LMFDB - Embedded newform 117.4.b.e.64.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,4,Mod(64,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("117.64");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	117.b (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:	$6.90322347067$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.1362828.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 23x^{2} + 48 $ x^4 + 23*x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.4
Root			$4.54739i$ of defining polynomial
Character	$\chi$	$=$	117.64
Dual form			117.4.b.e.64.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.54739i q^{2} -12.6788 q^{4} +12.9118i q^{5} +16.7289i q^{7} -21.2762i q^{8} +O(q^{10})$	\(q+4.54739i q^{2} -12.6788 q^{4} +12.9118i q^{5} +16.7289i q^{7} -21.2762i q^{8} -58.7151 q^{10} -24.9280i q^{11} +(33.7151 - 32.5621i) q^{13} -76.0727 q^{14} -4.67878 q^{16} -134.145 q^{17} +14.9376i q^{19} -163.706i q^{20} +113.358 q^{22} +72.0000 q^{23} -41.7151 q^{25} +(148.073 + 153.316i) q^{26} -212.101i q^{28} +206.145 q^{29} +249.142i q^{31} -191.486i q^{32} -610.012i q^{34} -216.000 q^{35} +293.955i q^{37} -67.9273 q^{38} +274.715 q^{40} +250.506i q^{41} -432.145 q^{43} +316.057i q^{44} +327.412i q^{46} +159.889i q^{47} +63.1454 q^{49} -189.695i q^{50} +(-427.467 + 412.848i) q^{52} +194.581 q^{53} +321.866 q^{55} +355.927 q^{56} +937.424i q^{58} -232.647i q^{59} -185.006 q^{61} -1132.94 q^{62} +833.333 q^{64} +(420.436 + 435.324i) q^{65} +39.4393i q^{67} +1700.80 q^{68} -982.237i q^{70} +920.460i q^{71} -549.078i q^{73} -1336.73 q^{74} -189.391i q^{76} +417.018 q^{77} +933.140 q^{79} -60.4116i q^{80} -1139.15 q^{82} -1095.38i q^{83} -1732.06i q^{85} -1965.13i q^{86} -530.375 q^{88} +532.114i q^{89} +(544.727 + 564.015i) q^{91} -912.872 q^{92} -727.079 q^{94} -192.872 q^{95} +362.661i q^{97} +287.147i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 14 q^{4}+O(q^{10}) $ 4 * q - 14 * q^4	$ 4 q - 14 q^{4} - 88 q^{10} - 12 q^{13} - 84 q^{14} + 18 q^{16} - 96 q^{17} + 380 q^{22} + 288 q^{23} - 20 q^{25} + 372 q^{26} + 384 q^{29} - 864 q^{35} - 492 q^{38} + 952 q^{40} - 1288 q^{43} - 188 q^{49} - 1306 q^{52} - 984 q^{53} - 328 q^{55} + 1644 q^{56} + 288 q^{61} - 1668 q^{62} + 1314 q^{64} + 360 q^{65} + 4380 q^{68} - 3144 q^{74} - 1416 q^{77} + 4320 q^{79} - 3088 q^{82} + 1036 q^{88} - 24 q^{91} - 1008 q^{92} - 1660 q^{94} + 1872 q^{95}+O(q^{100}) $ 4 * q - 14 * q^4 - 88 * q^10 - 12 * q^13 - 84 * q^14 + 18 * q^16 - 96 * q^17 + 380 * q^22 + 288 * q^23 - 20 * q^25 + 372 * q^26 + 384 * q^29 - 864 * q^35 - 492 * q^38 + 952 * q^40 - 1288 * q^43 - 188 * q^49 - 1306 * q^52 - 984 * q^53 - 328 * q^55 + 1644 * q^56 + 288 * q^61 - 1668 * q^62 + 1314 * q^64 + 360 * q^65 + 4380 * q^68 - 3144 * q^74 - 1416 * q^77 + 4320 * q^79 - 3088 * q^82 + 1036 * q^88 - 24 * q^91 - 1008 * q^92 - 1660 * q^94 + 1872 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$-1$	$1$

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 $

$2$			4.54739i			1.60775i	0.594801	+	0.803873i	$0.297231\pi$
$2$			4.54739i			1.60775i	−0.594801	+	0.803873i	$0.702769\pi$
$3$		0			0
$3$		0			0
$4$	−12.6788			−1.58485
$5$			12.9118i			1.15487i	0.816437	+	0.577434i	$0.195946\pi$
$5$			12.9118i			1.15487i	−0.816437	+	0.577434i	$0.804054\pi$
$6$		0			0
$7$			16.7289i			0.903273i	0.892202	+	0.451637i	$0.149160\pi$
$7$			16.7289i			0.903273i	−0.892202	+	0.451637i	$0.850840\pi$
$8$		−	21.2762i		−	0.940286i
$9$		0			0
$10$	−58.7151			−1.85674
$11$		−	24.9280i		−	0.683280i	−0.939831	−	0.341640i	$-0.889018\pi$
$11$		−	24.9280i		−	0.683280i	0.939831	−	0.341640i	$-0.110982\pi$
$12$		0			0
$13$	33.7151	−	32.5621i	0.719299	−	0.694700i
$13$	33.7151	−	32.5621i	0.719299	−	0.694700i
$14$	−76.0727			−1.45223
$15$		0			0
$16$	−4.67878			−0.0731059
$17$	−134.145			−1.91383			−0.956913	−	0.290376i	$-0.906220\pi$
$17$	−134.145			−1.91383			−0.956913	+	0.290376i	$0.906220\pi$
$18$		0			0
$19$			14.9376i			0.180365i	0.995925	+	0.0901824i	$0.0287450\pi$
$19$			14.9376i			0.180365i	−0.995925	+	0.0901824i	$0.971255\pi$
$20$		−	163.706i		−	1.83029i
$21$		0			0
$22$	113.358			1.09854
$23$	72.0000			0.652741			0.326370	−	0.945242i	$-0.394174\pi$
$23$	72.0000			0.652741			0.326370	+	0.945242i	$0.394174\pi$
$24$		0			0
$25$	−41.7151			−0.333721
$26$	148.073	+	153.316i	1.11690	+	1.15645i
$27$		0			0
$28$		−	212.101i		−	1.43155i
$29$	206.145			1.32001			0.660004	−	0.751262i	$-0.270555\pi$
$29$	206.145			1.32001			0.660004	+	0.751262i	$0.270555\pi$
$30$		0			0
$31$			249.142i			1.44346i	0.692176	+	0.721728i	$0.256652\pi$
$31$			249.142i			1.44346i	−0.692176	+	0.721728i	$0.743348\pi$
$32$		−	191.486i		−	1.05782i
$33$		0			0
$34$		−	610.012i		−	3.07694i
$35$	−216.000			−1.04316
$36$		0			0
$37$			293.955i			1.30610i	0.757313	+	0.653052i	$0.226512\pi$
$37$			293.955i			1.30610i	−0.757313	+	0.653052i	$0.773488\pi$
$38$	−67.9273			−0.289981
$39$		0			0
$40$	274.715			1.08591
$41$			250.506i			0.954208i	0.878847	+	0.477104i	$0.158314\pi$
$41$			250.506i			0.954208i	−0.878847	+	0.477104i	$0.841686\pi$
$42$		0			0
$43$	−432.145			−1.53259			−0.766297	−	0.642486i	$-0.777903\pi$
$43$	−432.145			−1.53259			−0.766297	+	0.642486i	$0.777903\pi$
$44$			316.057i			1.08290i
$45$		0			0
$46$			327.412i			1.04944i
$47$			159.889i			0.496217i	0.968732	+	0.248109i	$0.0798090\pi$
$47$			159.889i			0.496217i	−0.968732	+	0.248109i	$0.920191\pi$
$48$		0			0
$49$	63.1454			0.184097
$50$		−	189.695i		−	0.536539i
$51$		0			0
$52$	−427.467	+	412.848i	−1.13998	+	1.10099i
$53$	194.581			0.504298			0.252149	−	0.967688i	$-0.418863\pi$
$53$	194.581			0.504298			0.252149	+	0.967688i	$0.418863\pi$
$54$		0			0
$55$	321.866			0.789099
$56$	355.927			0.849336
$57$		0			0
$58$			937.424i			2.12224i
$59$		−	232.647i		−	0.513358i	−0.966497	−	0.256679i	$-0.917372\pi$
$59$		−	232.647i		−	0.513358i	0.966497	−	0.256679i	$-0.0826283\pi$
