LMFDB - Embedded newform 72.4.d.c.37.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,4,Mod(37,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("72.37");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	72.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$4.24813752041$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-10}, \sqrt{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 64 $ x^4 - 6*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.3
Root			$2.34521 - 1.58114i$ of defining polynomial
Character	$\chi$	$=$	72.37
Dual form			72.4.d.c.37.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.34521 - 1.58114i) q^{2} +(3.00000 - 7.41620i) q^{4} -6.32456i q^{5} +10.0000 q^{7} +(-4.69042 - 22.1359i) q^{8} +O(q^{10})$	\(q+(2.34521 - 1.58114i) q^{2} +(3.00000 - 7.41620i) q^{4} -6.32456i q^{5} +10.0000 q^{7} +(-4.69042 - 22.1359i) q^{8} +(-10.0000 - 14.8324i) q^{10} -37.9473i q^{11} +59.3296i q^{13} +(23.4521 - 15.8114i) q^{14} +(-46.0000 - 44.4972i) q^{16} +75.0467 q^{17} +118.659i q^{19} +(-46.9042 - 18.9737i) q^{20} +(-60.0000 - 88.9944i) q^{22} -150.093 q^{23} +85.0000 q^{25} +(93.8083 + 139.140i) q^{26} +(30.0000 - 74.1620i) q^{28} +246.658i q^{29} +62.0000 q^{31} +(-178.236 - 31.6228i) q^{32} +(176.000 - 118.659i) q^{34} -63.2456i q^{35} +59.3296i q^{37} +(187.617 + 278.280i) q^{38} +(-140.000 + 29.6648i) q^{40} +375.233 q^{41} -118.659i q^{43} +(-281.425 - 113.842i) q^{44} +(-352.000 + 237.318i) q^{46} -450.280 q^{47} -243.000 q^{49} +(199.343 - 134.397i) q^{50} +(440.000 + 177.989i) q^{52} +132.816i q^{53} -240.000 q^{55} +(-46.9042 - 221.359i) q^{56} +(390.000 + 578.463i) q^{58} -733.648i q^{59} -533.966i q^{61} +(145.403 - 98.0306i) q^{62} +(-468.000 + 207.654i) q^{64} +375.233 q^{65} -711.955i q^{67} +(225.140 - 556.561i) q^{68} +(-100.000 - 148.324i) q^{70} +30.0000 q^{73} +(93.8083 + 139.140i) q^{74} +(880.000 + 355.978i) q^{76} -379.473i q^{77} +94.0000 q^{79} +(-281.425 + 290.930i) q^{80} +(880.000 - 593.296i) q^{82} +670.403i q^{83} -474.637i q^{85} +(-187.617 - 278.280i) q^{86} +(-840.000 + 177.989i) q^{88} -750.467 q^{89} +593.296i q^{91} +(-450.280 + 1113.12i) q^{92} +(-1056.00 + 711.955i) q^{94} +750.467 q^{95} +130.000 q^{97} +(-569.886 + 384.217i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{4} + 40 q^{7}+O(q^{10}) $ 4 * q + 12 * q^4 + 40 * q^7	$ 4 q + 12 q^{4} + 40 q^{7} - 40 q^{10} - 184 q^{16} - 240 q^{22} + 340 q^{25} + 120 q^{28} + 248 q^{31} + 704 q^{34} - 560 q^{40} - 1408 q^{46} - 972 q^{49} + 1760 q^{52} - 960 q^{55} + 1560 q^{58} - 1872 q^{64} - 400 q^{70} + 120 q^{73} + 3520 q^{76} + 376 q^{79} + 3520 q^{82} - 3360 q^{88} - 4224 q^{94} + 520 q^{97}+O(q^{100}) $ 4 * q + 12 * q^4 + 40 * q^7 - 40 * q^10 - 184 * q^16 - 240 * q^22 + 340 * q^25 + 120 * q^28 + 248 * q^31 + 704 * q^34 - 560 * q^40 - 1408 * q^46 - 972 * q^49 + 1760 * q^52 - 960 * q^55 + 1560 * q^58 - 1872 * q^64 - 400 * q^70 + 120 * q^73 + 3520 * q^76 + 376 * q^79 + 3520 * q^82 - 3360 * q^88 - 4224 * q^94 + 520 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$.

$n$	$37$	$55$	$65$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	2.34521	−	1.58114i	0.829156	−	0.559017i
$2$	2.34521	−	1.58114i	0.829156	−	0.559017i
$3$		0			0
$3$		0			0
$4$	3.00000	−	7.41620i	0.375000	−	0.927025i
$5$		−	6.32456i		−	0.565685i	−0.959166	−	0.282843i	$-0.908723\pi$
$5$		−	6.32456i		−	0.565685i	0.959166	−	0.282843i	$-0.0912774\pi$
$6$		0			0
$7$	10.0000			0.539949			0.269975	−	0.962867i	$-0.412985\pi$
$7$	10.0000			0.539949			0.269975	+	0.962867i	$0.412985\pi$
$8$	−4.69042	−	22.1359i	−0.207289	−	0.978280i
$9$		0			0
$10$	−10.0000	−	14.8324i	−0.316228	−	0.469042i
$11$		−	37.9473i		−	1.04014i	−0.854123	−	0.520071i	$-0.825906\pi$
$11$		−	37.9473i		−	1.04014i	0.854123	−	0.520071i	$-0.174094\pi$
$12$		0			0
$13$			59.3296i			1.26577i	0.774244	+	0.632887i	$0.218130\pi$
$13$			59.3296i			1.26577i	−0.774244	+	0.632887i	$0.781870\pi$
$14$	23.4521	−	15.8114i	0.447702	−	0.301841i
$15$		0			0
$16$	−46.0000	−	44.4972i	−0.718750	−	0.695269i
$17$	75.0467			1.07068			0.535338	−	0.844638i	$-0.320184\pi$
$17$	75.0467			1.07068			0.535338	+	0.844638i	$0.320184\pi$
$18$		0			0
$19$			118.659i			1.43275i	0.697715	+	0.716376i	$0.254200\pi$
$19$			118.659i			1.43275i	−0.697715	+	0.716376i	$0.745800\pi$
$20$	−46.9042	−	18.9737i	−0.524404	−	0.212132i
$21$		0			0
$22$	−60.0000	−	88.9944i	−0.581456	−	0.862439i
$23$	−150.093			−1.36072			−0.680361	−	0.732877i	$-0.738177\pi$
$23$	−150.093			−1.36072			−0.680361	+	0.732877i	$0.738177\pi$
$24$		0			0
$25$	85.0000			0.680000
$26$	93.8083	+	139.140i	0.707589	+	1.04952i
$27$		0			0
$28$	30.0000	−	74.1620i	0.202481	−	0.500546i
$29$			246.658i			1.57942i	0.613480	+	0.789710i	$0.289769\pi$
$29$			246.658i			1.57942i	−0.613480	+	0.789710i	$0.710231\pi$
$30$		0			0
$31$	62.0000			0.359211			0.179605	−	0.983739i	$-0.442518\pi$
$31$	62.0000			0.359211			0.179605	+	0.983739i	$0.442518\pi$
$32$	−178.236	−	31.6228i	−0.984623	−	0.174693i
$33$		0			0
$34$	176.000	−	118.659i	0.887757	−	0.598526i
$35$		−	63.2456i		−	0.305441i
$36$		0			0
$37$			59.3296i			0.263614i	0.991275	+	0.131807i	$0.0420779\pi$
$37$			59.3296i			0.263614i	−0.991275	+	0.131807i	$0.957922\pi$
$38$	187.617	+	278.280i	0.800933	+	1.18797i
$39$		0			0
$40$	−140.000	+	29.6648i	−0.553399	+	0.117260i
