LMFDB - Embedded newform 1764.4.k.d.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(361,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1764.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.k (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:	$104.079369250$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1764.361
Dual form			1764.4.k.d.1549.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.00000 - 6.92820i) q^{5} +O(q^{10})$	$q+(-4.00000 - 6.92820i) q^{5} +(-20.0000 + 34.6410i) q^{11} -12.0000 q^{13} +(-29.0000 + 50.2295i) q^{17} +(-13.0000 - 22.5167i) q^{19} +(-32.0000 - 55.4256i) q^{23} +(30.5000 - 52.8275i) q^{25} +62.0000 q^{29} +(-126.000 + 218.238i) q^{31} +(-13.0000 - 22.5167i) q^{37} -6.00000 q^{41} +416.000 q^{43} +(-198.000 - 342.946i) q^{47} +(-225.000 + 389.711i) q^{53} +320.000 q^{55} +(137.000 - 237.291i) q^{59} +(288.000 + 498.831i) q^{61} +(48.0000 + 83.1384i) q^{65} +(238.000 - 412.228i) q^{67} +448.000 q^{71} +(79.0000 - 136.832i) q^{73} +(468.000 + 810.600i) q^{79} -530.000 q^{83} +464.000 q^{85} +(-195.000 - 337.750i) q^{89} +(-104.000 + 180.133i) q^{95} +214.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{5}+O(q^{10}) $ 2 * q - 8 * q^5	$ 2 q - 8 q^{5} - 40 q^{11} - 24 q^{13} - 58 q^{17} - 26 q^{19} - 64 q^{23} + 61 q^{25} + 124 q^{29} - 252 q^{31} - 26 q^{37} - 12 q^{41} + 832 q^{43} - 396 q^{47} - 450 q^{53} + 640 q^{55} + 274 q^{59} + 576 q^{61} + 96 q^{65} + 476 q^{67} + 896 q^{71} + 158 q^{73} + 936 q^{79} - 1060 q^{83} + 928 q^{85} - 390 q^{89} - 208 q^{95} + 428 q^{97}+O(q^{100}) $ 2 * q - 8 * q^5 - 40 * q^11 - 24 * q^13 - 58 * q^17 - 26 * q^19 - 64 * q^23 + 61 * q^25 + 124 * q^29 - 252 * q^31 - 26 * q^37 - 12 * q^41 + 832 * q^43 - 396 * q^47 - 450 * q^53 + 640 * q^55 + 274 * q^59 + 576 * q^61 + 96 * q^65 + 476 * q^67 + 896 * q^71 + 158 * q^73 + 936 * q^79 - 1060 * q^83 + 928 * q^85 - 390 * q^89 - 208 * q^95 + 428 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	−4.00000	−	6.92820i	−0.357771	−	0.619677i	0.629817	−	0.776743i	$-0.283130\pi$
$5$	−4.00000	−	6.92820i	−0.357771	−	0.619677i	−0.987588	+	0.157066i	$0.949796\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−20.0000	+	34.6410i	−0.548202	+	0.949514i	0.450195	+	0.892930i	$0.351354\pi$
$11$	−20.0000	+	34.6410i	−0.548202	+	0.949514i	−0.998398	+	0.0565844i	$0.981979\pi$
$12$		0			0
$13$	−12.0000			−0.256015			−0.128008	−	0.991773i	$-0.540858\pi$
$13$	−12.0000			−0.256015			−0.128008	+	0.991773i	$0.540858\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−29.0000	+	50.2295i	−0.413737	+	0.716614i	−0.995295	−	0.0968912i	$-0.969110\pi$
$17$	−29.0000	+	50.2295i	−0.413737	+	0.716614i	0.581558	+	0.813505i	$0.302443\pi$
$18$		0			0
$19$	−13.0000	−	22.5167i	−0.156969	−	0.271878i	0.776805	−	0.629741i	$-0.216839\pi$
$19$	−13.0000	−	22.5167i	−0.156969	−	0.271878i	−0.933774	+	0.357863i	$0.883505\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−32.0000	−	55.4256i	−0.290107	−	0.502480i	0.683728	−	0.729737i	$-0.260358\pi$
$23$	−32.0000	−	55.4256i	−0.290107	−	0.502480i	−0.973835	+	0.227257i	$0.927024\pi$
$24$		0			0
$25$	30.5000	−	52.8275i	0.244000	−	0.422620i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	62.0000			0.397004			0.198502	−	0.980101i	$-0.436392\pi$
$29$	62.0000			0.397004			0.198502	+	0.980101i	$0.436392\pi$
$30$		0			0
$31$	−126.000	+	218.238i	−0.730009	+	1.26441i	0.226870	+	0.973925i	$0.427151\pi$
$31$	−126.000	+	218.238i	−0.730009	+	1.26441i	−0.956879	+	0.290487i	$0.906183\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−13.0000	−	22.5167i	−0.0577618	−	0.100046i	0.835699	−	0.549188i	$-0.185063\pi$
$37$	−13.0000	−	22.5167i	−0.0577618	−	0.100046i	−0.893460	+	0.449142i	$0.851730\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.00000			−0.0228547			−0.0114273	−	0.999935i	$-0.503638\pi$
$41$	−6.00000			−0.0228547			−0.0114273	+	0.999935i	$0.503638\pi$
$42$		0			0
$43$	416.000			1.47534			0.737668	−	0.675164i	$-0.235927\pi$
$43$	416.000			1.47534			0.737668	+	0.675164i	$0.235927\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−198.000	−	342.946i	−0.614495	−	1.06434i	−0.990473	−	0.137708i	$-0.956026\pi$
$47$	−198.000	−	342.946i	−0.614495	−	1.06434i	0.375978	−	0.926629i	$-0.377307\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−225.000	+	389.711i	−0.583134	+	1.01002i	0.411971	+	0.911197i	$0.364841\pi$
$53$	−225.000	+	389.711i	−0.583134	+	1.01002i	−0.995105	+	0.0988214i	$0.968493\pi$
$54$		0			0
$55$	320.000			0.784523
$56$		0			0
$57$		0			0
$58$		0			0
$59$	137.000	−	237.291i	0.302303	−	0.523604i	−0.674354	−	0.738408i	$-0.735578\pi$
$59$	137.000	−	237.291i	0.302303	−	0.523604i	0.976657	+	0.214804i	$0.0689112\pi$
$60$		0			0
$61$	288.000	+	498.831i	0.604502	+	1.04703i	0.992130	+	0.125212i	$0.0399612\pi$
$61$	288.000	+	498.831i	0.604502	+	1.04703i	−0.387628	+	0.921816i	$0.626705\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	48.0000	+	83.1384i	0.0915949	+	0.158647i
$66$		0			0
$67$	238.000	−	412.228i	0.433975	−	0.751667i	−0.563236	−	0.826296i	$-0.690444\pi$
$67$	238.000	−	412.228i	0.433975	−	0.751667i	0.997211	+	0.0746290i	$0.0237773\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	448.000			0.748843			0.374421	−	0.927259i	$-0.377841\pi$
$71$	448.000			0.748843			0.374421	+	0.927259i	$0.377841\pi$
$72$		0			0
$73$	79.0000	−	136.832i	0.126661	−	0.219383i	−0.795720	−	0.605665i	$-0.792907\pi$
$73$	79.0000	−	136.832i	0.126661	−	0.219383i	0.922381	+	0.386281i	$0.126241\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	468.000	+	810.600i	0.666508	+	1.15443i	0.978874	+	0.204464i	$0.0655450\pi$
