LMFDB - Embedded newform 624.4.q.c.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(289,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("624.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$36.8171918436$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	624.289
Dual form			624.4.q.c.529.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50000 + 2.59808i) q^{3} +7.00000 q^{5} +(-5.00000 + 8.66025i) q^{7} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})$	\(q+(1.50000 + 2.59808i) q^{3} +7.00000 q^{5} +(-5.00000 + 8.66025i) q^{7} +(-4.50000 + 7.79423i) q^{9} +(-11.0000 - 19.0526i) q^{11} +(-45.5000 + 11.2583i) q^{13} +(10.5000 + 18.1865i) q^{15} +(-18.5000 + 32.0429i) q^{17} +(15.0000 - 25.9808i) q^{19} -30.0000 q^{21} +(-81.0000 - 140.296i) q^{23} -76.0000 q^{25} -27.0000 q^{27} +(56.5000 + 97.8609i) q^{29} -196.000 q^{31} +(33.0000 - 57.1577i) q^{33} +(-35.0000 + 60.6218i) q^{35} +(-6.50000 - 11.2583i) q^{37} +(-97.5000 - 101.325i) q^{39} +(-142.500 - 246.817i) q^{41} +(-123.000 + 213.042i) q^{43} +(-31.5000 + 54.5596i) q^{45} +462.000 q^{47} +(121.500 + 210.444i) q^{49} -111.000 q^{51} -537.000 q^{53} +(-77.0000 - 133.368i) q^{55} +90.0000 q^{57} +(288.000 - 498.831i) q^{59} +(317.500 - 549.926i) q^{61} +(-45.0000 - 77.9423i) q^{63} +(-318.500 + 78.8083i) q^{65} +(101.000 + 174.937i) q^{67} +(243.000 - 420.888i) q^{69} +(-543.000 + 940.504i) q^{71} -805.000 q^{73} +(-114.000 - 197.454i) q^{75} +220.000 q^{77} -884.000 q^{79} +(-40.5000 - 70.1481i) q^{81} -518.000 q^{83} +(-129.500 + 224.301i) q^{85} +(-169.500 + 293.583i) q^{87} +(-97.0000 - 168.009i) q^{89} +(130.000 - 450.333i) q^{91} +(-294.000 - 509.223i) q^{93} +(105.000 - 181.865i) q^{95} +(601.000 - 1040.96i) q^{97} +198.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} + 14 q^{5} - 10 q^{7} - 9 q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 + 14 * q^5 - 10 * q^7 - 9 * q^9	$ 2 q + 3 q^{3} + 14 q^{5} - 10 q^{7} - 9 q^{9} - 22 q^{11} - 91 q^{13} + 21 q^{15} - 37 q^{17} + 30 q^{19} - 60 q^{21} - 162 q^{23} - 152 q^{25} - 54 q^{27} + 113 q^{29} - 392 q^{31} + 66 q^{33} - 70 q^{35} - 13 q^{37} - 195 q^{39} - 285 q^{41} - 246 q^{43} - 63 q^{45} + 924 q^{47} + 243 q^{49} - 222 q^{51} - 1074 q^{53} - 154 q^{55} + 180 q^{57} + 576 q^{59} + 635 q^{61} - 90 q^{63} - 637 q^{65} + 202 q^{67} + 486 q^{69} - 1086 q^{71} - 1610 q^{73} - 228 q^{75} + 440 q^{77} - 1768 q^{79} - 81 q^{81} - 1036 q^{83} - 259 q^{85} - 339 q^{87} - 194 q^{89} + 260 q^{91} - 588 q^{93} + 210 q^{95} + 1202 q^{97} + 396 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 + 14 * q^5 - 10 * q^7 - 9 * q^9 - 22 * q^11 - 91 * q^13 + 21 * q^15 - 37 * q^17 + 30 * q^19 - 60 * q^21 - 162 * q^23 - 152 * q^25 - 54 * q^27 + 113 * q^29 - 392 * q^31 + 66 * q^33 - 70 * q^35 - 13 * q^37 - 195 * q^39 - 285 * q^41 - 246 * q^43 - 63 * q^45 + 924 * q^47 + 243 * q^49 - 222 * q^51 - 1074 * q^53 - 154 * q^55 + 180 * q^57 + 576 * q^59 + 635 * q^61 - 90 * q^63 - 637 * q^65 + 202 * q^67 + 486 * q^69 - 1086 * q^71 - 1610 * q^73 - 228 * q^75 + 440 * q^77 - 1768 * q^79 - 81 * q^81 - 1036 * q^83 - 259 * q^85 - 339 * q^87 - 194 * q^89 + 260 * q^91 - 588 * q^93 + 210 * q^95 + 1202 * q^97 + 396 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$1$

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
$3$	1.50000	+	2.59808i	0.288675	+	0.500000i
$3$	1.50000	+	2.59808i	0.288675	+	0.500000i
$4$		0			0
$5$	7.00000			0.626099			0.313050	−	0.949737i	$-0.398649\pi$
$5$	7.00000			0.626099			0.313050	+	0.949737i	$0.398649\pi$
$6$		0			0
$7$	−5.00000	+	8.66025i	−0.269975	+	0.467610i	−0.968855	−	0.247629i	$-0.920349\pi$
$7$	−5.00000	+	8.66025i	−0.269975	+	0.467610i	0.698880	+	0.715239i	$0.253682\pi$
$8$		0			0
$9$	−4.50000	+	7.79423i	−0.166667	+	0.288675i
$10$		0			0
$11$	−11.0000	−	19.0526i	−0.301511	−	0.522233i	0.674967	−	0.737848i	$-0.264158\pi$
$11$	−11.0000	−	19.0526i	−0.301511	−	0.522233i	−0.976478	+	0.215615i	$0.930824\pi$
$12$		0			0
$13$	−45.5000	+	11.2583i	−0.970725	+	0.240192i
$13$	−45.5000	+	11.2583i	−0.970725	+	0.240192i
$14$		0			0
$15$	10.5000	+	18.1865i	0.180739	+	0.313050i
$16$		0			0
$17$	−18.5000	+	32.0429i	−0.263936	+	0.457150i	−0.967284	−	0.253695i	$-0.918354\pi$
$17$	−18.5000	+	32.0429i	−0.263936	+	0.457150i	0.703348	+	0.710845i	$0.251687\pi$
$18$		0			0
$19$	15.0000	−	25.9808i	0.181118	−	0.313705i	−0.761144	−	0.648583i	$-0.775362\pi$
$19$	15.0000	−	25.9808i	0.181118	−	0.313705i	0.942261	+	0.334878i	$0.108695\pi$
$20$		0			0
$21$	−30.0000			−0.311740
$22$		0			0
$23$	−81.0000	−	140.296i	−0.734333	−	1.27190i	−0.955015	−	0.296557i	$-0.904162\pi$
$23$	−81.0000	−	140.296i	−0.734333	−	1.27190i	0.220682	−	0.975346i	$-0.429172\pi$
$24$		0			0
$25$	−76.0000			−0.608000
$26$		0			0
$27$	−27.0000			−0.192450
$28$		0			0
$29$	56.5000	+	97.8609i	0.361786	+	0.626631i	0.988255	−	0.152815i	$-0.0488339\pi$
$29$	56.5000	+	97.8609i	0.361786	+	0.626631i	−0.626469	+	0.779446i	$0.715501\pi$
$30$		0			0
$31$	−196.000			−1.13557			−0.567785	−	0.823177i	$-0.692199\pi$
$31$	−196.000			−1.13557			−0.567785	+	0.823177i	$0.692199\pi$
$32$		0			0
$33$	33.0000	−	57.1577i	0.174078	−	0.301511i
$34$		0			0
$35$	−35.0000	+	60.6218i	−0.169031	+	0.292770i
$36$		0			0
$37$	−6.50000	−	11.2583i	−0.0288809	−	0.0500232i	0.851224	−	0.524803i	$-0.175861\pi$
$37$	−6.50000	−	11.2583i	−0.0288809	−	0.0500232i	−0.880105	+	0.474780i	$0.842528\pi$
