LMFDB - Embedded newform 1045.2.b.c.419.12

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 26 x^{18} + 281 x^{16} + 1640 x^{14} + 5623 x^{12} + 11551 x^{10} + 13894 x^{8} + 9095 x^{6} + \cdots + 4 $ x^20 + 26x^18 + 281x^16 + 1640x^14 + 5623x^12 + 11551x^10 + 13894x^8 + 9095x^6 + 2753x^4 + 276*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			419.12
Root			$0.386431i$ of defining polynomial
Character	$\chi$	$=$	1045.419
Dual form			1045.2.b.c.419.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.386431i q^{2} -0.261531i q^{3} +1.85067 q^{4} +(0.0276061 + 2.23590i) q^{5} +0.101064 q^{6} +4.13791i q^{7} +1.48802i q^{8} +2.93160 q^{9} +O(q^{10})$	\(q+0.386431i q^{2} -0.261531i q^{3} +1.85067 q^{4} +(0.0276061 + 2.23590i) q^{5} +0.101064 q^{6} +4.13791i q^{7} +1.48802i q^{8} +2.93160 q^{9} +(-0.864019 + 0.0106678i) q^{10} +1.00000 q^{11} -0.484009i q^{12} -0.244900i q^{13} -1.59902 q^{14} +(0.584757 - 0.00721985i) q^{15} +3.12633 q^{16} -4.03954i q^{17} +1.13286i q^{18} +1.00000 q^{19} +(0.0510897 + 4.13791i) q^{20} +1.08219 q^{21} +0.386431i q^{22} -0.483419i q^{23} +0.389163 q^{24} +(-4.99848 + 0.123449i) q^{25} +0.0946368 q^{26} -1.55130i q^{27} +7.65791i q^{28} -5.24273 q^{29} +(0.00278997 + 0.225968i) q^{30} -9.76294 q^{31} +4.18414i q^{32} -0.261531i q^{33} +1.56100 q^{34} +(-9.25195 + 0.114231i) q^{35} +5.42543 q^{36} -1.52969i q^{37} +0.386431i q^{38} -0.0640490 q^{39} +(-3.32705 + 0.0410783i) q^{40} +6.06547 q^{41} +0.418193i q^{42} +7.60200i q^{43} +1.85067 q^{44} +(0.0809299 + 6.55476i) q^{45} +0.186808 q^{46} -9.01914i q^{47} -0.817633i q^{48} -10.1223 q^{49} +(-0.0477043 - 1.93156i) q^{50} -1.05647 q^{51} -0.453229i q^{52} -5.86595i q^{53} +0.599470 q^{54} +(0.0276061 + 2.23590i) q^{55} -6.15728 q^{56} -0.261531i q^{57} -2.02595i q^{58} +12.7234 q^{59} +(1.08219 - 0.0133616i) q^{60} +7.58454 q^{61} -3.77270i q^{62} +12.1307i q^{63} +4.63577 q^{64} +(0.547571 - 0.00676072i) q^{65} +0.101064 q^{66} +2.72263i q^{67} -7.47585i q^{68} -0.126429 q^{69} +(-0.0441425 - 3.57523i) q^{70} +2.89186 q^{71} +4.36227i q^{72} +7.24330i q^{73} +0.591117 q^{74} +(0.0322857 + 1.30726i) q^{75} +1.85067 q^{76} +4.13791i q^{77} -0.0247505i q^{78} -5.91489 q^{79} +(0.0863056 + 6.99015i) q^{80} +8.38909 q^{81} +2.34388i q^{82} +4.94874i q^{83} +2.00278 q^{84} +(9.03199 - 0.111516i) q^{85} -2.93765 q^{86} +1.37114i q^{87} +1.48802i q^{88} +3.41496 q^{89} +(-2.53296 + 0.0312738i) q^{90} +1.01337 q^{91} -0.894649i q^{92} +2.55331i q^{93} +3.48527 q^{94} +(0.0276061 + 2.23590i) q^{95} +1.09428 q^{96} -12.5567i q^{97} -3.91157i q^{98} +2.93160 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 12 q^{4} - 8 q^{6} - 10 q^{9}+O(q^{10}) $ 20 * q - 12 * q^4 - 8 * q^6 - 10 * q^9	$ 20 q - 12 q^{4} - 8 q^{6} - 10 q^{9} - 6 q^{10} + 20 q^{11} + 24 q^{14} - 6 q^{15} - 4 q^{16} + 20 q^{19} - 6 q^{20} - 30 q^{21} + 38 q^{24} + 2 q^{25} + 8 q^{26} + 50 q^{29} - 20 q^{30} - 50 q^{31} + 28 q^{34} + 6 q^{35} - 12 q^{36} + 48 q^{39} + 12 q^{40} - 34 q^{41} - 12 q^{44} - 18 q^{45} - 36 q^{46} - 6 q^{49} + 26 q^{50} - 40 q^{51} - 6 q^{54} - 40 q^{56} + 30 q^{59} - 30 q^{60} - 14 q^{61} + 36 q^{64} + 30 q^{65} - 8 q^{66} - 12 q^{69} - 54 q^{70} - 40 q^{71} + 50 q^{74} - 8 q^{75} - 12 q^{76} + 106 q^{79} + 8 q^{80} - 30 q^{84} - 22 q^{85} + 56 q^{86} + 36 q^{89} - 64 q^{90} - 56 q^{91} + 28 q^{94} + 66 q^{96} - 10 q^{99}+O(q^{100}) $ 20 * q - 12 * q^4 - 8 * q^6 - 10 * q^9 - 6 * q^10 + 20 * q^11 + 24 * q^14 - 6 * q^15 - 4 * q^16 + 20 * q^19 - 6 * q^20 - 30 * q^21 + 38 * q^24 + 2 * q^25 + 8 * q^26 + 50 * q^29 - 20 * q^30 - 50 * q^31 + 28 * q^34 + 6 * q^35 - 12 * q^36 + 48 * q^39 + 12 * q^40 - 34 * q^41 - 12 * q^44 - 18 * q^45 - 36 * q^46 - 6 * q^49 + 26 * q^50 - 40 * q^51 - 6 * q^54 - 40 * q^56 + 30 * q^59 - 30 * q^60 - 14 * q^61 + 36 * q^64 + 30 * q^65 - 8 * q^66 - 12 * q^69 - 54 * q^70 - 40 * q^71 + 50 * q^74 - 8 * q^75 - 12 * q^76 + 106 * q^79 + 8 * q^80 - 30 * q^84 - 22 * q^85 + 56 * q^86 + 36 * q^89 - 64 * q^90 - 56 * q^91 + 28 * q^94 + 66 * q^96 - 10 * 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 $

$2$			0.386431i			0.273248i	0.990623	+	0.136624i	$0.0436252\pi$
$2$			0.386431i			0.273248i	−0.990623	+	0.136624i	$0.956375\pi$
$3$		−	0.261531i		−	0.150995i	−0.997146	−	0.0754976i	$-0.975945\pi$
$3$		−	0.261531i		−	0.150995i	0.997146	−	0.0754976i	$-0.0240545\pi$
$4$	1.85067			0.925336
$5$	0.0276061	+	2.23590i	0.0123458	+	0.999924i
$5$	0.0276061	+	2.23590i	0.0123458	+	0.999924i
$6$	0.101064			0.0412591
$7$			4.13791i			1.56398i	0.623289	+	0.781992i	$0.285796\pi$
$7$			4.13791i			1.56398i	−0.623289	+	0.781992i	$0.714204\pi$
$8$			1.48802i			0.526094i
$9$	2.93160			0.977200
$10$	−0.864019	+	0.0106678i	−0.273227	+	0.00337346i
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$		−	0.484009i		−	0.139721i
$13$		−	0.244900i		−	0.0679230i	−0.999423	−	0.0339615i	$-0.989188\pi$
$13$		−	0.244900i		−	0.0679230i	0.999423	−	0.0339615i	$-0.0108124\pi$
$14$	−1.59902			−0.427355
$15$	0.584757	−	0.00721985i	0.150984	−	0.00186416i
$16$	3.12633			0.781582
$17$		−	4.03954i		−	0.979732i	−0.871798	−	0.489866i	$-0.837046\pi$
$17$		−	4.03954i		−	0.979732i	0.871798	−	0.489866i	$-0.162954\pi$
$18$			1.13286i			0.267018i
$19$	1.00000			0.229416
$19$	1.00000			0.229416
$20$	0.0510897	+	4.13791i	0.0114240	+	0.925265i
$21$	1.08219			0.236154
$22$			0.386431i			0.0823873i
$23$		−	0.483419i		−	0.100800i	−0.998729	−	0.0503999i	$-0.983950\pi$
$23$		−	0.483419i		−	0.100800i	0.998729	−	0.0503999i	$-0.0160496\pi$
$24$	0.389163			0.0794376
$25$	−4.99848	+	0.123449i	−0.999695	+	0.0246897i
$26$	0.0946368			0.0185598
$27$		−	1.55130i		−	0.298548i
$28$			7.65791i			1.44721i
$29$	−5.24273			−0.973550			−0.486775	−	0.873527i	$-0.661827\pi$
$29$	−5.24273			−0.973550			−0.486775	+	0.873527i	$0.661827\pi$
$30$	0.00278997	+	0.225968i	0.000509376	+	0.0412559i
$31$	−9.76294			−1.75348			−0.876738	−	0.480969i	$-0.840285\pi$
$31$	−9.76294			−1.75348			−0.876738	+	0.480969i	$0.840285\pi$
$32$			4.18414i			0.739659i
$33$		−	0.261531i		−	0.0455268i
$34$	1.56100			0.267709
$35$	−9.25195	+	0.114231i	−1.56386	+	0.0193086i
$36$	5.42543			0.904238
$37$		−	1.52969i		−	0.251479i	−0.992063	−	0.125739i	$-0.959870\pi$
$37$		−	1.52969i		−	0.251479i	0.992063	−	0.125739i	$-0.0401303\pi$
$38$			0.386431i			0.0626873i
$39$	−0.0640490			−0.0102560
$40$	−3.32705	+	0.0410783i	−0.526053	+	0.00649505i
$41$	6.06547			0.947268			0.473634	−	0.880722i	$-0.342942\pi$
$41$	6.06547			0.947268			0.473634	+	0.880722i	$0.342942\pi$
$42$			0.418193i			0.0645285i
$43$			7.60200i			1.15929i	0.814868	+	0.579647i	$0.196810\pi$
$43$			7.60200i			1.15929i	−0.814868	+	0.579647i	$0.803190\pi$
$44$	1.85067			0.278999
$45$	0.0809299	+	6.55476i	0.0120643	+	0.977126i
$46$	0.186808			0.0275433
$47$		−	9.01914i		−	1.31558i	−0.753203	−	0.657788i	$-0.771492\pi$
$47$		−	9.01914i		−	1.31558i	0.753203	−	0.657788i	$-0.228508\pi$
$48$		−	0.817633i		−	0.118015i
$49$	−10.1223			−1.44604
$50$	−0.0477043	−	1.93156i	−0.00674641	−	0.273164i
$51$	−1.05647			−0.147935
$52$		−	0.453229i		−	0.0628516i
$53$		−	5.86595i		−	0.805751i	−0.915255	−	0.402875i	$-0.868011\pi$
$53$		−	5.86595i		−	0.805751i	0.915255	−	0.402875i	$-0.131989\pi$
$54$	0.599470			0.0815775
$55$	0.0276061	+	2.23590i	0.00372240	+	0.301488i