$79$	468.000	+	810.600i	0.666508	+	1.15443i	−0.312366	+	0.949962i	$0.601122\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−530.000			−0.700904			−0.350452	−	0.936581i	$-0.613972\pi$
$83$	−530.000			−0.700904			−0.350452	+	0.936581i	$0.613972\pi$
$84$		0			0
$85$	464.000			0.592093
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−195.000	−	337.750i	−0.232247	−	0.402263i	0.726222	−	0.687460i	$-0.241274\pi$
$89$	−195.000	−	337.750i	−0.232247	−	0.402263i	−0.958469	+	0.285197i	$0.907941\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−104.000	+	180.133i	−0.112318	+	0.194540i
$96$		0			0
$97$	214.000			0.224004			0.112002	−	0.993708i	$-0.464274\pi$
$97$	214.000			0.224004			0.112002	+	0.993708i	$0.464274\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	716.000	−	1240.15i	0.705393	−	1.22178i	−0.261157	−	0.965296i	$-0.584104\pi$
$101$	716.000	−	1240.15i	0.705393	−	1.22178i	0.966550	−	0.256480i	$-0.0825627\pi$
$102$		0			0
$103$	−382.000	−	661.643i	−0.365433	−	0.632948i	0.623413	−	0.781893i	$-0.285746\pi$
$103$	−382.000	−	661.643i	−0.365433	−	0.632948i	−0.988846	+	0.148945i	$0.952412\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	162.000	+	280.592i	0.146366	+	0.253513i	0.929882	−	0.367859i	$-0.119909\pi$
$107$	162.000	+	280.592i	0.146366	+	0.253513i	−0.783516	+	0.621372i	$0.786576\pi$
$108$		0			0
$109$	667.000	−	1155.28i	0.586119	−	1.01519i	−0.408615	−	0.912707i	$-0.633988\pi$
$109$	667.000	−	1155.28i	0.586119	−	1.01519i	0.994735	−	0.102482i	$-0.0326784\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1798.00			−1.49683			−0.748414	−	0.663232i	$-0.769184\pi$
$113$	−1798.00			−1.49683			−0.748414	+	0.663232i	$0.769184\pi$
$114$		0			0
$115$	−256.000	+	443.405i	−0.207584	+	0.359545i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−134.500	−	232.961i	−0.101052	−	0.175027i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1488.00			−1.06473
$126$		0			0
$127$	−384.000			−0.268303			−0.134152	−	0.990961i	$-0.542831\pi$
$127$	−384.000			−0.268303			−0.134152	+	0.990961i	$0.542831\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−907.000	−	1570.97i	−0.604923	−	1.04776i	−0.992064	−	0.125737i	$-0.959870\pi$
$131$	−907.000	−	1570.97i	−0.604923	−	1.04776i	0.387140	−	0.922021i	$-0.373463\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	833.000	−	1442.80i	0.519474	−	0.899756i	−0.480269	−	0.877121i	$-0.659461\pi$
$137$	833.000	−	1442.80i	0.519474	−	0.899756i	0.999744	−	0.0226350i	$-0.00720557\pi$
$138$		0			0
$139$	1126.00			0.687094			0.343547	−	0.939135i	$-0.388372\pi$
$139$	1126.00			0.687094			0.343547	+	0.939135i	$0.388372\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	240.000	−	415.692i	0.140348	−	0.243090i
$144$		0			0
$145$	−248.000	−	429.549i	−0.142036	−	0.246014i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1347.00	+	2333.07i	0.740608	+	1.28277i	0.952219	+	0.305416i	$0.0987956\pi$
$149$	1347.00	+	2333.07i	0.740608	+	1.28277i	−0.211611	+	0.977354i	$0.567871\pi$
$150$		0			0
$151$	1324.00	−	2293.24i	0.713547	−	1.23590i	−0.249970	−	0.968253i	$-0.580421\pi$
$151$	1324.00	−	2293.24i	0.713547	−	1.23590i	0.963517	−	0.267646i	$-0.0862458\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2016.00			1.04470
$156$		0			0
$157$	278.000	−	481.510i	0.141317	−	0.244769i	−0.786676	−	0.617367i	$-0.788200\pi$
$157$	278.000	−	481.510i	0.141317	−	0.244769i	0.927993	+	0.372598i	$0.121533\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	164.000	+	284.056i	0.0788066	+	0.136497i	0.902735	−	0.430196i	$-0.141556\pi$
$163$	164.000	+	284.056i	0.0788066	+	0.136497i	−0.823929	+	0.566693i	$0.808222\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4268.00			1.97765			0.988826	−	0.149077i	$-0.0476302\pi$
$167$	4268.00			1.97765			0.988826	+	0.149077i	$0.0476302\pi$
$168$		0			0
$169$	−2053.00			−0.934456
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1738.00	−	3010.30i	−0.763802	−	1.32294i	−0.940878	−	0.338746i	$-0.889997\pi$
$173$	−1738.00	−	3010.30i	−0.763802	−	1.32294i	0.177076	−	0.984197i	$-0.443336\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1134.00	−	1964.15i	0.473515	−	0.820152i	−0.526026	−	0.850469i	$-0.676318\pi$
$179$	1134.00	−	1964.15i	0.473515	−	0.820152i	0.999540	+	0.0303171i	$0.00965173\pi$
$180$		0			0
$181$	−276.000			−0.113342			−0.0566710	−	0.998393i	$-0.518049\pi$
$181$	−276.000			−0.113342			−0.0566710	+	0.998393i	$0.518049\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−104.000	+	180.133i	−0.0413310	+	0.0715874i
$186$		0			0
$187$	−1160.00	−	2009.18i	−0.453624	−	0.785699i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1500.00	−	2598.08i	−0.568252	−	0.984242i	−0.996739	−	0.0806937i	$-0.974286\pi$
$191$	−1500.00	−	2598.08i	−0.568252	−	0.984242i	0.428487	−	0.903548i	$-0.359047\pi$
$192$		0			0
$193$	−1639.00	+	2838.83i	−0.611284	+	1.05877i	0.379740	+	0.925093i	$0.376013\pi$
$193$	−1639.00	+	2838.83i	−0.611284	+	1.05877i	−0.991024	+	0.133682i	$0.957320\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2362.00			0.854241			0.427121	−	0.904195i	$-0.359528\pi$
$197$	2362.00			0.854241			0.427121	+	0.904195i	$0.359528\pi$
$198$		0			0
$199$	518.000	−	897.202i	0.184523	−	0.319603i	−0.758893	−	0.651216i	$-0.774259\pi$
$199$	518.000	−	897.202i	0.184523	−	0.319603i	0.943416	+	0.331613i	$0.107593\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	24.0000	+	41.5692i	0.00817674	+	0.0141625i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1040.00			0.344202
