LMFDB - Embedded newform 4536.2.k.a.3401.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4536,2,Mod(3401,4536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4536, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4536.3401");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4536 = 2^{3} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4536.k (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$36.2201423569$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3401.10
Character	$\chi$	$=$	4536.3401
Dual form			4536.2.k.a.3401.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.53716 q^{5} +(-2.50698 - 0.845621i) q^{7} +O(q^{10})$	$q-2.53716 q^{5} +(-2.50698 - 0.845621i) q^{7} -6.56241i q^{11} -5.93389i q^{13} +3.52698 q^{17} +0.261355i q^{19} -3.34825i q^{23} +1.43717 q^{25} -1.16408i q^{29} -1.95560i q^{31} +(6.36059 + 2.14547i) q^{35} -1.16528 q^{37} +5.71538 q^{41} -5.34728 q^{43} +7.59534 q^{47} +(5.56985 + 4.23990i) q^{49} -3.54792i q^{53} +16.6499i q^{55} -10.9580 q^{59} -6.39356i q^{61} +15.0552i q^{65} -9.08328 q^{67} -10.5889i q^{71} -2.72923i q^{73} +(-5.54931 + 16.4518i) q^{77} +1.30537 q^{79} +9.06482 q^{83} -8.94851 q^{85} +14.2599 q^{89} +(-5.01782 + 14.8761i) q^{91} -0.663100i q^{95} +11.4972i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q + 48 q^{25} - 24 q^{43} - 12 q^{49} + 24 q^{79} - 12 q^{91}+O(q^{100}) $ 48 * q + 48 * q^25 - 24 * q^43 - 12 * q^49 + 24 * q^79 - 12 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4536\mathbb{Z}\right)^\times$.

$n$	$1135$	$2269$	$2593$	$3809$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.53716			−1.13465			−0.567326	−	0.823494i	$-0.692022\pi$
$5$	−2.53716			−1.13465			−0.567326	+	0.823494i	$0.692022\pi$
$6$		0			0
$7$	−2.50698	−	0.845621i	−0.947548	−	0.319615i
$7$	−2.50698	−	0.845621i	−0.947548	−	0.319615i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	6.56241i		−	1.97864i	−0.145757	−	0.989320i	$-0.546562\pi$
$11$		−	6.56241i		−	1.97864i	0.145757	−	0.989320i	$-0.453438\pi$
$12$		0			0
$13$		−	5.93389i		−	1.64576i	−0.568212	−	0.822882i	$-0.692365\pi$
$13$		−	5.93389i		−	1.64576i	0.568212	−	0.822882i	$-0.307635\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.52698			0.855419			0.427709	−	0.903916i	$-0.359321\pi$
$17$	3.52698			0.855419			0.427709	+	0.903916i	$0.359321\pi$
$18$		0			0
$19$			0.261355i			0.0599590i	0.999551	+	0.0299795i	$0.00954420\pi$
$19$			0.261355i			0.0599590i	−0.999551	+	0.0299795i	$0.990456\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	3.34825i		−	0.698158i	−0.937093	−	0.349079i	$-0.886495\pi$
$23$		−	3.34825i		−	0.698158i	0.937093	−	0.349079i	$-0.113505\pi$
$24$		0			0
$25$	1.43717			0.287434
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	1.16408i		−	0.216165i	−0.994142	−	0.108082i	$-0.965529\pi$
$29$		−	1.16408i		−	0.216165i	0.994142	−	0.108082i	$-0.0344710\pi$
$30$		0			0
$31$		−	1.95560i		−	0.351236i	−0.984458	−	0.175618i	$-0.943808\pi$
$31$		−	1.95560i		−	0.351236i	0.984458	−	0.175618i	$-0.0561924\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.36059	+	2.14547i	1.07514	+	0.362651i
$36$		0			0
$37$	−1.16528			−0.191570			−0.0957851	−	0.995402i	$-0.530536\pi$
$37$	−1.16528			−0.191570			−0.0957851	+	0.995402i	$0.530536\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.71538			0.892592			0.446296	−	0.894885i	$-0.352743\pi$
$41$	5.71538			0.892592			0.446296	+	0.894885i	$0.352743\pi$
$42$		0			0
$43$	−5.34728			−0.815453			−0.407726	−	0.913104i	$-0.633678\pi$
$43$	−5.34728			−0.815453			−0.407726	+	0.913104i	$0.633678\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.59534			1.10789			0.553947	−	0.832552i	$-0.313121\pi$
$47$	7.59534			1.10789			0.553947	+	0.832552i	$0.313121\pi$
$48$		0			0
$49$	5.56985	+	4.23990i	0.795693	+	0.605700i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	3.54792i		−	0.487344i	−0.969858	−	0.243672i	$-0.921648\pi$
$53$		−	3.54792i		−	0.487344i	0.969858	−	0.243672i	$-0.0783521\pi$
$54$		0			0
$55$			16.6499i			2.24507i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.9580			−1.42661			−0.713304	−	0.700855i	$-0.752802\pi$
$59$	−10.9580			−1.42661			−0.713304	+	0.700855i	$0.752802\pi$
$60$		0			0
$61$		−	6.39356i		−	0.818612i	−0.912397	−	0.409306i	$-0.865771\pi$
$61$		−	6.39356i		−	0.818612i	0.912397	−	0.409306i	$-0.134229\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			15.0552i			1.86737i
$66$		0			0
$67$	−9.08328			−1.10970			−0.554850	−	0.831951i	$-0.687224\pi$
$67$	−9.08328			−1.10970			−0.554850	+	0.831951i	$0.687224\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.5889i		−	1.25667i	−0.777944	−	0.628334i	$-0.783737\pi$
$71$		−	10.5889i		−	1.25667i	0.777944	−	0.628334i	$-0.216263\pi$
$72$		0			0
$73$		−	2.72923i		−	0.319433i	−0.987163	−	0.159716i	$-0.948942\pi$
$73$		−	2.72923i		−	0.319433i	0.987163	−	0.159716i	$-0.0510579\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5.54931	+	16.4518i	−0.632403	+	1.87486i
$78$		0			0
$79$	1.30537			0.146866			0.0734330	−	0.997300i	$-0.476604\pi$
$79$	1.30537			0.146866			0.0734330	+	0.997300i	$0.476604\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	9.06482			0.994994			0.497497	−	0.867466i	$-0.334253\pi$
