LMFDB - Embedded newform 1045.2.b.b.419.16

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			419.16
Root			$2.18657i$ of defining polynomial
Character	$\chi$	$=$	1045.419
Dual form			1045.2.b.b.419.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.18657i q^{2} -1.58567i q^{3} -2.78109 q^{4} +(-0.558280 - 2.16525i) q^{5} +3.46718 q^{6} -2.56330i q^{7} -1.70792i q^{8} +0.485645 q^{9} +O(q^{10})$	\(q+2.18657i q^{2} -1.58567i q^{3} -2.78109 q^{4} +(-0.558280 - 2.16525i) q^{5} +3.46718 q^{6} -2.56330i q^{7} -1.70792i q^{8} +0.485645 q^{9} +(4.73448 - 1.22072i) q^{10} -1.00000 q^{11} +4.40990i q^{12} +2.38273i q^{13} +5.60485 q^{14} +(-3.43338 + 0.885249i) q^{15} -1.82771 q^{16} -3.12660i q^{17} +1.06190i q^{18} -1.00000 q^{19} +(1.55263 + 6.02177i) q^{20} -4.06456 q^{21} -2.18657i q^{22} -0.112593i q^{23} -2.70820 q^{24} +(-4.37665 + 2.41764i) q^{25} -5.21000 q^{26} -5.52709i q^{27} +7.12879i q^{28} +0.414965 q^{29} +(-1.93566 - 7.50733i) q^{30} -8.03180 q^{31} -7.41224i q^{32} +1.58567i q^{33} +6.83654 q^{34} +(-5.55020 + 1.43104i) q^{35} -1.35062 q^{36} -3.76807i q^{37} -2.18657i q^{38} +3.77822 q^{39} +(-3.69807 + 0.953496i) q^{40} -5.68248 q^{41} -8.88745i q^{42} -9.38188i q^{43} +2.78109 q^{44} +(-0.271126 - 1.05154i) q^{45} +0.246193 q^{46} -3.73767i q^{47} +2.89814i q^{48} +0.429469 q^{49} +(-5.28633 - 9.56985i) q^{50} -4.95776 q^{51} -6.62659i q^{52} +4.22916i q^{53} +12.0854 q^{54} +(0.558280 + 2.16525i) q^{55} -4.37791 q^{56} +1.58567i q^{57} +0.907350i q^{58} -11.0901 q^{59} +(9.54856 - 2.46196i) q^{60} +0.894337 q^{61} -17.5621i q^{62} -1.24486i q^{63} +12.5520 q^{64} +(5.15921 - 1.33023i) q^{65} -3.46718 q^{66} +11.3404i q^{67} +8.69537i q^{68} -0.178536 q^{69} +(-3.12907 - 12.1359i) q^{70} +2.55185 q^{71} -0.829441i q^{72} -10.7963i q^{73} +8.23916 q^{74} +(3.83358 + 6.93993i) q^{75} +2.78109 q^{76} +2.56330i q^{77} +8.26136i q^{78} +2.50419 q^{79} +(1.02037 + 3.95745i) q^{80} -7.30721 q^{81} -12.4251i q^{82} -14.4319i q^{83} +11.3039 q^{84} +(-6.76989 + 1.74552i) q^{85} +20.5142 q^{86} -0.657998i q^{87} +1.70792i q^{88} -3.86607 q^{89} +(2.29928 - 0.592836i) q^{90} +6.10766 q^{91} +0.313132i q^{92} +12.7358i q^{93} +8.17268 q^{94} +(0.558280 + 2.16525i) q^{95} -11.7534 q^{96} +4.00346i q^{97} +0.939064i q^{98} -0.485645 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			2.18657i			1.54614i	0.634321	+	0.773070i	$0.281280\pi$
$2$			2.18657i			1.54614i	−0.634321	+	0.773070i	$0.718720\pi$
$3$		−	1.58567i		−	0.915488i	−0.889084	−	0.457744i	$-0.848658\pi$
$3$		−	1.58567i		−	0.915488i	0.889084	−	0.457744i	$-0.151342\pi$
$4$	−2.78109			−1.39055
$5$	−0.558280	−	2.16525i	−0.249670	−	0.968331i
$5$	−0.558280	−	2.16525i	−0.249670	−	0.968331i
$6$	3.46718			1.41547
$7$		−	2.56330i		−	0.968838i	−0.874836	−	0.484419i	$-0.839031\pi$
$7$		−	2.56330i		−	0.968838i	0.874836	−	0.484419i	$-0.160969\pi$
$8$		−	1.70792i		−	0.603840i
$9$	0.485645			0.161882
$10$	4.73448	−	1.22072i	1.49717	−	0.386025i
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$			4.40990i			1.27303i
$13$			2.38273i			0.660850i	0.943832	+	0.330425i	$0.107192\pi$
$13$			2.38273i			0.660850i	−0.943832	+	0.330425i	$0.892808\pi$
$14$	5.60485			1.49796
$15$	−3.43338	+	0.885249i	−0.886495	+	0.228570i
$16$	−1.82771			−0.456926
$17$		−	3.12660i		−	0.758312i	−0.925333	−	0.379156i	$-0.876214\pi$
$17$		−	3.12660i		−	0.758312i	0.925333	−	0.379156i	$-0.123786\pi$
$18$			1.06190i			0.250292i
$19$	−1.00000			−0.229416
$19$	−1.00000			−0.229416
$20$	1.55263	+	6.02177i	0.347178	+	1.34651i
$21$	−4.06456			−0.886960
$22$		−	2.18657i		−	0.466179i
$23$		−	0.112593i		−	0.0234773i	−0.999931	−	0.0117386i	$-0.996263\pi$
$23$		−	0.112593i		−	0.0234773i	0.999931	−	0.0117386i	$-0.00373661\pi$
$24$	−2.70820			−0.552808
$25$	−4.37665	+	2.41764i	−0.875329	+	0.483527i
$26$	−5.21000			−1.02177
$27$		−	5.52709i		−	1.06369i
$28$			7.12879i			1.34721i
$29$	0.414965			0.0770570			0.0385285	−	0.999258i	$-0.487733\pi$
$29$	0.414965			0.0770570			0.0385285	+	0.999258i	$0.487733\pi$
$30$	−1.93566	−	7.50733i	−0.353401	−	1.37065i
$31$	−8.03180			−1.44255			−0.721276	−	0.692647i	$-0.756444\pi$
$31$	−8.03180			−1.44255			−0.721276	+	0.692647i	$0.756444\pi$
$32$		−	7.41224i		−	1.31031i
$33$			1.58567i			0.276030i
$34$	6.83654			1.17246
$35$	−5.55020	+	1.43104i	−0.938156	+	0.241890i
$36$	−1.35062			−0.225104
$37$		−	3.76807i		−	0.619468i	−0.950823	−	0.309734i	$-0.899760\pi$
$37$		−	3.76807i		−	0.619468i	0.950823	−	0.309734i	$-0.100240\pi$
$38$		−	2.18657i		−	0.354709i
$39$	3.77822			0.605000
$40$	−3.69807	+	0.953496i	−0.584717	+	0.150761i
$41$	−5.68248			−0.887454			−0.443727	−	0.896162i	$-0.646344\pi$
$41$	−5.68248			−0.887454			−0.443727	+	0.896162i	$0.646344\pi$
$42$		−	8.88745i		−	1.37136i
$43$		−	9.38188i		−	1.43072i	−0.698754	−	0.715362i	$-0.746262\pi$
$43$		−	9.38188i		−	1.43072i	0.698754	−	0.715362i	$-0.253738\pi$
$44$	2.78109			0.419266
$45$	−0.271126	−	1.05154i	−0.0404171	−	0.156755i
$46$	0.246193			0.0362992
$47$		−	3.73767i		−	0.545195i	−0.962128	−	0.272598i	$-0.912117\pi$
$47$		−	3.73767i		−	0.545195i	0.962128	−	0.272598i	$-0.0878828\pi$
$48$			2.89814i			0.418311i
$49$	0.429469			0.0613527
$50$	−5.28633	−	9.56985i	−0.747600	−	1.35338i
$51$	−4.95776			−0.694226
$52$		−	6.62659i		−	0.918943i
$53$			4.22916i			0.580920i	0.956887	+	0.290460i	$0.0938083\pi$
$53$			4.22916i			0.580920i	−0.956887	+	0.290460i	$0.906192\pi$
$54$	12.0854			1.64461
$55$	0.558280	+	2.16525i	0.0752785	+	0.291963i
$56$	−4.37791			−0.585023
$57$			1.58567i			0.210027i
$58$			0.907350i			0.119141i
$59$	−11.0901			−1.44381			−0.721905	−	0.691993i	$-0.756733\pi$
$59$	−11.0901			−1.44381			−0.721905	+	0.691993i	$0.756733\pi$
$60$	9.54856	−	2.46196i	1.23271	−	0.317838i
$61$	0.894337			0.114508			0.0572540	−	0.998360i	$-0.481765\pi$
$61$	0.894337			0.114508			0.0572540	+	0.998360i	$0.481765\pi$
$62$		−	17.5621i		−	2.23039i
$63$		−	1.24486i		−	0.156837i
$64$	12.5520			1.56900
$65$	5.15921	−	1.33023i	0.639921	−	0.164995i
$66$	−3.46718			−0.426781
$67$			11.3404i			1.38545i	0.721203	+	0.692723i	$0.243589\pi$
$67$			11.3404i			1.38545i	−0.721203	+	0.692723i	$0.756411\pi$
$68$			8.69537i			1.05447i
$69$	−0.178536			−0.0214932
$70$	−3.12907	−	12.1359i	−0.373996	−	1.45052i
$71$	2.55185			0.302849			0.151424	−	0.988469i	$-0.451614\pi$
$71$	2.55185			0.302849			0.151424	+	0.988469i	$0.451614\pi$
$72$		−	0.829441i		−	0.0977506i
$73$		−	10.7963i		−	1.26361i	−0.775127	−	0.631806i	$-0.782314\pi$
$73$		−	10.7963i		−	1.26361i	0.775127	−	0.631806i	$-0.217686\pi$
$74$	8.23916			0.957783
$75$	3.83358	+	6.93993i	0.442663	+	0.801354i
$76$	2.78109			0.319013
$77$			2.56330i			0.292116i
$78$			8.26136i			0.935414i
$79$	2.50419			0.281744			0.140872	−	0.990028i	$-0.455009\pi$
