LMFDB - Embedded newform 432.5.q.a.305.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,5,Mod(17,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("432.17");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	432.q (of order $6$, degree $2$, not 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:	$44.6558240522$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	6.0.39400128.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 11x^{4} + 14x^{3} + 98x^{2} + 20x + 4 $ x^6 - x^5 + 11x^4 + 14x^3 + 98x^2 + 20x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			305.1
Root			$-1.28901 - 2.23263i$ of defining polynomial
Character	$\chi$	$=$	432.305
Dual form			432.5.q.a.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-13.8760 + 8.01130i) q^{5} +(36.2418 - 62.7727i) q^{7} +(83.2749 + 48.0788i) q^{11} +(-76.9530 - 133.286i) q^{13} +72.7905i q^{17} +190.660 q^{19} +(-12.5186 + 7.22762i) q^{23} +(-184.138 + 318.937i) q^{25} +(-620.301 - 358.131i) q^{29} +(-151.284 - 262.031i) q^{31} +1161.38i q^{35} +826.277 q^{37} +(-481.613 + 278.059i) q^{41} +(446.340 - 773.084i) q^{43} +(-3425.50 - 1977.72i) q^{47} +(-1426.44 - 2470.67i) q^{49} -1966.96i q^{53} -1540.69 q^{55} +(4689.48 - 2707.47i) q^{59} +(856.210 - 1483.00i) q^{61} +(2135.60 + 1232.99i) q^{65} +(-2317.24 - 4013.58i) q^{67} -6697.12i q^{71} -4823.86 q^{73} +(6036.07 - 3484.93i) q^{77} +(-2864.40 + 4961.28i) q^{79} +(2452.78 + 1416.12i) q^{83} +(-583.147 - 1010.04i) q^{85} -14277.7i q^{89} -11155.7 q^{91} +(-2645.60 + 1527.44i) q^{95} +(-3582.65 + 6205.34i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{5} - 12 q^{7} + 483 q^{11} - 6 q^{13} + 258 q^{19} - 282 q^{23} - 273 q^{25} + 1056 q^{29} - 1290 q^{31} + 12 q^{37} - 7629 q^{41} + 285 q^{43} - 9642 q^{47} - 1863 q^{49} - 2016 q^{55} + 6225 q^{59}+ \cdots - 28959 q^{97}+O(q^{100}) $ 6 * q + 12 * q^5 - 12 * q^7 + 483 * q^11 - 6 * q^13 + 258 * q^19 - 282 * q^23 - 273 * q^25 + 1056 * q^29 - 1290 * q^31 + 12 * q^37 - 7629 * q^41 + 285 * q^43 - 9642 * q^47 - 1863 * q^49 - 2016 * q^55 + 6225 * q^59 + 3630 * q^61 + 7158 * q^65 + 5055 * q^67 - 14622 * q^73 - 2580 * q^77 - 4764 * q^79 - 1866 * q^83 + 12366 * q^85 - 34836 * q^91 - 13362 * q^95 - 28959 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

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			0
$3$		0			0
$4$		0			0
$5$	−13.8760	+	8.01130i	−0.555039	+	0.320452i	−0.751152	−	0.660129i	$-0.770501\pi$
$5$	−13.8760	+	8.01130i	−0.555039	+	0.320452i	0.196113	+	0.980581i	$0.437168\pi$
$6$		0			0
$7$	36.2418	−	62.7727i	0.739630	−	1.28108i	−0.213033	−	0.977045i	$-0.568334\pi$
$7$	36.2418	−	62.7727i	0.739630	−	1.28108i	0.952662	−	0.304031i	$-0.0983326\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	83.2749	+	48.0788i	0.688222	+	0.397345i	0.802946	−	0.596052i	$-0.203265\pi$
$11$	83.2749	+	48.0788i	0.688222	+	0.397345i	−0.114723	+	0.993397i	$0.536598\pi$
$12$		0			0
$13$	−76.9530	−	133.286i	−0.455343	−	0.788677i	0.543365	−	0.839497i	$-0.317150\pi$
$13$	−76.9530	−	133.286i	−0.455343	−	0.788677i	−0.998708	+	0.0508193i	$0.983817\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			72.7905i			0.251870i	0.992038	+	0.125935i	$0.0401931\pi$
$17$			72.7905i			0.251870i	−0.992038	+	0.125935i	$0.959807\pi$
$18$		0			0
$19$	190.660			0.528145			0.264072	−	0.964503i	$-0.414934\pi$
$19$	190.660			0.528145			0.264072	+	0.964503i	$0.414934\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−12.5186	+	7.22762i	−0.0236647	+	0.0136628i	−0.511786	−	0.859113i	$-0.671016\pi$
$23$	−12.5186	+	7.22762i	−0.0236647	+	0.0136628i	0.488121	+	0.872776i	$0.337682\pi$
$24$		0			0
$25$	−184.138	+	318.937i	−0.294621	+	0.510298i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−620.301	−	358.131i	−0.737575	−	0.425839i	0.0836117	−	0.996498i	$-0.473354\pi$
$29$	−620.301	−	358.131i	−0.737575	−	0.425839i	−0.821187	+	0.570659i	$0.806688\pi$
$30$		0			0
$31$	−151.284	−	262.031i	−0.157423	−	0.272665i	0.776515	−	0.630098i	$-0.216985\pi$
$31$	−151.284	−	262.031i	−0.157423	−	0.272665i	−0.933939	+	0.357433i	$0.883652\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			1161.38i			0.948063i
$36$		0			0
$37$	826.277			0.603562			0.301781	−	0.953377i	$-0.402419\pi$
$37$	826.277			0.603562			0.301781	+	0.953377i	$0.402419\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−481.613	+	278.059i	−0.286504	+	0.165413i	−0.636364	−	0.771389i	$-0.719562\pi$
$41$	−481.613	+	278.059i	−0.286504	+	0.165413i	0.349860	+	0.936802i	$0.386229\pi$
$42$		0			0
$43$	446.340	−	773.084i	0.241395	−	0.418109i	−0.719717	−	0.694268i	$-0.755728\pi$
$43$	446.340	−	773.084i	0.241395	−	0.418109i	0.961112	+	0.276159i	$0.0890616\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3425.50	−	1977.72i	−1.55070	−	0.895299i	−0.998085	−	0.0618654i	$-0.980295\pi$
$47$	−3425.50	−	1977.72i	−1.55070	−	0.895299i	−0.552619	−	0.833434i	$-0.686372\pi$
$48$		0			0
$49$	−1426.44	−	2470.67i	−0.594104	−	1.02902i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	1966.96i		−	0.700234i	−0.936706	−	0.350117i	$-0.886142\pi$
$53$		−	1966.96i		−	0.700234i	0.936706	−	0.350117i	$-0.113858\pi$
$54$		0			0
$55$	−1540.69			−0.509320
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4689.48	−	2707.47i	1.34716	−	0.777785i	0.359316	−	0.933216i	$-0.383010\pi$
$59$	4689.48	−	2707.47i	1.34716	−	0.777785i	0.987847	+	0.155431i	$0.0496765\pi$
$60$		0			0
$61$	856.210	−	1483.00i	0.230102	−	0.398549i	−0.727736	−	0.685858i	$-0.759427\pi$
$61$	856.210	−	1483.00i	0.230102	−	0.398549i	0.957838	+	0.287309i	$0.0927606\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2135.60	+	1232.99i	0.505467	+	0.291831i
$66$		0			0
$67$	−2317.24	−	4013.58i	−0.516205	−	0.894093i	−0.999823	−	0.0188141i	$-0.994011\pi$
$67$	−2317.24	−	4013.58i	−0.516205	−	0.894093i	0.483618	−	0.875279i	$-0.339322\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	6697.12i		−	1.32853i	−0.747497	−	0.664265i	$-0.768744\pi$
