LMFDB - Embedded newform 1062.5.d.b.235.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1062,5,Mod(235,1062)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1062, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("1062.235");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 1062 = 2 \cdot 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1062.d (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:	$109.778900795$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 354)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			235.19
Character	$\chi$	$=$	1062.235
Dual form			1062.5.d.b.235.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} -13.0356 q^{5} +71.9157 q^{7} +22.6274i q^{8} +O(q^{10})$	\(q-2.82843i q^{2} -8.00000 q^{4} -13.0356 q^{5} +71.9157 q^{7} +22.6274i q^{8} +36.8702i q^{10} +76.6106i q^{11} -118.606i q^{13} -203.408i q^{14} +64.0000 q^{16} -31.5143 q^{17} +461.021 q^{19} +104.285 q^{20} +216.687 q^{22} +901.986i q^{23} -455.073 q^{25} -335.469 q^{26} -575.325 q^{28} +1161.22 q^{29} -457.352i q^{31} -181.019i q^{32} +89.1358i q^{34} -937.463 q^{35} +440.417i q^{37} -1303.96i q^{38} -294.962i q^{40} -2196.09 q^{41} -1247.28i q^{43} -612.885i q^{44} +2551.20 q^{46} +1319.13i q^{47} +2770.86 q^{49} +1287.14i q^{50} +948.848i q^{52} -53.4409 q^{53} -998.664i q^{55} +1627.27i q^{56} -3284.41i q^{58} +(-729.257 - 3403.75i) q^{59} +235.014i q^{61} -1293.59 q^{62} -512.000 q^{64} +1546.10i q^{65} +5947.06i q^{67} +252.114 q^{68} +2651.55i q^{70} -7826.66 q^{71} +8267.19i q^{73} +1245.69 q^{74} -3688.17 q^{76} +5509.50i q^{77} +519.406 q^{79} -834.278 q^{80} +6211.47i q^{82} +6918.96i q^{83} +410.807 q^{85} -3527.85 q^{86} -1733.50 q^{88} -1209.41i q^{89} -8529.63i q^{91} -7215.88i q^{92} +3731.07 q^{94} -6009.69 q^{95} +10730.3i q^{97} -7837.18i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 320 q^{4} - 80 q^{7}+O(q^{10}) $ 40 * q - 320 * q^4 - 80 * q^7	$ 40 q - 320 q^{4} - 80 q^{7} + 2560 q^{16} - 480 q^{17} - 792 q^{19} - 1024 q^{22} + 3400 q^{25} - 768 q^{26} + 640 q^{28} - 1608 q^{29} + 5760 q^{35} - 6264 q^{41} + 7040 q^{46} + 17912 q^{49} + 1104 q^{53} - 13584 q^{59} + 12288 q^{62} - 20480 q^{64} + 3840 q^{68} - 35352 q^{71} - 4608 q^{74} + 6336 q^{76} - 15720 q^{79} - 26872 q^{85} - 18432 q^{86} + 8192 q^{88} - 18432 q^{94} + 19128 q^{95}+O(q^{100}) $ 40 * q - 320 * q^4 - 80 * q^7 + 2560 * q^16 - 480 * q^17 - 792 * q^19 - 1024 * q^22 + 3400 * q^25 - 768 * q^26 + 640 * q^28 - 1608 * q^29 + 5760 * q^35 - 6264 * q^41 + 7040 * q^46 + 17912 * q^49 + 1104 * q^53 - 13584 * q^59 + 12288 * q^62 - 20480 * q^64 + 3840 * q^68 - 35352 * q^71 - 4608 * q^74 + 6336 * q^76 - 15720 * q^79 - 26872 * q^85 - 18432 * q^86 + 8192 * q^88 - 18432 * q^94 + 19128 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1062\mathbb{Z}\right)^\times$.

$n$	$119$	$415$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	2.82843i		−	0.707107i
$2$		−	2.82843i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−8.00000			−0.500000
$5$	−13.0356			−0.521424			−0.260712	−	0.965417i	$-0.583957\pi$
$5$	−13.0356			−0.521424			−0.260712	+	0.965417i	$0.583957\pi$
$6$		0			0
$7$	71.9157			1.46767			0.733833	−	0.679330i	$-0.237729\pi$
$7$	71.9157			1.46767			0.733833	+	0.679330i	$0.237729\pi$
$8$			22.6274i			0.353553i
$9$		0			0
$10$			36.8702i			0.368702i
$11$			76.6106i			0.633145i	0.948568	+	0.316573i	$0.102532\pi$
$11$			76.6106i			0.633145i	−0.948568	+	0.316573i	$0.897468\pi$
$12$		0			0
$13$		−	118.606i		−	0.701811i	−0.936411	−	0.350905i	$-0.885874\pi$
$13$		−	118.606i		−	0.701811i	0.936411	−	0.350905i	$-0.114126\pi$
$14$		−	203.408i		−	1.03780i
$15$		0			0
$16$	64.0000			0.250000
$17$	−31.5143			−0.109046			−0.0545230	−	0.998513i	$-0.517364\pi$
$17$	−31.5143			−0.109046			−0.0545230	+	0.998513i	$0.517364\pi$
$18$		0			0
$19$	461.021			1.27707			0.638534	−	0.769594i	$-0.279541\pi$
$19$	461.021			1.27707			0.638534	+	0.769594i	$0.279541\pi$
$20$	104.285			0.260712
$21$		0			0
$22$	216.687			0.447701
$23$			901.986i			1.70508i	0.522665	+	0.852538i	$0.324938\pi$
$23$			901.986i			1.70508i	−0.522665	+	0.852538i	$0.675062\pi$
$24$		0			0
$25$	−455.073			−0.728117
$26$	−335.469			−0.496255
$27$		0			0
$28$	−575.325			−0.733833
$29$	1161.22			1.38076			0.690378	−	0.723449i	$-0.257444\pi$
$29$	1161.22			1.38076			0.690378	+	0.723449i	$0.257444\pi$
$30$		0			0
$31$		−	457.352i		−	0.475912i	−0.971276	−	0.237956i	$-0.923523\pi$
$31$		−	457.352i		−	0.475912i	0.971276	−	0.237956i	$-0.0764774\pi$
$32$		−	181.019i		−	0.176777i
$33$		0			0
$34$			89.1358i			0.0771071i
$35$	−937.463			−0.765276
$36$		0			0
$37$			440.417i			0.321707i	0.986978	+	0.160853i	$0.0514247\pi$
$37$			440.417i			0.321707i	−0.986978	+	0.160853i	$0.948575\pi$
$38$		−	1303.96i		−	0.903023i
$39$		0			0
$40$		−	294.962i		−	0.184351i
$41$	−2196.09			−1.30642			−0.653208	−	0.757179i	$-0.726577\pi$
$41$	−2196.09			−1.30642			−0.653208	+	0.757179i	$0.726577\pi$
$42$		0			0
$43$		−	1247.28i		−	0.674572i	−0.941402	−	0.337286i	$-0.890491\pi$
$43$		−	1247.28i		−	0.674572i	0.941402	−	0.337286i	$-0.109509\pi$
$44$		−	612.885i		−	0.316573i
$45$		0			0
$46$	2551.20			1.20567
$47$			1319.13i			0.597162i	0.954384	+	0.298581i	$0.0965133\pi$
$47$			1319.13i			0.597162i	−0.954384	+	0.298581i	$0.903487\pi$
$48$		0			0
$49$	2770.86			1.15405
$50$			1287.14i			0.514857i
$51$		0			0
$52$			948.848i			0.350905i
$53$	−53.4409			−0.0190249			−0.00951244	−	0.999955i	$-0.503028\pi$
$53$	−53.4409			−0.0190249			−0.00951244	+	0.999955i	$0.503028\pi$
$54$		0			0
$55$		−	998.664i		−	0.330137i
$56$			1627.27i			0.518899i
$57$		0			0
$58$		−	3284.41i		−	0.976341i
$59$	−729.257	−	3403.75i	−0.209496	−	0.977809i
$59$	−729.257	−	3403.75i	−0.209496	−	0.977809i
$60$		0			0
