LMFDB - Embedded newform 832.4.f.j.129.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [832,4,Mod(129,832)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(832, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("832.129"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 832 = 2^{6} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	832.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$49.0895891248$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{217})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 109x^{2} + 2916 $ x^4 + 109*x^2 + 2916
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.1
Root			$7.86546i$ of defining polynomial
Character	$\chi$	$=$	832.129
Dual form			832.4.f.j.129.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.86546 q^{3} -2.13454i q^{5} +3.86546i q^{7} +7.40362 q^{9} +53.1928i q^{11} +(34.8655 + 31.3273i) q^{13} +12.5201i q^{15} +43.3273 q^{17} -148.386i q^{19} -22.6727i q^{21} -122.655 q^{23} +120.444 q^{25} +114.942 q^{27} -83.8836 q^{29} +190.269i q^{31} -312.000i q^{33} +8.25098 q^{35} -131.713i q^{37} +(-204.502 - 183.749i) q^{39} -387.695i q^{41} -74.6365 q^{43} -15.8033i q^{45} +298.789i q^{47} +328.058 q^{49} -254.135 q^{51} -100.386 q^{53} +113.542 q^{55} +870.349i q^{57} +479.309i q^{59} +479.811 q^{61} +28.6184i q^{63} +(66.8694 - 74.4217i) q^{65} +415.273i q^{67} +719.426 q^{69} +293.946i q^{71} -106.727i q^{73} -706.458 q^{75} -205.614 q^{77} -906.044 q^{79} -874.084 q^{81} +22.1605i q^{83} -92.4839i q^{85} +492.016 q^{87} -665.433i q^{89} +(-121.094 + 134.771i) q^{91} -1116.02i q^{93} -316.735 q^{95} +1254.24i q^{97} +393.819i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{3} + 118 q^{9} + 110 q^{13} + 26 q^{17} - 196 q^{23} - 78 q^{25} + 666 q^{27} - 748 q^{29} - 350 q^{35} - 52 q^{39} + 438 q^{43} + 1106 q^{49} - 1046 q^{51} - 48 q^{53} - 960 q^{55} + 564 q^{61}+ \cdots + 324 q^{95}+O(q^{100}) $ 4 * q + 6 * q^3 + 118 * q^9 + 110 * q^13 + 26 * q^17 - 196 * q^23 - 78 * q^25 + 666 * q^27 - 748 * q^29 - 350 * q^35 - 52 * q^39 + 438 * q^43 + 1106 * q^49 - 1046 * q^51 - 48 * q^53 - 960 * q^55 + 564 * q^61 - 1294 * q^65 + 1876 * q^69 - 4240 * q^75 - 1176 * q^77 - 1444 * q^79 - 668 * q^81 - 4160 * q^87 - 1162 * q^91 + 324 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/832\mathbb{Z}\right)^\times$.

$n$	$261$	$703$	$769$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−5.86546			−1.12881			−0.564404	−	0.825499i	$-0.690894\pi$
$3$	−5.86546			−1.12881			−0.564404	+	0.825499i	$0.690894\pi$
$4$		0			0
$5$		−	2.13454i		−	0.190919i	−0.995433	−	0.0954595i	$-0.969568\pi$
$5$		−	2.13454i		−	0.190919i	0.995433	−	0.0954595i	$-0.0304321\pi$
$6$		0			0
$7$			3.86546i			0.208715i	0.994540	+	0.104358i	$0.0332786\pi$
$7$			3.86546i			0.208715i	−0.994540	+	0.104358i	$0.966721\pi$
$8$		0			0
$9$	7.40362			0.274208
$10$		0			0
$11$			53.1928i			1.45802i	0.684503	+	0.729010i	$0.260019\pi$
$11$			53.1928i			1.45802i	−0.684503	+	0.729010i	$0.739981\pi$
$12$		0			0
$13$	34.8655	+	31.3273i	0.743841	+	0.668356i
$13$	34.8655	+	31.3273i	0.743841	+	0.668356i
$14$		0			0
$15$			12.5201i			0.215511i
$16$		0			0
$17$	43.3273			0.618142			0.309071	−	0.951039i	$-0.399982\pi$
$17$	43.3273			0.618142			0.309071	+	0.951039i	$0.399982\pi$
$18$		0			0
$19$		−	148.386i		−	1.79168i	−0.444374	−	0.895841i	$-0.646574\pi$
$19$		−	148.386i		−	1.79168i	0.444374	−	0.895841i	$-0.353426\pi$
$20$		0			0
$21$		−	22.6727i		−	0.235599i
$22$		0			0
$23$	−122.655			−1.11197			−0.555984	−	0.831193i	$-0.687658\pi$
$23$	−122.655			−1.11197			−0.555984	+	0.831193i	$0.687658\pi$
$24$		0			0
$25$	120.444			0.963550
$26$		0			0
$27$	114.942			0.819280
$28$		0			0
$29$	−83.8836			−0.537131			−0.268565	−	0.963261i	$-0.586550\pi$
$29$	−83.8836			−0.537131			−0.268565	+	0.963261i	$0.586550\pi$
$30$		0			0
$31$			190.269i			1.10237i	0.834384	+	0.551183i	$0.185823\pi$
$31$			190.269i			1.10237i	−0.834384	+	0.551183i	$0.814177\pi$
$32$		0			0
$33$		−	312.000i		−	1.64583i
$34$		0			0
$35$	8.25098			0.0398477
$36$		0			0
$37$		−	131.713i		−	0.585228i	−0.956231	−	0.292614i	$-0.905475\pi$
$37$		−	131.713i		−	0.585228i	0.956231	−	0.292614i	$-0.0945252\pi$
$38$		0			0
$39$	−204.502	−	183.749i	−0.839654	−	0.754446i
$40$		0			0
$41$		−	387.695i		−	1.47677i	−0.674377	−	0.738387i	$-0.735588\pi$
$41$		−	387.695i		−	1.47677i	0.674377	−	0.738387i	$-0.264412\pi$
$42$		0			0
$43$	−74.6365			−0.264697			−0.132348	−	0.991203i	$-0.542252\pi$
$43$	−74.6365			−0.264697			−0.132348	+	0.991203i	$0.542252\pi$
$44$		0			0
$45$		−	15.8033i		−	0.0523516i
$46$		0			0
$47$			298.789i			0.927295i	0.886020	+	0.463648i	$0.153460\pi$
$47$			298.789i			0.927295i	−0.886020	+	0.463648i	$0.846540\pi$
$48$		0			0
$49$	328.058			0.956438
$50$		0			0
$51$	−254.135			−0.697764
$52$		0			0
$53$	−100.386			−0.260170			−0.130085	−	0.991503i	$-0.541525\pi$
$53$	−100.386			−0.260170			−0.130085	+	0.991503i	$0.541525\pi$
$54$		0			0
$55$	113.542			0.278364
$56$		0			0
$57$			870.349i			2.02247i
$58$		0			0
$59$			479.309i			1.05764i	0.848734	+	0.528820i	$0.177365\pi$
$59$			479.309i			1.05764i	−0.848734	+	0.528820i	$0.822635\pi$
$60$		0			0
$61$	479.811			1.00711			0.503553	−	0.863964i	$-0.332026\pi$
$61$	479.811			1.00711			0.503553	+	0.863964i	$0.332026\pi$
$62$		0			0
$63$			28.6184i			0.0572314i
$64$		0			0
$65$	66.8694	−	74.4217i	0.127602	−	0.142014i
$66$		0			0
$67$			415.273i			0.757219i	0.925557	+	0.378609i	$0.123598\pi$
$67$			415.273i			0.757219i	−0.925557	+	0.378609i	$0.876402\pi$
$68$		0			0
$69$	719.426			1.25520
