LMFDB - Embedded newform 1520.2.q.h.961.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(881,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("1520.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.q (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:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 5x^{2} + 4x + 16 $ x^4 - x^3 + 5x^2 + 4x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.1
Root			$1.28078 - 2.21837i$ of defining polynomial
Character	$\chi$	$=$	1520.961
Dual form			1520.2.q.h.881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.28078 + 2.21837i) q^{3} +(-0.500000 + 0.866025i) q^{5} -0.438447 q^{7} +(-1.78078 - 3.08440i) q^{9} +O(q^{10})$	\(q+(-1.28078 + 2.21837i) q^{3} +(-0.500000 + 0.866025i) q^{5} -0.438447 q^{7} +(-1.78078 - 3.08440i) q^{9} -1.00000 q^{11} +(1.00000 + 1.73205i) q^{13} +(-1.28078 - 2.21837i) q^{15} +(-2.56155 + 4.43674i) q^{17} +(2.50000 + 3.57071i) q^{19} +(0.561553 - 0.972638i) q^{21} +(-2.34233 - 4.05703i) q^{23} +(-0.500000 - 0.866025i) q^{25} +1.43845 q^{27} +(-1.00000 - 1.73205i) q^{29} -10.2462 q^{31} +(1.28078 - 2.21837i) q^{33} +(0.219224 - 0.379706i) q^{35} -4.68466 q^{37} -5.12311 q^{39} +(3.06155 - 5.30277i) q^{41} +(1.56155 - 2.70469i) q^{43} +3.56155 q^{45} +(1.43845 + 2.49146i) q^{47} -6.80776 q^{49} +(-6.56155 - 11.3649i) q^{51} +(3.78078 + 6.54850i) q^{53} +(0.500000 - 0.866025i) q^{55} +(-11.1231 + 0.972638i) q^{57} +(7.28078 - 12.6107i) q^{59} +(-2.56155 - 4.43674i) q^{61} +(0.780776 + 1.35234i) q^{63} -2.00000 q^{65} +(4.71922 + 8.17394i) q^{67} +12.0000 q^{69} +(8.12311 - 14.0696i) q^{71} +(0.842329 - 1.45896i) q^{73} +2.56155 q^{75} +0.438447 q^{77} +(-5.56155 + 9.63289i) q^{79} +(3.50000 - 6.06218i) q^{81} +10.8078 q^{83} +(-2.56155 - 4.43674i) q^{85} +5.12311 q^{87} +(1.34233 + 2.32498i) q^{89} +(-0.438447 - 0.759413i) q^{91} +(13.1231 - 22.7299i) q^{93} +(-4.34233 + 0.379706i) q^{95} +(-0.842329 + 1.45896i) q^{97} +(1.78078 + 3.08440i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} - 2 q^{5} - 10 q^{7} - 3 q^{9}+O(q^{10}) $ 4 * q - q^3 - 2 * q^5 - 10 * q^7 - 3 * q^9	$ 4 q - q^{3} - 2 q^{5} - 10 q^{7} - 3 q^{9} - 4 q^{11} + 4 q^{13} - q^{15} - 2 q^{17} + 10 q^{19} - 6 q^{21} + 3 q^{23} - 2 q^{25} + 14 q^{27} - 4 q^{29} - 8 q^{31} + q^{33} + 5 q^{35} + 6 q^{37} - 4 q^{39} + 4 q^{41} - 2 q^{43} + 6 q^{45} + 14 q^{47} + 14 q^{49} - 18 q^{51} + 11 q^{53} + 2 q^{55} - 28 q^{57} + 25 q^{59} - 2 q^{61} - q^{63} - 8 q^{65} + 23 q^{67} + 48 q^{69} + 16 q^{71} - 9 q^{73} + 2 q^{75} + 10 q^{77} - 14 q^{79} + 14 q^{81} + 2 q^{83} - 2 q^{85} + 4 q^{87} - 7 q^{89} - 10 q^{91} + 36 q^{93} - 5 q^{95} + 9 q^{97} + 3 q^{99}+O(q^{100}) $ 4 * q - q^3 - 2 * q^5 - 10 * q^7 - 3 * q^9 - 4 * q^11 + 4 * q^13 - q^15 - 2 * q^17 + 10 * q^19 - 6 * q^21 + 3 * q^23 - 2 * q^25 + 14 * q^27 - 4 * q^29 - 8 * q^31 + q^33 + 5 * q^35 + 6 * q^37 - 4 * q^39 + 4 * q^41 - 2 * q^43 + 6 * q^45 + 14 * q^47 + 14 * q^49 - 18 * q^51 + 11 * q^53 + 2 * q^55 - 28 * q^57 + 25 * q^59 - 2 * q^61 - q^63 - 8 * q^65 + 23 * q^67 + 48 * q^69 + 16 * q^71 - 9 * q^73 + 2 * q^75 + 10 * q^77 - 14 * q^79 + 14 * q^81 + 2 * q^83 - 2 * q^85 + 4 * q^87 - 7 * q^89 - 10 * q^91 + 36 * q^93 - 5 * q^95 + 9 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.28078	+	2.21837i	−0.739457	+	1.28078i	0.213284	+	0.976990i	$0.431584\pi$
$3$	−1.28078	+	2.21837i	−0.739457	+	1.28078i	−0.952740	+	0.303786i	$0.901749\pi$
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	−0.438447			−0.165717			−0.0828587	−	0.996561i	$-0.526405\pi$
$7$	−0.438447			−0.165717			−0.0828587	+	0.996561i	$0.526405\pi$
$8$		0			0
$9$	−1.78078	−	3.08440i	−0.593592	−	1.02813i
$10$		0			0
$11$	−1.00000			−0.301511			−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000			−0.301511			−0.150756	+	0.988571i	$0.548171\pi$
$12$		0			0
$13$	1.00000	+	1.73205i	0.277350	+	0.480384i	0.970725	−	0.240192i	$-0.0772105\pi$
$13$	1.00000	+	1.73205i	0.277350	+	0.480384i	−0.693375	+	0.720577i	$0.743877\pi$
$14$		0			0
$15$	−1.28078	−	2.21837i	−0.330695	−	0.572781i
$16$		0			0
$17$	−2.56155	+	4.43674i	−0.621268	+	1.07607i	0.367982	+	0.929833i	$0.380049\pi$
$17$	−2.56155	+	4.43674i	−0.621268	+	1.07607i	−0.989250	+	0.146235i	$0.953285\pi$
$18$		0			0
$19$	2.50000	+	3.57071i	0.573539	+	0.819178i
$19$	2.50000	+	3.57071i	0.573539	+	0.819178i
$20$		0			0
$21$	0.561553	−	0.972638i	0.122541	−	0.212247i
$22$		0			0
$23$	−2.34233	−	4.05703i	−0.488409	−	0.845950i	0.511502	−	0.859282i	$-0.329089\pi$
$23$	−2.34233	−	4.05703i	−0.488409	−	0.845950i	−0.999911	+	0.0133324i	$0.995756\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$	1.43845			0.276829
$28$		0			0
$29$	−1.00000	−	1.73205i	−0.185695	−	0.321634i	0.758115	−	0.652121i	$-0.226120\pi$
$29$	−1.00000	−	1.73205i	−0.185695	−	0.321634i	−0.943811	+	0.330487i	$0.892787\pi$
$30$		0			0
$31$	−10.2462			−1.84027			−0.920137	−	0.391597i	$-0.871923\pi$
$31$	−10.2462			−1.84027			−0.920137	+	0.391597i	$0.871923\pi$
$32$		0			0
$33$	1.28078	−	2.21837i	0.222955	−	0.386169i
$34$		0			0
$35$	0.219224	−	0.379706i	0.0370556	−	0.0641821i
$36$		0			0
$37$	−4.68466			−0.770153			−0.385077	−	0.922885i	$-0.625825\pi$
$37$	−4.68466			−0.770153			−0.385077	+	0.922885i	$0.625825\pi$
$38$		0			0
$39$	−5.12311			−0.820353
$40$		0			0
$41$	3.06155	−	5.30277i	0.478134	−	0.828153i	−0.521552	−	0.853220i	$-0.674647\pi$