$41$	375.233			1.42931			0.714654	−	0.699479i	$-0.246584\pi$
$41$	375.233			1.42931			0.714654	+	0.699479i	$0.246584\pi$
$42$		0			0
$43$		−	118.659i		−	0.420822i	−0.977613	−	0.210411i	$-0.932520\pi$
$43$		−	118.659i		−	0.420822i	0.977613	−	0.210411i	$-0.0674802\pi$
$44$	−281.425	−	113.842i	−0.964237	−	0.390053i
$45$		0			0
$46$	−352.000	+	237.318i	−1.12825	+	0.760667i
$47$	−450.280			−1.39745			−0.698724	−	0.715391i	$-0.746249\pi$
$47$	−450.280			−1.39745			−0.698724	+	0.715391i	$0.746249\pi$
$48$		0			0
$49$	−243.000			−0.708455
$50$	199.343	−	134.397i	0.563826	−	0.380132i
$51$		0			0
$52$	440.000	+	177.989i	1.17340	+	0.474665i
$53$			132.816i			0.344220i	0.985078	+	0.172110i	$0.0550584\pi$
$53$			132.816i			0.344220i	−0.985078	+	0.172110i	$0.944942\pi$
$54$		0			0
$55$	−240.000			−0.588393
$56$	−46.9042	−	221.359i	−0.111926	−	0.528221i
$57$		0			0
$58$	390.000	+	578.463i	0.882923	+	1.30959i
$59$		−	733.648i		−	1.61886i	−0.587215	−	0.809431i	$-0.699776\pi$
$59$		−	733.648i		−	1.61886i	0.587215	−	0.809431i	$-0.300224\pi$
$60$		0			0
$61$		−	533.966i		−	1.12078i	−0.828230	−	0.560388i	$-0.810652\pi$
$61$		−	533.966i		−	1.12078i	0.828230	−	0.560388i	$-0.189348\pi$
$62$	145.403	−	98.0306i	0.297842	−	0.200805i
$63$		0			0
$64$	−468.000	+	207.654i	−0.914062	+	0.405573i
$65$	375.233			0.716030
$66$		0			0
$67$		−	711.955i		−	1.29820i	−0.760705	−	0.649098i	$-0.775146\pi$
$67$		−	711.955i		−	1.29820i	0.760705	−	0.649098i	$-0.224854\pi$
$68$	225.140	−	556.561i	0.401503	−	0.992543i
$69$		0			0
$70$	−100.000	−	148.324i	−0.170747	−	0.253259i
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	30.0000			0.0480991			0.0240496	−	0.999711i	$-0.492344\pi$
$73$	30.0000			0.0480991			0.0240496	+	0.999711i	$0.492344\pi$
$74$	93.8083	+	139.140i	0.147365	+	0.218577i
$75$		0			0
$76$	880.000	+	355.978i	1.32820	+	0.537282i
$77$		−	379.473i		−	0.561623i
$78$		0			0
$79$	94.0000			0.133871			0.0669356	−	0.997757i	$-0.478678\pi$
$79$	94.0000			0.133871			0.0669356	+	0.997757i	$0.478678\pi$
$80$	−281.425	+	290.930i	−0.393303	+	0.406586i
$81$		0			0
$82$	880.000	−	593.296i	1.18512	−	0.799007i
$83$			670.403i			0.886582i	0.896378	+	0.443291i	$0.146189\pi$
$83$			670.403i			0.886582i	−0.896378	+	0.443291i	$0.853811\pi$
$84$		0			0
$85$		−	474.637i		−	0.605666i
$86$	−187.617	−	278.280i	−0.235247	−	0.348927i
$87$		0			0
$88$	−840.000	+	177.989i	−1.01755	+	0.215610i
$89$	−750.467			−0.893812			−0.446906	−	0.894581i	$-0.647474\pi$
$89$	−750.467			−0.893812			−0.446906	+	0.894581i	$0.647474\pi$
$90$		0			0
$91$			593.296i			0.683454i
$92$	−450.280	+	1113.12i	−0.510271	+	1.26142i
$93$		0			0
$94$	−1056.00	+	711.955i	−1.15870	+	0.781197i
$95$	750.467			0.810487
$96$		0			0
$97$	130.000			0.136077			0.0680387	−	0.997683i	$-0.478326\pi$
$97$	130.000			0.136077			0.0680387	+	0.997683i	$0.478326\pi$
$98$	−569.886	+	384.217i	−0.587420	+	0.396038i
$99$		0			0
$100$	255.000	−	630.377i	0.255000	−	0.630377i
$101$			562.885i			0.554546i	0.960791	+	0.277273i	$0.0894307\pi$
$101$			562.885i			0.554546i	−0.960791	+	0.277273i	$0.910569\pi$
$102$		0			0
$103$	−570.000			−0.545279			−0.272640	−	0.962116i	$-0.587897\pi$
$103$	−570.000			−0.545279			−0.272640	+	0.962116i	$0.587897\pi$
$104$	1313.32	−	278.280i	1.23828	−	0.262381i
$105$		0			0
$106$	210.000	+	311.480i	0.192425	+	0.285412i
$107$			1290.21i			1.16569i	0.812582	+	0.582847i	$0.198061\pi$
$107$			1290.21i			1.16569i	−0.812582	+	0.582847i	$0.801939\pi$
$108$		0			0
$109$			1720.56i			1.51192i	0.654616	+	0.755961i	$0.272830\pi$
$109$			1720.56i			1.51192i	−0.654616	+	0.755961i	$0.727170\pi$
$110$	−562.850	+	379.473i	−0.487869	+	0.328921i
$111$		0			0
$112$	−460.000	−	444.972i	−0.388089	−	0.375410i
$113$	−1350.84			−1.12457			−0.562285	−	0.826944i	$-0.690077\pi$
$113$	−1350.84			−1.12457			−0.562285	+	0.826944i	$0.690077\pi$
$114$		0			0
$115$			949.273i			0.769741i
$116$	1829.26	+	739.973i	1.46416	+	0.592282i
$117$		0			0
$118$	−1160.00	−	1720.56i	−0.904972	−	1.34229i
$119$	750.467			0.578111
$120$		0			0
$121$	−109.000			−0.0818933
$122$	−844.275	−	1252.26i	−0.626533	−	0.929299i
$123$		0			0
$124$	186.000	−	459.804i	0.134704	−	0.332997i
$125$		−	1328.16i		−	0.950352i
$126$		0			0
$127$	2530.00			1.76773			0.883863	−	0.467746i	$-0.154934\pi$
$127$	2530.00			1.76773			0.883863	+	0.467746i	$0.154934\pi$
$128$	−769.228	+	1226.96i	−0.531178	+	0.847260i
$129$		0			0
$130$	880.000	−	593.296i	0.593701	−	0.400273i
$131$			252.982i			0.168726i	0.996435	+	0.0843632i	$0.0268856\pi$
$131$			252.982i			0.168726i	−0.996435	+	0.0843632i	$0.973114\pi$
$132$		0			0
$133$			1186.59i			0.773613i
$134$	−1125.70	−	1669.68i	−0.725714	−	1.07641i
$135$		0			0
$136$	−352.000	−	1661.23i	−0.221939	−	1.04742i
$137$	−675.420			−0.421204			−0.210602	−	0.977572i	$-0.567542\pi$
$137$	−675.420			−0.421204			−0.210602	+	0.977572i	$0.567542\pi$
$138$		0			0
$139$		−	237.318i		−	0.144814i	−0.997375	−	0.0724068i	$-0.976932\pi$
$139$		−	237.318i		−	0.144814i	0.997375	−	0.0724068i	$-0.0230680\pi$
$140$	−469.042	−	189.737i	−0.283152	−	0.114541i
$141$		0			0
$142$		0			0
$143$	2251.40			1.31658
$144$		0			0
$145$	1560.00			0.893455
$146$	70.3562	−	47.4342i	0.0398817	−	0.0268882i
$147$		0			0
$148$	440.000	+	177.989i	0.244377	+	0.0988553i
$149$		−	2940.92i		−	1.61698i	−0.588513	−	0.808488i	$-0.700286\pi$
