LMFDB - Embedded newform 624.4.c.c.337.1

Newspace parameters

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.c (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$36.8171918436$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.1362828.1
Defining polynomial:	$ x^{4} + 23x^{2} + 48 $ x^4 + 23*x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$4.54739i$ of defining polynomial
Character	$\chi$	$=$	624.337
Dual form			624.4.c.c.337.4

$q$-expansion

$f(q)$	$=$	$q-3.00000 q^{3} -12.9118i q^{5} -16.7289i q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -12.9118i q^{5} -16.7289i q^{7} +9.00000 q^{9} -24.9280i q^{11} +(33.7151 - 32.5621i) q^{13} +38.7355i q^{15} +134.145 q^{17} -14.9376i q^{19} +50.1866i q^{21} +72.0000 q^{23} -41.7151 q^{25} -27.0000 q^{27} -206.145 q^{29} -249.142i q^{31} +74.7841i q^{33} -216.000 q^{35} +293.955i q^{37} +(-101.145 + 97.6863i) q^{39} -250.506i q^{41} +432.145 q^{43} -116.206i q^{45} +159.889i q^{47} +63.1454 q^{49} -402.436 q^{51} -194.581 q^{53} -321.866 q^{55} +44.8129i q^{57} -232.647i q^{59} -185.006 q^{61} -150.560i q^{63} +(-420.436 - 435.324i) q^{65} -39.4393i q^{67} -216.000 q^{69} +920.460i q^{71} -549.078i q^{73} +125.145 q^{75} -417.018 q^{77} -933.140 q^{79} +81.0000 q^{81} -1095.38i q^{83} -1732.06i q^{85} +618.436 q^{87} -532.114i q^{89} +(-544.727 - 564.015i) q^{91} +747.425i q^{93} -192.872 q^{95} +362.661i q^{97} -224.352i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 + 36 * q^9	$ 4 q - 12 q^{3} + 36 q^{9} - 12 q^{13} + 96 q^{17} + 288 q^{23} - 20 q^{25} - 108 q^{27} - 384 q^{29} - 864 q^{35} + 36 q^{39} + 1288 q^{43} - 188 q^{49} - 288 q^{51} + 984 q^{53} + 328 q^{55} + 288 q^{61} - 360 q^{65} - 864 q^{69} + 60 q^{75} + 1416 q^{77} - 4320 q^{79} + 324 q^{81} + 1152 q^{87} + 24 q^{91} + 1872 q^{95}+O(q^{100}) $ 4 * q - 12 * q^3 + 36 * q^9 - 12 * q^13 + 96 * q^17 + 288 * q^23 - 20 * q^25 - 108 * q^27 - 384 * q^29 - 864 * q^35 + 36 * q^39 + 1288 * q^43 - 188 * q^49 - 288 * q^51 + 984 * q^53 + 328 * q^55 + 288 * q^61 - 360 * q^65 - 864 * q^69 + 60 * q^75 + 1416 * q^77 - 4320 * q^79 + 324 * q^81 + 1152 * q^87 + 24 * q^91 + 1872 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−3.00000			−0.577350
$3$	−3.00000			−0.577350
$4$		0			0
$5$		−	12.9118i		−	1.15487i	−0.816437	−	0.577434i	$-0.804054\pi$
$5$		−	12.9118i		−	1.15487i	0.816437	−	0.577434i	$-0.195946\pi$
$6$		0			0
$7$		−	16.7289i		−	0.903273i	−0.892202	−	0.451637i	$-0.850840\pi$
$7$		−	16.7289i		−	0.903273i	0.892202	−	0.451637i	$-0.149160\pi$
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$		−	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
$14$		0			0
$15$			38.7355i			0.666764i
$16$		0			0
$17$	134.145			1.91383			0.956913	−	0.290376i	$-0.0937804\pi$
$17$	134.145			1.91383			0.956913	+	0.290376i	$0.0937804\pi$
$18$		0			0
$19$		−	14.9376i		−	0.180365i	−0.995925	−	0.0901824i	$-0.971255\pi$
$19$		−	14.9376i		−	0.180365i	0.995925	−	0.0901824i	$-0.0287450\pi$
$20$		0			0
$21$			50.1866i			0.521505i
$22$		0			0
$23$	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
$26$		0			0
$27$	−27.0000			−0.192450
$28$		0			0
$29$	−206.145			−1.32001			−0.660004	−	0.751262i	$-0.729445\pi$
$29$	−206.145			−1.32001			−0.660004	+	0.751262i	$0.729445\pi$
$30$		0			0
$31$		−	249.142i		−	1.44346i	−0.692176	−	0.721728i	$-0.743348\pi$
$31$		−	249.142i		−	1.44346i	0.692176	−	0.721728i	$-0.256652\pi$
$32$		0			0
$33$			74.7841i			0.394492i
$34$		0			0
$35$	−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	$0.773488\pi$