$70$		0			0
$71$			293.946i			0.491337i	0.969354	+	0.245669i	$0.0790075\pi$
$71$			293.946i			0.491337i	−0.969354	+	0.245669i	$0.920993\pi$
$72$		0			0
$73$		−	106.727i		−	0.171116i	−0.996333	−	0.0855579i	$-0.972733\pi$
$73$		−	106.727i		−	0.171116i	0.996333	−	0.0855579i	$-0.0272673\pi$
$74$		0			0
$75$	−706.458			−1.08766
$76$		0			0
$77$	−205.614			−0.304311
$78$		0			0
$79$	−906.044			−1.29035			−0.645177	−	0.764033i	$-0.723216\pi$
$79$	−906.044			−1.29035			−0.645177	+	0.764033i	$0.723216\pi$
$80$		0			0
$81$	−874.084			−1.19902
$82$		0			0
$83$			22.1605i			0.0293064i	0.999893	+	0.0146532i	$0.00466442\pi$
$83$			22.1605i			0.0293064i	−0.999893	+	0.0146532i	$0.995336\pi$
$84$		0			0
$85$		−	92.4839i		−	0.118015i
$86$		0			0
$87$	492.016			0.606317
$88$		0			0
$89$		−	665.433i		−	0.792537i	−0.918135	−	0.396269i	$-0.870305\pi$
$89$		−	665.433i		−	0.792537i	0.918135	−	0.396269i	$-0.129695\pi$
$90$		0			0
$91$	−121.094	+	134.771i	−0.139496	+	0.155251i
$92$		0			0
$93$		−	1116.02i		−	1.24436i
$94$		0			0
$95$	−316.735			−0.342066
$96$		0			0
$97$			1254.24i			1.31287i	0.754381	+	0.656437i	$0.227937\pi$
$97$			1254.24i			1.31287i	−0.754381	+	0.656437i	$0.772063\pi$
$98$		0			0
$99$			393.819i			0.399801i
$100$		0			0
$101$	−1757.21			−1.73118			−0.865588	−	0.500757i	$-0.833055\pi$
$101$	−1757.21			−1.73118			−0.865588	+	0.500757i	$0.833055\pi$
$102$		0			0
$103$	−86.2769			−0.0825351			−0.0412676	−	0.999148i	$-0.513140\pi$
$103$	−86.2769			−0.0825351			−0.0412676	+	0.999148i	$0.513140\pi$
$104$		0			0
$105$	−48.3958			−0.0449804
$106$		0			0
$107$	−2100.93			−1.89817			−0.949087	−	0.315013i	$-0.897991\pi$
$107$	−2100.93			−1.89817			−0.949087	+	0.315013i	$0.897991\pi$
$108$		0			0
$109$			2166.87i			1.90411i	0.305920	+	0.952057i	$0.401036\pi$
$109$			2166.87i			1.90411i	−0.305920	+	0.952057i	$0.598964\pi$
$110$		0			0
$111$			772.556i			0.660611i
$112$		0			0
$113$	−356.779			−0.297017			−0.148509	−	0.988911i	$-0.547447\pi$
$113$	−356.779			−0.297017			−0.148509	+	0.988911i	$0.547447\pi$
$114$		0			0
$115$			261.811i			0.212296i
$116$		0			0
$117$	258.131	+	231.935i	0.203967	+	0.183269i
$118$		0			0
$119$			167.480i			0.129016i
$120$		0			0
$121$	−1498.47			−1.12582
$122$		0			0
$123$			2274.01i			1.66699i
$124$		0			0
$125$		−	523.909i		−	0.374879i
$126$		0			0
$127$	−997.666			−0.697075			−0.348538	−	0.937295i	$-0.613322\pi$
$127$	−997.666			−0.697075			−0.348538	+	0.937295i	$0.613322\pi$
$128$		0			0
$129$	437.777			0.298792
$130$		0			0
$131$	1680.88			1.12106			0.560530	−	0.828134i	$-0.310597\pi$
$131$	1680.88			1.12106			0.560530	+	0.828134i	$0.310597\pi$
$132$		0			0
$133$	573.578			0.373951
$134$		0			0
$135$		−	245.348i		−	0.156416i
$136$		0			0
$137$			1578.58i			0.984433i	0.870473	+	0.492217i	$0.163813\pi$
$137$			1578.58i			0.984433i	−0.870473	+	0.492217i	$0.836187\pi$
$138$		0			0
$139$	−1724.07			−1.05204			−0.526021	−	0.850472i	$-0.676316\pi$
$139$	−1724.07			−1.05204			−0.526021	+	0.850472i	$0.676316\pi$
$140$		0			0
$141$		−	1752.54i		−	1.04674i
$142$		0			0
$143$	−1666.39	+	1854.59i	−0.974477	+	1.08454i
$144$		0			0
$145$			179.053i			0.102548i
$146$		0			0
$147$	−1924.21			−1.07964
$148$		0			0
$149$			2729.67i			1.50083i	0.660970	+	0.750413i	$0.270145\pi$
$149$			2729.67i			1.50083i	−0.660970	+	0.750413i	$0.729855\pi$
$150$		0			0
$151$		−	484.316i		−	0.261014i	−0.991447	−	0.130507i	$-0.958340\pi$
$151$		−	484.316i		−	0.261014i	0.991447	−	0.130507i	$-0.0416604\pi$
$152$		0			0
$153$	320.779			0.169500
$154$		0			0
$155$	406.137			0.210463
$156$		0			0
$157$	−235.775			−0.119853			−0.0599264	−	0.998203i	$-0.519087\pi$
$157$	−235.775			−0.119853			−0.0599264	+	0.998203i	$0.519087\pi$
$158$		0			0
$159$	588.807			0.293682
$160$		0			0
$161$		−	474.116i		−	0.232085i
$162$		0			0
$163$		−	1455.22i		−	0.699274i	−0.936885	−	0.349637i	$-0.886305\pi$
$163$		−	1455.22i		−	0.699274i	0.936885	−	0.349637i	$-0.113695\pi$
$164$		0			0
$165$	−665.977			−0.314219
$166$		0			0
$167$			1630.52i			0.755528i	0.925902	+	0.377764i	$0.123307\pi$
$167$			1630.52i			0.755528i	−0.925902	+	0.377764i	$0.876693\pi$
$168$		0			0
$169$	234.201	+	2184.48i	0.106600	+	0.994302i
$170$		0			0
$171$		−	1098.59i		−	0.491294i
$172$		0			0
$173$	960.241			0.421999			0.210999	−	0.977486i	$-0.432328\pi$
$173$	960.241			0.421999			0.210999	+	0.977486i	$0.432328\pi$
$174$		0			0
$175$			465.570i			0.201108i
$176$		0			0
$177$		−	2811.37i		−	1.19387i
$178$		0			0
$179$	2100.32			0.877012			0.438506	−	0.898728i	$-0.355508\pi$
$179$	2100.32			0.877012			0.438506	+	0.898728i	$0.355508\pi$
$180$		0			0
$181$	−3385.80			−1.39041			−0.695206	−	0.718811i	$-0.744687\pi$
$181$	−3385.80			−1.39041			−0.695206	+	0.718811i	$0.744687\pi$
$182$		0			0
$183$	−2814.31			−1.13683
$184$		0			0
$185$	−281.146			−0.111731
$186$		0			0
$187$			2304.70i			0.901263i
$188$		0			0
$189$			444.303i			0.170996i
$190$		0			0
$191$	−3728.07			−1.41232			−0.706162	−	0.708050i	$-0.749575\pi$
$191$	−3728.07			−1.41232			−0.706162	+	0.708050i	$0.749575\pi$
$192$		0			0
$193$		−	2573.20i		−	0.959705i	−0.877349	−	0.479852i	$-0.840690\pi$
$193$		−	2573.20i		−	0.959705i	0.877349	−	0.479852i	$-0.159310\pi$
$194$		0			0
$195$	−392.220	+	436.518i	−0.144038	+	0.160306i
$196$		0			0
$197$			1941.54i			0.702178i	0.936342	+	0.351089i	$0.114188\pi$
$197$			1941.54i			0.702178i	−0.936342	+	0.351089i	$0.885812\pi$
$198$		0			0
$199$	2290.69			0.815994			0.407997	−	0.912983i	$-0.366227\pi$
$199$	2290.69			0.815994			0.407997	+	0.912983i	$0.366227\pi$