$38$		0			0
$39$	−97.5000	−	101.325i	−0.400320	−	0.416025i
$40$		0			0
$41$	−142.500	−	246.817i	−0.542799	−	0.940156i	−0.998742	−	0.0501465i	$-0.984031\pi$
$41$	−142.500	−	246.817i	−0.542799	−	0.940156i	0.455943	−	0.890009i	$-0.349302\pi$
$42$		0			0
$43$	−123.000	+	213.042i	−0.436217	+	0.755550i	−0.997394	−	0.0721459i	$-0.977015\pi$
$43$	−123.000	+	213.042i	−0.436217	+	0.755550i	0.561177	+	0.827696i	$0.310349\pi$
$44$		0			0
$45$	−31.5000	+	54.5596i	−0.104350	+	0.180739i
$46$		0			0
$47$	462.000			1.43382			0.716911	−	0.697165i	$-0.245555\pi$
$47$	462.000			1.43382			0.716911	+	0.697165i	$0.245555\pi$
$48$		0			0
$49$	121.500	+	210.444i	0.354227	+	0.613540i
$50$		0			0
$51$	−111.000			−0.304767
$52$		0			0
$53$	−537.000			−1.39175			−0.695874	−	0.718164i	$-0.744983\pi$
$53$	−537.000			−1.39175			−0.695874	+	0.718164i	$0.744983\pi$
$54$		0			0
$55$	−77.0000	−	133.368i	−0.188776	−	0.326970i
$56$		0			0
$57$	90.0000			0.209137
$58$		0			0
$59$	288.000	−	498.831i	0.635498	−	1.10072i	−0.350911	−	0.936409i	$-0.614128\pi$
$59$	288.000	−	498.831i	0.635498	−	1.10072i	0.986409	−	0.164307i	$-0.0525387\pi$
$60$		0			0
$61$	317.500	−	549.926i	0.666421	−	1.15428i	−0.312476	−	0.949926i	$-0.601159\pi$
$61$	317.500	−	549.926i	0.666421	−	1.15428i	0.978898	−	0.204350i	$-0.0655082\pi$
$62$		0			0
$63$	−45.0000	−	77.9423i	−0.0899915	−	0.155870i
$64$		0			0
$65$	−318.500	+	78.8083i	−0.607770	+	0.150384i
$66$		0			0
$67$	101.000	+	174.937i	0.184166	+	0.318985i	0.943295	−	0.331955i	$-0.107708\pi$
$67$	101.000	+	174.937i	0.184166	+	0.318985i	−0.759129	+	0.650940i	$0.774375\pi$
$68$		0			0
$69$	243.000	−	420.888i	0.423968	−	0.734333i
$70$		0			0
$71$	−543.000	+	940.504i	−0.907637	+	1.57207i	−0.0902997	+	0.995915i	$0.528783\pi$
$71$	−543.000	+	940.504i	−0.907637	+	1.57207i	−0.817338	+	0.576159i	$0.804551\pi$
$72$		0			0
$73$	−805.000			−1.29066			−0.645330	−	0.763904i	$-0.723280\pi$
$73$	−805.000			−1.29066			−0.645330	+	0.763904i	$0.723280\pi$
$74$		0			0
$75$	−114.000	−	197.454i	−0.175514	−	0.304000i
$76$		0			0
$77$	220.000			0.325602
$78$		0			0
$79$	−884.000			−1.25896			−0.629480	−	0.777017i	$-0.716732\pi$
$79$	−884.000			−1.25896			−0.629480	+	0.777017i	$0.716732\pi$
$80$		0			0
$81$	−40.5000	−	70.1481i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−518.000			−0.685035			−0.342517	−	0.939511i	$-0.611280\pi$
$83$	−518.000			−0.685035			−0.342517	+	0.939511i	$0.611280\pi$
$84$		0			0
$85$	−129.500	+	224.301i	−0.165250	+	0.286221i
$86$		0			0
$87$	−169.500	+	293.583i	−0.208877	+	0.361786i
$88$		0			0
$89$	−97.0000	−	168.009i	−0.115528	−	0.200100i	0.802463	−	0.596702i	$-0.203523\pi$
$89$	−97.0000	−	168.009i	−0.115528	−	0.200100i	−0.917991	+	0.396602i	$0.870189\pi$
$90$		0			0
$91$	130.000	−	450.333i	0.149755	−	0.518766i
$92$		0			0
$93$	−294.000	−	509.223i	−0.327811	−	0.567785i
$94$		0			0
$95$	105.000	−	181.865i	0.113398	−	0.196410i
$96$		0			0
$97$	601.000	−	1040.96i	0.629096	−	1.08963i	−0.358638	−	0.933477i	$-0.616759\pi$
$97$	601.000	−	1040.96i	0.629096	−	1.08963i	0.987733	−	0.156149i	$-0.0499081\pi$
$98$		0			0
$99$	198.000			0.201008
$100$		0			0
$101$	214.500	+	371.525i	0.211322	+	0.366021i	0.952129	−	0.305698i	$-0.0988897\pi$
$101$	214.500	+	371.525i	0.211322	+	0.366021i	−0.740806	+	0.671719i	$0.765556\pi$
$102$		0			0
$103$	1302.00			1.24553			0.622766	−	0.782408i	$-0.286009\pi$
$103$	1302.00			1.24553			0.622766	+	0.782408i	$0.286009\pi$
$104$		0			0
$105$	−210.000			−0.195180
$106$		0			0
$107$	−669.000	−	1158.74i	−0.604436	−	1.04691i	−0.992140	−	0.125130i	$-0.960065\pi$
$107$	−669.000	−	1158.74i	−0.604436	−	1.04691i	0.387704	−	0.921784i	$-0.373268\pi$
$108$		0			0
$109$	−1034.00			−0.908617			−0.454308	−	0.890844i	$-0.650114\pi$
$109$	−1034.00			−0.908617			−0.454308	+	0.890844i	$0.650114\pi$
$110$		0			0
$111$	19.5000	−	33.7750i	0.0166744	−	0.0288809i
$112$		0			0
$113$	−538.500	+	932.709i	−0.448299	+	0.776477i	−0.998275	−	0.0587032i	$-0.981303\pi$
$113$	−538.500	+	932.709i	−0.448299	+	0.776477i	0.549976	+	0.835180i	$0.314637\pi$
$114$		0			0
$115$	−567.000	−	982.073i	−0.459765	−	0.796337i
$116$		0			0
$117$	117.000	−	405.300i	0.0924500	−	0.320256i
$118$		0			0
$119$	−185.000	−	320.429i	−0.142512	−	0.246838i
$120$		0			0
$121$	423.500	−	733.524i	0.318182	−	0.551107i
$122$		0			0
$123$	427.500	−	740.452i	0.313385	−	0.542799i
$124$		0			0
$125$	−1407.00			−1.00677
$126$		0			0
$127$	−494.000	−	855.633i	−0.345161	−	0.597836i	0.640222	−	0.768190i	$-0.278842\pi$
$127$	−494.000	−	855.633i	−0.345161	−	0.597836i	−0.985383	+	0.170354i	$0.945509\pi$
$128$		0			0
$129$	−738.000			−0.503700
$130$		0			0
$131$	−560.000			−0.373492			−0.186746	−	0.982408i	$-0.559794\pi$
$131$	−560.000			−0.373492			−0.186746	+	0.982408i	$0.559794\pi$
$132$		0			0
$133$	150.000	+	259.808i	0.0977944	+	0.169385i
$134$		0			0
$135$	−189.000			−0.120493
$136$		0			0
$137$	259.500	−	449.467i	0.161829	−	0.280296i	−0.773696	−	0.633557i	$-0.781594\pi$
$137$	259.500	−	449.467i	0.161829	−	0.280296i	0.935525	+	0.353261i	$0.114927\pi$
$138$		0			0
$139$	−174.000	+	301.377i	−0.106176	+	0.183903i	−0.914218	−	0.405222i	$-0.867194\pi$
$139$	−174.000	+	301.377i	−0.106176	+	0.183903i	0.808042	+	0.589125i	$0.200527\pi$
$140$		0			0
$141$	693.000	+	1200.31i	0.413909	+	0.716911i
$142$		0			0
$143$	715.000	+	743.050i	0.418121	+	0.434524i
$144$		0			0
$145$	395.500	+	685.026i	0.226514	+	0.392333i