$41$	3.06155	−	5.30277i	0.478134	−	0.828153i	0.999686	+	0.0250670i	$0.00797991\pi$
$42$		0			0
$43$	1.56155	−	2.70469i	0.238135	−	0.412461i	−0.722044	−	0.691847i	$-0.756797\pi$
$43$	1.56155	−	2.70469i	0.238135	−	0.412461i	0.960179	+	0.279385i	$0.0901307\pi$
$44$		0			0
$45$	3.56155			0.530925
$46$		0			0
$47$	1.43845	+	2.49146i	0.209819	+	0.363417i	0.951657	−	0.307161i	$-0.0993792\pi$
$47$	1.43845	+	2.49146i	0.209819	+	0.363417i	−0.741838	+	0.670579i	$0.766046\pi$
$48$		0			0
$49$	−6.80776			−0.972538
$50$		0			0
$51$	−6.56155	−	11.3649i	−0.918801	−	1.59141i
$52$		0			0
$53$	3.78078	+	6.54850i	0.519330	+	0.899505i	0.999748	+	0.0224656i	$0.00715162\pi$
$53$	3.78078	+	6.54850i	0.519330	+	0.899505i	−0.480418	+	0.877040i	$0.659515\pi$
$54$		0			0
$55$	0.500000	−	0.866025i	0.0674200	−	0.116775i
$56$		0			0
$57$	−11.1231	+	0.972638i	−1.47329	+	0.128829i
$58$		0			0
$59$	7.28078	−	12.6107i	0.947876	−	1.64177i	0.197989	−	0.980204i	$-0.436559\pi$
$59$	7.28078	−	12.6107i	0.947876	−	1.64177i	0.749887	−	0.661566i	$-0.230108\pi$
$60$		0			0
$61$	−2.56155	−	4.43674i	−0.327973	−	0.568066i	0.654136	−	0.756377i	$-0.273032\pi$
$61$	−2.56155	−	4.43674i	−0.327973	−	0.568066i	−0.982110	+	0.188310i	$0.939699\pi$
$62$		0			0
$63$	0.780776	+	1.35234i	0.0983686	+	0.170379i
$64$		0			0
$65$	−2.00000			−0.248069
$66$		0			0
$67$	4.71922	+	8.17394i	0.576545	+	0.998605i	0.995872	+	0.0907698i	$0.0289328\pi$
$67$	4.71922	+	8.17394i	0.576545	+	0.998605i	−0.419327	+	0.907835i	$0.637734\pi$
$68$		0			0
$69$	12.0000			1.44463
$70$		0			0
$71$	8.12311	−	14.0696i	0.964035	−	1.66976i	0.251850	−	0.967766i	$-0.418961\pi$
$71$	8.12311	−	14.0696i	0.964035	−	1.66976i	0.712185	−	0.701992i	$-0.247706\pi$
$72$		0			0
$73$	0.842329	−	1.45896i	0.0985872	−	0.170758i	−0.812513	−	0.582943i	$-0.801901\pi$
$73$	0.842329	−	1.45896i	0.0985872	−	0.170758i	0.911100	+	0.412185i	$0.135234\pi$
$74$		0			0
$75$	2.56155			0.295783
$76$		0			0
$77$	0.438447			0.0499657
$78$		0			0
$79$	−5.56155	+	9.63289i	−0.625724	+	1.08379i	0.362677	+	0.931915i	$0.381863\pi$
$79$	−5.56155	+	9.63289i	−0.625724	+	1.08379i	−0.988400	+	0.151870i	$0.951470\pi$
$80$		0			0
$81$	3.50000	−	6.06218i	0.388889	−	0.673575i
$82$		0			0
$83$	10.8078			1.18631			0.593153	−	0.805090i	$-0.297883\pi$
$83$	10.8078			1.18631			0.593153	+	0.805090i	$0.297883\pi$
$84$		0			0
$85$	−2.56155	−	4.43674i	−0.277839	−	0.481232i
$86$		0			0
$87$	5.12311			0.549255
$88$		0			0
$89$	1.34233	+	2.32498i	0.142287	+	0.246448i	0.928357	−	0.371689i	$-0.121221\pi$
$89$	1.34233	+	2.32498i	0.142287	+	0.246448i	−0.786071	+	0.618137i	$0.787888\pi$
$90$		0			0
$91$	−0.438447	−	0.759413i	−0.0459618	−	0.0796081i
$92$		0			0
$93$	13.1231	−	22.7299i	1.36080	−	2.35698i
$94$		0			0
$95$	−4.34233	+	0.379706i	−0.445514	+	0.0389571i
$96$		0			0
$97$	−0.842329	+	1.45896i	−0.0855256	+	0.148135i	−0.905615	−	0.424101i	$-0.860590\pi$
$97$	−0.842329	+	1.45896i	−0.0855256	+	0.148135i	0.820089	+	0.572235i	$0.193924\pi$
$98$		0			0
$99$	1.78078	+	3.08440i	0.178975	+	0.309993i
$100$		0			0
$101$	−5.00000	−	8.66025i	−0.497519	−	0.861727i	0.502477	−	0.864590i	$-0.332422\pi$
$101$	−5.00000	−	8.66025i	−0.497519	−	0.861727i	−0.999996	+	0.00286291i	$0.999089\pi$
$102$		0			0
$103$	5.80776			0.572256			0.286128	−	0.958191i	$-0.407632\pi$
$103$	5.80776			0.572256			0.286128	+	0.958191i	$0.407632\pi$
$104$		0			0
$105$	0.561553	+	0.972638i	0.0548019	+	0.0949197i
$106$		0			0
$107$	−2.24621			−0.217149			−0.108575	−	0.994088i	$-0.534629\pi$
$107$	−2.24621			−0.217149			−0.108575	+	0.994088i	$0.534629\pi$
$108$		0			0
$109$	7.12311	−	12.3376i	0.682270	−	1.18173i	−0.292017	−	0.956413i	$-0.594326\pi$
$109$	7.12311	−	12.3376i	0.682270	−	1.18173i	0.974286	−	0.225313i	$-0.0723404\pi$
$110$		0			0
$111$	6.00000	−	10.3923i	0.569495	−	0.986394i
$112$		0			0
$113$	−16.8078			−1.58114			−0.790571	−	0.612371i	$-0.790216\pi$
$113$	−16.8078			−1.58114			−0.790571	+	0.612371i	$0.790216\pi$
$114$		0			0
$115$	4.68466			0.436847
$116$		0			0
$117$	3.56155	−	6.16879i	0.329266	−	0.570305i
$118$		0			0
$119$	1.12311	−	1.94528i	0.102955	−	0.178323i
$120$		0			0
$121$	−10.0000			−0.909091
$122$		0			0
$123$	7.84233	+	13.5833i	0.707119	+	1.22477i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−7.78078	−	13.4767i	−0.690432	−	1.19586i	−0.971696	−	0.236233i	$-0.924087\pi$
$127$	−7.78078	−	13.4767i	−0.690432	−	1.19586i	0.281264	−	0.959630i	$-0.409246\pi$
$128$		0			0
$129$	4.00000	+	6.92820i	0.352180	+	0.609994i
$130$		0			0
$131$	−8.06155	+	13.9630i	−0.704341	+	1.21995i	0.262588	+	0.964908i	$0.415424\pi$
$131$	−8.06155	+	13.9630i	−0.704341	+	1.21995i	−0.966929	+	0.255046i	$0.917909\pi$
$132$		0			0
$133$	−1.09612	−	1.56557i	−0.0950455	−	0.135752i
$134$		0			0
$135$	−0.719224	+	1.24573i	−0.0619009	+	0.107216i
$136$		0			0
$137$	2.71922	+	4.70983i	0.232319	+	0.402388i	0.958490	−	0.285126i	$-0.0920354\pi$
$137$	2.71922	+	4.70983i	0.232319	+	0.402388i	−0.726171	+	0.687514i	$0.758702\pi$
$138$		0			0
$139$	4.40388	+	7.62775i	0.373532	+	0.646977i	0.990106	−	0.140320i	$-0.0448131\pi$
$139$	4.40388	+	7.62775i	0.373532	+	0.646977i	−0.616574	+	0.787297i	$0.711480\pi$
$140$		0			0
$141$	−7.36932			−0.620608
$142$		0			0
$143$	−1.00000	−	1.73205i	−0.0836242	−	0.144841i
$144$		0			0
$145$	2.00000			0.166091
$146$		0			0
$147$	8.71922	−	15.1021i	0.719149	−	1.24560i
