LMFDB - Embedded newform 1620.4.i.h.541.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(541,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1620.541");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.i (of order $3$, 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:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			541.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1620.541
Dual form			1620.4.i.h.1081.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50000 + 4.33013i) q^{5} +(-8.50000 + 14.7224i) q^{7} +O(q^{10})$	$q+(2.50000 + 4.33013i) q^{5} +(-8.50000 + 14.7224i) q^{7} +(-15.0000 + 25.9808i) q^{11} +(30.5000 + 52.8275i) q^{13} +120.000 q^{17} -43.0000 q^{19} +(45.0000 + 77.9423i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(-45.0000 + 77.9423i) q^{29} +(-4.00000 - 6.92820i) q^{31} -85.0000 q^{35} +317.000 q^{37} +(-15.0000 - 25.9808i) q^{41} +(110.000 - 190.526i) q^{43} +(-90.0000 + 155.885i) q^{47} +(27.0000 + 46.7654i) q^{49} +630.000 q^{53} -150.000 q^{55} +(-420.000 - 727.461i) q^{59} +(-299.500 + 518.749i) q^{61} +(-152.500 + 264.138i) q^{65} +(-53.5000 - 92.6647i) q^{67} +210.000 q^{71} -421.000 q^{73} +(-255.000 - 441.673i) q^{77} +(-176.500 + 305.707i) q^{79} +(-675.000 + 1169.13i) q^{83} +(300.000 + 519.615i) q^{85} -1020.00 q^{89} -1037.00 q^{91} +(-107.500 - 186.195i) q^{95} +(498.500 - 863.427i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5 q^{5} - 17 q^{7}+O(q^{10}) $ 2 * q + 5 * q^5 - 17 * q^7	$ 2 q + 5 q^{5} - 17 q^{7} - 30 q^{11} + 61 q^{13} + 240 q^{17} - 86 q^{19} + 90 q^{23} - 25 q^{25} - 90 q^{29} - 8 q^{31} - 170 q^{35} + 634 q^{37} - 30 q^{41} + 220 q^{43} - 180 q^{47} + 54 q^{49} + 1260 q^{53} - 300 q^{55} - 840 q^{59} - 599 q^{61} - 305 q^{65} - 107 q^{67} + 420 q^{71} - 842 q^{73} - 510 q^{77} - 353 q^{79} - 1350 q^{83} + 600 q^{85} - 2040 q^{89} - 2074 q^{91} - 215 q^{95} + 997 q^{97}+O(q^{100}) $ 2 * q + 5 * q^5 - 17 * q^7 - 30 * q^11 + 61 * q^13 + 240 * q^17 - 86 * q^19 + 90 * q^23 - 25 * q^25 - 90 * q^29 - 8 * q^31 - 170 * q^35 + 634 * q^37 - 30 * q^41 + 220 * q^43 - 180 * q^47 + 54 * q^49 + 1260 * q^53 - 300 * q^55 - 840 * q^59 - 599 * q^61 - 305 * q^65 - 107 * q^67 + 420 * q^71 - 842 * q^73 - 510 * q^77 - 353 * q^79 - 1350 * q^83 + 600 * q^85 - 2040 * q^89 - 2074 * q^91 - 215 * q^95 + 997 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$6$		0			0
$7$	−8.50000	+	14.7224i	−0.458957	+	0.794937i	−0.998906	−	0.0467610i	$-0.985110\pi$
$7$	−8.50000	+	14.7224i	−0.458957	+	0.794937i	0.539949	+	0.841698i	$0.318443\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−15.0000	+	25.9808i	−0.411152	+	0.712136i	−0.995016	−	0.0997155i	$-0.968207\pi$
$11$	−15.0000	+	25.9808i	−0.411152	+	0.712136i	0.583864	+	0.811851i	$0.301540\pi$
$12$		0			0
$13$	30.5000	+	52.8275i	0.650706	+	1.12706i	0.982952	+	0.183863i	$0.0588603\pi$
$13$	30.5000	+	52.8275i	0.650706	+	1.12706i	−0.332246	+	0.943193i	$0.607806\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	120.000			1.71202			0.856008	−	0.516962i	$-0.172937\pi$
$17$	120.000			1.71202			0.856008	+	0.516962i	$0.172937\pi$
$18$		0			0
$19$	−43.0000			−0.519204			−0.259602	−	0.965716i	$-0.583591\pi$
$19$	−43.0000			−0.519204			−0.259602	+	0.965716i	$0.583591\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	45.0000	+	77.9423i	0.407963	+	0.706613i	0.994661	−	0.103193i	$-0.0329059\pi$
$23$	45.0000	+	77.9423i	0.407963	+	0.706613i	−0.586698	+	0.809806i	$0.699573\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−45.0000	+	77.9423i	−0.288148	+	0.499087i	−0.973368	−	0.229250i	$-0.926373\pi$
$29$	−45.0000	+	77.9423i	−0.288148	+	0.499087i	0.685220	+	0.728336i	$0.259706\pi$
$30$		0			0
$31$	−4.00000	−	6.92820i	−0.0231749	−	0.0401401i	0.854205	−	0.519936i	$-0.174044\pi$
$31$	−4.00000	−	6.92820i	−0.0231749	−	0.0401401i	−0.877380	+	0.479796i	$0.840711\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−85.0000			−0.410503
$36$		0			0
$37$	317.000			1.40850			0.704250	−	0.709952i	$-0.251284\pi$
$37$	317.000			1.40850			0.704250	+	0.709952i	$0.251284\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−15.0000	−	25.9808i	−0.0571367	−	0.0989637i	0.836042	−	0.548665i	$-0.184864\pi$
$41$	−15.0000	−	25.9808i	−0.0571367	−	0.0989637i	−0.893179	+	0.449701i	$0.851530\pi$
$42$		0			0
$43$	110.000	−	190.526i	0.390113	−	0.675695i	−0.602351	−	0.798231i	$-0.705769\pi$
$43$	110.000	−	190.526i	0.390113	−	0.675695i	0.992464	+	0.122536i	$0.0391027\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−90.0000	+	155.885i	−0.279316	+	0.483789i	−0.971215	−	0.238205i	$-0.923441\pi$
$47$	−90.0000	+	155.885i	−0.279316	+	0.483789i	0.691899	+	0.721994i	$0.256774\pi$
$48$		0			0
$49$	27.0000	+	46.7654i	0.0787172	+	0.136342i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	630.000			1.63278			0.816388	−	0.577503i	$-0.195973\pi$
$53$	630.000			1.63278			0.816388	+	0.577503i	$0.195973\pi$
$54$		0			0
$55$	−150.000			−0.367745
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−420.000	−	727.461i	−0.926769	−	1.60521i	−0.788692	−	0.614789i	$-0.789241\pi$
$59$	−420.000	−	727.461i	−0.926769	−	1.60521i	−0.138077	−	0.990422i	$-0.544092\pi$
$60$		0			0
$61$	−299.500	+	518.749i	−0.628640	+	1.08884i	0.359185	+	0.933266i	$0.383055\pi$
$61$	−299.500	+	518.749i	−0.628640	+	1.08884i	−0.987825	+	0.155570i	$0.950279\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−152.500	+	264.138i	−0.291005	+	0.504035i
$66$		0			0
$67$	−53.5000	−	92.6647i	−0.0975532	−	0.168967i	0.813118	−	0.582099i	$-0.197768\pi$
$67$	−53.5000	−	92.6647i	−0.0975532	−	0.168967i	−0.910671	+	0.413132i	$0.864435\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	210.000			0.351020			0.175510	−	0.984478i	$-0.443843\pi$
$71$	210.000			0.351020			0.175510	+	0.984478i	$0.443843\pi$
$72$		0			0
$73$	−421.000			−0.674991			−0.337495	−	0.941327i	$-0.609580\pi$