$83$	9.06482			0.994994			0.497497	+	0.867466i	$0.334253\pi$
$84$		0			0
$85$	−8.94851			−0.970602
$86$		0			0
$87$		0			0
$88$		0			0
$89$	14.2599			1.51155			0.755776	−	0.654830i	$-0.227260\pi$
$89$	14.2599			1.51155			0.755776	+	0.654830i	$0.227260\pi$
$90$		0			0
$91$	−5.01782	+	14.8761i	−0.526011	+	1.55944i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	0.663100i		−	0.0680326i
$96$		0			0
$97$			11.4972i			1.16736i	0.811983	+	0.583681i	$0.198388\pi$
$97$			11.4972i			1.16736i	−0.811983	+	0.583681i	$0.801612\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.18382			0.217298			0.108649	−	0.994080i	$-0.465348\pi$
$101$	2.18382			0.217298			0.108649	+	0.994080i	$0.465348\pi$
$102$		0			0
$103$		−	11.2650i		−	1.10997i	−0.831860	−	0.554986i	$-0.812724\pi$
$103$		−	11.2650i		−	1.10997i	0.831860	−	0.554986i	$-0.187276\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			7.82576i			0.756545i	0.925694	+	0.378272i	$0.123482\pi$
$107$			7.82576i			0.756545i	−0.925694	+	0.378272i	$0.876518\pi$
$108$		0			0
$109$	−17.5453			−1.68053			−0.840266	−	0.542174i	$-0.817601\pi$
$109$	−17.5453			−1.68053			−0.840266	+	0.542174i	$0.817601\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	4.40070i		−	0.413983i	−0.978343	−	0.206992i	$-0.933633\pi$
$113$		−	4.40070i		−	0.413983i	0.978343	−	0.206992i	$-0.0663673\pi$
$114$		0			0
$115$			8.49503i			0.792166i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−8.84206	−	2.98249i	−0.810550	−	0.273405i
$120$		0			0
$121$	−32.0652			−2.91502
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.03946			0.808514
$126$		0			0
$127$	19.5689			1.73646			0.868228	−	0.496165i	$-0.165259\pi$
$127$	19.5689			1.73646			0.868228	+	0.496165i	$0.165259\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.0193557			−0.00169112			−0.000845559	−	1.00000i	$-0.500269\pi$
$131$	−0.0193557			−0.00169112			−0.000845559		1.00000i	$0.500269\pi$
$132$		0			0
$133$	0.221008	−	0.655211i	0.0191638	−	0.0568140i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	4.23813i		−	0.362087i	−0.983475	−	0.181044i	$-0.942052\pi$
$137$		−	4.23813i		−	0.362087i	0.983475	−	0.181044i	$-0.0579476\pi$
$138$		0			0
$139$			20.3693i			1.72770i	0.503747	+	0.863851i	$0.331954\pi$
$139$			20.3693i			1.72770i	−0.503747	+	0.863851i	$0.668046\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−38.9406			−3.25638
$144$		0			0
$145$			2.95346i			0.245272i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			7.39435i			0.605769i	0.953027	+	0.302885i	$0.0979497\pi$
$149$			7.39435i			0.605769i	−0.953027	+	0.302885i	$0.902050\pi$
$150$		0			0
$151$	−13.8838			−1.12985			−0.564923	−	0.825144i	$-0.691094\pi$
$151$	−13.8838			−1.12985			−0.564923	+	0.825144i	$0.691094\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			4.96167i			0.398531i
$156$		0			0
$157$		−	2.15598i		−	0.172066i	−0.996292	−	0.0860329i	$-0.972581\pi$
$157$		−	2.15598i		−	0.172066i	0.996292	−	0.0860329i	$-0.0274190\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.83135	+	8.39397i	−0.223142	+	0.661538i
$162$		0			0
$163$	−17.1866			−1.34616			−0.673079	−	0.739571i	$-0.735029\pi$
$163$	−17.1866			−1.34616			−0.673079	+	0.739571i	$0.735029\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−10.8020			−0.835885			−0.417943	−	0.908473i	$-0.637249\pi$
$167$	−10.8020			−0.835885			−0.417943	+	0.908473i	$0.637249\pi$
$168$		0			0
$169$	−22.2110			−1.70854
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−19.1098			−1.45289			−0.726444	−	0.687225i	$-0.758829\pi$
$173$	−19.1098			−1.45289			−0.726444	+	0.687225i	$0.758829\pi$
$174$		0			0
$175$	−3.60295	−	1.21530i	−0.272357	−	0.0918681i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	14.9224i		−	1.11536i	−0.830057	−	0.557678i	$-0.811692\pi$
$179$		−	14.9224i		−	1.11536i	0.830057	−	0.557678i	$-0.188308\pi$
$180$		0			0
$181$		−	18.5179i		−	1.37642i	−0.725511	−	0.688211i	$-0.758396\pi$
$181$		−	18.5179i		−	1.37642i	0.725511	−	0.688211i	$-0.241604\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.95649			0.217365
$186$		0			0
$187$		−	23.1455i		−	1.69257i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	5.86809i		−	0.424600i	−0.977205	−	0.212300i	$-0.931905\pi$
$191$		−	5.86809i		−	0.424600i	0.977205	−	0.212300i	$-0.0680954\pi$
$192$		0			0
$193$	16.3915			1.17989			0.589945	−	0.807444i	$-0.299150\pi$
$193$	16.3915			1.17989			0.589945	+	0.807444i	$0.299150\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			6.90103i			0.491678i	0.969311	+	0.245839i	$0.0790635\pi$
$197$			6.90103i			0.491678i	−0.969311	+	0.245839i	$0.920936\pi$
$198$		0			0
$199$		−	10.3555i		−	0.734078i	−0.930205	−	0.367039i	$-0.880371\pi$
$199$		−	10.3555i		−	0.734078i	0.930205	−	0.367039i	$-0.119629\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.984373	+	2.91833i	−0.0690895	+	0.204826i
$204$		0			0
$205$	−14.5008			−1.01278
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.71512			0.118637
$210$		0			0
$211$	−6.86662			−0.472718			−0.236359	−	0.971666i	$-0.575954\pi$
