LMFDB - Embedded newform 440.2.b.d.89.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [440,2,Mod(89,440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(440, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("440.89");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 440 = 2^{3} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	440.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:	$3.51341768894$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.47985531136.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 19x^{6} + 91x^{4} + 45x^{2} + 4 $ x^8 + 19x^6 + 91x^4 + 45*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			89.4
Root			$-0.339102i$ of defining polynomial
Character	$\chi$	$=$	440.89
Dual form			440.2.b.d.89.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.339102i q^{3} +(0.422782 - 2.19574i) q^{5} +4.05237i q^{7} +2.88501 q^{9} +O(q^{10})$	$q-0.339102i q^{3} +(0.422782 - 2.19574i) q^{5} +4.05237i q^{7} +2.88501 q^{9} +1.00000 q^{11} +4.00000i q^{13} +(-0.744578 - 0.143366i) q^{15} -7.74350i q^{17} +7.06529 q^{19} +1.37417 q^{21} -2.72619i q^{23} +(-4.64251 - 1.85663i) q^{25} -1.99562i q^{27} +4.73057 q^{29} +0.219729 q^{31} -0.339102i q^{33} +(8.89793 + 1.71327i) q^{35} -1.32618i q^{37} +1.35641 q^{39} -7.79587 q^{41} +11.1177i q^{43} +(1.21973 - 6.33472i) q^{45} +3.01292i q^{47} -9.42170 q^{49} -2.62583 q^{51} +5.03945i q^{53} +(0.422782 - 2.19574i) q^{55} -2.39585i q^{57} -10.9503 q^{59} +12.7306 q^{61} +11.6911i q^{63} +(8.78294 + 1.69113i) q^{65} -4.70034i q^{67} -0.924456 q^{69} -2.52860 q^{71} -4.10474i q^{73} +(-0.629588 + 1.57428i) q^{75} +4.05237i q^{77} -13.9006 q^{79} +7.97831 q^{81} +3.63067i q^{83} +(-17.0027 - 3.27381i) q^{85} -1.60415i q^{87} -9.88501 q^{89} -16.2095 q^{91} -0.0745107i q^{93} +(2.98708 - 15.5135i) q^{95} -12.4962i q^{97} +2.88501 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 14 q^{9}+O(q^{10}) $ 8 * q - 14 * q^9	$ 8 q - 14 q^{9} + 8 q^{11} - 12 q^{15} + 18 q^{19} - 14 q^{21} - 2 q^{25} - 6 q^{29} - 30 q^{31} + 30 q^{35} - 8 q^{39} + 20 q^{41} - 22 q^{45} - 18 q^{49} - 46 q^{51} - 12 q^{59} + 58 q^{61} - 8 q^{65} + 60 q^{69} - 2 q^{71} + 26 q^{75} + 40 q^{79} + 88 q^{81} - 26 q^{85} - 42 q^{89} + 8 q^{91} + 28 q^{95} - 14 q^{99}+O(q^{100}) $ 8 * q - 14 * q^9 + 8 * q^11 - 12 * q^15 + 18 * q^19 - 14 * q^21 - 2 * q^25 - 6 * q^29 - 30 * q^31 + 30 * q^35 - 8 * q^39 + 20 * q^41 - 22 * q^45 - 18 * q^49 - 46 * q^51 - 12 * q^59 + 58 * q^61 - 8 * q^65 + 60 * q^69 - 2 * q^71 + 26 * q^75 + 40 * q^79 + 88 * q^81 - 26 * q^85 - 42 * q^89 + 8 * q^91 + 28 * q^95 - 14 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/440\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$221$	$321$
$\chi(n)$	$1$	$-1$	$1$	$1$

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			0
$2$		0			0
$3$		−	0.339102i		−	0.195781i	−0.995197	−	0.0978903i	$-0.968791\pi$
$3$		−	0.339102i		−	0.195781i	0.995197	−	0.0978903i	$-0.0312094\pi$
$4$		0			0
$5$	0.422782	−	2.19574i	0.189074	−	0.981963i
$5$	0.422782	−	2.19574i	0.189074	−	0.981963i
$6$		0			0
$7$			4.05237i			1.53165i	0.643048	+	0.765826i	$0.277670\pi$
$7$			4.05237i			1.53165i	−0.643048	+	0.765826i	$0.722330\pi$
$8$		0			0
$9$	2.88501			0.961670
$10$		0			0
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$		0			0
$13$			4.00000i			1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$			4.00000i			1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$	−0.744578	−	0.143366i	−0.192249	−	0.0370170i
$16$		0			0
$17$		−	7.74350i		−	1.87807i	−0.343816	−	0.939037i	$-0.611720\pi$
$17$		−	7.74350i		−	1.87807i	0.343816	−	0.939037i	$-0.388280\pi$
$18$		0			0
$19$	7.06529			1.62089			0.810445	−	0.585815i	$-0.199226\pi$
$19$	7.06529			1.62089			0.810445	+	0.585815i	$0.199226\pi$
$20$		0			0
$21$	1.37417			0.299868
$22$		0			0
$23$		−	2.72619i		−	0.568450i	−0.958758	−	0.284225i	$-0.908264\pi$
$23$		−	2.72619i		−	0.568450i	0.958758	−	0.284225i	$-0.0917363\pi$
$24$		0			0
$25$	−4.64251	−	1.85663i	−0.928502	−	0.371327i
$26$		0			0
$27$		−	1.99562i		−	0.384057i
$28$		0			0
$29$	4.73057			0.878445			0.439223	−	0.898378i	$-0.355254\pi$
$29$	4.73057			0.878445			0.439223	+	0.898378i	$0.355254\pi$
$30$		0			0
$31$	0.219729			0.0394646			0.0197323	−	0.999805i	$-0.493719\pi$
$31$	0.219729			0.0394646			0.0197323	+	0.999805i	$0.493719\pi$
$32$		0			0
$33$		−	0.339102i		−	0.0590301i
$34$		0			0
$35$	8.89793	+	1.71327i	1.50403	+	0.289595i
$36$		0			0
$37$		−	1.32618i		−	0.218022i	−0.994041	−	0.109011i	$-0.965232\pi$
$37$		−	1.32618i		−	0.218022i	0.994041	−	0.109011i	$-0.0347684\pi$
$38$		0			0
$39$	1.35641			0.217199
$40$		0			0
$41$	−7.79587			−1.21751			−0.608755	−	0.793358i	$-0.708331\pi$
$41$	−7.79587			−1.21751			−0.608755	+	0.793358i	$0.708331\pi$
$42$		0			0
$43$			11.1177i			1.69543i	0.530454	+	0.847714i	$0.322022\pi$
$43$			11.1177i			1.69543i	−0.530454	+	0.847714i	$0.677978\pi$
$44$		0			0
$45$	1.21973	−	6.33472i	0.181827	−	0.944324i
$46$		0			0
$47$			3.01292i			0.439480i	0.975558	+	0.219740i	$0.0705209\pi$
$47$			3.01292i			0.439480i	−0.975558	+	0.219740i	$0.929479\pi$
$48$		0			0
$49$	−9.42170			−1.34596
$50$		0			0
$51$	−2.62583			−0.367690
$52$		0			0
$53$			5.03945i			0.692221i	0.938194	+	0.346111i	$0.112498\pi$
$53$			5.03945i			0.692221i	−0.938194	+	0.346111i	$0.887502\pi$
$54$		0			0
$55$	0.422782	−	2.19574i	0.0570079	−	0.296073i
$56$		0			0
$57$		−	2.39585i		−	0.317339i
$58$		0			0
$59$	−10.9503			−1.42561			−0.712804	−	0.701363i	$-0.752575\pi$
$59$	−10.9503			−1.42561			−0.712804	+	0.701363i	$0.752575\pi$
$60$		0			0
$61$	12.7306			1.62998			0.814991	−	0.579473i	$-0.196742\pi$