$148$		0			0
$149$	−8.00000	+	13.8564i	−0.655386	+	1.13516i	0.326411	+	0.945228i	$0.394160\pi$
$149$	−8.00000	+	13.8564i	−0.655386	+	1.13516i	−0.981797	+	0.189933i	$0.939173\pi$
$150$		0			0
$151$	−20.4924			−1.66765			−0.833825	−	0.552029i	$-0.813854\pi$
$151$	−20.4924			−1.66765			−0.833825	+	0.552029i	$0.813854\pi$
$152$		0			0
$153$	18.2462			1.47512
$154$		0			0
$155$	5.12311	−	8.87348i	0.411498	−	0.712735i
$156$		0			0
$157$	−1.21922	+	2.11176i	−0.0973046	+	0.168537i	−0.910568	−	0.413359i	$-0.864355\pi$
$157$	−1.21922	+	2.11176i	−0.0973046	+	0.168537i	0.813263	+	0.581896i	$0.197689\pi$
$158$		0			0
$159$	−19.3693			−1.53609
$160$		0			0
$161$	1.02699	+	1.77879i	0.0809380	+	0.140189i
$162$		0			0
$163$	−17.0540			−1.33577			−0.667885	−	0.744264i	$-0.732800\pi$
$163$	−17.0540			−1.33577			−0.667885	+	0.744264i	$0.732800\pi$
$164$		0			0
$165$	1.28078	+	2.21837i	0.0997083	+	0.172700i
$166$		0			0
$167$	0.342329	+	0.592932i	0.0264902	+	0.0458824i	0.878967	−	0.476883i	$-0.158234\pi$
$167$	0.342329	+	0.592932i	0.0264902	+	0.0458824i	−0.852476	+	0.522766i	$0.824900\pi$
$168$		0			0
$169$	4.50000	−	7.79423i	0.346154	−	0.599556i
$170$		0			0
$171$	6.56155	−	14.0696i	0.501774	−	1.07593i
$172$		0			0
$173$	2.90388	−	5.02967i	0.220778	−	0.382399i	−0.734266	−	0.678861i	$-0.762474\pi$
$173$	2.90388	−	5.02967i	0.220778	−	0.382399i	0.955044	+	0.296463i	$0.0958070\pi$
$174$		0			0
$175$	0.219224	+	0.379706i	0.0165717	+	0.0287031i
$176$		0			0
$177$	18.6501	+	32.3029i	1.40183	+	2.42804i
$178$		0			0
$179$	−11.4924			−0.858984			−0.429492	−	0.903071i	$-0.641307\pi$
$179$	−11.4924			−0.858984			−0.429492	+	0.903071i	$0.641307\pi$
$180$		0			0
$181$	−10.6847	−	18.5064i	−0.794184	−	1.37557i	−0.923356	−	0.383945i	$-0.874565\pi$
$181$	−10.6847	−	18.5064i	−0.794184	−	1.37557i	0.129171	−	0.991622i	$-0.458768\pi$
$182$		0			0
$183$	13.1231			0.970088
$184$		0			0
$185$	2.34233	−	4.05703i	0.172211	−	0.298279i
$186$		0			0
$187$	2.56155	−	4.43674i	0.187319	−	0.324447i
$188$		0			0
$189$	−0.630683			−0.0458754
$190$		0			0
$191$	5.36932			0.388510			0.194255	−	0.980951i	$-0.437771\pi$
$191$	5.36932			0.388510			0.194255	+	0.980951i	$0.437771\pi$
$192$		0			0
$193$	−12.8078	+	22.1837i	−0.921923	+	1.59682i	−0.125487	+	0.992095i	$0.540049\pi$
$193$	−12.8078	+	22.1837i	−0.921923	+	1.59682i	−0.796436	+	0.604722i	$0.793284\pi$
$194$		0			0
$195$	2.56155	−	4.43674i	0.183437	−	0.317722i
$196$		0			0
$197$	14.4384			1.02870			0.514348	−	0.857581i	$-0.328034\pi$
$197$	14.4384			1.02870			0.514348	+	0.857581i	$0.328034\pi$
$198$		0			0
$199$	−1.43845	−	2.49146i	−0.101969	−	0.176615i	0.810527	−	0.585701i	$-0.199181\pi$
$199$	−1.43845	−	2.49146i	−0.101969	−	0.176615i	−0.912496	+	0.409086i	$0.865848\pi$
$200$		0			0
$201$	−24.1771			−1.70532
$202$		0			0
$203$	0.438447	+	0.759413i	0.0307730	+	0.0533003i
$204$		0			0
$205$	3.06155	+	5.30277i	0.213828	+	0.370361i
$206$		0			0
$207$	−8.34233	+	14.4493i	−0.579832	+	1.00430i
$208$		0			0
$209$	−2.50000	−	3.57071i	−0.172929	−	0.246991i
$210$		0			0
$211$	−1.65767	+	2.87117i	−0.114119	+	0.197659i	−0.917427	−	0.397904i	$-0.869738\pi$
$211$	−1.65767	+	2.87117i	−0.114119	+	0.197659i	0.803308	+	0.595563i	$0.203071\pi$
$212$		0			0
$213$	20.8078	+	36.0401i	1.42572	+	2.46943i
$214$		0			0
$215$	1.56155	+	2.70469i	0.106497	+	0.184458i
$216$		0			0
$217$	4.49242			0.304966
$218$		0			0
$219$	2.15767	+	3.73720i	0.145802	+	0.252536i
$220$		0			0
$221$	−10.2462			−0.689235
$222$		0			0
$223$	5.65767	−	9.79937i	0.378866	−	0.656215i	−0.612032	−	0.790833i	$-0.709648\pi$
$223$	5.65767	−	9.79937i	0.378866	−	0.656215i	0.990897	+	0.134619i	$0.0429809\pi$
$224$		0			0
$225$	−1.78078	+	3.08440i	−0.118718	+	0.205626i
$226$		0			0
$227$	−19.9309			−1.32286			−0.661429	−	0.750008i	$-0.730050\pi$
$227$	−19.9309			−1.32286			−0.661429	+	0.750008i	$0.730050\pi$
$228$		0			0
$229$	12.8769			0.850929			0.425465	−	0.904975i	$-0.360111\pi$
$229$	12.8769			0.850929			0.425465	+	0.904975i	$0.360111\pi$
$230$		0			0
$231$	−0.561553	+	0.972638i	−0.0369475	+	0.0639949i
$232$		0			0
$233$	−1.15767	+	2.00514i	−0.0758415	+	0.131361i	−0.901452	−	0.432879i	$-0.857498\pi$
$233$	−1.15767	+	2.00514i	−0.0758415	+	0.131361i	0.825610	+	0.564241i	$0.190831\pi$
$234$		0			0
$235$	−2.87689			−0.187668
$236$		0			0
$237$	−14.2462	−	24.6752i	−0.925391	−	1.60282i
$238$		0			0
$239$	−18.2462			−1.18025			−0.590125	−	0.807312i	$-0.700921\pi$
$239$	−18.2462			−1.18025			−0.590125	+	0.807312i	$0.700921\pi$
$240$		0			0
$241$	4.28078	+	7.41452i	0.275749	+	0.477611i	0.970324	−	0.241809i	$-0.0777408\pi$
$241$	4.28078	+	7.41452i	0.275749	+	0.477611i	−0.694575	+	0.719421i	$0.744407\pi$
$242$		0			0
$243$	11.1231	+	19.2658i	0.713548	+	1.23590i
$244$		0			0
$245$	3.40388	−	5.89570i	0.217466	−	0.376662i
$246$		0			0
$247$	−3.68466	+	7.90084i	−0.234449	+	0.502718i
$248$		0			0
$249$	−13.8423	+	23.9756i	−0.877222	+	1.51939i
$250$		0			0
$251$	12.9654	+	22.4568i	0.818371	+	1.41746i	0.906882	+	0.421385i	$0.138456\pi$
$251$	12.9654	+	22.4568i	0.818371	+	1.41746i	−0.0885109	+	0.996075i	$0.528211\pi$
$252$		0			0
$253$	2.34233	+	4.05703i	0.147261	+	0.255063i
$254$		0			0
$255$	13.1231			0.821801
$256$		0			0
$257$	−6.52699	−	11.3051i	−0.407142	−	0.705191i	0.587426	−	0.809278i	$-0.300141\pi$
$257$	−6.52699	−	11.3051i	−0.407142	−	0.705191i	−0.994568	+	0.104087i	$0.966808\pi$
