LMFDB - Embedded newform 1045.2.b.b.419.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(419,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1045.419");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$8.34436701122$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 19x^{14} + 144x^{12} + 552x^{10} + 1119x^{8} + 1146x^{6} + 524x^{4} + 83x^{2} + 4 $ x^16 + 19x^14 + 144x^12 + 552x^10 + 1119x^8 + 1146x^6 + 524x^4 + 83*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			419.5
Root			$-1.19140i$ of defining polynomial
Character	$\chi$	$=$	1045.419
Dual form			1045.2.b.b.419.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.19140i q^{2} +2.62369i q^{3} +0.580557 q^{4} +(0.0644373 - 2.23514i) q^{5} +3.12588 q^{6} +0.593512i q^{7} -3.07449i q^{8} -3.88377 q^{9} +O(q^{10})$	\(q-1.19140i q^{2} +2.62369i q^{3} +0.580557 q^{4} +(0.0644373 - 2.23514i) q^{5} +3.12588 q^{6} +0.593512i q^{7} -3.07449i q^{8} -3.88377 q^{9} +(-2.66295 - 0.0767708i) q^{10} -1.00000 q^{11} +1.52321i q^{12} -6.20418i q^{13} +0.707112 q^{14} +(5.86432 + 0.169064i) q^{15} -2.50184 q^{16} -4.18279i q^{17} +4.62714i q^{18} -1.00000 q^{19} +(0.0374095 - 1.29763i) q^{20} -1.55719 q^{21} +1.19140i q^{22} -2.93682i q^{23} +8.06651 q^{24} +(-4.99170 - 0.288052i) q^{25} -7.39169 q^{26} -2.31875i q^{27} +0.344568i q^{28} +3.45618 q^{29} +(0.201423 - 6.98678i) q^{30} -6.32876 q^{31} -3.16827i q^{32} -2.62369i q^{33} -4.98339 q^{34} +(1.32658 + 0.0382443i) q^{35} -2.25475 q^{36} -1.18743i q^{37} +1.19140i q^{38} +16.2779 q^{39} +(-6.87190 - 0.198111i) q^{40} +2.97689 q^{41} +1.85525i q^{42} -2.99812i q^{43} -0.580557 q^{44} +(-0.250260 + 8.68078i) q^{45} -3.49894 q^{46} +7.94453i q^{47} -6.56406i q^{48} +6.64774 q^{49} +(-0.343187 + 5.94712i) q^{50} +10.9744 q^{51} -3.60188i q^{52} +4.96564i q^{53} -2.76257 q^{54} +(-0.0644373 + 2.23514i) q^{55} +1.82474 q^{56} -2.62369i q^{57} -4.11770i q^{58} +14.9325 q^{59} +(3.40458 + 0.0981512i) q^{60} -5.71439 q^{61} +7.54010i q^{62} -2.30507i q^{63} -8.77837 q^{64} +(-13.8672 - 0.399781i) q^{65} -3.12588 q^{66} -2.49490i q^{67} -2.42835i q^{68} +7.70532 q^{69} +(0.0455644 - 1.58049i) q^{70} +6.87624 q^{71} +11.9406i q^{72} +6.57621i q^{73} -1.41471 q^{74} +(0.755762 - 13.0967i) q^{75} -0.580557 q^{76} -0.593512i q^{77} -19.3935i q^{78} +11.2479 q^{79} +(-0.161212 + 5.59196i) q^{80} -5.56762 q^{81} -3.54667i q^{82} -10.7653i q^{83} -0.904040 q^{84} +(-9.34912 - 0.269528i) q^{85} -3.57197 q^{86} +9.06795i q^{87} +3.07449i q^{88} -6.38146 q^{89} +(10.3423 + 0.298160i) q^{90} +3.68226 q^{91} -1.70499i q^{92} -16.6047i q^{93} +9.46514 q^{94} +(-0.0644373 + 2.23514i) q^{95} +8.31258 q^{96} -7.26873i q^{97} -7.92015i q^{98} +3.88377 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9}+O(q^{10}) $ 16 * q - 6 * q^4 + 3 * q^5 - 8 * q^6 - 8 * q^9	$ 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9} + 10 q^{10} - 16 q^{11} + 4 q^{14} + 3 q^{15} - 18 q^{16} - 16 q^{19} - 2 q^{20} - 10 q^{21} + 10 q^{24} - 7 q^{25} - 24 q^{26} + 2 q^{29} + 4 q^{30} - 32 q^{31} - 16 q^{34} - 18 q^{35} + 18 q^{36} + 40 q^{39} - 28 q^{40} + 6 q^{41} + 6 q^{44} + 16 q^{45} + 38 q^{49} - 30 q^{50} - 16 q^{51} + 18 q^{54} - 3 q^{55} + 12 q^{56} + 24 q^{59} - 20 q^{60} - 42 q^{61} + 62 q^{64} - 20 q^{65} + 8 q^{66} + 30 q^{69} - 18 q^{70} - 46 q^{71} - 2 q^{74} - 25 q^{75} + 6 q^{76} + 74 q^{79} - 22 q^{80} - 56 q^{81} + 34 q^{84} - 18 q^{85} + 8 q^{86} + 14 q^{89} - 4 q^{90} - 24 q^{91} + 64 q^{94} - 3 q^{95} + 54 q^{96} + 8 q^{99}+O(q^{100}) $ 16 * q - 6 * q^4 + 3 * q^5 - 8 * q^6 - 8 * q^9 + 10 * q^10 - 16 * q^11 + 4 * q^14 + 3 * q^15 - 18 * q^16 - 16 * q^19 - 2 * q^20 - 10 * q^21 + 10 * q^24 - 7 * q^25 - 24 * q^26 + 2 * q^29 + 4 * q^30 - 32 * q^31 - 16 * q^34 - 18 * q^35 + 18 * q^36 + 40 * q^39 - 28 * q^40 + 6 * q^41 + 6 * q^44 + 16 * q^45 + 38 * q^49 - 30 * q^50 - 16 * q^51 + 18 * q^54 - 3 * q^55 + 12 * q^56 + 24 * q^59 - 20 * q^60 - 42 * q^61 + 62 * q^64 - 20 * q^65 + 8 * q^66 + 30 * q^69 - 18 * q^70 - 46 * q^71 - 2 * q^74 - 25 * q^75 + 6 * q^76 + 74 * q^79 - 22 * q^80 - 56 * q^81 + 34 * q^84 - 18 * q^85 + 8 * q^86 + 14 * q^89 - 4 * q^90 - 24 * q^91 + 64 * q^94 - 3 * q^95 + 54 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times$.

$n$	$496$	$761$	$837$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	1.19140i		−	0.842450i	−0.906956	−	0.421225i	$-0.861600\pi$
$2$		−	1.19140i		−	0.842450i	0.906956	−	0.421225i	$-0.138400\pi$
$3$			2.62369i			1.51479i	0.652956	+	0.757395i	$0.273528\pi$
$3$			2.62369i			1.51479i	−0.652956	+	0.757395i	$0.726472\pi$
$4$	0.580557			0.290279
$5$	0.0644373	−	2.23514i	0.0288172	−	0.999585i
$5$	0.0644373	−	2.23514i	0.0288172	−	0.999585i
$6$	3.12588			1.27613
$7$			0.593512i			0.224326i	0.993690	+	0.112163i	$0.0357779\pi$
$7$			0.593512i			0.224326i	−0.993690	+	0.112163i	$0.964222\pi$
$8$		−	3.07449i		−	1.08699i
$9$	−3.88377			−1.29459
$10$	−2.66295	−	0.0767708i	−0.842100	−	0.0242771i
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$			1.52321i			0.439712i
$13$		−	6.20418i		−	1.72073i	−0.509678	−	0.860366i	$-0.670235\pi$
$13$		−	6.20418i		−	1.72073i	0.509678	−	0.860366i	$-0.329765\pi$
$14$	0.707112			0.188984
$15$	5.86432	+	0.169064i	1.51416	+	0.0436521i
$16$	−2.50184			−0.625460
$17$		−	4.18279i		−	1.01448i	−0.861806	−	0.507238i	$-0.830666\pi$
$17$		−	4.18279i		−	1.01448i	0.861806	−	0.507238i	$-0.169334\pi$
$18$			4.62714i			1.09063i
$19$	−1.00000			−0.229416
$19$	−1.00000			−0.229416
$20$	0.0374095	−	1.29763i	0.00836502	−	0.290158i
$21$	−1.55719			−0.339807
$22$			1.19140i			0.254008i
$23$		−	2.93682i		−	0.612370i	−0.951972	−	0.306185i	$-0.900947\pi$
$23$		−	2.93682i		−	0.612370i	0.951972	−	0.306185i	$-0.0990525\pi$
$24$	8.06651			1.64657
$25$	−4.99170	−	0.288052i	−0.998339	−	0.0576105i
$26$	−7.39169			−1.44963
$27$		−	2.31875i		−	0.446245i
$28$			0.344568i			0.0651171i
$29$	3.45618			0.641796			0.320898	−	0.947114i	$-0.396015\pi$
$29$	3.45618			0.641796			0.320898	+	0.947114i	$0.396015\pi$
$30$	0.201423	−	6.98678i	0.0367747	−	1.27560i
$31$	−6.32876			−1.13668			−0.568339	−	0.822794i	$-0.692414\pi$
$31$	−6.32876			−1.13668			−0.568339	+	0.822794i	$0.692414\pi$
$32$		−	3.16827i		−	0.560077i
$33$		−	2.62369i		−	0.456727i
$34$	−4.98339			−0.854645
$35$	1.32658	+	0.0382443i	0.224233	+	0.00646446i
$36$	−2.25475			−0.375792
$37$		−	1.18743i		−	0.195212i	−0.995225	−	0.0976061i	$-0.968881\pi$
$37$		−	1.18743i		−	0.195212i	0.995225	−	0.0976061i	$-0.0311185\pi$
$38$			1.19140i			0.193271i
$39$	16.2779			2.60655
$40$	−6.87190	−	0.198111i	−1.08654	−	0.0313242i
$41$	2.97689			0.464911			0.232456	−	0.972607i	$-0.425324\pi$
$41$	2.97689			0.464911			0.232456	+	0.972607i	$0.425324\pi$
$42$			1.85525i			0.286271i
$43$		−	2.99812i		−	0.457209i	−0.973519	−	0.228605i	$-0.926584\pi$
$43$		−	2.99812i		−	0.457209i	0.973519	−	0.228605i	$-0.0734163\pi$
$44$	−0.580557			−0.0875223
$45$	−0.250260	+	8.68078i	−0.0373065	+	1.29405i
$46$	−3.49894			−0.515890
$47$			7.94453i			1.15883i	0.815033	+	0.579414i	$0.196719\pi$
$47$			7.94453i			1.15883i	−0.815033	+	0.579414i	$0.803281\pi$
$48$		−	6.56406i		−	0.947440i
$49$	6.64774			0.949678
$50$	−0.343187	+	5.94712i	−0.0485339	+	0.841050i
$51$	10.9744			1.53672