$73$	−421.000			−0.674991			−0.337495	+	0.941327i	$0.609580\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−255.000	−	441.673i	−0.377402	−	0.653679i
$78$		0			0
$79$	−176.500	+	305.707i	−0.251365	+	0.435376i	−0.963902	−	0.266258i	$-0.914213\pi$
$79$	−176.500	+	305.707i	−0.251365	+	0.435376i	0.712537	+	0.701634i	$0.247546\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−675.000	+	1169.13i	−0.892661	+	1.54613i	−0.0559884	+	0.998431i	$0.517831\pi$
$83$	−675.000	+	1169.13i	−0.892661	+	1.54613i	−0.836673	+	0.547703i	$0.815502\pi$
$84$		0			0
$85$	300.000	+	519.615i	0.382818	+	0.663061i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1020.00			−1.21483			−0.607415	−	0.794385i	$-0.707793\pi$
$89$	−1020.00			−1.21483			−0.607415	+	0.794385i	$0.707793\pi$
$90$		0			0
$91$	−1037.00			−1.19458
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−107.500	−	186.195i	−0.116098	−	0.201087i
$96$		0			0
$97$	498.500	−	863.427i	0.521804	−	0.903791i	−0.477874	−	0.878428i	$-0.658592\pi$
$97$	498.500	−	863.427i	0.521804	−	0.903791i	0.999678	−	0.0253630i	$-0.00807417\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−480.000	+	831.384i	−0.472889	+	0.819068i	−0.999519	−	0.0310272i	$-0.990122\pi$
$101$	−480.000	+	831.384i	−0.472889	+	0.819068i	0.526630	+	0.850095i	$0.323455\pi$
$102$		0			0
$103$	−590.500	−	1022.78i	−0.564890	−	0.978419i	−0.997060	−	0.0766263i	$-0.975585\pi$
$103$	−590.500	−	1022.78i	−0.564890	−	0.978419i	0.432170	−	0.901792i	$-0.357748\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−330.000			−0.298152			−0.149076	−	0.988826i	$-0.547630\pi$
$107$	−330.000			−0.298152			−0.149076	+	0.988826i	$0.547630\pi$
$108$		0			0
$109$	1454.00			1.27769			0.638844	−	0.769336i	$-0.279413\pi$
$109$	1454.00			1.27769			0.638844	+	0.769336i	$0.279413\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	615.000	+	1065.21i	0.511985	+	0.886784i	0.999903	+	0.0138951i	$0.00442309\pi$
$113$	615.000	+	1065.21i	0.511985	+	0.886784i	−0.487918	+	0.872889i	$0.662244\pi$
$114$		0			0
$115$	−225.000	+	389.711i	−0.182447	+	0.316007i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1020.00	+	1766.69i	−0.785742	+	1.36094i
$120$		0			0
$121$	215.500	+	373.257i	0.161908	+	0.280433i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−125.000			−0.0894427
$126$		0			0
$127$	1280.00			0.894344			0.447172	−	0.894448i	$-0.352431\pi$
$127$	1280.00			0.894344			0.447172	+	0.894448i	$0.352431\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1110.00	−	1922.58i	−0.740314	−	1.28226i	−0.952352	−	0.305000i	$-0.901343\pi$
$131$	−1110.00	−	1922.58i	−0.740314	−	1.28226i	0.212038	−	0.977261i	$-0.431990\pi$
$132$		0			0
$133$	365.500	−	633.065i	0.238292	−	0.412734i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−585.000	+	1013.25i	−0.364817	+	0.631882i	−0.988747	−	0.149599i	$-0.952202\pi$
$137$	−585.000	+	1013.25i	−0.364817	+	0.631882i	0.623930	+	0.781480i	$0.285535\pi$
$138$		0			0
$139$	696.500	+	1206.37i	0.425010	+	0.736139i	0.996421	−	0.0845251i	$-0.0269373\pi$
$139$	696.500	+	1206.37i	0.425010	+	0.736139i	−0.571412	+	0.820664i	$0.693604\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1830.00			−1.07016
$144$		0			0
$145$	−450.000			−0.257727
$146$		0			0
$147$		0			0
$148$		0			0
$149$	690.000	+	1195.12i	0.379376	+	0.657098i	0.990972	−	0.134072i	$-0.0428054\pi$
$149$	690.000	+	1195.12i	0.379376	+	0.657098i	−0.611596	+	0.791170i	$0.709472\pi$
$150$		0			0
$151$	1329.50	−	2302.76i	0.716511	−	1.24103i	−0.245863	−	0.969305i	$-0.579071\pi$
$151$	1329.50	−	2302.76i	0.716511	−	1.24103i	0.962374	−	0.271729i	$-0.0875954\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	20.0000	−	34.6410i	0.0103641	−	0.0179512i
$156$		0			0
$157$	−925.000	−	1602.15i	−0.470210	−	0.814428i	0.529209	−	0.848491i	$-0.322489\pi$
$157$	−925.000	−	1602.15i	−0.470210	−	0.814428i	−0.999420	+	0.0340630i	$0.989155\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1530.00			−0.748950
$162$		0			0
$163$	1121.00			0.538672			0.269336	−	0.963046i	$-0.413196\pi$
$163$	1121.00			0.538672			0.269336	+	0.963046i	$0.413196\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	150.000	+	259.808i	0.0695051	+	0.120386i	0.898684	−	0.438598i	$-0.144525\pi$
$167$	150.000	+	259.808i	0.0695051	+	0.120386i	−0.829178	+	0.558984i	$0.811191\pi$
$168$		0			0
$169$	−762.000	+	1319.82i	−0.346837	+	0.600739i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−810.000	+	1402.96i	−0.355972	+	0.616562i	−0.987284	−	0.158967i	$-0.949183\pi$
$173$	−810.000	+	1402.96i	−0.355972	+	0.616562i	0.631312	+	0.775529i	$0.282517\pi$
$174$		0			0
$175$	−212.500	−	368.061i	−0.0917914	−	0.158987i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	630.000			0.263064			0.131532	−	0.991312i	$-0.458010\pi$
$179$	630.000			0.263064			0.131532	+	0.991312i	$0.458010\pi$
$180$		0			0
$181$	−2299.00			−0.944107			−0.472053	−	0.881570i	$-0.656487\pi$
$181$	−2299.00			−0.944107			−0.472053	+	0.881570i	$0.656487\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	792.500	+	1372.65i	0.314950	+	0.545509i
$186$		0			0
$187$	−1800.00	+	3117.69i	−0.703899	+	1.21919i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	450.000	−	779.423i	0.170476	−	0.295273i	−0.768111	−	0.640317i	$-0.778803\pi$
$191$	450.000	−	779.423i	0.170476	−	0.295273i	0.938586	+	0.345045i	$0.112136\pi$
$192$		0			0
$193$	−1730.50	−	2997.31i	−0.645410	−	1.11788i	−0.984207	−	0.177023i	$-0.943353\pi$
$193$	−1730.50	−	2997.31i	−0.645410	−	1.11788i	0.338797	−	0.940860i	$-0.389980\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4560.00			−1.64917			−0.824585	−	0.565738i	$-0.808591\pi$
$197$	−4560.00			−1.64917			−0.824585	+	0.565738i	$0.808591\pi$
$198$		0			0
$199$	−2077.00			−0.739872			−0.369936	−	0.929057i	$-0.620620\pi$