$61$			235.014i			0.0631589i	0.999501	+	0.0315794i	$0.0100537\pi$
$61$			235.014i			0.0631589i	−0.999501	+	0.0315794i	$0.989946\pi$
$62$	−1293.59			−0.336521
$63$		0			0
$64$	−512.000			−0.125000
$65$			1546.10i			0.365941i
$66$		0			0
$67$			5947.06i			1.32481i	0.749147	+	0.662404i	$0.230464\pi$
$67$			5947.06i			1.32481i	−0.749147	+	0.662404i	$0.769536\pi$
$68$	252.114			0.0545230
$69$		0			0
$70$			2651.55i			0.541132i
$71$	−7826.66			−1.55260			−0.776300	−	0.630363i	$-0.782906\pi$
$71$	−7826.66			−1.55260			−0.776300	+	0.630363i	$0.782906\pi$
$72$		0			0
$73$			8267.19i			1.55136i	0.631127	+	0.775680i	$0.282593\pi$
$73$			8267.19i			1.55136i	−0.631127	+	0.775680i	$0.717407\pi$
$74$	1245.69			0.227481
$75$		0			0
$76$	−3688.17			−0.638534
$77$			5509.50i			0.929246i
$78$		0			0
$79$	519.406			0.0832248			0.0416124	−	0.999134i	$-0.486751\pi$
$79$	519.406			0.0832248			0.0416124	+	0.999134i	$0.486751\pi$
$80$	−834.278			−0.130356
$81$		0			0
$82$			6211.47i			0.923776i
$83$			6918.96i			1.00435i	0.864766	+	0.502174i	$0.167466\pi$
$83$			6918.96i			1.00435i	−0.864766	+	0.502174i	$0.832534\pi$
$84$		0			0
$85$	410.807			0.0568591
$86$	−3527.85			−0.476994
$87$		0			0
$88$	−1733.50			−0.223851
$89$		−	1209.41i		−	0.152684i	−0.997082	−	0.0763422i	$-0.975676\pi$
$89$		−	1209.41i		−	0.152684i	0.997082	−	0.0763422i	$-0.0243241\pi$
$90$		0			0
$91$		−	8529.63i		−	1.03002i
$92$		−	7215.88i		−	0.852538i
$93$		0			0
$94$	3731.07			0.422257
$95$	−6009.69			−0.665893
$96$		0			0
$97$			10730.3i			1.14043i	0.821497	+	0.570213i	$0.193139\pi$
$97$			10730.3i			1.14043i	−0.821497	+	0.570213i	$0.806861\pi$
$98$		−	7837.18i		−	0.816033i
$99$		0			0
$100$	3640.59			0.364059
$101$		−	3057.38i		−	0.299714i	−0.988708	−	0.149857i	$-0.952119\pi$
$101$		−	3057.38i		−	0.299714i	0.988708	−	0.149857i	$-0.0478813\pi$
$102$		0			0
$103$			413.512i			0.0389775i	0.999810	+	0.0194887i	$0.00620385\pi$
$103$			413.512i			0.0389775i	−0.999810	+	0.0194887i	$0.993796\pi$
$104$	2683.75			0.248128
$105$		0			0
$106$			151.154i			0.0134526i
$107$	488.624			0.0426783			0.0213391	−	0.999772i	$-0.493207\pi$
$107$	488.624			0.0426783			0.0213391	+	0.999772i	$0.493207\pi$
$108$		0			0
$109$			12377.5i			1.04179i	0.853620	+	0.520896i	$0.174402\pi$
$109$			12377.5i			1.04179i	−0.853620	+	0.520896i	$0.825598\pi$
$110$	−2824.65			−0.233442
$111$		0			0
$112$	4602.60			0.366917
$113$		−	58.5659i		−	0.00458657i	−0.999997	−	0.00229328i	$-0.999270\pi$
$113$		−	58.5659i		−	0.00458657i	0.999997	−	0.00229328i	$-0.000729976\pi$
$114$		0			0
$115$		−	11757.9i		−	0.889067i
$116$	−9289.72			−0.690378
$117$		0			0
$118$	−9627.27	+	2062.65i	−0.691416	+	0.148136i
$119$	−2266.37			−0.160043
$120$		0			0
$121$	8771.82			0.599127
$122$	664.720			0.0446601
$123$		0			0
$124$			3658.81i			0.237956i
$125$	14079.4			0.901081
$126$		0			0
$127$	6961.62			0.431621			0.215811	−	0.976435i	$-0.430761\pi$
$127$	6961.62			0.431621			0.215811	+	0.976435i	$0.430761\pi$
$128$			1448.15i			0.0883883i
$129$		0			0
$130$	4373.03			0.258759
$131$		−	13594.7i		−	0.792185i	−0.918211	−	0.396092i	$-0.870366\pi$
$131$		−	13594.7i		−	0.792185i	0.918211	−	0.396092i	$-0.129634\pi$
$132$		0			0
$133$	33154.6			1.87431
$134$	16820.8			0.936780
$135$		0			0
$136$		−	713.087i		−	0.0385536i
$137$	21151.7			1.12695			0.563475	−	0.826133i	$-0.309464\pi$
$137$	21151.7			1.12695			0.563475	+	0.826133i	$0.309464\pi$
$138$		0			0
$139$	20763.3			1.07465			0.537324	−	0.843376i	$-0.319435\pi$
$139$	20763.3			1.07465			0.537324	+	0.843376i	$0.319435\pi$
$140$	7499.71			0.382638
$141$		0			0
$142$			22137.1i			1.09785i
$143$	9086.48			0.444348
$144$		0			0
$145$	−15137.1			−0.719959
$146$	23383.2			1.09698
$147$		0			0
$148$		−	3523.33i		−	0.160853i
$149$			6501.17i			0.292832i	0.989223	+	0.146416i	$0.0467738\pi$
$149$			6501.17i			0.292832i	−0.989223	+	0.146416i	$0.953226\pi$
$150$		0			0
$151$		−	4605.38i		−	0.201982i	−0.994887	−	0.100991i	$-0.967799\pi$
$151$		−	4605.38i		−	0.201982i	0.994887	−	0.100991i	$-0.0322013\pi$
$152$			10431.7i			0.451511i
$153$		0			0
$154$	15583.2			0.657076
$155$			5961.85i			0.248152i
$156$		0			0
$157$			8564.91i			0.347475i	0.984792	+	0.173737i	$0.0555844\pi$
$157$			8564.91i			0.347475i	−0.984792	+	0.173737i	$0.944416\pi$
$158$		−	1469.10i		−	0.0588488i
$159$		0			0
$160$			2359.69i			0.0921756i
$161$			64866.9i			2.50248i
$162$		0			0
$163$	−1698.22			−0.0639173			−0.0319587	−	0.999489i	$-0.510174\pi$
$163$	−1698.22			−0.0639173			−0.0319587	+	0.999489i	$0.510174\pi$
$164$	17568.7			0.653208
$165$		0			0
$166$	19569.8			0.710182
$167$	40686.7			1.45888			0.729439	−	0.684046i	$-0.239781\pi$
$167$	40686.7			1.45888			0.729439	+	0.684046i	$0.239781\pi$
$168$		0			0
$169$	14493.6			0.507462
$170$		−	1161.94i		−	0.0402055i
$171$		0			0
$172$			9978.26i			0.337286i
$173$		−	373.836i		−	0.0124908i	−0.999980	−	0.00624538i	$-0.998012\pi$
$173$		−	373.836i		−	0.0124908i	0.999980	−	0.00624538i	$-0.00198798\pi$
$174$		0			0
$175$	−32726.9			−1.06863
$176$			4903.08i			0.158286i
$177$		0			0
$178$	−3420.74			−0.107964
$179$			48418.3i			1.51114i	0.655071	+	0.755568i	$0.272639\pi$
$179$			48418.3i			1.51114i	−0.655071	+	0.755568i	$0.727361\pi$
$180$		0			0
$181$	−46967.3			−1.43363			−0.716817	−	0.697262i	$-0.754402\pi$
$181$	−46967.3			−1.43363			−0.716817	+	0.697262i	$0.754402\pi$
$182$	−24125.4			−0.728337
$183$		0			0
$184$	−20409.6			−0.602836
$185$		−	5741.09i		−	0.167746i
$186$		0			0
$187$		−	2414.33i		−	0.0690419i
$188$		−	10553.0i		−	0.298581i
$189$		0			0
$190$			16998.0i			0.470858i
$191$		−	35697.0i		−	0.978509i	−0.872141	−	0.489255i	$-0.837269\pi$
$191$		−	35697.0i		−	0.978509i	0.872141	−	0.489255i	$-0.162731\pi$
$192$		0			0
$193$	25170.3			0.675730			0.337865	−	0.941195i	$-0.390295\pi$
