LMFDB - Embedded newform 600.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,4,Mod(1,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	600.a (trivial)

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:	yes
Analytic conductor:	$35.4011460034$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} +4.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} +4.00000 q^{7} +9.00000 q^{9} -28.0000 q^{11} -16.0000 q^{13} +108.000 q^{17} +32.0000 q^{19} -12.0000 q^{21} -28.0000 q^{23} -27.0000 q^{27} -238.000 q^{29} -180.000 q^{31} +84.0000 q^{33} -40.0000 q^{37} +48.0000 q^{39} +422.000 q^{41} +276.000 q^{43} +60.0000 q^{47} -327.000 q^{49} -324.000 q^{51} +220.000 q^{53} -96.0000 q^{57} -804.000 q^{59} -358.000 q^{61} +36.0000 q^{63} -884.000 q^{67} +84.0000 q^{69} -64.0000 q^{71} -152.000 q^{73} -112.000 q^{77} -932.000 q^{79} +81.0000 q^{81} -1292.00 q^{83} +714.000 q^{87} -1146.00 q^{89} -64.0000 q^{91} +540.000 q^{93} +824.000 q^{97} -252.000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.00000	0.215980	0.107990	−	0.994152i	$-0.465559\pi$
$7$	4.00000	0.215980	0.107990	+	0.994152i	$0.465559\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−28.0000	−0.767483	−0.383742	−	0.923440i	$-0.625365\pi$
$11$	−28.0000	−0.767483	−0.383742	+	0.923440i	$0.625365\pi$
$12$	0	0
$13$	−16.0000	−0.341354	−0.170677	−	0.985327i	$-0.554595\pi$
$13$	−16.0000	−0.341354	−0.170677	+	0.985327i	$0.554595\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	108.000	1.54081	0.770407	−	0.637552i	$-0.220053\pi$
$17$	108.000	1.54081	0.770407	+	0.637552i	$0.220053\pi$
$18$	0	0
$19$	32.0000	0.386384	0.193192	−	0.981161i	$-0.438116\pi$
$19$	32.0000	0.386384	0.193192	+	0.981161i	$0.438116\pi$
$20$	0	0
$21$	−12.0000	−0.124696
$22$	0	0
$23$	−28.0000	−0.253844	−0.126922	−	0.991913i	$-0.540510\pi$
$23$	−28.0000	−0.253844	−0.126922	+	0.991913i	$0.540510\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−238.000	−1.52398	−0.761991	−	0.647587i	$-0.775778\pi$
$29$	−238.000	−1.52398	−0.761991	+	0.647587i	$0.775778\pi$
$30$	0	0
$31$	−180.000	−1.04287	−0.521435	−	0.853291i	$-0.674603\pi$
$31$	−180.000	−1.04287	−0.521435	+	0.853291i	$0.674603\pi$
$32$	0	0
$33$	84.0000	0.443107
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−40.0000	−0.177729	−0.0888643	−	0.996044i	$-0.528324\pi$
$37$	−40.0000	−0.177729	−0.0888643	+	0.996044i	$0.528324\pi$
$38$	0	0
$39$	48.0000	0.197081
$40$	0	0
$41$	422.000	1.60745	0.803724	−	0.595003i	$-0.202849\pi$
$41$	422.000	1.60745	0.803724	+	0.595003i	$0.202849\pi$
$42$	0	0
$43$	276.000	0.978828	0.489414	−	0.872052i	$-0.337211\pi$
$43$	276.000	0.978828	0.489414	+	0.872052i	$0.337211\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	60.0000	0.186211	0.0931053	−	0.995656i	$-0.470321\pi$
$47$	60.0000	0.186211	0.0931053	+	0.995656i	$0.470321\pi$
$48$	0	0
$49$	−327.000	−0.953353
$50$	0	0
$51$	−324.000	−0.889590
$52$	0	0
$53$	220.000	0.570176	0.285088	−	0.958501i	$-0.407977\pi$
$53$	220.000	0.570176	0.285088	+	0.958501i	$0.407977\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−96.0000	−0.223079
$58$	0	0
$59$	−804.000	−1.77410	−0.887050	−	0.461674i	$-0.847249\pi$
$59$	−804.000	−1.77410	−0.887050	+	0.461674i	$0.847249\pi$
$60$	0	0
$61$	−358.000	−0.751430	−0.375715	−	0.926735i	$-0.622603\pi$
$61$	−358.000	−0.751430	−0.375715	+	0.926735i	$0.622603\pi$
$62$	0	0
$63$	36.0000	0.0719932
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−884.000	−1.61191	−0.805954	−	0.591979i	$-0.798347\pi$
$67$	−884.000	−1.61191	−0.805954	+	0.591979i	$0.798347\pi$
$68$	0	0
$69$	84.0000	0.146557
$70$	0	0
$71$	−64.0000	−0.106978	−0.0534888	−	0.998568i	$-0.517034\pi$
$71$	−64.0000	−0.106978	−0.0534888	+	0.998568i	$0.517034\pi$
$72$	0	0
$73$	−152.000	−0.243702	−0.121851	−	0.992548i	$-0.538883\pi$
$73$	−152.000	−0.243702	−0.121851	+	0.992548i	$0.538883\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−112.000	−0.165761
$78$	0	0
$79$	−932.000	−1.32732	−0.663659	−	0.748035i	$-0.730998\pi$
$79$	−932.000	−1.32732	−0.663659	+	0.748035i	$0.730998\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1292.00	−1.70862	−0.854310	−	0.519764i	$-0.826020\pi$
$83$	−1292.00	−1.70862	−0.854310	+	0.519764i	$0.826020\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	714.000	0.879872
$88$	0	0
$89$	−1146.00	−1.36490	−0.682448	−	0.730934i	$-0.739085\pi$
$89$	−1146.00	−1.36490	−0.682448	+	0.730934i	$0.739085\pi$