$210$		0			0
$211$	3524.00			1.14977			0.574887	−	0.818233i	$-0.305046\pi$
$211$	3524.00			1.14977			0.574887	+	0.818233i	$0.305046\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1664.00	−	2882.13i	−0.527832	−	0.914232i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	348.000	−	602.754i	0.105923	−	0.183464i
$222$		0			0
$223$	−1336.00			−0.401189			−0.200595	−	0.979674i	$-0.564287\pi$
$223$	−1336.00			−0.401189			−0.200595	+	0.979674i	$0.564287\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	645.000	−	1117.17i	0.188591	−	0.326649i	−0.756190	−	0.654352i	$-0.772941\pi$
$227$	645.000	−	1117.17i	0.188591	−	0.326649i	0.944781	+	0.327703i	$0.106275\pi$
$228$		0			0
$229$	−2762.00	−	4783.92i	−0.797022	−	1.38048i	−0.921547	−	0.388267i	$-0.873074\pi$
$229$	−2762.00	−	4783.92i	−0.797022	−	1.38048i	0.124525	−	0.992216i	$-0.460259\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3157.00	+	5468.08i	0.887648	+	1.53745i	0.842648	+	0.538464i	$0.180995\pi$
$233$	3157.00	+	5468.08i	0.887648	+	1.53745i	0.0449994	+	0.998987i	$0.485671\pi$
$234$		0			0
$235$	−1584.00	+	2743.57i	−0.439697	+	0.761577i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3960.00			1.07176			0.535881	−	0.844294i	$-0.319980\pi$
$239$	3960.00			1.07176			0.535881	+	0.844294i	$0.319980\pi$
$240$		0			0
$241$	3509.00	−	6077.77i	0.937903	−	1.62450i	0.168529	−	0.985697i	$-0.446098\pi$
$241$	3509.00	−	6077.77i	0.937903	−	1.62450i	0.769374	−	0.638798i	$-0.220568\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	156.000	+	270.200i	0.0401864	+	0.0696049i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2394.00			0.602024			0.301012	−	0.953620i	$-0.402676\pi$
$251$	2394.00			0.602024			0.301012	+	0.953620i	$0.402676\pi$
$252$		0			0
$253$	2560.00			0.636149
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1383.00	−	2395.43i	−0.335678	−	0.581411i	0.647937	−	0.761694i	$-0.275632\pi$
$257$	−1383.00	−	2395.43i	−0.335678	−	0.581411i	−0.983615	+	0.180283i	$0.942299\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3984.00	−	6900.49i	0.934084	−	1.61788i	0.157823	−	0.987467i	$-0.449552\pi$
$263$	3984.00	−	6900.49i	0.934084	−	1.61788i	0.776260	−	0.630413i	$-0.217114\pi$
$264$		0			0
$265$	3600.00			0.834514
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1450.00	+	2511.47i	−0.328654	+	0.569246i	−0.982245	−	0.187602i	$-0.939928\pi$
$269$	−1450.00	+	2511.47i	−0.328654	+	0.569246i	0.653591	+	0.756848i	$0.273262\pi$
$270$		0			0
$271$	−1320.00	−	2286.31i	−0.295883	−	0.512484i	0.679307	−	0.733854i	$-0.262281\pi$
$271$	−1320.00	−	2286.31i	−0.295883	−	0.512484i	−0.975190	+	0.221370i	$0.928947\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1220.00	+	2113.10i	0.267523	+	0.463363i
$276$		0			0
$277$	761.000	−	1318.09i	0.165069	−	0.285908i	−0.771611	−	0.636095i	$-0.780549\pi$
$277$	761.000	−	1318.09i	0.165069	−	0.285908i	0.936680	+	0.350187i	$0.113882\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4534.00			0.962547			0.481274	−	0.876570i	$-0.340174\pi$
$281$	4534.00			0.962547			0.481274	+	0.876570i	$0.340174\pi$
$282$		0			0
$283$	−2417.00	+	4186.37i	−0.507688	+	0.879342i	0.492272	+	0.870441i	$0.336166\pi$
$283$	−2417.00	+	4186.37i	−0.507688	+	0.879342i	−0.999960	+	0.00890034i	$0.997167\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	774.500	+	1341.47i	0.157643	+	0.273046i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4656.00			0.928350			0.464175	−	0.885744i	$-0.346351\pi$
$293$	4656.00			0.928350			0.464175	+	0.885744i	$0.346351\pi$
$294$		0			0
$295$	−2192.00			−0.432621
$296$		0			0
$297$		0			0
$298$		0			0
$299$	384.000	+	665.108i	0.0742719	+	0.128643i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2304.00	−	3990.65i	0.432546	−	0.749192i
$306$		0			0
$307$	−7238.00			−1.34558			−0.672792	−	0.739831i	$-0.734905\pi$
$307$	−7238.00			−1.34558			−0.672792	+	0.739831i	$0.734905\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	548.000	−	949.164i	0.0999171	−	0.173062i	−0.811733	−	0.584029i	$-0.801476\pi$
$311$	548.000	−	949.164i	0.0999171	−	0.173062i	0.911650	+	0.410967i	$0.134809\pi$
$312$		0			0
$313$	1909.00	+	3306.48i	0.344738	+	0.597104i	0.985306	−	0.170797i	$-0.0546344\pi$
$313$	1909.00	+	3306.48i	0.344738	+	0.597104i	−0.640568	+	0.767901i	$0.721301\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	999.000	+	1730.32i	0.177001	+	0.306575i	0.940852	−	0.338818i	$-0.110027\pi$
$317$	999.000	+	1730.32i	0.177001	+	0.306575i	−0.763851	+	0.645393i	$0.776694\pi$
$318$		0			0
$319$	−1240.00	+	2147.74i	−0.217638	+	0.376961i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1508.00			0.259775
$324$		0			0
$325$	−366.000	+	633.931i	−0.0624678	+	0.108197i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	3968.00	+	6872.78i	0.658915	+	1.14127i	0.980897	+	0.194529i	$0.0623178\pi$
$331$	3968.00	+	6872.78i	0.658915	+	1.14127i	−0.321981	+	0.946746i	$0.604349\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3808.00			−0.621055
$336$		0			0
$337$	2766.00			0.447103			0.223551	−	0.974692i	$-0.428235\pi$
$337$	2766.00			0.447103			0.223551	+	0.974692i	$0.428235\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5040.00	−	8729.54i	−0.800385	−	1.38631i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4176.00	−	7233.04i	0.646050	−	1.11899i	−0.338008	−	0.941143i	$-0.609753\pi$
$347$	4176.00	−	7233.04i	0.646050	−	1.11899i	0.984058	−	0.177848i	$-0.0569137\pi$
$348$		0			0
$349$	−5924.00			−0.908609			−0.454304	−	0.890847i	$-0.650112\pi$
