LMFDB - Embedded newform 688.2.i.e.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [688,2,Mod(49,688)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(688, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("688.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 688 = 2^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	688.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$5.49370765906$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 43)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			49.1
Root			$0.809017 - 1.40126i$ of defining polynomial
Character	$\chi$	$=$	688.49
Dual form			688.2.i.e.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.30902 + 2.26728i) q^{3} +(-0.618034 + 1.07047i) q^{5} +(-2.11803 - 3.66854i) q^{7} +(-1.92705 - 3.33775i) q^{9} +O(q^{10})$	\(q+(-1.30902 + 2.26728i) q^{3} +(-0.618034 + 1.07047i) q^{5} +(-2.11803 - 3.66854i) q^{7} +(-1.92705 - 3.33775i) q^{9} +3.61803 q^{11} +(-0.690983 - 1.19682i) q^{13} +(-1.61803 - 2.80252i) q^{15} +(-3.04508 - 5.27424i) q^{17} +(-0.618034 + 1.07047i) q^{19} +11.0902 q^{21} +(2.19098 - 3.79489i) q^{23} +(1.73607 + 3.00696i) q^{25} +2.23607 q^{27} +(-1.50000 - 2.59808i) q^{29} +(-4.73607 + 8.20311i) q^{33} +5.23607 q^{35} +(2.42705 - 4.20378i) q^{37} +3.61803 q^{39} +9.47214 q^{41} +(6.50000 - 0.866025i) q^{43} +4.76393 q^{45} -1.14590 q^{47} +(-5.47214 + 9.47802i) q^{49} +15.9443 q^{51} +(0.690983 - 1.19682i) q^{53} +(-2.23607 + 3.87298i) q^{55} +(-1.61803 - 2.80252i) q^{57} -5.09017 q^{59} +(-1.42705 - 2.47172i) q^{61} +(-8.16312 + 14.1389i) q^{63} +1.70820 q^{65} +(-1.92705 + 3.33775i) q^{67} +(5.73607 + 9.93516i) q^{69} +(-5.39919 - 9.35167i) q^{71} +(-0.927051 - 1.60570i) q^{73} -9.09017 q^{75} +(-7.66312 - 13.2729i) q^{77} +(-0.690983 - 1.19682i) q^{79} +(2.85410 - 4.94345i) q^{81} +(8.01722 - 13.8862i) q^{83} +7.52786 q^{85} +7.85410 q^{87} +(0.927051 - 1.60570i) q^{89} +(-2.92705 + 5.06980i) q^{91} +(-0.763932 - 1.32317i) q^{95} -9.23607 q^{97} +(-6.97214 - 12.0761i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} + 2 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) $ 4 * q - 3 * q^3 + 2 * q^5 - 4 * q^7 - q^9	$ 4 q - 3 q^{3} + 2 q^{5} - 4 q^{7} - q^{9} + 10 q^{11} - 5 q^{13} - 2 q^{15} - q^{17} + 2 q^{19} + 22 q^{21} + 11 q^{23} - 2 q^{25} - 6 q^{29} - 10 q^{33} + 12 q^{35} + 3 q^{37} + 10 q^{39} + 20 q^{41} + 26 q^{43} + 28 q^{45} - 18 q^{47} - 4 q^{49} + 28 q^{51} + 5 q^{53} - 2 q^{57} + 2 q^{59} + q^{61} - 17 q^{63} - 20 q^{65} - q^{67} + 14 q^{69} + 3 q^{71} + 3 q^{73} - 14 q^{75} - 15 q^{77} - 5 q^{79} - 2 q^{81} + 3 q^{83} + 48 q^{85} + 18 q^{87} - 3 q^{89} - 5 q^{91} - 12 q^{95} - 28 q^{97} - 10 q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 + 2 * q^5 - 4 * q^7 - q^9 + 10 * q^11 - 5 * q^13 - 2 * q^15 - q^17 + 2 * q^19 + 22 * q^21 + 11 * q^23 - 2 * q^25 - 6 * q^29 - 10 * q^33 + 12 * q^35 + 3 * q^37 + 10 * q^39 + 20 * q^41 + 26 * q^43 + 28 * q^45 - 18 * q^47 - 4 * q^49 + 28 * q^51 + 5 * q^53 - 2 * q^57 + 2 * q^59 + q^61 - 17 * q^63 - 20 * q^65 - q^67 + 14 * q^69 + 3 * q^71 + 3 * q^73 - 14 * q^75 - 15 * q^77 - 5 * q^79 - 2 * q^81 + 3 * q^83 + 48 * q^85 + 18 * q^87 - 3 * q^89 - 5 * q^91 - 12 * q^95 - 28 * q^97 - 10 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/688\mathbb{Z}\right)^\times$.

$n$	$431$	$433$	$517$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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
$3$	−1.30902	+	2.26728i	−0.755761	+	1.30902i	0.189234	+	0.981932i	$0.439400\pi$
$3$	−1.30902	+	2.26728i	−0.755761	+	1.30902i	−0.944995	+	0.327085i	$0.893934\pi$
$4$		0			0
$5$	−0.618034	+	1.07047i	−0.276393	+	0.478727i	−0.970486	−	0.241159i	$-0.922473\pi$
$5$	−0.618034	+	1.07047i	−0.276393	+	0.478727i	0.694092	+	0.719886i	$0.255806\pi$
$6$		0			0
$7$	−2.11803	−	3.66854i	−0.800542	−	1.38658i	−0.919260	−	0.393651i	$-0.871212\pi$
$7$	−2.11803	−	3.66854i	−0.800542	−	1.38658i	0.118718	−	0.992928i	$-0.462121\pi$
$8$		0			0
$9$	−1.92705	−	3.33775i	−0.642350	−	1.11258i
$10$		0			0
$11$	3.61803			1.09088			0.545439	−	0.838150i	$-0.316363\pi$
$11$	3.61803			1.09088			0.545439	+	0.838150i	$0.316363\pi$
$12$		0			0
$13$	−0.690983	−	1.19682i	−0.191644	−	0.331937i	0.754151	−	0.656701i	$-0.228049\pi$
$13$	−0.690983	−	1.19682i	−0.191644	−	0.331937i	−0.945795	+	0.324763i	$0.894715\pi$
$14$		0			0
$15$	−1.61803	−	2.80252i	−0.417775	−	0.723607i
$16$		0			0
$17$	−3.04508	−	5.27424i	−0.738542	−	1.27919i	−0.953152	−	0.302492i	$-0.902182\pi$
$17$	−3.04508	−	5.27424i	−0.738542	−	1.27919i	0.214610	−	0.976700i	$-0.431152\pi$
$18$		0			0
$19$	−0.618034	+	1.07047i	−0.141787	+	0.245582i	−0.928170	−	0.372158i	$-0.878618\pi$
$19$	−0.618034	+	1.07047i	−0.141787	+	0.245582i	0.786383	+	0.617740i	$0.211951\pi$
$20$		0			0
$21$	11.0902			2.42007
$22$		0			0
$23$	2.19098	−	3.79489i	0.456852	−	0.791290i	−0.541941	−	0.840417i	$-0.682310\pi$
$23$	2.19098	−	3.79489i	0.456852	−	0.791290i	0.998793	+	0.0491264i	$0.0156437\pi$
$24$		0			0
$25$	1.73607	+	3.00696i	0.347214	+	0.601392i
$26$		0			0
$27$	2.23607			0.430331
$28$		0			0
$29$	−1.50000	−	2.59808i	−0.278543	−	0.482451i	0.692480	−	0.721437i	$-0.256518\pi$
$29$	−1.50000	−	2.59808i	−0.278543	−	0.482451i	−0.971023	+	0.238987i	$0.923185\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$		0			0
$33$	−4.73607	+	8.20311i	−0.824444	+	1.42798i
$34$		0			0
$35$	5.23607			0.885057
$36$		0			0
$37$	2.42705	−	4.20378i	0.399005	−	0.691096i	−0.594599	−	0.804023i	$-0.702689\pi$
$37$	2.42705	−	4.20378i	0.399005	−	0.691096i	0.993603	+	0.112926i	$0.0360224\pi$
$38$		0			0
$39$	3.61803			0.579349
$40$		0			0
$41$	9.47214			1.47930			0.739650	−	0.672992i	$-0.234991\pi$
$41$	9.47214			1.47930			0.739650	+	0.672992i	$0.234991\pi$
$42$		0			0
$43$	6.50000	−	0.866025i	0.991241	−	0.132068i
$43$	6.50000	−	0.866025i	0.991241	−	0.132068i
$44$		0			0
$45$	4.76393			0.710165
$46$		0			0
$47$	−1.14590			−0.167146			−0.0835732	−	0.996502i	$-0.526633\pi$
$47$	−1.14590			−0.167146			−0.0835732	+	0.996502i	$0.526633\pi$
$48$		0			0
$49$	−5.47214	+	9.47802i	−0.781734	+	1.35400i
$50$		0			0
$51$	15.9443			2.23264
$52$		0			0
$53$	0.690983	−	1.19682i	0.0949138	−	0.164396i	−0.814659	−	0.579941i	$-0.803076\pi$
$53$	0.690983	−	1.19682i	0.0949138	−	0.164396i	0.909573	+	0.415545i	$0.136409\pi$
$54$		0			0
$55$	−2.23607	+	3.87298i	−0.301511	+	0.522233i
$56$		0			0
$57$	−1.61803	−	2.80252i	−0.214314	−	0.371202i
$58$		0			0
$59$	−5.09017			−0.662684			−0.331342	−	0.943511i	$-0.607501\pi$
