LMFDB - Embedded newform 624.4.q.c.529.1

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			529.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	624.529
Dual form			624.4.q.c.289.1

$q$-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 $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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	0.698880	−	0.715239i	$-0.253682\pi$
$7$	−5.00000	−	8.66025i	−0.269975	−	0.467610i	−0.968855	+	0.247629i	$0.920349\pi$
$8$		0			0
$9$	−4.50000	−	7.79423i	−0.166667	−	0.288675i
$10$		0			0
$11$	−11.0000	+	19.0526i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	$-0.930824\pi$
$11$	−11.0000	+	19.0526i	−0.301511	+	0.522233i	0.674967	+	0.737848i	$0.264158\pi$
$12$		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	0.703348	−	0.710845i	$-0.251687\pi$
$17$	−18.5000	−	32.0429i	−0.263936	−	0.457150i	−0.967284	+	0.253695i	$0.918354\pi$
$18$		0			0
$19$	15.0000	+	25.9808i	0.181118	+	0.313705i	0.942261	−	0.334878i	$-0.108695\pi$
$19$	15.0000	+	25.9808i	0.181118	+	0.313705i	−0.761144	+	0.648583i	$0.775362\pi$
$20$		0			0
$21$	−30.0000			−0.311740
$22$		0			0
$23$	−81.0000	+	140.296i	−0.734333	+	1.27190i	0.220682	+	0.975346i	$0.429172\pi$
$23$	−81.0000	+	140.296i	−0.734333	+	1.27190i	−0.955015	+	0.296557i	$0.904162\pi$
$24$		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	−0.626469	−	0.779446i	$-0.715501\pi$
$29$	56.5000	−	97.8609i	0.361786	−	0.626631i	0.988255	+	0.152815i	$0.0488339\pi$
$30$		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	−0.880105	−	0.474780i	$-0.842528\pi$
$37$	−6.50000	+	11.2583i	−0.0288809	+	0.0500232i	0.851224	+	0.524803i	$0.175861\pi$
$38$		0			0
$39$	−97.5000	+	101.325i	−0.400320	+	0.416025i
$40$		0			0
$41$	−142.500	+	246.817i	−0.542799	+	0.940156i	0.455943	+	0.890009i	$0.349302\pi$
$41$	−142.500	+	246.817i	−0.542799	+	0.940156i	−0.998742	+	0.0501465i	$0.984031\pi$
$42$		0			0
$43$	−123.000	−	213.042i	−0.436217	−	0.755550i	0.561177	−	0.827696i	$-0.310349\pi$
$43$	−123.000	−	213.042i	−0.436217	−	0.755550i	−0.997394	+	0.0721459i	$0.977015\pi$
$44$		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	0.986409	+	0.164307i	$0.0525387\pi$
$59$	288.000	+	498.831i	0.635498	+	1.10072i	−0.350911	+	0.936409i	$0.614128\pi$
$60$		0			0
$61$	317.500	+	549.926i	0.666421	+	1.15428i	0.978898	+	0.204350i	$0.0655082\pi$
$61$	317.500	+	549.926i	0.666421	+	1.15428i	−0.312476	+	0.949926i	$0.601159\pi$
$62$		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	−0.759129	−	0.650940i	$-0.774375\pi$
$67$	101.000	−	174.937i	0.184166	−	0.318985i	0.943295	+	0.331955i	$0.107708\pi$
$68$		0			0
$69$	243.000	+	420.888i	0.423968	+	0.734333i
$70$		0			0
$71$	−543.000	−	940.504i	−0.907637	−	1.57207i	−0.817338	−	0.576159i	$-0.804551\pi$
$71$	−543.000	−	940.504i	−0.907637	−	1.57207i	−0.0902997	−	0.995915i	$-0.528783\pi$
$72$		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	−0.917991	−	0.396602i	$-0.870189\pi$
$89$	−97.0000	+	168.009i	−0.115528	+	0.200100i	0.802463	+	0.596702i	$0.203523\pi$