$71$		−	6697.12i		−	1.32853i	0.747497	−	0.664265i	$-0.231256\pi$
$72$		0			0
$73$	−4823.86			−0.905208			−0.452604	−	0.891712i	$-0.649505\pi$
$73$	−4823.86			−0.905208			−0.452604	+	0.891712i	$0.649505\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6036.07	−	3484.93i	1.01806	−	0.587777i
$78$		0			0
$79$	−2864.40	+	4961.28i	−0.458964	+	0.794950i	−0.998907	−	0.0467524i	$-0.985113\pi$
$79$	−2864.40	+	4961.28i	−0.458964	+	0.794950i	0.539942	+	0.841702i	$0.318446\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2452.78	+	1416.12i	0.356044	+	0.205562i	0.667344	−	0.744750i	$-0.267431\pi$
$83$	2452.78	+	1416.12i	0.356044	+	0.205562i	−0.311300	+	0.950312i	$0.600765\pi$
$84$		0			0
$85$	−583.147	−	1010.04i	−0.0807124	−	0.139798i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	14277.7i		−	1.80251i	−0.433290	−	0.901255i	$-0.642647\pi$
$89$		−	14277.7i		−	1.80251i	0.433290	−	0.901255i	$-0.357353\pi$
$90$		0			0
$91$	−11155.7			−1.34714
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2645.60	+	1527.44i	−0.293141	+	0.169245i
$96$		0			0
$97$	−3582.65	+	6205.34i	−0.380769	+	0.659511i	−0.991172	−	0.132580i	$-0.957674\pi$
$97$	−3582.65	+	6205.34i	−0.380769	+	0.659511i	0.610403	+	0.792091i	$0.291007\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1696.72	+	979.600i	0.166328	+	0.0960298i	0.580853	−	0.814008i	$-0.302719\pi$
$101$	1696.72	+	979.600i	0.166328	+	0.0960298i	−0.414525	+	0.910038i	$0.636052\pi$
$102$		0			0
$103$	−2577.74	−	4464.77i	−0.242976	−	0.420848i	0.718584	−	0.695440i	$-0.244790\pi$
$103$	−2577.74	−	4464.77i	−0.242976	−	0.420848i	−0.961561	+	0.274592i	$0.911457\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	9117.08i		−	0.796321i	−0.917316	−	0.398161i	$-0.869649\pi$
$107$		−	9117.08i		−	0.796321i	0.917316	−	0.398161i	$-0.130351\pi$
$108$		0			0
$109$	−16161.1			−1.36024			−0.680122	−	0.733099i	$-0.738073\pi$
$109$	−16161.1			−1.36024			−0.680122	+	0.733099i	$0.738073\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−18272.5	+	10549.6i	−1.43100	+	0.826189i	−0.997197	−	0.0748185i	$-0.976162\pi$
$113$	−18272.5	+	10549.6i	−1.43100	+	0.826189i	−0.433804	+	0.901007i	$0.642829\pi$
$114$		0			0
$115$	115.805	−	200.581i	0.00875654	−	0.0151668i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4569.26	+	2638.06i	0.322665	+	0.186291i
$120$		0			0
$121$	−2697.36	−	4671.97i	−0.184233	−	0.319102i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	15914.9i		−	1.01855i
$126$		0			0
$127$	20660.9			1.28098			0.640489	−	0.767968i	$-0.278732\pi$
$127$	20660.9			1.28098			0.640489	+	0.767968i	$0.278732\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2772.53	−	1600.72i	0.161560	−	0.0932768i	−0.417040	−	0.908888i	$-0.636933\pi$
$131$	2772.53	−	1600.72i	0.161560	−	0.0932768i	0.578600	+	0.815611i	$0.303599\pi$
$132$		0			0
$133$	6909.88	−	11968.3i	0.390631	−	0.676594i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3353.18	+	1935.96i	0.178655	+	0.103147i	0.586661	−	0.809833i	$-0.300442\pi$
$137$	3353.18	+	1935.96i	0.178655	+	0.103147i	−0.408005	+	0.912980i	$0.633776\pi$
$138$		0			0
$139$	−5839.62	−	10114.5i	−0.302242	−	0.523499i	0.674401	−	0.738365i	$-0.264402\pi$
$139$	−5839.62	−	10114.5i	−0.302242	−	0.523499i	−0.976644	+	0.214866i	$0.931068\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	14799.2i		−	0.723714i
$144$		0			0
$145$	11476.4			0.545844
$146$		0			0
$147$		0			0
$148$		0			0
$149$	13069.8	−	7545.88i	0.588705	−	0.339889i	−0.175880	−	0.984412i	$-0.556277\pi$
$149$	13069.8	−	7545.88i	0.588705	−	0.339889i	0.764585	+	0.644522i	$0.222944\pi$
$150$		0			0
$151$	15127.7	−	26201.9i	0.663465	−	1.14915i	−0.316234	−	0.948681i	$-0.602419\pi$
$151$	15127.7	−	26201.9i	0.663465	−	1.14915i	0.979699	−	0.200474i	$-0.0642481\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4198.42	+	2423.96i	0.174752	+	0.100893i
$156$		0			0
$157$	−10311.4	−	17859.8i	−0.418328	−	0.724565i	0.577443	−	0.816431i	$-0.304050\pi$
$157$	−10311.4	−	17859.8i	−0.418328	−	0.724565i	−0.995771	+	0.0918653i	$0.970717\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			1047.77i			0.0404216i
$162$		0			0
$163$	−39790.7			−1.49764			−0.748818	−	0.662776i	$-0.769378\pi$
$163$	−39790.7			−1.49764			−0.748818	+	0.662776i	$0.769378\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−23773.0	+	13725.3i	−0.852414	+	0.492141i	−0.861465	−	0.507818i	$-0.830452\pi$
$167$	−23773.0	+	13725.3i	−0.852414	+	0.492141i	0.00905084	+	0.999959i	$0.497119\pi$
$168$		0			0
$169$	2436.98	−	4220.97i	0.0853253	−	0.147788i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8729.26	−	5039.84i	−0.291666	−	0.168393i	0.347027	−	0.937855i	$-0.387191\pi$
$173$	−8729.26	−	5039.84i	−0.291666	−	0.168393i	−0.638693	+	0.769462i	$0.720525\pi$
$174$		0			0
$175$	13347.0	+	23117.7i	0.435821	+	0.754864i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	1215.45i		−	0.0379343i	−0.999820	−	0.0189672i	$-0.993962\pi$
$179$		−	1215.45i		−	0.0379343i	0.999820	−	0.0189672i	$-0.00603779\pi$
$180$		0			0
$181$	28359.9			0.865661			0.432831	−	0.901475i	$-0.357515\pi$
$181$	28359.9			0.865661			0.432831	+	0.901475i	$0.357515\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−11465.4	+	6619.55i	−0.335001	+	0.193413i
$186$		0			0
$187$	−3499.68	+	6061.62i	−0.100079	+	0.173343i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8445.34	−	4875.92i	−0.231500	−	0.133656i	0.379764	−	0.925083i	$-0.376005\pi$
$191$	−8445.34	−	4875.92i	−0.231500	−	0.133656i	−0.611264	+	0.791427i	$0.709339\pi$
$192$		0			0
$193$	26701.4	+	46248.2i	0.716836	+	1.24160i	0.962247	+	0.272177i	$0.0877436\pi$
$193$	26701.4	+	46248.2i	0.716836	+	1.24160i	−0.245411	+	0.969419i	$0.578923\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			68537.7i			1.76603i	0.469349	+	0.883013i	$0.344489\pi$
$197$			68537.7i			1.76603i	−0.469349	+	0.883013i	$0.655511\pi$