$193$	25170.3			0.675730			0.337865	+	0.941195i	$0.390295\pi$
$194$	30349.8			0.806403
$195$		0			0
$196$	−22166.9			−0.577023
$197$	68867.3			1.77452			0.887260	−	0.461270i	$-0.152606\pi$
$197$	68867.3			1.77452			0.887260	+	0.461270i	$0.152606\pi$
$198$		0			0
$199$	−37635.2			−0.950361			−0.475180	−	0.879888i	$-0.657617\pi$
$199$	−37635.2			−0.950361			−0.475180	+	0.879888i	$0.657617\pi$
$200$		−	10297.1i		−	0.257428i
$201$		0			0
$202$	−8647.57			−0.211929
$203$	83509.6			2.02649
$204$		0			0
$205$	28627.3			0.681196
$206$	1169.59			0.0275612
$207$		0			0
$208$		−	7590.79i		−	0.175453i
$209$			35319.1i			0.808569i
$210$		0			0
$211$		−	46880.1i		−	1.05299i	−0.850179	−	0.526494i	$-0.823506\pi$
$211$		−	46880.1i		−	1.05299i	0.850179	−	0.526494i	$-0.176494\pi$
$212$	427.527			0.00951244
$213$		0			0
$214$		−	1382.04i		−	0.0301781i
$215$			16259.1i			0.351738i
$216$		0			0
$217$		−	32890.7i		−	0.698480i
$218$	35008.9			0.736658
$219$		0			0
$220$			7989.31i			0.165068i
$221$			3737.78i			0.0765296i
$222$		0			0
$223$	53767.4			1.08121			0.540604	−	0.841277i	$-0.318196\pi$
$223$	53767.4			1.08121			0.540604	+	0.841277i	$0.318196\pi$
$224$		−	13018.1i		−	0.259449i
$225$		0			0
$226$	−165.649			−0.00324319
$227$		−	13393.3i		−	0.259918i	−0.991519	−	0.129959i	$-0.958515\pi$
$227$		−	13393.3i		−	0.259918i	0.991519	−	0.129959i	$-0.0414846\pi$
$228$		0			0
$229$			5622.87i			0.107223i	0.998562	+	0.0536114i	$0.0170732\pi$
$229$			5622.87i			0.107223i	−0.998562	+	0.0536114i	$0.982927\pi$
$230$	−33256.4			−0.628666
$231$		0			0
$232$			26275.3i			0.488171i
$233$			104467.i			1.92428i	0.272549	+	0.962142i	$0.412133\pi$
$233$			104467.i			1.92428i	−0.272549	+	0.962142i	$0.587867\pi$
$234$		0			0
$235$		−	17195.7i		−	0.311374i
$236$	5834.05	+	27230.0i	0.104748	+	0.488905i
$237$		0			0
$238$			6410.26i			0.113168i
$239$	84162.7			1.47341			0.736706	−	0.676213i	$-0.236380\pi$
$239$	84162.7			1.47341			0.736706	+	0.676213i	$0.236380\pi$
$240$		0			0
$241$	−53479.0			−0.920766			−0.460383	−	0.887720i	$-0.652288\pi$
$241$	−53479.0			−0.920766			−0.460383	+	0.887720i	$0.652288\pi$
$242$		−	24810.5i		−	0.423647i
$243$		0			0
$244$		−	1880.11i		−	0.0315794i
$245$	−36119.8			−0.601747
$246$		0			0
$247$		−	54679.9i		−	0.896260i
$248$	10348.7			0.168260
$249$		0			0
$250$		−	39822.5i		−	0.637161i
$251$	−8610.91			−0.136679			−0.0683395	−	0.997662i	$-0.521770\pi$
$251$	−8610.91			−0.136679			−0.0683395	+	0.997662i	$0.521770\pi$
$252$		0			0
$253$	−69101.6			−1.07956
$254$		−	19690.4i		−	0.305202i
$255$		0			0
$256$	4096.00			0.0625000
$257$	24174.3			0.366005			0.183003	−	0.983112i	$-0.441418\pi$
$257$	24174.3			0.366005			0.183003	+	0.983112i	$0.441418\pi$
$258$		0			0
$259$			31672.9i			0.472159i
$260$		−	12368.8i		−	0.182970i
$261$		0			0
$262$	−38451.6			−0.560159
$263$	−15619.3			−0.225814			−0.112907	−	0.993606i	$-0.536016\pi$
$263$	−15619.3			−0.225814			−0.112907	+	0.993606i	$0.536016\pi$
$264$		0			0
$265$	696.634			0.00992002
$266$		−	93775.5i		−	1.32534i
$267$		0			0
$268$		−	47576.5i		−	0.662404i
$269$		−	91681.8i		−	1.26701i	−0.773741	−	0.633503i	$-0.781617\pi$
$269$		−	91681.8i		−	1.26701i	0.773741	−	0.633503i	$-0.218383\pi$
$270$		0			0
$271$	−23909.2			−0.325557			−0.162779	−	0.986663i	$-0.552046\pi$
$271$	−23909.2			−0.325557			−0.162779	+	0.986663i	$0.552046\pi$
$272$	−2016.91			−0.0272615
$273$		0			0
$274$		−	59826.1i		−	0.796874i
$275$		−	34863.4i		−	0.461004i
$276$		0			0
$277$	145442.			1.89553			0.947767	−	0.318964i	$-0.103335\pi$
$277$	145442.			1.89553			0.947767	+	0.318964i	$0.103335\pi$
$278$		−	58727.4i		−	0.759891i
$279$		0			0
$280$		−	21212.4i		−	0.270566i
$281$	50468.8			0.639161			0.319581	−	0.947559i	$-0.396458\pi$
$281$	50468.8			0.639161			0.319581	+	0.947559i	$0.396458\pi$
$282$		0			0
$283$			148907.i			1.85927i	0.368488	+	0.929633i	$0.379876\pi$
$283$			148907.i			1.85927i	−0.368488	+	0.929633i	$0.620124\pi$
$284$	62613.3			0.776300
$285$		0			0
$286$		−	25700.4i		−	0.314202i
$287$	−157933.			−1.91738
$288$		0			0
$289$	−82527.9			−0.988109
$290$			42814.3i			0.509088i
$291$		0			0
$292$		−	66137.5i		−	0.775680i
$293$	−63017.4			−0.734049			−0.367024	−	0.930211i	$-0.619623\pi$
$293$	−63017.4			−0.734049			−0.367024	+	0.930211i	$0.619623\pi$
$294$		0			0
$295$	9506.29	+	44370.0i	0.109236	+	0.509853i
$296$	−9965.49			−0.113741
$297$		0			0
$298$	18388.1			0.207064
$299$	106981.			1.19664
$300$		0			0
$301$		−	89699.2i		−	0.990046i
$302$	−13026.0			−0.142823
$303$		0			0
$304$	29505.4			0.319267
$305$		−	3063.55i		−	0.0329325i
$306$		0			0
$307$	9766.49			0.103624			0.0518122	−	0.998657i	$-0.483500\pi$
$307$	9766.49			0.103624			0.0518122	+	0.998657i	$0.483500\pi$
$308$		−	44076.0i		−	0.464623i
$309$		0			0
$310$	16862.7			0.175470
$311$	−57858.3			−0.598198			−0.299099	−	0.954222i	$-0.596686\pi$
$311$	−57858.3			−0.598198			−0.299099	+	0.954222i	$0.596686\pi$
$312$		0			0
$313$			167799.i			1.71278i	0.516331	+	0.856389i	$0.327297\pi$
$313$			167799.i			1.71278i	−0.516331	+	0.856389i	$0.672703\pi$
$314$	24225.2			0.245702
$315$		0			0
$316$	−4155.25			−0.0416124
$317$	−45806.8			−0.455839			−0.227919	−	0.973680i	$-0.573192\pi$
$317$	−45806.8			−0.455839			−0.227919	+	0.973680i	$0.573192\pi$
$318$		0			0
$319$			88961.4i			0.874219i
$320$	6674.22			0.0651780
$321$		0			0
$322$	183471.			1.76952
$323$	−14528.8			−0.139259
$324$		0			0
$325$			53974.4i			0.511001i
$326$			4803.29i			0.0451964i
$327$		0			0
$328$		−	49691.7i		−	0.461888i
$329$			94866.2i			0.876435i
$330$		0			0
$331$	161647.			1.47540			0.737702	−	0.675127i	$-0.235911\pi$
$331$	161647.			1.47540			0.737702	+	0.675127i	$0.235911\pi$
$332$		−	55351.7i		−	0.502174i
$333$		0			0
$334$		−	115079.i		−	1.03158i
$335$		−	77523.5i		−	0.690786i