$258$		0			0
$259$	2.05398			0.127628
$260$		0			0
$261$	−3.56155	+	6.16879i	−0.220455	+	0.381839i
$262$		0			0
$263$	−1.09612	+	1.89853i	−0.0675895	+	0.117068i	−0.897840	−	0.440322i	$-0.854864\pi$
$263$	−1.09612	+	1.89853i	−0.0675895	+	0.117068i	0.830250	+	0.557391i	$0.188197\pi$
$264$		0			0
$265$	−7.56155			−0.464502
$266$		0			0
$267$	−6.87689			−0.420859
$268$		0			0
$269$	−2.00000	+	3.46410i	−0.121942	+	0.211210i	−0.920534	−	0.390664i	$-0.872246\pi$
$269$	−2.00000	+	3.46410i	−0.121942	+	0.211210i	0.798591	+	0.601874i	$0.205579\pi$
$270$		0			0
$271$	−4.12311	+	7.14143i	−0.250461	+	0.433811i	−0.963653	−	0.267158i	$-0.913915\pi$
$271$	−4.12311	+	7.14143i	−0.250461	+	0.433811i	0.713192	+	0.700969i	$0.247249\pi$
$272$		0			0
$273$	2.24621			0.135947
$274$		0			0
$275$	0.500000	+	0.866025i	0.0301511	+	0.0522233i
$276$		0			0
$277$	4.24621			0.255130			0.127565	−	0.991830i	$-0.459284\pi$
$277$	4.24621			0.255130			0.127565	+	0.991830i	$0.459284\pi$
$278$		0			0
$279$	18.2462	+	31.6034i	1.09237	+	1.89204i
$280$		0			0
$281$	−4.18466	−	7.24804i	−0.249636	−	0.432382i	0.713789	−	0.700361i	$-0.246978\pi$
$281$	−4.18466	−	7.24804i	−0.249636	−	0.432382i	−0.963425	+	0.267979i	$0.913644\pi$
$282$		0			0
$283$	−9.71922	+	16.8342i	−0.577748	+	1.00069i	0.417989	+	0.908452i	$0.362735\pi$
$283$	−9.71922	+	16.8342i	−0.577748	+	1.00069i	−0.995737	+	0.0922367i	$0.970598\pi$
$284$		0			0
$285$	4.71922	−	10.1192i	0.279543	−	0.599410i
$286$		0			0
$287$	−1.34233	+	2.32498i	−0.0792352	+	0.137239i
$288$		0			0
$289$	−4.62311	−	8.00745i	−0.271947	−	0.471027i
$290$		0			0
$291$	−2.15767	−	3.73720i	−0.126485	−	0.219078i
$292$		0			0
$293$	23.5616			1.37648			0.688240	−	0.725483i	$-0.258383\pi$
$293$	23.5616			1.37648			0.688240	+	0.725483i	$0.258383\pi$
$294$		0			0
$295$	7.28078	+	12.6107i	0.423903	+	0.734222i
$296$		0			0
$297$	−1.43845			−0.0834672
$298$		0			0
$299$	4.68466	−	8.11407i	0.270921	−	0.469249i
$300$		0			0
$301$	−0.684658	+	1.18586i	−0.0394631	+	0.0683520i
$302$		0			0
$303$	25.6155			1.47157
$304$		0			0
$305$	5.12311			0.293348
$306$		0			0
$307$	−11.9654	+	20.7247i	−0.682903	+	1.18282i	0.291187	+	0.956666i	$0.405950\pi$
$307$	−11.9654	+	20.7247i	−0.682903	+	1.18282i	−0.974091	+	0.226157i	$0.927384\pi$
$308$		0			0
$309$	−7.43845	+	12.8838i	−0.423158	+	0.732932i
$310$		0			0
$311$	−4.00000			−0.226819			−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000			−0.226819			−0.113410	+	0.993548i	$0.536177\pi$
$312$		0			0
$313$	−2.15767	−	3.73720i	−0.121959	−	0.211239i	0.798581	−	0.601887i	$-0.205584\pi$
$313$	−2.15767	−	3.73720i	−0.121959	−	0.211239i	−0.920540	+	0.390648i	$0.872251\pi$
$314$		0			0
$315$	−1.56155			−0.0879835
$316$		0			0
$317$	−9.34233	−	16.1814i	−0.524717	−	0.908837i	−0.999586	−	0.0287805i	$-0.990838\pi$
$317$	−9.34233	−	16.1814i	−0.524717	−	0.908837i	0.474868	−	0.880057i	$-0.342496\pi$
$318$		0			0
$319$	1.00000	+	1.73205i	0.0559893	+	0.0969762i
$320$		0			0
$321$	2.87689	−	4.98293i	0.160573	−	0.278120i
$322$		0			0
$323$	−22.2462	+	1.94528i	−1.23781	+	0.108238i
$324$		0			0
$325$	1.00000	−	1.73205i	0.0554700	−	0.0960769i
$326$		0			0
$327$	18.2462	+	31.6034i	1.00902	+	1.74767i
$328$		0			0
$329$	−0.630683	−	1.09238i	−0.0347707	−	0.0602246i
$330$		0			0
$331$	−23.4924			−1.29126			−0.645630	−	0.763650i	$-0.723405\pi$
$331$	−23.4924			−1.29126			−0.645630	+	0.763650i	$0.723405\pi$
$332$		0			0
$333$	8.34233	+	14.4493i	0.457157	+	0.791819i
$334$		0			0
$335$	−9.43845			−0.515677
$336$		0			0
$337$	−10.5270	+	18.2333i	−0.573442	+	0.993230i	0.422767	+	0.906238i	$0.361059\pi$
$337$	−10.5270	+	18.2333i	−0.573442	+	0.993230i	−0.996209	+	0.0869917i	$0.972275\pi$
$338$		0			0
$339$	21.5270	−	37.2858i	1.16919	−	2.02509i
$340$		0			0
$341$	10.2462			0.554863
$342$		0			0
$343$	6.05398			0.326884
$344$		0			0
$345$	−6.00000	+	10.3923i	−0.323029	+	0.559503i
$346$		0			0
$347$	0.842329	−	1.45896i	0.0452186	−	0.0783209i	−0.842530	−	0.538649i	$-0.818935\pi$
$347$	0.842329	−	1.45896i	0.0452186	−	0.0783209i	0.887749	+	0.460328i	$0.152268\pi$
$348$		0			0
$349$	−14.2462			−0.762582			−0.381291	−	0.924455i	$-0.624520\pi$
$349$	−14.2462			−0.762582			−0.381291	+	0.924455i	$0.624520\pi$
$350$		0			0
$351$	1.43845	+	2.49146i	0.0767786	+	0.132984i
$352$		0			0
$353$	24.1771			1.28682			0.643408	−	0.765523i	$-0.277520\pi$
$353$	24.1771			1.28682			0.643408	+	0.765523i	$0.277520\pi$
$354$		0			0
$355$	8.12311	+	14.0696i	0.431130	+	0.746739i
$356$		0			0
$357$	2.87689	+	4.98293i	0.152261	+	0.263724i
$358$		0			0
$359$	7.56155	−	13.0970i	0.399083	−	0.691233i	−0.594530	−	0.804074i	$-0.702662\pi$
$359$	7.56155	−	13.0970i	0.399083	−	0.691233i	0.993613	+	0.112841i	$0.0359950\pi$
$360$		0			0
$361$	−6.50000	+	17.8536i	−0.342105	+	0.939662i
$362$		0			0
$363$	12.8078	−	22.1837i	0.672233	−	1.16434i
$364$		0			0
$365$	0.842329	+	1.45896i	0.0440895	+	0.0763653i
$366$		0			0
$367$	−8.87689	−	15.3752i	−0.463370	−	0.802581i	0.535756	−	0.844373i	$-0.320027\pi$
$367$	−8.87689	−	15.3752i	−0.463370	−	0.802581i	−0.999126	+	0.0417921i	$0.986693\pi$
$368$		0			0
$369$	−21.8078			−1.13527
$370$		0			0
$371$	−1.65767	−	2.87117i	−0.0860620	−	0.149064i
$372$		0			0
$373$	−35.8078			−1.85406			−0.927028	−	0.374992i	$-0.877645\pi$
$373$	−35.8078			−1.85406			−0.927028	+	0.374992i	$0.877645\pi$
