LMFDB - Embedded newform 112.4.p.b.31.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,4,Mod(31,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 1]))

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

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

chi := DirichletCharacter("112.31");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	112.p (of order $6$, degree $2$, 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:	$6.60821392064$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	112.31
Dual form			112.4.p.b.47.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{3} +(-4.50000 + 2.59808i) q^{5} +(14.0000 - 12.1244i) q^{7} +(13.0000 + 22.5167i) q^{9} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{3} +(-4.50000 + 2.59808i) q^{5} +(14.0000 - 12.1244i) q^{7} +(13.0000 + 22.5167i) q^{9} +(22.5000 + 12.9904i) q^{11} +69.2820i q^{13} -5.19615i q^{15} +(31.5000 + 18.1865i) q^{17} +(8.50000 + 14.7224i) q^{19} +(3.50000 + 18.1865i) q^{21} +(121.500 - 70.1481i) q^{23} +(-49.0000 + 84.8705i) q^{25} -53.0000 q^{27} +90.0000 q^{29} +(-8.50000 + 14.7224i) q^{31} +(-22.5000 + 12.9904i) q^{33} +(-31.5000 + 90.9327i) q^{35} +(-99.5000 - 172.339i) q^{37} +(-60.0000 - 34.6410i) q^{39} -187.061i q^{41} +252.879i q^{43} +(-117.000 - 67.5500i) q^{45} +(-283.500 - 491.036i) q^{47} +(49.0000 - 339.482i) q^{49} +(-31.5000 + 18.1865i) q^{51} +(166.500 - 288.386i) q^{53} -135.000 q^{55} -17.0000 q^{57} +(-400.500 + 693.686i) q^{59} +(-310.500 + 179.267i) q^{61} +(455.000 + 157.617i) q^{63} +(-180.000 - 311.769i) q^{65} +(-187.500 - 108.253i) q^{67} +140.296i q^{69} -488.438i q^{71} +(349.500 + 201.784i) q^{73} +(-49.0000 - 84.8705i) q^{75} +(472.500 - 90.9327i) q^{77} +(1033.50 - 596.692i) q^{79} +(-324.500 + 562.050i) q^{81} -468.000 q^{83} -189.000 q^{85} +(-45.0000 + 77.9423i) q^{87} +(-166.500 + 96.1288i) q^{89} +(840.000 + 969.948i) q^{91} +(-8.50000 - 14.7224i) q^{93} +(-76.5000 - 44.1673i) q^{95} +1392.57i q^{97} +675.500i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 9 q^{5} + 28 q^{7} + 26 q^{9}+O(q^{10}) $ 2 * q - q^3 - 9 * q^5 + 28 * q^7 + 26 * q^9	$ 2 q - q^{3} - 9 q^{5} + 28 q^{7} + 26 q^{9} + 45 q^{11} + 63 q^{17} + 17 q^{19} + 7 q^{21} + 243 q^{23} - 98 q^{25} - 106 q^{27} + 180 q^{29} - 17 q^{31} - 45 q^{33} - 63 q^{35} - 199 q^{37} - 120 q^{39} - 234 q^{45} - 567 q^{47} + 98 q^{49} - 63 q^{51} + 333 q^{53} - 270 q^{55} - 34 q^{57} - 801 q^{59} - 621 q^{61} + 910 q^{63} - 360 q^{65} - 375 q^{67} + 699 q^{73} - 98 q^{75} + 945 q^{77} + 2067 q^{79} - 649 q^{81} - 936 q^{83} - 378 q^{85} - 90 q^{87} - 333 q^{89} + 1680 q^{91} - 17 q^{93} - 153 q^{95}+O(q^{100}) $ 2 * q - q^3 - 9 * q^5 + 28 * q^7 + 26 * q^9 + 45 * q^11 + 63 * q^17 + 17 * q^19 + 7 * q^21 + 243 * q^23 - 98 * q^25 - 106 * q^27 + 180 * q^29 - 17 * q^31 - 45 * q^33 - 63 * q^35 - 199 * q^37 - 120 * q^39 - 234 * q^45 - 567 * q^47 + 98 * q^49 - 63 * q^51 + 333 * q^53 - 270 * q^55 - 34 * q^57 - 801 * q^59 - 621 * q^61 + 910 * q^63 - 360 * q^65 - 375 * q^67 + 699 * q^73 - 98 * q^75 + 945 * q^77 + 2067 * q^79 - 649 * q^81 - 936 * q^83 - 378 * q^85 - 90 * q^87 - 333 * q^89 + 1680 * q^91 - 17 * q^93 - 153 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$1$

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
$3$	−0.500000	+	0.866025i	−0.0962250	+	0.166667i	−0.910119	−	0.414346i	$-0.864010\pi$
$3$	−0.500000	+	0.866025i	−0.0962250	+	0.166667i	0.813894	+	0.581013i	$0.197344\pi$
$4$		0			0
$5$	−4.50000	+	2.59808i	−0.402492	+	0.232379i	−0.687559	−	0.726129i	$-0.741318\pi$
$5$	−4.50000	+	2.59808i	−0.402492	+	0.232379i	0.285067	+	0.958508i	$0.407984\pi$
$6$		0			0
$7$	14.0000	−	12.1244i	0.755929	−	0.654654i
$7$	14.0000	−	12.1244i	0.755929	−	0.654654i
$8$		0			0
$9$	13.0000	+	22.5167i	0.481481	+	0.833950i
$10$		0			0
$11$	22.5000	+	12.9904i	0.616728	+	0.356068i	0.775594	−	0.631232i	$-0.217451\pi$
$11$	22.5000	+	12.9904i	0.616728	+	0.356068i	−0.158866	+	0.987300i	$0.550784\pi$
$12$		0			0
$13$			69.2820i			1.47811i	0.673647	+	0.739053i	$0.264727\pi$
$13$			69.2820i			1.47811i	−0.673647	+	0.739053i	$0.735273\pi$
$14$		0			0
$15$		−	5.19615i		−	0.0894427i
$16$		0			0
$17$	31.5000	+	18.1865i	0.449404	+	0.259464i	0.707579	−	0.706635i	$-0.249788\pi$
$17$	31.5000	+	18.1865i	0.449404	+	0.259464i	−0.258174	+	0.966098i	$0.583121\pi$
$18$		0			0
$19$	8.50000	+	14.7224i	0.102633	+	0.177766i	0.912769	−	0.408477i	$-0.133940\pi$
$19$	8.50000	+	14.7224i	0.102633	+	0.177766i	−0.810135	+	0.586243i	$0.800606\pi$
$20$		0			0
$21$	3.50000	+	18.1865i	0.0363696	+	0.188982i
$22$		0			0
$23$	121.500	−	70.1481i	1.10150	−	0.635951i	0.164885	−	0.986313i	$-0.447275\pi$
$23$	121.500	−	70.1481i	1.10150	−	0.635951i	0.936615	+	0.350361i	$0.113941\pi$
$24$		0			0
$25$	−49.0000	+	84.8705i	−0.392000	+	0.678964i
$26$		0			0
$27$	−53.0000			−0.377772
$28$		0			0
$29$	90.0000			0.576296			0.288148	−	0.957586i	$-0.406961\pi$
$29$	90.0000			0.576296			0.288148	+	0.957586i	$0.406961\pi$
$30$		0			0
$31$	−8.50000	+	14.7224i	−0.0492466	+	0.0852976i	−0.889598	−	0.456745i	$-0.849015\pi$
$31$	−8.50000	+	14.7224i	−0.0492466	+	0.0852976i	0.840351	+	0.542042i	$0.182349\pi$
$32$		0			0
$33$	−22.5000	+	12.9904i	−0.118689	+	0.0685253i
$34$		0			0
$35$	−31.5000	+	90.9327i	−0.152128	+	0.439155i
$36$		0			0
$37$	−99.5000	−	172.339i	−0.442100	−	0.765740i	0.555745	−	0.831353i	$-0.312433\pi$
$37$	−99.5000	−	172.339i	−0.442100	−	0.765740i	−0.997845	+	0.0656131i	$0.979100\pi$
$38$		0			0
$39$	−60.0000	−	34.6410i	−0.246351	−	0.142231i
$40$		0			0
$41$		−	187.061i		−	0.712539i	−0.934383	−	0.356269i	$-0.884049\pi$
$41$		−	187.061i		−	0.712539i	0.934383	−	0.356269i	$-0.115951\pi$
$42$		0			0
$43$			252.879i			0.896831i	0.893825	+	0.448416i	$0.148012\pi$
$43$			252.879i			0.896831i	−0.893825	+	0.448416i	$0.851988\pi$
$44$		0			0
$45$	−117.000	−	67.5500i	−0.387585	−	0.223772i
$46$		0			0
$47$	−283.500	−	491.036i	−0.879845	−	1.52394i	−0.851510	−	0.524338i	$-0.824313\pi$
$47$	−283.500	−	491.036i	−0.879845	−	1.52394i	−0.0283351	−	0.999598i	$-0.509021\pi$
$48$		0			0
$49$	49.0000	−	339.482i	0.142857	−	0.989743i
$50$		0			0
$51$	−31.5000	+	18.1865i	−0.0864879	+	0.0499338i
$52$		0			0
$53$	166.500	−	288.386i	0.431520	−	0.747414i	−0.565485	−	0.824759i	$-0.691311\pi$
$53$	166.500	−	288.386i	0.431520	−	0.747414i	0.997004	+	0.0773449i	$0.0246443\pi$
$54$		0			0
$55$	−135.000			−0.330971
$56$		0			0
$57$	−17.0000			−0.0395036
$58$		0			0
$59$	−400.500	+	693.686i	−0.883740	+	1.53068i	−0.0365889	+	0.999330i	$0.511649\pi$
$59$	−400.500	+	693.686i	−0.883740	+	1.53068i	−0.847151	+	0.531352i	$0.821684\pi$
$60$		0			0
$61$	−310.500	+	179.267i	−0.651729	+	0.376276i	−0.789118	−	0.614241i	$-0.789462\pi$
