LMFDB - Embedded newform 464.4.a.l.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [464,4,Mod(1,464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(464, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("464.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 464 = 2^{4} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	464.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:	$27.3768862427$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.13458092.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 $ x^5 - x^4 - 14x^3 + 18x^2 + 20*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-3.68360$ of defining polynomial
Character	$\chi$	$=$	464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.90549 q^{3} -6.52855 q^{5} -5.22706 q^{7} -23.3691 q^{9} +O(q^{10})$	$q+1.90549 q^{3} -6.52855 q^{5} -5.22706 q^{7} -23.3691 q^{9} +21.1299 q^{11} +83.4615 q^{13} -12.4401 q^{15} +11.3273 q^{17} +7.68096 q^{19} -9.96011 q^{21} -153.169 q^{23} -82.3780 q^{25} -95.9778 q^{27} -29.0000 q^{29} -270.530 q^{31} +40.2628 q^{33} +34.1251 q^{35} -298.404 q^{37} +159.035 q^{39} -184.710 q^{41} -208.337 q^{43} +152.566 q^{45} +553.098 q^{47} -315.678 q^{49} +21.5840 q^{51} -321.465 q^{53} -137.948 q^{55} +14.6360 q^{57} -104.930 q^{59} +464.230 q^{61} +122.152 q^{63} -544.883 q^{65} -745.813 q^{67} -291.862 q^{69} +509.252 q^{71} -0.374979 q^{73} -156.970 q^{75} -110.447 q^{77} -610.912 q^{79} +448.081 q^{81} -791.431 q^{83} -73.9507 q^{85} -55.2592 q^{87} -342.011 q^{89} -436.258 q^{91} -515.491 q^{93} -50.1455 q^{95} +601.476 q^{97} -493.787 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 8 q^{3} + 10 q^{5} - 40 q^{7} + 33 q^{9}+O(q^{10}) $ 5 * q - 8 * q^3 + 10 * q^5 - 40 * q^7 + 33 * q^9	$ 5 q - 8 q^{3} + 10 q^{5} - 40 q^{7} + 33 q^{9} - 12 q^{11} + 14 q^{13} + 74 q^{15} + 66 q^{17} - 214 q^{19} - 164 q^{23} + 207 q^{25} - 362 q^{27} - 145 q^{29} - 420 q^{31} - 576 q^{33} + 52 q^{35} + 378 q^{37} + 374 q^{39} - 1158 q^{41} + 204 q^{43} - 1506 q^{45} - 248 q^{47} - 283 q^{49} - 228 q^{51} - 554 q^{53} - 546 q^{55} + 44 q^{57} - 440 q^{59} + 618 q^{61} - 804 q^{63} - 1656 q^{65} - 1164 q^{67} - 1968 q^{69} + 692 q^{71} - 1950 q^{73} - 3074 q^{75} - 1616 q^{77} - 272 q^{79} + 1801 q^{81} - 512 q^{83} - 1628 q^{85} + 232 q^{87} + 866 q^{89} - 2580 q^{91} - 40 q^{93} - 2244 q^{95} + 1562 q^{97} + 238 q^{99}+O(q^{100}) $ 5 * q - 8 * q^3 + 10 * q^5 - 40 * q^7 + 33 * q^9 - 12 * q^11 + 14 * q^13 + 74 * q^15 + 66 * q^17 - 214 * q^19 - 164 * q^23 + 207 * q^25 - 362 * q^27 - 145 * q^29 - 420 * q^31 - 576 * q^33 + 52 * q^35 + 378 * q^37 + 374 * q^39 - 1158 * q^41 + 204 * q^43 - 1506 * q^45 - 248 * q^47 - 283 * q^49 - 228 * q^51 - 554 * q^53 - 546 * q^55 + 44 * q^57 - 440 * q^59 + 618 * q^61 - 804 * q^63 - 1656 * q^65 - 1164 * q^67 - 1968 * q^69 + 692 * q^71 - 1950 * q^73 - 3074 * q^75 - 1616 * q^77 - 272 * q^79 + 1801 * q^81 - 512 * q^83 - 1628 * q^85 + 232 * q^87 + 866 * q^89 - 2580 * q^91 - 40 * q^93 - 2244 * q^95 + 1562 * q^97 + 238 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.90549	0.366712	0.183356	−	0.983047i	$-0.441304\pi$
$3$	1.90549	0.366712	0.183356	+	0.983047i	$0.441304\pi$
$4$	0	0
$5$	−6.52855	−0.583931	−0.291966	−	0.956429i	$-0.594309\pi$
$5$	−6.52855	−0.583931	−0.291966	+	0.956429i	$0.594309\pi$
$6$	0	0
$7$	−5.22706	−0.282235	−0.141117	−	0.989993i	$-0.545070\pi$
$7$	−5.22706	−0.282235	−0.141117	+	0.989993i	$0.545070\pi$
$8$	0	0
$9$	−23.3691	−0.865523
$10$	0	0
$11$	21.1299	0.579173	0.289586	−	0.957152i	$-0.406482\pi$
$11$	21.1299	0.579173	0.289586	+	0.957152i	$0.406482\pi$
$12$	0	0
$13$	83.4615	1.78062	0.890310	−	0.455355i	$-0.150488\pi$
$13$	83.4615	1.78062	0.890310	+	0.455355i	$0.150488\pi$
$14$	0	0
$15$	−12.4401	−0.214134
$16$	0	0
$17$	11.3273	0.161604	0.0808020	−	0.996730i	$-0.474252\pi$
$17$	11.3273	0.161604	0.0808020	+	0.996730i	$0.474252\pi$
$18$	0	0
$19$	7.68096	0.0927438	0.0463719	−	0.998924i	$-0.485234\pi$
$19$	7.68096	0.0927438	0.0463719	+	0.998924i	$0.485234\pi$
$20$	0	0
$21$	−9.96011	−0.103499
$22$	0	0
$23$	−153.169	−1.38861	−0.694303	−	0.719683i	$-0.744287\pi$
$23$	−153.169	−1.38861	−0.694303	+	0.719683i	$0.744287\pi$
$24$	0	0
$25$	−82.3780	−0.659024
$26$	0	0
$27$	−95.9778	−0.684109
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−270.530	−1.56737	−0.783687	−	0.621156i	$-0.786663\pi$
$31$	−270.530	−1.56737	−0.783687	+	0.621156i	$0.786663\pi$
$32$	0	0
$33$	40.2628	0.212389
$34$	0	0
$35$	34.1251	0.164806
$36$	0	0
$37$	−298.404	−1.32587	−0.662936	−	0.748676i	$-0.730690\pi$
$37$	−298.404	−1.32587	−0.662936	+	0.748676i	$0.730690\pi$
$38$	0	0
$39$	159.035	0.652974
$40$	0	0
$41$	−184.710	−0.703584	−0.351792	−	0.936078i	$-0.614427\pi$
$41$	−184.710	−0.703584	−0.351792	+	0.936078i	$0.614427\pi$
$42$	0	0
$43$	−208.337	−0.738861	−0.369430	−	0.929258i	$-0.620447\pi$
$43$	−208.337	−0.738861	−0.369430	+	0.929258i	$0.620447\pi$
$44$	0	0
