LMFDB - Embedded newform 1452.3.e.h.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 = [4,0,-4,0,0,0,0,0,-28,0,0,0,-48,0,32,0,0,0,56,0,0,0,0,0,-108, 0,92,0,0,0,24] 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:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 11x^{2} - 10x + 3 $ x^4 - 2x^3 + 11x^2 - 10*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 132)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			485.1
Root			$0.500000 + 0.244099i$ of defining polynomial
Character	$\chi$	$=$	1452.485
Dual form			1452.3.e.h.485.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 2.82843i) q^{3} -3.80482i q^{5} +9.38083 q^{7} +(-7.00000 + 5.65685i) q^{9} -2.61917 q^{13} +(-10.7617 + 3.80482i) q^{15} +24.5802i q^{17} +32.7617 q^{19} +(-9.38083 - 26.5330i) q^{21} -28.3850i q^{23} +10.5233 q^{25} +(23.0000 + 14.1421i) q^{27} +7.60964i q^{29} +24.7617 q^{31} -35.6924i q^{35} +48.7617 q^{37} +(2.61917 + 7.40813i) q^{39} +26.3315i q^{41} +11.2383 q^{43} +(21.5233 + 26.6338i) q^{45} -13.1657i q^{47} +39.0000 q^{49} +(69.5233 - 24.5802i) q^{51} +60.5749i q^{53} +(-32.7617 - 92.6640i) q^{57} -107.682i q^{59} +21.3808 q^{61} +(-65.6658 + 53.0660i) q^{63} +9.96547i q^{65} -41.0467 q^{67} +(-80.2850 + 28.3850i) q^{69} +92.7647i q^{71} -67.8083 q^{73} +(-10.5233 - 29.7645i) q^{75} +57.3808 q^{79} +(17.0000 - 79.1960i) q^{81} +143.374i q^{83} +93.5233 q^{85} +(21.5233 - 7.60964i) q^{87} -41.5508i q^{89} -24.5700 q^{91} +(-24.7617 - 70.0366i) q^{93} -124.652i q^{95} -127.523 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 28 q^{9} - 48 q^{13} + 32 q^{15} + 56 q^{19} - 108 q^{25} + 92 q^{27} + 24 q^{31} + 120 q^{37} + 48 q^{39} + 120 q^{43} - 64 q^{45} + 156 q^{49} + 128 q^{51} - 56 q^{57} + 48 q^{61} + 136 q^{67}+ \cdots - 360 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 - 28 * q^9 - 48 * q^13 + 32 * q^15 + 56 * q^19 - 108 * q^25 + 92 * q^27 + 24 * q^31 + 120 * q^37 + 48 * q^39 + 120 * q^43 - 64 * q^45 + 156 * q^49 + 128 * q^51 - 56 * q^57 + 48 * q^61 + 136 * q^67 - 96 * q^69 + 104 * q^73 + 108 * q^75 + 192 * q^79 + 68 * q^81 + 224 * q^85 - 64 * q^87 + 352 * q^91 - 24 * q^93 - 360 * 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
$3$	−1.00000	−	2.82843i	−0.333333	−	0.942809i
$3$	−1.00000	−	2.82843i	−0.333333	−	0.942809i
$4$		0			0
$5$		−	3.80482i		−	0.760964i	−0.924788	−	0.380482i	$-0.875758\pi$
$5$		−	3.80482i		−	0.760964i	0.924788	−	0.380482i	$-0.124242\pi$
$6$		0			0
$7$	9.38083			1.34012			0.670059	−	0.742307i	$-0.266269\pi$
$7$	9.38083			1.34012			0.670059	+	0.742307i	$0.266269\pi$
$8$		0			0
$9$	−7.00000	+	5.65685i	−0.777778	+	0.628539i
$10$		0			0
$11$		0			0
$11$		0			0
$12$		0			0
$13$	−2.61917			−0.201474			−0.100737	−	0.994913i	$-0.532120\pi$
$13$	−2.61917			−0.201474			−0.100737	+	0.994913i	$0.532120\pi$
$14$		0			0
$15$	−10.7617	+	3.80482i	−0.717444	+	0.253655i
$16$		0			0
$17$			24.5802i			1.44589i	0.690903	+	0.722947i	$0.257213\pi$
$17$			24.5802i			1.44589i	−0.690903	+	0.722947i	$0.742787\pi$
$18$		0			0
$19$	32.7617			1.72430			0.862149	−	0.506655i	$-0.169118\pi$
$19$	32.7617			1.72430			0.862149	+	0.506655i	$0.169118\pi$
$20$		0			0
$21$	−9.38083	−	26.5330i	−0.446706	−	1.26348i
$22$		0			0
$23$		−	28.3850i		−	1.23413i	−0.786912	−	0.617066i	$-0.788321\pi$
$23$		−	28.3850i		−	1.23413i	0.786912	−	0.617066i	$-0.211679\pi$
$24$		0			0
$25$	10.5233			0.420933
$26$		0			0
$27$	23.0000	+	14.1421i	0.851852	+	0.523783i
$28$		0			0
$29$			7.60964i			0.262402i	0.991356	+	0.131201i	$0.0418833\pi$
$29$			7.60964i			0.262402i	−0.991356	+	0.131201i	$0.958117\pi$
$30$		0			0
$31$	24.7617			0.798763			0.399382	−	0.916785i	$-0.369225\pi$
$31$	24.7617			0.798763			0.399382	+	0.916785i	$0.369225\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	35.6924i		−	1.01978i
$36$		0			0
$37$	48.7617			1.31788			0.658941	−	0.752194i	$-0.271005\pi$
$37$	48.7617			1.31788			0.658941	+	0.752194i	$0.271005\pi$
$38$		0			0
$39$	2.61917	+	7.40813i	0.0671582	+	0.189952i
$40$		0			0
$41$			26.3315i			0.642231i	0.947040	+	0.321116i	$0.104058\pi$
$41$			26.3315i			0.642231i	−0.947040	+	0.321116i	$0.895942\pi$
$42$		0			0
$43$	11.2383			0.261357			0.130678	−	0.991425i	$-0.458284\pi$
$43$	11.2383			0.261357			0.130678	+	0.991425i	$0.458284\pi$
$44$		0			0
$45$	21.5233	+	26.6338i	0.478296	+	0.591861i
$46$		0			0
$47$		−	13.1657i		−	0.280122i	−0.990143	−	0.140061i	$-0.955270\pi$
$47$		−	13.1657i		−	0.280122i	0.990143	−	0.140061i	$-0.0447299\pi$
$48$		0			0
$49$	39.0000			0.795918
$50$		0			0
$51$	69.5233	−	24.5802i	1.36320	−	0.481965i
$52$		0			0
$53$			60.5749i			1.14292i	0.820629	+	0.571461i	$0.193623\pi$
$53$			60.5749i			1.14292i	−0.820629	+	0.571461i	$0.806377\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−32.7617	−	92.6640i	−0.574766	−	1.62568i
$58$		0			0
$59$		−	107.682i		−	1.82511i	−0.408949	−	0.912557i	$-0.634105\pi$
$59$		−	107.682i		−	1.82511i	0.408949	−	0.912557i	$-0.365895\pi$
$60$		0			0
$61$	21.3808			0.350505			0.175253	−	0.984523i	$-0.443926\pi$
$61$	21.3808			0.350505			0.175253	+	0.984523i	$0.443926\pi$
$62$		0			0
$63$	−65.6658	+	53.0660i	−1.04231	+	0.842317i
$64$		0			0
$65$			9.96547i			0.153315i
$66$		0			0
$67$	−41.0467			−0.612637			−0.306318	−	0.951929i	$-0.599097\pi$
$67$	−41.0467			−0.612637			−0.306318	+	0.951929i	$0.599097\pi$
$68$		0			0
$69$	−80.2850	+	28.3850i	−1.16355	+	0.411377i
$70$		0			0
$71$			92.7647i			1.30655i	0.757123	+	0.653273i	$0.226605\pi$
$71$			92.7647i			1.30655i	−0.757123	+	0.653273i	$0.773395\pi$
$72$		0			0