$211$	−6.86662			−0.472718			−0.236359	+	0.971666i	$0.575954\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	13.5669			0.925255
$216$		0			0
$217$	−1.65370	+	4.90264i	−0.112260	+	0.332813i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	20.9287i		−	1.40782i
$222$		0			0
$223$			0.539306i			0.0361146i	0.999837	+	0.0180573i	$0.00574813\pi$
$223$			0.539306i			0.0361146i	−0.999837	+	0.0180573i	$0.994252\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.22104			0.0810429			0.0405215	−	0.999179i	$-0.487098\pi$
$227$	1.22104			0.0810429			0.0405215	+	0.999179i	$0.487098\pi$
$228$		0			0
$229$			12.0145i			0.793941i	0.917831	+	0.396970i	$0.129938\pi$
$229$			12.0145i			0.793941i	−0.917831	+	0.396970i	$0.870062\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			5.13142i			0.336171i	0.985772	+	0.168085i	$0.0537584\pi$
$233$			5.13142i			0.336171i	−0.985772	+	0.168085i	$0.946242\pi$
$234$		0			0
$235$	−19.2706			−1.25707
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	12.2376i		−	0.791586i	−0.918340	−	0.395793i	$-0.870470\pi$
$239$		−	12.2376i		−	0.791586i	0.918340	−	0.395793i	$-0.129530\pi$
$240$		0			0
$241$			28.7890i			1.85446i	0.374490	+	0.927231i	$0.377818\pi$
$241$			28.7890i			1.85446i	−0.374490	+	0.927231i	$0.622182\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−14.1316	−	10.7573i	−0.902834	−	0.687259i
$246$		0			0
$247$	1.55085			0.0986784
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.82335			0.304447			0.152223	−	0.988346i	$-0.451357\pi$
$251$	4.82335			0.304447			0.152223	+	0.988346i	$0.451357\pi$
$252$		0			0
$253$	−21.9726			−1.38140
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.30045			0.268255			0.134127	−	0.990964i	$-0.457177\pi$
$257$	4.30045			0.268255			0.134127	+	0.990964i	$0.457177\pi$
$258$		0			0
$259$	2.92132	+	0.985382i	0.181522	+	0.0612287i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	22.2215i		−	1.37024i	−0.728432	−	0.685118i	$-0.759751\pi$
$263$		−	22.2215i		−	1.37024i	0.728432	−	0.685118i	$-0.240249\pi$
$264$		0			0
$265$			9.00163i			0.552966i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−21.6564			−1.32042			−0.660208	−	0.751083i	$-0.729532\pi$
$269$	−21.6564			−1.32042			−0.660208	+	0.751083i	$0.729532\pi$
$270$		0			0
$271$			17.0808i			1.03758i	0.854901	+	0.518792i	$0.173618\pi$
$271$			17.0808i			1.03758i	−0.854901	+	0.518792i	$0.826382\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	9.43129i		−	0.568728i
$276$		0			0
$277$	3.20960			0.192846			0.0964230	−	0.995340i	$-0.469260\pi$
$277$	3.20960			0.192846			0.0964230	+	0.995340i	$0.469260\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			20.8538i			1.24403i	0.783005	+	0.622016i	$0.213686\pi$
$281$			20.8538i			1.24403i	−0.783005	+	0.622016i	$0.786314\pi$
$282$		0			0
$283$		−	11.5463i		−	0.686358i	−0.939270	−	0.343179i	$-0.888496\pi$
$283$		−	11.5463i		−	0.686358i	0.939270	−	0.343179i	$-0.111504\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−14.3283	−	4.83304i	−0.845773	−	0.285286i
$288$		0			0
$289$	−4.56040			−0.268259
$290$		0			0
$291$		0			0
$292$		0			0
$293$	28.3201			1.65448			0.827238	−	0.561852i	$-0.189911\pi$
$293$	28.3201			1.65448			0.827238	+	0.561852i	$0.189911\pi$
$294$		0			0
$295$	27.8021			1.61870
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−19.8681			−1.14900
$300$		0			0
$301$	13.4055	+	4.52178i	0.772680	+	0.260631i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			16.2215i			0.928839i
$306$		0			0
$307$		−	24.2818i		−	1.38584i	−0.721016	−	0.692918i	$-0.756325\pi$
$307$		−	24.2818i		−	1.38584i	0.721016	−	0.692918i	$-0.243675\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	32.6238			1.84993			0.924964	−	0.380054i	$-0.124095\pi$
$311$	32.6238			1.84993			0.924964	+	0.380054i	$0.124095\pi$
$312$		0			0
$313$			14.5089i			0.820089i	0.912065	+	0.410045i	$0.134487\pi$
$313$			14.5089i			0.820089i	−0.912065	+	0.410045i	$0.865513\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			16.1868i			0.909139i	0.890711	+	0.454570i	$0.150207\pi$
$317$			16.1868i			0.909139i	−0.890711	+	0.454570i	$0.849793\pi$
$318$		0			0
$319$	−7.63919			−0.427712
$320$		0			0
$321$		0			0
$322$		0			0
$323$			0.921796i			0.0512901i
$324$		0			0
$325$		−	8.52800i		−	0.473048i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−19.0413	−	6.42278i	−1.04978	−	0.354099i
$330$		0			0
$331$	−29.2108			−1.60557			−0.802787	−	0.596266i	$-0.796650\pi$
$331$	−29.2108			−1.60557			−0.802787	+	0.596266i	$0.796650\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	23.0457			1.25912
$336$		0			0
$337$	16.9362			0.922575			0.461288	−	0.887251i	$-0.347388\pi$
$337$	16.9362			0.922575			0.461288	+	0.887251i	$0.347388\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−12.8334			−0.694970
$342$		0			0
$343$	−10.3781	−	15.3393i	−0.560366	−	0.828245i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			20.4825i			1.09956i	0.835309	+	0.549780i	$0.185289\pi$
$347$			20.4825i			1.09956i	−0.835309	+	0.549780i	$0.814711\pi$
$348$		0			0
$349$			6.90611i			0.369676i	0.982769	+	0.184838i	$0.0591760\pi$