$90$	0	0
$91$	−64.0000	−0.0737255
$92$	0	0
$93$	540.000	0.602101
$94$	0	0
$95$	0	0
$96$	0	0
$97$	824.000	0.862521	0.431260	−	0.902227i	$-0.358069\pi$
$97$	824.000	0.862521	0.431260	+	0.902227i	$0.358069\pi$
$98$	0	0
$99$	−252.000	−0.255828
$100$	0	0
$101$	−1290.00	−1.27089	−0.635445	−	0.772147i	$-0.719183\pi$
$101$	−1290.00	−1.27089	−0.635445	+	0.772147i	$0.719183\pi$
$102$	0	0
$103$	−1604.00	−1.53444	−0.767218	−	0.641387i	$-0.778359\pi$
$103$	−1604.00	−1.53444	−0.767218	+	0.641387i	$0.778359\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	892.000	0.805915	0.402957	−	0.915219i	$-0.367982\pi$
$107$	892.000	0.805915	0.402957	+	0.915219i	$0.367982\pi$
$108$	0	0
$109$	966.000	0.848863	0.424431	−	0.905460i	$-0.360474\pi$
$109$	966.000	0.848863	0.424431	+	0.905460i	$0.360474\pi$
$110$	0	0
$111$	120.000	0.102612
$112$	0	0
$113$	1124.00	0.935726	0.467863	−	0.883801i	$-0.345024\pi$
$113$	1124.00	0.935726	0.467863	+	0.883801i	$0.345024\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−144.000	−0.113785
$118$	0	0
$119$	432.000	0.332785
$120$	0	0
$121$	−547.000	−0.410969
$122$	0	0
$123$	−1266.00	−0.928060
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1884.00	−1.31636	−0.658181	−	0.752860i	$-0.728674\pi$
$127$	−1884.00	−1.31636	−0.658181	+	0.752860i	$0.728674\pi$
$128$	0	0
$129$	−828.000	−0.565127
$130$	0	0
$131$	−588.000	−0.392166	−0.196083	−	0.980587i	$-0.562822\pi$
$131$	−588.000	−0.392166	−0.196083	+	0.980587i	$0.562822\pi$
$132$	0	0
$133$	128.000	0.0834512
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1060.00	0.661036	0.330518	−	0.943800i	$-0.392777\pi$
$137$	1060.00	0.661036	0.330518	+	0.943800i	$0.392777\pi$
$138$	0	0
$139$	−2864.00	−1.74764	−0.873818	−	0.486254i	$-0.838363\pi$
$139$	−2864.00	−1.74764	−0.873818	+	0.486254i	$0.838363\pi$
$140$	0	0
$141$	−180.000	−0.107509
$142$	0	0
$143$	448.000	0.261984
$144$	0	0
$145$	0	0
$146$	0	0
$147$	981.000	0.550418
$148$	0	0
$149$	342.000	0.188038	0.0940192	−	0.995570i	$-0.470028\pi$
$149$	342.000	0.188038	0.0940192	+	0.995570i	$0.470028\pi$
$150$	0	0
$151$	1636.00	0.881694	0.440847	−	0.897582i	$-0.354678\pi$
$151$	1636.00	0.881694	0.440847	+	0.897582i	$0.354678\pi$
$152$	0	0
$153$	972.000	0.513605
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2072.00	−1.05327	−0.526636	−	0.850091i	$-0.676547\pi$
$157$	−2072.00	−1.05327	−0.526636	+	0.850091i	$0.676547\pi$
$158$	0	0
$159$	−660.000	−0.329191
$160$	0	0
$161$	−112.000	−0.0548251
$162$	0	0
$163$	772.000	0.370968	0.185484	−	0.982647i	$-0.440615\pi$
$163$	772.000	0.370968	0.185484	+	0.982647i	$0.440615\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1044.00	−0.483755	−0.241878	−	0.970307i	$-0.577763\pi$
$167$	−1044.00	−0.483755	−0.241878	+	0.970307i	$0.577763\pi$
$168$	0	0
$169$	−1941.00	−0.883477
$170$	0	0
$171$	288.000	0.128795
$172$	0	0
$173$	4404.00	1.93543	0.967717	−	0.252041i	$-0.0811018\pi$
$173$	4404.00	1.93543	0.967717	+	0.252041i	$0.0811018\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2412.00	1.02428
$178$	0	0
$179$	3452.00	1.44142	0.720711	−	0.693235i	$-0.243815\pi$
$179$	3452.00	1.44142	0.720711	+	0.693235i	$0.243815\pi$
$180$	0	0
$181$	526.000	0.216007	0.108004	−	0.994151i	$-0.465554\pi$
$181$	526.000	0.216007	0.108004	+	0.994151i	$0.465554\pi$
$182$	0	0
$183$	1074.00	0.433838
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3024.00	−1.18255
$188$	0	0
$189$	−108.000	−0.0415653
$190$	0	0
$191$	72.0000	0.0272761	0.0136381	−	0.999907i	$-0.495659\pi$
$191$	72.0000	0.0272761	0.0136381	+	0.999907i	$0.495659\pi$
$192$	0	0
$193$	208.000	0.0775760	0.0387880	−	0.999247i	$-0.487650\pi$
$193$	208.000	0.0775760	0.0387880	+	0.999247i	$0.487650\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	372.000	0.134538	0.0672688	−	0.997735i	$-0.478571\pi$
$197$	372.000	0.134538	0.0672688	+	0.997735i	$0.478571\pi$
$198$	0	0
$199$	4348.00	1.54885	0.774426	−	0.632665i	$-0.218039\pi$
$199$	4348.00	1.54885	0.774426	+	0.632665i	$0.218039\pi$
$200$	0	0
$201$	2652.00	0.930635
$202$	0	0
$203$	−952.000	−0.329149
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−252.000	−0.0846145
$208$	0	0
$209$	−896.000	−0.296544
$210$	0	0
$211$	−416.000	−0.135728	−0.0678640	−	0.997695i	$-0.521618\pi$
$211$	−416.000	−0.135728	−0.0678640	+	0.997695i	$0.521618\pi$
$212$	0	0
$213$	192.000	0.0617635
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−720.000	−0.225239
$218$	0	0
