LMFDB - Embedded newform 5488.2.a.e.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5488,2,Mod(1,5488)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5488.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5488, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5488 = 2^{4} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5488.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,1,0,0,0,0,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$43.8219006293$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 686)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	5488.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.445042 q^{3} +3.04892 q^{5} -2.80194 q^{9} +4.40581 q^{11} -5.71379 q^{13} +1.35690 q^{15} -7.54288 q^{17} +1.39612 q^{19} -7.18598 q^{23} +4.29590 q^{25} -2.58211 q^{27} -0.692021 q^{29} +7.13706 q^{31} +1.96077 q^{33} +1.66487 q^{37} -2.54288 q^{39} -0.506041 q^{41} -0.850855 q^{43} -8.54288 q^{45} +0.158834 q^{47} -3.35690 q^{51} -6.41119 q^{53} +13.4330 q^{55} +0.621334 q^{57} -10.9269 q^{59} -5.70410 q^{61} -17.4209 q^{65} -10.9390 q^{67} -3.19806 q^{69} -6.18060 q^{71} -1.54527 q^{73} +1.91185 q^{75} +7.84117 q^{79} +7.25667 q^{81} +3.21313 q^{83} -22.9976 q^{85} -0.307979 q^{87} -13.9487 q^{89} +3.17629 q^{93} +4.25667 q^{95} -1.55496 q^{97} -12.3448 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} - 4 q^{9} - 9 q^{13} - 4 q^{17} + 13 q^{19} - 7 q^{23} - q^{25} - 2 q^{27} + 3 q^{29} + 16 q^{31} - 7 q^{33} + 6 q^{37} + 11 q^{39} - 11 q^{41} + 11 q^{43} - 7 q^{45} - 8 q^{47} - 6 q^{51}+ \cdots - 14 q^{99}+O(q^{100}) $ 3 * q + q^3 - 4 * q^9 - 9 * q^13 - 4 * q^17 + 13 * q^19 - 7 * q^23 - q^25 - 2 * q^27 + 3 * q^29 + 16 * q^31 - 7 * q^33 + 6 * q^37 + 11 * q^39 - 11 * q^41 + 11 * q^43 - 7 * q^45 - 8 * q^47 - 6 * q^51 - 3 * q^53 + 21 * q^55 + 9 * q^57 - 4 * q^59 - 31 * q^61 - 14 * q^65 - 23 * q^67 - 14 * q^69 - 7 * q^71 - 27 * q^73 + 2 * q^75 + 32 * q^79 - 5 * q^81 - 11 * q^83 - 28 * q^85 - 6 * q^87 - 10 * q^89 + 17 * q^93 - 14 * q^95 - 5 * q^97 - 14 * 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
$3$	0.445042	0.256945	0.128473	−	0.991713i	$-0.458993\pi$
$3$	0.445042	0.256945	0.128473	+	0.991713i	$0.458993\pi$
$4$	0	0
$5$	3.04892	1.36352	0.681759	−	0.731577i	$-0.261215\pi$
$5$	3.04892	1.36352	0.681759	+	0.731577i	$0.261215\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.80194	−0.933979
$10$	0	0
$11$	4.40581	1.32840	0.664201	−	0.747554i	$-0.268772\pi$
$11$	4.40581	1.32840	0.664201	+	0.747554i	$0.268772\pi$
$12$	0	0
$13$	−5.71379	−1.58472	−0.792360	−	0.610053i	$-0.791148\pi$
$13$	−5.71379	−1.58472	−0.792360	+	0.610053i	$0.791148\pi$
$14$	0	0
$15$	1.35690	0.350349
$16$	0	0
$17$	−7.54288	−1.82942	−0.914708	−	0.404115i	$-0.867580\pi$
$17$	−7.54288	−1.82942	−0.914708	+	0.404115i	$0.867580\pi$
$18$	0	0
$19$	1.39612	0.320293	0.160146	−	0.987093i	$-0.448803\pi$
$19$	1.39612	0.320293	0.160146	+	0.987093i	$0.448803\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.18598	−1.49838	−0.749190	−	0.662355i	$-0.769557\pi$
$23$	−7.18598	−1.49838	−0.749190	+	0.662355i	$0.769557\pi$
$24$	0	0
$25$	4.29590	0.859179
$26$	0	0
$27$	−2.58211	−0.496926
$28$	0	0
$29$	−0.692021	−0.128505	−0.0642526	−	0.997934i	$-0.520466\pi$
$29$	−0.692021	−0.128505	−0.0642526	+	0.997934i	$0.520466\pi$
$30$	0	0
$31$	7.13706	1.28185	0.640927	−	0.767602i	$-0.278550\pi$
$31$	7.13706	1.28185	0.640927	+	0.767602i	$0.278550\pi$
$32$	0	0
$33$	1.96077	0.341326
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.66487	0.273704	0.136852	−	0.990592i	$-0.456302\pi$
$37$	1.66487	0.273704	0.136852	+	0.990592i	$0.456302\pi$
$38$	0	0
$39$	−2.54288	−0.407186
$40$	0	0
$41$	−0.506041	−0.0790303	−0.0395151	−	0.999219i	$-0.512581\pi$
$41$	−0.506041	−0.0790303	−0.0395151	+	0.999219i	$0.512581\pi$
$42$	0	0
$43$	−0.850855	−0.129754	−0.0648771	−	0.997893i	$-0.520666\pi$
$43$	−0.850855	−0.129754	−0.0648771	+	0.997893i	$0.520666\pi$
$44$	0	0
$45$	−8.54288	−1.27350
$46$	0	0
$47$	0.158834	0.0231683	0.0115841	−	0.999933i	$-0.496313\pi$
$47$	0.158834	0.0231683	0.0115841	+	0.999933i	$0.496313\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−3.35690	−0.470059
$52$	0	0
$53$	−6.41119	−0.880645	−0.440322	−	0.897840i	$-0.645136\pi$
$53$	−6.41119	−0.880645	−0.440322	+	0.897840i	$0.645136\pi$
$54$	0	0
$55$	13.4330	1.81130
$56$	0	0
$57$	0.621334	0.0822977
$58$	0	0
$59$	−10.9269	−1.42256	−0.711282	−	0.702907i	$-0.751885\pi$
$59$	−10.9269	−1.42256	−0.711282	+	0.702907i	$0.751885\pi$
$60$	0	0
$61$	−5.70410	−0.730336	−0.365168	−	0.930942i	$-0.618988\pi$
$61$	−5.70410	−0.730336	−0.365168	+	0.930942i	$0.618988\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−17.4209	−2.16079