$59$	−5.09017			−0.662684			−0.331342	+	0.943511i	$0.607501\pi$
$60$		0			0
$61$	−1.42705	−	2.47172i	−0.182715	−	0.316472i	0.760089	−	0.649819i	$-0.225155\pi$
$61$	−1.42705	−	2.47172i	−0.182715	−	0.316472i	−0.942804	+	0.333347i	$0.891822\pi$
$62$		0			0
$63$	−8.16312	+	14.1389i	−1.02846	+	1.78134i
$64$		0			0
$65$	1.70820			0.211877
$66$		0			0
$67$	−1.92705	+	3.33775i	−0.235427	+	0.407771i	−0.959397	−	0.282061i	$-0.908982\pi$
$67$	−1.92705	+	3.33775i	−0.235427	+	0.407771i	0.723970	+	0.689832i	$0.242315\pi$
$68$		0			0
$69$	5.73607	+	9.93516i	0.690541	+	1.19605i
$70$		0			0
$71$	−5.39919	−	9.35167i	−0.640766	−	1.10984i	−0.985262	−	0.171051i	$-0.945284\pi$
$71$	−5.39919	−	9.35167i	−0.640766	−	1.10984i	0.344497	−	0.938788i	$-0.388050\pi$
$72$		0			0
$73$	−0.927051	−	1.60570i	−0.108503	−	0.187933i	0.806661	−	0.591015i	$-0.201272\pi$
$73$	−0.927051	−	1.60570i	−0.108503	−	0.187933i	−0.915164	+	0.403082i	$0.867939\pi$
$74$		0			0
$75$	−9.09017			−1.04964
$76$		0			0
$77$	−7.66312	−	13.2729i	−0.873293	−	1.51259i
$78$		0			0
$79$	−0.690983	−	1.19682i	−0.0777417	−	0.134653i	0.824534	−	0.565813i	$-0.191438\pi$
$79$	−0.690983	−	1.19682i	−0.0777417	−	0.134653i	−0.902275	+	0.431161i	$0.858104\pi$
$80$		0			0
$81$	2.85410	−	4.94345i	0.317122	−	0.549272i
$82$		0			0
$83$	8.01722	−	13.8862i	0.880004	−	1.52421i	0.0286698	−	0.999589i	$-0.490873\pi$
$83$	8.01722	−	13.8862i	0.880004	−	1.52421i	0.851335	−	0.524623i	$-0.175794\pi$
$84$		0			0
$85$	7.52786			0.816511
$86$		0			0
$87$	7.85410			0.842048
$88$		0			0
$89$	0.927051	−	1.60570i	0.0982672	−	0.170204i	−0.812700	−	0.582682i	$-0.802003\pi$
$89$	0.927051	−	1.60570i	0.0982672	−	0.170204i	0.910968	+	0.412478i	$0.135337\pi$
$90$		0			0
$91$	−2.92705	+	5.06980i	−0.306838	+	0.531460i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.763932	−	1.32317i	−0.0783778	−	0.135754i
$96$		0			0
$97$	−9.23607			−0.937781			−0.468890	−	0.883256i	$-0.655346\pi$
$97$	−9.23607			−0.937781			−0.468890	+	0.883256i	$0.655346\pi$
$98$		0			0
$99$	−6.97214	−	12.0761i	−0.700726	−	1.21369i
$100$		0			0
$101$	−1.88197	−	3.25966i	−0.187263	−	0.324348i	0.757074	−	0.653329i	$-0.226628\pi$
$101$	−1.88197	−	3.25966i	−0.187263	−	0.324348i	−0.944337	+	0.328981i	$0.893295\pi$
$102$		0			0
$103$	−8.20820	−	14.2170i	−0.808778	−	1.40085i	−0.913710	−	0.406366i	$-0.866796\pi$
$103$	−8.20820	−	14.2170i	−0.808778	−	1.40085i	0.104932	−	0.994479i	$-0.466537\pi$
$104$		0			0
$105$	−6.85410	+	11.8717i	−0.668892	+	1.15855i
$106$		0			0
$107$	−7.52786			−0.727746			−0.363873	−	0.931449i	$-0.618546\pi$
$107$	−7.52786			−0.727746			−0.363873	+	0.931449i	$0.618546\pi$
$108$		0			0
$109$	6.92705	−	11.9980i	0.663491	−	1.14920i	−0.316201	−	0.948692i	$-0.602407\pi$
$109$	6.92705	−	11.9980i	0.663491	−	1.14920i	0.979692	−	0.200508i	$-0.0642593\pi$
$110$		0			0
$111$	6.35410	+	11.0056i	0.603105	+	1.04461i
$112$		0			0
$113$	−15.6180			−1.46922			−0.734611	−	0.678489i	$-0.762635\pi$
$113$	−15.6180			−1.46922			−0.734611	+	0.678489i	$0.762635\pi$
$114$		0			0
$115$	2.70820	+	4.69075i	0.252541	+	0.437414i
$116$		0			0
$117$	−2.66312	+	4.61266i	−0.246205	+	0.426440i
$118$		0			0
$119$	−12.8992	+	22.3420i	−1.18247	+	2.04809i
$120$		0			0
$121$	2.09017			0.190015
$122$		0			0
$123$	−12.3992	+	21.4760i	−1.11800	+	1.93643i
$124$		0			0
$125$	−10.4721			−0.936656
$126$		0			0
$127$	14.6525			1.30020			0.650098	−	0.759850i	$-0.274728\pi$
$127$	14.6525			1.30020			0.650098	+	0.759850i	$0.274728\pi$
$128$		0			0
$129$	−6.54508	+	15.8710i	−0.576263	+	1.39736i
$130$		0			0
$131$	−9.94427			−0.868835			−0.434418	−	0.900712i	$-0.643046\pi$
$131$	−9.94427			−0.868835			−0.434418	+	0.900712i	$0.643046\pi$
$132$		0			0
$133$	5.23607			0.454025
$134$		0			0
$135$	−1.38197	+	2.39364i	−0.118941	+	0.206011i
$136$		0			0
$137$	3.70820			0.316813			0.158407	−	0.987374i	$-0.449364\pi$
$137$	3.70820			0.316813			0.158407	+	0.987374i	$0.449364\pi$
$138$		0			0
$139$	−2.64590	+	4.58283i	−0.224422	+	0.388711i	−0.956146	−	0.292891i	$-0.905383\pi$
$139$	−2.64590	+	4.58283i	−0.224422	+	0.388711i	0.731724	+	0.681601i	$0.238716\pi$
$140$		0			0
$141$	1.50000	−	2.59808i	0.126323	−	0.218797i
$142$		0			0
$143$	−2.50000	−	4.33013i	−0.209061	−	0.362103i
$144$		0			0
$145$	3.70820			0.307950
$146$		0			0
$147$	−14.3262	−	24.8138i	−1.18161	−	2.04661i
$148$		0			0
$149$	−4.50000	+	7.79423i	−0.368654	+	0.638528i	−0.989355	−	0.145519i	$-0.953515\pi$
$149$	−4.50000	+	7.79423i	−0.368654	+	0.638528i	0.620701	+	0.784047i	$0.286848\pi$
$150$		0			0
$151$	8.85410			0.720537			0.360268	−	0.932849i	$-0.382685\pi$
$151$	8.85410			0.720537			0.360268	+	0.932849i	$0.382685\pi$
$152$		0			0
$153$	−11.7361	+	20.3275i	−0.948805	+	1.64338i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	2.57295	+	4.45648i	0.205344	+	0.355666i	0.950242	−	0.311512i	$-0.100835\pi$
$157$	2.57295	+	4.45648i	0.205344	+	0.355666i	−0.744898	+	0.667178i	$0.767502\pi$
$158$		0			0
$159$	1.80902	+	3.13331i	0.143464	+	0.248488i
$160$		0			0
$161$	−18.5623			−1.46291
$162$		0			0
$163$	7.00000	+	12.1244i	0.548282	+	0.949653i	0.998392	+	0.0566798i	$0.0180514\pi$
$163$	7.00000	+	12.1244i	0.548282	+	0.949653i	−0.450110	+	0.892973i	$0.648615\pi$
$164$		0			0
$165$	−5.85410	−	10.1396i	−0.455741	−	0.789367i
$166$		0			0
$167$	7.11803	−	12.3288i	0.550810	−	0.954031i	−0.447406	−	0.894331i	$-0.647652\pi$
$167$	7.11803	−	12.3288i	0.550810	−	0.954031i	0.998216	−	0.0597001i	$-0.0190144\pi$
$168$		0			0
$169$	5.54508	−	9.60437i	0.426545	−	0.738798i
$170$		0			0
$171$	4.76393			0.364307
$172$		0			0
$173$	−3.76393			−0.286166			−0.143083	−	0.989711i	$-0.545702\pi$
$173$	−3.76393			−0.286166			−0.143083	+	0.989711i	$0.545702\pi$
$174$		0			0
$175$	7.35410	−	12.7377i	0.555918	−	0.962878i
$176$		0			0
$177$	6.66312	−	11.5409i	0.500831	−	0.867464i
$178$		0			0
$179$	4.82624	+	8.35929i	0.360730	+	0.624803i	0.988081	−	0.153934i	$-0.0491943\pi$
$179$	4.82624	+	8.35929i	0.360730	+	0.624803i	−0.627351	+	0.778736i	$0.715861\pi$
$180$		0			0
$181$	9.69098	+	16.7853i	0.720325	+	1.24764i	0.960869	+	0.277002i	$0.0893407\pi$
$181$	9.69098	+	16.7853i	0.720325	+	1.24764i	−0.240544	+	0.970638i	$0.577326\pi$
$182$		0			0
$183$	7.47214			0.552356
$184$		0			0
$185$	3.00000	+	5.19615i	0.220564	+	0.382029i
$186$		0			0
$187$	−11.0172	−	19.0824i	−0.805659	−	1.39544i
$188$		0			0
$189$	−4.73607	−	8.20311i	−0.344498	−	0.596688i
$190$		0			0
$191$	−7.26393	+	12.5815i	−0.525600	+	0.910365i	0.473956	+	0.880549i	$0.342826\pi$
$191$	−7.26393	+	12.5815i	−0.525600	+	0.910365i	−0.999555	+	0.0298167i	$0.990508\pi$