$45$	152.566	0.505406
$46$	0	0
$47$	553.098	1.71654	0.858272	−	0.513194i	$-0.171538\pi$
$47$	553.098	1.71654	0.858272	+	0.513194i	$0.171538\pi$
$48$	0	0
$49$	−315.678	−0.920344
$50$	0	0
$51$	21.5840	0.0592620
$52$	0	0
$53$	−321.465	−0.833144	−0.416572	−	0.909103i	$-0.636769\pi$
$53$	−321.465	−0.833144	−0.416572	+	0.909103i	$0.636769\pi$
$54$	0	0
$55$	−137.948	−0.338197
$56$	0	0
$57$	14.6360	0.0340102
$58$	0	0
$59$	−104.930	−0.231538	−0.115769	−	0.993276i	$-0.536933\pi$
$59$	−104.930	−0.231538	−0.115769	+	0.993276i	$0.536933\pi$
$60$	0	0
$61$	464.230	0.974403	0.487201	−	0.873290i	$-0.338018\pi$
$61$	464.230	0.974403	0.487201	+	0.873290i	$0.338018\pi$
$62$	0	0
$63$	122.152	0.244281
$64$	0	0
$65$	−544.883	−1.03976
$66$	0	0
$67$	−745.813	−1.35993	−0.679967	−	0.733243i	$-0.738006\pi$
$67$	−745.813	−1.35993	−0.679967	+	0.733243i	$0.738006\pi$
$68$	0	0
$69$	−291.862	−0.509218
$70$	0	0
$71$	509.252	0.851227	0.425613	−	0.904905i	$-0.360058\pi$
$71$	509.252	0.851227	0.425613	+	0.904905i	$0.360058\pi$
$72$	0	0
$73$	−0.374979	−0.000601205 0	−0.000300603	−	1.00000i	$-0.500096\pi$
$73$	−0.374979	−0.000601205 0	−0.000300603		1.00000i	$0.500096\pi$
$74$	0	0
$75$	−156.970	−0.241672
$76$	0	0
$77$	−110.447	−0.163463
$78$	0	0
$79$	−610.912	−0.870037	−0.435018	−	0.900422i	$-0.643258\pi$
$79$	−610.912	−0.870037	−0.435018	+	0.900422i	$0.643258\pi$
$80$	0	0
$81$	448.081	0.614652
$82$	0	0
$83$	−791.431	−1.04664	−0.523319	−	0.852137i	$-0.675306\pi$
$83$	−791.431	−1.04664	−0.523319	+	0.852137i	$0.675306\pi$
$84$	0	0
$85$	−73.9507	−0.0943656
$86$	0	0
$87$	−55.2592	−0.0680966
$88$	0	0
$89$	−342.011	−0.407338	−0.203669	−	0.979040i	$-0.565287\pi$
$89$	−342.011	−0.407338	−0.203669	+	0.979040i	$0.565287\pi$
$90$	0	0
$91$	−436.258	−0.502553
$92$	0	0
$93$	−515.491	−0.574774
$94$	0	0
$95$	−50.1455	−0.0541560
$96$	0	0
$97$	601.476	0.629594	0.314797	−	0.949159i	$-0.398063\pi$
$97$	601.476	0.629594	0.314797	+	0.949159i	$0.398063\pi$
$98$	0	0
$99$	−493.787	−0.501287
$100$	0	0
$101$	402.327	0.396367	0.198183	−	0.980165i	$-0.436496\pi$
$101$	402.327	0.396367	0.198183	+	0.980165i	$0.436496\pi$
$102$	0	0
$103$	1338.38	1.28033	0.640166	−	0.768236i	$-0.278866\pi$
$103$	1338.38	1.28033	0.640166	+	0.768236i	$0.278866\pi$
$104$	0	0
$105$	65.0251	0.0604362
$106$	0	0
$107$	−500.501	−0.452199	−0.226099	−	0.974104i	$-0.572597\pi$
$107$	−500.501	−0.452199	−0.226099	+	0.974104i	$0.572597\pi$
$108$	0	0
$109$	1274.80	1.12022	0.560108	−	0.828420i	$-0.310760\pi$
$109$	1274.80	1.12022	0.560108	+	0.828420i	$0.310760\pi$
$110$	0	0
$111$	−568.605	−0.486212
$112$	0	0
$113$	335.278	0.279118	0.139559	−	0.990214i	$-0.455432\pi$
$113$	335.278	0.279118	0.139559	+	0.990214i	$0.455432\pi$
$114$	0	0
$115$	999.972	0.810851
$116$	0	0
$117$	−1950.42	−1.54117
$118$	0	0
$119$	−59.2083	−0.0456103
$120$	0	0
$121$	−884.528	−0.664559
$122$	0	0
$123$	−351.964	−0.258012
$124$	0	0
$125$	1353.88	0.968756
$126$	0	0
$127$	−755.312	−0.527741	−0.263870	−	0.964558i	$-0.584999\pi$
$127$	−755.312	−0.527741	−0.263870	+	0.964558i	$0.584999\pi$
$128$	0	0
$129$	−396.983	−0.270949
$130$	0	0
$131$	253.351	0.168973	0.0844863	−	0.996425i	$-0.473075\pi$
$131$	253.351	0.168973	0.0844863	+	0.996425i	$0.473075\pi$
$132$	0	0
$133$	−40.1488	−0.0261755
$134$	0	0
$135$	626.596	0.399473
$136$	0	0
$137$	−2477.49	−1.54501	−0.772504	−	0.635010i	$-0.780996\pi$
$137$	−2477.49	−1.54501	−0.772504	+	0.635010i	$0.780996\pi$
$138$	0	0
$139$	−423.114	−0.258187	−0.129094	−	0.991632i	$-0.541207\pi$
$139$	−423.114	−0.258187	−0.129094	+	0.991632i	$0.541207\pi$
$140$	0	0
$141$	1053.92	0.629477
$142$	0	0
$143$	1763.53	1.03129
$144$	0	0
$145$	189.328	0.108433
$146$	0	0
$147$	−601.521	−0.337501
$148$	0	0
$149$	−1263.82	−0.694873	−0.347437	−	0.937703i	$-0.612948\pi$
$149$	−1263.82	−0.694873	−0.347437	+	0.937703i	$0.612948\pi$
$150$	0	0
$151$	−3369.67	−1.81603	−0.908013	−	0.418943i	$-0.862401\pi$
$151$	−3369.67	−1.81603	−0.908013	+	0.418943i	$0.862401\pi$
$152$	0	0
$153$	−264.708	−0.139872
$154$	0	0
$155$	1766.17	0.915238
$156$	0	0
$157$	−3688.61	−1.87505	−0.937527	−	0.347914i	$-0.886890\pi$
$157$	−3688.61	−1.87505	−0.937527	+	0.347914i	$0.886890\pi$
$158$	0	0
$159$	−612.548	−0.305523
$160$	0	0
$161$	800.624	0.391913
$162$	0	0
$163$	1975.81	0.949432	0.474716	−	0.880139i	$-0.342551\pi$
$163$	1975.81	0.949432	0.474716	+	0.880139i	$0.342551\pi$
$164$	0	0
$165$	−262.857	−0.124021
$166$	0	0
$167$	1608.44	0.745297	0.372649	−	0.927973i	$-0.378450\pi$
$167$	1608.44	0.745297	0.372649	+	0.927973i	$0.378450\pi$
$168$	0	0
$169$	4768.82	2.17061
$170$	0	0
$171$	−179.497	−0.0802719
$172$	0	0
$173$	4445.71	1.95376	0.976881	−	0.213784i	$-0.0685788\pi$
$173$	4445.71	1.95376	0.976881	+	0.213784i	$0.0685788\pi$
$174$	0	0
$175$	430.595	0.186000
$176$	0	0
$177$	−199.944	−0.0849078
$178$	0	0
$179$	−1461.35	−0.610203	−0.305101	−	0.952320i	$-0.598690\pi$
$179$	−1461.35	−0.610203	−0.305101	+	0.952320i	$0.598690\pi$
$180$	0	0