$61$	−310.500	+	179.267i	−0.651729	+	0.376276i	0.137390	+	0.990517i	$0.456129\pi$
$62$		0			0
$63$	455.000	+	157.617i	0.909914	+	0.315204i
$64$		0			0
$65$	−180.000	−	311.769i	−0.343481	−	0.594926i
$66$		0			0
$67$	−187.500	−	108.253i	−0.341892	−	0.197391i	0.319216	−	0.947682i	$-0.396580\pi$
$67$	−187.500	−	108.253i	−0.341892	−	0.197391i	−0.661108	+	0.750290i	$0.729914\pi$
$68$		0			0
$69$			140.296i			0.244778i
$70$		0			0
$71$		−	488.438i		−	0.816436i	−0.912884	−	0.408218i	$-0.866150\pi$
$71$		−	488.438i		−	0.816436i	0.912884	−	0.408218i	$-0.133850\pi$
$72$		0			0
$73$	349.500	+	201.784i	0.560355	+	0.323521i	0.753288	−	0.657691i	$-0.228467\pi$
$73$	349.500	+	201.784i	0.560355	+	0.323521i	−0.192933	+	0.981212i	$0.561800\pi$
$74$		0			0
$75$	−49.0000	−	84.8705i	−0.0754404	−	0.130667i
$76$		0			0
$77$	472.500	−	90.9327i	0.699304	−	0.134581i
$78$		0			0
$79$	1033.50	−	596.692i	1.47187	−	0.849785i	0.472371	−	0.881400i	$-0.343398\pi$
$79$	1033.50	−	596.692i	1.47187	−	0.849785i	0.999500	+	0.0316144i	$0.0100649\pi$
$80$		0			0
$81$	−324.500	+	562.050i	−0.445130	+	0.770988i
$82$		0			0
$83$	−468.000			−0.618912			−0.309456	−	0.950914i	$-0.600147\pi$
$83$	−468.000			−0.618912			−0.309456	+	0.950914i	$0.600147\pi$
$84$		0			0
$85$	−189.000			−0.241176
$86$		0			0
$87$	−45.0000	+	77.9423i	−0.0554541	+	0.0960493i
$88$		0			0
$89$	−166.500	+	96.1288i	−0.198303	+	0.114490i	−0.595864	−	0.803086i	$-0.703190\pi$
$89$	−166.500	+	96.1288i	−0.198303	+	0.114490i	0.397561	+	0.917576i	$0.369857\pi$
$90$		0			0
$91$	840.000	+	969.948i	0.967648	+	1.11734i
$92$		0			0
$93$	−8.50000	−	14.7224i	−0.00947752	−	0.0164155i
$94$		0			0
$95$	−76.5000	−	44.1673i	−0.0826183	−	0.0476997i
$96$		0			0
$97$			1392.57i			1.45767i	0.684690	+	0.728835i	$0.259938\pi$
$97$			1392.57i			1.45767i	−0.684690	+	0.728835i	$0.740062\pi$
$98$		0			0
$99$			675.500i			0.685760i
$100$		0			0
$101$	1453.50	+	839.179i	1.43197	+	0.826746i	0.997271	−	0.0738307i	$-0.0235225\pi$
$101$	1453.50	+	839.179i	1.43197	+	0.826746i	0.434696	+	0.900577i	$0.356856\pi$
$102$		0			0
$103$	−665.500	−	1152.68i	−0.636638	−	1.10269i	−0.986166	−	0.165763i	$-0.946991\pi$
$103$	−665.500	−	1152.68i	−0.636638	−	1.10269i	0.349528	−	0.936926i	$-0.386342\pi$
$104$		0			0
$105$	−63.0000	−	72.7461i	−0.0585540	−	0.0676123i
$106$		0			0
$107$	1435.50	−	828.786i	1.29696	−	0.748802i	0.317084	−	0.948397i	$-0.397296\pi$
$107$	1435.50	−	828.786i	1.29696	−	0.748802i	0.979878	+	0.199595i	$0.0639628\pi$
$108$		0			0
$109$	−395.500	+	685.026i	−0.347542	+	0.601960i	−0.985812	−	0.167852i	$-0.946317\pi$
$109$	−395.500	+	685.026i	−0.347542	+	0.601960i	0.638271	+	0.769812i	$0.279650\pi$
$110$		0			0
$111$	199.000			0.170164
$112$		0			0
$113$	990.000			0.824171			0.412086	−	0.911145i	$-0.364800\pi$
$113$	990.000			0.824171			0.412086	+	0.911145i	$0.364800\pi$
$114$		0			0
$115$	−364.500	+	631.333i	−0.295563	+	0.511931i
$116$		0			0
$117$	−1560.00	+	900.666i	−1.23267	+	0.711681i
$118$		0			0
$119$	661.500	−	127.306i	0.509577	−	0.0980680i
$120$		0			0
$121$	−328.000	−	568.113i	−0.246431	−	0.426831i
$122$		0			0
$123$	162.000	+	93.5307i	0.118756	+	0.0685641i
$124$		0			0
$125$		−	1158.74i		−	0.829128i
$126$		0			0
$127$			1784.01i			1.24650i	0.782023	+	0.623250i	$0.214188\pi$
$127$			1784.01i			1.24650i	−0.782023	+	0.623250i	$0.785812\pi$
$128$		0			0
$129$	−219.000	−	126.440i	−0.149472	−	0.0862976i
$130$		0			0
$131$	−1309.50	−	2268.12i	−0.873371	−	1.51272i	−0.858488	−	0.512833i	$-0.828596\pi$
$131$	−1309.50	−	2268.12i	−0.873371	−	1.51272i	−0.0148822	−	0.999889i	$-0.504737\pi$
$132$		0			0
$133$	297.500	+	103.057i	0.193959	+	0.0671893i
$134$		0			0
$135$	238.500	−	137.698i	0.152050	−	0.0877864i
$136$		0			0
$137$	1030.50	−	1784.88i	0.642639	−	1.11308i	−0.342202	−	0.939626i	$-0.611173\pi$
$137$	1030.50	−	1784.88i	0.642639	−	1.11308i	0.984841	−	0.173457i	$-0.0554939\pi$
$138$		0			0
$139$	−1108.00			−0.676110			−0.338055	−	0.941126i	$-0.609769\pi$
$139$	−1108.00			−0.676110			−0.338055	+	0.941126i	$0.609769\pi$
$140$		0			0
$141$	567.000			0.338653
$142$		0			0
$143$	−900.000	+	1558.85i	−0.526306	+	0.911589i
$144$		0			0
$145$	−405.000	+	233.827i	−0.231955	+	0.133919i
$146$		0			0
$147$	269.500	+	212.176i	0.151211	+	0.119048i
$148$		0			0
$149$	−1381.50	−	2392.83i	−0.759576	−	1.31562i	−0.943067	−	0.332603i	$-0.892073\pi$
$149$	−1381.50	−	2392.83i	−0.759576	−	1.31562i	0.183490	−	0.983022i	$-0.441260\pi$
$150$		0			0
$151$	−715.500	−	413.094i	−0.385606	−	0.222630i	0.294648	−	0.955606i	$-0.404797\pi$
$151$	−715.500	−	413.094i	−0.385606	−	0.222630i	−0.680255	+	0.732976i	$0.738131\pi$
$152$		0			0
$153$			945.700i			0.499708i
$154$		0			0
$155$		−	88.3346i		−	0.0457755i
$156$		0			0
$157$	−760.500	−	439.075i	−0.386589	−	0.223197i	0.294092	−	0.955777i	$-0.404983\pi$
$157$	−760.500	−	439.075i	−0.386589	−	0.223197i	−0.680681	+	0.732580i	$0.738316\pi$
$158$		0			0
$159$	166.500	+	288.386i	0.0830460	+	0.143840i
$160$		0			0
$161$	850.500	−	2455.18i	0.416328	−	1.20184i
$162$		0			0
$163$	2101.50	−	1213.30i	1.00983	−	0.583025i	0.0986877	−	0.995118i	$-0.468536\pi$
$163$	2101.50	−	1213.30i	1.00983	−	0.583025i	0.911142	+	0.412093i	$0.135202\pi$
$164$		0			0
$165$	67.5000	−	116.913i	0.0318477	−	0.0551618i
$166$		0			0
$167$	−2700.00			−1.25109			−0.625546	−	0.780188i	$-0.715124\pi$
$167$	−2700.00			−1.25109			−0.625546	+	0.780188i	$0.715124\pi$
$168$		0			0
$169$	−2603.00			−1.18480
$170$		0			0
$171$	−221.000	+	382.783i	−0.0988321	+	0.171182i
$172$		0			0
$173$	1201.50	−	693.686i	0.528025	−	0.304855i	−0.212187	−	0.977229i	$-0.568059\pi$
$173$	1201.50	−	693.686i	0.528025	−	0.304855i	0.740212	+	0.672374i	$0.234725\pi$
$174$		0			0
$175$	343.000	+	1782.28i	0.148162	+	0.769873i
$176$		0			0
$177$	−400.500	−	693.686i	−0.170076	−	0.294580i
$178$		0			0
$179$	−2209.50	−	1275.66i	−0.922602	−	0.532665i	−0.0381379	−	0.999272i	$-0.512143\pi$
$179$	−2209.50	−	1275.66i	−0.922602	−	0.532665i	−0.884464	+	0.466608i	$0.845476\pi$
$180$		0			0
$181$		−	879.882i		−	0.361332i	−0.983545	−	0.180666i	$-0.942175\pi$
$181$		−	879.882i		−	0.361332i	0.983545	−	0.180666i	$-0.0578253\pi$
$182$		0			0
$183$		−	358.535i		−	0.144829i
$184$		0			0
$185$	895.500	+	517.017i	0.355884	+	0.205470i
$186$		0			0
$187$	472.500	+	818.394i	0.184773	+	0.320037i
$188$		0			0
$189$	−742.000	+	642.591i	−0.285569	+	0.247310i
$190$		0			0
$191$	−976.500	+	563.783i	−0.369932	+	0.213580i	−0.673429	−	0.739252i	$-0.735179\pi$
$191$	−976.500	+	563.783i	−0.369932	+	0.213580i	0.303497	+	0.952833i	$0.401846\pi$
$192$		0			0
$193$	−657.500	+	1138.82i	−0.245222	+	0.424737i	−0.962194	−	0.272365i	$-0.912194\pi$
$193$	−657.500	+	1138.82i	−0.245222	+	0.424737i	0.716972	+	0.697102i	$0.245528\pi$
