LMFDB - Embedded newform 2100.3.j.g.601.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2100,3,Mod(601,2100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2100.601"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2100.j (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-48,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$57.2208555157$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 56 x^{14} + 2052 x^{12} - 43310 x^{10} + 663499 x^{8} - 6680748 x^{6} + 49052709 x^{4} + \cdots + 601475625 $ x^16 - 56x^14 + 2052x^12 - 43310x^10 + 663499x^8 - 6680748x^6 + 49052709x^4 - 213293925*x^2 + 601475625
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			601.14
Root			$2.77207 - 1.60046i$ of defining polynomial
Character	$\chi$	$=$	2100.601
Dual form			2100.3.j.g.601.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +(4.90840 - 4.99075i) q^{7} -3.00000 q^{9} +3.52580 q^{11} +1.16482i q^{13} -3.02322i q^{17} -16.3188i q^{19} +(8.64424 + 8.50160i) q^{21} +16.0449 q^{23} -5.19615i q^{27} +0.746113 q^{29} +29.1033i q^{31} +6.10687i q^{33} -29.5364 q^{37} -2.01752 q^{39} -39.0763i q^{41} +27.0897 q^{43} -45.4066i q^{47} +(-0.815245 - 48.9932i) q^{49} +5.23636 q^{51} -33.1354 q^{53} +28.2649 q^{57} -60.9074i q^{59} +29.9928i q^{61} +(-14.7252 + 14.9723i) q^{63} -68.8999 q^{67} +27.7905i q^{69} +79.4959 q^{71} -62.5252i q^{73} +(17.3061 - 17.5964i) q^{77} +114.310 q^{79} +9.00000 q^{81} -1.64214i q^{83} +1.29231i q^{87} -36.0788i q^{89} +(5.81332 + 5.71739i) q^{91} -50.4084 q^{93} +84.0105i q^{97} -10.5774 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9} - 24 q^{11} - 12 q^{21} + 32 q^{29} - 72 q^{39} + 88 q^{49} + 24 q^{51} + 168 q^{71} - 16 q^{79} + 144 q^{81} - 568 q^{91} + 72 q^{99}+O(q^{100}) $ 16 * q - 48 * q^9 - 24 * q^11 - 12 * q^21 + 32 * q^29 - 72 * q^39 + 88 * q^49 + 24 * q^51 + 168 * q^71 - 16 * q^79 + 144 * q^81 - 568 * q^91 + 72 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$			1.73205i			0.577350i
$3$			1.73205i			0.577350i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	4.90840	−	4.99075i	0.701200	−	0.712965i
$7$	4.90840	−	4.99075i	0.701200	−	0.712965i
$8$		0			0
$9$	−3.00000			−0.333333
$10$		0			0
$11$	3.52580			0.320528			0.160264	−	0.987074i	$-0.448766\pi$
$11$	3.52580			0.320528			0.160264	+	0.987074i	$0.448766\pi$
$12$		0			0
$13$			1.16482i			0.0896014i	0.998996	+	0.0448007i	$0.0142653\pi$
$13$			1.16482i			0.0896014i	−0.998996	+	0.0448007i	$0.985735\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.02322i		−	0.177836i	−0.996039	−	0.0889181i	$-0.971659\pi$
$17$		−	3.02322i		−	0.177836i	0.996039	−	0.0889181i	$-0.0283409\pi$
$18$		0			0
$19$		−	16.3188i		−	0.858882i	−0.903095	−	0.429441i	$-0.858711\pi$
$19$		−	16.3188i		−	0.858882i	0.903095	−	0.429441i	$-0.141289\pi$
$20$		0			0
$21$	8.64424	+	8.50160i	0.411630	+	0.404838i
$22$		0			0
$23$	16.0449			0.697602			0.348801	−	0.937197i	$-0.386589\pi$
$23$	16.0449			0.697602			0.348801	+	0.937197i	$0.386589\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	5.19615i		−	0.192450i
$28$		0			0
$29$	0.746113			0.0257280			0.0128640	−	0.999917i	$-0.495905\pi$
$29$	0.746113			0.0257280			0.0128640	+	0.999917i	$0.495905\pi$
$30$		0			0
$31$			29.1033i			0.938816i	0.882981	+	0.469408i	$0.155533\pi$
$31$			29.1033i			0.938816i	−0.882981	+	0.469408i	$0.844467\pi$
$32$		0			0
$33$			6.10687i			0.185057i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−29.5364			−0.798281			−0.399141	−	0.916890i	$-0.630691\pi$
$37$	−29.5364			−0.798281			−0.399141	+	0.916890i	$0.630691\pi$
$38$		0			0
$39$	−2.01752			−0.0517314
$40$		0			0
$41$		−	39.0763i		−	0.953080i	−0.879153	−	0.476540i	$-0.841891\pi$
$41$		−	39.0763i		−	0.953080i	0.879153	−	0.476540i	$-0.158109\pi$
$42$		0			0
$43$	27.0897			0.629993			0.314997	−	0.949093i	$-0.397997\pi$
$43$	27.0897			0.629993			0.314997	+	0.949093i	$0.397997\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	45.4066i		−	0.966099i	−0.875593	−	0.483049i	$-0.839529\pi$
$47$		−	45.4066i		−	0.966099i	0.875593	−	0.483049i	$-0.160471\pi$
$48$		0			0
$49$	−0.815245	−	48.9932i	−0.0166377	−	0.999862i
$50$		0			0
$51$	5.23636			0.102674
$52$		0			0
$53$	−33.1354			−0.625197			−0.312599	−	0.949885i	$-0.601199\pi$
$53$	−33.1354			−0.625197			−0.312599	+	0.949885i	$0.601199\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	28.2649			0.495876
$58$		0			0
$59$		−	60.9074i		−	1.03233i	−0.856490	−	0.516164i	$-0.827359\pi$
$59$		−	60.9074i		−	1.03233i	0.856490	−	0.516164i	$-0.172641\pi$
$60$		0			0
$61$			29.9928i			0.491685i	0.969310	+	0.245843i	$0.0790646\pi$
$61$			29.9928i			0.491685i	−0.969310	+	0.245843i	$0.920935\pi$
$62$		0			0
$63$	−14.7252	+	14.9723i	−0.233733	+	0.237655i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−68.8999			−1.02836			−0.514178	−	0.857683i	$-0.671903\pi$
$67$	−68.8999			−1.02836			−0.514178	+	0.857683i	$0.671903\pi$
$68$		0			0
$69$			27.7905i			0.402761i
$70$		0			0
$71$	79.4959			1.11966			0.559830	−	0.828607i	$-0.310866\pi$
$71$	79.4959			1.11966			0.559830	+	0.828607i	$0.310866\pi$
$72$		0			0
$73$		−	62.5252i		−	0.856510i	−0.903658	−	0.428255i	$-0.859129\pi$
$73$		−	62.5252i		−	0.856510i	0.903658	−	0.428255i	$-0.140871\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	17.3061	−	17.5964i	0.224754	−	0.228525i