$66$	0	0
$67$	−10.9390	−1.33641	−0.668206	−	0.743976i	$-0.732937\pi$
$67$	−10.9390	−1.33641	−0.668206	+	0.743976i	$0.732937\pi$
$68$	0	0
$69$	−3.19806	−0.385001
$70$	0	0
$71$	−6.18060	−0.733503	−0.366751	−	0.930319i	$-0.619530\pi$
$71$	−6.18060	−0.733503	−0.366751	+	0.930319i	$0.619530\pi$
$72$	0	0
$73$	−1.54527	−0.180860	−0.0904301	−	0.995903i	$-0.528824\pi$
$73$	−1.54527	−0.180860	−0.0904301	+	0.995903i	$0.528824\pi$
$74$	0	0
$75$	1.91185	0.220762
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.84117	0.882200	0.441100	−	0.897458i	$-0.354588\pi$
$79$	7.84117	0.882200	0.441100	+	0.897458i	$0.354588\pi$
$80$	0	0
$81$	7.25667	0.806296
$82$	0	0
$83$	3.21313	0.352687	0.176343	−	0.984329i	$-0.443573\pi$
$83$	3.21313	0.352687	0.176343	+	0.984329i	$0.443573\pi$
$84$	0	0
$85$	−22.9976	−2.49444
$86$	0	0
$87$	−0.307979	−0.0330188
$88$	0	0
$89$	−13.9487	−1.47856	−0.739279	−	0.673399i	$-0.764834\pi$
$89$	−13.9487	−1.47856	−0.739279	+	0.673399i	$0.764834\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.17629	0.329366
$94$	0	0
$95$	4.25667	0.436725
$96$	0	0
$97$	−1.55496	−0.157882	−0.0789410	−	0.996879i	$-0.525154\pi$
$97$	−1.55496	−0.157882	−0.0789410	+	0.996879i	$0.525154\pi$
$98$	0	0
$99$	−12.3448	−1.24070
$100$	0	0
$101$	5.00969	0.498483	0.249241	−	0.968441i	$-0.419819\pi$
$101$	5.00969	0.498483	0.249241	+	0.968441i	$0.419819\pi$
$102$	0	0
$103$	−2.89546	−0.285298	−0.142649	−	0.989773i	$-0.545562\pi$
$103$	−2.89546	−0.285298	−0.142649	+	0.989773i	$0.545562\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.99330	−0.966088	−0.483044	−	0.875596i	$-0.660469\pi$
$107$	−9.99330	−0.966088	−0.483044	+	0.875596i	$0.660469\pi$
$108$	0	0
$109$	−14.6015	−1.39857	−0.699284	−	0.714844i	$-0.746498\pi$
$109$	−14.6015	−1.39857	−0.699284	+	0.714844i	$0.746498\pi$
$110$	0	0
$111$	0.740939	0.0703268
$112$	0	0
$113$	16.5405	1.55600	0.777999	−	0.628266i	$-0.216235\pi$
$113$	16.5405	1.55600	0.777999	+	0.628266i	$0.216235\pi$
$114$	0	0
$115$	−21.9095	−2.04307
$116$	0	0
$117$	16.0097	1.48010
$118$	0	0
$119$	0	0
$120$	0	0
$121$	8.41119	0.764654
$122$	0	0
$123$	−0.225209	−0.0203064
$124$	0	0
$125$	−2.14675	−0.192011
$126$	0	0
$127$	6.32304	0.561079	0.280540	−	0.959842i	$-0.409487\pi$
$127$	6.32304	0.561079	0.280540	+	0.959842i	$0.409487\pi$
$128$	0	0
$129$	−0.378666	−0.0333397
$130$	0	0
$131$	6.41550	0.560525	0.280263	−	0.959923i	$-0.409578\pi$
$131$	6.41550	0.560525	0.280263	+	0.959923i	$0.409578\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−7.87263	−0.677568
$136$	0	0
$137$	−2.21014	−0.188825	−0.0944127	−	0.995533i	$-0.530097\pi$
$137$	−2.21014	−0.188825	−0.0944127	+	0.995533i	$0.530097\pi$
$138$	0	0
$139$	14.5579	1.23479	0.617394	−	0.786654i	$-0.288188\pi$
$139$	14.5579	1.23479	0.617394	+	0.786654i	$0.288188\pi$
$140$	0	0
$141$	0.0706876	0.00595297
$142$	0	0
$143$	−25.1739	−2.10515
$144$	0	0
$145$	−2.10992	−0.175219
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.0532	0.823593	0.411796	−	0.911276i	$-0.364902\pi$
$149$	10.0532	0.823593	0.411796	+	0.911276i	$0.364902\pi$
$150$	0	0
$151$	4.67994	0.380848	0.190424	−	0.981702i	$-0.439014\pi$
$151$	4.67994	0.380848	0.190424	+	0.981702i	$0.439014\pi$
$152$	0	0
$153$	21.1347	1.70864
$154$	0	0
$155$	21.7603	1.74783
$156$	0	0
$157$	−13.0030	−1.03775	−0.518876	−	0.854850i	$-0.673649\pi$
$157$	−13.0030	−1.03775	−0.518876	+	0.854850i	$0.673649\pi$
$158$	0	0
$159$	−2.85325	−0.226277
$160$	0	0
$161$	0	0
$162$	0	0
$163$	19.1226	1.49780	0.748898	−	0.662685i	$-0.230583\pi$
$163$	19.1226	1.49780	0.748898	+	0.662685i	$0.230583\pi$
$164$	0	0
$165$	5.97823	0.465405
$166$	0	0
$167$	−22.3177	−1.72699	−0.863496	−	0.504355i	$-0.831730\pi$
$167$	−22.3177	−1.72699	−0.863496	+	0.504355i	$0.831730\pi$
$168$	0	0
$169$	19.6474	1.51134
$170$	0	0
$171$	−3.91185	−0.299147
$172$	0	0
$173$	1.95108	0.148338	0.0741690	−	0.997246i	$-0.476370\pi$
$173$	1.95108	0.148338	0.0741690	+	0.997246i	$0.476370\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.86294	−0.365521
$178$	0	0
$179$	−1.78448	−0.133378	−0.0666891	−	0.997774i	$-0.521244\pi$
$179$	−1.78448	−0.133378	−0.0666891	+	0.997774i	$0.521244\pi$
$180$	0	0
$181$	−10.3394	−0.768524	−0.384262	−	0.923224i	$-0.625544\pi$
$181$	−10.3394	−0.768524	−0.384262	+	0.923224i	$0.625544\pi$
$182$	0	0
$183$	−2.53856	−0.187656
$184$	0	0
$185$	5.07606	0.373200
$186$	0	0
$187$	−33.2325	−2.43020
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.1468	1.45777	0.728884	−	0.684637i	$-0.240039\pi$
$191$	20.1468	1.45777	0.728884	+	0.684637i	$0.240039\pi$
$192$	0	0
$193$	−13.9172	−1.00178	−0.500892	−	0.865510i	$-0.666995\pi$
$193$	−13.9172	−1.00178	−0.500892	+	0.865510i	$0.666995\pi$