$192$		0			0
$193$	2.70820			0.194941			0.0974704	−	0.995238i	$-0.468925\pi$
$193$	2.70820			0.194941			0.0974704	+	0.995238i	$0.468925\pi$
$194$		0			0
$195$	−2.23607	+	3.87298i	−0.160128	+	0.277350i
$196$		0			0
$197$	1.47214	+	2.54981i	0.104885	+	0.181667i	0.913691	−	0.406409i	$-0.133219\pi$
$197$	1.47214	+	2.54981i	0.104885	+	0.181667i	−0.808806	+	0.588076i	$0.799886\pi$
$198$		0			0
$199$	−1.94427			−0.137826			−0.0689129	−	0.997623i	$-0.521953\pi$
$199$	−1.94427			−0.137826			−0.0689129	+	0.997623i	$0.521953\pi$
$200$		0			0
$201$	−5.04508	−	8.73834i	−0.355853	−	0.616355i
$202$		0			0
$203$	−6.35410	+	11.0056i	−0.445971	+	0.772444i
$204$		0			0
$205$	−5.85410	+	10.1396i	−0.408868	+	0.708181i
$206$		0			0
$207$	−16.8885			−1.17383
$208$		0			0
$209$	−2.23607	+	3.87298i	−0.154672	+	0.267900i
$210$		0			0
$211$	9.23607			0.635837			0.317919	−	0.948118i	$-0.397016\pi$
$211$	9.23607			0.635837			0.317919	+	0.948118i	$0.397016\pi$
$212$		0			0
$213$	28.2705			1.93706
$214$		0			0
$215$	−3.09017	+	7.49326i	−0.210748	+	0.511036i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	4.85410			0.328010
$220$		0			0
$221$	−4.20820	+	7.28882i	−0.283074	+	0.490299i
$222$		0			0
$223$	−2.76393			−0.185087			−0.0925433	−	0.995709i	$-0.529500\pi$
$223$	−2.76393			−0.185087			−0.0925433	+	0.995709i	$0.529500\pi$
$224$		0			0
$225$	6.69098	−	11.5891i	0.446066	−	0.772608i
$226$		0			0
$227$	−0.736068	+	1.27491i	−0.0488545	+	0.0846186i	−0.889419	−	0.457094i	$-0.848890\pi$
$227$	−0.736068	+	1.27491i	−0.0488545	+	0.0846186i	0.840564	+	0.541712i	$0.182224\pi$
$228$		0			0
$229$	1.14590	+	1.98475i	0.0757231	+	0.131156i	0.901400	−	0.432986i	$-0.142540\pi$
$229$	1.14590	+	1.98475i	0.0757231	+	0.131156i	−0.825677	+	0.564143i	$0.809207\pi$
$230$		0			0
$231$	40.1246			2.64001
$232$		0			0
$233$	−9.01722	−	15.6183i	−0.590738	−	1.02319i	−0.994133	−	0.108162i	$-0.965503\pi$
$233$	−9.01722	−	15.6183i	−0.590738	−	1.02319i	0.403395	−	0.915026i	$-0.367830\pi$
$234$		0			0
$235$	0.708204	−	1.22665i	0.0461981	−	0.0800175i
$236$		0			0
$237$	3.61803			0.235017
$238$		0			0
$239$	−4.14590	+	7.18091i	−0.268176	+	0.464494i	−0.968391	−	0.249438i	$-0.919754\pi$
$239$	−4.14590	+	7.18091i	−0.268176	+	0.464494i	0.700215	+	0.713932i	$0.253087\pi$
$240$		0			0
$241$	−2.13525	−	3.69837i	−0.137544	−	0.238233i	0.789022	−	0.614364i	$-0.210587\pi$
$241$	−2.13525	−	3.69837i	−0.137544	−	0.238233i	−0.926566	+	0.376131i	$0.877254\pi$
$242$		0			0
$243$	10.8262	+	18.7516i	0.694503	+	1.20292i
$244$		0			0
$245$	−6.76393	−	11.7155i	−0.432132	−	0.748474i
$246$		0			0
$247$	1.70820			0.108690
$248$		0			0
$249$	20.9894	+	36.3546i	1.33015	+	2.30388i
$250$		0			0
$251$	−14.5902	−	25.2709i	−0.920923	−	1.59509i	−0.797990	−	0.602671i	$-0.794103\pi$
$251$	−14.5902	−	25.2709i	−0.920923	−	1.59509i	−0.122934	−	0.992415i	$-0.539230\pi$
$252$		0			0
$253$	7.92705	−	13.7301i	0.498369	−	0.863201i
$254$		0			0
$255$	−9.85410	+	17.0678i	−0.617088	+	1.06883i
$256$		0			0
$257$	3.43769			0.214437			0.107219	−	0.994235i	$-0.465805\pi$
$257$	3.43769			0.214437			0.107219	+	0.994235i	$0.465805\pi$
$258$		0			0
$259$	−20.5623			−1.27768
$260$		0			0
$261$	−5.78115	+	10.0133i	−0.357844	+	0.619805i
$262$		0			0
$263$	13.7533	−	23.8214i	0.848064	−	1.46889i	−0.0348693	−	0.999392i	$-0.511101\pi$
$263$	13.7533	−	23.8214i	0.848064	−	1.46889i	0.882933	−	0.469498i	$-0.155565\pi$
$264$		0			0
$265$	0.854102	+	1.47935i	0.0524671	+	0.0908756i
$266$		0			0
$267$	2.42705	+	4.20378i	0.148533	+	0.257267i
$268$		0			0
$269$	11.5623			0.704966			0.352483	−	0.935818i	$-0.385337\pi$
$269$	11.5623			0.704966			0.352483	+	0.935818i	$0.385337\pi$
$270$		0			0
$271$	6.92705	+	11.9980i	0.420788	+	0.728827i	0.996017	−	0.0891660i	$-0.0284202\pi$
$271$	6.92705	+	11.9980i	0.420788	+	0.728827i	−0.575228	+	0.817993i	$0.695087\pi$
$272$		0			0
$273$	−7.66312	−	13.2729i	−0.463793	−	0.803313i
$274$		0			0
$275$	6.28115	+	10.8793i	0.378768	+	0.656045i
$276$		0			0
$277$	6.23607	−	10.8012i	0.374689	−	0.648980i	−0.615591	−	0.788065i	$-0.711083\pi$
$277$	6.23607	−	10.8012i	0.374689	−	0.648980i	0.990280	+	0.139085i	$0.0444161\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3.73607	+	6.47106i	−0.222875	+	0.386031i	−0.955680	−	0.294408i	$-0.904878\pi$
$281$	−3.73607	+	6.47106i	−0.222875	+	0.386031i	0.732805	+	0.680439i	$0.238211\pi$
$282$		0			0
$283$	5.38197	+	9.32184i	0.319925	+	0.554126i	0.980472	−	0.196659i	$-0.0630091\pi$
$283$	5.38197	+	9.32184i	0.319925	+	0.554126i	−0.660547	+	0.750784i	$0.729676\pi$
$284$		0			0
$285$	4.00000			0.236940
$286$		0			0
$287$	−20.0623	−	34.7489i	−1.18424	−	2.05116i
$288$		0			0
$289$	−10.0451	+	17.3986i	−0.590887	+	1.02345i
$290$		0			0
$291$	12.0902	−	20.9408i	0.708738	−	1.22757i
$292$		0			0
$293$	−12.0902			−0.706315			−0.353158	−	0.935564i	$-0.614892\pi$
$293$	−12.0902			−0.706315			−0.353158	+	0.935564i	$0.614892\pi$
$294$		0			0
$295$	3.14590	−	5.44886i	0.183161	−	0.317245i
$296$		0			0
$297$	8.09017			0.469439
$298$		0			0
$299$	−6.05573			−0.350212
$300$		0			0
$301$	−16.9443	−	22.0113i	−0.976652	−	1.26871i
$302$		0			0
$303$	9.85410			0.566103
$304$		0			0
$305$	3.52786			0.202005
$306$		0			0
$307$	−11.6074	+	20.1046i	−0.662469	+	1.14743i	0.317496	+	0.948260i	$0.397158\pi$
$307$	−11.6074	+	20.1046i	−0.662469	+	1.14743i	−0.979965	+	0.199170i	$0.936175\pi$
$308$		0			0
$309$	42.9787			2.44497
$310$		0			0
$311$	−14.9164	+	25.8360i	−0.845832	+	1.46502i	0.0390649	+	0.999237i	$0.487562\pi$
$311$	−14.9164	+	25.8360i	−0.845832	+	1.46502i	−0.884897	+	0.465787i	$0.845771\pi$
$312$		0			0
$313$	−6.56231	+	11.3662i	−0.370923	+	0.642458i	−0.989708	−	0.143103i	$-0.954292\pi$
$313$	−6.56231	+	11.3662i	−0.370923	+	0.642458i	0.618784	+	0.785561i	$0.287625\pi$
$314$		0			0
$315$	−10.0902	−	17.4767i	−0.568517	−	0.984700i
$316$		0			0
$317$	33.3050			1.87059			0.935296	−	0.353866i	$-0.115133\pi$
$317$	33.3050			1.87059			0.935296	+	0.353866i	$0.115133\pi$
$318$		0			0
$319$	−5.42705	−	9.39993i	−0.303857	−	0.526295i
$320$		0			0
$321$	9.85410	−	17.0678i	0.550002	−	0.952632i
$322$		0			0
$323$	7.52786			0.418862
$324$		0			0
$325$	2.39919	−	4.15551i	0.133083	−	0.230506i
$326$		0			0
$327$	18.1353	+	31.4112i	1.00288	+	1.73704i
$328$		0			0
$329$	2.42705	+	4.20378i	0.133808	+	0.231762i
$330$		0			0
$331$	−2.73607	−	4.73901i	−0.150388	−	0.260479i	0.780982	−	0.624553i	$-0.214719\pi$
$331$	−2.73607	−	4.73901i	−0.150388	−	0.260479i	−0.931370	+	0.364074i	$0.881386\pi$
$332$		0			0
$333$	−18.7082			−1.02520
