LMFDB - Embedded newform 1452.3.e.d.485.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1452,3,Mod(485,1452)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1452, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1452.485"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1452 = 2^{2} \cdot 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1452.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,5,0,0,0,-4,0,7,0,0,0,0,0,-11,0,0,0,-8,0,-10,0,0,0,28,0,-10, 0,0,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	no
Analytic conductor:	$39.5641343851$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-11}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 3 $ x^2 - x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 132)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			485.1
Root			$0.500000 + 1.65831i$ of defining polynomial
Character	$\chi$	$=$	1452.485
Dual form			1452.3.e.d.485.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50000 - 1.65831i) q^{3} -3.31662i q^{5} -2.00000 q^{7} +(3.50000 - 8.29156i) q^{9} +(-5.50000 - 8.29156i) q^{15} -6.63325i q^{17} -4.00000 q^{19} +(-5.00000 + 3.31662i) q^{21} +3.31662i q^{23} +14.0000 q^{25} +(-5.00000 - 26.5330i) q^{27} -39.7995i q^{29} +9.00000 q^{31} +6.63325i q^{35} -51.0000 q^{37} -79.5990i q^{41} -22.0000 q^{43} +(-27.5000 - 11.6082i) q^{45} +39.7995i q^{47} -45.0000 q^{49} +(-11.0000 - 16.5831i) q^{51} +66.3325i q^{53} +(-10.0000 + 6.63325i) q^{57} -69.6491i q^{59} +80.0000 q^{61} +(-7.00000 + 16.5831i) q^{63} +1.00000 q^{67} +(5.50000 + 8.29156i) q^{69} -49.7494i q^{71} -108.000 q^{73} +(35.0000 - 23.2164i) q^{75} -122.000 q^{79} +(-56.5000 - 58.0409i) q^{81} +86.2322i q^{83} -22.0000 q^{85} +(-66.0000 - 99.4987i) q^{87} -96.1821i q^{89} +(22.5000 - 14.9248i) q^{93} +13.2665i q^{95} +79.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5 q^{3} - 4 q^{7} + 7 q^{9} - 11 q^{15} - 8 q^{19} - 10 q^{21} + 28 q^{25} - 10 q^{27} + 18 q^{31} - 102 q^{37} - 44 q^{43} - 55 q^{45} - 90 q^{49} - 22 q^{51} - 20 q^{57} + 160 q^{61} - 14 q^{63}+ \cdots + 158 q^{97}+O(q^{100}) $ 2 * q + 5 * q^3 - 4 * q^7 + 7 * q^9 - 11 * q^15 - 8 * q^19 - 10 * q^21 + 28 * q^25 - 10 * q^27 + 18 * q^31 - 102 * q^37 - 44 * q^43 - 55 * q^45 - 90 * q^49 - 22 * q^51 - 20 * q^57 + 160 * q^61 - 14 * q^63 + 2 * q^67 + 11 * q^69 - 216 * q^73 + 70 * q^75 - 244 * q^79 - 113 * q^81 - 44 * q^85 - 132 * q^87 + 45 * q^93 + 158 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1452\mathbb{Z}\right)^\times$.

$n$	$485$	$727$	$1333$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	2.50000	−	1.65831i	0.833333	−	0.552771i
$3$	2.50000	−	1.65831i	0.833333	−	0.552771i
$4$		0			0
$5$		−	3.31662i		−	0.663325i	−0.943398	−	0.331662i	$-0.892390\pi$
$5$		−	3.31662i		−	0.663325i	0.943398	−	0.331662i	$-0.107610\pi$
$6$		0			0
$7$	−2.00000			−0.285714			−0.142857	−	0.989743i	$-0.545629\pi$
$7$	−2.00000			−0.285714			−0.142857	+	0.989743i	$0.545629\pi$
$8$		0			0
$9$	3.50000	−	8.29156i	0.388889	−	0.921285i
$10$		0			0
$11$		0			0
$11$		0			0
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	−5.50000	−	8.29156i	−0.366667	−	0.552771i
$16$		0			0
$17$		−	6.63325i		−	0.390191i	−0.980784	−	0.195096i	$-0.937498\pi$
$17$		−	6.63325i		−	0.390191i	0.980784	−	0.195096i	$-0.0625017\pi$
$18$		0			0
$19$	−4.00000			−0.210526			−0.105263	−	0.994444i	$-0.533568\pi$
$19$	−4.00000			−0.210526			−0.105263	+	0.994444i	$0.533568\pi$
$20$		0			0
$21$	−5.00000	+	3.31662i	−0.238095	+	0.157935i
$22$		0			0
$23$			3.31662i			0.144201i	0.997397	+	0.0721005i	$0.0229702\pi$
$23$			3.31662i			0.144201i	−0.997397	+	0.0721005i	$0.977030\pi$
$24$		0			0
$25$	14.0000			0.560000
$26$		0			0
$27$	−5.00000	−	26.5330i	−0.185185	−	0.982704i
$28$		0			0
$29$		−	39.7995i		−	1.37240i	−0.727415	−	0.686198i	$-0.759278\pi$
$29$		−	39.7995i		−	1.37240i	0.727415	−	0.686198i	$-0.240722\pi$
$30$		0			0
$31$	9.00000			0.290323			0.145161	−	0.989408i	$-0.453630\pi$
$31$	9.00000			0.290323			0.145161	+	0.989408i	$0.453630\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			6.63325i			0.189521i
$36$		0			0
$37$	−51.0000			−1.37838			−0.689189	−	0.724581i	$-0.742033\pi$
$37$	−51.0000			−1.37838			−0.689189	+	0.724581i	$0.742033\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	79.5990i		−	1.94144i	−0.240216	−	0.970719i	$-0.577218\pi$
$41$		−	79.5990i		−	1.94144i	0.240216	−	0.970719i	$-0.422782\pi$
$42$		0			0
$43$	−22.0000			−0.511628			−0.255814	−	0.966726i	$-0.582343\pi$
$43$	−22.0000			−0.511628			−0.255814	+	0.966726i	$0.582343\pi$
$44$		0			0
$45$	−27.5000	−	11.6082i	−0.611111	−	0.257960i
$46$		0			0
$47$			39.7995i			0.846798i	0.905943	+	0.423399i	$0.139163\pi$
$47$			39.7995i			0.846798i	−0.905943	+	0.423399i	$0.860837\pi$
$48$		0			0
$49$	−45.0000			−0.918367
$50$		0			0
$51$	−11.0000	−	16.5831i	−0.215686	−	0.325159i
$52$		0			0
$53$			66.3325i			1.25156i	0.780001	+	0.625778i	$0.215219\pi$
$53$			66.3325i			1.25156i	−0.780001	+	0.625778i	$0.784781\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−10.0000	+	6.63325i	−0.175439	+	0.116373i
$58$		0			0
$59$		−	69.6491i		−	1.18049i	−0.807223	−	0.590247i	$-0.799030\pi$
$59$		−	69.6491i		−	1.18049i	0.807223	−	0.590247i	$-0.200970\pi$
$60$		0			0
$61$	80.0000			1.31148			0.655738	−	0.754989i	$-0.272358\pi$
$61$	80.0000			1.31148			0.655738	+	0.754989i	$0.272358\pi$
$62$		0			0
$63$	−7.00000	+	16.5831i	−0.111111	+	0.263224i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	1.00000			0.0149254			0.00746269	−	0.999972i	$-0.497625\pi$