$194$	0	0
$195$	−7.75302	−0.555205
$196$	0	0
$197$	4.08815	0.291268	0.145634	−	0.989339i	$-0.453478\pi$
$197$	4.08815	0.291268	0.145634	+	0.989339i	$0.453478\pi$
$198$	0	0
$199$	−8.45473	−0.599340	−0.299670	−	0.954043i	$-0.596877\pi$
$199$	−8.45473	−0.599340	−0.299670	+	0.954043i	$0.596877\pi$
$200$	0	0
$201$	−4.86831	−0.343384
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.54288	−0.107759
$206$	0	0
$207$	20.1347	1.39946
$208$	0	0
$209$	6.15106	0.425478
$210$	0	0
$211$	−4.04221	−0.278277	−0.139139	−	0.990273i	$-0.544433\pi$
$211$	−4.04221	−0.278277	−0.139139	+	0.990273i	$0.544433\pi$
$212$	0	0
$213$	−2.75063	−0.188470
$214$	0	0
$215$	−2.59419	−0.176922
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−0.687710	−0.0464711
$220$	0	0
$221$	43.0984	2.89911
$222$	0	0
$223$	2.87800	0.192725	0.0963626	−	0.995346i	$-0.469279\pi$
$223$	2.87800	0.192725	0.0963626	+	0.995346i	$0.469279\pi$
$224$	0	0
$225$	−12.0368	−0.802456
$226$	0	0
$227$	−25.1250	−1.66760	−0.833802	−	0.552064i	$-0.813840\pi$
$227$	−25.1250	−1.66760	−0.833802	+	0.552064i	$0.813840\pi$
$228$	0	0
$229$	5.87800	0.388429	0.194215	−	0.980959i	$-0.437784\pi$
$229$	5.87800	0.388429	0.194215	+	0.980959i	$0.437784\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.7506	1.16288	0.581441	−	0.813588i	$-0.302489\pi$
$233$	17.7506	1.16288	0.581441	+	0.813588i	$0.302489\pi$
$234$	0	0
$235$	0.484271	0.0315903
$236$	0	0
$237$	3.48965	0.226677
$238$	0	0
$239$	10.7681	0.696530	0.348265	−	0.937396i	$-0.386771\pi$
$239$	10.7681	0.696530	0.348265	+	0.937396i	$0.386771\pi$
$240$	0	0
$241$	−18.7899	−1.21036	−0.605181	−	0.796088i	$-0.706899\pi$
$241$	−18.7899	−1.21036	−0.605181	+	0.796088i	$0.706899\pi$
$242$	0	0
$243$	10.9758	0.704100
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−7.97716	−0.507575
$248$	0	0
$249$	1.42998	0.0906211
$250$	0	0
$251$	−2.76271	−0.174381	−0.0871903	−	0.996192i	$-0.527789\pi$
$251$	−2.76271	−0.174381	−0.0871903	+	0.996192i	$0.527789\pi$
$252$	0	0
$253$	−31.6601	−1.99045
$254$	0	0
$255$	−10.2349	−0.640934
$256$	0	0
$257$	−2.89546	−0.180614	−0.0903069	−	0.995914i	$-0.528785\pi$
$257$	−2.89546	−0.180614	−0.0903069	+	0.995914i	$0.528785\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.93900	0.120021
$262$	0	0
$263$	−20.0519	−1.23645	−0.618227	−	0.786000i	$-0.712149\pi$
$263$	−20.0519	−1.23645	−0.618227	+	0.786000i	$0.712149\pi$
$264$	0	0
$265$	−19.5472	−1.20077
$266$	0	0
$267$	−6.20775	−0.379908
$268$	0	0
$269$	−17.5961	−1.07285	−0.536427	−	0.843947i	$-0.680226\pi$
$269$	−17.5961	−1.07285	−0.536427	+	0.843947i	$0.680226\pi$
$270$	0	0
$271$	14.3787	0.873442	0.436721	−	0.899597i	$-0.356140\pi$
$271$	14.3787	0.873442	0.436721	+	0.899597i	$0.356140\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.9269	1.14134
$276$	0	0
$277$	3.95407	0.237577	0.118788	−	0.992920i	$-0.462099\pi$
$277$	3.95407	0.237577	0.118788	+	0.992920i	$0.462099\pi$
$278$	0	0
$279$	−19.9976	−1.19723
$280$	0	0
$281$	9.94438	0.593232	0.296616	−	0.954997i	$-0.404142\pi$
$281$	9.94438	0.593232	0.296616	+	0.954997i	$0.404142\pi$
$282$	0	0
$283$	21.0054	1.24864	0.624320	−	0.781169i	$-0.285376\pi$
$283$	21.0054	1.24864	0.624320	+	0.781169i	$0.285376\pi$
$284$	0	0
$285$	1.89440	0.112214
$286$	0	0
$287$	0	0
$288$	0	0
$289$	39.8950	2.34676
$290$	0	0
$291$	−0.692021	−0.0405670
$292$	0	0
$293$	18.7972	1.09814	0.549071	−	0.835776i	$-0.314982\pi$
$293$	18.7972	1.09814	0.549071	+	0.835776i	$0.314982\pi$
$294$	0	0
$295$	−33.3153	−1.93969
$296$	0	0
$297$	−11.3763	−0.660118
$298$	0	0
$299$	41.0592	2.37451
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2.22952	0.128083
$304$	0	0
$305$	−17.3913	−0.995825
$306$	0	0
$307$	−13.4601	−0.768209	−0.384104	−	0.923290i	$-0.625490\pi$
$307$	−13.4601	−0.768209	−0.384104	+	0.923290i	$0.625490\pi$
$308$	0	0
$309$	−1.28860	−0.0733060
$310$	0	0
$311$	20.7966	1.17926	0.589632	−	0.807672i	$-0.299273\pi$
$311$	20.7966	1.17926	0.589632	+	0.807672i	$0.299273\pi$
$312$	0	0
$313$	−25.8931	−1.46356	−0.731781	−	0.681539i	$-0.761311\pi$
$313$	−25.8931	−1.46356	−0.731781	+	0.681539i	$0.761311\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.51573	−0.422125	−0.211063	−	0.977473i	$-0.567692\pi$
$317$	−7.51573	−0.422125	−0.211063	+	0.977473i	$0.567692\pi$
$318$	0	0
$319$	−3.04892	−0.170707
$320$	0	0
$321$	−4.44743	−0.248232
$322$	0	0
$323$	−10.5308	−0.585949
$324$	0	0
$325$	−24.5459	−1.36156
$326$	0	0
$327$	−6.49827	−0.359355
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−16.9148	−0.929724	−0.464862	−	0.885383i	$-0.653896\pi$
$331$	−16.9148	−0.929724	−0.464862	+	0.885383i	$0.653896\pi$
$332$	0	0
$333$	−4.66487	−0.255634
$334$	0	0
$335$	−33.3521	−1.82222
$336$	0	0