$73$	−67.8083			−0.928881			−0.464441	−	0.885604i	$-0.653745\pi$
$73$	−67.8083			−0.928881			−0.464441	+	0.885604i	$0.653745\pi$
$74$		0			0
$75$	−10.5233	−	29.7645i	−0.140311	−	0.396859i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	57.3808			0.726340			0.363170	−	0.931723i	$-0.381694\pi$
$79$	57.3808			0.726340			0.363170	+	0.931723i	$0.381694\pi$
$80$		0			0
$81$	17.0000	−	79.1960i	0.209877	−	0.977728i
$82$		0			0
$83$			143.374i			1.72740i	0.504007	+	0.863700i	$0.331859\pi$
$83$			143.374i			1.72740i	−0.504007	+	0.863700i	$0.668141\pi$
$84$		0			0
$85$	93.5233			1.10027
$86$		0			0
$87$	21.5233	−	7.60964i	0.247395	−	0.0874672i
$88$		0			0
$89$		−	41.5508i		−	0.466863i	−0.972373	−	0.233431i	$-0.925005\pi$
$89$		−	41.5508i		−	0.466863i	0.972373	−	0.233431i	$-0.0749954\pi$
$90$		0			0
$91$	−24.5700			−0.270000
$92$		0			0
$93$	−24.7617	−	70.0366i	−0.266254	−	0.753081i
$94$		0			0
$95$		−	124.652i		−	1.31213i
$96$		0			0
$97$	−127.523			−1.31467			−0.657337	−	0.753597i	$-0.728317\pi$
$97$	−127.523			−1.31467			−0.657337	+	0.753597i	$0.728317\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	86.6041i		−	0.857466i	−0.903431	−	0.428733i	$-0.858960\pi$
$101$		−	86.6041i		−	0.857466i	0.903431	−	0.428733i	$-0.141040\pi$
$102$		0			0
$103$	117.047			1.13638			0.568188	−	0.822899i	$-0.307645\pi$
$103$	117.047			1.13638			0.568188	+	0.822899i	$0.307645\pi$
$104$		0			0
$105$	−100.953	+	35.6924i	−0.961460	+	0.339928i
$106$		0			0
$107$		−	119.398i		−	1.11587i	−0.829883	−	0.557937i	$-0.811593\pi$
$107$		−	119.398i		−	1.11587i	0.829883	−	0.557937i	$-0.188407\pi$
$108$		0			0
$109$	108.712			0.997362			0.498681	−	0.866786i	$-0.333818\pi$
$109$	108.712			0.997362			0.498681	+	0.866786i	$0.333818\pi$
$110$		0			0
$111$	−48.7617	−	137.919i	−0.439294	−	1.24251i
$112$		0			0
$113$			45.0533i			0.398702i	0.979928	+	0.199351i	$0.0638834\pi$
$113$			45.0533i			0.398702i	−0.979928	+	0.199351i	$0.936117\pi$
$114$		0			0
$115$	−108.000			−0.939130
$116$		0			0
$117$	18.3342	−	14.8163i	0.156702	−	0.126635i
$118$		0			0
$119$			230.583i			1.93767i
$120$		0			0
$121$		0			0
$122$		0			0
$123$	74.4767	−	26.3315i	0.605501	−	0.214077i
$124$		0			0
$125$		−	135.160i		−	1.08128i
$126$		0			0
$127$	−56.4275			−0.444311			−0.222155	−	0.975011i	$-0.571309\pi$
$127$	−56.4275			−0.444311			−0.222155	+	0.975011i	$0.571309\pi$
$128$		0			0
$129$	−11.2383	−	31.7868i	−0.0871189	−	0.246409i
$130$		0			0
$131$		−	170.310i		−	1.30008i	−0.759901	−	0.650039i	$-0.774753\pi$
$131$		−	170.310i		−	1.30008i	0.759901	−	0.650039i	$-0.225247\pi$
$132$		0			0
$133$	307.332			2.31076
$134$		0			0
$135$	53.8083	−	87.5109i	0.398580	−	0.648229i
$136$		0			0
$137$		−	268.026i		−	1.95640i	−0.207671	−	0.978199i	$-0.566588\pi$
$137$		−	268.026i		−	1.95640i	0.207671	−	0.978199i	$-0.433412\pi$
$138$		0			0
$139$	−18.9533			−0.136355			−0.0681775	−	0.997673i	$-0.521718\pi$
$139$	−18.9533			−0.136355			−0.0681775	+	0.997673i	$0.521718\pi$
$140$		0			0
$141$	−37.2383	+	13.1657i	−0.264102	+	0.0933740i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	28.9533			0.199678
$146$		0			0
$147$	−39.0000	−	110.309i	−0.265306	−	0.750399i
$148$		0			0
$149$			71.9894i			0.483150i	0.970382	+	0.241575i	$0.0776640\pi$
$149$			71.9894i			0.483150i	−0.970382	+	0.241575i	$0.922336\pi$
$150$		0			0
$151$	−292.712			−1.93849			−0.969247	−	0.246092i	$-0.920854\pi$
$151$	−292.712			−1.93849			−0.969247	+	0.246092i	$0.920854\pi$
$152$		0			0
$153$	−139.047	−	172.061i	−0.908802	−	1.12458i
$154$		0			0
$155$		−	94.2137i		−	0.607831i
$156$		0			0
$157$	163.808			1.04337			0.521683	−	0.853140i	$-0.325305\pi$
$157$	163.808			1.04337			0.521683	+	0.853140i	$0.325305\pi$
$158$		0			0
$159$	171.332	−	60.5749i	1.07756	−	0.380974i
$160$		0			0
$161$		−	266.275i		−	1.65388i
$162$		0			0
$163$	81.0467			0.497219			0.248609	−	0.968604i	$-0.420026\pi$
$163$	81.0467			0.497219			0.248609	+	0.968604i	$0.420026\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	215.363i		−	1.28960i	−0.764351	−	0.644801i	$-0.776940\pi$
$167$		−	215.363i		−	1.28960i	0.764351	−	0.644801i	$-0.223060\pi$
$168$		0			0
$169$	−162.140			−0.959408
$170$		0			0
$171$	−229.332	+	185.328i	−1.34112	+	1.08379i
$172$		0			0
$173$		−	78.9944i		−	0.456615i	−0.973589	−	0.228308i	$-0.926681\pi$
$173$		−	78.9944i		−	0.456615i	0.973589	−	0.228308i	$-0.0733192\pi$
$174$		0			0
$175$	98.7175			0.564100
$176$		0			0
$177$	−304.570	+	107.682i	−1.72073	+	0.608371i
$178$		0			0
$179$			198.997i			1.11172i	0.831277	+	0.555859i	$0.187611\pi$
$179$			198.997i			1.11172i	−0.831277	+	0.555859i	$0.812389\pi$
$180$		0			0
$181$	−214.000			−1.18232			−0.591160	−	0.806554i	$-0.701330\pi$
$181$	−214.000			−1.18232			−0.591160	+	0.806554i	$0.701330\pi$
$182$		0			0
$183$	−21.3808	−	60.4741i	−0.116835	−	0.330460i
$184$		0			0
$185$		−	185.529i		−	1.00286i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	215.759	+	132.665i	1.14158	+	0.701931i
$190$		0			0
$191$		−	137.818i		−	0.721560i	−0.932651	−	0.360780i	$-0.882510\pi$
$191$		−	137.818i		−	0.721560i	0.932651	−	0.360780i	$-0.117490\pi$
$192$		0			0
$193$	86.5700			0.448549			0.224275	−	0.974526i	$-0.427999\pi$
$193$	86.5700			0.448549			0.224275	+	0.974526i	$0.427999\pi$
$194$		0			0
$195$	28.1866	−	9.96547i	0.144547	−	0.0511050i
$196$		0			0
$197$		−	101.823i		−	0.516870i	−0.966029	−	0.258435i	$-0.916793\pi$
$197$		−	101.823i		−	0.516870i	0.966029	−	0.258435i	$-0.0832068\pi$
$198$		0			0
$199$	190.285			0.956206			0.478103	−	0.878304i	$-0.341325\pi$
$199$	190.285			0.956206			0.478103	+	0.878304i	$0.341325\pi$