$374$		0			0
$375$	−1.28078	+	2.21837i	−0.0661390	+	0.114556i
$376$		0			0
$377$	2.00000	−	3.46410i	0.103005	−	0.178410i
$378$		0			0
$379$	0.492423			0.0252940			0.0126470	−	0.999920i	$-0.495974\pi$
$379$	0.492423			0.0252940			0.0126470	+	0.999920i	$0.495974\pi$
$380$		0			0
$381$	39.8617			2.04218
$382$		0			0
$383$	2.56155	−	4.43674i	0.130889	−	0.226707i	−0.793130	−	0.609052i	$-0.791550\pi$
$383$	2.56155	−	4.43674i	0.130889	−	0.226707i	0.924020	+	0.382345i	$0.124883\pi$
$384$		0			0
$385$	−0.219224	+	0.379706i	−0.0111727	+	0.0193516i
$386$		0			0
$387$	−11.1231			−0.565419
$388$		0			0
$389$	9.56155	+	16.5611i	0.484790	+	0.839681i	0.999847	−	0.0174749i	$-0.00556271\pi$
$389$	9.56155	+	16.5611i	0.484790	+	0.839681i	−0.515057	+	0.857156i	$0.672229\pi$
$390$		0			0
$391$	24.0000			1.21373
$392$		0			0
$393$	−20.6501	−	35.7670i	−1.04166	−	1.80421i
$394$		0			0
$395$	−5.56155	−	9.63289i	−0.279832	−	0.484683i
$396$		0			0
$397$	−14.1501	+	24.5087i	−0.710173	+	1.23006i	0.254619	+	0.967041i	$0.418050\pi$
$397$	−14.1501	+	24.5087i	−0.710173	+	1.23006i	−0.964792	+	0.263014i	$0.915283\pi$
$398$		0			0
$399$	4.87689	−	0.426450i	0.244150	−	0.0213492i
$400$		0			0
$401$	−13.9654	+	24.1888i	−0.697401	+	1.20793i	0.271964	+	0.962307i	$0.412327\pi$
$401$	−13.9654	+	24.1888i	−0.697401	+	1.20793i	−0.969365	+	0.245626i	$0.921007\pi$
$402$		0			0
$403$	−10.2462	−	17.7470i	−0.510400	−	0.884039i
$404$		0			0
$405$	3.50000	+	6.06218i	0.173916	+	0.301232i
$406$		0			0
$407$	4.68466			0.232210
$408$		0			0
$409$	−10.5000	−	18.1865i	−0.519192	−	0.899266i	−0.999751	−	0.0223042i	$-0.992900\pi$
$409$	−10.5000	−	18.1865i	−0.519192	−	0.899266i	0.480560	−	0.876962i	$-0.340434\pi$
$410$		0			0
$411$	−13.9309			−0.687159
$412$		0			0
$413$	−3.19224	+	5.52911i	−0.157080	+	0.272070i
$414$		0			0
$415$	−5.40388	+	9.35980i	−0.265266	+	0.459454i
$416$		0			0
$417$	−22.5616			−1.10484
$418$		0			0
$419$	17.5616			0.857938			0.428969	−	0.903319i	$-0.358877\pi$
$419$	17.5616			0.857938			0.428969	+	0.903319i	$0.358877\pi$
$420$		0			0
$421$	−16.2462	+	28.1393i	−0.791792	+	1.37142i	0.133065	+	0.991107i	$0.457518\pi$
$421$	−16.2462	+	28.1393i	−0.791792	+	1.37142i	−0.924856	+	0.380316i	$0.875815\pi$
$422$		0			0
$423$	5.12311	−	8.87348i	0.249094	−	0.431443i
$424$		0			0
$425$	5.12311			0.248507
$426$		0			0
$427$	1.12311	+	1.94528i	0.0543509	+	0.0941385i
$428$		0			0
$429$	5.12311			0.247346
$430$		0			0
$431$	−3.68466	−	6.38202i	−0.177484	−	0.307411i	0.763534	−	0.645767i	$-0.223462\pi$
$431$	−3.68466	−	6.38202i	−0.177484	−	0.307411i	−0.941018	+	0.338356i	$0.890129\pi$
$432$		0			0
$433$	18.3693	+	31.8166i	0.882773	+	1.52901i	0.848245	+	0.529604i	$0.177659\pi$
$433$	18.3693	+	31.8166i	0.882773	+	1.52901i	0.0345280	+	0.999404i	$0.489007\pi$
$434$		0			0
$435$	−2.56155	+	4.43674i	−0.122817	+	0.212725i
$436$		0			0
$437$	8.63068	−	18.5064i	0.412862	−	0.885280i
$438$		0			0
$439$	−8.24621	+	14.2829i	−0.393570	+	0.681684i	−0.992918	−	0.118806i	$-0.962094\pi$
$439$	−8.24621	+	14.2829i	−0.393570	+	0.681684i	0.599347	+	0.800489i	$0.295427\pi$
$440$		0			0
$441$	12.1231	+	20.9978i	0.577291	+	0.999897i
$442$		0			0
$443$	16.9654	+	29.3850i	0.806052	+	1.39612i	0.915578	+	0.402139i	$0.131733\pi$
$443$	16.9654	+	29.3850i	0.806052	+	1.39612i	−0.109526	+	0.993984i	$0.534933\pi$
$444$		0			0
$445$	−2.68466			−0.127265
$446$		0			0
$447$	−20.4924	−	35.4939i	−0.969258	−	1.67880i
$448$		0			0
$449$	29.0000			1.36859			0.684297	−	0.729203i	$-0.260109\pi$
$449$	29.0000			1.36859			0.684297	+	0.729203i	$0.260109\pi$
$450$		0			0
$451$	−3.06155	+	5.30277i	−0.144163	+	0.249697i
$452$		0			0
$453$	26.2462	−	45.4598i	1.23315	−	2.13589i
$454$		0			0
$455$	0.876894			0.0411094
$456$		0			0
$457$	7.05398			0.329971			0.164986	−	0.986296i	$-0.447242\pi$
$457$	7.05398			0.329971			0.164986	+	0.986296i	$0.447242\pi$
$458$		0			0
$459$	−3.68466	+	6.38202i	−0.171985	+	0.297887i
$460$		0			0
$461$	19.4924	−	33.7619i	0.907853	−	1.57245i	0.0908110	−	0.995868i	$-0.471054\pi$
$461$	19.4924	−	33.7619i	0.907853	−	1.57245i	0.817042	−	0.576579i	$-0.195613\pi$
$462$		0			0
$463$	19.5616			0.909102			0.454551	−	0.890721i	$-0.349800\pi$
$463$	19.5616			0.909102			0.454551	+	0.890721i	$0.349800\pi$
$464$		0			0
$465$	13.1231	+	22.7299i	0.608569	+	1.05407i
$466$		0			0
$467$	−16.5616			−0.766377			−0.383189	−	0.923670i	$-0.625174\pi$
$467$	−16.5616			−0.766377			−0.383189	+	0.923670i	$0.625174\pi$
$468$		0			0
$469$	−2.06913	−	3.58384i	−0.0955436	−	0.165486i
$470$		0			0
$471$	−3.12311	−	5.40938i	−0.143905	−	0.249251i
$472$		0			0
$473$	−1.56155	+	2.70469i	−0.0718003	+	0.124362i
$474$		0			0
$475$	1.84233	−	3.95042i	0.0845319	−	0.181258i
$476$		0			0
$477$	13.4654	−	23.3228i	0.616540	−	1.06788i
$478$		0			0
$479$	14.6847	+	25.4346i	0.670959	+	1.16214i	0.977632	+	0.210321i	$0.0674508\pi$
$479$	14.6847	+	25.4346i	0.670959	+	1.16214i	−0.306673	+	0.951815i	$0.599216\pi$
$480$		0			0
$481$	−4.68466	−	8.11407i	−0.213602	−	0.369970i
$482$		0			0
$483$	−5.26137			−0.239400
$484$		0			0
$485$	−0.842329	−	1.45896i	−0.0382482	−	0.0662478i
$486$		0			0
$487$	6.93087			0.314068			0.157034	−	0.987593i	$-0.449807\pi$
$487$	6.93087			0.314068			0.157034	+	0.987593i	$0.449807\pi$
$488$		0			0
$489$	21.8423	−	37.8320i	0.987744	−	1.71082i
$490$		0			0