$78$		0			0
$79$	114.310			1.44696			0.723479	−	0.690346i	$-0.242542\pi$
$79$	114.310			1.44696			0.723479	+	0.690346i	$0.242542\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$		−	1.64214i		−	0.0197848i	−0.999951	−	0.00989238i	$-0.996851\pi$
$83$		−	1.64214i		−	0.0197848i	0.999951	−	0.00989238i	$-0.00314889\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			1.29231i			0.0148541i
$88$		0			0
$89$		−	36.0788i		−	0.405380i	−0.979243	−	0.202690i	$-0.935032\pi$
$89$		−	36.0788i		−	0.405380i	0.979243	−	0.202690i	$-0.0649684\pi$
$90$		0			0
$91$	5.81332	+	5.71739i	0.0638826	+	0.0628285i
$92$		0			0
$93$	−50.4084			−0.542025
$94$		0			0
$95$		0			0
$96$		0			0
$97$			84.0105i			0.866088i	0.901373	+	0.433044i	$0.142561\pi$
$97$			84.0105i			0.866088i	−0.901373	+	0.433044i	$0.857439\pi$
$98$		0			0
$99$	−10.5774			−0.106843
$100$		0			0
$101$		−	156.860i		−	1.55307i	−0.630077	−	0.776533i	$-0.716977\pi$
$101$		−	156.860i		−	1.55307i	0.630077	−	0.776533i	$-0.283023\pi$
$102$		0			0
$103$		−	32.9749i		−	0.320144i	−0.987105	−	0.160072i	$-0.948827\pi$
$103$		−	32.9749i		−	0.320144i	0.987105	−	0.160072i	$-0.0511727\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−161.109			−1.50569			−0.752846	−	0.658197i	$-0.771319\pi$
$107$	−161.109			−1.50569			−0.752846	+	0.658197i	$0.771319\pi$
$108$		0			0
$109$	164.005			1.50464			0.752319	−	0.658799i	$-0.228935\pi$
$109$	164.005			1.50464			0.752319	+	0.658799i	$0.228935\pi$
$110$		0			0
$111$		−	51.1586i		−	0.460888i
$112$		0			0
$113$	−176.021			−1.55771			−0.778855	−	0.627204i	$-0.784199\pi$
$113$	−176.021			−1.55771			−0.778855	+	0.627204i	$0.784199\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	3.49445i		−	0.0298671i
$118$		0			0
$119$	−15.0881	−	14.8391i	−0.126791	−	0.124699i
$120$		0			0
$121$	−108.569			−0.897262
$122$		0			0
$123$	67.6821			0.550261
$124$		0			0
$125$		0			0
$126$		0			0
$127$	101.066			0.795797			0.397898	−	0.917430i	$-0.369740\pi$
$127$	101.066			0.795797			0.397898	+	0.917430i	$0.369740\pi$
$128$		0			0
$129$			46.9208i			0.363727i
$130$		0			0
$131$		−	45.4781i		−	0.347161i	−0.984820	−	0.173581i	$-0.944466\pi$
$131$		−	45.4781i		−	0.347161i	0.984820	−	0.173581i	$-0.0555337\pi$
$132$		0			0
$133$	−81.4429	−	80.0989i	−0.612352	−	0.602248i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	119.405			0.871566			0.435783	−	0.900052i	$-0.356471\pi$
$137$	119.405			0.871566			0.435783	+	0.900052i	$0.356471\pi$
$138$		0			0
$139$		−	246.336i		−	1.77220i	−0.463491	−	0.886102i	$-0.653403\pi$
$139$		−	246.336i		−	1.77220i	0.463491	−	0.886102i	$-0.346597\pi$
$140$		0			0
$141$	78.6466			0.557777
$142$		0			0
$143$			4.10692i			0.0287197i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	84.8587	−	1.41205i	0.577270	−	0.00960575i
$148$		0			0
$149$	280.690			1.88382			0.941912	−	0.335859i	$-0.109026\pi$
$149$	280.690			1.88382			0.941912	+	0.335859i	$0.109026\pi$
$150$		0			0
$151$	−106.610			−0.706027			−0.353014	−	0.935618i	$-0.614843\pi$
$151$	−106.610			−0.706027			−0.353014	+	0.935618i	$0.614843\pi$
$152$		0			0
$153$			9.06965i			0.0592787i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			187.250i			1.19268i	0.802733	+	0.596338i	$0.203378\pi$
$157$			187.250i			1.19268i	−0.802733	+	0.596338i	$0.796622\pi$
$158$		0			0
$159$		−	57.3923i		−	0.360958i
$160$		0			0
$161$	78.7545	−	80.0759i	0.489159	−	0.497366i
$162$		0			0
$163$	172.413			1.05775			0.528873	−	0.848701i	$-0.322615\pi$
$163$	172.413			1.05775			0.528873	+	0.848701i	$0.322615\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	36.6537i		−	0.219483i	−0.993960	−	0.109741i	$-0.964998\pi$
$167$		−	36.6537i		−	0.219483i	0.993960	−	0.109741i	$-0.0350023\pi$
$168$		0			0
$169$	167.643			0.991972
$170$		0			0
$171$			48.9563i			0.286294i
$172$		0			0
$173$		−	61.9160i		−	0.357896i	−0.983859	−	0.178948i	$-0.942731\pi$
$173$		−	61.9160i		−	0.357896i	0.983859	−	0.178948i	$-0.0572694\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	105.495			0.596015
$178$		0			0
$179$	102.785			0.574218			0.287109	−	0.957898i	$-0.407306\pi$
$179$	102.785			0.574218			0.287109	+	0.957898i	$0.407306\pi$
$180$		0			0
$181$			44.5372i			0.246062i	0.992403	+	0.123031i	$0.0392615\pi$
$181$			44.5372i			0.246062i	−0.992403	+	0.123031i	$0.960739\pi$
$182$		0			0
$183$	−51.9490			−0.283875
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	10.6593i		−	0.0570014i
$188$		0			0
$189$	−25.9327	−	25.5048i	−0.137210	−	0.134946i
$190$		0			0
$191$	−114.488			−0.599412			−0.299706	−	0.954032i	$-0.596889\pi$
$191$	−114.488			−0.599412			−0.299706	+	0.954032i	$0.596889\pi$
$192$		0			0
$193$	129.978			0.673462			0.336731	−	0.941601i	$-0.390679\pi$
$193$	129.978			0.673462			0.336731	+	0.941601i	$0.390679\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−13.2224			−0.0671186			−0.0335593	−	0.999437i	$-0.510684\pi$
$197$	−13.2224			−0.0671186			−0.0335593	+	0.999437i	$0.510684\pi$
$198$		0			0
$199$		−	42.9039i		−	0.215598i	−0.994173	−	0.107799i	$-0.965620\pi$
$199$		−	42.9039i		−	0.215598i	0.994173	−	0.107799i	$-0.0343802\pi$
$200$		0			0
$201$		−	119.338i		−	0.593722i
$202$		0			0
$203$	3.66222	−	3.72367i	0.0180405	−	0.0183432i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−48.1346			−0.232534