$200$		0			0
$201$	41.0467	+	116.097i	0.204212	+	0.577599i
$202$		0			0
$203$			71.3848i			0.351649i
$204$		0			0
$205$	100.187			0.488715
$206$		0			0
$207$	160.570	+	198.695i	0.775700	+	0.959880i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	221.047			1.04761			0.523807	−	0.851837i	$-0.324511\pi$
$211$	221.047			1.04761			0.523807	+	0.851837i	$0.324511\pi$
$212$		0			0
$213$	262.378	−	92.7647i	1.23182	−	0.435515i
$214$		0			0
$215$		−	42.7599i		−	0.198883i
$216$		0			0
$217$	232.285			1.07044
$218$		0			0
$219$	67.8083	+	191.791i	0.309627	+	0.875757i
$220$		0			0
$221$		−	64.3797i		−	0.291311i
$222$		0			0
$223$	66.5700			0.298520			0.149260	−	0.988798i	$-0.452311\pi$
$223$	66.5700			0.298520			0.149260	+	0.988798i	$0.452311\pi$
$224$		0			0
$225$	−73.6633	+	59.5289i	−0.327392	+	0.264573i
$226$		0			0
$227$			392.137i			1.72747i	0.503943	+	0.863737i	$0.331882\pi$
$227$			392.137i			1.72747i	−0.503943	+	0.863737i	$0.668118\pi$
$228$		0			0
$229$	−135.808			−0.593049			−0.296525	−	0.955025i	$-0.595828\pi$
$229$	−135.808			−0.593049			−0.296525	+	0.955025i	$0.595828\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			45.0533i			0.193362i	0.995315	+	0.0966809i	$0.0308226\pi$
$233$			45.0533i			0.193362i	−0.995315	+	0.0966809i	$0.969177\pi$
$234$		0			0
$235$	−50.0933			−0.213163
$236$		0			0
$237$	−57.3808	−	162.298i	−0.242113	−	0.684800i
$238$		0			0
$239$			382.776i			1.60157i	0.598951	+	0.800786i	$0.295585\pi$
$239$			382.776i			1.60157i	−0.598951	+	0.800786i	$0.704415\pi$
$240$		0			0
$241$	−71.5233			−0.296777			−0.148389	−	0.988929i	$-0.547409\pi$
$241$	−71.5233			−0.296777			−0.148389	+	0.988929i	$0.547409\pi$
$242$		0			0
$243$	−241.000	+	31.1127i	−0.991770	+	0.128036i
$244$		0			0
$245$		−	148.388i		−	0.605666i
$246$		0			0
$247$	−85.8083			−0.347402
$248$		0			0
$249$	405.523	−	143.374i	1.62861	−	0.575800i
$250$		0			0
$251$			76.6386i			0.305333i	0.988278	+	0.152667i	$0.0487861\pi$
$251$			76.6386i			0.305333i	−0.988278	+	0.152667i	$0.951214\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−93.5233	−	264.524i	−0.366758	−	1.03735i
$256$		0			0
$257$		−	341.225i		−	1.32772i	−0.747855	−	0.663862i	$-0.768916\pi$
$257$		−	341.225i		−	1.32772i	0.747855	−	0.663862i	$-0.231084\pi$
$258$		0			0
$259$	457.425			1.76612
$260$		0			0
$261$	−43.0467	−	53.2675i	−0.164930	−	0.204090i
$262$		0			0
$263$			229.978i			0.874442i	0.899354	+	0.437221i	$0.144037\pi$
$263$			229.978i			0.874442i	−0.899354	+	0.437221i	$0.855963\pi$
$264$		0			0
$265$	230.477			0.869723
$266$		0			0
$267$	−117.523	+	41.5508i	−0.440162	+	0.155621i
$268$		0			0
$269$			72.2916i			0.268742i	0.990931	+	0.134371i	$0.0429014\pi$
$269$			72.2916i			0.268742i	−0.990931	+	0.134371i	$0.957099\pi$
$270$		0			0
$271$	−431.759			−1.59321			−0.796604	−	0.604502i	$-0.793372\pi$
$271$	−431.759			−1.59321			−0.796604	+	0.604502i	$0.793372\pi$
$272$		0			0
$273$	24.5700	+	69.4944i	0.0899999	+	0.254558i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−432.142			−1.56008			−0.780041	−	0.625729i	$-0.784802\pi$
$277$	−432.142			−1.56008			−0.780041	+	0.625729i	$0.784802\pi$
$278$		0			0
$279$	−173.332	+	140.073i	−0.621260	+	0.502054i
$280$		0			0
$281$		−	77.2432i		−	0.274887i	−0.990510	−	0.137443i	$-0.956111\pi$
$281$		−	77.2432i		−	0.274887i	0.990510	−	0.137443i	$-0.0438885\pi$
$282$		0			0
$283$	94.3834			0.333510			0.166755	−	0.985998i	$-0.446671\pi$
$283$	94.3834			0.333510			0.166755	+	0.985998i	$0.446671\pi$
$284$		0			0
$285$	−352.570	+	124.652i	−1.23709	+	0.437377i
$286$		0			0
$287$			247.011i			0.860666i
$288$		0			0
$289$	−315.187			−1.09061
$290$		0			0
$291$	127.523	+	360.690i	0.438224	+	1.23949i
$292$		0			0
$293$		−	167.412i		−	0.571373i	−0.958323	−	0.285686i	$-0.907778\pi$
$293$		−	167.412i		−	0.571373i	0.958323	−	0.285686i	$-0.0922215\pi$
$294$		0			0
$295$	−409.710			−1.38885
$296$		0			0
$297$		0			0
$298$		0			0
$299$			74.3452i			0.248646i
$300$		0			0
$301$	105.425			0.350249
$302$		0			0
$303$	−244.953	+	86.6041i	−0.808427	+	0.285822i
$304$		0			0
$305$		−	81.3503i		−	0.266722i
$306$		0			0
$307$	163.140			0.531401			0.265700	−	0.964056i	$-0.414397\pi$
$307$	163.140			0.531401			0.265700	+	0.964056i	$0.414397\pi$
$308$		0			0
$309$	−117.047	−	331.058i	−0.378792	−	1.07138i
$310$		0			0
$311$		−	394.128i		−	1.26729i	−0.773623	−	0.633646i	$-0.781558\pi$
$311$		−	394.128i		−	1.26729i	0.773623	−	0.633646i	$-0.218442\pi$
$312$		0			0
$313$	409.233			1.30745			0.653727	−	0.756730i	$-0.273204\pi$
$313$	409.233			1.30745			0.653727	+	0.756730i	$0.273204\pi$
$314$		0			0
$315$	201.907	+	249.847i	0.640974	+	0.793164i
$316$		0			0
$317$		−	349.137i		−	1.10138i	−0.834710	−	0.550689i	$-0.814365\pi$
$317$		−	349.137i		−	1.10138i	0.834710	−	0.550689i	$-0.185635\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−337.710	+	119.398i	−1.05206	+	0.371958i
$322$		0			0
$323$			805.288i			2.49315i
$324$		0			0
$325$	−27.5624			−0.0848073
$326$		0			0
$327$	−108.712	−	307.485i	−0.332454	−	0.940322i
$328$		0			0
$329$		−	123.506i		−	0.375397i
$330$		0			0
$331$	254.000			0.767372			0.383686	−	0.923464i	$-0.374655\pi$
$331$	254.000			0.767372			0.383686	+	0.923464i	$0.374655\pi$
$332$		0			0
$333$	−341.332	+	275.838i	−1.02502	+	0.828341i
$334$		0			0
$335$			156.175i			0.466195i
$336$		0			0
$337$	−367.425			−1.09028			−0.545141	−	0.838344i	$-0.683524\pi$
$337$	−367.425			−1.09028			−0.545141	+	0.838344i	$0.683524\pi$
$338$		0			0
$339$	127.430	−	45.0533i	0.375900	−	0.132901i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−93.8083			−0.273494