$349$			6.90611i			0.369676i	−0.982769	+	0.184838i	$0.940824\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	21.0049			1.11798			0.558990	−	0.829175i	$-0.311189\pi$
$353$	21.0049			1.11798			0.558990	+	0.829175i	$0.311189\pi$
$354$		0			0
$355$			26.8656i			1.42588i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			2.61864i			0.138206i	0.997610	+	0.0691032i	$0.0220138\pi$
$359$			2.61864i			0.138206i	−0.997610	+	0.0691032i	$0.977986\pi$
$360$		0			0
$361$	18.9317			0.996405
$362$		0			0
$363$		0			0
$364$		0			0
$365$			6.92450i			0.362445i
$366$		0			0
$367$			27.5344i			1.43728i	0.695381	+	0.718641i	$0.255236\pi$
$367$			27.5344i			1.43728i	−0.695381	+	0.718641i	$0.744764\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.00020	+	8.89455i	−0.155762	+	0.461782i
$372$		0			0
$373$	5.35537			0.277290			0.138645	−	0.990342i	$-0.455725\pi$
$373$	5.35537			0.277290			0.138645	+	0.990342i	$0.455725\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.90754			−0.355756
$378$		0			0
$379$	21.5330			1.10608			0.553038	−	0.833156i	$-0.313468\pi$
$379$	21.5330			1.10608			0.553038	+	0.833156i	$0.313468\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.71896			0.190030			0.0950149	−	0.995476i	$-0.469710\pi$
$383$	3.71896			0.190030			0.0950149	+	0.995476i	$0.469710\pi$
$384$		0			0
$385$	14.0795	−	41.7408i	0.717557	−	2.12731i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			8.92025i			0.452275i	0.974095	+	0.226137i	$0.0726098\pi$
$389$			8.92025i			0.452275i	−0.974095	+	0.226137i	$0.927390\pi$
$390$		0			0
$391$		−	11.8092i		−	0.597217i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.31194			−0.166642
$396$		0			0
$397$			13.3591i			0.670476i	0.942134	+	0.335238i	$0.108817\pi$
$397$			13.3591i			0.670476i	−0.942134	+	0.335238i	$0.891183\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			21.5890i			1.07810i	0.842273	+	0.539051i	$0.181217\pi$
$401$			21.5890i			1.07810i	−0.842273	+	0.539051i	$0.818783\pi$
$402$		0			0
$403$	−11.6043			−0.578052
$404$		0			0
$405$		0			0
$406$		0			0
$407$			7.64702i			0.379049i
$408$		0			0
$409$			29.8818i			1.47756i	0.673948	+	0.738779i	$0.264597\pi$
$409$			29.8818i			1.47756i	−0.673948	+	0.738779i	$0.735403\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	27.4714	+	9.26630i	1.35178	+	0.455965i
$414$		0			0
$415$	−22.9989			−1.12897
$416$		0			0
$417$		0			0
$418$		0			0
$419$	18.6088			0.909098			0.454549	−	0.890722i	$-0.349800\pi$
$419$	18.6088			0.909098			0.454549	+	0.890722i	$0.349800\pi$
$420$		0			0
$421$	19.5802			0.954281			0.477140	−	0.878827i	$-0.341673\pi$
$421$	19.5802			0.954281			0.477140	+	0.878827i	$0.341673\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5.06887			0.245876
$426$		0			0
$427$	−5.40653	+	16.0285i	−0.261641	+	0.775674i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			13.3632i			0.643683i	0.946794	+	0.321842i	$0.104302\pi$
$431$			13.3632i			0.643683i	−0.946794	+	0.321842i	$0.895698\pi$
$432$		0			0
$433$		−	3.05246i		−	0.146692i	−0.997307	−	0.0733458i	$-0.976632\pi$
$433$		−	3.05246i		−	0.146692i	0.997307	−	0.0733458i	$-0.0233677\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.875082			0.0418608
$438$		0			0
$439$		−	19.4151i		−	0.926631i	−0.886193	−	0.463316i	$-0.846660\pi$
$439$		−	19.4151i		−	0.926631i	0.886193	−	0.463316i	$-0.153340\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	32.9959i		−	1.56768i	−0.620962	−	0.783841i	$-0.713258\pi$
$443$		−	32.9959i		−	1.56768i	0.620962	−	0.783841i	$-0.286742\pi$
$444$		0			0
$445$	−36.1797			−1.71508
$446$		0			0
$447$		0			0
$448$		0			0
$449$			9.72985i			0.459180i	0.973287	+	0.229590i	$0.0737385\pi$
$449$			9.72985i			0.459180i	−0.973287	+	0.229590i	$0.926262\pi$
$450$		0			0
$451$		−	37.5066i		−	1.76612i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	12.7310	−	37.7430i	0.596839	−	1.76942i
$456$		0			0
$457$	−14.2963			−0.668753			−0.334376	−	0.942440i	$-0.608526\pi$
$457$	−14.2963			−0.668753			−0.334376	+	0.942440i	$0.608526\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7.91785			0.368771			0.184386	−	0.982854i	$-0.440970\pi$
$461$	7.91785			0.368771			0.184386	+	0.982854i	$0.440970\pi$
$462$		0			0
$463$	−17.6313			−0.819397			−0.409698	−	0.912221i	$-0.634366\pi$
$463$	−17.6313			−0.819397			−0.409698	+	0.912221i	$0.634366\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	25.1745			1.16494			0.582469	−	0.812853i	$-0.302087\pi$
$467$	25.1745			1.16494			0.582469	+	0.812853i	$0.302087\pi$
$468$		0			0
$469$	22.7716	+	7.68101i	1.05149	+	0.354676i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			35.0911i			1.61349i
$474$		0			0
$475$			0.375612i			0.0172342i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−22.6516			−1.03498			−0.517488	−	0.855690i	$-0.673133\pi$
$479$	−22.6516			−1.03498			−0.517488	+	0.855690i	$0.673133\pi$
$480$		0			0
$481$			6.91462i			0.315279i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	29.1702i		−	1.32455i
$486$		0			0
$487$	−6.17391			−0.279767			−0.139883	−	0.990168i	$-0.544673\pi$
$487$	−6.17391			−0.279767			−0.139883	+	0.990168i	$0.544673\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	24.0908i		−	1.08720i	−0.839344	−	0.543601i	$-0.817061\pi$