$61$	12.7306			1.62998			0.814991	+	0.579473i	$0.196742\pi$
$62$		0			0
$63$			11.6911i			1.47294i
$64$		0			0
$65$	8.78294	+	1.69113i	1.08939	+	0.209758i
$66$		0			0
$67$		−	4.70034i		−	0.574238i	−0.957895	−	0.287119i	$-0.907302\pi$
$67$		−	4.70034i		−	0.574238i	0.957895	−	0.287119i	$-0.0926976\pi$
$68$		0			0
$69$	−0.924456			−0.111291
$70$		0			0
$71$	−2.52860			−0.300090			−0.150045	−	0.988679i	$-0.547942\pi$
$71$	−2.52860			−0.300090			−0.150045	+	0.988679i	$0.547942\pi$
$72$		0			0
$73$		−	4.10474i		−	0.480423i	−0.970721	−	0.240212i	$-0.922783\pi$
$73$		−	4.10474i		−	0.480423i	0.970721	−	0.240212i	$-0.0772169\pi$
$74$		0			0
$75$	−0.629588	+	1.57428i	−0.0726986	+	0.181783i
$76$		0			0
$77$			4.05237i			0.461810i
$78$		0			0
$79$	−13.9006			−1.56394			−0.781970	−	0.623316i	$-0.785785\pi$
$79$	−13.9006			−1.56394			−0.781970	+	0.623316i	$0.785785\pi$
$80$		0			0
$81$	7.97831			0.886479
$82$		0			0
$83$			3.63067i			0.398518i	0.979947	+	0.199259i	$0.0638535\pi$
$83$			3.63067i			0.398518i	−0.979947	+	0.199259i	$0.936147\pi$
$84$		0			0
$85$	−17.0027	−	3.27381i	−1.84420	−	0.355094i
$86$		0			0
$87$		−	1.60415i		−	0.171983i
$88$		0			0
$89$	−9.88501			−1.04781			−0.523904	−	0.851777i	$-0.675525\pi$
$89$	−9.88501			−1.04781			−0.523904	+	0.851777i	$0.675525\pi$
$90$		0			0
$91$	−16.2095			−1.69922
$92$		0			0
$93$		−	0.0745107i		−	0.00772640i
$94$		0			0
$95$	2.98708	−	15.5135i	0.306468	−	1.59165i
$96$		0			0
$97$		−	12.4962i		−	1.26880i	−0.773006	−	0.634399i	$-0.781248\pi$
$97$		−	12.4962i		−	1.26880i	0.773006	−	0.634399i	$-0.218752\pi$
$98$		0			0
$99$	2.88501			0.289954
$100$		0			0
$101$	−0.643593			−0.0640399			−0.0320199	−	0.999487i	$-0.510194\pi$
$101$	−0.643593			−0.0640399			−0.0320199	+	0.999487i	$0.510194\pi$
$102$		0			0
$103$			4.36933i			0.430523i	0.976556	+	0.215261i	$0.0690604\pi$
$103$			4.36933i			0.430523i	−0.976556	+	0.215261i	$0.930940\pi$
$104$		0			0
$105$	0.580972	−	3.01731i	0.0566971	−	0.294459i
$106$		0			0
$107$		−	8.98708i		−	0.868814i	−0.900717	−	0.434407i	$-0.856958\pi$
$107$		−	8.98708i		−	0.868814i	0.900717	−	0.434407i	$-0.143042\pi$
$108$		0			0
$109$	−18.2095			−1.74415			−0.872076	−	0.489371i	$-0.837227\pi$
$109$	−18.2095			−1.74415			−0.872076	+	0.489371i	$0.837227\pi$
$110$		0			0
$111$	−0.449710			−0.0426845
$112$		0			0
$113$			10.4173i			0.979979i	0.871728	+	0.489989i	$0.162999\pi$
$113$			10.4173i			0.979979i	−0.871728	+	0.489989i	$0.837001\pi$
$114$		0			0
$115$	−5.98600	−	1.15258i	−0.558197	−	0.107479i
$116$		0			0
$117$			11.5400i			1.06688i
$118$		0			0
$119$	31.3795			2.87656
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$			2.64359i			0.238365i
$124$		0			0
$125$	−6.03945	+	9.40878i	−0.540185	+	0.841547i
$126$		0			0
$127$			9.77002i			0.866949i	0.901166	+	0.433475i	$0.142713\pi$
$127$			9.77002i			0.866949i	−0.901166	+	0.433475i	$0.857287\pi$
$128$		0			0
$129$	3.77002			0.331932
$130$		0			0
$131$	−12.0870			−1.05604			−0.528022	−	0.849231i	$-0.677066\pi$
$131$	−12.0870			−1.05604			−0.528022	+	0.849231i	$0.677066\pi$
$132$		0			0
$133$			28.6312i			2.48264i
$134$		0			0
$135$	−4.38185	−	0.843711i	−0.377130	−	0.0726151i
$136$		0			0
$137$			4.49621i			0.384137i	0.981381	+	0.192069i	$0.0615196\pi$
$137$			4.49621i			0.384137i	−0.981381	+	0.192069i	$0.938480\pi$
$138$		0			0
$139$	8.33472			0.706942			0.353471	−	0.935446i	$-0.385001\pi$
$139$	8.33472			0.706942			0.353471	+	0.935446i	$0.385001\pi$
$140$		0			0
$141$	1.02169			0.0860416
$142$		0			0
$143$			4.00000i			0.334497i
$144$		0			0
$145$	2.00000	−	10.3871i	0.166091	−	0.862601i
$146$		0			0
$147$			3.19492i			0.263512i
$148$		0			0
$149$	11.8136			0.967810			0.483905	−	0.875121i	$-0.339218\pi$
$149$	11.8136			0.967810			0.483905	+	0.875121i	$0.339218\pi$
$150$		0			0
$151$	−15.7700			−1.28335			−0.641673	−	0.766978i	$-0.721759\pi$
$151$	−15.7700			−1.28335			−0.641673	+	0.766978i	$0.721759\pi$
$152$		0			0
$153$		−	22.3401i		−	1.80609i
$154$		0			0
$155$	0.0928976	−	0.482468i	0.00746172	−	0.0387528i
$156$		0			0
$157$			0.0302288i			0.00241252i	0.999999	+	0.00120626i	$0.000383964\pi$
$157$			0.0302288i			0.00241252i	−0.999999	+	0.00120626i	$0.999616\pi$
$158$		0			0
$159$	1.70889			0.135523
$160$		0			0
$161$	11.0475			0.870668
$162$		0			0
$163$		−	14.0782i		−	1.10269i	−0.834277	−	0.551345i	$-0.814115\pi$
$163$		−	14.0782i		−	1.10269i	0.834277	−	0.551345i	$-0.185885\pi$
$164$		0			0
$165$	−0.744578	−	0.143366i	−0.0579653	−	0.0111610i
$166$		0			0
$167$		−	12.7306i		−	0.985121i	−0.870278	−	0.492561i	$-0.836061\pi$
$167$		−	12.7306i		−	0.985121i	0.870278	−	0.492561i	$-0.163939\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$		0			0
$171$	20.3834			1.55876
$172$		0			0
$173$			7.32180i			0.556666i	0.960485	+	0.278333i	$0.0897819\pi$
$173$			7.32180i			0.556666i	−0.960485	+	0.278333i	$0.910218\pi$
$174$		0			0
$175$	7.52377	−	18.8132i	0.568743	−	1.42214i
$176$		0			0
$177$			3.71327i			0.279106i
$178$		0			0
$179$	−6.56254			−0.490507			−0.245254	−	0.969459i	$-0.578871\pi$
$179$	−6.56254			−0.490507			−0.245254	+	0.969459i	$0.578871\pi$
$180$		0			0
$181$	−7.59390			−0.564450			−0.282225	−	0.959348i	$-0.591072\pi$
$181$	−7.59390			−0.564450			−0.282225	+	0.959348i	$0.591072\pi$
$182$		0			0
$183$		−	4.31696i		−	0.319119i
$184$		0			0
$185$	−2.91194	−	0.560684i	−0.214090	−	0.0412223i
$186$		0			0
$187$		−	7.74350i		−	0.566261i
$188$		0			0
$189$	8.08698			0.588241
$190$		0			0
$191$	−12.6414			−0.914702			−0.457351	−	0.889286i	$-0.651202\pi$
$191$	−12.6414			−0.914702			−0.457351	+	0.889286i	$0.651202\pi$
$192$		0			0
$193$		−	4.42170i		−	0.318281i	−0.987256	−	0.159140i	$-0.949128\pi$