$146$		0			0
$147$	−364.500	+	631.333i	−0.204513	+	0.354227i
$148$		0			0
$149$	322.500	−	558.586i	0.177317	−	0.307122i	−0.763644	−	0.645638i	$-0.776592\pi$
$149$	322.500	−	558.586i	0.177317	−	0.307122i	0.940961	+	0.338516i	$0.109925\pi$
$150$		0			0
$151$	−2914.00			−1.57045			−0.785225	−	0.619211i	$-0.787453\pi$
$151$	−2914.00			−1.57045			−0.785225	+	0.619211i	$0.787453\pi$
$152$		0			0
$153$	−166.500	−	288.386i	−0.0879786	−	0.152383i
$154$		0			0
$155$	−1372.00			−0.710979
$156$		0			0
$157$	−2079.00			−1.05683			−0.528415	−	0.848986i	$-0.677213\pi$
$157$	−2079.00			−1.05683			−0.528415	+	0.848986i	$0.677213\pi$
$158$		0			0
$159$	−805.500	−	1395.17i	−0.401763	−	0.695874i
$160$		0			0
$161$	1620.00			0.793006
$162$		0			0
$163$	850.000	−	1472.24i	0.408449	−	0.707454i	−0.586267	−	0.810118i	$-0.699403\pi$
$163$	850.000	−	1472.24i	0.408449	−	0.707454i	0.994716	+	0.102664i	$0.0327365\pi$
$164$		0			0
$165$	231.000	−	400.104i	0.108990	−	0.188776i
$166$		0			0
$167$	1840.00	+	3186.97i	0.852596	+	1.47674i	0.878858	+	0.477084i	$0.158306\pi$
$167$	1840.00	+	3186.97i	0.852596	+	1.47674i	−0.0262621	+	0.999655i	$0.508360\pi$
$168$		0			0
$169$	1943.50	−	1024.51i	0.884615	−	0.466321i
$170$		0			0
$171$	135.000	+	233.827i	0.0603726	+	0.104568i
$172$		0			0
$173$	−2073.00	+	3590.54i	−0.911025	+	1.57794i	−0.0984052	+	0.995146i	$0.531374\pi$
$173$	−2073.00	+	3590.54i	−0.911025	+	1.57794i	−0.812619	+	0.582795i	$0.801959\pi$
$174$		0			0
$175$	380.000	−	658.179i	0.164145	−	0.284307i
$176$		0			0
$177$	1728.00			0.733810
$178$		0			0
$179$	1837.00	+	3181.78i	0.767060	+	1.32859i	0.939150	+	0.343506i	$0.111615\pi$
$179$	1837.00	+	3181.78i	0.767060	+	1.32859i	−0.172090	+	0.985081i	$0.555052\pi$
$180$		0			0
$181$	−3283.00			−1.34820			−0.674098	−	0.738642i	$-0.735467\pi$
$181$	−3283.00			−1.34820			−0.674098	+	0.738642i	$0.735467\pi$
$182$		0			0
$183$	1905.00			0.769517
$184$		0			0
$185$	−45.5000	−	78.8083i	−0.0180823	−	0.0313195i
$186$		0			0
$187$	814.000			0.318319
$188$		0			0
$189$	135.000	−	233.827i	0.0519566	−	0.0899915i
$190$		0			0
$191$	−298.000	+	516.151i	−0.112893	+	0.195536i	−0.916935	−	0.399036i	$-0.869345\pi$
$191$	−298.000	+	516.151i	−0.112893	+	0.195536i	0.804043	+	0.594572i	$0.202678\pi$
$192$		0			0
$193$	196.500	+	340.348i	0.0732869	+	0.126937i	0.900340	−	0.435187i	$-0.143318\pi$
$193$	196.500	+	340.348i	0.0732869	+	0.126937i	−0.827053	+	0.562124i	$0.809984\pi$
$194$		0			0
$195$	−682.500	−	709.275i	−0.250640	−	0.260473i
$196$		0			0
$197$	1761.00	+	3050.14i	0.636884	+	1.10311i	0.986113	+	0.166077i	$0.0531101\pi$
$197$	1761.00	+	3050.14i	0.636884	+	1.10311i	−0.349229	+	0.937037i	$0.613557\pi$
$198$		0			0
$199$	1009.00	−	1747.64i	0.359428	−	0.622547i	−0.628438	−	0.777860i	$-0.716305\pi$
$199$	1009.00	−	1747.64i	0.359428	−	0.622547i	0.987865	+	0.155313i	$0.0496386\pi$
$200$		0			0
$201$	−303.000	+	524.811i	−0.106328	+	0.184166i
$202$		0			0
$203$	−1130.00			−0.390692
$204$		0			0
$205$	−997.500	−	1727.72i	−0.339846	−	0.588630i
$206$		0			0
$207$	1458.00			0.489556
$208$		0			0
$209$	−660.000			−0.218436
$210$		0			0
$211$	80.0000	+	138.564i	0.0261016	+	0.0452092i	0.878781	−	0.477225i	$-0.158357\pi$
$211$	80.0000	+	138.564i	0.0261016	+	0.0452092i	−0.852680	+	0.522434i	$0.825024\pi$
$212$		0			0
$213$	−3258.00			−1.04805
$214$		0			0
$215$	−861.000	+	1491.30i	−0.273115	+	0.473049i
$216$		0			0
$217$	980.000	−	1697.41i	0.306575	−	0.531003i
$218$		0			0
$219$	−1207.50	−	2091.45i	−0.372581	−	0.645330i
$220$		0			0
$221$	481.000	−	1666.23i	0.146405	−	0.507163i
$222$		0			0
$223$	2036.00	+	3526.46i	0.611393	+	1.05896i	0.991006	+	0.133818i	$0.0427239\pi$
$223$	2036.00	+	3526.46i	0.611393	+	1.05896i	−0.379613	+	0.925145i	$0.623943\pi$
$224$		0			0
$225$	342.000	−	592.361i	0.101333	−	0.175514i
$226$		0			0
$227$	−2897.00	+	5017.75i	−0.847051	+	1.46714i	0.0367765	+	0.999324i	$0.488291\pi$
$227$	−2897.00	+	5017.75i	−0.847051	+	1.46714i	−0.883828	+	0.467812i	$0.845042\pi$
$228$		0			0
$229$	6482.00			1.87049			0.935246	−	0.353999i	$-0.115178\pi$
$229$	6482.00			1.87049			0.935246	+	0.353999i	$0.115178\pi$
$230$		0			0
$231$	330.000	+	571.577i	0.0939931	+	0.162801i
$232$		0			0
$233$	6890.00			1.93725			0.968624	−	0.248530i	$-0.0799474\pi$
$233$	6890.00			1.93725			0.968624	+	0.248530i	$0.0799474\pi$
$234$		0			0
$235$	3234.00			0.897714
$236$		0			0
$237$	−1326.00	−	2296.70i	−0.363430	−	0.629480i
$238$		0			0
$239$	−2466.00			−0.667415			−0.333708	−	0.942677i	$-0.608300\pi$
$239$	−2466.00			−0.667415			−0.333708	+	0.942677i	$0.608300\pi$
$240$		0			0
$241$	1808.50	−	3132.41i	0.483385	−	0.837247i	−0.516433	−	0.856327i	$-0.672741\pi$
$241$	1808.50	−	3132.41i	0.483385	−	0.837247i	0.999818	+	0.0190805i	$0.00607389\pi$
$242$		0			0
$243$	121.500	−	210.444i	0.0320750	−	0.0555556i
$244$		0			0
$245$	850.500	+	1473.11i	0.221781	+	0.384137i
$246$		0			0
$247$	−390.000	+	1351.00i	−0.100466	+	0.348024i
$248$		0			0
$249$	−777.000	−	1345.80i	−0.197753	−	0.342517i
$250$		0			0
$251$	2430.00	−	4208.88i	0.611077	−	1.05842i	−0.379983	−	0.924994i	$-0.624070\pi$
$251$	2430.00	−	4208.88i	0.611077	−	1.05842i	0.991059	−	0.133422i	$-0.0425966\pi$
$252$		0			0
$253$	−1782.00	+	3086.51i	−0.442820	+	0.766986i
$254$		0			0
$255$	−777.000			−0.190814
$256$		0			0
$257$	−282.500	−	489.304i	−0.0685676	−	0.118763i	0.829703	−	0.558204i	$-0.188510\pi$