$349$	−5924.00			−0.908609			−0.454304	+	0.890847i	$0.650112\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1113.00	−	1927.77i	0.167816	−	0.290666i	−0.769836	−	0.638242i	$-0.779662\pi$
$353$	1113.00	−	1927.77i	0.167816	−	0.290666i	0.937652	+	0.347576i	$0.112995\pi$
$354$		0			0
$355$	−1792.00	−	3103.84i	−0.267914	−	0.464041i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1940.00	+	3360.18i	0.285207	+	0.493993i	0.972659	−	0.232237i	$-0.0746043\pi$
$359$	1940.00	+	3360.18i	0.285207	+	0.493993i	−0.687452	+	0.726229i	$0.741271\pi$
$360$		0			0
$361$	3091.50	−	5354.64i	0.450722	−	0.780673i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1264.00			−0.181262
$366$		0			0
$367$	1292.00	−	2237.81i	0.183765	−	0.318291i	−0.759394	−	0.650630i	$-0.774505\pi$
$367$	1292.00	−	2237.81i	0.183765	−	0.318291i	0.943160	+	0.332340i	$0.107838\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−5267.00	−	9122.71i	−0.731139	−	1.26637i	−0.956397	−	0.292071i	$-0.905656\pi$
$373$	−5267.00	−	9122.71i	−0.731139	−	1.26637i	0.225257	−	0.974299i	$-0.427678\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−744.000			−0.101639
$378$		0			0
$379$	−4472.00			−0.606098			−0.303049	−	0.952975i	$-0.598005\pi$
$379$	−4472.00			−0.606098			−0.303049	+	0.952975i	$0.598005\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1234.00	+	2137.35i	0.164633	+	0.285153i	0.936525	−	0.350601i	$-0.114023\pi$
$383$	1234.00	+	2137.35i	0.164633	+	0.285153i	−0.771892	+	0.635754i	$0.780689\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−523.000	+	905.863i	−0.0681675	+	0.118070i	−0.898095	−	0.439802i	$-0.855049\pi$
$389$	−523.000	+	905.863i	−0.0681675	+	0.118070i	0.829927	+	0.557872i	$0.188382\pi$
$390$		0			0
$391$	3712.00			0.480112
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3744.00	−	6484.80i	0.476914	−	0.826040i
$396$		0			0
$397$	−1062.00	−	1839.44i	−0.134258	−	0.232541i	0.791056	−	0.611744i	$-0.209532\pi$
$397$	−1062.00	−	1839.44i	−0.134258	−	0.232541i	−0.925314	+	0.379203i	$0.876198\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5799.00	+	10044.2i	0.722165	+	1.25083i	0.960130	+	0.279553i	$0.0901863\pi$
$401$	5799.00	+	10044.2i	0.722165	+	1.25083i	−0.237965	+	0.971274i	$0.576480\pi$
$402$		0			0
$403$	1512.00	−	2618.86i	0.186894	−	0.323709i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1040.00			0.126661
$408$		0			0
$409$	1385.00	−	2398.89i	0.167442	−	0.290018i	−0.770078	−	0.637950i	$-0.779783\pi$
$409$	1385.00	−	2398.89i	0.167442	−	0.290018i	0.937520	+	0.347932i	$0.113116\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	2120.00	+	3671.95i	0.250763	+	0.434335i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−9438.00			−1.10042			−0.550211	−	0.835026i	$-0.685453\pi$
$419$	−9438.00			−1.10042			−0.550211	+	0.835026i	$0.685453\pi$
$420$		0			0
$421$	5550.00			0.642495			0.321248	−	0.946995i	$-0.395898\pi$
$421$	5550.00			0.642495			0.321248	+	0.946995i	$0.395898\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1769.00	+	3064.00i	0.201904	+	0.349708i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1500.00	+	2598.08i	−0.167639	+	0.290359i	−0.937589	−	0.347744i	$-0.886948\pi$
$431$	−1500.00	+	2598.08i	−0.167639	+	0.290359i	0.769950	+	0.638104i	$0.220281\pi$
$432$		0			0
$433$	12926.0			1.43460			0.717302	−	0.696762i	$-0.245377\pi$
$433$	12926.0			1.43460			0.717302	+	0.696762i	$0.245377\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−832.000	+	1441.07i	−0.0910754	+	0.157747i
$438$		0			0
$439$	204.000	+	353.338i	0.0221786	+	0.0384144i	0.876902	−	0.480670i	$-0.159606\pi$
$439$	204.000	+	353.338i	0.0221786	+	0.0384144i	−0.854723	+	0.519084i	$0.826273\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7226.00	−	12515.8i	−0.774983	−	1.34231i	−0.934804	−	0.355164i	$-0.884425\pi$
$443$	−7226.00	−	12515.8i	−0.774983	−	1.34231i	0.159821	−	0.987146i	$-0.448908\pi$
$444$		0			0
$445$	−1560.00	+	2702.00i	−0.166182	+	0.287836i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−10258.0			−1.07818			−0.539092	−	0.842247i	$-0.681233\pi$
$449$	−10258.0			−1.07818			−0.539092	+	0.842247i	$0.681233\pi$
$450$		0			0
$451$	120.000	−	207.846i	0.0125290	−	0.0217009i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	2749.00	+	4761.41i	0.281385	+	0.487373i	0.971726	−	0.236111i	$-0.0758730\pi$
$457$	2749.00	+	4761.41i	0.281385	+	0.487373i	−0.690341	+	0.723484i	$0.742540\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16316.0			1.64840			0.824199	−	0.566300i	$-0.191625\pi$
$461$	16316.0			1.64840			0.824199	+	0.566300i	$0.191625\pi$
$462$		0			0
$463$	8944.00			0.897760			0.448880	−	0.893592i	$-0.351823\pi$
$463$	8944.00			0.897760			0.448880	+	0.893592i	$0.351823\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4711.00	−	8159.69i	−0.466807	−	0.808534i	0.532474	−	0.846447i	$-0.321263\pi$
$467$	−4711.00	−	8159.69i	−0.466807	−	0.808534i	−0.999281	+	0.0379124i	$0.987929\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−8320.00	+	14410.7i	−0.808782	+	1.40085i
$474$		0			0
$475$	−1586.00			−0.153201
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6910.00	−	11968.5i	0.659136	−	1.14166i	−0.321704	−	0.946840i	$-0.604256\pi$
$479$	6910.00	−	11968.5i	0.659136	−	1.14166i	0.980840	−	0.194816i	$-0.0624110\pi$
$480$		0			0
$481$	156.000	+	270.200i	0.0147879	+	0.0256134i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−856.000	−	1482.64i	−0.0801422	−	0.138810i
$486$		0			0
$487$	6632.00	−	11487.0i	0.617094	−	1.06884i	−0.372920	−	0.927864i	$-0.621643\pi$