$199$	−2077.00			−0.739872			−0.369936	+	0.929057i	$0.620620\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−765.000	−	1325.02i	−0.264495	−	0.458119i
$204$		0			0
$205$	75.0000	−	129.904i	0.0255523	−	0.0442579i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	645.000	−	1117.17i	0.213472	−	0.369744i
$210$		0			0
$211$	2010.50	+	3482.29i	0.655965	+	1.13616i	0.981651	+	0.190686i	$0.0610713\pi$
$211$	2010.50	+	3482.29i	0.655965	+	1.13616i	−0.325686	+	0.945478i	$0.605595\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1100.00			0.348927
$216$		0			0
$217$	136.000			0.0425451
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3660.00	+	6339.31i	1.11402	+	1.92954i
$222$		0			0
$223$	−40.0000	+	69.2820i	−0.0120117	+	0.0208048i	−0.871969	−	0.489562i	$-0.837157\pi$
$223$	−40.0000	+	69.2820i	−0.0120117	+	0.0208048i	0.859957	+	0.510366i	$0.170490\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1875.00	+	3247.60i	−0.548230	+	0.949562i	0.450166	+	0.892945i	$0.351365\pi$
$227$	−1875.00	+	3247.60i	−0.548230	+	0.949562i	−0.998396	+	0.0566168i	$0.981969\pi$
$228$		0			0
$229$	617.000	+	1068.68i	0.178046	+	0.308385i	0.941211	−	0.337819i	$-0.109689\pi$
$229$	617.000	+	1068.68i	0.178046	+	0.308385i	−0.763165	+	0.646203i	$0.776356\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5880.00			−1.65327			−0.826634	−	0.562739i	$-0.809747\pi$
$233$	−5880.00			−1.65327			−0.826634	+	0.562739i	$0.809747\pi$
$234$		0			0
$235$	−900.000			−0.249828
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2565.00	+	4442.71i	0.694209	+	1.20241i	0.970446	+	0.241316i	$0.0775792\pi$
$239$	2565.00	+	4442.71i	0.694209	+	1.20241i	−0.276237	+	0.961090i	$0.589087\pi$
$240$		0			0
$241$	3615.50	−	6262.23i	0.966369	−	1.67380i	0.260476	−	0.965480i	$-0.416120\pi$
$241$	3615.50	−	6262.23i	0.966369	−	1.67380i	0.705892	−	0.708319i	$-0.250546\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−135.000	+	233.827i	−0.0352034	+	0.0609741i
$246$		0			0
$247$	−1311.50	−	2271.58i	−0.337849	−	0.585172i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−7530.00			−1.89358			−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−7530.00			−1.89358			−0.946792	+	0.321847i	$0.895696\pi$
$252$		0			0
$253$	−2700.00			−0.670939
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2280.00	+	3949.08i	0.553395	+	0.958508i	0.998026	+	0.0627945i	$0.0200013\pi$
$257$	2280.00	+	3949.08i	0.553395	+	0.958508i	−0.444632	+	0.895714i	$0.646665\pi$
$258$		0			0
$259$	−2694.50	+	4667.01i	−0.646440	+	1.11967i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1050.00	+	1818.65i	−0.246182	+	0.426399i	−0.962463	−	0.271412i	$-0.912509\pi$
$263$	−1050.00	+	1818.65i	−0.246182	+	0.426399i	0.716282	+	0.697811i	$0.245843\pi$
$264$		0			0
$265$	1575.00	+	2727.98i	0.365100	+	0.632372i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3120.00			−0.707174			−0.353587	−	0.935402i	$-0.615038\pi$
$269$	−3120.00			−0.707174			−0.353587	+	0.935402i	$0.615038\pi$
$270$		0			0
$271$	3449.00			0.773106			0.386553	−	0.922267i	$-0.373666\pi$
$271$	3449.00			0.773106			0.386553	+	0.922267i	$0.373666\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−375.000	−	649.519i	−0.0822304	−	0.142427i
$276$		0			0
$277$	−1885.00	+	3264.92i	−0.408876	+	0.708194i	−0.994764	−	0.102198i	$-0.967413\pi$
$277$	−1885.00	+	3264.92i	−0.408876	+	0.708194i	0.585888	+	0.810392i	$0.300746\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3360.00	+	5819.69i	−0.713312	+	1.23549i	0.250294	+	0.968170i	$0.419473\pi$
$281$	−3360.00	+	5819.69i	−0.713312	+	1.23549i	−0.963607	+	0.267324i	$0.913861\pi$
$282$		0			0
$283$	50.0000	+	86.6025i	0.0105024	+	0.0181908i	0.871229	−	0.490877i	$-0.163324\pi$
$283$	50.0000	+	86.6025i	0.0105024	+	0.0181908i	−0.860726	+	0.509068i	$0.829990\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	510.000			0.104893
$288$		0			0
$289$	9487.00			1.93100
$290$		0			0
$291$		0			0
$292$		0			0
$293$	435.000	+	753.442i	0.0867337	+	0.150227i	0.906129	−	0.423002i	$-0.139024\pi$
$293$	435.000	+	753.442i	0.0867337	+	0.150227i	−0.819395	+	0.573229i	$0.805690\pi$
$294$		0			0
$295$	2100.00	−	3637.31i	0.414463	−	0.717872i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2745.00	+	4754.48i	−0.530928	+	0.919594i
$300$		0			0
$301$	1870.00	+	3238.94i	0.358090	+	0.620230i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2995.00			−0.562273
$306$		0			0
$307$	3440.00			0.639515			0.319758	−	0.947499i	$-0.396399\pi$
$307$	3440.00			0.639515			0.319758	+	0.947499i	$0.396399\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2940.00	−	5092.23i	−0.536052	−	0.928469i	−0.999112	−	0.0421419i	$-0.986582\pi$
$311$	−2940.00	−	5092.23i	−0.536052	−	0.928469i	0.463060	−	0.886327i	$-0.346751\pi$
$312$		0			0
$313$	−920.500	+	1594.35i	−0.166229	+	0.287917i	−0.937091	−	0.349085i	$-0.886492\pi$
$313$	−920.500	+	1594.35i	−0.166229	+	0.287917i	0.770862	+	0.637002i	$0.219826\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1710.00	−	2961.81i	0.302975	−	0.524769i	−0.673833	−	0.738884i	$-0.735353\pi$
$317$	1710.00	−	2961.81i	0.302975	−	0.524769i	0.976808	+	0.214115i	$0.0686867\pi$
$318$		0			0
$319$	−1350.00	−	2338.27i	−0.236945	−	0.410401i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−5160.00			−0.888886
$324$		0			0
$325$	−1525.00			−0.260282
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1530.00	−	2650.04i	−0.256388	−	0.444077i
$330$		0			0
$331$	2820.50	−	4885.25i	0.468365	−	0.811231i	−0.530982	−	0.847383i	$-0.678177\pi$
$331$	2820.50	−	4885.25i	0.468365	−	0.811231i	0.999346	+	0.0361519i	$0.0115100\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	267.500	−	463.324i	0.0436271	−	0.0755644i
$336$		0			0
$337$	528.500	+	915.389i	0.0854280	+	0.147966i	0.905574	−	0.424189i	$-0.139441\pi$
$337$	528.500	+	915.389i	0.0854280	+	0.147966i	−0.820146	+	0.572155i	$0.806108\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	240.000			0.0381136