$60$		0			0
$61$	−185.006			−0.388321			−0.194160	−	0.980970i	$-0.562198\pi$
$61$	−185.006			−0.388321			−0.194160	+	0.980970i	$0.562198\pi$
$62$	−1132.94			−2.32071
$63$		0			0
$64$	833.333			1.62760
$65$	420.436	+	435.324i	0.802287	+	0.830696i
$66$		0			0
$67$			39.4393i			0.0719145i	0.999353	+	0.0359573i	$0.0114480\pi$
$67$			39.4393i			0.0719145i	−0.999353	+	0.0359573i	$0.988552\pi$
$68$	1700.80			3.03312
$69$		0			0
$70$		−	982.237i		−	1.67714i
$71$			920.460i			1.53857i	0.638905	+	0.769286i	$0.279388\pi$
$71$			920.460i			1.53857i	−0.638905	+	0.769286i	$0.720612\pi$
$72$		0			0
$73$		−	549.078i		−	0.880338i	−0.897915	−	0.440169i	$-0.854918\pi$
$73$		−	549.078i		−	0.880338i	0.897915	−	0.440169i	$-0.145082\pi$
$74$	−1336.73			−2.09988
$75$		0			0
$76$		−	189.391i		−	0.285851i
$77$	417.018			0.617189
$78$		0			0
$79$	933.140			1.32894			0.664471	−	0.747314i	$-0.268657\pi$
$79$	933.140			1.32894			0.664471	+	0.747314i	$0.268657\pi$
$80$		−	60.4116i		−	0.0844277i
$81$		0			0
$82$	−1139.15			−1.53412
$83$		−	1095.38i		−	1.44860i	−0.689484	−	0.724301i	$-0.742163\pi$
$83$		−	1095.38i		−	1.44860i	0.689484	−	0.724301i	$-0.257837\pi$
$84$		0			0
$85$		−	1732.06i		−	2.21022i
$86$		−	1965.13i		−	2.46402i
$87$		0			0
$88$	−530.375			−0.642479
$89$			532.114i			0.633753i	0.948467	+	0.316876i	$0.102634\pi$
$89$			532.114i			0.633753i	−0.948467	+	0.316876i	$0.897366\pi$
$90$		0			0
$91$	544.727	+	564.015i	0.627504	+	0.649724i
$92$	−912.872			−1.03449
$93$		0			0
$94$	−727.079			−0.797792
$95$	−192.872			−0.208298
$96$		0			0
$97$			362.661i			0.379615i	0.981821	+	0.189808i	$0.0607864\pi$
$97$			362.661i			0.379615i	−0.981821	+	0.189808i	$0.939214\pi$
$98$			287.147i			0.295982i
$99$		0			0
$100$	528.897			0.528897
$101$	1490.58			1.46850			0.734249	−	0.678880i	$-0.237534\pi$
$101$	1490.58			1.46850			0.734249	+	0.678880i	$0.237534\pi$
$102$		0			0
$103$	−628.436			−0.601181			−0.300591	−	0.953753i	$-0.597184\pi$
$103$	−628.436			−0.601181			−0.300591	+	0.953753i	$0.597184\pi$
$104$	−692.799	−	717.331i	−0.653217	−	0.676347i
$105$		0			0
$106$			884.838i			0.810784i
$107$	−477.454			−0.431376			−0.215688	−	0.976462i	$-0.569199\pi$
$107$	−477.454			−0.431376			−0.215688	+	0.976462i	$0.569199\pi$
$108$		0			0
$109$			378.207i			0.332345i	0.986097	+	0.166173i	$0.0531409\pi$
$109$			378.207i			0.332345i	−0.986097	+	0.166173i	$0.946859\pi$
$110$			1463.65i			1.26867i
$111$		0			0
$112$		−	78.2706i		−	0.0660346i
$113$	−13.2732			−0.0110499			−0.00552495	−	0.999985i	$-0.501759\pi$
$113$	−13.2732			−0.0110499			−0.00552495	+	0.999985i	$0.501759\pi$
$114$		0			0
$115$			929.651i			0.753830i
$116$	−2613.67			−2.09201
$117$		0			0
$118$	1057.94			0.825349
$119$		−	2244.10i		−	1.72871i
$120$		0			0
$121$	709.593			0.533128
$122$		−	841.294i		−	0.624321i
$123$		0			0
$124$		−	3158.81i		−	2.28766i
$125$			1075.36i			0.769465i
$126$		0			0
$127$	−145.988			−0.102003			−0.0510015	−	0.998699i	$-0.516241\pi$
$127$	−145.988			−0.102003			−0.0510015	+	0.998699i	$0.516241\pi$
$128$			2257.60i			1.55895i
$129$		0			0
$130$	−1979.59	+	1911.89i	−1.33555	+	1.28987i
$131$	−317.163			−0.211532			−0.105766	−	0.994391i	$-0.533729\pi$
$131$	−317.163			−0.211532			−0.105766	+	0.994391i	$0.533729\pi$
$132$		0			0
$133$	−249.890			−0.162919
$134$	−179.346			−0.115620
$135$		0			0
$136$			2854.11i			1.79954i
$137$		−	443.149i		−	0.276356i	−0.990407	−	0.138178i	$-0.955875\pi$
$137$		−	443.149i		−	0.276356i	0.990407	−	0.138178i	$-0.0441247\pi$
$138$		0			0
$139$	785.018			0.479024			0.239512	−	0.970893i	$-0.423013\pi$
$139$	785.018			0.479024			0.239512	+	0.970893i	$0.423013\pi$
$140$	2738.62			1.65325
$141$		0			0
$142$	−4185.69			−2.47363
$143$	−811.709	−	840.452i	−0.474675	−	0.491483i
$144$		0			0
$145$			2661.71i			1.52444i
$146$	2496.87			1.41536
$147$		0			0
$148$		−	3726.99i		−	2.06997i
$149$			135.420i			0.0744566i	0.999307	+	0.0372283i	$0.0118529\pi$
$149$			135.420i			0.0744566i	−0.999307	+	0.0372283i	$0.988147\pi$
$150$		0			0
$151$		−	2373.74i		−	1.27929i	−0.768672	−	0.639643i	$-0.779082\pi$
$151$		−	2373.74i		−	1.27929i	0.768672	−	0.639643i	$-0.220918\pi$
$152$	317.817			0.169594
$153$		0			0
$154$			1896.34i			0.992283i
$155$	−3216.87			−1.66700
$156$		0			0
$157$	−1166.73			−0.593089			−0.296544	−	0.955019i	$-0.595834\pi$
$157$	−1166.73			−0.593089			−0.296544	+	0.955019i	$0.595834\pi$
$158$			4243.35i			2.13660i
$159$		0			0
$160$	2472.44			1.22165
$161$			1204.48i			0.589603i
$162$		0			0
$163$		−	2309.19i		−	1.10963i	−0.831974	−	0.554815i	$-0.812789\pi$
$163$		−	2309.19i		−	1.10963i	0.831974	−	0.554815i	$-0.187211\pi$
$164$		−	3176.12i		−	1.51227i
$165$		0			0
$166$	4981.14			2.32898
$167$			600.788i			0.278386i	0.990265	+	0.139193i	$0.0444508\pi$
$167$			600.788i			0.278386i	−0.990265	+	0.139193i	$0.955549\pi$
$168$		0			0
$169$	76.4186	−	2195.67i	0.0347831	−	0.999395i
$170$	7876.36			3.55347
$171$		0			0
$172$	5479.08			2.42893
$173$	3430.36			1.50755			0.753773	−	0.657135i	$-0.228232\pi$
$173$	3430.36			1.50755			0.753773	+	0.657135i	$0.228232\pi$
$174$		0			0
$175$		−	697.846i		−	0.301441i
$176$			116.633i			0.0499519i
$177$		0			0
$178$	−2419.73			−1.01891
$179$	978.837			0.408725			0.204362	−	0.978895i	$-0.434488\pi$
$179$	978.837			0.408725			0.204362	+	0.978895i	$0.434488\pi$
$180$		0			0
$181$	3839.09			1.57656			0.788279	−	0.615318i	$-0.210972\pi$
$181$	3839.09			1.57656			0.788279	+	0.615318i	$0.210972\pi$
$182$	−2564.80	+	2477.09i	−1.04459	+	1.00887i
$183$		0			0
$184$		−	1531.89i		−	0.613763i
$185$	−3795.49			−1.50838
$186$		0			0
$187$			3343.98i			1.30768i
$188$		−	2027.20i		−	0.786429i
$189$		0			0
$190$		−	877.065i		−	0.334890i
$191$	487.709			0.184761			0.0923806	−	0.995724i	$-0.470552\pi$
$191$	487.709			0.184761			0.0923806	+	0.995724i	$0.470552\pi$
$192$		0			0
$193$			4245.61i			1.58345i	0.610878	+	0.791725i	$0.290817\pi$