$487$	6632.00	−	11487.0i	0.617094	−	1.06884i	0.990013	−	0.140974i	$-0.0450234\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5940.00			0.545964			0.272982	−	0.962019i	$-0.411990\pi$
$491$	5940.00			0.545964			0.272982	+	0.962019i	$0.411990\pi$
$492$		0			0
$493$	−1798.00	+	3114.23i	−0.164255	+	0.284498i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	4126.00	+	7146.44i	0.370151	+	0.641120i	0.989588	−	0.143926i	$-0.0459728\pi$
$499$	4126.00	+	7146.44i	0.370151	+	0.641120i	−0.619438	+	0.785046i	$0.712639\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4704.00			−0.416980			−0.208490	−	0.978024i	$-0.566855\pi$
$503$	−4704.00			−0.416980			−0.208490	+	0.978024i	$0.566855\pi$
$504$		0			0
$505$	−11456.0			−1.00948
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−5394.00	−	9342.68i	−0.469715	−	0.813570i	0.529686	−	0.848194i	$-0.322310\pi$
$509$	−5394.00	−	9342.68i	−0.469715	−	0.813570i	−0.999400	+	0.0346241i	$0.988977\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3056.00	+	5293.15i	−0.261482	+	0.452901i
$516$		0			0
$517$	15840.0			1.34747
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−7293.00	+	12631.8i	−0.613267	+	1.06221i	0.377419	+	0.926043i	$0.376812\pi$
$521$	−7293.00	+	12631.8i	−0.613267	+	1.06221i	−0.990686	+	0.136167i	$0.956522\pi$
$522$		0			0
$523$	−13.0000	−	22.5167i	−0.00108690	−	0.00188257i	0.865481	−	0.500941i	$-0.167013\pi$
$523$	−13.0000	−	22.5167i	−0.00108690	−	0.00188257i	−0.866568	+	0.499058i	$0.833679\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7308.00	−	12657.8i	−0.604064	−	1.04627i
$528$		0			0
$529$	4035.50	−	6989.69i	0.331676	−	0.574479i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	72.0000			0.00585116
$534$		0			0
$535$	1296.00	−	2244.74i	0.104731	−	0.181399i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−5607.00	−	9711.61i	−0.445589	−	0.771783i	0.552504	−	0.833510i	$-0.313672\pi$
$541$	−5607.00	−	9711.61i	−0.445589	−	0.771783i	−0.998093	+	0.0617272i	$0.980339\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−10672.0			−0.838786
$546$		0			0
$547$	−5424.00			−0.423973			−0.211987	−	0.977273i	$-0.567993\pi$
$547$	−5424.00			−0.423973			−0.211987	+	0.977273i	$0.567993\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−806.000	−	1396.03i	−0.0623172	−	0.107936i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8809.00	+	15257.6i	−0.670106	+	1.16066i	0.307767	+	0.951462i	$0.400418\pi$
$557$	−8809.00	+	15257.6i	−0.670106	+	1.16066i	−0.977874	+	0.209197i	$0.932915\pi$
$558$		0			0
$559$	−4992.00			−0.377709
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1781.00	+	3084.78i	−0.133322	+	0.230920i	−0.924955	−	0.380076i	$-0.875898\pi$
$563$	−1781.00	+	3084.78i	−0.133322	+	0.230920i	0.791633	+	0.610997i	$0.209231\pi$
$564$		0			0
$565$	7192.00	+	12456.9i	0.535522	+	0.927551i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1419.00	+	2457.78i	0.104548	+	0.181082i	0.913553	−	0.406719i	$-0.133327\pi$
$569$	1419.00	+	2457.78i	0.104548	+	0.181082i	−0.809006	+	0.587801i	$0.799994\pi$
$570$		0			0
$571$	180.000	−	311.769i	0.0131922	−	0.0228496i	−0.859354	−	0.511381i	$-0.829134\pi$
$571$	180.000	−	311.769i	0.0131922	−	0.0228496i	0.872546	+	0.488532i	$0.162467\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3904.00			−0.283144
$576$		0			0
$577$	−11009.0	+	19068.1i	−0.794299	+	1.37577i	0.128984	+	0.991647i	$0.458828\pi$
$577$	−11009.0	+	19068.1i	−0.794299	+	1.37577i	−0.923283	+	0.384120i	$0.874505\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−9000.00	−	15588.5i	−0.639351	−	1.10739i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1454.00			−0.102237			−0.0511184	−	0.998693i	$-0.516279\pi$
$587$	−1454.00			−0.102237			−0.0511184	+	0.998693i	$0.516279\pi$
$588$		0			0
$589$	6552.00			0.458354
$590$		0			0
$591$		0			0
$592$		0			0
$593$	6909.00	+	11966.7i	0.478446	+	0.828693i	0.999695	−	0.0247118i	$-0.00786682\pi$
$593$	6909.00	+	11966.7i	0.478446	+	0.828693i	−0.521248	+	0.853405i	$0.674533\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	3348.00	−	5798.91i	0.228373	−	0.395554i	−0.728953	−	0.684564i	$-0.759993\pi$
$599$	3348.00	−	5798.91i	0.228373	−	0.395554i	0.957326	+	0.289010i	$0.0933260\pi$
$600$		0			0
$601$	10010.0			0.679395			0.339698	−	0.940535i	$-0.389675\pi$
$601$	10010.0			0.679395			0.339698	+	0.940535i	$0.389675\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1076.00	+	1863.69i	−0.0723068	+	0.125239i
$606$		0			0
$607$	−1440.00	−	2494.15i	−0.0962896	−	0.166779i	0.813856	−	0.581066i	$-0.197364\pi$
$607$	−1440.00	−	2494.15i	−0.0962896	−	0.166779i	−0.910146	+	0.414287i	$0.864031\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2376.00	+	4115.35i	0.157320	+	0.272487i
$612$		0			0
$613$	−3261.00	+	5648.22i	−0.214862	+	0.372152i	−0.953230	−	0.302246i	$-0.902264\pi$
$613$	−3261.00	+	5648.22i	−0.214862	+	0.372152i	0.738368	+	0.674398i	$0.235597\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6614.00			−0.431555			−0.215778	−	0.976443i	$-0.569229\pi$
$617$	−6614.00			−0.431555			−0.215778	+	0.976443i	$0.569229\pi$
$618$		0			0
$619$	−2633.00	+	4560.49i	−0.170968	+	0.296125i	−0.938759	−	0.344576i	$-0.888023\pi$
$619$	−2633.00	+	4560.49i	−0.170968	+	0.296125i	0.767791	+	0.640701i	$0.221356\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	2139.50	+	3705.72i	0.136928	+	0.237166i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1508.00			0.0955928
$630$		0			0
$631$	3344.00			0.210971			0.105485	−	0.994421i	$-0.466360\pi$