$491$	−4.58854	+	7.94759i	−0.207078	+	0.358670i	−0.950793	−	0.309827i	$-0.899729\pi$
$491$	−4.58854	+	7.94759i	−0.207078	+	0.358670i	0.743715	+	0.668497i	$0.233062\pi$
$492$		0			0
$493$	10.2462			0.461466
$494$		0			0
$495$	−3.56155			−0.160080
$496$		0			0
$497$	−3.56155	+	6.16879i	−0.159757	+	0.276708i
$498$		0			0
$499$	14.5000	−	25.1147i	0.649109	−	1.12429i	−0.334227	−	0.942493i	$-0.608475\pi$
$499$	14.5000	−	25.1147i	0.649109	−	1.12429i	0.983336	−	0.181797i	$-0.0581915\pi$
$500$		0			0
$501$	−1.75379			−0.0783535
$502$		0			0
$503$	13.7116	+	23.7493i	0.611372	+	1.05893i	0.991009	+	0.133792i	$0.0427154\pi$
$503$	13.7116	+	23.7493i	0.611372	+	1.05893i	−0.379637	+	0.925135i	$0.623951\pi$
$504$		0			0
$505$	10.0000			0.444994
$506$		0			0
$507$	11.5270	+	19.9653i	0.511931	+	0.886691i
$508$		0			0
$509$	0.438447	+	0.759413i	0.0194338	+	0.0336604i	0.875579	−	0.483075i	$-0.160480\pi$
$509$	0.438447	+	0.759413i	0.0194338	+	0.0336604i	−0.856145	+	0.516736i	$0.827147\pi$
$510$		0			0
$511$	−0.369317	+	0.639676i	−0.0163376	+	0.0282976i
$512$		0			0
$513$	3.59612	+	5.13628i	0.158772	+	0.226772i
$514$		0			0
$515$	−2.90388	+	5.02967i	−0.127960	+	0.221634i
$516$		0			0
$517$	−1.43845	−	2.49146i	−0.0632628	−	0.109574i
$518$		0			0
$519$	7.43845	+	12.8838i	0.326512	+	0.565535i
$520$		0			0
$521$	−22.8078			−0.999226			−0.499613	−	0.866249i	$-0.666524\pi$
$521$	−22.8078			−0.999226			−0.499613	+	0.866249i	$0.666524\pi$
$522$		0			0
$523$	7.31534	+	12.6705i	0.319878	+	0.554044i	0.980462	−	0.196707i	$-0.0630249\pi$
$523$	7.31534	+	12.6705i	0.319878	+	0.554044i	−0.660585	+	0.750752i	$0.729692\pi$
$524$		0			0
$525$	−1.12311			−0.0490163
$526$		0			0
$527$	26.2462	−	45.4598i	1.14330	−	1.98026i
$528$		0			0
$529$	0.526988	−	0.912769i	0.0229125	−	0.0396856i
$530$		0			0
$531$	−51.8617			−2.25061
$532$		0			0
$533$	12.2462			0.530442
$534$		0			0
$535$	1.12311	−	1.94528i	0.0485561	−	0.0841016i
$536$		0			0
$537$	14.7192	−	25.4944i	0.635181	−	1.10017i
$538$		0			0
$539$	6.80776			0.293231
$540$		0			0
$541$	14.2462	+	24.6752i	0.612492	+	1.06087i	0.990819	+	0.135196i	$0.0431664\pi$
$541$	14.2462	+	24.6752i	0.612492	+	1.06087i	−0.378326	+	0.925672i	$0.623500\pi$
$542$		0			0
$543$	54.7386			2.34906
$544$		0			0
$545$	7.12311	+	12.3376i	0.305120	+	0.528484i
$546$		0			0
$547$	−2.43845	−	4.22351i	−0.104260	−	0.180584i	0.809175	−	0.587567i	$-0.199914\pi$
$547$	−2.43845	−	4.22351i	−0.104260	−	0.180584i	−0.913436	+	0.406983i	$0.866581\pi$
$548$		0			0
$549$	−9.12311	+	15.8017i	−0.389365	+	0.674399i
$550$		0			0
$551$	3.68466	−	7.90084i	0.156972	−	0.336587i
$552$		0			0
$553$	2.43845	−	4.22351i	0.103693	−	0.179602i
$554$		0			0
$555$	6.00000	+	10.3923i	0.254686	+	0.441129i
$556$		0			0
$557$	−20.8348	−	36.0868i	−0.882797	−	1.52905i	−0.848218	−	0.529647i	$-0.822324\pi$
$557$	−20.8348	−	36.0868i	−0.882797	−	1.52905i	−0.0345785	−	0.999402i	$-0.511009\pi$
$558$		0			0
$559$	6.24621			0.264187
$560$		0			0
$561$	6.56155	+	11.3649i	0.277029	+	0.479828i
$562$		0			0
$563$	−21.3002			−0.897696			−0.448848	−	0.893608i	$-0.648166\pi$
$563$	−21.3002			−0.897696			−0.448848	+	0.893608i	$0.648166\pi$
$564$		0			0
$565$	8.40388	−	14.5560i	0.353554	−	0.612373i
$566$		0			0
$567$	−1.53457	+	2.65794i	−0.0644457	+	0.111623i
$568$		0			0
$569$	−11.1771			−0.468568			−0.234284	−	0.972168i	$-0.575274\pi$
$569$	−11.1771			−0.468568			−0.234284	+	0.972168i	$0.575274\pi$
$570$		0			0
$571$	4.80776			0.201199			0.100599	−	0.994927i	$-0.467924\pi$
$571$	4.80776			0.201199			0.100599	+	0.994927i	$0.467924\pi$
$572$		0			0
$573$	−6.87689	+	11.9111i	−0.287286	+	0.497595i
$574$		0			0
$575$	−2.34233	+	4.05703i	−0.0976819	+	0.169190i
$576$		0			0
$577$	−16.3153			−0.679217			−0.339608	−	0.940567i	$-0.610295\pi$
$577$	−16.3153			−0.679217			−0.339608	+	0.940567i	$0.610295\pi$
$578$		0			0
$579$	−32.8078	−	56.8247i	−1.36344	−	2.36155i
$580$		0			0
$581$	−4.73863			−0.196592
$582$		0			0
$583$	−3.78078	−	6.54850i	−0.156584	−	0.271211i
$584$		0			0
$585$	3.56155	+	6.16879i	0.147252	+	0.255048i
$586$		0			0
$587$	13.3693	−	23.1563i	0.551811	−	0.955764i	−0.446333	−	0.894867i	$-0.647270\pi$
$587$	13.3693	−	23.1563i	0.551811	−	0.955764i	0.998144	−	0.0608975i	$-0.0193963\pi$
$588$		0			0
$589$	−25.6155	−	36.5863i	−1.05547	−	1.50751i
$590$		0			0
$591$	−18.4924	+	32.0298i	−0.760677	+	1.31753i
$592$		0			0
$593$	−12.7732	−	22.1238i	−0.524532	−	0.908517i	−0.999592	−	0.0285632i	$-0.990907\pi$
$593$	−12.7732	−	22.1238i	−0.524532	−	0.908517i	0.475060	−	0.879954i	$-0.342427\pi$
$594$		0			0
$595$	1.12311	+	1.94528i	0.0460428	+	0.0797485i
$596$		0			0
$597$	7.36932			0.301606
$598$		0			0
$599$	−16.2462	−	28.1393i	−0.663802	−	1.14974i	−0.979609	−	0.200915i	$-0.935608\pi$
$599$	−16.2462	−	28.1393i	−0.663802	−	1.14974i	0.315806	−	0.948824i	$-0.397725\pi$
$600$		0			0
$601$	−11.6307			−0.474425			−0.237213	−	0.971458i	$-0.576234\pi$
$601$	−11.6307			−0.474425			−0.237213	+	0.971458i	$0.576234\pi$
$602$		0			0
$603$	16.8078	−	29.1119i	0.684465	−	1.18553i
$604$		0			0
$605$	5.00000	−	8.66025i	0.203279	−	0.352089i
$606$		0			0
$607$	−10.1922			−0.413690			−0.206845	−	0.978374i	$-0.566320\pi$
$607$	−10.1922			−0.413690			−0.206845	+	0.978374i	$0.566320\pi$
$608$		0			0
$609$	−2.24621			−0.0910211
$610$		0			0
$611$	−2.87689	+	4.98293i	−0.116387	+	0.201588i
$612$		0			0