$193$			4245.61i			1.58345i	−0.610878	+	0.791725i	$0.709183\pi$
$194$	−1649.16			−0.610325
$195$		0			0
$196$	−800.606			−0.291766
$197$		−	2712.71i		−	0.981079i	−0.871419	−	0.490539i	$-0.836800\pi$
$197$		−	2712.71i		−	0.981079i	0.871419	−	0.490539i	$-0.163200\pi$
$198$		0			0
$199$	−3116.90			−1.11031			−0.555153	−	0.831748i	$-0.687340\pi$
$199$	−3116.90			−1.11031			−0.555153	+	0.831748i	$0.687340\pi$
$200$			887.541i			0.313793i
$201$		0			0
$202$			6778.26i			2.36097i
$203$			3448.58i			1.19233i
$204$		0			0
$205$	−3234.49			−1.10198
$206$		−	2857.75i		−	0.966546i
$207$		0			0
$208$	−157.746	+	152.351i	−0.0525851	+	0.0507867i
$209$	372.366			0.123240
$210$		0			0
$211$	−1051.22			−0.342981			−0.171491	−	0.985186i	$-0.554858\pi$
$211$	−1051.22			−0.342981			−0.171491	+	0.985186i	$0.554858\pi$
$212$	−2467.06			−0.799236
$213$		0			0
$214$		−	2171.17i		−	0.693542i
$215$		−	5579.78i		−	1.76994i
$216$		0			0
$217$	−4167.85			−1.30384
$218$	−1719.85			−0.534327
$219$		0			0
$220$	−4080.87			−1.25060
$221$	−4522.73	+	4368.06i	−1.37661	+	1.32953i
$222$		0			0
$223$		−	5496.12i		−	1.65044i	−0.564814	−	0.825218i	$-0.691052\pi$
$223$		−	5496.12i		−	1.65044i	0.564814	−	0.825218i	$-0.308948\pi$
$224$	3203.35			0.955502
$225$		0			0
$226$		−	60.3585i		−	0.0177654i
$227$		−	921.570i		−	0.269457i	−0.990883	−	0.134729i	$-0.956984\pi$
$227$		−	921.570i		−	0.269457i	0.990883	−	0.134729i	$-0.0430162\pi$
$228$		0			0
$229$			192.941i			0.0556764i	0.999612	+	0.0278382i	$0.00886232\pi$
$229$			192.941i			0.0556764i	−0.999612	+	0.0278382i	$0.991138\pi$
$230$	−4227.49			−1.21197
$231$		0			0
$232$		−	4386.00i		−	1.24119i
$233$	913.779			0.256926			0.128463	−	0.991714i	$-0.458996\pi$
$233$	913.779			0.256926			0.128463	+	0.991714i	$0.458996\pi$
$234$		0			0
$235$	−2064.46			−0.573066
$236$			2949.68i			0.813594i
$237$		0			0
$238$	10204.8			2.77932
$239$		−	1976.86i		−	0.535032i	−0.963553	−	0.267516i	$-0.913797\pi$
$239$		−	1976.86i		−	0.535032i	0.963553	−	0.267516i	$-0.0862028\pi$
$240$		0			0
$241$			3904.45i			1.04360i	0.853068	+	0.521800i	$0.174739\pi$
$241$			3904.45i			1.04360i	−0.853068	+	0.521800i	$0.825261\pi$
$242$			3226.80i			0.857134i
$243$		0			0
$244$	2345.65			0.615429
$245$			815.322i			0.212608i
$246$		0			0
$247$	486.401	+	503.624i	0.125299	+	0.129736i
$248$	5300.80			1.35726
$249$		0			0
$250$	−4890.08			−1.23710
$251$	−942.035			−0.236895			−0.118448	−	0.992960i	$-0.537792\pi$
$251$	−942.035			−0.236895			−0.118448	+	0.992960i	$0.537792\pi$
$252$		0			0
$253$		−	1794.82i		−	0.446005i
$254$		−	663.866i		−	0.163995i
$255$		0			0
$256$	−3599.54			−0.878794
$257$	−812.616			−0.197236			−0.0986179	−	0.995125i	$-0.531442\pi$
$257$	−812.616			−0.197236			−0.0986179	+	0.995125i	$0.531442\pi$
$258$		0			0
$259$	−4917.52			−1.17977
$260$	−5330.62	−	5519.37i	−1.27150	−	1.31653i
$261$		0			0
$262$		−	1442.26i		−	0.340089i
$263$	2608.29			0.611536			0.305768	−	0.952106i	$-0.401087\pi$
$263$	2608.29			0.611536			0.305768	+	0.952106i	$0.401087\pi$
$264$		0			0
$265$			2512.40i			0.582398i
$266$		−	1136.35i		−	0.261932i
$267$		0			0
$268$		−	500.042i		−	0.113974i
$269$	4791.02			1.08592			0.542962	−	0.839757i	$-0.317303\pi$
$269$	4791.02			1.08592			0.542962	+	0.839757i	$0.317303\pi$
$270$		0			0
$271$		−	3663.62i		−	0.821214i	−0.911812	−	0.410607i	$-0.865317\pi$
$271$		−	3663.62i		−	0.821214i	0.911812	−	0.410607i	$-0.134683\pi$
$272$	627.637			0.139912
$273$		0			0
$274$	2015.17			0.444311
$275$			1039.88i			0.228025i
$276$		0			0
$277$	624.326			0.135423			0.0677114	−	0.997705i	$-0.478430\pi$
$277$	624.326			0.135423			0.0677114	+	0.997705i	$0.478430\pi$
$278$			3569.78i			0.770149i
$279$		0			0
$280$			4595.67i			0.980871i
$281$		−	5535.12i		−	1.17508i	−0.809195	−	0.587540i	$-0.800096\pi$
$281$		−	5535.12i		−	1.17508i	0.809195	−	0.587540i	$-0.199904\pi$
$282$		0			0
$283$	−175.151			−0.0367903			−0.0183952	−	0.999831i	$-0.505856\pi$
$283$	−175.151			−0.0367903			−0.0183952	+	0.999831i	$0.505856\pi$
$284$		−	11670.3i		−	2.43840i
$285$		0			0
$286$	3821.86	−	3691.16i	0.790180	−	0.763157i
$287$	−4190.69			−0.861911
$288$		0			0
$289$	13082.0			2.66273
$290$	−12103.8			−2.45091
$291$		0			0
$292$			6961.64i			1.39520i
$293$			7774.33i			1.55011i	0.631895	+	0.775054i	$0.282277\pi$
$293$			7774.33i			1.55011i	−0.631895	+	0.775054i	$0.717723\pi$
$294$		0			0
$295$	3003.90			0.592861
$296$	6254.25			1.22811
$297$		0			0
$298$	−615.808			−0.119707
$299$	2427.49	−	2344.47i	0.469516	−	0.453459i
$300$		0			0
$301$		−	7229.30i		−	1.38435i
$302$	10794.3			2.05677
$303$		0			0
$304$		−	69.8899i		−	0.0131857i
$305$		−	2388.76i		−	0.448459i
$306$		0			0
$307$			8022.85i			1.49149i	0.666230	+	0.745746i	$0.267907\pi$
$307$			8022.85i			1.49149i	−0.666230	+	0.745746i	$0.732093\pi$
$308$	−5287.27			−0.978150
$309$		0			0
$310$		−	14628.4i		−	2.68012i
$311$	−9264.87			−1.68927			−0.844635	−	0.535343i	$-0.820182\pi$
$311$	−9264.87			−1.68927			−0.844635	+	0.535343i	$0.820182\pi$
$312$		0			0
$313$	7423.57			1.34059			0.670296	−	0.742094i	$-0.266167\pi$
$313$	7423.57			1.34059			0.670296	+	0.742094i	$0.266167\pi$
$314$		−	5305.56i		−	0.953536i
$315$		0			0
$316$	−11831.1			−2.10617
$317$			2641.04i			0.467935i	0.972244	+	0.233968i	$0.0751710\pi$
$317$			2641.04i			0.467935i	−0.972244	+	0.233968i	$0.924829\pi$
$318$		0			0
$319$		−	5138.80i		−	0.901936i
$320$			10759.8i			1.87967i
$321$		0			0
$322$	−5477.23			−0.947932
$323$		−	2003.82i		−	0.345187i
$324$		0			0
$325$	−1406.43	+	1358.33i	−0.240045	+	0.231836i
$326$	10500.8			1.78400
$327$		0			0
$328$	5329.84			0.897229
$329$	−2674.76			−0.448220
$330$		0			0
$331$		−	10779.5i		−	1.79001i	−0.446052	−	0.895007i	$-0.647170\pi$
$331$		−	10779.5i		−	1.79001i	0.446052	−	0.895007i	$-0.352830\pi$
$332$			13888.1i			2.29581i
$333$		0			0
$334$	−2732.02			−0.447573
$335$	−509.233			−0.0830518
$336$		0			0