$491$		−	24.0908i		−	1.08720i	0.839344	−	0.543601i	$-0.182939\pi$
$492$		0			0
$493$		−	4.10570i		−	0.184911i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8.95417	+	26.5460i	−0.401649	+	1.19075i
$498$		0			0
$499$	14.3717			0.643365			0.321682	−	0.946848i	$-0.395752\pi$
$499$	14.3717			0.643365			0.321682	+	0.946848i	$0.395752\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−18.4937			−0.824593			−0.412296	−	0.911050i	$-0.635273\pi$
$503$	−18.4937			−0.824593			−0.412296	+	0.911050i	$0.635273\pi$
$504$		0			0
$505$	−5.54069			−0.246557
$506$		0			0
$507$		0			0
$508$		0			0
$509$	14.9777			0.663877			0.331938	−	0.943301i	$-0.392297\pi$
$509$	14.9777			0.663877			0.331938	+	0.943301i	$0.392297\pi$
$510$		0			0
$511$	−2.30790	+	6.84212i	−0.102095	+	0.302678i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			28.5810i			1.25943i
$516$		0			0
$517$		−	49.8437i		−	2.19212i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−15.4043			−0.674873			−0.337436	−	0.941348i	$-0.609560\pi$
$521$	−15.4043			−0.674873			−0.337436	+	0.941348i	$0.609560\pi$
$522$		0			0
$523$		−	16.9319i		−	0.740381i	−0.928956	−	0.370191i	$-0.879292\pi$
$523$		−	16.9319i		−	0.740381i	0.928956	−	0.370191i	$-0.120708\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	6.89737i		−	0.300454i
$528$		0			0
$529$	11.7892			0.512576
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	33.9144i		−	1.46900i
$534$		0			0
$535$		−	19.8552i		−	0.858415i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	27.8240	−	36.5516i	1.19846	−	1.57439i
$540$		0			0
$541$	6.44265			0.276991			0.138496	−	0.990363i	$-0.455773\pi$
$541$	6.44265			0.276991			0.138496	+	0.990363i	$0.455773\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	44.5151			1.90682
$546$		0			0
$547$	5.78382			0.247298			0.123649	−	0.992326i	$-0.460540\pi$
$547$	5.78382			0.247298			0.123649	+	0.992326i	$0.460540\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	0.304239			0.0129610
$552$		0			0
$553$	−3.27254	−	1.10385i	−0.139162	−	0.0469405i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	3.86933i		−	0.163949i	−0.996634	−	0.0819745i	$-0.973877\pi$
$557$		−	3.86933i		−	0.163949i	0.996634	−	0.0819745i	$-0.0261226\pi$
$558$		0			0
$559$			31.7302i			1.34204i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−24.5425			−1.03434			−0.517171	−	0.855882i	$-0.673015\pi$
$563$	−24.5425			−1.03434			−0.517171	+	0.855882i	$0.673015\pi$
$564$		0			0
$565$			11.1653i			0.469727i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	13.8702i		−	0.581469i	−0.956804	−	0.290735i	$-0.906100\pi$
$569$		−	13.8702i		−	0.581469i	0.956804	−	0.290735i	$-0.0938996\pi$
$570$		0			0
$571$	6.65780			0.278620			0.139310	−	0.990249i	$-0.455512\pi$
$571$	6.65780			0.278620			0.139310	+	0.990249i	$0.455512\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	4.81200i		−	0.200674i
$576$		0			0
$577$		−	20.4063i		−	0.849525i	−0.905305	−	0.424763i	$-0.860358\pi$
$577$		−	20.4063i		−	0.849525i	0.905305	−	0.424763i	$-0.139642\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−22.7253	−	7.66541i	−0.942804	−	0.318015i
$582$		0			0
$583$	−23.2829			−0.964279
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.87675			0.283834			0.141917	−	0.989879i	$-0.454673\pi$
$587$	6.87675			0.283834			0.141917	+	0.989879i	$0.454673\pi$
$588$		0			0
$589$	0.511106			0.0210598
$590$		0			0
$591$		0			0
$592$		0			0
$593$	13.7068			0.562871			0.281435	−	0.959580i	$-0.409189\pi$
$593$	13.7068			0.562871			0.281435	+	0.959580i	$0.409189\pi$
$594$		0			0
$595$	22.4337	+	7.56705i	0.919692	+	0.310219i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	17.7409i		−	0.724872i	−0.932009	−	0.362436i	$-0.881945\pi$
$599$		−	17.7409i		−	0.724872i	0.932009	−	0.362436i	$-0.118055\pi$
$600$		0			0
$601$		−	7.29042i		−	0.297382i	−0.988884	−	0.148691i	$-0.952494\pi$
$601$		−	7.29042i		−	0.297382i	0.988884	−	0.148691i	$-0.0475061\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	81.3545			3.30753
$606$		0			0
$607$		−	6.92869i		−	0.281227i	−0.990065	−	0.140613i	$-0.955093\pi$
$607$		−	6.92869i		−	0.281227i	0.990065	−	0.140613i	$-0.0449075\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	45.0699i		−	1.82333i
$612$		0			0
$613$	−4.60656			−0.186057			−0.0930287	−	0.995663i	$-0.529655\pi$
$613$	−4.60656			−0.186057			−0.0930287	+	0.995663i	$0.529655\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			29.4008i			1.18363i	0.806073	+	0.591816i	$0.201589\pi$
$617$			29.4008i			1.18363i	−0.806073	+	0.591816i	$0.798411\pi$
$618$		0			0
$619$			33.4329i			1.34378i	0.740650	+	0.671891i	$0.234518\pi$
$619$			33.4329i			1.34378i	−0.740650	+	0.671891i	$0.765482\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−35.7493	−	12.0585i	−1.43227	−	0.483114i
$624$		0			0
$625$	−30.1204			−1.20482
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4.10991			−0.163873
$630$		0			0
$631$	−26.3381			−1.04850			−0.524251	−	0.851564i	$-0.675655\pi$
$631$	−26.3381			−1.04850			−0.524251	+	0.851564i	$0.675655\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−49.6493			−1.97027