$257$	−282.500	−	489.304i	−0.0685676	−	0.118763i	−0.898271	+	0.439442i	$0.855176\pi$
$258$		0			0
$259$	130.000			0.0311884
$260$		0			0
$261$	−1017.00			−0.241190
$262$		0			0
$263$	−249.000	−	431.281i	−0.0583802	−	0.101118i	0.835358	−	0.549706i	$-0.185260\pi$
$263$	−249.000	−	431.281i	−0.0583802	−	0.101118i	−0.893738	+	0.448588i	$0.851927\pi$
$264$		0			0
$265$	−3759.00			−0.871372
$266$		0			0
$267$	291.000	−	504.027i	0.0667000	−	0.115528i
$268$		0			0
$269$	−2773.00	+	4802.98i	−0.628523	+	1.08863i	0.359325	+	0.933213i	$0.383007\pi$
$269$	−2773.00	+	4802.98i	−0.628523	+	1.08863i	−0.987848	+	0.155422i	$0.950326\pi$
$270$		0			0
$271$	−1128.00	−	1953.75i	−0.252845	−	0.437941i	0.711463	−	0.702724i	$-0.248033\pi$
$271$	−1128.00	−	1953.75i	−0.252845	−	0.437941i	−0.964308	+	0.264783i	$0.914700\pi$
$272$		0			0
$273$	1365.00	−	337.750i	0.302614	−	0.0748775i
$274$		0			0
$275$	836.000	+	1447.99i	0.183319	+	0.317518i
$276$		0			0
$277$	−1154.50	+	1999.65i	−0.250423	+	0.433745i	−0.963642	−	0.267196i	$-0.913903\pi$
$277$	−1154.50	+	1999.65i	−0.250423	+	0.433745i	0.713219	+	0.700941i	$0.247236\pi$
$278$		0			0
$279$	882.000	−	1527.67i	0.189262	−	0.327811i
$280$		0			0
$281$	5833.00			1.23832			0.619159	−	0.785265i	$-0.287473\pi$
$281$	5833.00			1.23832			0.619159	+	0.785265i	$0.287473\pi$
$282$		0			0
$283$	825.000	+	1428.94i	0.173290	+	0.300148i	0.939568	−	0.342362i	$-0.111227\pi$
$283$	825.000	+	1428.94i	0.173290	+	0.300148i	−0.766278	+	0.642509i	$0.777893\pi$
$284$		0			0
$285$	630.000			0.130940
$286$		0			0
$287$	2850.00			0.586168
$288$		0			0
$289$	1772.00	+	3069.19i	0.360676	+	0.624709i
$290$		0			0
$291$	3606.00			0.726417
$292$		0			0
$293$	−1495.50	+	2590.28i	−0.298184	+	0.516471i	−0.975721	−	0.219019i	$-0.929714\pi$
$293$	−1495.50	+	2590.28i	−0.298184	+	0.516471i	0.677536	+	0.735489i	$0.263048\pi$
$294$		0			0
$295$	2016.00	−	3491.81i	0.397885	−	0.689157i
$296$		0			0
$297$	297.000	+	514.419i	0.0580259	+	0.100504i
$298$		0			0
$299$	5265.00	+	5471.55i	1.01834	+	1.05829i
$300$		0			0
$301$	−1230.00	−	2130.42i	−0.235535	−	0.407959i
$302$		0			0
$303$	−643.500	+	1114.57i	−0.122007	+	0.211322i
$304$		0			0
$305$	2222.50	−	3849.48i	0.417246	−	0.722691i
$306$		0			0
$307$	2422.00			0.450263			0.225132	−	0.974328i	$-0.427719\pi$
$307$	2422.00			0.450263			0.225132	+	0.974328i	$0.427719\pi$
$308$		0			0
$309$	1953.00	+	3382.70i	0.359554	+	0.622766i
$310$		0			0
$311$	3402.00			0.620288			0.310144	−	0.950690i	$-0.399623\pi$
$311$	3402.00			0.620288			0.310144	+	0.950690i	$0.399623\pi$
$312$		0			0
$313$	2310.00			0.417153			0.208577	−	0.978006i	$-0.433117\pi$
$313$	2310.00			0.417153			0.208577	+	0.978006i	$0.433117\pi$
$314$		0			0
$315$	−315.000	−	545.596i	−0.0563436	−	0.0975900i
$316$		0			0
$317$	−257.000			−0.0455349			−0.0227674	−	0.999741i	$-0.507248\pi$
$317$	−257.000			−0.0455349			−0.0227674	+	0.999741i	$0.507248\pi$
$318$		0			0
$319$	1243.00	−	2152.94i	0.218165	−	0.377873i
$320$		0			0
$321$	2007.00	−	3476.23i	0.348971	−	0.604436i
$322$		0			0
$323$	555.000	+	961.288i	0.0956069	+	0.165596i
$324$		0			0
$325$	3458.00	−	855.633i	0.590201	−	0.146037i
$326$		0			0
$327$	−1551.00	−	2686.41i	−0.262295	−	0.454308i
$328$		0			0
$329$	−2310.00	+	4001.04i	−0.387096	+	0.670469i
$330$		0			0
$331$	514.000	−	890.274i	0.0853535	−	0.147837i	−0.820188	−	0.572094i	$-0.806131\pi$
$331$	514.000	−	890.274i	0.0853535	−	0.147837i	0.905542	+	0.424257i	$0.139465\pi$
$332$		0			0
$333$	117.000			0.0192539
$334$		0			0
$335$	707.000	+	1224.56i	0.115306	+	0.199716i
$336$		0			0
$337$	2487.00			0.402005			0.201002	−	0.979591i	$-0.435580\pi$
$337$	2487.00			0.402005			0.201002	+	0.979591i	$0.435580\pi$
$338$		0			0
$339$	−3231.00			−0.517651
$340$		0			0
$341$	2156.00	+	3734.30i	0.342387	+	0.593032i
$342$		0			0
$343$	−5860.00			−0.922479
$344$		0			0
$345$	1701.00	−	2946.22i	0.265446	−	0.459765i
$346$		0			0
$347$	−1425.00	+	2468.17i	−0.220455	+	0.381840i	−0.954946	−	0.296779i	$-0.904088\pi$
$347$	−1425.00	+	2468.17i	−0.220455	+	0.381840i	0.734491	+	0.678618i	$0.237421\pi$
$348$		0			0
$349$	1009.00	+	1747.64i	0.154758	+	0.268049i	0.932971	−	0.359952i	$-0.117207\pi$
$349$	1009.00	+	1747.64i	0.154758	+	0.268049i	−0.778213	+	0.628001i	$0.783874\pi$
$350$		0			0
$351$	1228.50	−	303.975i	0.186816	−	0.0462250i
$352$		0			0
$353$	2643.50	+	4578.68i	0.398582	+	0.690364i	0.993551	−	0.113385i	$-0.0361692\pi$
$353$	2643.50	+	4578.68i	0.398582	+	0.690364i	−0.594970	+	0.803748i	$0.702836\pi$
$354$		0			0
$355$	−3801.00	+	6583.53i	−0.568271	+	0.984274i
$356$		0			0
$357$	555.000	−	961.288i	0.0822793	−	0.142512i
$358$		0			0
$359$	7278.00			1.06997			0.534983	−	0.844863i	$-0.320318\pi$
$359$	7278.00			1.06997			0.534983	+	0.844863i	$0.320318\pi$
$360$		0			0
$361$	2979.50	+	5160.65i	0.434393	+	0.752390i
$362$		0			0
$363$	2541.00			0.367405
$364$		0			0
$365$	−5635.00			−0.808080
$366$		0			0
$367$	−2101.00	−	3639.04i	−0.298832	−	0.517592i	0.677037	−	0.735949i	$-0.263264\pi$
$367$	−2101.00	−	3639.04i	−0.298832	−	0.517592i	−0.975869	+	0.218357i	$0.929930\pi$
$368$		0			0
$369$	2565.00			0.361866
$370$		0			0
$371$	2685.00	−	4650.56i	0.375737	−	0.650795i
$372$		0			0
$373$	791.500	−	1370.92i	0.109872	−	0.190304i	−0.805846	−	0.592125i	$-0.798289\pi$
$373$	791.500	−	1370.92i	0.109872	−	0.190304i	0.915718	+	0.401821i	$0.131623\pi$
$374$		0			0
$375$	−2110.50	−	3655.49i	−0.290629	−	0.503384i