$200$		0			0
$201$		−	2435.77i		−	0.854755i
$202$		0			0
$203$		−	324.249i		−	0.112107i
$204$		0			0
$205$	−827.550			−0.281944
$206$		0			0
$207$	−908.088			−0.304911
$208$		0			0
$209$	7893.04			2.61231
$210$		0			0
$211$	−1175.23			−0.383440			−0.191720	−	0.981450i	$-0.561407\pi$
$211$	−1175.23			−0.383440			−0.191720	+	0.981450i	$0.561407\pi$
$212$		0			0
$213$		−	1724.13i		−	0.554625i
$214$		0			0
$215$			159.315i			0.0505357i
$216$		0			0
$217$	−735.478			−0.230080
$218$		0			0
$219$			626.003i			0.193157i
$220$		0			0
$221$	1510.63	+	1357.33i	0.459800	+	0.413139i
$222$		0			0
$223$			4350.88i			1.30653i	0.757129	+	0.653265i	$0.226601\pi$
$223$			4350.88i			1.30653i	−0.757129	+	0.653265i	$0.773399\pi$
$224$		0			0
$225$	891.720			0.264213
$226$		0			0
$227$		−	3645.85i		−	1.06601i	−0.846113	−	0.533003i	$-0.821063\pi$
$227$		−	3645.85i		−	1.06601i	0.846113	−	0.533003i	$-0.178937\pi$
$228$		0			0
$229$		−	3239.15i		−	0.934713i	−0.884069	−	0.467357i	$-0.845206\pi$
$229$		−	3239.15i		−	0.934713i	0.884069	−	0.467357i	$-0.154794\pi$
$230$		0			0
$231$	1206.02			0.343509
$232$		0			0
$233$	−2182.68			−0.613700			−0.306850	−	0.951758i	$-0.599275\pi$
$233$	−2182.68			−0.613700			−0.306850	+	0.951758i	$0.599275\pi$
$234$		0			0
$235$	637.777			0.177038
$236$		0			0
$237$	5314.36			1.45656
$238$		0			0
$239$		−	58.8048i		−	0.0159153i	−0.999968	−	0.00795767i	$-0.997467\pi$
$239$		−	58.8048i		−	0.0159153i	0.999968	−	0.00795767i	$-0.00253303\pi$
$240$		0			0
$241$			202.261i			0.0540614i	0.999635	+	0.0270307i	$0.00860518\pi$
$241$			202.261i			0.0540614i	−0.999635	+	0.0270307i	$0.991395\pi$
$242$		0			0
$243$	2023.48			0.534182
$244$		0			0
$245$		−	700.253i		−	0.182602i
$246$		0			0
$247$	4648.52	−	5173.53i	1.19748	−	1.33273i
$248$		0			0
$249$		−	129.981i		−	0.0330813i
$250$		0			0
$251$	−113.992			−0.0286658			−0.0143329	−	0.999897i	$-0.504562\pi$
$251$	−113.992			−0.0286658			−0.0143329	+	0.999897i	$0.504562\pi$
$252$		0			0
$253$		−	6524.34i		−	1.62127i
$254$		0			0
$255$			542.460i			0.133216i
$256$		0			0
$257$	−4234.73			−1.02784			−0.513920	−	0.857838i	$-0.671807\pi$
$257$	−4234.73			−1.02784			−0.513920	+	0.857838i	$0.671807\pi$
$258$		0			0
$259$	509.131			0.122146
$260$		0			0
$261$	−621.042			−0.147286
$262$		0			0
$263$	228.481			0.0535695			0.0267847	−	0.999641i	$-0.491473\pi$
$263$	228.481			0.0535695			0.0267847	+	0.999641i	$0.491473\pi$
$264$		0			0
$265$			214.277i			0.0496714i
$266$		0			0
$267$			3903.07i			0.894623i
$268$		0			0
$269$	−1219.30			−0.276365			−0.138183	−	0.990407i	$-0.544126\pi$
$269$	−1219.30			−0.276365			−0.138183	+	0.990407i	$0.544126\pi$
$270$		0			0
$271$		−	5891.13i		−	1.32052i	−0.751037	−	0.660260i	$-0.770446\pi$
$271$		−	5891.13i		−	1.32052i	0.751037	−	0.660260i	$-0.229554\pi$
$272$		0			0
$273$	710.274	−	790.494i	0.157464	−	0.175249i
$274$		0			0
$275$			6406.73i			1.40488i
$276$		0			0
$277$	6139.39			1.33170			0.665849	−	0.746087i	$-0.268069\pi$
$277$	6139.39			1.33170			0.665849	+	0.746087i	$0.268069\pi$
$278$		0			0
$279$			1408.68i			0.302278i
$280$		0			0
$281$			1852.24i			0.393221i	0.980482	+	0.196611i	$0.0629935\pi$
$281$			1852.24i			0.393221i	−0.980482	+	0.196611i	$0.937006\pi$
$282$		0			0
$283$	172.290			0.0361892			0.0180946	−	0.999836i	$-0.494240\pi$
$283$	172.290			0.0361892			0.0180946	+	0.999836i	$0.494240\pi$
$284$		0			0
$285$	1857.80			0.386127
$286$		0			0
$287$	1498.62			0.308225
$288$		0			0
$289$	−3035.75			−0.617900
$290$		0			0
$291$		−	7356.70i		−	1.48198i
$292$		0			0
$293$			5725.00i			1.14150i	0.821125	+	0.570748i	$0.193347\pi$
$293$			5725.00i			1.14150i	−0.821125	+	0.570748i	$0.806653\pi$
$294$		0			0
$295$	1023.10			0.201924
$296$		0			0
$297$			6114.07i			1.19453i
$298$		0			0
$299$	−4276.41	−	3842.44i	−0.827127	−	0.743190i
$300$		0			0
$301$		−	288.504i		−	0.0552462i
$302$		0			0
$303$	10306.8			1.95417
$304$		0			0
$305$		−	1024.18i		−	0.192276i
$306$		0			0
$307$			2115.94i			0.393364i	0.980467	+	0.196682i	$0.0630167\pi$
$307$			2115.94i			0.393364i	−0.980467	+	0.196682i	$0.936983\pi$
$308$		0			0
$309$	506.054			0.0931663
$310$		0			0
$311$	−3912.14			−0.713303			−0.356651	−	0.934238i	$-0.616082\pi$
$311$	−3912.14			−0.713303			−0.356651	+	0.934238i	$0.616082\pi$
$312$		0			0
$313$	3751.20			0.677413			0.338706	−	0.940892i	$-0.390011\pi$
$313$	3751.20			0.677413			0.338706	+	0.940892i	$0.390011\pi$
$314$		0			0
$315$	61.0871			0.0109266
$316$		0			0
$317$		−	8277.14i		−	1.46653i	−0.679942	−	0.733266i	$-0.737995\pi$
$317$		−	8277.14i		−	1.46653i	0.679942	−	0.733266i	$-0.262005\pi$
$318$		0			0
$319$		−	4462.00i		−	0.783147i
$320$		0			0
$321$	12322.9			2.14268
$322$		0			0
$323$		−	6429.14i		−	1.10751i
$324$		0			0
$325$	4199.33	+	3773.18i	0.716728	+	0.643994i
$326$		0			0
$327$		−	12709.7i		−	2.14938i
$328$		0			0
$329$	−1154.96			−0.193541
$330$		0			0
$331$			887.123i			0.147313i	0.997284	+	0.0736567i	$0.0234669\pi$
$331$			887.123i			0.147313i	−0.997284	+	0.0736567i	$0.976533\pi$
$332$		0			0
$333$		−	975.152i		−	0.160474i
$334$		0			0
$335$	886.417			0.144568
$336$		0			0
$337$	−6563.94			−1.06101			−0.530506	−	0.847681i	$-0.677998\pi$
$337$	−6563.94			−1.06101			−0.530506	+	0.847681i	$0.677998\pi$
$338$		0			0
$339$	2092.67			0.335275
$340$		0			0
$341$	−10120.9			−1.60727
$342$		0			0
$343$			2593.95i			0.408338i
$344$		0			0
$345$		−	1535.64i		−	0.239641i
$346$		0			0
$347$	8654.04			1.33883			0.669414	−	0.742890i	$-0.266545\pi$
$347$	8654.04			1.33883			0.669414	+	0.742890i	$0.266545\pi$
$348$		0			0