$193$		−	4.42170i		−	0.318281i	0.987256	−	0.159140i	$-0.0508723\pi$
$194$		0			0
$195$	0.573464	−	2.97831i	0.0410666	−	0.213281i
$196$		0			0
$197$		−	0.678204i		−	0.0483200i	−0.999708	−	0.0241600i	$-0.992309\pi$
$197$		−	0.678204i		−	0.0483200i	0.999708	−	0.0241600i	$-0.00769112\pi$
$198$		0			0
$199$	−4.42170			−0.313446			−0.156723	−	0.987643i	$-0.550093\pi$
$199$	−4.42170			−0.313446			−0.156723	+	0.987643i	$0.550093\pi$
$200$		0			0
$201$	−1.59390			−0.112425
$202$		0			0
$203$			19.1700i			1.34547i
$204$		0			0
$205$	−3.29595	+	17.1177i	−0.230199	+	1.19555i
$206$		0			0
$207$		−	7.86509i		−	0.546661i
$208$		0			0
$209$	7.06529			0.488717
$210$		0			0
$211$	11.1700			0.768977			0.384488	−	0.923130i	$-0.374378\pi$
$211$	11.1700			0.768977			0.384488	+	0.923130i	$0.374378\pi$
$212$		0			0
$213$			0.857454i			0.0587518i
$214$		0			0
$215$	24.4114	+	4.70034i	1.66485	+	0.320561i
$216$		0			0
$217$			0.890425i			0.0604460i
$218$		0			0
$219$	−1.39192			−0.0940576
$220$		0			0
$221$	30.9740			2.08354
$222$		0			0
$223$		−	19.4487i		−	1.30238i	−0.758915	−	0.651190i	$-0.774270\pi$
$223$		−	19.4487i		−	1.30238i	0.758915	−	0.651190i	$-0.225730\pi$
$224$		0			0
$225$	−13.3937	−	5.35641i	−0.892913	−	0.357094i
$226$		0			0
$227$		−	22.6047i		−	1.50032i	−0.661254	−	0.750162i	$-0.729976\pi$
$227$		−	22.6047i		−	1.50032i	0.661254	−	0.750162i	$-0.270024\pi$
$228$		0			0
$229$	17.0550			1.12703			0.563514	−	0.826106i	$-0.309449\pi$
$229$	17.0550			1.12703			0.563514	+	0.826106i	$0.309449\pi$
$230$		0			0
$231$	1.37417			0.0904135
$232$		0			0
$233$			14.5006i			0.949965i	0.879995	+	0.474983i	$0.157546\pi$
$233$			14.5006i			0.949965i	−0.879995	+	0.474983i	$0.842454\pi$
$234$		0			0
$235$	6.61558	+	1.27381i	0.431553	+	0.0830941i
$236$		0			0
$237$			4.71372i			0.306189i
$238$		0			0
$239$	−13.9006			−0.899155			−0.449578	−	0.893241i	$-0.648426\pi$
$239$	−13.9006			−0.899155			−0.449578	+	0.893241i	$0.648426\pi$
$240$		0			0
$241$	15.0572			0.969920			0.484960	−	0.874536i	$-0.338834\pi$
$241$	15.0572			0.969920			0.484960	+	0.874536i	$0.338834\pi$
$242$		0			0
$243$		−	8.69231i		−	0.557612i
$244$		0			0
$245$	−3.98332	+	20.6876i	−0.254485	+	1.32168i
$246$		0			0
$247$			28.2612i			1.79822i
$248$		0			0
$249$	1.23117			0.0780220
$250$		0			0
$251$	2.51084			0.158483			0.0792415	−	0.996855i	$-0.474750\pi$
$251$	2.51084			0.158483			0.0792415	+	0.996855i	$0.474750\pi$
$252$		0			0
$253$		−	2.72619i		−	0.171394i
$254$		0			0
$255$	−1.11015	+	5.76564i	−0.0695206	+	0.361058i
$256$		0			0
$257$		−	6.64359i		−	0.414416i	−0.978297	−	0.207208i	$-0.933562\pi$
$257$		−	6.64359i		−	0.414416i	0.978297	−	0.207208i	$-0.0664377\pi$
$258$		0			0
$259$	5.37417			0.333934
$260$		0			0
$261$	13.6478			0.844775
$262$		0			0
$263$			13.4088i			0.826821i	0.910545	+	0.413410i	$0.135662\pi$
$263$			13.4088i			0.826821i	−0.910545	+	0.413410i	$0.864338\pi$
$264$		0			0
$265$	11.0653	+	2.13059i	0.679736	+	0.130881i
$266$		0			0
$267$			3.35203i			0.205141i
$268$		0			0
$269$	19.4611			1.18657			0.593284	−	0.804994i	$-0.297831\pi$
$269$	19.4611			1.18657			0.593284	+	0.804994i	$0.297831\pi$
$270$		0			0
$271$	0.0530465			0.00322235			0.00161117	−	0.999999i	$-0.499487\pi$
$271$	0.0530465			0.00322235			0.00161117	+	0.999999i	$0.499487\pi$
$272$		0			0
$273$			5.49666i			0.332673i
$274$		0			0
$275$	−4.64251	−	1.85663i	−0.279954	−	0.111959i
$276$		0			0
$277$		−	4.09597i		−	0.246103i	−0.992400	−	0.123052i	$-0.960732\pi$
$277$		−	4.09597i		−	0.246103i	0.992400	−	0.123052i	$-0.0392681\pi$
$278$		0			0
$279$	0.633922			0.0379519
$280$		0			0
$281$	17.5400			1.04635			0.523176	−	0.852225i	$-0.324747\pi$
$281$	17.5400			1.04635			0.523176	+	0.852225i	$0.324747\pi$
$282$		0			0
$283$			15.1264i			0.899173i	0.893237	+	0.449586i	$0.148429\pi$
$283$			15.1264i			0.899173i	−0.893237	+	0.449586i	$0.851571\pi$
$284$		0			0
$285$	−5.26066	−	1.01292i	−0.311615	−	0.0600004i
$286$		0			0
$287$		−	31.5917i		−	1.86480i
$288$		0			0
$289$	−42.9617			−2.52716
$290$		0			0
$291$	−4.23749			−0.248406
$292$		0			0
$293$		−	13.3564i		−	0.780290i	−0.920754	−	0.390145i	$-0.872425\pi$
$293$		−	13.3564i		−	0.780290i	0.920754	−	0.390145i	$-0.127575\pi$
$294$		0			0
$295$	−4.62959	+	24.0440i	−0.269545	+	1.39989i
$296$		0			0
$297$		−	1.99562i		−	0.115797i
$298$		0			0
$299$	10.9048			0.630639
$300$		0			0
$301$	−45.0529			−2.59680
$302$		0			0
$303$			0.218243i			0.0125378i
$304$		0			0
$305$	5.38225	−	27.9530i	0.308187	−	1.60058i
$306$		0			0
$307$			15.7959i			0.901518i	0.892646	+	0.450759i	$0.148847\pi$
$307$			15.7959i			0.901518i	−0.892646	+	0.450759i	$0.851153\pi$
$308$		0			0
$309$	1.48165			0.0842880
$310$		0			0
$311$	−0.829968			−0.0470631			−0.0235316	−	0.999723i	$-0.507491\pi$
$311$	−0.829968			−0.0470631			−0.0235316	+	0.999723i	$0.507491\pi$
$312$		0			0
$313$		−	24.5050i		−	1.38510i	−0.721368	−	0.692552i	$-0.756487\pi$
$313$		−	24.5050i		−	1.38510i	0.721368	−	0.692552i	$-0.243513\pi$
$314$		0			0
$315$	25.6706	+	4.94279i	1.44638	+	0.278495i
$316$		0			0
$317$			11.4655i			0.643968i	0.946745	+	0.321984i	$0.104350\pi$
$317$			11.4655i			0.643968i	−0.946745	+	0.321984i	$0.895650\pi$
$318$		0			0
$319$	4.73057			0.264861
$320$		0			0
$321$	−3.04753			−0.170097
$322$		0			0
$323$		−	54.7101i		−	3.04415i
$324$		0			0
$325$	7.42654	−	18.5700i	0.411950	−	1.03008i
$326$		0			0
$327$			6.17487i			0.341471i
$328$		0			0
$329$	−12.2095			−0.673130
$330$		0			0
$331$	−24.4114			−1.34177			−0.670887	−	0.741559i	$-0.734087\pi$
$331$	−24.4114			−1.34177			−0.670887	+	0.741559i	$0.734087\pi$
$332$		0			0
$333$		−	3.82604i		−	0.209666i
$334$		0			0
$335$	−10.3207	−	1.98722i	−0.563881	−	0.108573i