$613$	−15.3423	+	26.5737i	−0.619671	+	1.07330i	0.369875	+	0.929082i	$0.379401\pi$
$613$	−15.3423	+	26.5737i	−0.619671	+	1.07330i	−0.989546	+	0.144220i	$0.953933\pi$
$614$		0			0
$615$	−15.6847			−0.632466
$616$		0			0
$617$	−3.59612	−	6.22866i	−0.144774	−	0.250756i	0.784514	−	0.620111i	$-0.212912\pi$
$617$	−3.59612	−	6.22866i	−0.144774	−	0.250756i	−0.929289	+	0.369354i	$0.879579\pi$
$618$		0			0
$619$	−26.0540			−1.04720			−0.523599	−	0.851965i	$-0.675411\pi$
$619$	−26.0540			−1.04720			−0.523599	+	0.851965i	$0.675411\pi$
$620$		0			0
$621$	−3.36932	−	5.83583i	−0.135206	−	0.234184i
$622$		0			0
$623$	−0.588540	−	1.01938i	−0.0235794	−	0.0408407i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$	11.1231	−	0.972638i	0.444214	−	0.0388434i
$628$		0			0
$629$	12.0000	−	20.7846i	0.478471	−	0.828737i
$630$		0			0
$631$	2.12311	+	3.67733i	0.0845195	+	0.146392i	0.905186	−	0.425015i	$-0.139731\pi$
$631$	2.12311	+	3.67733i	0.0845195	+	0.146392i	−0.820667	+	0.571407i	$0.806398\pi$
$632$		0			0
$633$	−4.24621	−	7.35465i	−0.168772	−	0.292321i
$634$		0			0
$635$	15.5616			0.617541
$636$		0			0
$637$	−6.80776	−	11.7914i	−0.269733	−	0.467192i
$638$		0			0
$639$	−57.8617			−2.28898
$640$		0			0
$641$	−3.71922	+	6.44188i	−0.146900	+	0.254439i	−0.930080	−	0.367356i	$-0.880263\pi$
$641$	−3.71922	+	6.44188i	−0.146900	+	0.254439i	0.783180	+	0.621795i	$0.213596\pi$
$642$		0			0
$643$	−11.2808	+	19.5389i	−0.444870	+	0.770538i	−0.998043	−	0.0625284i	$-0.980084\pi$
$643$	−11.2808	+	19.5389i	−0.444870	+	0.770538i	0.553173	+	0.833067i	$0.313417\pi$
$644$		0			0
$645$	−8.00000			−0.315000
$646$		0			0
$647$	−3.17708			−0.124904			−0.0624520	−	0.998048i	$-0.519892\pi$
$647$	−3.17708			−0.124904			−0.0624520	+	0.998048i	$0.519892\pi$
$648$		0			0
$649$	−7.28078	+	12.6107i	−0.285795	+	0.495012i
$650$		0			0
$651$	−5.75379	+	9.96585i	−0.225509	+	0.390593i
$652$		0			0
$653$	−5.06913			−0.198370			−0.0991852	−	0.995069i	$-0.531624\pi$
$653$	−5.06913			−0.198370			−0.0991852	+	0.995069i	$0.531624\pi$
$654$		0			0
$655$	−8.06155	−	13.9630i	−0.314991	−	0.545580i
$656$		0			0
$657$	−6.00000			−0.234082
$658$		0			0
$659$	9.46543	+	16.3946i	0.368721	+	0.638643i	0.989366	−	0.145448i	$-0.0464624\pi$
$659$	9.46543	+	16.3946i	0.368721	+	0.638643i	−0.620645	+	0.784092i	$0.713129\pi$
$660$		0			0
$661$	19.9309	+	34.5213i	0.775221	+	1.34272i	0.934670	+	0.355516i	$0.115695\pi$
$661$	19.9309	+	34.5213i	0.775221	+	1.34272i	−0.159449	+	0.987206i	$0.550972\pi$
$662$		0			0
$663$	13.1231	−	22.7299i	0.509659	−	0.882756i
$664$		0			0
$665$	1.90388	−	0.166481i	0.0738294	−	0.00645586i
$666$		0			0
$667$	−4.68466	+	8.11407i	−0.181391	+	0.314178i
$668$		0			0
$669$	14.4924	+	25.1016i	0.560309	+	0.970484i
$670$		0			0
$671$	2.56155	+	4.43674i	0.0988876	+	0.171278i
$672$		0			0
$673$	46.1080			1.77733			0.888665	−	0.458556i	$-0.151633\pi$
$673$	46.1080			1.77733			0.888665	+	0.458556i	$0.151633\pi$
$674$		0			0
$675$	−0.719224	−	1.24573i	−0.0276829	−	0.0479482i
$676$		0			0
$677$	−23.1771			−0.890768			−0.445384	−	0.895340i	$-0.646933\pi$
$677$	−23.1771			−0.890768			−0.445384	+	0.895340i	$0.646933\pi$
$678$		0			0
$679$	0.369317	−	0.639676i	0.0141731	−	0.0245485i
$680$		0			0
$681$	25.5270	−	44.2140i	0.978196	−	1.69429i
$682$		0			0
$683$	−28.0000			−1.07139			−0.535695	−	0.844411i	$-0.679950\pi$
$683$	−28.0000			−1.07139			−0.535695	+	0.844411i	$0.679950\pi$
$684$		0			0
$685$	−5.43845			−0.207792
$686$		0			0
$687$	−16.4924	+	28.5657i	−0.629225	+	1.08985i
$688$		0			0
$689$	−7.56155	+	13.0970i	−0.288072	+	0.498956i
$690$		0			0
$691$	21.0691			0.801507			0.400754	−	0.916186i	$-0.368748\pi$
$691$	21.0691			0.801507			0.400754	+	0.916186i	$0.368748\pi$
$692$		0			0
$693$	−0.780776	−	1.35234i	−0.0296592	−	0.0513713i
$694$		0			0
$695$	−8.80776			−0.334098
$696$		0			0
$697$	15.6847	+	27.1666i	0.594099	+	1.02901i
$698$		0			0
$699$	−2.96543	−	5.13628i	−0.112163	−	0.194272i
$700$		0			0
$701$	−3.24621	+	5.62260i	−0.122608	+	0.212363i	−0.920795	−	0.390046i	$-0.872459\pi$
$701$	−3.24621	+	5.62260i	−0.122608	+	0.212363i	0.798188	+	0.602409i	$0.205792\pi$
$702$		0			0
$703$	−11.7116	−	16.7276i	−0.441713	−	0.630892i
$704$		0			0
$705$	3.68466	−	6.38202i	0.138772	−	0.240361i
$706$		0			0
$707$	2.19224	+	3.79706i	0.0824475	+	0.142803i
$708$		0			0
$709$	20.0000	+	34.6410i	0.751116	+	1.30097i	0.947282	+	0.320400i	$0.103817\pi$
$709$	20.0000	+	34.6410i	0.751116	+	1.30097i	−0.196167	+	0.980571i	$0.562849\pi$
$710$		0			0
$711$	39.6155			1.48570
$712$		0			0
$713$	24.0000	+	41.5692i	0.898807	+	1.55678i
$714$		0			0
$715$	2.00000			0.0747958
$716$		0			0
$717$	23.3693	−	40.4768i	0.872743	−	1.51164i
$718$		0			0
$719$	7.43845	−	12.8838i	0.277407	−	0.480483i	−0.693332	−	0.720618i	$-0.743858\pi$
$719$	7.43845	−	12.8838i	0.277407	−	0.480483i	0.970740	+	0.240134i	$0.0771915\pi$
$720$		0			0
$721$	−2.54640			−0.0948328
$722$		0			0
$723$	−21.9309			−0.815618
$724$		0			0
$725$	−1.00000	+	1.73205i	−0.0371391	+	0.0643268i
$726$		0			0
$727$	−4.00000	+	6.92820i	−0.148352	+	0.256953i	−0.930618	−	0.365991i	$-0.880730\pi$
$727$	−4.00000	+	6.92820i	−0.148352	+	0.256953i	0.782267	+	0.622944i	$0.214063\pi$
$728$		0			0
$729$	−35.9848			−1.33277
$730$		0			0
$731$	8.00000	+	13.8564i	0.295891	+	0.512498i
$732$		0			0
$733$	−26.9309			−0.994714			−0.497357	−	0.867546i	$-0.665696\pi$
$733$	−26.9309			−0.994714			−0.497357	+	0.867546i	$0.665696\pi$