$38$		0			0
$39$	−101.145	+	97.6863i	−0.415288	+	0.401085i
$40$		0			0
$41$		−	250.506i		−	0.954208i	−0.878847	−	0.477104i	$-0.841686\pi$
$41$		−	250.506i		−	0.954208i	0.878847	−	0.477104i	$-0.158314\pi$
$42$		0			0
$43$	432.145			1.53259			0.766297	−	0.642486i	$-0.222097\pi$
$43$	432.145			1.53259			0.766297	+	0.642486i	$0.222097\pi$
$44$		0			0
$45$		−	116.206i		−	0.384956i
$46$		0			0
$47$			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
$50$		0			0
$51$	−402.436			−1.10495
$52$		0			0
$53$	−194.581			−0.504298			−0.252149	−	0.967688i	$-0.581137\pi$
$53$	−194.581			−0.504298			−0.252149	+	0.967688i	$0.581137\pi$
$54$		0			0
$55$	−321.866			−0.789099
$56$		0			0
$57$			44.8129i			0.104134i
$58$		0			0
$59$		−	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	$0.562198\pi$
$62$		0			0
$63$		−	150.560i		−	0.301091i
$64$		0			0
$65$	−420.436	−	435.324i	−0.802287	−	0.830696i
$66$		0			0
$67$		−	39.4393i		−	0.0719145i	−0.999353	−	0.0359573i	$-0.988552\pi$
$67$		−	39.4393i		−	0.0719145i	0.999353	−	0.0359573i	$-0.0114480\pi$
$68$		0			0
$69$	−216.000			−0.376860
$70$		0			0
$71$			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	$-0.145082\pi$
$74$		0			0
$75$	125.145			0.192674
$76$		0			0
$77$	−417.018			−0.617189
$78$		0			0
$79$	−933.140			−1.32894			−0.664471	−	0.747314i	$-0.731343\pi$
$79$	−933.140			−1.32894			−0.664471	+	0.747314i	$0.731343\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$		−	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
$86$		0			0
$87$	618.436			0.762107
$88$		0			0
$89$		−	532.114i		−	0.633753i	−0.948467	−	0.316876i	$-0.897366\pi$
$89$		−	532.114i		−	0.633753i	0.948467	−	0.316876i	$-0.102634\pi$
$90$		0			0
$91$	−544.727	−	564.015i	−0.627504	−	0.649724i
$92$		0			0
$93$			747.425i			0.833380i
$94$		0			0
$95$	−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	$0.939214\pi$
$98$		0			0
$99$		−	224.352i		−	0.227760i
$100$		0			0
$101$	−1490.58			−1.46850			−0.734249	−	0.678880i	$-0.762466\pi$
$101$	−1490.58			−1.46850			−0.734249	+	0.678880i	$0.762466\pi$
$102$		0			0
$103$	628.436			0.601181			0.300591	−	0.953753i	$-0.402816\pi$
$103$	628.436			0.601181			0.300591	+	0.953753i	$0.402816\pi$
$104$		0			0
$105$	648.000			0.602270
$106$		0			0
$107$	−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	$0.946859\pi$
$110$		0			0
$111$		−	881.864i		−	0.754079i
$112$		0			0
$113$	13.2732			0.0110499			0.00552495	−	0.999985i	$-0.498241\pi$
$113$	13.2732			0.0110499			0.00552495	+	0.999985i	$0.498241\pi$
$114$		0			0
$115$		−	929.651i		−	0.753830i
$116$		0			0
$117$	303.436	−	293.059i	0.239766	−	0.231567i
$118$		0			0
$119$		−	2244.10i		−	1.72871i
$120$		0			0
$121$	709.593			0.533128
$122$		0			0
$123$			751.519i			0.550912i
$124$		0			0
$125$		−	1075.36i		−	0.769465i
$126$		0			0
$127$	145.988			0.102003			0.0510015	−	0.998699i	$-0.483759\pi$
$127$	145.988			0.102003			0.0510015	+	0.998699i	$0.483759\pi$
$128$		0			0
$129$	−1296.44			−0.884844
$130$		0			0
$131$	−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
$134$		0			0
$135$			348.619i			0.222255i
$136$		0			0
$137$			443.149i			0.276356i	0.990407	+	0.138178i	$0.0441247\pi$
$137$			443.149i			0.276356i	−0.990407	+	0.138178i	$0.955875\pi$
$138$		0			0
$139$	−785.018			−0.479024			−0.239512	−	0.970893i	$-0.576987\pi$
$139$	−785.018			−0.479024			−0.239512	+	0.970893i	$0.576987\pi$
$140$		0			0
$141$		−	479.667i		−	0.286491i
$142$		0			0
$143$	−811.709	−	840.452i	−0.474675	−	0.491483i
$144$		0			0
$145$			2661.71i			1.52444i
$146$		0			0