$219$	456.000	0.140701
$220$	0	0
$221$	−1728.00	−0.525963
$222$	0	0
$223$	−5748.00	−1.72607	−0.863037	−	0.505141i	$-0.831441\pi$
$223$	−5748.00	−1.72607	−0.863037	+	0.505141i	$0.831441\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1148.00	0.335663	0.167831	−	0.985816i	$-0.446324\pi$
$227$	1148.00	0.335663	0.167831	+	0.985816i	$0.446324\pi$
$228$	0	0
$229$	3234.00	0.933226	0.466613	−	0.884462i	$-0.345474\pi$
$229$	3234.00	0.933226	0.466613	+	0.884462i	$0.345474\pi$
$230$	0	0
$231$	336.000	0.0957021
$232$	0	0
$233$	−228.000	−0.0641063	−0.0320532	−	0.999486i	$-0.510205\pi$
$233$	−228.000	−0.0641063	−0.0320532	+	0.999486i	$0.510205\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2796.00	0.766328
$238$	0	0
$239$	4760.00	1.28828	0.644140	−	0.764908i	$-0.277216\pi$
$239$	4760.00	1.28828	0.644140	+	0.764908i	$0.277216\pi$
$240$	0	0
$241$	3230.00	0.863330	0.431665	−	0.902034i	$-0.357926\pi$
$241$	3230.00	0.863330	0.431665	+	0.902034i	$0.357926\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−512.000	−0.131894
$248$	0	0
$249$	3876.00	0.986472
$250$	0	0
$251$	1708.00	0.429514	0.214757	−	0.976668i	$-0.431104\pi$
$251$	1708.00	0.429514	0.214757	+	0.976668i	$0.431104\pi$
$252$	0	0
$253$	784.000	0.194821
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6372.00	1.54659	0.773297	−	0.634044i	$-0.218606\pi$
$257$	6372.00	1.54659	0.773297	+	0.634044i	$0.218606\pi$
$258$	0	0
$259$	−160.000	−0.0383858
$260$	0	0
$261$	−2142.00	−0.507994
$262$	0	0
$263$	−3036.00	−0.711817	−0.355908	−	0.934521i	$-0.615828\pi$
$263$	−3036.00	−0.711817	−0.355908	+	0.934521i	$0.615828\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3438.00	0.788023
$268$	0	0
$269$	−114.000	−0.0258390	−0.0129195	−	0.999917i	$-0.504113\pi$
$269$	−114.000	−0.0258390	−0.0129195	+	0.999917i	$0.504113\pi$
$270$	0	0
$271$	−5236.00	−1.17367	−0.586835	−	0.809707i	$-0.699626\pi$
$271$	−5236.00	−1.17367	−0.586835	+	0.809707i	$0.699626\pi$
$272$	0	0
$273$	192.000	0.0425655
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−5712.00	−1.23899	−0.619496	−	0.785000i	$-0.712663\pi$
$277$	−5712.00	−1.23899	−0.619496	+	0.785000i	$0.712663\pi$
$278$	0	0
$279$	−1620.00	−0.347623
$280$	0	0
$281$	3222.00	0.684016	0.342008	−	0.939697i	$-0.388893\pi$
$281$	3222.00	0.684016	0.342008	+	0.939697i	$0.388893\pi$
$282$	0	0
$283$	−4620.00	−0.970426	−0.485213	−	0.874396i	$-0.661258\pi$
$283$	−4620.00	−0.970426	−0.485213	+	0.874396i	$0.661258\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1688.00	0.347176
$288$	0	0
$289$	6751.00	1.37411
$290$	0	0
$291$	−2472.00	−0.497977
$292$	0	0
$293$	−5404.00	−1.07749	−0.538746	−	0.842468i	$-0.681102\pi$
$293$	−5404.00	−1.07749	−0.538746	+	0.842468i	$0.681102\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	756.000	0.147702
$298$	0	0
$299$	448.000	0.0866505
$300$	0	0
$301$	1104.00	0.211407
$302$	0	0
$303$	3870.00	0.733748
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−9700.00	−1.80328	−0.901642	−	0.432483i	$-0.857638\pi$
$307$	−9700.00	−1.80328	−0.901642	+	0.432483i	$0.857638\pi$
$308$	0	0
$309$	4812.00	0.885907
$310$	0	0
$311$	9672.00	1.76350	0.881750	−	0.471716i	$-0.156365\pi$
$311$	9672.00	1.76350	0.881750	+	0.471716i	$0.156365\pi$
$312$	0	0
$313$	4048.00	0.731011	0.365506	−	0.930809i	$-0.380896\pi$
$313$	4048.00	0.731011	0.365506	+	0.930809i	$0.380896\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−84.0000	−0.0148830	−0.00744150	−	0.999972i	$-0.502369\pi$
$317$	−84.0000	−0.0148830	−0.00744150	+	0.999972i	$0.502369\pi$
$318$	0	0
$319$	6664.00	1.16963
$320$	0	0
$321$	−2676.00	−0.465295
$322$	0	0
$323$	3456.00	0.595347
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2898.00	−0.490091
$328$	0	0
$329$	240.000	0.0402177
$330$	0	0
$331$	5416.00	0.899366	0.449683	−	0.893188i	$-0.351537\pi$
$331$	5416.00	0.899366	0.449683	+	0.893188i	$0.351537\pi$
$332$	0	0
$333$	−360.000	−0.0592429
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−8216.00	−1.32805	−0.664027	−	0.747709i	$-0.731154\pi$
$337$	−8216.00	−1.32805	−0.664027	+	0.747709i	$0.731154\pi$
$338$	0	0
$339$	−3372.00	−0.540242
$340$	0	0
$341$	5040.00	0.800385
$342$	0	0
$343$	−2680.00	−0.421885
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3836.00	0.593450	0.296725	−	0.954963i	$-0.404105\pi$
$347$	3836.00	0.593450	0.296725	+	0.954963i	$0.404105\pi$
$348$	0	0
$349$	−2038.00	−0.312583	−0.156292	−	0.987711i	$-0.549954\pi$