$149$		−	2940.92i		−	1.61698i	0.588513	−	0.808488i	$-0.299714\pi$
$150$		0			0
$151$	−2262.00			−1.21907			−0.609533	−	0.792761i	$-0.708643\pi$
$151$	−2262.00			−1.21907			−0.609533	+	0.792761i	$0.708643\pi$
$152$	2626.63	−	556.561i	1.40163	−	0.296994i
$153$		0			0
$154$	−600.000	−	889.944i	−0.313957	−	0.465673i
$155$		−	392.122i		−	0.203200i
$156$		0			0
$157$		−	1720.56i		−	0.874621i	−0.899311	−	0.437310i	$-0.855931\pi$
$157$		−	1720.56i		−	0.874621i	0.899311	−	0.437310i	$-0.144069\pi$
$158$	220.450	−	148.627i	0.111000	−	0.0748363i
$159$		0			0
$160$	−200.000	+	1127.26i	−0.0988212	+	0.556987i
$161$	−1500.93			−0.734721
$162$		0			0
$163$			2017.21i			0.969324i	0.874702	+	0.484662i	$0.161057\pi$
$163$			2017.21i			0.969324i	−0.874702	+	0.484662i	$0.838943\pi$
$164$	1125.70	−	2782.80i	0.535990	−	1.32500i
$165$		0			0
$166$	1060.00	+	1572.23i	0.495614	+	0.735115i
$167$	1050.65			0.486838			0.243419	−	0.969921i	$-0.421731\pi$
$167$	1050.65			0.486838			0.243419	+	0.969921i	$0.421731\pi$
$168$		0			0
$169$	−1323.00			−0.602185
$170$	−750.467	−	1113.12i	−0.338577	−	0.502191i
$171$		0			0
$172$	−880.000	−	355.978i	−0.390113	−	0.157808i
$173$			1423.02i			0.625379i	0.949855	+	0.312690i	$0.101230\pi$
$173$			1423.02i			0.625379i	−0.949855	+	0.312690i	$0.898770\pi$
$174$		0			0
$175$	850.000			0.367165
$176$	−1688.55	+	1745.58i	−0.723177	+	0.747601i
$177$		0			0
$178$	−1760.00	+	1186.59i	−0.741110	+	0.499656i
$179$		−	126.491i		−	0.0528178i	−0.999651	−	0.0264089i	$-0.991593\pi$
$179$		−	126.491i		−	0.0528178i	0.999651	−	0.0264089i	$-0.00840719\pi$
$180$		0			0
$181$		−	59.3296i		−	0.0243643i	−0.999926	−	0.0121821i	$-0.996122\pi$
$181$		−	59.3296i		−	0.0243643i	0.999926	−	0.0121821i	$-0.00387779\pi$
$182$	938.083	+	1391.40i	0.382062	+	0.566690i
$183$		0			0
$184$	704.000	+	3322.46i	0.282063	+	1.33117i
$185$	375.233			0.149123
$186$		0			0
$187$		−	2847.82i		−	1.11365i
$188$	−1350.84	+	3339.37i	−0.524043	+	1.29547i
$189$		0			0
$190$	1760.00	−	1186.59i	0.672020	−	0.453076i
$191$	−3752.33			−1.42151			−0.710757	−	0.703437i	$-0.751648\pi$
$191$	−3752.33			−1.42151			−0.710757	+	0.703437i	$0.751648\pi$
$192$		0			0
$193$	−1350.00			−0.503498			−0.251749	−	0.967793i	$-0.581006\pi$
$193$	−1350.00			−0.503498			−0.251749	+	0.967793i	$0.581006\pi$
$194$	304.877	−	205.548i	0.112829	−	0.0760695i
$195$		0			0
$196$	−729.000	+	1802.14i	−0.265671	+	0.656755i
$197$			1448.32i			0.523801i	0.965095	+	0.261900i	$0.0843492\pi$
$197$			1448.32i			0.523801i	−0.965095	+	0.261900i	$0.915651\pi$
$198$		0			0
$199$	−3194.00			−1.13777			−0.568886	−	0.822416i	$-0.692625\pi$
$199$	−3194.00			−1.13777			−0.568886	+	0.822416i	$0.692625\pi$
$200$	−398.685	−	1881.56i	−0.140957	−	0.665230i
$201$		0			0
$202$	890.000	+	1320.08i	0.310001	+	0.459806i
$203$			2466.58i			0.852807i
$204$		0			0
$205$		−	2373.18i		−	0.808538i
$206$	−1336.77	+	901.249i	−0.452122	+	0.304820i
$207$		0			0
$208$	2640.00	−	2729.16i	0.880053	−	0.909775i
$209$	4502.80			1.49026
$210$		0			0
$211$		−	5458.32i		−	1.78088i	−0.455097	−	0.890442i	$-0.650396\pi$
$211$		−	5458.32i		−	1.78088i	0.455097	−	0.890442i	$-0.349604\pi$
$212$	984.987	+	398.447i	0.319100	+	0.129082i
$213$		0			0
$214$	2040.00	+	3025.81i	0.651643	+	0.966542i
$215$	−750.467			−0.238053
$216$		0			0
$217$	620.000			0.193955
$218$	2720.44	+	4035.07i	0.845190	+	1.25362i
$219$		0			0
$220$	−720.000	+	1779.89i	−0.220647	+	0.545455i
$221$			4452.49i			1.35523i
$222$		0			0
$223$	5330.00			1.60055			0.800276	−	0.599632i	$-0.204686\pi$
$223$	5330.00			1.60055			0.800276	+	0.599632i	$0.204686\pi$
$224$	−1782.36	−	316.228i	−0.531646	−	0.0943253i
$225$		0			0
$226$	−3168.00	+	2135.87i	−0.932443	+	0.628653i
$227$		−	4768.71i		−	1.39432i	−0.716915	−	0.697160i	$-0.754447\pi$
$227$		−	4768.71i		−	1.39432i	0.716915	−	0.697160i	$-0.245553\pi$
$228$		0			0
$229$		−	3619.10i		−	1.04435i	−0.852837	−	0.522177i	$-0.825120\pi$
$229$		−	3619.10i		−	1.04435i	0.852837	−	0.522177i	$-0.174880\pi$
$230$	1500.93	+	2226.24i	0.430298	+	0.638235i
$231$		0			0
$232$	5460.00	−	1156.93i	1.54511	−	0.327396i
$233$	150.093			0.0422015			0.0211007	−	0.999777i	$-0.493283\pi$
$233$	150.093			0.0422015			0.0211007	+	0.999777i	$0.493283\pi$
$234$		0			0
$235$			2847.82i			0.790516i
$236$	−5440.88	−	2200.95i	−1.50073	−	0.607073i
$237$		0			0
$238$	1760.00	−	1186.59i	0.479344	−	0.323174i
$239$	−2251.40			−0.609334			−0.304667	−	0.952459i	$-0.598545\pi$
$239$	−2251.40			−0.609334			−0.304667	+	0.952459i	$0.598545\pi$
$240$		0			0
$241$	−1162.00			−0.310585			−0.155293	−	0.987869i	$-0.549632\pi$
$241$	−1162.00			−0.310585			−0.155293	+	0.987869i	$0.549632\pi$
$242$	−255.628	+	172.344i	−0.0679023	+	0.0457798i
$243$		0			0
$244$	−3960.00	−	1601.90i	−1.03899	−	0.420291i
$245$			1536.87i			0.400763i
$246$		0			0
$247$	−7040.00			−1.81354
$248$	−290.806	−	1372.43i	−0.0744604	−	0.351408i
$249$		0			0
$250$	−2100.00	−	3114.80i	−0.531263	−	0.787990i
$251$		−	645.105i		−	0.162226i	−0.996705	−	0.0811128i	$-0.974153\pi$
$251$		−	645.105i		−	0.162226i	0.996705	−	0.0811128i	$-0.0258474\pi$
$252$		0			0
$253$			5695.64i			1.41534i
$254$	5933.38	−	4000.28i	1.46572	−	0.988189i
$255$		0			0
$256$	136.000	+	4093.74i	0.0332031	+	0.999449i
$257$	1801.12			0.437162			0.218581	−	0.975819i	$-0.429857\pi$