$67$	1.00000			0.0149254			0.00746269	+	0.999972i	$0.497625\pi$
$68$		0			0
$69$	5.50000	+	8.29156i	0.0797101	+	0.120168i
$70$		0			0
$71$		−	49.7494i		−	0.700695i	−0.936620	−	0.350348i	$-0.886063\pi$
$71$		−	49.7494i		−	0.700695i	0.936620	−	0.350348i	$-0.113937\pi$
$72$		0			0
$73$	−108.000			−1.47945			−0.739726	−	0.672908i	$-0.765045\pi$
$73$	−108.000			−1.47945			−0.739726	+	0.672908i	$0.765045\pi$
$74$		0			0
$75$	35.0000	−	23.2164i	0.466667	−	0.309552i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−122.000			−1.54430			−0.772152	−	0.635438i	$-0.780820\pi$
$79$	−122.000			−1.54430			−0.772152	+	0.635438i	$0.780820\pi$
$80$		0			0
$81$	−56.5000	−	58.0409i	−0.697531	−	0.716555i
$82$		0			0
$83$			86.2322i			1.03894i	0.854488	+	0.519471i	$0.173871\pi$
$83$			86.2322i			1.03894i	−0.854488	+	0.519471i	$0.826129\pi$
$84$		0			0
$85$	−22.0000			−0.258824
$86$		0			0
$87$	−66.0000	−	99.4987i	−0.758621	−	1.14366i
$88$		0			0
$89$		−	96.1821i		−	1.08070i	−0.841441	−	0.540349i	$-0.818292\pi$
$89$		−	96.1821i		−	1.08070i	0.841441	−	0.540349i	$-0.181708\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	22.5000	−	14.9248i	0.241935	−	0.160482i
$94$		0			0
$95$			13.2665i			0.139647i
$96$		0			0
$97$	79.0000			0.814433			0.407216	−	0.913332i	$-0.366499\pi$
$97$	79.0000			0.814433			0.407216	+	0.913332i	$0.366499\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	46.4327i		−	0.459730i	−0.973223	−	0.229865i	$-0.926172\pi$
$101$		−	46.4327i		−	0.459730i	0.973223	−	0.229865i	$-0.0738285\pi$
$102$		0			0
$103$	−114.000			−1.10680			−0.553398	−	0.832917i	$-0.686669\pi$
$103$	−114.000			−1.10680			−0.553398	+	0.832917i	$0.686669\pi$
$104$		0			0
$105$	11.0000	+	16.5831i	0.104762	+	0.157935i
$106$		0			0
$107$			165.831i			1.54982i	0.632069	+	0.774912i	$0.282206\pi$
$107$			165.831i			1.54982i	−0.632069	+	0.774912i	$0.717794\pi$
$108$		0			0
$109$	−6.00000			−0.0550459			−0.0275229	−	0.999621i	$-0.508762\pi$
$109$	−6.00000			−0.0550459			−0.0275229	+	0.999621i	$0.508762\pi$
$110$		0			0
$111$	−127.500	+	84.5739i	−1.14865	+	0.761927i
$112$		0			0
$113$		−	3.31662i		−	0.0293507i	−0.999892	−	0.0146753i	$-0.995329\pi$
$113$		−	3.31662i		−	0.0293507i	0.999892	−	0.0146753i	$-0.00467147\pi$
$114$		0			0
$115$	11.0000			0.0956522
$116$		0			0
$117$		0			0
$118$		0			0
$119$			13.2665i			0.111483i
$120$		0			0
$121$		0			0
$122$		0			0
$123$	−132.000	−	198.997i	−1.07317	−	1.61787i
$124$		0			0
$125$		−	129.348i		−	1.03479i
$126$		0			0
$127$	228.000			1.79528			0.897638	−	0.440734i	$-0.145282\pi$
$127$	228.000			1.79528			0.897638	+	0.440734i	$0.145282\pi$
$128$		0			0
$129$	−55.0000	+	36.4829i	−0.426357	+	0.282813i
$130$		0			0
$131$		−	72.9657i		−	0.556990i	−0.960438	−	0.278495i	$-0.910164\pi$
$131$		−	72.9657i		−	0.556990i	0.960438	−	0.278495i	$-0.0898356\pi$
$132$		0			0
$133$	8.00000			0.0601504
$134$		0			0
$135$	−88.0000	+	16.5831i	−0.651852	+	0.122838i
$136$		0			0
$137$			135.982i			0.992567i	0.868161	+	0.496283i	$0.165302\pi$
$137$			135.982i			0.992567i	−0.868161	+	0.496283i	$0.834698\pi$
$138$		0			0
$139$	−110.000			−0.791367			−0.395683	−	0.918387i	$-0.629492\pi$
$139$	−110.000			−0.791367			−0.395683	+	0.918387i	$0.629492\pi$
$140$		0			0
$141$	66.0000	+	99.4987i	0.468085	+	0.705665i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−132.000			−0.910345
$146$		0			0
$147$	−112.500	+	74.6241i	−0.765306	+	0.507647i
$148$		0			0
$149$			192.364i			1.29104i	0.763745	+	0.645518i	$0.223358\pi$
$149$			192.364i			1.29104i	−0.763745	+	0.645518i	$0.776642\pi$
$150$		0			0
$151$	242.000			1.60265			0.801325	−	0.598230i	$-0.204129\pi$
$151$	242.000			1.60265			0.801325	+	0.598230i	$0.204129\pi$
$152$		0			0
$153$	−55.0000	−	23.2164i	−0.359477	−	0.151741i
$154$		0			0
$155$		−	29.8496i		−	0.192578i
$156$		0			0
$157$	113.000			0.719745			0.359873	−	0.933001i	$-0.382820\pi$
$157$	113.000			0.719745			0.359873	+	0.933001i	$0.382820\pi$
$158$		0			0
$159$	110.000	+	165.831i	0.691824	+	1.04296i
$160$		0			0
$161$		−	6.63325i		−	0.0412003i
$162$		0			0
$163$	46.0000			0.282209			0.141104	−	0.989995i	$-0.454935\pi$
$163$	46.0000			0.282209			0.141104	+	0.989995i	$0.454935\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	278.596i		−	1.66824i	−0.551581	−	0.834121i	$-0.685975\pi$
$167$		−	278.596i		−	1.66824i	0.551581	−	0.834121i	$-0.314025\pi$
$168$		0			0
$169$	−169.000			−1.00000
$170$		0			0
$171$	−14.0000	+	33.1662i	−0.0818713	+	0.193955i
$172$		0			0
$173$		−	86.2322i		−	0.498452i	−0.968445	−	0.249226i	$-0.919824\pi$
$173$		−	86.2322i		−	0.498452i	0.968445	−	0.249226i	$-0.0801762\pi$
$174$		0			0
$175$	−28.0000			−0.160000
$176$		0			0
$177$	−115.500	−	174.123i	−0.652542	−	0.983745i
$178$		0			0
$179$		−	222.214i		−	1.24142i	−0.784041	−	0.620709i	$-0.786845\pi$
$179$		−	222.214i		−	1.24142i	0.784041	−	0.620709i	$-0.213155\pi$
$180$		0			0
$181$	−137.000			−0.756906			−0.378453	−	0.925620i	$-0.623544\pi$
$181$	−137.000			−0.756906			−0.378453	+	0.925620i	$0.623544\pi$
$182$		0			0
$183$	200.000	−	132.665i	1.09290	−	0.724945i
$184$		0			0
$185$			169.148i			0.914313i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	10.0000	+	53.0660i	0.0529101	+	0.280772i
$190$		0			0
$191$			29.8496i			0.156281i	0.996942	+	0.0781404i	$0.0248982\pi$
$191$			29.8496i			0.156281i	−0.996942	+	0.0781404i	$0.975102\pi$
$192$		0			0
$193$	272.000			1.40933			0.704663	−	0.709542i	$-0.251098\pi$