$90$		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	0.987733	+	0.156149i	$0.0499081\pi$
$97$	601.000	+	1040.96i	0.629096	+	1.08963i	−0.358638	+	0.933477i	$0.616759\pi$
$98$		0			0
$99$	198.000			0.201008
$100$		0			0
$101$	214.500	−	371.525i	0.211322	−	0.366021i	−0.740806	−	0.671719i	$-0.765556\pi$
$101$	214.500	−	371.525i	0.211322	−	0.366021i	0.952129	+	0.305698i	$0.0988897\pi$
$102$		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	0.387704	+	0.921784i	$0.373268\pi$
$107$	−669.000	+	1158.74i	−0.604436	+	1.04691i	−0.992140	+	0.125130i	$0.960065\pi$
$108$		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	0.549976	−	0.835180i	$-0.314637\pi$
$113$	−538.500	−	932.709i	−0.448299	−	0.776477i	−0.998275	+	0.0587032i	$0.981303\pi$
$114$		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	−0.985383	−	0.170354i	$-0.945509\pi$
$127$	−494.000	+	855.633i	−0.345161	+	0.597836i	0.640222	+	0.768190i	$0.278842\pi$
$128$		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	0.935525	−	0.353261i	$-0.114927\pi$
$137$	259.500	+	449.467i	0.161829	+	0.280296i	−0.773696	+	0.633557i	$0.781594\pi$
$138$		0			0
$139$	−174.000	−	301.377i	−0.106176	−	0.183903i	0.808042	−	0.589125i	$-0.200527\pi$
$139$	−174.000	−	301.377i	−0.106176	−	0.183903i	−0.914218	+	0.405222i	$0.867194\pi$
$140$		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	0.940961	−	0.338516i	$-0.109925\pi$
$149$	322.500	+	558.586i	0.177317	+	0.307122i	−0.763644	+	0.645638i	$0.776592\pi$
$150$		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	0.994716	−	0.102664i	$-0.0327365\pi$
$163$	850.000	+	1472.24i	0.408449	+	0.707454i	−0.586267	+	0.810118i	$0.699403\pi$
$164$		0			0
$165$	231.000	+	400.104i	0.108990	+	0.188776i
$166$		0			0
$167$	1840.00	−	3186.97i	0.852596	−	1.47674i	−0.0262621	−	0.999655i	$-0.508360\pi$
$167$	1840.00	−	3186.97i	0.852596	−	1.47674i	0.878858	−	0.477084i	$-0.158306\pi$
$168$		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	−0.812619	−	0.582795i	$-0.801959\pi$
$173$	−2073.00	−	3590.54i	−0.911025	−	1.57794i	−0.0984052	−	0.995146i	$-0.531374\pi$
$174$		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	−0.172090	−	0.985081i	$-0.555052\pi$
$179$	1837.00	−	3181.78i	0.767060	−	1.32859i	0.939150	−	0.343506i	$-0.111615\pi$
$180$		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	0.804043	−	0.594572i	$-0.202678\pi$
$191$	−298.000	−	516.151i	−0.112893	−	0.195536i	−0.916935	+	0.399036i	$0.869345\pi$
$192$		0			0
$193$	196.500	−	340.348i	0.0732869	−	0.126937i	−0.827053	−	0.562124i	$-0.809984\pi$
$193$	196.500	−	340.348i	0.0732869	−	0.126937i	0.900340	+	0.435187i	$0.143318\pi$
$194$		0			0
$195$	−682.500	+	709.275i	−0.250640	+	0.260473i
$196$		0			0
$197$	1761.00	−	3050.14i	0.636884	−	1.10311i	−0.349229	−	0.937037i	$-0.613557\pi$
$197$	1761.00	−	3050.14i	0.636884	−	1.10311i	0.986113	−	0.166077i	$-0.0531101\pi$
$198$		0			0
$199$	1009.00	+	1747.64i	0.359428	+	0.622547i	0.987865	−	0.155313i	$-0.0496386\pi$
$199$	1009.00	+	1747.64i	0.359428	+	0.622547i	−0.628438	+	0.777860i	$0.716305\pi$
$200$		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	−0.852680	−	0.522434i	$-0.825024\pi$