$257$	1801.12			0.437162			0.218581	+	0.975819i	$0.429857\pi$
$258$		0			0
$259$			593.296i			0.142338i
$260$	1125.70	−	2782.80i	0.268511	−	0.663778i
$261$		0			0
$262$	400.000	+	593.296i	0.0943209	+	0.139901i
$263$	−6604.11			−1.54839			−0.774195	−	0.632947i	$-0.781845\pi$
$263$	−6604.11			−1.54839			−0.774195	+	0.632947i	$0.781845\pi$
$264$		0			0
$265$	840.000			0.194720
$266$	1876.17	+	2782.80i	0.432463	+	0.641446i
$267$		0			0
$268$	−5280.00	−	2135.87i	−1.20346	−	0.486824i
$269$		−	6052.60i		−	1.37187i	−0.727662	−	0.685936i	$-0.759393\pi$
$269$		−	6052.60i		−	1.37187i	0.727662	−	0.685936i	$-0.240607\pi$
$270$		0			0
$271$	6402.00			1.43503			0.717516	−	0.696542i	$-0.245279\pi$
$271$	6402.00			1.43503			0.717516	+	0.696542i	$0.245279\pi$
$272$	−3452.15	−	3339.37i	−0.769548	−	0.744407i
$273$		0			0
$274$	−1584.00	+	1067.93i	−0.349244	+	0.235460i
$275$		−	3225.52i		−	0.707296i
$276$		0			0
$277$			2076.54i			0.450422i	0.974310	+	0.225211i	$0.0723072\pi$
$277$			2076.54i			0.450422i	−0.974310	+	0.225211i	$0.927693\pi$
$278$	−375.233	−	556.561i	−0.0809532	−	0.120073i
$279$		0			0
$280$	−1400.00	+	296.648i	−0.298807	+	0.0633147i
$281$	9005.60			1.91185			0.955923	−	0.293616i	$-0.0948587\pi$
$281$	9005.60			1.91185			0.955923	+	0.293616i	$0.0948587\pi$
$282$		0			0
$283$			5221.00i			1.09667i	0.836260	+	0.548333i	$0.184737\pi$
$283$			5221.00i			1.09667i	−0.836260	+	0.548333i	$0.815263\pi$
$284$		0			0
$285$		0			0
$286$	5280.00	−	3559.78i	1.09165	−	0.735993i
$287$	3752.33			0.771753
$288$		0			0
$289$	719.000			0.146346
$290$	3658.52	−	2466.58i	0.740814	−	0.499456i
$291$		0			0
$292$	90.0000	−	222.486i	0.0180372	−	0.0445891i
$293$			5293.65i			1.05549i	0.849403	+	0.527745i	$0.176962\pi$
$293$			5293.65i			1.05549i	−0.849403	+	0.527745i	$0.823038\pi$
$294$		0			0
$295$	−4640.00			−0.915767
$296$	1313.32	−	278.280i	0.257888	−	0.0546443i
$297$		0			0
$298$	−4650.00	−	6897.06i	−0.903917	−	1.34073i
$299$		−	8904.97i		−	1.72237i
$300$		0			0
$301$		−	1186.59i		−	0.227223i
$302$	−5304.86	+	3576.54i	−1.01080	+	0.681479i
$303$		0			0
$304$	5280.00	−	5458.32i	0.996147	−	1.02979i
$305$	−3377.10			−0.634007
$306$		0			0
$307$			4983.69i			0.926495i	0.886229	+	0.463247i	$0.153316\pi$
$307$			4983.69i			0.926495i	−0.886229	+	0.463247i	$0.846684\pi$
$308$	−2814.25	−	1138.42i	−0.520639	−	0.210609i
$309$		0			0
$310$	−620.000	−	919.609i	−0.113592	−	0.168485i
$311$	−1500.93			−0.273666			−0.136833	−	0.990594i	$-0.543692\pi$
$311$	−1500.93			−0.273666			−0.136833	+	0.990594i	$0.543692\pi$
$312$		0			0
$313$	6350.00			1.14672			0.573360	−	0.819304i	$-0.305640\pi$
$313$	6350.00			1.14672			0.573360	+	0.819304i	$0.305640\pi$
$314$	−2720.44	−	4035.07i	−0.488928	−	0.725197i
$315$		0			0
$316$	282.000	−	697.123i	0.0502017	−	0.124102i
$317$			1739.25i			0.308158i	0.988059	+	0.154079i	$0.0492411\pi$
$317$			1739.25i			0.308158i	−0.988059	+	0.154079i	$0.950759\pi$
$318$		0			0
$319$	9360.00			1.64282
$320$	1313.32	+	2959.89i	0.229427	+	0.517072i
$321$		0			0
$322$	−3520.00	+	2373.18i	−0.609199	+	0.410722i
$323$			8904.97i			1.53401i
$324$		0			0
$325$			5043.01i			0.860727i
$326$	3189.48	+	4730.77i	0.541868	+	0.803721i
$327$		0			0
$328$	−1760.00	−	8306.14i	−0.296280	−	1.39826i
$329$	−4502.80			−0.754551
$330$		0			0
$331$		−	3322.46i		−	0.551718i	−0.961198	−	0.275859i	$-0.911038\pi$
$331$		−	3322.46i		−	0.551718i	0.961198	−	0.275859i	$-0.0889623\pi$
$332$	4971.84	+	2011.21i	0.821883	+	0.332468i
$333$		0			0
$334$	2464.00	−	1661.23i	0.403665	−	0.272151i
$335$	−4502.80			−0.734371
$336$		0			0
$337$	−1430.00			−0.231149			−0.115574	−	0.993299i	$-0.536871\pi$
$337$	−1430.00			−0.231149			−0.115574	+	0.993299i	$0.536871\pi$
$338$	−3102.71	+	2091.85i	−0.499305	+	0.336632i
$339$		0			0
$340$	−3520.00	−	1423.91i	−0.561467	−	0.227125i
$341$		−	2352.73i		−	0.373630i
$342$		0			0
$343$	−5860.00			−0.922479
$344$	−2626.63	+	556.561i	−0.411682	+	0.0872318i
$345$		0			0
$346$	2250.00	+	3337.29i	0.349598	+	0.518537i
$347$			10106.6i			1.56355i	0.623559	+	0.781776i	$0.285686\pi$
$347$			10106.6i			1.56355i	−0.623559	+	0.781776i	$0.714314\pi$
$348$		0			0
$349$			1127.26i			0.172897i	0.996256	+	0.0864484i	$0.0275518\pi$
$349$			1127.26i			0.172897i	−0.996256	+	0.0864484i	$0.972448\pi$
$350$	1993.43	−	1343.97i	0.304438	−	0.205252i
$351$		0			0
$352$	−1200.00	+	6763.57i	−0.181705	+	1.02415i
$353$	1350.84			0.203677			0.101838	−	0.994801i	$-0.467528\pi$
$353$	1350.84			0.203677			0.101838	+	0.994801i	$0.467528\pi$
$354$		0			0
$355$		0			0
$356$	−2251.40	+	5565.61i	−0.335180	+	0.828586i
$357$		0			0
$358$	−200.000	−	296.648i	−0.0295261	−	0.0437942i
$359$	12007.5			1.76526			0.882632	−	0.470065i	$-0.155769\pi$
$359$	12007.5			1.76526			0.882632	+	0.470065i	$0.155769\pi$
$360$		0			0
$361$	−7221.00			−1.05278
$362$	−93.8083	−	139.140i	−0.0136200	−	0.0202018i
$363$		0			0
$364$	4400.00	+	1779.89i	0.633579	+	0.256295i
$365$		−	189.737i		−	0.0272090i
$366$		0			0
$367$	5070.00			0.721122			0.360561	−	0.932736i	$-0.382585\pi$
$367$	5070.00			0.721122			0.360561	+	0.932736i	$0.382585\pi$
$368$	6904.29	+	6678.73i	0.978019	+	0.946068i
$369$		0			0
$370$	880.000	−	593.296i	0.123646	−	0.0833621i
$371$			1328.16i			0.185861i
$372$		0			0
$373$			2432.51i			0.337670i	0.985644	+	0.168835i	$0.0540004\pi$