$376$		0			0
$377$	−3672.50	−	3816.57i	−0.501707	−	0.521389i
$378$		0			0
$379$	−1026.00	−	1777.08i	−0.139056	−	0.240851i	0.788084	−	0.615568i	$-0.211073\pi$
$379$	−1026.00	−	1777.08i	−0.139056	−	0.240851i	−0.927139	+	0.374717i	$0.877740\pi$
$380$		0			0
$381$	1482.00	−	2566.90i	0.199279	−	0.345161i
$382$		0			0
$383$	−3436.00	+	5951.33i	−0.458411	+	0.793991i	−0.998877	−	0.0473746i	$-0.984915\pi$
$383$	−3436.00	+	5951.33i	−0.458411	+	0.793991i	0.540466	+	0.841366i	$0.318248\pi$
$384$		0			0
$385$	1540.00			0.203859
$386$		0			0
$387$	−1107.00	−	1917.38i	−0.145406	−	0.251850i
$388$		0			0
$389$	−11653.0			−1.51884			−0.759422	−	0.650598i	$-0.774518\pi$
$389$	−11653.0			−1.51884			−0.759422	+	0.650598i	$0.774518\pi$
$390$		0			0
$391$	5994.00			0.775268
$392$		0			0
$393$	−840.000	−	1454.92i	−0.107818	−	0.186746i
$394$		0			0
$395$	−6188.00			−0.788233
$396$		0			0
$397$	−3067.00	+	5312.20i	−0.387729	+	0.671566i	−0.992144	−	0.125104i	$-0.960074\pi$
$397$	−3067.00	+	5312.20i	−0.387729	+	0.671566i	0.604415	+	0.796670i	$0.293407\pi$
$398$		0			0
$399$	−450.000	+	779.423i	−0.0564616	+	0.0977944i
$400$		0			0
$401$	5397.50	+	9348.74i	0.672165	+	1.16422i	0.977289	+	0.211912i	$0.0679689\pi$
$401$	5397.50	+	9348.74i	0.672165	+	1.16422i	−0.305124	+	0.952313i	$0.598698\pi$
$402$		0			0
$403$	8918.00	−	2206.63i	1.10233	−	0.272755i
$404$		0			0
$405$	−283.500	−	491.036i	−0.0347833	−	0.0602464i
$406$		0			0
$407$	−143.000	+	247.683i	−0.0174158	+	0.0301651i
$408$		0			0
$409$	4244.50	−	7351.69i	0.513147	−	0.888796i	−0.486737	−	0.873549i	$-0.661813\pi$
$409$	4244.50	−	7351.69i	0.513147	−	0.888796i	0.999884	−	0.0152477i	$-0.00485367\pi$
$410$		0			0
$411$	1557.00			0.186864
$412$		0			0
$413$	2880.00	+	4988.31i	0.343137	+	0.594331i
$414$		0			0
$415$	−3626.00			−0.428900
$416$		0			0
$417$	−1044.00			−0.122602
$418$		0			0
$419$	748.000	+	1295.57i	0.0872129	+	0.151057i	0.906332	−	0.422566i	$-0.138871\pi$
$419$	748.000	+	1295.57i	0.0872129	+	0.151057i	−0.819119	+	0.573623i	$0.805537\pi$
$420$		0			0
$421$	−11695.0			−1.35387			−0.676935	−	0.736043i	$-0.736692\pi$
$421$	−11695.0			−1.35387			−0.676935	+	0.736043i	$0.736692\pi$
$422$		0			0
$423$	−2079.00	+	3600.93i	−0.238970	+	0.413909i
$424$		0			0
$425$	1406.00	−	2435.26i	0.160473	−	0.277947i
$426$		0			0
$427$	3175.00	+	5499.26i	0.359834	+	0.623250i
$428$		0			0
$429$	−858.000	+	2972.20i	−0.0965609	+	0.334497i
$430$		0			0
$431$	5295.00	+	9171.21i	0.591766	+	1.02497i	0.993995	+	0.109430i	$0.0349024\pi$
$431$	5295.00	+	9171.21i	0.591766	+	1.02497i	−0.402228	+	0.915539i	$0.631764\pi$
$432$		0			0
$433$	6974.50	−	12080.2i	0.774072	−	1.34073i	−0.161243	−	0.986915i	$-0.551550\pi$
$433$	6974.50	−	12080.2i	0.774072	−	1.34073i	0.935315	−	0.353817i	$-0.115116\pi$
$434$		0			0
$435$	−1186.50	+	2055.08i	−0.130778	+	0.226514i
$436$		0			0
$437$	−4860.00			−0.532003
$438$		0			0
$439$	−5363.00	−	9288.99i	−0.583057	−	1.00988i	−0.995115	−	0.0987266i	$-0.968523\pi$
$439$	−5363.00	−	9288.99i	−0.583057	−	1.00988i	0.412058	−	0.911158i	$-0.364810\pi$
$440$		0			0
$441$	−2187.00			−0.236152
$442$		0			0
$443$	−16228.0			−1.74044			−0.870221	−	0.492662i	$-0.836024\pi$
$443$	−16228.0			−1.74044			−0.870221	+	0.492662i	$0.836024\pi$
$444$		0			0
$445$	−679.000	−	1176.06i	−0.0723319	−	0.125282i
$446$		0			0
$447$	1935.00			0.204748
$448$		0			0
$449$	−3769.00	+	6528.10i	−0.396147	+	0.686147i	−0.993247	−	0.116020i	$-0.962986\pi$
$449$	−3769.00	+	6528.10i	−0.396147	+	0.686147i	0.597100	+	0.802167i	$0.296320\pi$
$450$		0			0
$451$	−3135.00	+	5429.98i	−0.327320	+	0.566935i
$452$		0			0
$453$	−4371.00	−	7570.79i	−0.453350	−	0.785225i
$454$		0			0
$455$	910.000	−	3152.33i	0.0937614	−	0.324799i
$456$		0			0
$457$	−7769.50	−	13457.2i	−0.795278	−	1.37746i	−0.922663	−	0.385608i	$-0.873992\pi$
$457$	−7769.50	−	13457.2i	−0.795278	−	1.37746i	0.127385	−	0.991853i	$-0.459342\pi$
$458$		0			0
$459$	499.500	−	865.159i	0.0507945	−	0.0879786i
$460$		0			0
$461$	−2405.50	+	4166.45i	−0.243027	+	0.420935i	−0.961575	−	0.274543i	$-0.911474\pi$
$461$	−2405.50	+	4166.45i	−0.243027	+	0.420935i	0.718548	+	0.695477i	$0.244807\pi$
$462$		0			0
$463$	−562.000			−0.0564111			−0.0282056	−	0.999602i	$-0.508979\pi$
$463$	−562.000			−0.0564111			−0.0282056	+	0.999602i	$0.508979\pi$
$464$		0			0
$465$	−2058.00	−	3564.56i	−0.205242	−	0.355489i
$466$		0			0
$467$	−4914.00			−0.486922			−0.243461	−	0.969911i	$-0.578283\pi$
$467$	−4914.00			−0.486922			−0.243461	+	0.969911i	$0.578283\pi$
$468$		0			0
$469$	−2020.00			−0.198880
$470$		0			0
$471$	−3118.50	−	5401.40i	−0.305080	−	0.528415i
$472$		0			0
$473$	5412.00			0.526097
$474$		0			0
$475$	−1140.00	+	1974.54i	−0.110120	+	0.190733i
$476$		0			0
$477$	2416.50	−	4185.50i	0.231958	−	0.401763i
$478$		0			0
$479$	−1800.00	−	3117.69i	−0.171700	−	0.297392i	0.767315	−	0.641271i	$-0.221593\pi$
$479$	−1800.00	−	3117.69i	−0.171700	−	0.297392i	−0.939014	+	0.343878i	$0.888259\pi$
$480$		0			0
$481$	422.500	+	439.075i	0.0400506	+	0.0416218i
$482$		0			0
$483$	2430.00	+	4208.88i	0.228921	+	0.396503i
$484$		0			0
$485$	4207.00	−	7286.74i	0.393876	−	0.682214i
$486$		0			0
$487$	−8565.00	+	14835.0i	−0.796955	+	1.38037i	0.124635	+	0.992203i	$0.460224\pi$
$487$	−8565.00	+	14835.0i	−0.796955	+	1.38037i	−0.921590	+	0.388164i	$0.873109\pi$
$488$		0			0
$489$	5100.00			0.471636
$490$		0			0
$491$	−5919.00	−	10252.0i	−0.544034	−	0.942295i	−0.998667	−	0.0516158i	$-0.983563\pi$