$336$		0			0
$337$			14.3871i			0.783715i	0.920026	+	0.391857i	$0.128167\pi$
$337$			14.3871i			0.783715i	−0.920026	+	0.391857i	$0.871833\pi$
$338$		0			0
$339$	3.53253			0.191861
$340$		0			0
$341$	0.219729			0.0118990
$342$		0			0
$343$		−	9.81363i		−	0.529886i
$344$		0			0
$345$	−0.390843	+	2.02986i	−0.0210423	+	0.109284i
$346$		0			0
$347$			1.66528i			0.0893969i	0.999001	+	0.0446985i	$0.0142327\pi$
$347$			1.66528i			0.0893969i	−0.999001	+	0.0446985i	$0.985767\pi$
$348$		0			0
$349$	−6.73866			−0.360712			−0.180356	−	0.983601i	$-0.557725\pi$
$349$	−6.73866			−0.360712			−0.180356	+	0.983601i	$0.557725\pi$
$350$		0			0
$351$	7.98247			0.426073
$352$		0			0
$353$			23.1486i			1.23207i	0.787717	+	0.616037i	$0.211263\pi$
$353$			23.1486i			1.23207i	−0.787717	+	0.616037i	$0.788737\pi$
$354$		0			0
$355$	−1.06905	+	5.55214i	−0.0567391	+	0.294677i
$356$		0			0
$357$		−	10.6409i		−	0.563174i
$358$		0			0
$359$	−20.9481			−1.10560			−0.552800	−	0.833314i	$-0.686441\pi$
$359$	−20.9481			−1.10560			−0.552800	+	0.833314i	$0.686441\pi$
$360$		0			0
$361$	30.9184			1.62728
$362$		0			0
$363$		−	0.339102i		−	0.0177982i
$364$		0			0
$365$	−9.01292	−	1.73541i	−0.471758	−	0.0908354i
$366$		0			0
$367$			13.4833i			0.703822i	0.936033	+	0.351911i	$0.114468\pi$
$367$			13.4833i			0.703822i	−0.936033	+	0.351911i	$0.885532\pi$
$368$		0			0
$369$	−22.4912			−1.17084
$370$		0			0
$371$	−20.4217			−1.06024
$372$		0			0
$373$			7.47823i			0.387208i	0.981080	+	0.193604i	$0.0620177\pi$
$373$			7.47823i			0.387208i	−0.981080	+	0.193604i	$0.937982\pi$
$374$		0			0
$375$	3.19053	+	2.04799i	0.164758	+	0.105758i
$376$		0			0
$377$			18.9223i			0.974548i
$378$		0			0
$379$	−7.33807			−0.376931			−0.188466	−	0.982080i	$-0.560351\pi$
$379$	−7.33807			−0.376931			−0.188466	+	0.982080i	$0.560351\pi$
$380$		0			0
$381$	3.31303			0.169732
$382$		0			0
$383$		−	20.2316i		−	1.03379i	−0.856050	−	0.516894i	$-0.827088\pi$
$383$		−	20.2316i		−	1.03379i	0.856050	−	0.516894i	$-0.172912\pi$
$384$		0			0
$385$	8.89793	+	1.71327i	0.453481	+	0.0873162i
$386$		0			0
$387$			32.0746i			1.63044i
$388$		0			0
$389$	0.406104			0.0205903			0.0102952	−	0.999947i	$-0.496723\pi$
$389$	0.406104			0.0205903			0.0102952	+	0.999947i	$0.496723\pi$
$390$		0			0
$391$	−21.1103			−1.06759
$392$		0			0
$393$			4.09872i			0.206753i
$394$		0			0
$395$	−5.87692	+	30.5221i	−0.295700	+	1.53573i
$396$		0			0
$397$			6.74833i			0.338689i	0.985557	+	0.169345i	$0.0541651\pi$
$397$			6.74833i			0.338689i	−0.985557	+	0.169345i	$0.945835\pi$
$398$		0			0
$399$	9.70889			0.486052
$400$		0			0
$401$	−21.1700			−1.05718			−0.528590	−	0.848877i	$-0.677279\pi$
$401$	−21.1700			−1.05718			−0.528590	+	0.848877i	$0.677279\pi$
$402$		0			0
$403$			0.878918i			0.0437820i
$404$		0			0
$405$	3.37308	−	17.5183i	0.167610	−	0.870490i
$406$		0			0
$407$		−	1.32618i		−	0.0657362i
$408$		0			0
$409$	−3.25167			−0.160785			−0.0803923	−	0.996763i	$-0.525617\pi$
$409$	−3.25167			−0.160785			−0.0803923	+	0.996763i	$0.525617\pi$
$410$		0			0
$411$	1.52467			0.0752066
$412$		0			0
$413$		−	44.3747i		−	2.18354i
$414$		0			0
$415$	7.97199	+	1.53498i	0.391330	+	0.0753492i
$416$		0			0
$417$		−	2.82632i		−	0.138405i
$418$		0			0
$419$	−0.0516929			−0.00252536			−0.00126268	−	0.999999i	$-0.500402\pi$
$419$	−0.0516929			−0.00252536			−0.00126268	+	0.999999i	$0.500402\pi$
$420$		0			0
$421$	3.25167			0.158477			0.0792383	−	0.996856i	$-0.474751\pi$
$421$	3.25167			0.158477			0.0792383	+	0.996856i	$0.474751\pi$
$422$		0			0
$423$			8.69231i			0.422635i
$424$		0			0
$425$	−14.3768	+	35.9493i	−0.697379	+	1.74380i
$426$		0			0
$427$			51.5890i			2.49657i
$428$		0			0
$429$	1.35641			0.0654880
$430$		0			0
$431$	8.28303			0.398979			0.199490	−	0.979900i	$-0.436072\pi$
$431$	8.28303			0.398979			0.199490	+	0.979900i	$0.436072\pi$
$432$		0			0
$433$		−	12.2263i		−	0.587557i	−0.955873	−	0.293779i	$-0.905087\pi$
$433$		−	12.2263i		−	0.587557i	0.955873	−	0.293779i	$-0.0949129\pi$
$434$		0			0
$435$	−3.52228	−	0.678204i	−0.168880	−	0.0325174i
$436$		0			0
$437$		−	19.2613i		−	0.921395i
$438$		0			0
$439$	13.6706			0.652463			0.326232	−	0.945290i	$-0.394221\pi$
$439$	13.6706			0.652463			0.326232	+	0.945290i	$0.394221\pi$
$440$		0			0
$441$	−27.1817			−1.29437
$442$		0			0
$443$		−	5.27381i		−	0.250566i	−0.992121	−	0.125283i	$-0.960016\pi$
$443$		−	5.27381i		−	0.250566i	0.992121	−	0.125283i	$-0.0399839\pi$
$444$		0			0
$445$	−4.17920	+	21.7049i	−0.198113	+	1.02891i
$446$		0			0
$447$		−	4.00602i		−	0.189478i
$448$		0			0
$449$	−10.9245			−0.515557			−0.257778	−	0.966204i	$-0.582990\pi$
$449$	−10.9245			−0.515557			−0.257778	+	0.966204i	$0.582990\pi$
$450$		0			0
$451$	−7.79587			−0.367093
$452$		0			0
$453$			5.34764i			0.251254i
$454$		0			0
$455$	−6.85307	+	35.5917i	−0.321277	+	1.66857i
$456$		0			0
$457$		−	35.4843i		−	1.65988i	−0.557850	−	0.829942i	$-0.688374\pi$
$457$		−	35.4843i		−	1.65988i	0.557850	−	0.829942i	$-0.311626\pi$
$458$		0			0
$459$	−15.4531			−0.721287
$460$		0			0
$461$	−37.1657			−1.73098			−0.865490	−	0.500927i	$-0.832993\pi$
$461$	−37.1657			−1.73098			−0.865490	+	0.500927i	$0.832993\pi$
$462$		0			0
$463$		−	0.0309055i		−	0.00143630i	−1.00000		0.000718151i	$-0.999771\pi$
$463$		−	0.0309055i		−	0.00143630i	1.00000		0.000718151i	$-0.000228595\pi$
$464$		0			0
$465$	−0.163606	−	0.0315018i	−0.00758704	−	0.00146086i
$466$		0			0
$467$		−	35.3219i		−	1.63450i	−0.576283	−	0.817250i	$-0.695497\pi$
$467$		−	35.3219i		−	1.63450i	0.576283	−	0.817250i	$-0.304503\pi$
$468$		0			0
$469$	19.0475			0.879533
$470$		0			0
$471$	0.0102506			0.000472324
$472$		0			0
$473$			11.1177i			0.511191i