$52$		−	3.60188i		−	0.499492i
$53$			4.96564i			0.682083i	0.940048	+	0.341041i	$0.110780\pi$
$53$			4.96564i			0.682083i	−0.940048	+	0.341041i	$0.889220\pi$
$54$	−2.76257			−0.375939
$55$	−0.0644373	+	2.23514i	−0.00868872	+	0.301386i
$56$	1.82474			0.243842
$57$		−	2.62369i		−	0.347517i
$58$		−	4.11770i		−	0.540681i
$59$	14.9325			1.94405			0.972025	−	0.234879i	$-0.0754695\pi$
$59$	14.9325			1.94405			0.972025	+	0.234879i	$0.0754695\pi$
$60$	3.40458	+	0.0981512i	0.439529	+	0.0126713i
$61$	−5.71439			−0.731653			−0.365826	−	0.930683i	$-0.619214\pi$
$61$	−5.71439			−0.731653			−0.365826	+	0.930683i	$0.619214\pi$
$62$			7.54010i			0.957594i
$63$		−	2.30507i		−	0.290411i
$64$	−8.77837			−1.09730
$65$	−13.8672	−	0.399781i	−1.72002	−	0.0495867i
$66$	−3.12588			−0.384769
$67$		−	2.49490i		−	0.304801i	−0.988319	−	0.152400i	$-0.951300\pi$
$67$		−	2.49490i		−	0.304801i	0.988319	−	0.152400i	$-0.0487003\pi$
$68$		−	2.42835i		−	0.294481i
$69$	7.70532			0.927612
$70$	0.0455644	−	1.58049i	0.00544598	−	0.188905i
$71$	6.87624			0.816059			0.408029	−	0.912969i	$-0.366216\pi$
$71$	6.87624			0.816059			0.408029	+	0.912969i	$0.366216\pi$
$72$			11.9406i			1.40721i
$73$			6.57621i			0.769688i	0.922982	+	0.384844i	$0.125745\pi$
$73$			6.57621i			0.769688i	−0.922982	+	0.384844i	$0.874255\pi$
$74$	−1.41471			−0.164456
$75$	0.755762	−	13.0967i	0.0872679	−	1.51228i
$76$	−0.580557			−0.0665945
$77$		−	0.593512i		−	0.0676369i
$78$		−	19.3935i		−	2.19589i
$79$	11.2479			1.26549			0.632747	−	0.774359i	$-0.281927\pi$
$79$	11.2479			1.26549			0.632747	+	0.774359i	$0.281927\pi$
$80$	−0.161212	+	5.59196i	−0.0180240	+	0.625200i
$81$	−5.56762			−0.618624
$82$		−	3.54667i		−	0.391664i
$83$		−	10.7653i		−	1.18165i	−0.806800	−	0.590825i	$-0.798803\pi$
$83$		−	10.7653i		−	1.18165i	0.806800	−	0.590825i	$-0.201197\pi$
$84$	−0.904040			−0.0986389
$85$	−9.34912	−	0.269528i	−1.01405	−	0.0292344i
$86$	−3.57197			−0.385176
$87$			9.06795i			0.972187i
$88$			3.07449i			0.327741i
$89$	−6.38146			−0.676434			−0.338217	−	0.941068i	$-0.609824\pi$
$89$	−6.38146			−0.676434			−0.338217	+	0.941068i	$0.609824\pi$
$90$	10.3423	+	0.298160i	1.09018	+	0.0314289i
$91$	3.68226			0.386005
$92$		−	1.70499i		−	0.177758i
$93$		−	16.6047i		−	1.72183i
$94$	9.46514			0.976255
$95$	−0.0644373	+	2.23514i	−0.00661112	+	0.229320i
$96$	8.31258			0.848399
$97$		−	7.26873i		−	0.738028i	−0.929424	−	0.369014i	$-0.879695\pi$
$97$		−	7.26873i		−	0.738028i	0.929424	−	0.369014i	$-0.120305\pi$
$98$		−	7.92015i		−	0.800056i
$99$	3.88377			0.390334
$100$	−2.89797	−	0.167231i	−0.289797	−	0.0167231i
$101$	18.0364			1.79469			0.897346	−	0.441327i	$-0.145492\pi$
$101$	18.0364			1.79469			0.897346	+	0.441327i	$0.145492\pi$
$102$		−	13.0749i		−	1.29461i
$103$		−	0.578081i		−	0.0569600i	−0.999594	−	0.0284800i	$-0.990933\pi$
$103$		−	0.578081i		−	0.0569600i	0.999594	−	0.0284800i	$-0.00906669\pi$
$104$	−19.0747			−1.87043
$105$	−0.100341	+	3.48054i	−0.00979231	+	0.339666i
$106$	5.91608			0.574621
$107$			10.3566i			1.00121i	0.865674	+	0.500607i	$0.166890\pi$
$107$			10.3566i			1.00121i	−0.865674	+	0.500607i	$0.833110\pi$
$108$		−	1.34617i		−	0.129535i
$109$	−15.6298			−1.49706			−0.748530	−	0.663101i	$-0.769240\pi$
$109$	−15.6298			−1.49706			−0.748530	+	0.663101i	$0.769240\pi$
$110$	2.66295	+	0.0767708i	0.253903	+	0.00731981i
$111$	3.11545			0.295706
$112$		−	1.48487i		−	0.140307i
$113$		−	19.5381i		−	1.83799i	−0.394272	−	0.918994i	$-0.629003\pi$
$113$		−	19.5381i		−	1.83799i	0.394272	−	0.918994i	$-0.370997\pi$
$114$	−3.12588			−0.292765
$115$	−6.56420	−	0.189241i	−0.612115	−	0.0176468i
$116$	2.00651			0.186300
$117$			24.0957i			2.22764i
$118$		−	17.7907i		−	1.63776i
$119$	2.48254			0.227574
$120$	0.519784	−	18.0298i	0.0474496	−	1.64589i
$121$	1.00000			0.0909091
$122$			6.80815i			0.616381i
$123$			7.81044i			0.704244i
$124$	−3.67421			−0.329953
$125$	−0.965489	+	11.1386i	−0.0863559	+	0.996264i
$126$	−2.74626			−0.244657
$127$		−	0.648142i		−	0.0575133i	−0.999586	−	0.0287567i	$-0.990845\pi$
$127$		−	0.648142i		−	0.0575133i	0.999586	−	0.0287567i	$-0.00915479\pi$
$128$			4.12203i			0.364340i
$129$	7.86616			0.692576
$130$	−0.476300	+	16.5215i	−0.0417743	+	1.44903i
$131$	−4.50122			−0.393273			−0.196637	−	0.980476i	$-0.563002\pi$
$131$	−4.50122			−0.393273			−0.196637	+	0.980476i	$0.563002\pi$
$132$		−	1.52321i		−	0.132578i
$133$		−	0.593512i		−	0.0514640i
$134$	−2.97244			−0.256779
$135$	−5.18274	−	0.149414i	−0.446059	−	0.0128595i
$136$	−12.8599			−1.10273
$137$			2.69795i			0.230501i	0.993336	+	0.115251i	$0.0367671\pi$
$137$			2.69795i			0.230501i	−0.993336	+	0.115251i	$0.963233\pi$
$138$		−	9.18015i		−	0.781466i
$139$	12.2824			1.04178			0.520889	−	0.853625i	$-0.325601\pi$
$139$	12.2824			1.04178			0.520889	+	0.853625i	$0.325601\pi$
$140$	0.770157	+	0.0222030i	0.0650901	+	0.00187650i
$141$	−20.8440			−1.75538
$142$		−	8.19237i		−	0.687489i
$143$			6.20418i			0.518820i
$144$	9.71658			0.809715
$145$	0.222707	−	7.72504i	0.0184948	−	0.641529i
$146$	7.83493			0.648423
$147$			17.4417i			1.43856i
$148$		−	0.689371i		−	0.0566660i
$149$	2.10871			0.172753			0.0863763	−	0.996263i	$-0.472471\pi$
$149$	2.10871			0.172753			0.0863763	+	0.996263i	$0.472471\pi$
$150$	−15.6034	−	0.900417i	−1.27402	−	0.0735188i
$151$	−14.8928			−1.21196			−0.605979	−	0.795481i	$-0.707219\pi$
$151$	−14.8928			−1.21196			−0.605979	+	0.795481i	$0.707219\pi$
$152$			3.07449i			0.249374i
$153$			16.2450i			1.31333i
$154$	−0.707112			−0.0569807
$155$	−0.407808	+	14.1457i	−0.0327559	+	1.13621i
$156$	9.45025			0.756625
$157$			12.8778i			1.02776i	0.857863	+	0.513879i	$0.171792\pi$
$157$			12.8778i			1.02776i	−0.857863	+	0.513879i	$0.828208\pi$
$158$		−	13.4008i		−	1.06611i
$159$	−13.0283			−1.03321
$160$	−7.08153	−	0.204155i	−0.559844	−	0.0161398i
$161$	1.74304			0.137371
$162$			6.63328i			0.521160i
$163$			24.2526i			1.89961i	0.312842	+	0.949805i	$0.398719\pi$
$163$			24.2526i			1.89961i	−0.312842	+	0.949805i	$0.601281\pi$
$164$	1.72825			0.134954
$165$	−5.86432	−	0.169064i	−0.456537	−	0.0131616i
$166$	−12.8259			−0.995480
$167$			15.0986i			1.16836i	0.811622	+	0.584182i	$0.198585\pi$
$167$			15.0986i			1.16836i	−0.811622	+	0.584182i	$0.801415\pi$
$168$			4.78757i			0.369369i
$169$	−25.4919			−1.96092
$170$	−0.321116	+	11.1386i	−0.0246285	+	0.854290i
$171$	3.88377			0.297000
$172$		−	1.74058i		−	0.132718i
$173$			16.0183i			1.21785i	0.793229	+	0.608924i	$0.208398\pi$
$173$			16.0183i			1.21785i	−0.793229	+	0.608924i	$0.791602\pi$
$174$	10.8036			0.819018
$175$	0.170963	−	2.96263i	0.0129236	−	0.223954i
$176$	2.50184			0.188583
$177$			39.1784i			2.94483i
$178$			7.60290i			0.569861i
$179$	20.2259			1.51176			0.755878	−	0.654712i	$-0.227210\pi$
$179$	20.2259			1.51176			0.755878	+	0.654712i	$0.227210\pi$
$180$	−0.145290	+	5.03969i	−0.0108293	+	0.375636i
$181$	−2.90647			−0.216036			−0.108018	−	0.994149i	$-0.534450\pi$
$181$	−2.90647			−0.216036			−0.108018	+	0.994149i	$0.534450\pi$
$182$		−	4.38705i		−	0.325190i
$183$		−	14.9928i		−	1.10830i
$184$	−9.02921			−0.665643
$185$	−2.65407	−	0.0765147i	−0.195131	−	0.00562547i
$186$	−19.7829			−1.45055
$187$			4.18279i			0.305876i
$188$			4.61226i			0.336383i
$189$	1.37621			0.100104