$147$	−189.436			−0.106289
$148$		0			0
$149$		−	135.420i		−	0.0744566i	−0.999307	−	0.0372283i	$-0.988147\pi$
$149$		−	135.420i		−	0.0744566i	0.999307	−	0.0372283i	$-0.0118529\pi$
$150$		0			0
$151$			2373.74i			1.27929i	0.768672	+	0.639643i	$0.220918\pi$
$151$			2373.74i			1.27929i	−0.768672	+	0.639643i	$0.779082\pi$
$152$		0			0
$153$	1207.31			0.637942
$154$		0			0
$155$	−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	$0.595834\pi$
$158$		0			0
$159$	583.744			0.291157
$160$		0			0
$161$		−	1204.48i		−	0.589603i
$162$		0			0
$163$			2309.19i			1.10963i	0.831974	+	0.554815i	$0.187211\pi$
$163$			2309.19i			1.10963i	−0.831974	+	0.554815i	$0.812789\pi$
$164$		0			0
$165$	965.599			0.455587
$166$		0			0
$167$			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
$170$		0			0
$171$		−	134.439i		−	0.0601216i
$172$		0			0
$173$	−3430.36			−1.50755			−0.753773	−	0.657135i	$-0.771768\pi$
$173$	−3430.36			−1.50755			−0.753773	+	0.657135i	$0.771768\pi$
$174$		0			0
$175$			697.846i			0.301441i
$176$		0			0
$177$			697.942i			0.296387i
$178$		0			0
$179$	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	$0.210972\pi$
$182$		0			0
$183$	555.018			0.224197
$184$		0			0
$185$	3795.49			1.50838
$186$		0			0
$187$		−	3343.98i		−	1.30768i
$188$		0			0
$189$			451.679i			0.173835i
$190$		0			0
$191$	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	$0.709183\pi$
$194$		0			0
$195$	1261.31	+	1305.97i	0.463201	+	0.479603i
$196$		0			0
$197$			2712.71i			0.981079i	0.871419	+	0.490539i	$0.163200\pi$
$197$			2712.71i			0.981079i	−0.871419	+	0.490539i	$0.836800\pi$
$198$		0			0
$199$	3116.90			1.11031			0.555153	−	0.831748i	$-0.312660\pi$
$199$	3116.90			1.11031			0.555153	+	0.831748i	$0.312660\pi$
$200$		0			0
$201$			118.318i			0.0415199i
$202$		0			0
$203$			3448.58i			1.19233i
$204$		0			0
$205$	−3234.49			−1.10198
$206$		0			0
$207$	648.000			0.217580
$208$		0			0
$209$	−372.366			−0.123240
$210$		0			0
$211$	1051.22			0.342981			0.171491	−	0.985186i	$-0.445142\pi$
$211$	1051.22			0.342981			0.171491	+	0.985186i	$0.445142\pi$
$212$		0			0
$213$		−	2761.38i		−	0.888294i
$214$		0			0
$215$		−	5579.78i		−	1.76994i
$216$		0			0
$217$	−4167.85			−1.30384
$218$		0			0
$219$			1647.23i			0.508264i
$220$		0			0
$221$	4522.73	−	4368.06i	1.37661	−	1.32953i
$222$		0			0
$223$			5496.12i			1.65044i	0.564814	+	0.825218i	$0.308948\pi$
$223$			5496.12i			1.65044i	−0.564814	+	0.825218i	$0.691052\pi$
$224$		0			0
$225$	−375.436			−0.111240
$226$		0			0
$227$		−	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	$0.991138\pi$
$230$		0			0
$231$	1251.05			0.356334
$232$		0			0
$233$	−913.779			−0.256926			−0.128463	−	0.991714i	$-0.541004\pi$
$233$	−913.779			−0.256926			−0.128463	+	0.991714i	$0.541004\pi$
$234$		0			0
$235$	2064.46			0.573066
$236$		0			0
$237$	2799.42			0.767265
$238$		0			0
$239$		−	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	$0.825261\pi$
$242$		0			0
$243$	−243.000			−0.0641500
$244$		0			0
$245$		−	815.322i		−	0.212608i
$246$		0			0
$247$	−486.401	−	503.624i	−0.125299	−	0.129736i
$248$		0			0
$249$			3286.15i			0.836350i
$250$		0			0
$251$	−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
$254$		0			0
$255$			5196.18i			1.27607i
$256$		0			0
$257$	812.616			0.197236			0.0986179	−	0.995125i	$-0.468558\pi$
$257$	812.616			0.197236			0.0986179	+	0.995125i	$0.468558\pi$
$258$		0			0
$259$	4917.52			1.17977
$260$		0			0
$261$	−1855.31			−0.440003
$262$		0			0
$263$	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
$266$		0			0