$194$		0			0
$195$	360.000			0.132206
$196$		0			0
$197$	3258.00			1.17829			0.589144	−	0.808028i	$-0.299465\pi$
$197$	3258.00			1.17829			0.589144	+	0.808028i	$0.299465\pi$
$198$		0			0
$199$	−876.500	+	1518.14i	−0.312228	+	0.540795i	−0.978844	−	0.204606i	$-0.934409\pi$
$199$	−876.500	+	1518.14i	−0.312228	+	0.540795i	0.666616	+	0.745401i	$0.267742\pi$
$200$		0			0
$201$	187.500	−	108.253i	0.0657972	−	0.0379880i
$202$		0			0
$203$	1260.00	−	1091.19i	0.435639	−	0.377274i
$204$		0			0
$205$	486.000	+	841.777i	0.165579	+	0.286791i
$206$		0			0
$207$	3159.00	+	1823.85i	1.06070	+	0.612398i
$208$		0			0
$209$			441.673i			0.146178i
$210$		0			0
$211$		−	3543.78i		−	1.15623i	−0.815957	−	0.578113i	$-0.803789\pi$
$211$		−	3543.78i		−	1.15623i	0.815957	−	0.578113i	$-0.196211\pi$
$212$		0			0
$213$	423.000	+	244.219i	0.136073	+	0.0785616i
$214$		0			0
$215$	−657.000	−	1137.96i	−0.208405	−	0.360968i
$216$		0			0
$217$	59.5000	+	309.171i	0.0186135	+	0.0967184i
$218$		0			0
$219$	−349.500	+	201.784i	−0.107840	+	0.0622616i
$220$		0			0
$221$	−1260.00	+	2182.38i	−0.383515	+	0.664267i
$222$		0			0
$223$	2896.00			0.869644			0.434822	−	0.900517i	$-0.356811\pi$
$223$	2896.00			0.869644			0.434822	+	0.900517i	$0.356811\pi$
$224$		0			0
$225$	−2548.00			−0.754963
$226$		0			0
$227$	1849.50	−	3203.43i	0.540774	−	0.936647i	−0.458086	−	0.888908i	$-0.651465\pi$
$227$	1849.50	−	3203.43i	0.540774	−	0.936647i	0.998860	−	0.0477397i	$-0.0152018\pi$
$228$		0			0
$229$	−3304.50	+	1907.85i	−0.953570	+	0.550544i	−0.894188	−	0.447692i	$-0.852246\pi$
$229$	−3304.50	+	1907.85i	−0.953570	+	0.550544i	−0.0593818	+	0.998235i	$0.518913\pi$
$230$		0			0
$231$	−157.500	+	454.663i	−0.0448603	+	0.129501i
$232$		0			0
$233$	2830.50	+	4902.57i	0.795846	+	1.37845i	0.922300	+	0.386474i	$0.126307\pi$
$233$	2830.50	+	4902.57i	0.795846	+	1.37845i	−0.126454	+	0.991972i	$0.540360\pi$
$234$		0			0
$235$	2551.50	+	1473.11i	0.708262	+	0.408915i
$236$		0			0
$237$			1193.38i			0.327083i
$238$		0			0
$239$			3481.42i			0.942236i	0.882070	+	0.471118i	$0.156149\pi$
$239$			3481.42i			0.942236i	−0.882070	+	0.471118i	$0.843851\pi$
$240$		0			0
$241$	2101.50	+	1213.30i	0.561699	+	0.324297i	0.753827	−	0.657073i	$-0.228206\pi$
$241$	2101.50	+	1213.30i	0.561699	+	0.324297i	−0.192128	+	0.981370i	$0.561539\pi$
$242$		0			0
$243$	−1040.00	−	1801.33i	−0.274552	−	0.475537i
$244$		0			0
$245$	661.500	+	1654.97i	0.172497	+	0.431561i
$246$		0			0
$247$	−1020.00	+	588.897i	−0.262757	+	0.151703i
$248$		0			0
$249$	234.000	−	405.300i	0.0595548	−	0.103152i
$250$		0			0
$251$	6120.00			1.53901			0.769504	−	0.638642i	$-0.220504\pi$
$251$	6120.00			1.53901			0.769504	+	0.638642i	$0.220504\pi$
$252$		0			0
$253$	3645.00			0.905768
$254$		0			0
$255$	94.5000	−	163.679i	0.0232071	−	0.0401959i
$256$		0			0
$257$	3091.50	−	1784.88i	0.750360	−	0.433220i	−0.0754641	−	0.997149i	$-0.524044\pi$
$257$	3091.50	−	1784.88i	0.750360	−	0.433220i	0.825824	+	0.563928i	$0.190710\pi$
$258$		0			0
$259$	−3482.50	−	1206.37i	−0.835490	−	0.289422i
$260$		0			0
$261$	1170.00	+	2026.50i	0.277476	+	0.480602i
$262$		0			0
$263$	−3433.50	−	1982.33i	−0.805014	−	0.464775i	0.0402074	−	0.999191i	$-0.487198\pi$
$263$	−3433.50	−	1982.33i	−0.805014	−	0.464775i	−0.845221	+	0.534416i	$0.820531\pi$
$264$		0			0
$265$			1730.32i			0.401104i
$266$		0			0
$267$		−	192.258i		−	0.0440673i
$268$		0			0
$269$	−5332.50	−	3078.72i	−1.20866	−	0.697817i	−0.246190	−	0.969222i	$-0.579179\pi$
$269$	−5332.50	−	3078.72i	−1.20866	−	0.697817i	−0.962465	+	0.271404i	$0.912512\pi$
$270$		0			0
$271$	2874.50	+	4978.78i	0.644330	+	1.11601i	0.984456	+	0.175632i	$0.0561970\pi$
$271$	2874.50	+	4978.78i	0.644330	+	1.11601i	−0.340126	+	0.940380i	$0.610470\pi$
$272$		0			0
$273$	−1260.00	+	242.487i	−0.279336	+	0.0537582i
$274$		0			0
$275$	−2205.00	+	1273.06i	−0.483515	+	0.279157i
$276$		0			0
$277$	98.5000	−	170.607i	0.0213657	−	0.0370064i	−0.855145	−	0.518389i	$-0.826532\pi$
$277$	98.5000	−	170.607i	0.0213657	−	0.0370064i	0.876511	+	0.481383i	$0.159865\pi$
$278$		0			0
$279$	−442.000			−0.0948453
$280$		0			0
$281$	−3618.00			−0.768085			−0.384042	−	0.923315i	$-0.625468\pi$
$281$	−3618.00			−0.768085			−0.384042	+	0.923315i	$0.625468\pi$
$282$		0			0
$283$	1781.50	−	3085.65i	0.374202	−	0.648137i	−0.616005	−	0.787742i	$-0.711250\pi$
$283$	1781.50	−	3085.65i	0.374202	−	0.648137i	0.990207	+	0.139605i	$0.0445833\pi$
$284$		0			0
$285$	76.5000	−	44.1673i	0.0158999	−	0.00917981i
$286$		0			0
$287$	−2268.00	−	2618.86i	−0.466466	−	0.538629i
$288$		0			0
$289$	−1795.00	−	3109.03i	−0.365357	−	0.632817i
$290$		0			0
$291$	−1206.00	−	696.284i	−0.242945	−	0.140264i
$292$		0			0
$293$		−	6048.32i		−	1.20596i	−0.797756	−	0.602981i	$-0.793980\pi$
$293$		−	6048.32i		−	1.20596i	0.797756	−	0.602981i	$-0.206020\pi$
$294$		0			0
$295$		−	4162.12i		−	0.821450i
$296$		0			0
$297$	−1192.50	−	688.490i	−0.232983	−	0.134513i
$298$		0			0
$299$	4860.00	+	8417.77i	0.940004	+	1.62813i
$300$		0			0
$301$	3066.00	+	3540.31i	0.587114	+	0.677941i
$302$		0			0
$303$	−1453.50	+	839.179i	−0.275582	+	0.159107i
$304$		0			0
$305$	931.500	−	1613.41i	0.174877	−	0.302896i
$306$		0			0
$307$	−7316.00			−1.36009			−0.680043	−	0.733173i	$-0.738039\pi$
$307$	−7316.00			−1.36009			−0.680043	+	0.733173i	$0.738039\pi$
$308$		0			0
$309$	1331.00			0.245042
$310$		0			0
$311$	1273.50	−	2205.77i	0.232198	−	0.402179i	−0.726257	−	0.687424i	$-0.758742\pi$
$311$	1273.50	−	2205.77i	0.232198	−	0.402179i	0.958455	+	0.285245i	$0.0920750\pi$
$312$		0			0
$313$	7957.50	−	4594.26i	1.43701	−	0.829659i	0.439370	−	0.898306i	$-0.355202\pi$
$313$	7957.50	−	4594.26i	1.43701	−	0.829659i	0.997641	+	0.0686473i	$0.0218683\pi$
$314$		0			0
$315$	−2457.00	+	472.850i	−0.439480	+	0.0845780i
$316$		0			0
$317$	−247.500	−	428.683i	−0.0438517	−	0.0759534i	0.843266	−	0.537496i	$-0.180630\pi$
$317$	−247.500	−	428.683i	−0.0438517	−	0.0759534i	−0.887118	+	0.461542i	$0.847296\pi$
$318$		0			0
$319$	2025.00	+	1169.13i	0.355418	+	0.205200i
$320$		0			0
$321$			1657.57i			0.288214i
$322$		0			0
$323$			618.342i			0.106519i
$324$		0			0
$325$	−5880.00	−	3394.82i	−1.00358	−	0.579418i
$326$		0			0
$327$	−395.500	−	685.026i	−0.0668844	−	0.115847i
$328$		0			0
$329$	−9922.50	−	3437.25i	−1.66275	−	0.575994i
$330$		0			0
$331$	−7954.50	+	4592.53i	−1.32090	+	0.762624i	−0.983873	−	0.178870i	$-0.942756\pi$
$331$	−7954.50	+	4592.53i	−1.32090	+	0.762624i	−0.337030	+	0.941494i	$0.609422\pi$
$332$		0			0
$333$	2587.00	−	4480.82i	0.425726	−	0.737379i
$334$		0			0
$335$	1125.00			0.183479
$336$		0			0
$337$	−5194.00			−0.839570			−0.419785	−	0.907623i	$-0.637895\pi$
$337$	−5194.00			−0.839570			−0.419785	+	0.907623i	$0.637895\pi$
$338$		0			0