$336$		0			0
$337$		−	62124.9i		−	0.547024i	−0.961869	−	0.273512i	$-0.911815\pi$
$337$		−	62124.9i		−	0.547024i	0.961869	−	0.273512i	$-0.0881853\pi$
$338$		−	40994.1i		−	0.358829i
$339$		0			0
$340$	−3286.46			−0.0284296
$341$	35038.0			0.301321
$342$		0			0
$343$	26598.9			0.226087
$344$	28222.8			0.238497
$345$		0			0
$346$	−1057.37			−0.00883230
$347$		−	14358.0i		−	0.119243i	−0.998221	−	0.0596217i	$-0.981011\pi$
$347$		−	14358.0i		−	0.119243i	0.998221	−	0.0596217i	$-0.0189894\pi$
$348$		0			0
$349$			169000.i			1.38751i	0.720212	+	0.693754i	$0.244045\pi$
$349$			169000.i			1.38751i	−0.720212	+	0.693754i	$0.755955\pi$
$350$			92565.7i			0.755638i
$351$		0			0
$352$	13868.0			0.111925
$353$			187823.i			1.50730i	0.657278	+	0.753648i	$0.271708\pi$
$353$			187823.i			1.50730i	−0.657278	+	0.753648i	$0.728292\pi$
$354$		0			0
$355$	102025.			0.809563
$356$			9675.30i			0.0763422i
$357$		0			0
$358$	136948.			1.06853
$359$	203966.			1.58259			0.791297	−	0.611432i	$-0.209406\pi$
$359$	203966.			1.58259			0.791297	+	0.611432i	$0.209406\pi$
$360$		0			0
$361$	82219.6			0.630900
$362$			132843.i			1.01373i
$363$		0			0
$364$			68237.1i			0.515012i
$365$		−	107768.i		−	0.808915i
$366$		0			0
$367$		−	21743.1i		−	0.161432i	−0.996737	−	0.0807159i	$-0.974279\pi$
$367$		−	21743.1i		−	0.161432i	0.996737	−	0.0807159i	$-0.0257207\pi$
$368$			57727.1i			0.426269i
$369$		0			0
$370$	−16238.3			−0.118614
$371$	−3843.24			−0.0279222
$372$		0			0
$373$	47541.9			0.341711			0.170855	−	0.985296i	$-0.445347\pi$
$373$	47541.9			0.341711			0.170855	+	0.985296i	$0.445347\pi$
$374$	−6828.75			−0.0488200
$375$		0			0
$376$	−29848.5			−0.211129
$377$		−	137727.i		−	0.969029i
$378$		0			0
$379$	−87143.6			−0.606676			−0.303338	−	0.952883i	$-0.598101\pi$
$379$	−87143.6			−0.606676			−0.303338	+	0.952883i	$0.598101\pi$
$380$	48077.5			0.332947
$381$		0			0
$382$	−100966.			−0.691910
$383$	20889.8			0.142408			0.0712042	−	0.997462i	$-0.477316\pi$
$383$	20889.8			0.142408			0.0712042	+	0.997462i	$0.477316\pi$
$384$		0			0
$385$		−	71819.6i		−	0.484531i
$386$		−	71192.2i		−	0.477813i
$387$		0			0
$388$		−	85842.2i		−	0.570213i
$389$	226227.			1.49501			0.747506	−	0.664255i	$-0.231251\pi$
$389$	226227.			1.49501			0.747506	+	0.664255i	$0.231251\pi$
$390$		0			0
$391$		−	28425.4i		−	0.185932i
$392$			62697.5i			0.408017i
$393$		0			0
$394$		−	194786.i		−	1.25477i
$395$	−6770.77			−0.0433954
$396$		0			0
$397$			258128.i			1.63777i	0.573956	+	0.818886i	$0.305408\pi$
$397$			258128.i			1.63777i	−0.573956	+	0.818886i	$0.694592\pi$
$398$			106449.i			0.672006i
$399$		0			0
$400$	−29124.7			−0.182029
$401$			121353.i			0.754678i	0.926075	+	0.377339i	$0.123161\pi$
$401$			121353.i			0.754678i	−0.926075	+	0.377339i	$0.876839\pi$
$402$		0			0
$403$	−54244.6			−0.334000
$404$			24459.0i			0.149857i
$405$		0			0
$406$		−	236201.i		−	1.43294i
$407$	−33740.6			−0.203687
$408$		0			0
$409$			127039.i			0.759434i	0.925103	+	0.379717i	$0.123979\pi$
$409$			127039.i			0.759434i	−0.925103	+	0.379717i	$0.876021\pi$
$410$		−	80970.2i		−	0.481678i
$411$		0			0
$412$		−	3308.10i		−	0.0194887i
$413$	−52445.0	−	244783.i	−0.307471	−	1.43510i
$414$		0			0
$415$		−	90192.7i		−	0.523691i
$416$	−21470.0			−0.124064
$417$		0			0
$418$	99897.5			0.571745
$419$		−	66337.2i		−	0.377858i	−0.981991	−	0.188929i	$-0.939498\pi$
$419$		−	66337.2i		−	0.377858i	0.981991	−	0.188929i	$-0.0605017\pi$
$420$		0			0
$421$		−	316689.i		−	1.78677i	−0.449290	−	0.893386i	$-0.648323\pi$
$421$		−	316689.i		−	1.78677i	0.449290	−	0.893386i	$-0.351677\pi$
$422$	−132597.			−0.744575
$423$		0			0
$424$		−	1209.23i		−	0.00672631i
$425$	14341.3			0.0793982
$426$		0			0
$427$			16901.2i			0.0926962i
$428$	−3908.99			−0.0213391
$429$		0			0
$430$	45987.6			0.248716
$431$		−	19159.7i		−	0.103142i	−0.998669	−	0.0515709i	$-0.983577\pi$
$431$		−	19159.7i		−	0.103142i	0.998669	−	0.0515709i	$-0.0164228\pi$
$432$		0			0
$433$	59077.3			0.315097			0.157549	−	0.987511i	$-0.449641\pi$
$433$	59077.3			0.315097			0.157549	+	0.987511i	$0.449641\pi$
$434$	−93029.1			−0.493900
$435$		0			0
$436$		−	99020.2i		−	0.520896i
$437$			415834.i			2.17750i
$438$		0			0
$439$	16622.3			0.0862507			0.0431253	−	0.999070i	$-0.486269\pi$
$439$	16622.3			0.0862507			0.0431253	+	0.999070i	$0.486269\pi$
$440$	22597.2			0.116721
$441$		0			0
$442$	10572.0			0.0541146
$443$		−	166081.i		−	0.846278i	−0.906065	−	0.423139i	$-0.860928\pi$
$443$		−	166081.i		−	0.846278i	0.906065	−	0.423139i	$-0.139072\pi$
$444$		0			0
$445$			15765.4i			0.0796132i
$446$		−	152077.i		−	0.764530i
$447$		0			0
$448$	−36820.8			−0.183458
$449$	−231721.			−1.14940			−0.574702	−	0.818363i	$-0.694882\pi$
$449$	−231721.			−1.14940			−0.574702	+	0.818363i	$0.694882\pi$
$450$		0			0
$451$		−	168243.i		−	0.827151i
$452$			468.527i			0.00229328i
$453$		0			0
$454$	−37882.1			−0.183790
$455$			111189.i			0.537079i
$456$		0			0
$457$		−	58668.2i		−	0.280912i	−0.990087	−	0.140456i	$-0.955143\pi$
$457$		−	58668.2i		−	0.280912i	0.990087	−	0.140456i	$-0.0448569\pi$
$458$	15903.9			0.0758179
$459$		0			0
$460$			94063.3i			0.444534i
$461$	−92384.8			−0.434709			−0.217355	−	0.976093i	$-0.569743\pi$
$461$	−92384.8			−0.434709			−0.217355	+	0.976093i	$0.569743\pi$
$462$		0			0
$463$			348577.i			1.62606i	0.582220	+	0.813031i	$0.302184\pi$
$463$			348577.i			1.62606i	−0.582220	+	0.813031i	$0.697816\pi$
$464$	74317.8			0.345189
$465$		0			0
$466$	295479.			1.36067
$467$		−	270751.i		−	1.24147i	−0.784021	−	0.620734i	$-0.786835\pi$
$467$		−	270751.i		−	1.24147i	0.784021	−	0.620734i	$-0.213165\pi$
$468$		0			0
$469$			427687.i			1.94438i
$470$	−48636.7			−0.220175
$471$		0			0
$472$	77018.2	−	16501.2i	0.345708	−	0.0740681i
$473$	95555.1			0.427102
$474$		0			0