$334$		0			0
$335$	−2.38197	−	4.12569i	−0.130141	−	0.225410i
$336$		0			0
$337$	−14.0000	−	24.2487i	−0.762629	−	1.32091i	−0.941491	−	0.337037i	$-0.890575\pi$
$337$	−14.0000	−	24.2487i	−0.762629	−	1.32091i	0.178863	−	0.983874i	$-0.442758\pi$
$338$		0			0
$339$	20.4443	−	35.4105i	1.11038	−	1.92324i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	16.7082			0.902158
$344$		0			0
$345$	−14.1803			−0.763444
$346$		0			0
$347$	5.04508	−	8.73834i	0.270834	−	0.469099i	−0.698241	−	0.715862i	$-0.746034\pi$
$347$	5.04508	−	8.73834i	0.270834	−	0.469099i	0.969076	+	0.246764i	$0.0793671\pi$
$348$		0			0
$349$	−10.3541	+	17.9338i	−0.554242	+	0.959976i	0.443720	+	0.896166i	$0.353659\pi$
$349$	−10.3541	+	17.9338i	−0.554242	+	0.959976i	−0.997962	+	0.0638103i	$0.979675\pi$
$350$		0			0
$351$	−1.54508	−	2.67617i	−0.0824705	−	0.142843i
$352$		0			0
$353$	8.61803	+	14.9269i	0.458692	+	0.794477i	0.998892	−	0.0470591i	$-0.0149849\pi$
$353$	8.61803	+	14.9269i	0.458692	+	0.794477i	−0.540200	+	0.841536i	$0.681652\pi$
$354$		0			0
$355$	13.3475			0.708413
$356$		0			0
$357$	−33.7705	−	58.4922i	−1.78732	−	3.09574i
$358$		0			0
$359$	−2.67376	−	4.63109i	−0.141116	−	0.244420i	0.786801	−	0.617206i	$-0.211736\pi$
$359$	−2.67376	−	4.63109i	−0.141116	−	0.244420i	−0.927917	+	0.372787i	$0.878402\pi$
$360$		0			0
$361$	8.73607	+	15.1313i	0.459793	+	0.796385i
$362$		0			0
$363$	−2.73607	+	4.73901i	−0.143606	+	0.248733i
$364$		0			0
$365$	2.29180			0.119958
$366$		0			0
$367$	9.59017	−	16.6107i	0.500603	−	0.867069i	−0.499397	−	0.866373i	$-0.666445\pi$
$367$	9.59017	−	16.6107i	0.500603	−	0.867069i	1.00000		0.000696189i	$-0.000221604\pi$
$368$		0			0
$369$	−18.2533	−	31.6156i	−0.950228	−	1.64584i
$370$		0			0
$371$	−5.85410			−0.303930
$372$		0			0
$373$	−3.00000	−	5.19615i	−0.155334	−	0.269047i	0.777847	−	0.628454i	$-0.216312\pi$
$373$	−3.00000	−	5.19615i	−0.155334	−	0.269047i	−0.933181	+	0.359408i	$0.882979\pi$
$374$		0			0
$375$	13.7082	−	23.7433i	0.707889	−	1.22610i
$376$		0			0
$377$	−2.07295	+	3.59045i	−0.106762	+	0.184918i
$378$		0			0
$379$	27.3607			1.40542			0.702712	−	0.711475i	$-0.251972\pi$
$379$	27.3607			1.40542			0.702712	+	0.711475i	$0.251972\pi$
$380$		0			0
$381$	−19.1803	+	33.2213i	−0.982639	+	1.70198i
$382$		0			0
$383$	−38.1246			−1.94808			−0.974038	−	0.226383i	$-0.927310\pi$
$383$	−38.1246			−1.94808			−0.974038	+	0.226383i	$0.927310\pi$
$384$		0			0
$385$	18.9443			0.965489
$386$		0			0
$387$	−15.4164	−	20.0265i	−0.783660	−	1.01800i
$388$		0			0
$389$	−35.0689			−1.77806			−0.889031	−	0.457846i	$-0.848621\pi$
$389$	−35.0689			−1.77806			−0.889031	+	0.457846i	$0.848621\pi$
$390$		0			0
$391$	−26.6869			−1.34962
$392$		0			0
$393$	13.0172	−	22.5465i	0.656632	−	1.13732i
$394$		0			0
$395$	1.70820			0.0859491
$396$		0			0
$397$	−16.2082	+	28.0734i	−0.813466	+	1.40897i	0.0969574	+	0.995289i	$0.469089\pi$
$397$	−16.2082	+	28.0734i	−0.813466	+	1.40897i	−0.910424	+	0.413677i	$0.864244\pi$
$398$		0			0
$399$	−6.85410	+	11.8717i	−0.343134	+	0.594326i
$400$		0			0
$401$	12.7082	+	22.0113i	0.634617	+	1.09919i	0.986596	+	0.163182i	$0.0521756\pi$
$401$	12.7082	+	22.0113i	0.634617	+	1.09919i	−0.351979	+	0.936008i	$0.614491\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	3.52786	+	6.11044i	0.175301	+	0.303630i
$406$		0			0
$407$	8.78115	−	15.2094i	0.435266	−	0.753902i
$408$		0			0
$409$	8.90983			0.440563			0.220281	−	0.975436i	$-0.429302\pi$
$409$	8.90983			0.440563			0.220281	+	0.975436i	$0.429302\pi$
$410$		0			0
$411$	−4.85410	+	8.40755i	−0.239435	+	0.414714i
$412$		0			0
$413$	10.7812	+	18.6735i	0.530506	+	0.918863i
$414$		0			0
$415$	9.90983	+	17.1643i	0.486454	+	0.842564i
$416$		0			0
$417$	−6.92705	−	11.9980i	−0.339219	−	0.587545i
$418$		0			0
$419$	10.2361			0.500065			0.250032	−	0.968237i	$-0.419559\pi$
$419$	10.2361			0.500065			0.250032	+	0.968237i	$0.419559\pi$
$420$		0			0
$421$	3.82624	+	6.62724i	0.186479	+	0.322992i	0.944074	−	0.329734i	$-0.106959\pi$
$421$	3.82624	+	6.62724i	0.186479	+	0.322992i	−0.757595	+	0.652725i	$0.773626\pi$
$422$		0			0
$423$	2.20820	+	3.82472i	0.107367	+	0.185964i
$424$		0			0
$425$	10.5729	−	18.3129i	0.512863	−	0.888305i
$426$		0			0
$427$	−6.04508	+	10.4704i	−0.292542	+	0.506698i
$428$		0			0
$429$	13.0902			0.631999
$430$		0			0
$431$	24.7984			1.19450			0.597248	−	0.802057i	$-0.296261\pi$
$431$	24.7984			1.19450			0.597248	+	0.802057i	$0.296261\pi$
$432$		0			0
$433$	17.3541	−	30.0582i	0.833985	−	1.44450i	−0.0608693	−	0.998146i	$-0.519387\pi$
$433$	17.3541	−	30.0582i	0.833985	−	1.44450i	0.894854	−	0.446359i	$-0.147279\pi$
$434$		0			0
$435$	−4.85410	+	8.40755i	−0.232736	+	0.403111i
$436$		0			0
$437$	2.70820	+	4.69075i	0.129551	+	0.224389i
$438$		0			0
$439$	0.572949	+	0.992377i	0.0273454	+	0.0473636i	0.879374	−	0.476131i	$-0.157961\pi$
$439$	0.572949	+	0.992377i	0.0273454	+	0.0473636i	−0.852029	+	0.523495i	$0.824628\pi$
$440$		0			0
$441$	42.1803			2.00859
$442$		0			0
$443$	1.23607	+	2.14093i	0.0587274	+	0.101719i	0.893894	−	0.448278i	$-0.147962\pi$
$443$	1.23607	+	2.14093i	0.0587274	+	0.101719i	−0.835167	+	0.549996i	$0.814629\pi$
$444$		0			0
$445$	1.14590	+	1.98475i	0.0543208	+	0.0940863i
$446$		0			0
$447$	−11.7812	−	20.4056i	−0.557229	−	0.965150i
$448$		0			0
$449$	16.4164	−	28.4341i	0.774738	−	1.34189i	−0.160203	−	0.987084i	$-0.551215\pi$
$449$	16.4164	−	28.4341i	0.774738	−	1.34189i	0.934942	−	0.354802i	$-0.115452\pi$
$450$		0			0
$451$	34.2705			1.61374
$452$		0			0
$453$	−11.5902	+	20.0748i	−0.544554	+	0.943195i
$454$		0			0
$455$	−3.61803	−	6.26662i	−0.169616	−	0.293784i
$456$		0			0
$457$	−15.7082			−0.734799			−0.367399	−	0.930063i	$-0.619752\pi$
$457$	−15.7082			−0.734799			−0.367399	+	0.930063i	$0.619752\pi$
$458$		0			0
$459$	−6.80902	−	11.7936i	−0.317818	−	0.550476i
$460$		0			0
$461$	15.7984	−	27.3636i	0.735804	−	1.27445i	−0.218566	−	0.975822i	$-0.570138\pi$
$461$	15.7984	−	27.3636i	0.735804	−	1.27445i	0.954370	−	0.298627i	$-0.0965287\pi$
$462$		0			0
$463$	−14.9164	+	25.8360i	−0.693224	+	1.20070i	0.277551	+	0.960711i	$0.410477\pi$
$463$	−14.9164	+	25.8360i	−0.693224	+	1.20070i	−0.970776	+	0.239989i	$0.922856\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.52786	+	13.0386i	−0.348348	+	0.603356i	−0.985956	−	0.167004i	$-0.946591\pi$
$467$	−7.52786	+	13.0386i	−0.348348	+	0.603356i	0.637608	+	0.770361i	$0.279924\pi$
$468$		0			0
$469$	16.3262			0.753876
$470$		0			0
$471$	−13.4721			−0.620763
$472$		0			0
$473$	23.5172	−	3.13331i	1.08132	−	0.144070i
$474$		0			0