$56$	−6.15728			−0.822802
$57$		−	0.261531i		−	0.0346407i
$58$		−	2.02595i		−	0.266020i
$59$	12.7234			1.65644			0.828220	−	0.560403i	$-0.189354\pi$
$59$	12.7234			1.65644			0.828220	+	0.560403i	$0.189354\pi$
$60$	1.08219	−	0.0133616i	0.139711	−	0.00172497i
$61$	7.58454			0.971101			0.485551	−	0.874209i	$-0.338619\pi$
$61$	7.58454			0.971101			0.485551	+	0.874209i	$0.338619\pi$
$62$		−	3.77270i		−	0.479133i
$63$			12.1307i			1.52833i
$64$	4.63577			0.579472
$65$	0.547571	−	0.00676072i	0.0679178	−	0.000838564i
$66$	0.101064			0.0124401
$67$			2.72263i			0.332623i	0.986073	+	0.166311i	$0.0531857\pi$
$67$			2.72263i			0.332623i	−0.986073	+	0.166311i	$0.946814\pi$
$68$		−	7.47585i		−	0.906581i
$69$	−0.126429			−0.0152203
$70$	−0.0441425	−	3.57523i	−0.00527604	−	0.427322i
$71$	2.89186			0.343201			0.171600	−	0.985167i	$-0.445106\pi$
$71$	2.89186			0.343201			0.171600	+	0.985167i	$0.445106\pi$
$72$			4.36227i			0.514099i
$73$			7.24330i			0.847764i	0.905717	+	0.423882i	$0.139333\pi$
$73$			7.24330i			0.847764i	−0.905717	+	0.423882i	$0.860667\pi$
$74$	0.591117			0.0687160
$75$	0.0322857	+	1.30726i	0.00372803	+	0.150949i
$76$	1.85067			0.212287
$77$			4.13791i			0.471559i
$78$		−	0.0247505i		−	0.00280244i
$79$	−5.91489			−0.665477			−0.332739	−	0.943019i	$-0.607973\pi$
$79$	−5.91489			−0.665477			−0.332739	+	0.943019i	$0.607973\pi$
$80$	0.0863056	+	6.99015i	0.00964926	+	0.781522i
$81$	8.38909			0.932121
$82$			2.34388i			0.258839i
$83$			4.94874i			0.543195i	0.962411	+	0.271598i	$0.0875520\pi$
$83$			4.94874i			0.543195i	−0.962411	+	0.271598i	$0.912448\pi$
$84$	2.00278			0.218522
$85$	9.03199	−	0.111516i	0.979657	−	0.0120956i
$86$	−2.93765			−0.316774
$87$			1.37114i			0.147001i
$88$			1.48802i			0.158623i
$89$	3.41496			0.361985			0.180993	−	0.983484i	$-0.442069\pi$
$89$	3.41496			0.361985			0.180993	+	0.983484i	$0.442069\pi$
$90$	−2.53296	+	0.0312738i	−0.266997	+	0.00329655i
$91$	1.01337			0.106230
$92$		−	0.894649i		−	0.0932736i
$93$			2.55331i			0.264766i
$94$	3.48527			0.359478
$95$	0.0276061	+	2.23590i	0.00283232	+	0.229398i
$96$	1.09428			0.111685
$97$		−	12.5567i		−	1.27494i	−0.770477	−	0.637468i	$-0.779982\pi$
$97$		−	12.5567i		−	1.27494i	0.770477	−	0.637468i	$-0.220018\pi$
$98$		−	3.91157i		−	0.395128i
$99$	2.93160			0.294637
$100$	−9.25054	+	0.228463i	−0.925054	+	0.0228463i
$101$	−11.5009			−1.14438			−0.572192	−	0.820120i	$-0.693907\pi$
$101$	−11.5009			−1.14438			−0.572192	+	0.820120i	$0.693907\pi$
$102$		−	0.408251i		−	0.0404228i
$103$		−	13.0487i		−	1.28573i	−0.765980	−	0.642864i	$-0.777746\pi$
$103$		−	13.0487i		−	1.28573i	0.765980	−	0.642864i	$-0.222254\pi$
$104$	0.364415			0.0357339
$105$	0.0298751	+	2.41967i	0.00291551	+	0.236136i
$106$	2.26678			0.220169
$107$		−	15.0538i		−	1.45531i	−0.685945	−	0.727654i	$-0.740611\pi$
$107$		−	15.0538i		−	1.45531i	0.685945	−	0.727654i	$-0.259389\pi$
$108$		−	2.87095i		−	0.276257i
$109$	3.97702			0.380929			0.190465	−	0.981694i	$-0.439001\pi$
$109$	3.97702			0.380929			0.190465	+	0.981694i	$0.439001\pi$
$110$	−0.864019	+	0.0106678i	−0.0823810	+	0.00101714i
$111$	−0.400061			−0.0379721
$112$			12.9365i			1.22238i
$113$		−	0.880484i		−	0.0828290i	−0.999142	−	0.0414145i	$-0.986814\pi$
$113$		−	0.880484i		−	0.0828290i	0.999142	−	0.0414145i	$-0.0131864\pi$
$114$	0.101064			0.00946548
$115$	1.08087	−	0.0133453i	0.100792	−	0.00124445i
$116$	−9.70257			−0.900861
$117$		−	0.717949i		−	0.0663744i
$118$			4.91670i			0.452619i
$119$	16.7152			1.53228
$120$	0.0107433	+	0.870129i	0.000980721	+	0.0794315i
$121$	1.00000			0.0909091
$122$			2.93090i			0.265351i
$123$		−	1.58631i		−	0.143033i
$124$	−18.0680			−1.62255
$125$	−0.414007	−	11.1727i	−0.0370299	−	0.999314i
$126$	−4.68768			−0.417611
$127$			14.6685i			1.30162i	0.759241	+	0.650809i	$0.225570\pi$
$127$			14.6685i			1.30162i	−0.759241	+	0.650809i	$0.774430\pi$
$128$			10.1597i			0.897998i
$129$	1.98816			0.175048
$130$	0.00261255	+	0.211598i	0.000229136	+	0.0185584i
$131$	1.37384			0.120033			0.0600165	−	0.998197i	$-0.480885\pi$
$131$	1.37384			0.120033			0.0600165	+	0.998197i	$0.480885\pi$
$132$		−	0.484009i		−	0.0421275i
$133$			4.13791i			0.358802i
$134$	−1.05211			−0.0908884
$135$	3.46855	−	0.0428253i	0.298525	−	0.00368581i
$136$	6.01090			0.515430
$137$			3.57696i			0.305600i	0.988257	+	0.152800i	$0.0488290\pi$
$137$			3.57696i			0.305600i	−0.988257	+	0.152800i	$0.951171\pi$
$138$		−	0.0488561i		−	0.00415891i
$139$	−7.87917			−0.668303			−0.334151	−	0.942519i	$-0.608450\pi$
$139$	−7.87917			−0.668303			−0.334151	+	0.942519i	$0.608450\pi$
$140$	−17.1223	+	0.211405i	−1.44710	+	0.0178670i
$141$	−2.35879			−0.198646
$142$			1.11750i			0.0937788i
$143$		−	0.244900i		−	0.0204796i
$144$	9.16515			0.763762
$145$	−0.144731	−	11.7222i	−0.0120193	−	0.973476i
$146$	−2.79903			−0.231650
$147$			2.64730i			0.218346i
$148$		−	2.83094i		−	0.232702i
$149$	−0.608700			−0.0498666			−0.0249333	−	0.999689i	$-0.507937\pi$
$149$	−0.608700			−0.0498666			−0.0249333	+	0.999689i	$0.507937\pi$
$150$	−0.505164	+	0.0124762i	−0.0412465	+	0.00101868i
$151$	−22.8471			−1.85927			−0.929636	−	0.368479i	$-0.879879\pi$
$151$	−22.8471			−1.85927			−0.929636	+	0.368479i	$0.879879\pi$
$152$			1.48802i			0.120694i
$153$		−	11.8423i		−	0.957394i
$154$	−1.59902			−0.128852
$155$	−0.269516	−	21.8289i	−0.0216481	−	1.75334i
$156$	−0.118534			−0.00949029
$157$		−	9.47273i		−	0.756006i	−0.925804	−	0.378003i	$-0.876611\pi$
$157$		−	9.47273i		−	0.756006i	0.925804	−	0.378003i	$-0.123389\pi$
$158$		−	2.28569i		−	0.181840i
$159$	−1.53413			−0.121664
$160$	−9.35531	+	0.115508i	−0.739603	+	0.00913168i
$161$	2.00034			0.157649
$162$			3.24180i			0.254700i
$163$			23.5045i			1.84101i	0.390727	+	0.920507i	$0.372224\pi$
$163$			23.5045i			1.84101i	−0.390727	+	0.920507i	$0.627776\pi$
$164$	11.2252			0.876540
$165$	0.584757	−	0.00721985i	0.0455233	−	0.000562064i
$166$	−1.91235			−0.148427
$167$			18.0231i			1.39467i	0.716746	+	0.697334i	$0.245631\pi$
$167$			18.0231i			1.39467i	−0.716746	+	0.697334i	$0.754369\pi$
$168$			1.61032i			0.124239i
$169$	12.9400			0.995386
$170$	0.0430931	+	3.49024i	0.00330509	+	0.267689i
$171$	2.93160			0.224185
$172$			14.0688i			1.07274i
$173$		−	18.1676i		−	1.38126i	−0.723210	−	0.690628i	$-0.757334\pi$
$173$		−	18.1676i		−	1.38126i	0.723210	−	0.690628i	$-0.242666\pi$
$174$	−0.529850			−0.0401678
$175$	−0.510820	−	20.6833i	−0.0386143	−	1.56351i
$176$	3.12633			0.235656
$177$		−	3.32756i		−	0.250115i
$178$			1.31965i			0.0989117i
$179$	21.8821			1.63555			0.817774	−	0.575540i	$-0.195208\pi$
$179$	21.8821			1.63555			0.817774	+	0.575540i	$0.195208\pi$
$180$	0.149775	+	12.1307i	0.0111636	+	0.904170i
$181$	−17.5731			−1.30620			−0.653100	−	0.757271i	$-0.726532\pi$
$181$	−17.5731			−1.30620			−0.653100	+	0.757271i	$0.726532\pi$
$182$			0.391599i			0.0290272i
$183$		−	1.98360i		−	0.146632i
$184$	0.719335			0.0530301
$185$	3.42022	−	0.0422286i	0.251460	−	0.00310471i
$186$	−0.986679			−0.0723468
$187$		−	4.03954i		−	0.295400i
$188$		−	16.6915i		−	1.21735i
$189$	6.41914			0.466924
$190$	−0.864019	+	0.0106678i	−0.0626825	+	0.000773925i
$191$	−20.5263			−1.48523			−0.742615	−	0.669719i	$-0.766415\pi$
$191$	−20.5263			−1.48523			−0.742615	+	0.669719i	$0.766415\pi$
$192$		−	1.21240i		−	0.0874974i