$337$	313.465			0.0506693			0.0253346	−	0.999679i	$-0.491935\pi$
$337$	313.465			0.0506693			0.0253346	+	0.999679i	$0.491935\pi$
$338$	9984.58	+	347.505i	1.60677	+	0.0559225i
$339$		0			0
$340$			21960.4i			3.50286i
$341$	6210.61			0.986286
$342$		0			0
$343$			6794.35i			1.06956i
$344$			9194.43i			1.44108i
$345$		0			0
$346$			15599.2i			2.42375i
$347$	2849.23			0.440792			0.220396	−	0.975410i	$-0.429265\pi$
$347$	2849.23			0.440792			0.220396	+	0.975410i	$0.429265\pi$
$348$		0			0
$349$			6466.94i			0.991883i	0.868356	+	0.495941i	$0.165177\pi$
$349$			6466.94i			0.991883i	−0.868356	+	0.495941i	$0.834823\pi$
$350$	3173.38			0.484641
$351$		0			0
$352$	−4773.38			−0.722789
$353$		−	2773.10i		−	0.418122i	−0.977903	−	0.209061i	$-0.932959\pi$
$353$		−	2773.10i		−	0.418122i	0.977903	−	0.209061i	$-0.0670408\pi$
$354$		0			0
$355$	−11884.8			−1.77685
$356$		−	6746.56i		−	1.00440i
$357$		0			0
$358$			4451.16i			0.657126i
$359$			1467.11i			0.215685i	0.994168	+	0.107843i	$0.0343942\pi$
$359$			1467.11i			0.215685i	−0.994168	+	0.107843i	$0.965606\pi$
$360$		0			0
$361$	6635.87			0.967469
$362$			17457.8i			2.53471i
$363$		0			0
$364$	−6906.47	−	7151.03i	−0.994498	−	1.02971i
$365$	7089.59			1.01667
$366$		0			0
$367$	4648.22			0.661130			0.330565	−	0.943783i	$-0.392761\pi$
$367$	4648.22			0.661130			0.330565	+	0.943783i	$0.392761\pi$
$368$	−336.872			−0.0477192
$369$		0			0
$370$		−	17259.6i		−	2.42509i
$371$			3255.12i			0.455519i
$372$		0			0
$373$	1763.72			0.244831			0.122416	−	0.992479i	$-0.460936\pi$
$373$	1763.72			0.244831			0.122416	+	0.992479i	$0.460936\pi$
$374$	−15206.4			−2.10242
$375$		0			0
$376$	3401.84			0.466586
$377$	6950.22	−	6712.53i	0.949481	−	0.917010i
$378$		0			0
$379$		−	1930.47i		−	0.261640i	−0.991406	−	0.130820i	$-0.958239\pi$
$379$		−	1930.47i		−	0.261640i	0.991406	−	0.130820i	$-0.0417611\pi$
$380$	2445.38			0.330120
$381$		0			0
$382$			2217.81i			0.297049i
$383$		−	8845.93i		−	1.18017i	−0.807340	−	0.590086i	$-0.799094\pi$
$383$		−	8845.93i		−	1.18017i	0.807340	−	0.590086i	$-0.200906\pi$
$384$		0			0
$385$			5384.46i			0.712772i
$386$	−19306.5			−2.54579
$387$		0			0
$388$		−	4598.10i		−	0.601632i
$389$	−1598.08			−0.208292			−0.104146	−	0.994562i	$-0.533211\pi$
$389$	−1598.08			−0.208292			−0.104146	+	0.994562i	$0.533211\pi$
$390$		0			0
$391$	−9658.47			−1.24923
$392$		−	1343.50i		−	0.173104i
$393$		0			0
$394$	12335.8			1.57733
$395$			12048.5i			1.53475i
$396$		0			0
$397$			3578.82i			0.452433i	0.974077	+	0.226217i	$0.0726357\pi$
$397$			3578.82i			0.452433i	−0.974077	+	0.226217i	$0.927364\pi$
$398$		−	14173.7i		−	1.78509i
$399$		0			0
$400$	195.176			0.0243970
$401$			3485.99i			0.434120i	0.976158	+	0.217060i	$0.0696467\pi$
$401$			3485.99i			0.434120i	−0.976158	+	0.217060i	$0.930353\pi$
$402$		0			0
$403$	8112.58	+	8399.84i	1.00277	+	1.03828i
$404$	−18898.8			−2.32735
$405$		0			0
$406$	−15682.0			−1.91696
$407$	7327.71			0.892435
$408$		0			0
$409$		−	14709.1i		−	1.77828i	−0.457637	−	0.889139i	$-0.651304\pi$
$409$		−	14709.1i		−	1.77828i	0.457637	−	0.889139i	$-0.348696\pi$
$410$		−	14708.5i		−	1.77171i
$411$		0			0
$412$	7967.80			0.952780
$413$	3891.92			0.463702
$414$		0			0
$415$	14143.4			1.67294
$416$	−6235.20	−	6455.98i	−0.734869	−	0.760891i
$417$		0			0
$418$			1693.29i			0.198138i
$419$	−3709.01			−0.432451			−0.216226	−	0.976343i	$-0.569375\pi$
$419$	−3709.01			−0.432451			−0.216226	+	0.976343i	$0.569375\pi$
$420$		0			0
$421$		−	794.029i		−	0.0919207i	−0.998943	−	0.0459603i	$-0.985365\pi$
$421$		−	794.029i		−	0.0919207i	0.998943	−	0.0459603i	$-0.0146348\pi$
$422$		−	4780.32i		−	0.551427i
$423$		0			0
$424$		−	4139.96i		−	0.474185i
$425$	5595.89			0.638684
$426$		0			0
$427$		−	3094.94i		−	0.350760i
$428$	6053.53			0.683664
$429$		0			0
$430$	25373.5			2.84562
$431$			2891.52i			0.323155i	0.986860	+	0.161577i	$0.0516582\pi$
$431$			2891.52i			0.323155i	−0.986860	+	0.161577i	$0.948342\pi$
$432$		0			0
$433$	−5560.94			−0.617186			−0.308593	−	0.951194i	$-0.599858\pi$
$433$	−5560.94			−0.617186			−0.308593	+	0.951194i	$0.599858\pi$
$434$		−	18952.9i		−	2.09624i
$435$		0			0
$436$		−	4795.20i		−	0.526717i
$437$			1075.51i			0.117731i
$438$		0			0
$439$	−15127.2			−1.64460			−0.822302	−	0.569051i	$-0.807311\pi$
$439$	−15127.2			−1.64460			−0.822302	+	0.569051i	$0.807311\pi$
$440$		−	6848.11i		−	0.741979i
$441$		0			0
$442$	−19863.3	−	20566.6i	−2.13755	−	2.21324i
$443$	−2357.89			−0.252883			−0.126441	−	0.991974i	$-0.540356\pi$
$443$	−2357.89			−0.252883			−0.126441	+	0.991974i	$0.540356\pi$
$444$		0			0
$445$	−6870.56			−0.731901
$446$	24993.0			2.65348
$447$		0			0
$448$			13940.7i			1.47017i
$449$			7165.06i			0.753096i	0.926397	+	0.376548i	$0.122889\pi$
$449$			7165.06i			0.753096i	−0.926397	+	0.376548i	$0.877111\pi$
$450$		0			0
$451$	6244.63			0.651992
$452$	168.288			0.0175124
$453$		0			0
$454$	4190.74			0.433219
$455$	−7282.47	+	7033.41i	−0.750346	+	0.724685i
$456$		0			0
$457$		−	8020.96i		−	0.821017i	−0.911857	−	0.410508i	$-0.865351\pi$
$457$		−	8020.96i		−	0.821017i	0.911857	−	0.410508i	$-0.134649\pi$
$458$	−877.378			−0.0895135
$459$		0			0
$460$		−	11786.8i		−	1.19471i
$461$		−	4146.59i		−	0.418928i	−0.977816	−	0.209464i	$-0.932828\pi$
$461$		−	4146.59i		−	0.418928i	0.977816	−	0.209464i	$-0.0671720\pi$
$462$		0			0
$463$			7118.21i			0.714495i	0.934010	+	0.357248i	$0.116285\pi$
$463$			7118.21i			0.714495i	−0.934010	+	0.357248i	$0.883715\pi$
$464$	−964.509			−0.0965004
$465$		0			0
$466$			4155.31i			0.413071i
$467$	−2128.22			−0.210883			−0.105441	−	0.994426i	$-0.533626\pi$
$467$	−2128.22			−0.210883			−0.105441	+	0.994426i	$0.533626\pi$
$468$		0			0
$469$	−659.774			−0.0649585
$470$		−	9387.91i		−	0.921344i
$471$		0			0
$472$	−4949.86			−0.482703
$473$			10772.5i			1.04719i
$474$		0			0
$475$		−	623.125i		−	0.0601915i
$476$			28452.4i			2.73974i
$477$		0			0
$478$	8989.57			0.860196