$349$			10173.5i			1.56039i	0.625535	+	0.780196i	$0.284881\pi$
$349$			10173.5i			1.56039i	−0.625535	+	0.780196i	$0.715119\pi$
$350$		0			0
$351$	4007.50	+	3600.82i	0.609414	+	0.547571i
$352$		0			0
$353$			921.610i			0.138958i	0.997583	+	0.0694792i	$0.0221338\pi$
$353$			921.610i			0.138958i	−0.997583	+	0.0694792i	$0.977866\pi$
$354$		0			0
$355$	627.439			0.0938056
$356$		0			0
$357$		−	982.347i		−	0.145634i
$358$		0			0
$359$			2056.77i			0.302374i	0.988505	+	0.151187i	$0.0483096\pi$
$359$			2056.77i			0.302374i	−0.988505	+	0.151187i	$0.951690\pi$
$360$		0			0
$361$	−15159.3			−2.21013
$362$		0			0
$363$	8789.21			1.27084
$364$		0			0
$365$	−227.813			−0.0326693
$366$		0			0
$367$	2211.81			0.314593			0.157296	−	0.987551i	$-0.449722\pi$
$367$	2211.81			0.314593			0.157296	+	0.987551i	$0.449722\pi$
$368$		0			0
$369$		−	2870.34i		−	0.404944i
$370$		0			0
$371$		−	388.036i		−	0.0543014i
$372$		0			0
$373$	−10546.1			−1.46396			−0.731981	−	0.681325i	$-0.761404\pi$
$373$	−10546.1			−1.46396			−0.731981	+	0.681325i	$0.761404\pi$
$374$		0			0
$375$			3072.97i			0.423167i
$376$		0			0
$377$	−2924.64	−	2627.85i	−0.399540	−	0.358994i
$378$		0			0
$379$		−	12239.0i		−	1.65877i	−0.558680	−	0.829384i	$-0.688692\pi$
$379$		−	12239.0i		−	1.65877i	0.558680	−	0.829384i	$-0.311308\pi$
$380$		0			0
$381$	5851.77			0.786864
$382$		0			0
$383$		−	7234.70i		−	0.965211i	−0.875838	−	0.482605i	$-0.839691\pi$
$383$		−	7234.70i		−	0.965211i	0.875838	−	0.482605i	$-0.160309\pi$
$384$		0			0
$385$			438.892i			0.0580988i
$386$		0			0
$387$	−552.580			−0.0725820
$388$		0			0
$389$	−11781.1			−1.53554			−0.767768	−	0.640728i	$-0.778633\pi$
$389$	−11781.1			−1.53554			−0.767768	+	0.640728i	$0.778633\pi$
$390$		0			0
$391$	−5314.29			−0.687354
$392$		0			0
$393$	−9859.12			−1.26546
$394$		0			0
$395$			1933.99i			0.246353i
$396$		0			0
$397$		−	11724.0i		−	1.48214i	−0.671425	−	0.741072i	$-0.734317\pi$
$397$		−	11724.0i		−	1.48214i	0.671425	−	0.741072i	$-0.265683\pi$
$398$		0			0
$399$	−3364.30			−0.422120
$400$		0			0
$401$			1707.99i			0.212701i	0.994329	+	0.106350i	$0.0339165\pi$
$401$			1707.99i			0.212701i	−0.994329	+	0.106350i	$0.966084\pi$
$402$		0			0
$403$	−5960.62	+	6633.82i	−0.736773	+	0.819985i
$404$		0			0
$405$			1865.77i			0.228915i
$406$		0			0
$407$	7006.17			0.853275
$408$		0			0
$409$			9607.06i			1.16146i	0.814095	+	0.580732i	$0.197234\pi$
$409$			9607.06i			1.16146i	−0.814095	+	0.580732i	$0.802766\pi$
$410$		0			0
$411$		−	9259.11i		−	1.11124i
$412$		0			0
$413$	−1852.75			−0.220746
$414$		0			0
$415$	47.3024			0.00559514
$416$		0			0
$417$	10112.5			1.18755
$418$		0			0
$419$	2708.67			0.315817			0.157908	−	0.987454i	$-0.449525\pi$
$419$	2708.67			0.315817			0.157908	+	0.987454i	$0.449525\pi$
$420$		0			0
$421$			10957.4i			1.26848i	0.773137	+	0.634239i	$0.218686\pi$
$421$			10957.4i			1.26848i	−0.773137	+	0.634239i	$0.781314\pi$
$422$		0			0
$423$			2212.12i			0.254272i
$424$		0			0
$425$	5218.50			0.595611
$426$		0			0
$427$			1854.69i			0.210199i
$428$		0			0
$429$	9774.12	−	10878.0i	1.10000	−	1.22423i
$430$		0			0
$431$			9513.13i			1.06318i	0.847001	+	0.531591i	$0.178406\pi$
$431$			9513.13i			1.06318i	−0.847001	+	0.531591i	$0.821594\pi$
$432$		0			0
$433$	7271.33			0.807016			0.403508	−	0.914976i	$-0.367791\pi$
$433$	7271.33			0.807016			0.403508	+	0.914976i	$0.367791\pi$
$434$		0			0
$435$		−	1050.23i		−	0.115758i
$436$		0			0
$437$			18200.2i			1.99229i
$438$		0			0
$439$	9615.25			1.04535			0.522677	−	0.852531i	$-0.324933\pi$
$439$	9615.25			1.04535			0.522677	+	0.852531i	$0.324933\pi$
$440$		0			0
$441$	2428.82			0.262263
$442$		0			0
$443$	4782.54			0.512924			0.256462	−	0.966554i	$-0.417443\pi$
$443$	4782.54			0.512924			0.256462	+	0.966554i	$0.417443\pi$
$444$		0			0
$445$	−1420.39			−0.151310
$446$		0			0
$447$		−	16010.7i		−	1.69414i
$448$		0			0
$449$		−	6868.72i		−	0.721949i	−0.932576	−	0.360974i	$-0.882444\pi$
$449$		−	6868.72i		−	0.721949i	0.932576	−	0.360974i	$-0.117556\pi$
$450$		0			0
$451$	20622.6			2.15317
$452$		0			0
$453$			2840.73i			0.294634i
$454$		0			0
$455$	287.674	+	258.481i	0.0296404	+	0.0266325i
$456$		0			0
$457$			8401.47i			0.859966i	0.902837	+	0.429983i	$0.141480\pi$
$457$			8401.47i			0.859966i	−0.902837	+	0.429983i	$0.858520\pi$
$458$		0			0
$459$	4980.12			0.506431
$460$		0			0
$461$			15930.5i			1.60945i	0.593650	+	0.804723i	$0.297686\pi$
$461$			15930.5i			1.60945i	−0.593650	+	0.804723i	$0.702314\pi$
$462$		0			0
$463$		−	11572.2i		−	1.16156i	−0.814060	−	0.580781i	$-0.802747\pi$
$463$		−	11572.2i		−	1.16156i	0.814060	−	0.580781i	$-0.197253\pi$
$464$		0			0
$465$	−2382.18			−0.237572
$466$		0			0
$467$	−10862.4			−1.07634			−0.538169	−	0.842837i	$-0.680884\pi$
$467$	−10862.4			−1.07634			−0.538169	+	0.842837i	$0.680884\pi$
$468$		0			0
$469$	−1605.22			−0.158043
$470$		0			0
$471$	1382.93			0.135291
$472$		0			0
$473$		−	3970.12i		−	0.385933i
$474$		0			0
$475$		−	17872.1i		−	1.72638i
$476$		0			0
$477$	−743.216			−0.0713407
$478$		0			0
$479$		−	2085.72i		−	0.198954i	−0.995040	−	0.0994769i	$-0.968283\pi$
$479$		−	2085.72i		−	0.198954i	0.995040	−	0.0994769i	$-0.0317169\pi$
$480$		0			0
$481$	4126.21	−	4592.23i	0.391141	−	0.435317i
$482$		0			0
$483$			2780.91i			0.261979i
$484$		0			0
$485$	2677.23			0.250653
$486$		0			0
$487$			11829.6i			1.10072i	0.834929	+	0.550358i	$0.185509\pi$
$487$			11829.6i			1.10072i	−0.834929	+	0.550358i	$0.814491\pi$
$488$		0			0
$489$			8535.54i			0.789347i
$490$		0			0
$491$	2700.04			0.248169			0.124085	−	0.992272i	$-0.460401\pi$