$337$	20.6722	1.12608	0.563042	−	0.826428i	$-0.309631\pi$
$337$	20.6722	1.12608	0.563042	+	0.826428i	$0.309631\pi$
$338$	0	0
$339$	7.36121	0.399806
$340$	0	0
$341$	31.4446	1.70282
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−9.75063	−0.524956
$346$	0	0
$347$	14.3599	0.770879	0.385439	−	0.922733i	$-0.374050\pi$
$347$	14.3599	0.770879	0.385439	+	0.922733i	$0.374050\pi$
$348$	0	0
$349$	4.80731	0.257330	0.128665	−	0.991688i	$-0.458931\pi$
$349$	4.80731	0.257330	0.128665	+	0.991688i	$0.458931\pi$
$350$	0	0
$351$	14.7536	0.787490
$352$	0	0
$353$	−29.1239	−1.55011	−0.775055	−	0.631894i	$-0.782278\pi$
$353$	−29.1239	−1.55011	−0.775055	+	0.631894i	$0.782278\pi$
$354$	0	0
$355$	−18.8442	−1.00014
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9.98361	0.526915	0.263457	−	0.964671i	$-0.415137\pi$
$359$	9.98361	0.526915	0.263457	+	0.964671i	$0.415137\pi$
$360$	0	0
$361$	−17.0508	−0.897412
$362$	0	0
$363$	3.74333	0.196474
$364$	0	0
$365$	−4.71140	−0.246606
$366$	0	0
$367$	−23.6160	−1.23274	−0.616371	−	0.787456i	$-0.711398\pi$
$367$	−23.6160	−1.23274	−0.616371	+	0.787456i	$0.711398\pi$
$368$	0	0
$369$	1.41789	0.0738127
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−26.3980	−1.36684	−0.683419	−	0.730026i	$-0.739508\pi$
$373$	−26.3980	−1.36684	−0.683419	+	0.730026i	$0.739508\pi$
$374$	0	0
$375$	−0.955395	−0.0493364
$376$	0	0
$377$	3.95407	0.203645
$378$	0	0
$379$	26.3913	1.35563	0.677816	−	0.735232i	$-0.262927\pi$
$379$	26.3913	1.35563	0.677816	+	0.735232i	$0.262927\pi$
$380$	0	0
$381$	2.81402	0.144167
$382$	0	0
$383$	28.9081	1.47714	0.738568	−	0.674179i	$-0.235502\pi$
$383$	28.9081	1.47714	0.738568	+	0.674179i	$0.235502\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.38404	0.121188
$388$	0	0
$389$	3.46250	0.175556	0.0877779	−	0.996140i	$-0.472023\pi$
$389$	3.46250	0.175556	0.0877779	+	0.996140i	$0.472023\pi$
$390$	0	0
$391$	54.2030	2.74116
$392$	0	0
$393$	2.85517	0.144024
$394$	0	0
$395$	23.9071	1.20290
$396$	0	0
$397$	38.2368	1.91905	0.959525	−	0.281622i	$-0.0908724\pi$
$397$	38.2368	1.91905	0.959525	+	0.281622i	$0.0908724\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9.49934	0.474374	0.237187	−	0.971464i	$-0.423775\pi$
$401$	9.49934	0.474374	0.237187	+	0.971464i	$0.423775\pi$
$402$	0	0
$403$	−40.7797	−2.03138
$404$	0	0
$405$	22.1250	1.09940
$406$	0	0
$407$	7.33513	0.363589
$408$	0	0
$409$	−9.25906	−0.457831	−0.228916	−	0.973446i	$-0.573518\pi$
$409$	−9.25906	−0.457831	−0.228916	+	0.973446i	$0.573518\pi$
$410$	0	0
$411$	−0.983607	−0.0485177
$412$	0	0
$413$	0	0
$414$	0	0
$415$	9.79656	0.480894
$416$	0	0
$417$	6.47889	0.317273
$418$	0	0
$419$	−26.8310	−1.31078	−0.655390	−	0.755291i	$-0.727496\pi$
$419$	−26.8310	−1.31078	−0.655390	+	0.755291i	$0.727496\pi$
$420$	0	0
$421$	14.7560	0.719164	0.359582	−	0.933114i	$-0.382919\pi$
$421$	14.7560	0.719164	0.359582	+	0.933114i	$0.382919\pi$
$422$	0	0
$423$	−0.445042	−0.0216387
$424$	0	0
$425$	−32.4034	−1.57180
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−11.2034	−0.540907
$430$	0	0
$431$	31.8582	1.53455	0.767277	−	0.641316i	$-0.221611\pi$
$431$	31.8582	1.53455	0.767277	+	0.641316i	$0.221611\pi$
$432$	0	0
$433$	16.0194	0.769842	0.384921	−	0.922949i	$-0.374229\pi$
$433$	16.0194	0.769842	0.384921	+	0.922949i	$0.374229\pi$
$434$	0	0
$435$	−0.939001	−0.0450217
$436$	0	0
$437$	−10.0325	−0.479921
$438$	0	0
$439$	2.35690	0.112489	0.0562443	−	0.998417i	$-0.482087\pi$
$439$	2.35690	0.112489	0.0562443	+	0.998417i	$0.482087\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13.1830	−0.626343	−0.313172	−	0.949697i	$-0.601391\pi$
$443$	−13.1830	−0.626343	−0.313172	+	0.949697i	$0.601391\pi$
$444$	0	0
$445$	−42.5284	−2.01604
$446$	0	0
$447$	4.47411	0.211618
$448$	0	0
$449$	−15.3448	−0.724167	−0.362083	−	0.932146i	$-0.617934\pi$
$449$	−15.3448	−0.724167	−0.362083	+	0.932146i	$0.617934\pi$
$450$	0	0
$451$	−2.22952	−0.104984
$452$	0	0
$453$	2.08277	0.0978570
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.8159	−1.06728	−0.533642	−	0.845710i	$-0.679177\pi$
$457$	−22.8159	−1.06728	−0.533642	+	0.845710i	$0.679177\pi$
$458$	0	0
$459$	19.4765	0.909085
$460$	0	0
$461$	0.440730	0.0205268	0.0102634	−	0.999947i	$-0.496733\pi$
$461$	0.440730	0.0205268	0.0102634	+	0.999947i	$0.496733\pi$
$462$	0	0
$463$	25.5133	1.18571	0.592853	−	0.805311i	$-0.298002\pi$
$463$	25.5133	1.18571	0.592853	+	0.805311i	$0.298002\pi$
$464$	0	0
$465$	9.68425	0.449096
$466$	0	0
$467$	−21.4101	−0.990742	−0.495371	−	0.868681i	$-0.664968\pi$
$467$	−21.4101	−0.990742	−0.495371	+	0.868681i	$0.664968\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−5.78687	−0.266645
$472$	0	0
$473$	−3.74871	−0.172366
$474$	0	0
$475$	5.99761	0.275189