$211$	80.0000	−	138.564i	0.0261016	−	0.0452092i	0.878781	+	0.477225i	$0.158357\pi$
$212$		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	−0.379613	−	0.925145i	$-0.623943\pi$
$223$	2036.00	−	3526.46i	0.611393	−	1.05896i	0.991006	−	0.133818i	$-0.0427239\pi$
$224$		0			0
$225$	342.000	+	592.361i	0.101333	+	0.175514i
$226$		0			0
$227$	−2897.00	−	5017.75i	−0.847051	−	1.46714i	−0.883828	−	0.467812i	$-0.845042\pi$
$227$	−2897.00	−	5017.75i	−0.847051	−	1.46714i	0.0367765	−	0.999324i	$-0.488291\pi$
$228$		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	0.999818	−	0.0190805i	$-0.00607389\pi$
$241$	1808.50	+	3132.41i	0.483385	+	0.837247i	−0.516433	+	0.856327i	$0.672741\pi$
$242$		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	0.991059	+	0.133422i	$0.0425966\pi$
$251$	2430.00	+	4208.88i	0.611077	+	1.05842i	−0.379983	+	0.924994i	$0.624070\pi$
$252$		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	−0.898271	−	0.439442i	$-0.855176\pi$
$257$	−282.500	+	489.304i	−0.0685676	+	0.118763i	0.829703	+	0.558204i	$0.188510\pi$
$258$		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	−0.893738	−	0.448588i	$-0.851927\pi$
$263$	−249.000	+	431.281i	−0.0583802	+	0.101118i	0.835358	+	0.549706i	$0.185260\pi$
$264$		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	−0.987848	−	0.155422i	$-0.950326\pi$
$269$	−2773.00	−	4802.98i	−0.628523	−	1.08863i	0.359325	−	0.933213i	$-0.383007\pi$
$270$		0			0
$271$	−1128.00	+	1953.75i	−0.252845	+	0.437941i	−0.964308	−	0.264783i	$-0.914700\pi$
$271$	−1128.00	+	1953.75i	−0.252845	+	0.437941i	0.711463	+	0.702724i	$0.248033\pi$
$272$		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	0.713219	−	0.700941i	$-0.247236\pi$
$277$	−1154.50	−	1999.65i	−0.250423	−	0.433745i	−0.963642	+	0.267196i	$0.913903\pi$
$278$		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	−0.766278	−	0.642509i	$-0.777893\pi$
$283$	825.000	−	1428.94i	0.173290	−	0.300148i	0.939568	+	0.342362i	$0.111227\pi$
$284$		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	0.677536	−	0.735489i	$-0.263048\pi$
$293$	−1495.50	−	2590.28i	−0.298184	−	0.516471i	−0.975721	+	0.219019i	$0.929714\pi$
$294$		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	0.905542	−	0.424257i	$-0.139465\pi$
$331$	514.000	+	890.274i	0.0853535	+	0.147837i	−0.820188	+	0.572094i	$0.806131\pi$
$332$		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	0.734491	−	0.678618i	$-0.237421\pi$
$347$	−1425.00	−	2468.17i	−0.220455	−	0.381840i	−0.954946	+	0.296779i	$0.904088\pi$
$348$		0			0
$349$	1009.00	−	1747.64i	0.154758	−	0.268049i	−0.778213	−	0.628001i	$-0.783874\pi$
$349$	1009.00	−	1747.64i	0.154758	−	0.268049i	0.932971	+	0.359952i	$0.117207\pi$
$350$		0			0
$351$	1228.50	+	303.975i	0.186816	+	0.0462250i
$352$		0			0
$353$	2643.50	−	4578.68i	0.398582	−	0.690364i	−0.594970	−	0.803748i	$-0.702836\pi$
$353$	2643.50	−	4578.68i	0.398582	−	0.690364i	0.993551	+	0.113385i	$0.0361692\pi$
$354$		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	−0.975869	−	0.218357i	$-0.929930\pi$