$373$			2432.51i			0.337670i	−0.985644	+	0.168835i	$0.946000\pi$
$374$	−4502.80	−	6678.73i	−0.622551	−	0.923393i
$375$		0			0
$376$	2112.00	+	9967.37i	0.289676	+	1.36710i
$377$	−14634.1			−1.99919
$378$		0			0
$379$		−	7712.85i		−	1.04534i	−0.852536	−	0.522668i	$-0.824937\pi$
$379$		−	7712.85i		−	1.04534i	0.852536	−	0.522668i	$-0.175063\pi$
$380$	2251.40	−	5565.61i	0.303933	−	0.751341i
$381$		0			0
$382$	−8800.00	+	5932.96i	−1.17866	+	0.794651i
$383$	5853.64			0.780958			0.390479	−	0.920612i	$-0.372309\pi$
$383$	5853.64			0.780958			0.390479	+	0.920612i	$0.372309\pi$
$384$		0			0
$385$	−2400.00			−0.317702
$386$	−3166.03	+	2134.54i	−0.417479	+	0.281464i
$387$		0			0
$388$	390.000	−	964.106i	0.0510290	−	0.126147i
$389$		−	2156.67i		−	0.281099i	−0.990074	−	0.140550i	$-0.955113\pi$
$389$		−	2156.67i		−	0.281099i	0.990074	−	0.140550i	$-0.0448870\pi$
$390$		0			0
$391$	−11264.0			−1.45689
$392$	1139.77	+	5379.03i	0.146855	+	0.693067i
$393$		0			0
$394$	2290.00	+	3396.62i	0.292814	+	0.434313i
$395$		−	594.508i		−	0.0757290i
$396$		0			0
$397$		−	12162.6i		−	1.53759i	−0.639498	−	0.768793i	$-0.720858\pi$
$397$		−	12162.6i		−	1.53759i	0.639498	−	0.768793i	$-0.279142\pi$
$398$	−7490.59	+	5050.16i	−0.943391	+	0.636034i
$399$		0			0
$400$	−3910.00	−	3782.26i	−0.488750	−	0.472783i
$401$	−3377.10			−0.420559			−0.210280	−	0.977641i	$-0.567437\pi$
$401$	−3377.10			−0.420559			−0.210280	+	0.977641i	$0.567437\pi$
$402$		0			0
$403$			3678.43i			0.454680i
$404$	4174.47	+	1688.66i	0.514078	+	0.207955i
$405$		0			0
$406$	3900.00	+	5784.63i	0.476733	+	0.707110i
$407$	2251.40			0.274196
$408$		0			0
$409$	3526.00			0.426282			0.213141	−	0.977021i	$-0.431631\pi$
$409$	3526.00			0.426282			0.213141	+	0.977021i	$0.431631\pi$
$410$	−3752.33	−	5565.61i	−0.451987	−	0.670404i
$411$		0			0
$412$	−1710.00	+	4227.23i	−0.204480	+	0.505487i
$413$		−	7336.48i		−	0.874104i
$414$		0			0
$415$	4240.00			0.501526
$416$	1876.17	−	10574.7i	0.221122	−	1.24631i
$417$		0			0
$418$	10560.0	−	7119.55i	1.23566	−	0.833083i
$419$			2466.58i			0.287590i	0.989608	+	0.143795i	$0.0459306\pi$
$419$			2466.58i			0.287590i	−0.989608	+	0.143795i	$0.954069\pi$
$420$		0			0
$421$		−	13586.5i		−	1.57284i	−0.617694	−	0.786418i	$-0.711933\pi$
$421$		−	13586.5i		−	1.57284i	0.617694	−	0.786418i	$-0.288067\pi$
$422$	−8630.36	−	12800.9i	−0.995544	−	1.47663i
$423$		0			0
$424$	2940.00	−	622.961i	0.336743	−	0.0713529i
$425$	6378.97			0.728059
$426$		0			0
$427$		−	5339.66i		−	0.605163i
$428$	9568.45	+	3870.63i	1.08063	+	0.437135i
$429$		0			0
$430$	−1760.00	+	1186.59i	−0.197383	+	0.133076i
$431$	6754.20			0.754845			0.377423	−	0.926041i	$-0.376810\pi$
$431$	6754.20			0.754845			0.377423	+	0.926041i	$0.376810\pi$
$432$		0			0
$433$	7790.00			0.864581			0.432290	−	0.901734i	$-0.357706\pi$
$433$	7790.00			0.864581			0.432290	+	0.901734i	$0.357706\pi$
$434$	1454.03	−	980.306i	0.160819	−	0.108424i
$435$		0			0
$436$	12760.0	+	5161.67i	1.40159	+	0.566971i
$437$		−	17809.9i		−	1.94958i
$438$		0			0
$439$	−9354.00			−1.01695			−0.508476	−	0.861076i	$-0.669791\pi$
$439$	−9354.00			−1.01695			−0.508476	+	0.861076i	$0.669791\pi$
$440$	1125.70	+	5312.63i	0.121967	+	0.575613i
$441$		0			0
$442$	7040.00	+	10442.0i	0.757599	+	1.12370i
$443$		−	6488.99i		−	0.695940i	−0.937506	−	0.347970i	$-0.886871\pi$
$443$		−	6488.99i		−	0.695940i	0.937506	−	0.347970i	$-0.113129\pi$
$444$		0			0
$445$			4746.37i			0.505617i
$446$	12500.0	−	8427.47i	1.32711	−	0.894736i
$447$		0			0
$448$	−4680.00	+	2076.54i	−0.493547	+	0.218989i
$449$	−10131.3			−1.06487			−0.532434	−	0.846472i	$-0.678722\pi$
$449$	−10131.3			−1.06487			−0.532434	+	0.846472i	$0.678722\pi$
$450$		0			0
$451$		−	14239.1i		−	1.48668i
$452$	−4052.52	+	10018.1i	−0.421713	+	1.04250i
$453$		0			0
$454$	−7540.00	−	11183.6i	−0.779449	−	1.15611i
$455$	3752.33			0.386620
$456$		0			0
$457$	−12010.0			−1.22933			−0.614665	−	0.788788i	$-0.710709\pi$
$457$	−12010.0			−1.22933			−0.614665	+	0.788788i	$0.710709\pi$
$458$	−5722.31	−	8487.55i	−0.583812	−	0.865933i
$459$		0			0
$460$	7040.00	+	2847.82i	0.713569	+	0.288653i
$461$			9316.07i			0.941199i	0.882347	+	0.470599i	$0.155962\pi$
$461$			9316.07i			0.941199i	−0.882347	+	0.470599i	$0.844038\pi$
$462$		0			0
$463$	14770.0			1.48255			0.741274	−	0.671202i	$-0.234222\pi$
$463$	14770.0			1.48255			0.741274	+	0.671202i	$0.234222\pi$
$464$	10975.6	−	11346.3i	1.09812	−	1.13521i
$465$		0			0
$466$	352.000	−	237.318i	0.0349916	−	0.0235913i
$467$			8740.54i			0.866089i	0.901372	+	0.433045i	$0.142561\pi$
$467$			8740.54i			0.866089i	−0.901372	+	0.433045i	$0.857439\pi$
$468$		0			0
$469$		−	7119.55i		−	0.700960i
$470$	4502.80	+	6678.73i	0.441912	+	0.655461i
$471$		0			0
$472$	−16240.0	+	3441.12i	−1.58370	+	0.335572i
$473$	−4502.80			−0.437714
$474$		0			0
$475$			10086.0i			0.974271i
$476$	2251.40	−	5565.61i	0.216791	−	0.535923i
$477$		0			0
$478$	−5280.00	+	3559.78i	−0.505233	+	0.340628i
$479$	−3001.87			−0.286344			−0.143172	−	0.989698i	$-0.545730\pi$
$479$	−3001.87			−0.286344			−0.143172	+	0.989698i	$0.545730\pi$
$480$		0			0
$481$	−3520.00			−0.333676
$482$	−2725.13	+	1837.28i	−0.257524	+	0.173622i
$483$		0			0
$484$	−327.000	+	808.366i	−0.0307100	+	0.0759171i
$485$		−	822.192i		−	0.0769770i
$486$		0			0
$487$	−9910.00			−0.922105			−0.461052	−	0.887373i	$-0.652528\pi$