$631$	3344.00			0.210971			0.105485	+	0.994421i	$0.466360\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1536.00	+	2660.43i	0.0959910	+	0.166261i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−2441.00	+	4227.94i	−0.150411	+	0.260520i	−0.931379	−	0.364052i	$-0.881393\pi$
$641$	−2441.00	+	4227.94i	−0.150411	+	0.260520i	0.780967	+	0.624572i	$0.214726\pi$
$642$		0			0
$643$	−15898.0			−0.975048			−0.487524	−	0.873110i	$-0.662100\pi$
$643$	−15898.0			−0.975048			−0.487524	+	0.873110i	$0.662100\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3066.00	−	5310.47i	0.186301	−	0.322683i	−0.757713	−	0.652588i	$-0.773683\pi$
$647$	3066.00	−	5310.47i	0.186301	−	0.322683i	0.944014	+	0.329905i	$0.107017\pi$
$648$		0			0
$649$	5480.00	+	9491.64i	0.331447	+	0.574082i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−12099.0	−	20956.1i	−0.725070	−	1.25586i	−0.958945	−	0.283591i	$-0.908474\pi$
$653$	−12099.0	−	20956.1i	−0.725070	−	1.25586i	0.233876	−	0.972266i	$-0.424859\pi$
$654$		0			0
$655$	−7256.00	+	12567.8i	−0.432848	+	0.749715i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−17456.0			−1.03185			−0.515925	−	0.856634i	$-0.672552\pi$
$659$	−17456.0			−1.03185			−0.515925	+	0.856634i	$0.672552\pi$
$660$		0			0
$661$	328.000	−	568.113i	0.0193006	−	0.0334297i	−0.856214	−	0.516622i	$-0.827189\pi$
$661$	328.000	−	568.113i	0.0193006	−	0.0334297i	0.875514	+	0.483192i	$0.160523\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1984.00	−	3436.39i	−0.115174	−	0.199487i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−23040.0			−1.32556
$672$		0			0
$673$	−18214.0			−1.04324			−0.521618	−	0.853179i	$-0.674671\pi$
$673$	−18214.0			−1.04324			−0.521618	+	0.853179i	$0.674671\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15126.0	+	26199.0i	0.858699	+	1.48731i	0.873170	+	0.487415i	$0.162060\pi$
$677$	15126.0	+	26199.0i	0.858699	+	1.48731i	−0.0144709	+	0.999895i	$0.504606\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−5418.00	+	9384.25i	−0.303534	+	0.525737i	−0.976934	−	0.213542i	$-0.931500\pi$
$683$	−5418.00	+	9384.25i	−0.303534	+	0.525737i	0.673400	+	0.739279i	$0.264833\pi$
$684$		0			0
$685$	−13328.0			−0.743411
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2700.00	−	4676.54i	0.149291	−	0.258580i
$690$		0			0
$691$	−4789.00	−	8294.79i	−0.263650	−	0.456655i	0.703559	−	0.710637i	$-0.251593\pi$
$691$	−4789.00	−	8294.79i	−0.263650	−	0.456655i	−0.967209	+	0.253982i	$0.918260\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4504.00	−	7801.16i	−0.245822	−	0.425777i
$696$		0			0
$697$	174.000	−	301.377i	0.00945584	−	0.0163780i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−12442.0			−0.670368			−0.335184	−	0.942153i	$-0.608798\pi$
$701$	−12442.0			−0.670368			−0.335184	+	0.942153i	$0.608798\pi$
$702$		0			0
$703$	−338.000	+	585.433i	−0.0181336	+	0.0314083i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	12587.0	+	21801.3i	0.666734	+	1.15482i	0.978812	+	0.204761i	$0.0656418\pi$
$709$	12587.0	+	21801.3i	0.666734	+	1.15482i	−0.312078	+	0.950057i	$0.601025\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	16128.0			0.847123
$714$		0			0
$715$	−3840.00			−0.200850
$716$		0			0
$717$		0			0
$718$		0			0
$719$	17094.0	+	29607.7i	0.886646	+	1.53572i	0.843815	+	0.536634i	$0.180304\pi$
$719$	17094.0	+	29607.7i	0.886646	+	1.53572i	0.0428311	+	0.999082i	$0.486362\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1891.00	−	3275.31i	0.0968689	−	0.167782i
$726$		0			0
$727$	−5204.00			−0.265482			−0.132741	−	0.991151i	$-0.542378\pi$
$727$	−5204.00			−0.265482			−0.132741	+	0.991151i	$0.542378\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12064.0	+	20895.5i	−0.610401	+	1.05725i
$732$		0			0
$733$	16440.0	+	28474.9i	0.828411	+	1.43485i	0.899284	+	0.437365i	$0.144088\pi$
$733$	16440.0	+	28474.9i	0.828411	+	1.43485i	−0.0708733	+	0.997485i	$0.522579\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	9520.00	+	16489.1i	0.475812	+	0.824131i
$738$		0			0
$739$	1956.00	−	3387.89i	0.0973648	−	0.168641i	−0.813228	−	0.581945i	$-0.802292\pi$
$739$	1956.00	−	3387.89i	0.0973648	−	0.168641i	0.910593	+	0.413304i	$0.135625\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	16008.0			0.790413			0.395206	−	0.918592i	$-0.370673\pi$
$743$	16008.0			0.790413			0.395206	+	0.918592i	$0.370673\pi$
$744$		0			0
$745$	10776.0	−	18664.6i	0.529936	−	0.917876i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−4980.00	−	8625.61i	−0.241974	−	0.419112i	0.719302	−	0.694697i	$-0.244462\pi$
$751$	−4980.00	−	8625.61i	−0.241974	−	0.419112i	−0.961277	+	0.275585i	$0.911128\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−21184.0			−1.02115
$756$		0			0
$757$	12378.0			0.594301			0.297151	−	0.954831i	$-0.403964\pi$
$757$	12378.0			0.594301			0.297151	+	0.954831i	$0.403964\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	17335.0	+	30025.1i	0.825747	+	1.43024i	0.901347	+	0.433097i	$0.142579\pi$
$761$	17335.0	+	30025.1i	0.825747	+	1.43024i	−0.0756005	+	0.997138i	$0.524087\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1644.00	+	2847.49i	−0.0773943	+	0.134051i
$768$		0			0
$769$	−10898.0			−0.511043			−0.255521	−	0.966803i	$-0.582247\pi$
$769$	−10898.0			−0.511043			−0.255521	+	0.966803i	$0.582247\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12904.0	+	22350.4i	−0.600420	+	1.03996i	0.392337	+	0.919821i	$0.371667\pi$
$773$	−12904.0	+	22350.4i	−0.600420	+	1.03996i	−0.992757	+	0.120136i	$0.961667\pi$
$774$		0			0
$775$	7686.00	+	13312.5i	0.356244	+	0.617033i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	78.0000	+	135.100i	0.00358747	+	0.00621368i
$780$		0			0