$367$	−2101.00	+	3639.04i	−0.298832	+	0.517592i	0.677037	+	0.735949i	$0.263264\pi$
$368$		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	0.915718	−	0.401821i	$-0.131623\pi$
$373$	791.500	+	1370.92i	0.109872	+	0.190304i	−0.805846	+	0.592125i	$0.798289\pi$
$374$		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	−0.927139	−	0.374717i	$-0.877740\pi$
$379$	−1026.00	+	1777.08i	−0.139056	+	0.240851i	0.788084	+	0.615568i	$0.211073\pi$
$380$		0			0
$381$	1482.00	+	2566.90i	0.199279	+	0.345161i
$382$		0			0
$383$	−3436.00	−	5951.33i	−0.458411	−	0.793991i	0.540466	−	0.841366i	$-0.318248\pi$
$383$	−3436.00	−	5951.33i	−0.458411	−	0.793991i	−0.998877	+	0.0473746i	$0.984915\pi$
$384$		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	0.604415	−	0.796670i	$-0.293407\pi$
$397$	−3067.00	−	5312.20i	−0.387729	−	0.671566i	−0.992144	+	0.125104i	$0.960074\pi$
$398$		0			0
$399$	−450.000	−	779.423i	−0.0564616	−	0.0977944i
$400$		0			0
$401$	5397.50	−	9348.74i	0.672165	−	1.16422i	−0.305124	−	0.952313i	$-0.598698\pi$
$401$	5397.50	−	9348.74i	0.672165	−	1.16422i	0.977289	−	0.211912i	$-0.0679689\pi$
$402$		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	0.999884	+	0.0152477i	$0.00485367\pi$
$409$	4244.50	+	7351.69i	0.513147	+	0.888796i	−0.486737	+	0.873549i	$0.661813\pi$
$410$		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	−0.819119	−	0.573623i	$-0.805537\pi$
$419$	748.000	−	1295.57i	0.0872129	−	0.151057i	0.906332	+	0.422566i	$0.138871\pi$
$420$		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	−0.402228	−	0.915539i	$-0.631764\pi$
$431$	5295.00	−	9171.21i	0.591766	−	1.02497i	0.993995	−	0.109430i	$-0.0349024\pi$
$432$		0			0
$433$	6974.50	+	12080.2i	0.774072	+	1.34073i	0.935315	+	0.353817i	$0.115116\pi$
$433$	6974.50	+	12080.2i	0.774072	+	1.34073i	−0.161243	+	0.986915i	$0.551550\pi$
$434$		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	0.412058	+	0.911158i	$0.364810\pi$
$439$	−5363.00	+	9288.99i	−0.583057	+	1.00988i	−0.995115	+	0.0987266i	$0.968523\pi$
$440$		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	0.597100	−	0.802167i	$-0.296320\pi$
$449$	−3769.00	−	6528.10i	−0.396147	−	0.686147i	−0.993247	+	0.116020i	$0.962986\pi$
$450$		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	0.127385	+	0.991853i	$0.459342\pi$
$457$	−7769.50	+	13457.2i	−0.795278	+	1.37746i	−0.922663	+	0.385608i	$0.873992\pi$
$458$		0			0
$459$	499.500	+	865.159i	0.0507945	+	0.0879786i
$460$		0			0
$461$	−2405.50	−	4166.45i	−0.243027	−	0.420935i	0.718548	−	0.695477i	$-0.244807\pi$
$461$	−2405.50	−	4166.45i	−0.243027	−	0.420935i	−0.961575	+	0.274543i	$0.911474\pi$
$462$		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	−0.939014	−	0.343878i	$-0.888259\pi$
$479$	−1800.00	+	3117.69i	−0.171700	+	0.297392i	0.767315	+	0.641271i	$0.221593\pi$
$480$		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	−0.921590	−	0.388164i	$-0.873109\pi$
$487$	−8565.00	−	14835.0i	−0.796955	−	1.38037i	0.124635	−	0.992203i	$-0.460224\pi$
$488$		0			0
$489$	5100.00			0.471636
$490$		0			0