$190$	2.66295	+	0.0767708i	0.193191	+	0.00556954i
$191$	10.9110			0.789494			0.394747	−	0.918790i	$-0.370832\pi$
$191$	10.9110			0.789494			0.394747	+	0.918790i	$0.370832\pi$
$192$		−	23.0318i		−	1.66217i
$193$		−	5.31006i		−	0.382226i	−0.981568	−	0.191113i	$-0.938790\pi$
$193$		−	5.31006i		−	0.382226i	0.981568	−	0.191113i	$-0.0612098\pi$
$194$	−8.65999			−0.621751
$195$	1.04890	−	36.3833i	0.0751135	−	2.60547i
$196$	3.85940			0.275671
$197$		−	3.51694i		−	0.250572i	−0.992121	−	0.125286i	$-0.960015\pi$
$197$		−	3.51694i		−	0.250572i	0.992121	−	0.125286i	$-0.0399848\pi$
$198$		−	4.62714i		−	0.328837i
$199$	−16.7061			−1.18427			−0.592134	−	0.805840i	$-0.701714\pi$
$199$	−16.7061			−1.18427			−0.592134	+	0.805840i	$0.701714\pi$
$200$	−0.885613	+	15.3469i	−0.0626223	+	1.08519i
$201$	6.54586			0.461709
$202$		−	21.4887i		−	1.51194i
$203$			2.05128i			0.143972i
$204$	6.37125			0.446077
$205$	0.191822	−	6.65375i	0.0133975	−	0.464718i
$206$	−0.688727			−0.0479859
$207$			11.4060i			0.792768i
$208$			15.5219i			1.07625i
$209$	1.00000			0.0691714
$210$	4.14673	+	0.119547i	0.286152	+	0.00824952i
$211$	17.6892			1.21778			0.608888	−	0.793256i	$-0.291616\pi$
$211$	17.6892			1.21778			0.608888	+	0.793256i	$0.291616\pi$
$212$			2.88284i			0.197994i
$213$			18.0411i			1.23616i
$214$	12.3389			0.843473
$215$	−6.70122	−	0.193191i	−0.457019	−	0.0131755i
$216$	−7.12898			−0.485066
$217$		−	3.75619i		−	0.254987i
$218$			18.6214i			1.26120i
$219$	−17.2540			−1.16592
$220$	−0.0374095	+	1.29763i	−0.00252215	+	0.0874860i
$221$	−25.9508			−1.74564
$222$		−	3.71176i		−	0.249117i
$223$		−	12.1908i		−	0.816355i	−0.912903	−	0.408178i	$-0.866164\pi$
$223$		−	12.1908i		−	0.816355i	0.912903	−	0.408178i	$-0.133836\pi$
$224$	1.88041			0.125640
$225$	19.3866	+	1.11873i	1.29244	+	0.0745821i
$226$	−23.2777			−1.54841
$227$		−	29.9229i		−	1.98605i	−0.117899	−	0.993026i	$-0.537616\pi$
$227$		−	29.9229i		−	1.98605i	0.117899	−	0.993026i	$-0.462384\pi$
$228$		−	1.52321i		−	0.100877i
$229$	−10.0952			−0.667109			−0.333554	−	0.942731i	$-0.608248\pi$
$229$	−10.0952			−0.667109			−0.333554	+	0.942731i	$0.608248\pi$
$230$	−0.225462	+	7.82062i	−0.0148665	+	0.515676i
$231$	1.55719			0.102456
$232$		−	10.6260i		−	0.697629i
$233$			4.27564i			0.280107i	0.990144	+	0.140053i	$0.0447274\pi$
$233$			4.27564i			0.280107i	−0.990144	+	0.140053i	$0.955273\pi$
$234$	28.7076			1.87668
$235$	17.7571	+	0.511924i	1.15835	+	0.0333942i
$236$	8.66919			0.564316
$237$			29.5112i			1.91696i
$238$		−	2.95770i		−	0.191719i
$239$	25.8196			1.67013			0.835066	−	0.550149i	$-0.185429\pi$
$239$	25.8196			1.67013			0.835066	+	0.550149i	$0.185429\pi$
$240$	−14.6716	−	0.422970i	−0.947047	−	0.0273026i
$241$	16.4521			1.05977			0.529885	−	0.848070i	$-0.322235\pi$
$241$	16.4521			1.05977			0.529885	+	0.848070i	$0.322235\pi$
$242$		−	1.19140i		−	0.0765863i
$243$		−	21.5640i		−	1.38333i
$244$	−3.31753			−0.212383
$245$	0.428362	−	14.8586i	0.0273671	−	0.949283i
$246$	9.30538			0.593290
$247$			6.20418i			0.394763i
$248$			19.4577i			1.23556i
$249$	28.2450			1.78995
$250$	13.2705	+	1.15029i	0.839302	+	0.0727505i
$251$	−16.3977			−1.03501			−0.517507	−	0.855679i	$-0.673140\pi$
$251$	−16.3977			−1.03501			−0.517507	+	0.855679i	$0.673140\pi$
$252$		−	1.33822i		−	0.0843001i
$253$			2.93682i			0.184636i
$254$	−0.772199			−0.0484521
$255$	0.707158	−	24.5292i	0.0442840	−	1.53608i
$256$	−12.6457			−0.790358
$257$		−	13.1408i		−	0.819703i	−0.912152	−	0.409851i	$-0.865581\pi$
$257$		−	13.1408i		−	0.819703i	0.912152	−	0.409851i	$-0.134419\pi$
$258$		−	9.37177i		−	0.583461i
$259$	0.704753			0.0437912
$260$	−8.05071	−	0.232096i	−0.499284	−	0.0143940i
$261$	−13.4230			−0.830864
$262$			5.36277i			0.331313i
$263$			28.1361i			1.73495i	0.497483	+	0.867474i	$0.334258\pi$
$263$			28.1361i			1.73495i	−0.497483	+	0.867474i	$0.665742\pi$
$264$	−8.06651			−0.496459
$265$	11.0989	+	0.319972i	0.681800	+	0.0196557i
$266$	−0.707112			−0.0433558
$267$		−	16.7430i		−	1.02466i
$268$		−	1.44843i		−	0.0884772i
$269$	−24.0542			−1.46661			−0.733305	−	0.679900i	$-0.762023\pi$
$269$	−24.0542			−1.46661			−0.733305	+	0.679900i	$0.762023\pi$
$270$	−0.178013	+	6.17474i	−0.0108335	+	0.375782i
$271$	−8.63562			−0.524577			−0.262288	−	0.964990i	$-0.584477\pi$
$271$	−8.63562			−0.524577			−0.262288	+	0.964990i	$0.584477\pi$
$272$			10.4647i			0.634514i
$273$			9.66111i			0.584717i
$274$	3.21435			0.194186
$275$	4.99170	+	0.288052i	0.301011	+	0.0173702i
$276$	4.47338			0.269266
$277$		−	0.949604i		−	0.0570562i	−0.999593	−	0.0285281i	$-0.990918\pi$
$277$		−	0.949604i		−	0.0570562i	0.999593	−	0.0285281i	$-0.00908200\pi$
$278$		−	14.6333i		−	0.877645i
$279$	24.5795			1.47153
$280$	0.117581	−	4.07855i	0.00702683	−	0.243740i
$281$	25.0916			1.49684			0.748420	−	0.663225i	$-0.230813\pi$
$281$	25.0916			1.49684			0.748420	+	0.663225i	$0.230813\pi$
$282$			24.8336i			1.47882i
$283$			24.6821i			1.46720i	0.679582	+	0.733599i	$0.262161\pi$
$283$			24.6821i			1.46720i	−0.679582	+	0.733599i	$0.737839\pi$
$284$	3.99205			0.236885
$285$	−5.86432	−	0.169064i	−0.347373	−	0.0100145i
$286$	7.39169			0.437080
$287$			1.76682i			0.104292i
$288$			12.3049i			0.725070i
$289$	−0.495755			−0.0291620
$290$	−9.20364	−	0.265333i	−0.540456	−	0.0155809i
$291$	19.0709			1.11796
$292$			3.81787i			0.223424i
$293$		−	12.2030i		−	0.712905i	−0.934313	−	0.356453i	$-0.883986\pi$
$293$		−	12.2030i		−	0.712905i	0.934313	−	0.356453i	$-0.116014\pi$
$294$	20.7800			1.21192
$295$	0.962211	−	33.3763i	0.0560221	−	1.94324i
$296$	−3.65074			−0.212195
$297$			2.31875i			0.134548i
$298$		−	2.51233i		−	0.145535i
$299$	−18.2206			−1.05372
$300$	0.438763	−	7.60338i	0.0253320	−	0.438981i
$301$	1.77942			0.102564
$302$			17.7433i			1.02101i
$303$			47.3221i			2.71858i
$304$	2.50184			0.143490
$305$	−0.368220	+	12.7725i	−0.0210842	+	0.731349i
$306$	19.3544			1.10642
$307$		−	22.3727i		−	1.27688i	−0.769673	−	0.638439i	$-0.779581\pi$
$307$		−	22.3727i		−	1.27688i	0.769673	−	0.638439i	$-0.220419\pi$
$308$		−	0.344568i		−	0.0196336i
$309$	1.51671			0.0862824
$310$	16.8532	+	0.485864i	0.957196	+	0.0275952i
$311$	−25.1644			−1.42694			−0.713470	−	0.700686i	$-0.752878\pi$
$311$	−25.1644			−1.42694			−0.713470	+	0.700686i	$0.752878\pi$
$312$		−	50.0461i		−	2.83330i
$313$			20.4497i			1.15588i	0.816078	+	0.577941i	$0.196144\pi$
$313$			20.4497i			1.15588i	−0.816078	+	0.577941i	$0.803856\pi$
$314$	15.3426			0.865834
$315$	−5.15214	−	0.148532i	−0.290290	−	0.00836884i
$316$	6.53008			0.367346
$317$			24.3410i			1.36713i	0.729891	+	0.683563i	$0.239571\pi$
$317$			24.3410i			1.36713i	−0.729891	+	0.683563i	$0.760429\pi$
$318$			15.5220i			0.870430i
$319$	−3.45618			−0.193509
$320$	−0.565654	+	19.6209i	−0.0316210	+	1.09684i
$321$	−27.1727			−1.51663
$322$		−	2.07666i		−	0.115728i
$323$			4.18279i			0.232737i
$324$	−3.23232			−0.179573
$325$	−1.78713	+	30.9694i	−0.0991322	+	1.71787i
$326$	28.8946			1.60033
$327$		−	41.0077i		−	2.26773i
$328$		−	9.15239i		−	0.505356i
$329$	−4.71517			−0.259956
$330$	−0.201423	+	6.98678i	−0.0110880	+	0.384609i
$331$	−10.8627			−0.597068			−0.298534	−	0.954399i	$-0.596498\pi$
$331$	−10.8627			−0.597068			−0.298534	+	0.954399i	$0.596498\pi$
$332$		−	6.24990i		−	0.343008i
$333$			4.61171i			0.252720i