$476$	0	0
$477$	17.9638	0.822504
$478$	0	0
$479$	−32.6437	−1.49153	−0.745764	−	0.666210i	$-0.767915\pi$
$479$	−32.6437	−1.49153	−0.745764	+	0.666210i	$0.767915\pi$
$480$	0	0
$481$	−9.51275	−0.433744
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.74094	−0.215275
$486$	0	0
$487$	−41.4010	−1.87606	−0.938030	−	0.346555i	$-0.887352\pi$
$487$	−41.4010	−1.87606	−0.938030	+	0.346555i	$0.887352\pi$
$488$	0	0
$489$	8.51035	0.384851
$490$	0	0
$491$	−1.97046	−0.0889256	−0.0444628	−	0.999011i	$-0.514158\pi$
$491$	−1.97046	−0.0889256	−0.0444628	+	0.999011i	$0.514158\pi$
$492$	0	0
$493$	5.21983	0.235089
$494$	0	0
$495$	−37.6383	−1.69172
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−4.32006	−0.193392	−0.0966962	−	0.995314i	$-0.530828\pi$
$499$	−4.32006	−0.193392	−0.0966962	+	0.995314i	$0.530828\pi$
$500$	0	0
$501$	−9.93230	−0.443742
$502$	0	0
$503$	29.8649	1.33161	0.665804	−	0.746127i	$-0.268089\pi$
$503$	29.8649	1.33161	0.665804	+	0.746127i	$0.268089\pi$
$504$	0	0
$505$	15.2741	0.679690
$506$	0	0
$507$	8.74392	0.388331
$508$	0	0
$509$	−37.9017	−1.67996	−0.839981	−	0.542615i	$-0.817434\pi$
$509$	−37.9017	−1.67996	−0.839981	+	0.542615i	$0.817434\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−3.60494	−0.159162
$514$	0	0
$515$	−8.82802	−0.389009
$516$	0	0
$517$	0.699791	0.0307768
$518$	0	0
$519$	0.868313	0.0381147
$520$	0	0
$521$	−4.97823	−0.218100	−0.109050	−	0.994036i	$-0.534781\pi$
$521$	−4.97823	−0.218100	−0.109050	+	0.994036i	$0.534781\pi$
$522$	0	0
$523$	−16.8810	−0.738154	−0.369077	−	0.929399i	$-0.620326\pi$
$523$	−16.8810	−0.738154	−0.369077	+	0.929399i	$0.620326\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−53.8340	−2.34505
$528$	0	0
$529$	28.6383	1.24514
$530$	0	0
$531$	30.6165	1.32865
$532$	0	0
$533$	2.89141	0.125241
$534$	0	0
$535$	−30.4687	−1.31728
$536$	0	0
$537$	−0.794168	−0.0342709
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−11.3720	−0.488919	−0.244459	−	0.969660i	$-0.578610\pi$
$541$	−11.3720	−0.488919	−0.244459	+	0.969660i	$0.578610\pi$
$542$	0	0
$543$	−4.60148	−0.197469
$544$	0	0
$545$	−44.5187	−1.90697
$546$	0	0
$547$	−20.8528	−0.891600	−0.445800	−	0.895133i	$-0.647081\pi$
$547$	−20.8528	−0.891600	−0.445800	+	0.895133i	$0.647081\pi$
$548$	0	0
$549$	15.9825	0.682118
$550$	0	0
$551$	−0.966148	−0.0411593
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.25906	0.0958918
$556$	0	0
$557$	22.1672	0.939254	0.469627	−	0.882865i	$-0.344388\pi$
$557$	22.1672	0.939254	0.469627	+	0.882865i	$0.344388\pi$
$558$	0	0
$559$	4.86161	0.205624
$560$	0	0
$561$	−14.7899	−0.624428
$562$	0	0
$563$	19.0358	0.802262	0.401131	−	0.916021i	$-0.368617\pi$
$563$	19.0358	0.802262	0.401131	+	0.916021i	$0.368617\pi$
$564$	0	0
$565$	50.4306	2.12163
$566$	0	0
$567$	0	0
$568$	0	0
$569$	16.4209	0.688399	0.344200	−	0.938897i	$-0.388150\pi$
$569$	16.4209	0.688399	0.344200	+	0.938897i	$0.388150\pi$
$570$	0	0
$571$	27.0726	1.13295	0.566477	−	0.824078i	$-0.308306\pi$
$571$	27.0726	1.13295	0.566477	+	0.824078i	$0.308306\pi$
$572$	0	0
$573$	8.96615	0.374566
$574$	0	0
$575$	−30.8702	−1.28738
$576$	0	0
$577$	10.1056	0.420702	0.210351	−	0.977626i	$-0.432539\pi$
$577$	10.1056	0.420702	0.210351	+	0.977626i	$0.432539\pi$
$578$	0	0
$579$	−6.19375	−0.257403
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−28.2465	−1.16985
$584$	0	0
$585$	48.8122	2.01814
$586$	0	0
$587$	−10.4964	−0.433231	−0.216615	−	0.976257i	$-0.569502\pi$
$587$	−10.4964	−0.433231	−0.216615	+	0.976257i	$0.569502\pi$
$588$	0	0
$589$	9.96423	0.410569
$590$	0	0
$591$	1.81940	0.0748400
$592$	0	0
$593$	−16.0218	−0.657935	−0.328968	−	0.944341i	$-0.606701\pi$
$593$	−16.0218	−0.657935	−0.328968	+	0.944341i	$0.606701\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3.76271	−0.153997
$598$	0	0
$599$	27.7356	1.13324	0.566622	−	0.823978i	$-0.308250\pi$
$599$	27.7356	1.13324	0.566622	+	0.823978i	$0.308250\pi$
$600$	0	0
$601$	−11.7885	−0.480864	−0.240432	−	0.970666i	$-0.577289\pi$
$601$	−11.7885	−0.480864	−0.240432	+	0.970666i	$0.577289\pi$
$602$	0	0
$603$	30.6504	1.24818
$604$	0	0
$605$	25.6450	1.04262
$606$	0	0
$607$	−26.7875	−1.08727	−0.543635	−	0.839322i	$-0.682952\pi$
$607$	−26.7875	−1.08727	−0.543635	+	0.839322i	$0.682952\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.907542	−0.0367152
$612$	0	0
$613$	−3.90276	−0.157631	−0.0788154	−	0.996889i	$-0.525114\pi$
$613$	−3.90276	−0.157631	−0.0788154	+	0.996889i	$0.525114\pi$
$614$	0	0
$615$	−0.686645	−0.0276882
$616$	0	0
$617$	−45.5883	−1.83532	−0.917659	−	0.397370i	$-0.869923\pi$
$617$	−45.5883	−1.83532	−0.917659	+	0.397370i	$0.869923\pi$
$618$	0	0
$619$	−45.7434	−1.83858	−0.919292	−	0.393576i	$-0.871238\pi$