$344$		0			0
$345$	108.000	+	305.470i	0.313043	+	0.885421i
$346$		0			0
$347$			386.883i			1.11494i	0.830198	+	0.557468i	$0.188227\pi$
$347$			386.883i			1.11494i	−0.830198	+	0.557468i	$0.811773\pi$
$348$		0			0
$349$	412.811			1.18284			0.591420	−	0.806364i	$-0.298568\pi$
$349$	412.811			1.18284			0.591420	+	0.806364i	$0.298568\pi$
$350$		0			0
$351$	−60.2409	−	37.0406i	−0.171626	−	0.105529i
$352$		0			0
$353$			259.812i			0.736012i	0.929823	+	0.368006i	$0.119959\pi$
$353$			259.812i			0.736012i	−0.929823	+	0.368006i	$0.880041\pi$
$354$		0			0
$355$	352.953			0.994235
$356$		0			0
$357$	652.187	−	230.583i	1.82685	−	0.645890i
$358$		0			0
$359$			233.481i			0.650364i	0.945651	+	0.325182i	$0.105426\pi$
$359$			233.481i			0.650364i	−0.945651	+	0.325182i	$0.894574\pi$
$360$		0			0
$361$	712.327			1.97320
$362$		0			0
$363$		0			0
$364$		0			0
$365$			257.999i			0.706845i
$366$		0			0
$367$	524.280			1.42856			0.714278	−	0.699862i	$-0.246755\pi$
$367$	524.280			1.42856			0.714278	+	0.699862i	$0.246755\pi$
$368$		0			0
$369$	−148.953	−	184.320i	−0.403668	−	0.499513i
$370$		0			0
$371$			568.243i			1.53165i
$372$		0			0
$373$	534.521			1.43303			0.716516	−	0.697571i	$-0.245736\pi$
$373$	534.521			1.43303			0.716516	+	0.697571i	$0.245736\pi$
$374$		0			0
$375$	−382.290	+	135.160i	−1.01944	+	0.360427i
$376$		0			0
$377$		−	19.9309i		−	0.0528672i
$378$		0			0
$379$	−465.233			−1.22753			−0.613764	−	0.789489i	$-0.710345\pi$
$379$	−465.233			−1.22753			−0.613764	+	0.789489i	$0.710345\pi$
$380$		0			0
$381$	56.4275	+	159.601i	0.148104	+	0.418900i
$382$		0			0
$383$		−	140.111i		−	0.365826i	−0.983129	−	0.182913i	$-0.941447\pi$
$383$		−	140.111i		−	0.365826i	0.983129	−	0.182913i	$-0.0585527\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−78.6684	+	63.5736i	−0.203277	+	0.164273i
$388$		0			0
$389$			651.104i			1.67379i	0.547363	+	0.836895i	$0.315632\pi$
$389$			651.104i			1.67379i	−0.547363	+	0.836895i	$0.684368\pi$
$390$		0			0
$391$	697.710			1.78442
$392$		0			0
$393$	−481.710	+	170.310i	−1.22573	+	0.433359i
$394$		0			0
$395$		−	218.324i		−	0.552719i
$396$		0			0
$397$	−129.617			−0.326490			−0.163245	−	0.986586i	$-0.552196\pi$
$397$	−129.617			−0.326490			−0.163245	+	0.986586i	$0.552196\pi$
$398$		0			0
$399$	−307.332	−	869.265i	−0.770255	−	2.17861i
$400$		0			0
$401$		−	328.299i		−	0.818701i	−0.912377	−	0.409350i	$-0.865755\pi$
$401$		−	328.299i		−	0.818701i	0.912377	−	0.409350i	$-0.134245\pi$
$402$		0			0
$403$	−64.8550			−0.160930
$404$		0			0
$405$	−301.327	−	64.6820i	−0.744016	−	0.159709i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−245.715			−0.600770			−0.300385	−	0.953818i	$-0.597115\pi$
$409$	−245.715			−0.600770			−0.300385	+	0.953818i	$0.597115\pi$
$410$		0			0
$411$	−758.093	+	268.026i	−1.84451	+	0.652133i
$412$		0			0
$413$		−	1010.14i		−	2.44587i
$414$		0			0
$415$	545.513			1.31449
$416$		0			0
$417$	18.9533	+	53.6082i	0.0454517	+	0.128557i
$418$		0			0
$419$		−	324.859i		−	0.775320i	−0.921803	−	0.387660i	$-0.873284\pi$
$419$		−	324.859i		−	0.775320i	0.921803	−	0.387660i	$-0.126716\pi$
$420$		0			0
$421$	420.378			0.998523			0.499262	−	0.866451i	$-0.333605\pi$
$421$	420.378			0.998523			0.499262	+	0.866451i	$0.333605\pi$
$422$		0			0
$423$	74.4767	+	92.1602i	0.176068	+	0.217873i
$424$		0			0
$425$			258.666i			0.608625i
$426$		0			0
$427$	200.570			0.469719
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	119.941i		−	0.278285i	−0.990272	−	0.139142i	$-0.955565\pi$
$431$		−	119.941i		−	0.278285i	0.990272	−	0.139142i	$-0.0444345\pi$
$432$		0			0
$433$	−395.327			−0.912994			−0.456497	−	0.889725i	$-0.650896\pi$
$433$	−395.327			−0.912994			−0.456497	+	0.889725i	$0.650896\pi$
$434$		0			0
$435$	−28.9533	−	81.8924i	−0.0665594	−	0.188258i
$436$		0			0
$437$		−	929.941i		−	2.12801i
$438$		0			0
$439$	73.9508			0.168453			0.0842264	−	0.996447i	$-0.473158\pi$
$439$	73.9508			0.168453			0.0842264	+	0.996447i	$0.473158\pi$
$440$		0			0
$441$	−273.000	+	220.617i	−0.619048	+	0.500266i
$442$		0			0
$443$			86.0619i			0.194271i	0.995271	+	0.0971354i	$0.0309680\pi$
$443$			86.0619i			0.194271i	−0.995271	+	0.0971354i	$0.969032\pi$
$444$		0			0
$445$	−158.093			−0.355266
$446$		0			0
$447$	203.617	−	71.9894i	0.455518	−	0.161050i
$448$		0			0
$449$		−	202.438i		−	0.450863i	−0.974259	−	0.225432i	$-0.927621\pi$
$449$		−	202.438i		−	0.450863i	0.974259	−	0.225432i	$-0.0723792\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	292.712	+	827.916i	0.646164	+	1.82763i
$454$		0			0
$455$			93.4844i			0.205460i
$456$		0			0
$457$	888.948			1.94518			0.972591	−	0.232522i	$-0.0746978\pi$
$457$	888.948			1.94518			0.972591	+	0.232522i	$0.0746978\pi$
$458$		0			0
$459$	−347.617	+	565.345i	−0.757335	+	1.23169i
$460$		0			0
$461$			657.807i			1.42691i	0.700699	+	0.713457i	$0.252872\pi$
$461$			657.807i			1.42691i	−0.700699	+	0.713457i	$0.747128\pi$
$462$		0			0
$463$	371.907			0.803254			0.401627	−	0.915803i	$-0.368445\pi$
$463$	371.907			0.803254			0.401627	+	0.915803i	$0.368445\pi$
$464$		0			0
$465$	−266.477	+	94.2137i	−0.573068	+	0.202610i
$466$		0			0
$467$			588.778i			1.26077i	0.776284	+	0.630384i	$0.217102\pi$
$467$			588.778i			1.26077i	−0.776284	+	0.630384i	$0.782898\pi$
$468$		0			0
$469$	−385.052			−0.821006
$470$		0			0
$471$	−163.808	−	463.320i	−0.347788	−	0.983694i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	344.762			0.725814
$476$		0			0
$477$	−342.663	−	424.024i	−0.718372	−	0.888940i
$478$		0			0
