0.709512	−	0.704693i	$-0.248915\pi$
$821$	33381.4	1.41902	0.709512	+	0.704693i	$0.248915\pi$
$822$	0	0
$823$	−3953.76	−0.167460	−0.0837299	−	0.996488i	$-0.526683\pi$
$823$	−3953.76	−0.167460	−0.0837299	+	0.996488i	$0.526683\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33403.5	1.40454	0.702270	−	0.711910i	$-0.252170\pi$
$827$	33403.5	1.40454	0.702270	+	0.711910i	$0.252170\pi$
$828$	0	0
$829$	41612.3	1.74337	0.871686	−	0.490065i	$-0.163027\pi$
$829$	41612.3	1.74337	0.871686	+	0.490065i	$0.163027\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−17644.5	−0.733909
$834$	0	0
$835$	2232.05	0.0925070
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−10843.5	−0.446199	−0.223099	−	0.974796i	$-0.571617\pi$
$839$	−10843.5	−0.446199	−0.223099	+	0.974796i	$0.571617\pi$
$840$	0	0
$841$	−13888.0	−0.569438
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1186.36	−0.0482984
$846$	0	0
$847$	−11771.4	−0.477531
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4011.33	0.161583
$852$	0	0
$853$	13530.7	0.543119	0.271560	−	0.962422i	$-0.412461\pi$
$853$	13530.7	0.543119	0.271560	+	0.962422i	$0.412461\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3510.58	0.139929	0.0699645	−	0.997549i	$-0.477711\pi$
$857$	3510.58	0.139929	0.0699645	+	0.997549i	$0.477711\pi$
$858$	0	0
$859$	−4945.74	−0.196445	−0.0982226	−	0.995164i	$-0.531316\pi$
$859$	−4945.74	−0.196445	−0.0982226	+	0.995164i	$0.531316\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38957.6	1.53665	0.768326	−	0.640059i	$-0.221090\pi$
$863$	38957.6	1.53665	0.768326	+	0.640059i	$0.221090\pi$
$864$	0	0
$865$	3439.96	0.135216
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−5740.54	−0.224090
$870$	0	0
$871$	−6675.95	−0.259708
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3831.45	0.148030
$876$	0	0
$877$	544.671	0.0209718	0.0104859	−	0.999945i	$-0.496662\pi$
$877$	544.671	0.0209718	0.0104859	+	0.999945i	$0.496662\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	19771.1	0.756079	0.378040	−	0.925789i	$-0.376598\pi$
$881$	19771.1	0.756079	0.378040	+	0.925789i	$0.376598\pi$
$882$	0	0
$883$	−12249.6	−0.466854	−0.233427	−	0.972374i	$-0.574994\pi$
$883$	−12249.6	−0.466854	−0.233427	+	0.972374i	$0.574994\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21817.3	0.825877	0.412938	−	0.910759i	$-0.364503\pi$
$887$	21817.3	0.825877	0.412938	+	0.910759i	$0.364503\pi$
$888$	0	0
$889$	−26211.4	−0.988866
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−634.054	−0.0237602
$894$	0	0
$895$	4931.13	0.184167
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−10626.3	−0.394225
$900$	0	0
$901$	−8362.40	−0.309203
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4919.58	−0.180699
$906$	0	0
$907$	−35592.0	−1.30299	−0.651496	−	0.758652i	$-0.725858\pi$
$907$	−35592.0	−1.30299	−0.651496	+	0.758652i	$0.725858\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−952.654	−0.0346464	−0.0173232	−	0.999850i	$-0.505514\pi$
$911$	−952.654	−0.0346464	−0.0173232	+	0.999850i	$0.505514\pi$
$912$	0	0
$913$	−7462.60	−0.270511
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3984.54	−0.143491
$918$	0	0
$919$	−42326.8	−1.51929	−0.759647	−	0.650335i	$-0.774628\pi$
$919$	−42326.8	−1.51929	−0.759647	+	0.650335i	$0.774628\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1035.06	0.0369118
$924$	0	0
$925$	−3663.94	−0.130237
$926$	0	0
$927$	0	0
$928$	0	0
$929$	27446.4	0.969309	0.484654	−	0.874706i	$-0.338945\pi$
$929$	27446.4	0.969309	0.484654	+	0.874706i	$0.338945\pi$
$930$	0	0
$931$	3735.36	0.131495
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2165.38	0.0757387
$936$	0	0
$937$	15560.3	0.542511	0.271256	−	0.962507i	$-0.412561\pi$
$937$	15560.3	0.542511	0.271256	+	0.962507i	$0.412561\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−17725.4	−0.614060	−0.307030	−	0.951700i	$-0.599335\pi$