$79$	2.50419			0.281744			0.140872	+	0.990028i	$0.455009\pi$
$80$	1.02037	+	3.95745i	0.114081	+	0.442456i
$81$	−7.30721			−0.811913
$82$		−	12.4251i		−	1.37213i
$83$		−	14.4319i		−	1.58410i	−0.610455	−	0.792051i	$-0.709013\pi$
$83$		−	14.4319i		−	1.58410i	0.610455	−	0.792051i	$-0.290987\pi$
$84$	11.3039			1.23336
$85$	−6.76989	+	1.74552i	−0.734297	+	0.189328i
$86$	20.5142			2.21210
$87$		−	0.657998i		−	0.0705448i
$88$			1.70792i			0.182065i
$89$	−3.86607			−0.409803			−0.204902	−	0.978783i	$-0.565687\pi$
$89$	−3.86607			−0.409803			−0.204902	+	0.978783i	$0.565687\pi$
$90$	2.29928	−	0.592836i	0.242365	−	0.0624904i
$91$	6.10766			0.640256
$92$			0.313132i			0.0326463i
$93$			12.7358i			1.32064i
$94$	8.17268			0.842948
$95$	0.558280	+	2.16525i	0.0572783	+	0.222150i
$96$	−11.7534			−1.19957
$97$			4.00346i			0.406490i	0.979128	+	0.203245i	$0.0651488\pi$
$97$			4.00346i			0.406490i	−0.979128	+	0.203245i	$0.934851\pi$
$98$			0.939064i			0.0948598i
$99$	−0.485645			−0.0488092
$100$	12.1719	−	6.72367i	1.21719	−	0.672367i
$101$	−4.67039			−0.464722			−0.232361	−	0.972630i	$-0.574645\pi$
$101$	−4.67039			−0.464722			−0.232361	+	0.972630i	$0.574645\pi$
$102$		−	10.8405i		−	1.07337i
$103$			3.18981i			0.314302i	0.987575	+	0.157151i	$0.0502309\pi$
$103$			3.18981i			0.314302i	−0.987575	+	0.157151i	$0.949769\pi$
$104$	4.06950			0.399047
$105$	2.26916	+	8.80080i	0.221448	+	0.858870i
$106$	−9.24736			−0.898183
$107$			7.85167i			0.759049i	0.925182	+	0.379525i	$0.123912\pi$
$107$			7.85167i			0.759049i	−0.925182	+	0.379525i	$0.876088\pi$
$108$			15.3714i			1.47911i
$109$	18.9030			1.81058			0.905291	−	0.424792i	$-0.139653\pi$
$109$	18.9030			1.81058			0.905291	+	0.424792i	$0.139653\pi$
$110$	−4.73448	+	1.22072i	−0.451415	+	0.116391i
$111$	−5.97493			−0.567115
$112$			4.68497i			0.442688i
$113$			3.22744i			0.303612i	0.988410	+	0.151806i	$0.0485089\pi$
$113$			3.22744i			0.303612i	−0.988410	+	0.151806i	$0.951491\pi$
$114$	−3.46718			−0.324732
$115$	−0.243793	+	0.0628585i	−0.0227338	+	0.00586159i
$116$	−1.15406			−0.107151
$117$			1.15716i			0.106979i
$118$		−	24.2493i		−	2.23233i
$119$	−8.01443			−0.734682
$120$	1.51193	+	5.86393i	0.138020	+	0.535301i
$121$	1.00000			0.0909091
$122$			1.95553i			0.177045i
$123$			9.01055i			0.812454i
$124$	22.3372			2.00594
$125$	7.67819	+	8.12683i	0.686758	+	0.726886i
$126$	2.72197			0.242492
$127$		−	3.15497i		−	0.279958i	−0.990154	−	0.139979i	$-0.955297\pi$
$127$		−	3.15497i		−	0.279958i	0.990154	−	0.139979i	$-0.0447035\pi$
$128$			12.6213i			1.11558i
$129$	−14.8766			−1.30981
$130$	2.90864	+	11.2810i	0.255105	+	0.989407i
$131$	−2.30234			−0.201157			−0.100578	−	0.994929i	$-0.532069\pi$
$131$	−2.30234			−0.201157			−0.100578	+	0.994929i	$0.532069\pi$
$132$		−	4.40990i		−	0.383833i
$133$			2.56330i			0.222267i
$134$	−24.7965			−2.14209
$135$	−11.9675	+	3.08566i	−1.03000	+	0.265572i
$136$	−5.33998			−0.457899
$137$		−	7.00068i		−	0.598108i	−0.954236	−	0.299054i	$-0.903329\pi$
$137$		−	7.00068i		−	0.598108i	0.954236	−	0.299054i	$-0.0966711\pi$
$138$		−	0.390381i		−	0.0332315i
$139$	6.53732			0.554489			0.277244	−	0.960799i	$-0.410579\pi$
$139$	6.53732			0.554489			0.277244	+	0.960799i	$0.410579\pi$
$140$	15.4356	−	3.97986i	1.30455	−	0.336360i
$141$	−5.92672			−0.499120
$142$			5.57980i			0.468246i
$143$		−	2.38273i		−	0.199254i
$144$	−0.887616			−0.0739680
$145$	−0.231667	−	0.898504i	−0.0192389	−	0.0746167i
$146$	23.6069			1.95372
$147$		−	0.680997i		−	0.0561677i
$148$			10.4794i			0.861399i
$149$	−7.66541			−0.627975			−0.313988	−	0.949427i	$-0.601665\pi$
$149$	−7.66541			−0.627975			−0.313988	+	0.949427i	$0.601665\pi$
$150$	−15.1746	+	8.38239i	−1.23900	+	0.684419i
$151$	−3.41971			−0.278292			−0.139146	−	0.990272i	$-0.544436\pi$
$151$	−3.41971			−0.278292			−0.139146	+	0.990272i	$0.544436\pi$
$152$			1.70792i			0.138530i
$153$		−	1.51842i		−	0.122757i
$154$	−5.60485			−0.451652
$155$	4.48399	+	17.3909i	0.360163	+	1.39687i
$156$	−10.5076			−0.841281
$157$		−	0.0138709i		−	0.00110702i	−1.00000		0.000553510i	$-0.999824\pi$
$157$		−	0.0138709i		−	0.00110702i	1.00000		0.000553510i	$-0.000176188\pi$
$158$			5.47560i			0.435615i
$159$	6.70606			0.531825
$160$	−16.0494	+	4.13811i	−1.26882	+	0.327146i
$161$	−0.288611			−0.0227457
$162$		−	15.9777i		−	1.25533i
$163$			7.46593i			0.584777i	0.956300	+	0.292388i	$0.0944500\pi$
$163$			7.46593i			0.584777i	−0.956300	+	0.292388i	$0.905550\pi$
$164$	15.8035			1.23405
$165$	3.43338	−	0.885249i	0.267288	−	0.0689165i
$166$	31.5563			2.44924
$167$		−	22.6732i		−	1.75450i	−0.480030	−	0.877252i	$-0.659374\pi$
$167$		−	22.6732i		−	1.75450i	0.480030	−	0.877252i	$-0.340626\pi$
$168$			6.94193i			0.535581i
$169$	7.32261			0.563278
$170$	−3.81670	−	14.8028i	−0.292728	−	1.13533i
$171$	−0.485645			−0.0371382
$172$			26.0919i			1.98949i
$173$			9.16297i			0.696648i	0.937374	+	0.348324i	$0.113249\pi$
$173$			9.16297i			0.696648i	−0.937374	+	0.348324i	$0.886751\pi$
$174$	1.43876			0.109072
$175$	6.19714	+	11.2187i	0.468459	+	0.848052i
$176$	1.82771			0.137769
$177$			17.5853i			1.32179i
$178$		−	8.45345i		−	0.633613i
$179$	17.7874			1.32949			0.664745	−	0.747070i	$-0.268540\pi$
$179$	17.7874			1.32949			0.664745	+	0.747070i	$0.268540\pi$
$180$	0.754026	+	2.92444i	0.0562018	+	0.217975i
$181$	15.6404			1.16254			0.581272	−	0.813709i	$-0.302555\pi$
$181$	15.6404			1.16254			0.581272	+	0.813709i	$0.302555\pi$
$182$			13.3548i			0.989926i
$183$		−	1.41812i		−	0.104831i
$184$	−0.192300			−0.0141765
$185$	−8.15884	+	2.10364i	−0.599850	+	0.154663i
$186$	−27.8477			−2.04189
$187$			3.12660i			0.228640i
$188$			10.3948i			0.758119i
$189$	−14.1676			−1.03054
$190$	−4.73448	+	1.22072i	−0.343475	+	0.0885603i
$191$	11.3902			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$191$	11.3902			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$192$		−	19.9033i		−	1.43640i
$193$			25.8821i			1.86303i	0.363699	+	0.931517i	$0.381514\pi$
$193$			25.8821i			1.86303i	−0.363699	+	0.931517i	$0.618486\pi$
$194$	−8.75385			−0.628490
$195$	−2.10931	−	8.18081i	−0.151051	−	0.585840i
$196$	−1.19439			−0.0853138
$197$		−	14.9365i		−	1.06418i	−0.846687	−	0.532091i	$-0.821407\pi$
$197$		−	14.9365i		−	1.06418i	0.846687	−	0.532091i	$-0.178593\pi$
$198$		−	1.06190i		−	0.0754658i
$199$	4.99400			0.354015			0.177008	−	0.984209i	$-0.443358\pi$
$199$	4.99400			0.354015			0.177008	+	0.984209i	$0.443358\pi$
$200$	4.12912	+	7.47495i	0.291973	+	0.528559i
$201$	17.9821			1.26836
$202$		−	10.2122i		−	0.718524i
$203$		−	1.06368i		−	0.0746558i
$204$	13.7880			0.965353
$205$	3.17242	+	12.3040i	0.221571	+	0.859350i
$206$	−6.97475			−0.485954
$207$		−	0.0546803i		−	0.00380054i
$208$		−	4.35493i		−	0.301960i
$209$	1.00000			0.0691714
$210$	−19.2436	+	4.96169i	−1.32793	+	0.342389i