$193$	272.000			1.40933			0.704663	+	0.709542i	$0.251098\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			126.032i			0.639755i	0.947459	+	0.319878i	$0.103642\pi$
$197$			126.032i			0.639755i	−0.947459	+	0.319878i	$0.896358\pi$
$198$		0			0
$199$	122.000			0.613065			0.306533	−	0.951860i	$-0.400831\pi$
$199$	122.000			0.613065			0.306533	+	0.951860i	$0.400831\pi$
$200$		0			0
$201$	2.50000	−	1.65831i	0.0124378	−	0.00825031i
$202$		0			0
$203$			79.5990i			0.392113i
$204$		0			0
$205$	−264.000			−1.28780
$206$		0			0
$207$	27.5000	+	11.6082i	0.132850	+	0.0560782i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−164.000			−0.777251			−0.388626	−	0.921396i	$-0.627050\pi$
$211$	−164.000			−0.777251			−0.388626	+	0.921396i	$0.627050\pi$
$212$		0			0
$213$	−82.5000	−	124.373i	−0.387324	−	0.583913i
$214$		0			0
$215$			72.9657i			0.339376i
$216$		0			0
$217$	−18.0000			−0.0829493
$218$		0			0
$219$	−270.000	+	179.098i	−1.23288	+	0.817798i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	63.0000			0.282511			0.141256	−	0.989973i	$-0.454886\pi$
$223$	63.0000			0.282511			0.141256	+	0.989973i	$0.454886\pi$
$224$		0			0
$225$	49.0000	−	116.082i	0.217778	−	0.515919i
$226$		0			0
$227$		−	59.6992i		−	0.262992i	−0.991317	−	0.131496i	$-0.958022\pi$
$227$		−	59.6992i		−	0.262992i	0.991317	−	0.131496i	$-0.0419781\pi$
$228$		0			0
$229$	125.000			0.545852			0.272926	−	0.962035i	$-0.412009\pi$
$229$	125.000			0.545852			0.272926	+	0.962035i	$0.412009\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			66.3325i			0.284689i	0.989817	+	0.142344i	$0.0454641\pi$
$233$			66.3325i			0.284689i	−0.989817	+	0.142344i	$0.954536\pi$
$234$		0			0
$235$	132.000			0.561702
$236$		0			0
$237$	−305.000	+	202.314i	−1.28692	+	0.853646i
$238$		0			0
$239$			192.364i			0.804871i	0.915448	+	0.402436i	$0.131836\pi$
$239$			192.364i			0.804871i	−0.915448	+	0.402436i	$0.868164\pi$
$240$		0			0
$241$	44.0000			0.182573			0.0912863	−	0.995825i	$-0.470902\pi$
$241$	44.0000			0.182573			0.0912863	+	0.995825i	$0.470902\pi$
$242$		0			0
$243$	−237.500	−	51.4077i	−0.977366	−	0.211554i
$244$		0			0
$245$			149.248i			0.609176i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	143.000	+	215.581i	0.574297	+	0.865786i
$250$		0			0
$251$		−	301.813i		−	1.20244i	−0.799083	−	0.601221i	$-0.794681\pi$
$251$		−	301.813i		−	1.20244i	0.799083	−	0.601221i	$-0.205319\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−55.0000	+	36.4829i	−0.215686	+	0.143070i
$256$		0			0
$257$			66.3325i			0.258103i	0.991638	+	0.129052i	$0.0411933\pi$
$257$			66.3325i			0.258103i	−0.991638	+	0.129052i	$0.958807\pi$
$258$		0			0
$259$	102.000			0.393822
$260$		0			0
$261$	−330.000	−	139.298i	−1.26437	−	0.533710i
$262$		0			0
$263$		−	378.095i		−	1.43762i	−0.695204	−	0.718812i	$-0.744686\pi$
$263$		−	378.095i		−	1.43762i	0.695204	−	0.718812i	$-0.255314\pi$
$264$		0			0
$265$	220.000			0.830189
$266$		0			0
$267$	−159.500	−	240.455i	−0.597378	−	0.900582i
$268$		0			0
$269$		−	331.662i		−	1.23295i	−0.787376	−	0.616473i	$-0.788561\pi$
$269$		−	331.662i		−	1.23295i	0.787376	−	0.616473i	$-0.211439\pi$
$270$		0			0
$271$	−44.0000			−0.162362			−0.0811808	−	0.996699i	$-0.525869\pi$
$271$	−44.0000			−0.162362			−0.0811808	+	0.996699i	$0.525869\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	274.000			0.989170			0.494585	−	0.869129i	$-0.335320\pi$
$277$	274.000			0.989170			0.494585	+	0.869129i	$0.335320\pi$
$278$		0			0
$279$	31.5000	−	74.6241i	0.112903	−	0.267470i
$280$		0			0
$281$		−	484.227i		−	1.72323i	−0.507564	−	0.861614i	$-0.669454\pi$
$281$		−	484.227i		−	1.72323i	0.507564	−	0.861614i	$-0.330546\pi$
$282$		0			0
$283$	224.000			0.791519			0.395760	−	0.918354i	$-0.370481\pi$
$283$	224.000			0.791519			0.395760	+	0.918354i	$0.370481\pi$
$284$		0			0
$285$	22.0000	+	33.1662i	0.0771930	+	0.116373i
$286$		0			0
$287$			159.198i			0.554697i
$288$		0			0
$289$	245.000			0.847751
$290$		0			0
$291$	197.500	−	131.007i	0.678694	−	0.450195i
$292$		0			0
$293$		−	331.662i		−	1.13195i	−0.824421	−	0.565977i	$-0.808499\pi$
$293$		−	331.662i		−	1.13195i	0.824421	−	0.565977i	$-0.191501\pi$
$294$		0			0
$295$	−231.000			−0.783051
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	44.0000			0.146179
$302$		0			0
$303$	−77.0000	−	116.082i	−0.254125	−	0.383108i
$304$		0			0
$305$		−	265.330i		−	0.869934i
$306$		0			0
$307$	380.000			1.23779			0.618893	−	0.785476i	$-0.287582\pi$
$307$	380.000			1.23779			0.618893	+	0.785476i	$0.287582\pi$
$308$		0			0
$309$	−285.000	+	189.048i	−0.922330	+	0.611805i
$310$		0			0
$311$			371.462i			1.19441i	0.802088	+	0.597206i	$0.203722\pi$
$311$			371.462i			1.19441i	−0.802088	+	0.597206i	$0.796278\pi$
$312$		0			0
$313$	−571.000			−1.82428			−0.912141	−	0.409878i	$-0.865571\pi$
$313$	−571.000			−1.82428			−0.912141	+	0.409878i	$0.865571\pi$
$314$		0			0
$315$	55.0000	+	23.2164i	0.174603	+	0.0737028i
$316$		0			0
$317$		−	89.5489i		−	0.282489i	−0.989975	−	0.141244i	$-0.954890\pi$
$317$		−	89.5489i		−	0.282489i	0.989975	−	0.141244i	$-0.0451103\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	275.000	+	414.578i	0.856698	+	1.29152i
$322$		0			0
$323$			26.5330i			0.0821455i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−15.0000	+	9.94987i	−0.0458716	+	0.0304278i
$328$		0			0
$329$		−	79.5990i		−	0.241942i
$330$		0			0
$331$	527.000			1.59215			0.796073	−	0.605201i	$-0.206907\pi$
$331$	527.000			1.59215			0.796073	+	0.605201i	$0.206907\pi$
$332$		0			0