$941$	−17725.4	−0.614060	−0.307030	+	0.951700i	$0.599335\pi$
$942$	0	0
$943$	−31707.2	−1.09494
$944$	0	0
$945$	0	0
$946$	0	0
$947$	588.382	0.0201899	0.0100950	−	0.999949i	$-0.496787\pi$
$947$	588.382	0.0201899	0.0100950	+	0.999949i	$0.496787\pi$
$948$	0	0
$949$	22078.6	0.755219
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−28644.2	−0.973638	−0.486819	−	0.873503i	$-0.661843\pi$
$953$	−28644.2	−0.973638	−0.486819	+	0.873503i	$0.661843\pi$
$954$	0	0
$955$	5241.26	0.177595
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8916.47	−0.300238
$960$	0	0
$961$	−19037.8	−0.639045
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2327.18	0.0776318
$966$	0	0
$967$	44523.4	1.48064	0.740318	−	0.672257i	$-0.234675\pi$
$967$	44523.4	1.48064	0.740318	+	0.672257i	$0.234675\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−18242.7	−0.602922	−0.301461	−	0.953478i	$-0.597474\pi$
$971$	−18242.7	−0.602922	−0.301461	+	0.953478i	$0.597474\pi$
$972$	0	0
$973$	−36796.4	−1.21237
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−23434.6	−0.767390	−0.383695	−	0.923460i	$-0.625349\pi$
$977$	−23434.6	−0.767390	−0.383695	+	0.923460i	$0.625349\pi$
$978$	0	0
$979$	−19338.6	−0.631321
$980$	0	0
$981$	0	0
$982$	0	0
$983$	59519.4	1.93121	0.965603	−	0.260022i	$-0.0837299\pi$
$983$	59519.4	1.93121	0.965603	+	0.260022i	$0.0837299\pi$
$984$	0	0
$985$	−977.974	−0.0316354
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7202.47	−0.231573
$990$	0	0
$991$	8466.93	0.271404	0.135702	−	0.990750i	$-0.456671\pi$
$991$	8466.93	0.271404	0.135702	+	0.990750i	$0.456671\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−4049.69	−0.129029
$996$	0	0
$997$	27474.3	0.872738	0.436369	−	0.899768i	$-0.356264\pi$
$997$	27474.3	0.872738	0.436369	+	0.899768i	$0.356264\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.4.a.a.1.1		2
3.2	odd	2		152.4.a.a.1.1	✓	2
12.11	even	2		304.4.a.e.1.2		2
24.5	odd	2		1216.4.a.m.1.2		2
24.11	even	2		1216.4.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.a.a.1.1	✓	2	3.2	odd	2
304.4.a.e.1.2		2	12.11	even	2
1216.4.a.k.1.1		2	24.11	even	2
1216.4.a.m.1.2		2	24.5	odd	2
1368.4.a.a.1.1		2	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:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1368.a (trivial)

\(f(q)\)	\(=\)	\(q-1.27492 q^{5} +12.0997 q^{7} +O(q^{10})\)	\(q-1.27492 q^{5} +12.0997 q^{7} -18.9244 q^{11} -55.9244 q^{13} +89.7492 q^{17} -19.0000 q^{19} +135.072 q^{23} -123.375 q^{25} +102.474 q^{29} -103.698 q^{31} -15.4261 q^{35} +29.6977 q^{37} -234.743 q^{41} -53.3231 q^{43} +33.3713 q^{47} -196.598 q^{49} -93.1752 q^{53} +24.1271 q^{55} -637.320 q^{59} -125.571 q^{61} +71.2990 q^{65} +119.375 q^{67} -18.5083 q^{71} -394.794 q^{73} -228.979 q^{77} +303.341 q^{79} +394.337 q^{83} -114.423 q^{85} +1021.88 q^{89} -676.667 q^{91} +24.2234 q^{95} +1322.82 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{5} - 6 q^{7}+O(q^{10}) \) 2 * q + 5 * q^5 - 6 * q^7	\( 2 q + 5 q^{5} - 6 q^{7} + 15 q^{11} - 59 q^{13} + 104 q^{17} - 38 q^{19} + 21 q^{23} - 209 q^{25} + 137 q^{29} + 4 q^{31} - 129 q^{35} - 152 q^{37} + 210 q^{41} + 67 q^{43} - 273 q^{47} - 212 q^{49} - 209 q^{53} + 237 q^{55} - 799 q^{59} + 149 q^{61} + 52 q^{65} + 201 q^{67} - 792 q^{71} - 246 q^{73} - 843 q^{77} - 254 q^{79} - 374 q^{83} - 25 q^{85} + 564 q^{89} - 621 q^{91} - 95 q^{95} - 178 q^{97}+O(q^{100}) \) 2 * q + 5 * q^5 - 6 * q^7 + 15 * q^11 - 59 * q^13 + 104 * q^17 - 38 * q^19 + 21 * q^23 - 209 * q^25 + 137 * q^29 + 4 * q^31 - 129 * q^35 - 152 * q^37 + 210 * q^41 + 67 * q^43 - 273 * q^47 - 212 * q^49 - 209 * q^53 + 237 * q^55 - 799 * q^59 + 149 * q^61 + 52 * q^65 + 201 * q^67 - 792 * q^71 - 246 * q^73 - 843 * q^77 - 254 * q^79 - 374 * q^83 - 25 * q^85 + 564 * q^89 - 621 * q^91 - 95 * q^95 - 178 * q^97

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