$198$		0			0
$199$	8237.42			0.208010			0.104005	−	0.994577i	$-0.466834\pi$
$199$	8237.42			0.208010			0.104005	+	0.994577i	$0.466834\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−44961.7	+	25958.7i	−1.09107	+	0.629927i
$204$		0			0
$205$	4455.24	−	7716.70i	0.106014	−	0.183622i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	15877.2	+	9166.71i	0.363481	+	0.209856i
$210$		0			0
$211$	20393.7	+	35322.9i	0.458069	+	0.793399i	0.998859	−	0.0477587i	$-0.0152079\pi$
$211$	20393.7	+	35322.9i	0.458069	+	0.793399i	−0.540790	+	0.841158i	$0.681875\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			14303.1i			0.309423i
$216$		0			0
$217$	−21931.2			−0.465740
$218$		0			0
$219$		0			0
$220$		0			0
$221$	9701.99	−	5601.45i	0.198644	−	0.114687i
$222$		0			0
$223$	−29959.8	+	51891.9i	−0.602461	+	1.04349i	0.389986	+	0.920821i	$0.372480\pi$
$223$	−29959.8	+	51891.9i	−0.602461	+	1.04349i	−0.992447	+	0.122672i	$0.960854\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	85572.3	+	49405.2i	1.66066	+	0.958785i	0.972399	+	0.233325i	$0.0749607\pi$
$227$	85572.3	+	49405.2i	1.66066	+	0.958785i	0.688265	+	0.725459i	$0.258373\pi$
$228$		0			0
$229$	2658.97	+	4605.47i	0.0507040	+	0.0878220i	0.890263	−	0.455446i	$-0.150520\pi$
$229$	2658.97	+	4605.47i	0.0507040	+	0.0878220i	−0.839559	+	0.543268i	$0.817187\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	18467.8i		−	0.340176i	−0.985429	−	0.170088i	$-0.945595\pi$
$233$		−	18467.8i		−	0.340176i	0.985429	−	0.170088i	$-0.0544052\pi$
$234$		0			0
$235$	63376.3			1.14760
$236$		0			0
$237$		0			0
$238$		0			0
$239$	22383.5	−	12923.1i	0.391861	−	0.226241i	−0.291105	−	0.956691i	$-0.594023\pi$
$239$	22383.5	−	12923.1i	0.391861	−	0.226241i	0.682966	+	0.730450i	$0.260690\pi$
$240$		0			0
$241$	−41536.2	+	71942.7i	−0.715142	+	1.23866i	0.247763	+	0.968821i	$0.420305\pi$
$241$	−41536.2	+	71942.7i	−0.715142	+	1.23866i	−0.962905	+	0.269841i	$0.913029\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	39586.6	+	22855.3i	0.659502	+	0.380764i
$246$		0			0
$247$	−14671.9	−	25412.4i	−0.240487	−	0.416536i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	49051.6i		−	0.778585i	−0.921114	−	0.389292i	$-0.872720\pi$
$251$		−	49051.6i		−	0.778585i	0.921114	−	0.389292i	$-0.127280\pi$
$252$		0			0
$253$	−1389.98			−0.0217154
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−84096.0	+	48552.9i	−1.27324	+	0.735104i	−0.975596	−	0.219574i	$-0.929533\pi$
$257$	−84096.0	+	48552.9i	−1.27324	+	0.735104i	−0.297641	+	0.954678i	$0.596200\pi$
$258$		0			0
$259$	29945.8	−	51867.7i	0.446413	−	0.773209i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	35921.0	+	20739.0i	0.519323	+	0.299831i	0.736658	−	0.676266i	$-0.236403\pi$
$263$	35921.0	+	20739.0i	0.519323	+	0.299831i	−0.217335	+	0.976097i	$0.569736\pi$
$264$		0			0
$265$	15757.9	+	27293.5i	0.224391	+	0.388657i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	115737.i		−	1.59944i	−0.600370	−	0.799722i	$-0.704980\pi$
$269$		−	115737.i		−	1.59944i	0.600370	−	0.799722i	$-0.295020\pi$
$270$		0			0
$271$	5077.71			0.0691400			0.0345700	−	0.999402i	$-0.488994\pi$
$271$	5077.71			0.0691400			0.0345700	+	0.999402i	$0.488994\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−30668.2	+	17706.3i	−0.405529	+	0.234133i
$276$		0			0
$277$	21192.9	−	36707.2i	0.276205	−	0.478401i	−0.694234	−	0.719750i	$-0.744257\pi$
$277$	21192.9	−	36707.2i	0.276205	−	0.478401i	0.970438	+	0.241349i	$0.0775899\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	35902.5	+	20728.3i	0.454687	+	0.262513i	0.709807	−	0.704396i	$-0.248782\pi$
$281$	35902.5	+	20728.3i	0.454687	+	0.262513i	−0.255121	+	0.966909i	$0.582115\pi$
$282$		0			0
$283$	19194.0	+	33245.0i	0.239658	+	0.415100i	0.960616	−	0.277879i	$-0.0896313\pi$
$283$	19194.0	+	33245.0i	0.239658	+	0.415100i	−0.720958	+	0.692979i	$0.756298\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			40309.6i			0.489378i
$288$		0			0
$289$	78222.5			0.936561
$290$		0			0
$291$		0			0
$292$		0			0
$293$	11239.5	−	6489.12i	0.130922	−	0.0755876i	−0.433109	−	0.901342i	$-0.642583\pi$
$293$	11239.5	−	6489.12i	0.130922	−	0.0755876i	0.564030	+	0.825754i	$0.309250\pi$
$294$		0			0
$295$	−43380.7	+	75137.6i	−0.498486	+	0.863402i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1926.69	+	1112.37i	0.0215511	+	0.0124425i
$300$		0			0
$301$	−32352.4	−	56036.0i	−0.357086	−	0.618492i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			27437.4i			0.294947i
$306$		0			0
$307$	126105.			1.33799			0.668997	−	0.743265i	$-0.266724\pi$
$307$	126105.			1.33799			0.668997	+	0.743265i	$0.266724\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−86432.4	+	49901.7i	−0.893626	+	0.515935i	−0.875127	−	0.483894i	$-0.839222\pi$
$311$	−86432.4	+	49901.7i	−0.893626	+	0.515935i	−0.0184989	+	0.999829i	$0.505889\pi$
$312$		0			0
$313$	1112.68	−	1927.21i	0.0113574	−	0.0196716i	−0.860291	−	0.509804i	$-0.829718\pi$
$313$	1112.68	−	1927.21i	0.0113574	−	0.0196716i	0.871648	+	0.490132i	$0.163051\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	145552.	+	84034.3i	1.44843	+	0.836254i	0.998388	−	0.0567515i	$-0.0180743\pi$
$317$	145552.	+	84034.3i	1.44843	+	0.836254i	0.450046	+	0.893005i	$0.351408\pi$
$318$		0			0
$319$	−34437.0	−	59646.6i	−0.338410	−	0.586144i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			13878.3i			0.133024i
$324$		0			0
$325$	56679.9			0.536615
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−248293.	+	143352.i	−2.29389	+	1.32438i
$330$		0			0
$331$	3379.43	−	5853.34i	0.0308452	−	0.0534254i	−0.850191	−	0.526475i	$-0.823513\pi$
$331$	3379.43	−	5853.34i	0.0308452	−	0.0534254i	0.881036	+	0.473049i	$0.156847\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	64308.1	+	37128.3i	0.573028	+	0.330838i
$336$		0			0
$337$	111046.	+	192338.i	0.977787	+	1.69358i	0.670413	+	0.741988i	$0.266117\pi$