$349$	−2038.00	−0.312583	−0.156292	+	0.987711i	$0.549954\pi$
$350$	0	0
$351$	432.000	0.0656936
$352$	0	0
$353$	5292.00	0.797917	0.398959	−	0.916969i	$-0.369372\pi$
$353$	5292.00	0.797917	0.398959	+	0.916969i	$0.369372\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1296.00	−0.192133
$358$	0	0
$359$	−3896.00	−0.572766	−0.286383	−	0.958115i	$-0.592453\pi$
$359$	−3896.00	−0.572766	−0.286383	+	0.958115i	$0.592453\pi$
$360$	0	0
$361$	−5835.00	−0.850707
$362$	0	0
$363$	1641.00	0.237273
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7652.00	1.08837	0.544184	−	0.838966i	$-0.316839\pi$
$367$	7652.00	1.08837	0.544184	+	0.838966i	$0.316839\pi$
$368$	0	0
$369$	3798.00	0.535816
$370$	0	0
$371$	880.000	0.123146
$372$	0	0
$373$	−1576.00	−0.218773	−0.109386	−	0.993999i	$-0.534889\pi$
$373$	−1576.00	−0.218773	−0.109386	+	0.993999i	$0.534889\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3808.00	0.520217
$378$	0	0
$379$	−5416.00	−0.734040	−0.367020	−	0.930213i	$-0.619622\pi$
$379$	−5416.00	−0.734040	−0.367020	+	0.930213i	$0.619622\pi$
$380$	0	0
$381$	5652.00	0.760002
$382$	0	0
$383$	8292.00	1.10627	0.553135	−	0.833092i	$-0.313431\pi$
$383$	8292.00	1.10627	0.553135	+	0.833092i	$0.313431\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2484.00	0.326276
$388$	0	0
$389$	−9642.00	−1.25673	−0.628366	−	0.777918i	$-0.716276\pi$
$389$	−9642.00	−1.25673	−0.628366	+	0.777918i	$0.716276\pi$
$390$	0	0
$391$	−3024.00	−0.391126
$392$	0	0
$393$	1764.00	0.226417
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−13032.0	−1.64750	−0.823750	−	0.566954i	$-0.808122\pi$
$397$	−13032.0	−1.64750	−0.823750	+	0.566954i	$0.808122\pi$
$398$	0	0
$399$	−384.000	−0.0481806
$400$	0	0
$401$	−13358.0	−1.66351	−0.831754	−	0.555144i	$-0.812663\pi$
$401$	−13358.0	−1.66351	−0.831754	+	0.555144i	$0.812663\pi$
$402$	0	0
$403$	2880.00	0.355988
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1120.00	0.136404
$408$	0	0
$409$	6410.00	0.774949	0.387474	−	0.921880i	$-0.373348\pi$
$409$	6410.00	0.774949	0.387474	+	0.921880i	$0.373348\pi$
$410$	0	0
$411$	−3180.00	−0.381649
$412$	0	0
$413$	−3216.00	−0.383170
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8592.00	1.00900
$418$	0	0
$419$	7644.00	0.891250	0.445625	−	0.895220i	$-0.352981\pi$
$419$	7644.00	0.891250	0.445625	+	0.895220i	$0.352981\pi$
$420$	0	0
$421$	14674.0	1.69873	0.849367	−	0.527803i	$-0.176984\pi$
$421$	14674.0	1.69873	0.849367	+	0.527803i	$0.176984\pi$
$422$	0	0
$423$	540.000	0.0620702
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1432.00	−0.162294
$428$	0	0
$429$	−1344.00	−0.151256
$430$	0	0
$431$	−9704.00	−1.08451	−0.542257	−	0.840213i	$-0.682430\pi$
$431$	−9704.00	−1.08451	−0.542257	+	0.840213i	$0.682430\pi$
$432$	0	0
$433$	1296.00	0.143838	0.0719189	−	0.997410i	$-0.477088\pi$
$433$	1296.00	0.143838	0.0719189	+	0.997410i	$0.477088\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−896.000	−0.0980812
$438$	0	0
$439$	−15684.0	−1.70514	−0.852570	−	0.522613i	$-0.824957\pi$
$439$	−15684.0	−1.70514	−0.852570	+	0.522613i	$0.824957\pi$
$440$	0	0
$441$	−2943.00	−0.317784
$442$	0	0
$443$	−5772.00	−0.619043	−0.309521	−	0.950892i	$-0.600169\pi$
$443$	−5772.00	−0.619043	−0.309521	+	0.950892i	$0.600169\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1026.00	−0.108564
$448$	0	0
$449$	−4782.00	−0.502620	−0.251310	−	0.967907i	$-0.580861\pi$
$449$	−4782.00	−0.502620	−0.251310	+	0.967907i	$0.580861\pi$
$450$	0	0
$451$	−11816.0	−1.23369
$452$	0	0
$453$	−4908.00	−0.509046
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15000.0	1.53538	0.767692	−	0.640819i	$-0.221405\pi$
$457$	15000.0	1.53538	0.767692	+	0.640819i	$0.221405\pi$
$458$	0	0
$459$	−2916.00	−0.296530
$460$	0	0
$461$	−3762.00	−0.380073	−0.190037	−	0.981777i	$-0.560861\pi$
$461$	−3762.00	−0.380073	−0.190037	+	0.981777i	$0.560861\pi$
$462$	0	0
$463$	5036.00	0.505492	0.252746	−	0.967533i	$-0.418666\pi$
$463$	5036.00	0.505492	0.252746	+	0.967533i	$0.418666\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2268.00	0.224733	0.112367	−	0.993667i	$-0.464157\pi$
$467$	2268.00	0.224733	0.112367	+	0.993667i	$0.464157\pi$
$468$	0	0
$469$	−3536.00	−0.348139
$470$	0	0
$471$	6216.00	0.608106
$472$	0	0
$473$	−7728.00	−0.751234
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1980.00	0.190059
$478$	0	0
$479$	16208.0	1.54606	0.773030	−	0.634370i	$-0.218740\pi$
$479$	16208.0	1.54606	0.773030	+	0.634370i	$0.218740\pi$