$487$	−9910.00			−0.922105			−0.461052	+	0.887373i	$0.652528\pi$
$488$	−11819.8	+	2504.52i	−1.09643	+	0.232325i
$489$		0			0
$490$	2430.00	+	3604.27i	0.224033	+	0.332295i
$491$		−	3389.96i		−	0.311582i	−0.987790	−	0.155791i	$-0.950207\pi$
$491$		−	3389.96i		−	0.311582i	0.987790	−	0.155791i	$-0.0497927\pi$
$492$		0			0
$493$			18510.8i			1.69105i
$494$	−16510.3	+	11131.2i	−1.50371	+	1.01380i
$495$		0			0
$496$	−2852.00	−	2758.83i	−0.258183	−	0.249748i
$497$		0			0
$498$		0			0
$499$			4271.73i			0.383224i	0.981471	+	0.191612i	$0.0613716\pi$
$499$			4271.73i			0.383224i	−0.981471	+	0.191612i	$0.938628\pi$
$500$	−9849.87	−	3984.47i	−0.880999	−	0.356382i
$501$		0			0
$502$	−1020.00	−	1512.90i	−0.0906869	−	0.134510i
$503$	900.560			0.0798290			0.0399145	−	0.999203i	$-0.487291\pi$
$503$	900.560			0.0798290			0.0399145	+	0.999203i	$0.487291\pi$
$504$		0			0
$505$	3560.00			0.313699
$506$	9005.60	+	13357.5i	0.791201	+	1.17354i
$507$		0			0
$508$	7590.00	−	18763.0i	0.662897	−	1.63873i
$509$			853.815i			0.0743510i	0.999309	+	0.0371755i	$0.0118361\pi$
$509$			853.815i			0.0743510i	−0.999309	+	0.0371755i	$0.988164\pi$
$510$		0			0
$511$	300.000			0.0259711
$512$	6791.72	+	9385.64i	0.586239	+	0.810138i
$513$		0			0
$514$	4224.00	−	2847.82i	0.362476	−	0.244381i
$515$			3605.00i			0.308457i
$516$		0			0
$517$			17086.9i			1.45354i
$518$	938.083	+	1391.40i	0.0795695	+	0.118021i
$519$		0			0
$520$	−1760.00	−	8306.14i	−0.148425	−	0.700478i
$521$	7879.90			0.662619			0.331310	−	0.943522i	$-0.392510\pi$
$521$	7879.90			0.662619			0.331310	+	0.943522i	$0.392510\pi$
$522$		0			0
$523$			10323.3i			0.863114i	0.902086	+	0.431557i	$0.142036\pi$
$523$			10323.3i			0.863114i	−0.902086	+	0.431557i	$0.857964\pi$
$524$	1876.17	+	758.947i	0.156414	+	0.0632724i
$525$		0			0
$526$	−15488.0	+	10442.0i	−1.28386	+	0.865576i
$527$	4652.89			0.384598
$528$		0			0
$529$	10361.0			0.851566
$530$	1969.97	−	1328.16i	0.161453	−	0.108852i
$531$		0			0
$532$	8800.00	+	3559.78i	0.717159	+	0.290105i
$533$			22262.4i			1.80918i
$534$		0			0
$535$	8160.00			0.659416
$536$	−15759.8	+	3339.37i	−1.27000	+	0.269102i
$537$		0			0
$538$	−9570.00	−	14194.6i	−0.766900	−	1.13750i
$539$			9221.20i			0.736893i
$540$		0			0
$541$		−	15603.7i		−	1.24003i	−0.784591	−	0.620014i	$-0.787127\pi$
$541$		−	15603.7i		−	1.24003i	0.784591	−	0.620014i	$-0.212873\pi$
$542$	15014.0	−	10122.5i	1.18987	−	0.802208i
$543$		0			0
$544$	−13376.0	−	2373.18i	−1.05421	−	0.187039i
$545$	10881.8			0.855273
$546$		0			0
$547$			2254.52i			0.176228i	0.996110	+	0.0881138i	$0.0280839\pi$
$547$			2254.52i			0.176228i	−0.996110	+	0.0881138i	$0.971916\pi$
$548$	−2026.26	+	5009.05i	−0.157952	+	0.390467i
$549$		0			0
$550$	−5100.00	−	7564.52i	−0.395390	−	0.586459i
$551$	−29268.2			−2.26292
$552$		0			0
$553$	940.000			0.0722837
$554$	3283.29	+	4869.91i	0.251794	+	0.373470i
$555$		0			0
$556$	−1760.00	−	711.955i	−0.134246	−	0.0543051i
$557$			284.605i			0.0216501i	0.999941	+	0.0108250i	$0.00344579\pi$
$557$			284.605i			0.0216501i	−0.999941	+	0.0108250i	$0.996554\pi$
$558$		0			0
$559$	7040.00			0.532666
$560$	−2814.25	+	2909.30i	−0.212364	+	0.219536i
$561$		0			0
$562$	21120.0	−	14239.1i	1.58522	−	1.06875i
$563$		−	18328.6i		−	1.37204i	−0.727584	−	0.686018i	$-0.759357\pi$
$563$		−	18328.6i		−	1.37204i	0.727584	−	0.686018i	$-0.240643\pi$
$564$		0			0
$565$			8543.46i			0.636152i
$566$	8255.13	+	12244.3i	0.613055	+	0.909307i
$567$		0			0
$568$		0			0
$569$	7879.90			0.580567			0.290283	−	0.956941i	$-0.406250\pi$
$569$	7879.90			0.580567			0.290283	+	0.956941i	$0.406250\pi$
$570$		0			0
$571$			8306.14i			0.608759i	0.952551	+	0.304379i	$0.0984491\pi$
$571$			8306.14i			0.608759i	−0.952551	+	0.304379i	$0.901551\pi$
$572$	6754.20	−	16696.8i	0.493719	−	1.22051i
$573$		0			0
$574$	8800.00	−	5932.96i	0.639904	−	0.431423i
$575$	−12757.9			−0.925291
$576$		0			0
$577$	−6330.00			−0.456709			−0.228355	−	0.973578i	$-0.573335\pi$
$577$	−6330.00			−0.456709			−0.228355	+	0.973578i	$0.573335\pi$
$578$	1686.20	−	1136.84i	0.121344	−	0.0818101i
$579$		0			0
$580$	4680.00	−	11569.3i	0.335046	−	0.828255i
$581$			6704.03i			0.478709i
$582$		0			0
$583$	5040.00			0.358037
$584$	−140.712	−	664.078i	−0.00997042	−	0.0470544i
$585$		0			0
$586$	8370.00	+	12414.7i	0.590037	+	0.875166i
$587$			1922.66i			0.135191i	0.997713	+	0.0675953i	$0.0215327\pi$
$587$			1922.66i			0.135191i	−0.997713	+	0.0675953i	$0.978467\pi$
$588$		0			0
$589$			7356.87i			0.514660i
$590$	−10881.8	+	7336.48i	−0.759314	+	0.511929i
$591$		0			0
$592$	2640.00	−	2729.16i	0.183283	−	0.189473i
$593$	10356.4			0.717180			0.358590	−	0.933495i	$-0.383258\pi$
$593$	10356.4			0.717180			0.358590	+	0.933495i	$0.383258\pi$
$594$		0			0
$595$		−	4746.37i		−	0.327029i
$596$	−21810.4	−	8822.75i	−1.49898	−	0.606366i
$597$		0			0
$598$	−14080.0	−	20884.0i	−0.962833	−	1.42811i
$599$	750.467			0.0511907			0.0255954	−	0.999672i	$-0.491852\pi$
$599$	750.467			0.0511907			0.0255954	+	0.999672i	$0.491852\pi$
$600$		0			0
$601$	18578.0			1.26092			0.630460	−	0.776222i	$-0.282866\pi$
$601$	18578.0			1.26092			0.630460	+	0.776222i	$0.282866\pi$
$602$	−1876.17	−	2782.80i	−0.127021	−	0.188403i
$603$		0			0
$604$	−6786.00	+	16775.4i	−0.457150	+	1.13010i
$605$			689.377i			0.0463259i
$606$		0			0