$475$	−209798.			−0.929855
$476$	18131.0			0.0800216
$477$		0			0
$478$		−	238048.i		−	1.04186i
$479$	−265996.			−1.15932			−0.579660	−	0.814858i	$-0.696815\pi$
$479$	−265996.			−1.15932			−0.579660	+	0.814858i	$0.696815\pi$
$480$		0			0
$481$	52236.1			0.225777
$482$			151262.i			0.651080i
$483$		0			0
$484$	−70174.6			−0.299564
$485$		−	139875.i		−	0.594645i
$486$		0			0
$487$	179950.			0.758743			0.379371	−	0.925245i	$-0.376140\pi$
$487$	179950.			0.758743			0.379371	+	0.925245i	$0.376140\pi$
$488$	−5317.76			−0.0223300
$489$		0			0
$490$			102162.i			0.425499i
$491$	−180154.			−0.747277			−0.373639	−	0.927574i	$-0.621890\pi$
$491$	−180154.			−0.747277			−0.373639	+	0.927574i	$0.621890\pi$
$492$		0			0
$493$	−36594.9			−0.150566
$494$	−154658.			−0.633751
$495$		0			0
$496$		−	29270.5i		−	0.118978i
$497$	−562859.			−2.27870
$498$		0			0
$499$	348732.			1.40052			0.700261	−	0.713887i	$-0.253067\pi$
$499$	348732.			1.40052			0.700261	+	0.713887i	$0.253067\pi$
$500$	−112635.			−0.450541
$501$		0			0
$502$			24355.3i			0.0966466i
$503$		−	337691.i		−	1.33470i	−0.744744	−	0.667350i	$-0.767428\pi$
$503$		−	337691.i		−	1.33470i	0.744744	−	0.667350i	$-0.232572\pi$
$504$		0			0
$505$			39854.7i			0.156278i
$506$			195449.i			0.763365i
$507$		0			0
$508$	−55692.9			−0.215811
$509$			423913.i			1.63622i	0.575062	+	0.818110i	$0.304978\pi$
$509$			423913.i			1.63622i	−0.575062	+	0.818110i	$0.695022\pi$
$510$		0			0
$511$			594541.i			2.27688i
$512$		−	11585.2i		−	0.0441942i
$513$		0			0
$514$		−	68375.2i		−	0.258805i
$515$		−	5390.38i		−	0.0203238i
$516$		0			0
$517$	−101059.			−0.378090
$518$	89584.4			0.333867
$519$		0			0
$520$	−34984.2			−0.129380
$521$	−124425.			−0.458387			−0.229194	−	0.973381i	$-0.573609\pi$
$521$	−124425.			−0.458387			−0.229194	+	0.973381i	$0.573609\pi$
$522$		0			0
$523$	−515652.			−1.88518			−0.942591	−	0.333950i	$-0.891618\pi$
$523$	−515652.			−1.88518			−0.942591	+	0.333950i	$0.891618\pi$
$524$			108757.i			0.396092i
$525$		0			0
$526$			44178.1i			0.159675i
$527$			14413.1i			0.0518963i
$528$		0			0
$529$	−533737.			−1.90729
$530$		−	1970.38i		−	0.00701452i
$531$		0			0
$532$	−265237.			−0.937154
$533$			260469.i			0.916857i
$534$		0			0
$535$	−6369.50			−0.0222535
$536$	−134567.			−0.468390
$537$		0			0
$538$	−259315.			−0.895908
$539$			212277.i			0.730678i
$540$		0			0
$541$		−	273373.i		−	0.934032i	−0.884249	−	0.467016i	$-0.845329\pi$
$541$		−	273373.i		−	0.934032i	0.884249	−	0.467016i	$-0.154671\pi$
$542$			67625.5i			0.230204i
$543$		0			0
$544$			5704.69i			0.0192768i
$545$		−	161348.i		−	0.543215i
$546$		0			0
$547$	−382095.			−1.27702			−0.638508	−	0.769615i	$-0.720448\pi$
$547$	−382095.			−1.27702			−0.638508	+	0.769615i	$0.720448\pi$
$548$	−169214.			−0.563475
$549$		0			0
$550$	−98608.7			−0.325979
$551$	535345.			1.76332
$552$		0			0
$553$	37353.4			0.122146
$554$		−	411373.i		−	1.34034i
$555$		0			0
$556$	−166106.			−0.537324
$557$	−67859.1			−0.218724			−0.109362	−	0.994002i	$-0.534881\pi$
$557$	−67859.1			−0.218724			−0.109362	+	0.994002i	$0.534881\pi$
$558$		0			0
$559$	−147935.			−0.473422
$560$	−59997.7			−0.191319
$561$		0			0
$562$		−	142747.i		−	0.451955i
$563$		−	181249.i		−	0.571818i	−0.958257	−	0.285909i	$-0.907704\pi$
$563$		−	181249.i		−	0.571818i	0.958257	−	0.285909i	$-0.0922956\pi$
$564$		0			0
$565$			763.441i			0.00239155i
$566$	421172.			1.31470
$567$		0			0
$568$		−	177097.i		−	0.548927i
$569$		−	162103.i		−	0.500687i	−0.968157	−	0.250344i	$-0.919456\pi$
$569$		−	162103.i		−	0.500687i	0.968157	−	0.250344i	$-0.0805436\pi$
$570$		0			0
$571$			646936.i			1.98422i	0.125386	+	0.992108i	$0.459983\pi$
$571$			646936.i			1.98422i	−0.125386	+	0.992108i	$0.540017\pi$
$572$	−72691.8			−0.222174
$573$		0			0
$574$			446702.i			1.35579i
$575$		−	410470.i		−	1.24150i
$576$		0			0
$577$	−249691.			−0.749984			−0.374992	−	0.927028i	$-0.622355\pi$
$577$	−249691.			−0.749984			−0.374992	+	0.927028i	$0.622355\pi$
$578$			233424.i			0.698699i
$579$		0			0
$580$	121097.			0.359979
$581$			497581.i			1.47405i
$582$		0			0
$583$		−	4094.14i		−	0.0120455i
$584$	−187065.			−0.548488
$585$		0			0
$586$			178240.i			0.519051i
$587$			124990.i			0.362744i	0.983415	+	0.181372i	$0.0580539\pi$
$587$			124990.i			0.362744i	−0.983415	+	0.181372i	$0.941946\pi$
$588$		0			0
$589$		−	210849.i		−	0.607772i
$590$	125497.	−	26887.9i	0.360521	−	0.0772417i
$591$		0			0
$592$			28186.7i			0.0804267i
$593$	562164.			1.59865			0.799326	−	0.600898i	$-0.205190\pi$
$593$	562164.			1.59865			0.799326	+	0.600898i	$0.205190\pi$
$594$		0			0
$595$	29543.5			0.0834503
$596$		−	52009.3i		−	0.146416i
$597$		0			0
$598$		−	302588.i		−	0.846153i
$599$	128620.			0.358471			0.179235	−	0.983806i	$-0.442638\pi$
$599$	128620.			0.358471			0.179235	+	0.983806i	$0.442638\pi$
$600$		0			0
$601$		−	443026.i		−	1.22654i	−0.789874	−	0.613269i	$-0.789854\pi$
$601$		−	443026.i		−	1.22654i	0.789874	−	0.613269i	$-0.210146\pi$
$602$	−253708.			−0.700068
$603$		0			0
$604$			36843.1i			0.100991i
$605$	−114346.			−0.312399
$606$		0			0
$607$	174622.			0.473938			0.236969	−	0.971517i	$-0.423846\pi$
$607$	174622.			0.473938			0.236969	+	0.971517i	$0.423846\pi$
$608$		−	83453.8i		−	0.225756i
$609$		0			0
$610$	−8665.02			−0.0232868
$611$	156457.			0.419095
$612$		0			0
$613$		−	528180.i		−	1.40560i	−0.711389	−	0.702799i	$-0.751933\pi$
$613$		−	528180.i		−	1.40560i	0.711389	−	0.702799i	$-0.248067\pi$
$614$		−	27623.8i		−	0.0732735i
$615$		0			0
$616$	−124666.			−0.328538
$617$	−274885.			−0.722072			−0.361036	−	0.932552i	$-0.617577\pi$
$617$	−274885.			−0.722072			−0.361036	+	0.932552i	$0.617577\pi$
$618$		0			0
$619$	−372699.			−0.972695			−0.486347	−	0.873766i	$-0.661671\pi$
$619$	−372699.			−0.972695			−0.486347	+	0.873766i	$0.661671\pi$