$480$	0	0
$481$	640.000	0.0606684
$482$	0	0
$483$	336.000	0.0316533
$484$	0	0
$485$	0	0
$486$	0	0
$487$	11572.0	1.07675	0.538375	−	0.842705i	$-0.319038\pi$
$487$	11572.0	1.07675	0.538375	+	0.842705i	$0.319038\pi$
$488$	0	0
$489$	−2316.00	−0.214178
$490$	0	0
$491$	5636.00	0.518023	0.259011	−	0.965874i	$-0.416603\pi$
$491$	5636.00	0.518023	0.259011	+	0.965874i	$0.416603\pi$
$492$	0	0
$493$	−25704.0	−2.34817
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−256.000	−0.0231050
$498$	0	0
$499$	−5560.00	−0.498797	−0.249399	−	0.968401i	$-0.580233\pi$
$499$	−5560.00	−0.498797	−0.249399	+	0.968401i	$0.580233\pi$
$500$	0	0
$501$	3132.00	0.279296
$502$	0	0
$503$	15172.0	1.34490	0.672451	−	0.740141i	$-0.265241\pi$
$503$	15172.0	1.34490	0.672451	+	0.740141i	$0.265241\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	5823.00	0.510076
$508$	0	0
$509$	−17342.0	−1.51016	−0.755079	−	0.655634i	$-0.772402\pi$
$509$	−17342.0	−1.51016	−0.755079	+	0.655634i	$0.772402\pi$
$510$	0	0
$511$	−608.000	−0.0526347
$512$	0	0
$513$	−864.000	−0.0743597
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1680.00	−0.142914
$518$	0	0
$519$	−13212.0	−1.11742
$520$	0	0
$521$	−4886.00	−0.410863	−0.205431	−	0.978672i	$-0.565860\pi$
$521$	−4886.00	−0.410863	−0.205431	+	0.978672i	$0.565860\pi$
$522$	0	0
$523$	18548.0	1.55076	0.775380	−	0.631495i	$-0.217558\pi$
$523$	18548.0	1.55076	0.775380	+	0.631495i	$0.217558\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−19440.0	−1.60687
$528$	0	0
$529$	−11383.0	−0.935563
$530$	0	0
$531$	−7236.00	−0.591367
$532$	0	0
$533$	−6752.00	−0.548708
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−10356.0	−0.832206
$538$	0	0
$539$	9156.00	0.731682
$540$	0	0
$541$	−15770.0	−1.25324	−0.626622	−	0.779323i	$-0.715563\pi$
$541$	−15770.0	−1.25324	−0.626622	+	0.779323i	$0.715563\pi$
$542$	0	0
$543$	−1578.00	−0.124712
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−7700.00	−0.601880	−0.300940	−	0.953643i	$-0.597300\pi$
$547$	−7700.00	−0.601880	−0.300940	+	0.953643i	$0.597300\pi$
$548$	0	0
$549$	−3222.00	−0.250477
$550$	0	0
$551$	−7616.00	−0.588843
$552$	0	0
$553$	−3728.00	−0.286674
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19236.0	−1.46330	−0.731648	−	0.681683i	$-0.761248\pi$
$557$	−19236.0	−1.46330	−0.731648	+	0.681683i	$0.761248\pi$
$558$	0	0
$559$	−4416.00	−0.334127
$560$	0	0
$561$	9072.00	0.682745
$562$	0	0
$563$	8388.00	0.627908	0.313954	−	0.949438i	$-0.398346\pi$
$563$	8388.00	0.627908	0.313954	+	0.949438i	$0.398346\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	324.000	0.0239977
$568$	0	0
$569$	−16758.0	−1.23468	−0.617339	−	0.786697i	$-0.711789\pi$
$569$	−16758.0	−1.23468	−0.617339	+	0.786697i	$0.711789\pi$
$570$	0	0
$571$	8056.00	0.590426	0.295213	−	0.955432i	$-0.404609\pi$
$571$	8056.00	0.590426	0.295213	+	0.955432i	$0.404609\pi$
$572$	0	0
$573$	−216.000	−0.0157479
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5728.00	0.413275	0.206638	−	0.978418i	$-0.433748\pi$
$577$	5728.00	0.413275	0.206638	+	0.978418i	$0.433748\pi$
$578$	0	0
$579$	−624.000	−0.0447885
$580$	0	0
$581$	−5168.00	−0.369027
$582$	0	0
$583$	−6160.00	−0.437601
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12172.0	0.855864	0.427932	−	0.903811i	$-0.359242\pi$
$587$	12172.0	0.855864	0.427932	+	0.903811i	$0.359242\pi$
$588$	0	0
$589$	−5760.00	−0.402948
$590$	0	0
$591$	−1116.00	−0.0776753
$592$	0	0
$593$	10708.0	0.741526	0.370763	−	0.928728i	$-0.379096\pi$
$593$	10708.0	0.741526	0.370763	+	0.928728i	$0.379096\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−13044.0	−0.894230
$598$	0	0
$599$	−9416.00	−0.642283	−0.321141	−	0.947031i	$-0.604066\pi$
$599$	−9416.00	−0.642283	−0.321141	+	0.947031i	$0.604066\pi$
$600$	0	0
$601$	9270.00	0.629170	0.314585	−	0.949229i	$-0.398135\pi$
$601$	9270.00	0.629170	0.314585	+	0.949229i	$0.398135\pi$
$602$	0	0
$603$	−7956.00	−0.537302
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−7996.00	−0.534675	−0.267337	−	0.963603i	$-0.586144\pi$
$607$	−7996.00	−0.534675	−0.267337	+	0.963603i	$0.586144\pi$
$608$	0	0
$609$	2856.00	0.190034
$610$	0	0
$611$	−960.000	−0.0635637
$612$	0	0
$613$	−232.000	−0.0152861	−0.00764306	−	0.999971i	$-0.502433\pi$
$613$	−232.000	−0.0152861	−0.00764306	+	0.999971i	$0.502433\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3740.00	0.244030	0.122015	−	0.992528i	$-0.461064\pi$