$607$	−8030.00			−0.536948			−0.268474	−	0.963287i	$-0.586519\pi$
$607$	−8030.00			−0.536948			−0.268474	+	0.963287i	$0.586519\pi$
$608$	3752.33	−	21149.3i	0.250291	−	1.41072i
$609$		0			0
$610$	−7920.00	+	5339.66i	−0.525691	+	0.354421i
$611$		−	26714.9i		−	1.76885i
$612$		0			0
$613$			3856.42i			0.254094i	0.991897	+	0.127047i	$0.0405499\pi$
$613$			3856.42i			0.254094i	−0.991897	+	0.127047i	$0.959450\pi$
$614$	7879.90	+	11687.8i	0.517926	+	0.768209i
$615$		0			0
$616$	−8400.00	+	1779.89i	−0.549425	+	0.116418i
$617$	300.187			0.0195868			0.00979340	−	0.999952i	$-0.496883\pi$
$617$	300.187			0.0195868			0.00979340	+	0.999952i	$0.496883\pi$
$618$		0			0
$619$		−	1423.91i		−	0.0924584i	−0.998931	−	0.0462292i	$-0.985280\pi$
$619$		−	1423.91i		−	0.0924584i	0.998931	−	0.0462292i	$-0.0147205\pi$
$620$	−2908.06	−	1176.37i	−0.188372	−	0.0762001i
$621$		0			0
$622$	−3520.00	+	2373.18i	−0.226912	+	0.152984i
$623$	−7504.67			−0.482613
$624$		0			0
$625$	2225.00			0.142400
$626$	14892.1	−	10040.2i	0.950810	−	0.641036i
$627$		0			0
$628$	−12760.0	−	5161.67i	−0.810795	−	0.327983i
$629$			4452.49i			0.282245i
$630$		0			0
$631$	−12902.0			−0.813979			−0.406989	−	0.913433i	$-0.633421\pi$
$631$	−12902.0			−0.813979			−0.406989	+	0.913433i	$0.633421\pi$
$632$	−440.899	−	2080.78i	−0.0277500	−	0.130964i
$633$		0			0
$634$	2750.00	+	4078.91i	0.172266	+	0.255511i
$635$		−	16001.1i		−	0.999977i
$636$		0			0
$637$		−	14417.1i		−	0.896744i
$638$	21951.1	−	14799.5i	1.36215	−	0.918364i
$639$		0			0
$640$	7760.00	+	4865.03i	0.479283	+	0.300480i
$641$	19887.4			1.22543			0.612717	−	0.790302i	$-0.290076\pi$
$641$	19887.4			1.22543			0.612717	+	0.790302i	$0.290076\pi$
$642$		0			0
$643$			29783.5i			1.82666i	0.407216	+	0.913332i	$0.366500\pi$
$643$			29783.5i			1.82666i	−0.407216	+	0.913332i	$0.633500\pi$
$644$	−4502.80	+	11131.2i	−0.275520	+	0.681105i
$645$		0			0
$646$	14080.0	+	20884.0i	0.857539	+	1.27194i
$647$	13958.7			0.848180			0.424090	−	0.905620i	$-0.360594\pi$
$647$	13958.7			0.848180			0.424090	+	0.905620i	$0.360594\pi$
$648$		0			0
$649$	−27840.0			−1.68385
$650$	7973.71	+	11826.9i	0.481161	+	0.713677i
$651$		0			0
$652$	14960.0	+	6051.62i	0.898587	+	0.363496i
$653$		−	18461.4i		−	1.10635i	−0.833064	−	0.553177i	$-0.813415\pi$
$653$		−	18461.4i		−	1.10635i	0.833064	−	0.553177i	$-0.186585\pi$
$654$		0			0
$655$	1600.00			0.0954461
$656$	−17260.7	−	16696.8i	−1.02731	−	0.993752i
$657$		0			0
$658$	−10560.0	+	7119.55i	−0.625641	+	0.421807i
$659$			5160.84i			0.305065i	0.988298	+	0.152532i	$0.0487428\pi$
$659$			5160.84i			0.305065i	−0.988298	+	0.152532i	$0.951257\pi$
$660$		0			0
$661$		−	13467.8i		−	0.792492i	−0.918144	−	0.396246i	$-0.870313\pi$
$661$		−	13467.8i		−	0.792492i	0.918144	−	0.396246i	$-0.129687\pi$
$662$	−5253.27	−	7791.85i	−0.308420	−	0.457461i
$663$		0			0
$664$	14840.0	−	3144.47i	0.867325	−	0.183779i
$665$	7504.67			0.437622
$666$		0			0
$667$		−	37021.7i		−	2.14915i
$668$	3151.96	−	7791.85i	0.182564	−	0.451311i
$669$		0			0
$670$	−10560.0	+	7119.55i	−0.608908	+	0.410526i
$671$	−20262.6			−1.16577
$672$		0			0
$673$	−15010.0			−0.859722			−0.429861	−	0.902895i	$-0.641437\pi$
$673$	−15010.0			−0.859722			−0.429861	+	0.902895i	$0.641437\pi$
$674$	−3353.65	+	2261.03i	−0.191658	+	0.129216i
$675$		0			0
$676$	−3969.00	+	9811.63i	−0.225819	+	0.558240i
$677$		−	6280.28i		−	0.356530i	−0.983983	−	0.178265i	$-0.942952\pi$
$677$		−	6280.28i		−	0.356530i	0.983983	−	0.178265i	$-0.0570485\pi$
$678$		0			0
$679$	1300.00			0.0734748
$680$	−10506.5	+	2226.24i	−0.592510	+	0.125548i
$681$		0			0
$682$	−3720.00	−	5517.65i	−0.208865	−	0.309797i
$683$			12510.0i			0.700850i	0.936591	+	0.350425i	$0.113963\pi$
$683$			12510.0i			0.700850i	−0.936591	+	0.350425i	$0.886037\pi$
$684$		0			0
$685$			4271.73i			0.238269i
$686$	−13742.9	+	9265.47i	−0.764879	+	0.515681i
$687$		0			0
$688$	−5280.00	+	5458.32i	−0.292584	+	0.302466i
$689$	−7879.90			−0.435704
$690$		0			0
$691$		−	28359.5i		−	1.56128i	−0.624978	−	0.780642i	$-0.714892\pi$
$691$		−	28359.5i		−	1.56128i	0.624978	−	0.780642i	$-0.285108\pi$
$692$	10553.4	+	4269.07i	0.579742	+	0.234517i
$693$		0			0
$694$	15980.0	+	23702.2i	0.874053	+	1.29643i
$695$	−1500.93			−0.0819189
$696$		0			0
$697$	28160.0			1.53032
$698$	1782.36	+	2643.66i	0.0966522	+	0.143358i
$699$		0			0
$700$	2550.00	−	6303.77i	0.137687	−	0.340372i
$701$			19802.2i			1.06693i	0.845822	+	0.533465i	$0.179110\pi$
$701$			19802.2i			1.06693i	−0.845822	+	0.533465i	$0.820890\pi$
$702$		0			0
$703$	−7040.00			−0.377694
$704$	7879.90	+	17759.4i	0.421853	+	0.950754i
$705$		0			0
$706$	3168.00	−	2135.87i	0.168880	−	0.113859i
$707$			5628.85i			0.299427i
$708$		0			0
$709$		−	12874.5i		−	0.681964i	−0.940070	−	0.340982i	$-0.889240\pi$
$709$		−	12874.5i		−	0.681964i	0.940070	−	0.340982i	$-0.110760\pi$
$710$		0			0
$711$		0			0
$712$	3520.00	+	16612.3i	0.185277	+	0.874398i
$713$	−9305.78			−0.488786
$714$		0			0
$715$		−	14239.1i		−	0.744772i
$716$	−938.083	−	379.473i	−0.0489634	−	0.0198067i
$717$		0			0
$718$	28160.0	−	18985.5i	1.46368	−	0.986813i
$719$	−1500.93			−0.0778517			−0.0389258	−	0.999242i	$-0.512394\pi$
$719$	−1500.93			−0.0778517			−0.0389258	+	0.999242i	$0.512394\pi$
$720$		0			0
$721$	−5700.00			−0.294423
$722$	−16934.7	+	11417.4i	−0.872917	+	0.588520i
$723$		0			0
$724$	−440.000	−	177.989i	−0.0225863	−	0.00913660i