$620$		−	47694.8i		−	0.124076i
$621$		0			0
$622$			163648.i			0.422990i
$623$		−	86975.7i		−	0.224090i
$624$		0			0
$625$	100888.			0.258272
$626$	474608.			1.21112
$627$		0			0
$628$		−	68519.3i		−	0.173737i
$629$		−	13879.4i		−	0.0350808i
$630$		0			0
$631$	−654254.			−1.64319			−0.821594	−	0.570072i	$-0.806915\pi$
$631$	−654254.			−1.64319			−0.821594	+	0.570072i	$0.806915\pi$
$632$			11752.8i			0.0294244i
$633$		0			0
$634$			129561.i			0.322327i
$635$	−90748.8			−0.225057
$636$		0			0
$637$		−	328641.i		−	0.809922i
$638$	251621.			0.618166
$639$		0			0
$640$		−	18877.6i		−	0.0460878i
$641$	−140139.			−0.341069			−0.170535	−	0.985352i	$-0.554549\pi$
$641$	−140139.			−0.341069			−0.170535	+	0.985352i	$0.554549\pi$
$642$		0			0
$643$	290047.			0.701531			0.350765	−	0.936463i	$-0.385921\pi$
$643$	290047.			0.701531			0.350765	+	0.936463i	$0.385921\pi$
$644$		−	518935.i		−	1.25124i
$645$		0			0
$646$			41093.5i			0.0984710i
$647$	−379096.			−0.905608			−0.452804	−	0.891610i	$-0.649576\pi$
$647$	−379096.			−0.905608			−0.452804	+	0.891610i	$0.649576\pi$
$648$		0			0
$649$	260764.	−	55868.8i	0.619095	−	0.132642i
$650$	152663.			0.361332
$651$		0			0
$652$	13585.8			0.0319587
$653$	−3248.56			−0.00761842			−0.00380921	−	0.999993i	$-0.501213\pi$
$653$	−3248.56			−0.00761842			−0.00380921	+	0.999993i	$0.501213\pi$
$654$		0			0
$655$			177215.i			0.413064i
$656$	−140549.			−0.326604
$657$		0			0
$658$	268322.			0.619733
$659$			530315.i			1.22113i	0.791965	+	0.610567i	$0.209058\pi$
$659$			530315.i			1.22113i	−0.791965	+	0.610567i	$0.790942\pi$
$660$		0			0
$661$	−241571.			−0.552894			−0.276447	−	0.961029i	$-0.589157\pi$
$661$	−241571.			−0.552894			−0.276447	+	0.961029i	$0.589157\pi$
$662$		−	457206.i		−	1.04327i
$663$		0			0
$664$	−156558.			−0.355091
$665$	−432191.			−0.977309
$666$		0			0
$667$			1.04740e6i			2.35429i
$668$	−325493.			−0.729439
$669$		0			0
$670$	−219269.			−0.488460
$671$	−18004.6			−0.0399887
$672$		0			0
$673$		−	720933.i		−	1.59171i	−0.605485	−	0.795857i	$-0.707021\pi$
$673$		−	720933.i		−	1.59171i	0.605485	−	0.795857i	$-0.292979\pi$
$674$	−175716.			−0.386804
$675$		0			0
$676$	−115949.			−0.253731
$677$	−53229.2			−0.116138			−0.0580688	−	0.998313i	$-0.518494\pi$
$677$	−53229.2			−0.116138			−0.0580688	+	0.998313i	$0.518494\pi$
$678$		0			0
$679$			771675.i			1.67377i
$680$			9295.51i			0.0201027i
$681$		0			0
$682$		−	99102.3i		−	0.213066i
$683$			116305.i			0.249320i	0.992200	+	0.124660i	$0.0397840\pi$
$683$			116305.i			0.249320i	−0.992200	+	0.124660i	$0.960216\pi$
$684$		0			0
$685$	−275725.			−0.587619
$686$		−	75233.2i		−	0.159868i
$687$		0			0
$688$		−	79826.1i		−	0.168643i
$689$			6338.41i			0.0133519i
$690$		0			0
$691$			363295.i			0.760858i	0.924810	+	0.380429i	$0.124224\pi$
$691$			363295.i			0.760858i	−0.924810	+	0.380429i	$0.875776\pi$
$692$			2990.69i			0.00624538i
$693$		0			0
$694$	−40610.5			−0.0843178
$695$	−270661.			−0.560347
$696$		0			0
$697$	69208.0			0.142459
$698$	478004.			0.981117
$699$		0			0
$700$	261815.			0.534317
$701$			795382.i			1.61860i	0.587395	+	0.809300i	$0.300153\pi$
$701$			795382.i			1.61860i	−0.587395	+	0.809300i	$0.699847\pi$
$702$		0			0
$703$			203042.i			0.410841i
$704$		−	39224.6i		−	0.0791432i
$705$		0			0
$706$	531243.			1.06582
$707$		−	219873.i		−	0.439880i
$708$		0			0
$709$	557121.			1.10830			0.554150	−	0.832417i	$-0.313044\pi$
$709$	557121.			1.10830			0.554150	+	0.832417i	$0.313044\pi$
$710$		−	288571.i		−	0.572447i
$711$		0			0
$712$	27365.9			0.0539821
$713$	412524.			0.811467
$714$		0			0
$715$	−118448.			−0.231694
$716$		−	387346.i		−	0.755568i
$717$		0			0
$718$		−	576904.i		−	1.11906i
$719$			755892.i			1.46218i	0.682279	+	0.731092i	$0.260989\pi$
$719$			755892.i			1.46218i	−0.682279	+	0.731092i	$0.739011\pi$
$720$		0			0
$721$			29738.0i			0.0572060i
$722$		−	232552.i		−	0.446114i
$723$		0			0
$724$	375738.			0.716817
$725$	−528438.			−1.00535
$726$		0			0
$727$	−496616.			−0.939620			−0.469810	−	0.882768i	$-0.655678\pi$
$727$	−496616.			−0.939620			−0.469810	+	0.882768i	$0.655678\pi$
$728$	193004.			0.364169
$729$		0			0
$730$	−304813.			−0.571990
$731$			39307.2i			0.0735593i
$732$		0			0
$733$	−456465.			−0.849570			−0.424785	−	0.905294i	$-0.639650\pi$
$733$	−456465.			−0.849570			−0.424785	+	0.905294i	$0.639650\pi$
$734$	−61498.7			−0.114150
$735$		0			0
$736$	163277.			0.301418
$737$	−455608.			−0.838796
$738$		0			0
$739$		−	264440.i		−	0.484214i	−0.970250	−	0.242107i	$-0.922161\pi$
$739$		−	264440.i		−	0.484214i	0.970250	−	0.242107i	$-0.0778386\pi$
$740$			45928.8i			0.0838728i
$741$		0			0
$742$			10870.3i			0.0197440i
$743$	100102.			0.181329			0.0906645	−	0.995881i	$-0.471101\pi$
$743$	100102.			0.181329			0.0906645	+	0.995881i	$0.471101\pi$
$744$		0			0
$745$		−	84746.5i		−	0.152690i
$746$		−	134469.i		−	0.241626i
$747$		0			0
$748$			19314.6i			0.0345210i
$749$	35139.7			0.0626375
$750$		0			0
$751$			112757.i			0.199923i	0.994991	+	0.0999616i	$0.0318720\pi$
$751$			112757.i			0.199923i	−0.994991	+	0.0999616i	$0.968128\pi$
$752$			84424.4i			0.149291i
$753$		0			0
$754$	−389551.			−0.685207
$755$			60033.9i			0.105318i
$756$		0			0
$757$	594305.			1.03709			0.518547	−	0.855049i	$-0.326473\pi$
$757$	594305.			1.03709			0.518547	+	0.855049i	$0.326473\pi$
$758$			246479.i			0.428985i
$759$		0			0
$760$		−	135984.i		−	0.235429i
$761$	493882.			0.852813			0.426407	−	0.904532i	$-0.359779\pi$
$761$	493882.			0.852813			0.426407	+	0.904532i	$0.359779\pi$
$762$		0			0
$763$			890138.i			1.52900i
$764$			285576.i			0.489255i
$765$		0			0
$766$		−	59085.1i		−	0.100698i
$767$	−403706.	+	86494.2i	−0.686237	+	0.147027i
$768$		0			0
$769$		−	526058.i		−	0.889572i	−0.895637	−	0.444786i	$-0.853280\pi$