$208$		0			0
$209$		−	57.5367i		−	0.275295i
$210$		0			0
$211$	356.420			1.68920			0.844598	−	0.535401i	$-0.179839\pi$
$211$	356.420			1.68920			0.844598	+	0.535401i	$0.179839\pi$
$212$		0			0
$213$			137.691i			0.646437i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	145.247	+	142.851i	0.669343	+	0.658297i
$218$		0			0
$219$	108.297			0.494506
$220$		0			0
$221$	3.52150			0.0159344
$222$		0			0
$223$			265.455i			1.19038i	0.803585	+	0.595191i	$0.202923\pi$
$223$			265.455i			1.19038i	−0.803585	+	0.595191i	$0.797077\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	245.334i		−	1.08077i	−0.841419	−	0.540383i	$-0.818279\pi$
$227$		−	245.334i		−	1.08077i	0.841419	−	0.540383i	$-0.181721\pi$
$228$		0			0
$229$		−	330.649i		−	1.44388i	−0.691955	−	0.721940i	$-0.743251\pi$
$229$		−	330.649i		−	1.44388i	0.691955	−	0.721940i	$-0.256749\pi$
$230$		0			0
$231$	30.4779	+	29.9750i	0.131939	+	0.129762i
$232$		0			0
$233$	323.715			1.38934			0.694668	−	0.719331i	$-0.255551\pi$
$233$	323.715			1.38934			0.694668	+	0.719331i	$0.255551\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			197.990i			0.835402i
$238$		0			0
$239$	−56.5257			−0.236509			−0.118255	−	0.992983i	$-0.537730\pi$
$239$	−56.5257			−0.236509			−0.118255	+	0.992983i	$0.537730\pi$
$240$		0			0
$241$		−	207.148i		−	0.859535i	−0.902940	−	0.429768i	$-0.858596\pi$
$241$		−	207.148i		−	0.859535i	0.902940	−	0.429768i	$-0.141404\pi$
$242$		0			0
$243$			15.5885i			0.0641500i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	19.0084			0.0769570
$248$		0			0
$249$	2.84426			0.0114227
$250$		0			0
$251$			292.441i			1.16510i	0.812793	+	0.582552i	$0.197946\pi$
$251$			292.441i			1.16510i	−0.812793	+	0.582552i	$0.802054\pi$
$252$		0			0
$253$	56.5710			0.223601
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	192.205i		−	0.747878i	−0.927453	−	0.373939i	$-0.878007\pi$
$257$		−	192.205i		−	0.747878i	0.927453	−	0.373939i	$-0.121993\pi$
$258$		0			0
$259$	−144.976	+	147.409i	−0.559755	+	0.569146i
$260$		0			0
$261$	−2.23834			−0.00857601
$262$		0			0
$263$	−99.1475			−0.376987			−0.188493	−	0.982074i	$-0.560360\pi$
$263$	−99.1475			−0.376987			−0.188493	+	0.982074i	$0.560360\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	62.4904			0.234046
$268$		0			0
$269$		−	325.971i		−	1.21179i	−0.795545	−	0.605894i	$-0.792815\pi$
$269$		−	325.971i		−	1.21179i	0.795545	−	0.605894i	$-0.207185\pi$
$270$		0			0
$271$		−	140.304i		−	0.517725i	−0.965914	−	0.258863i	$-0.916652\pi$
$271$		−	140.304i		−	0.517725i	0.965914	−	0.258863i	$-0.0833477\pi$
$272$		0			0
$273$	−9.90281	+	10.0690i	−0.0362740	+	0.0368827i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	31.4353			0.113485			0.0567424	−	0.998389i	$-0.481929\pi$
$277$	31.4353			0.113485			0.0567424	+	0.998389i	$0.481929\pi$
$278$		0			0
$279$		−	87.3099i		−	0.312939i
$280$		0			0
$281$	433.598			1.54305			0.771527	−	0.636197i	$-0.219493\pi$
$281$	433.598			1.54305			0.771527	+	0.636197i	$0.219493\pi$
$282$		0			0
$283$			368.878i			1.30346i	0.758452	+	0.651728i	$0.225956\pi$
$283$			368.878i			1.30346i	−0.758452	+	0.651728i	$0.774044\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−195.020	−	191.802i	−0.679513	−	0.668300i
$288$		0			0
$289$	279.860			0.968374
$290$		0			0
$291$	−145.511			−0.500036
$292$		0			0
$293$		−	26.1637i		−	0.0892961i	−0.999003	−	0.0446480i	$-0.985783\pi$
$293$		−	26.1637i		−	0.0892961i	0.999003	−	0.0446480i	$-0.0142166\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	18.3206i		−	0.0616856i
$298$		0			0
$299$			18.6893i			0.0625061i
$300$		0			0
$301$	132.967	−	135.198i	0.441751	−	0.449163i
$302$		0			0
$303$	271.689			0.896663
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	411.125i		−	1.33917i	−0.742736	−	0.669584i	$-0.766472\pi$
$307$		−	411.125i		−	1.33917i	0.742736	−	0.669584i	$-0.233528\pi$
$308$		0			0
$309$	57.1142			0.184835
$310$		0			0
$311$			342.548i			1.10144i	0.834689	+	0.550721i	$0.185647\pi$
$311$			342.548i			1.10144i	−0.834689	+	0.550721i	$0.814353\pi$
$312$		0			0
$313$			17.9251i			0.0572687i	0.999590	+	0.0286344i	$0.00911585\pi$
$313$			17.9251i			0.0572687i	−0.999590	+	0.0286344i	$0.990884\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−515.437			−1.62599			−0.812993	−	0.582274i	$-0.802163\pi$
$317$	−515.437			−1.62599			−0.812993	+	0.582274i	$0.802163\pi$
$318$		0			0
$319$	2.63065			0.00824655
$320$		0			0
$321$		−	279.049i		−	0.869311i
$322$		0			0
$323$	−49.3351			−0.152740
$324$		0			0
$325$		0			0
$326$		0			0
$327$			284.066i			0.868703i
$328$		0			0
$329$	−226.613	−	222.874i	−0.688794	−	0.677428i
$330$		0			0
$331$	−72.5855			−0.219291			−0.109646	−	0.993971i	$-0.534972\pi$
$331$	−72.5855			−0.219291			−0.109646	+	0.993971i	$0.534972\pi$
$332$		0			0
$333$	88.6092			0.266094
$334$		0			0
$335$		0			0
$336$		0			0
$337$	169.353			0.502531			0.251265	−	0.967918i	$-0.419153\pi$
$337$	169.353			0.502531			0.251265	+	0.967918i	$0.419153\pi$
$338$		0			0
$339$		−	304.878i		−	0.899344i
$340$		0			0
$341$			102.612i			0.300916i
$342$		0			0
$343$	−248.515	−	236.410i	−0.724532	−	0.689241i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−667.398			−1.92334			−0.961668	−	0.274215i	$-0.911582\pi$