$636$		0			0
$637$	25.1591	−	33.0509i	0.996840	−	1.30952i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			13.4959i			0.533057i	0.963827	+	0.266529i	$0.0858767\pi$
$641$			13.4959i			0.533057i	−0.963827	+	0.266529i	$0.914123\pi$
$642$		0			0
$643$			22.2014i			0.875539i	0.899087	+	0.437769i	$0.144231\pi$
$643$			22.2014i			0.875539i	−0.899087	+	0.437769i	$0.855769\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−11.6716			−0.458856			−0.229428	−	0.973326i	$-0.573686\pi$
$647$	−11.6716			−0.458856			−0.229428	+	0.973326i	$0.573686\pi$
$648$		0			0
$649$			71.9107i			2.82274i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	23.5518i		−	0.921651i	−0.887491	−	0.460826i	$-0.847553\pi$
$653$		−	23.5518i		−	0.921651i	0.887491	−	0.460826i	$-0.152447\pi$
$654$		0			0
$655$	0.0491085			0.00191883
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	10.9353i		−	0.425977i	−0.977055	−	0.212988i	$-0.931680\pi$
$659$		−	10.9353i		−	0.425977i	0.977055	−	0.212988i	$-0.0683197\pi$
$660$		0			0
$661$			11.5083i			0.447622i	0.974633	+	0.223811i	$0.0718499\pi$
$661$			11.5083i			0.447622i	−0.974633	+	0.223811i	$0.928150\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.560731	+	1.66237i	−0.0217442	+	0.0644641i
$666$		0			0
$667$	−3.89764			−0.150917
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−41.9572			−1.61974
$672$		0			0
$673$	29.1243			1.12266			0.561329	−	0.827592i	$-0.310290\pi$
$673$	29.1243			1.12266			0.561329	+	0.827592i	$0.310290\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	9.21943			0.354332			0.177166	−	0.984181i	$-0.443307\pi$
$677$	9.21943			0.354332			0.177166	+	0.984181i	$0.443307\pi$
$678$		0			0
$679$	9.72227	−	28.8232i	0.373106	−	1.10613i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			17.2370i			0.659554i	0.944059	+	0.329777i	$0.106974\pi$
$683$			17.2370i			0.659554i	−0.944059	+	0.329777i	$0.893026\pi$
$684$		0			0
$685$			10.7528i			0.410843i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−21.0530			−0.802054
$690$		0			0
$691$		−	15.3431i		−	0.583681i	−0.956467	−	0.291840i	$-0.905732\pi$
$691$		−	15.3431i		−	0.583681i	0.956467	−	0.291840i	$-0.0942676\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	51.6802i		−	1.96034i
$696$		0			0
$697$	20.1580			0.763540
$698$		0			0
$699$		0			0
$700$		0			0
$701$			13.1791i			0.497767i	0.968533	+	0.248883i	$0.0800636\pi$
$701$			13.1791i			0.497767i	−0.968533	+	0.248883i	$0.919936\pi$
$702$		0			0
$703$		−	0.304551i		−	0.0114864i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5.47478	−	1.84668i	−0.205900	−	0.0694516i
$708$		0			0
$709$	−16.4679			−0.618466			−0.309233	−	0.950986i	$-0.600072\pi$
$709$	−16.4679			−0.618466			−0.309233	+	0.950986i	$0.600072\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−6.54783			−0.245218
$714$		0			0
$715$	98.7984			3.69485
$716$		0			0
$717$		0			0
$718$		0			0
$719$	13.3816			0.499048			0.249524	−	0.968369i	$-0.419726\pi$
$719$	13.3816			0.499048			0.249524	+	0.968369i	$0.419726\pi$
$720$		0			0
$721$	−9.52591	+	28.2410i	−0.354763	+	1.05175i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	1.67298i		−	0.0621331i
$726$		0			0
$727$		−	8.26807i		−	0.306646i	−0.988176	−	0.153323i	$-0.951003\pi$
$727$		−	8.26807i		−	0.306646i	0.988176	−	0.153323i	$-0.0489974\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−18.8598			−0.697554
$732$		0			0
$733$			14.0611i			0.519359i	0.965695	+	0.259679i	$0.0836169\pi$
$733$			14.0611i			0.519359i	−0.965695	+	0.259679i	$0.916383\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			59.6082i			2.19570i
$738$		0			0
$739$	25.8860			0.952231			0.476115	−	0.879383i	$-0.342044\pi$
$739$	25.8860			0.952231			0.476115	+	0.879383i	$0.342044\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			42.2105i			1.54855i	0.632847	+	0.774277i	$0.281886\pi$
$743$			42.2105i			1.54855i	−0.632847	+	0.774277i	$0.718114\pi$
$744$		0			0
$745$		−	18.7606i		−	0.687337i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	6.61763	−	19.6190i	0.241803	−	0.716862i
$750$		0			0
$751$	−42.6314			−1.55564			−0.777820	−	0.628487i	$-0.783675\pi$
$751$	−42.6314			−1.55564			−0.777820	+	0.628487i	$0.783675\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	35.2253			1.28198
$756$		0			0
$757$	51.5092			1.87213			0.936067	−	0.351823i	$-0.114438\pi$
$757$	51.5092			1.87213			0.936067	+	0.351823i	$0.114438\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−30.9690			−1.12263			−0.561313	−	0.827604i	$-0.689703\pi$
$761$	−30.9690			−1.12263			−0.561313	+	0.827604i	$0.689703\pi$
$762$		0			0
$763$	43.9856	+	14.8367i	1.59238	+	0.537123i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			65.0234i			2.34786i
$768$		0			0
$769$		−	47.5741i		−	1.71557i	−0.514011	−	0.857783i	$-0.671841\pi$
$769$		−	47.5741i		−	1.71557i	0.514011	−	0.857783i	$-0.328159\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−27.9971			−1.00699			−0.503493	−	0.864000i	$-0.667952\pi$
$773$	−27.9971			−1.00699			−0.503493	+	0.864000i	$0.667952\pi$
$774$		0			0
$775$		−	2.81053i		−	0.100957i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			1.49374i			0.0535189i