$337$	111046.	+	192338.i	0.977787	+	1.69358i	0.307375	+	0.951589i	$0.400550\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	29094.2i		−	0.250206i
$342$		0			0
$343$	−32754.4			−0.278408
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−140770.	+	81273.8i	−1.16910	+	0.674981i	−0.953468	−	0.301494i	$-0.902515\pi$
$347$	−140770.	+	81273.8i	−1.16910	+	0.674981i	−0.215633	+	0.976475i	$0.569181\pi$
$348$		0			0
$349$	36348.5	−	62957.4i	0.298425	−	0.516887i	−0.677351	−	0.735660i	$-0.736872\pi$
$349$	36348.5	−	62957.4i	0.298425	−	0.516887i	0.975776	+	0.218773i	$0.0702054\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−111396.	−	64314.6i	−0.893966	−	0.516131i	−0.0187281	−	0.999825i	$-0.505962\pi$
$353$	−111396.	−	64314.6i	−0.893966	−	0.516131i	−0.875238	+	0.483693i	$0.839295\pi$
$354$		0			0
$355$	53652.7	+	92929.2i	0.425730	+	0.737387i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	95302.8i		−	0.739463i	−0.929139	−	0.369732i	$-0.879450\pi$
$359$		−	95302.8i		−	0.739463i	0.929139	−	0.369732i	$-0.120550\pi$
$360$		0			0
$361$	−93969.7			−0.721063
$362$		0			0
$363$		0			0
$364$		0			0
$365$	66935.7	−	38645.4i	0.502426	−	0.290076i
$366$		0			0
$367$	52073.0	−	90193.1i	0.386616	−	0.669639i	−0.605376	−	0.795940i	$-0.706977\pi$
$367$	52073.0	−	90193.1i	0.386616	−	0.669639i	0.991992	+	0.126301i	$0.0403104\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−123471.	−	71286.2i	−0.897053	−	0.517914i
$372$		0			0
$373$	−122915.	−	212896.i	−0.883463	−	1.53020i	−0.847465	−	0.530851i	$-0.821872\pi$
$373$	−122915.	−	212896.i	−0.883463	−	1.53020i	−0.0359975	−	0.999352i	$-0.511461\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			110237.i			0.775612i
$378$		0			0
$379$	−200830.			−1.39814			−0.699070	−	0.715053i	$-0.746403\pi$
$379$	−200830.			−1.39814			−0.699070	+	0.715053i	$0.746403\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	117206.	−	67668.8i	0.799008	−	0.461308i	−0.0441160	−	0.999026i	$-0.514047\pi$
$383$	117206.	−	67668.8i	0.799008	−	0.461308i	0.843124	+	0.537719i	$0.180714\pi$
$384$		0			0
$385$	−55837.6	+	96713.6i	−0.376708	+	0.652478i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	175090.	+	101088.i	1.15708	+	0.668039i	0.950602	−	0.310413i	$-0.100467\pi$
$389$	175090.	+	101088.i	1.15708	+	0.668039i	0.206475	+	0.978452i	$0.433801\pi$
$390$		0			0
$391$	−526.102	−	911.236i	−0.00344125	−	0.00596043i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	91790.2i		−	0.588304i
$396$		0			0
$397$	102384.			0.649608			0.324804	−	0.945781i	$-0.394702\pi$
$397$	102384.			0.649608			0.324804	+	0.945781i	$0.394702\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	78026.4	−	45048.6i	0.485236	−	0.280151i	−0.237360	−	0.971422i	$-0.576282\pi$
$401$	78026.4	−	45048.6i	0.485236	−	0.280151i	0.722596	+	0.691271i	$0.242949\pi$
$402$		0			0
$403$	−23283.5	+	40328.2i	−0.143363	+	0.248313i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	68808.1	+	39726.4i	0.415385	+	0.239823i
$408$		0			0
$409$	−74861.0	−	129663.i	−0.447517	−	0.775122i	0.550707	−	0.834699i	$-0.314358\pi$
$409$	−74861.0	−	129663.i	−0.447517	−	0.775122i	−0.998224	+	0.0595770i	$0.981025\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	392495.i		−	2.30109i
$414$		0			0
$415$	−45379.7			−0.263491
$416$		0			0
$417$		0			0
$418$		0			0
$419$	109423.	−	63175.3i	0.623276	−	0.359848i	−0.154868	−	0.987935i	$-0.549495\pi$
$419$	109423.	−	63175.3i	0.623276	−	0.359848i	0.778143	+	0.628087i	$0.216162\pi$
$420$		0			0
$421$	107645.	−	186446.i	0.607335	−	1.05194i	−0.384342	−	0.923191i	$-0.625572\pi$
$421$	107645.	−	186446.i	0.607335	−	1.05194i	0.991678	−	0.128745i	$-0.0410949\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−23215.6	−	13403.5i	−0.128529	−	0.0742063i
$426$		0			0
$427$	−62061.3	−	107493.i	−0.340381	−	0.589557i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			121972.i			0.656607i	0.944572	+	0.328304i	$0.106477\pi$
$431$			121972.i			0.656607i	−0.944572	+	0.328304i	$0.893523\pi$
$432$		0			0
$433$	−152419.			−0.812951			−0.406475	−	0.913662i	$-0.633242\pi$
$433$	−152419.			−0.812951			−0.406475	+	0.913662i	$0.633242\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2386.80	+	1378.02i	−0.0124984	+	0.00721593i
$438$		0			0
$439$	115564.	−	200162.i	0.599643	−	1.03861i	−0.393230	−	0.919440i	$-0.628642\pi$
$439$	115564.	−	200162.i	0.599643	−	1.03861i	0.992874	−	0.119173i	$-0.0380242\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−17481.4	−	10092.9i	−0.0890774	−	0.0514289i	0.454800	−	0.890594i	$-0.349711\pi$
$443$	−17481.4	−	10092.9i	−0.0890774	−	0.0514289i	−0.543877	+	0.839165i	$0.683044\pi$
$444$		0			0
$445$	114383.	+	198117.i	0.577618	+	1.00046i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	92962.8i		−	0.461123i	−0.973058	−	0.230561i	$-0.925944\pi$
$449$		−	92962.8i		−	0.461123i	0.973058	−	0.230561i	$-0.0740562\pi$
$450$		0			0
$451$	−53475.0			−0.262905
$452$		0			0
$453$		0			0
$454$		0			0
$455$	154796.	−	89371.5i	0.747716	−	0.431694i
$456$		0			0
$457$	−110443.	+	191293.i	−0.528819	+	0.915941i	0.470616	+	0.882338i	$0.344032\pi$
$457$	−110443.	+	191293.i	−0.528819	+	0.915941i	−0.999435	+	0.0336033i	$0.989302\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−91967.2	−	53097.3i	−0.432744	−	0.249845i	0.267771	−	0.963483i	$-0.413713\pi$
$461$	−91967.2	−	53097.3i	−0.432744	−	0.249845i	−0.700515	+	0.713638i	$0.747046\pi$
$462$		0			0
$463$	−34090.0	−	59045.5i	−0.159025	−	0.275439i	0.775493	−	0.631357i	$-0.217502\pi$
$463$	−34090.0	−	59045.5i	−0.159025	−	0.275439i	−0.934517	+	0.355918i	$0.884168\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	224350.i		−	1.02871i	−0.857578	−	0.514353i	$-0.828032\pi$
$467$		−	224350.i		−	1.02871i	0.857578	−	0.514353i	$-0.171968\pi$
$468$		0			0
$469$	−335925.			−1.52720
$470$		0			0
$471$		0			0
$472$		0			0
$473$	74337.9	−	42919.0i	0.332267	−	0.191835i
$474$		0			0