$617$	3740.00	0.244030	0.122015	+	0.992528i	$0.461064\pi$
$618$	0	0
$619$	−26000.0	−1.68825	−0.844126	−	0.536145i	$-0.819880\pi$
$619$	−26000.0	−1.68825	−0.844126	+	0.536145i	$0.819880\pi$
$620$	0	0
$621$	756.000	0.0488522
$622$	0	0
$623$	−4584.00	−0.294790
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2688.00	0.171210
$628$	0	0
$629$	−4320.00	−0.273847
$630$	0	0
$631$	11660.0	0.735622	0.367811	−	0.929901i	$-0.380107\pi$
$631$	11660.0	0.735622	0.367811	+	0.929901i	$0.380107\pi$
$632$	0	0
$633$	1248.00	0.0783626
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5232.00	0.325431
$638$	0	0
$639$	−576.000	−0.0356592
$640$	0	0
$641$	−7602.00	−0.468426	−0.234213	−	0.972185i	$-0.575251\pi$
$641$	−7602.00	−0.468426	−0.234213	+	0.972185i	$0.575251\pi$
$642$	0	0
$643$	29268.0	1.79505	0.897525	−	0.440963i	$-0.145363\pi$
$643$	29268.0	1.79505	0.897525	+	0.440963i	$0.145363\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17836.0	1.08378	0.541890	−	0.840449i	$-0.317709\pi$
$647$	17836.0	1.08378	0.541890	+	0.840449i	$0.317709\pi$
$648$	0	0
$649$	22512.0	1.36159
$650$	0	0
$651$	2160.00	0.130042
$652$	0	0
$653$	−19188.0	−1.14990	−0.574950	−	0.818189i	$-0.694978\pi$
$653$	−19188.0	−1.14990	−0.574950	+	0.818189i	$0.694978\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1368.00	−0.0812340
$658$	0	0
$659$	13860.0	0.819285	0.409643	−	0.912246i	$-0.365653\pi$
$659$	13860.0	0.819285	0.409643	+	0.912246i	$0.365653\pi$
$660$	0	0
$661$	−16558.0	−0.974329	−0.487165	−	0.873310i	$-0.661969\pi$
$661$	−16558.0	−0.974329	−0.487165	+	0.873310i	$0.661969\pi$
$662$	0	0
$663$	5184.00	0.303665
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6664.00	0.386853
$668$	0	0
$669$	17244.0	0.996549
$670$	0	0
$671$	10024.0	0.576710
$672$	0	0
$673$	−4640.00	−0.265764	−0.132882	−	0.991132i	$-0.542423\pi$
$673$	−4640.00	−0.265764	−0.132882	+	0.991132i	$0.542423\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−34260.0	−1.94493	−0.972466	−	0.233045i	$-0.925131\pi$
$677$	−34260.0	−1.94493	−0.972466	+	0.233045i	$0.925131\pi$
$678$	0	0
$679$	3296.00	0.186287
$680$	0	0
$681$	−3444.00	−0.193795
$682$	0	0
$683$	−19420.0	−1.08797	−0.543987	−	0.839094i	$-0.683086\pi$
$683$	−19420.0	−1.08797	−0.543987	+	0.839094i	$0.683086\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−9702.00	−0.538798
$688$	0	0
$689$	−3520.00	−0.194632
$690$	0	0
$691$	4608.00	0.253685	0.126843	−	0.991923i	$-0.459516\pi$
$691$	4608.00	0.253685	0.126843	+	0.991923i	$0.459516\pi$
$692$	0	0
$693$	−1008.00	−0.0552536
$694$	0	0
$695$	0	0
$696$	0	0
$697$	45576.0	2.47678
$698$	0	0
$699$	684.000	0.0370118
$700$	0	0
$701$	−2318.00	−0.124893	−0.0624463	−	0.998048i	$-0.519890\pi$
$701$	−2318.00	−0.124893	−0.0624463	+	0.998048i	$0.519890\pi$
$702$	0	0
$703$	−1280.00	−0.0686716
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5160.00	−0.274486
$708$	0	0
$709$	16834.0	0.891698	0.445849	−	0.895108i	$-0.352902\pi$
$709$	16834.0	0.891698	0.445849	+	0.895108i	$0.352902\pi$
$710$	0	0
$711$	−8388.00	−0.442440
$712$	0	0
$713$	5040.00	0.264726
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−14280.0	−0.743789
$718$	0	0
$719$	−7400.00	−0.383830	−0.191915	−	0.981412i	$-0.561470\pi$
$719$	−7400.00	−0.383830	−0.191915	+	0.981412i	$0.561470\pi$
$720$	0	0
$721$	−6416.00	−0.331407
$722$	0	0
$723$	−9690.00	−0.498444
$724$	0	0
$725$	0	0
$726$	0	0
$727$	20340.0	1.03765	0.518823	−	0.854882i	$-0.326370\pi$
$727$	20340.0	1.03765	0.518823	+	0.854882i	$0.326370\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	29808.0	1.50819
$732$	0	0
$733$	4896.00	0.246709	0.123355	−	0.992363i	$-0.460635\pi$
$733$	4896.00	0.246709	0.123355	+	0.992363i	$0.460635\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24752.0	1.23711
$738$	0	0
$739$	26040.0	1.29621	0.648103	−	0.761552i	$-0.275562\pi$
$739$	26040.0	1.29621	0.648103	+	0.761552i	$0.275562\pi$
$740$	0	0
$741$	1536.00	0.0761489
$742$	0	0
$743$	−6780.00	−0.334770	−0.167385	−	0.985892i	$-0.553532\pi$
$743$	−6780.00	−0.334770	−0.167385	+	0.985892i	$0.553532\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−11628.0	−0.569540
$748$	0	0
$749$	3568.00	0.174061
$750$	0	0
$751$	−20692.0	−1.00541	−0.502704	−	0.864458i	$-0.667662\pi$
$751$	−20692.0	−1.00541	−0.502704	+	0.864458i	$0.667662\pi$
$752$	0	0
$753$	−5124.00	−0.247980
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−10816.0	−0.519305	−0.259653	−	0.965702i	$-0.583608\pi$