$491$	−5919.00	+	10252.0i	−0.544034	+	0.942295i	0.454633	+	0.890679i	$0.349770\pi$
$491$	−5919.00	+	10252.0i	−0.544034	+	0.942295i	−0.998667	+	0.0516158i	$0.983563\pi$
$492$		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	0.900890	−	0.434047i	$-0.142915\pi$
$503$	841.000	+	1456.65i	0.0745494	+	0.129123i	−0.826341	+	0.563170i	$0.809581\pi$
$504$		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	0.320138	+	0.947371i	$0.396271\pi$
$509$	−7583.50	+	13135.0i	−0.660379	+	1.14381i	−0.980516	+	0.196438i	$0.937062\pi$
$510$		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	0.413435	+	0.910534i	$0.364329\pi$
$523$	−6959.00	+	12053.3i	−0.581828	+	1.00775i	−0.995263	+	0.0972214i	$0.969005\pi$
$524$		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	−0.827520	−	0.561436i	$-0.810249\pi$
$557$	952.500	−	1649.78i	0.0724573	−	0.125500i	0.899978	+	0.435936i	$0.143583\pi$
$558$		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	0.762105	−	0.647454i	$-0.224166\pi$
$563$	−2400.00	−	4156.92i	−0.179659	−	0.311178i	−0.941764	+	0.336275i	$0.890833\pi$
$564$		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	0.458148	+	0.888876i	$0.348513\pi$
$569$	−7339.00	+	12711.5i	−0.540715	+	0.936546i	−0.998863	+	0.0476701i	$0.984820\pi$
$570$		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	−0.999849	−	0.0173645i	$-0.994472\pi$
$587$	−6896.00	+	11944.2i	−0.484887	+	0.839848i	0.514963	+	0.857213i	$0.327806\pi$
$588$		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	−0.796866	−	0.604156i	$-0.793510\pi$
$601$	1838.50	−	3184.38i	0.124782	−	0.216129i	0.921648	+	0.388028i	$0.126844\pi$
$602$		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	0.622570	−	0.782564i	$-0.286089\pi$
$607$	−5480.00	−	9491.64i	−0.366435	−	0.634685i	−0.989005	+	0.147879i	$0.952755\pi$
$608$		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	−0.0169241	−	0.999857i	$-0.505387\pi$
$613$	13013.5	−	22540.0i	0.857439	−	1.48513i	0.874363	−	0.485272i	$-0.161279\pi$
$614$		0			0
$615$	2992.50	+	5183.16i	0.196210	+	0.339846i
$616$		0			0
$617$	−8840.50	−	15312.2i	−0.576832	−	0.999102i	−0.995840	−	0.0911193i	$-0.970956\pi$
$617$	−8840.50	−	15312.2i	−0.576832	−	0.999102i	0.419008	−	0.907982i	$-0.362378\pi$
$618$		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	0.960459	−	0.278422i	$-0.0898115\pi$
$631$	3790.00	+	6564.47i	0.239109	+	0.414148i	−0.721350	+	0.692571i	$0.756478\pi$
$632$		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	0.877905	+	0.478835i	$0.158941\pi$
$641$	13853.5	+	23995.0i	0.853635	+	1.47854i	−0.0242696	+	0.999705i	$0.507726\pi$
$642$		0			0
$643$	5608.00	+	9713.34i	0.343947	+	0.595734i	0.985162	−	0.171628i	$-0.0549026\pi$
$643$	5608.00	+	9713.34i	0.343947	+	0.595734i	−0.641215	+	0.767361i	$0.721569\pi$
$644$		0			0
$645$	−5166.00			−0.315366
$646$		0			0
$647$	−1268.00	+	2196.24i	−0.0770483	+	0.133452i	−0.901975	−	0.431788i	$-0.857883\pi$
$647$	−1268.00	+	2196.24i	−0.0770483	+	0.133452i	0.824927	+	0.565239i	$0.191216\pi$
$648$		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	0.468072	+	0.883690i	$0.344949\pi$