$475$	−4.29180			−0.196921
$476$		0			0
$477$	−5.32624			−0.243872
$478$		0			0
$479$	−6.90983	+	11.9682i	−0.315718	+	0.546840i	−0.979590	−	0.201007i	$-0.935579\pi$
$479$	−6.90983	+	11.9682i	−0.315718	+	0.546840i	0.663872	+	0.747846i	$0.268912\pi$
$480$		0			0
$481$	−6.70820			−0.305868
$482$		0			0
$483$	24.2984	−	42.0860i	1.10561	−	1.91498i
$484$		0			0
$485$	5.70820	−	9.88690i	0.259196	−	0.448941i
$486$		0			0
$487$	0.663119	+	1.14856i	0.0300488	+	0.0520460i	0.880659	−	0.473751i	$-0.157100\pi$
$487$	0.663119	+	1.14856i	0.0300488	+	0.0520460i	−0.850610	+	0.525797i	$0.823767\pi$
$488$		0			0
$489$	−36.6525			−1.65748
$490$		0			0
$491$	10.8262	+	18.7516i	0.488581	+	0.846248i	0.999914	−	0.0131354i	$-0.00418125\pi$
$491$	10.8262	+	18.7516i	0.488581	+	0.846248i	−0.511332	+	0.859383i	$0.670848\pi$
$492$		0			0
$493$	−9.13525	+	15.8227i	−0.411431	+	0.712620i
$494$		0			0
$495$	17.2361			0.774704
$496$		0			0
$497$	−22.8713	+	39.6143i	−1.02592	+	1.77694i
$498$		0			0
$499$	6.92705	+	11.9980i	0.310097	+	0.537104i	0.978383	−	0.206800i	$-0.0663050\pi$
$499$	6.92705	+	11.9980i	0.310097	+	0.537104i	−0.668286	+	0.743905i	$0.732972\pi$
$500$		0			0
$501$	18.6353	+	32.2772i	0.832562	+	1.44204i
$502$		0			0
$503$	−6.73607	−	11.6672i	−0.300346	−	0.520215i	0.675868	−	0.737023i	$-0.263769\pi$
$503$	−6.73607	−	11.6672i	−0.300346	−	0.520215i	−0.976214	+	0.216807i	$0.930436\pi$
$504$		0			0
$505$	4.65248			0.207032
$506$		0			0
$507$	14.5172	+	25.1446i	0.644732	+	1.11671i
$508$		0			0
$509$	−9.35410	−	16.2018i	−0.414613	−	0.718131i	0.580774	−	0.814064i	$-0.302750\pi$
$509$	−9.35410	−	16.2018i	−0.414613	−	0.718131i	−0.995388	+	0.0959332i	$0.969416\pi$
$510$		0			0
$511$	−3.92705	+	6.80185i	−0.173723	+	0.300896i
$512$		0			0
$513$	−1.38197	+	2.39364i	−0.0610153	+	0.105682i
$514$		0			0
$515$	20.2918			0.894163
$516$		0			0
$517$	−4.14590			−0.182336
$518$		0			0
$519$	4.92705	−	8.53390i	0.216274	−	0.374597i
$520$		0			0
$521$	11.0172	−	19.0824i	0.482673	−	0.836015i	−0.517129	−	0.855908i	$-0.672999\pi$
$521$	11.0172	−	19.0824i	0.482673	−	0.836015i	0.999802	+	0.0198930i	$0.00633256\pi$
$522$		0			0
$523$	17.9164	+	31.0321i	0.783430	+	1.35694i	0.929933	+	0.367730i	$0.119865\pi$
$523$	17.9164	+	31.0321i	0.783430	+	1.35694i	−0.146503	+	0.989210i	$0.546802\pi$
$524$		0			0
$525$	19.2533	+	33.3477i	0.840282	+	1.45541i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	1.89919	+	3.28949i	0.0825733	+	0.143021i
$530$		0			0
$531$	9.80902	+	16.9897i	0.425675	+	0.737291i
$532$		0			0
$533$	−6.54508	−	11.3364i	−0.283499	−	0.491035i
$534$		0			0
$535$	4.65248	−	8.05832i	0.201144	−	0.348392i
$536$		0			0
$537$	−25.2705			−1.09050
$538$		0			0
$539$	−19.7984	+	34.2918i	−0.852776	+	1.47705i
$540$		0			0
$541$	18.9721	+	32.8607i	0.815676	+	1.41279i	0.908842	+	0.417141i	$0.136968\pi$
$541$	18.9721	+	32.8607i	0.815676	+	1.41279i	−0.0931658	+	0.995651i	$0.529699\pi$
$542$		0			0
$543$	−50.7426			−2.17758
$544$		0			0
$545$	8.56231	+	14.8303i	0.366769	+	0.635262i
$546$		0			0
$547$	10.5000	−	18.1865i	0.448948	−	0.777600i	−0.549370	−	0.835579i	$-0.685132\pi$
$547$	10.5000	−	18.1865i	0.448948	−	0.777600i	0.998318	+	0.0579790i	$0.0184657\pi$
$548$		0			0
$549$	−5.50000	+	9.52628i	−0.234734	+	0.406572i
$550$		0			0
$551$	3.70820			0.157975
$552$		0			0
$553$	−2.92705	+	5.06980i	−0.124471	+	0.215590i
$554$		0			0
$555$	−15.7082			−0.666776
$556$		0			0
$557$	−38.5623			−1.63394			−0.816969	−	0.576682i	$-0.804347\pi$
$557$	−38.5623			−1.63394			−0.816969	+	0.576682i	$0.804347\pi$
$558$		0			0
$559$	−5.52786	−	7.18091i	−0.233804	−	0.303720i
$560$		0			0
$561$	57.6869			2.43554
$562$		0			0
$563$	1.32624			0.0558943			0.0279471	−	0.999609i	$-0.491103\pi$
$563$	1.32624			0.0558943			0.0279471	+	0.999609i	$0.491103\pi$
$564$		0			0
$565$	9.65248	−	16.7186i	0.406083	−	0.703356i
$566$		0			0
$567$	−24.1803			−1.01548
$568$		0			0
$569$	12.9721	−	22.4684i	0.543820	−	0.941924i	−0.454860	−	0.890563i	$-0.650311\pi$
$569$	12.9721	−	22.4684i	0.543820	−	0.941924i	0.998680	−	0.0513613i	$-0.0163560\pi$
$570$		0			0
$571$	−21.5902	+	37.3953i	−0.903520	+	1.56494i	−0.0806295	+	0.996744i	$0.525693\pi$
$571$	−21.5902	+	37.3953i	−0.903520	+	1.56494i	−0.822891	+	0.568199i	$0.807640\pi$
$572$		0			0
$573$	−19.0172	−	32.9388i	−0.794456	−	1.37604i
$574$		0			0
$575$	15.2148			0.634500
$576$		0			0
$577$	14.1074	+	24.4347i	0.587298	+	1.01723i	0.994585	+	0.103930i	$0.0331418\pi$
$577$	14.1074	+	24.4347i	0.587298	+	1.01723i	−0.407286	+	0.913301i	$0.633525\pi$
$578$		0			0
$579$	−3.54508	+	6.14027i	−0.147329	+	0.255181i
$580$		0			0
$581$	−67.9230			−2.81792
$582$		0			0
$583$	2.50000	−	4.33013i	0.103539	−	0.179336i
$584$		0			0
$585$	−3.29180	−	5.70156i	−0.136099	−	0.235730i
$586$		0			0
$587$	7.87132	+	13.6335i	0.324884	+	0.562716i	0.981489	−	0.191519i	$-0.0613413\pi$
$587$	7.87132	+	13.6335i	0.324884	+	0.562716i	−0.656605	+	0.754235i	$0.728008\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−7.70820			−0.317073
$592$		0			0
$593$	15.2361	+	26.3896i	0.625670	+	1.08369i	0.988411	+	0.151803i	$0.0485078\pi$
$593$	15.2361	+	26.3896i	0.625670	+	1.08369i	−0.362741	+	0.931890i	$0.618159\pi$
$594$		0			0
$595$	−15.9443	−	27.6163i	−0.653651	−	1.13216i
$596$		0			0
$597$	2.54508	−	4.40822i	0.104163	−	0.180416i
$598$		0			0
$599$	18.1180	−	31.3814i	0.740283	−	1.28221i	−0.212084	−	0.977252i	$-0.568025\pi$
$599$	18.1180	−	31.3814i	0.740283	−	1.28221i	0.952366	−	0.304956i	$-0.0986417\pi$
$600$		0			0
$601$	29.4164			1.19992			0.599960	−	0.800030i	$-0.295183\pi$
$601$	29.4164			1.19992			0.599960	+	0.800030i	$0.295183\pi$
$602$		0			0
$603$	14.8541			0.604906
$604$		0			0
$605$	−1.29180	+	2.23746i	−0.0525190	+	0.0909655i
$606$		0			0
$607$	−11.9271	+	20.6583i	−0.484104	+	0.838493i	−0.999833	−	0.0182588i	$-0.994188\pi$
$607$	−11.9271	+	20.6583i	−0.484104	+	0.838493i	0.515729	+	0.856752i	$0.327521\pi$
$608$		0			0
$609$	−16.6353	−	28.8131i	−0.674095	−	1.16757i
$610$		0			0
$611$	0.791796	+	1.37143i	0.0320326	+	0.0554822i
$612$		0			0
$613$	5.20163			0.210092			0.105046	−	0.994467i	$-0.466501\pi$
$613$	5.20163			0.210092			0.105046	+	0.994467i	$0.466501\pi$
$614$		0			0
$615$	−15.3262	−	26.5458i	−0.618014	−	1.07043i
$616$		0			0
$617$	−4.85410	−	8.40755i	−0.195419	−	0.338475i	0.751619	−	0.659598i	$-0.229273\pi$
$617$	−4.85410	−	8.40755i	−0.195419	−	0.338475i	−0.947038	+	0.321122i	$0.895940\pi$
$618$		0			0
$619$	−15.1353	−	26.2150i	−0.608337	−	1.05367i	−0.991514	−	0.129996i	$-0.958503\pi$