$333$	−178.500	+	422.870i	−0.536036	+	1.26988i
$334$		0			0
$335$		−	3.31662i		−	0.00990037i
$336$		0			0
$337$	−306.000			−0.908012			−0.454006	−	0.890999i	$-0.650006\pi$
$337$	−306.000			−0.908012			−0.454006	+	0.890999i	$0.650006\pi$
$338$		0			0
$339$	−5.50000	−	8.29156i	−0.0162242	−	0.0244589i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	188.000			0.548105
$344$		0			0
$345$	27.5000	−	18.2414i	0.0797101	−	0.0528737i
$346$		0			0
$347$		−	305.129i		−	0.879336i	−0.898160	−	0.439668i	$-0.855096\pi$
$347$		−	305.129i		−	0.879336i	0.898160	−	0.439668i	$-0.144904\pi$
$348$		0			0
$349$	−246.000			−0.704871			−0.352436	−	0.935836i	$-0.614646\pi$
$349$	−246.000			−0.704871			−0.352436	+	0.935836i	$0.614646\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			348.246i			0.986531i	0.869879	+	0.493266i	$0.164197\pi$
$353$			348.246i			0.986531i	−0.869879	+	0.493266i	$0.835803\pi$
$354$		0			0
$355$	−165.000			−0.464789
$356$		0			0
$357$	22.0000	+	33.1662i	0.0616246	+	0.0929027i
$358$		0			0
$359$			636.792i			1.77379i	0.461967	+	0.886897i	$0.347144\pi$
$359$			636.792i			1.77379i	−0.461967	+	0.886897i	$0.652856\pi$
$360$		0			0
$361$	−345.000			−0.955679
$362$		0			0
$363$		0			0
$364$		0			0
$365$			358.195i			0.981357i
$366$		0			0
$367$	503.000			1.37057			0.685286	−	0.728274i	$-0.259677\pi$
$367$	503.000			1.37057			0.685286	+	0.728274i	$0.259677\pi$
$368$		0			0
$369$	−660.000	−	278.596i	−1.78862	−	0.755004i
$370$		0			0
$371$		−	132.665i		−	0.357588i
$372$		0			0
$373$	446.000			1.19571			0.597855	−	0.801604i	$-0.296020\pi$
$373$	446.000			1.19571			0.597855	+	0.801604i	$0.296020\pi$
$374$		0			0
$375$	−214.500	−	323.371i	−0.572000	−	0.862322i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	263.000			0.693931			0.346966	−	0.937878i	$-0.387212\pi$
$379$	263.000			0.693931			0.346966	+	0.937878i	$0.387212\pi$
$380$		0			0
$381$	570.000	−	378.095i	1.49606	−	0.992376i
$382$		0			0
$383$			215.581i			0.562874i	0.959580	+	0.281437i	$0.0908110\pi$
$383$			215.581i			0.562874i	−0.959580	+	0.281437i	$0.909189\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−77.0000	+	182.414i	−0.198966	+	0.471355i
$388$		0			0
$389$			89.5489i			0.230203i	0.993354	+	0.115101i	$0.0367193\pi$
$389$			89.5489i			0.230203i	−0.993354	+	0.115101i	$0.963281\pi$
$390$		0			0
$391$	22.0000			0.0562660
$392$		0			0
$393$	−121.000	−	182.414i	−0.307888	−	0.464159i
$394$		0			0
$395$			404.628i			1.02438i
$396$		0			0
$397$	406.000			1.02267			0.511335	−	0.859381i	$-0.329151\pi$
$397$	406.000			1.02267			0.511335	+	0.859381i	$0.329151\pi$
$398$		0			0
$399$	20.0000	−	13.2665i	0.0501253	−	0.0332494i
$400$		0			0
$401$		−	384.728i		−	0.959423i	−0.877426	−	0.479711i	$-0.840741\pi$
$401$		−	384.728i		−	0.959423i	0.877426	−	0.479711i	$-0.159259\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−192.500	+	187.389i	−0.475309	+	0.462690i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	386.000			0.943765			0.471883	−	0.881661i	$-0.343575\pi$
$409$	386.000			0.943765			0.471883	+	0.881661i	$0.343575\pi$
$410$		0			0
$411$	225.500	+	339.954i	0.548662	+	0.827139i
$412$		0			0
$413$			139.298i			0.337284i
$414$		0			0
$415$	286.000			0.689157
$416$		0			0
$417$	−275.000	+	182.414i	−0.659472	+	0.437445i
$418$		0			0
$419$		−	198.997i		−	0.474934i	−0.971396	−	0.237467i	$-0.923683\pi$
$419$		−	198.997i		−	0.474934i	0.971396	−	0.237467i	$-0.0763172\pi$
$420$		0			0
$421$	170.000			0.403800			0.201900	−	0.979406i	$-0.435288\pi$
$421$	170.000			0.403800			0.201900	+	0.979406i	$0.435288\pi$
$422$		0			0
$423$	330.000	+	139.298i	0.780142	+	0.329310i
$424$		0			0
$425$		−	92.8655i		−	0.218507i
$426$		0			0
$427$	−160.000			−0.374707
$428$		0			0
$429$		0			0
$430$		0			0
$431$			696.491i			1.61599i	0.589190	+	0.807994i	$0.299447\pi$
$431$			696.491i			1.61599i	−0.589190	+	0.807994i	$0.700553\pi$
$432$		0			0
$433$	95.0000			0.219400			0.109700	−	0.993965i	$-0.465011\pi$
$433$	95.0000			0.219400			0.109700	+	0.993965i	$0.465011\pi$
$434$		0			0
$435$	−330.000	+	218.897i	−0.758621	+	0.503212i
$436$		0			0
$437$		−	13.2665i		−	0.0303581i
$438$		0			0
$439$	248.000			0.564920			0.282460	−	0.959279i	$-0.408850\pi$
$439$	248.000			0.564920			0.282460	+	0.959279i	$0.408850\pi$
$440$		0			0
$441$	−157.500	+	373.120i	−0.357143	+	0.846078i
$442$		0			0
$443$		−	374.779i		−	0.846001i	−0.906129	−	0.423001i	$-0.860977\pi$
$443$		−	374.779i		−	0.846001i	0.906129	−	0.423001i	$-0.139023\pi$
$444$		0			0
$445$	−319.000			−0.716854
$446$		0			0
$447$	319.000	+	480.911i	0.713647	+	1.07586i
$448$		0			0
$449$		−	799.307i		−	1.78019i	−0.455772	−	0.890096i	$-0.650637\pi$
$449$		−	799.307i		−	1.78019i	0.455772	−	0.890096i	$-0.349363\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	605.000	−	401.312i	1.33554	−	0.885898i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−352.000			−0.770241			−0.385120	−	0.922866i	$-0.625840\pi$
$457$	−352.000			−0.770241			−0.385120	+	0.922866i	$0.625840\pi$
$458$		0			0
$459$	−176.000	+	33.1662i	−0.383442	+	0.0722576i
$460$		0			0
$461$			530.660i			1.15111i	0.817764	+	0.575553i	$0.195213\pi$
$461$			530.660i			1.15111i	−0.817764	+	0.575553i	$0.804787\pi$
$462$		0			0
$463$	519.000			1.12095			0.560475	−	0.828171i	$-0.310619\pi$
$463$	519.000			1.12095			0.560475	+	0.828171i	$0.310619\pi$
$464$		0			0
$465$	−49.5000	−	74.6241i	−0.106452	−	0.160482i
$466$		0			0
$467$			606.942i			1.29966i	0.760079	+	0.649831i	$0.225160\pi$
$467$			606.942i			1.29966i	−0.760079	+	0.649831i	$0.774840\pi$