$334$	17.9885			0.984288
$335$	−5.57645	−	0.160765i	−0.304674	−	0.00878351i
$336$	3.89585			0.212536
$337$		−	5.15444i		−	0.280780i	−0.990096	−	0.140390i	$-0.955164\pi$
$337$		−	5.15444i		−	0.280780i	0.990096	−	0.140390i	$-0.0448357\pi$
$338$			30.3711i			1.65197i
$339$	51.2620			2.78417
$340$	−5.42770	−	0.156476i	−0.294359	−	0.00848612i
$341$	6.32876			0.342721
$342$		−	4.62714i		−	0.250207i
$343$			8.10009i			0.437364i
$344$	−9.21768			−0.496984
$345$	0.496510	−	17.2225i	0.0267312	−	0.927227i
$346$	19.0842			1.02598
$347$			7.51141i			0.403234i	0.979464	+	0.201617i	$0.0646195\pi$
$347$			7.51141i			0.403234i	−0.979464	+	0.201617i	$0.935380\pi$
$348$			5.26447i			0.282205i
$349$	25.7772			1.37982			0.689910	−	0.723895i	$-0.257650\pi$
$349$	25.7772			1.37982			0.689910	+	0.723895i	$0.257650\pi$
$350$	−3.52969	−	0.203685i	−0.188670	−	0.0108874i
$351$	−14.3860			−0.767867
$352$			3.16827i			0.168869i
$353$		−	19.8985i		−	1.05909i	−0.848282	−	0.529544i	$-0.822363\pi$
$353$		−	19.8985i		−	1.05909i	0.848282	−	0.529544i	$-0.177637\pi$
$354$	46.6773			2.48087
$355$	0.443086	−	15.3693i	0.0235165	−	0.815720i
$356$	−3.70480			−0.196354
$357$			6.51342i			0.344727i
$358$		−	24.0972i		−	1.27358i
$359$	3.74841			0.197833			0.0989166	−	0.995096i	$-0.468462\pi$
$359$	3.74841			0.197833			0.0989166	+	0.995096i	$0.468462\pi$
$360$	26.6889	+	0.769420i	1.40663	+	0.0405520i
$361$	1.00000			0.0526316
$362$			3.46278i			0.181999i
$363$			2.62369i			0.137708i
$364$	2.13776			0.112049
$365$	14.6988	+	0.423753i	0.769368	+	0.0221803i
$366$	−17.8625			−0.933688
$367$			5.99508i			0.312941i	0.987683	+	0.156470i	$0.0500116\pi$
$367$			5.99508i			0.312941i	−0.987683	+	0.156470i	$0.949988\pi$
$368$			7.34745i			0.383012i
$369$	−11.5616			−0.601870
$370$	−0.0911599	+	3.16207i	−0.00473918	+	0.164388i
$371$	−2.94716			−0.153009
$372$		−	9.63999i		−	0.499810i
$373$			3.18013i			0.164661i	0.996605	+	0.0823303i	$0.0262363\pi$
$373$			3.18013i			0.164661i	−0.996605	+	0.0823303i	$0.973764\pi$
$374$	4.98339			0.257685
$375$	−29.2242	−	2.53315i	−1.50913	−	0.130811i
$376$	24.4253			1.25964
$377$		−	21.4428i		−	1.10436i
$378$		−	1.63962i		−	0.0843329i
$379$	13.2357			0.679871			0.339936	−	0.940449i	$-0.389595\pi$
$379$	13.2357			0.679871			0.339936	+	0.940449i	$0.389595\pi$
$380$	−0.0374095	+	1.29763i	−0.00191907	+	0.0665668i
$381$	1.70053			0.0871207
$382$		−	12.9994i		−	0.665109i
$383$		−	11.2296i		−	0.573803i	−0.957960	−	0.286902i	$-0.907375\pi$
$383$		−	11.2296i		−	0.573803i	0.957960	−	0.286902i	$-0.0926253\pi$
$384$	−10.8150			−0.551899
$385$	−1.32658	−	0.0382443i	−0.0676088	−	0.00194911i
$386$	−6.32642			−0.322006
$387$			11.6440i			0.591899i
$388$		−	4.21992i		−	0.214234i
$389$	3.95920			0.200740			0.100370	−	0.994950i	$-0.467997\pi$
$389$	3.95920			0.200740			0.100370	+	0.994950i	$0.467997\pi$
$390$	−43.3472	−	1.24967i	−2.19497	−	0.0632793i
$391$	−12.2841			−0.621234
$392$		−	20.4384i		−	1.03229i
$393$		−	11.8098i		−	0.595727i
$394$	−4.19009			−0.211094
$395$	0.724787	−	25.1407i	0.0364680	−	1.26497i
$396$	2.25475			0.113306
$397$			20.6143i			1.03460i	0.855804	+	0.517301i	$0.173063\pi$
$397$			20.6143i			1.03460i	−0.855804	+	0.517301i	$0.826937\pi$
$398$			19.9038i			0.997686i
$399$	1.55719			0.0779572
$400$	12.4884	+	0.720661i	0.624421	+	0.0360330i
$401$	22.6731			1.13224			0.566119	−	0.824323i	$-0.308444\pi$
$401$	22.6731			1.13224			0.566119	+	0.824323i	$0.308444\pi$
$402$		−	7.79876i		−	0.388967i
$403$			39.2648i			1.95592i
$404$	10.4712			0.520961
$405$	−0.358762	+	12.4444i	−0.0178270	+	0.618367i
$406$	2.44390			0.121289
$407$			1.18743i			0.0588587i
$408$		−	33.7405i		−	1.67041i
$409$	−15.2085			−0.752014			−0.376007	−	0.926617i	$-0.622703\pi$
$409$	−15.2085			−0.752014			−0.376007	+	0.926617i	$0.622703\pi$
$410$	−7.92731	−	0.228538i	−0.391502	−	0.0112867i
$411$	−7.07859			−0.349161
$412$		−	0.335609i		−	0.0165343i
$413$			8.86263i			0.436101i
$414$	13.5891			0.667867
$415$	−24.0620	−	0.693689i	−1.18116	−	0.0340518i
$416$	−19.6565			−0.963741
$417$			32.2252i			1.57807i
$418$		−	1.19140i		−	0.0582735i
$419$	−31.1217			−1.52040			−0.760198	−	0.649691i	$-0.774898\pi$
$419$	−31.1217			−1.52040			−0.760198	+	0.649691i	$0.774898\pi$
$420$	−0.0582539	+	2.02066i	−0.00284250	+	0.0985979i
$421$	−3.76841			−0.183661			−0.0918306	−	0.995775i	$-0.529272\pi$
$421$	−3.76841			−0.183661			−0.0918306	+	0.995775i	$0.529272\pi$
$422$		−	21.0750i		−	1.02591i
$423$		−	30.8548i		−	1.50021i
$424$	15.2668			0.741421
$425$	−1.20486	+	20.8792i	−0.0584445	+	1.01279i
$426$	21.4943			1.04140
$427$		−	3.39156i		−	0.164129i
$428$			6.01263i			0.290631i
$429$	−16.2779			−0.785904
$430$	−0.230168	+	7.98386i	−0.0110997	+	0.385016i
$431$	−18.6048			−0.896161			−0.448081	−	0.893993i	$-0.647892\pi$
$431$	−18.6048			−0.896161			−0.448081	+	0.893993i	$0.647892\pi$
$432$			5.80115i			0.279108i
$433$		−	8.39226i		−	0.403306i	−0.979457	−	0.201653i	$-0.935369\pi$
$433$		−	8.39226i		−	0.403306i	0.979457	−	0.201653i	$-0.0646314\pi$
$434$	−4.47514			−0.214814
$435$	20.2681	+	0.584314i	0.971783	+	0.0280157i
$436$	−9.07397			−0.434564
$437$			2.93682i			0.140487i
$438$			20.5565i			0.982225i
$439$	14.5168			0.692848			0.346424	−	0.938078i	$-0.387396\pi$
$439$	14.5168			0.692848			0.346424	+	0.938078i	$0.387396\pi$
$440$	6.87190	+	0.198111i	0.327605	+	0.00944459i
$441$	−25.8183			−1.22944
$442$			30.9179i			1.47061i
$443$		−	16.5006i		−	0.783969i	−0.919972	−	0.391984i	$-0.871789\pi$
$443$		−	16.5006i		−	0.783969i	0.919972	−	0.391984i	$-0.128211\pi$
$444$	1.80870			0.0858371
$445$	−0.411204	+	14.2635i	−0.0194929	+	0.676153i
$446$	−14.5241			−0.687738
$447$			5.53262i			0.261684i
$448$		−	5.21006i		−	0.246152i
$449$	8.58589			0.405193			0.202597	−	0.979262i	$-0.435062\pi$
$449$	8.58589			0.405193			0.202597	+	0.979262i	$0.435062\pi$
$450$	1.33286	−	23.0973i	0.0628316	−	1.08882i
$451$	−2.97689			−0.140176
$452$		−	11.3430i		−	0.533529i
$453$		−	39.0741i		−	1.83586i
$454$	−35.6502			−1.67315
$455$	0.237274	−	8.23035i	0.0111236	−	0.385845i
$456$	−8.06651			−0.377749
$457$		−	16.7002i		−	0.781200i	−0.920560	−	0.390600i	$-0.872267\pi$
$457$		−	16.7002i		−	0.781200i	0.920560	−	0.390600i	$-0.127733\pi$
$458$			12.0274i			0.562006i
$459$	−9.69887			−0.452705
$460$	−3.81090	−	0.109865i	−0.177684	−	0.00512249i
$461$	−26.2293			−1.22162			−0.610811	−	0.791777i	$-0.709156\pi$
$461$	−26.2293			−1.22162			−0.610811	+	0.791777i	$0.709156\pi$
$462$		−	1.85525i		−	0.0863138i
$463$			2.29006i			0.106428i	0.998583	+	0.0532141i	$0.0169466\pi$
$463$			2.29006i			0.106428i	−0.998583	+	0.0532141i	$0.983053\pi$
$464$	−8.64680			−0.401417
$465$	−37.1139	−	1.06996i	−1.72111	−	0.0496183i
$466$	5.09402			0.235976
$467$		−	4.65931i		−	0.215607i	−0.994172	−	0.107804i	$-0.965618\pi$
$467$		−	4.65931i		−	0.215607i	0.994172	−	0.107804i	$-0.0343818\pi$
$468$			13.9889i			0.646638i
$469$	1.48075			0.0683748
$470$	0.609908	−	21.1559i	0.0281329	−	0.975849i
$471$	−33.7873			−1.55684
$472$		−	45.9098i		−	2.11317i
$473$			2.99812i			0.137854i
$474$	35.1597			1.61494
$475$	4.99170	+	0.288052i	0.229035	+	0.0132168i
$476$	1.44125			0.0660598
$477$		−	19.2854i		−	0.883019i
$478$		−	30.7616i		−	1.40700i