$479$		−	547.165i		−	1.14231i	−0.820843	−	0.571154i	$-0.806496\pi$
$479$		−	547.165i		−	1.14231i	0.820843	−	0.571154i	$-0.193504\pi$
$480$		0			0
$481$	−127.715			−0.265520
$482$		0			0
$483$	−753.140	+	266.275i	−1.55930	+	0.551294i
$484$		0			0
$485$			485.204i			1.00042i
$486$		0			0
$487$	33.9016			0.0696132			0.0348066	−	0.999394i	$-0.488918\pi$
$487$	33.9016			0.0696132			0.0348066	+	0.999394i	$0.488918\pi$
$488$		0			0
$489$	−81.0467	−	229.235i	−0.165740	−	0.468782i
$490$		0			0
$491$			367.494i			0.748460i	0.927336	+	0.374230i	$0.122093\pi$
$491$			367.494i			0.748460i	−0.927336	+	0.374230i	$0.877907\pi$
$492$		0			0
$493$	−187.047			−0.379405
$494$		0			0
$495$		0			0
$496$		0			0
$497$			870.210i			1.75093i
$498$		0			0
$499$	−182.953			−0.366640			−0.183320	−	0.983053i	$-0.558684\pi$
$499$	−182.953			−0.366640			−0.183320	+	0.983053i	$0.558684\pi$
$500$		0			0
$501$	−609.140	+	215.363i	−1.21585	+	0.429867i
$502$		0			0
$503$			154.486i			0.307130i	0.988139	+	0.153565i	$0.0490754\pi$
$503$			154.486i			0.307130i	−0.988139	+	0.153565i	$0.950925\pi$
$504$		0			0
$505$	−329.513			−0.652501
$506$		0			0
$507$	162.140	+	458.601i	0.319803	+	0.904539i
$508$		0			0
$509$			843.639i			1.65744i	0.559660	+	0.828722i	$0.310932\pi$
$509$			843.639i			1.65744i	−0.559660	+	0.828722i	$0.689068\pi$
$510$		0			0
$511$	−636.098			−1.24481
$512$		0			0
$513$	753.518	+	463.320i	1.46885	+	0.903158i
$514$		0			0
$515$		−	445.342i		−	0.864741i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−223.430	+	78.9944i	−0.430501	+	0.152205i
$520$		0			0
$521$			352.942i			0.677431i	0.940889	+	0.338716i	$0.109992\pi$
$521$			352.942i			0.677431i	−0.940889	+	0.338716i	$0.890008\pi$
$522$		0			0
$523$	−1029.13			−1.96775			−0.983877	−	0.178849i	$-0.942763\pi$
$523$	−1029.13			−1.96775			−0.983877	+	0.178849i	$0.942763\pi$
$524$		0			0
$525$	−98.7175	−	279.215i	−0.188033	−	0.531839i
$526$		0			0
$527$			608.647i			1.15493i
$528$		0			0
$529$	−276.710			−0.523081
$530$		0			0
$531$	609.140	+	753.772i	1.14716	+	1.41953i
$532$		0			0
$533$		−	68.9666i		−	0.129393i
$534$		0			0
$535$	−454.290			−0.849140
$536$		0			0
$537$	562.850	−	198.997i	1.04814	−	0.370573i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	81.4691			0.150590			0.0752949	−	0.997161i	$-0.476010\pi$
$541$	81.4691			0.150590			0.0752949	+	0.997161i	$0.476010\pi$
$542$		0			0
$543$	214.000	+	605.283i	0.394107	+	1.11470i
$544$		0			0
$545$		−	413.632i		−	0.758957i
$546$		0			0
$547$	378.668			0.692264			0.346132	−	0.938186i	$-0.387495\pi$
$547$	378.668			0.692264			0.346132	+	0.938186i	$0.387495\pi$
$548$		0			0
$549$	−149.666	+	120.948i	−0.272615	+	0.220306i
$550$		0			0
$551$			249.305i			0.452458i
$552$		0			0
$553$	538.280			0.973381
$554$		0			0
$555$	−524.757	+	185.529i	−0.945507	+	0.334287i
$556$		0			0
$557$		−	5.79599i		−	0.0104057i	−0.999986	−	0.00520286i	$-0.998344\pi$
$557$		−	5.79599i		−	0.0104057i	0.999986	−	0.00520286i	$-0.00165613\pi$
$558$		0			0
$559$	−29.4351			−0.0526567
$560$		0			0
$561$		0			0
$562$		0			0
$563$			276.178i			0.490548i	0.969454	+	0.245274i	$0.0788779\pi$
$563$			276.178i			0.490548i	−0.969454	+	0.245274i	$0.921122\pi$
$564$		0			0
$565$	171.420			0.303398
$566$		0			0
$567$	159.474	−	742.924i	0.281259	−	1.31027i
$568$		0			0
$569$		−	533.697i		−	0.937956i	−0.883210	−	0.468978i	$-0.844622\pi$
$569$		−	533.697i		−	0.937956i	0.883210	−	0.468978i	$-0.155378\pi$
$570$		0			0
$571$	−407.140			−0.713030			−0.356515	−	0.934290i	$-0.616035\pi$
$571$	−407.140			−0.713030			−0.356515	+	0.934290i	$0.616035\pi$
$572$		0			0
$573$	−389.808	+	137.818i	−0.680294	+	0.240520i
$574$		0			0
$575$		−	298.705i		−	0.519487i
$576$		0			0
$577$	−384.093			−0.665673			−0.332836	−	0.942985i	$-0.608006\pi$
$577$	−384.093			−0.665673			−0.332836	+	0.942985i	$0.608006\pi$
$578$		0			0
$579$	−86.5700	−	244.857i	−0.149516	−	0.422896i
$580$		0			0
$581$			1344.97i			2.31492i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−56.3732	−	69.7583i	−0.0963645	−	0.119245i
$586$		0			0
$587$		−	259.875i		−	0.442717i	−0.975193	−	0.221358i	$-0.928951\pi$
$587$		−	259.875i		−	0.442717i	0.975193	−	0.221358i	$-0.0710490\pi$
$588$		0			0
$589$	811.233			1.37731
$590$		0			0
$591$	−288.000	+	101.823i	−0.487310	+	0.172290i
$592$		0			0
$593$			532.613i			0.898167i	0.893490	+	0.449083i	$0.148249\pi$
$593$			532.613i			0.898167i	−0.893490	+	0.449083i	$0.851751\pi$
$594$		0			0
$595$	877.327			1.47450
$596$		0			0
$597$	−190.285	−	538.207i	−0.318735	−	0.901520i
$598$		0			0
$599$			243.144i			0.405916i	0.979187	+	0.202958i	$0.0650556\pi$
$599$			243.144i			0.405916i	−0.979187	+	0.202958i	$0.934944\pi$
$600$		0			0
$601$	−892.948			−1.48577			−0.742885	−	0.669419i	$-0.766543\pi$
$601$	−892.948			−1.48577			−0.742885	+	0.669419i	$0.766543\pi$
$602$		0			0
$603$	287.327	−	232.195i	0.476495	−	0.385066i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	159.573			0.262887			0.131444	−	0.991324i	$-0.458039\pi$
$607$	159.573			0.262887			0.131444	+	0.991324i	$0.458039\pi$
$608$		0			0
$609$	201.907	−	71.3848i	0.331538	−	0.117216i
$610$		0			0
$611$			34.4833i			0.0564375i
$612$		0			0
$613$	−400.712			−0.653691			−0.326845	−	0.945078i	$-0.605986\pi$
$613$	−400.712			−0.653691			−0.326845	+	0.945078i	$0.605986\pi$
$614$		0			0
$615$	−100.187	−	283.371i	−0.162905	−	0.460765i
$616$		0			0
$617$		−	40.4664i		−	0.0655858i	−0.999462	−	0.0327929i	$-0.989560\pi$
$617$		−	40.4664i		−	0.0655858i	0.999462	−	0.0327929i	$-0.0104402\pi$
$618$		0			0
$619$	726.187			1.17316			0.586580	−	0.809891i	$-0.300474\pi$