$468$		0			0
$469$	−2.00000			−0.00426439
$470$		0			0
$471$	282.500	−	187.389i	0.599788	−	0.397854i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−56.0000			−0.117895
$476$		0			0
$477$	550.000	+	232.164i	1.15304	+	0.486716i
$478$		0			0
$479$			39.7995i			0.0830887i	0.999137	+	0.0415444i	$0.0132278\pi$
$479$			39.7995i			0.0830887i	−0.999137	+	0.0415444i	$0.986772\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−11.0000	−	16.5831i	−0.0227743	−	0.0343336i
$484$		0			0
$485$		−	262.013i		−	0.540234i
$486$		0			0
$487$	137.000			0.281314			0.140657	−	0.990058i	$-0.455078\pi$
$487$	137.000			0.281314			0.140657	+	0.990058i	$0.455078\pi$
$488$		0			0
$489$	115.000	−	76.2824i	0.235174	−	0.155997i
$490$		0			0
$491$			92.8655i			0.189135i	0.995518	+	0.0945677i	$0.0301469\pi$
$491$			92.8655i			0.189135i	−0.995518	+	0.0945677i	$0.969853\pi$
$492$		0			0
$493$	−264.000			−0.535497
$494$		0			0
$495$		0			0
$496$		0			0
$497$			99.4987i			0.200199i
$498$		0			0
$499$	818.000			1.63928			0.819639	−	0.572880i	$-0.194174\pi$
$499$	818.000			1.63928			0.819639	+	0.572880i	$0.194174\pi$
$500$		0			0
$501$	−462.000	−	696.491i	−0.922156	−	1.39020i
$502$		0			0
$503$			550.560i			1.09455i	0.836952	+	0.547276i	$0.184335\pi$
$503$			550.560i			1.09455i	−0.836952	+	0.547276i	$0.815665\pi$
$504$		0			0
$505$	−154.000			−0.304950
$506$		0			0
$507$	−422.500	+	280.255i	−0.833333	+	0.552771i
$508$		0			0
$509$		−	580.409i		−	1.14029i	−0.821543	−	0.570147i	$-0.806886\pi$
$509$		−	580.409i		−	1.14029i	0.821543	−	0.570147i	$-0.193114\pi$
$510$		0			0
$511$	216.000			0.422701
$512$		0			0
$513$	20.0000	+	106.132i	0.0389864	+	0.206885i
$514$		0			0
$515$			378.095i			0.734165i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−143.000	−	215.581i	−0.275530	−	0.415377i
$520$		0			0
$521$			766.140i			1.47052i	0.677786	+	0.735259i	$0.262940\pi$
$521$			766.140i			1.47052i	−0.677786	+	0.735259i	$0.737060\pi$
$522$		0			0
$523$	520.000			0.994264			0.497132	−	0.867675i	$-0.334386\pi$
$523$	520.000			0.994264			0.497132	+	0.867675i	$0.334386\pi$
$524$		0			0
$525$	−70.0000	+	46.4327i	−0.133333	+	0.0884433i
$526$		0			0
$527$		−	59.6992i		−	0.113281i
$528$		0			0
$529$	518.000			0.979206
$530$		0			0
$531$	−577.500	−	243.772i	−1.08757	−	0.459081i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	550.000			1.02804
$536$		0			0
$537$	−368.500	−	555.535i	−0.686220	−	1.03452i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	572.000			1.05730			0.528651	−	0.848839i	$-0.322698\pi$
$541$	572.000			1.05730			0.528651	+	0.848839i	$0.322698\pi$
$542$		0			0
$543$	−342.500	+	227.189i	−0.630755	+	0.418396i
$544$		0			0
$545$			19.8997i			0.0365133i
$546$		0			0
$547$	748.000			1.36746			0.683729	−	0.729736i	$-0.260357\pi$
$547$	748.000			1.36746			0.683729	+	0.729736i	$0.260357\pi$
$548$		0			0
$549$	280.000	−	663.325i	0.510018	−	1.20824i
$550$		0			0
$551$			159.198i			0.288926i
$552$		0			0
$553$	244.000			0.441230
$554$		0			0
$555$	280.500	+	422.870i	0.505405	+	0.761927i
$556$		0			0
$557$			378.095i			0.678807i	0.940641	+	0.339403i	$0.110225\pi$
$557$			378.095i			0.678807i	−0.940641	+	0.339403i	$0.889775\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	1008.25i		−	1.79086i	−0.445203	−	0.895430i	$-0.646868\pi$
$563$		−	1008.25i		−	1.79086i	0.445203	−	0.895430i	$-0.353132\pi$
$564$		0			0
$565$	−11.0000			−0.0194690
$566$		0			0
$567$	113.000	+	116.082i	0.199295	+	0.204730i
$568$		0			0
$569$			265.330i			0.466309i	0.972440	+	0.233155i	$0.0749048\pi$
$569$			265.330i			0.466309i	−0.972440	+	0.233155i	$0.925095\pi$
$570$		0			0
$571$	776.000			1.35902			0.679510	−	0.733667i	$-0.262193\pi$
$571$	776.000			1.35902			0.679510	+	0.733667i	$0.262193\pi$
$572$		0			0
$573$	49.5000	+	74.6241i	0.0863874	+	0.130234i
$574$		0			0
$575$			46.4327i			0.0807526i
$576$		0			0
$577$	−391.000			−0.677643			−0.338821	−	0.940851i	$-0.610028\pi$
$577$	−391.000			−0.677643			−0.338821	+	0.940851i	$0.610028\pi$
$578$		0			0
$579$	680.000	−	451.061i	1.17444	−	0.779034i
$580$		0			0
$581$		−	172.464i		−	0.296841i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$			1074.59i			1.83064i	0.402726	+	0.915321i	$0.368063\pi$
$587$			1074.59i			1.83064i	−0.402726	+	0.915321i	$0.631937\pi$
$588$		0			0
$589$	−36.0000			−0.0611205
$590$		0			0
$591$	209.000	+	315.079i	0.353638	+	0.533129i
$592$		0			0
$593$		−	39.7995i		−	0.0671155i	−0.999437	−	0.0335578i	$-0.989316\pi$
$593$		−	39.7995i		−	0.0671155i	0.999437	−	0.0335578i	$-0.0106838\pi$
$594$		0			0
$595$	44.0000			0.0739496
$596$		0			0
$597$	305.000	−	202.314i	0.510888	−	0.338885i
$598$		0			0
$599$			835.789i			1.39531i	0.716435	+	0.697654i	$0.245773\pi$
$599$			835.789i			1.39531i	−0.716435	+	0.697654i	$0.754227\pi$
$600$		0			0
$601$	−310.000			−0.515807			−0.257903	−	0.966171i	$-0.583032\pi$
$601$	−310.000			−0.515807			−0.257903	+	0.966171i	$0.583032\pi$
$602$		0			0
$603$	3.50000	−	8.29156i	0.00580431	−	0.0137505i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	374.000			0.616145			0.308072	−	0.951363i	$-0.400316\pi$
$607$	374.000			0.616145			0.308072	+	0.951363i	$0.400316\pi$
$608$		0			0
$609$	132.000	+	198.997i	0.216749	+	0.326761i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−216.000			−0.352365			−0.176183	−	0.984357i	$-0.556375\pi$
$613$	−216.000			−0.352365			−0.176183	+	0.984357i	$0.556375\pi$
$614$		0			0
$615$	−660.000	+	437.794i	−1.07317	+	0.711861i
$616$		0			0