$267$			1596.34i			0.365897i
$268$		0			0
$269$	−4791.02			−1.08592			−0.542962	−	0.839757i	$-0.682697\pi$
$269$	−4791.02			−1.08592			−0.542962	+	0.839757i	$0.682697\pi$
$270$		0			0
$271$			3663.62i			0.821214i	0.911812	+	0.410607i	$0.134683\pi$
$271$			3663.62i			0.821214i	−0.911812	+	0.410607i	$0.865317\pi$
$272$		0			0
$273$	1634.18	+	1692.05i	0.362290	+	0.375118i
$274$		0			0
$275$			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	$0.478430\pi$
$278$		0			0
$279$		−	2242.27i		−	0.481152i
$280$		0			0
$281$			5535.12i			1.17508i	0.809195	+	0.587540i	$0.199904\pi$
$281$			5535.12i			1.17508i	−0.809195	+	0.587540i	$0.800096\pi$
$282$		0			0
$283$	175.151			0.0367903			0.0183952	−	0.999831i	$-0.494144\pi$
$283$	175.151			0.0367903			0.0183952	+	0.999831i	$0.494144\pi$
$284$		0			0
$285$	578.616			0.120261
$286$		0			0
$287$	−4190.69			−0.861911
$288$		0			0
$289$	13082.0			2.66273
$290$		0			0
$291$		−	1087.98i		−	0.219171i
$292$		0			0
$293$		−	7774.33i		−	1.55011i	−0.631895	−	0.775054i	$-0.717723\pi$
$293$		−	7774.33i		−	1.55011i	0.631895	−	0.775054i	$-0.282277\pi$
$294$		0			0
$295$	−3003.90			−0.592861
$296$		0			0
$297$			673.057i			0.131497i
$298$		0			0
$299$	2427.49	−	2344.47i	0.469516	−	0.453459i
$300$		0			0
$301$		−	7229.30i		−	1.38435i
$302$		0			0
$303$	4471.74			0.847838
$304$		0			0
$305$			2388.76i			0.448459i
$306$		0			0
$307$		−	8022.85i		−	1.49149i	−0.666230	−	0.745746i	$-0.732093\pi$
$307$		−	8022.85i		−	1.49149i	0.666230	−	0.745746i	$-0.267907\pi$
$308$		0			0
$309$	−1885.31			−0.347092
$310$		0			0
$311$	−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	$0.266167\pi$
$314$		0			0
$315$	−1944.00			−0.347721
$316$		0			0
$317$		−	2641.04i		−	0.467935i	−0.972244	−	0.233968i	$-0.924829\pi$
$317$		−	2641.04i		−	0.467935i	0.972244	−	0.233968i	$-0.0751710\pi$
$318$		0			0
$319$			5138.80i			0.901936i
$320$		0			0
$321$	1432.36			0.249055
$322$		0			0
$323$		−	2003.82i		−	0.345187i
$324$		0			0
$325$	−1406.43	+	1358.33i	−0.240045	+	0.231836i
$326$		0			0
$327$		−	1134.62i		−	0.191880i
$328$		0			0
$329$	2674.76			0.448220
$330$		0			0
$331$			10779.5i			1.79001i	0.446052	+	0.895007i	$0.352830\pi$
$331$			10779.5i			1.79001i	−0.446052	+	0.895007i	$0.647170\pi$
$332$		0			0
$333$			2645.59i			0.435368i
$334$		0			0
$335$	−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	$0.491935\pi$
$338$		0			0
$339$	−39.8196			−0.00637966
$340$		0			0
$341$	−6210.61			−0.986286
$342$		0			0
$343$		−	6794.35i		−	1.06956i
$344$		0			0
$345$			2788.95i			0.435224i
$346$		0			0
$347$	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	$0.834823\pi$
$350$		0			0
$351$	−910.308	+	879.177i	−0.138429	+	0.133695i
$352$		0			0
$353$			2773.10i			0.418122i	0.977903	+	0.209061i	$0.0670408\pi$
$353$			2773.10i			0.418122i	−0.977903	+	0.209061i	$0.932959\pi$
$354$		0			0
$355$	11884.8			1.77685
$356$		0			0
$357$			6732.30i			0.998070i
$358$		0			0
$359$			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
$362$		0			0
$363$	−2128.78			−0.307801
$364$		0			0
$365$	−7089.59			−1.01667
$366$		0			0
$367$	−4648.22			−0.661130			−0.330565	−	0.943783i	$-0.607239\pi$
$367$	−4648.22			−0.661130			−0.330565	+	0.943783i	$0.607239\pi$
$368$		0			0
$369$		−	2254.56i		−	0.318069i
$370$		0			0
$371$			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	$0.460936\pi$
$374$		0			0
$375$			3226.08i			0.444251i
$376$		0			0
$377$	−6950.22	+	6712.53i	−0.949481	+	0.917010i
$378$		0			0
$379$			1930.47i			0.261640i	0.991406	+	0.130820i	$0.0417611\pi$