$479$	−6.28392			−0.287120			−0.143560	−	0.989642i	$-0.545855\pi$
$479$	−6.28392			−0.287120			−0.143560	+	0.989642i	$0.545855\pi$
$480$	0.535640	−	18.5798i	0.0244485	−	0.848047i
$481$	−7.36703			−0.335908
$482$		−	19.6010i		−	0.892803i
$483$			4.57320i			0.208088i
$484$	0.580557			0.0263890
$485$	−16.2466	−	0.468377i	−0.737721	−	0.0212679i
$486$	−25.6914			−1.16539
$487$		−	16.2881i		−	0.738086i	−0.929412	−	0.369043i	$-0.879685\pi$
$487$		−	16.2881i		−	0.738086i	0.929412	−	0.369043i	$-0.120315\pi$
$488$			17.5688i			0.795303i
$489$	−63.6314			−2.87751
$490$	−17.7026	−	0.510352i	−0.799723	−	0.0230554i
$491$	25.4955			1.15060			0.575299	−	0.817943i	$-0.304886\pi$
$491$	25.4955			1.15060			0.575299	+	0.817943i	$0.304886\pi$
$492$			4.53441i			0.204427i
$493$		−	14.4565i		−	0.651087i
$494$	7.39169			0.332568
$495$	0.250260	−	8.68078i	0.0112483	−	0.390172i
$496$	15.8335			0.710946
$497$			4.08113i			0.183064i
$498$		−	33.6511i		−	1.50794i
$499$	31.7364			1.42072			0.710358	−	0.703840i	$-0.248533\pi$
$499$	31.7364			1.42072			0.710358	+	0.703840i	$0.248533\pi$
$500$	−0.560522	+	6.46658i	−0.0250673	+	0.289194i
$501$	−39.6141			−1.76983
$502$			19.5363i			0.871947i
$503$			17.0580i			0.760580i	0.924867	+	0.380290i	$0.124176\pi$
$503$			17.0580i			0.760580i	−0.924867	+	0.380290i	$0.875824\pi$
$504$	−7.08689			−0.315675
$505$	1.16222	−	40.3139i	0.0517180	−	1.79395i
$506$	3.49894			0.155547
$507$		−	66.8830i		−	2.97038i
$508$		−	0.376284i		−	0.0166949i
$509$	7.23190			0.320548			0.160274	−	0.987073i	$-0.448762\pi$
$509$	7.23190			0.320548			0.160274	+	0.987073i	$0.448762\pi$
$510$	−29.2242	−	0.842511i	−1.29407	−	0.0373070i
$511$	−3.90306			−0.172661
$512$			23.3102i			1.03018i
$513$			2.31875i			0.102376i
$514$	−15.6560			−0.690558
$515$	−1.29209	−	0.0372499i	−0.0569363	−	0.00164143i
$516$	4.56675			0.201040
$517$		−	7.94453i		−	0.349400i
$518$		−	0.839646i		−	0.0368919i
$519$	−42.0271			−1.84478
$520$	−1.22912	+	42.6346i	−0.0539005	+	1.86965i
$521$	5.85852			0.256666			0.128333	−	0.991731i	$-0.459037\pi$
$521$	5.85852			0.256666			0.128333	+	0.991731i	$0.459037\pi$
$522$			15.9922i			0.699961i
$523$			16.2875i			0.712201i	0.934448	+	0.356100i	$0.115894\pi$
$523$			16.2875i			0.712201i	−0.934448	+	0.356100i	$0.884106\pi$
$524$	−2.61322			−0.114159
$525$	7.77304	+	0.448553i	0.339243	+	0.0195765i
$526$	33.5215			1.46161
$527$			26.4719i			1.15313i
$528$			6.56406i			0.285664i
$529$	14.3751			0.625004
$530$	0.381216	−	13.2233i	0.0165590	−	0.574382i
$531$	−57.9946			−2.51675
$532$		−	0.344568i		−	0.0149389i
$533$		−	18.4691i		−	0.799987i
$534$	−19.9477			−0.863221
$535$	23.1485	+	0.667354i	1.00080	+	0.0288522i
$536$	−7.67054			−0.331317
$537$			53.0666i			2.28999i
$538$			28.6583i			1.23555i
$539$	−6.64774			−0.286339
$540$	−3.00888	−	0.0867435i	−0.129481	−	0.00373285i
$541$	−21.8632			−0.939970			−0.469985	−	0.882674i	$-0.655741\pi$
$541$	−21.8632			−0.939970			−0.469985	+	0.882674i	$0.655741\pi$
$542$			10.2885i			0.441929i
$543$		−	7.62568i		−	0.327249i
$544$	−13.2522			−0.568184
$545$	−1.00714	+	34.9347i	−0.0431411	+	1.49644i
$546$	11.5103			0.492595
$547$		−	10.3432i		−	0.442242i	−0.975246	−	0.221121i	$-0.929028\pi$
$547$		−	10.3432i		−	0.442242i	0.975246	−	0.221121i	$-0.0709716\pi$
$548$			1.56631i			0.0669096i
$549$	22.1934			0.947192
$550$	0.343187	−	5.94712i	0.0146335	−	0.253586i
$551$	−3.45618			−0.147238
$552$		−	23.6899i		−	1.00831i
$553$			6.67579i			0.283883i
$554$	−1.13136			−0.0480669
$555$	0.200751	−	6.96347i	0.00852142	−	0.295583i
$556$	7.13062			0.302406
$557$		−	2.64883i		−	0.112234i	−0.998424	−	0.0561172i	$-0.982128\pi$
$557$		−	2.64883i		−	0.112234i	0.998424	−	0.0561172i	$-0.0178721\pi$
$558$		−	29.2841i		−	1.23969i
$559$	−18.6009			−0.786734
$560$	−3.31889	−	0.0956810i	−0.140249	−	0.00404326i
$561$	−10.9744			−0.463338
$562$		−	29.8942i		−	1.26101i
$563$		−	22.2276i		−	0.936781i	−0.883522	−	0.468390i	$-0.844834\pi$
$563$		−	22.2276i		−	0.936781i	0.883522	−	0.468390i	$-0.155166\pi$
$564$	−12.1012			−0.509550
$565$	−43.6703	−	1.25898i	−1.83722	−	0.0529657i
$566$	29.4064			1.23604
$567$		−	3.30445i		−	0.138774i
$568$		−	21.1409i		−	0.887052i
$569$	−5.31396			−0.222773			−0.111386	−	0.993777i	$-0.535529\pi$
$569$	−5.31396			−0.222773			−0.111386	+	0.993777i	$0.535529\pi$
$570$	−0.201423	+	6.98678i	−0.00843669	+	0.292644i
$571$	21.4533			0.897795			0.448897	−	0.893583i	$-0.351817\pi$
$571$	21.4533			0.897795			0.448897	+	0.893583i	$0.351817\pi$
$572$			3.60188i			0.150602i
$573$			28.6272i			1.19592i
$574$	2.10499			0.0878606
$575$	−0.845959	+	14.6597i	−0.0352789	+	0.611353i
$576$	34.0932			1.42055
$577$		−	27.1761i		−	1.13136i	−0.824626	−	0.565679i	$-0.808614\pi$
$577$		−	27.1761i		−	1.13136i	0.824626	−	0.565679i	$-0.191386\pi$
$578$			0.590644i			0.0245676i
$579$	13.9320			0.578993
$580$	0.129294	−	4.48483i	0.00536864	−	0.186222i
$581$	6.38935			0.265075
$582$		−	22.7212i		−	0.941823i
$583$		−	4.96564i		−	0.205656i
$584$	20.2185			0.836647
$585$	53.8571	+	1.55266i	2.22672	+	0.0641945i
$586$	−14.5387			−0.600587
$587$		−	2.28446i		−	0.0942899i	−0.998888	−	0.0471450i	$-0.984988\pi$
$587$		−	2.28446i		−	0.0942899i	0.998888	−	0.0471450i	$-0.0150123\pi$
$588$			10.1259i			0.417584i
$589$	6.32876			0.260772
$590$	−39.7646	−	1.14638i	−1.63708	−	0.0471958i
$591$	9.22738			0.379564
$592$			2.97076i			0.122097i
$593$			24.9685i			1.02533i	0.858588	+	0.512667i	$0.171342\pi$
$593$			24.9685i			1.02533i	−0.858588	+	0.512667i	$0.828658\pi$
$594$	2.76257			0.113350
$595$	0.159968	−	5.54881i	0.00655804	−	0.227479i
$596$	1.22423			0.0501464
$597$		−	43.8318i		−	1.79392i
$598$			21.7081i			0.887709i
$599$	41.4871			1.69512			0.847559	−	0.530701i	$-0.178071\pi$
$599$	41.4871			1.69512			0.847559	+	0.530701i	$0.178071\pi$
$600$	−40.2656	−	2.32358i	−1.64384	−	0.0948597i
$601$	3.47752			0.141851			0.0709255	−	0.997482i	$-0.477405\pi$
$601$	3.47752			0.141851			0.0709255	+	0.997482i	$0.477405\pi$
$602$		−	2.12001i		−	0.0864050i
$603$			9.68964i			0.394593i
$604$	−8.64612			−0.351806
$605$	0.0644373	−	2.23514i	0.00261975	−	0.0908713i
$606$	56.3797			2.29027
$607$		−	28.6420i		−	1.16254i	−0.813710	−	0.581271i	$-0.802556\pi$
$607$		−	28.6420i		−	1.16254i	0.813710	−	0.581271i	$-0.197444\pi$
$608$			3.16827i			0.128490i
$609$	−5.38194			−0.218087
$610$	15.2172	+	0.438698i	0.616125	+	0.0177624i
$611$	49.2893			1.99403
$612$			9.43117i			0.381232i
$613$		−	20.2961i		−	0.819750i	−0.912142	−	0.409875i	$-0.865572\pi$
$613$		−	20.2961i		−	0.819750i	0.912142	−	0.409875i	$-0.134428\pi$
$614$	−26.6549			−1.07570
$615$	17.4574	+	0.503283i	0.703951	+	0.0202943i
$616$	−1.82474			−0.0735210
$617$		−	16.3976i		−	0.660142i	−0.943956	−	0.330071i	$-0.892927\pi$
$617$		−	16.3976i		−	0.660142i	0.943956	−	0.330071i	$-0.107073\pi$
$618$		−	1.80701i		−	0.0726886i
$619$	2.23701			0.0899131			0.0449566	−	0.998989i	$-0.485685\pi$
$619$	2.23701			0.0899131			0.0449566	+	0.998989i	$0.485685\pi$
$620$	−0.236756	+	8.21236i	−0.00950834	+	0.329816i
$621$	−6.80977			−0.273267
$622$			29.9809i			1.20213i
$623$		−	3.78747i		−	0.151742i
$624$	−40.7246			−1.63029
$625$	24.8341	+	2.87574i	0.993362	+	0.115030i
$626$	24.3638			0.973773
$627$			2.62369i			0.104780i
$628$			7.47628i			0.298336i