$474$		0			0
$475$	−32.8007	−	13.1177i	−1.50500	−	0.601880i
$476$		0			0
$477$			14.5389i			0.665688i
$478$		0			0
$479$	35.0787			1.60279			0.801394	−	0.598137i	$-0.204092\pi$
$479$	35.0787			1.60279			0.801394	+	0.598137i	$0.204092\pi$
$480$		0			0
$481$	5.30471			0.241874
$482$		0			0
$483$		−	3.74624i		−	0.170460i
$484$		0			0
$485$	−27.4384	−	5.28317i	−1.24591	−	0.239896i
$486$		0			0
$487$		−	17.0663i		−	0.773346i	−0.922217	−	0.386673i	$-0.873624\pi$
$487$		−	17.0663i		−	0.773346i	0.922217	−	0.386673i	$-0.126376\pi$
$488$		0			0
$489$	−4.77395			−0.215885
$490$		0			0
$491$	15.0122			0.677493			0.338747	−	0.940878i	$-0.389997\pi$
$491$	15.0122			0.677493			0.338747	+	0.940878i	$0.389997\pi$
$492$		0			0
$493$		−	36.6312i		−	1.64979i
$494$		0			0
$495$	1.21973	−	6.33472i	0.0548228	−	0.284724i
$496$		0			0
$497$		−	10.2468i		−	0.459633i
$498$		0			0
$499$	25.4967			1.14139			0.570694	−	0.821163i	$-0.306674\pi$
$499$	25.4967			1.14139			0.570694	+	0.821163i	$0.306674\pi$
$500$		0			0
$501$	−4.31696			−0.192868
$502$		0			0
$503$			36.5788i			1.63097i	0.578779	+	0.815484i	$0.303529\pi$
$503$			36.5788i			1.63097i	−0.578779	+	0.815484i	$0.696471\pi$
$504$		0			0
$505$	−0.272099	+	1.41316i	−0.0121083	+	0.0628848i
$506$		0			0
$507$			1.01731i			0.0451801i
$508$		0			0
$509$	14.1327			0.626423			0.313212	−	0.949683i	$-0.398595\pi$
$509$	14.1327			0.626423			0.313212	+	0.949683i	$0.398595\pi$
$510$		0			0
$511$	16.6339			0.735841
$512$		0			0
$513$		−	14.0996i		−	0.622514i
$514$		0			0
$515$	9.59390	+	1.84727i	0.422758	+	0.0814006i
$516$		0			0
$517$			3.01292i			0.132508i
$518$		0			0
$519$	2.48283			0.108984
$520$		0			0
$521$	19.8551			0.869866			0.434933	−	0.900463i	$-0.356772\pi$
$521$	19.8551			0.869866			0.434933	+	0.900463i	$0.356772\pi$
$522$		0			0
$523$			7.79587i			0.340889i	0.985367	+	0.170445i	$0.0545204\pi$
$523$			7.79587i			0.340889i	−0.985367	+	0.170445i	$0.945480\pi$
$524$		0			0
$525$	−6.37958	−	2.55132i	−0.278428	−	0.111349i
$526$		0			0
$527$		−	1.70147i		−	0.0741174i
$528$		0			0
$529$	15.5679			0.676864
$530$		0			0
$531$	−31.5917			−1.37096
$532$		0			0
$533$		−	31.1835i		−	1.35071i
$534$		0			0
$535$	−19.7332	−	3.79957i	−0.853143	−	0.164270i
$536$		0			0
$537$			2.22537i			0.0960317i
$538$		0			0
$539$	−9.42170			−0.405821
$540$		0			0
$541$	4.21757			0.181327			0.0906637	−	0.995882i	$-0.471101\pi$
$541$	4.21757			0.181327			0.0906637	+	0.995882i	$0.471101\pi$
$542$		0			0
$543$			2.57510i			0.110508i
$544$		0			0
$545$	−7.69863	+	39.9832i	−0.329773	+	1.71269i
$546$		0			0
$547$			20.4136i			0.872823i	0.899747	+	0.436412i	$0.143751\pi$
$547$			20.4136i			0.872823i	−0.899747	+	0.436412i	$0.856249\pi$
$548$		0			0
$549$	36.7278			1.56751
$550$		0			0
$551$	33.4229			1.42386
$552$		0			0
$553$		−	56.3304i		−	2.39541i
$554$		0			0
$555$	−0.190129	+	0.987443i	−0.00807053	+	0.0419146i
$556$		0			0
$557$			15.4782i			0.655834i	0.944707	+	0.327917i	$0.106347\pi$
$557$			15.4782i			0.655834i	−0.944707	+	0.327917i	$0.893653\pi$
$558$		0			0
$559$	−44.4707			−1.88091
$560$		0			0
$561$	−2.62583			−0.110863
$562$		0			0
$563$		−	1.08305i		−	0.0456452i	−0.999740	−	0.0228226i	$-0.992735\pi$
$563$		−	1.08305i		−	0.0456452i	0.999740	−	0.0228226i	$-0.00726529\pi$
$564$		0			0
$565$	22.8737	+	4.40425i	0.962303	+	0.185288i
$566$		0			0
$567$			32.3311i			1.35778i
$568$		0			0
$569$	−19.0572			−0.798920			−0.399460	−	0.916751i	$-0.630802\pi$
$569$	−19.0572			−0.798920			−0.399460	+	0.916751i	$0.630802\pi$
$570$		0			0
$571$	23.9617			1.00277			0.501384	−	0.865225i	$-0.332824\pi$
$571$	23.9617			1.00277			0.501384	+	0.865225i	$0.332824\pi$
$572$		0			0
$573$			4.28673i			0.179081i
$574$		0			0
$575$	−5.06154	+	12.6564i	−0.211081	+	0.527807i
$576$		0			0
$577$			1.57392i			0.0655230i	0.999463	+	0.0327615i	$0.0104302\pi$
$577$			1.57392i			0.0655230i	−0.999463	+	0.0327615i	$0.989570\pi$
$578$		0			0
$579$	−1.49941			−0.0623132
$580$		0			0
$581$	−14.7128			−0.610390
$582$		0			0
$583$			5.03945i			0.208713i
$584$		0			0
$585$	25.3389	+	4.87892i	1.04763	+	0.201718i
$586$		0			0
$587$			23.7488i			0.980220i	0.871661	+	0.490110i	$0.163043\pi$
$587$			23.7488i			0.980220i	−0.871661	+	0.490110i	$0.836957\pi$
$588$		0			0
$589$	1.55245			0.0639677
$590$		0			0
$591$	−0.229980			−0.00946012
$592$		0			0
$593$			14.1306i			0.580274i	0.956985	+	0.290137i	$0.0937008\pi$
$593$			14.1306i			0.580274i	−0.956985	+	0.290137i	$0.906299\pi$
$594$		0			0
$595$	13.2667	−	68.9011i	0.543881	−	2.82467i
$596$		0			0
$597$			1.49941i			0.0613666i
$598$		0			0
$599$	−23.8395			−0.974054			−0.487027	−	0.873387i	$-0.661919\pi$
$599$	−23.8395			−0.974054			−0.487027	+	0.873387i	$0.661919\pi$
$600$		0			0
$601$	28.2353			1.15174			0.575871	−	0.817540i	$-0.304663\pi$
$601$	28.2353			1.15174			0.575871	+	0.817540i	$0.304663\pi$
$602$		0			0
$603$		−	13.5605i		−	0.552228i
$604$		0			0
$605$	0.422782	−	2.19574i	0.0171885	−	0.0892694i
$606$		0			0
$607$		−	29.6699i		−	1.20427i	−0.798396	−	0.602133i	$-0.794318\pi$
$607$		−	29.6699i		−	1.20427i	0.798396	−	0.602133i	$-0.205682\pi$
$608$		0			0
$609$	6.50059			0.263417
$610$		0			0
$611$	−12.0517			−0.487559
$612$		0			0
$613$			9.34023i			0.377248i	0.982049	+	0.188624i	$0.0604028\pi$
$613$			9.34023i			0.377248i	−0.982049	+	0.188624i	$0.939597\pi$
$614$		0			0
$615$	5.80463	+	1.11766i	0.234065	+	0.0450685i
$616$		0			0
$617$			17.7754i			0.715609i	0.933796	+	0.357805i	$0.116475\pi$
$617$			17.7754i			0.715609i	−0.933796	+	0.357805i	$0.883525\pi$
$618$		0			0
$619$	8.64143			0.347328			0.173664	−	0.984805i	$-0.444439\pi$
$619$	8.64143			0.347328			0.173664	+	0.984805i	$0.444439\pi$