$479$			3715.30i			0.354397i	0.984175	+	0.177199i	$0.0567035\pi$
$479$			3715.30i			0.354397i	−0.984175	+	0.177199i	$0.943296\pi$
$480$		0			0
$481$	9571.78	+	9910.71i	0.907350	+	0.939479i
$482$	−17755.0			−1.67784
$483$		0			0
$484$	−8996.77			−0.844926
$485$	−4682.62			−0.438405
$486$		0			0
$487$			8139.28i			0.757343i	0.925531	+	0.378671i	$0.123619\pi$
$487$			8139.28i			0.757343i	−0.925531	+	0.378671i	$0.876381\pi$
$488$			3936.23i			0.365133i
$489$		0			0
$490$	−3707.59			−0.341820
$491$	18081.7			1.66194			0.830972	−	0.556315i	$-0.187785\pi$
$491$	18081.7			1.66194			0.830972	+	0.556315i	$0.187785\pi$
$492$		0			0
$493$	−27653.4			−2.52626
$494$	−2290.18	+	2211.86i	−0.208583	+	0.201450i
$495$		0			0
$496$		−	1165.68i		−	0.105525i
$497$	−15398.3			−1.38975
$498$		0			0
$499$		−	11031.5i		−	0.989659i	−0.868990	−	0.494829i	$-0.835231\pi$
$499$		−	11031.5i		−	0.989659i	0.868990	−	0.494829i	$-0.164769\pi$
$500$		−	13634.2i		−	1.21948i
$501$		0			0
$502$		−	4283.80i		−	0.380868i
$503$	−8016.14			−0.710581			−0.355290	−	0.934756i	$-0.615618\pi$
$503$	−8016.14			−0.710581			−0.355290	+	0.934756i	$0.615618\pi$
$504$		0			0
$505$			19246.1i			1.69592i
$506$	8161.74			0.717063
$507$		0			0
$508$	1850.95			0.161659
$509$			20173.9i			1.75676i	0.477959	+	0.878382i	$0.341377\pi$
$509$			20173.9i			1.75676i	−0.477959	+	0.878382i	$0.658623\pi$
$510$		0			0
$511$	9185.44			0.795186
$512$			1692.30i			0.146074i
$513$		0			0
$514$		−	3695.29i		−	0.317105i
$515$		−	8114.25i		−	0.694285i
$516$		0			0
$517$	3985.72			0.339056
$518$		−	22361.9i		−	1.89677i
$519$		0			0
$520$	9262.05	−	8945.30i	0.781092	−	0.754380i
$521$	−9746.95			−0.819619			−0.409810	−	0.912171i	$-0.634405\pi$
$521$	−9746.95			−0.819619			−0.409810	+	0.912171i	$0.634405\pi$
$522$		0			0
$523$	−18929.3			−1.58264			−0.791320	−	0.611402i	$-0.790606\pi$
$523$	−18929.3			−1.58264			−0.791320	+	0.611402i	$0.790606\pi$
$524$	4021.24			0.335245
$525$		0			0
$526$			11860.9i			0.983195i
$527$		−	33421.2i		−	2.76252i
$528$		0			0
$529$	−6983.00			−0.573929
$530$	−11424.9			−0.936349
$531$		0			0
$532$	3168.30			0.258201
$533$	8157.02	+	8445.85i	0.662889	+	0.686361i
$534$		0			0
$535$		−	6164.80i		−	0.498182i
$536$	839.120			0.0676203
$537$		0			0
$538$			21786.6i			1.74589i
$539$		−	1574.09i		−	0.125790i
$540$		0			0
$541$		−	11366.8i		−	0.903321i	−0.892190	−	0.451661i	$-0.850832\pi$
$541$		−	11366.8i		−	0.903321i	0.892190	−	0.451661i	$-0.149168\pi$
$542$	16659.9			1.32030
$543$		0			0
$544$			25687.0i			2.02449i
$545$	−4883.34			−0.383815
$546$		0			0
$547$	17495.4			1.36755			0.683775	−	0.729693i	$-0.260337\pi$
$547$	17495.4			1.36755			0.683775	+	0.729693i	$0.260337\pi$
$548$			5618.59i			0.437983i
$549$		0			0
$550$	−4728.72			−0.366606
$551$			3079.33i			0.238083i
$552$		0			0
$553$			15610.4i			1.20040i
$554$			2839.05i			0.217725i
$555$		0			0
$556$	−9953.06			−0.759180
$557$			11873.1i			0.903192i	0.892223	+	0.451596i	$0.149145\pi$
$557$			11873.1i			0.903192i	−0.892223	+	0.451596i	$0.850855\pi$
$558$		0			0
$559$	−14569.8	+	14071.6i	−1.10239	+	1.06469i
$560$	1010.62			0.0762613
$561$		0			0
$562$	25170.4			1.88923
$563$	−2829.31			−0.211796			−0.105898	−	0.994377i	$-0.533772\pi$
$563$	−2829.31			−0.211796			−0.105898	+	0.994377i	$0.533772\pi$
$564$		0			0
$565$		−	171.381i		−	0.0127612i
$566$		−	796.481i		−	0.0591495i
$567$		0			0
$568$	19583.9			1.44670
$569$	16136.8			1.18891			0.594453	−	0.804130i	$-0.297368\pi$
$569$	16136.8			1.18891			0.594453	+	0.804130i	$0.297368\pi$
$570$		0			0
$571$	17840.5			1.30754			0.653769	−	0.756695i	$-0.273187\pi$
$571$	17840.5			1.30754			0.653769	+	0.756695i	$0.273187\pi$
$572$	10291.5	+	10655.9i	0.752288	+	0.778926i
$573$		0			0
$574$		−	19056.7i		−	1.38573i
$575$	−3003.49			−0.217833
$576$		0			0
$577$		−	8516.17i		−	0.614442i	−0.951638	−	0.307221i	$-0.900601\pi$
$577$		−	8516.17i		−	0.614442i	0.951638	−	0.307221i	$-0.0993990\pi$
$578$			59488.9i			4.28099i
$579$		0			0
$580$		−	33747.3i		−	2.41600i
$581$	18324.5			1.30848
$582$		0			0
$583$		−	4850.53i		−	0.344577i
$584$	−11682.3			−0.827770
$585$		0			0
$586$	−35352.9			−2.49218
$587$		−	20688.3i		−	1.45468i	−0.686277	−	0.727340i	$-0.740756\pi$
$587$		−	20688.3i		−	1.45468i	0.686277	−	0.727340i	$-0.259244\pi$
$588$		0			0
$589$	−3721.59			−0.260349
$590$			13659.9i			0.953169i
$591$		0			0
$592$		−	1375.35i		−	0.0954839i
$593$		−	11435.9i		−	0.791933i	−0.918265	−	0.395966i	$-0.870410\pi$
$593$		−	11435.9i		−	0.791933i	0.918265	−	0.395966i	$-0.129590\pi$
$594$		0			0
$595$	28975.4			1.99643
$596$		−	1716.96i		−	0.118002i
$597$		0			0
$598$	10661.2	+	11038.7i	0.729047	+	0.754863i
$599$	−1260.80			−0.0860016			−0.0430008	−	0.999075i	$-0.513692\pi$
$599$	−1260.80			−0.0860016			−0.0430008	+	0.999075i	$0.513692\pi$
$600$		0			0
$601$	6261.10			0.424951			0.212476	−	0.977166i	$-0.431847\pi$
$601$	6261.10			0.424951			0.212476	+	0.977166i	$0.431847\pi$
$602$	32874.5			2.22569
$603$		0			0
$604$			30096.1i			2.02747i
$605$			9162.14i			0.615692i
$606$		0			0
$607$	3230.33			0.216005			0.108003	−	0.994151i	$-0.465555\pi$
$607$	3230.33			0.216005			0.108003	+	0.994151i	$0.465555\pi$
$608$	2860.35			0.190794
$609$		0			0
$610$	10862.6			0.721009
$611$	5206.33	+	5390.68i	0.344722	+	0.356929i
$612$		0			0
$613$		−	14868.5i		−	0.979660i	−0.871818	−	0.489830i	$-0.837059\pi$
$613$		−	14868.5i		−	0.979660i	0.871818	−	0.489830i	$-0.162941\pi$
$614$	−36483.0			−2.39794
$615$		0			0
$616$		−	8872.57i		−	0.580334i
$617$		−	19952.8i		−	1.30190i	−0.759121	−	0.650949i	$-0.774371\pi$
$617$		−	19952.8i		−	1.30190i	0.759121	−	0.650949i	$-0.225629\pi$
$618$		0			0
$619$			8316.48i			0.540012i	0.962859	+	0.270006i	$0.0870257\pi$
$619$			8316.48i			0.540012i	−0.962859	+	0.270006i	$0.912974\pi$
$620$	40786.0			2.64194
$621$		0			0
$622$		−	42131.0i		−	2.71592i
$623$	−8901.66			−0.572452
$624$		0			0
$625$	−19099.2			−1.22235