$629$	−4.96677			−0.198038
$630$	−0.176962	+	6.13828i	−0.00705032	+	0.244555i
$631$	−1.68491			−0.0670752			−0.0335376	−	0.999437i	$-0.510677\pi$
$631$	−1.68491			−0.0670752			−0.0335376	+	0.999437i	$0.510677\pi$
$632$		−	34.5817i		−	1.37558i
$633$			46.4111i			1.84468i
$634$	29.0000			1.15174
$635$	−1.44869	−	0.0417645i	−0.0574894	−	0.00165737i
$636$	−7.56369			−0.299920
$637$		−	41.2438i		−	1.63414i
$638$			4.11770i			0.163021i
$639$	−26.7057			−1.05646
$640$	9.21332	+	0.265613i	0.364189	+	0.0104993i
$641$	32.2141			1.27238			0.636191	−	0.771532i	$-0.280509\pi$
$641$	32.2141			1.27238			0.636191	+	0.771532i	$0.280509\pi$
$642$			32.3736i			1.27768i
$643$			11.7245i			0.462370i	0.972910	+	0.231185i	$0.0742602\pi$
$643$			11.7245i			0.462370i	−0.972910	+	0.231185i	$0.925740\pi$
$644$	1.01193			0.0398758
$645$	0.506873	−	17.5820i	0.0199581	−	0.692289i
$646$	4.98339			0.196069
$647$			35.0212i			1.37682i	0.725320	+	0.688412i	$0.241692\pi$
$647$			35.0212i			1.37682i	−0.725320	+	0.688412i	$0.758308\pi$
$648$			17.1176i			0.672441i
$649$	−14.9325			−0.586153
$650$	36.8971	+	2.12919i	1.44722	+	0.0835139i
$651$	9.85510			0.386252
$652$			14.0800i			0.551416i
$653$			38.1276i			1.49205i	0.665919	+	0.746024i	$0.268039\pi$
$653$			38.1276i			1.49205i	−0.665919	+	0.746024i	$0.731961\pi$
$654$	−48.8567			−1.91045
$655$	−0.290046	+	10.0609i	−0.0113330	+	0.393110i
$656$	−7.44769			−0.290783
$657$		−	25.5405i		−	0.996431i
$658$			5.61767i			0.219000i
$659$	5.49755			0.214154			0.107077	−	0.994251i	$-0.465851\pi$
$659$	5.49755			0.214154			0.107077	+	0.994251i	$0.465851\pi$
$660$	−3.40458	−	0.0981512i	−0.132523	−	0.00382053i
$661$	16.4155			0.638489			0.319245	−	0.947672i	$-0.396571\pi$
$661$	16.4155			0.638489			0.319245	+	0.947672i	$0.396571\pi$
$662$			12.9419i			0.503000i
$663$		−	68.0870i		−	2.64428i
$664$	−33.0979			−1.28445
$665$	−1.32658	−	0.0382443i	−0.0514426	−	0.00148305i
$666$	5.49441			0.212904
$667$		−	10.1502i		−	0.393016i
$668$			8.76560i			0.339151i
$669$	31.9849			1.23661
$670$	−0.191536	+	6.64381i	−0.00739966	+	0.256673i
$671$	5.71439			0.220602
$672$			4.93361i			0.190318i
$673$		−	18.3464i		−	0.707203i	−0.935396	−	0.353601i	$-0.884957\pi$
$673$		−	18.3464i		−	0.707203i	0.935396	−	0.353601i	$-0.115043\pi$
$674$	−6.14102			−0.236543
$675$	−0.667923	+	11.5745i	−0.0257084	+	0.445503i
$676$	−14.7995			−0.569212
$677$		−	45.8970i		−	1.76397i	−0.471281	−	0.881983i	$-0.656208\pi$
$677$		−	45.8970i		−	1.76397i	0.471281	−	0.881983i	$-0.343792\pi$
$678$		−	61.0737i		−	2.34552i
$679$	4.31408			0.165559
$680$	−0.828659	+	28.7437i	−0.0317776	+	1.10227i
$681$	78.5085			3.00845
$682$		−	7.54010i		−	0.288725i
$683$		−	21.2340i		−	0.812497i	−0.913763	−	0.406248i	$-0.866837\pi$
$683$		−	21.2340i		−	0.812497i	0.913763	−	0.406248i	$-0.133163\pi$
$684$	2.25475			0.0862127
$685$	6.03029	+	0.173848i	0.230406	+	0.00664241i
$686$	9.65048			0.368457
$687$		−	26.4867i		−	1.01053i
$688$			7.50081i			0.285966i
$689$	30.8077			1.17368
$690$	−20.5189	−	0.591544i	−0.781142	−	0.0225197i
$691$	−15.6597			−0.595721			−0.297861	−	0.954609i	$-0.596273\pi$
$691$	−15.6597			−0.595721			−0.297861	+	0.954609i	$0.596273\pi$
$692$			9.29953i			0.353515i
$693$			2.30507i			0.0875622i
$694$	8.94912			0.339704
$695$	0.791442	−	27.4528i	0.0300211	−	1.04134i
$696$	27.8793			1.05676
$697$		−	12.4517i		−	0.471642i
$698$		−	30.7110i		−	1.16243i
$699$	−11.2180			−0.424303
$700$	0.0992535	−	1.71998i	0.00375143	−	0.0650090i
$701$	10.2248			0.386185			0.193092	−	0.981181i	$-0.438148\pi$
$701$	10.2248			0.386185			0.193092	+	0.981181i	$0.438148\pi$
$702$			17.1395i			0.646889i
$703$			1.18743i			0.0447848i
$704$	8.77837			0.330847
$705$	−1.34313	+	46.5893i	−0.0505853	+	1.75465i
$706$	−23.7071			−0.892229
$707$			10.7048i			0.402597i
$708$			22.7453i			0.854821i
$709$	41.8962			1.57344			0.786722	−	0.617307i	$-0.211776\pi$
$709$	41.8962			1.57344			0.786722	+	0.617307i	$0.211776\pi$
$710$	−18.3111	−	0.527894i	−0.687203	−	0.0198115i
$711$	−43.6845			−1.63830
$712$			19.6197i			0.735280i
$713$			18.5864i			0.696067i
$714$	7.76011			0.290415
$715$	13.8672	+	0.399781i	0.518604	+	0.0149509i
$716$	11.7423			0.438831
$717$			67.7428i			2.52990i
$718$		−	4.46586i		−	0.166665i
$719$	35.8827			1.33820			0.669099	−	0.743173i	$-0.266680\pi$
$719$	35.8827			1.33820			0.669099	+	0.743173i	$0.266680\pi$
$720$	0.626110	−	21.7179i	0.0233337	−	0.809378i
$721$	0.343098			0.0127776
$722$		−	1.19140i		−	0.0443395i
$723$			43.1652i			1.60533i
$724$	−1.68737			−0.0627107
$725$	−17.2522	−	0.995560i	−0.640730	−	0.0369742i
$726$	3.12588			0.116012
$727$			19.3626i			0.718119i	0.933315	+	0.359060i	$0.116903\pi$
$727$			19.3626i			0.718119i	−0.933315	+	0.359060i	$0.883097\pi$
$728$		−	11.3210i		−	0.419586i
$729$	39.8745			1.47683
$730$	0.504861	−	17.5122i	0.0186857	−	0.648154i
$731$	−12.5405			−0.463828
$732$		−	8.70419i		−	0.321716i
$733$		−	34.6357i		−	1.27930i	−0.768667	−	0.639649i	$-0.779080\pi$
$733$		−	34.6357i		−	1.27930i	0.768667	−	0.639649i	$-0.220920\pi$
$734$	7.14256			0.263637
$735$	38.9845	+	1.12389i	1.43797	+	0.0414554i
$736$	−9.30465			−0.342974
$737$			2.49490i			0.0919009i
$738$			13.7745i			0.507045i
$739$	−31.1405			−1.14552			−0.572760	−	0.819723i	$-0.694127\pi$
$739$	−31.1405			−1.14552			−0.572760	+	0.819723i	$0.694127\pi$
$740$	−1.54084	−	0.0444212i	−0.0566424	−	0.00163296i
$741$	−16.2779			−0.597983
$742$			3.51126i			0.128903i
$743$			9.65172i			0.354087i	0.984203	+	0.177044i	$0.0566534\pi$
$743$			9.65172i			0.354087i	−0.984203	+	0.177044i	$0.943347\pi$
$744$	−51.0510			−1.87162
$745$	0.135880	−	4.71327i	0.00497825	−	0.172681i
$746$	3.78881			0.138718
$747$			41.8101i			1.52975i
$748$			2.42835i			0.0887893i
$749$	−6.14679			−0.224599
$750$	−3.01800	+	34.8178i	−0.110202	+	1.27137i
$751$	−19.2926			−0.703998			−0.351999	−	0.936000i	$-0.614498\pi$
$751$	−19.2926			−0.703998			−0.351999	+	0.936000i	$0.614498\pi$
$752$		−	19.8759i		−	0.724801i
$753$		−	43.0226i		−	1.56783i
$754$	−25.5470			−0.930366
$755$	−0.959651	+	33.2875i	−0.0349253	+	1.21146i
$756$	0.798968			0.0290582
$757$			54.6809i			1.98741i	0.112018	+	0.993706i	$0.464268\pi$
$757$			54.6809i			1.98741i	−0.112018	+	0.993706i	$0.535732\pi$
$758$		−	15.7690i		−	0.572757i
$759$	−7.70532			−0.279686
$760$	6.87190	+	0.198111i	0.249270	+	0.00718626i
$761$	−20.7495			−0.752170			−0.376085	−	0.926585i	$-0.622730\pi$
$761$	−20.7495			−0.752170			−0.376085	+	0.926585i	$0.622730\pi$
$762$		−	2.02601i		−	0.0733948i
$763$		−	9.27644i		−	0.335830i
$764$	6.33447			0.229173
$765$	36.3099	+	1.04678i	1.31279	+	0.0378466i
$766$	−13.3789			−0.483400
$767$		−	92.6441i		−	3.34519i
$768$		−	33.1785i		−	1.19723i
$769$	33.6641			1.21396			0.606980	−	0.794717i	$-0.292381\pi$
$769$	33.6641			1.21396			0.606980	+	0.794717i	$0.292381\pi$
$770$	−0.0455644	+	1.58049i	−0.00164203	+	0.0569570i
$771$	34.4776			1.24168
$772$		−	3.08279i		−	0.110952i
$773$		−	23.8308i		−	0.857136i	−0.903510	−	0.428568i	$-0.859018\pi$
$773$		−	23.8308i		−	0.857136i	0.903510	−	0.428568i	$-0.140982\pi$
$774$	13.8727			0.498645
$775$	31.5912	+	1.82301i	1.13479	+	0.0654846i
$776$	−22.3476			−0.802232
$777$			1.84906i			0.0663346i
$778$		−	4.71701i		−	0.169113i