$769$		−	526058.i		−	0.889572i	0.895637	−	0.444786i	$-0.146720\pi$
$770$	−203137.			−0.342615
$771$		0			0
$772$	−201362.			−0.337865
$773$			820920.i			1.37386i	0.726725	+	0.686929i	$0.241042\pi$
$773$			820920.i			1.37386i	−0.726725	+	0.686929i	$0.758958\pi$
$774$		0			0
$775$			208128.i			0.346520i
$776$	−242798.			−0.403202
$777$		0			0
$778$		−	639866.i		−	1.05713i
$779$	−1.01244e6			−1.66838
$780$		0			0
$781$		−	599605.i		−	0.983022i
$782$	−80399.2			−0.131474
$783$		0			0
$784$	177335.			0.288511
$785$		−	111649.i		−	0.181182i
$786$		0			0
$787$	38812.3			0.0626643			0.0313321	−	0.999509i	$-0.490025\pi$
$787$	38812.3			0.0626643			0.0313321	+	0.999509i	$0.490025\pi$
$788$	−550939.			−0.887260
$789$		0			0
$790$			19150.6i			0.0306852i
$791$		−	4211.80i		−	0.00673155i
$792$		0			0
$793$	27874.1			0.0443256
$794$	730095.			1.15808
$795$		0			0
$796$	301082.			0.475180
$797$		−	209085.i		−	0.329160i	−0.986364	−	0.164580i	$-0.947373\pi$
$797$		−	209085.i		−	0.329160i	0.986364	−	0.164580i	$-0.0526269\pi$
$798$		0			0
$799$		−	41571.5i		−	0.0651181i
$800$			82377.1i			0.128714i
$801$		0			0
$802$	343238.			0.533638
$803$	−633354.			−0.982236
$804$		0			0
$805$		−	845578.i		−	1.30485i
$806$			153427.i			0.236174i
$807$		0			0
$808$	69180.6			0.105965
$809$		−	1.28940e6i		−	1.97011i	−0.172233	−	0.985056i	$-0.555098\pi$
$809$		−	1.28940e6i		−	1.97011i	0.172233	−	0.985056i	$-0.444902\pi$
$810$		0			0
$811$			448438.i			0.681806i	0.940099	+	0.340903i	$0.110733\pi$
$811$			448438.i			0.681806i	−0.940099	+	0.340903i	$0.889267\pi$
$812$	−668077.			−1.01324
$813$		0			0
$814$			95432.8i			0.144029i
$815$	22137.3			0.0333280
$816$		0			0
$817$		−	575024.i		−	0.861473i
$818$	359320.			0.537001
$819$		0			0
$820$	−229018.			−0.340598
$821$			259012.i			0.384268i	0.981369	+	0.192134i	$0.0615408\pi$
$821$			259012.i			0.384268i	−0.981369	+	0.192134i	$0.938459\pi$
$822$		0			0
$823$		−	837577.i		−	1.23659i	−0.785947	−	0.618294i	$-0.787824\pi$
$823$		−	837577.i		−	1.23659i	0.785947	−	0.618294i	$-0.212176\pi$
$824$	−9356.71			−0.0137806
$825$		0			0
$826$	−692352.	+	148337.i	−1.01477	+	0.217415i
$827$	−739452.			−1.08118			−0.540592	−	0.841285i	$-0.681800\pi$
$827$	−739452.			−1.08118			−0.540592	+	0.841285i	$0.681800\pi$
$828$		0			0
$829$	1.21521e6			1.76824			0.884122	−	0.467257i	$-0.154758\pi$
$829$	1.21521e6			1.76824			0.884122	+	0.467257i	$0.154758\pi$
$830$	−255104.			−0.370306
$831$		0			0
$832$			60726.3i			0.0877264i
$833$	−87321.8			−0.125844
$834$		0			0
$835$	−530375.			−0.760694
$836$		−	282553.i		−	0.404285i
$837$		0			0
$838$	−187630.			−0.267186
$839$		−	1.25452e6i		−	1.78219i	−0.453818	−	0.891095i	$-0.649938\pi$
$839$		−	1.25452e6i		−	1.78219i	0.453818	−	0.891095i	$-0.350062\pi$
$840$		0			0
$841$	641140.			0.906485
$842$	−895732.			−1.26344
$843$		0			0
$844$			375040.i			0.526494i
$845$	−188933.			−0.264602
$846$		0			0
$847$	630831.			0.879319
$848$	−3420.22			−0.00475622
$849$		0			0
$850$		−	40563.3i		−	0.0561430i
$851$	−397250.			−0.548535
$852$		0			0
$853$	−860547.			−1.18270			−0.591352	−	0.806413i	$-0.701406\pi$
$853$	−860547.			−1.18270			−0.591352	+	0.806413i	$0.701406\pi$
$854$	47803.8			0.0655461
$855$		0			0
$856$			11056.3i			0.0150891i
$857$			250066.i			0.340481i	0.985403	+	0.170240i	$0.0544544\pi$
$857$			250066.i			0.340481i	−0.985403	+	0.170240i	$0.945546\pi$
$858$		0			0
$859$			1.05336e6i			1.42755i	0.700374	+	0.713776i	$0.253017\pi$
$859$			1.05336e6i			1.42755i	−0.700374	+	0.713776i	$0.746983\pi$
$860$		−	130073.i		−	0.175869i
$861$		0			0
$862$	−54191.9			−0.0729323
$863$			284135.i			0.381507i	0.981638	+	0.190754i	$0.0610931\pi$
$863$			284135.i			0.381507i	−0.981638	+	0.190754i	$0.938907\pi$
$864$		0			0
$865$			4873.18i			0.00651298i
$866$		−	167096.i		−	0.222808i
$867$		0			0
$868$			263126.i			0.349240i
$869$			39792.0i			0.0526934i
$870$		0			0
$871$	705357.			0.929764
$872$	−280071.			−0.368329
$873$		0			0
$874$	1.17616e6			1.53972
$875$	1.01253e6			1.32249
$876$		0			0
$877$	650765.			0.846106			0.423053	−	0.906105i	$-0.360958\pi$
$877$	650765.			0.846106			0.423053	+	0.906105i	$0.360958\pi$
$878$		−	47015.0i		−	0.0609884i
$879$		0			0
$880$		−	63914.5i		−	0.0825342i
$881$		−	804733.i		−	1.03681i	−0.855135	−	0.518406i	$-0.826526\pi$
$881$		−	804733.i		−	1.03681i	0.855135	−	0.518406i	$-0.173474\pi$
$882$		0			0
$883$	−1.12964e6			−1.44883			−0.724415	−	0.689364i	$-0.757890\pi$
$883$	−1.12964e6			−1.44883			−0.724415	+	0.689364i	$0.757890\pi$
$884$		−	29902.3i		−	0.0382648i
$885$		0			0
$886$	−469748.			−0.598409
$887$		−	163030.i		−	0.207214i	−0.994618	−	0.103607i	$-0.966962\pi$
$887$		−	163030.i		−	0.207214i	0.994618	−	0.103607i	$-0.0330385\pi$
$888$		0			0
$889$	500649.			0.633476
$890$	44591.3			0.0562951
$891$		0			0
$892$	−430139.			−0.540604
$893$			608147.i			0.762616i
$894$		0			0
$895$		−	631161.i		−	0.787942i
$896$			104145.i			0.129725i
$897$		0			0
$898$			655406.i			0.812751i
$899$		−	531084.i		−	0.657118i
$900$		0			0
$901$	1684.15			0.00207459
$902$	−475864.			−0.584884
$903$		0			0
$904$	1325.19			0.00162160
$905$	612246.			0.747530
$906$		0			0
$907$	32431.9			0.0394237			0.0197119	−	0.999806i	$-0.493725\pi$
$907$	32431.9			0.0394237			0.0197119	+	0.999806i	$0.493725\pi$
$908$			107147.i			0.129959i
$909$		0			0
$910$	314489.			0.379772
$911$	542118.			0.653216			0.326608	−	0.945160i	$-0.394094\pi$
$911$	542118.			0.653216			0.326608	+	0.945160i	$0.394094\pi$
$912$		0			0
$913$	−530065.			−0.635899
$914$	−165939.			−0.198635
$915$		0			0
$916$		−	44983.0i		−	0.0536114i
$917$		−	977671.i		−	1.16266i
$918$		0			0
$919$		−	631129.i		−	0.747286i	−0.927573	−	0.373643i	$-0.878109\pi$
$919$		−	631129.i		−	0.747286i	0.927573	−	0.373643i	$-0.121891\pi$
$920$	266051.			0.314333