$181$	3789.62	1.55624	0.778122	−	0.628113i	$-0.216172\pi$
$181$	3789.62	1.55624	0.778122	+	0.628113i	$0.216172\pi$
$182$	0	0
$183$	884.585	0.357325
$184$	0	0
$185$	1948.14	0.774218
$186$	0	0
$187$	239.344	0.0935966
$188$	0	0
$189$	501.682	0.193079
$190$	0	0
$191$	4782.10	1.81163	0.905813	−	0.423678i	$-0.139261\pi$
$191$	4782.10	1.81163	0.905813	+	0.423678i	$0.139261\pi$
$192$	0	0
$193$	−3557.27	−1.32673	−0.663363	−	0.748298i	$-0.730871\pi$
$193$	−3557.27	−1.32673	−0.663363	+	0.748298i	$0.730871\pi$
$194$	0	0
$195$	−1038.27	−0.381292
$196$	0	0
$197$	−2290.24	−0.828290	−0.414145	−	0.910211i	$-0.635919\pi$
$197$	−2290.24	−0.828290	−0.414145	+	0.910211i	$0.635919\pi$
$198$	0	0
$199$	2788.00	0.993146	0.496573	−	0.867995i	$-0.334591\pi$
$199$	2788.00	0.993146	0.496573	+	0.867995i	$0.334591\pi$
$200$	0	0
$201$	−1421.14	−0.498703
$202$	0	0
$203$	151.585	0.0524097
$204$	0	0
$205$	1205.89	0.410845
$206$	0	0
$207$	3579.42	1.20187
$208$	0	0
$209$	162.298	0.0537147
$210$	0	0
$211$	−628.449	−0.205044	−0.102522	−	0.994731i	$-0.532691\pi$
$211$	−628.449	−0.205044	−0.102522	+	0.994731i	$0.532691\pi$
$212$	0	0
$213$	970.374	0.312155
$214$	0	0
$215$	1360.14	0.431444
$216$	0	0
$217$	1414.08	0.442367
$218$	0	0
$219$	−0.714519	−0.000220469 0
$220$	0	0
$221$	945.391	0.287755
$222$	0	0
$223$	−136.439	−0.0409714	−0.0204857	−	0.999790i	$-0.506521\pi$
$223$	−136.439	−0.0409714	−0.0204857	+	0.999790i	$0.506521\pi$
$224$	0	0
$225$	1925.10	0.570400
$226$	0	0
$227$	−4180.45	−1.22232	−0.611159	−	0.791508i	$-0.709296\pi$
$227$	−4180.45	−1.22232	−0.611159	+	0.791508i	$0.709296\pi$
$228$	0	0
$229$	−1352.95	−0.390417	−0.195209	−	0.980762i	$-0.562538\pi$
$229$	−1352.95	−0.390417	−0.195209	+	0.980762i	$0.562538\pi$
$230$	0	0
$231$	−210.456	−0.0599436
$232$	0	0
$233$	2175.34	0.611635	0.305818	−	0.952090i	$-0.401070\pi$
$233$	2175.34	0.611635	0.305818	+	0.952090i	$0.401070\pi$
$234$	0	0
$235$	−3610.93	−1.00234
$236$	0	0
$237$	−1164.09	−0.319053
$238$	0	0
$239$	4512.18	1.22121	0.610604	−	0.791936i	$-0.290927\pi$
$239$	4512.18	1.22121	0.610604	+	0.791936i	$0.290927\pi$
$240$	0	0
$241$	1950.53	0.521346	0.260673	−	0.965427i	$-0.416056\pi$
$241$	1950.53	0.521346	0.260673	+	0.965427i	$0.416056\pi$
$242$	0	0
$243$	3445.21	0.909509
$244$	0	0
$245$	2060.92	0.537417
$246$	0	0
$247$	641.064	0.165141
$248$	0	0
$249$	−1508.06	−0.383814
$250$	0	0
$251$	−27.4143	−0.00689393	−0.00344696	−	0.999994i	$-0.501097\pi$
$251$	−27.4143	−0.00689393	−0.00344696	+	0.999994i	$0.501097\pi$
$252$	0	0
$253$	−3236.44	−0.804243
$254$	0	0
$255$	−140.912	−0.0346050
$256$	0	0
$257$	4458.31	1.08211	0.541053	−	0.840988i	$-0.318026\pi$
$257$	4458.31	1.08211	0.541053	+	0.840988i	$0.318026\pi$
$258$	0	0
$259$	1559.77	0.374207
$260$	0	0
$261$	677.704	0.160724
$262$	0	0
$263$	−4641.08	−1.08814	−0.544071	−	0.839039i	$-0.683118\pi$
$263$	−4641.08	−1.08814	−0.544071	+	0.839039i	$0.683118\pi$
$264$	0	0
$265$	2098.70	0.486499
$266$	0	0
$267$	−651.699	−0.149376
$268$	0	0
$269$	−235.021	−0.0532694	−0.0266347	−	0.999645i	$-0.508479\pi$
$269$	−235.021	−0.0532694	−0.0266347	+	0.999645i	$0.508479\pi$
$270$	0	0
$271$	−3816.09	−0.855392	−0.427696	−	0.903923i	$-0.640675\pi$
$271$	−3816.09	−0.855392	−0.427696	+	0.903923i	$0.640675\pi$
$272$	0	0
$273$	−831.286	−0.184292
$274$	0	0
$275$	−1740.64	−0.381689
$276$	0	0
$277$	−2024.79	−0.439198	−0.219599	−	0.975590i	$-0.570475\pi$
$277$	−2024.79	−0.439198	−0.219599	+	0.975590i	$0.570475\pi$
$278$	0	0
$279$	6322.04	1.35660
$280$	0	0
$281$	5451.81	1.15739	0.578697	−	0.815543i	$-0.303561\pi$
$281$	5451.81	1.15739	0.578697	+	0.815543i	$0.303561\pi$
$282$	0	0
$283$	−5026.80	−1.05587	−0.527937	−	0.849284i	$-0.677034\pi$
$283$	−5026.80	−1.05587	−0.527937	+	0.849284i	$0.677034\pi$
$284$	0	0
$285$	−95.5517	−0.0198596
$286$	0	0
$287$	965.493	0.198576
$288$	0	0
$289$	−4784.69	−0.973884
$290$	0	0
$291$	1146.11	0.230880
$292$	0	0
$293$	6160.34	1.22830	0.614148	−	0.789191i	$-0.289500\pi$
$293$	6160.34	1.22830	0.614148	+	0.789191i	$0.289500\pi$
$294$	0	0
$295$	685.043	0.135203
$296$	0	0
$297$	−2028.00	−0.396217
$298$	0	0
$299$	−12783.7	−2.47258
$300$	0	0
$301$	1088.99	0.208532
$302$	0	0
$303$	766.629	0.145352
$304$	0	0
$305$	−3030.75	−0.568984
$306$	0	0
$307$	4329.54	0.804885	0.402443	−	0.915445i	$-0.368161\pi$
$307$	4329.54	0.804885	0.402443	+	0.915445i	$0.368161\pi$
$308$	0	0
$309$	2550.26	0.469513
$310$	0	0
$311$	6411.83	1.16907	0.584536	−	0.811368i	$-0.301276\pi$
$311$	6411.83	1.16907	0.584536	+	0.811368i	$0.301276\pi$
$312$	0	0
$313$	−19.5263	−0.00352617	−0.00176309	−	0.999998i	$-0.500561\pi$
$313$	−19.5263	−0.00352617	−0.00176309	+	0.999998i	$0.500561\pi$
$314$	0	0
$315$	−797.474	−0.142643
$316$	0	0
$317$	−8198.11	−1.45253	−0.726265	−	0.687415i	$-0.758745\pi$
$317$	−8198.11	−1.45253	−0.726265	+	0.687415i	$0.758745\pi$
$318$	0	0
$319$	−612.767	−0.107550
$320$	0	0
$321$	−953.699	−0.165826
$322$	0	0
$323$	87.0043	0.0149878