$475$	−35107.8	+	60808.5i	−0.155603	+	0.269511i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−94542.7	−	54584.3i	−0.412057	−	0.237901i	0.279616	−	0.960112i	$-0.409793\pi$
$479$	−94542.7	−	54584.3i	−0.412057	−	0.237901i	−0.691673	+	0.722211i	$0.743126\pi$
$480$		0			0
$481$	−63584.5	−	110132.i	−0.274828	−	0.476016i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	114807.i		−	0.488073i
$486$		0			0
$487$	−106309.			−0.448240			−0.224120	−	0.974562i	$-0.571951\pi$
$487$	−106309.			−0.448240			−0.224120	+	0.974562i	$0.571951\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−64472.7	+	37223.3i	−0.267432	+	0.154402i	−0.627720	−	0.778439i	$-0.716012\pi$
$491$	−64472.7	+	37223.3i	−0.267432	+	0.154402i	0.360288	+	0.932841i	$0.382678\pi$
$492$		0			0
$493$	26068.5	−	45152.0i	0.107256	−	0.185773i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−420397.	−	242716.i	−1.70195	−	0.982621i
$498$		0			0
$499$	−55603.4	−	96308.0i	−0.223306	−	0.386777i	0.732504	−	0.680763i	$-0.238352\pi$
$499$	−55603.4	−	96308.0i	−0.223306	−	0.386777i	−0.955810	+	0.293985i	$0.905018\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	115897.i		−	0.458077i	−0.973417	−	0.229038i	$-0.926442\pi$
$503$		−	115897.i		−	0.458077i	0.973417	−	0.229038i	$-0.0735581\pi$
$504$		0			0
$505$	−31391.5			−0.123092
$506$		0			0
$507$		0			0
$508$		0			0
$509$	170299.	−	98322.1i	0.657319	−	0.379503i	−0.133936	−	0.990990i	$-0.542762\pi$
$509$	170299.	−	98322.1i	0.657319	−	0.379503i	0.791255	+	0.611487i	$0.209428\pi$
$510$		0			0
$511$	−174825.	+	302807.i	−0.669519	+	1.15964i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	71537.3	+	41302.1i	0.269723	+	0.155725i
$516$		0			0
$517$	−190172.	−	329388.i	−0.711486	−	1.23233i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			299306.i			1.10266i	0.834289	+	0.551328i	$0.185879\pi$
$521$			299306.i			1.10266i	−0.834289	+	0.551328i	$0.814121\pi$
$522$		0			0
$523$	162892.			0.595520			0.297760	−	0.954641i	$-0.403760\pi$
$523$	162892.			0.595520			0.297760	+	0.954641i	$0.403760\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	19073.4	−	11012.0i	0.0686763	−	0.0396503i
$528$		0			0
$529$	−139816.	+	242168.i	−0.499627	+	0.865379i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	74123.1	+	42795.0i	0.260915	+	0.150639i
$534$		0			0
$535$	73039.7	+	126508.i	0.255183	+	0.441989i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	274327.i		−	0.944257i
$540$		0			0
$541$	−62918.7			−0.214974			−0.107487	−	0.994207i	$-0.534280\pi$
$541$	−62918.7			−0.214974			−0.107487	+	0.994207i	$0.534280\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	224251.	−	129471.i	0.754989	−	0.435893i
$546$		0			0
$547$	−7403.57	+	12823.4i	−0.0247438	+	0.0428575i	−0.878132	−	0.478418i	$-0.841210\pi$
$547$	−7403.57	+	12823.4i	−0.0247438	+	0.0428575i	0.853388	+	0.521276i	$0.174544\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−118267.	−	68281.3i	−0.389547	−	0.224905i
$552$		0			0
$553$	207622.	+	359612.i	0.678927	+	1.17594i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			254392.i			0.819962i	0.912094	+	0.409981i	$0.134465\pi$
$557$			254392.i			0.819962i	−0.912094	+	0.409981i	$0.865535\pi$
$558$		0			0
$559$	−137389.			−0.439671
$560$		0			0
$561$		0			0
$562$		0			0
$563$	80806.7	−	46653.8i	0.254936	−	0.147187i	−0.367087	−	0.930187i	$-0.619645\pi$
$563$	80806.7	−	46653.8i	0.254936	−	0.147187i	0.622022	+	0.783000i	$0.286311\pi$
$564$		0			0
$565$	169032.	−	292772.i	0.529508	−	0.917134i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	136853.	+	79012.1i	0.422698	+	0.244045i	0.696231	−	0.717818i	$-0.254859\pi$
$569$	136853.	+	79012.1i	0.422698	+	0.244045i	−0.273533	+	0.961863i	$0.588192\pi$
$570$		0			0
$571$	−219188.	−	379644.i	−0.672270	−	1.16441i	−0.977259	−	0.212050i	$-0.931986\pi$
$571$	−219188.	−	379644.i	−0.672270	−	1.16441i	0.304988	−	0.952356i	$-0.401347\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	5323.52i		−	0.0161014i
$576$		0			0
$577$	214770.			0.645094			0.322547	−	0.946554i	$-0.395461\pi$
$577$	214770.			0.645094			0.322547	+	0.946554i	$0.395461\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	177787.	−	102645.i	0.526681	−	0.304079i
$582$		0			0
$583$	94568.9	−	163798.i	0.278235	−	0.481917i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	568470.	+	328206.i	1.64980	+	0.952512i	0.977150	+	0.212550i	$0.0681767\pi$
$587$	568470.	+	328206.i	1.64980	+	0.952512i	0.672649	+	0.739962i	$0.265157\pi$
$588$		0			0
$589$	−28843.8	−	49959.0i	−0.0831423	−	0.144007i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	118369.i		−	0.336612i	−0.985735	−	0.168306i	$-0.946170\pi$
$593$		−	118369.i		−	0.336612i	0.985735	−	0.168306i	$-0.0538296\pi$
$594$		0			0
$595$	−84537.3			−0.238789
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−306693.	+	177069.i	−0.854772	+	0.493503i	−0.862258	−	0.506469i	$-0.830950\pi$
$599$	−306693.	+	177069.i	−0.854772	+	0.493503i	0.00748620	+	0.999972i	$0.497617\pi$
$600$		0			0
$601$	297820.	−	515839.i	0.824526	−	1.42812i	−0.0777554	−	0.996972i	$-0.524775\pi$
$601$	297820.	−	515839.i	0.824526	−	1.42812i	0.902281	−	0.431148i	$-0.141891\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	74857.1	+	43218.8i	0.204514	+	0.118076i
$606$		0			0
$607$	4860.81	+	8419.17i	0.0131926	+	0.0228503i	0.872546	−	0.488531i	$-0.162467\pi$
$607$	4860.81	+	8419.17i	0.0131926	+	0.0228503i	−0.859354	+	0.511382i	$0.829134\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			608765.i			1.63067i
$612$		0			0
$613$	194450.			0.517473			0.258736	−	0.965948i	$-0.416694\pi$
$613$	194450.			0.517473			0.258736	+	0.965948i	$0.416694\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−422308.	+	243819.i	−1.10932	+	0.640469i	−0.938654	−	0.344860i	$-0.887927\pi$
$617$	−422308.	+	243819.i	−1.10932	+	0.640469i	−0.170670	+	0.985328i	$0.554593\pi$
$618$		0			0