$620$		0			0
$621$	−5.44044			−0.218317
$622$		0			0
$623$		−	40.0577i		−	1.60488i
$624$		0			0
$625$	18.1058	+	17.2389i	0.724233	+	0.689556i
$626$		0			0
$627$		−	2.39585i		−	0.0956812i
$628$		0			0
$629$	−10.2693			−0.409462
$630$		0			0
$631$	20.0414			0.797837			0.398919	−	0.916986i	$-0.369386\pi$
$631$	20.0414			0.797837			0.398919	+	0.916986i	$0.369386\pi$
$632$		0			0
$633$		−	3.78778i		−	0.150551i
$634$		0			0
$635$	21.4524	+	4.13059i	0.851312	+	0.163917i
$636$		0			0
$637$		−	37.6868i		−	1.49321i
$638$		0			0
$639$	−7.29504			−0.288587
$640$		0			0
$641$	48.0113			1.89633			0.948166	−	0.317777i	$-0.102936\pi$
$641$	48.0113			1.89633			0.948166	+	0.317777i	$0.102936\pi$
$642$		0			0
$643$			35.9396i			1.41732i	0.705550	+	0.708660i	$0.250700\pi$
$643$			35.9396i			1.41732i	−0.705550	+	0.708660i	$0.749300\pi$
$644$		0			0
$645$	1.59390	−	8.27797i	0.0627596	−	0.325945i
$646$		0			0
$647$		−	32.9172i		−	1.29411i	−0.762443	−	0.647055i	$-0.776000\pi$
$647$		−	32.9172i		−	1.29411i	0.762443	−	0.647055i	$-0.224000\pi$
$648$		0			0
$649$	−10.9503			−0.429837
$650$		0			0
$651$	0.301945			0.0118341
$652$		0			0
$653$		−	3.34461i		−	0.130885i	−0.997856	−	0.0654424i	$-0.979154\pi$
$653$		−	3.34461i		−	0.130885i	0.997856	−	0.0654424i	$-0.0208459\pi$
$654$		0			0
$655$	−5.11015	+	26.5398i	−0.199670	+	1.03700i
$656$		0			0
$657$		−	11.8422i		−	0.462009i
$658$		0			0
$659$	8.65168			0.337022			0.168511	−	0.985700i	$-0.446104\pi$
$659$	8.65168			0.337022			0.168511	+	0.985700i	$0.446104\pi$
$660$		0			0
$661$	−29.0550			−1.13011			−0.565055	−	0.825053i	$-0.691145\pi$
$661$	−29.0550			−1.13011			−0.565055	+	0.825053i	$0.691145\pi$
$662$		0			0
$663$		−	10.5033i		−	0.407916i
$664$		0			0
$665$	62.8665	+	12.1047i	2.43786	+	0.469402i
$666$		0			0
$667$		−	12.8964i		−	0.499352i
$668$		0			0
$669$	−6.59508			−0.254981
$670$		0			0
$671$	12.7306			0.491458
$672$		0			0
$673$			5.71765i			0.220399i	0.993909	+	0.110200i	$0.0351490\pi$
$673$			5.71765i			0.220399i	−0.993909	+	0.110200i	$0.964851\pi$
$674$		0			0
$675$	−3.70513	+	9.26468i	−0.142611	+	0.356598i
$676$		0			0
$677$			10.0701i			0.387026i	0.981098	+	0.193513i	$0.0619882\pi$
$677$			10.0701i			0.387026i	−0.981098	+	0.193513i	$0.938012\pi$
$678$		0			0
$679$	50.6393			1.94336
$680$		0			0
$681$	−7.66528			−0.293734
$682$		0			0
$683$			3.79120i			0.145066i	0.997366	+	0.0725331i	$0.0231083\pi$
$683$			3.79120i			0.145066i	−0.997366	+	0.0725331i	$0.976892\pi$
$684$		0			0
$685$	9.87249	+	1.90092i	0.377209	+	0.0726303i
$686$		0			0
$687$		−	5.78340i		−	0.220650i
$688$		0			0
$689$	−20.1578			−0.767950
$690$		0			0
$691$	27.5637			1.04857			0.524287	−	0.851542i	$-0.324332\pi$
$691$	27.5637			1.04857			0.524287	+	0.851542i	$0.324332\pi$
$692$		0			0
$693$			11.6911i			0.444109i
$694$		0			0
$695$	3.52377	−	18.3008i	0.133664	−	0.694191i
$696$		0			0
$697$			60.3673i			2.28657i
$698$		0			0
$699$	4.91718			0.185985
$700$		0			0
$701$	13.6830			0.516801			0.258401	−	0.966038i	$-0.416805\pi$
$701$	13.6830			0.516801			0.258401	+	0.966038i	$0.416805\pi$
$702$		0			0
$703$		−	9.36984i		−	0.353390i
$704$		0			0
$705$	0.431951	−	2.24336i	0.0162682	−	0.0844897i
$706$		0			0
$707$		−	2.60808i		−	0.0980868i
$708$		0			0
$709$	−14.3584			−0.539241			−0.269621	−	0.962967i	$-0.586898\pi$
$709$	−14.3584			−0.539241			−0.269621	+	0.962967i	$0.586898\pi$
$710$		0			0
$711$	−40.1034			−1.50399
$712$		0			0
$713$		−	0.599025i		−	0.0224336i
$714$		0			0
$715$	8.78294	+	1.69113i	0.328463	+	0.0632446i
$716$		0			0
$717$			4.71372i			0.176037i
$718$		0			0
$719$	−48.8332			−1.82117			−0.910585	−	0.413323i	$-0.864368\pi$
$719$	−48.8332			−1.82117			−0.910585	+	0.413323i	$0.864368\pi$
$720$		0			0
$721$	−17.7061			−0.659411
$722$		0			0
$723$		−	5.10593i		−	0.189891i
$724$		0			0
$725$	−21.9617	−	8.78294i	−0.815639	−	0.326190i
$726$		0			0
$727$			32.2566i			1.19633i	0.801373	+	0.598165i	$0.204103\pi$
$727$			32.2566i			1.19633i	−0.801373	+	0.598165i	$0.795897\pi$
$728$		0			0
$729$	20.9874			0.777310
$730$		0			0
$731$	86.0896			3.18414
$732$		0			0
$733$		−	18.8176i		−	0.695042i	−0.937672	−	0.347521i	$-0.887024\pi$
$733$		−	18.8176i		−	0.695042i	0.937672	−	0.347521i	$-0.112976\pi$
$734$		0			0
$735$	7.01519	+	1.35075i	0.258759	+	0.0498232i
$736$		0			0
$737$		−	4.70034i		−	0.173139i
$738$		0			0
$739$	14.3089			0.526360			0.263180	−	0.964747i	$-0.415229\pi$
$739$	14.3089			0.526360			0.263180	+	0.964747i	$0.415229\pi$
$740$		0			0
$741$	9.58342			0.352056
$742$		0			0
$743$			43.8448i			1.60851i	0.594284	+	0.804255i	$0.297435\pi$
$743$			43.8448i			1.60851i	−0.594284	+	0.804255i	$0.702565\pi$
$744$		0			0
$745$	4.99458	−	25.9396i	0.182987	−	0.950353i
$746$		0			0
$747$			10.4745i			0.383243i
$748$		0			0
$749$	36.4190			1.33072
$750$		0			0
$751$	23.0903			0.842578			0.421289	−	0.906926i	$-0.361578\pi$
$751$	23.0903			0.842578			0.421289	+	0.906926i	$0.361578\pi$
$752$		0			0
$753$		−	0.851432i		−	0.0310279i
$754$		0			0
$755$	−6.66728	+	34.6268i	−0.242647	+	1.26020i
$756$		0			0
$757$			19.0270i			0.691549i	0.938318	+	0.345775i	$0.112384\pi$
$757$			19.0270i			0.691549i	−0.938318	+	0.345775i	$0.887616\pi$
$758$		0			0
$759$	−0.924456			−0.0335556
$760$		0			0
$761$	−35.9537			−1.30332			−0.651659	−	0.758512i	$-0.725927\pi$
$761$	−35.9537			−1.30332			−0.651659	+	0.758512i	$0.725927\pi$
$762$		0			0
$763$		−	73.7915i		−	2.67143i
$764$		0			0
$765$	−49.0529	−	9.44497i	−1.77351	−	0.341484i
$766$		0			0
$767$		−	43.8012i		−	1.58157i
$768$		0			0
$769$	17.4665			0.629858			0.314929	−	0.949115i	$-0.398019\pi$
$769$	17.4665			0.629858			0.314929	+	0.949115i	$0.398019\pi$