$379$			1930.47i			0.261640i	−0.991406	+	0.130820i	$0.958239\pi$
$380$		0			0
$381$	−437.965			−0.0588914
$382$		0			0
$383$		−	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
$386$		0			0
$387$	3889.31			0.510865
$388$		0			0
$389$	1598.08			0.208292			0.104146	−	0.994562i	$-0.466789\pi$
$389$	1598.08			0.208292			0.104146	+	0.994562i	$0.466789\pi$
$390$		0			0
$391$	9658.47			1.24923
$392$		0			0
$393$	951.489			0.122128
$394$		0			0
$395$			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	$0.927364\pi$
$398$		0			0
$399$	749.669			0.0940611
$400$		0			0
$401$		−	3485.99i		−	0.434120i	−0.976158	−	0.217060i	$-0.930353\pi$
$401$		−	3485.99i		−	0.434120i	0.976158	−	0.217060i	$-0.0696467\pi$
$402$		0			0
$403$	−8112.58	−	8399.84i	−1.00277	−	1.03828i
$404$		0			0
$405$		−	1045.86i		−	0.128319i
$406$		0			0
$407$	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	$-0.348696\pi$
$410$		0			0
$411$		−	1329.45i		−	0.159554i
$412$		0			0
$413$	−3891.92			−0.463702
$414$		0			0
$415$	−14143.4			−1.67294
$416$		0			0
$417$	2355.05			0.276565
$418$		0			0
$419$	−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	$-0.0146348\pi$
$422$		0			0
$423$			1439.00i			0.165406i
$424$		0			0
$425$	−5595.89			−0.638684
$426$		0			0
$427$			3094.94i			0.350760i
$428$		0			0
$429$	2435.13	+	2521.35i	0.274054	+	0.283758i
$430$		0			0
$431$			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	$0.599858\pi$
$434$		0			0
$435$		−	7985.14i		−	0.880133i
$436$		0			0
$437$		−	1075.51i		−	0.117731i
$438$		0			0
$439$	15127.2			1.64460			0.822302	−	0.569051i	$-0.192689\pi$
$439$	15127.2			1.64460			0.822302	+	0.569051i	$0.192689\pi$
$440$		0			0
$441$	568.308			0.0613658
$442$		0			0
$443$	−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
$446$		0			0
$447$			406.260i			0.0429876i
$448$		0			0
$449$		−	7165.06i		−	0.753096i	−0.926397	−	0.376548i	$-0.877111\pi$
$449$		−	7165.06i		−	0.753096i	0.926397	−	0.376548i	$-0.122889\pi$
$450$		0			0
$451$	−6244.63			−0.651992
$452$		0			0
$453$		−	7121.22i		−	0.738596i
$454$		0			0
$455$	−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	$-0.134649\pi$
$458$		0			0
$459$	−3621.92			−0.368316
$460$		0			0
$461$			4146.59i			0.418928i	0.977816	+	0.209464i	$0.0671720\pi$
$461$			4146.59i			0.418928i	−0.977816	+	0.209464i	$0.932828\pi$
$462$		0			0
$463$		−	7118.21i		−	0.714495i	−0.934010	−	0.357248i	$-0.883715\pi$
$463$		−	7118.21i		−	0.714495i	0.934010	−	0.357248i	$-0.116285\pi$
$464$		0			0
$465$	9650.62			0.962444
$466$		0			0
$467$	−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
$470$		0			0
$471$	3500.18			0.342420
$472$		0			0
$473$		−	10772.5i		−	1.04719i
$474$		0			0
$475$			623.125i			0.0601915i
$476$		0			0
$477$	−1751.23			−0.168099
$478$		0			0
$479$			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
$482$		0			0
$483$			3613.43i			0.340408i
$484$		0			0
$485$	4682.62			0.438405
$486$		0			0
$487$		−	8139.28i		−	0.757343i	−0.925531	−	0.378671i	$-0.876381\pi$
$487$		−	8139.28i		−	0.757343i	0.925531	−	0.378671i	$-0.123619\pi$
$488$		0			0
$489$		−	6927.57i		−	0.640645i
$490$		0			0
$491$	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
$494$		0			0
$495$	−2896.80			−0.263033
$496$		0			0
$497$	15398.3			1.38975
$498$		0			0
$499$			11031.5i			0.989659i	0.868990	+	0.494829i	$0.164769\pi$
$499$			11031.5i			0.989659i	−0.868990	+	0.494829i	$0.835231\pi$
$500$		0			0
$501$		−	1802.37i		−	0.160726i
$502$		0			0
$503$	−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
$506$		0			0