$619$	8628.23	−	14944.5i	0.0225185	−	0.0390033i	−0.854547	−	0.519375i	$-0.826165\pi$
$619$	8628.23	−	14944.5i	0.0225185	−	0.0390033i	0.877065	+	0.480372i	$0.159498\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−896249.	−	517449.i	−2.30915	−	1.33319i
$624$		0			0
$625$	12412.5	+	21499.1i	0.0317760	+	0.0550376i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			60145.1i			0.152019i
$630$		0			0
$631$	429836.			1.07955			0.539777	−	0.841808i	$-0.318509\pi$
$631$	429836.			1.07955			0.539777	+	0.841808i	$0.318509\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−286690.	+	165521.i	−0.710993	+	0.410492i
$636$		0			0
$637$	−219538.	+	380251.i	−0.541042	+	0.937113i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	27979.5	+	16154.0i	0.0680964	+	0.0393155i	0.533662	−	0.845698i	$-0.320816\pi$
$641$	27979.5	+	16154.0i	0.0680964	+	0.0393155i	−0.465565	+	0.885014i	$0.654149\pi$
$642$		0			0
$643$	138524.	+	239931.i	0.335045	+	0.580315i	0.983494	−	0.180943i	$-0.0579150\pi$
$643$	138524.	+	239931.i	0.335045	+	0.580315i	−0.648448	+	0.761259i	$0.724582\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			125452.i			0.299689i	0.988710	+	0.149844i	$0.0478773\pi$
$647$			125452.i			0.299689i	−0.988710	+	0.149844i	$0.952123\pi$
$648$		0			0
$649$	520687.			1.23620
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−245936.	+	141991.i	−0.576760	+	0.332992i	−0.759845	−	0.650105i	$-0.774725\pi$
$653$	−245936.	+	141991.i	−0.576760	+	0.332992i	0.183085	+	0.983097i	$0.441392\pi$
$654$		0			0
$655$	−25647.8	+	44423.2i	−0.0597815	+	0.103545i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	311139.	+	179636.i	0.716446	+	0.413640i	0.813443	−	0.581644i	$-0.197590\pi$
$659$	311139.	+	179636.i	0.716446	+	0.413640i	−0.0969973	+	0.995285i	$0.530924\pi$
$660$		0			0
$661$	−98902.1	−	171303.i	−0.226362	−	0.392070i	0.730366	−	0.683057i	$-0.239350\pi$
$661$	−98902.1	−	171303.i	−0.226362	−	0.392070i	−0.956727	+	0.290987i	$0.906016\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			221429.i			0.500715i
$666$		0			0
$667$	10353.7			0.0232726
$668$		0			0
$669$		0			0
$670$		0			0
$671$	142602.	−	82331.1i	0.316723	−	0.182860i
$672$		0			0
$673$	−119036.	+	206176.i	−0.262814	+	0.455207i	−0.966989	−	0.254820i	$-0.917984\pi$
$673$	−119036.	+	206176.i	−0.262814	+	0.455207i	0.704175	+	0.710027i	$0.251317\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−216636.	−	125075.i	−0.472666	−	0.272894i	0.244689	−	0.969602i	$-0.421314\pi$
$677$	−216636.	−	125075.i	−0.472666	−	0.272894i	−0.717355	+	0.696708i	$0.754647\pi$
$678$		0			0
$679$	259684.	+	449786.i	0.563256	+	0.975588i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	267358.i		−	0.573129i	−0.958061	−	0.286564i	$-0.907487\pi$
$683$		−	267358.i		−	0.573129i	0.958061	−	0.286564i	$-0.0925132\pi$
$684$		0			0
$685$	−62038.2			−0.132214
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−262169.	+	151363.i	−0.552259	+	0.318847i
$690$		0			0
$691$	168343.	−	291579.i	0.352565	−	0.610660i	−0.634133	−	0.773224i	$-0.718643\pi$
$691$	168343.	−	291579.i	0.352565	−	0.610660i	0.986698	+	0.162564i	$0.0519763\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	162061.	+	93565.9i	0.335513	+	0.193708i
$696$		0			0
$697$	−20240.1	−	35056.9i	−0.0416627	−	0.0721618i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			635795.i			1.29384i	0.762557	+	0.646921i	$0.223944\pi$
$701$			635795.i			1.29384i	−0.762557	+	0.646921i	$0.776056\pi$
$702$		0			0
$703$	157538.			0.318768
$704$		0			0
$705$		0			0
$706$		0			0
$707$	122984.	−	71005.0i	0.246043	−	0.142053i
$708$		0			0
$709$	−215513.	+	373280.i	−0.428728	+	0.742578i	−0.996760	−	0.0804279i	$-0.974371\pi$
$709$	−215513.	+	373280.i	−0.428728	+	0.742578i	0.568033	+	0.823006i	$0.307705\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3787.73	+	2186.84i	0.00745074	+	0.00430169i
$714$		0			0
$715$	118561.	+	205354.i	0.231916	+	0.401690i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			798370.i			1.54435i	0.635409	+	0.772176i	$0.280832\pi$
$719$			798370.i			1.54435i	−0.635409	+	0.772176i	$0.719168\pi$
$720$		0			0
$721$	−373688.			−0.718850
$722$		0			0
$723$		0			0
$724$		0			0
$725$	228442.	−	131891.i	0.434610	−	0.250922i
$726$		0			0
$727$	158557.	−	274628.i	0.299996	−	0.519609i	−0.676139	−	0.736775i	$-0.736348\pi$
$727$	158557.	−	274628.i	0.299996	−	0.519609i	0.976135	+	0.217166i	$0.0696812\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	56273.2	+	32489.3i	0.105309	+	0.0608004i
$732$		0			0
$733$	488922.	+	846837.i	0.909979	+	1.57613i	0.814091	+	0.580738i	$0.197236\pi$
$733$	488922.	+	846837.i	0.909979	+	1.57613i	0.0958883	+	0.995392i	$0.469431\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	445641.i		−	0.820446i
$738$		0			0
$739$	392565.			0.718825			0.359412	−	0.933179i	$-0.382977\pi$
$739$	392565.			0.718825			0.359412	+	0.933179i	$0.382977\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	899450.	−	519298.i	1.62929	−	0.940673i	0.644990	−	0.764191i	$-0.276861\pi$
$743$	899450.	−	519298.i	1.62929	−	0.940673i	0.984304	−	0.176482i	$-0.0564719\pi$
$744$		0			0
$745$	−120905.	+	209413.i	−0.217836	+	0.377304i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−572304.	−	330420.i	−1.02015	−	0.588983i
$750$		0			0
$751$	159783.	+	276752.i	0.283302	+	0.490694i	0.972196	−	0.234168i	$-0.0752367\pi$
$751$	159783.	+	276752.i	0.283302	+	0.490694i	−0.688894	+	0.724862i	$0.741903\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			484769.i			0.850435i
$756$		0			0
$757$	500321.			0.873085			0.436543	−	0.899684i	$-0.356203\pi$
$757$	500321.			0.873085			0.436543	+	0.899684i	$0.356203\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	783645.	−	452437.i	1.35316	−	0.781249i	0.364471	−	0.931215i	$-0.381250\pi$
$761$	783645.	−	452437.i	1.35316	−	0.781249i	0.988691	+	0.149966i	$0.0479165\pi$
$762$		0			0