$342$		0			0
$343$	−6749.00			−1.06242
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−645.000	−	1117.17i	−0.0997851	−	0.172833i	0.811811	−	0.583921i	$-0.198482\pi$
$347$	−645.000	−	1117.17i	−0.0997851	−	0.172833i	−0.911596	+	0.411088i	$0.865149\pi$
$348$		0			0
$349$	−1733.50	+	3002.51i	−0.265880	+	0.460518i	−0.967794	−	0.251744i	$-0.918996\pi$
$349$	−1733.50	+	3002.51i	−0.265880	+	0.460518i	0.701914	+	0.712262i	$0.252329\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3870.00	−	6703.04i	0.583511	−	1.01067i	−0.411548	−	0.911388i	$-0.635012\pi$
$353$	3870.00	−	6703.04i	0.583511	−	1.01067i	0.995059	−	0.0992825i	$-0.0316548\pi$
$354$		0			0
$355$	525.000	+	909.327i	0.0784904	+	0.135949i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−8130.00			−1.19522			−0.597611	−	0.801786i	$-0.703883\pi$
$359$	−8130.00			−1.19522			−0.597611	+	0.801786i	$0.703883\pi$
$360$		0			0
$361$	−5010.00			−0.730427
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1052.50	−	1822.98i	−0.150933	−	0.261423i
$366$		0			0
$367$	−6056.50	+	10490.2i	−0.861435	+	1.49205i	0.00910851	+	0.999959i	$0.497101\pi$
$367$	−6056.50	+	10490.2i	−0.861435	+	1.49205i	−0.870544	+	0.492091i	$0.836233\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−5355.00	+	9275.13i	−0.749374	+	1.29795i
$372$		0			0
$373$	−2174.50	−	3766.34i	−0.301853	−	0.522826i	0.674702	−	0.738090i	$-0.264272\pi$
$373$	−2174.50	−	3766.34i	−0.301853	−	0.522826i	−0.976556	+	0.215264i	$0.930939\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5490.00			−0.749998
$378$		0			0
$379$	−7663.00			−1.03858			−0.519290	−	0.854598i	$-0.673804\pi$
$379$	−7663.00			−1.03858			−0.519290	+	0.854598i	$0.673804\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1980.00	−	3429.46i	−0.264160	−	0.457539i	0.703183	−	0.711009i	$-0.251761\pi$
$383$	−1980.00	−	3429.46i	−0.264160	−	0.457539i	−0.967343	+	0.253470i	$0.918428\pi$
$384$		0			0
$385$	1275.00	−	2208.36i	0.168779	−	0.292334i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	660.000	−	1143.15i	0.0860240	−	0.148998i	−0.819803	−	0.572645i	$-0.805917\pi$
$389$	660.000	−	1143.15i	0.0860240	−	0.148998i	0.905827	+	0.423648i	$0.139250\pi$
$390$		0			0
$391$	5400.00	+	9353.07i	0.698439	+	1.20973i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1765.00			−0.224827
$396$		0			0
$397$	14390.0			1.81918			0.909589	−	0.415510i	$-0.136397\pi$
$397$	14390.0			1.81918			0.909589	+	0.415510i	$0.136397\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4305.00	+	7456.48i	0.536113	+	0.928576i	0.999109	+	0.0422147i	$0.0134414\pi$
$401$	4305.00	+	7456.48i	0.536113	+	0.928576i	−0.462995	+	0.886361i	$0.653225\pi$
$402$		0			0
$403$	244.000	−	422.620i	0.0301601	−	0.0522388i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4755.00	+	8235.90i	−0.579107	+	1.00304i
$408$		0			0
$409$	−3548.50	−	6146.18i	−0.429003	−	0.743054i	0.567782	−	0.823179i	$-0.307802\pi$
$409$	−3548.50	−	6146.18i	−0.429003	−	0.743054i	−0.996785	+	0.0801246i	$0.974468\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	14280.0			1.70139
$414$		0			0
$415$	−6750.00			−0.798420
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4440.00	−	7690.31i	−0.517681	−	0.896649i	−0.999789	−	0.0205376i	$-0.993462\pi$
$419$	−4440.00	−	7690.31i	−0.517681	−	0.896649i	0.482108	−	0.876112i	$-0.339871\pi$
$420$		0			0
$421$	2739.50	−	4744.95i	0.317138	−	0.549299i	−0.662752	−	0.748839i	$-0.730612\pi$
$421$	2739.50	−	4744.95i	0.317138	−	0.549299i	0.979890	+	0.199540i	$0.0639449\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1500.00	+	2598.08i	−0.171202	+	0.296530i
$426$		0			0
$427$	−5091.50	−	8818.74i	−0.577037	−	0.999458i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	12510.0			1.39811			0.699055	−	0.715068i	$-0.253604\pi$
$431$	12510.0			1.39811			0.699055	+	0.715068i	$0.253604\pi$
$432$		0			0
$433$	−6790.00			−0.753595			−0.376797	−	0.926296i	$-0.622975\pi$
$433$	−6790.00			−0.753595			−0.376797	+	0.926296i	$0.622975\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1935.00	−	3351.52i	−0.211816	−	0.366876i
$438$		0			0
$439$	5588.00	−	9678.70i	0.607519	−	1.05225i	−0.384129	−	0.923279i	$-0.625498\pi$
$439$	5588.00	−	9678.70i	0.607519	−	1.05225i	0.991648	−	0.128974i	$-0.0411683\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6930.00	+	12003.1i	−0.743238	+	1.28733i	0.207776	+	0.978176i	$0.433377\pi$
$443$	−6930.00	+	12003.1i	−0.743238	+	1.28733i	−0.951014	+	0.309149i	$0.899956\pi$
$444$		0			0
$445$	−2550.00	−	4416.73i	−0.271644	−	0.470501i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	4740.00			0.498206			0.249103	−	0.968477i	$-0.419864\pi$
$449$	4740.00			0.498206			0.249103	+	0.968477i	$0.419864\pi$
$450$		0			0
$451$	900.000			0.0939675
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2592.50	−	4490.34i	−0.267117	−	0.462660i
$456$		0			0
$457$	845.000	−	1463.58i	0.0864933	−	0.149811i	−0.819533	−	0.573032i	$-0.805767\pi$
$457$	845.000	−	1463.58i	0.0864933	−	0.149811i	0.906027	+	0.423221i	$0.139101\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7350.00	−	12730.6i	0.742568	−	1.28616i	−0.208755	−	0.977968i	$-0.566941\pi$
$461$	7350.00	−	12730.6i	0.742568	−	1.28616i	0.951323	−	0.308197i	$-0.0997255\pi$
$462$		0			0
$463$	165.500	+	286.654i	0.0166122	+	0.0287731i	0.874212	−	0.485544i	$-0.161379\pi$
$463$	165.500	+	286.654i	0.0166122	+	0.0287731i	−0.857600	+	0.514318i	$0.828045\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	8580.00			0.850182			0.425091	−	0.905151i	$-0.360242\pi$
$467$	8580.00			0.850182			0.425091	+	0.905151i	$0.360242\pi$
$468$		0			0
$469$	1819.00			0.179091
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3300.00	+	5715.77i	0.320791	+	0.555626i
$474$		0			0