$339$	−495.000	+	857.365i	−0.0793059	+	0.137362i
$340$		0			0
$341$	−382.500	+	220.836i	−0.0607435	+	0.0350703i
$342$		0			0
$343$	−3430.00	−	5346.84i	−0.539949	−	0.841698i
$344$		0			0
$345$	−364.500	−	631.333i	−0.0568812	−	0.0985212i
$346$		0			0
$347$	3892.50	+	2247.34i	0.602191	+	0.347675i	0.769903	−	0.638161i	$-0.220305\pi$
$347$	3892.50	+	2247.34i	0.602191	+	0.347675i	−0.167712	+	0.985836i	$0.553638\pi$
$348$		0			0
$349$		−	9436.21i		−	1.44730i	−0.690165	−	0.723652i	$-0.742462\pi$
$349$		−	9436.21i		−	1.44730i	0.690165	−	0.723652i	$-0.257538\pi$
$350$		0			0
$351$		−	3671.95i		−	0.558388i
$352$		0			0
$353$	5395.50	+	3115.09i	0.813523	+	0.469688i	0.848178	−	0.529712i	$-0.177700\pi$
$353$	5395.50	+	3115.09i	0.813523	+	0.469688i	−0.0346551	+	0.999399i	$0.511033\pi$
$354$		0			0
$355$	1269.00	+	2197.97i	0.189723	+	0.328609i
$356$		0			0
$357$	−220.500	+	636.529i	−0.0326893	+	0.0943660i
$358$		0			0
$359$	−5836.50	+	3369.70i	−0.858046	+	0.495393i	−0.863358	−	0.504593i	$-0.831643\pi$
$359$	−5836.50	+	3369.70i	−0.858046	+	0.495393i	0.00531114	+	0.999986i	$0.498309\pi$
$360$		0			0
$361$	3285.00	−	5689.79i	0.478933	−	0.829536i
$362$		0			0
$363$	656.000			0.0948514
$364$		0			0
$365$	−2097.00			−0.300718
$366$		0			0
$367$	6245.50	−	10817.5i	0.888317	−	1.53861i	0.0464536	−	0.998920i	$-0.485208\pi$
$367$	6245.50	−	10817.5i	0.888317	−	1.53861i	0.841864	−	0.539690i	$-0.181459\pi$
$368$		0			0
$369$	4212.00	−	2431.80i	0.594222	−	0.343074i
$370$		0			0
$371$	−1165.50	−	6056.12i	−0.163099	−	0.847487i
$372$		0			0
$373$	3888.50	+	6735.08i	0.539783	+	0.934931i	0.998915	+	0.0465632i	$0.0148269\pi$
$373$	3888.50	+	6735.08i	0.539783	+	0.934931i	−0.459133	+	0.888368i	$0.651840\pi$
$374$		0			0
$375$	1003.50	+	579.371i	0.138188	+	0.0797829i
$376$		0			0
$377$			6235.38i			0.851826i
$378$		0			0
$379$			6231.92i			0.844623i	0.906451	+	0.422312i	$0.138781\pi$
$379$			6231.92i			0.844623i	−0.906451	+	0.422312i	$0.861219\pi$
$380$		0			0
$381$	−1545.00	−	892.006i	−0.207750	−	0.119945i
$382$		0			0
$383$	3640.50	+	6305.53i	0.485694	+	0.841247i	0.999865	−	0.0164409i	$-0.00523355\pi$
$383$	3640.50	+	6305.53i	0.485694	+	0.841247i	−0.514171	+	0.857688i	$0.671900\pi$
$384$		0			0
$385$	−1890.00	+	1636.79i	−0.250190	+	0.216671i
$386$		0			0
$387$	−5694.00	+	3287.43i	−0.747913	+	0.431808i
$388$		0			0
$389$	−5791.50	+	10031.2i	−0.754860	+	1.30746i	0.190583	+	0.981671i	$0.438962\pi$
$389$	−5791.50	+	10031.2i	−0.754860	+	1.30746i	−0.945444	+	0.325786i	$0.894371\pi$
$390$		0			0
$391$	5103.00			0.660025
$392$		0			0
$393$	2619.00			0.336160
$394$		0			0
$395$	−3100.50	+	5370.22i	−0.394945	+	0.684064i
$396$		0			0
$397$	−2260.50	+	1305.10i	−0.285771	+	0.164990i	−0.636033	−	0.771662i	$-0.719426\pi$
$397$	−2260.50	+	1305.10i	−0.285771	+	0.164990i	0.350262	+	0.936652i	$0.386093\pi$
$398$		0			0
$399$	−238.000	+	206.114i	−0.0298619	+	0.0258612i
$400$		0			0
$401$	3802.50	+	6586.12i	0.473536	+	0.820188i	0.999541	−	0.0302934i	$-0.00964418\pi$
$401$	3802.50	+	6586.12i	0.473536	+	0.820188i	−0.526005	+	0.850481i	$0.676311\pi$
$402$		0			0
$403$	−1020.00	−	588.897i	−0.126079	−	0.0727917i
$404$		0			0
$405$		−	3372.30i		−	0.413756i
$406$		0			0
$407$		−	5170.17i		−	0.629671i
$408$		0			0
$409$	−4228.50	−	2441.33i	−0.511212	−	0.295149i	0.222119	−	0.975019i	$-0.428702\pi$
$409$	−4228.50	−	2441.33i	−0.511212	−	0.295149i	−0.733332	+	0.679871i	$0.762036\pi$
$410$		0			0
$411$	1030.50	+	1784.88i	0.123676	+	0.214213i
$412$		0			0
$413$	2803.50	+	14567.4i	0.334022	+	1.73563i
$414$		0			0
$415$	2106.00	−	1215.90i	0.249107	−	0.143822i
$416$		0			0
$417$	554.000	−	959.556i	0.0650587	−	0.112685i
$418$		0			0
$419$	12780.0			1.49008			0.745040	−	0.667019i	$-0.232430\pi$
$419$	12780.0			1.49008			0.745040	+	0.667019i	$0.232430\pi$
$420$		0			0
$421$	9802.00			1.13473			0.567364	−	0.823467i	$-0.307963\pi$
$421$	9802.00			1.13473			0.567364	+	0.823467i	$0.307963\pi$
$422$		0			0
$423$	7371.00	−	12766.9i	0.847258	−	1.46749i
$424$		0			0
$425$	−3087.00	+	1782.28i	−0.352333	+	0.203420i
$426$		0			0
$427$	−2173.50	+	6274.35i	−0.246330	+	0.711094i
$428$		0			0
$429$	−900.000	−	1558.85i	−0.101288	−	0.175435i
$430$		0			0
$431$	−2083.50	−	1202.91i	−0.232851	−	0.134436i	0.379036	−	0.925382i	$-0.376256\pi$
$431$	−2083.50	−	1202.91i	−0.232851	−	0.134436i	−0.611886	+	0.790946i	$0.709589\pi$
$432$		0			0
$433$			8390.05i			0.931178i	0.885001	+	0.465589i	$0.154158\pi$
$433$			8390.05i			0.931178i	−0.885001	+	0.465589i	$0.845842\pi$
$434$		0			0
$435$		−	467.654i		−	0.0515455i
$436$		0			0
$437$	2065.50	+	1192.52i	0.226101	+	0.130540i
$438$		0			0
$439$	752.500	+	1303.37i	0.0818106	+	0.141700i	0.904028	−	0.427474i	$-0.140596\pi$
$439$	752.500	+	1303.37i	0.0818106	+	0.141700i	−0.822217	+	0.569174i	$0.807263\pi$
$440$		0			0
$441$	8281.00	−	3309.95i	0.894180	−	0.357407i
$442$		0			0
$443$	5611.50	−	3239.80i	0.601829	−	0.347466i	−0.167932	−	0.985799i	$-0.553709\pi$
$443$	5611.50	−	3239.80i	0.601829	−	0.347466i	0.769761	+	0.638332i	$0.220375\pi$
$444$		0			0
$445$	499.500	−	865.159i	0.0532103	−	0.0921629i
$446$		0			0
$447$	2763.00			0.292361
$448$		0			0
$449$	−18090.0			−1.90138			−0.950690	−	0.310142i	$-0.899623\pi$
$449$	−18090.0			−1.90138			−0.950690	+	0.310142i	$0.899623\pi$
$450$		0			0
$451$	2430.00	−	4208.88i	0.253712	−	0.439443i
$452$		0			0
$453$	715.500	−	413.094i	0.0742100	−	0.0428452i
$454$		0			0
$455$	−6300.00	−	2182.38i	−0.649118	−	0.224861i
$456$		0			0
$457$	5868.50	+	10164.5i	0.600693	+	1.04043i	0.992716	+	0.120476i	$0.0384421\pi$
$457$	5868.50	+	10164.5i	0.600693	+	1.04043i	−0.392023	+	0.919955i	$0.628225\pi$
$458$		0			0
$459$	−1669.50	−	963.886i	−0.169773	−	0.0980182i
$460$		0			0
$461$		−	956.092i		−	0.0965936i	−0.998833	−	0.0482968i	$-0.984621\pi$
$461$		−	956.092i		−	0.0965936i	0.998833	−	0.0482968i	$-0.0153793\pi$
$462$		0			0
$463$		−	4922.49i		−	0.494098i	−0.969003	−	0.247049i	$-0.920539\pi$
$463$		−	4922.49i		−	0.494098i	0.969003	−	0.247049i	$-0.0794609\pi$
$464$		0			0
$465$	76.5000	+	44.1673i	0.00762925	+	0.00440475i
$466$		0			0
$467$	4198.50	+	7272.02i	0.416024	+	0.720575i	0.995535	−	0.0943890i	$-0.0300897\pi$
$467$	4198.50	+	7272.02i	0.416024	+	0.720575i	−0.579511	+	0.814964i	$0.696756\pi$
$468$		0			0
$469$	−3937.50	+	757.772i	−0.387669	+	0.0746070i
$470$		0			0
$471$	760.500	−	439.075i	0.0743991	−	0.0429544i
$472$		0			0
$473$	−3285.00	+	5689.79i	−0.319333	+	0.553101i
$474$		0			0
$475$	−1666.00			−0.160929
$476$		0			0
$477$	8658.00			0.831075
$478$		0			0
$479$	−7870.50	+	13632.1i	−0.750756	+	1.30035i	0.196700	+	0.980464i	$0.436977\pi$
$479$	−7870.50	+	13632.1i	−0.750756	+	1.30035i	−0.947457	+	0.319885i	$0.896356\pi$
$480$		0			0