$921$		0			0
$922$			261304.i			0.307386i
$923$			928289.i			1.08963i
$924$		0			0
$925$		−	200422.i		−	0.234240i
$926$	985926.			1.14980
$927$		0			0
$928$		−	210202.i		−	0.244085i
$929$		−	43280.6i		−	0.0501490i	−0.999686	−	0.0250745i	$-0.992018\pi$
$929$		−	43280.6i		−	0.0501490i	0.999686	−	0.0250745i	$-0.00798230\pi$
$930$		0			0
$931$	1.27743e6			1.47379
$932$		−	835739.i		−	0.962142i
$933$		0			0
$934$	−765799.			−0.877851
$935$			31472.2i			0.0360001i
$936$		0			0
$937$		−	399631.i		−	0.455176i	−0.973757	−	0.227588i	$-0.926916\pi$
$937$		−	399631.i		−	0.455176i	0.973757	−	0.227588i	$-0.0730840\pi$
$938$	1.20968e6			1.37488
$939$		0			0
$940$			137565.i			0.155687i
$941$			927158.i			1.04707i	0.852005	+	0.523533i	$0.175386\pi$
$941$			927158.i			1.04707i	−0.852005	+	0.523533i	$0.824614\pi$
$942$		0			0
$943$		−	1.98084e6i		−	2.22754i
$944$	−46672.4	−	217840.i	−0.0523741	−	0.244452i
$945$		0			0
$946$		−	270271.i		−	0.302007i
$947$	−528847.			−0.589698			−0.294849	−	0.955544i	$-0.595269\pi$
$947$	−528847.			−0.589698			−0.294849	+	0.955544i	$0.595269\pi$
$948$		0			0
$949$	980539.			1.08876
$950$			593400.i			0.657507i
$951$		0			0
$952$		−	51282.1i		−	0.0565838i
$953$	594717.			0.654824			0.327412	−	0.944882i	$-0.393824\pi$
$953$	594717.			0.654824			0.327412	+	0.944882i	$0.393824\pi$
$954$		0			0
$955$			465331.i			0.510218i
$956$	−673302.			−0.736706
$957$		0			0
$958$			752349.i			0.819763i
$959$	1.52114e6			1.65399
$960$		0			0
$961$	714351.			0.773508
$962$		−	147746.i		−	0.159649i
$963$		0			0
$964$	427832.			0.460383
$965$	−328109.			−0.352341
$966$		0			0
$967$		−	1.36543e6i		−	1.46021i	−0.683336	−	0.730105i	$-0.739471\pi$
$967$		−	1.36543e6i		−	1.46021i	0.683336	−	0.730105i	$-0.260529\pi$
$968$			198484.i			0.211823i
$969$		0			0
$970$	−395628.			−0.420478
$971$	−185537.			−0.196785			−0.0983924	−	0.995148i	$-0.531370\pi$
$971$	−185537.			−0.196785			−0.0983924	+	0.995148i	$0.531370\pi$
$972$		0			0
$973$	1.49320e6			1.57722
$974$		−	508976.i		−	0.536512i
$975$		0			0
$976$			15040.9i			0.0157897i
$977$		−	1.42229e6i		−	1.49004i	−0.667040	−	0.745022i	$-0.732439\pi$
$977$		−	1.42229e6i		−	1.49004i	0.667040	−	0.745022i	$-0.267561\pi$
$978$		0			0
$979$	92653.8			0.0966714
$980$	288959.			0.300873
$981$		0			0
$982$			509554.i			0.528405i
$983$			688290.i			0.712303i	0.934428	+	0.356151i	$0.115911\pi$
$983$			688290.i			0.712303i	−0.934428	+	0.356151i	$0.884089\pi$
$984$		0			0
$985$	−897727.			−0.925277
$986$			103506.i			0.106466i
$987$		0			0
$988$			437439.i			0.448130i
$989$	1.12503e6			1.15020
$990$		0			0
$991$		−	817163.i		−	0.832073i	−0.909348	−	0.416036i	$-0.863419\pi$
$991$		−	817163.i		−	0.832073i	0.909348	−	0.416036i	$-0.136581\pi$
$992$	−82789.5			−0.0841302
$993$		0			0
$994$			1.59201e6i			1.61128i
$995$	490598.			0.495541
$996$		0			0
$997$	−1.58760e6			−1.59716			−0.798582	−	0.601886i	$-0.794416\pi$
$997$	−1.58760e6			−1.59716			−0.798582	+	0.601886i	$0.794416\pi$
$998$		−	986362.i		−	0.990319i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1062.5.d.b.235.19		40
3.2	odd	2		354.5.d.a.235.32	yes	40
59.58	odd	2	inner	1062.5.d.b.235.20		40
177.176	even	2		354.5.d.a.235.31	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.5.d.a.235.31	✓	40	177.176	even	2
354.5.d.a.235.32	yes	40	3.2	odd	2
1062.5.d.b.235.19		40	1.1	even	1	trivial
1062.5.d.b.235.20		40	59.58	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:	\( N \)	\(=\)	\( 1062 = 2 \cdot 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1062.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} -13.0356 q^{5} +71.9157 q^{7} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} -13.0356 q^{5} +71.9157 q^{7} +22.6274i q^{8} +36.8702i q^{10} +76.6106i q^{11} -118.606i q^{13} -203.408i q^{14} +64.0000 q^{16} -31.5143 q^{17} +461.021 q^{19} +104.285 q^{20} +216.687 q^{22} +901.986i q^{23} -455.073 q^{25} -335.469 q^{26} -575.325 q^{28} +1161.22 q^{29} -457.352i q^{31} -181.019i q^{32} +89.1358i q^{34} -937.463 q^{35} +440.417i q^{37} -1303.96i q^{38} -294.962i q^{40} -2196.09 q^{41} -1247.28i q^{43} -612.885i q^{44} +2551.20 q^{46} +1319.13i q^{47} +2770.86 q^{49} +1287.14i q^{50} +948.848i q^{52} -53.4409 q^{53} -998.664i q^{55} +1627.27i q^{56} -3284.41i q^{58} +(-729.257 - 3403.75i) q^{59} +235.014i q^{61} -1293.59 q^{62} -512.000 q^{64} +1546.10i q^{65} +5947.06i q^{67} +252.114 q^{68} +2651.55i q^{70} -7826.66 q^{71} +8267.19i q^{73} +1245.69 q^{74} -3688.17 q^{76} +5509.50i q^{77} +519.406 q^{79} -834.278 q^{80} +6211.47i q^{82} +6918.96i q^{83} +410.807 q^{85} -3527.85 q^{86} -1733.50 q^{88} -1209.41i q^{89} -8529.63i q^{91} -7215.88i q^{92} +3731.07 q^{94} -6009.69 q^{95} +10730.3i q^{97} -7837.18i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} - 80 q^{7}+O(q^{10}) \) 40 * q - 320 * q^4 - 80 * q^7	\( 40 q - 320 q^{4} - 80 q^{7} + 2560 q^{16} - 480 q^{17} - 792 q^{19} - 1024 q^{22} + 3400 q^{25} - 768 q^{26} + 640 q^{28} - 1608 q^{29} + 5760 q^{35} - 6264 q^{41} + 7040 q^{46} + 17912 q^{49} + 1104 q^{53} - 13584 q^{59} + 12288 q^{62} - 20480 q^{64} + 3840 q^{68} - 35352 q^{71} - 4608 q^{74} + 6336 q^{76} - 15720 q^{79} - 26872 q^{85} - 18432 q^{86} + 8192 q^{88} - 18432 q^{94} + 19128 q^{95}+O(q^{100}) \) 40 * q - 320 * q^4 - 80 * q^7 + 2560 * q^16 - 480 * q^17 - 792 * q^19 - 1024 * q^22 + 3400 * q^25 - 768 * q^26 + 640 * q^28 - 1608 * q^29 + 5760 * q^35 - 6264 * q^41 + 7040 * q^46 + 17912 * q^49 + 1104 * q^53 - 13584 * q^59 + 12288 * q^62 - 20480 * q^64 + 3840 * q^68 - 35352 * q^71 - 4608 * q^74 + 6336 * q^76 - 15720 * q^79 - 26872 * q^85 - 18432 * q^86 + 8192 * q^88 - 18432 * q^94 + 19128 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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