$491$	2700.04			0.248169			0.124085	+	0.992272i	$0.460401\pi$
$492$		0			0
$493$	−3634.45			−0.332023
$494$		0			0
$495$	840.622			0.0763296
$496$		0			0
$497$	−1136.24			−0.102550
$498$		0			0
$499$			3013.67i			0.270361i	0.990821	+	0.135181i	$0.0431615\pi$
$499$			3013.67i			0.270361i	−0.990821	+	0.135181i	$0.956839\pi$
$500$		0			0
$501$		−	9563.74i		−	0.852847i
$502$		0			0
$503$	−13902.0			−1.23233			−0.616163	−	0.787619i	$-0.711314\pi$
$503$	−13902.0			−1.23233			−0.616163	+	0.787619i	$0.711314\pi$
$504$		0			0
$505$			3750.83i			0.330515i
$506$		0			0
$507$	−1373.69	−	12813.0i	−0.120331	−	1.12238i
$508$		0			0
$509$			1630.66i			0.142000i	0.997476	+	0.0709999i	$0.0226190\pi$
$509$			1630.66i			0.142000i	−0.997476	+	0.0709999i	$0.977381\pi$
$510$		0			0
$511$	412.549			0.0357145
$512$		0			0
$513$		−	17055.7i		−	1.46789i
$514$		0			0
$515$			184.162i			0.0157575i
$516$		0			0
$517$	−15893.4			−1.35202
$518$		0			0
$519$	−5632.25			−0.476355
$520$		0			0
$521$	8654.86			0.727785			0.363893	−	0.931441i	$-0.381447\pi$
$521$	8654.86			0.727785			0.363893	+	0.931441i	$0.381447\pi$
$522$		0			0
$523$	−5093.00			−0.425816			−0.212908	−	0.977072i	$-0.568293\pi$
$523$	−5093.00			−0.425816			−0.212908	+	0.977072i	$0.568293\pi$
$524$		0			0
$525$		−	2730.78i		−	0.227012i
$526$		0			0
$527$			8243.85i			0.681418i
$528$		0			0
$529$	2877.15			0.236472
$530$		0			0
$531$			3548.62i			0.290013i
$532$		0			0
$533$	12145.4	−	13517.2i	0.987011	−	1.09849i
$534$		0			0
$535$			4484.52i			0.362398i
$536$		0			0
$537$	−12319.3			−0.989979
$538$		0			0
$539$			17450.3i			1.39451i
$540$		0			0
$541$		−	5004.07i		−	0.397674i	−0.980033	−	0.198837i	$-0.936284\pi$
$541$		−	5004.07i		−	0.397674i	0.980033	−	0.198837i	$-0.0637165\pi$
$542$		0			0
$543$	19859.3			1.56951
$544$		0			0
$545$	4625.27			0.363532
$546$		0			0
$547$	10856.3			0.848598			0.424299	−	0.905522i	$-0.360521\pi$
$547$	10856.3			0.848598			0.424299	+	0.905522i	$0.360521\pi$
$548$		0			0
$549$	3552.34			0.276157
$550$		0			0
$551$			12447.1i			0.962368i
$552$		0			0
$553$		−	3502.28i		−	0.269316i
$554$		0			0
$555$	1649.05			0.126123
$556$		0			0
$557$		−	23083.8i		−	1.75600i	−0.478658	−	0.878001i	$-0.658877\pi$
$557$		−	23083.8i		−	1.75600i	0.478658	−	0.878001i	$-0.341123\pi$
$558$		0			0
$559$	−2602.24	−	2338.16i	−0.196892	−	0.176912i
$560$		0			0
$561$		−	13518.1i		−	1.01735i
$562$		0			0
$563$	−15495.9			−1.15999			−0.579996	−	0.814619i	$-0.696946\pi$
$563$	−15495.9			−1.15999			−0.579996	+	0.814619i	$0.696946\pi$
$564$		0			0
$565$			761.559i			0.0567062i
$566$		0			0
$567$		−	3378.74i		−	0.250253i
$568$		0			0
$569$	−18165.5			−1.33838			−0.669188	−	0.743094i	$-0.733358\pi$
$569$	−18165.5			−1.33838			−0.669188	+	0.743094i	$0.733358\pi$
$570$		0			0
$571$	−8289.75			−0.607557			−0.303779	−	0.952743i	$-0.598248\pi$
$571$	−8289.75			−0.607557			−0.303779	+	0.952743i	$0.598248\pi$
$572$		0			0
$573$	21866.9			1.59424
$574$		0			0
$575$	−14773.0			−1.07144
$576$		0			0
$577$		−	5336.46i		−	0.385026i	−0.981294	−	0.192513i	$-0.938336\pi$
$577$		−	5336.46i		−	0.385026i	0.981294	−	0.192513i	$-0.0616637\pi$
$578$		0			0
$579$			15093.0i			1.08332i
$580$		0			0
$581$	−85.6604			−0.00611668
$582$		0			0
$583$		−	5339.78i		−	0.379333i
$584$		0			0
$585$	495.075	−	550.990i	0.0349895	−	0.0389413i
$586$		0			0
$587$			6877.72i			0.483601i	0.970326	+	0.241800i	$0.0777379\pi$
$587$			6877.72i			0.483601i	−0.970326	+	0.241800i	$0.922262\pi$
$588$		0			0
$589$	28233.2			1.97509
$590$		0			0
$591$		−	11388.0i		−	0.792624i
$592$		0			0
$593$			8358.37i			0.578814i	0.957206	+	0.289407i	$0.0934582\pi$
$593$			8358.37i			0.578814i	−0.957206	+	0.289407i	$0.906542\pi$
$594$		0			0
$595$	357.493			0.0246315
$596$		0			0
$597$	−13436.0			−0.921101
$598$		0			0
$599$	−14896.0			−1.01609			−0.508043	−	0.861332i	$-0.669631\pi$
$599$	−14896.0			−1.01609			−0.508043	+	0.861332i	$0.669631\pi$
$600$		0			0
$601$	7661.60			0.520005			0.260003	−	0.965608i	$-0.416277\pi$
$601$	7661.60			0.520005			0.260003	+	0.965608i	$0.416277\pi$
$602$		0			0
$603$			3074.52i			0.207636i
$604$		0			0
$605$			3198.54i			0.214941i
$606$		0			0
$607$	22718.2			1.51912			0.759558	−	0.650440i	$-0.225415\pi$
$607$	22718.2			1.51912			0.759558	+	0.650440i	$0.225415\pi$
$608$		0			0
$609$			1901.87i			0.126548i
$610$		0			0
$611$	−9360.26	+	10417.4i	−0.619763	+	0.689761i
$612$		0			0
$613$		−	12080.9i		−	0.795995i	−0.917387	−	0.397997i	$-0.869705\pi$
$613$		−	12080.9i		−	0.795995i	0.917387	−	0.397997i	$-0.130295\pi$
$614$		0			0
$615$	4853.96			0.318261
$616$		0			0
$617$			21104.3i			1.37703i	0.725223	+	0.688514i	$0.241737\pi$
$617$			21104.3i			1.37703i	−0.725223	+	0.688514i	$0.758263\pi$
$618$		0			0
$619$			12516.1i			0.812706i	0.913716	+	0.406353i	$0.133200\pi$
$619$			12516.1i			0.812706i	−0.913716	+	0.406353i	$0.866800\pi$
$620$		0			0
$621$	−14098.1			−0.911013
$622$		0			0
$623$	2572.21			0.165415
$624$		0			0
$625$	13937.2			0.891978
$626$		0			0
$627$	−46296.3			−2.94880
$628$		0			0
$629$		−	5706.76i		−	0.361754i
$630$		0			0
$631$		−	10069.7i		−	0.635288i	−0.948210	−	0.317644i	$-0.897108\pi$
$631$		−	10069.7i		−	0.635288i	0.948210	−	0.317644i	$-0.102892\pi$
$632$		0			0
$633$	6893.24			0.432831
$634$		0			0
$635$			2129.56i			0.133085i
$636$		0			0
$637$	11437.9	+	10277.2i	0.711438	+	0.639241i
$638$		0			0
$639$			2176.26i			0.134729i
$640$		0			0
$641$	3388.87			0.208818			0.104409	−	0.994534i	$-0.466705\pi$
$641$	3388.87			0.208818			0.104409	+	0.994534i	$0.466705\pi$
$642$		0			0
