LMFDB - Embedded newform 1568.4.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,4,Mod(1,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1568.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1568.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:	$92.5149948890$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1568.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{3} -4.00000 q^{5} +37.0000 q^{9} +O(q^{10})$

$q+8.00000 q^{3} -4.00000 q^{5} +37.0000 q^{9} +40.0000 q^{11} +36.0000 q^{13} -32.0000 q^{15} -40.0000 q^{17} -72.0000 q^{19} +176.000 q^{23} -109.000 q^{25} +80.0000 q^{27} +162.000 q^{29} -16.0000 q^{31} +320.000 q^{33} -54.0000 q^{37} +288.000 q^{39} +472.000 q^{41} +72.0000 q^{43} -148.000 q^{45} +144.000 q^{47} -320.000 q^{51} +486.000 q^{53} -160.000 q^{55} -576.000 q^{57} -648.000 q^{59} +684.000 q^{61} -144.000 q^{65} -216.000 q^{67} +1408.00 q^{69} +608.000 q^{71} -1008.00 q^{73} -872.000 q^{75} -1008.00 q^{79} -359.000 q^{81} +216.000 q^{83} +160.000 q^{85} +1296.00 q^{87} +1040.00 q^{89} -128.000 q^{93} +288.000 q^{95} +936.000 q^{97} +1480.00 q^{99} +O(q^{100})$

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
$3$	8.00000	1.53960	0.769800	−	0.638285i	$-0.220356\pi$
$3$	8.00000	1.53960	0.769800	+	0.638285i	$0.220356\pi$
$4$	0	0
$5$	−4.00000	−0.357771	−0.178885	−	0.983870i	$-0.557249\pi$
$5$	−4.00000	−0.357771	−0.178885	+	0.983870i	$0.557249\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	37.0000	1.37037
$10$	0	0
$11$	40.0000	1.09640	0.548202	−	0.836346i	$-0.315312\pi$
$11$	40.0000	1.09640	0.548202	+	0.836346i	$0.315312\pi$
$12$	0	0
$13$	36.0000	0.768046	0.384023	−	0.923323i	$-0.374538\pi$
$13$	36.0000	0.768046	0.384023	+	0.923323i	$0.374538\pi$
$14$	0	0
$15$	−32.0000	−0.550824
$16$	0	0
$17$	−40.0000	−0.570672	−0.285336	−	0.958428i	$-0.592105\pi$
$17$	−40.0000	−0.570672	−0.285336	+	0.958428i	$0.592105\pi$
$18$	0	0
$19$	−72.0000	−0.869365	−0.434682	−	0.900584i	$-0.643139\pi$
$19$	−72.0000	−0.869365	−0.434682	+	0.900584i	$0.643139\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	176.000	1.59559	0.797794	−	0.602930i	$-0.206000\pi$
$23$	176.000	1.59559	0.797794	+	0.602930i	$0.206000\pi$
$24$	0	0
$25$	−109.000	−0.872000
$26$	0	0
$27$	80.0000	0.570222
$28$	0	0
$29$	162.000	1.03733	0.518666	−	0.854977i	$-0.326429\pi$
$29$	162.000	1.03733	0.518666	+	0.854977i	$0.326429\pi$
$30$	0	0
$31$	−16.0000	−0.0926995	−0.0463498	−	0.998925i	$-0.514759\pi$
$31$	−16.0000	−0.0926995	−0.0463498	+	0.998925i	$0.514759\pi$
$32$	0	0
$33$	320.000	1.68803
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−54.0000	−0.239934	−0.119967	−	0.992778i	$-0.538279\pi$
$37$	−54.0000	−0.239934	−0.119967	+	0.992778i	$0.538279\pi$
$38$	0	0
$39$	288.000	1.18248
$40$	0	0
$41$	472.000	1.79790	0.898951	−	0.438048i	$-0.144330\pi$
$41$	472.000	1.79790	0.898951	+	0.438048i	$0.144330\pi$
$42$	0	0
$43$	72.0000	0.255346	0.127673	−	0.991816i	$-0.459249\pi$
$43$	72.0000	0.255346	0.127673	+	0.991816i	$0.459249\pi$
$44$	0	0
$45$	−148.000	−0.490279
$46$	0	0
$47$	144.000	0.446906	0.223453	−	0.974715i	$-0.428267\pi$
$47$	144.000	0.446906	0.223453	+	0.974715i	$0.428267\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−320.000	−0.878607