$507$	−229.256	+	6587.01i	−0.0200821	+	0.577001i
$508$		0			0
$509$		−	20173.9i		−	1.75676i	−0.477959	−	0.878382i	$-0.658623\pi$
$509$		−	20173.9i		−	1.75676i	0.477959	−	0.878382i	$-0.341377\pi$
$510$		0			0
$511$	−9185.44			−0.795186
$512$		0			0
$513$			403.316i			0.0347112i
$514$		0			0
$515$		−	8114.25i		−	0.694285i
$516$		0			0
$517$	3985.72			0.339056
$518$		0			0
$519$	10291.1			0.870382
$520$		0			0
$521$	9746.95			0.819619			0.409810	−	0.912171i	$-0.365595\pi$
$521$	9746.95			0.819619			0.409810	+	0.912171i	$0.365595\pi$
$522$		0			0
$523$	18929.3			1.58264			0.791320	−	0.611402i	$-0.209394\pi$
$523$	18929.3			1.58264			0.791320	+	0.611402i	$0.209394\pi$
$524$		0			0
$525$		−	2093.54i		−	0.174037i
$526$		0			0
$527$		−	33421.2i		−	2.76252i
$528$		0			0
$529$	−6983.00			−0.573929
$530$		0			0
$531$		−	2093.83i		−	0.171119i
$532$		0			0
$533$	−8157.02	−	8445.85i	−0.662889	−	0.686361i
$534$		0			0
$535$			6164.80i			0.498182i
$536$		0			0
$537$	−2936.51			−0.235977
$538$		0			0
$539$		−	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	$-0.149168\pi$
$542$		0			0
$543$	−11517.3			−0.910227
$544$		0			0
$545$	4883.34			0.383815
$546$		0			0
$547$	−17495.4			−1.36755			−0.683775	−	0.729693i	$-0.739663\pi$
$547$	−17495.4			−1.36755			−0.683775	+	0.729693i	$0.739663\pi$
$548$		0			0
$549$	−1665.05			−0.129440
$550$		0			0
$551$			3079.33i			0.238083i
$552$		0			0
$553$			15610.4i			1.20040i
$554$		0			0
$555$	−11386.5			−0.870862
$556$		0			0
$557$		−	11873.1i		−	0.903192i	−0.892223	−	0.451596i	$-0.850855\pi$
$557$		−	11873.1i		−	0.903192i	0.892223	−	0.451596i	$-0.149145\pi$
$558$		0			0
$559$	14569.8	−	14071.6i	1.10239	−	1.06469i
$560$		0			0
$561$			10031.9i			0.754989i
$562$		0			0
$563$	−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
$566$		0			0
$567$		−	1355.04i		−	0.100364i
$568$		0			0
$569$	−16136.8			−1.18891			−0.594453	−	0.804130i	$-0.702632\pi$
$569$	−16136.8			−1.18891			−0.594453	+	0.804130i	$0.702632\pi$
$570$		0			0
$571$	−17840.5			−1.30754			−0.653769	−	0.756695i	$-0.726813\pi$
$571$	−17840.5			−1.30754			−0.653769	+	0.756695i	$0.726813\pi$
$572$		0			0
$573$	−1463.13			−0.106672
$574$		0			0
$575$	−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	$-0.0993990\pi$
$578$		0			0
$579$		−	12736.8i		−	0.914205i
$580$		0			0
$581$	−18324.5			−1.30848
$582$		0			0
$583$			4850.53i			0.344577i
$584$		0			0
$585$	−3783.92	−	3917.91i	−0.267429	−	0.276899i
$586$		0			0
$587$		−	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
$590$		0			0
$591$		−	8138.13i		−	0.566426i
$592$		0			0
$593$			11435.9i			0.791933i	0.918265	+	0.395966i	$0.129590\pi$
$593$			11435.9i			0.791933i	−0.918265	+	0.395966i	$0.870410\pi$
$594$		0			0
$595$	−28975.4			−1.99643
$596$		0			0
$597$	−9350.69			−0.641035
$598$		0			0
$599$	−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	$0.431847\pi$
$602$		0			0
$603$		−	354.954i		−	0.0239715i
$604$		0			0
$605$		−	9162.14i		−	0.615692i
$606$		0			0
$607$	−3230.33			−0.216005			−0.108003	−	0.994151i	$-0.534445\pi$
$607$	−3230.33			−0.216005			−0.108003	+	0.994151i	$0.534445\pi$
$608$		0			0
$609$		−	10345.7i		−	0.688391i
$610$		0			0
$611$	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	$-0.162941\pi$
$614$		0			0
$615$	9703.48			0.636231
$616$		0			0
$617$			19952.8i			1.30190i	0.759121	+	0.650949i	$0.225629\pi$
$617$			19952.8i			1.30190i	−0.759121	+	0.650949i	$0.774371\pi$
$618$		0			0
$619$		−	8316.48i		−	0.540012i	−0.962859	−	0.270006i	$-0.912974\pi$