$619$	−15.1353	−	26.2150i	−0.608337	−	1.05367i	0.383177	−	0.923675i	$-0.374830\pi$
$620$		0			0
$621$	4.89919	−	8.48564i	0.196598	−	0.340517i
$622$		0			0
$623$	−7.85410			−0.314668
$624$		0			0
$625$	−2.20820	+	3.82472i	−0.0883282	+	0.152989i
$626$		0			0
$627$	−5.85410	−	10.1396i	−0.233790	−	0.404937i
$628$		0			0
$629$	−29.5623			−1.17873
$630$		0			0
$631$	−13.5451	−	23.4608i	−0.539221	−	0.933959i	−0.998946	−	0.0458973i	$-0.985385\pi$
$631$	−13.5451	−	23.4608i	−0.539221	−	0.933959i	0.459725	−	0.888061i	$-0.347948\pi$
$632$		0			0
$633$	−12.0902	+	20.9408i	−0.480541	+	0.832322i
$634$		0			0
$635$	−9.05573	+	15.6850i	−0.359366	+	0.622439i
$636$		0			0
$637$	15.1246			0.599259
$638$		0			0
$639$	−20.8090	+	36.0423i	−0.823192	+	1.42581i
$640$		0			0
$641$	10.8197			0.427351			0.213675	−	0.976905i	$-0.431457\pi$
$641$	10.8197			0.427351			0.213675	+	0.976905i	$0.431457\pi$
$642$		0			0
$643$	−9.43769			−0.372186			−0.186093	−	0.982532i	$-0.559583\pi$
$643$	−9.43769			−0.372186			−0.186093	+	0.982532i	$0.559583\pi$
$644$		0			0
$645$	−12.9443	−	16.8151i	−0.509680	−	0.662094i
$646$		0			0
$647$	21.6738			0.852084			0.426042	−	0.904703i	$-0.359908\pi$
$647$	21.6738			0.852084			0.426042	+	0.904703i	$0.359908\pi$
$648$		0			0
$649$	−18.4164			−0.722907
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−32.8885			−1.28703			−0.643514	−	0.765434i	$-0.722524\pi$
$653$	−32.8885			−1.28703			−0.643514	+	0.765434i	$0.722524\pi$
$654$		0			0
$655$	6.14590	−	10.6450i	0.240140	−	0.415935i
$656$		0			0
$657$	−3.57295	+	6.18853i	−0.139394	+	0.241438i
$658$		0			0
$659$	−14.6803	−	25.4271i	−0.571865	−	0.990499i	−0.996375	−	0.0850756i	$-0.972887\pi$
$659$	−14.6803	−	25.4271i	−0.571865	−	0.990499i	0.424510	−	0.905423i	$-0.360447\pi$
$660$		0			0
$661$	−43.3951			−1.68787			−0.843937	−	0.536442i	$-0.819768\pi$
$661$	−43.3951			−1.68787			−0.843937	+	0.536442i	$0.819768\pi$
$662$		0			0
$663$	−11.0172	−	19.0824i	−0.427873	−	0.741098i
$664$		0			0
$665$	−3.23607	+	5.60503i	−0.125489	+	0.217354i
$666$		0			0
$667$	−13.1459			−0.509011
$668$		0			0
$669$	3.61803	−	6.26662i	0.139881	−	0.242281i
$670$		0			0
$671$	−5.16312	−	8.94278i	−0.199320	−	0.345232i
$672$		0			0
$673$	−6.91641	−	11.9796i	−0.266608	−	0.461778i	0.701376	−	0.712792i	$-0.252570\pi$
$673$	−6.91641	−	11.9796i	−0.266608	−	0.461778i	−0.967984	+	0.251013i	$0.919236\pi$
$674$		0			0
$675$	3.88197	+	6.72376i	0.149417	+	0.258798i
$676$		0			0
$677$	−31.7639			−1.22079			−0.610394	−	0.792098i	$-0.708989\pi$
$677$	−31.7639			−1.22079			−0.610394	+	0.792098i	$0.708989\pi$
$678$		0			0
$679$	19.5623	+	33.8829i	0.750732	+	1.30031i
$680$		0			0
$681$	−1.92705	−	3.33775i	−0.0738448	−	0.127903i
$682$		0			0
$683$	−5.56231	+	9.63420i	−0.212836	+	0.368642i	−0.952601	−	0.304223i	$-0.901603\pi$
$683$	−5.56231	+	9.63420i	−0.212836	+	0.368642i	0.739765	+	0.672865i	$0.234937\pi$
$684$		0			0
$685$	−2.29180	+	3.96951i	−0.0875650	+	0.151667i
$686$		0			0
$687$	−6.00000			−0.228914
$688$		0			0
$689$	−1.90983			−0.0727587
$690$		0			0
$691$	7.86475	−	13.6221i	0.299189	−	0.518211i	−0.676762	−	0.736202i	$-0.736617\pi$
$691$	7.86475	−	13.6221i	0.299189	−	0.518211i	0.975951	+	0.217992i	$0.0699506\pi$
$692$		0			0
$693$	−29.5344	+	51.1552i	−1.12192	+	1.94322i
$694$		0			0
$695$	−3.27051	−	5.66469i	−0.124058	−	0.214874i
$696$		0			0
$697$	−28.8435	−	49.9583i	−1.09252	−	1.89231i
$698$		0			0
$699$	47.2148			1.78583
$700$		0			0
$701$	7.25329	+	12.5631i	0.273953	+	0.474500i	0.969870	−	0.243621i	$-0.0783354\pi$
$701$	7.25329	+	12.5631i	0.273953	+	0.474500i	−0.695917	+	0.718122i	$0.745002\pi$
$702$		0			0
$703$	3.00000	+	5.19615i	0.113147	+	0.195977i
$704$		0			0
$705$	1.85410	+	3.21140i	0.0698295	+	0.120948i
$706$		0			0
$707$	−7.97214	+	13.8081i	−0.299823	+	0.519309i
$708$		0			0
$709$	28.8885			1.08493			0.542466	−	0.840078i	$-0.317491\pi$
$709$	28.8885			1.08493			0.542466	+	0.840078i	$0.317491\pi$
$710$		0			0
$711$	−2.66312	+	4.61266i	−0.0998748	+	0.172988i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	6.18034			0.231132
$716$		0			0
$717$	−10.8541	−	18.7999i	−0.405354	−	0.702093i
$718$		0			0
$719$	−5.39919	+	9.35167i	−0.201356	+	0.348758i	−0.948965	−	0.315380i	$-0.897868\pi$
$719$	−5.39919	+	9.35167i	−0.201356	+	0.348758i	0.747610	+	0.664138i	$0.231201\pi$
$720$		0			0
$721$	−34.7705	+	60.2243i	−1.29492	+	2.24287i
$722$		0			0
$723$	11.1803			0.415801
$724$		0			0
$725$	5.20820	−	9.02087i	0.193428	−	0.335027i
$726$		0			0
$727$	17.9787			0.666794			0.333397	−	0.942787i	$-0.391805\pi$
$727$	17.9787			0.666794			0.333397	+	0.942787i	$0.391805\pi$
$728$		0			0
$729$	−39.5623			−1.46527
$730$		0			0
$731$	−24.3607	−	31.6455i	−0.901012	−	1.17045i
$732$		0			0
$733$	24.5410			0.906443			0.453222	−	0.891398i	$-0.350275\pi$
$733$	24.5410			0.906443			0.453222	+	0.891398i	$0.350275\pi$
$734$		0			0
$735$	35.4164			1.30635
$736$		0			0
$737$	−6.97214	+	12.0761i	−0.256822	+	0.444829i
$738$		0			0
$739$	−49.5410			−1.82240			−0.911198	−	0.411969i	$-0.864841\pi$
$739$	−49.5410			−1.82240			−0.911198	+	0.411969i	$0.864841\pi$
$740$		0			0
$741$	−2.23607	+	3.87298i	−0.0821440	+	0.142278i
$742$		0			0
$743$	16.6803	−	28.8912i	0.611942	−	1.05992i	−0.378970	−	0.925409i	$-0.623722\pi$
$743$	16.6803	−	28.8912i	0.611942	−	1.05992i	0.990913	−	0.134506i	$-0.0429449\pi$
$744$		0			0
$745$	−5.56231	−	9.63420i	−0.203787	−	0.352970i
$746$		0			0
$747$	−61.7984			−2.26108
$748$		0			0
$749$	15.9443	+	27.6163i	0.582591	+	1.00908i
$750$		0			0
$751$	−18.9894	+	32.8905i	−0.692931	+	1.20019i	0.277942	+	0.960598i	$0.410348\pi$
$751$	−18.9894	+	32.8905i	−0.692931	+	1.20019i	−0.970873	+	0.239595i	$0.922985\pi$
$752$		0			0
$753$	76.3951			2.78399
$754$		0			0
$755$	−5.47214	+	9.47802i	−0.199151	+	0.344940i
$756$		0			0
$757$	15.4894	+	26.8284i	0.562970	+	0.975093i	0.997235	+	0.0743080i	$0.0236748\pi$
$757$	15.4894	+	26.8284i	0.562970	+	0.975093i	−0.434265	+	0.900785i	$0.642992\pi$
$758$		0			0
$759$	20.7533	+	35.9458i	0.753297	+	1.30475i
$760$		0			0
$761$	−0.545085	−	0.944115i	−0.0197593	−	0.0342241i	0.855977	−	0.517014i	$-0.172957\pi$
$761$	−0.545085	−	0.944115i	−0.0197593	−	0.0342241i	−0.875736	+	0.482790i	$0.839623\pi$
$762$		0			0
$763$	−58.6869			−2.12461
$764$		0			0
$765$	−14.5066	−	25.1261i	−0.524486	−	0.908437i
$766$		0			0
$767$	3.51722	+	6.09201i	0.126999	+	0.219970i
$768$		0			0
$769$	−9.94427	+	17.2240i	−0.358600	+	0.621113i	−0.987727	−	0.156189i	$-0.950079\pi$
$769$	−9.94427	+	17.2240i	−0.358600	+	0.621113i	0.629128	+	0.777302i	$0.283412\pi$