$211$	16.4193			1.13035			0.565177	−	0.824970i	$-0.308808\pi$
$211$	16.4193			1.13035			0.565177	+	0.824970i	$0.308808\pi$
$212$		−	11.7617i		−	0.807797i
$213$		−	4.04639i		−	0.277254i
$214$	−17.1682			−1.17360
$215$	−20.3142	+	5.23772i	−1.38541	+	0.357209i
$216$	−9.43981			−0.642298
$217$			20.5879i			1.39760i
$218$			41.3328i			2.79941i
$219$	−17.1194			−1.15682
$220$	−1.55263	−	6.02177i	−0.104678	−	0.405988i
$221$	7.44984			0.501131
$222$		−	13.0646i		−	0.876839i
$223$		−	23.0816i		−	1.54566i	−0.634613	−	0.772830i	$-0.718841\pi$
$223$		−	23.0816i		−	1.54566i	0.634613	−	0.772830i	$-0.281159\pi$
$224$	−18.9998			−1.26948
$225$	−2.12550	+	1.17411i	−0.141700	+	0.0782742i
$226$	−7.05702			−0.469426
$227$		−	23.2627i		−	1.54400i	−0.635622	−	0.772000i	$-0.719256\pi$
$227$		−	23.2627i		−	1.54400i	0.635622	−	0.772000i	$-0.280744\pi$
$228$		−	4.40990i		−	0.292053i
$229$	19.5223			1.29007			0.645035	−	0.764153i	$-0.276843\pi$
$229$	19.5223			1.29007			0.645035	+	0.764153i	$0.276843\pi$
$230$	−0.137445	−	0.533070i	−0.00906283	−	0.0351496i
$231$	4.06456			0.267428
$232$		−	0.708725i		−	0.0465301i
$233$		−	17.2292i		−	1.12872i	−0.825529	−	0.564360i	$-0.809123\pi$
$233$		−	17.2292i		−	1.12872i	0.825529	−	0.564360i	$-0.190877\pi$
$234$	−2.53021			−0.165405
$235$	−8.09300	+	2.08667i	−0.527929	+	0.136119i
$236$	30.8426			2.00768
$237$		−	3.97083i		−	0.257933i
$238$		−	17.5241i		−	1.13592i
$239$	−7.03024			−0.454748			−0.227374	−	0.973807i	$-0.573014\pi$
$239$	−7.03024			−0.454748			−0.227374	+	0.973807i	$0.573014\pi$
$240$	6.27521	−	1.61797i	0.405063	−	0.104440i
$241$	27.1296			1.74757			0.873786	−	0.486310i	$-0.161658\pi$
$241$	27.1296			1.74757			0.873786	+	0.486310i	$0.161658\pi$
$242$			2.18657i			0.140558i
$243$		−	4.99442i		−	0.320392i
$244$	−2.48723			−0.159229
$245$	−0.239764	−	0.929909i	−0.0153180	−	0.0594097i
$246$	−19.7022			−1.25617
$247$		−	2.38273i		−	0.151609i
$248$			13.7176i			0.871071i
$249$	−22.8842			−1.45023
$250$	−17.7699	+	16.7889i	−1.12387	+	1.06182i
$251$	−30.6080			−1.93196			−0.965981	−	0.258611i	$-0.916735\pi$
$251$	−30.6080			−1.93196			−0.965981	+	0.258611i	$0.916735\pi$
$252$			3.46206i			0.218089i
$253$			0.112593i			0.00707867i
$254$	6.89856			0.432854
$255$	2.76782	+	10.7348i	0.173328	+	0.672240i
$256$	−2.49345			−0.155841
$257$			13.3809i			0.834677i	0.908751	+	0.417338i	$0.137037\pi$
$257$			13.3809i			0.834677i	−0.908751	+	0.417338i	$0.862963\pi$
$258$		−	32.5287i		−	2.02515i
$259$	−9.65872			−0.600164
$260$	−14.3482	+	3.69949i	−0.889840	+	0.229433i
$261$	0.201526			0.0124741
$262$		−	5.03424i		−	0.311016i
$263$		−	4.12008i		−	0.254055i	−0.991899	−	0.127028i	$-0.959456\pi$
$263$		−	4.12008i		−	0.254055i	0.991899	−	0.127028i	$-0.0405437\pi$
$264$	2.70820			0.166678
$265$	9.15721	−	2.36106i	0.562523	−	0.145039i
$266$	−5.60485			−0.343655
$267$			6.13032i			0.375170i
$268$		−	31.5386i		−	1.92653i
$269$	−21.9499			−1.33831			−0.669155	−	0.743123i	$-0.733344\pi$
$269$	−21.9499			−1.33831			−0.669155	+	0.743123i	$0.733344\pi$
$270$	−6.74702	−	26.1679i	−0.410611	−	1.59253i
$271$	−17.1825			−1.04377			−0.521883	−	0.853017i	$-0.674770\pi$
$271$	−17.1825			−1.04377			−0.521883	+	0.853017i	$0.674770\pi$
$272$			5.71451i			0.346493i
$273$		−	9.68474i		−	0.586147i
$274$	15.3075			0.924759
$275$	4.37665	−	2.41764i	0.263922	−	0.145789i
$276$	0.496525			0.0298873
$277$		−	18.7728i		−	1.12795i	−0.825793	−	0.563974i	$-0.809272\pi$
$277$		−	18.7728i		−	1.12795i	0.825793	−	0.563974i	$-0.190728\pi$
$278$			14.2943i			0.857317i
$279$	−3.90060			−0.233523
$280$	2.44410	+	9.47929i	0.146063	+	0.566496i
$281$	−13.6904			−0.816700			−0.408350	−	0.912825i	$-0.633896\pi$
$281$	−13.6904			−0.816700			−0.408350	+	0.912825i	$0.633896\pi$
$282$		−	12.9592i		−	0.771709i
$283$			11.3808i			0.676521i	0.941053	+	0.338260i	$0.109838\pi$
$283$			11.3808i			0.676521i	−0.941053	+	0.338260i	$0.890162\pi$
$284$	−7.09693			−0.421125
$285$	3.43338	−	0.885249i	0.203376	−	0.0524376i
$286$	5.21000			0.308074
$287$			14.5659i			0.859800i
$288$		−	3.59972i		−	0.212115i
$289$	7.22436			0.424962
$290$	1.96464	−	0.506555i	0.115368	−	0.0297460i
$291$	6.34818			0.372137
$292$			30.0255i			1.75711i
$293$		−	13.0476i		−	0.762251i	−0.924523	−	0.381125i	$-0.875537\pi$
$293$		−	13.0476i		−	0.762251i	0.924523	−	0.381125i	$-0.124463\pi$
$294$	1.48905			0.0868430
$295$	6.19139	+	24.0129i	0.360476	+	1.39808i
$296$	−6.43556			−0.374059
$297$			5.52709i			0.320714i
$298$		−	16.7610i		−	0.970937i
$299$	0.268279			0.0155150
$300$	−10.6615	−	19.3006i	−0.615544	−	1.11432i
$301$	−24.0486			−1.38614
$302$		−	7.47744i		−	0.430278i
$303$			7.40571i			0.425447i
$304$	1.82771			0.104826
$305$	−0.499290	−	1.93647i	−0.0285893	−	0.110882i
$306$	3.32013			0.189799
$307$			15.8229i			0.903062i	0.892255	+	0.451531i	$0.149122\pi$
$307$			15.8229i			0.903062i	−0.892255	+	0.451531i	$0.850878\pi$
$308$		−	7.12879i		−	0.406201i
$309$	5.05800			0.287739
$310$	−38.0264	+	9.80456i	−2.15975	+	0.556862i
$311$	4.96330			0.281443			0.140721	−	0.990049i	$-0.455058\pi$
$311$	4.96330			0.281443			0.140721	+	0.990049i	$0.455058\pi$
$312$		−	6.45289i		−	0.365323i
$313$			18.5334i			1.04757i	0.851851	+	0.523785i	$0.175480\pi$
$313$			18.5334i			1.04757i	−0.851851	+	0.523785i	$0.824520\pi$
$314$	0.0303298			0.00171161
$315$	−2.69543	+	0.694978i	−0.151870	+	0.0391576i
$316$	−6.96440			−0.391778
$317$		−	7.76167i		−	0.435939i	−0.975956	−	0.217969i	$-0.930057\pi$
$317$		−	7.76167i		−	0.435939i	0.975956	−	0.217969i	$-0.0699433\pi$
$318$			14.6633i			0.822276i
$319$	−0.414965			−0.0232336
$320$	−7.00752	−	27.1782i	−0.391732	−	1.51931i
$321$	12.4502			0.694901
$322$		−	0.631068i		−	0.0351680i
$323$			3.12660i			0.173969i
$324$	20.3220			1.12900
$325$	−5.76057	−	10.4284i	−0.319539	−	0.578461i
$326$	−16.3248			−0.904147
$327$		−	29.9740i		−	1.65757i
$328$			9.70520i			0.535880i
$329$	−9.58079			−0.528206
$330$	1.93566	+	7.50733i	0.106555	+	0.413265i
$331$	−6.12570			−0.336699			−0.168349	−	0.985727i	$-0.553844\pi$
$331$	−6.12570			−0.336699			−0.168349	+	0.985727i	$0.553844\pi$
$332$			40.1364i			2.20277i
$333$		−	1.82995i		−	0.100280i
$334$	49.5765			2.71271
$335$	24.5548	−	6.33110i	1.34157	−	0.345905i
$336$	7.42882			0.405275
$337$			14.7031i			0.800931i	0.916312	+	0.400465i	$0.131152\pi$
$337$			14.7031i			0.800931i	−0.916312	+	0.400465i	$0.868848\pi$
$338$			16.0114i			0.870906i
$339$	5.11766			0.277953
$340$	18.8277	−	4.85445i	1.02107	−	0.263270i
$341$	8.03180			0.434946
$342$		−	1.06190i		−	0.0574208i
$343$		−	19.0440i		−	1.02828i
$344$	−16.0235			−0.863928
$345$	0.0996730	+	0.386575i	0.00536621	+	0.0208125i
$346$	−20.0355			−1.07711
$347$			20.4892i			1.09992i	0.835192	+	0.549959i	$0.185357\pi$
$347$			20.4892i			1.09992i	−0.835192	+	0.549959i	$0.814643\pi$