$491$	−5919.00	−	10252.0i	−0.544034	−	0.942295i	0.454633	−	0.890679i	$-0.349770\pi$
$492$		0			0
$493$	−4181.00			−0.381953
$494$		0			0
$495$	1386.00			0.125851
$496$		0			0
$497$	−5430.00	−	9405.04i	−0.490078	−	0.848840i
$498$		0			0
$499$	−8976.00			−0.805252			−0.402626	−	0.915364i	$-0.631903\pi$
$499$	−8976.00			−0.805252			−0.402626	+	0.915364i	$0.631903\pi$
$500$		0			0
$501$	−5520.00	+	9560.92i	−0.492246	+	0.852596i
$502$		0			0
$503$	841.000	−	1456.65i	0.0745494	−	0.129123i	−0.826341	−	0.563170i	$-0.809581\pi$
$503$	841.000	−	1456.65i	0.0745494	−	0.129123i	0.900890	+	0.434047i	$0.142915\pi$
$504$		0			0
$505$	1501.50	+	2600.67i	0.132309	+	0.229165i
$506$		0			0
$507$	5577.00	+	3512.60i	0.488527	+	0.307692i
$508$		0			0
$509$	−7583.50	−	13135.0i	−0.660379	−	1.14381i	−0.980516	−	0.196438i	$-0.937062\pi$
$509$	−7583.50	−	13135.0i	−0.660379	−	1.14381i	0.320138	−	0.947371i	$-0.396271\pi$
$510$		0			0
$511$	4025.00	−	6971.50i	0.348445	−	0.603525i
$512$		0			0
$513$	−405.000	+	701.481i	−0.0348561	+	0.0603726i
$514$		0			0
$515$	9114.00			0.779827
$516$		0			0
$517$	−5082.00	−	8802.28i	−0.432314	−	0.748789i
$518$		0			0
$519$	−12438.0			−1.05196
$520$		0			0
$521$	−6783.00			−0.570381			−0.285191	−	0.958471i	$-0.592057\pi$
$521$	−6783.00			−0.570381			−0.285191	+	0.958471i	$0.592057\pi$
$522$		0			0
$523$	−6959.00	−	12053.3i	−0.581828	−	1.00775i	−0.995263	−	0.0972214i	$-0.969005\pi$
$523$	−6959.00	−	12053.3i	−0.581828	−	1.00775i	0.413435	−	0.910534i	$-0.364329\pi$
$524$		0			0
$525$	2280.00			0.189538
$526$		0			0
$527$	3626.00	−	6280.42i	0.299717	−	0.519126i
$528$		0			0
$529$	−7038.50	+	12191.0i	−0.578491	+	1.00198i
$530$		0			0
$531$	2592.00	+	4489.48i	0.211833	+	0.366905i
$532$		0			0
$533$	9262.50	+	9625.87i	0.752727	+	0.782257i
$534$		0			0
$535$	−4683.00	−	8111.19i	−0.378437	−	0.655472i
$536$		0			0
$537$	−5511.00	+	9545.33i	−0.442863	+	0.767060i
$538$		0			0
$539$	2673.00	−	4629.77i	0.213607	−	0.369978i
$540$		0			0
$541$	−1335.00			−0.106093			−0.0530463	−	0.998592i	$-0.516893\pi$
$541$	−1335.00			−0.106093			−0.0530463	+	0.998592i	$0.516893\pi$
$542$		0			0
$543$	−4924.50	−	8529.48i	−0.389191	−	0.674098i
$544$		0			0
$545$	−7238.00			−0.568884
$546$		0			0
$547$	3806.00			0.297501			0.148750	−	0.988875i	$-0.452475\pi$
$547$	3806.00			0.297501			0.148750	+	0.988875i	$0.452475\pi$
$548$		0			0
$549$	2857.50	+	4949.34i	0.222140	+	0.384759i
$550$		0			0
$551$	3390.00			0.262103
$552$		0			0
$553$	4420.00	−	7655.66i	0.339887	−	0.588702i
$554$		0			0
$555$	136.500	−	236.425i	0.0104398	−	0.0180823i
$556$		0			0
$557$	952.500	+	1649.78i	0.0724573	+	0.125500i	0.899978	−	0.435936i	$-0.143583\pi$
$557$	952.500	+	1649.78i	0.0724573	+	0.125500i	−0.827520	+	0.561436i	$0.810249\pi$
$558$		0			0
$559$	3198.00	−	11078.2i	0.241970	−	0.838207i
$560$		0			0
$561$	1221.00	+	2114.83i	0.0918907	+	0.159159i
$562$		0			0
$563$	−2400.00	+	4156.92i	−0.179659	+	0.311178i	−0.941764	−	0.336275i	$-0.890833\pi$
$563$	−2400.00	+	4156.92i	−0.179659	+	0.311178i	0.762105	+	0.647454i	$0.224166\pi$
$564$		0			0
$565$	−3769.50	+	6528.97i	−0.280680	+	0.486152i
$566$		0			0
$567$	810.000			0.0599944
$568$		0			0
$569$	−7339.00	−	12711.5i	−0.540715	−	0.936546i	−0.998863	−	0.0476701i	$-0.984820\pi$
$569$	−7339.00	−	12711.5i	−0.540715	−	0.936546i	0.458148	−	0.888876i	$-0.348513\pi$
$570$		0			0
$571$	586.000			0.0429481			0.0214740	−	0.999769i	$-0.493164\pi$
$571$	586.000			0.0429481			0.0214740	+	0.999769i	$0.493164\pi$
$572$		0			0
$573$	−1788.00			−0.130357
$574$		0			0
$575$	6156.00	+	10662.5i	0.446475	+	0.773317i
$576$		0			0
$577$	8939.00			0.644949			0.322474	−	0.946578i	$-0.395485\pi$
$577$	8939.00			0.644949			0.322474	+	0.946578i	$0.395485\pi$
$578$		0			0
$579$	−589.500	+	1021.04i	−0.0423122	+	0.0732869i
$580$		0			0
$581$	2590.00	−	4486.01i	0.184942	−	0.320329i
$582$		0			0
$583$	5907.00	+	10231.2i	0.419628	+	0.726816i
$584$		0			0
$585$	819.000	−	2837.10i	0.0578829	−	0.200512i
$586$		0			0
$587$	−6896.00	−	11944.2i	−0.484887	−	0.839848i	0.514963	−	0.857213i	$-0.327806\pi$
$587$	−6896.00	−	11944.2i	−0.484887	−	0.839848i	−0.999849	+	0.0173645i	$0.994472\pi$
$588$		0			0
$589$	−2940.00	+	5092.23i	−0.205672	+	0.356234i
$590$		0			0
$591$	−5283.00	+	9150.42i	−0.367705	+	0.636884i
$592$		0			0
$593$	9569.00			0.662650			0.331325	−	0.943517i	$-0.392504\pi$
$593$	9569.00			0.662650			0.331325	+	0.943517i	$0.392504\pi$
$594$		0			0
$595$	−1295.00	−	2243.01i	−0.0892266	−	0.154545i
$596$		0			0
$597$	6054.00			0.415031
$598$		0			0
$599$	5192.00			0.354156			0.177078	−	0.984197i	$-0.443336\pi$
$599$	5192.00			0.354156			0.177078	+	0.984197i	$0.443336\pi$
$600$		0			0
$601$	1838.50	+	3184.38i	0.124782	+	0.216129i	0.921648	−	0.388028i	$-0.126844\pi$
$601$	1838.50	+	3184.38i	0.124782	+	0.216129i	−0.796866	+	0.604156i	$0.793510\pi$
$602$		0			0
$603$	−1818.00			−0.122777
$604$		0			0
$605$	2964.50	−	5134.66i	0.199213	−	0.345048i
$606$		0			0
$607$	−5480.00	+	9491.64i	−0.366435	+	0.634685i	−0.989005	−	0.147879i	$-0.952755\pi$
$607$	−5480.00	+	9491.64i	−0.366435	+	0.634685i	0.622570	+	0.782564i	$0.286089\pi$
$608$		0			0
$609$	−1695.00	−	2935.83i	−0.112783	−	0.195346i
$610$		0			0
$611$	−21021.0	+	5201.35i	−1.39185	+	0.344393i
$612$		0			0
$613$	13013.5	+	22540.0i	0.857439	+	1.48513i	0.874363	+	0.485272i	$0.161279\pi$