$619$	−45.7434	−1.83858	−0.919292	+	0.393576i	$0.871238\pi$
$620$	0	0
$621$	18.5550	0.744585
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−28.0248	−1.12099
$626$	0	0
$627$	2.73748	0.109324
$628$	0	0
$629$	−12.5579	−0.500718
$630$	0	0
$631$	19.4832	0.775614	0.387807	−	0.921741i	$-0.373233\pi$
$631$	19.4832	0.775614	0.387807	+	0.921741i	$0.373233\pi$
$632$	0	0
$633$	−1.79895	−0.0715020
$634$	0	0
$635$	19.2784	0.765041
$636$	0	0
$637$	0	0
$638$	0	0
$639$	17.3177	0.685076
$640$	0	0
$641$	3.11231	0.122929	0.0614644	−	0.998109i	$-0.480423\pi$
$641$	3.11231	0.122929	0.0614644	+	0.998109i	$0.480423\pi$
$642$	0	0
$643$	−19.4644	−0.767602	−0.383801	−	0.923416i	$-0.625385\pi$
$643$	−19.4644	−0.767602	−0.383801	+	0.923416i	$0.625385\pi$
$644$	0	0
$645$	−1.15452	−0.0454592
$646$	0	0
$647$	13.0084	0.511411	0.255706	−	0.966755i	$-0.417692\pi$
$647$	13.0084	0.511411	0.255706	+	0.966755i	$0.417692\pi$
$648$	0	0
$649$	−48.1420	−1.88974
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27.9571	1.09404	0.547022	−	0.837118i	$-0.315761\pi$
$653$	27.9571	1.09404	0.547022	+	0.837118i	$0.315761\pi$
$654$	0	0
$655$	19.5603	0.764286
$656$	0	0
$657$	4.32975	0.168920
$658$	0	0
$659$	12.3655	0.481692	0.240846	−	0.970563i	$-0.422575\pi$
$659$	12.3655	0.481692	0.240846	+	0.970563i	$0.422575\pi$
$660$	0	0
$661$	29.6233	1.15221	0.576105	−	0.817375i	$-0.304572\pi$
$661$	29.6233	1.15221	0.576105	+	0.817375i	$0.304572\pi$
$662$	0	0
$663$	19.1806	0.744913
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.97285	0.192550
$668$	0	0
$669$	1.28083	0.0495198
$670$	0	0
$671$	−25.1312	−0.970180
$672$	0	0
$673$	29.6528	1.14303	0.571516	−	0.820591i	$-0.306356\pi$
$673$	29.6528	1.14303	0.571516	+	0.820591i	$0.306356\pi$
$674$	0	0
$675$	−11.0925	−0.426949
$676$	0	0
$677$	19.0868	0.733566	0.366783	−	0.930307i	$-0.380459\pi$
$677$	19.0868	0.733566	0.366783	+	0.930307i	$0.380459\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−11.1817	−0.428482
$682$	0	0
$683$	−8.31336	−0.318102	−0.159051	−	0.987270i	$-0.550843\pi$
$683$	−8.31336	−0.318102	−0.159051	+	0.987270i	$0.550843\pi$
$684$	0	0
$685$	−6.73855	−0.257467
$686$	0	0
$687$	2.61596	0.0998050
$688$	0	0
$689$	36.6322	1.39558
$690$	0	0
$691$	42.4868	1.61627	0.808137	−	0.588995i	$-0.200476\pi$
$691$	42.4868	1.61627	0.808137	+	0.588995i	$0.200476\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	44.3860	1.68366
$696$	0	0
$697$	3.81700	0.144579
$698$	0	0
$699$	7.89977	0.298797
$700$	0	0
$701$	14.7356	0.556554	0.278277	−	0.960501i	$-0.410237\pi$
$701$	14.7356	0.556554	0.278277	+	0.960501i	$0.410237\pi$
$702$	0	0
$703$	2.32437	0.0876653
$704$	0	0
$705$	0.215521	0.00811698
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−44.8732	−1.68525	−0.842625	−	0.538502i	$-0.818991\pi$
$709$	−44.8732	−1.68525	−0.842625	+	0.538502i	$0.818991\pi$
$710$	0	0
$711$	−21.9705	−0.823957
$712$	0	0
$713$	−51.2868	−1.92071
$714$	0	0
$715$	−76.7531	−2.87040
$716$	0	0
$717$	4.79225	0.178970
$718$	0	0
$719$	0.0639828	0.00238616	0.00119308	−	0.999999i	$-0.499620\pi$
$719$	0.0639828	0.00238616	0.00119308	+	0.999999i	$0.499620\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−8.36227	−0.310996
$724$	0	0
$725$	−2.97285	−0.110409
$726$	0	0
$727$	9.80625	0.363694	0.181847	−	0.983327i	$-0.441792\pi$
$727$	9.80625	0.363694	0.181847	+	0.983327i	$0.441792\pi$
$728$	0	0
$729$	−16.8853	−0.625381
$730$	0	0
$731$	6.41789	0.237374
$732$	0	0
$733$	2.92692	0.108108	0.0540541	−	0.998538i	$-0.482786\pi$
$733$	2.92692	0.108108	0.0540541	+	0.998538i	$0.482786\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−48.1952	−1.77529
$738$	0	0
$739$	−46.3116	−1.70360	−0.851799	−	0.523869i	$-0.824488\pi$
$739$	−46.3116	−1.70360	−0.851799	+	0.523869i	$0.824488\pi$
$740$	0	0
$741$	−3.55017	−0.130419
$742$	0	0
$743$	−20.6692	−0.758279	−0.379139	−	0.925340i	$-0.623780\pi$
$743$	−20.6692	−0.758279	−0.379139	+	0.925340i	$0.623780\pi$
$744$	0	0
$745$	30.6515	1.12298
$746$	0	0
$747$	−9.00298	−0.329402
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−26.2185	−0.956727	−0.478363	−	0.878162i	$-0.658770\pi$
$751$	−26.2185	−0.956727	−0.478363	+	0.878162i	$0.658770\pi$
$752$	0	0
$753$	−1.22952	−0.0448062
$754$	0	0
$755$	14.2687	0.519293
$756$	0	0
$757$	−10.5603	−0.383822	−0.191911	−	0.981412i	$-0.561468\pi$
$757$	−10.5603	−0.383822	−0.191911	+	0.981412i	$0.561468\pi$
$758$	0	0
$759$	−14.0901	−0.511437
$760$	0	0
$761$	−40.4819	−1.46747	−0.733733	−	0.679437i	$-0.762224\pi$
$761$	−40.4819	−1.46747	−0.733733	+	0.679437i	$0.762224\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	64.4379	2.32976
$766$	0	0
$767$	62.4341	2.25437
$768$	0	0
$769$	−23.8670	−0.860666	−0.430333	−	0.902670i	$-0.641604\pi$