$619$		−	8316.48i		−	0.540012i	0.962859	−	0.270006i	$-0.0870257\pi$
$620$		0			0
$621$	−1944.00			−0.125620
$622$		0			0
$623$	−8901.66			−0.572452
$624$		0			0
$625$	−19099.2			−1.22235
$626$		0			0
$627$	1117.10			0.0711525
$628$		0			0
$629$			39432.6i			2.49965i
$630$		0			0
$631$			12605.9i			0.795299i	0.917537	+	0.397649i	$0.130174\pi$
$631$			12605.9i			0.795299i	−0.917537	+	0.397649i	$0.869826\pi$
$632$		0			0
$633$	−3153.66			−0.198020
$634$		0			0
$635$		−	1884.98i		−	0.117800i
$636$		0			0
$637$	2128.95	−	2056.15i	0.132421	−	0.127892i
$638$		0			0
$639$			8284.14i			0.512857i
$640$		0			0
$641$	9224.04			0.568374			0.284187	−	0.958769i	$-0.408276\pi$
$641$	9224.04			0.568374			0.284187	+	0.958769i	$0.408276\pi$
$642$		0			0
$643$			4439.16i			0.272260i	0.990691	+	0.136130i	$0.0434665\pi$
$643$			4439.16i			0.272260i	−0.990691	+	0.136130i	$0.956533\pi$
$644$		0			0
$645$			16739.4i			1.02188i
$646$		0			0
$647$	−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
$650$		0			0
$651$	12503.6			0.752770
$652$		0			0
$653$	27112.8			1.62482			0.812410	−	0.583087i	$-0.198155\pi$
$653$	27112.8			1.62482			0.812410	+	0.583087i	$0.198155\pi$
$654$		0			0
$655$			4095.15i			0.244291i
$656$		0			0
$657$		−	4941.70i		−	0.293446i
$658$		0			0
$659$	−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	$-0.0286977\pi$
$662$		0			0
$663$	−13568.2	+	13104.2i	−0.794788	+	0.767607i
$664$		0			0
$665$			3226.53i			0.188150i
$666$		0			0
$667$	−14842.5			−0.861623
$668$		0			0
$669$		−	16488.4i		−	0.952880i
$670$		0			0
$671$			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	$0.537659\pi$
$674$		0			0
$675$	1126.31			0.0642246
$676$		0			0
$677$	22889.5			1.29943			0.649715	−	0.760178i	$-0.274888\pi$
$677$	22889.5			1.29943			0.649715	+	0.760178i	$0.274888\pi$
$678$		0			0
$679$	6066.91			0.342896
$680$		0			0
$681$			2764.71i			0.155571i
$682$		0			0
$683$			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
$686$		0			0
$687$		−	578.823i		−	0.0321448i
$688$		0			0
$689$	−6560.34	+	6335.98i	−0.362742	+	0.350336i
$690$		0			0
$691$		−	30317.8i		−	1.66910i	−0.550935	−	0.834548i	$-0.685729\pi$
$691$		−	30317.8i		−	1.66910i	0.550935	−	0.834548i	$-0.314271\pi$
$692$		0			0
$693$	−3753.16			−0.205730
$694$		0			0
$695$			10136.0i			0.553210i
$696$		0			0
$697$		−	33604.3i		−	1.82619i
$698$		0			0
$699$	2741.34			0.148336
$700$		0			0
$701$	−9606.16			−0.517574			−0.258787	−	0.965934i	$-0.583323\pi$
$701$	−9606.16			−0.517574			−0.258787	+	0.965934i	$0.583323\pi$
$702$		0			0
$703$	4390.99			0.235575
$704$		0			0
$705$	−6193.38			−0.330860
$706$		0			0
$707$			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	$-0.212753\pi$
$710$		0			0
$711$	−8398.26			−0.442981
$712$		0			0
$713$		−	17938.2i		−	0.942203i
$714$		0			0
$715$	−10851.8	+	10480.6i	−0.567598	+	0.548187i
$716$		0			0
$717$			5930.59i			0.308901i
$718$		0			0
$719$	−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
$722$		0			0
$723$		−	11713.3i		−	0.602522i
$724$		0			0
$725$	8599.38			0.440514
$726$		0			0
$727$	15733.5			0.802644			0.401322	−	0.915937i	$-0.368551\pi$
$727$	15733.5			0.802644			0.401322	+	0.915937i	$0.368551\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	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	$0.856466\pi$
$734$		0			0
$735$			2445.96i			0.122749i
$736$		0			0
$737$	−983.144			−0.0491378
$738$		0			0
$739$		−	38749.2i		−	1.92884i	−0.264377	−	0.964419i	$-0.585166\pi$
$739$		−	38749.2i		−	1.92884i	0.264377	−	0.964419i	$-0.414834\pi$
$740$		0			0
$741$	1459.20	+	1510.87i	0.0723417	+	0.0749033i