$481$	11940.0	−	6893.56i	1.13184	−	0.653471i
$482$		0			0
$483$	1701.00	+	1964.15i	0.160245	+	0.185035i
$484$		0			0
$485$	−3618.00	−	6266.56i	−0.338732	−	0.586701i
$486$		0			0
$487$	934.500	+	539.534i	0.0869533	+	0.0502025i	0.542846	−	0.839832i	$-0.317347\pi$
$487$	934.500	+	539.534i	0.0869533	+	0.0502025i	−0.455893	+	0.890035i	$0.650680\pi$
$488$		0			0
$489$			2426.60i			0.224407i
$490$		0			0
$491$		−	1278.25i		−	0.117488i	−0.998273	−	0.0587442i	$-0.981290\pi$
$491$		−	1278.25i		−	0.117488i	0.998273	−	0.0587442i	$-0.0187096\pi$
$492$		0			0
$493$	2835.00	+	1636.79i	0.258990	+	0.149528i
$494$		0			0
$495$	−1755.00	−	3039.75i	−0.159356	−	0.276013i
$496$		0			0
$497$	−5922.00	−	6838.14i	−0.534483	−	0.617168i
$498$		0			0
$499$	−10450.5	+	6033.60i	−0.937532	+	0.541285i	−0.889186	−	0.457546i	$-0.848728\pi$
$499$	−10450.5	+	6033.60i	−0.937532	+	0.541285i	−0.0483464	+	0.998831i	$0.515395\pi$
$500$		0			0
$501$	1350.00	−	2338.27i	0.120386	−	0.208515i
$502$		0			0
$503$	2808.00			0.248912			0.124456	−	0.992225i	$-0.460282\pi$
$503$	2808.00			0.248912			0.124456	+	0.992225i	$0.460282\pi$
$504$		0			0
$505$	−8721.00			−0.768474
$506$		0			0
$507$	1301.50	−	2254.26i	0.114007	−	0.197466i
$508$		0			0
$509$	−11236.5	+	6487.40i	−0.978485	+	0.564929i	−0.901813	−	0.432127i	$-0.857763\pi$
$509$	−11236.5	+	6487.40i	−0.978485	+	0.564929i	−0.0766729	+	0.997056i	$0.524430\pi$
$510$		0			0
$511$	7339.50	−	1412.49i	0.635382	−	0.122279i
$512$		0			0
$513$	−450.500	−	780.289i	−0.0387720	−	0.0671552i
$514$		0			0
$515$	5989.50	+	3458.04i	0.512483	+	0.295882i
$516$		0			0
$517$		−	14731.1i		−	1.25314i
$518$		0			0
$519$			1387.37i			0.117339i
$520$		0			0
$521$	373.500	+	215.640i	0.0314075	+	0.0181332i	0.515622	−	0.856816i	$-0.327561\pi$
$521$	373.500	+	215.640i	0.0314075	+	0.0181332i	−0.484214	+	0.874950i	$0.660894\pi$
$522$		0			0
$523$	−9395.50	−	16273.5i	−0.785538	−	1.36059i	−0.928677	−	0.370890i	$-0.879053\pi$
$523$	−9395.50	−	16273.5i	−0.785538	−	1.36059i	0.143139	−	0.989703i	$-0.454281\pi$
$524$		0			0
$525$	−1715.00	−	594.093i	−0.142569	−	0.0493874i
$526$		0			0
$527$	−535.500	+	309.171i	−0.0442633	+	0.0255554i
$528$		0			0
$529$	3758.00	−	6509.05i	0.308868	−	0.534976i
$530$		0			0
$531$	−20826.0			−1.70202
$532$		0			0
$533$	12960.0			1.05321
$534$		0			0
$535$	−4306.50	+	7459.08i	−0.348012	+	0.602774i
$536$		0			0
$537$	2209.50	−	1275.66i	0.177555	−	0.102511i
$538$		0			0
$539$	5512.50	−	7001.82i	0.440520	−	0.559535i
$540$		0			0
$541$	−6273.50	−	10866.0i	−0.498556	−	0.863524i	0.501443	−	0.865191i	$-0.332803\pi$
$541$	−6273.50	−	10866.0i	−0.498556	−	0.863524i	−0.999999	+	0.00166651i	$0.999470\pi$
$542$		0			0
$543$	762.000	+	439.941i	0.0602220	+	0.0347692i
$544$		0			0
$545$		−	4110.16i		−	0.323045i
$546$		0			0
$547$		−	2151.21i		−	0.168152i	−0.996459	−	0.0840758i	$-0.973206\pi$
$547$		−	2151.21i		−	0.168152i	0.996459	−	0.0840758i	$-0.0267938\pi$
$548$		0			0
$549$	−8073.00	−	4660.95i	−0.627591	−	0.362340i
$550$		0			0
$551$	765.000	+	1325.02i	0.0591472	+	0.102446i
$552$		0			0
$553$	7234.50	−	20884.2i	0.556315	−	1.60594i
$554$		0			0
$555$	−895.500	+	517.017i	−0.0684898	+	0.0395426i
$556$		0			0
$557$	−1759.50	+	3047.54i	−0.133846	+	0.231829i	−0.925156	−	0.379587i	$-0.876066\pi$
$557$	−1759.50	+	3047.54i	−0.133846	+	0.231829i	0.791310	+	0.611415i	$0.209400\pi$
$558$		0			0
$559$	−17520.0			−1.32561
$560$		0			0
$561$	−945.000			−0.0711193
$562$		0			0
$563$	−4288.50	+	7427.90i	−0.321028	+	0.556037i	−0.980700	−	0.195517i	$-0.937362\pi$
$563$	−4288.50	+	7427.90i	−0.321028	+	0.556037i	0.659672	+	0.751553i	$0.270695\pi$
$564$		0			0
$565$	−4455.00	+	2572.10i	−0.331723	+	0.191520i
$566$		0			0
$567$	2271.50	+	11803.1i	0.168243	+	0.874219i
$568$		0			0
$569$	−5431.50	−	9407.63i	−0.400176	−	0.693126i	0.593571	−	0.804782i	$-0.297718\pi$
$569$	−5431.50	−	9407.63i	−0.400176	−	0.693126i	−0.993747	+	0.111656i	$0.964384\pi$
$570$		0			0
$571$	−1261.50	−	728.327i	−0.0924556	−	0.0533792i	0.453059	−	0.891480i	$-0.350333\pi$
$571$	−1261.50	−	728.327i	−0.0924556	−	0.0533792i	−0.545515	+	0.838101i	$0.683666\pi$
$572$		0			0
$573$		−	1127.57i		−	0.0822072i
$574$		0			0
$575$			13749.0i			0.997172i
$576$		0			0
$577$	565.500	+	326.492i	0.0408008	+	0.0235564i	0.520262	−	0.854007i	$-0.325834\pi$
$577$	565.500	+	326.492i	0.0408008	+	0.0235564i	−0.479461	+	0.877563i	$0.659168\pi$
$578$		0			0
$579$	−657.500	−	1138.82i	−0.0471930	−	0.0817407i
$580$		0			0
$581$	−6552.00	+	5674.20i	−0.467853	+	0.405173i
$582$		0			0
$583$	7492.50	−	4325.80i	0.532260	−	0.307301i
$584$		0			0
$585$	4680.00	−	8106.00i	0.330759	−	0.572892i
$586$		0			0
$587$	9684.00			0.680922			0.340461	−	0.940259i	$-0.389417\pi$
$587$	9684.00			0.680922			0.340461	+	0.940259i	$0.389417\pi$
$588$		0			0
$589$	−289.000			−0.0202174
$590$		0			0
$591$	−1629.00	+	2821.51i	−0.113381	+	0.196381i
$592$		0			0
$593$	6187.50	−	3572.35i	0.428483	−	0.247385i	−0.270217	−	0.962799i	$-0.587096\pi$
$593$	6187.50	−	3572.35i	0.428483	−	0.247385i	0.698700	+	0.715415i	$0.253762\pi$
$594$		0			0
$595$	−2646.00	+	2291.50i	−0.182312	+	0.157887i
$596$		0			0
$597$	−876.500	−	1518.14i	−0.0600884	−	0.104076i
$598$		0			0
$599$	−14773.5	−	8529.48i	−1.00773	−	0.581812i	−0.0972021	−	0.995265i	$-0.530989\pi$
$599$	−14773.5	−	8529.48i	−1.00773	−	0.581812i	−0.910526	+	0.413453i	$0.864323\pi$
$600$		0			0
$601$			26070.8i			1.76947i	0.466095	+	0.884735i	$0.345661\pi$
$601$			26070.8i			1.76947i	−0.466095	+	0.884735i	$0.654339\pi$
$602$		0			0
$603$		−	5629.17i		−	0.380161i
$604$		0			0
$605$	2952.00	+	1704.34i	0.198373	+	0.114531i
$606$		0			0
$607$	−6147.50	−	10647.8i	−0.411070	−	0.711994i	0.583937	−	0.811799i	$-0.301511\pi$
$607$	−6147.50	−	10647.8i	−0.411070	−	0.711994i	−0.995007	+	0.0998051i	$0.968178\pi$
$608$		0			0
$609$	315.000	+	1636.79i	0.0209597	+	0.108910i
$610$		0			0
$611$	34020.0	−	19641.5i	2.25254	−	1.30050i
$612$		0			0
$613$	−527.500	+	913.657i	−0.0347562	+	0.0601994i	−0.882880	−	0.469598i	$-0.844399\pi$
$613$	−527.500	+	913.657i	−0.0347562	+	0.0601994i	0.848124	+	0.529798i	$0.177732\pi$
$614$		0			0
$615$	−972.000			−0.0637314
$616$		0			0
$617$	1494.00			0.0974816			0.0487408	−	0.998811i	$-0.484479\pi$
$617$	1494.00			0.0974816			0.0487408	+	0.998811i	$0.484479\pi$
$618$		0			0
$619$	2799.50	−	4848.88i	0.181779	−	0.314851i	−0.760707	−	0.649095i	$-0.775148\pi$
$619$	2799.50	−	4848.88i	0.181779	−	0.314851i	0.942487	+	0.334244i	$0.108481\pi$
$620$		0			0
$621$	−6439.50	+	3717.85i	−0.416116	+	0.240245i
$622$		0			0
$623$	−1165.50	+	3364.51i	−0.0749515	+	0.216366i
$624$		0			0
$625$	−3114.50	−	5394.47i	−0.199328	−	0.345246i
$626$		0			0
$627$	−382.500	−	220.836i	−0.0243630	−	0.0140660i
$628$		0			0