$613$	13013.5	+	22540.0i	0.857439	+	1.48513i	−0.0169241	+	0.999857i	$0.505387\pi$
$614$		0			0
$615$	2992.50	−	5183.16i	0.196210	−	0.339846i
$616$		0			0
$617$	−8840.50	+	15312.2i	−0.576832	+	0.999102i	0.419008	+	0.907982i	$0.362378\pi$
$617$	−8840.50	+	15312.2i	−0.576832	+	0.999102i	−0.995840	+	0.0911193i	$0.970956\pi$
$618$		0			0
$619$	−3192.00			−0.207265			−0.103633	−	0.994616i	$-0.533047\pi$
$619$	−3192.00			−0.207265			−0.103633	+	0.994616i	$0.533047\pi$
$620$		0			0
$621$	2187.00	+	3788.00i	0.141323	+	0.244778i
$622$		0			0
$623$	1940.00			0.124758
$624$		0			0
$625$	−349.000			−0.0223360
$626$		0			0
$627$	−990.000	−	1714.73i	−0.0630571	−	0.109218i
$628$		0			0
$629$	481.000			0.0304908
$630$		0			0
$631$	3790.00	−	6564.47i	0.239109	−	0.414148i	−0.721350	−	0.692571i	$-0.756478\pi$
$631$	3790.00	−	6564.47i	0.239109	−	0.414148i	0.960459	+	0.278422i	$0.0898115\pi$
$632$		0			0
$633$	−240.000	+	415.692i	−0.0150697	+	0.0261016i
$634$		0			0
$635$	−3458.00	−	5989.43i	−0.216105	−	0.374304i
$636$		0			0
$637$	−7897.50	−	8207.32i	−0.491225	−	0.510496i
$638$		0			0
$639$	−4887.00	−	8464.53i	−0.302546	−	0.524025i
$640$		0			0
$641$	13853.5	−	23995.0i	0.853635	−	1.47854i	−0.0242696	−	0.999705i	$-0.507726\pi$
$641$	13853.5	−	23995.0i	0.853635	−	1.47854i	0.877905	−	0.478835i	$-0.158941\pi$
$642$		0			0
$643$	5608.00	−	9713.34i	0.343947	−	0.595734i	−0.641215	−	0.767361i	$-0.721569\pi$
$643$	5608.00	−	9713.34i	0.343947	−	0.595734i	0.985162	+	0.171628i	$0.0549026\pi$
$644$		0			0
$645$	−5166.00			−0.315366
$646$		0			0
$647$	−1268.00	−	2196.24i	−0.0770483	−	0.133452i	0.824927	−	0.565239i	$-0.191216\pi$
$647$	−1268.00	−	2196.24i	−0.0770483	−	0.133452i	−0.901975	+	0.431788i	$0.857883\pi$
$648$		0			0
$649$	−12672.0			−0.766440
$650$		0			0
$651$	5880.00			0.354002
$652$		0			0
$653$	−8865.00	−	15354.6i	−0.531262	−	0.920173i	−0.999334	−	0.0364829i	$-0.988385\pi$
$653$	−8865.00	−	15354.6i	−0.531262	−	0.920173i	0.468072	−	0.883690i	$-0.344949\pi$
$654$		0			0
$655$	−3920.00			−0.233843
$656$		0			0
$657$	3622.50	−	6274.35i	0.215110	−	0.372581i
$658$		0			0
$659$	9460.00	−	16385.2i	0.559195	−	0.968554i	−0.438369	−	0.898795i	$-0.644444\pi$
$659$	9460.00	−	16385.2i	0.559195	−	0.968554i	0.997564	−	0.0697586i	$-0.0222229\pi$
$660$		0			0
$661$	−2620.50	−	4538.84i	−0.154199	−	0.267081i	0.778568	−	0.627560i	$-0.215946\pi$
$661$	−2620.50	−	4538.84i	−0.154199	−	0.267081i	−0.932767	+	0.360480i	$0.882613\pi$
$662$		0			0
$663$	5050.50	−	1249.67i	0.295845	−	0.0732026i
$664$		0			0
$665$	1050.00	+	1818.65i	0.0612290	+	0.106052i
$666$		0			0
$667$	9153.00	−	15853.5i	0.531343	−	0.920313i
$668$		0			0
$669$	−6108.00	+	10579.4i	−0.352988	+	0.611393i
$670$		0			0
$671$	−13970.0			−0.803735
$672$		0			0
$673$	−10233.5	−	17724.9i	−0.586140	−	1.01522i	−0.994732	−	0.102508i	$-0.967313\pi$
$673$	−10233.5	−	17724.9i	−0.586140	−	1.01522i	0.408592	−	0.912717i	$-0.366020\pi$
$674$		0			0
$675$	2052.00			0.117010
$676$		0			0
$677$	−70.0000			−0.00397388			−0.00198694	−	0.999998i	$-0.500632\pi$
$677$	−70.0000			−0.00397388			−0.00198694	+	0.999998i	$0.500632\pi$
$678$		0			0
$679$	6010.00	+	10409.6i	0.339680	+	0.588343i
$680$		0			0
$681$	−17382.0			−0.978091
$682$		0			0
$683$	3216.00	−	5570.28i	0.180171	−	0.312065i	−0.761768	−	0.647850i	$-0.775668\pi$
$683$	3216.00	−	5570.28i	0.180171	−	0.312065i	0.941939	+	0.335785i	$0.109002\pi$
$684$		0			0
$685$	1816.50	−	3146.27i	0.101321	−	0.175493i
$686$		0			0
$687$	9723.00	+	16840.7i	0.539964	+	0.935246i
$688$		0			0
$689$	24433.5	−	6045.72i	1.35100	−	0.334287i
$690$		0			0
$691$	−3333.00	−	5772.93i	−0.183492	−	0.317818i	0.759575	−	0.650420i	$-0.225407\pi$
$691$	−3333.00	−	5772.93i	−0.183492	−	0.317818i	−0.943067	+	0.332601i	$0.892074\pi$
$692$		0			0
$693$	−990.000	+	1714.73i	−0.0542669	+	0.0939931i
$694$		0			0
$695$	−1218.00	+	2109.64i	−0.0664768	+	0.115141i
$696$		0			0
$697$	10545.0			0.573056
$698$		0			0
$699$	10335.0	+	17900.7i	0.559235	+	0.968624i
$700$		0			0
$701$	−14054.0			−0.757221			−0.378611	−	0.925556i	$-0.623598\pi$
$701$	−14054.0			−0.757221			−0.378611	+	0.925556i	$0.623598\pi$
$702$		0			0
$703$	−390.000			−0.0209234
$704$		0			0
$705$	4851.00	+	8402.18i	0.259148	+	0.448857i
$706$		0			0
$707$	−4290.00			−0.228207
$708$		0			0
$709$	35.5000	−	61.4878i	0.00188044	−	0.00325701i	−0.865084	−	0.501628i	$-0.832735\pi$
$709$	35.5000	−	61.4878i	0.00188044	−	0.00325701i	0.866964	+	0.498371i	$0.166068\pi$
$710$		0			0
$711$	3978.00	−	6890.10i	0.209827	−	0.363430i
$712$		0			0
$713$	15876.0	+	27498.0i	0.833886	+	1.44433i
$714$		0			0
$715$	5005.00	+	5201.35i	0.261785	+	0.272055i
$716$		0			0
$717$	−3699.00	−	6406.86i	−0.192666	−	0.333708i
$718$		0			0
$719$	1968.00	−	3408.68i	0.102078	−	0.176804i	−0.810463	−	0.585790i	$-0.800784\pi$
$719$	1968.00	−	3408.68i	0.102078	−	0.176804i	0.912541	+	0.408986i	$0.134118\pi$
$720$		0			0
$721$	−6510.00	+	11275.7i	−0.336262	+	0.582423i
$722$		0			0
$723$	10851.0			0.558165
$724$		0			0
$725$	−4294.00	−	7437.43i	−0.219966	−	0.380992i
$726$		0			0
$727$	−34202.0			−1.74482			−0.872409	−	0.488777i	$-0.837443\pi$
$727$	−34202.0			−1.74482			−0.872409	+	0.488777i	$0.837443\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	−4551.00	−	7882.56i	−0.230267	−	0.398833i
$732$		0			0
$733$	−27363.0			−1.37882			−0.689410	−	0.724371i	$-0.742130\pi$
$733$	−27363.0			−1.37882			−0.689410	+	0.724371i	$0.742130\pi$