$347$	−667.398			−1.92334			−0.961668	+	0.274215i	$0.911582\pi$
$348$		0			0
$349$		−	44.2062i		−	0.126665i	−0.997992	−	0.0633326i	$-0.979827\pi$
$349$		−	44.2062i		−	0.126665i	0.997992	−	0.0633326i	$-0.0201729\pi$
$350$		0			0
$351$	6.05257			0.0172438
$352$		0			0
$353$			485.027i			1.37401i	0.726651	+	0.687007i	$0.241076\pi$
$353$			485.027i			1.37401i	−0.726651	+	0.687007i	$0.758924\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	25.7022	−	26.1334i	0.0719948	−	0.0732028i
$358$		0			0
$359$	478.427			1.33267			0.666333	−	0.745655i	$-0.267863\pi$
$359$	478.427			1.33267			0.666333	+	0.745655i	$0.267863\pi$
$360$		0			0
$361$	94.6984			0.262322
$362$		0			0
$363$		−	188.047i		−	0.518034i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			333.456i			0.908600i	0.890849	+	0.454300i	$0.150111\pi$
$367$			333.456i			0.908600i	−0.890849	+	0.454300i	$0.849889\pi$
$368$		0			0
$369$			117.229i			0.317693i
$370$		0			0
$371$	−162.642	+	165.371i	−0.438388	+	0.445744i
$372$		0			0
$373$	130.786			0.350632			0.175316	−	0.984512i	$-0.443905\pi$
$373$	130.786			0.350632			0.175316	+	0.984512i	$0.443905\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			0.869086i			0.00230527i
$378$		0			0
$379$	−422.198			−1.11398			−0.556989	−	0.830520i	$-0.688043\pi$
$379$	−422.198			−1.11398			−0.556989	+	0.830520i	$0.688043\pi$
$380$		0			0
$381$			175.052i			0.459453i
$382$		0			0
$383$		−	693.416i		−	1.81049i	−0.424894	−	0.905243i	$-0.639689\pi$
$383$		−	693.416i		−	1.81049i	0.424894	−	0.905243i	$-0.360311\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−81.2692			−0.209998
$388$		0			0
$389$	−54.2476			−0.139454			−0.0697269	−	0.997566i	$-0.522213\pi$
$389$	−54.2476			−0.139454			−0.0697269	+	0.997566i	$0.522213\pi$
$390$		0			0
$391$		−	48.5070i		−	0.124059i
$392$		0			0
$393$	78.7704			0.200434
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	144.476i		−	0.363921i	−0.983306	−	0.181960i	$-0.941756\pi$
$397$		−	144.476i		−	0.363921i	0.983306	−	0.181960i	$-0.0582442\pi$
$398$		0			0
$399$	138.735	−	141.063i	0.347708	−	0.353542i
$400$		0			0
$401$	−28.0405			−0.0699265			−0.0349632	−	0.999389i	$-0.511131\pi$
$401$	−28.0405			−0.0699265			−0.0349632	+	0.999389i	$0.511131\pi$
$402$		0			0
$403$	−33.9000			−0.0841192
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−104.140			−0.255871
$408$		0			0
$409$			138.179i			0.337846i	0.985629	+	0.168923i	$0.0540289\pi$
$409$			138.179i			0.337846i	−0.985629	+	0.168923i	$0.945971\pi$
$410$		0			0
$411$			206.815i			0.503199i
$412$		0			0
$413$	−303.974	−	298.958i	−0.736014	−	0.723868i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	426.667			1.02318
$418$		0			0
$419$			354.161i			0.845253i	0.906304	+	0.422627i	$0.138892\pi$
$419$			354.161i			0.845253i	−0.906304	+	0.422627i	$0.861108\pi$
$420$		0			0
$421$	407.258			0.967359			0.483680	−	0.875245i	$-0.339300\pi$
$421$	407.258			0.967359			0.483680	+	0.875245i	$0.339300\pi$
$422$		0			0
$423$			136.220i			0.322033i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	149.687	+	147.217i	0.350554	+	0.344770i
$428$		0			0
$429$	−7.11339			−0.0165813
$430$		0			0
$431$	−807.246			−1.87296			−0.936480	−	0.350720i	$-0.885937\pi$
$431$	−807.246			−1.87296			−0.936480	+	0.350720i	$0.885937\pi$
$432$		0			0
$433$			377.791i			0.872497i	0.899826	+	0.436249i	$0.143693\pi$
$433$			377.791i			0.872497i	−0.899826	+	0.436249i	$0.856307\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	261.832i		−	0.599158i
$438$		0			0
$439$			598.595i			1.36354i	0.731566	+	0.681771i	$0.238790\pi$
$439$			598.595i			1.36354i	−0.731566	+	0.681771i	$0.761210\pi$
$440$		0			0
$441$	2.44574	+	146.980i	0.00554589	+	0.333287i
$442$		0			0
$443$	−612.914			−1.38355			−0.691776	−	0.722112i	$-0.743172\pi$
$443$	−612.914			−1.38355			−0.691776	+	0.722112i	$0.743172\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			486.169i			1.08763i
$448$		0			0
$449$	113.063			0.251810			0.125905	−	0.992042i	$-0.459816\pi$
$449$	113.063			0.251810			0.125905	+	0.992042i	$0.459816\pi$
$450$		0			0
$451$		−	137.775i		−	0.305489i
$452$		0			0
$453$		−	184.654i		−	0.407625i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−17.5009			−0.0382952			−0.0191476	−	0.999817i	$-0.506095\pi$
$457$	−17.5009			−0.0382952			−0.0191476	+	0.999817i	$0.506095\pi$
$458$		0			0
$459$	−15.7091			−0.0342246
$460$		0			0
$461$		−	784.933i		−	1.70267i	−0.524619	−	0.851337i	$-0.675792\pi$
$461$		−	784.933i		−	1.70267i	0.524619	−	0.851337i	$-0.324208\pi$
$462$		0			0
$463$	−741.378			−1.60125			−0.800624	−	0.599167i	$-0.795498\pi$
$463$	−741.378			−1.60125			−0.800624	+	0.599167i	$0.795498\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	115.641i		−	0.247625i	−0.992306	−	0.123813i	$-0.960488\pi$
$467$		−	115.641i		−	0.247625i	0.992306	−	0.123813i	$-0.0395122\pi$
$468$		0			0
$469$	−338.188	+	343.862i	−0.721084	+	0.733182i
$470$		0			0
$471$	−324.327			−0.688592
$472$		0			0
$473$	95.5130			0.201930
$474$		0			0
$475$		0			0
$476$		0			0
$477$	99.4063			0.208399
$478$		0			0
$479$			547.469i			1.14294i	0.820623	+	0.571470i	$0.193627\pi$
$479$			547.469i			1.14294i	−0.820623	+	0.571470i	$0.806373\pi$
$480$		0			0
$481$		−	34.4045i		−	0.0715271i
$482$		0			0