$725$			20965.9i			1.07401i
$726$		0			0
$727$	23850.0			1.21671			0.608355	−	0.793665i	$-0.291830\pi$
$727$	23850.0			1.21671			0.608355	+	0.793665i	$0.291830\pi$
$728$	13133.2	−	2782.80i	0.668609	−	0.141673i
$729$		0			0
$730$	−300.000	−	444.972i	−0.0152103	−	0.0225605i
$731$		−	8904.97i		−	0.450564i
$732$		0			0
$733$			2195.19i			0.110616i	0.998469	+	0.0553079i	$0.0176140\pi$
$733$			2195.19i			0.110616i	−0.998469	+	0.0553079i	$0.982386\pi$
$734$	11890.2	−	8016.37i	0.597923	−	0.403120i
$735$		0			0
$736$	26752.0	+	4746.37i	1.33980	+	0.237708i
$737$	−27016.8			−1.35031
$738$		0			0
$739$		−	2610.50i		−	0.129944i	−0.997887	−	0.0649722i	$-0.979304\pi$
$739$		−	2610.50i		−	0.129944i	0.997887	−	0.0649722i	$-0.0206959\pi$
$740$	1125.70	−	2782.80i	0.0559210	−	0.138240i
$741$		0			0
$742$	2100.00	+	3114.80i	0.103899	+	0.154108i
$743$	2851.77			0.140809			0.0704047	−	0.997519i	$-0.477571\pi$
$743$	2851.77			0.140809			0.0704047	+	0.997519i	$0.477571\pi$
$744$		0			0
$745$	−18600.0			−0.914700
$746$	3846.14	+	5704.75i	0.188763	+	0.279981i
$747$		0			0
$748$	−21120.0	−	8543.46i	−1.03238	−	0.417620i
$749$			12902.1i			0.629416i
$750$		0			0
$751$	−20578.0			−0.999869			−0.499935	−	0.866063i	$-0.666643\pi$
$751$	−20578.0			−0.999869			−0.499935	+	0.866063i	$0.666643\pi$
$752$	20712.9	+	20036.2i	1.00442	+	0.971602i
$753$		0			0
$754$	−34320.0	+	23138.5i	−1.65764	+	1.11758i
$755$			14306.1i			0.689608i
$756$		0			0
$757$		−	20468.7i		−	0.982758i	−0.870946	−	0.491379i	$-0.836493\pi$
$757$		−	20468.7i		−	0.982758i	0.870946	−	0.491379i	$-0.163507\pi$
$758$	−12195.1	−	18088.2i	−0.584361	−	0.866747i
$759$		0			0
$760$	−3520.00	−	16612.3i	−0.168005	−	0.792883i
$761$	22138.8			1.05457			0.527286	−	0.849688i	$-0.323210\pi$
$761$	22138.8			1.05457			0.527286	+	0.849688i	$0.323210\pi$
$762$		0			0
$763$			17205.6i			0.816362i
$764$	−11257.0	+	27828.0i	−0.533068	+	1.31778i
$765$		0			0
$766$	13728.0	−	9255.42i	0.647536	−	0.436569i
$767$	43527.1			2.04911
$768$		0			0
$769$	−6854.00			−0.321406			−0.160703	−	0.987003i	$-0.551376\pi$
$769$	−6854.00			−0.321406			−0.160703	+	0.987003i	$0.551376\pi$
$770$	−5628.50	+	3794.73i	−0.263425	+	0.177601i
$771$		0			0
$772$	−4050.00	+	10011.9i	−0.188812	+	0.466755i
$773$		−	38902.3i		−	1.81012i	−0.425288	−	0.905058i	$-0.639827\pi$
$773$		−	38902.3i		−	1.81012i	0.425288	−	0.905058i	$-0.360173\pi$
$774$		0			0
$775$	5270.00			0.244263
$776$	−609.754	−	2877.67i	−0.0282073	−	0.133122i
$777$		0			0
$778$	−3410.00	−	5057.85i	−0.157139	−	0.233075i
$779$			44524.9i			2.04784i
$780$		0			0
$781$		0			0
$782$	−26416.4	+	17809.9i	−1.20799	+	0.814428i
$783$

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:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	72.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.34521 - 1.58114i) q^{2} +(3.00000 - 7.41620i) q^{4} -6.32456i q^{5} +10.0000 q^{7} +(-4.69042 - 22.1359i) q^{8} +O(q^{10})\)	\(q+(2.34521 - 1.58114i) q^{2} +(3.00000 - 7.41620i) q^{4} -6.32456i q^{5} +10.0000 q^{7} +(-4.69042 - 22.1359i) q^{8} +(-10.0000 - 14.8324i) q^{10} -37.9473i q^{11} +59.3296i q^{13} +(23.4521 - 15.8114i) q^{14} +(-46.0000 - 44.4972i) q^{16} +75.0467 q^{17} +118.659i q^{19} +(-46.9042 - 18.9737i) q^{20} +(-60.0000 - 88.9944i) q^{22} -150.093 q^{23} +85.0000 q^{25} +(93.8083 + 139.140i) q^{26} +(30.0000 - 74.1620i) q^{28} +246.658i q^{29} +62.0000 q^{31} +(-178.236 - 31.6228i) q^{32} +(176.000 - 118.659i) q^{34} -63.2456i q^{35} +59.3296i q^{37} +(187.617 + 278.280i) q^{38} +(-140.000 + 29.6648i) q^{40} +375.233 q^{41} -118.659i q^{43} +(-281.425 - 113.842i) q^{44} +(-352.000 + 237.318i) q^{46} -450.280 q^{47} -243.000 q^{49} +(199.343 - 134.397i) q^{50} +(440.000 + 177.989i) q^{52} +132.816i q^{53} -240.000 q^{55} +(-46.9042 - 221.359i) q^{56} +(390.000 + 578.463i) q^{58} -733.648i q^{59} -533.966i q^{61} +(145.403 - 98.0306i) q^{62} +(-468.000 + 207.654i) q^{64} +375.233 q^{65} -711.955i q^{67} +(225.140 - 556.561i) q^{68} +(-100.000 - 148.324i) q^{70} +30.0000 q^{73} +(93.8083 + 139.140i) q^{74} +(880.000 + 355.978i) q^{76} -379.473i q^{77} +94.0000 q^{79} +(-281.425 + 290.930i) q^{80} +(880.000 - 593.296i) q^{82} +670.403i q^{83} -474.637i q^{85} +(-187.617 - 278.280i) q^{86} +(-840.000 + 177.989i) q^{88} -750.467 q^{89} +593.296i q^{91} +(-450.280 + 1113.12i) q^{92} +(-1056.00 + 711.955i) q^{94} +750.467 q^{95} +130.000 q^{97} +(-569.886 + 384.217i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{4} + 40 q^{7}+O(q^{10}) \) 4 * q + 12 * q^4 + 40 * q^7	\( 4 q + 12 q^{4} + 40 q^{7} - 40 q^{10} - 184 q^{16} - 240 q^{22} + 340 q^{25} + 120 q^{28} + 248 q^{31} + 704 q^{34} - 560 q^{40} - 1408 q^{46} - 972 q^{49} + 1760 q^{52} - 960 q^{55} + 1560 q^{58} - 1872 q^{64} - 400 q^{70} + 120 q^{73} + 3520 q^{76} + 376 q^{79} + 3520 q^{82} - 3360 q^{88} - 4224 q^{94} + 520 q^{97}+O(q^{100}) \) 4 * q + 12 * q^4 + 40 * q^7 - 40 * q^10 - 184 * q^16 - 240 * q^22 + 340 * q^25 + 120 * q^28 + 248 * q^31 + 704 * q^34 - 560 * q^40 - 1408 * q^46 - 972 * q^49 + 1760 * q^52 - 960 * q^55 + 1560 * q^58 - 1872 * q^64 - 400 * q^70 + 120 * q^73 + 3520 * q^76 + 376 * q^79 + 3520 * q^82 - 3360 * q^88 - 4224 * q^94 + 520 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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