$475$	537.500	−	930.977i	0.0519204	−	0.0899288i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−7395.00	+	12808.5i	−0.705399	+	1.22179i	0.261148	+	0.965299i	$0.415899\pi$
$479$	−7395.00	+	12808.5i	−0.705399	+	1.22179i	−0.966547	+	0.256488i	$0.917434\pi$
$480$		0			0
$481$	9668.50	+	16746.3i	0.916519	+	1.58746i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4985.00			0.466716
$486$		0			0
$487$	13097.0			1.21865			0.609324	−	0.792921i	$-0.291441\pi$
$487$	13097.0			1.21865			0.609324	+	0.792921i	$0.291441\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	795.000	+	1376.98i	0.0730710	+	0.126563i	0.900246	−	0.435382i	$-0.143387\pi$
$491$	795.000	+	1376.98i	0.0730710	+	0.126563i	−0.827175	+	0.561945i	$0.810053\pi$
$492$		0			0
$493$	−5400.00	+	9353.07i	−0.493314	+	0.854445i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1785.00	+	3091.71i	−0.161103	+	0.279039i
$498$		0			0
$499$	−8632.00	−	14951.1i	−0.774392	−	1.34129i	−0.935136	−	0.354289i	$-0.884723\pi$
$499$	−8632.00	−	14951.1i	−0.774392	−	1.34129i	0.160744	−	0.986996i	$-0.448611\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	14730.0			1.30572			0.652861	−	0.757478i	$-0.273569\pi$
$503$	14730.0			1.30572			0.652861	+	0.757478i	$0.273569\pi$
$504$		0			0
$505$	−4800.00			−0.422965
$506$		0			0
$507$		0			0
$508$		0			0
$509$	435.000	+	753.442i	0.0378802	+	0.0656105i	0.884344	−	0.466836i	$-0.154606\pi$
$509$	435.000	+	753.442i	0.0378802	+	0.0656105i	−0.846464	+	0.532446i	$0.821273\pi$
$510$		0			0
$511$	3578.50	−	6198.14i	0.309792	−	0.536575i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2952.50	−	5113.88i	0.252627	−	0.437562i
$516$		0			0
$517$	−2700.00	−	4676.54i	−0.229683	−	0.397822i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6990.00			0.587788			0.293894	−	0.955838i	$-0.405049\pi$
$521$	6990.00			0.587788			0.293894	+	0.955838i	$0.405049\pi$
$522$		0			0
$523$	12119.0			1.01324			0.506622	−	0.862168i	$-0.330894\pi$
$523$	12119.0			1.01324			0.506622	+	0.862168i	$0.330894\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−480.000	−	831.384i	−0.0396758	−	0.0687204i
$528$		0			0
$529$	2033.50	−	3522.13i	0.167132	−	0.289482i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	915.000	−	1584.83i	0.0743584	−	0.128793i
$534$		0			0
$535$	−825.000	−	1428.94i	−0.0666689	−	0.115474i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1620.00			−0.129459
$540$		0			0
$541$	−21511.0			−1.70948			−0.854741	−	0.519054i	$-0.826284\pi$
$541$	−21511.0			−1.70948			−0.854741	+	0.519054i	$0.826284\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3635.00	+	6296.00i	0.285700	+	0.494846i
$546$		0			0
$547$	5403.50	−	9359.14i	0.422371	−	0.731568i	−0.573800	−	0.818996i	$-0.694531\pi$
$547$	5403.50	−	9359.14i	0.422371	−	0.731568i	0.996171	+	0.0874274i	$0.0278646\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1935.00	−	3351.52i	0.149608	−	0.259128i
$552$		0			0
$553$	−3000.50	−	5197.02i	−0.230731	−	0.399638i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	20460.0			1.55641			0.778203	−	0.628013i	$-0.216132\pi$
$557$	20460.0			1.55641			0.778203	+	0.628013i	$0.216132\pi$
$558$		0			0
$559$	13420.0			1.01539
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2490.00	−	4312.81i	−0.186396	−	0.322847i	0.757650	−	0.652661i	$-0.226347\pi$
$563$	−2490.00	−	4312.81i	−0.186396	−	0.322847i	−0.944046	+	0.329814i	$0.893014\pi$
$564$		0			0
$565$	−3075.00	+	5326.06i	−0.228967	+	0.396582i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	7920.00	−	13717.8i	0.583521	−	1.01069i	−0.411537	−	0.911393i	$-0.635008\pi$
$569$	7920.00	−	13717.8i	0.583521	−	1.01069i	0.995058	−	0.0992955i	$-0.0316589\pi$
$570$		0			0
$571$	12195.5	+	21123.2i	0.893810	+	1.54813i	0.835270	+	0.549840i	$0.185311\pi$
$571$	12195.5	+	21123.2i	0.893810	+	1.54813i	0.0585406	+	0.998285i	$0.481355\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2250.00			−0.163185
$576$		0			0
$577$	7673.00			0.553607			0.276803	−	0.960927i	$-0.410725\pi$
$577$	7673.00			0.553607			0.276803	+	0.960927i	$0.410725\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−11475.0	−	19875.3i	−0.819386	−	1.41922i
$582$		0			0
$583$	−9450.00	+	16367.9i	−0.671319	+	1.16276i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	9465.00	−	16393.9i	0.665524	−	1.15272i	−0.313619	−	0.949549i	$-0.601542\pi$
$587$	9465.00	−	16393.9i	0.665524	−	1.15272i	0.979143	−	0.203172i	$-0.0651250\pi$
$588$		0			0
$589$	172.000	+	297.913i	0.0120325	+	0.0208409i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−14190.0			−0.982653			−0.491327	−	0.870975i	$-0.663488\pi$
$593$	−14190.0			−0.982653			−0.491327	+	0.870975i	$0.663488\pi$
$594$		0			0
$595$	−10200.0			−0.702789
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6435.00	−	11145.7i	−0.438943	−	0.760272i	0.558665	−	0.829393i	$-0.311314\pi$
$599$	−6435.00	−	11145.7i	−0.438943	−	0.760272i	−0.997608	+	0.0691215i	$0.977980\pi$
$600$		0			0
$601$	−9799.00	+	16972.4i	−0.665074	+	1.15194i	0.314191	+	0.949360i	$0.398267\pi$
$601$	−9799.00	+	16972.4i	−0.665074	+	1.15194i	−0.979265	+	0.202582i	$0.935067\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1077.50	+	1866.28i	−0.0724076	+	0.125414i
$606$		0			0
$607$	7581.50	+	13131.5i	0.506958	+	0.878077i	0.999968	+	0.00805329i	$0.00256347\pi$
$607$	7581.50	+	13131.5i	0.506958	+	0.878077i	−0.493009	+	0.870024i	$0.664103\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−10980.0			−0.727010
$612$		0			0
$613$	−29599.0			−1.95023			−0.975116	−	0.221695i	$-0.928841\pi$
$613$	−29599.0			−1.95023			−0.975116	+	0.221695i	$0.928841\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1245.00	−	2156.40i	−0.0812347	−	0.140703i	0.822546	−	0.568699i	$-0.192553\pi$
$617$	−1245.00	−	2156.40i	−0.0812347	−	0.140703i	−0.903781	+	0.427996i	$0.859220\pi$
$618$		0			0
$619$	−1856.50	+	3215.55i	−0.120548	+	0.208795i	−0.919984	−	0.391956i	$-0.871798\pi$