$769$	−23.8670	−0.860666	−0.430333	+	0.902670i	$0.641604\pi$
$770$	0	0
$771$	−1.28860	−0.0464078
$772$	0	0
$773$	−27.8297	−1.00096	−0.500482	−	0.865747i	$-0.666844\pi$
$773$	−27.8297	−1.00096	−0.500482	+	0.865747i	$0.666844\pi$
$774$	0	0
$775$	30.6601	1.10134
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.706496	−0.0253128
$780$	0	0
$781$	−27.2306	−0.974387
$782$	0	0
$783$	1.78687	0.0638576
$784$	0	0
$785$	−39.6450	−1.41499
$786$	0	0
$787$	−3.91617	−0.139596	−0.0697981	−	0.997561i	$-0.522236\pi$
$787$	−3.91617	−0.139596	−0.0697981	+	0.997561i	$0.522236\pi$
$788$	0	0
$789$	−8.92394	−0.317701
$790$	0	0
$791$	0	0
$792$	0	0
$793$	32.5921	1.15738
$794$	0	0
$795$	−8.69932	−0.308533
$796$	0	0
$797$	−10.3110	−0.365233	−0.182617	−	0.983184i	$-0.558457\pi$
$797$	−10.3110	−0.365233	−0.182617	+	0.983184i	$0.558457\pi$
$798$	0	0
$799$	−1.19806	−0.0423844
$800$	0	0
$801$	39.0834	1.38094
$802$	0	0
$803$	−6.80817	−0.240255
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−7.83100	−0.275664
$808$	0	0
$809$	−39.5200	−1.38945	−0.694725	−	0.719275i	$-0.744474\pi$
$809$	−39.5200	−1.38945	−0.694725	+	0.719275i	$0.744474\pi$
$810$	0	0
$811$	27.8769	0.978892	0.489446	−	0.872034i	$-0.337199\pi$
$811$	27.8769	0.978892	0.489446	+	0.872034i	$0.337199\pi$
$812$	0	0
$813$	6.39911	0.224427
$814$	0	0
$815$	58.3032	2.04227
$816$	0	0
$817$	−1.18790	−0.0415593
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.76032	0.235937	0.117968	−	0.993017i	$-0.462362\pi$
$821$	6.76032	0.235937	0.117968	+	0.993017i	$0.462362\pi$
$822$	0	0
$823$	4.70901	0.164146	0.0820728	−	0.996626i	$-0.473846\pi$
$823$	4.70901	0.164146	0.0820728	+	0.996626i	$0.473846\pi$
$824$	0	0
$825$	8.42327	0.293261
$826$	0	0
$827$	23.4010	0.813733	0.406867	−	0.913488i	$-0.366621\pi$
$827$	23.4010	0.813733	0.406867	+	0.913488i	$0.366621\pi$
$828$	0	0
$829$	18.6746	0.648594	0.324297	−	0.945955i	$-0.394872\pi$
$829$	18.6746	0.648594	0.324297	+	0.945955i	$0.394872\pi$
$830$	0	0
$831$	1.75973	0.0610442
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−68.0447	−2.35478
$836$	0	0
$837$	−18.4286	−0.636987
$838$	0	0
$839$	−21.0334	−0.726153	−0.363076	−	0.931759i	$-0.618274\pi$
$839$	−21.0334	−0.726153	−0.363076	+	0.931759i	$0.618274\pi$
$840$	0	0
$841$	−28.5211	−0.983486
$842$	0	0
$843$	4.42566	0.152428
$844$	0	0
$845$	59.9033	2.06074
$846$	0	0
$847$	0	0
$848$	0	0
$849$	9.34827	0.320832
$850$	0	0
$851$	−11.9638	−0.410112
$852$	0	0
$853$	5.53750	0.189600	0.0948002	−	0.995496i	$-0.469779\pi$
$853$	5.53750	0.189600	0.0948002	+	0.995496i	$0.469779\pi$
$854$	0	0
$855$	−11.9269	−0.407892
$856$	0	0
$857$	−29.8955	−1.02121	−0.510605	−	0.859816i	$-0.670578\pi$
$857$	−29.8955	−1.02121	−0.510605	+	0.859816i	$0.670578\pi$
$858$	0	0
$859$	33.5603	1.14506	0.572532	−	0.819882i	$-0.305961\pi$
$859$	33.5603	1.14506	0.572532	+	0.819882i	$0.305961\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31.6980	1.07901	0.539506	−	0.841982i	$-0.318611\pi$
$863$	31.6980	1.07901	0.539506	+	0.841982i	$0.318611\pi$
$864$	0	0
$865$	5.94869	0.202262
$866$	0	0
$867$	17.7549	0.602989
$868$	0	0
$869$	34.5467	1.17192
$870$	0	0
$871$	62.5032	2.11784
$872$	0	0
$873$	4.35690	0.147459
$874$	0	0
$875$	0	0
$876$	0	0
$877$	39.4644	1.33262	0.666309	−	0.745675i	$-0.267873\pi$
$877$	39.4644	1.33262	0.666309	+	0.745675i	$0.267873\pi$
$878$	0	0
$879$	8.36552	0.282162
$880$	0	0
$881$	7.31634	0.246494	0.123247	−	0.992376i	$-0.460669\pi$
$881$	7.31634	0.246494	0.123247	+	0.992376i	$0.460669\pi$
$882$	0	0
$883$	44.8321	1.50872	0.754360	−	0.656461i	$-0.227948\pi$
$883$	44.8321	1.50872	0.754360	+	0.656461i	$0.227948\pi$
$884$	0	0
$885$	−14.8267	−0.498394
$886$	0	0
$887$	−23.0790	−0.774919	−0.387459	−	0.921887i	$-0.626647\pi$
$887$	−23.0790	−0.774919	−0.387459	+	0.921887i	$0.626647\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	31.9715	1.07109
$892$	0	0
$893$	0.221751	0.00742063
$894$	0	0
$895$	−5.44073	−0.181864
$896$	0	0
$897$	18.2731	0.610120
$898$	0	0
$899$	−4.93900	−0.164725
$900$	0	0
$901$	48.3588	1.61107
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−31.5241	−1.04790
$906$	0	0
$907$	44.6031	1.48102	0.740511	−	0.672044i	$-0.234583\pi$
$907$	44.6031	1.48102	0.740511	+	0.672044i	$0.234583\pi$
$908$	0	0
$909$	−14.0368	−0.465572
$910$	0	0
$911$	−7.19460	−0.238368	−0.119184	−	0.992872i	$-0.538028\pi$
$911$	−7.19460	−0.238368	−0.119184	+	0.992872i	$0.538028\pi$
$912$	0	0
$913$	14.1564	0.468510
$914$	0	0
$915$	−7.73987	−0.255872
$916$	0	0
$917$	0	0
$918$	0	0
$919$	45.6469	1.50575	0.752877	−	0.658161i	$-0.228666\pi$
$919$	45.6469	1.50575	0.752877	+	0.658161i	$0.228666\pi$
$920$	0	0
$921$	−5.99031	−0.197387
$922$	0	0
$923$	35.3147	1.16240