$483$	138.696	+	136.407i	0.287154	+	0.282416i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	742.778			1.52521			0.762605	−	0.646864i	$-0.223920\pi$
$487$	742.778			1.52521			0.762605	+	0.646864i	$0.223920\pi$
$488$		0			0
$489$			298.628i			0.610691i
$490$		0			0
$491$	−51.4480			−0.104782			−0.0523910	−	0.998627i	$-0.516684\pi$
$491$	−51.4480			−0.104782			−0.0523910	+	0.998627i	$0.516684\pi$
$492$		0			0
$493$		−	2.25566i		−	0.00457538i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	390.198	−	396.745i	0.785106	−	0.798279i
$498$		0			0
$499$	−541.199			−1.08457			−0.542284	−	0.840195i	$-0.682440\pi$
$499$	−541.199			−1.08457			−0.542284	+	0.840195i	$0.682440\pi$
$500$		0			0
$501$	63.4860			0.126719
$502$		0			0
$503$		−	896.037i		−	1.78139i	−0.454605	−	0.890693i	$-0.650220\pi$
$503$		−	896.037i		−	1.78139i	0.454605	−	0.890693i	$-0.349780\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			290.367i			0.572715i
$508$		0			0
$509$			70.0126i			0.137549i	0.997632	+	0.0687746i	$0.0219089\pi$
$509$			70.0126i			0.137549i	−0.997632	+	0.0687746i	$0.978091\pi$
$510$		0			0
$511$	−312.048	−	306.899i	−0.610661	−	0.600584i
$512$		0			0
$513$	−84.7947			−0.165292
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	160.095i		−	0.309661i
$518$		0			0
$519$	107.242			0.206631
$520$		0			0
$521$			156.236i			0.299877i	0.988695	+	0.149938i	$0.0479075\pi$
$521$			156.236i			0.299877i	−0.988695	+	0.149938i	$0.952092\pi$
$522$		0			0
$523$		−	186.792i		−	0.357155i	−0.983926	−	0.178578i	$-0.942850\pi$
$523$		−	186.792i		−	0.357155i	0.983926	−	0.178578i	$-0.0571495\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	87.9855			0.166955
$528$		0			0
$529$	−271.563			−0.513351
$530$		0			0
$531$			182.722i			0.344109i
$532$		0			0
$533$	45.5168			0.0853973
$534$		0			0
$535$		0			0
$536$		0			0
$537$			178.029i			0.331525i
$538$		0			0
$539$	−2.87439	−	172.740i	−0.00533283	−	0.320483i
$540$		0			0
$541$	0.599842			0.00110877			0.000554383	−	1.00000i	$-0.499824\pi$
$541$	0.599842			0.00110877			0.000554383		1.00000i	$0.499824\pi$
$542$		0			0
$543$	−77.1407			−0.142064
$544$		0			0
$545$		0			0
$546$		0			0
$547$	702.970			1.28514			0.642568	−	0.766228i	$-0.277869\pi$
$547$	702.970			1.28514			0.642568	+	0.766228i	$0.277869\pi$
$548$		0			0
$549$		−	89.9784i		−	0.163895i
$550$		0			0
$551$		−	12.1756i		−	0.0220973i
$552$		0			0
$553$	561.078	−	570.492i	1.01461	−	1.03163i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−796.707			−1.43035			−0.715177	−	0.698944i	$-0.753654\pi$
$557$	−796.707			−1.43035			−0.715177	+	0.698944i	$0.753654\pi$
$558$		0			0
$559$			31.5546i			0.0564483i
$560$		0			0
$561$	18.4624			0.0329098
$562$		0			0
$563$			251.122i			0.446043i	0.974814	+	0.223022i	$0.0715920\pi$
$563$			251.122i			0.446043i	−0.974814	+	0.223022i	$0.928408\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	44.1756	−	44.9168i	0.0779111	−	0.0792183i
$568$		0			0
$569$	−298.888			−0.525287			−0.262644	−	0.964893i	$-0.584594\pi$
$569$	−298.888			−0.525287			−0.262644	+	0.964893i	$0.584594\pi$
$570$		0			0
$571$	−533.229			−0.933851			−0.466925	−	0.884297i	$-0.654638\pi$
$571$	−533.229			−0.933851			−0.466925	+	0.884297i	$0.654638\pi$
$572$		0			0
$573$		−	198.299i		−	0.346071i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			909.660i			1.57653i	0.615334	+	0.788267i	$0.289021\pi$
$577$			909.660i			1.57653i	−0.615334	+	0.788267i	$0.710979\pi$
$578$		0			0
$579$			225.129i			0.388824i
$580$		0			0
$581$	−8.19549	−	8.06025i	−0.0141058	−	0.0138731i
$582$		0			0
$583$	−116.829			−0.200393
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	376.620i		−	0.641601i	−0.947147	−	0.320801i	$-0.896048\pi$
$587$		−	376.620i		−	0.641601i	0.947147	−	0.320801i	$-0.103952\pi$
$588$		0			0
$589$	474.929			0.806332
$590$		0			0
$591$		−	22.9018i		−	0.0387510i
$592$		0			0
$593$		−	347.411i		−	0.585853i	−0.956135	−	0.292926i	$-0.905371\pi$
$593$		−	347.411i		−	0.585853i	0.956135	−	0.292926i	$-0.0946291\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	74.3118			0.124475
$598$		0			0
$599$	−249.800			−0.417029			−0.208515	−	0.978019i	$-0.566863\pi$
$599$	−249.800			−0.417029			−0.208515	+	0.978019i	$0.566863\pi$
$600$		0			0
$601$			1091.62i			1.81635i	0.418593	+	0.908174i	$0.362523\pi$
$601$			1091.62i			1.81635i	−0.418593	+	0.908174i	$0.637477\pi$
$602$		0			0
$603$	206.700			0.342786
$604$		0			0
$605$		0			0
$606$		0			0
$607$			218.884i			0.360599i	0.983612	+	0.180300i	$0.0577067\pi$
$607$			218.884i			0.360599i	−0.983612	+	0.180300i	$0.942293\pi$
$608$		0			0
$609$	6.44958	+	6.34315i	0.0105904	+	0.0104157i
$610$		0			0
$611$	52.8905			0.0865638
$612$		0			0
$613$	−498.607			−0.813389			−0.406694	−	0.913564i	$-0.633319\pi$
$613$	−498.607			−0.813389			−0.406694	+	0.913564i	$0.633319\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−705.265			−1.14306			−0.571528	−	0.820583i	$-0.693649\pi$
$617$	−705.265			−1.14306			−0.571528	+	0.820583i	$0.693649\pi$
$618$		0			0
$619$			1065.24i			1.72090i	0.509536	+	0.860450i	$0.329817\pi$
$619$			1065.24i			1.72090i	−0.509536	+	0.860450i	$0.670183\pi$
$620$		0			0
$621$		−	83.3715i		−	0.134254i
$622$		0			0
$623$	−180.061	−	177.089i	−0.289022	−	0.284252i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	99.6565			0.158942
$628$		0			0