$619$	−1856.50	+	3215.55i	−0.120548	+	0.208795i	0.799436	+	0.600751i	$0.205132\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	8670.00	−	15016.9i	0.557554	−	0.965712i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	38040.0			2.41137
$630$		0			0
$631$	19409.0			1.22450			0.612250	−	0.790664i	$-0.290264\pi$
$631$	19409.0			1.22450			0.612250	+	0.790664i	$0.290264\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3200.00	+	5542.56i	0.199981	+	0.346378i
$636$		0			0
$637$	−1647.00	+	2852.69i	−0.102444	+	0.177437i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	3240.00	−	5611.84i	0.199645	−	0.345795i	−0.748768	−	0.662832i	$-0.769355\pi$
$641$	3240.00	−	5611.84i	0.199645	−	0.345795i	0.948413	+	0.317037i	$0.102688\pi$
$642$		0			0
$643$	−15130.0	−	26205.9i	−0.927945	−	1.60725i	−0.786754	−	0.617266i	$-0.788240\pi$
$643$	−15130.0	−	26205.9i	−0.927945	−	1.60725i	−0.141191	−	0.989982i	$-0.545093\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−21510.0			−1.30703			−0.653513	−	0.756916i	$-0.726705\pi$
$647$	−21510.0			−1.30703			−0.653513	+	0.756916i	$0.726705\pi$
$648$		0			0
$649$	25200.0			1.52417
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−270.000	−	467.654i	−0.0161806	−	0.0280256i	0.857822	−	0.513947i	$-0.171817\pi$
$653$	−270.000	−	467.654i	−0.0161806	−	0.0280256i	−0.874002	+	0.485922i	$0.838484\pi$
$654$		0			0
$655$	5550.00	−	9612.88i	0.331079	−	0.573445i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	14640.0	−	25357.2i	0.865392	−	1.49890i	−0.00126511	−	0.999999i	$-0.500403\pi$
$659$	14640.0	−	25357.2i	0.865392	−	1.49890i	0.866657	−	0.498904i	$-0.166264\pi$
$660$		0			0
$661$	10884.5	+	18852.5i	0.640481	+	1.10935i	0.985325	+	0.170686i	$0.0545983\pi$
$661$	10884.5	+	18852.5i	0.640481	+	1.10935i	−0.344844	+	0.938660i	$0.612068\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3655.00			0.213135
$666$		0			0
$667$	−8100.00			−0.470215
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−8985.00	−	15562.5i	−0.516933	−	0.895354i
$672$		0			0
$673$	7275.50	−	12601.5i	0.416716	−	0.721773i	−0.578891	−	0.815405i	$-0.696514\pi$
$673$	7275.50	−	12601.5i	0.416716	−	0.721773i	0.995607	+	0.0936316i	$0.0298476\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−750.000	+	1299.04i	−0.0425773	+	0.0737461i	−0.886529	−	0.462674i	$-0.846890\pi$
$677$	−750.000	+	1299.04i	−0.0425773	+	0.0737461i	0.843951	+	0.536420i	$0.180224\pi$
$678$		0			0
$679$	8474.50	+	14678.3i	0.478971	+	0.829602i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−16530.0			−0.926066			−0.463033	−	0.886341i	$-0.653239\pi$
$683$	−16530.0			−0.926066			−0.463033	+	0.886341i	$0.653239\pi$
$684$		0			0
$685$	−5850.00			−0.326302
$686$		0			0
$687$		0			0
$688$		0			0
$689$	19215.0	+	33281.4i	1.06246	+	1.84023i
$690$		0			0
$691$	12986.0	−	22492.4i	0.714921	−	1.23828i	−0.248068	−	0.968743i	$-0.579796\pi$
$691$	12986.0	−	22492.4i	0.714921	−	1.23828i	0.962990	−	0.269538i	$-0.0868709\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3482.50	+	6031.87i	−0.190070	+	0.329211i
$696$		0			0
$697$	−1800.00	−	3117.69i	−0.0978190	−	0.169428i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	10230.0			0.551187			0.275593	−	0.961274i	$-0.411126\pi$
$701$	10230.0			0.551187			0.275593	+	0.961274i	$0.411126\pi$
$702$		0			0
$703$	−13631.0			−0.731299
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−8160.00	−	14133.5i	−0.434071	−	0.751833i
$708$		0			0
$709$	7311.50	−	12663.9i	0.387291	−	0.670807i	−0.604793	−	0.796382i	$-0.706744\pi$
$709$	7311.50	−	12663.9i	0.387291	−	0.670807i	0.992084	+	0.125575i	$0.0400776\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	360.000	−	623.538i	0.0189090	−	0.0327513i
$714$		0			0
$715$	−4575.00	−	7924.13i	−0.239294	−	0.414470i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6690.00			−0.347003			−0.173501	−	0.984834i	$-0.555508\pi$
$719$	−6690.00			−0.347003			−0.173501	+	0.984834i	$0.555508\pi$
$720$		0			0
$721$	20077.0			1.03704
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1125.00	−	1948.56i	−0.0576296	−	0.0998174i
$726$		0			0
$727$	−5860.00	+	10149.8i	−0.298948	+	0.517794i	−0.975896	−	0.218238i	$-0.929969\pi$
$727$	−5860.00	+	10149.8i	−0.298948	+	0.517794i	0.676947	+	0.736031i	$0.263302\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	13200.0	−	22863.1i	0.667879	−	1.15680i
$732$		0			0
$733$	2375.00	+	4113.62i	0.119676	+	0.207285i	0.919639	−	0.392764i	$-0.128481\pi$
$733$	2375.00	+	4113.62i	0.119676	+	0.207285i	−0.799963	+	0.600049i	$0.795148\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3210.00			0.160437
$738$		0			0
$739$	−30724.0			−1.52936			−0.764682	−	0.644407i	$-0.777104\pi$
$739$	−30724.0			−1.52936			−0.764682	+	0.644407i	$0.777104\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	480.000	+	831.384i	0.0237005	+	0.0410505i	0.877632	−	0.479334i	$-0.159122\pi$
$743$	480.000	+	831.384i	0.0237005	+	0.0410505i	−0.853932	+	0.520385i	$0.825789\pi$
$744$		0			0
$745$	−3450.00	+	5975.58i	−0.169662	+	0.293863i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2805.00	−	4858.40i	0.136839	−	0.237012i
$750$		0			0
$751$	−11390.5	−	19728.9i	−0.553456	−	0.958613i	−0.998022	−	0.0628674i	$-0.979975\pi$
$751$	−11390.5	−	19728.9i	−0.553456	−	0.958613i	0.444566	−	0.895746i	$-0.353358\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	13295.0			0.640867
$756$		0			0
$757$	32387.0			1.55499			0.777494	−	0.628891i	$-0.216491\pi$
$757$	32387.0			1.55499			0.777494	+	0.628891i	$0.216491\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	12645.0	+	21901.8i	0.602340	+	1.04328i	0.992466	+	0.122522i	$0.0390983\pi$
$761$	12645.0	+	21901.8i	0.602340	+	1.04328i	−0.390126	+	0.920762i	$0.627568\pi$
$762$		0			0
$763$	−12359.0	+	21406.4i	−0.586403	+	1.01568i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	25620.0	−	44375.1i	1.20611	−	2.08904i
$768$		0			0