$763$	−585707.	+	1.01447e6i	−1.00608	+	1.74258i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−721738.	−	416696.i	−1.22684	−	0.708318i
$768$		0			0
$769$	−71688.6	−	124168.i	−0.121227	−	0.209970i	0.799025	−	0.601298i	$-0.205349\pi$
$769$	−71688.6	−	124168.i	−0.121227	−	0.209970i	−0.920252	+	0.391327i	$0.872016\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			710063.i			1.18833i	0.804342	+	0.594166i	$0.202518\pi$
$773$			710063.i			1.18833i	−0.804342	+	0.594166i	$0.797482\pi$
$774$		0			0
$775$	111428.			0.185521
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−91824.5	+	53014.9i	−0.151316	+	0.0873621i
$780$		0			0
$781$	321990.	−	557702.i	0.527885	−	0.914324i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	286161.	+	165215.i	0.464377	+	0.268108i
$786$		0			0
$787$	−300855.	−	521097.i	−0.485745	−	0.841335i	0.514121	−	0.857718i	$-0.328118\pi$
$787$	−300855.	−	521097.i	−0.485745	−	0.841335i	−0.999866	+	0.0163827i	$0.994785\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1.52935e6i			2.44429i
$792$		0			0
$793$	−263552.			−0.419102
$794$		0			0
$795$		0			0
$796$		0			0
$797$	372227.	−	214905.i	0.585991	−	0.338322i	−0.177519	−	0.984117i	$-0.556807\pi$
$797$	372227.	−	214905.i	0.585991	−	0.338322i	0.763511	+	0.645795i	$0.223474\pi$
$798$		0			0
$799$	143959.	−	249344.i	0.225499	−	0.390576i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−401706.	−	231925.i	−0.622984	−	0.359680i
$804$		0			0
$805$	−8393.99	−	14538.8i	−0.0129532	−	0.0224356i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			629639.i			0.962042i	0.876709	+	0.481021i	$0.159734\pi$
$809$			629639.i			0.962042i	−0.876709	+	0.481021i	$0.840266\pi$
$810$		0			0
$811$	−577760.			−0.878428			−0.439214	−	0.898383i	$-0.644743\pi$
$811$	−577760.			−0.878428			−0.439214	+	0.898383i	$0.644743\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	552135.	−	318775.i	0.831246	−	0.479920i
$816$		0			0
$817$	85099.3	−	147396.i	0.127492	−	0.220822i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−38728.3	−	22359.8i	−0.0574569	−	0.0331727i	0.470996	−	0.882135i	$-0.343894\pi$
$821$	−38728.3	−	22359.8i	−0.0574569	−	0.0331727i	−0.528453	+	0.848962i	$0.677228\pi$
$822$		0			0
$823$	−146289.	−	253380.i	−0.215979	−	0.374087i	0.737596	−	0.675242i	$-0.235961\pi$
$823$	−146289.	−	253380.i	−0.215979	−	0.374087i	−0.953575	+	0.301155i	$0.902628\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	806491.i		−	1.17920i	−0.807694	−	0.589601i	$-0.799285\pi$
$827$		−	806491.i		−	1.17920i	0.807694	−	0.589601i	$-0.200715\pi$
$828$		0			0
$829$	204904.			0.298154			0.149077	−	0.988826i	$-0.452370\pi$
$829$	204904.			0.298154			0.149077	+	0.988826i	$0.452370\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	179842.	−	103832.i	0.259179	−	0.149637i
$834$		0			0
$835$	219915.	−	380905.i	0.315415	−	0.546315i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	667796.	+	385552.i	0.948681	+	0.547721i	0.892671	−	0.450709i	$-0.148829\pi$
$839$	667796.	+	385552.i	0.948681	+	0.547721i	0.0560100	+	0.998430i	$0.482162\pi$
$840$		0			0
$841$	−97125.1	−	168226.i	−0.137322	−	0.237848i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			78093.4i			0.109371i
$846$		0			0
$847$	−391030.			−0.545058
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−10343.8	+	5972.02i	−0.0142831	+	0.00824635i
$852$		0			0
$853$	−541184.	+	937358.i	−0.743784	+	1.28827i	0.206977	+	0.978346i	$0.433637\pi$
$853$	−541184.	+	937358.i	−0.743784	+	1.28827i	−0.950761	+	0.309925i	$0.899696\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	461890.	+	266672.i	0.628893	+	0.363092i	0.780323	−	0.625376i	$-0.215055\pi$
$857$	461890.	+	266672.i	0.628893	+	0.363092i	−0.151430	+	0.988468i	$0.548388\pi$
$858$		0			0
$859$	565525.	+	979517.i	0.766417	+	1.32747i	0.939494	+	0.342565i	$0.111296\pi$
$859$	565525.	+	979517.i	0.766417	+	1.32747i	−0.173077	+	0.984908i	$0.555371\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			110025.i			0.147730i	0.997268	+	0.0738649i	$0.0235334\pi$
$863$			110025.i			0.147730i	−0.997268	+	0.0738649i	$0.976467\pi$
$864$		0			0
$865$	161503.			0.215848
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−477065.	+	275433.i	−0.631739	+	0.364735i
$870$		0			0
$871$	−356638.	+	617715.i	−0.470101	+	0.814238i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−999020.	−	576784.i	−1.30484	−	0.753351i
$876$		0			0
$877$	204377.	+	353991.i	0.265725	+	0.460249i	0.967753	−	0.251900i	$-0.0810554\pi$
$877$	204377.	+	353991.i	0.265725	+	0.460249i	−0.702028	+	0.712149i	$0.747722\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			235809.i			0.303815i	0.988395	+	0.151907i	$0.0485416\pi$
$881$			235809.i			0.303815i	−0.988395	+	0.151907i	$0.951458\pi$
$882$		0			0
$883$	273719.			0.351061			0.175531	−	0.984474i	$-0.443836\pi$
$883$	273719.			0.351061			0.175531	+	0.984474i	$0.443836\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	322071.	−	185948.i	0.409359	−	0.236344i	−0.281155	−	0.959662i	$-0.590718\pi$
$887$	322071.	−	185948.i	0.409359	−	0.236344i	0.690514	+	0.723319i	$0.257384\pi$
$888$		0			0
$889$	748789.	−	1.29694e6i	0.947449	−	1.64103i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−653108.	−	377072.i	−0.818996	−	0.472848i
$894$		0			0
$895$	9737.36	+	16865.6i	0.0121561	+	0.0210550i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			216718.i			0.268148i
$900$		0			0
$901$	143176.			0.176368
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−393522.	+	227200.i	−0.480476	+	0.277403i
$906$		0			0
$907$	−71965.4	+	124648.i	−0.0874801	+	0.151520i	−0.906445	−	0.422323i	$-0.861215\pi$
$907$	−71965.4	+	124648.i	−0.0874801	+	0.151520i	0.818965	+	0.573843i	$0.194548\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−772201.	−	445830.i	−0.930451	−	0.537196i	−0.0434969	−	0.999054i	$-0.513850\pi$
$911$	−772201.	−	445830.i	−0.930451	−	0.537196i	−0.886954	+	0.461857i	$0.847183\pi$
$912$		0			0