$770$		0			0
$771$	−2.25285			−0.0811346
$772$		0			0
$773$			10.5361i			0.378958i	0.981885	+	0.189479i	$0.0606798\pi$
$773$			10.5361i			0.378958i	−0.981885	+	0.189479i	$0.939320\pi$
$774$		0			0
$775$	−1.02010	−	0.407957i	−0.0366430	−	0.0146543i
$776$		0			0
$777$		−	1.82239i		−	0.0653779i
$778$		0			0
$779$	−55.0801			−1.97345
$780$		0			0
$781$	−2.52860			−0.0904805
$782$		0			0
$783$		−	9.44042i		−	0.337373i
$784$		0			0
$785$	0.0663744	+	0.0127802i	0.00236900	+	0.000456144i
$786$		0			0
$787$		−	37.8748i		−	1.35009i	−0.737777	−	0.675045i	$-0.764124\pi$
$787$		−	37.8748i		−	1.35009i	0.737777	−	0.675045i	$-0.235876\pi$
$788$		0			0
$789$	4.54694			0.161875
$790$		0			0
$791$	−42.2148			−1.50099
$792$		0			0
$793$			50.9223i			1.80830i
$794$		0			0
$795$	0.722486	−	3.75226i	0.0256239	−	0.133079i
$796$		0			0
$797$		−	29.9574i		−	1.06114i	−0.847640	−	0.530572i	$-0.821977\pi$
$797$		−	29.9574i		−	1.06114i	0.847640	−	0.530572i	$-0.178023\pi$
$798$		0			0
$799$	23.3306			0.825376
$800$		0			0
$801$	−28.5184			−1.00765
$802$		0			0
$803$		−	4.10474i		−	0.144853i
$804$		0			0
$805$	4.67070	−	24.2575i	0.164620	−	0.854963i
$806$		0			0
$807$		−	6.59931i		−	0.232307i
$808$		0			0
$809$	2.59589			0.0912667			0.0456333	−	0.998958i	$-0.485469\pi$
$809$	2.59589			0.0912667			0.0456333	+	0.998958i	$0.485469\pi$
$810$		0			0
$811$	39.2012			1.37654			0.688271	−	0.725454i	$-0.258370\pi$
$811$	39.2012			1.37654			0.688271	+	0.725454i	$0.258370\pi$
$812$		0			0
$813$		−	0.0179882i		−	0.000630873i
$814$		0			0
$815$	−30.9120	−	5.95201i	−1.08280	−	0.208490i
$816$		0			0
$817$			78.5495i			2.74810i
$818$		0			0
$819$	−46.7645			−1.63408
$820$		0			0
$821$	−35.7578			−1.24796			−0.623979	−	0.781441i	$-0.714485\pi$
$821$	−35.7578			−1.24796			−0.623979	+	0.781441i	$0.714485\pi$
$822$		0			0
$823$		−	17.3093i		−	0.603365i	−0.953408	−	0.301683i	$-0.902452\pi$
$823$		−	17.3093i		−	0.603365i	0.953408	−	0.301683i	$-0.0975483\pi$
$824$		0			0
$825$	−0.629588	+	1.57428i	−0.0219194	+	0.0548095i
$826$		0			0
$827$			25.9957i			0.903958i	0.892029	+	0.451979i	$0.149282\pi$
$827$			25.9957i			0.903958i	−0.892029	+	0.451979i	$0.850718\pi$
$828$		0			0
$829$	46.6902			1.62162			0.810808	−	0.585312i	$-0.199028\pi$
$829$	46.6902			1.62162			0.810808	+	0.585312i	$0.199028\pi$
$830$		0			0
$831$	−1.38895			−0.0481822
$832$		0			0
$833$			72.9569i			2.52781i
$834$		0			0
$835$	−27.9530	−	5.38225i	−0.967353	−	0.186261i
$836$		0			0
$837$		−	0.438496i		−	0.0151566i
$838$		0			0
$839$	9.45364			0.326376			0.163188	−	0.986595i	$-0.447822\pi$
$839$	9.45364			0.326376			0.163188	+	0.986595i	$0.447822\pi$
$840$		0			0
$841$	−6.62168			−0.228334
$842$		0			0
$843$		−	5.94786i		−	0.204855i
$844$		0			0
$845$	−1.26835	+	6.58721i	−0.0436324	+	0.226607i
$846$		0			0
$847$			4.05237i			0.139241i
$848$		0			0
$849$	5.12940			0.176041
$850$		0			0
$851$	−3.61542			−0.123935
$852$		0			0
$853$			21.5733i			0.738656i	0.929299	+	0.369328i	$0.120412\pi$
$853$			21.5733i			0.738656i	−0.929299	+	0.369328i	$0.879588\pi$
$854$		0			0
$855$	8.61775	−	44.7566i	0.294721	−	1.53065i
$856$		0			0
$857$		−	19.5783i		−	0.668782i	−0.942434	−	0.334391i	$-0.891469\pi$
$857$		−	19.5783i		−	0.668782i	0.942434	−	0.334391i	$-0.108531\pi$
$858$		0			0
$859$	28.3598			0.967622			0.483811	−	0.875172i	$-0.339252\pi$
$859$	28.3598			0.967622			0.483811	+	0.875172i	$0.339252\pi$
$860$		0			0
$861$	−10.7128			−0.365092
$862$		0			0
$863$		−	9.05630i		−	0.308280i	−0.988049	−	0.154140i	$-0.950739\pi$
$863$		−	9.05630i		−	0.308280i	0.988049	−	0.154140i	$-0.0492607\pi$
$864$		0			0
$865$	16.0767	+	3.09552i	0.546625	+	0.105251i
$866$		0			0
$867$			14.5684i			0.494769i
$868$		0			0
$869$	−13.9006			−0.471546
$870$		0			0
$871$	18.8014			0.637060
$872$		0			0
$873$		−	36.0517i		−	1.22016i
$874$		0			0
$875$	−38.1278	−	24.4741i	−1.28896	−	0.827375i
$876$		0			0
$877$		−	53.2624i		−	1.79854i	−0.437392	−	0.899271i	$-0.644098\pi$
$877$		−	53.2624i		−	1.79854i	0.437392	−	0.899271i	$-0.355902\pi$
$878$		0			0
$879$	−4.52918			−0.152766
$880$		0			0
$881$	33.7557			1.13726			0.568629	−	0.822594i	$-0.307474\pi$
$881$	33.7557			1.13726			0.568629	+	0.822594i	$0.307474\pi$
$882$		0			0
$883$		−	0.157109i		−	0.00528714i	−0.999997	−	0.00264357i	$-0.999159\pi$
$883$		−	0.157109i		−	0.00528714i	0.999997	−	0.00264357i	$-0.000841476\pi$
$884$		0			0
$885$	8.15336	+	1.56990i	0.274072	+	0.0527717i
$886$		0			0
$887$			8.63050i			0.289784i	0.989447	+	0.144892i	$0.0462835\pi$
$887$			8.63050i			0.289784i	−0.989447	+	0.144892i	$0.953717\pi$
$888$		0			0
$889$	−39.5917			−1.32786
$890$		0			0
$891$	7.97831			0.267284
$892$		0			0
$893$			21.2872i			0.712348i
$894$		0			0
$895$	−2.77452	+	14.4096i	−0.0927420	+	0.481660i
$896$		0			0
$897$		−	3.69783i		−	0.123467i
$898$		0			0
$899$	1.03945			0.0346675
$900$		0			0
$901$	39.0229			1.30004
$902$		0			0
$903$			15.2775i			0.508404i
$904$		0			0
$905$	−3.21056	+	16.6742i	−0.106723	+	0.554269i
$906$		0			0
$907$			6.69596i			0.222336i	0.993802	+	0.111168i	$0.0354591\pi$
$907$			6.69596i			0.222336i	−0.993802	+	0.111168i	$0.964541\pi$
$908$		0			0
$909$	−1.85677			−0.0615852
$910$		0			0
$911$	−29.9345			−0.991776			−0.495888	−	0.868387i	$-0.665157\pi$
$911$	−29.9345			−0.991776			−0.495888	+	0.868387i	$0.665157\pi$
$912$		0			0
$913$			3.63067i			0.120158i
$914$		0			0
$915$	−9.47891	−	1.82513i	−0.313363	−	0.0603370i
$916$		0			0
$917$		−	48.9809i		−	1.61749i
$918$		0			0
$919$	−35.3618			−1.16648			−0.583238	−	0.812301i	$-0.698215\pi$
$919$	−35.3618			−1.16648			−0.583238	+	0.812301i	$0.698215\pi$
$920$		0			0
$921$	5.35641			0.176500
$922$		0			0