$643$		−	24633.3i		−	1.51080i	−0.655264	−	0.755400i	$-0.727443\pi$
$643$		−	24633.3i		−	1.51080i	0.655264	−	0.755400i	$-0.272557\pi$
$644$		0			0
$645$		−	934.453i		−	0.0570451i
$646$		0			0
$647$	−1908.12			−0.115945			−0.0579723	−	0.998318i	$-0.518463\pi$
$647$	−1908.12			−0.115945			−0.0579723	+	0.998318i	$0.518463\pi$
$648$		0			0
$649$	−25495.8			−1.54206
$650$		0			0
$651$	4313.91			0.259717
$652$		0			0
$653$	10331.4			0.619141			0.309570	−	0.950877i	$-0.399815\pi$
$653$	10331.4			0.619141			0.309570	+	0.950877i	$0.399815\pi$
$654$		0			0
$655$		−	3587.90i		−	0.214032i
$656$		0			0
$657$		−	790.166i		−	0.0469213i
$658$		0			0
$659$	−19039.8			−1.12547			−0.562737	−	0.826636i	$-0.690251\pi$
$659$	−19039.8			−1.12547			−0.562737	+	0.826636i	$0.690251\pi$
$660$		0			0
$661$			28643.0i			1.68545i	0.538343	+	0.842726i	$0.319050\pi$
$661$			28643.0i			1.68545i	−0.538343	+	0.842726i	$0.680950\pi$
$662$		0			0
$663$	−8860.52	−	7961.35i	−0.519026	−	0.466355i
$664$		0			0
$665$		−	1224.33i		−	0.0713945i
$666$		0			0
$667$	10288.7			0.597272
$668$		0			0
$669$		−	25519.9i		−	1.47482i
$670$		0			0
$671$			25522.5i			1.46838i
$672$		0			0
$673$	18372.8			1.05233			0.526166	−	0.850382i	$-0.323629\pi$
$673$	18372.8			1.05233			0.526166	+	0.850382i	$0.323629\pi$
$674$		0			0
$675$	13844.0			0.789417
$676$		0			0
$677$	12837.4			0.728779			0.364389	−	0.931247i	$-0.381278\pi$
$677$	12837.4			0.728779			0.364389	+	0.931247i	$0.381278\pi$
$678$		0			0
$679$	−4848.22			−0.274017
$680$		0			0
$681$			21384.6i			1.20332i
$682$		0			0
$683$			12224.7i			0.684871i	0.939541	+	0.342436i	$0.111252\pi$
$683$			12224.7i			0.684871i	−0.939541	+	0.342436i	$0.888748\pi$
$684$		0			0
$685$	3369.55			0.187947
$686$		0			0
$687$			18999.1i			1.05511i
$688$		0			0
$689$	−3499.99	−	3144.81i	−0.193525	−	0.173886i
$690$		0			0
$691$			7662.99i			0.421873i	0.977500	+	0.210936i	$0.0676513\pi$
$691$			7662.99i			0.421873i	−0.977500	+	0.210936i	$0.932349\pi$
$692$		0			0
$693$	−1522.29			−0.0834446
$694$		0			0
$695$			3680.10i			0.200855i
$696$		0			0
$697$		−	16797.8i		−	0.912856i
$698$		0			0
$699$	12802.4			0.692750
$700$		0			0
$701$	−15215.8			−0.819818			−0.409909	−	0.912126i	$-0.634440\pi$
$701$	−15215.8			−0.819818			−0.409909	+	0.912126i	$0.634440\pi$
$702$		0			0
$703$	−19544.3			−1.04854
$704$		0			0
$705$	−3740.86			−0.199842
$706$		0			0
$707$		−	6792.42i		−	0.361323i
$708$		0			0
$709$			15410.9i			0.816314i	0.912912	+	0.408157i	$0.133828\pi$
$709$			15410.9i			0.816314i	−0.912912	+	0.408157i	$0.866172\pi$
$710$		0			0
$711$	−6708.01			−0.353825
$712$		0			0
$713$		−	23337.4i		−	1.22579i
$714$		0			0
$715$	3958.70	+	3556.97i	0.207059	+	0.186046i
$716$		0			0
$717$			344.917i			0.0179654i
$718$		0			0
$719$	−1155.79			−0.0599494			−0.0299747	−	0.999551i	$-0.509543\pi$
$719$	−1155.79			−0.0599494			−0.0299747	+	0.999551i	$0.509543\pi$
$720$		0			0
$721$		−	333.500i		−	0.0172263i
$722$		0			0
$723$		−	1186.36i		−	0.0610249i
$724$		0			0
$725$	−10103.2			−0.517552
$726$		0			0
$727$	−18048.8			−0.920759			−0.460379	−	0.887722i	$-0.652287\pi$
$727$	−18048.8			−0.920759			−0.460379	+	0.887722i	$0.652287\pi$
$728$		0			0
$729$	11731.6			0.596029
$730$		0			0
$731$	−3233.80			−0.163620
$732$		0			0
$733$		−	7855.68i		−	0.395847i	−0.980217	−	0.197924i	$-0.936580\pi$
$733$		−	7855.68i		−	0.395847i	0.980217	−	0.197924i	$-0.0634198\pi$
$734$		0			0
$735$			4107.31i			0.206123i
$736$		0			0
$737$	−22089.5			−1.10404
$738$		0			0
$739$		−	302.801i		−	0.0150727i	−0.999972	−	0.00753635i	$-0.997601\pi$
$739$		−	302.801i		−	0.0150727i	0.999972	−	0.00753635i	$-0.00239892\pi$
$740$		0			0
$741$	−27265.7	+	30345.1i	−1.35173	+	1.50439i
$742$		0			0
$743$		−	17199.9i		−	0.849265i	−0.905366	−	0.424633i	$-0.860403\pi$
$743$		−	17199.9i		−	0.849265i	0.905366	−	0.424633i	$-0.139597\pi$
$744$		0			0
$745$	5826.58			0.286536
$746$		0			0
$747$			164.068i			0.00803604i
$748$		0			0
$749$		−	8121.07i		−	0.396178i
$750$		0			0
$751$	−12999.7			−0.631644			−0.315822	−	0.948819i	$-0.602280\pi$
$751$	−12999.7			−0.631644			−0.315822	+	0.948819i	$0.602280\pi$
$752$		0			0
$753$	668.616			0.0323582
$754$		0			0
$755$	−1033.79			−0.0498325
$756$		0			0
$757$	−6046.81			−0.290324			−0.145162	−	0.989408i	$-0.546370\pi$
$757$	−6046.81			−0.290324			−0.145162	+	0.989408i	$0.546370\pi$
$758$		0			0
$759$			38268.2i			1.83010i
$760$		0			0
$761$			26902.3i			1.28148i	0.767758	+	0.640740i	$0.221372\pi$
$761$			26902.3i			1.28148i	−0.767758	+	0.640740i	$0.778628\pi$
$762$		0			0
$763$	−8375.95			−0.397418
$764$		0			0
$765$		−	684.715i		−	0.0323607i
$766$		0			0
$767$	−15015.5	+	16711.3i	−0.706880	+	0.786716i
$768$		0			0
$769$		−	2068.29i		−	0.0969887i	−0.998823	−	0.0484943i	$-0.984558\pi$
$769$		−	2068.29i		−	0.0969887i	0.998823	−	0.0484943i	$-0.0154423\pi$
$770$		0			0
$771$	24838.6			1.16024
$772$		0			0
$773$		−	8389.72i		−	0.390371i	−0.980766	−	0.195186i	$-0.937469\pi$
$773$		−	8389.72i		−	0.390371i	0.980766	−	0.195186i	$-0.0625310\pi$
$774$		0			0
$775$			22916.7i			1.06218i
$776$		0			0
$777$	−2986.29			−0.137880
$778$		0			0
$779$	−57528.3			−2.64591
$780$		0			0
$781$	−15635.8			−0.716379
$782$		0			0
$783$	−9641.73			−0.440060
$784$		0			0
$785$			503.271i			0.0228822i
$786$		0			0
$787$			21458.3i			0.971928i	0.873979	+	0.485964i	$0.161531\pi$
$787$			21458.3i			0.971928i	−0.873979	+	0.485964i	$0.838469\pi$
$788$		0			0
$789$	−1340.15			−0.0604697
$790$		0			0
$791$		−	1379.11i		−	0.0619920i
$792$		0			0
$793$	16728.8	+	15031.2i	0.749128	+	0.673106i