$193$			14.2453i			1.02540i	0.858569	+	0.512699i	$0.171354\pi$
$193$			14.2453i			1.02540i	−0.858569	+	0.512699i	$0.828646\pi$
$194$	4.85228			0.348373
$195$	−0.00176814	−	0.143207i	−0.000126619	−	0.0102553i
$196$	−18.7331			−1.33808
$197$			3.26945i			0.232939i	0.993194	+	0.116469i	$0.0371577\pi$
$197$			3.26945i			0.232939i	−0.993194	+	0.116469i	$0.962842\pi$
$198$			1.13286i			0.0805089i
$199$	14.9387			1.05897			0.529487	−	0.848318i	$-0.322384\pi$
$199$	14.9387			1.05897			0.529487	+	0.848318i	$0.322384\pi$
$200$	−0.183694	−	7.43782i	−0.0129891	−	0.525933i
$201$	0.712054			0.0502244
$202$		−	4.44431i		−	0.312700i
$203$		−	21.6940i		−	1.52262i
$204$	−1.95517			−0.136889
$205$	0.167444	+	13.5618i	0.0116948	+	0.947195i
$206$	5.04243			0.351322
$207$		−	1.41719i		−	0.0985016i
$208$		−	0.765637i		−	0.0530874i
$209$	1.00000			0.0691714
$210$	−0.935036	+	0.0115446i	−0.0645236	+	0.000796657i
$211$	7.07216			0.486867			0.243434	−	0.969918i	$-0.421726\pi$
$211$	7.07216			0.486867			0.243434	+	0.969918i	$0.421726\pi$
$212$		−	10.8560i		−	0.745590i
$213$		−	0.756312i		−	0.0518217i
$214$	5.81725			0.397659
$215$	−16.9973	+	0.209861i	−1.15921	+	0.0143124i
$216$	2.30836			0.157064
$217$		−	40.3982i		−	2.74241i
$218$			1.53684i			0.104088i
$219$	1.89435			0.128008
$220$	0.0510897	+	4.13791i	0.00344447	+	0.278978i
$221$	−0.989282			−0.0665463
$222$		−	0.154596i		−	0.0103758i
$223$		−	6.07334i		−	0.406701i	−0.979106	−	0.203351i	$-0.934817\pi$
$223$		−	6.07334i		−	0.406701i	0.979106	−	0.203351i	$-0.0651831\pi$
$224$	−17.3136			−1.15681
$225$	−14.6535	+	0.361902i	−0.976903	+	0.0241268i
$226$	0.340246			0.0226328
$227$			1.60412i			0.106469i	0.998582	+	0.0532345i	$0.0169531\pi$
$227$			1.60412i			0.106469i	−0.998582	+	0.0532345i	$0.983047\pi$
$228$		−	0.484009i		−	0.0320542i
$229$	−5.47550			−0.361831			−0.180916	−	0.983499i	$-0.557906\pi$
$229$	−5.47550			−0.361831			−0.180916	+	0.983499i	$0.557906\pi$
$230$	0.00515703	+	0.417683i	0.000340044	+	0.0275412i
$231$	1.08219			0.0712031
$232$		−	7.80127i		−	0.512179i
$233$		−	15.8903i		−	1.04101i	−0.853859	−	0.520505i	$-0.825744\pi$
$233$		−	15.8903i		−	1.04101i	0.853859	−	0.520505i	$-0.174256\pi$
$234$	0.277437			0.0181366
$235$	20.1659	−	0.248983i	1.31548	−	0.0162419i
$236$	23.5468			1.53276
$237$			1.54693i			0.100484i
$238$			6.45928i			0.418693i
$239$	29.9052			1.93440			0.967202	−	0.254008i	$-0.0817491\pi$
$239$	29.9052			1.93440			0.967202	+	0.254008i	$0.0817491\pi$
$240$	1.82814	−	0.0225716i	0.118006	−	0.00145699i
$241$	−7.02083			−0.452251			−0.226126	−	0.974098i	$-0.572606\pi$
$241$	−7.02083			−0.452251			−0.226126	+	0.974098i	$0.572606\pi$
$242$			0.386431i			0.0248407i
$243$		−	6.84791i		−	0.439294i
$244$	14.0365			0.898595
$245$	−0.279437	−	22.6325i	−0.0178526	−	1.44593i
$246$	0.612999			0.0390834
$247$		−	0.244900i		−	0.0155826i
$248$		−	14.5274i		−	0.922492i
$249$	1.29425			0.0820198
$250$	4.31746	−	0.159985i	0.273060	−	0.0101183i
$251$	11.8694			0.749191			0.374596	−	0.927188i	$-0.377782\pi$
$251$	11.8694			0.749191			0.374596	+	0.927188i	$0.377782\pi$
$252$			22.4500i			1.41421i
$253$		−	0.483419i		−	0.0303923i
$254$	−5.66835			−0.355664
$255$	−0.0291648	−	2.36215i	−0.00182637	−	0.147923i
$256$	5.34553			0.334096
$257$		−	24.0606i		−	1.50086i	−0.660950	−	0.750430i	$-0.729846\pi$
$257$		−	24.0606i		−	1.50086i	0.660950	−	0.750430i	$-0.270154\pi$
$258$			0.768286i			0.0478314i
$259$	6.32970			0.393309
$260$	1.01337	−	0.0125119i	0.0628468	−	0.000775953i
$261$	−15.3696			−0.951354
$262$			0.530894i			0.0327987i
$263$			8.07318i			0.497814i	0.968527	+	0.248907i	$0.0800713\pi$
$263$			8.07318i			0.497814i	−0.968527	+	0.248907i	$0.919929\pi$
$264$	0.389163			0.0239513
$265$	13.1157	−	0.161936i	0.805689	−	0.00994764i
$266$	−1.59902			−0.0980419
$267$		−	0.893120i		−	0.0546581i
$268$			5.03870i			0.307788i
$269$	−9.12119			−0.556129			−0.278064	−	0.960562i	$-0.589693\pi$
$269$	−9.12119			−0.556129			−0.278064	+	0.960562i	$0.589693\pi$
$270$	0.0165490	+	1.34035i	0.00100714	+	0.0815713i
$271$	17.0227			1.03406			0.517028	−	0.855969i	$-0.327038\pi$
$271$	17.0227			1.03406			0.517028	+	0.855969i	$0.327038\pi$
$272$		−	12.6289i		−	0.765740i
$273$		−	0.265029i		−	0.0160403i
$274$	−1.38225			−0.0835045
$275$	−4.99848	+	0.123449i	−0.301419	+	0.00744423i
$276$	−0.233979			−0.0140839
$277$		−	26.3075i		−	1.58066i	−0.612680	−	0.790331i	$-0.709908\pi$
$277$		−	26.3075i		−	1.58066i	0.612680	−	0.790331i	$-0.290092\pi$
$278$		−	3.04475i		−	0.182612i
$279$	−28.6210			−1.71350
$280$	−0.169978	−	13.7671i	−0.0101581	−	0.822739i
$281$	−0.467398			−0.0278826			−0.0139413	−	0.999903i	$-0.504438\pi$
$281$	−0.467398			−0.0278826			−0.0139413	+	0.999903i	$0.504438\pi$
$282$		−	0.911508i		−	0.0542795i
$283$		−	20.3573i		−	1.21011i	−0.796182	−	0.605057i	$-0.793150\pi$
$283$		−	20.3573i		−	1.21011i	0.796182	−	0.605057i	$-0.206850\pi$
$284$	5.35188			0.317576
$285$	0.584757	−	0.00721985i	0.0346380	−	0.000427667i
$286$	0.0946368			0.00559599
$287$			25.0984i			1.48151i
$288$			12.2662i			0.722795i
$289$	0.682144			0.0401261
$290$	4.52982	−	0.0559285i	0.266000	−	0.00328424i
$291$	−3.28396			−0.192509
$292$			13.4050i			0.784467i
$293$		−	29.4685i		−	1.72157i	−0.508970	−	0.860784i	$-0.669973\pi$
$293$		−	29.4685i		−	1.72157i	0.508970	−	0.860784i	$-0.330027\pi$
$294$	−1.02300			−0.0596625
$295$	0.351242	+	28.4481i	0.0204501	+	1.65631i
$296$	2.27620			0.132301
$297$		−	1.55130i		−	0.0900155i
$298$		−	0.235220i		−	0.0136259i
$299$	−0.118389			−0.00684662
$300$	0.0597502	+	2.41931i	0.00344968	+	0.139679i
$301$	−31.4564			−1.81312
$302$		−	8.82883i		−	0.508042i
$303$			3.00785i			0.172797i
$304$	3.12633			0.179307
$305$	0.209379	+	16.9583i	0.0119890	+	0.971027i
$306$	4.57623			0.261606
$307$		−	13.9125i		−	0.794028i	−0.917813	−	0.397014i	$-0.870046\pi$
$307$		−	13.9125i		−	0.794028i	0.917813	−	0.397014i	$-0.129954\pi$
$308$			7.65791i			0.436350i
$309$	−3.41265			−0.194139
$310$	8.43537	−	0.104149i	0.479097	−	0.00591528i
$311$	8.17133			0.463354			0.231677	−	0.972793i	$-0.425579\pi$
$311$	8.17133			0.463354			0.231677	+	0.972793i	$0.425579\pi$
$312$		−	0.0953060i		−	0.00539564i
$313$			24.9259i			1.40889i	0.709757	+	0.704447i	$0.248805\pi$
$313$			24.9259i			1.40889i	−0.709757	+	0.704447i	$0.751195\pi$
$314$	3.66055			0.206577
$315$	−27.1230	+	0.334881i	−1.52821	+	0.0188684i
$316$	−10.9465			−0.615790
$317$		−	22.6159i		−	1.27023i	−0.772416	−	0.635117i	$-0.780952\pi$
$317$		−	22.6159i		−	1.27023i	0.772416	−	0.635117i	$-0.219048\pi$
$318$		−	0.592835i		−	0.0332445i
$319$	−5.24273			−0.293537
$320$	0.127975	+	10.3651i	0.00715404	+	0.579428i
$321$	−3.93704			−0.219744
$322$			0.772994i			0.0430773i
$323$		−	4.03954i		−	0.224766i
$324$	15.5255			0.862525
$325$	0.0302326	+	1.22413i	0.00167700	+	0.0679023i
$326$	−9.08285			−0.503053
$327$		−	1.04012i		−	0.0575185i
$328$			9.02553i			0.498351i
$329$	37.3204			2.05754
$330$	0.00278997	+	0.225968i	0.000153583	+	0.0124391i
$331$	−19.8439			−1.09072			−0.545358	−	0.838203i	$-0.683606\pi$
$331$	−19.8439			−1.09072			−0.545358	+	0.838203i	$0.683606\pi$
$332$			9.15850i			0.502638i
$333$		−	4.48443i		−	0.245745i
$334$	−6.96467			−0.381090
$335$	−6.08753	+	0.0751612i	−0.332597	+	0.00410649i
$336$	3.38329			0.184574