$770$		0			0
$771$	−4.50000	+	7.79423i	−0.162064	+	0.280702i
$772$		0			0
$773$	15.5967			0.560976			0.280488	−	0.959858i	$-0.409504\pi$
$773$	15.5967			0.560976			0.280488	+	0.959858i	$0.409504\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	26.9164	−	46.6206i	0.965621	−	1.67250i
$778$		0			0
$779$	−5.85410	+	10.1396i	−0.209745	+	0.363289i
$780$		0			0
$781$	−19.5344	−	33.8346i	−0.698997	−	1.21070i
$782$		0			0
$783$	−3.35410	−	5.80948i	−0.119866	−	0.207614i
$784$		0			0
$785$	−6.36068			−0.227022
$786$		0			0
$787$	5.22542	+	9.05070i	0.186266	+	0.322623i	0.944002	−	0.329938i	$-0.107028\pi$
$787$	5.22542	+	9.05070i	0.186266	+	0.322623i	−0.757736	+	0.652561i	$0.773695\pi$
$788$		0			0
$789$	36.0066	+	62.3652i	1.28187	+	2.22026i
$790$		0			0
$791$	33.0795	+	57.2954i	1.17617	+	2.03719i
$792$		0			0
$793$	−1.97214	+	3.41584i	−0.0700326	+	0.121300i
$794$		0			0
$795$	−4.47214			−0.158610
$796$		0			0
$797$	4.11803	−	7.13264i	0.145868	−	0.252651i	−0.783828	−	0.620978i	$-0.786736\pi$
$797$	4.11803	−	7.13264i	0.145868	−	0.252651i	0.929697	+	0.368326i	$0.120069\pi$
$798$		0			0
$799$	3.48936	+	6.04374i	0.123445	+	0.213812i
$800$		0			0
$801$	−7.14590			−0.252488
$802$		0			0
$803$	−3.35410	−	5.80948i	−0.118364	−	0.205012i
$804$		0			0
$805$	11.4721	−	19.8703i	0.404340	−	0.700337i
$806$		0			0
$807$	−15.1353	+	26.2150i	−0.532786	+	0.922813i
$808$		0			0
$809$	4.74265			0.166743			0.0833713	−	0.996519i	$-0.473431\pi$
$809$	4.74265			0.166743			0.0833713	+	0.996519i	$0.473431\pi$
$810$		0			0
$811$	23.6976	−	41.0454i	0.832134	−	1.44130i	−0.0642089	−	0.997936i	$-0.520452\pi$
$811$	23.6976	−	41.0454i	0.832134	−	1.44130i	0.896343	−	0.443362i	$-0.146214\pi$
$812$		0			0
$813$	−36.2705			−1.27206
$814$		0			0
$815$	−17.3050			−0.606166
$816$		0			0
$817$	−3.09017	+	7.49326i	−0.108111	+	0.262156i
$818$		0			0
$819$	22.5623			0.788391
$820$		0			0
$821$	0.326238			0.0113858			0.00569289	−	0.999984i	$-0.498188\pi$
$821$	0.326238			0.0113858			0.00569289	+	0.999984i	$0.498188\pi$
$822$		0			0
$823$	26.0066	−	45.0447i	0.906532	−	1.57016i	0.0876853	−	0.996148i	$-0.472053\pi$
$823$	26.0066	−	45.0447i	0.906532	−	1.57016i	0.818847	−	0.574012i	$-0.194614\pi$
$824$		0			0
$825$	−32.8885			−1.14503
$826$		0			0
$827$	4.59017	−	7.95041i	0.159616	−	0.276463i	−0.775114	−	0.631821i	$-0.782308\pi$
$827$	4.59017	−	7.95041i	0.159616	−	0.276463i	0.934730	+	0.355358i	$0.115641\pi$
$828$		0			0
$829$	−14.0795	+	24.3865i	−0.489002	+	0.846977i	−0.999920	−	0.0126530i	$-0.995972\pi$
$829$	−14.0795	+	24.3865i	−0.489002	+	0.846977i	0.510918	+	0.859630i	$0.329306\pi$
$830$		0			0
$831$	16.3262	+	28.2779i	0.566351	+	0.980949i
$832$		0			0
$833$	66.6525			2.30937
$834$		0			0
$835$	8.79837	+	15.2392i	0.304480	+	0.527375i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−25.8885			−0.893772			−0.446886	−	0.894591i	$-0.647467\pi$
$839$	−25.8885			−0.893772			−0.446886	+	0.894591i	$0.647467\pi$
$840$		0			0
$841$	10.0000	−	17.3205i	0.344828	−	0.597259i
$842$		0			0
$843$	−9.78115	−	16.9415i	−0.336881	−	0.583495i
$844$		0			0
$845$	6.85410	+	11.8717i	0.235788	+	0.408397i
$846$		0			0
$847$	−4.42705	−	7.66788i	−0.152115	−	0.263471i
$848$		0			0
$849$	−28.1803			−0.967147
$850$		0			0
$851$	−10.6353	−	18.4208i	−0.364572	−	0.631457i
$852$		0			0
$853$	−5.62868	−	9.74915i	−0.192722	−	0.333805i	0.753429	−	0.657529i	$-0.228398\pi$
$853$	−5.62868	−	9.74915i	−0.192722	−	0.333805i	−0.946151	+	0.323724i	$0.895065\pi$
$854$		0			0
$855$	−2.94427	+	5.09963i	−0.100692	+	0.174404i
$856$		0			0
$857$	18.0902	−	31.3331i	0.617948	−	1.07032i	−0.371911	−	0.928268i	$-0.621297\pi$
$857$	18.0902	−	31.3331i	0.617948	−	1.07032i	0.989860	−	0.142050i	$-0.0453693\pi$
$858$		0			0
$859$	49.4164			1.68607			0.843033	−	0.537862i	$-0.180768\pi$
$859$	49.4164			1.68607			0.843033	+	0.537862i	$0.180768\pi$
$860$		0			0
$861$	105.048			3.58001
$862$		0			0
$863$	4.75329	−	8.23294i	0.161804	−	0.280252i	−0.773712	−	0.633538i	$-0.781602\pi$
$863$	4.75329	−	8.23294i	0.161804	−	0.280252i	0.935516	+	0.353285i	$0.114935\pi$
$864$		0			0
$865$	2.32624	−	4.02916i	0.0790945	−	0.136996i
$866$		0			0
$867$	−26.2984	−	45.5501i	−0.893140	−	1.54696i
$868$		0			0
$869$	−2.50000	−	4.33013i	−0.0848067	−	0.146889i
$870$		0			0
$871$	5.32624			0.180473
$872$		0			0
$873$	17.7984	+	30.8277i	0.602384	+	1.04336i
$874$		0			0
$875$	22.1803	+	38.4175i	0.749832	+	1.29875i
$876$		0			0
$877$	3.82624	+	6.62724i	0.129203	+	0.223786i	0.923368	−	0.383916i	$-0.125425\pi$
$877$	3.82624	+	6.62724i	0.129203	+	0.223786i	−0.794165	+	0.607702i	$0.792091\pi$
$878$		0			0
$879$	15.8262	−	27.4118i	0.533806	−	0.924579i
$880$		0			0
$881$	33.7639			1.13754			0.568768	−	0.822498i	$-0.307420\pi$
$881$	33.7639			1.13754			0.568768	+	0.822498i	$0.307420\pi$
$882$		0			0
$883$	6.56231	−	11.3662i	0.220839	−	0.382505i	−0.734224	−	0.678907i	$-0.762454\pi$
$883$	6.56231	−	11.3662i	0.220839	−	0.382505i	0.955063	+	0.296403i	$0.0957871\pi$
$884$		0			0
$885$	8.23607	+	14.2653i	0.276852	+	0.479522i
$886$		0			0
$887$	52.9574			1.77814			0.889068	−	0.457775i	$-0.151353\pi$
$887$	52.9574			1.77814			0.889068	+	0.457775i	$0.151353\pi$
$888$		0			0
$889$	−31.0344	−	53.7532i	−1.04086	−	1.80283i
$890$		0			0
$891$	10.3262	−	17.8856i	0.345942	−	0.599189i
$892$		0			0
$893$	0.708204	−	1.22665i	0.0236991	−	0.0410481i
$894$		0			0
$895$	−11.9311			−0.398813
$896$		0			0
$897$	7.92705	−	13.7301i	0.264677	−	0.458433i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−8.41641			−0.280391
$902$		0			0
$903$	72.0861	−	9.60437i	2.39888	−	0.319613i
$904$		0			0
$905$	−23.9574			−0.796372
$906$		0			0
$907$	21.7082			0.720809			0.360405	−	0.932796i	$-0.382639\pi$
$907$	21.7082			0.720809			0.360405	+	0.932796i	$0.382639\pi$
$908$		0			0
$909$	−7.25329	+	12.5631i	−0.240576	+	0.416691i
$910$		0			0
$911$	42.9787			1.42395			0.711974	−	0.702206i	$-0.247801\pi$
$911$	42.9787			1.42395			0.711974	+	0.702206i	$0.247801\pi$
$912$		0			0
$913$	29.0066	−	50.2409i	0.959978	−	1.66273i
$914$		0			0
$915$	−4.61803	+	7.99867i	−0.152667	+	0.264428i
$916$		0			0
$917$	21.0623	+	36.4810i	0.695539	+	1.20471i
$918$		0			0
$919$	−29.3951			−0.969656			−0.484828	−	0.874610i	$-0.661118\pi$
$919$	−29.3951			−0.969656			−0.484828	+	0.874610i	$0.661118\pi$
$920$		0			0
$921$	−30.3885	−	52.6345i	−1.00134	−	1.73437i
$922$		0			0
$923$	−7.46149	+	12.9237i	−0.245598	+	0.425388i
$924$		0			0
$925$	16.8541			0.554159
$926$		0			0
$927$	−31.6353	+	54.7939i	−1.03904	+	1.79967i
$928$		0			0