$769$	−8141.50	−	14101.5i	−0.381782	−	0.661265i	0.609535	−	0.792759i	$-0.291356\pi$
$769$	−8141.50	−	14101.5i	−0.381782	−	0.661265i	−0.991317	+	0.131494i	$0.958023\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	31050.0			1.44475			0.722374	−	0.691502i	$-0.243051\pi$
$773$	31050.0			1.44475			0.722374	+	0.691502i	$0.243051\pi$
$774$		0			0
$775$	200.000			0.00926995
$776$		0			0
$777$		0			0
$778$		0			0
$779$	645.000	+	1117.17i	0.0296656	+	0.0513824i
$780$		0			0
$781$	−3150.00	+	5455.96i	−0.144322	+	0.249974i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4625.00	−	8010.73i	0.210284	−	0.364223i
$786$		0			0
$787$	2526.50	+	4376.03i	0.114435	+	0.198206i	0.917554	−	0.397612i	$-0.130161\pi$
$787$	2526.50	+	4376.03i	0.114435	+	0.198206i	−0.803119	+	0.595819i	$0.796828\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−20910.0			−0.939917
$792$		0			0
$793$	−36539.0			−1.63624
$794$		0			0
$795$		0			0
$796$		0			0
$797$	21840.0	+	37828.0i	0.970656	+	1.68122i	0.693584	+	0.720375i	$0.256030\pi$
$797$	21840.0	+	37828.0i	0.970656	+	1.68122i	0.277071	+	0.960849i	$0.410636\pi$
$798$		0			0
$799$	−10800.0	+	18706.1i	−0.478193	+	0.828255i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	6315.00	−	10937.9i	0.277524	−	0.480685i
$804$		0			0
$805$	−3825.00	−	6625.09i	−0.167470	−	0.290067i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1920.00			−0.0834408			−0.0417204	−	0.999129i	$-0.513284\pi$
$809$	−1920.00			−0.0834408			−0.0417204	+	0.999129i	$0.513284\pi$
$810$		0			0
$811$	37268.0			1.61363			0.806817	−	0.590802i	$-0.201189\pi$
$811$	37268.0			1.61363			0.806817	+	0.590802i	$0.201189\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2802.50	+	4854.07i	0.120451	+	0.208627i
$816$		0			0
$817$	−4730.00	+	8192.60i	−0.202548	+	0.350824i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2640.00	−	4572.61i	0.112225	−	0.194379i	−0.804442	−	0.594031i	$-0.797536\pi$
$821$	2640.00	−	4572.61i	0.112225	−	0.194379i	0.916667	+	0.399652i	$0.130869\pi$
$822$		0			0
$823$	−2255.50	−	3906.64i	−0.0955307	−	0.165464i	0.814299	−	0.580445i	$-0.197121\pi$
$823$	−2255.50	−	3906.64i	−0.0955307	−	0.165464i	−0.909830	+	0.414981i	$0.863788\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	10140.0			0.426363			0.213182	−	0.977013i	$-0.431617\pi$
$827$	10140.0			0.426363			0.213182	+	0.977013i	$0.431617\pi$
$828$		0			0
$829$	−11923.0			−0.499521			−0.249760	−	0.968308i	$-0.580352\pi$
$829$	−11923.0			−0.499521			−0.249760	+	0.968308i	$0.580352\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3240.00	+	5611.84i	0.134765	+	0.233420i
$834$		0			0
$835$	−750.000	+	1299.04i	−0.0310836	+	0.0538384i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	10650.0	−	18446.3i	0.438235	−	0.759045i	−0.559319	−	0.828953i	$-0.688937\pi$
$839$	10650.0	−	18446.3i	0.438235	−	0.759045i	0.997553	+	0.0699080i	$0.0222706\pi$
$840$		0			0
$841$	8144.50	+	14106.7i	0.333942	+	0.578404i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−7620.00			−0.310220
$846$		0			0
$847$	−7327.00			−0.297236
$848$		0			0
$849$		0			0
$850$		0			0
$851$	14265.0	+	24707.7i	0.574616	+	0.995263i
$852$		0			0
$853$	20385.5	−	35308.7i	0.818272	−	1.41729i	−0.0886820	−	0.996060i	$-0.528265\pi$
$853$	20385.5	−	35308.7i	0.818272	−	1.41729i	0.906954	−	0.421229i	$-0.138401\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	12585.0	−	21797.9i	0.501628	−	0.868846i	−0.498370	−	0.866964i	$-0.666068\pi$
$857$	12585.0	−	21797.9i	0.501628	−	0.868846i	0.999998	−	0.00188116i	$-0.000598793\pi$
$858$		0			0
$859$	21378.5	+	37028.6i	0.849156	+	1.47078i	0.881962	+	0.471320i	$0.156222\pi$
$859$	21378.5	+	37028.6i	0.849156	+	1.47078i	−0.0328065	+	0.999462i	$0.510445\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−15300.0			−0.603497			−0.301749	−	0.953388i	$-0.597570\pi$
$863$	−15300.0			−0.603497			−0.301749	+	0.953388i	$0.597570\pi$
$864$		0			0
$865$	−8100.00			−0.318391
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−5295.00	−	9171.21i	−0.206698	−	0.358011i
$870$		0			0
$871$	3263.50	−	5652.55i	0.126957	−	0.219896i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1062.50	−	1840.30i	0.0410503	−	0.0711013i
$876$		0			0
$877$	11871.5	+	20562.0i	0.457095	+	0.791711i	0.998806	−	0.0488534i	$-0.0155567\pi$
$877$	11871.5	+	20562.0i	0.457095	+	0.791711i	−0.541711	+	0.840565i	$0.682223\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	27810.0			1.06350			0.531750	−	0.846902i	$-0.321535\pi$
$881$	27810.0			1.06350			0.531750	+	0.846902i	$0.321535\pi$
$882$		0			0
$883$	−42991.0			−1.63846			−0.819231	−	0.573463i	$-0.805600\pi$
$883$	−42991.0			−1.63846			−0.819231	+	0.573463i	$0.805600\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−22800.0	−	39490.8i	−0.863077	−	1.49489i	−0.868944	−	0.494910i	$-0.835201\pi$
$887$	−22800.0	−	39490.8i	−0.863077	−	1.49489i	0.00586749	−	0.999983i	$-0.498132\pi$
$888$		0			0
$889$	−10880.0	+	18844.7i	−0.410465	+	0.710946i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3870.00	−	6703.04i	0.145022	−	0.251185i
$894$		0			0
$895$	1575.00	+	2727.98i	0.0588228	+	0.101884i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	720.000			0.0267112
$900$		0			0
$901$	75600.0			2.79534
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−5747.50	−	9954.96i	−0.211109	−	0.365651i
$906$		0			0
$907$	2496.50	−	4324.06i	0.0913946	−	0.158300i	−0.816704	−	0.577057i	$-0.804201\pi$
$907$	2496.50	−	4324.06i	0.0913946	−	0.158300i	0.908098	+	0.418757i	$0.137534\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−2460.00	+	4260.84i	−0.0894659	+	0.154959i	−0.907285	−	0.420515i	$-0.861849\pi$
$911$	−2460.00	+	4260.84i	−0.0894659	+	0.154959i	0.817820	+	0.575475i	$0.195183\pi$
$912$		0			0
$913$	−20250.0	−	35074.0i	−0.734039	−	1.27139i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	37740.0			1.35909