$348$			1.82995i			0.0980958i
$349$	33.4349			1.78973			0.894864	−	0.446340i	$-0.147273\pi$
$349$	33.4349			1.78973			0.894864	+	0.446340i	$0.147273\pi$
$350$	−24.5304	+	13.5505i	−1.31121	+	0.724304i
$351$	13.1695			0.702938
$352$			7.41224i			0.395074i
$353$		−	9.19893i		−	0.489610i	−0.969572	−	0.244805i	$-0.921276\pi$
$353$		−	9.19893i		−	0.489610i	0.969572	−	0.244805i	$-0.0787239\pi$
$354$	−38.4514			−2.04367
$355$	−1.42465	−	5.52540i	−0.0756123	−	0.293258i
$356$	10.7519			0.569850
$357$			12.7083i			0.672592i
$358$			38.8933i			2.05558i
$359$	−9.36150			−0.494081			−0.247041	−	0.969005i	$-0.579458\pi$
$359$	−9.36150			−0.494081			−0.247041	+	0.969005i	$0.579458\pi$
$360$	−1.79595	+	0.463060i	−0.0946549	+	0.0244054i
$361$	1.00000			0.0526316
$362$			34.1989i			1.79745i
$363$		−	1.58567i		−	0.0832262i
$364$	−16.9860			−0.890307
$365$	−23.3767	+	6.02736i	−1.22359	+	0.315486i
$366$	3.10083			0.162083
$367$		−	7.40964i		−	0.386780i	−0.981122	−	0.193390i	$-0.938052\pi$
$367$		−	7.40964i		−	0.386780i	0.981122	−	0.193390i	$-0.0619483\pi$
$368$			0.205787i			0.0107274i
$369$	−2.75967			−0.143663
$370$	−4.59976	−	17.8399i	−0.239130	−	0.927451i
$371$	10.8406			0.562818
$372$		−	35.4194i		−	1.83641i
$373$			4.32900i			0.224147i	0.993700	+	0.112073i	$0.0357492\pi$
$373$			4.32900i			0.224147i	−0.993700	+	0.112073i	$0.964251\pi$
$374$	−6.83654			−0.353509
$375$	12.8865	−	12.1751i	0.665455	−	0.628719i
$376$	−6.38363			−0.329211
$377$			0.988748i			0.0509231i
$378$		−	30.9785i		−	1.59336i
$379$	7.89046			0.405306			0.202653	−	0.979251i	$-0.435044\pi$
$379$	7.89046			0.405306			0.202653	+	0.979251i	$0.435044\pi$
$380$	−1.55263	−	6.02177i	−0.0796482	−	0.308910i
$381$	−5.00274			−0.256298
$382$			24.9054i			1.27427i
$383$			9.15223i			0.467657i	0.972278	+	0.233829i	$0.0751255\pi$
$383$			9.15223i			0.467657i	−0.972278	+	0.233829i	$0.924875\pi$
$384$	20.0133			1.02130
$385$	5.55020	−	1.43104i	0.282865	−	0.0729326i
$386$	−56.5930			−2.88051
$387$		−	4.55626i		−	0.231608i
$388$		−	11.1340i		−	0.565243i
$389$	5.65500			0.286720			0.143360	−	0.989671i	$-0.454209\pi$
$389$	5.65500			0.286720			0.143360	+	0.989671i	$0.454209\pi$
$390$	17.8879	−	4.61215i	0.905791	−	0.233545i
$391$	−0.352034			−0.0178031
$392$		−	0.733497i		−	0.0370472i
$393$			3.65076i			0.184157i
$394$	32.6597			1.64537
$395$	−1.39804	−	5.42222i	−0.0703431	−	0.272821i
$396$	1.35062			0.0678714
$397$			11.3674i			0.570514i	0.958451	+	0.285257i	$0.0920790\pi$
$397$			11.3674i			0.570514i	−0.958451	+	0.285257i	$0.907921\pi$
$398$			10.9197i			0.547357i
$399$	4.06456			0.203483
$400$	7.99922	−	4.41873i	0.399961	−	0.220936i
$401$	−24.6506			−1.23099			−0.615497	−	0.788140i	$-0.711044\pi$
$401$	−24.6506			−1.23099			−0.615497	+	0.788140i	$0.711044\pi$
$402$			39.3192i			1.96106i
$403$		−	19.1376i		−	0.953311i
$404$	12.9888			0.646217
$405$	4.07947	+	15.8220i	0.202711	+	0.786200i
$406$	2.32581			0.115428
$407$			3.76807i			0.186776i
$408$			8.46745i			0.419201i
$409$	−36.0852			−1.78430			−0.892149	−	0.451742i	$-0.850803\pi$
$409$	−36.0852			−1.78430			−0.892149	+	0.451742i	$0.850803\pi$
$410$	−26.9036	+	6.93671i	−1.32867	+	0.342580i
$411$	−11.1008			−0.547561
$412$		−	8.87117i		−	0.437051i
$413$			28.4273i			1.39882i
$414$	0.119562			0.00587617
$415$	−31.2486	+	8.05702i	−1.53394	+	0.395503i
$416$	17.6614			0.865919
$417$		−	10.3661i		−	0.507628i
$418$			2.18657i			0.106949i
$419$	33.2037			1.62211			0.811053	−	0.584973i	$-0.198895\pi$
$419$	33.2037			1.62211			0.811053	+	0.584973i	$0.198895\pi$
$420$	−6.31075	−	24.4759i	−0.307933	−	1.19430i
$421$	2.43091			0.118476			0.0592378	−	0.998244i	$-0.481133\pi$
$421$	2.43091			0.118476			0.0592378	+	0.998244i	$0.481133\pi$
$422$			35.9020i			1.74768i
$423$		−	1.81518i		−	0.0882571i
$424$	7.22306			0.350783
$425$	7.55898	+	13.6840i	0.366665	+	0.663773i
$426$	8.84773			0.428674
$427$		−	2.29246i		−	0.110940i
$428$		−	21.8362i		−	1.05549i
$429$	−3.77822			−0.182414
$430$	−11.4526	−	44.4183i	−0.552295	−	2.14204i
$431$	24.4764			1.17899			0.589494	−	0.807772i	$-0.299327\pi$
$431$	24.4764			1.17899			0.589494	+	0.807772i	$0.299327\pi$
$432$			10.1019i			0.486028i
$433$		−	27.9437i		−	1.34289i	−0.741055	−	0.671445i	$-0.765674\pi$
$433$		−	27.9437i		−	1.34289i	0.741055	−	0.671445i	$-0.234326\pi$
$434$	−45.0170			−2.16088
$435$	−1.42473	+	0.367347i	−0.0683107	+	0.0176129i
$436$	−52.5711			−2.51770
$437$			0.112593i			0.00538606i
$438$		−	37.4328i		−	1.78861i
$439$	17.1989			0.820860			0.410430	−	0.911892i	$-0.365379\pi$
$439$	17.1989			0.820860			0.410430	+	0.911892i	$0.365379\pi$
$440$	3.69807	−	0.953496i	0.176299	−	0.0454561i
$441$	0.208569			0.00993188
$442$			16.2896i			0.774818i
$443$			32.5404i			1.54604i	0.634381	+	0.773021i	$0.281255\pi$
$443$			32.5404i			1.54604i	−0.634381	+	0.773021i	$0.718745\pi$
$444$	16.6168			0.788600
$445$	2.15835	+	8.37103i	0.102316	+	0.396825i
$446$	50.4696			2.38981
$447$			12.1548i			0.574904i
$448$		−	32.1746i		−	1.52010i
$449$	35.3343			1.66753			0.833763	−	0.552122i	$-0.186182\pi$
$449$	35.3343			1.66753			0.833763	+	0.552122i	$0.186182\pi$
$450$	−2.56728	−	4.64755i	−0.121023	−	0.219088i
$451$	5.68248			0.267578
$452$		−	8.97581i		−	0.422186i
$453$			5.42254i			0.254773i
$454$	50.8656			2.38724
$455$	−3.40978	−	13.2246i	−0.159853	−	0.619980i
$456$	2.70820			0.126823
$457$			30.5173i			1.42754i	0.700380	+	0.713770i	$0.253014\pi$
$457$			30.5173i			1.42754i	−0.700380	+	0.713770i	$0.746986\pi$
$458$			42.6869i			1.99463i
$459$	−17.2810			−0.806608
$460$	0.678010	−	0.174815i	0.0316124	−	0.00815081i
$461$	−10.2194			−0.475965			−0.237982	−	0.971269i	$-0.576486\pi$
$461$	−10.2194			−0.475965			−0.237982	+	0.971269i	$0.576486\pi$
$462$			8.88745i			0.413482i
$463$		−	26.1814i		−	1.21675i	−0.793648	−	0.608377i	$-0.791821\pi$
$463$		−	26.1814i		−	1.21675i	0.793648	−	0.608377i	$-0.208179\pi$
$464$	−0.758434			−0.0352094
$465$	27.5762	−	7.11014i	1.27882	−	0.329725i
$466$	37.6728			1.74516
$467$			21.8569i			1.01142i	0.862705	+	0.505708i	$0.168769\pi$
$467$			21.8569i			1.01142i	−0.862705	+	0.505708i	$0.831231\pi$
$468$		−	3.21817i		−	0.148760i
$469$	29.0688			1.34227
$470$	−4.56264	−	17.6959i	−0.210459	−	0.816252i
$471$	−0.0219947			−0.00101346
$472$			18.9410i			0.871829i
$473$			9.38188i			0.431379i
$474$	8.68250			0.398801
$475$	4.37665	−	2.41764i	0.200814	−	0.110929i
$476$	22.2889			1.02161
$477$			2.05387i			0.0940403i
$478$		−	15.3721i		−	0.703104i
$479$	35.9834			1.64413			0.822063	−	0.569397i	$-0.192823\pi$
$479$	35.9834			1.64413			0.822063	+	0.569397i	$0.192823\pi$
$480$	6.56168	+	25.4491i	0.299498	+	1.16159i
$481$	8.97829			0.409375
$482$			59.3208i			2.70199i
$483$			0.457642i			0.0208234i
$484$	−2.78109			−0.126413
$485$	8.66851	−	2.23505i	0.393617	−	0.101489i