$780$		0			0
$781$	−69.4885			−2.48649
$782$		0			0
$783$		0			0
$784$		0			0
$785$			5.47006i			0.195235i
$786$		0			0
$787$		−	10.2328i		−	0.364761i	−0.983228	−	0.182381i	$-0.941620\pi$
$787$		−	10.2328i		−	0.364761i	0.983228	−	0.182381i	$-0.0583803\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.72133	+	11.0324i	−0.132315	+	0.392269i
$792$		0			0
$793$	−37.9387			−1.34724
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−35.5114			−1.25788			−0.628940	−	0.777454i	$-0.716511\pi$
$797$	−35.5114			−1.25788			−0.628940	+	0.777454i	$0.716511\pi$
$798$		0			0
$799$	26.7886			0.947713
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−17.9103			−0.632042
$804$		0			0
$805$	7.18358	−	21.2968i	0.253188	−	0.750615i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	18.4932i		−	0.650188i	−0.945682	−	0.325094i	$-0.894604\pi$
$809$		−	18.4932i		−	0.650188i	0.945682	−	0.325094i	$-0.105396\pi$
$810$		0			0
$811$		−	35.6320i		−	1.25121i	−0.780140	−	0.625605i	$-0.784852\pi$
$811$		−	35.6320i		−	1.25121i	0.780140	−	0.625605i	$-0.215148\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	43.6051			1.52742
$816$		0			0
$817$		−	1.39754i		−	0.0488938i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			19.7903i			0.690685i	0.938477	+	0.345343i	$0.112237\pi$
$821$			19.7903i			0.690685i	−0.938477	+	0.345343i	$0.887763\pi$
$822$		0			0
$823$	6.95877			0.242568			0.121284	−	0.992618i	$-0.461299\pi$
$823$	6.95877			0.242568			0.121284	+	0.992618i	$0.461299\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			4.92505i			0.171261i	0.996327	+	0.0856304i	$0.0272904\pi$
$827$			4.92505i			0.171261i	−0.996327	+	0.0856304i	$0.972710\pi$
$828$		0			0
$829$			42.9274i			1.49093i	0.666544	+	0.745466i	$0.267773\pi$
$829$			42.9274i			1.49093i	−0.666544	+	0.745466i	$0.732227\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	19.6448	+	14.9541i	0.680651	+	0.518128i
$834$		0			0
$835$	27.4064			0.948438
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−17.6928			−0.610823			−0.305411	−	0.952220i	$-0.598794\pi$
$839$	−17.6928			−0.610823			−0.305411	+	0.952220i	$0.598794\pi$
$840$		0			0
$841$	27.6449			0.953273
$842$		0			0
$843$		0			0
$844$		0			0
$845$	56.3529			1.93860
$846$		0			0
$847$	80.3867	+	27.1150i	2.76212	+	0.931683i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			3.90163i			0.133746i
$852$		0			0
$853$			27.2197i			0.931985i	0.884789	+	0.465993i	$0.154303\pi$
$853$			27.2197i			0.931985i	−0.884789	+	0.465993i	$0.845697\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−25.6606			−0.876548			−0.438274	−	0.898841i	$-0.644410\pi$
$857$	−25.6606			−0.876548			−0.438274	+	0.898841i	$0.644410\pi$
$858$		0			0
$859$		−	19.4090i		−	0.662228i	−0.943591	−	0.331114i	$-0.892576\pi$
$859$		−	19.4090i		−	0.662228i	0.943591	−	0.331114i	$-0.107424\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	22.6418i		−	0.770737i	−0.922763	−	0.385368i	$-0.874074\pi$
$863$		−	22.6418i		−	0.770737i	0.922763	−	0.385368i	$-0.125926\pi$
$864$		0			0
$865$	48.4845			1.64852
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	8.56639i		−	0.290595i
$870$		0			0
$871$			53.8992i			1.82630i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−22.6617	−	7.64396i	−0.766106	−	0.258413i
$876$		0			0
$877$	35.2790			1.19129			0.595644	−	0.803249i	$-0.296897\pi$
$877$	35.2790			1.19129			0.595644	+	0.803249i	$0.296897\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.975907			−0.0328791			−0.0164396	−	0.999865i	$-0.505233\pi$
$881$	−0.975907			−0.0328791			−0.0164396	+	0.999865i	$0.505233\pi$
$882$		0			0
$883$	3.26917			0.110016			0.0550082	−	0.998486i	$-0.482481\pi$
$883$	3.26917			0.110016			0.0550082	+	0.998486i	$0.482481\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	24.3305			0.816939			0.408470	−	0.912772i	$-0.366063\pi$
$887$	24.3305			0.816939			0.408470	+	0.912772i	$0.366063\pi$
$888$		0			0
$889$	−49.0587	−	16.5479i	−1.64538	−	0.554997i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			1.98508i			0.0664282i
$894$		0			0
$895$			37.8606i			1.26554i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.27648			−0.0759249
$900$		0			0
$901$		−	12.5135i		−	0.416884i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			46.9827i			1.56176i
$906$		0			0
$907$	44.1017			1.46437			0.732186	−	0.681105i	$-0.238500\pi$
$907$	44.1017			1.46437			0.732186	+	0.681105i	$0.238500\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	23.2343i		−	0.769785i	−0.922961	−	0.384893i	$-0.874238\pi$
$911$		−	23.2343i		−	0.769785i	0.922961	−	0.384893i	$-0.125762\pi$
$912$		0			0
$913$		−	59.4871i		−	1.96874i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	0.0485243	+	0.0163676i	0.00160241	+	0.000540506i
$918$		0			0
$919$	2.91416			0.0961291			0.0480646	−	0.998844i	$-0.484695\pi$
$919$	2.91416			0.0961291			0.0480646	+	0.998844i	$0.484695\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−62.8331			−2.06818
$924$		0			0
$925$	−1.67470			−0.0550638
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−16.6953			−0.547757			−0.273878	−	0.961764i	$-0.588307\pi$