$923$		−	10.1144i		−	0.332920i
$924$		0			0
$925$	−2.46223	+	6.15680i	−0.0809576	+	0.202434i
$926$		0			0
$927$			12.6056i			0.414021i
$928$		0			0
$929$	27.2134			0.892843			0.446421	−	0.894823i	$-0.352698\pi$
$929$	27.2134			0.892843			0.446421	+	0.894823i	$0.352698\pi$
$930$		0			0
$931$	−66.5671			−2.18165
$932$		0			0
$933$			0.281444i			0.00921405i
$934$		0			0
$935$	−17.0027	−	3.27381i	−0.556047	−	0.107065i
$936$		0			0
$937$			16.3659i			0.534651i	0.963606	+	0.267326i	$0.0861399\pi$
$937$			16.3659i			0.534651i	−0.963606	+	0.267326i	$0.913860\pi$
$938$		0			0
$939$	−8.30968			−0.271176
$940$		0			0
$941$	−18.0134			−0.587221			−0.293611	−	0.955925i	$-0.594857\pi$
$941$	−18.0134			−0.587221			−0.293611	+	0.955925i	$0.594857\pi$
$942$		0			0
$943$			21.2530i			0.692094i
$944$		0			0
$945$	3.41903	−	17.7569i	0.111221	−	0.577631i
$946$		0			0
$947$		−	47.9751i		−	1.55898i	−0.626414	−	0.779491i	$-0.715478\pi$
$947$		−	47.9751i		−	1.55898i	0.626414	−	0.779491i	$-0.284522\pi$
$948$		0			0
$949$	16.4190			0.532982
$950$		0			0
$951$	3.88798			0.126076
$952$		0			0
$953$			40.7788i			1.32096i	0.750845	+	0.660478i	$0.229646\pi$
$953$			40.7788i			1.32096i	−0.750845	+	0.660478i	$0.770354\pi$
$954$		0			0
$955$	−5.34457	+	27.7572i	−0.172946	+	0.898203i
$956$		0			0
$957$		−	1.60415i		−	0.0518547i
$958$		0			0
$959$	−18.2203			−0.588364
$960$		0			0
$961$	−30.9517			−0.998443
$962$		0			0
$963$		−	25.9278i		−	0.835512i
$964$		0			0
$965$	−9.70889	−	1.86941i	−0.312540	−	0.0601786i
$966$		0			0
$967$		−	0.835313i		−	0.0268618i	−0.999910	−	0.0134309i	$-0.995725\pi$
$967$		−	0.835313i		−	0.0268618i	0.999910	−	0.0134309i	$-0.00427532\pi$
$968$		0			0
$969$	−18.5523			−0.595985
$970$		0			0
$971$	27.3542			0.877839			0.438920	−	0.898526i	$-0.355361\pi$
$971$	27.3542			0.877839			0.438920	+	0.898526i	$0.355361\pi$
$972$		0			0
$973$			33.7754i			1.08279i
$974$		0			0
$975$	−6.29714	−	2.51835i	−0.201670	−	0.0806518i
$976$		0			0
$977$			4.40765i			0.141013i	0.997511	+	0.0705066i	$0.0224616\pi$
$977$			4.40765i			0.141013i	−0.997511	+	0.0705066i	$0.977538\pi$
$978$		0			0
$979$	−9.88501			−0.315926
$980$		0			0
$981$	−52.5345			−1.67730
$982$		0			0
$983$			28.3968i			0.905718i	0.891582	+	0.452859i	$0.149596\pi$
$983$			28.3968i			0.905718i	−0.891582	+	0.452859i	$0.850404\pi$
$984$		0			0
$985$	−1.48916	−	0.286732i	−0.0474485	−	0.00913605i
$986$		0			0
$987$			4.14026i			0.131786i
$988$		0			0
$989$	30.3089			0.963766
$990$		0			0
$991$	−6.76451			−0.214882			−0.107441	−	0.994211i	$-0.534266\pi$
$991$	−6.76451			−0.214882			−0.107441	+	0.994211i	$0.534266\pi$
$992$		0			0
$993$			8.27797i			0.262693i
$994$		0			0
$995$	−1.86941	+	9.70889i	−0.0592644	+	0.307792i
$996$		0			0
$997$			33.5747i			1.06332i	0.846958	+	0.531660i	$0.178432\pi$
$997$			33.5747i			1.06332i	−0.846958	+	0.531660i	$0.821568\pi$
$998$		0			0
$999$	−2.64655			−0.0837330

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	440.2.b.d.89.4	✓	8
3.2	odd	2		3960.2.d.f.3169.4		8
4.3	odd	2		880.2.b.j.529.5		8
5.2	odd	4		2200.2.a.y.1.2		4
5.3	odd	4		2200.2.a.x.1.3		4
5.4	even	2	inner	440.2.b.d.89.5	yes	8
15.14	odd	2		3960.2.d.f.3169.3		8
20.3	even	4		4400.2.a.ce.1.2		4
20.7	even	4		4400.2.a.cb.1.3		4
20.19	odd	2		880.2.b.j.529.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.b.d.89.4	✓	8	1.1	even	1	trivial
440.2.b.d.89.5	yes	8	5.4	even	2	inner
880.2.b.j.529.4		8	20.19	odd	2
880.2.b.j.529.5		8	4.3	odd	2
2200.2.a.x.1.3		4	5.3	odd	4
2200.2.a.y.1.2		4	5.2	odd	4
3960.2.d.f.3169.3		8	15.14	odd	2
3960.2.d.f.3169.4		8	3.2	odd	2
4400.2.a.cb.1.3		4	20.7	even	4
4400.2.a.ce.1.2		4	20.3	even	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:	\( N \)	\(=\)	\( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	440.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.339102i q^{3} +(0.422782 - 2.19574i) q^{5} +4.05237i q^{7} +2.88501 q^{9} +O(q^{10})\)	\(q-0.339102i q^{3} +(0.422782 - 2.19574i) q^{5} +4.05237i q^{7} +2.88501 q^{9} +1.00000 q^{11} +4.00000i q^{13} +(-0.744578 - 0.143366i) q^{15} -7.74350i q^{17} +7.06529 q^{19} +1.37417 q^{21} -2.72619i q^{23} +(-4.64251 - 1.85663i) q^{25} -1.99562i q^{27} +4.73057 q^{29} +0.219729 q^{31} -0.339102i q^{33} +(8.89793 + 1.71327i) q^{35} -1.32618i q^{37} +1.35641 q^{39} -7.79587 q^{41} +11.1177i q^{43} +(1.21973 - 6.33472i) q^{45} +3.01292i q^{47} -9.42170 q^{49} -2.62583 q^{51} +5.03945i q^{53} +(0.422782 - 2.19574i) q^{55} -2.39585i q^{57} -10.9503 q^{59} +12.7306 q^{61} +11.6911i q^{63} +(8.78294 + 1.69113i) q^{65} -4.70034i q^{67} -0.924456 q^{69} -2.52860 q^{71} -4.10474i q^{73} +(-0.629588 + 1.57428i) q^{75} +4.05237i q^{77} -13.9006 q^{79} +7.97831 q^{81} +3.63067i q^{83} +(-17.0027 - 3.27381i) q^{85} -1.60415i q^{87} -9.88501 q^{89} -16.2095 q^{91} -0.0745107i q^{93} +(2.98708 - 15.5135i) q^{95} -12.4962i q^{97} +2.88501 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 14 q^{9}+O(q^{10}) \) 8 * q - 14 * q^9	\( 8 q - 14 q^{9} + 8 q^{11} - 12 q^{15} + 18 q^{19} - 14 q^{21} - 2 q^{25} - 6 q^{29} - 30 q^{31} + 30 q^{35} - 8 q^{39} + 20 q^{41} - 22 q^{45} - 18 q^{49} - 46 q^{51} - 12 q^{59} + 58 q^{61} - 8 q^{65} + 60 q^{69} - 2 q^{71} + 26 q^{75} + 40 q^{79} + 88 q^{81} - 26 q^{85} - 42 q^{89} + 8 q^{91} + 28 q^{95} - 14 q^{99}+O(q^{100}) \) 8 * q - 14 * q^9 + 8 * q^11 - 12 * q^15 + 18 * q^19 - 14 * q^21 - 2 * q^25 - 6 * q^29 - 30 * q^31 + 30 * q^35 - 8 * q^39 + 20 * q^41 - 22 * q^45 - 18 * q^49 - 46 * q^51 - 12 * q^59 + 58 * q^61 - 8 * q^65 + 60 * q^69 - 2 * q^71 + 26 * q^75 + 40 * q^79 + 88 * q^81 - 26 * q^85 - 42 * q^89 + 8 * q^91 + 28 * q^95 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more