$629$		−	7238.24i		−	0.458836i
$630$		0			0
$631$			19007.5i			1.19917i	0.800310	+	0.599586i	$0.204668\pi$
$631$			19007.5i			1.19917i	−0.800310	+	0.599586i	$0.795332\pi$
$632$		0			0
$633$	3069.00	+	1771.89i	0.192704	+	0.111258i
$634$		0			0
$635$	−4635.00	−	8028.06i	−0.289660	−	0.501707i
$636$		0			0
$637$	23520.0	+	3394.82i	1.46295	+	0.211158i
$638$		0			0
$639$	10998.0	−	6349.70i	0.680867	−	0.393099i
$640$		0			0
$641$	−11011.5	+	19072.5i	−0.678515	+	1.17522i	0.296913	+	0.954904i	$0.404043\pi$
$641$	−11011.5	+	19072.5i	−0.678515	+	1.17522i	−0.975428	+	0.220318i	$0.929291\pi$
$642$		0			0
$643$	10036.0			0.615523			0.307761	−	0.951464i	$-0.400420\pi$
$643$	10036.0			0.615523			0.307761	+	0.951464i	$0.400420\pi$
$644$		0			0
$645$	1314.00			0.0802150
$646$		0			0
$647$	−13846.5	+	23982.8i	−0.841363	+	1.45728i	0.0473790	+	0.998877i	$0.484913\pi$
$647$	−13846.5	+	23982.8i	−0.841363	+	1.45728i	−0.888742	+	0.458407i	$0.848420\pi$
$648$		0			0
$649$	−18022.5	+	10405.3i	−1.09005	+	0.629343i
$650$		0			0
$651$	−297.500	−	103.057i	−0.0179108	−	0.00620449i
$652$		0			0
$653$	−643.500	−	1114.57i	−0.0385637	−	0.0667943i	0.846099	−	0.533025i	$-0.178945\pi$
$653$	−643.500	−	1114.57i	−0.0385637	−	0.0667943i	−0.884663	+	0.466231i	$0.845612\pi$
$654$		0			0
$655$	11785.5	+	6804.36i	0.703050	+	0.405906i
$656$		0			0
$657$			10492.8i			0.623077i
$658$		0			0
$659$			5767.73i			0.340939i	0.985363	+	0.170470i	$0.0545284\pi$
$659$			5767.73i			0.340939i	−0.985363	+	0.170470i	$0.945472\pi$
$660$		0			0
$661$	9259.50	+	5345.97i	0.544861	+	0.314575i	0.747047	−	0.664772i	$-0.231471\pi$
$661$	9259.50	+	5345.97i	0.544861	+	0.314575i	−0.202186	+	0.979347i	$0.564805\pi$
$662$		0			0
$663$	−1260.00	−	2182.38i	−0.0738075	−	0.127838i
$664$		0			0
$665$	−1606.50	+	309.171i	−0.0936803	+	0.0180288i
$666$		0			0
$667$	10935.0	−	6313.33i	0.634790	−	0.366496i
$668$		0			0
$669$	−1448.00	+	2508.01i	−0.0836815	+	0.144941i
$670$		0			0
$671$	−9315.00			−0.535919
$672$		0			0
$673$	14518.0			0.831542			0.415771	−	0.909469i	$-0.363512\pi$
$673$	14518.0			0.831542			0.415771	+	0.909469i	$0.363512\pi$
$674$		0			0
$675$	2597.00	−	4498.14i	0.148087	−	0.256494i
$676$		0			0
$677$	−7312.50	+	4221.87i	−0.415129	+	0.239675i	−0.692991	−	0.720946i	$-0.743707\pi$
$677$	−7312.50	+	4221.87i	−0.415129	+	0.239675i	0.277862	+	0.960621i	$0.410374\pi$
$678$		0			0
$679$	16884.0	+	19496.0i	0.954269	+	1.10189i
$680$		0			0
$681$	1849.50	+	3203.43i	0.104072	+	0.180258i
$682$		0			0
$683$	−23557.5	−	13600.9i	−1.31977	−	0.761969i	−0.336078	−	0.941834i	$-0.609101\pi$
$683$	−23557.5	−	13600.9i	−1.31977	−	0.761969i	−0.983691	+	0.179865i	$0.942434\pi$
$684$		0			0
$685$			10709.3i			0.597343i
$686$		0			0
$687$		−	3815.71i		−	0.211904i
$688$		0			0
$689$	19980.0	+	11535.5i	1.10476	+	0.637832i
$690$		0			0
$691$	−9953.50	−	17240.0i	−0.547972	−	0.949116i	−0.998413	−	0.0563099i	$-0.982067\pi$
$691$	−9953.50	−	17240.0i	−0.547972	−	0.949116i	0.450441	−	0.892806i	$-0.351267\pi$
$692$		0			0
$693$	8190.00	+	9457.00i	0.448936	+	0.518386i
$694$		0			0
$695$	4986.00	−	2878.67i	0.272129	−	0.157114i
$696$		0			0
$697$	3402.00	−	5892.44i	0.184878	−	0.320218i
$698$		0			0
$699$	−5661.00			−0.306321
$700$		0			0
$701$	12294.0			0.662394			0.331197	−	0.943562i	$-0.392548\pi$
$701$	12294.0			0.662394			0.331197	+	0.943562i	$0.392548\pi$
$702$		0			0
$703$	1691.50	−	2929.76i	0.0907484	−	0.157181i
$704$		0			0
$705$	−2551.50	+	1473.11i	−0.136305	+	0.0786957i
$706$		0			0
$707$	30523.5	−	5874.25i	1.62370	−	0.312481i
$708$		0			0
$709$	4080.50	+	7067.63i	0.216144	+	0.374373i	0.953626	−	0.300994i	$-0.0973185\pi$
$709$	4080.50	+	7067.63i	0.216144	+	0.374373i	−0.737482	+	0.675367i	$0.763985\pi$
$710$		0			0
$711$	26871.0	+	15514.0i	1.41736	+	0.818312i
$712$		0			0
$713$			2385.03i			0.125274i
$714$		0			0
$715$		−	9353.07i		−	0.489210i
$716$		0			0
$717$	−3015.00	−	1740.71i	−0.157039	−	0.0906667i
$718$		0			0
$719$	−6997.50	−	12120.0i	−0.362952	−	0.628652i	0.625493	−	0.780230i	$-0.284898\pi$
$719$	−6997.50	−	12120.0i	−0.362952	−	0.628652i	−0.988445	+	0.151578i	$0.951565\pi$
$720$		0			0
$721$	−23292.5	−	8068.76i	−1.20313	−	0.416777i
$722$		0			0
$723$	−2101.50	+	1213.30i	−0.108099	+	0.0624110i
$724$		0			0
$725$	−4410.00	+	7638.34i	−0.225908	+	0.391284i
$726$		0			0
$727$	1232.00			0.0628506			0.0314253	−	0.999506i	$-0.489995\pi$
$727$	1232.00			0.0628506			0.0314253	+	0.999506i	$0.489995\pi$
$728$		0			0
$729$	−15443.0			−0.784586
$730$		0			0
$731$	−4599.00	+	7965.70i	−0.232695	+	0.403040i
$732$		0			0
$733$	−23902.5	+	13800.1i	−1.20445	+	0.695387i	−0.961541	−	0.274663i	$-0.911434\pi$
$733$	−23902.5	+	13800.1i	−1.20445	+	0.695387i	−0.242905	+	0.970050i	$0.578100\pi$
$734$		0			0
$735$	−1764.00	−	254.611i	−0.0885253	−	0.0127775i
$736$		0			0
$737$	−2812.50	−	4871.39i	−0.140570	−	0.243474i
$738$		0			0
$739$	20554.5	+	11867.1i	1.02315	+	0.590717i	0.915015	−	0.403419i	$-0.132178\pi$
$739$	20554.5	+	11867.1i	1.02315	+	0.590717i	0.108137	+	0.994136i	$0.465512\pi$
$740$		0			0
$741$		−	1177.79i		−	0.0583905i
$742$		0			0
$743$			11275.7i			0.556748i	0.960473	+	0.278374i	$0.0897954\pi$
$743$			11275.7i			0.556748i	−0.960473	+	0.278374i	$0.910205\pi$
$744$		0			0
$745$	12433.5	+	7178.48i	0.611447	+	0.353019i
$746$		0			0
$747$	−6084.00	−	10537.8i	−0.297995	−	0.516142i
$748$		0			0
$749$	10048.5	−	29007.5i	0.490206	−	1.41510i
$750$		0			0
$751$	−5392.50	+	3113.36i	−0.262017	+	0.151276i	−0.625255	−	0.780421i	$-0.715005\pi$
$751$	−5392.50	+	3113.36i	−0.262017	+	0.151276i	0.363237	+	0.931697i	$0.381672\pi$
$752$		0			0
$753$	−3060.00	+	5300.08i	−0.148091	+	0.256501i
$754$		0			0
$755$	4293.00			0.206938
$756$		0			0
$757$	−17062.0			−0.819193			−0.409596	−	0.912267i	$-0.634330\pi$
$757$	−17062.0			−0.819193			−0.409596	+	0.912267i	$0.634330\pi$
$758$		0			0
$759$	−1822.50	+	3156.66i	−0.0871575	+	0.150961i
$760$		0			0
$761$	23251.5	−	13424.3i	1.10758	−	0.639460i	0.169376	−	0.985551i	$-0.445825\pi$
$761$	23251.5	−	13424.3i	1.10758	−	0.639460i	0.938201	+	0.346091i	$0.112491\pi$
$762$		0			0
$763$	2768.50	+	14385.5i	0.131358	+	0.682558i
$764$		0			0
$765$	−2457.00	−	4255.65i	−0.116122	−	0.201129i
$766$		0			0
$767$	−48060.0	−	27747.5i	−2.26251	−	1.30626i
$768$		0			0
$769$		−	4330.13i		−	0.203054i	−0.994833	−	0.101527i	$-0.967627\pi$
$769$		−	4330.13i		−	0.203054i	0.994833	−	0.101527i	$-0.0323728\pi$
$770$		0			0
$771$			3569.76i			0.166747i
$772$		0			0
$773$	14593.5	+	8425.56i	0.679032	+	0.392039i	0.799490	−	0.600679i	$-0.205103\pi$
$773$	14593.5	+	8425.56i	0.679032	+	0.392039i	−0.120458	+	0.992718i	$0.538436\pi$
$774$		0			0
$775$	−833.000	−	1442.80i	−0.0386093	−	0.0668733i
$776$		0			0
$777$	2786.00	−	2412.75i	0.128632	−	0.111399i