$779$	−2.97689			−0.106658
$780$	0.608948	−	21.1226i	0.0218038	−	0.756311i
$781$	−6.87624			−0.246051
$782$			14.6353i			0.523359i
$783$		−	8.01403i		−	0.286398i
$784$	−16.6316			−0.593985
$785$	28.7836	+	0.829808i	1.02733	+	0.0296171i
$786$	−14.0703			−0.501870
$787$		−	1.93897i		−	0.0691170i	−0.999403	−	0.0345585i	$-0.988997\pi$
$787$		−	1.93897i		−	0.0691170i	0.999403	−	0.0345585i	$-0.0110025\pi$
$788$		−	2.04179i		−	0.0727356i
$789$	−73.8206			−2.62808
$790$	−29.9528	−	0.863514i	−1.06567	−	0.0307224i
$791$	11.5961			0.412309
$792$		−	11.9406i		−	0.424291i
$793$			35.4531i			1.25898i
$794$	24.5599			0.871600
$795$	−0.839509	+	29.1201i	−0.0297743	+	1.03278i
$796$	−9.69888			−0.343768
$797$			13.6929i			0.485027i	0.970148	+	0.242513i	$0.0779719\pi$
$797$			13.6929i			0.485027i	−0.970148	+	0.242513i	$0.922028\pi$
$798$		−	1.85525i		−	0.0656750i
$799$	33.2303			1.17560
$800$	−0.912629	+	15.8150i	−0.0322663	+	0.559146i
$801$	24.7842			0.875705
$802$		−	27.0128i		−	0.953854i
$803$		−	6.57621i		−	0.232070i
$804$	3.80025			0.134024
$805$	0.112317	−	3.89593i	0.00395864	−	0.137314i
$806$	46.7802			1.64776
$807$		−	63.1109i		−	2.22161i
$808$		−	55.4528i		−	1.95082i
$809$	3.53291			0.124211			0.0621053	−	0.998070i	$-0.480219\pi$
$809$	3.53291			0.124211			0.0621053	+	0.998070i	$0.480219\pi$
$810$	14.8263	+	0.427430i	0.520943	+	0.0150184i
$811$	7.38520			0.259330			0.129665	−	0.991558i	$-0.458610\pi$
$811$	7.38520			0.259330			0.129665	+	0.991558i	$0.458610\pi$
$812$			1.19089i			0.0417919i
$813$		−	22.6572i		−	0.794624i
$814$	1.41471			0.0495855
$815$	54.2079	+	1.56277i	1.89882	+	0.0547415i
$816$	−27.4561			−0.961156
$817$			2.99812i			0.104891i
$818$			18.1195i			0.633534i
$819$	−14.3010			−0.499719
$820$	0.111364	−	3.86289i	0.00388899	−	0.134898i
$821$	−19.0026			−0.663194			−0.331597	−	0.943421i	$-0.607587\pi$
$821$	−19.0026			−0.663194			−0.331597	+	0.943421i	$0.607587\pi$
$822$			8.43346i			0.294151i
$823$		−	2.13975i		−	0.0745869i	−0.999304	−	0.0372934i	$-0.988126\pi$
$823$		−	2.13975i		−	0.0745869i	0.999304	−	0.0372934i	$-0.0118736\pi$
$824$	−1.77730			−0.0619152
$825$	−0.755762	+	13.0967i	−0.0263122	+	0.455968i
$826$	10.5590			0.367393
$827$		−	21.2906i		−	0.740347i	−0.928963	−	0.370174i	$-0.879298\pi$
$827$		−	21.2906i		−	0.740347i	0.928963	−	0.370174i	$-0.120702\pi$
$828$			6.62181i			0.230124i
$829$	9.62092			0.334148			0.167074	−	0.985944i	$-0.446568\pi$
$829$	9.62092			0.334148			0.167074	+	0.985944i	$0.446568\pi$
$830$	−0.826463	+	28.6676i	−0.0286870	+	0.995067i
$831$	2.49147			0.0864282
$832$			54.4626i			1.88815i
$833$		−	27.8061i		−	0.963425i
$834$	38.3932			1.32945
$835$	33.7475	+	0.972912i	1.16788	+	0.0336690i
$836$	0.580557			0.0200790
$837$			14.6748i			0.507236i
$838$			37.0786i			1.28086i
$839$	−6.80136			−0.234809			−0.117404	−	0.993084i	$-0.537457\pi$
$839$	−6.80136			−0.234809			−0.117404	+	0.993084i	$0.537457\pi$
$840$	10.7009	+	0.308498i	0.369216	+	0.0106442i
$841$	−17.0548			−0.588098
$842$			4.48970i			0.154725i
$843$			65.8327i			2.26740i
$844$	10.2696			0.353494
$845$	−1.64263	+	56.9779i	−0.0565081	+	1.96010i
$846$	−36.7605			−1.26385
$847$			0.593512i			0.0203933i
$848$		−	12.4232i		−	0.426615i
$849$	−64.7583			−2.22250
$850$	24.8756	+	1.43548i	0.853226	+	0.0492365i
$851$	−3.48727			−0.119542
$852$			10.4739i			0.358831i
$853$		−	25.4028i		−	0.869774i	−0.900485	−	0.434887i	$-0.856788\pi$
$853$		−	25.4028i		−	0.869774i	0.900485	−	0.434887i	$-0.143212\pi$
$854$	−4.04071			−0.138270
$855$	0.250260	−	8.68078i	0.00855870	−	0.296876i
$856$	31.8414			1.08832
$857$			13.0542i			0.445924i	0.974827	+	0.222962i	$0.0715727\pi$
$857$			13.0542i			0.445924i	−0.974827	+	0.222962i	$0.928427\pi$
$858$			19.3935i			0.662084i
$859$	−30.4319			−1.03832			−0.519161	−	0.854676i	$-0.673756\pi$
$859$	−30.4319			−1.03832			−0.519161	+	0.854676i	$0.673756\pi$
$860$	−3.89044	−	0.112158i	−0.132663	−	0.00382457i
$861$	−4.63559			−0.157980
$862$			22.1658i			0.754971i
$863$		−	42.3921i		−	1.44304i	−0.692392	−	0.721521i	$-0.743443\pi$
$863$		−	42.3921i		−	1.44304i	0.692392	−	0.721521i	$-0.256557\pi$
$864$	−7.34645			−0.249931
$865$	35.8031	+	1.03217i	1.21734	+	0.0350950i
$866$	−9.99857			−0.339765
$867$		−	1.30071i		−	0.0441744i
$868$		−	2.18068i		−	0.0740172i
$869$	−11.2479			−0.381561
$870$	0.696154	−	24.1475i	0.0236018	−	0.818678i
$871$	−15.4788			−0.524480
$872$			48.0535i			1.62730i
$873$			28.2301i			0.955444i
$874$	3.49894			0.118353
$875$	−6.61087	−	0.573029i	−0.223488	−	0.0193719i
$876$	−10.0169			−0.338441
$877$			51.3776i			1.73490i	0.497525	+	0.867450i	$0.334242\pi$
$877$			51.3776i			1.73490i	−0.497525	+	0.867450i	$0.665758\pi$
$878$		−	17.2953i		−	0.583690i
$879$	32.0169			1.07990
$880$	0.161212	−	5.59196i	0.00543444	−	0.188505i
$881$	−18.5710			−0.625673			−0.312836	−	0.949807i	$-0.601279\pi$
$881$	−18.5710			−0.625673			−0.312836	+	0.949807i	$0.601279\pi$
$882$			30.7601i			1.03575i
$883$			9.41279i			0.316766i	0.987378	+	0.158383i	$0.0506280\pi$
$883$			9.41279i			0.316766i	−0.987378	+	0.158383i	$0.949372\pi$
$884$	−15.0659			−0.506722
$885$	87.5692	+	2.52455i	2.94360	+	0.0848617i
$886$	−19.6589			−0.660454
$887$			31.8246i			1.06856i	0.845306	+	0.534282i	$0.179418\pi$
$887$			31.8246i			1.06856i	−0.845306	+	0.534282i	$0.820582\pi$
$888$		−	9.57842i		−	0.321431i
$889$	0.384680			0.0129018
$890$	16.9935	+	0.489910i	0.569625	+	0.0164218i
$891$	5.56762			0.186522
$892$		−	7.07745i		−	0.236970i
$893$		−	7.94453i		−	0.265854i
$894$	6.59159			0.220456
$895$	1.30330	−	45.2078i	0.0435646	−	1.51113i
$896$	−2.44648			−0.0817310
$897$		−	47.8052i		−	1.59617i
$898$		−	10.2293i		−	0.341355i
$899$	−21.8733			−0.729515
$900$	11.2550	+	0.649488i	0.375168	+	0.0216496i
$901$	20.7702			0.691957
$902$			3.54667i			0.118091i
$903$			4.66865i			0.155363i
$904$	−60.0695			−1.99788
$905$	−0.187285	+	6.49636i	−0.00622556	+	0.215946i
$906$	−46.5531			−1.54662
$907$			22.8319i			0.758120i	0.925372	+	0.379060i	$0.123753\pi$
$907$			22.8319i			0.758120i	−0.925372	+	0.379060i	$0.876247\pi$
$908$		−	17.3719i		−	0.576508i
$909$	−70.0495			−2.32339
$910$	−9.80567	−	0.282690i	−0.325055	−	0.00937107i
$911$	25.2950			0.838061			0.419031	−	0.907972i	$-0.362370\pi$
$911$	25.2950			0.838061			0.419031	+	0.907972i	$0.362370\pi$
$912$			6.56406i			0.217358i
$913$			10.7653i			0.356281i
$914$	−19.8966			−0.658122
$915$	−33.5110	−	0.966096i	−1.10784	−	0.0319382i
$916$	−5.86084			−0.193647
$917$		−	2.67153i		−	0.0882216i
$918$			11.5553i			0.381381i
$919$	57.1547			1.88536			0.942680	−	0.333699i	$-0.108297\pi$
$919$	57.1547			1.88536			0.942680	+	0.333699i	$0.108297\pi$
$920$	−0.581818	+	20.1816i	−0.0191820	+	0.665366i
$921$	58.6991			1.93420
$922$			31.2497i			1.02915i
$923$		−	42.6614i		−	1.40422i
$924$	0.904040			0.0297407
$925$	−0.342042	+	5.92729i	−0.0112463	+	0.194888i
$926$	2.72839			0.0896604
$927$			2.24513i			0.0737399i
$928$		−	10.9501i		−	0.359455i
$929$	54.2551			1.78005			0.890027	−	0.455909i	$-0.150686\pi$
$929$	54.2551			1.78005			0.890027	+	0.455909i	$0.150686\pi$
$930$	−1.27476	+	44.2176i	−0.0418009	+	1.44995i
$931$	−6.64774			−0.217871
$932$			2.48226i			0.0813090i
$933$		−	66.0236i		−	2.16152i
$934$	−5.55112			−0.181638