$617$			305.129i			0.494537i	0.968947	+	0.247269i	$0.0795330\pi$
$617$			305.129i			0.494537i	−0.968947	+	0.247269i	$0.920467\pi$
$618$		0			0
$619$	−709.000			−1.14540			−0.572698	−	0.819767i	$-0.694103\pi$
$619$	−709.000			−1.14540			−0.572698	+	0.819767i	$0.694103\pi$
$620$		0			0
$621$	88.0000	−	16.5831i	0.141707	−	0.0267039i
$622$		0			0
$623$			192.364i			0.308771i
$624$		0			0
$625$	−79.0000			−0.126400
$626$		0			0
$627$		0			0
$628$		0			0
$629$			338.296i			0.537831i
$630$		0			0
$631$	685.000			1.08558			0.542789	−	0.839869i	$-0.317368\pi$
$631$	685.000			1.08558			0.542789	+	0.839869i	$0.317368\pi$
$632$		0			0
$633$	−410.000	+	271.963i	−0.647709	+	0.429642i
$634$		0			0
$635$		−	756.190i		−	1.19085i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−412.500	−	174.123i	−0.645540	−	0.272493i
$640$		0			0
$641$			288.546i			0.450150i	0.974341	+	0.225075i	$0.0722628\pi$
$641$			288.546i			0.450150i	−0.974341	+	0.225075i	$0.927737\pi$
$642$		0			0
$643$	479.000			0.744946			0.372473	−	0.928043i	$-0.378510\pi$
$643$	479.000			0.744946			0.372473	+	0.928043i	$0.378510\pi$
$644$		0			0
$645$	121.000	+	182.414i	0.187597	+	0.282813i
$646$		0			0
$647$			673.275i			1.04061i	0.853980	+	0.520305i	$0.174182\pi$
$647$			673.275i			1.04061i	−0.853980	+	0.520305i	$0.825818\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−45.0000	+	29.8496i	−0.0691244	+	0.0458520i
$652$		0			0
$653$		−	1243.73i		−	1.90465i	−0.305091	−	0.952323i	$-0.598687\pi$
$653$		−	1243.73i		−	1.90465i	0.305091	−	0.952323i	$-0.401313\pi$
$654$		0			0
$655$	−242.000			−0.369466
$656$		0			0
$657$	−378.000	+	895.489i	−0.575342	+	1.36300i
$658$		0			0
$659$			577.093i			0.875710i	0.899046	+	0.437855i	$0.144262\pi$
$659$			577.093i			0.875710i	−0.899046	+	0.437855i	$0.855738\pi$
$660$		0			0
$661$	−87.0000			−0.131619			−0.0658094	−	0.997832i	$-0.520963\pi$
$661$	−87.0000			−0.131619			−0.0658094	+	0.997832i	$0.520963\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	26.5330i		−	0.0398992i
$666$		0			0
$667$	132.000			0.197901
$668$		0			0
$669$	157.500	−	104.474i	0.235426	−	0.156164i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−890.000			−1.32244			−0.661218	−	0.750193i	$-0.729960\pi$
$673$	−890.000			−1.32244			−0.661218	+	0.750193i	$0.729960\pi$
$674$		0			0
$675$	−70.0000	−	371.462i	−0.103704	−	0.550314i
$676$		0			0
$677$		−	391.362i		−	0.578082i	−0.957317	−	0.289041i	$-0.906664\pi$
$677$		−	391.362i		−	0.578082i	0.957317	−	0.289041i	$-0.0933364\pi$
$678$		0			0
$679$	−158.000			−0.232695
$680$		0			0
$681$	−99.0000	−	149.248i	−0.145374	−	0.219160i
$682$		0			0
$683$		−	238.797i		−	0.349630i	−0.984601	−	0.174815i	$-0.944067\pi$
$683$		−	238.797i		−	0.349630i	0.984601	−	0.174815i	$-0.0559327\pi$
$684$		0			0
$685$	451.000			0.658394
$686$		0			0
$687$	312.500	−	207.289i	0.454876	−	0.301731i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−405.000			−0.586107			−0.293054	−	0.956096i	$-0.594671\pi$
$691$	−405.000			−0.586107			−0.293054	+	0.956096i	$0.594671\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			364.829i			0.524933i
$696$		0			0
$697$	−528.000			−0.757532
$698$		0			0
$699$	110.000	+	165.831i	0.157368	+	0.237241i
$700$		0			0
$701$		−	218.897i		−	0.312264i	−0.987736	−	0.156132i	$-0.950097\pi$
$701$		−	218.897i		−	0.312264i	0.987736	−	0.156132i	$-0.0499026\pi$
$702$		0			0
$703$	204.000			0.290185
$704$		0			0
$705$	330.000	−	218.897i	0.468085	−	0.310493i
$706$		0			0
$707$			92.8655i			0.131351i
$708$		0			0
$709$	−219.000			−0.308886			−0.154443	−	0.988002i	$-0.549358\pi$
$709$	−219.000			−0.308886			−0.154443	+	0.988002i	$0.549358\pi$
$710$		0			0
$711$	−427.000	+	1011.57i	−0.600563	+	1.42274i
$712$		0			0
$713$			29.8496i			0.0418648i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	319.000	+	480.911i	0.444909	+	0.670726i
$718$		0			0
$719$		−	69.6491i		−	0.0968694i	−0.998826	−	0.0484347i	$-0.984577\pi$
$719$		−	69.6491i		−	0.0968694i	0.998826	−	0.0484347i	$-0.0154233\pi$
$720$		0			0
$721$	228.000			0.316227
$722$		0			0
$723$	110.000	−	72.9657i	0.152144	−	0.100921i
$724$		0			0
$725$		−	557.193i		−	0.768542i
$726$		0			0
$727$	141.000			0.193948			0.0969739	−	0.995287i	$-0.469084\pi$
$727$	141.000			0.193948			0.0969739	+	0.995287i	$0.469084\pi$
$728$		0			0
$729$	−679.000	+	265.330i	−0.931413	+	0.363964i
$730$		0			0
$731$			145.931i			0.199633i
$732$		0			0
$733$	−1232.00			−1.68076			−0.840382	−	0.541995i	$-0.817669\pi$
$733$	−1232.00			−1.68076			−0.840382	+	0.541995i	$0.817669\pi$
$734$		0			0
$735$	247.500	+	373.120i	0.336735	+	0.507647i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−98.0000			−0.132612			−0.0663058	−	0.997799i	$-0.521121\pi$
$739$	−98.0000			−0.132612			−0.0663058	+	0.997799i	$0.521121\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1220.52i			1.64269i	0.570432	+	0.821344i	$0.306776\pi$
$743$			1220.52i			1.64269i	−0.570432	+	0.821344i	$0.693224\pi$
$744$		0			0
$745$	638.000			0.856376
$746$		0			0
$747$	715.000	+	301.813i	0.957162	+	0.404033i
$748$		0			0
$749$		−	331.662i		−	0.442807i
$750$		0			0
$751$	245.000			0.326232			0.163116	−	0.986607i	$-0.447846\pi$
$751$	245.000			0.326232			0.163116	+	0.986607i	$0.447846\pi$
$752$		0			0
$753$	−500.500	−	754.532i	−0.664675	−	1.00203i
$754$		0			0
$755$		−	802.623i		−	1.06308i
$756$		0			0
$757$	614.000			0.811096			0.405548	−	0.914074i	$-0.367081\pi$
$757$	614.000			0.811096			0.405548	+	0.914074i	$0.367081\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	12.0000			0.0157274