$778$		0			0
$779$	2754.00	−	1590.02i	0.126665	−	0.0731303i
$780$		0			0
$781$	6345.00	−	10989.9i	0.290707	−	0.503519i
$782$		0			0
$783$	−4770.00			−0.217709
$784$		0			0
$785$	4563.00			0.207466
$786$		0			0
$787$	−13532.5	+	23439.0i	−0.612937	+	1.06164i	0.377806	+	0.925885i	$0.376679\pi$
$787$	−13532.5	+	23439.0i	−0.612937	+	1.06164i	−0.990743	+	0.135753i	$0.956655\pi$
$788$		0			0
$789$	3433.50	−	1982.33i	0.154925	−	0.0894460i
$790$		0			0
$791$	13860.0	−	12003.1i	0.623015	−	0.539547i
$792$		0			0
$793$	−12420.0	−	21512.1i	−0.556175	−	0.963324i
$794$		0			0
$795$	−1498.50	−	865.159i	−0.0668507	−	0.0385963i
$796$		0			0
$797$		−	13717.8i		−	0.609675i	−0.952404	−	0.304837i	$-0.901398\pi$
$797$		−	13717.8i		−	0.609675i	0.952404	−	0.304837i	$-0.0986021\pi$
$798$		0			0
$799$		−	20623.5i		−	0.913151i
$800$		0			0
$801$	−4329.00	−	2499.35i	−0.190958	−	0.110250i
$802$		0			0
$803$	5242.50	+	9080.28i	0.230391	+	0.399049i
$804$		0			0
$805$	2551.50	+	13258.0i	0.111712	+	0.580475i
$806$		0			0
$807$	5332.50	−	3078.72i	0.232606	−	0.134295i
$808$		0			0
$809$	11164.5	−	19337.5i	0.485195	−	0.840383i	−0.514660	−	0.857394i	$-0.672082\pi$
$809$	11164.5	−	19337.5i	0.485195	−	0.840383i	0.999855	+	0.0170116i	$0.00541521\pi$
$810$		0			0
$811$	−43252.0			−1.87273			−0.936364	−	0.351029i	$-0.885832\pi$
$811$	−43252.0			−1.87273			−0.936364	+	0.351029i	$0.885832\pi$
$812$		0			0
$813$	−5749.00			−0.248003
$814$		0			0
$815$	−6304.50	+	10919.7i	−0.270966	+	0.469326i
$816$		0			0
$817$	−3723.00	+	2149.48i	−0.159426	+	0.0920448i
$818$		0			0
$819$	−10920.0	+	31523.3i	−0.465904	+	1.34495i
$820$		0			0
$821$	−14845.5	−	25713.2i	−0.631074	−	1.09305i	−0.987333	−	0.158664i	$-0.949281\pi$
$821$	−14845.5	−	25713.2i	−0.631074	−	1.09305i	0.356259	−	0.934387i	$-0.384052\pi$
$822$		0			0
$823$	10858.5	+	6269.16i	0.459907	+	0.265527i	0.712005	−	0.702174i	$-0.247787\pi$
$823$	10858.5	+	6269.16i	0.459907	+	0.265527i	−0.252098	+	0.967702i	$0.581121\pi$
$824$		0			0
$825$		−	2546.11i		−	0.107448i
$826$		0			0
$827$		−	6910.88i		−	0.290586i	−0.989389	−	0.145293i	$-0.953587\pi$
$827$		−	6910.88i		−	0.290586i	0.989389	−	0.145293i	$-0.0464125\pi$
$828$		0			0
$829$	−37546.5	−	21677.5i	−1.57303	−	0.908191i	−0.995795	−	0.0916146i	$-0.970797\pi$
$829$	−37546.5	−	21677.5i	−1.57303	−	0.908191i	−0.577238	−	0.816576i	$-0.695869\pi$
$830$		0			0
$831$	98.5000	+	170.607i	0.00411183	+	0.00712189i
$832$		0			0
$833$	7717.50	−	9802.54i	0.321003	−	0.407729i
$834$		0			0
$835$	12150.0	−	7014.81i	0.503555	−	0.290727i
$836$		0			0
$837$	450.500	−	780.289i	0.0186040	−	0.0322231i
$838$		0			0
$839$	9648.00			0.397004			0.198502	−	0.980101i	$-0.436392\pi$
$839$	9648.00			0.397004			0.198502	+	0.980101i	$0.436392\pi$
$840$		0			0
$841$	−16289.0			−0.667883
$842$		0			0
$843$	1809.00	−	3133.28i	0.0739090	−	0.128014i
$844$		0			0
$845$	11713.5	−	6762.79i	0.476872	−	0.275322i
$846$		0			0
$847$	−11480.0	−	3976.79i	−0.465711	−	0.161327i
$848$		0			0
$849$	1781.50	+	3085.65i	0.0720152	+	0.124734i
$850$		0			0
$851$	−24178.5	−	13959.5i	−0.973946	−	0.562308i
$852$		0			0
$853$		−	8119.85i		−	0.325930i	−0.986632	−	0.162965i	$-0.947894\pi$
$853$		−	8119.85i		−	0.325930i	0.986632	−	0.162965i	$-0.0521058\pi$
$854$		0			0
$855$		−	2296.70i		−	0.0918660i
$856$		0			0
$857$	1201.50	+	693.686i	0.0478908	+	0.0276498i	0.523754	−	0.851869i	$-0.324531\pi$
$857$	1201.50	+	693.686i	0.0478908	+	0.0276498i	−0.475863	+	0.879519i	$0.657864\pi$
$858$		0			0
$859$	−507.500	−	879.016i	−0.0201579	−	0.0349146i	0.855770	−	0.517356i	$-0.173084\pi$
$859$	−507.500	−	879.016i	−0.0201579	−	0.0349146i	−0.875928	+	0.482441i	$0.839750\pi$
$860$		0			0
$861$	3402.00	−	654.715i	0.134657	−	0.0259148i
$862$		0			0
$863$	−2794.50	+	1613.41i	−0.110227	+	0.0636396i	−0.554100	−	0.832450i	$-0.686937\pi$
$863$	−2794.50	+	1613.41i	−0.110227	+	0.0636396i	0.443873	+	0.896090i	$0.353604\pi$
$864$		0			0
$865$	−3604.50	+	6243.18i	−0.141684	+	0.245404i
$866$		0			0
$867$	3590.00			0.140626
$868$		0			0
$869$	31005.0			1.21033
$870$		0			0
$871$	7500.00	−	12990.4i	0.291766	−	0.505353i
$872$		0			0
$873$	−31356.0	+	18103.4i	−1.21562	+	0.701841i
$874$		0			0
$875$	−14049.0	−	16222.4i	−0.542792	−	0.626762i
$876$		0			0
$877$	7770.50	+	13458.9i	0.299192	+	0.518215i	0.975951	−	0.217989i	$-0.0699497\pi$
$877$	7770.50	+	13458.9i	0.299192	+	0.518215i	−0.676760	+	0.736204i	$0.736616\pi$
$878$		0			0
$879$	5238.00	+	3024.16i	0.200994	+	0.116044i
$880$		0			0
$881$		−	28724.3i		−	1.09846i	−0.835670	−	0.549232i	$-0.814920\pi$
$881$		−	28724.3i		−	1.09846i	0.835670	−	0.549232i	$-0.185080\pi$
$882$		0			0
$883$			45826.6i			1.74653i	0.487244	+	0.873266i	$0.338002\pi$
$883$			45826.6i			1.74653i	−0.487244	+	0.873266i	$0.661998\pi$
$884$		0			0
$885$	3604.50	+	2081.06i	0.136908	+	0.0790441i
$886$		0			0
$887$	3064.50	+	5307.87i	0.116004	+	0.200925i	0.918181	−	0.396162i	$-0.129658\pi$
$887$	3064.50	+	5307.87i	0.116004	+	0.200925i	−0.802176	+	0.597087i	$0.796325\pi$
$888$		0			0
$889$	21630.0	+	24976.2i	0.816026	+	0.942265i
$890$		0			0
$891$	−14602.5	+	8430.76i	−0.549048	+	0.316993i
$892$		0			0
$893$	4819.50	−	8347.62i	0.180603	−	0.312813i
$894$		0			0
$895$	13257.0			0.495120
$896$		0			0
$897$	−9720.00			−0.361808
$898$		0			0
$899$	−765.000	+	1325.02i	−0.0283806	+	0.0491567i
$900$		0			0
$901$	10489.5	−	6056.12i	0.387853	−	0.223927i
$902$		0			0
$903$	−4599.00	+	885.078i	−0.169485	+	0.0326174i
$904$		0			0
$905$	2286.00	+	3959.47i	0.0839660	+	0.145433i
$906$		0			0
$907$	6352.50	+	3667.62i	0.232559	+	0.134268i	0.611752	−	0.791049i	$-0.290465\pi$
$907$	6352.50	+	3667.62i	0.232559	+	0.134268i	−0.379193	+	0.925318i	$0.623798\pi$
$908$		0			0
$909$			43637.3i			1.59225i
$910$		0			0
$911$		−	2774.75i		−	0.100913i	−0.998726	−	0.0504563i	$-0.983932\pi$
$911$		−	2774.75i		−	0.100913i	0.998726	−	0.0504563i	$-0.0160676\pi$
$912$		0			0
$913$	−10530.0	−	6079.50i	−0.381700	−	0.220375i
$914$		0			0
$915$	931.500	+	1613.41i	0.0336551	+	0.0582924i
$916$		0			0
$917$	−45832.5	−	15876.8i	−1.65052	−	0.571755i
$918$		0			0
$919$	1027.50	−	593.227i	0.0368815	−	0.0212935i	−0.481446	−	0.876476i	$-0.659888\pi$
$919$	1027.50	−	593.227i	0.0368815	−	0.0212935i	0.518327	+	0.855182i	$0.326555\pi$
$920$		0			0
$921$	3658.00	−	6335.84i	0.130874	−	0.226681i
$922$		0			0
$923$	33840.0			1.20678
$924$		0			0
$925$	19502.0			0.693213
$926$		0			0
$927$	17303.0	−	29969.7i	0.613058	−	1.06185i
$928$		0			0
$929$	22009.5	−	12707.2i	0.777296	−	0.448772i	−0.0581749	−	0.998306i	$-0.518528\pi$
$929$	22009.5	−	12707.2i	0.777296	−	0.448772i	0.835471	+	0.549534i	$0.185195\pi$
$930$		0			0
$931$	5414.50	−	2164.20i	0.190605	−	0.0761855i
$932$		0			0