$626$			33757.9i			2.15533i
$627$		0			0
$628$	14792.7			0.939955
$629$		−	39432.6i		−	2.49965i
$630$		0			0
$631$		−	12605.9i		−	0.795299i	−0.917537	−	0.397649i	$-0.869826\pi$
$631$		−	12605.9i		−	0.795299i	0.917537	−	0.397649i	$-0.130174\pi$
$632$		−	19853.7i		−	1.24959i
$633$		0			0
$634$	−12009.8			−0.752321
$635$		−	1884.98i		−	0.117800i
$636$		0			0
$637$	2128.95	−	2056.15i	0.132421	−	0.127892i
$638$	23368.1			1.45008
$639$		0			0
$640$	−29149.8			−1.80038
$641$	−9224.04			−0.568374			−0.284187	−	0.958769i	$-0.591724\pi$
$641$	−9224.04			−0.568374			−0.284187	+	0.958769i	$0.591724\pi$
$642$		0			0
$643$		−	4439.16i		−	0.272260i	−0.990691	−	0.136130i	$-0.956533\pi$
$643$		−	4439.16i		−	0.272260i	0.990691	−	0.136130i	$-0.0434665\pi$
$644$		−	15271.3i		−	0.934431i
$645$		0			0
$646$	9112.13			0.554972
$647$	−9601.15			−0.583401			−0.291700	−	0.956510i	$-0.594221\pi$
$647$	−9601.15			−0.583401			−0.291700	+	0.956510i	$0.594221\pi$
$648$		0			0
$649$	−5799.44			−0.350767
$650$	−6176.87	−	6395.59i	−0.372733	−	0.385932i
$651$		0			0
$652$			29277.7i			1.75859i
$653$	−27112.8			−1.62482			−0.812410	−	0.583087i	$-0.801845\pi$
$653$	−27112.8			−1.62482			−0.812410	+	0.583087i	$0.801845\pi$
$654$		0			0
$655$		−	4095.15i		−	0.244291i
$656$		−	1172.06i		−	0.0697583i
$657$		0			0
$658$		−	12163.2i		−	0.720624i
$659$	−5587.26			−0.330271			−0.165136	−	0.986271i	$-0.552806\pi$
$659$	−5587.26			−0.330271			−0.165136	+	0.986271i	$0.552806\pi$
$660$		0			0
$661$		−	3060.13i		−	0.180069i	−0.995939	−	0.0900343i	$-0.971302\pi$
$661$		−	3060.13i		−	0.180069i	0.995939	−	0.0900343i	$-0.0286977\pi$
$662$	49018.6			2.87789
$663$		0			0
$664$	−23305.6			−1.36210
$665$		−	3226.53i		−	0.188150i
$666$		0			0
$667$	14842.5			0.861623
$668$		−	7617.26i		−	0.441199i
$669$		0			0
$670$		−	2315.68i		−	0.133526i
$671$			4611.83i			0.265332i
$672$		0			0
$673$	−4121.55			−0.236069			−0.118034	−	0.993010i	$-0.537659\pi$
$673$	−4121.55			−0.236069			−0.118034	+	0.993010i	$0.537659\pi$
$674$			1425.45i			0.0814633i
$675$		0			0
$676$	−968.894	+	27838.4i	−0.0551260	+	1.58389i
$677$	−22889.5			−1.29943			−0.649715	−	0.760178i	$-0.725112\pi$
$677$	−22889.5			−1.29943			−0.649715	+	0.760178i	$0.725112\pi$
$678$		0			0
$679$	−6066.91			−0.342896
$680$	−36851.8			−2.07824
$681$		0			0
$682$			28242.1i			1.58570i
$683$			19297.9i			1.08113i	0.841301	+	0.540566i	$0.181790\pi$
$683$			19297.9i			1.08113i	−0.841301	+	0.540566i	$0.818210\pi$
$684$		0			0
$685$	5721.87			0.319155
$686$	−30896.6			−1.71959
$687$		0			0
$688$	2021.91			0.112042
$689$	6560.34	−	6335.98i	0.362742	−	0.350336i
$690$		0			0
$691$			30317.8i			1.66910i	0.550935	+	0.834548i	$0.314271\pi$
$691$			30317.8i			1.66910i	−0.550935	+	0.834548i	$0.685729\pi$
$692$	−43492.8			−2.38923
$693$		0			0
$694$			12956.6i			0.708682i
$695$			10136.0i			0.553210i
$696$		0			0
$697$		−	33604.3i		−	1.82619i
$698$	−29407.7			−1.59470
$699$		0			0
$700$			8847.84i			0.477738i
$701$	9606.16			0.517574			0.258787	−	0.965934i	$-0.416677\pi$
$701$	9606.16			0.517574			0.258787	+	0.965934i	$0.416677\pi$
$702$		0			0
$703$	−4390.99			−0.235575
$704$		−	20773.4i		−	1.11211i
$705$		0			0
$706$	12610.4			0.672235
$707$			24935.7i			1.32646i
$708$		0			0
$709$		−	23398.8i		−	1.23944i	−0.784825	−	0.619718i	$-0.787247\pi$
$709$		−	23398.8i		−	1.23944i	0.784825	−	0.619718i	$-0.212753\pi$
$710$		−	54044.9i		−	2.85672i
$711$		0			0
$712$	11321.4			0.595909
$713$			17938.2i			0.942203i
$714$		0			0
$715$	10851.8	−	10480.6i	0.567598	−	0.548187i
$716$	−12410.5			−0.647766
$717$		0			0
$718$	−6671.51			−0.346767
$719$	−23588.7			−1.22352			−0.611758	−	0.791045i	$-0.709537\pi$
$719$	−23588.7			−1.22352			−0.611758	+	0.791045i	$0.709537\pi$
$720$		0			0
$721$		−	10513.0i		−	0.543031i
$722$			30175.9i			1.55544i
$723$		0			0
$724$	−48674.9			−2.49861
$725$	−8599.38			−0.440514
$726$		0			0
$727$	−15733.5			−0.802644			−0.401322	−	0.915937i	$-0.631449\pi$
$727$	−15733.5			−0.802644			−0.401322	+	0.915937i	$0.631449\pi$
$728$	12000.1	−	11589.7i	0.610926	−	0.590034i
$729$		0			0
$730$			32239.2i			1.63455i
$731$	57970.3			2.93312
$732$		0			0
$733$			17297.1i			0.871598i	0.900044	+	0.435799i	$0.143534\pi$
$733$			17297.1i			0.871598i	−0.900044	+	0.435799i	$0.856466\pi$
$734$			21137.3i			1.06293i
$735$		0			0
$736$		−	13787.0i		−	0.690484i
$737$	983.144			0.0491378
$738$		0			0
$739$			38749.2i			1.92884i	0.264377	+	0.964419i	$0.414834\pi$
$739$			38749.2i			1.92884i	−0.264377	+	0.964419i	$0.585166\pi$
$740$	48122.2			2.39055
$741$		0			0
$742$	−14802.3			−0.732359
$743$		−	3139.76i		−	0.155029i	−0.996991	−	0.0775144i	$-0.975302\pi$
$743$		−	3139.76i		−	0.155029i	0.996991	−	0.0775144i	$-0.0246984\pi$
$744$		0			0
$745$	−1748.52			−0.0859876
$746$			8020.33i			0.393626i
$747$		0			0
$748$		−	42397.6i		−	2.07247i
$749$		−	7987.25i		−	0.389650i
$750$		0			0
$751$	−40628.6			−1.97411			−0.987055	−	0.160380i	$-0.948728\pi$
$751$	−40628.6			−1.97411			−0.987055	+	0.160380i	$0.948728\pi$
$752$		−	748.086i		−	0.0362764i
$753$		0			0
$754$	30524.5	+	31605.4i	1.47432	+	1.52652i
$755$	30649.3			1.47741
$756$		0			0
$757$	24004.9			1.15254			0.576271	−	0.817259i	$-0.304507\pi$
$757$	24004.9			1.15254			0.576271	+	0.817259i	$0.304507\pi$
$758$	8778.62			0.420651
$759$		0			0
$760$			4103.60i			0.195859i
$761$		−	29540.1i		−	1.40713i	−0.710630	−	0.703566i	$-0.751590\pi$
$761$		−	29540.1i		−	1.40713i	0.710630	−	0.703566i	$-0.248410\pi$
$762$		0			0
$763$	−6326.97			−0.300199
$764$	−6183.56			−0.292818
$765$		0			0
$766$	40225.9			1.89742
$767$	−7575.49	−	7843.73i	−0.356630	−	0.369258i
$768$		0			0
$769$		−	7585.63i		−	0.355715i	−0.984056	−	0.177857i	$-0.943083\pi$
$769$		−	7585.63i		−	0.355715i	0.984056	−	0.177857i	$-0.0569166\pi$
$770$	−24485.2			−1.14596
$771$		0			0
$772$		−	53829.2i		−	2.50953i
$773$			3284.29i			0.152817i	0.997077	+	0.0764086i	$0.0243453\pi$