$337$			17.4056i			0.948144i	0.880486	+	0.474072i	$0.157216\pi$
$337$			17.4056i			0.948144i	−0.880486	+	0.474072i	$0.842784\pi$
$338$			5.00042i			0.271987i
$339$	−0.230274			−0.0125068
$340$	16.7152	−	0.206379i	0.906511	−	0.0111925i
$341$	−9.76294			−0.528693
$342$			1.13286i			0.0612581i
$343$		−	12.9199i		−	0.697607i
$344$	−11.3119			−0.609897
$345$	−0.00349021	−	0.282683i	−0.000187907	−	0.0152191i
$346$	7.02051			0.377425
$347$			3.85633i			0.207019i	0.994628	+	0.103509i	$0.0330071\pi$
$347$			3.85633i			0.207019i	−0.994628	+	0.103509i	$0.966993\pi$
$348$			2.53753i			0.136026i
$349$	5.00844			0.268096			0.134048	−	0.990975i	$-0.457202\pi$
$349$	5.00844			0.268096			0.134048	+	0.990975i	$0.457202\pi$
$350$	7.99264	−	0.197396i	0.427225	−	0.0105513i
$351$	−0.379913			−0.0202783
$352$			4.18414i			0.223016i
$353$		−	16.8399i		−	0.896296i	−0.893959	−	0.448148i	$-0.852084\pi$
$353$		−	16.8399i		−	0.896296i	0.893959	−	0.448148i	$-0.147916\pi$
$354$	1.28587			0.0683432
$355$	0.0798329	+	6.46590i	0.00423709	+	0.343175i
$356$	6.31998			0.334958
$357$		−	4.37156i		−	0.231368i
$358$			8.45593i			0.446910i
$359$	1.05288			0.0555691			0.0277846	−	0.999614i	$-0.491155\pi$
$359$	1.05288			0.0555691			0.0277846	+	0.999614i	$0.491155\pi$
$360$	−9.75360	+	0.120425i	−0.514060	+	0.00634696i
$361$	1.00000			0.0526316
$362$		−	6.79080i		−	0.356916i
$363$		−	0.261531i		−	0.0137268i
$364$	1.87542			0.0982988
$365$	−16.1953	+	0.199959i	−0.847700	+	0.0104663i
$366$	0.766522			0.0400667
$367$			5.43166i			0.283531i	0.989900	+	0.141765i	$0.0452778\pi$
$367$			5.43166i			0.283531i	−0.989900	+	0.141765i	$0.954722\pi$
$368$		−	1.51133i		−	0.0787833i
$369$	17.7815			0.925670
$370$	0.0163184	+	1.32168i	0.000848354	+	0.0687107i
$371$	24.2728			1.26018
$372$			4.72535i			0.244998i
$373$			23.1593i			1.19914i	0.800321	+	0.599572i	$0.204662\pi$
$373$			23.1593i			1.19914i	−0.800321	+	0.599572i	$0.795338\pi$
$374$	1.56100			0.0807174
$375$	−2.92200	+	0.108276i	−0.150892	+	0.00559133i
$376$	13.4206			0.692116
$377$			1.28394i			0.0661265i
$378$			2.48055i			0.127586i
$379$	20.8260			1.06976			0.534879	−	0.844929i	$-0.320357\pi$
$379$	20.8260			1.06976			0.534879	+	0.844929i	$0.320357\pi$
$380$	0.0510897	+	4.13791i	0.00262085	+	0.212270i
$381$	3.83627			0.196538
$382$		−	7.93198i		−	0.405835i
$383$			23.5549i			1.20360i	0.798647	+	0.601800i	$0.205549\pi$
$383$			23.5549i			1.20360i	−0.798647	+	0.601800i	$0.794451\pi$
$384$	2.65708			0.135593
$385$	−9.25195	+	0.114231i	−0.471523	+	0.00582177i
$386$	−5.50481			−0.280187
$387$			22.2860i			1.13286i
$388$		−	23.2382i		−	1.17974i
$389$	−16.9626			−0.860037			−0.430019	−	0.902820i	$-0.641493\pi$
$389$	−16.9626			−0.860037			−0.430019	+	0.902820i	$0.641493\pi$
$390$	0.0553396		0.000683263i	0.00280223		3.45984e-5i
$391$	−1.95279			−0.0987567
$392$		−	15.0622i		−	0.760755i
$393$		−	0.359302i		−	0.0181244i
$394$	−1.26342			−0.0636500
$395$	−0.163287	−	13.2251i	−0.00821585	−	0.665426i
$396$	5.42543			0.272638
$397$			11.2020i			0.562210i	0.959677	+	0.281105i	$0.0907009\pi$
$397$			11.2020i			0.562210i	−0.959677	+	0.281105i	$0.909299\pi$
$398$			5.77276i			0.289362i
$399$	1.08219			0.0541774
$400$	−15.6269	+	0.385941i	−0.781344	+	0.0192970i
$401$	−7.62060			−0.380555			−0.190277	−	0.981730i	$-0.560939\pi$
$401$	−7.62060			−0.380555			−0.190277	+	0.981730i	$0.560939\pi$
$402$			0.275159i			0.0137237i
$403$			2.39094i			0.119101i
$404$	−21.2844			−1.05894
$405$	0.231590	+	18.7571i	0.0115078	+	0.932050i
$406$	8.38321			0.416052
$407$		−	1.52969i		−	0.0758237i
$408$		−	1.57204i		−	0.0778275i
$409$	30.7305			1.51953			0.759763	−	0.650200i	$-0.225315\pi$
$409$	30.7305			1.51953			0.759763	+	0.650200i	$0.225315\pi$
$410$	−5.24068	+	0.0647054i	−0.258819	+	0.00319557i
$411$	0.935486			0.0461441
$412$		−	24.1489i		−	1.18973i
$413$			52.6482i			2.59065i
$414$	0.547646			0.0269153
$415$	−11.0649	+	0.136615i	−0.543154	+	0.00670618i
$416$	1.02470			0.0502399
$417$			2.06065i			0.100910i
$418$			0.386431i			0.0189009i
$419$	23.9749			1.17125			0.585625	−	0.810582i	$-0.300849\pi$
$419$	23.9749			1.17125			0.585625	+	0.810582i	$0.300849\pi$
$420$	0.0552890	+	4.47802i	0.00269783	+	0.218505i
$421$	−9.89352			−0.482181			−0.241090	−	0.970503i	$-0.577505\pi$
$421$	−9.89352			−0.482181			−0.241090	+	0.970503i	$0.577505\pi$
$422$			2.73290i			0.133035i
$423$		−	26.4405i		−	1.28558i
$424$	8.72864			0.423900
$425$	0.498675	+	20.1915i	0.0241893	+	0.979433i
$426$	0.292262			0.0141601
$427$			31.3842i			1.51879i
$428$		−	27.8597i		−	1.34665i
$429$	−0.0640490			−0.00309231
$430$	−0.0810968	−	6.56827i	−0.00391083	−	0.316750i
$431$	20.9302			1.00817			0.504086	−	0.863653i	$-0.331829\pi$
$431$	20.9302			1.00817			0.504086	+	0.863653i	$0.331829\pi$
$432$		−	4.84987i		−	0.233340i
$433$			4.10135i			0.197098i	0.995132	+	0.0985490i	$0.0314201\pi$
$433$			4.10135i			0.197098i	−0.995132	+	0.0985490i	$0.968580\pi$
$434$	15.6111			0.749356
$435$	−3.06572	+	0.0378517i	−0.146990	+	0.00181485i
$436$	7.36016			0.352488
$437$		−	0.483419i		−	0.0231251i
$438$			0.732035i			0.0349780i
$439$	−29.1785			−1.39262			−0.696308	−	0.717743i	$-0.745175\pi$
$439$	−29.1785			−1.39262			−0.696308	+	0.717743i	$0.745175\pi$
$440$	−3.32705	+	0.0410783i	−0.158611	+	0.00195833i
$441$	−29.6746			−1.41308
$442$		−	0.382289i		−	0.0181836i
$443$		−	19.1722i		−	0.910899i	−0.890262	−	0.455450i	$-0.849478\pi$
$443$		−	19.1722i		−	0.910899i	0.890262	−	0.455450i	$-0.150522\pi$
$444$	−0.740381			−0.0351369
$445$	0.0942737	+	7.63551i	0.00446900	+	0.361958i
$446$	2.34693			0.111130
$447$			0.159194i			0.00752962i
$448$			19.1824i			0.906284i
$449$	−3.01676			−0.142370			−0.0711849	−	0.997463i	$-0.522678\pi$
$449$	−3.01676			−0.142370			−0.0711849	+	0.997463i	$0.522678\pi$
$450$	−0.139850	−	5.66258i	−0.00659259	−	0.266936i
$451$	6.06547			0.285612
$452$		−	1.62949i		−	0.0766446i
$453$			5.97524i			0.280741i
$454$	−0.619880			−0.0290924
$455$	0.0279753	+	2.26580i	0.00131150	+	0.106222i
$456$	0.389163			0.0182242
$457$			6.32717i			0.295973i	0.988989	+	0.147986i	$0.0472792\pi$
$457$			6.32717i			0.295973i	−0.988989	+	0.147986i	$0.952721\pi$
$458$		−	2.11590i		−	0.0988695i
$459$	−6.26653			−0.292497
$460$	2.00034	−	0.0246977i	0.0932665	−	0.00115154i
$461$	−26.9740			−1.25631			−0.628153	−	0.778090i	$-0.716189\pi$
$461$	−26.9740			−1.25631			−0.628153	+	0.778090i	$0.716189\pi$
$462$			0.418193i			0.0194561i
$463$			5.17767i			0.240627i	0.992736	+	0.120313i	$0.0383899\pi$
$463$			5.17767i			0.240627i	−0.992736	+	0.120313i	$0.961610\pi$
$464$	−16.3905			−0.760909
$465$	−5.70895	+	0.0704869i	−0.264746	+	0.00326875i
$466$	6.14051			0.284454
$467$		−	12.5932i		−	0.582744i	−0.956610	−	0.291372i	$-0.905888\pi$
$467$		−	12.5932i		−	0.582744i	0.956610	−	0.291372i	$-0.0941117\pi$
$468$		−	1.32869i		−	0.0614186i
$469$	−11.2660			−0.520216
$470$	0.0962146	+	7.79271i	0.00443805	+	0.359451i
$471$	−2.47742			−0.114153
$472$			18.9326i			0.871443i
$473$			7.60200i			0.349540i
$474$	−0.597781			−0.0274570
$475$	−4.99848	+	0.123449i	−0.229346	+	0.00566421i
$476$	30.9344			1.41788
$477$		−	17.1966i		−	0.787380i
$478$			11.5563i			0.528571i
$479$	−36.1790			−1.65306			−0.826530	−	0.562893i	$-0.809688\pi$
$479$	−36.1790			−1.65306			−0.826530	+	0.562893i	$0.809688\pi$