$742$		0			0
$743$		−	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
$746$		0			0
$747$		−	9858.45i		−	0.482867i
$748$		0			0
$749$			7987.25i			0.389650i
$750$		0			0
$751$	40628.6			1.97411			0.987055	−	0.160380i	$-0.0512721\pi$
$751$	40628.6			1.97411			0.987055	+	0.160380i	$0.0512721\pi$
$752$		0			0
$753$	2826.11			0.136772
$754$		0			0
$755$	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	$0.304507\pi$
$758$		0			0
$759$			5384.46i			0.257501i
$760$		0			0
$761$			29540.1i			1.40713i	0.710630	+	0.703566i	$0.248410\pi$
$761$			29540.1i			1.40713i	−0.710630	+	0.703566i	$0.751590\pi$
$762$		0			0
$763$	6326.97			0.300199
$764$		0			0
$765$		−	15588.5i		−	0.736739i
$766$		0			0
$767$	−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	$-0.0569166\pi$
$770$		0			0
$771$	−2437.85			−0.113874
$772$		0			0
$773$		−	3284.29i		−	0.152817i	−0.997077	−	0.0764086i	$-0.975655\pi$
$773$		−	3284.29i		−	0.152817i	0.997077	−	0.0764086i	$-0.0243453\pi$
$774$		0			0
$775$			10393.0i			0.481712i
$776$		0			0
$777$	−14752.6			−0.681140
$778$		0			0
$779$	−3741.98			−0.172106
$780$		0			0
$781$	22945.3			1.05128
$782$		0			0
$783$	5565.92			0.254036
$784$		0			0
$785$			15064.6i			0.684939i
$786$		0			0
$787$			41624.1i			1.88531i	0.333772	+	0.942654i	$0.391679\pi$
$787$			41624.1i			1.88531i	−0.333772	+	0.942654i	$0.608321\pi$
$788$		0			0
$789$	−7824.87			−0.353071
$790$		0			0
$791$		−	222.046i		−	0.00998108i
$792$		0			0
$793$	−6237.49	+	6024.18i	−0.279319	+	0.269767i
$794$		0			0
$795$		−	7537.20i		−	0.336248i
$796$		0			0
$797$	30333.3			1.34813			0.674066	−	0.738671i	$-0.264546\pi$
$797$	30333.3			1.34813			0.674066	+	0.738671i	$0.264546\pi$

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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -12.9118i q^{5} -16.7289i q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -12.9118i q^{5} -16.7289i q^{7} +9.00000 q^{9} -24.9280i q^{11} +(33.7151 - 32.5621i) q^{13} +38.7355i q^{15} +134.145 q^{17} -14.9376i q^{19} +50.1866i q^{21} +72.0000 q^{23} -41.7151 q^{25} -27.0000 q^{27} -206.145 q^{29} -249.142i q^{31} +74.7841i q^{33} -216.000 q^{35} +293.955i q^{37} +(-101.145 + 97.6863i) q^{39} -250.506i q^{41} +432.145 q^{43} -116.206i q^{45} +159.889i q^{47} +63.1454 q^{49} -402.436 q^{51} -194.581 q^{53} -321.866 q^{55} +44.8129i q^{57} -232.647i q^{59} -185.006 q^{61} -150.560i q^{63} +(-420.436 - 435.324i) q^{65} -39.4393i q^{67} -216.000 q^{69} +920.460i q^{71} -549.078i q^{73} +125.145 q^{75} -417.018 q^{77} -933.140 q^{79} +81.0000 q^{81} -1095.38i q^{83} -1732.06i q^{85} +618.436 q^{87} -532.114i q^{89} +(-544.727 - 564.015i) q^{91} +747.425i q^{93} -192.872 q^{95} +362.661i q^{97} -224.352i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 + 36 * q^9	\( 4 q - 12 q^{3} + 36 q^{9} - 12 q^{13} + 96 q^{17} + 288 q^{23} - 20 q^{25} - 108 q^{27} - 384 q^{29} - 864 q^{35} + 36 q^{39} + 1288 q^{43} - 188 q^{49} - 288 q^{51} + 984 q^{53} + 328 q^{55} + 288 q^{61} - 360 q^{65} - 864 q^{69} + 60 q^{75} + 1416 q^{77} - 4320 q^{79} + 324 q^{81} + 1152 q^{87} + 24 q^{91} + 1872 q^{95}+O(q^{100}) \) 4 * q - 12 * q^3 + 36 * q^9 - 12 * q^13 + 96 * q^17 + 288 * q^23 - 20 * q^25 - 108 * q^27 - 384 * q^29 - 864 * q^35 + 36 * q^39 + 1288 * q^43 - 188 * q^49 - 288 * q^51 + 984 * q^53 + 328 * q^55 + 288 * q^61 - 360 * q^65 - 864 * q^69 + 60 * q^75 + 1416 * q^77 - 4320 * q^79 + 324 * q^81 + 1152 * q^87 + 24 * q^91 + 1872 * q^95

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