$324$	0	0
$325$	−6875.39	−1.17347
$326$	0	0
$327$	2429.11	0.410796
$328$	0	0
$329$	−2891.08	−0.484469
$330$	0	0
$331$	−2374.46	−0.394297	−0.197148	−	0.980374i	$-0.563168\pi$
$331$	−2374.46	−0.394297	−0.197148	+	0.980374i	$0.563168\pi$
$332$	0	0
$333$	6973.43	1.14757
$334$	0	0
$335$	4869.08	0.794108
$336$	0	0
$337$	−4985.34	−0.805842	−0.402921	−	0.915235i	$-0.632005\pi$
$337$	−4985.34	−0.805842	−0.402921	+	0.915235i	$0.632005\pi$
$338$	0	0
$339$	638.869	0.102356
$340$	0	0
$341$	−5716.26	−0.907780
$342$	0	0
$343$	3442.95	0.541988
$344$	0	0
$345$	1905.44	0.297348
$346$	0	0
$347$	2023.98	0.313121	0.156560	−	0.987668i	$-0.449959\pi$
$347$	2023.98	0.313121	0.156560	+	0.987668i	$0.449959\pi$
$348$	0	0
$349$	−2651.99	−0.406755	−0.203378	−	0.979100i	$-0.565192\pi$
$349$	−2651.99	−0.406755	−0.203378	+	0.979100i	$0.565192\pi$
$350$	0	0
$351$	−8010.45	−1.21814
$352$	0	0
$353$	−1729.22	−0.260729	−0.130364	−	0.991466i	$-0.541615\pi$
$353$	−1729.22	−0.260729	−0.130364	+	0.991466i	$0.541615\pi$
$354$	0	0
$355$	−3324.68	−0.497058
$356$	0	0
$357$	−112.821	−0.0167258
$358$	0	0
$359$	−6875.38	−1.01078	−0.505388	−	0.862892i	$-0.668651\pi$
$359$	−6875.38	−1.01078	−0.505388	+	0.862892i	$0.668651\pi$
$360$	0	0
$361$	−6800.00	−0.991399
$362$	0	0
$363$	−1685.46	−0.243701
$364$	0	0
$365$	2.44807	0.000351063 0
$366$	0	0
$367$	7435.15	1.05753	0.528763	−	0.848770i	$-0.322656\pi$
$367$	7435.15	1.05753	0.528763	+	0.848770i	$0.322656\pi$
$368$	0	0
$369$	4316.52	0.608968
$370$	0	0
$371$	1680.32	0.235142
$372$	0	0
$373$	1397.48	0.193991	0.0969954	−	0.995285i	$-0.469077\pi$
$373$	1397.48	0.193991	0.0969954	+	0.995285i	$0.469077\pi$
$374$	0	0
$375$	2579.80	0.355254
$376$	0	0
$377$	−2420.38	−0.330653
$378$	0	0
$379$	−7009.57	−0.950020	−0.475010	−	0.879980i	$-0.657556\pi$
$379$	−7009.57	−0.950020	−0.475010	+	0.879980i	$0.657556\pi$
$380$	0	0
$381$	−1439.24	−0.193529
$382$	0	0
$383$	11728.7	1.56477	0.782387	−	0.622792i	$-0.214002\pi$
$383$	11728.7	1.56477	0.782387	+	0.622792i	$0.214002\pi$
$384$	0	0
$385$	721.060	0.0954510
$386$	0	0
$387$	4868.64	0.639501
$388$	0	0
$389$	−4367.15	−0.569212	−0.284606	−	0.958645i	$-0.591863\pi$
$389$	−4367.15	−0.569212	−0.284606	+	0.958645i	$0.591863\pi$
$390$	0	0
$391$	−1734.99	−0.224404
$392$	0	0
$393$	482.758	0.0619642
$394$	0	0
$395$	3988.37	0.508042
$396$	0	0
$397$	1632.74	0.206410	0.103205	−	0.994660i	$-0.467090\pi$
$397$	1632.74	0.206410	0.103205	+	0.994660i	$0.467090\pi$
$398$	0	0
$399$	−76.5032	−0.00959887
$400$	0	0
$401$	5028.65	0.626232	0.313116	−	0.949715i	$-0.398627\pi$
$401$	5028.65	0.626232	0.313116	+	0.949715i	$0.398627\pi$
$402$	0	0
$403$	−22578.8	−2.79090
$404$	0	0
$405$	−2925.32	−0.358915
$406$	0	0
$407$	−6305.23	−0.767908
$408$	0	0
$409$	−2497.21	−0.301904	−0.150952	−	0.988541i	$-0.548234\pi$
$409$	−2497.21	−0.301904	−0.150952	+	0.988541i	$0.548234\pi$
$410$	0	0
$411$	−4720.82	−0.566572
$412$	0	0
$413$	548.477	0.0653482
$414$	0	0
$415$	5166.90	0.611164
$416$	0	0
$417$	−806.239	−0.0946803
$418$	0	0
$419$	−12909.4	−1.50517	−0.752585	−	0.658496i	$-0.771193\pi$
$419$	−12909.4	−1.50517	−0.752585	+	0.658496i	$0.771193\pi$
$420$	0	0
$421$	−4019.30	−0.465293	−0.232647	−	0.972561i	$-0.574739\pi$
$421$	−4019.30	−0.465293	−0.232647	+	0.972561i	$0.574739\pi$
$422$	0	0
$423$	−12925.4	−1.48571
$424$	0	0
$425$	−933.118	−0.106501
$426$	0	0
$427$	−2426.56	−0.275010
$428$	0	0
$429$	3360.39	0.378185
$430$	0	0
$431$	−8768.94	−0.980012	−0.490006	−	0.871719i	$-0.663005\pi$
$431$	−8768.94	−0.980012	−0.490006	+	0.871719i	$0.663005\pi$
$432$	0	0
$433$	4496.26	0.499021	0.249511	−	0.968372i	$-0.419730\pi$
$433$	4496.26	0.499021	0.249511	+	0.968372i	$0.419730\pi$
$434$	0	0
$435$	360.762	0.0397638
$436$	0	0
$437$	−1176.48	−0.128785
$438$	0	0
$439$	7316.62	0.795451	0.397725	−	0.917504i	$-0.369800\pi$
$439$	7316.62	0.795451	0.397725	+	0.917504i	$0.369800\pi$
$440$	0	0
$441$	7377.11	0.796578
$442$	0	0
$443$	12801.1	1.37291	0.686454	−	0.727173i	$-0.259166\pi$
$443$	12801.1	1.37291	0.686454	+	0.727173i	$0.259166\pi$
$444$	0	0
$445$	2232.84	0.237858
$446$	0	0
$447$	−2408.19	−0.254818
$448$	0	0
$449$	6412.22	0.673968	0.336984	−	0.941510i	$-0.390593\pi$
$449$	6412.22	0.673968	0.336984	+	0.941510i	$0.390593\pi$
$450$	0	0
$451$	−3902.91	−0.407496
$452$	0	0
$453$	−6420.87	−0.665957
$454$	0	0
$455$	2848.14	0.293456
$456$	0	0
$457$	−2153.32	−0.220412	−0.110206	−	0.993909i	$-0.535151\pi$
$457$	−2153.32	−0.220412	−0.110206	+	0.993909i	$0.535151\pi$
$458$	0	0
$459$	−1087.17	−0.110555
$460$	0	0
$461$	−1850.55	−0.186960	−0.0934800	−	0.995621i	$-0.529799\pi$
$461$	−1850.55	−0.186960	−0.0934800	+	0.995621i	$0.529799\pi$
$462$	0	0
$463$	−1892.04	−0.189914	−0.0949572	−	0.995481i	$-0.530271\pi$
$463$	−1892.04	−0.189914	−0.0949572	+	0.995481i	$0.530271\pi$
$464$	0	0
$465$	3365.41	0.335628
$466$	0	0
$467$	3739.40	0.370533	0.185267	−	0.982688i	$-0.440685\pi$
$467$	3739.40	0.370533	0.185267	+	0.982688i	$0.440685\pi$