$480$	0.0302089	+	2.44671i	0.00137884	+	0.111676i
$481$	−0.374620			−0.0170812
$482$		−	2.71306i		−	0.123577i
$483$		−	0.523153i		−	0.0238043i
$484$	1.85067			0.0841214
$485$	28.0754	−	0.346640i	1.27484	−	0.0157401i
$486$	2.64624			0.120036
$487$		−	13.3983i		−	0.607133i	−0.952810	−	0.303566i	$-0.901823\pi$
$487$		−	13.3983i		−	0.607133i	0.952810	−	0.303566i	$-0.0981774\pi$
$488$			11.2859i			0.510890i
$489$	6.14716			0.277984
$490$	8.74587	−	0.107983i	0.395098	−	0.00487818i
$491$	−26.7013			−1.20501			−0.602506	−	0.798115i	$-0.705831\pi$
$491$	−26.7013			−1.20501			−0.602506	+	0.798115i	$0.705831\pi$
$492$		−	2.93574i		−	0.132353i
$493$			21.1782i			0.953818i
$494$	0.0946368			0.00425791
$495$	0.0809299	+	6.55476i	0.00363753	+	0.294615i
$496$	−30.5221			−1.37048
$497$			11.9663i			0.536760i
$498$			0.500138i			0.0224117i
$499$	−4.18714			−0.187442			−0.0937211	−	0.995598i	$-0.529876\pi$
$499$	−4.18714			−0.187442			−0.0937211	+	0.995598i	$0.529876\pi$
$500$	−0.766190	−	20.6769i	−0.0342651	−	0.924701i
$501$	4.71360			0.210588
$502$			4.58671i			0.204715i
$503$			3.27450i			0.146003i	0.997332	+	0.0730014i	$0.0232577\pi$
$503$			3.27450i			0.146003i	−0.997332	+	0.0730014i	$0.976742\pi$
$504$	−18.0507			−0.804042
$505$	−0.317495	−	25.7149i	−0.0141283	−	1.14430i
$506$	0.186808			0.00830462
$507$		−	3.38422i		−	0.150299i
$508$			27.1466i			1.20443i
$509$	−4.77423			−0.211614			−0.105807	−	0.994387i	$-0.533743\pi$
$509$	−4.77423			−0.211614			−0.105807	+	0.994387i	$0.533743\pi$
$510$	0.912806	−	0.0112702i	0.0404197	−	0.000499052i
$511$	−29.9721			−1.32589
$512$			22.3851i			0.989289i
$513$		−	1.55130i		−	0.0684915i
$514$	9.29776			0.410107
$515$	29.1756	−	0.360224i	1.28563	−	0.0158734i
$516$	3.67943			0.161978
$517$		−	9.01914i		−	0.396661i
$518$			2.44599i			0.107471i
$519$	−4.75139			−0.208563
$520$	0.0100601	+	0.814795i	0.000441163	+	0.0357311i
$521$	−40.1413			−1.75862			−0.879311	−	0.476247i	$-0.841997\pi$
$521$	−40.1413			−1.75862			−0.879311	+	0.476247i	$0.841997\pi$
$522$		−	5.93928i		−	0.259955i
$523$		−	25.7012i		−	1.12384i	−0.827193	−	0.561918i	$-0.810064\pi$
$523$		−	25.7012i		−	1.12384i	0.827193	−	0.561918i	$-0.189936\pi$
$524$	2.54253			0.111071
$525$	−5.40932	+	0.133595i	−0.236082	+	0.00583058i
$526$	−3.11972			−0.136026
$527$			39.4377i			1.71794i
$528$		−	0.817633i		−	0.0355829i
$529$	22.7663			0.989839
$530$	0.0625770	+	5.06830i	0.00271817	+	0.220153i
$531$	37.2998			1.61867
$532$			7.65791i			0.332013i
$533$		−	1.48543i		−	0.0643413i
$534$	0.345129			0.0149352
$535$	33.6588	−	0.415576i	1.45520	−	0.0179669i
$536$	−4.05133			−0.174991
$537$		−	5.72286i		−	0.246960i
$538$		−	3.52471i		−	0.151961i
$539$	−10.1223			−0.435999
$540$	6.41914	−	0.0792555i	0.276236	−	0.00341061i
$541$	13.5359			0.581952			0.290976	−	0.956730i	$-0.406020\pi$
$541$	13.5359			0.581952			0.290976	+	0.956730i	$0.406020\pi$
$542$			6.57809i			0.282553i
$543$			4.59592i			0.197230i
$544$	16.9020			0.724667
$545$	0.109790	+	8.89221i	0.00470288	+	0.380900i
$546$	0.102415			0.00438297
$547$			16.5470i			0.707498i	0.935340	+	0.353749i	$0.115093\pi$
$547$			16.5470i			0.707498i	−0.935340	+	0.353749i	$0.884907\pi$
$548$			6.61977i			0.282783i
$549$	22.2349			0.948961
$550$	−0.0477043	−	1.93156i	−0.00203412	−	0.0823622i
$551$	−5.24273			−0.223348
$552$		−	0.188129i		−	0.00800729i
$553$		−	24.4753i		−	1.04080i
$554$	10.1660			0.431912
$555$	−0.0110441	−	0.894495i	−0.000468796	−	0.0379692i
$556$	−14.5818			−0.618404
$557$			20.7288i			0.878307i	0.898412	+	0.439153i	$0.144721\pi$
$557$			20.7288i			0.878307i	−0.898412	+	0.439153i	$0.855279\pi$
$558$		−	11.0600i		−	0.468209i
$559$	1.86173			0.0787427
$560$	−28.9246	+	0.357125i	−1.22229	+	0.0150913i
$561$	−1.05647			−0.0446040
$562$		−	0.180617i		−	0.00761886i
$563$		−	4.80258i		−	0.202405i	−0.994866	−	0.101202i	$-0.967731\pi$
$563$		−	4.80258i		−	0.202405i	0.994866	−	0.101202i	$-0.0322690\pi$
$564$	−4.36534			−0.183814
$565$	1.96867	−	0.0243067i	0.0828227	−	0.00102259i
$566$	7.86667			0.330661
$567$			34.7133i			1.45782i
$568$			4.30314i			0.180556i
$569$	−18.7387			−0.785565			−0.392783	−	0.919631i	$-0.628488\pi$
$569$	−18.7387			−0.785565			−0.392783	+	0.919631i	$0.628488\pi$
$570$	0.00278997	+	0.225968i	0.000116859	+	0.00946476i
$571$	−34.1368			−1.42858			−0.714291	−	0.699849i	$-0.753251\pi$
$571$	−34.1368			−1.42858			−0.714291	+	0.699849i	$0.753251\pi$
$572$		−	0.453229i		−	0.0189505i
$573$			5.36826i			0.224262i
$574$	−9.69878			−0.404819
$575$	0.0596774	+	2.41636i	0.00248872	+	0.100769i
$576$	13.5902			0.566260
$577$		−	43.9533i		−	1.82980i	−0.403679	−	0.914901i	$-0.632269\pi$
$577$		−	43.9533i		−	1.82980i	0.403679	−	0.914901i	$-0.367731\pi$
$578$			0.263601i			0.0109644i
$579$	3.72559			0.154830
$580$	−0.267850	−	21.6940i	−0.0111219	−	0.900792i
$581$	−20.4775			−0.849548
$582$		−	1.26902i		−	0.0526027i
$583$		−	5.86595i		−	0.242943i
$584$	−10.7782			−0.446003
$585$	1.60526	−	0.0198197i	0.0663693	−	0.000819445i
$586$	11.3875			0.470415
$587$			15.3671i			0.634267i	0.948381	+	0.317134i	$0.102720\pi$
$587$			15.3671i			0.634267i	−0.948381	+	0.317134i	$0.897280\pi$
$588$			4.89929i			0.202043i
$589$	−9.76294			−0.402275
$590$	−10.9932	+	0.135731i	−0.452584	+	0.00558794i
$591$	0.855064			0.0351726
$592$		−	4.78230i		−	0.196551i
$593$		−	16.1352i		−	0.662595i	−0.943526	−	0.331297i	$-0.892514\pi$
$593$		−	16.1352i		−	0.662595i	0.943526	−	0.331297i	$-0.107486\pi$
$594$	0.599470			0.0245965
$595$	0.461442	+	37.3736i	0.0189173	+	1.53217i
$596$	−1.12650			−0.0461434
$597$		−	3.90693i		−	0.159900i
$598$		−	0.0457492i		−	0.00187082i
$599$	9.65248			0.394390			0.197195	−	0.980364i	$-0.436817\pi$
$599$	9.65248			0.394390			0.197195	+	0.980364i	$0.436817\pi$
$600$	−1.94522	+	0.0480416i	−0.0794134	+	0.00196129i
$601$	29.8944			1.21942			0.609708	−	0.792626i	$-0.291287\pi$
$601$	29.8944			1.21942			0.609708	+	0.792626i	$0.291287\pi$
$602$		−	12.1557i		−	0.495430i
$603$			7.98168i			0.325039i
$604$	−42.2825			−1.72045
$605$	0.0276061	+	2.23590i	0.00112235	+	0.0909022i
$606$	−1.16233			−0.0472162
$607$			15.3867i			0.624528i	0.949995	+	0.312264i	$0.101087\pi$
$607$			15.3867i			0.624528i	−0.949995	+	0.312264i	$0.898913\pi$
$608$			4.18414i			0.169689i
$609$	−5.67365			−0.229908
$610$	−6.55319	+	0.0809106i	−0.265331	+	0.00327597i
$611$	−2.20879			−0.0893579
$612$		−	21.9162i		−	0.885911i
$613$		−	43.3990i		−	1.75287i	−0.481521	−	0.876435i	$-0.659915\pi$
$613$		−	43.3990i		−	1.75287i	0.481521	−	0.876435i	$-0.340085\pi$
$614$	5.37621			0.216966
$615$	3.54683	−	0.0437918i	0.143022	−	0.00176586i
$616$	−6.15728			−0.248084
$617$			32.5803i			1.31164i	0.754919	+	0.655818i	$0.227676\pi$
$617$			32.5803i			1.31164i	−0.754919	+	0.655818i	$0.772324\pi$
$618$		−	1.31875i		−	0.0530480i
$619$	4.04214			0.162467			0.0812336	−	0.996695i	$-0.474114\pi$
$619$	4.04214			0.162467			0.0812336	+	0.996695i	$0.474114\pi$
$620$	−0.498786	−	40.3982i	−0.0200317	−	1.62243i
$621$	−0.749927			−0.0300935
$622$			3.15765i			0.126610i
$623$			14.1308i			0.566139i
$624$	−0.200238			−0.00801594
$625$	24.9695	−	1.23411i	0.998781	−	0.0493644i
$626$	−9.63212			−0.384977
$627$		−	0.261531i		−	0.0104446i
$628$		−	17.5309i		−	0.699560i
$629$	−6.17922			−0.246382