$757$	−10816.0	−0.519305	−0.259653	+	0.965702i	$0.583608\pi$
$758$	0	0
$759$	−2352.00	−0.112480
$760$	0	0
$761$	13978.0	0.665837	0.332919	−	0.942956i	$-0.391967\pi$
$761$	13978.0	0.665837	0.332919	+	0.942956i	$0.391967\pi$
$762$	0	0
$763$	3864.00	0.183337
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12864.0	0.605596
$768$	0	0
$769$	2926.00	0.137210	0.0686048	−	0.997644i	$-0.478145\pi$
$769$	2926.00	0.137210	0.0686048	+	0.997644i	$0.478145\pi$
$770$	0	0
$771$	−19116.0	−0.892926
$772$	0	0
$773$	13916.0	0.647508	0.323754	−	0.946141i	$-0.395055\pi$
$773$	13916.0	0.647508	0.323754	+	0.946141i	$0.395055\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	480.000	0.0221620
$778$	0	0
$779$	13504.0	0.621092
$780$	0	0
$781$	1792.00	0.0821035
$782$	0	0
$783$	6426.00	0.293291
$784$	0	0
$785$	0	0
$786$	0	0
$787$	29996.0	1.35863	0.679315	−	0.733847i	$-0.262277\pi$
$787$	29996.0	1.35863	0.679315	+	0.733847i	$0.262277\pi$
$788$	0	0
$789$	9108.00	0.410968
$790$	0	0
$791$	4496.00	0.202098
$792$	0	0
$793$	5728.00	0.256503
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−8940.00	−0.397329	−0.198664	−	0.980068i	$-0.563660\pi$
$797$	−8940.00	−0.397329	−0.198664	+	0.980068i	$0.563660\pi$
$798$	0	0
$799$	6480.00	0.286916
$800$	0	0
$801$	−10314.0	−0.454965
$802$	0	0
$803$	4256.00	0.187037
$804$	0	0
$805$	0	0
$806$	0	0
$807$	342.000	0.0149182
$808$	0	0
$809$	−10698.0	−0.464922	−0.232461	−	0.972606i	$-0.574678\pi$
$809$	−10698.0	−0.464922	−0.232461	+	0.972606i	$0.574678\pi$
$810$	0	0
$811$	−6408.00	−0.277454	−0.138727	−	0.990331i	$-0.544301\pi$
$811$	−6408.00	−0.277454	−0.138727	+	0.990331i	$0.544301\pi$
$812$	0	0
$813$	15708.0	0.677618
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8832.00	0.378204
$818$	0	0
$819$	−576.000	−0.0245752
$820$	0	0
$821$	23130.0	0.983243	0.491622	−	0.870809i	$-0.336404\pi$
$821$	23130.0	0.983243	0.491622	+	0.870809i	$0.336404\pi$
$822$	0	0
$823$	11852.0	0.501986	0.250993	−	0.967989i	$-0.419243\pi$
$823$	11852.0	0.501986	0.250993	+	0.967989i	$0.419243\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32628.0	−1.37193	−0.685965	−	0.727634i	$-0.740620\pi$
$827$	−32628.0	−1.37193	−0.685965	+	0.727634i	$0.740620\pi$
$828$	0	0
$829$	36694.0	1.53732	0.768658	−	0.639660i	$-0.220925\pi$
$829$	36694.0	1.53732	0.768658	+	0.639660i	$0.220925\pi$
$830$	0	0
$831$	17136.0	0.715332
$832$	0	0
$833$	−35316.0	−1.46894
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4860.00	0.200700
$838$	0	0
$839$	−1704.00	−0.0701175	−0.0350588	−	0.999385i	$-0.511162\pi$
$839$	−1704.00	−0.0701175	−0.0350588	+	0.999385i	$0.511162\pi$
$840$	0	0
$841$	32255.0	1.32252
$842$	0	0
$843$	−9666.00	−0.394917
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2188.00	−0.0887610
$848$	0	0
$849$	13860.0	0.560276
$850$	0	0
$851$	1120.00	0.0451153
$852$	0	0
$853$	31880.0	1.27966	0.639830	−	0.768516i	$-0.279005\pi$
$853$	31880.0	1.27966	0.639830	+	0.768516i	$0.279005\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7972.00	−0.317758	−0.158879	−	0.987298i	$-0.550788\pi$
$857$	−7972.00	−0.317758	−0.158879	+	0.987298i	$0.550788\pi$
$858$	0	0
$859$	−6008.00	−0.238638	−0.119319	−	0.992856i	$-0.538071\pi$
$859$	−6008.00	−0.238638	−0.119319	+	0.992856i	$0.538071\pi$
$860$	0	0
$861$	−5064.00	−0.200442
$862$	0	0
$863$	1716.00	0.0676863	0.0338432	−	0.999427i	$-0.489225\pi$
$863$	1716.00	0.0676863	0.0338432	+	0.999427i	$0.489225\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−20253.0	−0.793342
$868$	0	0
$869$	26096.0	1.01870
$870$	0	0
$871$	14144.0	0.550231
$872$	0	0
$873$	7416.00	0.287507
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−9032.00	−0.347764	−0.173882	−	0.984767i	$-0.555631\pi$
$877$	−9032.00	−0.347764	−0.173882	+	0.984767i	$0.555631\pi$
$878$	0	0
$879$	16212.0	0.622090
$880$	0	0
$881$	27838.0	1.06457	0.532285	−	0.846565i	$-0.321334\pi$
$881$	27838.0	1.06457	0.532285	+	0.846565i	$0.321334\pi$
$882$	0	0
$883$	−4316.00	−0.164490	−0.0822452	−	0.996612i	$-0.526209\pi$
$883$	−4316.00	−0.164490	−0.0822452	+	0.996612i	$0.526209\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−43524.0	−1.64757	−0.823784	−	0.566904i	$-0.808141\pi$
$887$	−43524.0	−1.64757	−0.823784	+	0.566904i	$0.808141\pi$
$888$	0	0
$889$	−7536.00	−0.284307
$890$	0	0
$891$	−2268.00	−0.0852759
$892$	0	0
$893$	1920.00	0.0719489
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1344.00	−0.0500277
$898$	0	0
$899$	42840.0	1.58931
$900$	0	0