$468$	0	0
$469$	3898.41	0.383821
$470$	0	0
$471$	−7028.61	−0.687604
$472$	0	0
$473$	−4402.13	−0.427928
$474$	0	0
$475$	−632.742	−0.0611204
$476$	0	0
$477$	7512.35	0.721105
$478$	0	0
$479$	7260.30	0.692550	0.346275	−	0.938133i	$-0.387446\pi$
$479$	7260.30	0.692550	0.346275	+	0.938133i	$0.387446\pi$
$480$	0	0
$481$	−24905.2	−2.36087
$482$	0	0
$483$	1525.58	0.143719
$484$	0	0
$485$	−3926.77	−0.367640
$486$	0	0
$487$	−4756.40	−0.442573	−0.221287	−	0.975209i	$-0.571026\pi$
$487$	−4756.40	−0.442573	−0.221287	+	0.975209i	$0.571026\pi$
$488$	0	0
$489$	3764.88	0.348168
$490$	0	0
$491$	2007.83	0.184546	0.0922730	−	0.995734i	$-0.470587\pi$
$491$	2007.83	0.184546	0.0922730	+	0.995734i	$0.470587\pi$
$492$	0	0
$493$	−328.491	−0.0300091
$494$	0	0
$495$	3223.71	0.292717
$496$	0	0
$497$	−2661.89	−0.240246
$498$	0	0
$499$	8952.55	0.803149	0.401574	−	0.915826i	$-0.368463\pi$
$499$	8952.55	0.803149	0.401574	+	0.915826i	$0.368463\pi$
$500$	0	0
$501$	3064.86	0.273309
$502$	0	0
$503$	−20564.9	−1.82295	−0.911477	−	0.411351i	$-0.865057\pi$
$503$	−20564.9	−1.82295	−0.911477	+	0.411351i	$0.865057\pi$
$504$	0	0
$505$	−2626.61	−0.231451
$506$	0	0
$507$	9086.94	0.795987
$508$	0	0
$509$	13321.9	1.16008	0.580041	−	0.814587i	$-0.303036\pi$
$509$	13321.9	1.16008	0.580041	+	0.814587i	$0.303036\pi$
$510$	0	0
$511$	1.96004	0.000169681 0
$512$	0	0
$513$	−737.201	−0.0634468
$514$	0	0
$515$	−8737.66	−0.747626
$516$	0	0
$517$	11686.9	0.994176
$518$	0	0
$519$	8471.24	0.716467
$520$	0	0
$521$	−13047.0	−1.09712	−0.548558	−	0.836113i	$-0.684823\pi$
$521$	−13047.0	−1.09712	−0.548558	+	0.836113i	$0.684823\pi$
$522$	0	0
$523$	20207.4	1.68950	0.844750	−	0.535161i	$-0.179749\pi$
$523$	20207.4	1.68950	0.844750	+	0.535161i	$0.179749\pi$
$524$	0	0
$525$	820.494	0.0682082
$526$	0	0
$527$	−3064.36	−0.253294
$528$	0	0
$529$	11293.8	0.928228
$530$	0	0
$531$	2452.13	0.200402
$532$	0	0
$533$	−15416.2	−1.25281
$534$	0	0
$535$	3267.55	0.264053
$536$	0	0
$537$	−2784.58	−0.223768
$538$	0	0
$539$	−6670.23	−0.533038
$540$	0	0
$541$	−17866.0	−1.41981	−0.709906	−	0.704297i	$-0.751263\pi$
$541$	−17866.0	−1.41981	−0.709906	+	0.704297i	$0.751263\pi$
$542$	0	0
$543$	7221.08	0.570693
$544$	0	0
$545$	−8322.59	−0.654129
$546$	0	0
$547$	8027.79	0.627502	0.313751	−	0.949505i	$-0.398414\pi$
$547$	8027.79	0.627502	0.313751	+	0.949505i	$0.398414\pi$
$548$	0	0
$549$	−10848.6	−0.843367
$550$	0	0
$551$	−222.748	−0.0172221
$552$	0	0
$553$	3193.27	0.245555
$554$	0	0
$555$	3712.16	0.283915
$556$	0	0
$557$	17357.6	1.32040	0.660201	−	0.751089i	$-0.270471\pi$
$557$	17357.6	1.32040	0.660201	+	0.751089i	$0.270471\pi$
$558$	0	0
$559$	−17388.1	−1.31563
$560$	0	0
$561$	456.067	0.0343229
$562$	0	0
$563$	6073.90	0.454679	0.227340	−	0.973816i	$-0.426997\pi$
$563$	6073.90	0.454679	0.227340	+	0.973816i	$0.426997\pi$
$564$	0	0
$565$	−2188.88	−0.162986
$566$	0	0
$567$	−2342.15	−0.173476
$568$	0	0
$569$	5084.04	0.374577	0.187288	−	0.982305i	$-0.440030\pi$
$569$	5084.04	0.374577	0.187288	+	0.982305i	$0.440030\pi$
$570$	0	0
$571$	−15552.3	−1.13983	−0.569914	−	0.821704i	$-0.693024\pi$
$571$	−15552.3	−1.13983	−0.569914	+	0.821704i	$0.693024\pi$
$572$	0	0
$573$	9112.24	0.664344
$574$	0	0
$575$	12617.8	0.915125
$576$	0	0
$577$	−20714.3	−1.49454	−0.747269	−	0.664521i	$-0.768635\pi$
$577$	−20714.3	−1.49454	−0.747269	+	0.664521i	$0.768635\pi$
$578$	0	0
$579$	−6778.34	−0.486526
$580$	0	0
$581$	4136.86	0.295397
$582$	0	0
$583$	−6792.52	−0.482534
$584$	0	0
$585$	12733.4	0.899936
$586$	0	0
$587$	1015.03	0.0713711	0.0356856	−	0.999363i	$-0.488639\pi$
$587$	1015.03	0.0713711	0.0356856	+	0.999363i	$0.488639\pi$
$588$	0	0
$589$	−2077.93	−0.145364
$590$	0	0
$591$	−4364.03	−0.303744
$592$	0	0
$593$	3831.39	0.265323	0.132661	−	0.991161i	$-0.457648\pi$
$593$	3831.39	0.265323	0.132661	+	0.991161i	$0.457648\pi$
$594$	0	0
$595$	386.545	0.0266333
$596$	0	0
$597$	5312.50	0.364198
$598$	0	0
$599$	16703.5	1.13938	0.569688	−	0.821861i	$-0.307064\pi$
$599$	16703.5	1.13938	0.569688	+	0.821861i	$0.307064\pi$
$600$	0	0
$601$	17781.6	1.20686	0.603432	−	0.797414i	$-0.293799\pi$
$601$	17781.6	1.20686	0.603432	+	0.797414i	$0.293799\pi$
$602$	0	0
$603$	17429.0	1.17705
$604$	0	0
$605$	5774.69	0.388057
$606$	0	0
$607$	−7610.08	−0.508869	−0.254435	−	0.967090i	$-0.581889\pi$
$607$	−7610.08	−0.508869	−0.254435	+	0.967090i	$0.581889\pi$
$608$	0	0
$609$	288.843	0.0192192
$610$	0	0
$611$	46162.4	3.05651
$612$	0	0
$613$	4207.29	0.277212	0.138606	−	0.990348i	$-0.455738\pi$
$613$	4207.29	0.277212	0.138606	+	0.990348i	$0.455738\pi$
$614$	0	0
$615$	2297.81	0.150661
$616$	0	0
$617$	28539.4	1.86216	0.931082	−	0.364810i	$-0.118866\pi$
$617$	28539.4	1.86216	0.931082	+	0.364810i	$0.118866\pi$
$618$	0	0
$619$	−16799.9	−1.09086	−0.545432	−	0.838155i	$-0.683634\pi$
$619$	−16799.9	−1.09086	−0.545432	+	0.838155i	$0.683634\pi$
$620$	0	0
$621$	14700.8	0.949958
$622$	0	0
$623$	1787.71	0.114965
$624$	0	0
$625$	1458.39	0.0933369