$794$		0			0
$795$		−	1256.83i		−	0.0560695i
$796$		0			0
$797$	−6049.72			−0.268873			−0.134437	−	0.990922i	$-0.542922\pi$
$797$	−6049.72			−0.268873			−0.134437	+	0.990922i	$0.542922\pi$
$798$		0			0
$799$			12945.7i			0.573200i
$800$		0			0
$801$		−	4926.62i		−	0.217320i
$802$		0			0
$803$	5677.10			0.249490
$804$		0			0
$805$	−1012.02			−0.0443094
$806$		0			0
$807$	7151.78			0.311964
$808$		0			0
$809$	−33364.1			−1.44996			−0.724981	−	0.688769i	$-0.758152\pi$
$809$	−33364.1			−1.44996			−0.724981	+	0.688769i	$0.758152\pi$
$810$		0			0
$811$			2210.77i			0.0957221i	0.998854	+	0.0478611i	$0.0152405\pi$
$811$			2210.77i			0.0957221i	−0.998854	+	0.0478611i	$0.984760\pi$
$812$		0			0
$813$			34554.2i			1.49061i
$814$		0			0
$815$	−3106.23			−0.133505
$816$		0			0
$817$			11075.0i			0.474253i
$818$		0			0
$819$	−896.537	+	997.794i	−0.0382510	+	0.0425711i
$820$		0			0
$821$			25402.9i			1.07986i	0.841709	+	0.539932i	$0.181550\pi$
$821$			25402.9i			1.07986i	−0.841709	+	0.539932i	$0.818450\pi$
$822$		0			0
$823$	42030.9			1.78020			0.890100	−	0.455765i	$-0.150634\pi$
$823$	42030.9			1.78020			0.890100	+	0.455765i	$0.150634\pi$
$824$		0			0
$825$		−	37578.4i		−	1.58583i
$826$		0			0
$827$			27713.0i			1.16527i	0.812735	+	0.582633i	$0.197978\pi$
$827$			27713.0i			1.16527i	−0.812735	+	0.582633i	$0.802022\pi$
$828$		0			0
$829$	−23579.7			−0.987886			−0.493943	−	0.869494i	$-0.664445\pi$
$829$	−23579.7			−0.987886			−0.493943	+	0.869494i	$0.664445\pi$
$830$		0			0
$831$	−36010.4			−1.50323
$832$		0			0
$833$	14213.9			0.591214
$834$		0			0
$835$	3480.41			0.144245
$836$		0			0
$837$			21869.9i			0.903146i
$838$		0			0
$839$		−	8648.03i		−	0.355856i	−0.984044	−	0.177928i	$-0.943061\pi$
$839$		−	8648.03i		−	0.355856i	0.984044	−	0.177928i	$-0.0569394\pi$
$840$		0			0
$841$	−17352.5			−0.711491
$842$		0			0
$843$		−	10864.2i		−	0.443872i
$844$		0			0
$845$	4662.86	−	499.911i	0.189831	−	0.0203520i
$846$		0			0
$847$		−	5792.27i		−	0.234976i
$848$		0			0
$849$	−1010.56			−0.0408507
$850$		0			0
$851$			16155.2i			0.650755i
$852$		0			0
$853$		−	23117.7i		−	0.927942i	−0.885851	−	0.463971i	$-0.846424\pi$
$853$		−	23117.7i		−	0.927942i	0.885851	−	0.463971i	$-0.153576\pi$
$854$		0			0
$855$	−2344.98			−0.0937974
$856$		0			0
$857$	−3246.65			−0.129409			−0.0647045	−	0.997904i	$-0.520610\pi$
$857$	−3246.65			−0.129409			−0.0647045	+	0.997904i	$0.520610\pi$
$858$		0			0
$859$	−29452.5			−1.16986			−0.584928	−	0.811085i	$-0.698877\pi$
$859$	−29452.5			−1.16986			−0.584928	+	0.811085i	$0.698877\pi$
$860$		0			0
$861$	−8790.09			−0.347927
$862$		0			0
$863$		−	15485.4i		−	0.610809i	−0.952223	−	0.305405i	$-0.901208\pi$
$863$		−	15485.4i		−	0.610809i	0.952223	−	0.305405i	$-0.0987917\pi$
$864$		0			0
$865$		−	2049.67i		−	0.0805676i
$866$		0			0
$867$	17806.0			0.697491
$868$		0			0
$869$		−	48195.0i		−	1.88136i
$870$		0			0
$871$	−13009.4	+	14478.7i	−0.506092	+	0.563251i
$872$		0			0
$873$			9285.92i			0.360001i
$874$		0			0
$875$	2025.15			0.0782430
$876$		0			0
$877$			13702.1i			0.527578i	0.964580	+	0.263789i	$0.0849723\pi$
$877$			13702.1i			0.527578i	−0.964580	+	0.263789i	$0.915028\pi$
$878$		0			0
$879$		−	33579.8i		−	1.28853i
$880$		0			0
$881$	−16868.6			−0.645084			−0.322542	−	0.946555i	$-0.604537\pi$
$881$	−16868.6			−0.645084			−0.322542	+	0.946555i	$0.604537\pi$
$882$		0			0
$883$	15162.4			0.577867			0.288934	−	0.957349i	$-0.406699\pi$
$883$	15162.4			0.577867			0.288934	+	0.957349i	$0.406699\pi$
$884$		0			0
$885$	−6000.98			−0.227933
$886$		0			0
$887$	−13173.9			−0.498686			−0.249343	−	0.968415i	$-0.580215\pi$
$887$	−13173.9			−0.498686			−0.249343	+	0.968415i	$0.580215\pi$
$888$		0			0
$889$		−	3856.44i		−	0.145490i
$890$		0			0
$891$		−	46494.9i		−	1.74819i
$892$		0			0
$893$	44336.0			1.66142
$894$		0			0
$895$		−	4483.21i		−	0.167438i
$896$		0			0
$897$	25083.1	+	22537.7i	0.933668	+	0.838919i
$898$		0			0
$899$		−	15960.4i		−	0.592114i
$900$		0			0
$901$	−4349.43			−0.160822
$902$		0			0
$903$			1692.21i			0.0623624i
$904$		0			0
$905$			7227.12i			0.265456i
$906$		0			0
$907$	32190.1			1.17845			0.589225	−	0.807969i	$-0.299433\pi$
$907$	32190.1			1.17845			0.589225	+	0.807969i	$0.299433\pi$
$908$		0			0
$909$	−13009.7			−0.474703
$910$		0			0
$911$	13646.0			0.496281			0.248141	−	0.968724i	$-0.420180\pi$
$911$	13646.0			0.496281			0.248141	+	0.968724i	$0.420180\pi$
$912$		0			0
$913$	−1178.78			−0.0427293
$914$		0			0
$915$			6007.26i			0.217043i
$916$		0			0
$917$			6497.36i			0.233982i
$918$		0			0
$919$	34923.6			1.25356			0.626780	−	0.779196i	$-0.284372\pi$
$919$	34923.6			1.25356			0.626780	+	0.779196i	$0.284372\pi$
$920$		0			0
$921$		−	12410.9i		−	0.444033i
$922$		0			0
$923$	−9208.52	+	10248.6i	−0.328388	+	0.365477i
$924$		0			0
$925$		−	15864.0i		−	0.563897i
$926$		0			0
$927$	−638.761			−0.0226318
$928$		0			0
$929$			4287.49i			0.151419i	0.997130	+	0.0757094i	$0.0241221\pi$
$929$			4287.49i			0.151419i	−0.997130	+	0.0757094i	$0.975878\pi$
$930$		0			0
$931$		−	48679.1i		−	1.71363i
$932$		0			0
$933$	22946.5			0.805182
$934$		0			0
$935$	4919.47			0.172068
$936$		0			0
$937$	−14174.1			−0.494182			−0.247091	−	0.968992i	$-0.579475\pi$
$937$	−14174.1			−0.494182			−0.247091	+	0.968992i	$0.579475\pi$
$938$		0			0
$939$	−22002.5			−0.764669
$940$		0			0
$941$			11317.1i			0.392058i	0.980598	+	0.196029i	$0.0628047\pi$
$941$			11317.1i			0.392058i	−0.980598	+	0.196029i	$0.937195\pi$
$942$		0			0
$943$			47552.5i			1.64212i
$944$		0			0
$945$	948.382			0.0326464
$946$		0			0