$734$		0			0
$735$	8.71922	+	15.1021i	0.321613	+	0.557051i
$736$		0			0
$737$	−4.71922	−	8.17394i	−0.173835	−	0.301091i
$738$		0			0
$739$	9.37689	−	16.2413i	0.344935	−	0.597444i	−0.640407	−	0.768036i	$-0.721234\pi$
$739$	9.37689	−	16.2413i	0.344935	−	0.597444i	0.985342	+	0.170591i	$0.0545677\pi$
$740$		0			0
$741$	−12.8078	−	18.2931i	−0.470505	−	0.672016i
$742$		0			0
$743$	−9.21922	+	15.9682i	−0.338221	+	0.585815i	−0.984098	−	0.177626i	$-0.943158\pi$
$743$	−9.21922	+	15.9682i	−0.338221	+	0.585815i	0.645878	+	0.763441i	$0.276492\pi$
$744$		0			0
$745$	−8.00000	−	13.8564i	−0.293097	−	0.507659i
$746$		0			0
$747$	−19.2462	−	33.3354i	−0.704182	−	1.21968i
$748$		0			0
$749$	0.984845			0.0359855
$750$		0			0
$751$	−17.4384	−	30.2043i	−0.636338	−	1.10217i	−0.986230	−	0.165380i	$-0.947115\pi$
$751$	−17.4384	−	30.2043i	−0.636338	−	1.10217i	0.349892	−	0.936790i	$-0.386218\pi$
$752$		0			0
$753$	−66.4233			−2.42060
$754$		0			0
$755$	10.2462	−	17.7470i	0.372898	−	0.645878i
$756$		0			0
$757$	9.78078	−	16.9408i	0.355488	−	0.615724i	−0.631713	−	0.775202i	$-0.717648\pi$
$757$	9.78078	−	16.9408i	0.355488	−	0.615724i	0.987201	+	0.159478i	$0.0509812\pi$
$758$		0			0
$759$	−12.0000			−0.435572
$760$		0			0
$761$	−51.9848			−1.88445			−0.942225	−	0.334982i	$-0.891270\pi$
$761$	−51.9848			−1.88445			−0.942225	+	0.334982i	$0.891270\pi$
$762$		0			0
$763$	−3.12311	+	5.40938i	−0.113064	+	0.195833i
$764$		0			0
$765$	−9.12311	+	15.8017i	−0.329847	+	0.571311i
$766$		0			0
$767$	29.1231			1.05157
$768$		0			0
$769$	−13.2462	−	22.9431i	−0.477671	−	0.827350i	0.522002	−	0.852944i	$-0.325185\pi$
$769$	−13.2462	−	22.9431i	−0.477671	−	0.827350i	−0.999672	+	0.0255946i	$0.991852\pi$
$770$		0			0
$771$	33.4384			1.20426
$772$		0			0
$773$	7.15009	+	12.3843i	0.257171	+	0.445433i	0.965483	−	0.260466i	$-0.0838763\pi$
$773$	7.15009	+	12.3843i	0.257171	+	0.445433i	−0.708312	+	0.705900i	$0.750543\pi$
$774$		0			0
$775$	5.12311	+	8.87348i	0.184027	+	0.318745i
$776$		0			0
$777$	−2.63068	+	4.55648i	−0.0943752	+	0.163463i
$778$		0			0
$779$	26.5885	−	2.32498i	0.952633	−	0.0833011i
$780$		0			0
$781$	−8.12311	+	14.0696i	−0.290668	+	0.503451i
$782$		0			0
$783$	−1.43845	−	2.49146i	−0.0514059	−	0.0890376i
$784$		0			0
$785$	−1.21922	−	2.11176i	−0.0435160	−	0.0753718i
$786$		0			0
$787$	−8.56155			−0.305186			−0.152593	−	0.988289i	$-0.548762\pi$
$787$	−8.56155			−0.305186			−0.152593	+	0.988289i	$0.548762\pi$
$788$		0			0
$789$	−2.80776	−	4.86319i	−0.0999590	−	0.173134i
$790$		0			0

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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.28078 + 2.21837i) q^{3} +(-0.500000 + 0.866025i) q^{5} -0.438447 q^{7} +(-1.78078 - 3.08440i) q^{9} +O(q^{10})\)	\(q+(-1.28078 + 2.21837i) q^{3} +(-0.500000 + 0.866025i) q^{5} -0.438447 q^{7} +(-1.78078 - 3.08440i) q^{9} -1.00000 q^{11} +(1.00000 + 1.73205i) q^{13} +(-1.28078 - 2.21837i) q^{15} +(-2.56155 + 4.43674i) q^{17} +(2.50000 + 3.57071i) q^{19} +(0.561553 - 0.972638i) q^{21} +(-2.34233 - 4.05703i) q^{23} +(-0.500000 - 0.866025i) q^{25} +1.43845 q^{27} +(-1.00000 - 1.73205i) q^{29} -10.2462 q^{31} +(1.28078 - 2.21837i) q^{33} +(0.219224 - 0.379706i) q^{35} -4.68466 q^{37} -5.12311 q^{39} +(3.06155 - 5.30277i) q^{41} +(1.56155 - 2.70469i) q^{43} +3.56155 q^{45} +(1.43845 + 2.49146i) q^{47} -6.80776 q^{49} +(-6.56155 - 11.3649i) q^{51} +(3.78078 + 6.54850i) q^{53} +(0.500000 - 0.866025i) q^{55} +(-11.1231 + 0.972638i) q^{57} +(7.28078 - 12.6107i) q^{59} +(-2.56155 - 4.43674i) q^{61} +(0.780776 + 1.35234i) q^{63} -2.00000 q^{65} +(4.71922 + 8.17394i) q^{67} +12.0000 q^{69} +(8.12311 - 14.0696i) q^{71} +(0.842329 - 1.45896i) q^{73} +2.56155 q^{75} +0.438447 q^{77} +(-5.56155 + 9.63289i) q^{79} +(3.50000 - 6.06218i) q^{81} +10.8078 q^{83} +(-2.56155 - 4.43674i) q^{85} +5.12311 q^{87} +(1.34233 + 2.32498i) q^{89} +(-0.438447 - 0.759413i) q^{91} +(13.1231 - 22.7299i) q^{93} +(-4.34233 + 0.379706i) q^{95} +(-0.842329 + 1.45896i) q^{97} +(1.78078 + 3.08440i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} - 2 q^{5} - 10 q^{7} - 3 q^{9}+O(q^{10}) \) 4 * q - q^3 - 2 * q^5 - 10 * q^7 - 3 * q^9	\( 4 q - q^{3} - 2 q^{5} - 10 q^{7} - 3 q^{9} - 4 q^{11} + 4 q^{13} - q^{15} - 2 q^{17} + 10 q^{19} - 6 q^{21} + 3 q^{23} - 2 q^{25} + 14 q^{27} - 4 q^{29} - 8 q^{31} + q^{33} + 5 q^{35} + 6 q^{37} - 4 q^{39} + 4 q^{41} - 2 q^{43} + 6 q^{45} + 14 q^{47} + 14 q^{49} - 18 q^{51} + 11 q^{53} + 2 q^{55} - 28 q^{57} + 25 q^{59} - 2 q^{61} - q^{63} - 8 q^{65} + 23 q^{67} + 48 q^{69} + 16 q^{71} - 9 q^{73} + 2 q^{75} + 10 q^{77} - 14 q^{79} + 14 q^{81} + 2 q^{83} - 2 q^{85} + 4 q^{87} - 7 q^{89} - 10 q^{91} + 36 q^{93} - 5 q^{95} + 9 q^{97} + 3 q^{99}+O(q^{100}) \) 4 * q - q^3 - 2 * q^5 - 10 * q^7 - 3 * q^9 - 4 * q^11 + 4 * q^13 - q^15 - 2 * q^17 + 10 * q^19 - 6 * q^21 + 3 * q^23 - 2 * q^25 + 14 * q^27 - 4 * q^29 - 8 * q^31 + q^33 + 5 * q^35 + 6 * q^37 - 4 * q^39 + 4 * q^41 - 2 * q^43 + 6 * q^45 + 14 * q^47 + 14 * q^49 - 18 * q^51 + 11 * q^53 + 2 * q^55 - 28 * q^57 + 25 * q^59 - 2 * q^61 - q^63 - 8 * q^65 + 23 * q^67 + 48 * q^69 + 16 * q^71 - 9 * q^73 + 2 * q^75 + 10 * q^77 - 14 * q^79 + 14 * q^81 + 2 * q^83 - 2 * q^85 + 4 * q^87 - 7 * q^89 - 10 * q^91 + 36 * q^93 - 5 * q^95 + 9 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more