$630$	−0.129408	−	10.4812i	−0.00515575	−	0.417580i
$631$	39.0445			1.55434			0.777169	−	0.629292i	$-0.216655\pi$
$631$	39.0445			1.55434			0.777169	+	0.629292i	$0.216655\pi$
$632$		−	8.80146i		−	0.350103i
$633$		−	1.84959i		−	0.0735146i
$634$	8.73946			0.347088
$635$	−32.7972	+	0.404939i	−1.30152	+	0.0160695i
$636$	−2.83917			−0.112580
$637$			2.47895i			0.0982197i
$638$		−	2.02595i		−	0.0802082i
$639$	8.47778			0.335376
$640$	−22.7160	+	0.280469i	−0.897930	+	0.0110865i
$641$	30.5292			1.20583			0.602916	−	0.797805i	$-0.294006\pi$
$641$	30.5292			1.20583			0.602916	+	0.797805i	$0.294006\pi$
$642$		−	1.52139i		−	0.0600446i
$643$			27.4261i			1.08158i	0.841158	+	0.540790i	$0.181875\pi$
$643$			27.4261i			1.08158i	−0.841158	+	0.540790i	$0.818125\pi$
$644$	3.70198			0.145878
$645$	0.0548853	+	4.44532i	0.00216111	+	0.175034i
$646$	1.56100			0.0614167
$647$			42.2102i			1.65945i	0.558171	+	0.829726i	$0.311503\pi$
$647$			42.2102i			1.65945i	−0.558171	+	0.829726i	$0.688497\pi$
$648$			12.4831i			0.490383i
$649$	12.7234			0.499436
$650$	−0.473040	+	0.0116828i	−0.0185541	+	0.000458236i
$651$	−10.5654			−0.414090
$652$			43.4991i			1.70356i
$653$		−	28.8516i		−	1.12905i	−0.825415	−	0.564526i	$-0.809059\pi$
$653$		−	28.8516i		−	1.12905i	0.825415	−	0.564526i	$-0.190941\pi$
$654$	0.401932			0.0157168
$655$	0.0379263	+	3.07176i	0.00148190	+	0.120024i
$656$	18.9627			0.740367
$657$			21.2345i			0.828436i
$658$			14.4217i			0.562218i
$659$	11.5297			0.449135			0.224567	−	0.974459i	$-0.427903\pi$
$659$	11.5297			0.449135			0.224567	+	0.974459i	$0.427903\pi$
$660$	1.08219	−	0.0133616i	0.0421243	−	0.000520098i
$661$	16.8814			0.656610			0.328305	−	0.944572i	$-0.393523\pi$
$661$	16.8814			0.656610			0.328305	+	0.944572i	$0.393523\pi$
$662$		−	7.66827i		−	0.298036i
$663$			0.258728i			0.0100482i
$664$	−7.36381			−0.285771
$665$	−9.25195	+	0.114231i	−0.358775	+	0.00442970i
$666$	1.73292			0.0671493
$667$			2.53443i			0.0981337i
$668$			33.3548i			1.29054i
$669$	−1.58837			−0.0614099
$670$	−0.0290446	−	2.35241i	−0.00112209	−	0.0908814i
$671$	7.58454			0.292798
$672$			4.52805i			0.174673i
$673$			7.27270i			0.280342i	0.990127	+	0.140171i	$0.0447652\pi$
$673$			7.27270i			0.280342i	−0.990127	+	0.140171i	$0.955235\pi$
$674$	−6.72606			−0.259078
$675$	0.191506	+	7.75413i	0.00737106	+	0.298457i
$676$	23.9477			0.921067
$677$			0.515123i			0.0197978i	0.999951	+	0.00989888i	$0.00315096\pi$
$677$			0.515123i			0.0197978i	−0.999951	+	0.00989888i	$0.996849\pi$
$678$		−	0.0889850i		−	0.00341745i
$679$	51.9583			1.99398
$680$	0.165937	+	13.4398i	0.00636340	+	0.515391i
$681$	0.419527			0.0160763
$682$		−	3.77270i		−	0.144464i
$683$			29.0661i			1.11218i	0.831121	+	0.556091i	$0.187700\pi$
$683$			29.0661i			1.11218i	−0.831121	+	0.556091i	$0.812300\pi$
$684$	5.42543			0.207447
$685$	−7.99771	+	0.0987457i	−0.305577	+	0.00377288i
$686$	4.99263			0.190619
$687$			1.43201i			0.0546348i
$688$			23.7663i			0.906083i
$689$	−1.43657			−0.0547290
$690$	0.109237	−	0.00134872i	0.00415859	−	5.13450e-5i
$691$	−1.60956			−0.0612306			−0.0306153	−	0.999531i	$-0.509747\pi$
$691$	−1.60956			−0.0612306			−0.0306153	+	0.999531i	$0.509747\pi$
$692$		−	33.6222i		−	1.27813i
$693$			12.1307i			0.460807i
$694$	−1.49020			−0.0565673
$695$	−0.217513	−	17.6170i	−0.00825073	−	0.668252i
$696$	−2.04028			−0.0773365
$697$		−	24.5017i		−	0.928068i
$698$			1.93541i			0.0732565i
$699$	−4.15582			−0.157187
$700$	−0.945359	−	38.2779i	−0.0357312	−	1.44677i
$701$	−26.6908			−1.00810			−0.504048	−	0.863675i	$-0.668157\pi$
$701$	−26.6908			−1.00810			−0.504048	+	0.863675i	$0.668157\pi$
$702$		−	0.146810i		−	0.00554099i
$703$		−	1.52969i		−	0.0576932i
$704$	4.63577			0.174717
$705$	−0.0651168	−	5.27401i	−0.00245244	−	0.198631i
$706$	6.50744			0.244911
$707$		−	47.5898i		−	1.78980i
$708$		−	6.15822i		−	0.231440i
$709$	−34.7167			−1.30381			−0.651907	−	0.758299i	$-0.726031\pi$
$709$	−34.7167			−1.30381			−0.651907	+	0.758299i	$0.726031\pi$
$710$	−2.49862	+	0.0308499i	−0.0937717	+	0.00115777i
$711$	−17.3401			−0.650305
$712$			5.08153i			0.190438i
$713$			4.71959i			0.176750i
$714$	1.68930			0.0632206
$715$	0.547571	−	0.00676072i	0.0204780	−	0.000252837i
$716$	40.4966			1.51343
$717$		−	7.82114i		−	0.292086i
$718$			0.406867i			0.0151841i
$719$	−0.454145			−0.0169368			−0.00846838	−	0.999964i	$-0.502696\pi$
$719$	−0.454145			−0.0169368			−0.00846838	+	0.999964i	$0.502696\pi$
$720$	0.253014	+	20.4923i	0.00942926	+	0.763704i
$721$	53.9945			2.01086
$722$			0.386431i			0.0143815i
$723$			1.83617i			0.0682878i
$724$	−32.5221			−1.20867
$725$	26.2057	−	0.647208i	0.973254	−	0.0240367i
$726$	0.101064			0.00375083
$727$			34.4667i			1.27830i	0.769082	+	0.639150i	$0.220714\pi$
$727$			34.4667i			1.27830i	−0.769082	+	0.639150i	$0.779286\pi$
$728$			1.50792i			0.0558872i
$729$	23.3763			0.865790
$730$	−0.0772703	−	6.25835i	−0.00285990	−	0.231632i
$731$	30.7086			1.13580
$732$		−	3.67098i		−	0.135683i
$733$		−	33.3530i		−	1.23192i	−0.787777	−	0.615960i	$-0.788768\pi$
$733$		−	33.3530i		−	1.23192i	0.787777	−	0.615960i	$-0.211232\pi$
$734$	−2.09896			−0.0774741
$735$	−5.91910	+	0.0730816i	−0.218329	+	0.00269565i
$736$	2.02269			0.0745574
$737$			2.72263i			0.100290i
$738$			6.87133i			0.252937i
$739$	−14.6031			−0.537185			−0.268592	−	0.963254i	$-0.586558\pi$
$739$	−14.6031			−0.537185			−0.268592	+	0.963254i	$0.586558\pi$
$740$	6.32970	−	0.0781512i	0.232684	−	0.00287290i
$741$	−0.0640490			−0.00235290
$742$			9.37975i			0.344341i
$743$			29.2470i			1.07297i	0.843910	+	0.536484i	$0.180248\pi$
$743$			29.2470i			1.07297i	−0.843910	+	0.536484i	$0.819752\pi$
$744$	−3.79938			−0.139292
$745$	−0.0168038	−	1.36099i	−0.000615644	−	0.0498628i
$746$	−8.94946			−0.327663
$747$			14.5077i			0.530810i
$748$		−	7.47585i		−	0.273344i
$749$	62.2914			2.27608
$750$	−0.0418410	−	1.12915i	−0.00152782	−	0.0412308i
$751$	−26.1796			−0.955307			−0.477654	−	0.878548i	$-0.658513\pi$
$751$	−26.1796			−0.955307			−0.477654	+	0.878548i	$0.658513\pi$
$752$		−	28.1968i		−	1.02823i
$753$		−	3.10422i		−	0.113124i
$754$	−0.496155			−0.0180689
$755$	−0.630719	−	51.0838i	−0.0229542	−	1.85913i
$756$	11.8797			0.432061
$757$			35.4067i			1.28688i	0.765497	+	0.643439i	$0.222493\pi$
$757$			35.4067i			1.28688i	−0.765497	+	0.643439i	$0.777507\pi$
$758$			8.04778i			0.292309i
$759$	−0.126429			−0.00458909
$760$	−3.32705	+	0.0410783i	−0.120685	+	0.00149007i
$761$	3.04755			0.110474			0.0552368	−	0.998473i	$-0.482409\pi$
$761$	3.04755			0.110474			0.0552368	+	0.998473i	$0.482409\pi$
$762$			1.48245i			0.0537036i
$763$			16.4566i			0.595767i
$764$	−37.9874			−1.37434
$765$	26.4782	−	0.326919i	0.957321	−	0.0118198i
$766$	−9.10233			−0.328881
$767$		−	3.11595i		−	0.112510i
$768$		−	1.39802i		−	0.0504469i
$769$	−48.6008			−1.75259			−0.876295	−	0.481776i	$-0.839992\pi$
$769$	−48.6008			−1.75259			−0.876295	+	0.481776i	$0.839992\pi$
$770$	−0.0441425	−	3.57523i	−0.00159079	−	0.128843i
$771$	−6.29261			−0.226623
$772$			26.3633i			0.948837i
$773$		−	51.0426i		−	1.83587i	−0.396727	−	0.917937i	$-0.629854\pi$
$773$		−	51.0426i		−	1.83587i	0.396727	−	0.917937i	$-0.370146\pi$
$774$	−8.61200			−0.309552
$775$	48.7998	−	1.20522i	1.75294	−	0.0432928i
$776$	18.6845			0.670735
$777$		−	1.65542i		−	0.0593877i
$778$		−	6.55486i		−	0.235003i