$781$	−8960.00	+	15519.2i	−0.410517	+	0.711037i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4448.00			−0.202237
$786$		0			0
$787$	10527.0	−	18233.3i	0.476807	−	0.825854i	−0.522840	−	0.852431i	$-0.675127\pi$
$787$	10527.0	−	18233.3i	0.476807	−	0.825854i	0.999647	+	0.0265772i	$0.00846077\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−3456.00	−	5985.97i	−0.154762	−	0.268055i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	24276.0			1.07892			0.539461	−	0.842011i	$-0.318628\pi$
$797$	24276.0			1.07892			0.539461	+	0.842011i	$0.318628\pi$
$798$		0			0
$799$	22968.0			1.01696
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3160.00	+	5473.28i	0.138872	+	0.240533i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	10763.0	−	18642.1i	0.467747	−	0.810161i	−0.531574	−	0.847012i	$-0.678399\pi$
$809$	10763.0	−	18642.1i	0.467747	−	0.810161i	0.999321	+	0.0368510i	$0.0117327\pi$
$810$		0			0
$811$	−12806.0			−0.554475			−0.277238	−	0.960801i	$-0.589419\pi$
$811$	−12806.0			−0.554475			−0.277238	+	0.960801i	$0.589419\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1312.00	−	2272.45i	0.0563894	−	0.0976693i
$816$		0			0
$817$	−5408.00	−	9366.93i	−0.231581	−	0.401111i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6607.00	+	11443.7i	0.280860	+	0.486463i	0.971597	−	0.236643i	$-0.0760471\pi$
$821$	6607.00	+	11443.7i	0.280860	+	0.486463i	−0.690737	+	0.723106i	$0.742714\pi$
$822$		0			0
$823$	−16124.0	+	27927.6i	−0.682925	+	1.18286i	0.291159	+	0.956675i	$0.405959\pi$
$823$	−16124.0	+	27927.6i	−0.682925	+	1.18286i	−0.974084	+	0.226186i	$0.927374\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	14316.0			0.601954			0.300977	−	0.953631i	$-0.402687\pi$
$827$	14316.0			0.601954			0.300977	+	0.953631i	$0.402687\pi$
$828$		0			0
$829$	−12584.0	+	21796.1i	−0.527214	+	0.913161i	0.472283	+	0.881447i	$0.343430\pi$
$829$	−12584.0	+	21796.1i	−0.527214	+	0.913161i	−0.999497	+	0.0317144i	$0.989903\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−17072.0	−	29569.6i	−0.707546	−	1.22551i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−9356.00			−0.384988			−0.192494	−	0.981298i	$-0.561658\pi$
$839$	−9356.00			−0.384988			−0.192494	+	0.981298i	$0.561658\pi$
$840$		0			0
$841$	−20545.0			−0.842388
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8212.00	+	14223.6i	0.334321	+	0.579061i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−832.000	+	1441.07i	−0.0335142	+	0.0580483i
$852$		0			0
$853$	2372.00			0.0952119			0.0476059	−	0.998866i	$-0.484841\pi$
$853$	2372.00			0.0952119			0.0476059	+	0.998866i	$0.484841\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	5847.00	−	10127.3i	0.233057	−	0.403666i	−0.725649	−	0.688065i	$-0.758461\pi$
$857$	5847.00	−	10127.3i	0.233057	−	0.403666i	0.958706	+	0.284398i	$0.0917938\pi$
$858$		0			0
$859$	10253.0	+	17758.7i	0.407250	+	0.705378i	0.994580	−	0.103969i	$-0.0331544\pi$
$859$	10253.0	+	17758.7i	0.407250	+	0.705378i	−0.587330	+	0.809347i	$0.699821\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−14068.0	−	24366.5i	−0.554902	−	0.961118i	−0.997911	−	0.0646012i	$-0.979422\pi$
$863$	−14068.0	−	24366.5i	−0.554902	−	0.961118i	0.443009	−	0.896517i	$-0.353911\pi$
$864$		0			0
$865$	−13904.0	+	24082.4i	−0.546532	+	0.946621i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−37440.0			−1.46152
$870$		0			0
$871$	−2856.00	+	4946.74i	−0.111104	+	0.192438i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	18535.0	+	32103.6i	0.713663	+	1.23610i	0.963473	+	0.267806i	$0.0862985\pi$
$877$	18535.0	+	32103.6i	0.713663	+	1.23610i	−0.249810	+	0.968295i	$0.580368\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	6198.00			0.237021			0.118511	−	0.992953i	$-0.462188\pi$
$881$	6198.00			0.237021			0.118511	+	0.992953i	$0.462188\pi$
$882$		0			0
$883$	−31876.0			−1.21485			−0.607425	−	0.794377i	$-0.707798\pi$
$883$	−31876.0			−1.21485			−0.607425	+	0.794377i	$0.707798\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−66.0000	−	114.315i	−0.00249838	−	0.00432732i	0.864774	−	0.502162i	$-0.167462\pi$
$887$	−66.0000	−	114.315i	−0.00249838	−	0.00432732i	−0.867272	+	0.497835i	$0.834129\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−5148.00	+	8916.60i	−0.192913	+	0.334135i
$894$		0			0
$895$	−18144.0			−0.677639
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−7812.00	+	13530.8i	−0.289816	+	0.501976i
$900$		0			0
$901$	−13050.0	−	22603.3i	−0.482529	−	0.835765i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1104.00	+	1912.18i	0.0405505	+	0.0702355i
$906$		0			0
$907$	−19122.0	+	33120.3i	−0.700039	+	1.21250i	0.268413	+	0.963304i	$0.413501\pi$
$907$	−19122.0	+	33120.3i	−0.700039	+	1.21250i	−0.968452	+	0.249200i	$0.919832\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	7008.00			0.254869			0.127434	−	0.991847i	$-0.459326\pi$
$911$	7008.00			0.254869			0.127434	+	0.991847i	$0.459326\pi$
$912$		0			0
$913$	10600.0	−	18359.7i	0.384237	−	0.665519i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−18332.0	−	31752.0i	−0.658016	−	1.13972i	−0.981128	−	0.193357i	$-0.938062\pi$
$919$	−18332.0	−	31752.0i	−0.658016	−	1.13972i	0.323112	−	0.946361i	$-0.395271\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5376.00			−0.191715
$924$		0			0
$925$	−1586.00			−0.0563755
$926$		0			0
$927$		0			0
$928$		0			0
$929$	22755.0	+	39412.8i	0.803625	+	1.39192i	0.917216	+	0.398391i	$0.130431\pi$
$929$	22755.0	+	39412.8i	0.803625	+	1.39192i	−0.113591	+	0.993528i	$0.536235\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−9280.00	+	16073.4i	−0.324587	+	0.562200i