$629$			89.2949i			0.141963i
$630$		0			0
$631$	−22.6799			−0.0359427			−0.0179714	−	0.999839i	$-0.505721\pi$
$631$	−22.6799			−0.0359427			−0.0179714	+	0.999839i	$0.505721\pi$
$632$		0			0
$633$			617.338i			0.975258i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	57.0682	−	0.949612i	0.0895890	−	0.00149076i
$638$		0			0
$639$	−238.488			−0.373220
$640$		0			0
$641$	737.987			1.15131			0.575653	−	0.817694i	$-0.304748\pi$
$641$	737.987			1.15131			0.575653	+	0.817694i	$0.304748\pi$
$642$		0			0
$643$			1129.60i			1.75676i	0.477960	+	0.878381i	$0.341376\pi$
$643$			1129.60i			1.75676i	−0.477960	+	0.878381i	$0.658624\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			78.4769i			0.121293i	0.998159	+	0.0606467i	$0.0193163\pi$
$647$			78.4769i			0.121293i	−0.998159	+	0.0606467i	$0.980684\pi$
$648$		0			0
$649$		−	214.747i		−	0.330890i
$650$		0			0
$651$	−247.424	+	251.576i	−0.380068	+	0.386445i
$652$		0			0
$653$	−13.6361			−0.0208822			−0.0104411	−	0.999945i	$-0.503324\pi$
$653$	−13.6361			−0.0208822			−0.0104411	+	0.999945i	$0.503324\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			187.576i			0.285503i
$658$		0			0
$659$	−1028.83			−1.56120			−0.780601	−	0.625030i	$-0.785087\pi$
$659$	−1028.83			−1.56120			−0.780601	+	0.625030i	$0.785087\pi$
$660$		0			0
$661$		−	469.205i		−	0.709841i	−0.934896	−	0.354921i	$-0.884508\pi$
$661$		−	469.205i		−	0.709841i	0.934896	−	0.354921i	$-0.115492\pi$
$662$		0			0
$663$			6.09941i			0.00919971i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	11.9713			0.0179479
$668$		0			0
$669$	−459.782			−0.687267
$670$		0			0
$671$			105.749i			0.157599i
$672$		0			0
$673$	784.695			1.16597			0.582983	−	0.812485i	$-0.301886\pi$
$673$	784.695			1.16597			0.582983	+	0.812485i	$0.301886\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	1104.59i		−	1.63160i	−0.578335	−	0.815799i	$-0.696297\pi$
$677$		−	1104.59i		−	1.63160i	0.578335	−	0.815799i	$-0.303703\pi$
$678$		0			0
$679$	419.276	+	412.357i	0.617490	+	0.607301i
$680$		0			0
$681$	424.931			0.623981
$682$		0			0
$683$	333.712			0.488597			0.244299	−	0.969700i	$-0.421442\pi$
$683$	333.712			0.488597			0.244299	+	0.969700i	$0.421442\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	572.700			0.833625
$688$		0			0
$689$		−	38.5968i		−	0.0560185i
$690$		0			0
$691$			1092.94i			1.58168i	0.612020	+	0.790842i	$0.290357\pi$
$691$			1092.94i			1.58168i	−0.612020	+	0.790842i	$0.709643\pi$
$692$		0			0
$693$	−51.9182	+	52.7893i	−0.0749180	+	0.0761750i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−118.136			−0.169492
$698$		0			0
$699$			560.691i			0.802133i
$700$		0			0
$701$	−511.967			−0.730338			−0.365169	−	0.930941i	$-0.618989\pi$
$701$	−511.967			−0.730338			−0.365169	+	0.930941i	$0.618989\pi$
$702$		0			0
$703$			481.997i			0.685629i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−782.848	−	769.930i	−1.10728	−	1.08901i
$708$		0			0
$709$	−879.130			−1.23996			−0.619979	−	0.784619i	$-0.712859\pi$
$709$	−879.130			−1.23996			−0.619979	+	0.784619i	$0.712859\pi$
$710$		0			0
$711$	−342.929			−0.482320
$712$		0			0
$713$			466.958i			0.654920i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	97.9053i		−	0.136549i
$718$		0			0
$719$		−	242.715i		−	0.337574i	−0.985653	−	0.168787i	$-0.946015\pi$
$719$		−	242.715i		−	0.337574i	0.985653	−	0.168787i	$-0.0539849\pi$
$720$		0			0
$721$	−164.569	−	161.854i	−0.228252	−	0.224485i
$722$		0			0
$723$	358.791			0.496253
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	147.548i		−	0.202955i	−0.994838	−	0.101477i	$-0.967643\pi$
$727$		−	147.548i		−	0.202955i	0.994838	−	0.101477i	$-0.0323570\pi$
$728$		0			0
$729$	−27.0000			−0.0370370
$730$		0			0
$731$		−	81.8981i		−	0.112036i
$732$		0			0
$733$		−	179.234i		−	0.244521i	−0.992498	−	0.122260i	$-0.960986\pi$
$733$		−	179.234i		−	0.244521i	0.992498	−	0.122260i	$-0.0390143\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−242.928			−0.329617
$738$		0			0
$739$	−643.114			−0.870249			−0.435124	−	0.900370i	$-0.643296\pi$
$739$	−643.114			−0.870249			−0.435124	+	0.900370i	$0.643296\pi$
$740$		0			0
$741$			32.9235i			0.0444311i
$742$		0			0
$743$	−155.874			−0.209791			−0.104895	−	0.994483i	$-0.533451\pi$
$743$	−155.874			−0.209791			−0.104895	+	0.994483i	$0.533451\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			4.92641i			0.00659492i
$748$		0			0
$749$	−790.787	+	804.055i	−1.05579	+	1.07351i
$750$		0			0
$751$	72.5148			0.0965576			0.0482788	−	0.998834i	$-0.484626\pi$
$751$	72.5148			0.0965576			0.0482788	+	0.998834i	$0.484626\pi$
$752$		0			0
$753$	−506.523			−0.672674
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1413.10			−1.86671			−0.933354	−	0.358958i	$-0.883132\pi$
$757$	−1413.10			−1.86671			−0.933354	+	0.358958i	$0.883132\pi$
$758$		0			0
$759$			97.9838i			0.129096i
$760$		0			0
$761$		−	734.130i		−	0.964691i	−0.875981	−	0.482345i	$-0.839785\pi$
$761$		−	734.130i		−	0.964691i	0.875981	−	0.482345i	$-0.160215\pi$
$762$		0			0
$763$	805.004	−	818.511i	1.05505	−	1.07275i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	70.9460			0.0924980
$768$		0			0
$769$		−	1157.02i		−	1.50458i	−0.658834	−	0.752288i	$-0.728950\pi$
$769$		−	1157.02i		−	1.50458i	0.658834	−	0.752288i	$-0.271050\pi$
$770$		0			0