$924$	0	0
$925$	7.15213	0.235161
$926$	0	0
$927$	8.11290	0.266463
$928$	0	0
$929$	−54.6631	−1.79344	−0.896719	−	0.442601i	$-0.854056\pi$
$929$	−54.6631	−1.79344	−0.896719	+	0.442601i	$0.854056\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	9.25534	0.303006
$934$	0	0
$935$	−101.323	−3.31362
$936$	0	0
$937$	49.8316	1.62793	0.813964	−	0.580916i	$-0.197305\pi$
$937$	49.8316	1.62793	0.813964	+	0.580916i	$0.197305\pi$
$938$	0	0
$939$	−11.5235	−0.376055
$940$	0	0
$941$	25.0086	0.815258	0.407629	−	0.913148i	$-0.366356\pi$
$941$	25.0086	0.815258	0.407629	+	0.913148i	$0.366356\pi$
$942$	0	0
$943$	3.63640	0.118417
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9.35796	0.304093	0.152046	−	0.988373i	$-0.451414\pi$
$947$	9.35796	0.304093	0.152046	+	0.988373i	$0.451414\pi$
$948$	0	0
$949$	8.82935	0.286613
$950$	0	0
$951$	−3.34481	−0.108463
$952$	0	0
$953$	−32.2411	−1.04439	−0.522196	−	0.852825i	$-0.674887\pi$
$953$	−32.2411	−1.04439	−0.522196	+	0.852825i	$0.674887\pi$
$954$	0	0
$955$	61.4258	1.98769
$956$	0	0
$957$	−1.35690	−0.0438622
$958$	0	0
$959$	0	0
$960$	0	0
$961$	19.9377	0.643151
$962$	0	0
$963$	28.0006	0.902306
$964$	0	0
$965$	−42.4325	−1.36595
$966$	0	0
$967$	51.8859	1.66854	0.834269	−	0.551358i	$-0.185890\pi$
$967$	51.8859	1.66854	0.834269	+	0.551358i	$0.185890\pi$
$968$	0	0
$969$	−4.68664	−0.150557
$970$	0	0
$971$	−31.1976	−1.00118	−0.500589	−	0.865685i	$-0.666883\pi$
$971$	−31.1976	−1.00118	−0.500589	+	0.865685i	$0.666883\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−10.9239	−0.349846
$976$	0	0
$977$	9.30367	0.297651	0.148825	−	0.988863i	$-0.452451\pi$
$977$	9.30367	0.297651	0.148825	+	0.988863i	$0.452451\pi$
$978$	0	0
$979$	−61.4553	−1.96412
$980$	0	0
$981$	40.9124	1.30623
$982$	0	0
$983$	30.5109	0.973148	0.486574	−	0.873639i	$-0.338246\pi$
$983$	30.5109	0.973148	0.486574	+	0.873639i	$0.338246\pi$
$984$	0	0
$985$	12.4644	0.397149
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6.11423	0.194421
$990$	0	0
$991$	−50.5609	−1.60612	−0.803060	−	0.595898i	$-0.796796\pi$
$991$	−50.5609	−1.60612	−0.803060	+	0.595898i	$0.796796\pi$
$992$	0	0
$993$	−7.52781	−0.238888
$994$	0	0
$995$	−25.7778	−0.817210
$996$	0	0
$997$	−34.0907	−1.07966	−0.539831	−	0.841773i	$-0.681512\pi$
$997$	−34.0907	−1.07966	−0.539831	+	0.841773i	$0.681512\pi$
$998$	0	0
$999$	−4.29888	−0.136011

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5488.2.a.e.1.2		3
4.3	odd	2		686.2.a.a.1.2	✓	3
7.6	odd	2		5488.2.a.b.1.2		3
12.11	even	2		6174.2.a.k.1.1		3
28.3	even	6		686.2.c.c.667.2		6
28.11	odd	6		686.2.c.d.667.2		6
28.19	even	6		686.2.c.c.361.2		6
28.23	odd	6		686.2.c.d.361.2		6
28.27	even	2		686.2.a.b.1.2	yes	3
84.83	odd	2		6174.2.a.l.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
686.2.a.a.1.2	✓	3	4.3	odd	2
686.2.a.b.1.2	yes	3	28.27	even	2
686.2.c.c.361.2		6	28.19	even	6
686.2.c.c.667.2		6	28.3	even	6
686.2.c.d.361.2		6	28.23	odd	6
686.2.c.d.667.2		6	28.11	odd	6
5488.2.a.b.1.2		3	7.6	odd	2
5488.2.a.e.1.2		3	1.1	even	1	trivial
6174.2.a.k.1.1		3	12.11	even	2
6174.2.a.l.1.3		3	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5488 = 2^{4} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5488.a (trivial)

\(f(q)\)	\(=\)	\(q+0.445042 q^{3} +3.04892 q^{5} -2.80194 q^{9} +4.40581 q^{11} -5.71379 q^{13} +1.35690 q^{15} -7.54288 q^{17} +1.39612 q^{19} -7.18598 q^{23} +4.29590 q^{25} -2.58211 q^{27} -0.692021 q^{29} +7.13706 q^{31} +1.96077 q^{33} +1.66487 q^{37} -2.54288 q^{39} -0.506041 q^{41} -0.850855 q^{43} -8.54288 q^{45} +0.158834 q^{47} -3.35690 q^{51} -6.41119 q^{53} +13.4330 q^{55} +0.621334 q^{57} -10.9269 q^{59} -5.70410 q^{61} -17.4209 q^{65} -10.9390 q^{67} -3.19806 q^{69} -6.18060 q^{71} -1.54527 q^{73} +1.91185 q^{75} +7.84117 q^{79} +7.25667 q^{81} +3.21313 q^{83} -22.9976 q^{85} -0.307979 q^{87} -13.9487 q^{89} +3.17629 q^{93} +4.25667 q^{95} -1.55496 q^{97} -12.3448 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} - 4 q^{9} - 9 q^{13} - 4 q^{17} + 13 q^{19} - 7 q^{23} - q^{25} - 2 q^{27} + 3 q^{29} + 16 q^{31} - 7 q^{33} + 6 q^{37} + 11 q^{39} - 11 q^{41} + 11 q^{43} - 7 q^{45} - 8 q^{47} - 6 q^{51}+ \cdots - 14 q^{99}+O(q^{100}) \) 3 * q + q^3 - 4 * q^9 - 9 * q^13 - 4 * q^17 + 13 * q^19 - 7 * q^23 - q^25 - 2 * q^27 + 3 * q^29 + 16 * q^31 - 7 * q^33 + 6 * q^37 + 11 * q^39 - 11 * q^41 + 11 * q^43 - 7 * q^45 - 8 * q^47 - 6 * q^51 - 3 * q^53 + 21 * q^55 + 9 * q^57 - 4 * q^59 - 31 * q^61 - 14 * q^65 - 23 * q^67 - 14 * q^69 - 7 * q^71 - 27 * q^73 + 2 * q^75 + 32 * q^79 - 5 * q^81 - 11 * q^83 - 28 * q^85 - 6 * q^87 - 10 * q^89 + 17 * q^93 - 14 * q^95 - 5 * q^97 - 14 * q^99

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