$779$	6.06547			0.217318
$780$	−0.00327225	−	0.265029i	−0.000117165	−	0.00948956i
$781$	2.89186			0.103479
$782$		−	0.754617i		−	0.0269850i
$783$			8.13304i			0.290651i
$784$	−31.6457			−1.13020
$785$	21.1801	−	0.261505i	0.755949	−	0.00933351i
$786$	0.138845			0.00495245
$787$		−	3.01361i		−	0.107424i	−0.998556	−	0.0537118i	$-0.982895\pi$
$787$		−	3.01361i		−	0.107424i	0.998556	−	0.0537118i	$-0.0171052\pi$
$788$			6.05068i			0.215547i
$789$	2.11139			0.0751675
$790$	5.11058	−	0.0630990i	0.181826	−	0.00224496i
$791$	3.64337			0.129543
$792$			4.36227i			0.155007i
$793$		−	1.85745i		−	0.0659601i
$794$	−4.32878			−0.153623
$795$	−0.0423513	−	3.43016i	−0.00150205	−	0.121655i
$796$	27.6466			0.979907
$797$		−	45.9063i		−	1.62608i	−0.582206	−	0.813042i	$-0.697810\pi$
$797$		−	45.9063i		−	1.62608i	0.582206	−	0.813042i	$-0.302190\pi$
$798$			0.418193i			0.0148039i
$799$	−36.4331			−1.28891
$800$	−0.516527	−	20.9143i	−0.0182620	−	0.739433i
$801$	10.0113			0.353732
$802$		−	2.94483i		−	0.103986i
$803$			7.24330i			0.255611i
$804$	1.31778			0.0464744
$805$	0.0552216	+	4.47256i	0.00194631	+	0.157637i
$806$	−0.923933			−0.0325442
$807$			2.38548i			0.0839728i
$808$		−	17.1136i		−	0.602053i
$809$	−41.8470			−1.47126			−0.735631	−	0.677383i	$-0.763114\pi$
$809$	−41.8470			−1.47126			−0.735631	+	0.677383i	$0.763114\pi$
$810$	−7.24834	+	0.0894933i	−0.254681	+	0.00314448i
$811$	28.5357			1.00202			0.501011	−	0.865441i	$-0.332962\pi$
$811$	28.5357			1.00202			0.501011	+	0.865441i	$0.332962\pi$
$812$		−	40.1484i		−	1.40893i
$813$		−	4.45197i		−	0.156137i
$814$	0.591117			0.0207186
$815$	−52.5536	+	0.648866i	−1.84087	+	0.0227288i
$816$	−3.30286			−0.115623
$817$			7.60200i			0.265960i
$818$			11.8752i			0.415207i
$819$	2.97081			0.103808
$820$	0.309883	+	25.0984i	0.0108216	+	0.876474i
$821$	−18.4483			−0.643849			−0.321924	−	0.946765i	$-0.604330\pi$
$821$	−18.4483			−0.643849			−0.321924	+	0.946765i	$0.604330\pi$
$822$			0.361500i			0.0126088i
$823$			8.59026i			0.299438i	0.988729	+	0.149719i	$0.0478369\pi$
$823$			8.59026i			0.299438i	−0.988729	+	0.149719i	$0.952163\pi$
$824$	19.4167			0.676414
$825$	0.0322857	+	1.30726i	0.00112404	+	0.0455129i
$826$	−20.3449			−0.707888
$827$			22.3427i			0.776931i	0.921463	+	0.388466i	$0.126995\pi$
$827$			22.3427i			0.776931i	−0.921463	+	0.388466i	$0.873005\pi$
$828$		−	2.62275i		−	0.0911470i
$829$	15.8776			0.551450			0.275725	−	0.961237i	$-0.411082\pi$
$829$	15.8776			0.551450			0.275725	+	0.961237i	$0.411082\pi$
$830$	−0.0527923	−	4.27581i	−0.00183245	−	0.148415i
$831$	−6.88023			−0.238672
$832$		−	1.13530i		−	0.0393595i
$833$			40.8895i			1.41674i
$834$	−0.796298			−0.0275736
$835$	−40.2978	+	0.497546i	−1.39456	+	0.0172183i
$836$	1.85067			0.0640068
$837$			15.1452i			0.523496i
$838$			9.26463i			0.320042i
$839$	−50.1265			−1.73056			−0.865280	−	0.501289i	$-0.832859\pi$
$839$	−50.1265			−1.73056			−0.865280	+	0.501289i	$0.832859\pi$
$840$	−3.60052	+	0.0444547i	−0.124230	+	0.00153383i
$841$	−1.51378			−0.0521994
$842$		−	3.82316i		−	0.131755i
$843$			0.122239i			0.00421014i
$844$	13.0882			0.450516
$845$	0.357223	+	28.9326i	0.0122888	+	0.995311i
$846$	10.2174			0.351282
$847$			4.13791i			0.142180i
$848$		−	18.3389i		−	0.629760i
$849$	−5.32406			−0.182721
$850$	−7.80262	+	0.192703i	−0.267628	+	0.00660967i
$851$	−0.739478			−0.0253490
$852$		−	1.39969i		−	0.0479524i
$853$		−	31.6970i		−	1.08528i	−0.839964	−	0.542642i	$-0.817424\pi$
$853$		−	31.6970i		−	1.08528i	0.839964	−	0.542642i	$-0.182576\pi$
$854$	−12.1278			−0.415005
$855$	0.0809299	+	6.55476i	0.00276775	+	0.224168i
$856$	22.4003			0.765628
$857$			20.4944i			0.700075i	0.936736	+	0.350037i	$0.113831\pi$
$857$			20.4944i			0.700075i	−0.936736	+	0.350037i	$0.886169\pi$
$858$		−	0.0247505i		−	0.000844968i
$859$	−8.63383			−0.294583			−0.147291	−	0.989093i	$-0.547055\pi$
$859$	−8.63383			−0.294583			−0.147291	+	0.989093i	$0.547055\pi$
$860$	−31.4564	+	0.388384i	−1.07265	+	0.0132438i
$861$	6.56401			0.223701
$862$			8.08807i			0.275481i
$863$		−	49.4127i		−	1.68203i	−0.541014	−	0.841014i	$-0.681959\pi$
$863$		−	49.4127i		−	1.68203i	0.541014	−	0.841014i	$-0.318041\pi$
$864$	6.49086			0.220824
$865$	40.6209	−	0.501535i	1.38115	−	0.0170527i
$866$	−1.58489			−0.0538566
$867$		−	0.178402i		−	0.00605885i
$868$		−	74.7638i		−	2.53765i
$869$	−5.91489			−0.200649
$870$	−0.0146271	−	1.18469i	−0.000495904	−	0.0401647i
$871$	0.666773			0.0225927
$872$			5.91787i			0.200404i
$873$		−	36.8111i		−	1.24587i
$874$	0.186808			0.00631887
$875$	46.2315	−	1.71312i	1.56291	−	0.0579141i
$876$	3.50582			0.118451
$877$			12.9391i			0.436922i	0.975846	+	0.218461i	$0.0701037\pi$
$877$			12.9391i			0.436922i	−0.975846	+	0.218461i	$0.929896\pi$
$878$		−	11.2755i		−	0.380529i
$879$	−7.70694			−0.259949
$880$	0.0863056	+	6.99015i	0.00290936	+	0.235638i
$881$	−18.7876			−0.632970			−0.316485	−	0.948598i	$-0.602503\pi$
$881$	−18.7876			−0.632970			−0.316485	+	0.948598i	$0.602503\pi$
$882$		−	11.4672i		−	0.386120i
$883$			49.3212i			1.65979i	0.557919	+	0.829895i	$0.311600\pi$
$883$			49.3212i			1.65979i	−0.557919	+	0.829895i	$0.688400\pi$
$884$	−1.83084			−0.0615777
$885$	7.44008	−	0.0918607i	0.250096	−	0.00308787i
$886$	7.40873			0.248901
$887$			43.4343i			1.45838i	0.684310	+	0.729191i	$0.260103\pi$
$887$			43.4343i			1.45838i	−0.684310	+	0.729191i	$0.739897\pi$
$888$		−	0.595297i		−	0.0199769i
$889$	−60.6969			−2.03571
$890$	−2.95059	+	0.0364302i	−0.0989042	+	0.00122114i
$891$	8.38909			0.281045
$892$		−	11.2398i		−	0.376335i
$893$		−	9.01914i		−	0.301814i
$894$	−0.0615175			−0.00205745
$895$	0.604079	+	48.9262i	0.0201921	+	1.63542i
$896$	−42.0399			−1.40445
$897$			0.0309625i			0.00103381i
$898$		−	1.16577i		−	0.0389022i
$899$	51.1844			1.70710
$900$	−27.1189	+	0.669762i	−0.903963	+	0.0223254i
$901$	−23.6957			−0.789419
$902$			2.34388i			0.0780428i
$903$			8.22684i			0.273772i
$904$	1.31018			0.0435758
$905$	−0.485125	−	39.2917i	−0.0161261	−	1.30610i
$906$	−2.30901			−0.0767119
$907$			39.3136i			1.30539i	0.757622	+	0.652694i	$0.226361\pi$
$907$			39.3136i			1.30539i	−0.757622	+	0.652694i	$0.773639\pi$
$908$			2.96869i			0.0985196i
$909$	−33.7161			−1.11829
$910$	−0.875575	+	0.0108105i	−0.0290250	+	0.000358364i
$911$	2.30597			0.0764001			0.0382001	−	0.999270i	$-0.487838\pi$
$911$	2.30597			0.0764001			0.0382001	+	0.999270i	$0.487838\pi$
$912$		−	0.817633i		−	0.0270745i
$913$			4.94874i			0.163779i
$914$	−2.44501			−0.0808738
$915$	4.43512	−	0.0547593i	0.146620	−	0.00181029i
$916$	−10.1334			−0.334815
$917$			5.68483i			0.187730i
$918$		−	2.42158i		−	0.0799240i
$919$	−35.1934			−1.16092			−0.580462	−	0.814287i	$-0.697128\pi$
$919$	−35.1934			−1.16092			−0.580462	+	0.814287i	$0.697128\pi$
$920$	0.0198580	+	1.60836i	0.000654699	+	0.0530261i
$921$	−3.63855			−0.119894
$922$		−	10.4236i		−	0.343283i
$923$		−	0.708216i		−	0.0233112i
$924$	2.00278			0.0658868
$925$	0.188838	+	7.64609i	0.00620894	+	0.251402i
$926$	−2.00081			−0.0657507
$927$		−	38.2537i		−	1.25641i
$928$		−	21.9363i		−	0.720095i
$929$	−26.4097			−0.866474			−0.433237	−	0.901280i	$-0.642629\pi$
$929$	−26.4097			−0.866474			−0.433237	+	0.901280i	$0.642629\pi$
$930$	−0.0272383	−	2.20611i	−0.000893179	−	0.0723413i
$931$	−10.1223			−0.331745
$932$		−	29.4078i		−	0.963284i