$626$	0	0
$627$	309.256	0.0196978
$628$	0	0
$629$	−3380.10	−0.214266
$630$	0	0
$631$	16207.1	1.02249	0.511247	−	0.859434i	$-0.329184\pi$
$631$	16207.1	1.02249	0.511247	+	0.859434i	$0.329184\pi$
$632$	0	0
$633$	−1197.50	−0.0751919
$634$	0	0
$635$	4931.09	0.308164
$636$	0	0
$637$	−26346.9	−1.63878
$638$	0	0
$639$	−11900.8	−0.736756
$640$	0	0
$641$	10697.0	0.659133	0.329567	−	0.944132i	$-0.393097\pi$
$641$	10697.0	0.659133	0.329567	+	0.944132i	$0.393097\pi$
$642$	0	0
$643$	−7901.34	−0.484601	−0.242300	−	0.970201i	$-0.577902\pi$
$643$	−7901.34	−0.484601	−0.242300	+	0.970201i	$0.577902\pi$
$644$	0	0
$645$	2591.72	0.158216
$646$	0	0
$647$	312.370	0.0189807	0.00949036	−	0.999955i	$-0.496979\pi$
$647$	312.370	0.0189807	0.00949036	+	0.999955i	$0.496979\pi$
$648$	0	0
$649$	−2217.17	−0.134101
$650$	0	0
$651$	2694.50	0.162221
$652$	0	0
$653$	13732.3	0.822950	0.411475	−	0.911421i	$-0.365014\pi$
$653$	13732.3	0.822950	0.411475	+	0.911421i	$0.365014\pi$
$654$	0	0
$655$	−1654.02	−0.0986684
$656$	0	0
$657$	8.76293	0.000520357 0
$658$	0	0
$659$	−19869.0	−1.17449	−0.587244	−	0.809410i	$-0.699787\pi$
$659$	−19869.0	−1.17449	−0.587244	+	0.809410i	$0.699787\pi$
$660$	0	0
$661$	−11047.2	−0.650057	−0.325029	−	0.945704i	$-0.605374\pi$
$661$	−11047.2	−0.650057	−0.325029	+	0.945704i	$0.605374\pi$
$662$	0	0
$663$	1801.43	0.105523
$664$	0	0
$665$	262.114	0.0152847
$666$	0	0
$667$	4441.90	0.257858
$668$	0	0
$669$	−259.983	−0.0150247
$670$	0	0
$671$	9809.12	0.564347
$672$	0	0
$673$	−25673.8	−1.47051	−0.735253	−	0.677793i	$-0.762937\pi$
$673$	−25673.8	−1.47051	−0.735253	+	0.677793i	$0.762937\pi$
$674$	0	0
$675$	7906.46	0.450844
$676$	0	0
$677$	4268.79	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$677$	4268.79	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$678$	0	0
$679$	−3143.95	−0.177693
$680$	0	0
$681$	−7965.79	−0.448238
$682$	0	0
$683$	−7370.85	−0.412940	−0.206470	−	0.978453i	$-0.566198\pi$
$683$	−7370.85	−0.412940	−0.206470	+	0.978453i	$0.566198\pi$
$684$	0	0
$685$	16174.4	0.902178
$686$	0	0
$687$	−2578.04	−0.143171
$688$	0	0
$689$	−26830.0	−1.48351
$690$	0	0
$691$	−2112.00	−0.116272	−0.0581361	−	0.998309i	$-0.518516\pi$
$691$	−2112.00	−0.116272	−0.0581361	+	0.998309i	$0.518516\pi$
$692$	0	0
$693$	2581.05	0.141481
$694$	0	0
$695$	2762.32	0.150764
$696$	0	0
$697$	−2092.27	−0.113702
$698$	0	0
$699$	4145.08	0.224294
$700$	0	0
$701$	20945.5	1.12853	0.564266	−	0.825593i	$-0.309159\pi$
$701$	20945.5	1.12853	0.564266	+	0.825593i	$0.309159\pi$
$702$	0	0
$703$	−2292.02	−0.122966
$704$	0	0
$705$	−6880.58	−0.367571
$706$	0	0
$707$	−2102.99	−0.111868
$708$	0	0
$709$	7826.11	0.414550	0.207275	−	0.978283i	$-0.433541\pi$
$709$	7826.11	0.414550	0.207275	+	0.978283i	$0.433541\pi$
$710$	0	0
$711$	14276.5	0.753037
$712$	0	0
$713$	41436.8	2.17646
$714$	0	0
$715$	−11513.3	−0.602200
$716$	0	0
$717$	8597.91	0.447831
$718$	0	0
$719$	−23373.7	−1.21237	−0.606183	−	0.795325i	$-0.707300\pi$
$719$	−23373.7	−1.21237	−0.606183	+	0.795325i	$0.707300\pi$
$720$	0	0
$721$	−6995.78	−0.361354
$722$	0	0
$723$	3716.71	0.191184
$724$	0	0
$725$	2388.96	0.122378
$726$	0	0
$727$	−31240.8	−1.59375	−0.796875	−	0.604145i	$-0.793515\pi$
$727$	−31240.8	−1.59375	−0.796875	+	0.604145i	$0.793515\pi$
$728$	0	0
$729$	−5533.38	−0.281125
$730$	0	0
$731$	−2359.88	−0.119403
$732$	0	0
$733$	−4426.59	−0.223056	−0.111528	−	0.993761i	$-0.535574\pi$
$733$	−4426.59	−0.223056	−0.111528	+	0.993761i	$0.535574\pi$
$734$	0	0
$735$	3927.06	0.197077
$736$	0	0
$737$	−15758.9	−0.787636
$738$	0	0
$739$	32119.5	1.59883	0.799414	−	0.600781i	$-0.205144\pi$
$739$	32119.5	1.59883	0.799414	+	0.600781i	$0.205144\pi$
$740$	0	0
$741$	1221.54	0.0605593
$742$	0	0
$743$	19215.0	0.948763	0.474382	−	0.880319i	$-0.342672\pi$
$743$	19215.0	0.948763	0.474382	+	0.880319i	$0.342672\pi$
$744$	0	0
$745$	8250.91	0.405758
$746$	0	0
$747$	18495.0	0.905888
$748$	0	0
$749$	2616.15	0.127626
$750$	0	0
$751$	−3593.57	−0.174609	−0.0873045	−	0.996182i	$-0.527825\pi$
$751$	−3593.57	−0.174609	−0.0873045	+	0.996182i	$0.527825\pi$
$752$	0	0
$753$	−52.2377	−0.00252808
$754$	0	0
$755$	21999.1	1.06043
$756$	0	0
$757$	−32956.7	−1.58234	−0.791169	−	0.611598i	$-0.790527\pi$
$757$	−32956.7	−1.58234	−0.791169	+	0.611598i	$0.790527\pi$
$758$	0	0
$759$	−6167.01	−0.294925
$760$	0	0
$761$	15689.7	0.747373	0.373687	−	0.927555i	$-0.378094\pi$
$761$	15689.7	0.747373	0.373687	+	0.927555i	$0.378094\pi$
$762$	0	0
$763$	−6663.45	−0.316164
$764$	0	0
$765$	1728.16	0.0816756
$766$	0	0
$767$	−8757.64	−0.412282
$768$	0	0
$769$	2134.77	0.100106	0.0500532	−	0.998747i	$-0.484061\pi$
$769$	2134.77	0.100106	0.0500532	+	0.998747i	$0.484061\pi$
$770$	0	0
$771$	8495.25	0.396821
$772$	0	0
$773$	14780.2	0.687721	0.343861	−	0.939021i	$-0.388265\pi$
$773$	14780.2	0.687721	0.343861	+	0.939021i	$0.388265\pi$
$774$	0	0
$775$	22285.7	1.03294
$776$	0	0
$777$	2972.13	0.137226
$778$	0	0
$779$	−1418.75	−0.0652530
$780$	0	0