$486$	10.9207			0.495371
$487$		−	38.1167i		−	1.72723i	−0.504149	−	0.863617i	$-0.668194\pi$
$487$		−	38.1167i		−	1.72723i	0.504149	−	0.863617i	$-0.331806\pi$
$488$		−	1.52745i		−	0.0691445i
$489$	11.8385			0.535356
$490$	2.03331	−	0.524261i	0.0918557	−	0.0236837i
$491$	17.9270			0.809034			0.404517	−	0.914531i	$-0.367440\pi$
$491$	17.9270			0.809034			0.404517	+	0.914531i	$0.367440\pi$
$492$		−	25.0592i		−	1.12976i
$493$		−	1.29743i		−	0.0584333i
$494$	5.21000			0.234409
$495$	0.271126	+	1.05154i	0.0121862	+	0.0472634i
$496$	14.6798			0.659141
$497$		−	6.54116i		−	0.293411i
$498$		−	50.0379i		−	2.24225i
$499$	−23.3365			−1.04469			−0.522343	−	0.852735i	$-0.674942\pi$
$499$	−23.3365			−1.04469			−0.522343	+	0.852735i	$0.674942\pi$
$500$	−21.3538	−	22.6015i	−0.954969	−	1.01077i
$501$	−35.9522			−1.60623
$502$		−	66.9267i		−	2.98708i
$503$		−	35.8629i		−	1.59905i	−0.600636	−	0.799523i	$-0.705086\pi$
$503$		−	35.8629i		−	1.59905i	0.600636	−	0.799523i	$-0.294914\pi$
$504$	−2.12611			−0.0947045
$505$	2.60739	+	10.1126i	0.116027	+	0.450004i
$506$	−0.246193			−0.0109446
$507$		−	11.6113i		−	0.515674i
$508$			8.77425i			0.389295i
$509$	−8.02832			−0.355849			−0.177925	−	0.984044i	$-0.556938\pi$
$509$	−8.02832			−0.355849			−0.177925	+	0.984044i	$0.556938\pi$
$510$	−23.4724	+	6.05204i	−1.03938	+	0.267989i
$511$	−27.6742			−1.22424
$512$			19.7905i			0.874626i
$513$			5.52709i			0.244027i
$514$	−29.2583			−1.29053
$515$	6.90676	−	1.78081i	0.304348	−	0.0784718i
$516$	41.3732			1.82135
$517$			3.73767i			0.164383i
$518$		−	21.1195i		−	0.927937i
$519$	14.5295			0.637773
$520$	−2.27192	−	8.81150i	−0.0996303	−	0.386410i
$521$	−9.84553			−0.431341			−0.215670	−	0.976466i	$-0.569194\pi$
$521$	−9.84553			−0.431341			−0.215670	+	0.976466i	$0.569194\pi$
$522$			0.440650i			0.0192867i
$523$		−	37.0715i		−	1.62102i	−0.585722	−	0.810512i	$-0.699189\pi$
$523$		−	37.0715i		−	1.62102i	0.585722	−	0.810512i	$-0.300811\pi$
$524$	6.40303			0.279718
$525$	17.7891	−	9.82662i	0.776382	−	0.428869i
$526$	9.00885			0.392805
$527$			25.1122i			1.09391i
$528$		−	2.89814i		−	0.126125i
$529$	22.9873			0.999449
$530$	5.16262	+	20.0229i	0.224250	+	0.869739i
$531$	−5.38586			−0.233726
$532$		−	7.12879i		−	0.309072i
$533$		−	13.5398i		−	0.586474i
$534$	−13.4044			−0.580065
$535$	17.0009	−	4.38343i	0.735011	−	0.189512i
$536$	19.3684			0.836588
$537$		−	28.2049i		−	1.21713i
$538$		−	47.9950i		−	2.06921i
$539$	−0.429469			−0.0184985
$540$	33.2829	−	8.58152i	1.43227	−	0.369290i
$541$	41.3858			1.77932			0.889658	−	0.456628i	$-0.150943\pi$
$541$	41.3858			1.77932			0.889658	+	0.456628i	$0.150943\pi$
$542$		−	37.5709i		−	1.61381i
$543$		−	24.8006i		−	1.06430i
$544$	−23.1751			−0.993625
$545$	−10.5532	−	40.9299i	−0.452049	−	1.75324i
$546$	21.1764			0.906265
$547$		−	14.6183i		−	0.625035i	−0.949912	−	0.312517i	$-0.898828\pi$
$547$		−	14.6183i		−	0.625035i	0.949912	−	0.312517i	$-0.101172\pi$
$548$			19.4695i			0.831698i
$549$	0.434330			0.0185368
$550$	5.28633	+	9.56985i	0.225410	+	0.408060i
$551$	−0.414965			−0.0176781
$552$			0.304924i			0.0129784i
$553$		−	6.41901i		−	0.272964i
$554$	41.0480			1.74396
$555$	3.33568	+	12.9372i	0.141592	+	0.549155i
$556$	−18.1809			−0.771042
$557$			21.8651i			0.926455i	0.886239	+	0.463227i	$0.153309\pi$
$557$			21.8651i			0.926455i	−0.886239	+	0.463227i	$0.846691\pi$
$558$		−	8.52894i		−	0.361059i
$559$	22.3545			0.945493
$560$	10.1441	−	2.61552i	0.428668	−	0.110526i
$561$	4.95776			0.209317
$562$		−	29.9350i		−	1.26273i
$563$		−	17.0052i		−	0.716682i	−0.933591	−	0.358341i	$-0.883343\pi$
$563$		−	17.0052i		−	0.716682i	0.933591	−	0.358341i	$-0.116657\pi$
$564$	16.4828			0.694049
$565$	6.98822	−	1.80181i	0.293997	−	0.0758029i
$566$	−24.8850			−1.04600
$567$			18.7306i			0.786612i
$568$		−	4.35834i		−	0.182872i
$569$	27.6851			1.16062			0.580310	−	0.814396i	$-0.302932\pi$
$569$	27.6851			1.16062			0.580310	+	0.814396i	$0.302932\pi$
$570$	1.93566	+	7.50733i	0.0810759	+	0.314448i
$571$	−17.7807			−0.744097			−0.372049	−	0.928213i	$-0.621345\pi$
$571$	−17.7807			−0.744097			−0.372049	+	0.928213i	$0.621345\pi$
$572$			6.62659i			0.277072i
$573$		−	18.0611i		−	0.754512i
$574$	−31.8494			−1.32937
$575$	0.272209	+	0.492781i	0.0113519	+	0.0205504i
$576$	6.09581			0.253992
$577$			16.8688i			0.702258i	0.936327	+	0.351129i	$0.114202\pi$
$577$			16.8688i			0.702258i	−0.936327	+	0.351129i	$0.885798\pi$
$578$			15.7966i			0.657051i
$579$	41.0405			1.70558
$580$	0.644286	+	2.49882i	0.0267525	+	0.103758i
$581$	−36.9933			−1.53474
$582$			13.8807i			0.575375i
$583$		−	4.22916i		−	0.175154i
$584$	−18.4392			−0.763019
$585$	2.50554	−	0.646019i	0.103592	−	0.0267096i
$586$	28.5296			1.17855
$587$			24.4155i			1.00774i	0.863780	+	0.503869i	$0.168090\pi$
$587$			24.4155i			1.00774i	−0.863780	+	0.503869i	$0.831910\pi$
$588$			1.89392i			0.0781038i
$589$	8.03180			0.330944
$590$	−52.5059	+	13.5379i	−2.16163	+	0.557347i
$591$	−23.6844			−0.974245
$592$			6.88693i			0.283051i
$593$			9.90592i			0.406787i	0.979097	+	0.203394i	$0.0651971\pi$
$593$			9.90592i			0.406787i	−0.979097	+	0.203394i	$0.934803\pi$
$594$	−12.0854			−0.495869
$595$	4.47430	+	17.3533i	0.183428	+	0.711415i
$596$	21.3182			0.873229
$597$		−	7.91884i		−	0.324097i
$598$			0.586611i			0.0239883i
$599$	−35.2948			−1.44211			−0.721053	−	0.692880i	$-0.756341\pi$
$599$	−35.2948			−1.44211			−0.721053	+	0.692880i	$0.756341\pi$
$600$	11.8528	−	6.54743i	0.483889	−	0.267298i
$601$	3.55904			0.145176			0.0725881	−	0.997362i	$-0.476874\pi$
$601$	3.55904			0.145176			0.0725881	+	0.997362i	$0.476874\pi$
$602$		−	52.5840i		−	2.14316i
$603$			5.50739i			0.224278i
$604$	9.51053			0.386978
$605$	−0.558280	−	2.16525i	−0.0226973	−	0.0880301i
$606$	−16.1931			−0.657801
$607$		−	10.3955i		−	0.421940i	−0.977493	−	0.210970i	$-0.932338\pi$
$607$		−	10.3955i		−	0.421940i	0.977493	−	0.210970i	$-0.0676622\pi$
$608$			7.41224i			0.300606i
$609$	−1.68665			−0.0683465
$610$	4.23422	−	1.09173i	0.171439	−	0.0442030i
$611$	8.90585			0.360292
$612$			4.22286i			0.170699i
$613$		−	3.57767i		−	0.144501i	−0.997387	−	0.0722504i	$-0.976982\pi$
$613$		−	3.57767i		−	0.144501i	0.997387	−	0.0722504i	$-0.0230181\pi$
$614$	−34.5979			−1.39626
$615$	19.5101	−	5.03041i	0.786724	−	0.202846i
$616$	4.37791			0.176391
$617$			13.7403i			0.553164i	0.960990	+	0.276582i	$0.0892017\pi$
$617$			13.7403i			0.553164i	−0.960990	+	0.276582i	$0.910798\pi$
$618$			11.0597i			0.444885i
$619$	29.9620			1.20427			0.602137	−	0.798393i	$-0.294316\pi$
$619$	29.9620			1.20427			0.602137	+	0.798393i	$0.294316\pi$
$620$	−12.4704	−	48.3656i	−0.500823	−	1.94241i
$621$	−0.622312			−0.0249725
$622$			10.8526i			0.435150i
$623$			9.90993i			0.397033i
$624$	−6.90548			−0.276441
$625$	13.3101	−	21.1623i	0.532403	−	0.846491i