$734$		0			0
$735$	−2551.50	+	4419.33i	−0.128046	+	0.221781i
$736$		0			0
$737$	2222.00	−	3848.62i	0.111056	−	0.192355i
$738$		0			0
$739$	10888.0	+	18858.6i	0.541978	+	0.938733i	0.998790	+	0.0491701i	$0.0156576\pi$
$739$	10888.0	+	18858.6i	0.541978	+	0.938733i	−0.456813	+	0.889563i	$0.651009\pi$
$740$		0			0
$741$	−4095.00	+	1013.25i	−0.203014	+	0.0502330i
$742$		0			0
$743$	1242.00	+	2151.21i	0.0613251	+	0.106218i	0.895058	−	0.445950i	$-0.147134\pi$
$743$	1242.00	+	2151.21i	0.0613251	+	0.106218i	−0.833733	+	0.552168i	$0.813801\pi$
$744$		0			0
$745$	2257.50	−	3910.10i	0.111018	−	0.192289i
$746$		0			0
$747$	2331.00	−	4037.41i	0.114172	−	0.197753i
$748$		0			0
$749$	13380.0			0.652730
$750$		0			0
$751$	16453.0	+	28497.4i	0.799439	+	1.38467i	0.919982	+	0.391960i	$0.128203\pi$
$751$	16453.0	+	28497.4i	0.799439	+	1.38467i	−0.120543	+	0.992708i	$0.538464\pi$
$752$		0			0
$753$	14580.0			0.705611
$754$		0			0
$755$	−20398.0			−0.983257
$756$		0			0
$757$	1957.00	+	3389.62i	0.0939609	+	0.162745i	0.909174	−	0.416415i	$-0.136714\pi$
$757$	1957.00	+	3389.62i	0.0939609	+	0.162745i	−0.815214	+	0.579160i	$0.803380\pi$
$758$		0			0
$759$	−10692.0			−0.511324
$760$		0			0
$761$	16519.0	−	28611.7i	0.786877	−	1.36291i	−0.140995	−	0.990010i	$-0.545030\pi$
$761$	16519.0	−	28611.7i	0.786877	−	1.36291i	0.927871	−	0.372900i	$-0.121637\pi$
$762$		0			0
$763$	5170.00	−	8954.70i	0.245303	−	0.424878i
$764$		0			0
$765$	−1165.50	−	2018.71i	−0.0550833	−	0.0954071i
$766$		0			0
$767$	−7488.00	+	25939.2i	−0.352511	+	1.22113i
$768$		0			0
$769$	8793.00	+	15229.9i	0.412332	+	0.714181i	0.995144	−	0.0984263i	$-0.0313809\pi$
$769$	8793.00	+	15229.9i	0.412332	+	0.714181i	−0.582812	+	0.812607i	$0.698048\pi$
$770$		0			0
$771$	847.500	−	1467.91i	0.0395875	−	0.0685676i
$772$		0			0
$773$	−9157.00	+	15860.4i	−0.426073	+	0.737980i	−0.996520	−	0.0833544i	$-0.973437\pi$
$773$	−9157.00	+	15860.4i	−0.426073	+	0.737980i	0.570447	+	0.821334i	$0.306770\pi$
$774$		0			0
$775$	14896.0			0.690426
$776$		0			0
$777$	195.000	+	337.750i	0.00900333	+	0.0155942i
$778$		0			0
$779$	−8550.00			−0.393242
$780$		0			0
$781$	23892.0			1.09465
$782$		0			0
$783$	−1525.50	−	2642.24i	−0.0696257	−	0.120595i
$784$		0			0
$785$	−14553.0			−0.661680
$786$		0			0
$787$	−21034.0	+	36432.0i	−0.952708	+	1.65014i	−0.213180	+	0.977013

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.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.50000 + 2.59808i) q^{3} +7.00000 q^{5} +(-5.00000 + 8.66025i) q^{7} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)	\(q+(1.50000 + 2.59808i) q^{3} +7.00000 q^{5} +(-5.00000 + 8.66025i) q^{7} +(-4.50000 + 7.79423i) q^{9} +(-11.0000 - 19.0526i) q^{11} +(-45.5000 + 11.2583i) q^{13} +(10.5000 + 18.1865i) q^{15} +(-18.5000 + 32.0429i) q^{17} +(15.0000 - 25.9808i) q^{19} -30.0000 q^{21} +(-81.0000 - 140.296i) q^{23} -76.0000 q^{25} -27.0000 q^{27} +(56.5000 + 97.8609i) q^{29} -196.000 q^{31} +(33.0000 - 57.1577i) q^{33} +(-35.0000 + 60.6218i) q^{35} +(-6.50000 - 11.2583i) q^{37} +(-97.5000 - 101.325i) q^{39} +(-142.500 - 246.817i) q^{41} +(-123.000 + 213.042i) q^{43} +(-31.5000 + 54.5596i) q^{45} +462.000 q^{47} +(121.500 + 210.444i) q^{49} -111.000 q^{51} -537.000 q^{53} +(-77.0000 - 133.368i) q^{55} +90.0000 q^{57} +(288.000 - 498.831i) q^{59} +(317.500 - 549.926i) q^{61} +(-45.0000 - 77.9423i) q^{63} +(-318.500 + 78.8083i) q^{65} +(101.000 + 174.937i) q^{67} +(243.000 - 420.888i) q^{69} +(-543.000 + 940.504i) q^{71} -805.000 q^{73} +(-114.000 - 197.454i) q^{75} +220.000 q^{77} -884.000 q^{79} +(-40.5000 - 70.1481i) q^{81} -518.000 q^{83} +(-129.500 + 224.301i) q^{85} +(-169.500 + 293.583i) q^{87} +(-97.0000 - 168.009i) q^{89} +(130.000 - 450.333i) q^{91} +(-294.000 - 509.223i) q^{93} +(105.000 - 181.865i) q^{95} +(601.000 - 1040.96i) q^{97} +198.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} + 14 q^{5} - 10 q^{7} - 9 q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 + 14 * q^5 - 10 * q^7 - 9 * q^9	\( 2 q + 3 q^{3} + 14 q^{5} - 10 q^{7} - 9 q^{9} - 22 q^{11} - 91 q^{13} + 21 q^{15} - 37 q^{17} + 30 q^{19} - 60 q^{21} - 162 q^{23} - 152 q^{25} - 54 q^{27} + 113 q^{29} - 392 q^{31} + 66 q^{33} - 70 q^{35} - 13 q^{37} - 195 q^{39} - 285 q^{41} - 246 q^{43} - 63 q^{45} + 924 q^{47} + 243 q^{49} - 222 q^{51} - 1074 q^{53} - 154 q^{55} + 180 q^{57} + 576 q^{59} + 635 q^{61} - 90 q^{63} - 637 q^{65} + 202 q^{67} + 486 q^{69} - 1086 q^{71} - 1610 q^{73} - 228 q^{75} + 440 q^{77} - 1768 q^{79} - 81 q^{81} - 1036 q^{83} - 259 q^{85} - 339 q^{87} - 194 q^{89} + 260 q^{91} - 588 q^{93} + 210 q^{95} + 1202 q^{97} + 396 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 + 14 * q^5 - 10 * q^7 - 9 * q^9 - 22 * q^11 - 91 * q^13 + 21 * q^15 - 37 * q^17 + 30 * q^19 - 60 * q^21 - 162 * q^23 - 152 * q^25 - 54 * q^27 + 113 * q^29 - 392 * q^31 + 66 * q^33 - 70 * q^35 - 13 * q^37 - 195 * q^39 - 285 * q^41 - 246 * q^43 - 63 * q^45 + 924 * q^47 + 243 * q^49 - 222 * q^51 - 1074 * q^53 - 154 * q^55 + 180 * q^57 + 576 * q^59 + 635 * q^61 - 90 * q^63 - 637 * q^65 + 202 * q^67 + 486 * q^69 - 1086 * q^71 - 1610 * q^73 - 228 * q^75 + 440 * q^77 - 1768 * q^79 - 81 * q^81 - 1036 * q^83 - 259 * q^85 - 339 * q^87 - 194 * q^89 + 260 * q^91 - 588 * q^93 + 210 * q^95 + 1202 * q^97 + 396 * q^99

Introduction

L-functions

Modular forms

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