$781$	10760.4	0.493007
$782$	0	0
$783$	2783.36	0.127036
$784$	0	0
$785$	24081.3	1.09490
$786$	0	0
$787$	−35525.7	−1.60909	−0.804545	−	0.593892i	$-0.797591\pi$
$787$	−35525.7	−1.60909	−0.804545	+	0.593892i	$0.797591\pi$
$788$	0	0
$789$	−8843.53	−0.399034
$790$	0	0
$791$	−1752.52	−0.0787768
$792$	0	0
$793$	38745.3	1.73504
$794$	0	0
$795$	3999.05	0.178405
$796$	0	0
$797$	2190.00	0.0973321	0.0486660	−	0.998815i	$-0.484503\pi$
$797$	2190.00	0.0973321	0.0486660	+	0.998815i	$0.484503\pi$
$798$	0	0
$799$	6265.09	0.277400
$800$	0	0
$801$	7992.50	0.352561
$802$	0	0
$803$	−7.92326	−0.000348202 0
$804$	0	0
$805$	−5226.91	−0.228850
$806$	0	0
$807$	−447.830	−0.0195345
$808$	0	0
$809$	21464.6	0.932824	0.466412	−	0.884568i	$-0.345546\pi$
$809$	21464.6	0.932824	0.466412	+	0.884568i	$0.345546\pi$
$810$	0	0
$811$	32288.8	1.39804	0.699021	−	0.715101i	$-0.253620\pi$
$811$	32288.8	1.39804	0.699021	+	0.715101i	$0.253620\pi$
$812$	0	0
$813$	−7271.52	−0.313682
$814$	0	0
$815$	−12899.2	−0.554403
$816$	0	0
$817$	−1600.22	−0.0685248
$818$	0	0
$819$	10195.0	0.434971
$820$	0	0
$821$	−40334.4	−1.71459	−0.857296	−	0.514825i	$-0.827857\pi$
$821$	−40334.4	−1.71459	−0.857296	+	0.514825i	$0.827857\pi$
$822$	0	0
$823$	−8188.53	−0.346822	−0.173411	−	0.984850i	$-0.555479\pi$
$823$	−8188.53	−0.346822	−0.173411	+	0.984850i	$0.555479\pi$
$824$	0	0
$825$	−3316.77	−0.139970
$826$	0	0
$827$	7630.74	0.320855	0.160427	−	0.987048i	$-0.448713\pi$
$827$	7630.74	0.320855	0.160427	+	0.987048i	$0.448713\pi$
$828$	0	0
$829$	12637.7	0.529462	0.264731	−	0.964322i	$-0.414717\pi$
$829$	12637.7	0.529462	0.264731	+	0.964322i	$0.414717\pi$
$830$	0	0
$831$	−3858.21	−0.161059
$832$	0	0
$833$	−3575.77	−0.148731
$834$	0	0
$835$	−10500.8	−0.435203
$836$	0	0
$837$	25964.8	1.07225
$838$	0	0
$839$	−24059.1	−0.990005	−0.495002	−	0.868892i	$-0.664833\pi$
$839$	−24059.1	−0.990005	−0.495002	+	0.868892i	$0.664833\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	10388.4	0.424430
$844$	0	0
$845$	−31133.5	−1.26749
$846$	0	0
$847$	4623.48	0.187562
$848$	0	0
$849$	−9578.51	−0.387201
$850$	0	0
$851$	45706.2	1.84111
$852$	0	0
$853$	−18588.4	−0.746138	−0.373069	−	0.927804i	$-0.621695\pi$
$853$	−18588.4	−0.746138	−0.373069	+	0.927804i	$0.621695\pi$
$854$	0	0
$855$	1171.86	0.0468733
$856$	0	0
$857$	28616.2	1.14062	0.570310	−	0.821430i	$-0.306823\pi$
$857$	28616.2	1.14062	0.570310	+	0.821430i	$0.306823\pi$
$858$	0	0
$859$	31968.9	1.26981	0.634904	−	0.772591i	$-0.281040\pi$
$859$	31968.9	1.26981	0.634904	+	0.772591i	$0.281040\pi$
$860$	0	0
$861$	1839.74	0.0728200
$862$	0	0
$863$	−15417.8	−0.608145	−0.304072	−	0.952649i	$-0.598346\pi$
$863$	−15417.8	−0.608145	−0.304072	+	0.952649i	$0.598346\pi$
$864$	0	0
$865$	−29024.0	−1.14086
$866$	0	0
$867$	−9117.18	−0.357135
$868$	0	0
$869$	−12908.5	−0.503902
$870$	0	0
$871$	−62246.7	−2.42153
$872$	0	0
$873$	−14056.0	−0.544928
$874$	0	0
$875$	−7076.80	−0.273417
$876$	0	0
$877$	−41961.0	−1.61565	−0.807824	−	0.589424i	$-0.799355\pi$
$877$	−41961.0	−1.61565	−0.807824	+	0.589424i	$0.799355\pi$
$878$	0	0
$879$	11738.5	0.450430
$880$	0	0
$881$	−22884.6	−0.875145	−0.437573	−	0.899183i	$-0.644162\pi$
$881$	−22884.6	−0.875145	−0.437573	+	0.899183i	$0.644162\pi$
$882$	0	0
$883$	−29611.7	−1.12855	−0.564277	−	0.825585i	$-0.690845\pi$
$883$	−29611.7	−1.12855	−0.564277	+	0.825585i	$0.690845\pi$
$884$	0	0
$885$	1305.34	0.0495803
$886$	0	0
$887$	−27117.7	−1.02652	−0.513259	−	0.858234i	$-0.671562\pi$
$887$	−27117.7	−1.02652	−0.513259	+	0.858234i	$0.671562\pi$
$888$	0	0
$889$	3948.06	0.148947
$890$	0	0
$891$	9467.91	0.355990
$892$	0	0
$893$	4248.32	0.159199
$894$	0	0
$895$	9540.48	0.356316
$896$	0	0
$897$	−24359.2	−0.906724
$898$	0	0
$899$	7845.36	0.291054
$900$	0	0
$901$	−3641.32	−0.134639
$902$	0	0
$903$	2075.05	0.0764712
$904$	0	0
$905$	−24740.7	−0.908740
$906$	0	0
$907$	−10010.9	−0.366491	−0.183245	−	0.983067i	$-0.558660\pi$
$907$	−10010.9	−0.366491	−0.183245	+	0.983067i	$0.558660\pi$
$908$	0	0
$909$	−9402.02	−0.343064
$910$	0	0
$911$	−40394.1	−1.46906	−0.734532	−	0.678575i	$-0.762598\pi$
$911$	−40394.1	−1.46906	−0.734532	+	0.678575i	$0.762598\pi$
$912$	0	0
$913$	−16722.8	−0.606183
$914$	0	0
$915$	−5775.06	−0.208653
$916$	0	0
$917$	−1324.28	−0.0476899
$918$	0	0
$919$	−2392.98	−0.0858945	−0.0429472	−	0.999077i	$-0.513675\pi$
$919$	−2392.98	−0.0858945	−0.0429472	+	0.999077i	$0.513675\pi$
$920$	0	0
$921$	8249.89	0.295161
$922$	0	0
$923$	42502.9	1.51571
$924$	0	0
$925$	24581.9	0.873781
$926$	0	0
$927$	−31276.7	−1.10816
$928$	0	0
$929$	−21920.3	−0.774148	−0.387074	−	0.922049i	$-0.626514\pi$
$929$	−21920.3	−0.774148	−0.387074	+	0.922049i	$0.626514\pi$
$930$	0	0
$931$	−2424.71	−0.0853562
$932$	0	0
$933$	12217.7	0.428712
$934$	0	0
$935$	−1562.57	−0.0546540
$936$	0	0
$937$	4891.90	0.170556	0.0852782	−	0.996357i	$-0.472822\pi$
$937$	4891.90	0.170556	0.0852782	+	0.996357i	$0.472822\pi$
$938$	0	0
$939$	−37.2072	−0.00129309
$940$	0	0