$626$	−40.5246			−1.61969
$627$		−	1.58567i		−	0.0633256i
$628$			0.0385764i			0.00153936i
$629$	−11.7813			−0.469750
$630$	−1.51962	−	5.89375i	−0.0605431	−	0.234813i
$631$	−12.0729			−0.480616			−0.240308	−	0.970697i	$-0.577248\pi$
$631$	−12.0729			−0.480616			−0.240308	+	0.970697i	$0.577248\pi$
$632$		−	4.27696i		−	0.170128i
$633$		−	26.0357i		−	1.03482i
$634$	16.9714			0.674022
$635$	−6.83130	+	1.76135i	−0.271092	+	0.0698972i
$636$	−18.6502			−0.739528
$637$			1.02331i			0.0405449i
$638$		−	0.907350i		−	0.0359223i
$639$	1.23929			0.0490256
$640$	27.3284	−	7.04623i	1.08025	−	0.278527i
$641$	1.31259			0.0518441			0.0259220	−	0.999664i	$-0.491748\pi$
$641$	1.31259			0.0518441			0.0259220	+	0.999664i	$0.491748\pi$
$642$			27.2232i			1.07441i
$643$			24.3317i			0.959549i	0.877392	+	0.479774i	$0.159282\pi$
$643$			24.3317i			0.959549i	−0.877392	+	0.479774i	$0.840718\pi$
$644$	0.802653			0.0316290
$645$	8.30530	+	32.2116i	0.327021	+	1.26833i
$646$	−6.83654			−0.268980
$647$		−	41.6851i		−	1.63881i	−0.573216	−	0.819405i	$-0.694304\pi$
$647$		−	41.6851i		−	1.63881i	0.573216	−	0.819405i	$-0.305696\pi$
$648$			12.4801i			0.490265i
$649$	11.0901			0.435325
$650$	22.8023	−	12.5959i	0.894382	−	0.494051i
$651$	32.6457			1.27949
$652$		−	20.7635i		−	0.813160i
$653$			39.8201i			1.55828i	0.626849	+	0.779140i	$0.284344\pi$
$653$			39.8201i			1.55828i	−0.626849	+	0.779140i	$0.715656\pi$
$654$	65.5403			2.56283
$655$	1.28535	+	4.98516i	0.0502229	+	0.194786i
$656$	10.3859			0.405501
$657$		−	5.24317i		−	0.204556i
$658$		−	20.9491i		−	0.816680i
$659$	−3.42304			−0.133343			−0.0666714	−	0.997775i	$-0.521238\pi$
$659$	−3.42304			−0.133343			−0.0666714	+	0.997775i	$0.521238\pi$
$660$	−9.54856	+	2.46196i	−0.371677	+	0.0958317i
$661$	−19.2320			−0.748039			−0.374020	−	0.927421i	$-0.622021\pi$
$661$	−19.2320			−0.748039			−0.374020	+	0.927421i	$0.622021\pi$
$662$		−	13.3943i		−	0.520583i
$663$		−	11.8130i		−	0.458779i
$664$	−24.6484			−0.956544
$665$	5.55020	−	1.43104i	0.215228	−	0.0554934i
$666$	4.00131			0.155048
$667$		−	0.0467222i		−	0.00180909i
$668$			63.0563i			2.43972i
$669$	−36.5999			−1.41503
$670$	13.8434	+	53.6908i	0.534817	+	2.07426i
$671$	−0.894337			−0.0345255
$672$			30.1275i			1.16219i
$673$			13.2695i			0.511501i	0.966743	+	0.255750i	$0.0823225\pi$
$673$			13.2695i			0.511501i	−0.966743	+	0.255750i	$0.917677\pi$
$674$	−32.1495			−1.23835
$675$	13.3625	+	24.1901i	0.514322	+	0.931078i
$676$	−20.3649			−0.783264
$677$			28.6954i			1.10285i	0.834223	+	0.551427i	$0.185917\pi$
$677$			28.6954i			1.10285i	−0.834223	+	0.551427i	$0.814083\pi$
$678$			11.1901i			0.429754i
$679$	10.2621			0.393823
$680$	2.98120	+	11.5624i	0.114324	+	0.443398i
$681$	−36.8870			−1.41351
$682$			17.5621i			0.672487i
$683$		−	2.84703i		−	0.108939i	−0.998515	−	0.0544693i	$-0.982653\pi$
$683$		−	2.84703i		−	0.108939i	0.998515	−	0.0544693i	$-0.0173467\pi$
$684$	1.35062			0.0516424
$685$	−15.1582	+	3.90834i	−0.579167	+	0.149330i
$686$	41.6410			1.58986
$687$		−	30.9560i		−	1.18104i
$688$			17.1473i			0.653735i
$689$	−10.0769			−0.383901
$690$	−0.845274	+	0.217942i	−0.0321790	+	0.00829691i
$691$	−22.0715			−0.839640			−0.419820	−	0.907607i	$-0.637907\pi$
$691$	−22.0715			−0.839640			−0.419820	+	0.907607i	$0.637907\pi$
$692$		−	25.4831i		−	0.968721i
$693$			1.24486i			0.0472882i
$694$	−44.8011			−1.70063
$695$	−3.64966	−	14.1550i	−0.138439	−	0.536928i
$696$	−1.12381			−0.0425977
$697$			17.7669i			0.672968i
$698$			73.1077i			2.76717i
$699$	−27.3198			−1.03333
$700$	−17.2348	−	31.2002i	−0.651415	−	1.17926i
$701$	−14.2832			−0.539470			−0.269735	−	0.962935i	$-0.586936\pi$
$701$	−14.2832			−0.539470			−0.269735	+	0.962935i	$0.586936\pi$
$702$			28.7962i			1.08684i
$703$			3.76807i			0.142116i
$704$	−12.5520			−0.473071
$705$	3.30877	+	12.8328i	0.124615	+	0.483313i
$706$	20.1141			0.757005
$707$			11.9716i			0.450240i
$708$		−	48.9063i		−	1.83801i
$709$	−22.9789			−0.862990			−0.431495	−	0.902115i	$-0.642014\pi$
$709$	−22.9789			−0.862990			−0.431495	+	0.902115i	$0.642014\pi$
$710$	12.0817	−	3.11509i	0.453417	−	0.116907i
$711$	1.21615			0.0456092
$712$			6.60293i			0.247455i
$713$			0.904325i			0.0338672i
$714$	−27.7875			−1.03992
$715$	−5.15921	+	1.33023i	−0.192944	+	0.0497478i
$716$	−49.4683			−1.84872
$717$			11.1476i			0.416317i
$718$		−	20.4696i		−	0.763918i
$719$	−18.5378			−0.691342			−0.345671	−	0.938356i	$-0.612349\pi$
$719$	−18.5378			−0.691342			−0.345671	+	0.938356i	$0.612349\pi$
$720$	0.495538	+	1.92191i	0.0184676	+	0.0716255i
$721$	8.17646			0.304507
$722$			2.18657i			0.0813758i
$723$		−	43.0187i		−	1.59988i
$724$	−43.4975			−1.61657
$725$	−1.81615	+	1.00323i	−0.0674503	+	0.0372592i
$726$	3.46718			0.128679
$727$			5.95158i			0.220732i	0.993891	+	0.110366i	$0.0352023\pi$
$727$			5.95158i			0.220732i	−0.993891	+	0.110366i	$0.964798\pi$
$728$		−	10.4314i		−	0.386612i
$729$	−29.8412			−1.10523
$730$	−13.1793	−	51.1149i	−0.487786	−	1.89185i
$731$	−29.3334			−1.08494
$732$			3.94394i			0.145772i
$733$			15.6914i			0.579574i	0.957091	+	0.289787i	$0.0935845\pi$
$733$			15.6914i			0.579574i	−0.957091	+	0.289787i	$0.906416\pi$
$734$	16.2017			0.598016
$735$	−1.47453	+	0.380187i	−0.0543889	+	0.0140234i
$736$	−0.834568			−0.0307626
$737$		−	11.3404i		−	0.417728i
$738$		−	6.03421i		−	0.222122i
$739$	41.5745			1.52934			0.764671	−	0.644421i	$-0.222902\pi$
$739$	41.5745			1.52934			0.764671	+	0.644421i	$0.222902\pi$
$740$	22.6905	−	5.85042i	0.834119	−	0.215066i
$741$	−3.77822			−0.138797
$742$			23.7038i			0.870194i
$743$		−	8.39983i		−	0.308160i	−0.988058	−	0.154080i	$-0.950759\pi$
$743$		−	8.39983i		−	0.308160i	0.988058	−	0.154080i	$-0.0492413\pi$
$744$	21.7517			0.797455
$745$	4.27945	+	16.5976i	0.156787	+	0.608088i
$746$	−9.46566			−0.346562
$747$		−	7.00876i		−	0.256437i
$748$		−	8.69537i		−	0.317934i
$749$	20.1262			0.735396
$750$	26.6217	+	28.1772i	0.972087	+	1.02889i
$751$	−35.9666			−1.31244			−0.656220	−	0.754570i	$-0.727846\pi$
$751$	−35.9666			−1.31244			−0.656220	+	0.754570i	$0.727846\pi$
$752$			6.83136i			0.249114i
$753$			48.5343i			1.76869i
$754$	−2.16197			−0.0787342
$755$	1.90916	+	7.40454i	0.0694813	+	0.269479i
$756$	39.4015			1.43302
$757$			4.17679i			0.151808i	0.997115	+	0.0759040i	$0.0241843\pi$
$757$			4.17679i			0.151808i	−0.997115	+	0.0759040i	$0.975816\pi$
$758$			17.2531i			0.626659i
$759$	0.178536			0.00648044
$760$	3.69807	−	0.953496i	0.134143	−	0.0345869i
$761$	−19.7937			−0.717521			−0.358761	−	0.933430i	$-0.616801\pi$
$761$	−19.7937			−0.717521			−0.358761	+	0.933430i	$0.616801\pi$
$762$		−	10.9388i		−	0.396273i
$763$		−	48.4542i		−	1.75416i
$764$	−31.6771			−1.14604
$765$	−3.28776	+	0.847703i	−0.118869	+	0.0306488i
$766$	−20.0120			−0.723063
$767$		−	26.4247i		−	0.954141i