$901$	23760.0	0.878535
$902$	0	0
$903$	−3312.00	−0.122056
$904$	0	0
$905$	0	0
$906$	0	0
$907$	10556.0	0.386446	0.193223	−	0.981155i	$-0.438106\pi$
$907$	10556.0	0.386446	0.193223	+	0.981155i	$0.438106\pi$
$908$	0	0
$909$	−11610.0	−0.423630
$910$	0	0
$911$	−47472.0	−1.72647	−0.863237	−	0.504799i	$-0.831567\pi$
$911$	−47472.0	−1.72647	−0.863237	+	0.504799i	$0.831567\pi$
$912$	0	0
$913$	36176.0	1.31134
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2352.00	−0.0847000
$918$	0	0
$919$	11964.0	0.429441	0.214720	−	0.976676i	$-0.431116\pi$
$919$	11964.0	0.429441	0.214720	+	0.976676i	$0.431116\pi$
$920$	0	0
$921$	29100.0	1.04113
$922$	0	0
$923$	1024.00	0.0365172
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−14436.0	−0.511478
$928$	0	0
$929$	−15214.0	−0.537304	−0.268652	−	0.963237i	$-0.586578\pi$
$929$	−15214.0	−0.537304	−0.268652	+	0.963237i	$0.586578\pi$
$930$	0	0
$931$	−10464.0	−0.368361
$932$	0	0
$933$	−29016.0	−1.01816
$934$	0	0
$935$	0	0
$936$	0	0
$937$	39712.0	1.38456	0.692281	−	0.721628i	$-0.256606\pi$
$937$	39712.0	1.38456	0.692281	+	0.721628i	$0.256606\pi$
$938$	0	0
$939$	−12144.0	−0.422049
$940$	0	0
$941$	−36034.0	−1.24833	−0.624163	−	0.781294i	$-0.714560\pi$
$941$	−36034.0	−1.24833	−0.624163	+	0.781294i	$0.714560\pi$
$942$	0	0
$943$	−11816.0	−0.408040
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2532.00	−0.0868838	−0.0434419	−	0.999056i	$-0.513832\pi$
$947$	−2532.00	−0.0868838	−0.0434419	+	0.999056i	$0.513832\pi$
$948$	0	0
$949$	2432.00	0.0831887
$950$	0	0
$951$	252.000	0.00859270
$952$	0	0
$953$	55284.0	1.87914	0.939572	−	0.342351i	$-0.111223\pi$
$953$	55284.0	1.87914	0.939572	+	0.342351i	$0.111223\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−19992.0	−0.675287
$958$	0	0
$959$	4240.00	0.142770
$960$	0	0
$961$	2609.00	0.0875768
$962$	0	0
$963$	8028.00	0.268638
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4372.00	0.145392	0.0726960	−	0.997354i	$-0.476840\pi$
$967$	4372.00	0.145392	0.0726960	+	0.997354i	$0.476840\pi$
$968$	0	0
$969$	−10368.0	−0.343724
$970$	0	0
$971$	24300.0	0.803114	0.401557	−	0.915834i	$-0.368469\pi$
$971$	24300.0	0.803114	0.401557	+	0.915834i	$0.368469\pi$
$972$	0	0
$973$	−11456.0	−0.377454
$974$	0	0
$975$	0	0
$976$	0	0
$977$	46204.0	1.51300	0.756498	−	0.653996i	$-0.226909\pi$
$977$	46204.0	1.51300	0.756498	+	0.653996i	$0.226909\pi$
$978$	0	0
$979$	32088.0	1.04754
$980$	0	0
$981$	8694.00	0.282954
$982$	0	0
$983$	−25468.0	−0.826351	−0.413176	−	0.910651i	$-0.635580\pi$
$983$	−25468.0	−0.826351	−0.413176	+	0.910651i	$0.635580\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−720.000	−0.0232197
$988$	0	0
$989$	−7728.00	−0.248469
$990$	0	0
$991$	11668.0	0.374012	0.187006	−	0.982359i	$-0.440122\pi$
$991$	11668.0	0.374012	0.187006	+	0.982359i	$0.440122\pi$
$992$	0	0
$993$	−16248.0	−0.519249
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−7224.00	−0.229475	−0.114737	−	0.993396i	$-0.536603\pi$
$997$	−7224.00	−0.229475	−0.114737	+	0.993396i	$0.536603\pi$
$998$	0	0
$999$	1080.00	0.0342039

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.4.a.f.1.1		1
3.2	odd	2		1800.4.a.u.1.1		1
4.3	odd	2		1200.4.a.z.1.1		1
5.2	odd	4		120.4.f.c.49.2	yes	2
5.3	odd	4		120.4.f.c.49.1	✓	2
5.4	even	2		600.4.a.k.1.1		1
15.2	even	4		360.4.f.a.289.1		2
15.8	even	4		360.4.f.a.289.2		2
15.14	odd	2		1800.4.a.o.1.1		1
20.3	even	4		240.4.f.e.49.2		2
20.7	even	4		240.4.f.e.49.1		2
20.19	odd	2		1200.4.a.l.1.1		1
40.3	even	4		960.4.f.a.769.1		2
40.13	odd	4		960.4.f.b.769.2		2
40.27	even	4		960.4.f.a.769.2		2
40.37	odd	4		960.4.f.b.769.1		2
60.23	odd	4		720.4.f.b.289.2		2
60.47	odd	4		720.4.f.b.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.f.c.49.1	✓	2	5.3	odd	4
120.4.f.c.49.2	yes	2	5.2	odd	4
240.4.f.e.49.1		2	20.7	even	4
240.4.f.e.49.2		2	20.3	even	4
360.4.f.a.289.1		2	15.2	even	4
360.4.f.a.289.2		2	15.8	even	4
600.4.a.f.1.1		1	1.1	even	1	trivial
600.4.a.k.1.1		1	5.4	even	2
720.4.f.b.289.1		2	60.47	odd	4
720.4.f.b.289.2		2	60.23	odd	4
960.4.f.a.769.1		2	40.3	even	4
960.4.f.a.769.2		2	40.27	even	4
960.4.f.b.769.1		2	40.37	odd	4
960.4.f.b.769.2		2	40.13	odd	4
1200.4.a.l.1.1		1	20.19	odd	2
1200.4.a.z.1.1		1	4.3	odd	2
1800.4.a.o.1.1		1	15.14	odd	2
1800.4.a.u.1.1		1	3.2	odd	2

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:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	600.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more