$913$	136170.	+	235854.i	0.163358	+	0.282944i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	232053.i		−	0.275961i
$918$		0			0
$919$	582156.			0.689300			0.344650	−	0.938731i	$-0.387998\pi$
$919$	582156.			0.689300			0.344650	+	0.938731i	$0.387998\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−892636.	+	515364.i	−1.04778	+	0.604937i
$924$		0			0
$925$	−152149.	+	263530.i	−0.177822	+	0.307997i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	668027.	+	385685.i	0.774038	+	0.446891i	0.834313	−	0.551291i	$-0.185864\pi$
$929$	668027.	+	385685.i	0.774038	+	0.446891i	−0.0602751	+	0.998182i	$0.519198\pi$
$930$		0			0
$931$	−271966.	−	471059.i	−0.313773	−	0.543470i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	112148.i		−	0.128283i
$936$		0			0
$937$	−1.46097e6			−1.66403			−0.832015	−	0.554753i	$-0.812813\pi$
$937$	−1.46097e6			−1.66403			−0.832015	+	0.554753i	$0.812813\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−146658.	+	84673.2i	−0.165626	+	0.0956240i	−0.580522	−	0.814245i	$-0.697151\pi$
$941$	−146658.	+	84673.2i	−0.165626	+	0.0956240i	0.414896	+	0.909869i	$0.363818\pi$
$942$		0			0
$943$	4019.42	−	6961.83i	0.00452001	−	0.00782889i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	514322.	+	296944.i	0.573503	+	0.331112i	0.758547	−	0.651618i	$-0.225910\pi$
$947$	514322.	+	296944.i	0.573503	+	0.331112i	−0.185044	+	0.982730i	$0.559243\pi$
$948$		0			0
$949$	371210.	+	642955.i	0.412180	+	0.713917i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1.25477e6i		−	1.38159i	−0.723050	−	0.690796i	$-0.757260\pi$
$953$		−	1.25477e6i		−	1.38159i	0.723050	−	0.690796i	$-0.242740\pi$
$954$		0			0
$955$	156250.			0.171322
$956$		0			0
$957$		0			0
$958$		0			0
$959$	243051.	−	140326.i	0.264278	−	0.152581i
$960$		0			0
$961$	415987.	−	720510.i	0.450436	−	0.780178i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−741017.	−	427826.i	−0.795744	−	0.459423i
$966$		0			0
$967$	−528928.	−	916130.i	−0.565645	−	0.979725i	−0.996989	−	0.0775381i	$-0.975294\pi$
$967$	−528928.	−	916130.i	−0.565645	−	0.979725i	0.431345	−	0.902187i	$-0.358039\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	775745.i		−	0.822774i	−0.911461	−	0.411387i	$-0.865044\pi$
$971$		−	775745.i		−	0.822774i	0.911461	−	0.411387i	$-0.134956\pi$
$972$		0			0
$973$	−846555.			−0.894189
$974$		0			0
$975$		0			0
$976$		0			0
$977$	429995.	−	248258.i	0.450479	−	0.260084i	−0.257554	−	0.966264i	$-0.582916\pi$
$977$	429995.	−	248258.i	0.450479	−	0.260084i	0.708032	+	0.706180i	$0.249583\pi$
$978$		0			0
$979$	686453.	−	1.18897e6i	0.716218	−	1.24053i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	260643.	+	150482.i	0.269736	+	0.155732i	0.628768	−	0.777593i	$-0.283560\pi$
$983$	260643.	+	150482.i	0.269736	+	0.155732i	−0.359032	+	0.933325i	$0.616893\pi$
$984$		0			0
$985$	−549076.	−	951028.i	−0.565927	−	0.980214i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			12903.9i			0.0131925i
$990$		0			0
$991$	−1.70094e6			−1.73197			−0.865987	−	0.500067i	$-0.833309\pi$
$991$	−1.70094e6			−1.73197			−0.865987	+	0.500067i	$0.833309\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−114302.	+	65992.4i	−0.115454	+	0.0666573i
$996$		0			0
$997$	312919.	−	541991.i	0.314805	−	0.545258i	−0.664591	−	0.747207i	$-0.731394\pi$
$997$	312919.	−	541991.i	0.314805	−	0.545258i	0.979396	+	0.201949i	$0.0647276\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.5.q.a.305.1		6
3.2	odd	2		144.5.q.a.65.3		6
4.3	odd	2		27.5.d.a.8.1		6
9.2	odd	6		1296.5.e.c.161.5		6
9.4	even	3		144.5.q.a.113.3		6
9.5	odd	6	inner	432.5.q.a.17.1		6
9.7	even	3		1296.5.e.c.161.2		6
12.11	even	2		9.5.d.a.2.3	✓	6
36.7	odd	6		81.5.b.a.80.5		6
36.11	even	6		81.5.b.a.80.2		6
36.23	even	6		27.5.d.a.17.1		6
36.31	odd	6		9.5.d.a.5.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.5.d.a.2.3	✓	6	12.11	even	2
9.5.d.a.5.3	yes	6	36.31	odd	6
27.5.d.a.8.1		6	4.3	odd	2
27.5.d.a.17.1		6	36.23	even	6
81.5.b.a.80.2		6	36.11	even	6
81.5.b.a.80.5		6	36.7	odd	6
144.5.q.a.65.3		6	3.2	odd	2
144.5.q.a.113.3		6	9.4	even	3
432.5.q.a.17.1		6	9.5	odd	6	inner
432.5.q.a.305.1		6	1.1	even	1	trivial
1296.5.e.c.161.2		6	9.7	even	3
1296.5.e.c.161.5		6	9.2	odd	6

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:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	432.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-13.8760 + 8.01130i) q^{5} +(36.2418 - 62.7727i) q^{7} +(83.2749 + 48.0788i) q^{11} +(-76.9530 - 133.286i) q^{13} +72.7905i q^{17} +190.660 q^{19} +(-12.5186 + 7.22762i) q^{23} +(-184.138 + 318.937i) q^{25} +(-620.301 - 358.131i) q^{29} +(-151.284 - 262.031i) q^{31} +1161.38i q^{35} +826.277 q^{37} +(-481.613 + 278.059i) q^{41} +(446.340 - 773.084i) q^{43} +(-3425.50 - 1977.72i) q^{47} +(-1426.44 - 2470.67i) q^{49} -1966.96i q^{53} -1540.69 q^{55} +(4689.48 - 2707.47i) q^{59} +(856.210 - 1483.00i) q^{61} +(2135.60 + 1232.99i) q^{65} +(-2317.24 - 4013.58i) q^{67} -6697.12i q^{71} -4823.86 q^{73} +(6036.07 - 3484.93i) q^{77} +(-2864.40 + 4961.28i) q^{79} +(2452.78 + 1416.12i) q^{83} +(-583.147 - 1010.04i) q^{85} -14277.7i q^{89} -11155.7 q^{91} +(-2645.60 + 1527.44i) q^{95} +(-3582.65 + 6205.34i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{5} - 12 q^{7} + 483 q^{11} - 6 q^{13} + 258 q^{19} - 282 q^{23} - 273 q^{25} + 1056 q^{29} - 1290 q^{31} + 12 q^{37} - 7629 q^{41} + 285 q^{43} - 9642 q^{47} - 1863 q^{49} - 2016 q^{55} + 6225 q^{59}+ \cdots - 28959 q^{97}+O(q^{100}) \) 6 * q + 12 * q^5 - 12 * q^7 + 483 * q^11 - 6 * q^13 + 258 * q^19 - 282 * q^23 - 273 * q^25 + 1056 * q^29 - 1290 * q^31 + 12 * q^37 - 7629 * q^41 + 285 * q^43 - 9642 * q^47 - 1863 * q^49 - 2016 * q^55 + 6225 * q^59 + 3630 * q^61 + 7158 * q^65 + 5055 * q^67 - 14622 * q^73 - 2580 * q^77 - 4764 * q^79 - 1866 * q^83 + 12366 * q^85 - 34836 * q^91 - 13362 * q^95 - 28959 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more