$619$	726.187			1.17316			0.586580	+	0.809891i	$0.300474\pi$
$620$		0			0
$621$	401.425	−	652.856i	0.646417	−	1.05130i
$622$		0			0
$623$		−	389.781i		−	0.625651i
$624$		0			0
$625$	−251.176			−0.401882
$626$		0			0
$627$		0			0
$628$		0			0
$629$			1198.57i			1.90552i
$630$		0			0
$631$	−864.182			−1.36954			−0.684771	−	0.728758i	$-0.740098\pi$
$631$	−864.182			−1.36954			−0.684771	+	0.728758i	$0.740098\pi$
$632$		0			0
$633$	−221.047	−	625.214i	−0.349205	−	0.987700i
$634$		0			0
$635$			214.697i			0.338105i
$636$		0			0
$637$	−102.148			−0.160357
$638$		0			0
$639$	−524.757	−	649.353i	−0.821215	−	1.01620i
$640$		0			0
$641$		−	217.177i		−	0.338810i	−0.985547	−	0.169405i	$-0.945815\pi$
$641$		−	217.177i		−	0.338810i	0.985547	−	0.169405i	$-0.0541846\pi$
$642$		0			0
$643$	269.420			0.419004			0.209502	−	0.977808i	$-0.432816\pi$
$643$	269.420			0.419004			0.209502	+	0.977808i	$0.432816\pi$
$644$		0			0
$645$	−120.943	+	42.7599i	−0.187509	+	0.0662944i
$646$		0			0
$647$		−	747.069i		−	1.15467i	−0.816509	−	0.577333i	$-0.804093\pi$
$647$		−	747.069i		−	1.15467i	0.816509	−	0.577333i	$-0.195907\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−232.285	−	657.001i	−0.356813	−	1.00922i
$652$		0			0
$653$			100.312i			0.153617i	0.997046	+	0.0768086i	$0.0244730\pi$
$653$			100.312i			0.153617i	−0.997046	+	0.0768086i	$0.975527\pi$
$654$		0			0
$655$	−648.000			−0.989313
$656$		0			0
$657$	474.658	−	383.582i	0.722463	−	0.583838i
$658$		0			0
$659$		−	959.712i		−	1.45632i	−0.685409	−	0.728158i	$-0.740377\pi$
$659$		−	959.712i		−	1.45632i	0.685409	−	0.728158i	$-0.259623\pi$
$660$		0			0
$661$	198.658			0.300542			0.150271	−	0.988645i	$-0.451985\pi$
$661$	198.658			0.300542			0.150271	+	0.988645i	$0.451985\pi$
$662$		0			0
$663$	−182.093	+	64.3797i	−0.274651	+	0.0971036i
$664$		0			0
$665$		−	1169.34i		−	1.75841i
$666$		0			0
$667$	216.000			0.323838
$668$		0			0
$669$	−66.5700	−	188.288i	−0.0995067	−	0.281447i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	81.1450			0.120572			0.0602861	−	0.998181i	$-0.480799\pi$
$673$	81.1450			0.120572			0.0602861	+	0.998181i	$0.480799\pi$
$674$		0			0
$675$	242.036	+	148.822i	0.358573	+	0.220477i
$676$		0			0
$677$			740.909i			1.09440i	0.837002	+	0.547200i	$0.184306\pi$
$677$			740.909i			1.09440i	−0.837002	+	0.547200i	$0.815694\pi$
$678$		0			0
$679$	−1196.27			−1.76182
$680$		0			0
$681$	1109.13	−	392.137i	1.62868	−	0.575825i
$682$		0			0
$683$			169.039i			0.247494i	0.992314	+	0.123747i	$0.0394912\pi$
$683$			169.039i			0.247494i	−0.992314	+	0.123747i	$0.960509\pi$
$684$		0			0
$685$	−1019.79			−1.48875
$686$		0			0
$687$	135.808	+	384.124i	0.197683	+	0.559132i
$688$		0			0
$689$		−	158.656i		−	0.230270i
$690$		0			0
$691$	−573.233			−0.829571			−0.414785	−	0.909919i	$-0.636143\pi$
$691$	−573.233			−0.829571			−0.414785	+	0.909919i	$0.636143\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			72.1141i			0.103761i
$696$		0			0
$697$	−647.233			−0.928599
$698$		0			0
$699$	127.430	−	45.0533i	0.182303	−	0.0644540i
$700$		0			0
$701$		−	373.352i		−	0.532600i	−0.963890	−	0.266300i	$-0.914199\pi$
$701$		−	373.352i		−	0.532600i	0.963890	−	0.266300i	$-0.0858011\pi$
$702$		0			0
$703$	1597.51			2.27242
$704$		0			0
$705$	50.0933	+	141.685i	0.0710543	+	0.200972i
$706$		0			0
$707$		−	812.418i		−	1.14911i
$708$		0			0
$709$	−161.518			−0.227811			−0.113906	−	0.993492i	$-0.536336\pi$
$709$	−161.518			−0.227811			−0.113906	+	0.993492i	$0.536336\pi$
$710$		0			0
$711$	−401.666	+	324.595i	−0.564931	+	0.456533i
$712$		0			0
$713$		−	702.861i		−	0.985779i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	1082.65	−	382.776i	1.50998	−	0.533857i
$718$		0			0
$719$			591.499i			0.822669i	0.911485	+	0.411334i	$0.134937\pi$
$719$			591.499i			0.822669i	−0.911485	+	0.411334i	$0.865063\pi$
$720$		0			0
$721$	1097.99			1.52288
$722$		0			0
$723$	71.5233	+	202.299i	0.0989258	+	0.279804i
$724$		0			0
$725$			80.0788i			0.110453i
$726$		0			0
$727$	198.482			0.273015			0.136507	−	0.990639i	$-0.456412\pi$
$727$	198.482			0.273015			0.136507	+	0.990639i	$0.456412\pi$
$728$		0			0
$729$	329.000	+	650.538i	0.451303	+	0.892371i
$730$		0			0
$731$			276.241i			0.377894i
$732$		0			0
$733$	1210.24			1.65107			0.825536	−	0.564349i	$-0.190873\pi$
$733$	1210.24			1.65107			0.825536	+	0.564349i	$0.190873\pi$
$734$		0			0
$735$	−419.705	+	148.388i	−0.571027	+	0.201889i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−602.560			−0.815372			−0.407686	−	0.913122i	$-0.633664\pi$
$739$	−602.560			−0.815372			−0.407686	+	0.913122i	$0.633664\pi$
$740$		0			0
$741$	85.8083	+	242.703i	0.115801	+	0.327534i
$742$		0			0
$743$			1311.63i			1.76532i	0.470013	+	0.882660i	$0.344249\pi$
$743$			1311.63i			1.76532i	−0.470013	+	0.882660i	$0.655751\pi$
$744$		0			0
$745$	273.907			0.367660
$746$		0			0
$747$	−811.047	−	1003.62i	−1.08574	−	1.34353i
$748$		0			0
$749$		−	1120.06i		−	1.49540i
$750$		0			0
$751$	−1444.37			−1.92326			−0.961630	−	0.274350i	$-0.911537\pi$
$751$	−1444.37			−1.92326			−0.961630	+	0.274350i	$0.911537\pi$
$752$		0			0
$753$	216.767	−	76.6386i	0.287871	−	0.101778i
$754$		0			0
$755$			1113.72i			1.47512i
$756$		0			0
$757$	1053.52			1.39170			0.695851	−	0.718186i	$-0.255027\pi$
$757$	1053.52			1.39170			0.695851	+	0.718186i	$0.255027\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1246.59i		−	1.63809i	−0.573730	−	0.819044i	$-0.694504\pi$
$761$		−	1246.59i		−	1.63809i	0.573730	−	0.819044i	$-0.305496\pi$
$762$		0			0
$763$	1019.81			1.33658
$764$		0			0