$653$	−8865.00	+	15354.6i	−0.531262	+	0.920173i	−0.999334	+	0.0364829i	$0.988385\pi$
$654$		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	0.997564	+	0.0697586i	$0.0222229\pi$
$659$	9460.00	+	16385.2i	0.559195	+	0.968554i	−0.438369	+	0.898795i	$0.644444\pi$
$660$		0			0
$661$	−2620.50	+	4538.84i	−0.154199	+	0.267081i	−0.932767	−	0.360480i	$-0.882613\pi$
$661$	−2620.50	+	4538.84i	−0.154199	+	0.267081i	0.778568	+	0.627560i	$0.215946\pi$
$662$		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	0.408592	+	0.912717i	$0.366020\pi$
$673$	−10233.5	+	17724.9i	−0.586140	+	1.01522i	−0.994732	+	0.102508i	$0.967313\pi$
$674$		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	0.941939	−	0.335785i	$-0.109002\pi$
$683$	3216.00	+	5570.28i	0.180171	+	0.312065i	−0.761768	+	0.647850i	$0.775668\pi$
$684$		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	−0.943067	−	0.332601i	$-0.892074\pi$
$691$	−3333.00	+	5772.93i	−0.183492	+	0.317818i	0.759575	+	0.650420i	$0.225407\pi$
$692$		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	0.866964	−	0.498371i	$-0.166068\pi$
$709$	35.5000	+	61.4878i	0.00188044	+	0.00325701i	−0.865084	+	0.501628i	$0.832735\pi$
$710$		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	0.912541	−	0.408986i	$-0.134118\pi$
$719$	1968.00	+	3408.68i	0.102078	+	0.176804i	−0.810463	+	0.585790i	$0.800784\pi$
$720$		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	−0.456813	−	0.889563i	$-0.651009\pi$
$739$	10888.0	−	18858.6i	0.541978	−	0.938733i	0.998790	−	0.0491701i	$-0.0156576\pi$
$740$		0			0
$741$	−4095.00	−	1013.25i	−0.203014	−	0.0502330i
$742$		0			0
$743$	1242.00	−	2151.21i	0.0613251	−	0.106218i	−0.833733	−	0.552168i	$-0.813801\pi$
$743$	1242.00	−	2151.21i	0.0613251	−	0.106218i	0.895058	+	0.445950i	$0.147134\pi$
$744$		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	−0.120543	−	0.992708i	$-0.538464\pi$
$751$	16453.0	−	28497.4i	0.799439	−	1.38467i	0.919982	−	0.391960i	$-0.128203\pi$
$752$		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	−0.815214	−	0.579160i	$-0.803380\pi$
$757$	1957.00	−	3389.62i	0.0939609	−	0.162745i	0.909174	+	0.416415i	$0.136714\pi$
$758$		0			0
$759$	−10692.0			−0.511324
$760$		0			0
$761$	16519.0	+	28611.7i	0.786877	+	1.36291i	0.927871	+	0.372900i	$0.121637\pi$
$761$	16519.0	+	28611.7i	0.786877	+	1.36291i	−0.140995	+	0.990010i	$0.545030\pi$
$762$		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	−0.582812	−	0.812607i	$-0.698048\pi$
$769$	8793.00	−	15229.9i	0.412332	−	0.714181i	0.995144	+	0.0984263i	$0.0313809\pi$
$770$		0			0
$771$	847.500	+	1467.91i	0.0395875	+	0.0685676i
$772$		0			0
$773$	−9157.00	−	15860.4i	−0.426073	−	0.737980i	0.570447	−	0.821334i	$-0.306770\pi$
$773$	−9157.00	−	15860.4i	−0.426073	−	0.737980i	−0.996520	+	0.0833544i	$0.973437\pi$
$774$		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	−0.739528	−	0.673125i	$-0.764951\pi$
$787$	−21034.0	−	36432.0i	−0.952708	−	1.65014i	−0.213180

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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