$929$	−16.6953			−0.547757			−0.273878	+	0.961764i	$0.588307\pi$
$930$		0			0
$931$	−1.10812	+	1.45571i	−0.0363172	+	0.0477090i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			58.7238i			1.92047i
$936$		0			0
$937$		−	40.4272i		−	1.32070i	−0.750959	−	0.660349i	$-0.770408\pi$
$937$		−	40.4272i		−	1.32070i	0.750959	−	0.660349i	$-0.229592\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−55.9778			−1.82482			−0.912411	−	0.409274i	$-0.865782\pi$
$941$	−55.9778			−1.82482			−0.912411	+	0.409274i	$0.865782\pi$
$942$		0			0
$943$		−	19.1365i		−	0.623170i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			20.6112i			0.669774i	0.942258	+	0.334887i	$0.108698\pi$
$947$			20.6112i			0.669774i	−0.942258	+	0.334887i	$0.891302\pi$
$948$		0			0
$949$	−16.1950			−0.525711
$950$		0			0
$951$		0			0
$952$		0			0
$953$			17.3803i			0.563005i	0.959561	+	0.281502i	$0.0908327\pi$
$953$			17.3803i			0.563005i	−0.959561	+	0.281502i	$0.909167\pi$
$954$		0			0
$955$			14.8883i			0.481773i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3.58385	+	10.6249i	−0.115729	+	0.343095i
$960$		0			0
$961$	27.1756			0.876633
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−41.5879			−1.33876
$966$		0			0
$967$	−2.65984			−0.0855348			−0.0427674	−	0.999085i	$-0.513617\pi$
$967$	−2.65984			−0.0855348			−0.0427674	+	0.999085i	$0.513617\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−34.9474			−1.12151			−0.560757	−	0.827980i	$-0.689490\pi$
$971$	−34.9474			−1.12151			−0.560757	+	0.827980i	$0.689490\pi$
$972$		0			0
$973$	17.2247	−	51.0654i	0.552199	−	1.63708i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	15.8123i		−	0.505880i	−0.967482	−	0.252940i	$-0.918602\pi$
$977$		−	15.8123i		−	0.505880i	0.967482	−	0.252940i	$-0.0813975\pi$
$978$		0			0
$979$		−	93.5796i		−	2.99082i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	60.5664			1.93177			0.965885	−	0.258972i	$-0.0833839\pi$
$983$	60.5664			1.93177			0.965885	+	0.258972i	$0.0833839\pi$
$984$		0			0
$985$		−	17.5090i		−	0.557884i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			17.9040i			0.569315i
$990$		0			0
$991$	8.00131			0.254170			0.127085	−	0.991892i	$-0.459438\pi$
$991$	8.00131			0.254170			0.127085	+	0.991892i	$0.459438\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			26.2734i			0.832923i
$996$		0			0
$997$		−	43.6879i		−	1.38361i	−0.722084	−	0.691805i	$-0.756816\pi$
$997$		−	43.6879i		−	1.38361i	0.722084	−	0.691805i	$-0.243184\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4536.2.k.a.3401.10		48
3.2	odd	2	inner	4536.2.k.a.3401.39		48
7.6	odd	2	inner	4536.2.k.a.3401.40		48
9.2	odd	6		1512.2.bu.a.881.5		48
9.4	even	3		1512.2.bu.a.1385.20		48
9.5	odd	6		504.2.bu.a.209.12	yes	48
9.7	even	3		504.2.bu.a.41.13	yes	48
21.20	even	2	inner	4536.2.k.a.3401.9		48
36.7	odd	6		1008.2.cc.d.545.12		48
36.11	even	6		3024.2.cc.d.881.5		48
36.23	even	6		1008.2.cc.d.209.13		48
36.31	odd	6		3024.2.cc.d.2897.20		48
63.13	odd	6		1512.2.bu.a.1385.5		48
63.20	even	6		1512.2.bu.a.881.20		48
63.34	odd	6		504.2.bu.a.41.12	✓	48
63.41	even	6		504.2.bu.a.209.13	yes	48
252.83	odd	6		3024.2.cc.d.881.20		48
252.139	even	6		3024.2.cc.d.2897.5		48
252.167	odd	6		1008.2.cc.d.209.12		48
252.223	even	6		1008.2.cc.d.545.13		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.bu.a.41.12	✓	48	63.34	odd	6
504.2.bu.a.41.13	yes	48	9.7	even	3
504.2.bu.a.209.12	yes	48	9.5	odd	6
504.2.bu.a.209.13	yes	48	63.41	even	6
1008.2.cc.d.209.12		48	252.167	odd	6
1008.2.cc.d.209.13		48	36.23	even	6
1008.2.cc.d.545.12		48	36.7	odd	6
1008.2.cc.d.545.13		48	252.223	even	6
1512.2.bu.a.881.5		48	9.2	odd	6
1512.2.bu.a.881.20		48	63.20	even	6
1512.2.bu.a.1385.5		48	63.13	odd	6
1512.2.bu.a.1385.20		48	9.4	even	3
3024.2.cc.d.881.5		48	36.11	even	6
3024.2.cc.d.881.20		48	252.83	odd	6
3024.2.cc.d.2897.5		48	252.139	even	6
3024.2.cc.d.2897.20		48	36.31	odd	6
4536.2.k.a.3401.9		48	21.20	even	2	inner
4536.2.k.a.3401.10		48	1.1	even	1	trivial
4536.2.k.a.3401.39		48	3.2	odd	2	inner
4536.2.k.a.3401.40		48	7.6	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4536 = 2^{3} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4536.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.53716 q^{5} +(-2.50698 - 0.845621i) q^{7} +O(q^{10})\)	\(q-2.53716 q^{5} +(-2.50698 - 0.845621i) q^{7} -6.56241i q^{11} -5.93389i q^{13} +3.52698 q^{17} +0.261355i q^{19} -3.34825i q^{23} +1.43717 q^{25} -1.16408i q^{29} -1.95560i q^{31} +(6.36059 + 2.14547i) q^{35} -1.16528 q^{37} +5.71538 q^{41} -5.34728 q^{43} +7.59534 q^{47} +(5.56985 + 4.23990i) q^{49} -3.54792i q^{53} +16.6499i q^{55} -10.9580 q^{59} -6.39356i q^{61} +15.0552i q^{65} -9.08328 q^{67} -10.5889i q^{71} -2.72923i q^{73} +(-5.54931 + 16.4518i) q^{77} +1.30537 q^{79} +9.06482 q^{83} -8.94851 q^{85} +14.2599 q^{89} +(-5.01782 + 14.8761i) q^{91} -0.663100i q^{95} +11.4972i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q + 48 q^{25} - 24 q^{43} - 12 q^{49} + 24 q^{79} - 12 q^{91}+O(q^{100}) \) 48 * q + 48 * q^25 - 24 * q^43 - 12 * q^49 + 24 * q^79 - 12 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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