$765$	−654.663	+	529.048i	−0.855769	+	0.691566i
$766$		0			0
$767$			282.037i			0.367714i
$768$		0			0
$769$	381.233			0.495752			0.247876	−	0.968792i	$-0.420267\pi$
$769$	381.233			0.495752			0.247876	+	0.968792i	$0.420267\pi$
$770$		0			0
$771$	−965.130	+	341.225i	−1.25179	+	0.442574i
$772$		0			0
$773$			150.682i			0.194931i	0.995239	+	0.0974654i	$0.0310735\pi$
$773$			150.682i			0.194931i	−0.995239	+	0.0974654i	$0.968926\pi$
$774$		0			0
$775$	260.575			0.336226
$776$		0			0
$777$	−457.425	−	1293.79i	−0.588706	−	1.66511i
$778$		0			0
$779$			862.663i			1.10740i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−107.617	+	175.022i	−0.137441	+	0.223527i
$784$		0			0
$785$		−	623.262i		−	0.793964i
$786$		0			0
$787$	−172.565			−0.219269			−0.109635	−	0.993972i	$-0.534968\pi$
$787$	−172.565			−0.219269			−0.109635	+	0.993972i	$0.534968\pi$
$788$		0			0
$789$	650.477	−	229.978i	0.824432	−	0.291481i
$790$		0			0
$791$			422.638i			0.534308i
$792$		0			0
$793$	−56.0000			−0.0706179
$794$		0			0
$795$	−230.477	−	651.886i	−0.289908	−	0.819983i
$796$		0			0
$797$			81.5902i			0.102372i	0.998689	+	0.0511858i	$0.0163001\pi$
$797$			81.5902i			0.102372i	−0.998689	+	0.0511858i	$0.983700\pi$
$798$		0			0
$799$	323.617			0.405027
$800$		0			0
$801$	235.047	+	290.855i	0.293442	+	0.363115i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−1013.13			−1.25855
$806$		0			0
$807$	204.472	−	72.2916i	0.253372	−	0.0895807i
$808$		0			0
$809$		−	175.022i		−	0.216343i	−0.994132	−	0.108172i	$-0.965500\pi$
$809$		−	175.022i		−	0.216343i	0.994132	−	0.108172i	$-0.0344996\pi$
$810$		0			0
$811$	415.052			0.511778			0.255889	−	0.966706i	$-0.417632\pi$
$811$	415.052			0.511778			0.255889	+	0.966706i	$0.417632\pi$
$812$		0			0
$813$	431.759	+	1221.20i	0.531069	+	1.50209i
$814$		0			0
$815$		−	308.368i		−	0.378366i
$816$		0			0
$817$	368.187			0.450657
$818$		0			0
$819$	171.990	−	138.989i	0.210000	−	0.169705i
$820$		0			0
$821$		−	1180.93i		−	1.43841i	−0.694798	−	0.719205i	$-0.744506\pi$
$821$		−	1180.93i		−	1.43841i	0.694798	−	0.719205i	$-0.255494\pi$
$822$		0			0
$823$	−639.337			−0.776837			−0.388418	−	0.921483i	$-0.626978\pi$
$823$	−639.337			−0.776837			−0.388418	+	0.921483i	$0.626978\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1278.65i		−	1.54613i	−0.634326	−	0.773066i	$-0.718722\pi$
$827$		−	1278.65i		−	1.54613i	0.634326	−	0.773066i	$-0.281278\pi$
$828$		0			0
$829$	530.943			0.640462			0.320231	−	0.947339i	$-0.396239\pi$
$829$	530.943			0.640462			0.320231	+	0.947339i	$0.396239\pi$
$830$		0			0
$831$	432.142	+	1222.28i	0.520027	+	1.47086i
$832$		0			0
$833$			958.628i			1.15081i
$834$		0			0
$835$	−819.420			−0.981341
$836$		0			0
$837$	569.518	+	350.183i	0.680428	+	0.418378i
$838$		0			0
$839$		−	37.0790i		−	0.0441943i	−0.999756	−	0.0220971i	$-0.992966\pi$
$839$		−	37.0790i		−	0.0441943i	0.999756	−	0.0220971i	$-0.00703431\pi$
$840$		0			0
$841$	783.093			0.931145
$842$		0			0
$843$	−218.477	+	77.2432i	−0.259166	+	0.0916289i
$844$		0			0
$845$			616.914i			0.730075i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−94.3834	−	266.956i	−0.111170	−	0.314436i
$850$		0			0
$851$		−	1384.10i		−	1.62644i
$852$		0			0
$853$	−86.5309			−0.101443			−0.0507215	−	0.998713i	$-0.516152\pi$
$853$	−86.5309			−0.101443			−0.0507215	+	0.998713i	$0.516152\pi$
$854$		0			0
$855$	705.140	+	872.566i	0.824725	+	1.02055i
$856$		0			0
$857$		−	754.314i		−	0.880180i	−0.897954	−	0.440090i	$-0.854947\pi$
$857$		−	754.314i		−	0.880180i	0.897954	−	0.440090i	$-0.145053\pi$
$858$		0			0
$859$	18.1866			0.0211718			0.0105859	−	0.999944i	$-0.496630\pi$
$859$	18.1866			0.0211718			0.0105859	+	0.999944i	$0.496630\pi$
$860$		0			0
$861$	698.653	−	247.011i	0.811444	−	0.286889i
$862$		0			0
$863$			549.823i			0.637107i	0.947905	+	0.318553i	$0.103197\pi$
$863$			549.823i			0.637107i	−0.947905	+	0.318553i	$0.896803\pi$
$864$		0			0
$865$	−300.560			−0.347468
$866$		0			0
$867$	315.187	+	891.482i	0.363537	+	1.02824i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	107.508			0.123431
$872$		0			0
$873$	892.663	−	721.381i	1.02252	−	0.826324i
$874$		0			0
$875$		−	1267.91i		−	1.44904i
$876$		0			0
$877$	−222.334			−0.253517			−0.126758	−	0.991934i	$-0.540457\pi$
$877$	−222.334			−0.253517			−0.126758	+	0.991934i	$0.540457\pi$
$878$		0			0
$879$	−473.513	+	167.412i	−0.538695	+	0.190458i
$880$		0			0
$881$		−	266.817i		−	0.302857i	−0.988468	−	0.151429i	$-0.951613\pi$
$881$		−	266.817i		−	0.302857i	0.988468	−	0.151429i	$-0.0483874\pi$
$882$		0			0
$883$	105.430			0.119400			0.0596999	−	0.998216i	$-0.480986\pi$
$883$	105.430			0.119400			0.0596999	+	0.998216i	$0.480986\pi$
$884$		0			0
$885$	409.710	+	1158.83i	0.462949	+	1.30942i
$886$		0			0
$887$		−	182.027i		−	0.205216i	−0.994722	−	0.102608i	$-0.967281\pi$
$887$		−	182.027i		−	0.205216i	0.994722	−	0.102608i	$-0.0327188\pi$
$888$		0			0
$889$	−529.337			−0.595429
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	431.332i		−	0.483014i
$894$		0			0
$895$	757.150			0.845978
$896$		0			0
$897$	210.280	−	74.3452i	0.234426	−	0.0828820i
$898$		0			0
$899$			188.427i			0.209597i
$900$		0			0
$901$	−1488.94			−1.65255
$902$		0			0
$903$	−105.425	−	298.187i	−0.116750	−	0.330218i
$904$		0			0
$905$			814.232i			0.899704i
$906$		0			0
$907$	544.093			0.599882			0.299941	−	0.953958i	$-0.403033\pi$
$907$	544.093			0.599882			0.299941	+	0.953958i	$0.403033\pi$
$908$		0			0
$909$	489.907	+	606.229i	0.538951	+	0.666918i
$910$		0			0
$911$			362.480i			0.397893i	0.980010	+	0.198946i	$0.0637519\pi$