$768$			3.95379i			0.142670i
$769$	−49.1704			−1.77313			−0.886565	−	0.462604i	$-0.846915\pi$
$769$	−49.1704			−1.77313			−0.886565	+	0.462604i	$0.846915\pi$
$770$	3.12907	+	12.1359i	0.112764	+	0.437348i
$771$	21.2177			0.764136
$772$		−	71.9805i		−	2.59063i
$773$			18.6122i			0.669434i	0.942319	+	0.334717i	$0.108641\pi$
$773$			18.6122i			0.669434i	−0.942319	+	0.334717i	$0.891359\pi$
$774$	9.96260			0.358098
$775$	35.1523	−	19.4180i	1.26271	−	0.697513i
$776$	6.83758			0.245455
$777$			15.3156i			0.549443i
$778$			12.3651i			0.443309i
$779$	5.68248			0.203596
$780$	5.86618	+	22.7516i	0.210043	+	0.814638i
$781$	−2.55185			−0.0913123
$782$		−	0.769747i		−	0.0275261i
$783$		−	2.29355i		−	0.0819647i
$784$	−0.784943			−0.0280337
$785$	−0.0300341	+	0.00774386i	−0.00107196	+	0.000276390i
$786$	−7.98265			−0.284732
$787$		−	26.0136i		−	0.927284i	−0.886023	−	0.463642i	$-0.846542\pi$
$787$		−	26.0136i		−	0.927284i	0.886023	−	0.463642i	$-0.153458\pi$
$788$			41.5398i			1.47979i
$789$	−6.53310			−0.232584
$790$	11.8561	−	3.05692i	0.421820	−	0.108760i
$791$	8.27291			0.294151
$792$			0.829441i			0.0294729i
$793$			2.13096i			0.0756727i
$794$	−24.8557			−0.882094
$795$	−3.74386	−	14.5203i	−0.132781	−	0.514983i
$796$	−13.8888			−0.492274
$797$		−	34.3724i		−	1.21753i	−0.793350	−	0.608766i	$-0.791665\pi$
$797$		−	34.3724i		−	1.21753i	0.793350	−	0.608766i	$-0.208335\pi$
$798$			8.88745i			0.314612i
$799$	−11.6862			−0.413428
$800$	17.9201	+	32.4408i	0.633571	+	1.14695i
$801$	−1.87754			−0.0663396
$802$		−	53.9003i		−	1.90329i
$803$			10.7963i			0.380993i
$804$	−50.0099			−1.76371
$805$	0.161126	+	0.624915i	0.00567893	+	0.0220254i
$806$	41.8457			1.47395
$807$			34.8054i			1.22521i
$808$			7.97665i			0.280617i
$809$	−34.2274			−1.20337			−0.601686	−	0.798733i	$-0.705504\pi$
$809$	−34.2274			−1.20337			−0.601686	+	0.798733i	$0.705504\pi$
$810$	−34.5959	+	8.92005i	−1.21557	+	0.313419i
$811$	−18.2235			−0.639913			−0.319957	−	0.947432i	$-0.603668\pi$
$811$	−18.2235			−0.639913			−0.319957	+	0.947432i	$0.603668\pi$
$812$			2.95820i			0.103812i
$813$			27.2459i			0.955555i
$814$	−8.23916			−0.288782
$815$	16.1656	−	4.16808i	0.566258	−	0.146001i
$816$	9.06133			0.317210
$817$			9.38188i			0.328230i
$818$		−	78.9028i		−	2.75877i
$819$	2.96615			0.103646
$820$	−8.82278	−	34.2186i	−0.308105	−	1.19497i
$821$	−24.1056			−0.841290			−0.420645	−	0.907225i	$-0.638196\pi$
$821$	−24.1056			−0.841290			−0.420645	+	0.907225i	$0.638196\pi$
$822$		−	24.2726i		−	0.846606i
$823$		−	20.5286i		−	0.715581i	−0.933802	−	0.357791i	$-0.883530\pi$
$823$		−	20.5286i		−	0.715581i	0.933802	−	0.357791i	$-0.116470\pi$
$824$	5.44794			0.189788
$825$	−3.83358	−	6.93993i	−0.133468	−	0.241617i
$826$	−62.1584			−2.16277
$827$		−	48.8963i		−	1.70029i	−0.526548	−	0.850146i	$-0.676514\pi$
$827$		−	48.8963i		−	1.70029i	0.526548	−	0.850146i	$-0.323486\pi$
$828$			0.152071i			0.00528483i
$829$	20.5971			0.715366			0.357683	−	0.933843i	$-0.383567\pi$
$829$	20.5971			0.715366			0.357683	+	0.933843i	$0.383567\pi$
$830$	−17.6172	−	68.3274i	−0.611503	−	2.37168i
$831$	−29.7675			−1.03262
$832$			29.9080i			1.03687i
$833$		−	1.34278i		−	0.0465245i
$834$	22.6661			0.784863
$835$	−49.0932	+	12.6580i	−1.69894	+	0.438048i
$836$	−2.78109			−0.0961861
$837$			44.3924i			1.53443i
$838$			72.6022i			2.50800i
$839$	2.46052			0.0849468			0.0424734	−	0.999098i	$-0.486476\pi$
$839$	2.46052			0.0849468			0.0424734	+	0.999098i	$0.486476\pi$
$840$	15.0310	−	3.87554i	0.518620	−	0.133719i
$841$	−28.8278			−0.994062
$842$			5.31537i			0.183180i
$843$			21.7085i			0.747679i
$844$	−45.6637			−1.57181
$845$	−4.08807	−	15.8553i	−0.140634	−	0.545439i
$846$	3.96902			0.136458
$847$		−	2.56330i		−	0.0880762i
$848$		−	7.72966i		−	0.265438i
$849$	18.0463			0.619347
$850$	−29.9211	+	16.5283i	−1.02629	+	0.566914i
$851$	−0.424259			−0.0145434
$852$			11.2534i			0.385535i
$853$		−	51.5658i		−	1.76558i	−0.469769	−	0.882790i	$-0.655663\pi$
$853$		−	51.5658i		−	1.76558i	0.469769	−	0.882790i	$-0.344337\pi$
$854$	5.01262			0.171528
$855$	0.271126	+	1.05154i	0.00927231	+	0.0359621i
$856$	13.4100			0.458344
$857$		−	10.4031i		−	0.355362i	−0.984088	−	0.177681i	$-0.943140\pi$
$857$		−	10.4031i		−	0.355362i	0.984088	−	0.177681i	$-0.0568595\pi$
$858$		−	8.26136i		−	0.282038i
$859$	23.5719			0.804264			0.402132	−	0.915582i	$-0.368269\pi$
$859$	23.5719			0.804264			0.402132	+	0.915582i	$0.368269\pi$
$860$	56.4956	−	14.5666i	1.92648	−	0.496716i
$861$	23.0968			0.787136
$862$			53.5195i			1.82288i
$863$		−	52.9902i		−	1.80381i	−0.431937	−	0.901904i	$-0.642170\pi$
$863$		−	52.9902i		−	1.80381i	0.431937	−	0.901904i	$-0.357830\pi$
$864$	−40.9681			−1.39376
$865$	19.8402	−	5.11550i	0.674585	−	0.173932i
$866$	61.1009			2.07629
$867$		−	11.4555i		−	0.389048i
$868$		−	57.2570i		−	1.94343i
$869$	−2.50419			−0.0849490
$870$	−0.803230	−	3.11528i	−0.0272321	−	0.105618i
$871$	−27.0210			−0.915572
$872$		−	32.2848i		−	1.09330i
$873$			1.94426i			0.0658033i
$874$	−0.246193			−0.00832760
$875$	20.8316	−	19.6815i	0.704235	−	0.665357i
$876$	47.6106			1.60861
$877$			43.1720i			1.45781i	0.684613	+	0.728907i	$0.259971\pi$
$877$			43.1720i			1.45781i	−0.684613	+	0.728907i	$0.740029\pi$
$878$			37.6067i			1.26916i
$879$	−20.6893			−0.697832
$880$	−1.02037	−	3.95745i	−0.0343967	−	0.133406i
$881$	−39.8859			−1.34379			−0.671895	−	0.740647i	$-0.734519\pi$
$881$	−39.8859			−1.34379			−0.671895	+	0.740647i	$0.734519\pi$
$882$			0.456052i			0.0153561i
$883$		−	29.6852i		−	0.998987i	−0.866318	−	0.499494i	$-0.833519\pi$
$883$		−	29.6852i		−	0.998987i	0.866318	−	0.499494i	$-0.166481\pi$
$884$	−20.7187			−0.696845
$885$	38.0766	−	9.81750i	1.27993	−	0.330012i
$886$	−71.1519			−2.39039
$887$			35.2458i			1.18344i	0.806144	+	0.591720i	$0.201551\pi$
$887$			35.2458i			1.18344i	−0.806144	+	0.591720i	$0.798449\pi$
$888$			10.2047i			0.342447i
$889$	−8.08714			−0.271234
$890$	−18.3039	+	4.71939i	−0.613547	+	0.158194i
$891$	7.30721			0.244801
$892$			64.1922i			2.14931i
$893$			3.73767i			0.125076i
$894$	−26.5774			−0.888881
$895$	−9.93033	−	38.5142i	−0.331934	−	1.28739i
$896$	32.3523			1.08081
$897$		−	0.425402i		−	0.0142038i
$898$			77.2609i			2.57823i
$899$	−3.33291			−0.111159
$900$	5.91121	−	3.26532i	0.197040	−	0.108844i
$901$	13.2229			0.440519
$902$			12.4251i			0.413712i
$903$			38.1332i			1.26899i
$904$	5.51220			0.183333
$905$	−8.73174	−	33.8655i	−0.290253	−	1.12573i
$906$	−11.8568			−0.393915
$907$			46.6161i			1.54786i	0.633269	+	0.773931i	$0.281713\pi$
$907$			46.6161i			1.54786i	−0.633269	+	0.773931i	$0.718287\pi$
$908$			64.6958i			2.14701i
$909$	−2.26815			−0.0752299
$910$	28.9166	−	7.45573i	0.958576	−	0.247155i
$911$	−59.9934			−1.98767			−0.993835	−	0.110871i	$-0.964636\pi$