$929$	−17.6180	−	30.5153i	−0.578029	−	1.00118i	−0.995705	−	0.0925796i	$-0.970489\pi$
$929$	−17.6180	−	30.5153i	−0.578029	−	1.00118i	0.417676	−	0.908596i	$-0.362845\pi$
$930$		0			0
$931$	−6.76393	−	11.7155i	−0.221679	−	0.383959i
$932$		0			0
$933$	−39.0517	−	67.6395i	−1.27849	−	2.21442i
$934$		0			0
$935$	27.2361			0.890715
$936$		0			0
$937$	27.6631	+	47.9139i	0.903715	+	1.56528i	0.822633	+	0.568572i	$0.192504\pi$
$937$	27.6631	+	47.9139i	0.903715	+	1.56528i	0.0810814	+	0.996707i	$0.474163\pi$
$938$		0			0
$939$	−17.1803	−	29.7572i	−0.560659	−	0.971090i
$940$		0			0
$941$	13.1180	−	22.7211i	0.427636	−	0.740687i	−0.569027	−	0.822319i	$-0.692680\pi$
$941$	13.1180	−	22.7211i	0.427636	−	0.740687i	0.996663	+	0.0816322i	$0.0260133\pi$
$942$		0			0
$943$	20.7533	−	35.9458i	0.675820	−	1.17055i
$944$		0			0
$945$	11.7082			0.380868
$946$		0			0
$947$	−25.3607			−0.824111			−0.412056	−	0.911159i	$-0.635189\pi$
$947$	−25.3607			−0.824111			−0.412056	+	0.911159i	$0.635189\pi$
$948$		0			0
$949$	−1.28115	+	2.21902i	−0.0415880	+	0.0720325i
$950$		0			0
$951$	−43.5967	+	75.5118i	−1.41372	+	2.44864i
$952$		0			0
$953$	19.4443	+	33.6785i	0.629862	+	1.09095i	0.987579	+	0.157123i	$0.0502218\pi$
$953$	19.4443	+	33.6785i	0.629862	+	1.09095i	−0.357717	+	0.933830i	$0.616445\pi$
$954$		0			0
$955$	−8.97871	−	15.5516i	−0.290544	−	0.503238i
$956$		0			0
$957$	28.4164			0.918572
$958$		0			0
$959$	−7.85410	−	13.6037i	−0.253622	−	0.439287i
$960$		0			0
$961$	15.5000	+	26.8468i	0.500000	+	0.866025i
$962$		0			0
$963$	14.5066	+	25.1261i	0.467468	+	0.809678i
$964$		0			0
$965$	−1.67376	+	2.89904i	−0.0538803	+	0.0933234i
$966$		0			0
$967$	−38.1033			−1.22532			−0.612660	−	0.790346i	$-0.709901\pi$
$967$	−38.1033			−1.22532			−0.612660	+	0.790346i	$0.709901\pi$
$968$		0			0
$969$	−9.85410	+	17.0678i	−0.316559	+	0.548297i
$970$		0			0
$971$	−8.68034	−	15.0348i	−0.278565	−	0.482489i	0.692463	−	0.721453i	$-0.256526\pi$
$971$	−8.68034	−	15.0348i	−0.278565	−	0.482489i	−0.971028	+	0.238964i	$0.923192\pi$
$972$		0			0
$973$	22.4164			0.718637
$974$		0			0
$975$	6.28115	+	10.8793i	0.201158	+	0.348416i
$976$		0			0
$977$	−20.6180	+	35.7115i	−0.659629	+	1.14251i	0.321082	+	0.947051i	$0.395953\pi$
$977$	−20.6180	+	35.7115i	−0.659629	+	1.14251i	−0.980712	+	0.195460i	$0.937380\pi$
$978$		0			0
$979$	3.35410	−	5.80948i	0.107198	−	0.185672i
$980$		0			0
$981$	−53.3951			−1.70478
$982$		0			0
$983$	−5.23607	+	9.06914i	−0.167005	+	0.289261i	−0.937365	−	0.348348i	$-0.886743\pi$
$983$	−5.23607	+	9.06914i	−0.167005	+	0.289261i	0.770361	+	0.637608i	$0.220076\pi$
$984$		0			0
$985$	−3.63932			−0.115958
$986$		0			0
$987$	−12.7082			−0.404507
$988$		0			0
$989$	10.9549	−	26.5643i	0.348346	−	0.844694i
$990$		0			0
$991$	10.8197			0.343698			0.171849	−	0.985123i	$-0.445026\pi$
$991$	10.8197			0.343698			0.171849	+	0.985123i	$0.445026\pi$
$992$		0			0
$993$	14.3262			0.454629
$994$		0			0
$995$	1.20163	−	2.08128i	0.0380941	−	0.0659809i
$996$		0			0
$997$	−19.4508			−0.616015			−0.308007	−	0.951384i	$-0.599662\pi$
$997$	−19.4508			−0.616015			−0.308007	+	0.951384i	$0.599662\pi$
$998$		0			0
$999$	5.42705	−	9.39993i	0.171704	−	0.297401i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.i.e.49.1		4
4.3	odd	2		43.2.c.b.6.2	✓	4
12.11	even	2		387.2.h.d.307.1		4
43.36	even	3	inner	688.2.i.e.337.1		4
172.79	odd	6		43.2.c.b.36.2	yes	4
172.123	even	6		1849.2.a.h.1.1		2
172.135	odd	6		1849.2.a.e.1.2		2
516.251	even	6		387.2.h.d.208.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.c.b.6.2	✓	4	4.3	odd	2
43.2.c.b.36.2	yes	4	172.79	odd	6
387.2.h.d.208.1		4	516.251	even	6
387.2.h.d.307.1		4	12.11	even	2
688.2.i.e.49.1		4	1.1	even	1	trivial
688.2.i.e.337.1		4	43.36	even	3	inner
1849.2.a.e.1.2		2	172.135	odd	6
1849.2.a.h.1.1		2	172.123	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:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	688.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.30902 + 2.26728i) q^{3} +(-0.618034 + 1.07047i) q^{5} +(-2.11803 - 3.66854i) q^{7} +(-1.92705 - 3.33775i) q^{9} +O(q^{10})\)	\(q+(-1.30902 + 2.26728i) q^{3} +(-0.618034 + 1.07047i) q^{5} +(-2.11803 - 3.66854i) q^{7} +(-1.92705 - 3.33775i) q^{9} +3.61803 q^{11} +(-0.690983 - 1.19682i) q^{13} +(-1.61803 - 2.80252i) q^{15} +(-3.04508 - 5.27424i) q^{17} +(-0.618034 + 1.07047i) q^{19} +11.0902 q^{21} +(2.19098 - 3.79489i) q^{23} +(1.73607 + 3.00696i) q^{25} +2.23607 q^{27} +(-1.50000 - 2.59808i) q^{29} +(-4.73607 + 8.20311i) q^{33} +5.23607 q^{35} +(2.42705 - 4.20378i) q^{37} +3.61803 q^{39} +9.47214 q^{41} +(6.50000 - 0.866025i) q^{43} +4.76393 q^{45} -1.14590 q^{47} +(-5.47214 + 9.47802i) q^{49} +15.9443 q^{51} +(0.690983 - 1.19682i) q^{53} +(-2.23607 + 3.87298i) q^{55} +(-1.61803 - 2.80252i) q^{57} -5.09017 q^{59} +(-1.42705 - 2.47172i) q^{61} +(-8.16312 + 14.1389i) q^{63} +1.70820 q^{65} +(-1.92705 + 3.33775i) q^{67} +(5.73607 + 9.93516i) q^{69} +(-5.39919 - 9.35167i) q^{71} +(-0.927051 - 1.60570i) q^{73} -9.09017 q^{75} +(-7.66312 - 13.2729i) q^{77} +(-0.690983 - 1.19682i) q^{79} +(2.85410 - 4.94345i) q^{81} +(8.01722 - 13.8862i) q^{83} +7.52786 q^{85} +7.85410 q^{87} +(0.927051 - 1.60570i) q^{89} +(-2.92705 + 5.06980i) q^{91} +(-0.763932 - 1.32317i) q^{95} -9.23607 q^{97} +(-6.97214 - 12.0761i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} + 2 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) \) 4 * q - 3 * q^3 + 2 * q^5 - 4 * q^7 - q^9	\( 4 q - 3 q^{3} + 2 q^{5} - 4 q^{7} - q^{9} + 10 q^{11} - 5 q^{13} - 2 q^{15} - q^{17} + 2 q^{19} + 22 q^{21} + 11 q^{23} - 2 q^{25} - 6 q^{29} - 10 q^{33} + 12 q^{35} + 3 q^{37} + 10 q^{39} + 20 q^{41} + 26 q^{43} + 28 q^{45} - 18 q^{47} - 4 q^{49} + 28 q^{51} + 5 q^{53} - 2 q^{57} + 2 q^{59} + q^{61} - 17 q^{63} - 20 q^{65} - q^{67} + 14 q^{69} + 3 q^{71} + 3 q^{73} - 14 q^{75} - 15 q^{77} - 5 q^{79} - 2 q^{81} + 3 q^{83} + 48 q^{85} + 18 q^{87} - 3 q^{89} - 5 q^{91} - 12 q^{95} - 28 q^{97} - 10 q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 + 2 * q^5 - 4 * q^7 - q^9 + 10 * q^11 - 5 * q^13 - 2 * q^15 - q^17 + 2 * q^19 + 22 * q^21 + 11 * q^23 - 2 * q^25 - 6 * q^29 - 10 * q^33 + 12 * q^35 + 3 * q^37 + 10 * q^39 + 20 * q^41 + 26 * q^43 + 28 * q^45 - 18 * q^47 - 4 * q^49 + 28 * q^51 + 5 * q^53 - 2 * q^57 + 2 * q^59 + q^61 - 17 * q^63 - 20 * q^65 - q^67 + 14 * q^69 + 3 * q^71 + 3 * q^73 - 14 * q^75 - 15 * q^77 - 5 * q^79 - 2 * q^81 + 3 * q^83 + 48 * q^85 + 18 * q^87 - 3 * q^89 - 5 * q^91 - 12 * q^95 - 28 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more