$773$			3284.29i			0.152817i	−0.997077	+	0.0764086i	$0.975655\pi$
$774$		0			0
$775$		−	10393.0i		−	0.481712i
$776$	7716.07			0.356947
$777$		0			0
$778$		−	7267.08i		−	0.334881i
$779$	−3741.98			−0.172106
$780$		0			0
$781$	22945.3			1.05128
$782$		−	43920.8i		−	2.00845i
$783$		0			0
$784$	−295.443			−0.0134586
$785$		−	15064.6i		−	0.684939i
$786$		0			0
$787$		−	41624.1i		−	1.88531i	−0.333772	−	0.942654i	$-0.608321\pi$
$787$		−	41624.1i		−	1.88531i	0.333772	−	0.942654i	$-0.391679\pi$
$788$			34393.8i			1.55486i
$789$		0			0
$790$	−54789.4			−2.46749
$791$		−	222.046i		−	0.00998108i
$792$		0			0
$793$	−6237.49	+	6024.18i	−0.279319	+	0.269767i
$794$	−16274.3			−0.727398
$795$		0			0
$796$	39518.4			1.75967
$797$	−30333.3			−1.34813			−0.674066	−	0.738671i	$-0.735454\pi$
$797$	−30333.3			−1.34813			−0.674066	+	0.738671i	$0.735454\pi$
$798$		0			0
$799$		−	21448.4i		−	0.949674i
$800$			7987.87i			0.353017i
$801$		0			0
$802$	−15852.2			−0.697954
$803$	−13687.4			−0.601518
$804$		0			0
$805$	−15552.0			−0.680914
$806$	−38197.4	+	36891.1i	−1.66929	+	1.61220i
$807$		0			0
$808$		−	31714.0i		−	1.38081i
$809$	24853.9			1.08012			0.540060	−	0.841627i	$-0.318402\pi$
$809$	24853.9			1.08012			0.540060	+	0.841627i	$0.318402\pi$
$810$		0			0
$811$			17383.5i			0.752674i	0.926483	+	0.376337i	$0.122816\pi$
$811$			17383.5i			0.752674i	−0.926483	+	0.376337i	$0.877184\pi$
$812$		−	43723.7i		−	1.88966i
$813$		0			0
$814$			33322.0i			1.43481i
$815$	29815.8			1.28148
$816$		0			0
$817$		−	6455.23i		−	0.276426i
$818$	66887.8			2.85902
$819$		0			0
$820$	41009.4			1.74648
$821$		−	31169.4i		−	1.32499i	−0.749065	−	0.662496i	$-0.769497\pi$
$821$		−	31169.4i		−	1.32499i	0.749065	−	0.662496i	$-0.230503\pi$
$822$		0			0
$823$	−5512.79			−0.233492			−0.116746	−	0.993162i	$-0.537246\pi$
$823$	−5512.79			−0.233492			−0.116746	+	0.993162i	$0.537246\pi$
$824$			13370.8i			0.565282i
$825$		0			0
$826$			17698.1i			0.745516i
$827$			13335.8i			0.560738i	0.959892	+	0.280369i	$0.0904568\pi$
$827$			13335.8i			0.560738i	−0.959892	+	0.280369i	$0.909543\pi$
$828$		0			0
$829$	28338.5			1.18726			0.593628	−	0.804739i	$-0.297695\pi$
$829$	28338.5			1.18726			0.593628	+	0.804739i	$0.297695\pi$
$830$			64315.5i			2.68967i
$831$		0			0
$832$	28095.9	−	27135.1i	1.17073	−	1.13070i
$833$	−8470.66			−0.352330
$834$		0			0
$835$	−7757.27			−0.321499
$836$	−4721.15			−0.195316
$837$		0			0
$838$		−	16866.3i		−	0.695272i
$839$			27149.5i			1.11717i	0.829447	+	0.558585i	$0.188656\pi$
$839$			27149.5i			1.11717i	−0.829447	+	0.558585i	$0.811344\pi$
$840$		0			0
$841$	18106.9			0.742421
$842$	3610.76			0.147785
$843$		0			0
$844$	13328.2			0.543573
$845$	28350.1	+	986.703i	1.15417	+	0.0401700i
$846$		0			0
$847$			11870.7i			0.481560i
$848$	−910.404			−0.0368672
$849$		0			0
$850$			25446.7i			1.02684i
$851$			21164.7i			0.852547i
$852$		0			0
$853$			7978.22i			0.320245i	0.987097	+	0.160123i	$0.0511889\pi$
$853$			7978.22i			0.320245i	−0.987097	+	0.160123i	$0.948811\pi$
$854$	14073.9			0.563933
$855$		0			0
$856$			10158.4i			0.405616i
$857$	−13614.4			−0.542657			−0.271329	−	0.962487i	$-0.587463\pi$
$857$	−13614.4			−0.542657			−0.271329	+	0.962487i	$0.587463\pi$
$858$		0			0
$859$	−35007.7			−1.39051			−0.695255	−	0.718763i	$-0.744709\pi$
$859$	−35007.7			−1.39051			−0.695255	+	0.718763i	$0.744709\pi$
$860$			70744.8i			2.80509i
$861$		0			0
$862$	−13148.9			−0.519551
$863$		−	24461.5i		−	0.964867i	−0.875933	−	0.482434i	$-0.839753\pi$
$863$		−	24461.5i		−	0.964867i	0.875933	−	0.482434i	$-0.160247\pi$
$864$		0			0
$865$			44292.2i			1.74102i
$866$		−	25287.8i		−	0.992278i
$867$		0			0
$868$	52843.3			2.06638
$869$		−	23261.3i		−	0.908040i
$870$		0			0
$871$	1284.23	+	1329.70i	0.0499591	+	0.0517281i
$872$	8046.82			0.312500
$873$		0			0
$874$	−4890.77			−0.189282
$875$	−17989.5			−0.695037
$876$		0			0
$877$			43121.6i			1.66034i	0.557514	+	0.830168i	$0.311755\pi$
$877$			43121.6i			1.66034i	−0.557514	+	0.830168i	$0.688245\pi$
$878$		−	68789.3i		−	2.64411i
$879$		0			0
$880$	−1505.94			−0.0576878
$881$	−40824.5			−1.56120			−0.780598	−	0.625034i	$-0.785085\pi$
$881$	−40824.5			−1.56120			−0.780598	+	0.625034i	$0.785085\pi$
$882$		0			0
$883$	−8262.14			−0.314885			−0.157442	−	0.987528i	$-0.550325\pi$
$883$	−8262.14			−0.314885			−0.157442	+	0.987528i	$0.550325\pi$
$884$	57342.7	−	55381.6i	2.18172	−	2.10711i
$885$		0			0
$886$		−	10722.3i		−	0.406571i
$887$	40858.6			1.54667			0.773336	−	0.633997i	$-0.218587\pi$
$887$	40858.6			1.54667			0.773336	+	0.633997i	$0.218587\pi$
$888$		0			0
$889$		−	2442.22i		−	0.0921365i
$890$		−	31243.2i		−	1.17671i
$891$		0			0
$892$			69684.1i			2.61569i
$893$	−2388.37			−0.0895001
$894$		0			0
$895$			12638.6i			0.472023i
$896$	−37767.1			−1.40816
$897$		0			0
$898$	−32582.3			−1.21079
$899$			51359.4i			1.90537i
$900$		0			0
$901$	−26102.2			−0.965139
$902$			28396.8i			1.04824i
$903$		0			0
$904$			282.404i			0.0103901i
$905$			49569.6i			1.82072i
$906$		0			0
$907$	32729.3			1.19819			0.599095	−	0.800678i	$-0.295527\pi$
$907$	32729.3			1.19819			0.599095	+	0.800678i	$0.295527\pi$
$908$			11684.4i			0.427048i
$909$		0			0
$910$	−31983.7	−	33116.2i	−1.16511	−	1.20637i
$911$	−11065.9			−0.402447			−0.201223	−	0.979545i	$-0.564492\pi$
$911$	−11065.9			−0.402447			−0.201223	+	0.979545i	$0.564492\pi$
$912$		0			0
$913$	−27305.7			−0.989801
$914$	36474.5			1.31999
$915$		0			0
$916$		−	2446.26i		−	0.0882386i
$917$		−	5305.77i		−	0.191071i
$918$		0			0
$919$	50682.2			1.81921			0.909604	−	0.415477i	$-0.136385\pi$
$919$	50682.2			1.81921			0.909604	+	0.415477i	$0.136385\pi$
$920$	19779.5			0.708816
$921$		0			0
$922$	18856.2			0.673531
$923$	29972.1	+	31033.4i	1.06885	+	1.10669i
$924$		0			0
$925$		−	12262.3i		−	0.435874i
$926$	−32369.3			−1.14873
$927$		0			0
$928$		−	39474.0i		−	1.39633i
$929$			41045.5i			1.44958i	0.688970	+	0.724790i	$0.258063\pi$