$933$	1273.50	+	2205.77i	0.0446865	+	0.0773993i
$934$		0			0
$935$	−4252.50	−	2455.18i	−0.148740	−	0.0858749i
$936$		0			0
$937$			13870.3i			0.483588i	0.970328	+	0.241794i	$0.0777358\pi$
$937$			13870.3i			0.483588i	−0.970328	+	0.241794i	$0.922264\pi$
$938$		0			0
$939$			9188.53i			0.319336i
$940$		0			0
$941$	8851.50	+	5110.42i	0.306643	+	0.177040i	0.645423	−	0.763825i	$-0.276681\pi$
$941$	8851.50	+	5110.42i	0.306643	+	0.177040i	−0.338781	+	0.940865i	$0.610014\pi$
$942$		0			0
$943$	−13122.0	−	22728.0i	−0.453140	−	0.784862i
$944$		0			0
$945$	1669.50	−	4819.43i	0.0574697	−	0.165901i
$946$		0			0
$947$	−11686.5	+	6747.20i	−0.401014	+	0.231526i	−0.686921	−	0.726732i	$-0.741038\pi$
$947$	−11686.5	+	6747.20i	−0.401014	+	0.231526i	0.285907	+	0.958257i	$0.407705\pi$
$948$		0			0
$949$	−13980.0	+	24214.1i	−0.478198	+	0.828263i
$950$		0			0
$951$	495.000			0.0168785
$952$		0			0
$953$	6966.00			0.236780			0.118390	−	0.992967i	$-0.462227\pi$
$953$	6966.00			0.236780			0.118390	+	0.992967i	$0.462227\pi$
$954$		0			0
$955$	2929.50	−	5074.04i	0.0992632	−	0.171929i
$956$		0			0
$957$	−2025.00	+	1169.13i	−0.0684002	+	0.0394909i
$958$		0			0
$959$	−7213.50	−	37482.4i	−0.242895	−	1.26212i
$960$		0			0
$961$	14751.0	+	25549.5i	0.495150	+	0.857624i
$962$		0			0
$963$	37323.0	+	21548.4i	1.24893	+	0.721068i
$964$		0			0
$965$		−	6832.94i		−	0.227938i
$966$		0			0
$967$		−	15979.9i		−	0.531416i	−0.964054	−	0.265708i	$-0.914394\pi$
$967$		−	15979.9i		−	0.531416i	0.964054	−	0.265708i	$-0.0856057\pi$
$968$		0			0
$969$	−535.500	−	309.171i	−0.0177531	−	0.0102497i
$970$		0			0
$971$	11074.5	+	19181.6i	0.366012	+	0.633951i	0.988938	−	0.148329i	$-0.0473896\pi$
$971$	11074.5	+	19181.6i	0.366012	+	0.633951i	−0.622926	+	0.782281i	$0.714056\pi$
$972$		0			0
$973$	−15512.0	+	13433.8i	−0.511091	+	0.442618i
$974$		0			0
$975$	5880.00	−	3394.82i	0.193139	−	0.111509i
$976$		0			0
$977$	−13369.5	+	23156.7i	−0.437798	+	0.758288i	−0.997519	−	0.0703930i	$-0.977575\pi$
$977$	−13369.5	+	23156.7i	−0.437798	+	0.758288i	0.559722	+	0.828681i	$0.310908\pi$
$978$		0			0
$979$	−4995.00			−0.163065
$980$		0			0
$981$	−20566.0			−0.669339
$982$		0			0
$983$	2623.50	−	4544.04i	0.0851238	−	0.147439i	−0.820320	−	0.571905i	$-0.806205\pi$
$983$	2623.50	−	4544.04i	0.0851238	−	0.147439i	0.905444	+	0.424466i	$0.139538\pi$
$984$		0			0
$985$	−14661.0	+	8464.53i	−0.474252	+	0.273810i
$986$		0			0
$987$	7938.00	−	6874.51i	0.255997	−	0.221700i
$988$		0			0
$989$	17739.0	+	30724.8i	0.570341	+	0.987860i
$990$		0			0
$991$	30430.5	+	17569.1i	0.975436	+	0.563168i	0.900889	−	0.434050i	$-0.142916\pi$
$991$	30430.5	+	17569.1i	0.975436	+	0.563168i	0.0745466	+	0.997218i	$0.476249\pi$
$992$		0			0
$993$		−	9185.07i		−	0.293534i
$994$		0			0
$995$		−	9108.86i		−	0.290221i
$996$		0			0
$997$	19279.5	+	11131.0i	0.612425	+	0.353584i	0.773914	−	0.633291i	$-0.218296\pi$
$997$	19279.5	+	11131.0i	0.612425	+	0.353584i	−0.161489	+	0.986875i	$0.551630\pi$
$998$		0			0
$999$	5273.50	+	9133.97i	0.167013	+	0.289275i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.4.p.b.31.1	✓	2
4.3	odd	2		112.4.p.c.31.1	yes	2
7.3	odd	6		784.4.f.b.783.2		2
7.4	even	3		784.4.f.c.783.1		2
7.5	odd	6		112.4.p.c.47.1	yes	2
8.3	odd	2		448.4.p.b.255.1		2
8.5	even	2		448.4.p.c.255.1		2
28.3	even	6		784.4.f.c.783.2		2
28.11	odd	6		784.4.f.b.783.1		2
28.19	even	6	inner	112.4.p.b.47.1	yes	2
56.5	odd	6		448.4.p.b.383.1		2
56.19	even	6		448.4.p.c.383.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.4.p.b.31.1	✓	2	1.1	even	1	trivial
112.4.p.b.47.1	yes	2	28.19	even	6	inner
112.4.p.c.31.1	yes	2	4.3	odd	2
112.4.p.c.47.1	yes	2	7.5	odd	6
448.4.p.b.255.1		2	8.3	odd	2
448.4.p.b.383.1		2	56.5	odd	6
448.4.p.c.255.1		2	8.5	even	2
448.4.p.c.383.1		2	56.19	even	6
784.4.f.b.783.1		2	28.11	odd	6
784.4.f.b.783.2		2	7.3	odd	6
784.4.f.c.783.1		2	7.4	even	3
784.4.f.c.783.2		2	28.3	even	6

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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	112.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{3} +(-4.50000 + 2.59808i) q^{5} +(14.0000 - 12.1244i) q^{7} +(13.0000 + 22.5167i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{3} +(-4.50000 + 2.59808i) q^{5} +(14.0000 - 12.1244i) q^{7} +(13.0000 + 22.5167i) q^{9} +(22.5000 + 12.9904i) q^{11} +69.2820i q^{13} -5.19615i q^{15} +(31.5000 + 18.1865i) q^{17} +(8.50000 + 14.7224i) q^{19} +(3.50000 + 18.1865i) q^{21} +(121.500 - 70.1481i) q^{23} +(-49.0000 + 84.8705i) q^{25} -53.0000 q^{27} +90.0000 q^{29} +(-8.50000 + 14.7224i) q^{31} +(-22.5000 + 12.9904i) q^{33} +(-31.5000 + 90.9327i) q^{35} +(-99.5000 - 172.339i) q^{37} +(-60.0000 - 34.6410i) q^{39} -187.061i q^{41} +252.879i q^{43} +(-117.000 - 67.5500i) q^{45} +(-283.500 - 491.036i) q^{47} +(49.0000 - 339.482i) q^{49} +(-31.5000 + 18.1865i) q^{51} +(166.500 - 288.386i) q^{53} -135.000 q^{55} -17.0000 q^{57} +(-400.500 + 693.686i) q^{59} +(-310.500 + 179.267i) q^{61} +(455.000 + 157.617i) q^{63} +(-180.000 - 311.769i) q^{65} +(-187.500 - 108.253i) q^{67} +140.296i q^{69} -488.438i q^{71} +(349.500 + 201.784i) q^{73} +(-49.0000 - 84.8705i) q^{75} +(472.500 - 90.9327i) q^{77} +(1033.50 - 596.692i) q^{79} +(-324.500 + 562.050i) q^{81} -468.000 q^{83} -189.000 q^{85} +(-45.0000 + 77.9423i) q^{87} +(-166.500 + 96.1288i) q^{89} +(840.000 + 969.948i) q^{91} +(-8.50000 - 14.7224i) q^{93} +(-76.5000 - 44.1673i) q^{95} +1392.57i q^{97} +675.500i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 9 q^{5} + 28 q^{7} + 26 q^{9}+O(q^{10}) \) 2 * q - q^3 - 9 * q^5 + 28 * q^7 + 26 * q^9	\( 2 q - q^{3} - 9 q^{5} + 28 q^{7} + 26 q^{9} + 45 q^{11} + 63 q^{17} + 17 q^{19} + 7 q^{21} + 243 q^{23} - 98 q^{25} - 106 q^{27} + 180 q^{29} - 17 q^{31} - 45 q^{33} - 63 q^{35} - 199 q^{37} - 120 q^{39} - 234 q^{45} - 567 q^{47} + 98 q^{49} - 63 q^{51} + 333 q^{53} - 270 q^{55} - 34 q^{57} - 801 q^{59} - 621 q^{61} + 910 q^{63} - 360 q^{65} - 375 q^{67} + 699 q^{73} - 98 q^{75} + 945 q^{77} + 2067 q^{79} - 649 q^{81} - 936 q^{83} - 378 q^{85} - 90 q^{87} - 333 q^{89} + 1680 q^{91} - 17 q^{93} - 153 q^{95}+O(q^{100}) \) 2 * q - q^3 - 9 * q^5 + 28 * q^7 + 26 * q^9 + 45 * q^11 + 63 * q^17 + 17 * q^19 + 7 * q^21 + 243 * q^23 - 98 * q^25 - 106 * q^27 + 180 * q^29 - 17 * q^31 - 45 * q^33 - 63 * q^35 - 199 * q^37 - 120 * q^39 - 234 * q^45 - 567 * q^47 + 98 * q^49 - 63 * q^51 + 333 * q^53 - 270 * q^55 - 34 * q^57 - 801 * q^59 - 621 * q^61 + 910 * q^63 - 360 * q^65 - 375 * q^67 + 699 * q^73 - 98 * q^75 + 945 * q^77 + 2067 * q^79 - 649 * q^81 - 936 * q^83 - 378 * q^85 - 90 * q^87 - 333 * q^89 + 1680 * q^91 - 17 * q^93 - 153 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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