$936$		0			0
$937$	−3838.00			−0.133812			−0.0669061	−	0.997759i	$-0.521313\pi$
$937$	−3838.00			−0.133812			−0.0669061	+	0.997759i	$0.521313\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8416.00	+	14576.9i	−0.291556	+	0.504989i	−0.974178	−	0.225782i	$-0.927506\pi$
$941$	−8416.00	+	14576.9i	−0.291556	+	0.504989i	0.682622	+	0.730771i	$0.260839\pi$
$942$		0			0
$943$	192.000	+	332.554i	0.00663031	+	0.0114840i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−20464.0	−	35444.7i	−0.702208	−	1.21626i	−0.967690	−	0.252144i	$-0.918864\pi$
$947$	−20464.0	−	35444.7i	−0.702208	−	1.21626i	0.265482	−	0.964116i	$-0.414469\pi$
$948$		0			0
$949$	−948.000	+	1641.98i	−0.0324272	+	0.0561655i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	24070.0			0.818157			0.409079	−	0.912499i	$-0.365850\pi$
$953$	24070.0			0.818157			0.409079	+	0.912499i	$0.365850\pi$
$954$		0			0
$955$	−12000.0	+	20784.6i	−0.406608	+	0.704266i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−16856.5	−	29196.3i	−0.565825	−	0.980038i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	26224.0			0.874798
$966$		0			0
$967$	−17152.0			−0.570394			−0.285197	−	0.958469i	$-0.592059\pi$
$967$	−17152.0			−0.570394			−0.285197	+	0.958469i	$0.592059\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−16455.0	−	28500.9i	−0.543837	−	0.941954i	−0.998679	−	0.0513817i	$-0.983638\pi$
$971$	−16455.0	−	28500.9i	−0.543837	−	0.941954i	0.454842	−	0.890572i	$-0.349696\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3411.00	+	5908.03i	−0.111697	+	0.193464i	−0.916454	−	0.400139i	$-0.868962\pi$
$977$	−3411.00	+	5908.03i	−0.111697	+	0.193464i	0.804758	+	0.593603i	$0.202295\pi$
$978$		0			0
$979$	15600.0			0.509273
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−24210.0	+	41933.0i	−0.785533	+	1.36058i	0.143147	+	0.989701i	$0.454278\pi$
$983$	−24210.0	+	41933.0i	−0.785533	+	1.36058i	−0.928680	+	0.370882i	$0.879055\pi$
$984$		0			0
$985$	−9448.00	−	16364.4i	−0.305623	−	0.529354i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−13312.0	−	23057.1i	−0.428005	−	0.741326i
$990$		0			0
$991$	24608.0	−	42622.3i	0.788798	−	1.36624i	−0.137906	−	0.990445i	$-0.544037\pi$
$991$	24608.0	−	42622.3i	0.788798	−	1.36624i	0.926704	−	0.375793i	$-0.122630\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−8288.00			−0.264068
$996$		0			0
$997$	−17632.0	+	30539.5i	−0.560091	+	0.970107i	0.437397	+	0.899269i	$0.355901\pi$
$997$	−17632.0	+	30539.5i	−0.560091	+	0.970107i	−0.997488	+	0.0708379i	$0.977433\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.k.d.361.1		2
3.2	odd	2		196.4.e.f.165.1		2
7.2	even	3	inner	1764.4.k.d.1549.1		2
7.3	odd	6		1764.4.a.c.1.1		1
7.4	even	3		252.4.a.d.1.1		1
7.5	odd	6		1764.4.k.m.1549.1		2
7.6	odd	2		1764.4.k.m.361.1		2
21.2	odd	6		196.4.e.f.177.1		2
21.5	even	6		196.4.e.a.177.1		2
21.11	odd	6		28.4.a.a.1.1	✓	1
21.17	even	6		196.4.a.d.1.1		1
21.20	even	2		196.4.e.a.165.1		2
28.11	odd	6		1008.4.a.o.1.1		1
84.11	even	6		112.4.a.g.1.1		1
84.59	odd	6		784.4.a.a.1.1		1
105.32	even	12		700.4.e.a.449.2		2
105.53	even	12		700.4.e.a.449.1		2
105.74	odd	6		700.4.a.n.1.1		1
168.11	even	6		448.4.a.a.1.1		1
168.53	odd	6		448.4.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.4.a.a.1.1	✓	1	21.11	odd	6
112.4.a.g.1.1		1	84.11	even	6
196.4.a.d.1.1		1	21.17	even	6
196.4.e.a.165.1		2	21.20	even	2
196.4.e.a.177.1		2	21.5	even	6
196.4.e.f.165.1		2	3.2	odd	2
196.4.e.f.177.1		2	21.2	odd	6
252.4.a.d.1.1		1	7.4	even	3
448.4.a.a.1.1		1	168.11	even	6
448.4.a.p.1.1		1	168.53	odd	6
700.4.a.n.1.1		1	105.74	odd	6
700.4.e.a.449.1		2	105.53	even	12
700.4.e.a.449.2		2	105.32	even	12
784.4.a.a.1.1		1	84.59	odd	6
1008.4.a.o.1.1		1	28.11	odd	6
1764.4.a.c.1.1		1	7.3	odd	6
1764.4.k.d.361.1		2	1.1	even	1	trivial
1764.4.k.d.1549.1		2	7.2	even	3	inner
1764.4.k.m.361.1		2	7.6	odd	2
1764.4.k.m.1549.1		2	7.5	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.00000 - 6.92820i) q^{5} +O(q^{10})\)	\(q+(-4.00000 - 6.92820i) q^{5} +(-20.0000 + 34.6410i) q^{11} -12.0000 q^{13} +(-29.0000 + 50.2295i) q^{17} +(-13.0000 - 22.5167i) q^{19} +(-32.0000 - 55.4256i) q^{23} +(30.5000 - 52.8275i) q^{25} +62.0000 q^{29} +(-126.000 + 218.238i) q^{31} +(-13.0000 - 22.5167i) q^{37} -6.00000 q^{41} +416.000 q^{43} +(-198.000 - 342.946i) q^{47} +(-225.000 + 389.711i) q^{53} +320.000 q^{55} +(137.000 - 237.291i) q^{59} +(288.000 + 498.831i) q^{61} +(48.0000 + 83.1384i) q^{65} +(238.000 - 412.228i) q^{67} +448.000 q^{71} +(79.0000 - 136.832i) q^{73} +(468.000 + 810.600i) q^{79} -530.000 q^{83} +464.000 q^{85} +(-195.000 - 337.750i) q^{89} +(-104.000 + 180.133i) q^{95} +214.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{5}+O(q^{10}) \) 2 * q - 8 * q^5	\( 2 q - 8 q^{5} - 40 q^{11} - 24 q^{13} - 58 q^{17} - 26 q^{19} - 64 q^{23} + 61 q^{25} + 124 q^{29} - 252 q^{31} - 26 q^{37} - 12 q^{41} + 832 q^{43} - 396 q^{47} - 450 q^{53} + 640 q^{55} + 274 q^{59} + 576 q^{61} + 96 q^{65} + 476 q^{67} + 896 q^{71} + 158 q^{73} + 936 q^{79} - 1060 q^{83} + 928 q^{85} - 390 q^{89} - 208 q^{95} + 428 q^{97}+O(q^{100}) \) 2 * q - 8 * q^5 - 40 * q^11 - 24 * q^13 - 58 * q^17 - 26 * q^19 - 64 * q^23 + 61 * q^25 + 124 * q^29 - 252 * q^31 - 26 * q^37 - 12 * q^41 + 832 * q^43 - 396 * q^47 - 450 * q^53 + 640 * q^55 + 274 * q^59 + 576 * q^61 + 96 * q^65 + 476 * q^67 + 896 * q^71 + 158 * q^73 + 936 * q^79 - 1060 * q^83 + 928 * q^85 - 390 * q^89 - 208 * q^95 + 428 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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