$947$			40514.3i			1.39022i	0.718904	+	0.695110i	$0.244644\pi$
$947$			40514.3i			1.39022i	−0.718904	+	0.695110i	$0.755356\pi$
$948$		0			0
$949$	3343.47	−	3721.09i	0.114366	−	0.127283i
$950$		0			0
$951$			48549.3i			1.65543i
$952$		0			0
$953$	49479.9			1.68186			0.840930	−	0.541144i	$-0.182009\pi$
$953$	49479.9			1.68186			0.840930	+	0.541144i	$0.182009\pi$
$954$		0			0
$955$			7957.72i			0.269640i
$956$		0			0
$957$			26171.7i			0.884023i
$958$		0			0
$959$	−6101.95			−0.205466
$960$		0			0
$961$	−6411.32			−0.215210
$962$		0			0
$963$	−15554.5			−0.520495
$964$		0			0
$965$	−5492.60			−0.183226
$966$		0			0
$967$		−	46779.9i		−	1.55568i	−0.628465	−	0.777838i	$-0.716316\pi$
$967$		−	46779.9i		−	1.55568i	0.628465	−	0.777838i	$-0.283684\pi$
$968$		0			0
$969$			37709.9i			1.25017i
$970$		0			0
$971$	−24378.5			−0.805710			−0.402855	−	0.915264i	$-0.631982\pi$
$971$	−24378.5			−0.805710			−0.402855	+	0.915264i	$0.631982\pi$
$972$		0			0
$973$		−	6664.32i		−	0.219577i
$974$		0			0
$975$	−24631.0	−	22131.4i	−0.809049	−	0.726946i
$976$		0			0
$977$			24237.6i			0.793685i	0.917887	+	0.396842i	$0.129894\pi$
$977$			24237.6i			0.793685i	−0.917887	+	0.396842i	$0.870106\pi$
$978$		0			0
$979$	35396.2			1.15554
$980$		0			0
$981$			16042.7i			0.522124i
$982$		0			0
$983$		−	32953.2i		−	1.06922i	−0.845098	−	0.534611i	$-0.820458\pi$
$983$		−	32953.2i		−	1.06922i	0.845098	−	0.534611i	$-0.179542\pi$
$984$		0			0
$985$	4144.29			0.134059
$986$		0			0
$987$	6774.36			0.218470
$988$		0			0
$989$	9154.51			0.294334
$990$		0			0
$991$	−30254.8			−0.969804			−0.484902	−	0.874569i	$-0.661145\pi$
$991$	−30254.8			−0.969804			−0.484902	+	0.874569i	$0.661145\pi$
$992$		0			0
$993$		−	5203.39i		−	0.166288i
$994$		0			0
$995$		−	4889.57i		−	0.155789i
$996$		0			0
$997$	4794.46			0.152299			0.0761495	−	0.997096i	$-0.475737\pi$
$997$	4794.46			0.152299			0.0761495	+	0.997096i	$0.475737\pi$
$998$		0			0
$999$		−	15139.3i		−	0.479466i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	832.4.f.j.129.1		4
4.3	odd	2		832.4.f.h.129.3		4
8.3	odd	2		208.4.f.d.129.2		4
8.5	even	2		26.4.b.a.25.4	yes	4
13.12	even	2	inner	832.4.f.j.129.2		4
24.5	odd	2		234.4.b.b.181.2		4
40.13	odd	4		650.4.c.f.649.3		4
40.29	even	2		650.4.d.d.51.1		4
40.37	odd	4		650.4.c.e.649.2		4
52.51	odd	2		832.4.f.h.129.4		4
104.5	odd	4		338.4.a.i.1.2		2
104.21	odd	4		338.4.a.f.1.2		2
104.29	even	6		338.4.e.g.147.1		8
104.37	odd	12		338.4.c.i.191.1		4
104.45	odd	12		338.4.c.h.315.1		4
104.51	odd	2		208.4.f.d.129.1		4
104.61	even	6		338.4.e.g.23.3		8
104.69	even	6		338.4.e.g.23.1		8
104.77	even	2		26.4.b.a.25.2	✓	4
104.85	odd	12		338.4.c.i.315.1		4
104.93	odd	12		338.4.c.h.191.1		4
104.101	even	6		338.4.e.g.147.3		8
312.77	odd	2		234.4.b.b.181.3		4
520.77	odd	4		650.4.c.f.649.2		4
520.389	even	2		650.4.d.d.51.3		4
520.493	odd	4		650.4.c.e.649.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.4.b.a.25.2	✓	4	104.77	even	2
26.4.b.a.25.4	yes	4	8.5	even	2
208.4.f.d.129.1		4	104.51	odd	2
208.4.f.d.129.2		4	8.3	odd	2
234.4.b.b.181.2		4	24.5	odd	2
234.4.b.b.181.3		4	312.77	odd	2
338.4.a.f.1.2		2	104.21	odd	4
338.4.a.i.1.2		2	104.5	odd	4
338.4.c.h.191.1		4	104.93	odd	12
338.4.c.h.315.1		4	104.45	odd	12
338.4.c.i.191.1		4	104.37	odd	12
338.4.c.i.315.1		4	104.85	odd	12
338.4.e.g.23.1		8	104.69	even	6
338.4.e.g.23.3		8	104.61	even	6
338.4.e.g.147.1		8	104.29	even	6
338.4.e.g.147.3		8	104.101	even	6
650.4.c.e.649.2		4	40.37	odd	4
650.4.c.e.649.3		4	520.493	odd	4
650.4.c.f.649.2		4	520.77	odd	4
650.4.c.f.649.3		4	40.13	odd	4
650.4.d.d.51.1		4	40.29	even	2
650.4.d.d.51.3		4	520.389	even	2
832.4.f.h.129.3		4	4.3	odd	2
832.4.f.h.129.4		4	52.51	odd	2
832.4.f.j.129.1		4	1.1	even	1	trivial
832.4.f.j.129.2		4	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 832 = 2^{6} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	832.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.86546 q^{3} -2.13454i q^{5} +3.86546i q^{7} +7.40362 q^{9} +53.1928i q^{11} +(34.8655 + 31.3273i) q^{13} +12.5201i q^{15} +43.3273 q^{17} -148.386i q^{19} -22.6727i q^{21} -122.655 q^{23} +120.444 q^{25} +114.942 q^{27} -83.8836 q^{29} +190.269i q^{31} -312.000i q^{33} +8.25098 q^{35} -131.713i q^{37} +(-204.502 - 183.749i) q^{39} -387.695i q^{41} -74.6365 q^{43} -15.8033i q^{45} +298.789i q^{47} +328.058 q^{49} -254.135 q^{51} -100.386 q^{53} +113.542 q^{55} +870.349i q^{57} +479.309i q^{59} +479.811 q^{61} +28.6184i q^{63} +(66.8694 - 74.4217i) q^{65} +415.273i q^{67} +719.426 q^{69} +293.946i q^{71} -106.727i q^{73} -706.458 q^{75} -205.614 q^{77} -906.044 q^{79} -874.084 q^{81} +22.1605i q^{83} -92.4839i q^{85} +492.016 q^{87} -665.433i q^{89} +(-121.094 + 134.771i) q^{91} -1116.02i q^{93} -316.735 q^{95} +1254.24i q^{97} +393.819i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{3} + 118 q^{9} + 110 q^{13} + 26 q^{17} - 196 q^{23} - 78 q^{25} + 666 q^{27} - 748 q^{29} - 350 q^{35} - 52 q^{39} + 438 q^{43} + 1106 q^{49} - 1046 q^{51} - 48 q^{53} - 960 q^{55} + 564 q^{61}+ \cdots + 324 q^{95}+O(q^{100}) \) 4 * q + 6 * q^3 + 118 * q^9 + 110 * q^13 + 26 * q^17 - 196 * q^23 - 78 * q^25 + 666 * q^27 - 748 * q^29 - 350 * q^35 - 52 * q^39 + 438 * q^43 + 1106 * q^49 - 1046 * q^51 - 48 * q^53 - 960 * q^55 + 564 * q^61 - 1294 * q^65 + 1876 * q^69 - 4240 * q^75 - 1176 * q^77 - 1444 * q^79 - 668 * q^81 - 4160 * q^87 - 1162 * q^91 + 324 * q^95

Introduction

L-functions

Modular forms

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Database

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