$911$	−59.9934			−1.98767			−0.993835	+	0.110871i	$0.964636\pi$
$912$		−	2.89814i		−	0.0959671i
$913$			14.4319i			0.477625i
$914$	−66.7283			−2.20718
$915$	−3.07060	+	0.791711i	−0.101511	+	0.0261731i
$916$	−54.2933			−1.79390
$917$			5.90161i			0.194888i
$918$		−	37.7861i		−	1.24713i
$919$	−36.6285			−1.20826			−0.604132	−	0.796885i	$-0.706480\pi$
$919$	−36.6285			−1.20826			−0.604132	+	0.796885i	$0.706480\pi$
$920$	0.107357	+	0.416378i	0.00353946	+	0.0137276i
$921$	25.0900			0.826742
$922$		−	22.3454i		−	0.735908i
$923$			6.08036i			0.200137i
$924$	−11.3039			−0.371872
$925$	9.10983	+	16.4915i	0.299529	+	0.542238i
$926$	57.2475			1.88127
$927$			1.54912i			0.0508797i
$928$		−	3.07582i		−	0.100969i
$929$	−6.71768			−0.220400			−0.110200	−	0.993909i	$-0.535149\pi$
$929$	−6.71768			−0.220400			−0.110200	+	0.993909i	$0.535149\pi$
$930$	15.5468	+	60.2974i	0.509800	+	1.97723i
$931$	−0.429469			−0.0140753
$932$			47.9159i			1.56954i
$933$		−	7.87016i		−	0.257657i
$934$	−47.7916			−1.56379
$935$	6.76989	−	1.74552i	0.221399	−	0.0570846i
$936$	1.97633			0.0645985
$937$		−	20.3160i		−	0.663694i	−0.943333	−	0.331847i	$-0.892328\pi$
$937$		−	20.3160i		−	0.663694i	0.943333	−	0.331847i	$-0.107672\pi$
$938$			63.5611i			2.07534i
$939$	29.3879			0.959037
$940$	22.5074	−	5.80321i	0.734111	−	0.189280i
$941$	46.5700			1.51814			0.759070	−	0.651009i	$-0.225654\pi$
$941$	46.5700			1.51814			0.759070	+	0.651009i	$0.225654\pi$
$942$		−	0.0480931i		−	0.00156696i
$943$			0.639809i			0.0208350i
$944$	20.2695			0.659715
$945$	7.90949	+	30.6765i	0.257296	+	0.997906i
$946$	−20.5142			−0.666973
$947$		−	58.8813i		−	1.91338i	−0.291101	−	0.956692i	$-0.594022\pi$
$947$		−	58.8813i		−	1.91338i	0.291101	−	0.956692i	$-0.405978\pi$
$948$			11.0433i			0.358668i
$949$	25.7247			0.835058
$950$	5.28633	+	9.56985i	0.171511	+	0.310487i
$951$	−12.3075			−0.399097
$952$			13.6880i			0.443630i
$953$		−	45.4419i		−	1.47201i	−0.676977	−	0.736004i	$-0.736711\pi$
$953$		−	45.4419i		−	1.47201i	0.676977	−	0.736004i	$-0.263289\pi$
$954$	−4.49094			−0.145399
$955$	−6.35890	−	24.6626i	−0.205769	−	0.798063i
$956$	19.5517			0.632349
$957$			0.657998i			0.0212701i
$958$			78.6804i			2.54205i
$959$	−17.9449			−0.579470
$960$	−43.0957	+	11.1116i	−1.39091	+	0.358626i
$961$	33.5097			1.08096
$962$			19.6317i			0.632951i
$963$			3.81312i			0.122876i
$964$	−75.4500			−2.43008
$965$	56.0413	−	14.4494i	1.80403	−	0.465144i
$966$	−1.00067			−0.0321959
$967$		−	26.9945i		−	0.868085i	−0.900892	−	0.434043i	$-0.857087\pi$
$967$		−	26.9945i		−	0.868085i	0.900892	−	0.434043i	$-0.142913\pi$
$968$		−	1.70792i		−	0.0548945i
$969$	4.95776			0.159266
$970$	4.88710	+	18.9543i	0.156915	+	0.608586i
$971$	−11.6954			−0.375324			−0.187662	−	0.982234i	$-0.560091\pi$
$971$	−11.6954			−0.375324			−0.187662	+	0.982234i	$0.560091\pi$
$972$			13.8900i			0.445521i
$973$		−	16.7572i		−	0.537210i
$974$	83.3449			2.67054
$975$	−16.5360	+	9.13437i	−0.529574	+	0.292534i
$976$	−1.63458			−0.0523218
$977$		−	29.3847i		−	0.940099i	−0.882640	−	0.470049i	$-0.844236\pi$
$977$		−	29.3847i		−	0.940099i	0.882640	−	0.470049i	$-0.155764\pi$
$978$			25.8858i			0.827735i
$979$	3.86607			0.123560
$980$	0.666806	+	2.58616i	0.0213003	+	0.0826120i
$981$	9.18016			0.293100
$982$			39.1986i			1.25088i
$983$			1.15867i			0.0369558i	0.999829	+	0.0184779i	$0.00588203\pi$
$983$			1.15867i			0.0369558i	−0.999829	+	0.0184779i	$0.994118\pi$
$984$	15.3893			0.490592
$985$	−32.3413	+	8.33875i	−1.03048	+	0.265695i
$986$	2.83692			0.0903460
$987$			15.1920i			0.483566i
$988$			6.62659i			0.210820i
$989$	−1.05634			−0.0335895
$990$	−2.29928	+	0.592836i	−0.0730758	+	0.0188416i
$991$	−13.2320			−0.420327			−0.210163	−	0.977666i	$-0.567400\pi$
$991$	−13.2320			−0.420327			−0.210163	+	0.977666i	$0.567400\pi$
$992$			59.5336i			1.89019i
$993$			9.71335i			0.308244i
$994$	14.3027			0.453655
$995$	−2.78805	−	10.8133i	−0.0883871	−	0.342804i
$996$	63.6431			2.01661
$997$		−	21.9664i		−	0.695683i	−0.937553	−	0.347842i	$-0.886915\pi$
$997$		−	21.9664i		−	0.695683i	0.937553	−	0.347842i	$-0.113085\pi$
$998$		−	51.0270i		−	1.61523i
$999$	−20.8265			−0.658921

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.b.b.419.16	yes	16
5.2	odd	4		5225.2.a.z.1.1		16
5.3	odd	4		5225.2.a.z.1.16		16
5.4	even	2	inner	1045.2.b.b.419.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.b.b.419.1	✓	16	5.4	even	2	inner
1045.2.b.b.419.16	yes	16	1.1	even	1	trivial
5225.2.a.z.1.1		16	5.2	odd	4
5225.2.a.z.1.16		16	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)

\(f(q)\)	\(=\)	\(q+2.18657i q^{2} -1.58567i q^{3} -2.78109 q^{4} +(-0.558280 - 2.16525i) q^{5} +3.46718 q^{6} -2.56330i q^{7} -1.70792i q^{8} +0.485645 q^{9} +O(q^{10})\)	\(q+2.18657i q^{2} -1.58567i q^{3} -2.78109 q^{4} +(-0.558280 - 2.16525i) q^{5} +3.46718 q^{6} -2.56330i q^{7} -1.70792i q^{8} +0.485645 q^{9} +(4.73448 - 1.22072i) q^{10} -1.00000 q^{11} +4.40990i q^{12} +2.38273i q^{13} +5.60485 q^{14} +(-3.43338 + 0.885249i) q^{15} -1.82771 q^{16} -3.12660i q^{17} +1.06190i q^{18} -1.00000 q^{19} +(1.55263 + 6.02177i) q^{20} -4.06456 q^{21} -2.18657i q^{22} -0.112593i q^{23} -2.70820 q^{24} +(-4.37665 + 2.41764i) q^{25} -5.21000 q^{26} -5.52709i q^{27} +7.12879i q^{28} +0.414965 q^{29} +(-1.93566 - 7.50733i) q^{30} -8.03180 q^{31} -7.41224i q^{32} +1.58567i q^{33} +6.83654 q^{34} +(-5.55020 + 1.43104i) q^{35} -1.35062 q^{36} -3.76807i q^{37} -2.18657i q^{38} +3.77822 q^{39} +(-3.69807 + 0.953496i) q^{40} -5.68248 q^{41} -8.88745i q^{42} -9.38188i q^{43} +2.78109 q^{44} +(-0.271126 - 1.05154i) q^{45} +0.246193 q^{46} -3.73767i q^{47} +2.89814i q^{48} +0.429469 q^{49} +(-5.28633 - 9.56985i) q^{50} -4.95776 q^{51} -6.62659i q^{52} +4.22916i q^{53} +12.0854 q^{54} +(0.558280 + 2.16525i) q^{55} -4.37791 q^{56} +1.58567i q^{57} +0.907350i q^{58} -11.0901 q^{59} +(9.54856 - 2.46196i) q^{60} +0.894337 q^{61} -17.5621i q^{62} -1.24486i q^{63} +12.5520 q^{64} +(5.15921 - 1.33023i) q^{65} -3.46718 q^{66} +11.3404i q^{67} +8.69537i q^{68} -0.178536 q^{69} +(-3.12907 - 12.1359i) q^{70} +2.55185 q^{71} -0.829441i q^{72} -10.7963i q^{73} +8.23916 q^{74} +(3.83358 + 6.93993i) q^{75} +2.78109 q^{76} +2.56330i q^{77} +8.26136i q^{78} +2.50419 q^{79} +(1.02037 + 3.95745i) q^{80} -7.30721 q^{81} -12.4251i q^{82} -14.4319i q^{83} +11.3039 q^{84} +(-6.76989 + 1.74552i) q^{85} +20.5142 q^{86} -0.657998i q^{87} +1.70792i q^{88} -3.86607 q^{89} +(2.29928 - 0.592836i) q^{90} +6.10766 q^{91} +0.313132i q^{92} +12.7358i q^{93} +8.17268 q^{94} +(0.558280 + 2.16525i) q^{95} -11.7534 q^{96} +4.00346i q^{97} +0.939064i q^{98} -0.485645 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9}+O(q^{10}) \) 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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