$764$		0			0
$765$	−77.0000	+	182.414i	−0.100654	+	0.238450i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−424.000			−0.551365			−0.275683	−	0.961249i	$-0.588904\pi$
$769$	−424.000			−0.551365			−0.275683	+	0.961249i	$0.588904\pi$
$770$		0			0
$771$	110.000	+	165.831i	0.142672	+	0.215086i
$772$		0			0
$773$			756.190i			0.978254i	0.872213	+	0.489127i	$0.162685\pi$
$773$			756.190i			0.978254i	−0.872213	+	0.489127i	$0.837315\pi$
$774$		0			0
$775$	126.000			0.162581
$776$		0			0
$777$	255.000	−	169.148i	0.328185	−	0.217694i
$778$		0			0
$779$			318.396i			0.408724i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−1056.00	+	198.997i	−1.34866	+	0.254147i
$784$		0			0
$785$		−	374.779i		−	0.477425i
$786$		0			0
$787$	176.000			0.223634			0.111817	−	0.993729i	$-0.464333\pi$
$787$	176.000			0.223634			0.111817	+	0.993729i	$0.464333\pi$
$788$		0			0
$789$	−627.000	−	945.238i	−0.794677	−	1.19802i
$790$		0			0
$791$			6.63325i			0.00838590i
$792$		0			0
$793$		0			0
$794$		0			0
$795$	550.000	−	364.829i	0.691824	−	0.458904i
$796$		0			0
$797$		−	646.742i		−	0.811470i	−0.913991	−	0.405735i	$-0.867016\pi$
$797$		−	646.742i		−	0.811470i	0.913991	−	0.405735i	$-0.132984\pi$
$798$		0			0
$799$	264.000			0.330413
$800$		0			0
$801$	−797.500	−	336.637i	−0.995630	−	0.420271i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−22.0000			−0.0273292
$806$		0			0
$807$	−550.000	−	829.156i	−0.681537	−	1.02746i
$808$		0			0
$809$		−	1406.25i		−	1.73826i	−0.494587	−	0.869128i	$-0.664681\pi$
$809$		−	1406.25i		−	1.73826i	0.494587	−	0.869128i	$-0.335319\pi$
$810$		0			0
$811$	−458.000			−0.564735			−0.282367	−	0.959306i	$-0.591120\pi$
$811$	−458.000			−0.564735			−0.282367	+	0.959306i	$0.591120\pi$
$812$		0			0
$813$	−110.000	+	72.9657i	−0.135301	+	0.0897488i
$814$		0			0
$815$		−	152.565i		−	0.187196i
$816$		0			0
$817$	88.0000			0.107711
$818$		0			0
$819$		0			0
$820$		0			0
$821$			404.628i			0.492848i	0.969162	+	0.246424i	$0.0792556\pi$
$821$			404.628i			0.492848i	−0.969162	+	0.246424i	$0.920744\pi$
$822$		0			0
$823$	827.000			1.00486			0.502430	−	0.864618i	$-0.332439\pi$
$823$	827.000			1.00486			0.502430	+	0.864618i	$0.332439\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	106.132i		−	0.128334i	−0.997939	−	0.0641669i	$-0.979561\pi$
$827$		−	106.132i		−	0.128334i	0.997939	−	0.0641669i	$-0.0204390\pi$
$828$		0			0
$829$	−1181.00			−1.42461			−0.712304	−	0.701871i	$-0.752348\pi$
$829$	−1181.00			−1.42461			−0.712304	+	0.701871i	$0.752348\pi$
$830$		0			0
$831$	685.000	−	454.378i	0.824308	−	0.546784i
$832$		0			0
$833$			298.496i			0.358339i
$834$		0			0
$835$	−924.000			−1.10659
$836$		0			0
$837$	−45.0000	−	238.797i	−0.0537634	−	0.285301i
$838$		0			0
$839$			268.647i			0.320199i	0.987101	+	0.160099i	$0.0511814\pi$
$839$			268.647i			0.320199i	−0.987101	+	0.160099i	$0.948819\pi$
$840$		0			0
$841$	−743.000			−0.883472
$842$		0			0
$843$	−803.000	−	1210.57i	−0.952550	−	1.43602i
$844$		0			0
$845$			560.510i			0.663325i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	560.000	−	371.462i	0.659600	−	0.437529i
$850$		0			0
$851$		−	169.148i		−	0.198764i
$852$		0			0
$853$	894.000			1.04807			0.524033	−	0.851698i	$-0.324427\pi$
$853$	894.000			1.04807			0.524033	+	0.851698i	$0.324427\pi$
$854$		0			0
$855$	110.000	+	46.4327i	0.128655	+	0.0543073i
$856$		0			0
$857$		−	1140.92i		−	1.33129i	−0.746267	−	0.665647i	$-0.768156\pi$
$857$		−	1140.92i		−	1.33129i	0.746267	−	0.665647i	$-0.231844\pi$
$858$		0			0
$859$	−1277.00			−1.48661			−0.743306	−	0.668951i	$-0.766743\pi$
$859$	−1277.00			−1.48661			−0.743306	+	0.668951i	$0.766743\pi$
$860$		0			0
$861$	264.000	+	397.995i	0.306620	+	0.462247i
$862$		0			0
$863$		−	1127.65i		−	1.30667i	−0.757071	−	0.653333i	$-0.773370\pi$
$863$		−	1127.65i		−	1.30667i	0.757071	−	0.653333i	$-0.226630\pi$
$864$		0			0
$865$	−286.000			−0.330636
$866$		0			0
$867$	612.500	−	406.287i	0.706459	−	0.468612i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	276.500	−	655.033i	0.316724	−	0.750325i
$874$		0			0
$875$			258.697i			0.295653i
$876$		0			0
$877$	−876.000			−0.998860			−0.499430	−	0.866354i	$-0.666457\pi$
$877$	−876.000			−0.998860			−0.499430	+	0.866354i	$0.666457\pi$
$878$		0			0
$879$	−550.000	−	829.156i	−0.625711	−	0.943295i
$880$		0			0
$881$			1197.30i			1.35903i	0.733664	+	0.679513i	$0.237809\pi$
$881$			1197.30i			1.35903i	−0.733664	+	0.679513i	$0.762191\pi$
$882$		0			0
$883$	−206.000			−0.233296			−0.116648	−	0.993173i	$-0.537215\pi$
$883$	−206.000			−0.233296			−0.116648	+	0.993173i	$0.537215\pi$
$884$		0			0
$885$	−577.500	+	383.070i	−0.652542	+	0.432848i
$886$		0			0
$887$		−	975.088i		−	1.09931i	−0.835392	−	0.549655i	$-0.814759\pi$
$887$		−	975.088i		−	1.09931i	0.835392	−	0.549655i	$-0.185241\pi$
$888$		0			0
$889$	−456.000			−0.512936
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	159.198i		−	0.178273i
$894$		0			0
$895$	−737.000			−0.823464
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	358.195i		−	0.398438i
$900$		0			0
$901$	440.000			0.488346
$902$		0			0
$903$	110.000	−	72.9657i	0.121816	−	0.0808037i
$904$		0			0
$905$			454.378i			0.502075i
$906$		0			0
$907$	−170.000			−0.187431			−0.0937155	−	0.995599i	$-0.529874\pi$
$907$	−170.000			−0.187431			−0.0937155	+	0.995599i	$0.529874\pi$
$908$		0			0
$909$	−385.000	−	162.515i	−0.423542	−	0.178784i
$910$		0			0
$911$		−	1273.58i		−	1.39801i	−0.715118	−	0.699003i	$-0.753627\pi$