$771$	332.908			0.431788
$772$		0			0
$773$			1207.99i			1.56273i	0.624076	+	0.781364i	$0.285476\pi$
$773$			1207.99i			1.56273i	−0.624076	+	0.781364i	$0.714524\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−255.320	−	251.107i	−0.328597	−	0.323174i
$778$		0			0
$779$	−637.676			−0.818583
$780$		0			0
$781$	280.287			0.358882
$782$		0			0
$783$		−	3.87692i		−	0.00495136i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			945.463i			1.20135i	0.799493	+	0.600676i	$0.205102\pi$
$787$			945.463i			1.20135i	−0.799493	+	0.600676i	$0.794898\pi$
$788$		0			0
$789$		−	171.728i		−	0.217653i
$790$		0			0
$791$	−863.982	+	878.479i	−1.09227	+	1.11059i
$792$		0			0
$793$	−34.9361			−0.0440557
$794$		0			0
$795$		0			0
$796$		0			0
$797$			222.632i			0.279338i	0.990198	+	0.139669i	$0.0446039\pi$
$797$			222.632i			0.279338i	−0.990198	+	0.139669i	$0.955396\pi$
$798$		0			0
$799$	−137.274			−0.171807
$800$		0			0
$801$			108.236i			0.135127i
$802$		0			0
$803$		−	220.452i		−	0.274535i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	564.598			0.699626
$808$		0			0
$809$	−189.617			−0.234385			−0.117192	−	0.993109i	$-0.537389\pi$
$809$	−189.617			−0.234385			−0.117192	+	0.993109i	$0.537389\pi$
$810$		0			0
$811$			1586.30i			1.95598i	0.208648	+	0.977991i	$0.433094\pi$
$811$			1586.30i			1.95598i	−0.208648	+	0.977991i	$0.566906\pi$
$812$		0			0
$813$	243.013			0.298909
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	442.070i		−	0.541090i
$818$		0			0
$819$	−17.4400	−	17.1522i	−0.0212942	−	0.0209428i
$820$		0			0
$821$	1281.41			1.56079			0.780393	−	0.625289i	$-0.215019\pi$
$821$	1281.41			1.56079			0.780393	+	0.625289i	$0.215019\pi$
$822$		0			0
$823$	1314.96			1.59776			0.798879	−	0.601491i	$-0.205427\pi$
$823$	1314.96			1.59776			0.798879	+	0.601491i	$0.205427\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−885.845			−1.07115			−0.535577	−	0.844486i	$-0.679906\pi$
$827$	−885.845			−1.07115			−0.535577	+	0.844486i	$0.679906\pi$
$828$		0			0
$829$			614.223i			0.740920i	0.928848	+	0.370460i	$0.120800\pi$
$829$			614.223i			0.740920i	−0.928848	+	0.370460i	$0.879200\pi$
$830$		0			0
$831$			54.4475i			0.0655205i
$832$		0			0
$833$	−148.117	+	2.46466i	−0.177812	+	0.00295878i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	151.225			0.180675
$838$		0			0
$839$			1020.74i			1.21661i	0.793702	+	0.608307i	$0.208151\pi$
$839$			1020.74i			1.21661i	−0.793702	+	0.608307i	$0.791849\pi$
$840$		0			0
$841$	−840.443			−0.999338
$842$		0			0
$843$			751.014i			0.890882i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−532.898	+	541.840i	−0.629160	+	0.639716i
$848$		0			0
$849$	−638.916			−0.752551
$850$		0			0
$851$	−473.907			−0.556883
$852$		0			0
$853$			830.679i			0.973833i	0.873449	+	0.486916i	$0.161878\pi$
$853$			830.679i			0.973833i	−0.873449	+	0.486916i	$0.838122\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	605.854i		−	0.706947i	−0.935445	−	0.353473i	$-0.885000\pi$
$857$		−	605.854i		−	0.706947i	0.935445	−	0.353473i	$-0.115000\pi$
$858$		0			0
$859$		−	125.260i		−	0.145820i	−0.997339	−	0.0729102i	$-0.976771\pi$
$859$		−	125.260i		−	0.145820i	0.997339	−	0.0729102i	$-0.0232287\pi$
$860$		0			0
$861$	332.211	−	337.785i	0.385843	−	0.392317i
$862$		0			0
$863$	−830.505			−0.962346			−0.481173	−	0.876626i	$-0.659789\pi$
$863$	−830.505			−0.962346			−0.481173	+	0.876626i	$0.659789\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			484.732i			0.559091i
$868$		0			0
$869$	403.034			0.463790
$870$		0			0
$871$		−	80.2558i		−	0.0921422i
$872$		0			0
$873$		−	252.032i		−	0.288696i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1312.26			1.49630			0.748152	−	0.663527i	$-0.230941\pi$
$877$	1312.26			1.49630			0.748152	+	0.663527i	$0.230941\pi$
$878$		0			0
$879$	45.3169			0.0515551
$880$		0			0
$881$		−	721.319i		−	0.818750i	−0.912366	−	0.409375i	$-0.865747\pi$
$881$		−	721.319i		−	0.818750i	0.912366	−	0.409375i	$-0.134253\pi$
$882$		0			0
$883$	720.310			0.815753			0.407877	−	0.913037i	$-0.366269\pi$
$883$	720.310			0.815753			0.407877	+	0.913037i	$0.366269\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	209.253i		−	0.235911i	−0.993019	−	0.117956i	$-0.962366\pi$
$887$		−	209.253i		−	0.235911i	0.993019	−	0.117956i	$-0.0376341\pi$
$888$		0			0
$889$	496.073	−	504.396i	0.558012	−	0.567375i
$890$		0			0
$891$	31.7322			0.0356142
$892$		0			0
$893$	−740.980			−0.829764
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−32.3709			−0.0360879
$898$		0			0
$899$			21.7143i			0.0241539i
$900$		0			0
$901$			100.176i			0.111183i
$902$		0			0
$903$	234.170	+	230.306i	0.259324	+	0.255045i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−361.827			−0.398927			−0.199463	−	0.979905i	$-0.563920\pi$
$907$	−361.827			−0.398927			−0.199463	+	0.979905i	$0.563920\pi$
$908$		0			0
$909$			470.579i			0.517688i
$910$		0			0
$911$	−35.6156			−0.0390951			−0.0195476	−	0.999809i	$-0.506223\pi$
$911$	−35.6156			−0.0390951			−0.0195476	+	0.999809i	$0.506223\pi$
$912$		0			0
$913$		−	5.78985i		−	0.00634156i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−226.970	−	223.225i	−0.247514	−	0.243429i
$918$		0			0
$919$	−497.186			−0.541008			−0.270504	−	0.962719i	$-0.587190\pi$
$919$	−497.186			−0.541008			−0.270504	+	0.962719i	$0.587190\pi$