$941$	−40152.3	−1.39100	−0.695499	−	0.718527i	$-0.744817\pi$
$941$	−40152.3	−1.39100	−0.695499	+	0.718527i	$0.744817\pi$
$942$	0	0
$943$	28291.9	0.977001
$944$	0	0
$945$	−3275.26	−0.112745
$946$	0	0
$947$	16057.5	0.551002	0.275501	−	0.961301i	$-0.411156\pi$
$947$	16057.5	0.551002	0.275501	+	0.961301i	$0.411156\pi$
$948$	0	0
$949$	−31.2963	−0.00107052
$950$	0	0
$951$	−15621.4	−0.532659
$952$	0	0
$953$	−21912.3	−0.744815	−0.372408	−	0.928069i	$-0.621468\pi$
$953$	−21912.3	−0.744815	−0.372408	+	0.928069i	$0.621468\pi$
$954$	0	0
$955$	−31220.2	−1.05787
$956$	0	0
$957$	−1167.62	−0.0394397
$958$	0	0
$959$	12950.0	0.436055
$960$	0	0
$961$	43395.3	1.45666
$962$	0	0
$963$	11696.3	0.391388
$964$	0	0
$965$	23223.8	0.774717
$966$	0	0
$967$	−17618.1	−0.585895	−0.292948	−	0.956129i	$-0.594636\pi$
$967$	−17618.1	−0.585895	−0.292948	+	0.956129i	$0.594636\pi$
$968$	0	0
$969$	165.786	0.00549619
$970$	0	0
$971$	−24152.9	−0.798252	−0.399126	−	0.916896i	$-0.630686\pi$
$971$	−24152.9	−0.798252	−0.399126	+	0.916896i	$0.630686\pi$
$972$	0	0
$973$	2211.64	0.0728694
$974$	0	0
$975$	−13101.0	−0.430325
$976$	0	0
$977$	−43754.7	−1.43279	−0.716395	−	0.697695i	$-0.754209\pi$
$977$	−43754.7	−1.43279	−0.716395	+	0.697695i	$0.754209\pi$
$978$	0	0
$979$	−7226.66	−0.235919
$980$	0	0
$981$	−29790.9	−0.969572
$982$	0	0
$983$	51651.4	1.67591	0.837957	−	0.545736i	$-0.183750\pi$
$983$	51651.4	1.67591	0.837957	+	0.545736i	$0.183750\pi$
$984$	0	0
$985$	14952.0	0.483665
$986$	0	0
$987$	−5508.91	−0.177660
$988$	0	0
$989$	31910.7	1.02599
$990$	0	0
$991$	−23894.0	−0.765911	−0.382956	−	0.923767i	$-0.625094\pi$
$991$	−23894.0	−0.765911	−0.382956	+	0.923767i	$0.625094\pi$
$992$	0	0
$993$	−4524.51	−0.144593
$994$	0	0
$995$	−18201.6	−0.579929
$996$	0	0
$997$	5951.47	0.189052	0.0945261	−	0.995522i	$-0.469866\pi$
$997$	5951.47	0.189052	0.0945261	+	0.995522i	$0.469866\pi$
$998$	0	0
$999$	28640.1	0.907040

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.4.a.l.1.4		5
4.3	odd	2		29.4.a.b.1.5	✓	5
8.3	odd	2		1856.4.a.y.1.4		5
8.5	even	2		1856.4.a.bb.1.2		5
12.11	even	2		261.4.a.f.1.1		5
20.19	odd	2		725.4.a.c.1.1		5
28.27	even	2		1421.4.a.e.1.5		5
116.115	odd	2		841.4.a.b.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.4.a.b.1.5	✓	5	4.3	odd	2
261.4.a.f.1.1		5	12.11	even	2
464.4.a.l.1.4		5	1.1	even	1	trivial
725.4.a.c.1.1		5	20.19	odd	2
841.4.a.b.1.1		5	116.115	odd	2
1421.4.a.e.1.5		5	28.27	even	2
1856.4.a.y.1.4		5	8.3	odd	2
1856.4.a.bb.1.2		5	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	464.a (trivial)

\(f(q)\)	\(=\)	\(q+1.90549 q^{3} -6.52855 q^{5} -5.22706 q^{7} -23.3691 q^{9} +O(q^{10})\)	\(q+1.90549 q^{3} -6.52855 q^{5} -5.22706 q^{7} -23.3691 q^{9} +21.1299 q^{11} +83.4615 q^{13} -12.4401 q^{15} +11.3273 q^{17} +7.68096 q^{19} -9.96011 q^{21} -153.169 q^{23} -82.3780 q^{25} -95.9778 q^{27} -29.0000 q^{29} -270.530 q^{31} +40.2628 q^{33} +34.1251 q^{35} -298.404 q^{37} +159.035 q^{39} -184.710 q^{41} -208.337 q^{43} +152.566 q^{45} +553.098 q^{47} -315.678 q^{49} +21.5840 q^{51} -321.465 q^{53} -137.948 q^{55} +14.6360 q^{57} -104.930 q^{59} +464.230 q^{61} +122.152 q^{63} -544.883 q^{65} -745.813 q^{67} -291.862 q^{69} +509.252 q^{71} -0.374979 q^{73} -156.970 q^{75} -110.447 q^{77} -610.912 q^{79} +448.081 q^{81} -791.431 q^{83} -73.9507 q^{85} -55.2592 q^{87} -342.011 q^{89} -436.258 q^{91} -515.491 q^{93} -50.1455 q^{95} +601.476 q^{97} -493.787 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 8 q^{3} + 10 q^{5} - 40 q^{7} + 33 q^{9}+O(q^{10}) \) 5 * q - 8 * q^3 + 10 * q^5 - 40 * q^7 + 33 * q^9	\( 5 q - 8 q^{3} + 10 q^{5} - 40 q^{7} + 33 q^{9} - 12 q^{11} + 14 q^{13} + 74 q^{15} + 66 q^{17} - 214 q^{19} - 164 q^{23} + 207 q^{25} - 362 q^{27} - 145 q^{29} - 420 q^{31} - 576 q^{33} + 52 q^{35} + 378 q^{37} + 374 q^{39} - 1158 q^{41} + 204 q^{43} - 1506 q^{45} - 248 q^{47} - 283 q^{49} - 228 q^{51} - 554 q^{53} - 546 q^{55} + 44 q^{57} - 440 q^{59} + 618 q^{61} - 804 q^{63} - 1656 q^{65} - 1164 q^{67} - 1968 q^{69} + 692 q^{71} - 1950 q^{73} - 3074 q^{75} - 1616 q^{77} - 272 q^{79} + 1801 q^{81} - 512 q^{83} - 1628 q^{85} + 232 q^{87} + 866 q^{89} - 2580 q^{91} - 40 q^{93} - 2244 q^{95} + 1562 q^{97} + 238 q^{99}+O(q^{100}) \) 5 * q - 8 * q^3 + 10 * q^5 - 40 * q^7 + 33 * q^9 - 12 * q^11 + 14 * q^13 + 74 * q^15 + 66 * q^17 - 214 * q^19 - 164 * q^23 + 207 * q^25 - 362 * q^27 - 145 * q^29 - 420 * q^31 - 576 * q^33 + 52 * q^35 + 378 * q^37 + 374 * q^39 - 1158 * q^41 + 204 * q^43 - 1506 * q^45 - 248 * q^47 - 283 * q^49 - 228 * q^51 - 554 * q^53 - 546 * q^55 + 44 * q^57 - 440 * q^59 + 618 * q^61 - 804 * q^63 - 1656 * q^65 - 1164 * q^67 - 1968 * q^69 + 692 * q^71 - 1950 * q^73 - 3074 * q^75 - 1616 * q^77 - 272 * q^79 + 1801 * q^81 - 512 * q^83 - 1628 * q^85 + 232 * q^87 + 866 * q^89 - 2580 * q^91 - 40 * q^93 - 2244 * q^95 + 1562 * q^97 + 238 * q^99

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