$918$		0			0
$919$	5456.00			0.195840			0.0979199	−	0.995194i	$-0.468781\pi$
$919$	5456.00			0.195840			0.0979199	+	0.995194i	$0.468781\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	6405.00	+	11093.8i	0.228411	+	0.395619i
$924$		0			0
$925$	−3962.50	+	6863.25i	−0.140850	+	0.243959i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−7395.00	+	12808.5i	−0.261165	+	0.452351i	−0.966552	−	0.256472i	$-0.917440\pi$
$929$	−7395.00	+	12808.5i	−0.261165	+	0.452351i	0.705387	+	0.708822i	$0.250773\pi$
$930$		0			0
$931$	−1161.00	−	2010.91i	−0.0408703	−	0.0707894i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−18000.0			−0.629586
$936$		0			0
$937$	−14023.0			−0.488913			−0.244456	−	0.969660i	$-0.578610\pi$
$937$	−14023.0			−0.488913			−0.244456	+	0.969660i	$0.578610\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−17625.0	−	30527.4i	−0.610583	−	1.05756i	−0.991142	−	0.132804i	$-0.957602\pi$
$941$	−17625.0	−	30527.4i	−0.610583	−	1.05756i	0.380559	−	0.924757i	$-0.375731\pi$
$942$		0			0
$943$	1350.00	−	2338.27i	0.0466194	−	0.0807471i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−18960.0	+	32839.7i	−0.650599	+	1.12687i	0.332379	+	0.943146i	$0.392149\pi$
$947$	−18960.0	+	32839.7i	−0.650599	+	1.12687i	−0.982978	+	0.183724i	$0.941185\pi$
$948$		0			0
$949$	−12840.5	−	22240.4i	−0.439221	−	0.760752i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	49080.0			1.66827			0.834133	−	0.551564i	$-0.185969\pi$
$953$	49080.0			1.66827			0.834133	+	0.551564i	$0.185969\pi$
$954$		0			0
$955$	4500.00			0.152478
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−9945.00	−	17225.2i	−0.334870	−	0.580013i
$960$		0			0
$961$	14863.5	−	25744.3i	0.498926	−	0.864165i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	8652.50	−	14986.6i	0.288636	−	0.499932i
$966$		0			0
$967$	16536.5	+	28642.1i	0.549926	+	0.952499i	0.998279	+	0.0586426i	$0.0186772\pi$
$967$	16536.5	+	28642.1i	0.549926	+	0.952499i	−0.448354	+	0.893856i	$0.647989\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	6420.00			0.212181			0.106090	−	0.994356i	$-0.466167\pi$
$971$	6420.00			0.212181			0.106090	+	0.994356i	$0.466167\pi$
$972$		0			0
$973$	−23681.0			−0.780245
$974$		0			0
$975$		0			0
$976$		0			0
$977$	12555.0	+	21745.9i	0.411126	+	0.712091i	0.995013	−	0.0997438i	$-0.0318023\pi$
$977$	12555.0	+	21745.9i	0.411126	+	0.712091i	−0.583887	+	0.811835i	$0.698469\pi$
$978$		0			0
$979$	15300.0	−	26500.4i	0.499479	−	0.865123i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	26205.0	−	45388.4i	0.850264	−	1.47270i	−0.0307057	−	0.999528i	$-0.509775\pi$
$983$	26205.0	−	45388.4i	0.850264	−	1.47270i	0.880970	−	0.473172i	$-0.156891\pi$
$984$		0			0
$985$	−11400.0	−	19745.4i	−0.368766	−	0.638721i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	19800.0			0.636606
$990$		0			0
$991$	52619.0			1.68668			0.843339	−	0.537382i	$-0.180587\pi$
$991$	52619.0			1.68668			0.843339	+	0.537382i	$0.180587\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−5192.50	−	8993.67i	−0.165441	−	0.286551i
$996$		0			0
$997$	26945.0	−	46670.1i	0.855924	−	1.48250i	−0.0198602	−	0.999803i	$-0.506322\pi$
$997$	26945.0	−	46670.1i	0.855924	−	1.48250i	0.875785	−	0.482702i	$-0.160345\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.i.h.541.1		2
3.2	odd	2		1620.4.i.b.541.1		2
9.2	odd	6		540.4.a.d.1.1	yes	1
9.4	even	3	inner	1620.4.i.h.1081.1		2
9.5	odd	6		1620.4.i.b.1081.1		2
9.7	even	3		540.4.a.b.1.1	✓	1
36.7	odd	6		2160.4.a.a.1.1		1
36.11	even	6		2160.4.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.4.a.b.1.1	✓	1	9.7	even	3
540.4.a.d.1.1	yes	1	9.2	odd	6
1620.4.i.b.541.1		2	3.2	odd	2
1620.4.i.b.1081.1		2	9.5	odd	6
1620.4.i.h.541.1		2	1.1	even	1	trivial
1620.4.i.h.1081.1		2	9.4	even	3	inner
2160.4.a.a.1.1		1	36.7	odd	6
2160.4.a.k.1.1		1	36.11	even	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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.50000 + 4.33013i) q^{5} +(-8.50000 + 14.7224i) q^{7} +O(q^{10})\)	\(q+(2.50000 + 4.33013i) q^{5} +(-8.50000 + 14.7224i) q^{7} +(-15.0000 + 25.9808i) q^{11} +(30.5000 + 52.8275i) q^{13} +120.000 q^{17} -43.0000 q^{19} +(45.0000 + 77.9423i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(-45.0000 + 77.9423i) q^{29} +(-4.00000 - 6.92820i) q^{31} -85.0000 q^{35} +317.000 q^{37} +(-15.0000 - 25.9808i) q^{41} +(110.000 - 190.526i) q^{43} +(-90.0000 + 155.885i) q^{47} +(27.0000 + 46.7654i) q^{49} +630.000 q^{53} -150.000 q^{55} +(-420.000 - 727.461i) q^{59} +(-299.500 + 518.749i) q^{61} +(-152.500 + 264.138i) q^{65} +(-53.5000 - 92.6647i) q^{67} +210.000 q^{71} -421.000 q^{73} +(-255.000 - 441.673i) q^{77} +(-176.500 + 305.707i) q^{79} +(-675.000 + 1169.13i) q^{83} +(300.000 + 519.615i) q^{85} -1020.00 q^{89} -1037.00 q^{91} +(-107.500 - 186.195i) q^{95} +(498.500 - 863.427i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{5} - 17 q^{7}+O(q^{10}) \) 2 * q + 5 * q^5 - 17 * q^7	\( 2 q + 5 q^{5} - 17 q^{7} - 30 q^{11} + 61 q^{13} + 240 q^{17} - 86 q^{19} + 90 q^{23} - 25 q^{25} - 90 q^{29} - 8 q^{31} - 170 q^{35} + 634 q^{37} - 30 q^{41} + 220 q^{43} - 180 q^{47} + 54 q^{49} + 1260 q^{53} - 300 q^{55} - 840 q^{59} - 599 q^{61} - 305 q^{65} - 107 q^{67} + 420 q^{71} - 842 q^{73} - 510 q^{77} - 353 q^{79} - 1350 q^{83} + 600 q^{85} - 2040 q^{89} - 2074 q^{91} - 215 q^{95} + 997 q^{97}+O(q^{100}) \) 2 * q + 5 * q^5 - 17 * q^7 - 30 * q^11 + 61 * q^13 + 240 * q^17 - 86 * q^19 + 90 * q^23 - 25 * q^25 - 90 * q^29 - 8 * q^31 - 170 * q^35 + 634 * q^37 - 30 * q^41 + 220 * q^43 - 180 * q^47 + 54 * q^49 + 1260 * q^53 - 300 * q^55 - 840 * q^59 - 599 * q^61 - 305 * q^65 - 107 * q^67 + 420 * q^71 - 842 * q^73 - 510 * q^77 - 353 * q^79 - 1350 * q^83 + 600 * q^85 - 2040 * q^89 - 2074 * q^91 - 215 * q^95 + 997 * q^97

Introduction

L-functions

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