$935$	9.34912	+	0.269528i	0.305749	+	0.00881450i
$936$	74.0817			2.42144
$937$			1.88862i			0.0616985i	0.999524	+	0.0308493i	$0.00982118\pi$
$937$			1.88862i			0.0616985i	−0.999524	+	0.0308493i	$0.990179\pi$
$938$		−	1.76417i		−	0.0576024i
$939$	−53.6536			−1.75092
$940$	10.3090	+	0.297201i	0.336244	+	0.00969363i
$941$	53.4385			1.74204			0.871022	−	0.491244i	$-0.163458\pi$
$941$	53.4385			1.74204			0.871022	+	0.491244i	$0.163458\pi$
$942$			40.2543i			1.31156i
$943$		−	8.74258i		−	0.284698i
$944$	−37.3588			−1.21592
$945$	0.0886791	−	3.07602i	0.00288473	−	0.100063i
$946$	3.57197			0.116135
$947$			46.7907i			1.52049i	0.649634	+	0.760247i	$0.274922\pi$
$947$			46.7907i			1.52049i	−0.649634	+	0.760247i	$0.725078\pi$
$948$			17.1329i			0.556452i
$949$	40.8000			1.32443
$950$	0.343187	−	5.94712i	0.0111344	−	0.192950i
$951$	−63.8634			−2.07091
$952$		−	7.63252i		−	0.247371i
$953$			32.6342i			1.05713i	0.848894	+	0.528563i	$0.177269\pi$
$953$			32.6342i			1.05713i	−0.848894	+	0.528563i	$0.822731\pi$
$954$	−22.9767			−0.743899
$955$	0.703076	−	24.3876i	0.0227510	−	0.789166i
$956$	14.9898			0.484804
$957$		−	9.06795i		−	0.293125i
$958$			7.48668i			0.241884i
$959$	−1.60126			−0.0517075
$960$	−51.4792	−	1.48410i	−1.66148	−	0.0478992i
$961$	9.05315			0.292037
$962$			8.77711i			0.282985i
$963$		−	40.2229i		−	1.29616i
$964$	9.55136			0.307629
$965$	−11.8687	−	0.342166i	−0.382068	−	0.0110147i
$966$	5.44853			0.175303
$967$		−	20.5674i		−	0.661403i	−0.943735	−	0.330702i	$-0.892715\pi$
$967$		−	20.5674i		−	0.661403i	0.943735	−	0.330702i	$-0.107285\pi$
$968$		−	3.07449i		−	0.0988177i
$969$	−10.9744			−0.352548
$970$	−0.558026	+	19.3563i	−0.0179171	+	0.621493i
$971$	−52.0842			−1.67146			−0.835731	−	0.549139i	$-0.814956\pi$
$971$	−52.0842			−1.67146			−0.835731	+	0.549139i	$0.814956\pi$
$972$		−	12.5191i		−	0.401552i
$973$			7.28973i			0.233698i
$974$	−19.4058			−0.621801
$975$	−81.2542	−	4.68889i	−2.60222	−	0.150165i
$976$	14.2965			0.457619
$977$		−	37.1382i		−	1.18816i	−0.804408	−	0.594078i	$-0.797517\pi$
$977$		−	37.1382i		−	1.18816i	0.804408	−	0.594078i	$-0.202483\pi$
$978$			75.8107i			2.42416i
$979$	6.38146			0.203952
$980$	0.248689	−	8.62629i	0.00794408	−	0.275557i
$981$	60.7025			1.93808
$982$		−	30.3755i		−	0.969320i
$983$			30.9040i			0.985683i	0.870119	+	0.492842i	$0.164042\pi$
$983$			30.9040i			0.985683i	−0.870119	+	0.492842i	$0.835958\pi$
$984$	24.0131			0.765509
$985$	−7.86085	−	0.226622i	−0.250468	−	0.00722078i
$986$	−17.2235			−0.548508
$987$		−	12.3712i		−	0.393779i
$988$			3.60188i			0.114591i
$989$	−8.80495			−0.279981
$990$	−10.3423	−	0.298160i	−0.328700	−	0.00947616i
$991$	7.16186			0.227504			0.113752	−	0.993509i	$-0.463713\pi$
$991$	7.16186			0.227504			0.113752	+	0.993509i	$0.463713\pi$
$992$			20.0512i			0.636627i
$993$		−	28.5004i		−	0.904434i
$994$	4.86227			0.154222
$995$	−1.07650	+	37.3406i	−0.0341273	+	1.18378i
$996$	16.3978			0.519585
$997$		−	30.9756i		−	0.981007i	−0.871439	−	0.490503i	$-0.836813\pi$
$997$		−	30.9756i		−	0.981007i	0.871439	−	0.490503i	$-0.163187\pi$
$998$		−	37.8109i		−	1.19688i
$999$	−2.75336			−0.0871124

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.b.b.419.5	✓	16
5.2	odd	4		5225.2.a.z.1.12		16
5.3	odd	4		5225.2.a.z.1.5		16
5.4	even	2	inner	1045.2.b.b.419.12	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.b.b.419.5	✓	16	1.1	even	1	trivial
1045.2.b.b.419.12	yes	16	5.4	even	2	inner
5225.2.a.z.1.5		16	5.3	odd	4
5225.2.a.z.1.12		16	5.2	odd	4

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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.19140i q^{2} +2.62369i q^{3} +0.580557 q^{4} +(0.0644373 - 2.23514i) q^{5} +3.12588 q^{6} +0.593512i q^{7} -3.07449i q^{8} -3.88377 q^{9} +O(q^{10})\)	\(q-1.19140i q^{2} +2.62369i q^{3} +0.580557 q^{4} +(0.0644373 - 2.23514i) q^{5} +3.12588 q^{6} +0.593512i q^{7} -3.07449i q^{8} -3.88377 q^{9} +(-2.66295 - 0.0767708i) q^{10} -1.00000 q^{11} +1.52321i q^{12} -6.20418i q^{13} +0.707112 q^{14} +(5.86432 + 0.169064i) q^{15} -2.50184 q^{16} -4.18279i q^{17} +4.62714i q^{18} -1.00000 q^{19} +(0.0374095 - 1.29763i) q^{20} -1.55719 q^{21} +1.19140i q^{22} -2.93682i q^{23} +8.06651 q^{24} +(-4.99170 - 0.288052i) q^{25} -7.39169 q^{26} -2.31875i q^{27} +0.344568i q^{28} +3.45618 q^{29} +(0.201423 - 6.98678i) q^{30} -6.32876 q^{31} -3.16827i q^{32} -2.62369i q^{33} -4.98339 q^{34} +(1.32658 + 0.0382443i) q^{35} -2.25475 q^{36} -1.18743i q^{37} +1.19140i q^{38} +16.2779 q^{39} +(-6.87190 - 0.198111i) q^{40} +2.97689 q^{41} +1.85525i q^{42} -2.99812i q^{43} -0.580557 q^{44} +(-0.250260 + 8.68078i) q^{45} -3.49894 q^{46} +7.94453i q^{47} -6.56406i q^{48} +6.64774 q^{49} +(-0.343187 + 5.94712i) q^{50} +10.9744 q^{51} -3.60188i q^{52} +4.96564i q^{53} -2.76257 q^{54} +(-0.0644373 + 2.23514i) q^{55} +1.82474 q^{56} -2.62369i q^{57} -4.11770i q^{58} +14.9325 q^{59} +(3.40458 + 0.0981512i) q^{60} -5.71439 q^{61} +7.54010i q^{62} -2.30507i q^{63} -8.77837 q^{64} +(-13.8672 - 0.399781i) q^{65} -3.12588 q^{66} -2.49490i q^{67} -2.42835i q^{68} +7.70532 q^{69} +(0.0455644 - 1.58049i) q^{70} +6.87624 q^{71} +11.9406i q^{72} +6.57621i q^{73} -1.41471 q^{74} +(0.755762 - 13.0967i) q^{75} -0.580557 q^{76} -0.593512i q^{77} -19.3935i q^{78} +11.2479 q^{79} +(-0.161212 + 5.59196i) q^{80} -5.56762 q^{81} -3.54667i q^{82} -10.7653i q^{83} -0.904040 q^{84} +(-9.34912 - 0.269528i) q^{85} -3.57197 q^{86} +9.06795i q^{87} +3.07449i q^{88} -6.38146 q^{89} +(10.3423 + 0.298160i) q^{90} +3.68226 q^{91} -1.70499i q^{92} -16.6047i q^{93} +9.46514 q^{94} +(-0.0644373 + 2.23514i) q^{95} +8.31258 q^{96} -7.26873i q^{97} -7.92015i q^{98} +3.88377 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9}+O(q^{10}) \) 16 * q - 6 * q^4 + 3 * q^5 - 8 * q^6 - 8 * q^9	\( 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9} + 10 q^{10} - 16 q^{11} + 4 q^{14} + 3 q^{15} - 18 q^{16} - 16 q^{19} - 2 q^{20} - 10 q^{21} + 10 q^{24} - 7 q^{25} - 24 q^{26} + 2 q^{29} + 4 q^{30} - 32 q^{31} - 16 q^{34} - 18 q^{35} + 18 q^{36} + 40 q^{39} - 28 q^{40} + 6 q^{41} + 6 q^{44} + 16 q^{45} + 38 q^{49} - 30 q^{50} - 16 q^{51} + 18 q^{54} - 3 q^{55} + 12 q^{56} + 24 q^{59} - 20 q^{60} - 42 q^{61} + 62 q^{64} - 20 q^{65} + 8 q^{66} + 30 q^{69} - 18 q^{70} - 46 q^{71} - 2 q^{74} - 25 q^{75} + 6 q^{76} + 74 q^{79} - 22 q^{80} - 56 q^{81} + 34 q^{84} - 18 q^{85} + 8 q^{86} + 14 q^{89} - 4 q^{90} - 24 q^{91} + 64 q^{94} - 3 q^{95} + 54 q^{96} + 8 q^{99}+O(q^{100}) \) 16 * q - 6 * q^4 + 3 * q^5 - 8 * q^6 - 8 * q^9 + 10 * q^10 - 16 * q^11 + 4 * q^14 + 3 * q^15 - 18 * q^16 - 16 * q^19 - 2 * q^20 - 10 * q^21 + 10 * q^24 - 7 * q^25 - 24 * q^26 + 2 * q^29 + 4 * q^30 - 32 * q^31 - 16 * q^34 - 18 * q^35 + 18 * q^36 + 40 * q^39 - 28 * q^40 + 6 * q^41 + 6 * q^44 + 16 * q^45 + 38 * q^49 - 30 * q^50 - 16 * q^51 + 18 * q^54 - 3 * q^55 + 12 * q^56 + 24 * q^59 - 20 * q^60 - 42 * q^61 + 62 * q^64 - 20 * q^65 + 8 * q^66 + 30 * q^69 - 18 * q^70 - 46 * q^71 - 2 * q^74 - 25 * q^75 + 6 * q^76 + 74 * q^79 - 22 * q^80 - 56 * q^81 + 34 * q^84 - 18 * q^85 + 8 * q^86 + 14 * q^89 - 4 * q^90 - 24 * q^91 + 64 * q^94 - 3 * q^95 + 54 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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