$920$		0			0
$921$	712.089			0.773170
$922$		0			0
$923$			92.5983i			0.100323i
$924$		0			0
$925$		0			0
$926$		0			0
$927$			98.9246i			0.106715i
$928$		0			0
$929$			1484.19i			1.59763i	0.601579	+	0.798813i	$0.294538\pi$
$929$			1484.19i			1.59763i	−0.601579	+	0.798813i	$0.705462\pi$
$930$		0			0
$931$	−799.508	+	13.3038i	−0.858763	+	0.0142898i
$932$		0			0
$933$	−593.311			−0.635918
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	93.5018i		−	0.0997885i	−0.998755	−	0.0498942i	$-0.984112\pi$
$937$		−	93.5018i		−	0.0997885i	0.998755	−	0.0498942i	$-0.0158884\pi$
$938$		0			0
$939$	−31.0472			−0.0330641
$940$		0			0
$941$			774.652i			0.823222i	0.911360	+	0.411611i	$0.135034\pi$
$941$			774.652i			0.823222i	−0.911360	+	0.411611i	$0.864966\pi$
$942$		0			0
$943$		−	626.973i		−	0.664871i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1007.23			1.06360			0.531798	−	0.846871i	$-0.321517\pi$
$947$	1007.23			1.06360			0.531798	+	0.846871i	$0.321517\pi$
$948$		0			0
$949$	72.8305			0.0767444
$950$		0			0
$951$		−	892.764i		−	0.938763i
$952$		0			0
$953$	995.073			1.04415			0.522074	−	0.852900i	$-0.325159\pi$
$953$	995.073			1.04415			0.522074	+	0.852900i	$0.325159\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			4.55642i			0.00476115i
$958$		0			0
$959$	586.085	−	595.919i	0.611142	−	0.621396i
$960$		0			0
$961$	113.999			0.118625
$962$		0			0
$963$	483.327			0.501897
$964$		0			0
$965$		0			0
$966$		0			0
$967$	799.010			0.826277			0.413138	−	0.910668i	$-0.364433\pi$
$967$	799.010			0.826277			0.413138	+	0.910668i	$0.364433\pi$
$968$		0			0
$969$		−	85.4509i		−	0.0881846i
$970$		0			0
$971$			782.049i			0.805406i	0.915331	+	0.402703i	$0.131929\pi$
$971$			782.049i			0.805406i	−0.915331	+	0.402703i	$0.868071\pi$
$972$		0			0
$973$	−1229.40	−	1209.12i	−1.26352	−	1.24267i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1377.93			1.41036			0.705182	−	0.709027i	$-0.250865\pi$
$977$	1377.93			1.41036			0.705182	+	0.709027i	$0.250865\pi$
$978$		0			0
$979$		−	127.207i		−	0.129936i
$980$		0			0
$981$	−492.016			−0.501546
$982$		0			0
$983$			1443.06i			1.46801i	0.679142	+	0.734007i	$0.262352\pi$
$983$			1443.06i			1.46801i	−0.679142	+	0.734007i	$0.737648\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	386.029	−	392.506i	0.391113	−	0.397676i
$988$		0			0
$989$	434.650			0.439485
$990$		0			0
$991$	−654.050			−0.659989			−0.329995	−	0.943983i	$-0.607047\pi$
$991$	−654.050			−0.659989			−0.329995	+	0.943983i	$0.607047\pi$
$992$		0			0
$993$		−	125.722i		−	0.126608i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			1246.90i			1.25065i	0.780365	+	0.625324i	$0.215033\pi$
$997$			1246.90i			1.25065i	−0.780365	+	0.625324i	$0.784967\pi$
$998$		0			0
$999$			153.476i			0.153629i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.3.j.g.601.14		16
5.2	odd	4		420.3.p.a.349.15	yes	16
5.3	odd	4		420.3.p.a.349.1	✓	16
5.4	even	2	inner	2100.3.j.g.601.3		16
7.6	odd	2	inner	2100.3.j.g.601.6		16
15.2	even	4		1260.3.p.e.1189.4		16
15.8	even	4		1260.3.p.e.1189.14		16
20.3	even	4		1680.3.bd.b.769.9		16
20.7	even	4		1680.3.bd.b.769.7		16
35.13	even	4		420.3.p.a.349.16	yes	16
35.27	even	4		420.3.p.a.349.2	yes	16
35.34	odd	2	inner	2100.3.j.g.601.11		16
105.62	odd	4		1260.3.p.e.1189.13		16
105.83	odd	4		1260.3.p.e.1189.3		16
140.27	odd	4		1680.3.bd.b.769.10		16
140.83	odd	4		1680.3.bd.b.769.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.3.p.a.349.1	✓	16	5.3	odd	4
420.3.p.a.349.2	yes	16	35.27	even	4
420.3.p.a.349.15	yes	16	5.2	odd	4
420.3.p.a.349.16	yes	16	35.13	even	4
1260.3.p.e.1189.3		16	105.83	odd	4
1260.3.p.e.1189.4		16	15.2	even	4
1260.3.p.e.1189.13		16	105.62	odd	4
1260.3.p.e.1189.14		16	15.8	even	4
1680.3.bd.b.769.7		16	20.7	even	4
1680.3.bd.b.769.8		16	140.83	odd	4
1680.3.bd.b.769.9		16	20.3	even	4
1680.3.bd.b.769.10		16	140.27	odd	4
2100.3.j.g.601.3		16	5.4	even	2	inner
2100.3.j.g.601.6		16	7.6	odd	2	inner
2100.3.j.g.601.11		16	35.34	odd	2	inner
2100.3.j.g.601.14		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2100.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +(4.90840 - 4.99075i) q^{7} -3.00000 q^{9} +3.52580 q^{11} +1.16482i q^{13} -3.02322i q^{17} -16.3188i q^{19} +(8.64424 + 8.50160i) q^{21} +16.0449 q^{23} -5.19615i q^{27} +0.746113 q^{29} +29.1033i q^{31} +6.10687i q^{33} -29.5364 q^{37} -2.01752 q^{39} -39.0763i q^{41} +27.0897 q^{43} -45.4066i q^{47} +(-0.815245 - 48.9932i) q^{49} +5.23636 q^{51} -33.1354 q^{53} +28.2649 q^{57} -60.9074i q^{59} +29.9928i q^{61} +(-14.7252 + 14.9723i) q^{63} -68.8999 q^{67} +27.7905i q^{69} +79.4959 q^{71} -62.5252i q^{73} +(17.3061 - 17.5964i) q^{77} +114.310 q^{79} +9.00000 q^{81} -1.64214i q^{83} +1.29231i q^{87} -36.0788i q^{89} +(5.81332 + 5.71739i) q^{91} -50.4084 q^{93} +84.0105i q^{97} -10.5774 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9} - 24 q^{11} - 12 q^{21} + 32 q^{29} - 72 q^{39} + 88 q^{49} + 24 q^{51} + 168 q^{71} - 16 q^{79} + 144 q^{81} - 568 q^{91} + 72 q^{99}+O(q^{100}) \) 16 * q - 48 * q^9 - 24 * q^11 - 12 * q^21 + 32 * q^29 - 72 * q^39 + 88 * q^49 + 24 * q^51 + 168 * q^71 - 16 * q^79 + 144 * q^81 - 568 * q^91 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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