$911$			362.480i			0.397893i	−0.980010	+	0.198946i	$0.936248\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−230.093	+	81.3503i	−0.251468	+	0.0889074i
$916$		0			0
$917$		−	1597.65i		−	1.74226i
$918$		0			0
$919$	−1139.85			−1.24031			−0.620156	−	0.784478i	$-0.712931\pi$
$919$	−1139.85			−1.24031			−0.620156	+	0.784478i	$0.712931\pi$
$920$		0			0
$921$	−163.140	−	461.429i	−0.177134	−	0.501009i
$922$		0			0
$923$		−	242.966i		−	0.263236i
$924$		0			0
$925$	513.135			0.554740
$926$		0			0
$927$	−819.327	+	662.116i	−0.883847	+	0.714257i
$928$		0			0
$929$			1385.91i			1.49184i	0.666038	+	0.745918i	$0.267989\pi$
$929$			1385.91i			1.49184i	−0.666038	+	0.745918i	$0.732011\pi$
$930$		0			0
$931$	1277.70			1.37240
$932$		0			0
$933$	−1114.76	+	394.128i	−1.19481	+	0.422431i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	213.233			0.227570			0.113785	−	0.993505i	$-0.463702\pi$
$937$	213.233			0.227570			0.113785	+	0.993505i	$0.463702\pi$
$938$		0			0
$939$	−409.233	−	1157.49i	−0.435818	−	1.23268i
$940$		0			0
$941$			1039.25i			1.10441i	0.833709	+	0.552205i	$0.186213\pi$
$941$			1039.25i			1.10441i	−0.833709	+	0.552205i	$0.813787\pi$
$942$		0			0
$943$	747.420			0.792598
$944$		0			0
$945$	504.767	−	820.925i	0.534145	−	0.868704i
$946$		0			0
$947$			68.5492i			0.0723856i	0.999345	+	0.0361928i	$0.0115230\pi$
$947$			68.5492i			0.0723856i	−0.999345	+	0.0361928i	$0.988477\pi$
$948$		0			0
$949$	177.601			0.187146
$950$		0			0
$951$	−987.508	+	349.137i	−1.03839	+	0.367126i
$952$		0			0
$953$			139.996i			0.146901i	0.997299	+	0.0734504i	$0.0234010\pi$
$953$			139.996i			0.146901i	−0.997299	+	0.0734504i	$0.976599\pi$
$954$		0			0
$955$	−524.373			−0.549082
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	2514.31i		−	2.62181i
$960$		0			0
$961$	−347.860			−0.361977
$962$		0			0
$963$	675.420	+	835.789i	0.701371	+	0.867902i
$964$		0			0
$965$		−	329.383i		−	0.341330i
$966$		0			0
$967$	279.573			0.289113			0.144557	−	0.989497i	$-0.453824\pi$
$967$	279.573			0.289113			0.144557	+	0.989497i	$0.453824\pi$
$968$		0			0
$969$	2277.70	−	805.288i	2.35057	−	0.831051i
$970$		0			0
$971$			462.792i			0.476614i	0.971190	+	0.238307i	$0.0765924\pi$
$971$			462.792i			0.476614i	−0.971190	+	0.238307i	$0.923408\pi$
$972$		0			0
$973$	−177.798			−0.182732
$974$		0			0
$975$	27.5624	+	77.9581i	0.0282691	+	0.0799571i
$976$		0			0
$977$		−	615.047i		−	0.629526i	−0.949170	−	0.314763i	$-0.898075\pi$
$977$		−	615.047i		−	0.629526i	0.949170	−	0.314763i	$-0.101925\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−760.987	+	614.971i	−0.775726	+	0.626881i
$982$		0			0
$983$			1060.88i			1.07923i	0.841913	+	0.539613i	$0.181429\pi$
$983$			1060.88i			1.07923i	−0.841913	+	0.539613i	$0.818571\pi$
$984$		0			0
$985$	−387.420			−0.393320
$986$		0			0
$987$	−349.327	+	123.506i	−0.353928	+	0.125132i
$988$		0			0
$989$		−	319.001i		−	0.322549i
$990$		0			0
$991$	−1001.52			−1.01061			−0.505307	−	0.862940i	$-0.668621\pi$
$991$	−1001.52			−1.01061			−0.505307	+	0.862940i	$0.668621\pi$
$992$		0			0
$993$	−254.000	−	718.420i	−0.255791	−	0.723485i
$994$		0			0
$995$		−	724.001i		−	0.727639i
$996$		0			0
$997$	324.417			0.325394			0.162697	−	0.986676i	$-0.447981\pi$
$997$	324.417			0.325394			0.162697	+	0.986676i	$0.447981\pi$
$998$		0			0
$999$	1121.52	+	689.594i	1.12264	+	0.690284i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1452.3.e.h.485.1		4
3.2	odd	2	inner	1452.3.e.h.485.4		4
11.10	odd	2		132.3.e.b.89.1	✓	4
33.32	even	2		132.3.e.b.89.4	yes	4
44.43	even	2		528.3.i.c.353.3		4
132.131	odd	2		528.3.i.c.353.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.3.e.b.89.1	✓	4	11.10	odd	2
132.3.e.b.89.4	yes	4	33.32	even	2
528.3.i.c.353.2		4	132.131	odd	2
528.3.i.c.353.3		4	44.43	even	2
1452.3.e.h.485.1		4	1.1	even	1	trivial
1452.3.e.h.485.4		4	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)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 2.82843i) q^{3} -3.80482i q^{5} +9.38083 q^{7} +(-7.00000 + 5.65685i) q^{9} -2.61917 q^{13} +(-10.7617 + 3.80482i) q^{15} +24.5802i q^{17} +32.7617 q^{19} +(-9.38083 - 26.5330i) q^{21} -28.3850i q^{23} +10.5233 q^{25} +(23.0000 + 14.1421i) q^{27} +7.60964i q^{29} +24.7617 q^{31} -35.6924i q^{35} +48.7617 q^{37} +(2.61917 + 7.40813i) q^{39} +26.3315i q^{41} +11.2383 q^{43} +(21.5233 + 26.6338i) q^{45} -13.1657i q^{47} +39.0000 q^{49} +(69.5233 - 24.5802i) q^{51} +60.5749i q^{53} +(-32.7617 - 92.6640i) q^{57} -107.682i q^{59} +21.3808 q^{61} +(-65.6658 + 53.0660i) q^{63} +9.96547i q^{65} -41.0467 q^{67} +(-80.2850 + 28.3850i) q^{69} +92.7647i q^{71} -67.8083 q^{73} +(-10.5233 - 29.7645i) q^{75} +57.3808 q^{79} +(17.0000 - 79.1960i) q^{81} +143.374i q^{83} +93.5233 q^{85} +(21.5233 - 7.60964i) q^{87} -41.5508i q^{89} -24.5700 q^{91} +(-24.7617 - 70.0366i) q^{93} -124.652i q^{95} -127.523 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 28 q^{9} - 48 q^{13} + 32 q^{15} + 56 q^{19} - 108 q^{25} + 92 q^{27} + 24 q^{31} + 120 q^{37} + 48 q^{39} + 120 q^{43} - 64 q^{45} + 156 q^{49} + 128 q^{51} - 56 q^{57} + 48 q^{61} + 136 q^{67}+ \cdots - 360 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 - 28 * q^9 - 48 * q^13 + 32 * q^15 + 56 * q^19 - 108 * q^25 + 92 * q^27 + 24 * q^31 + 120 * q^37 + 48 * q^39 + 120 * q^43 - 64 * q^45 + 156 * q^49 + 128 * q^51 - 56 * q^57 + 48 * q^61 + 136 * q^67 - 96 * q^69 + 104 * q^73 + 108 * q^75 + 192 * q^79 + 68 * q^81 + 224 * q^85 - 64 * q^87 + 352 * q^91 - 24 * q^93 - 360 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more