$933$		−	2.13706i		−	0.0699642i
$934$	4.86640			0.159233
$935$	9.03199	−	0.111516i	0.295378	−	0.00364695i
$936$	1.06832			0.0349191
$937$			25.8042i			0.842987i	0.906831	+	0.421493i	$0.138494\pi$
$937$			25.8042i			0.842987i	−0.906831	+	0.421493i	$0.861506\pi$
$938$		−	4.35353i		−	0.142148i
$939$	6.51890			0.212736
$940$	37.3204	−	0.460785i	1.21726	−	0.0150292i
$941$	57.3538			1.86968			0.934841	−	0.355067i	$-0.115542\pi$
$941$	57.3538			1.86968			0.934841	+	0.355067i	$0.115542\pi$
$942$		−	0.957349i		−	0.0311921i
$943$		−	2.93216i		−	0.0954843i
$944$	39.7774			1.29464
$945$	0.177207	+	14.3525i	0.00576455	+	0.466888i
$946$	−2.93765			−0.0955111
$947$		−	59.3155i		−	1.92749i	−0.266818	−	0.963747i	$-0.585972\pi$
$947$		−	59.3155i		−	1.92749i	0.266818	−	0.963747i	$-0.414028\pi$
$948$			2.86286i			0.0929813i
$949$	1.77388			0.0575827
$950$	−0.0477043	−	1.93156i	−0.00154773	−	0.0626682i
$951$	−5.91476			−0.191799
$952$			24.8726i			0.806125i
$953$			1.58112i			0.0512174i	0.999672	+	0.0256087i	$0.00815239\pi$
$953$			1.58112i			0.0512174i	−0.999672	+	0.0256087i	$0.991848\pi$
$954$	6.64531			0.215150
$955$	−0.566650	−	45.8947i	−0.0183363	−	1.48512i
$956$	55.3446			1.78997
$957$			1.37114i			0.0443226i
$958$		−	13.9807i		−	0.451695i
$959$	−14.8011			−0.477953
$960$	2.71080	−	0.0334696i	0.0874908	−	0.00108023i
$961$	64.3150			2.07468
$962$		−	0.144765i		−	0.00466740i
$963$		−	44.1318i		−	1.42213i
$964$	−12.9932			−0.418484
$965$	−31.8510	+	0.393256i	−1.02532	+	0.0126594i
$966$	0.202162			0.00650446
$967$		−	38.7297i		−	1.24546i	−0.782436	−	0.622731i	$-0.786023\pi$
$967$		−	38.7297i		−	1.24546i	0.782436	−	0.622731i	$-0.213977\pi$
$968$			1.48802i			0.0478267i
$969$	−1.05647			−0.0339386
$970$	0.133952	+	10.8492i	0.00430095	+	0.348346i
$971$	−44.2299			−1.41941			−0.709703	−	0.704501i	$-0.751171\pi$
$971$	−44.2299			−1.41941			−0.709703	+	0.704501i	$0.751171\pi$
$972$		−	12.6732i		−	0.406494i
$973$		−	32.6033i		−	1.04521i
$974$	5.17750			0.165898
$975$	0.320147	−	0.00790676i	0.0102529	−	0.000253219i
$976$	23.7118			0.758995
$977$		−	0.637601i		−	0.0203987i	−0.999948	−	0.0101993i	$-0.996753\pi$
$977$		−	0.637601i		−	0.0203987i	0.999948	−	0.0101993i	$-0.00324660\pi$
$978$			2.37545i			0.0759585i
$979$	3.41496			0.109143
$980$	−0.517146	−	41.8852i	−0.0165196	−	1.33797i
$981$	11.6590			0.372244
$982$		−	10.3182i		−	0.329267i
$983$			27.4345i			0.875025i	0.899212	+	0.437512i	$0.144140\pi$
$983$			27.4345i			0.875025i	−0.899212	+	0.437512i	$0.855860\pi$
$984$	2.36046			0.0752486
$985$	−7.31016	+	0.0902567i	−0.232921	+	0.00287582i
$986$	−8.18390			−0.260629
$987$		−	9.76046i		−	0.310679i
$988$		−	0.453229i		−	0.0144191i
$989$	3.67495			0.116857
$990$	−2.53296	+	0.0312738i	−0.0805027	+	0.000993947i
$991$	−49.4562			−1.57103			−0.785514	−	0.618844i	$-0.787601\pi$
$991$	−49.4562			−1.57103			−0.785514	+	0.618844i	$0.787601\pi$
$992$		−	40.8495i		−	1.29697i
$993$			5.18979i			0.164693i
$994$	−4.62413			−0.146669
$995$	0.412398	+	33.4013i	0.0130739	+	1.05889i
$996$	2.39523			0.0758959
$997$		−	23.2499i		−	0.736331i	−0.929760	−	0.368165i	$-0.879986\pi$
$997$		−	23.2499i		−	0.736331i	0.929760	−	0.368165i	$-0.120014\pi$
$998$		−	1.61804i		−	0.0512182i
$999$	−2.37300			−0.0750784

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.b.c.419.12	yes	20
5.2	odd	4		5225.2.a.ba.1.9		20
5.3	odd	4		5225.2.a.ba.1.12		20
5.4	even	2	inner	1045.2.b.c.419.9	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.b.c.419.9	✓	20	5.4	even	2	inner
1045.2.b.c.419.12	yes	20	1.1	even	1	trivial
5225.2.a.ba.1.9		20	5.2	odd	4
5225.2.a.ba.1.12		20	5.3	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)

\(f(q)\)	\(=\)	\(q+0.386431i q^{2} -0.261531i q^{3} +1.85067 q^{4} +(0.0276061 + 2.23590i) q^{5} +0.101064 q^{6} +4.13791i q^{7} +1.48802i q^{8} +2.93160 q^{9} +O(q^{10})\)	\(q+0.386431i q^{2} -0.261531i q^{3} +1.85067 q^{4} +(0.0276061 + 2.23590i) q^{5} +0.101064 q^{6} +4.13791i q^{7} +1.48802i q^{8} +2.93160 q^{9} +(-0.864019 + 0.0106678i) q^{10} +1.00000 q^{11} -0.484009i q^{12} -0.244900i q^{13} -1.59902 q^{14} +(0.584757 - 0.00721985i) q^{15} +3.12633 q^{16} -4.03954i q^{17} +1.13286i q^{18} +1.00000 q^{19} +(0.0510897 + 4.13791i) q^{20} +1.08219 q^{21} +0.386431i q^{22} -0.483419i q^{23} +0.389163 q^{24} +(-4.99848 + 0.123449i) q^{25} +0.0946368 q^{26} -1.55130i q^{27} +7.65791i q^{28} -5.24273 q^{29} +(0.00278997 + 0.225968i) q^{30} -9.76294 q^{31} +4.18414i q^{32} -0.261531i q^{33} +1.56100 q^{34} +(-9.25195 + 0.114231i) q^{35} +5.42543 q^{36} -1.52969i q^{37} +0.386431i q^{38} -0.0640490 q^{39} +(-3.32705 + 0.0410783i) q^{40} +6.06547 q^{41} +0.418193i q^{42} +7.60200i q^{43} +1.85067 q^{44} +(0.0809299 + 6.55476i) q^{45} +0.186808 q^{46} -9.01914i q^{47} -0.817633i q^{48} -10.1223 q^{49} +(-0.0477043 - 1.93156i) q^{50} -1.05647 q^{51} -0.453229i q^{52} -5.86595i q^{53} +0.599470 q^{54} +(0.0276061 + 2.23590i) q^{55} -6.15728 q^{56} -0.261531i q^{57} -2.02595i q^{58} +12.7234 q^{59} +(1.08219 - 0.0133616i) q^{60} +7.58454 q^{61} -3.77270i q^{62} +12.1307i q^{63} +4.63577 q^{64} +(0.547571 - 0.00676072i) q^{65} +0.101064 q^{66} +2.72263i q^{67} -7.47585i q^{68} -0.126429 q^{69} +(-0.0441425 - 3.57523i) q^{70} +2.89186 q^{71} +4.36227i q^{72} +7.24330i q^{73} +0.591117 q^{74} +(0.0322857 + 1.30726i) q^{75} +1.85067 q^{76} +4.13791i q^{77} -0.0247505i q^{78} -5.91489 q^{79} +(0.0863056 + 6.99015i) q^{80} +8.38909 q^{81} +2.34388i q^{82} +4.94874i q^{83} +2.00278 q^{84} +(9.03199 - 0.111516i) q^{85} -2.93765 q^{86} +1.37114i q^{87} +1.48802i q^{88} +3.41496 q^{89} +(-2.53296 + 0.0312738i) q^{90} +1.01337 q^{91} -0.894649i q^{92} +2.55331i q^{93} +3.48527 q^{94} +(0.0276061 + 2.23590i) q^{95} +1.09428 q^{96} -12.5567i q^{97} -3.91157i q^{98} +2.93160 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{4} - 8 q^{6} - 10 q^{9}+O(q^{10}) \) 20 * q - 12 * q^4 - 8 * q^6 - 10 * q^9	\( 20 q - 12 q^{4} - 8 q^{6} - 10 q^{9} - 6 q^{10} + 20 q^{11} + 24 q^{14} - 6 q^{15} - 4 q^{16} + 20 q^{19} - 6 q^{20} - 30 q^{21} + 38 q^{24} + 2 q^{25} + 8 q^{26} + 50 q^{29} - 20 q^{30} - 50 q^{31} + 28 q^{34} + 6 q^{35} - 12 q^{36} + 48 q^{39} + 12 q^{40} - 34 q^{41} - 12 q^{44} - 18 q^{45} - 36 q^{46} - 6 q^{49} + 26 q^{50} - 40 q^{51} - 6 q^{54} - 40 q^{56} + 30 q^{59} - 30 q^{60} - 14 q^{61} + 36 q^{64} + 30 q^{65} - 8 q^{66} - 12 q^{69} - 54 q^{70} - 40 q^{71} + 50 q^{74} - 8 q^{75} - 12 q^{76} + 106 q^{79} + 8 q^{80} - 30 q^{84} - 22 q^{85} + 56 q^{86} + 36 q^{89} - 64 q^{90} - 56 q^{91} + 28 q^{94} + 66 q^{96} - 10 q^{99}+O(q^{100}) \) 20 * q - 12 * q^4 - 8 * q^6 - 10 * q^9 - 6 * q^10 + 20 * q^11 + 24 * q^14 - 6 * q^15 - 4 * q^16 + 20 * q^19 - 6 * q^20 - 30 * q^21 + 38 * q^24 + 2 * q^25 + 8 * q^26 + 50 * q^29 - 20 * q^30 - 50 * q^31 + 28 * q^34 + 6 * q^35 - 12 * q^36 + 48 * q^39 + 12 * q^40 - 34 * q^41 - 12 * q^44 - 18 * q^45 - 36 * q^46 - 6 * q^49 + 26 * q^50 - 40 * q^51 - 6 * q^54 - 40 * q^56 + 30 * q^59 - 30 * q^60 - 14 * q^61 + 36 * q^64 + 30 * q^65 - 8 * q^66 - 12 * q^69 - 54 * q^70 - 40 * q^71 + 50 * q^74 - 8 * q^75 - 12 * q^76 + 106 * q^79 + 8 * q^80 - 30 * q^84 - 22 * q^85 + 56 * q^86 + 36 * q^89 - 64 * q^90 - 56 * q^91 + 28 * q^94 + 66 * q^96 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more