$929$			41045.5i			1.44958i	−0.688970	+	0.724790i	$0.741937\pi$
$930$		0			0
$931$			943.243i			0.0332047i
$932$	−11585.6			−0.407188
$933$		0			0
$934$		−	9677.86i		−	0.339046i
$935$	−43176.9			−1.51020
$936$		0			0
$937$	−788.985			−0.0275080			−0.0137540	−	0.999905i	$-0.504378\pi$
$937$	−788.985			−0.0275080			−0.0137540	+	0.999905i	$0.504378\pi$
$938$		−	3000.25i		−	0.104437i
$939$		0			0
$940$	26174.8			0.908222
$941$		−	25676.2i		−	0.889499i	−0.895655	−	0.444750i	$-0.853293\pi$
$941$		−	25676.2i		−	0.889499i	0.895655	−	0.444750i	$-0.146707\pi$
$942$		0			0
$943$			18036.5i			0.622851i
$944$			1088.51i			0.0375295i
$945$		0			0
$946$	−48986.9			−1.68362
$947$			679.352i			0.0233115i	0.999932	+	0.0116557i	$0.00371022\pi$
$947$			679.352i			0.0233115i	−0.999932	+	0.0116557i	$0.996290\pi$
$948$		0			0
$949$	−17879.1	−	18512.2i	−0.611571	−	0.633227i
$950$	2833.60			0.0967726
$951$		0			0
$952$	−47746.0			−1.62548
$953$	−20238.4			−0.687917			−0.343958	−	0.938985i	$-0.611768\pi$
$953$	−20238.4			−0.687917			−0.343958	+	0.938985i	$0.611768\pi$
$954$		0			0
$955$			6297.21i			0.213375i
$956$			25064.2i			0.847944i
$957$		0			0
$958$	−16894.9			−0.569781
$959$	7413.38			0.249625
$960$		0			0
$961$	−32280.6			−1.08357
$962$	−45067.9	+	43526.6i	−1.51044	+	1.45879i
$963$		0			0
$964$		−	49503.6i		−	1.65395i
$965$	−54818.6			−1.82868
$966$		0			0
$967$			6161.63i			0.204906i	0.994738	+	0.102453i	$0.0326692\pi$
$967$			6161.63i			0.204906i	−0.994738	+	0.102453i	$0.967331\pi$
$968$		−	15097.5i		−	0.501293i
$969$		0			0
$970$		−	21293.7i		−	0.704845i
$971$	48569.5			1.60522			0.802610	−	0.596504i	$-0.203444\pi$
$971$	48569.5			1.60522			0.802610	+	0.596504i	$0.203444\pi$
$972$		0			0
$973$			13132.4i			0.432689i
$974$	−37012.5			−1.21761
$975$		0			0
$976$	865.602			0.0283886
$977$		−	44595.8i		−	1.46033i	−0.683269	−	0.730167i	$-0.739442\pi$
$977$		−	44595.8i		−	1.46033i	0.683269	−	0.730167i	$-0.260558\pi$
$978$		0			0
$979$	13264.6			0.433031
$980$		−	10337.3i		−	0.336951i
$981$		0			0
$982$			82224.4i			2.67198i
$983$			29599.0i			0.960389i	0.877162	+	0.480195i	$0.159434\pi$
$983$			29599.0i			0.960389i	−0.877162	+	0.480195i	$0.840566\pi$
$984$		0			0
$985$	35026.0			1.13302
$986$		−	125751.i		−	4.06159i
$987$		0			0
$988$	−6166.97	−	6385.34i	−0.198580	−	0.205612i
$989$	−31114.5			−1.00039
$990$		0			0
$991$	5718.45			0.183302			0.0916511	−	0.995791i	$-0.470786\pi$
$991$	5718.45			0.183302			0.0916511	+	0.995791i	$0.470786\pi$
$992$	47707.2			1.52692
$993$		0			0
$994$		−	70021.9i		−	2.23437i
$995$		−	40244.8i		−	1.28226i
$996$		0			0
$997$	−1491.44			−0.0473766			−0.0236883	−	0.999719i	$-0.507541\pi$
$997$	−1491.44			−0.0473766			−0.0236883	+	0.999719i	$0.507541\pi$
$998$	50164.8			1.59112
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.b.e.64.4		4
3.2	odd	2		39.4.b.b.25.1	✓	4
12.11	even	2		624.4.c.c.337.1		4
13.5	odd	4		1521.4.a.w.1.4		4
13.8	odd	4		1521.4.a.w.1.1		4
13.12	even	2	inner	117.4.b.e.64.1		4
39.5	even	4		507.4.a.l.1.1		4
39.8	even	4		507.4.a.l.1.4		4
39.38	odd	2		39.4.b.b.25.4	yes	4
156.155	even	2		624.4.c.c.337.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.b.b.25.1	✓	4	3.2	odd	2
39.4.b.b.25.4	yes	4	39.38	odd	2
117.4.b.e.64.1		4	13.12	even	2	inner
117.4.b.e.64.4		4	1.1	even	1	trivial
507.4.a.l.1.1		4	39.5	even	4
507.4.a.l.1.4		4	39.8	even	4
624.4.c.c.337.1		4	12.11	even	2
624.4.c.c.337.4		4	156.155	even	2
1521.4.a.w.1.1		4	13.8	odd	4
1521.4.a.w.1.4		4	13.5	odd	4

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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.54739i q^{2} -12.6788 q^{4} +12.9118i q^{5} +16.7289i q^{7} -21.2762i q^{8} +O(q^{10})\)	\(q+4.54739i q^{2} -12.6788 q^{4} +12.9118i q^{5} +16.7289i q^{7} -21.2762i q^{8} -58.7151 q^{10} -24.9280i q^{11} +(33.7151 - 32.5621i) q^{13} -76.0727 q^{14} -4.67878 q^{16} -134.145 q^{17} +14.9376i q^{19} -163.706i q^{20} +113.358 q^{22} +72.0000 q^{23} -41.7151 q^{25} +(148.073 + 153.316i) q^{26} -212.101i q^{28} +206.145 q^{29} +249.142i q^{31} -191.486i q^{32} -610.012i q^{34} -216.000 q^{35} +293.955i q^{37} -67.9273 q^{38} +274.715 q^{40} +250.506i q^{41} -432.145 q^{43} +316.057i q^{44} +327.412i q^{46} +159.889i q^{47} +63.1454 q^{49} -189.695i q^{50} +(-427.467 + 412.848i) q^{52} +194.581 q^{53} +321.866 q^{55} +355.927 q^{56} +937.424i q^{58} -232.647i q^{59} -185.006 q^{61} -1132.94 q^{62} +833.333 q^{64} +(420.436 + 435.324i) q^{65} +39.4393i q^{67} +1700.80 q^{68} -982.237i q^{70} +920.460i q^{71} -549.078i q^{73} -1336.73 q^{74} -189.391i q^{76} +417.018 q^{77} +933.140 q^{79} -60.4116i q^{80} -1139.15 q^{82} -1095.38i q^{83} -1732.06i q^{85} -1965.13i q^{86} -530.375 q^{88} +532.114i q^{89} +(544.727 + 564.015i) q^{91} -912.872 q^{92} -727.079 q^{94} -192.872 q^{95} +362.661i q^{97} +287.147i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 14 q^{4}+O(q^{10}) \) 4 * q - 14 * q^4	\( 4 q - 14 q^{4} - 88 q^{10} - 12 q^{13} - 84 q^{14} + 18 q^{16} - 96 q^{17} + 380 q^{22} + 288 q^{23} - 20 q^{25} + 372 q^{26} + 384 q^{29} - 864 q^{35} - 492 q^{38} + 952 q^{40} - 1288 q^{43} - 188 q^{49} - 1306 q^{52} - 984 q^{53} - 328 q^{55} + 1644 q^{56} + 288 q^{61} - 1668 q^{62} + 1314 q^{64} + 360 q^{65} + 4380 q^{68} - 3144 q^{74} - 1416 q^{77} + 4320 q^{79} - 3088 q^{82} + 1036 q^{88} - 24 q^{91} - 1008 q^{92} - 1660 q^{94} + 1872 q^{95}+O(q^{100}) \) 4 * q - 14 * q^4 - 88 * q^10 - 12 * q^13 - 84 * q^14 + 18 * q^16 - 96 * q^17 + 380 * q^22 + 288 * q^23 - 20 * q^25 + 372 * q^26 + 384 * q^29 - 864 * q^35 - 492 * q^38 + 952 * q^40 - 1288 * q^43 - 188 * q^49 - 1306 * q^52 - 984 * q^53 - 328 * q^55 + 1644 * q^56 + 288 * q^61 - 1668 * q^62 + 1314 * q^64 + 360 * q^65 + 4380 * q^68 - 3144 * q^74 - 1416 * q^77 + 4320 * q^79 - 3088 * q^82 + 1036 * q^88 - 24 * q^91 - 1008 * q^92 - 1660 * q^94 + 1872 * q^95

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