$911$		−	1273.58i		−	1.39801i	0.715118	−	0.699003i	$-0.246373\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−440.000	−	663.325i	−0.480874	−	0.724945i
$916$		0			0
$917$			145.931i			0.159140i
$918$		0			0
$919$	−470.000			−0.511425			−0.255713	−	0.966753i	$-0.582310\pi$
$919$	−470.000			−0.511425			−0.255713	+	0.966753i	$0.582310\pi$
$920$		0			0
$921$	950.000	−	630.159i	1.03149	−	0.684211i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−714.000			−0.771892
$926$		0			0
$927$	−399.000	+	945.238i	−0.430421	+	1.01967i
$928$		0			0
$929$		−	676.591i		−	0.728301i	−0.931340	−	0.364150i	$-0.881359\pi$
$929$		−	676.591i		−	0.728301i	0.931340	−	0.364150i	$-0.118641\pi$
$930$		0			0
$931$	180.000			0.193340
$932$		0			0
$933$	616.000	+	928.655i	0.660236	+	0.995343i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−760.000			−0.811099			−0.405550	−	0.914073i	$-0.632920\pi$
$937$	−760.000			−0.811099			−0.405550	+	0.914073i	$0.632920\pi$
$938$		0			0
$939$	−1427.50	+	946.896i	−1.52023	+	1.00841i
$940$		0			0
$941$			696.491i			0.740161i	0.929000	+	0.370080i	$0.120670\pi$
$941$			696.491i			0.740161i	−0.929000	+	0.370080i	$0.879330\pi$
$942$		0			0
$943$	264.000			0.279958
$944$		0			0
$945$	176.000	−	33.1662i	0.186243	−	0.0350966i
$946$		0			0
$947$		−	248.747i		−	0.262668i	−0.991338	−	0.131334i	$-0.958074\pi$
$947$		−	248.747i		−	0.262668i	0.991338	−	0.131334i	$-0.0419261\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−148.500	−	223.872i	−0.156151	−	0.235407i
$952$		0			0
$953$		−	1174.09i		−	1.23199i	−0.787751	−	0.615994i	$-0.788754\pi$
$953$		−	1174.09i		−	1.23199i	0.787751	−	0.615994i	$-0.211246\pi$
$954$		0			0
$955$	99.0000			0.103665
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	271.963i		−	0.283590i
$960$		0			0
$961$	−880.000			−0.915713
$962$		0			0
$963$	1375.00	+	580.409i	1.42783	+	0.602710i
$964$		0			0
$965$		−	902.122i		−	0.934841i
$966$		0			0
$967$	−556.000			−0.574974			−0.287487	−	0.957785i	$-0.592820\pi$
$967$	−556.000			−0.574974			−0.287487	+	0.957785i	$0.592820\pi$
$968$		0			0
$969$	44.0000	+	66.3325i	0.0454076	+	0.0684546i
$970$		0			0
$971$		−	1681.53i		−	1.73175i	−0.500261	−	0.865875i	$-0.666763\pi$
$971$		−	1681.53i		−	1.73175i	0.500261	−	0.865875i	$-0.333237\pi$
$972$		0			0
$973$	220.000			0.226105
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	301.813i		−	0.308918i	−0.987999	−	0.154459i	$-0.950637\pi$
$977$		−	301.813i		−	0.308918i	0.987999	−	0.154459i	$-0.0493634\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−21.0000	+	49.7494i	−0.0214067	+	0.0507129i
$982$		0			0
$983$		−	1920.33i		−	1.95354i	−0.214300	−	0.976768i	$-0.568747\pi$
$983$		−	1920.33i		−	1.95354i	0.214300	−	0.976768i	$-0.431253\pi$
$984$		0			0
$985$	418.000			0.424365
$986$		0			0
$987$	−132.000	−	198.997i	−0.133739	−	0.201619i
$988$		0			0
$989$		−	72.9657i		−	0.0737773i
$990$		0			0
$991$	−310.000			−0.312815			−0.156408	−	0.987693i	$-0.549991\pi$
$991$	−310.000			−0.312815			−0.156408	+	0.987693i	$0.549991\pi$
$992$		0			0
$993$	1317.50	−	873.931i	1.32679	−	0.880091i
$994$		0			0
$995$		−	404.628i		−	0.406662i
$996$		0			0
$997$	134.000			0.134403			0.0672016	−	0.997739i	$-0.478593\pi$
$997$	134.000			0.134403			0.0672016	+	0.997739i	$0.478593\pi$
$998$		0			0
$999$	255.000	+	1353.18i	0.255255	+	1.35454i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1452.3.e.d.485.1		2
3.2	odd	2	inner	1452.3.e.d.485.2		2
11.10	odd	2		132.3.e.a.89.1	✓	2
33.32	even	2		132.3.e.a.89.2	yes	2
44.43	even	2		528.3.i.b.353.2		2
132.131	odd	2		528.3.i.b.353.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.3.e.a.89.1	✓	2	11.10	odd	2
132.3.e.a.89.2	yes	2	33.32	even	2
528.3.i.b.353.1		2	132.131	odd	2
528.3.i.b.353.2		2	44.43	even	2
1452.3.e.d.485.1		2	1.1	even	1	trivial
1452.3.e.d.485.2		2	3.2	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1452.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.50000 - 1.65831i) q^{3} -3.31662i q^{5} -2.00000 q^{7} +(3.50000 - 8.29156i) q^{9} +(-5.50000 - 8.29156i) q^{15} -6.63325i q^{17} -4.00000 q^{19} +(-5.00000 + 3.31662i) q^{21} +3.31662i q^{23} +14.0000 q^{25} +(-5.00000 - 26.5330i) q^{27} -39.7995i q^{29} +9.00000 q^{31} +6.63325i q^{35} -51.0000 q^{37} -79.5990i q^{41} -22.0000 q^{43} +(-27.5000 - 11.6082i) q^{45} +39.7995i q^{47} -45.0000 q^{49} +(-11.0000 - 16.5831i) q^{51} +66.3325i q^{53} +(-10.0000 + 6.63325i) q^{57} -69.6491i q^{59} +80.0000 q^{61} +(-7.00000 + 16.5831i) q^{63} +1.00000 q^{67} +(5.50000 + 8.29156i) q^{69} -49.7494i q^{71} -108.000 q^{73} +(35.0000 - 23.2164i) q^{75} -122.000 q^{79} +(-56.5000 - 58.0409i) q^{81} +86.2322i q^{83} -22.0000 q^{85} +(-66.0000 - 99.4987i) q^{87} -96.1821i q^{89} +(22.5000 - 14.9248i) q^{93} +13.2665i q^{95} +79.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{3} - 4 q^{7} + 7 q^{9} - 11 q^{15} - 8 q^{19} - 10 q^{21} + 28 q^{25} - 10 q^{27} + 18 q^{31} - 102 q^{37} - 44 q^{43} - 55 q^{45} - 90 q^{49} - 22 q^{51} - 20 q^{57} + 160 q^{61} - 14 q^{63}+ \cdots + 158 q^{97}+O(q^{100}) \) 2 * q + 5 * q^3 - 4 * q^7 + 7 * q^9 - 11 * q^15 - 8 * q^19 - 10 * q^21 + 28 * q^25 - 10 * q^27 + 18 * q^31 - 102 * q^37 - 44 * q^43 - 55 * q^45 - 90 * q^49 - 22 * q^51 - 20 * q^57 + 160 * q^61 - 14 * q^63 + 2 * q^67 + 11 * q^69 - 216 * q^73 + 70 * q^75 - 244 * q^79 - 113 * q^81 - 44 * q^85 - 132 * q^87 + 45 * q^93 + 158 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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