$52$	0	0
$53$	486.000	1.25957	0.629785	−	0.776769i	$-0.283143\pi$
$53$	486.000	1.25957	0.629785	+	0.776769i	$0.283143\pi$
$54$	0	0
$55$	−160.000	−0.392262
$56$	0	0
$57$	−576.000	−1.33847
$58$	0	0
$59$	−648.000	−1.42987	−0.714936	−	0.699190i	$-0.753544\pi$
$59$	−648.000	−1.42987	−0.714936	+	0.699190i	$0.753544\pi$
$60$	0	0
$61$	684.000	1.43569	0.717846	−	0.696202i	$-0.245128\pi$
$61$	684.000	1.43569	0.717846	+	0.696202i	$0.245128\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−144.000	−0.274785
$66$	0	0
$67$	−216.000	−0.393860	−0.196930	−	0.980418i	$-0.563097\pi$
$67$	−216.000	−0.393860	−0.196930	+	0.980418i	$0.563097\pi$
$68$	0	0
$69$	1408.00	2.45657
$70$	0	0
$71$	608.000	1.01629	0.508143	−	0.861273i	$-0.330332\pi$
$71$	608.000	1.01629	0.508143	+	0.861273i	$0.330332\pi$
$72$	0	0
$73$	−1008.00	−1.61613	−0.808065	−	0.589093i	$-0.799485\pi$
$73$	−1008.00	−1.61613	−0.808065	+	0.589093i	$0.799485\pi$
$74$	0	0
$75$	−872.000	−1.34253
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1008.00	−1.43556	−0.717778	−	0.696272i	$-0.754841\pi$
$79$	−1008.00	−1.43556	−0.717778	+	0.696272i	$0.754841\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	0	0
$83$	216.000	0.285652	0.142826	−	0.989748i	$-0.454381\pi$
$83$	216.000	0.285652	0.142826	+	0.989748i	$0.454381\pi$
$84$	0	0
$85$	160.000	0.204170
$86$	0	0
$87$	1296.00	1.59708
$88$	0	0
$89$	1040.00	1.23865	0.619325	−	0.785135i	$-0.287406\pi$
$89$	1040.00	1.23865	0.619325	+	0.785135i	$0.287406\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−128.000	−0.142720
$94$	0	0
$95$	288.000	0.311033
$96$	0	0
$97$	936.000	0.979757	0.489878	−	0.871791i	$-0.337041\pi$
$97$	936.000	0.979757	0.489878	+	0.871791i	$0.337041\pi$
$98$	0	0
$99$	1480.00	1.50248
$100$	0	0
$101$	−148.000	−0.145807	−0.0729037	−	0.997339i	$-0.523227\pi$
$101$	−148.000	−0.145807	−0.0729037	+	0.997339i	$0.523227\pi$
$102$	0	0
$103$	1744.00	1.66836	0.834182	−	0.551490i	$-0.185940\pi$
$103$	1744.00	1.66836	0.834182	+	0.551490i	$0.185940\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−328.000	−0.296345	−0.148173	−	0.988962i	$-0.547339\pi$
$107$	−328.000	−0.296345	−0.148173	+	0.988962i	$0.547339\pi$
$108$	0	0
$109$	−1294.00	−1.13709	−0.568545	−	0.822652i	$-0.692493\pi$
$109$	−1294.00	−1.13709	−0.568545	+	0.822652i	$0.692493\pi$
$110$	0	0
$111$	−432.000	−0.369402
$112$	0	0
$113$	594.000	0.494503	0.247251	−	0.968951i	$-0.420473\pi$
$113$	594.000	0.494503	0.247251	+	0.968951i	$0.420473\pi$
$114$	0	0
$115$	−704.000	−0.570855
$116$	0	0
$117$	1332.00	1.05251
$118$	0	0
$119$	0	0
$120$	0	0
$121$	269.000	0.202104
$122$	0	0
$123$	3776.00	2.76805
$124$	0	0
$125$	936.000	0.669747
$126$	0	0
$127$	576.000	0.402455	0.201227	−	0.979545i	$-0.435507\pi$
$127$	576.000	0.402455	0.201227	+	0.979545i	$0.435507\pi$
$128$	0	0
$129$	576.000	0.393132
$130$	0	0
$131$	−2088.00	−1.39259	−0.696295	−	0.717755i	$-0.745170\pi$
$131$	−2088.00	−1.39259	−0.696295	+	0.717755i	$0.745170\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−320.000	−0.204009
$136$	0	0
$137$	2502.00	1.56029	0.780147	−	0.625596i	$-0.215144\pi$
$137$	2502.00	1.56029	0.780147	+	0.625596i	$0.215144\pi$
$138$	0	0
$139$	1960.00	1.19601	0.598004	−	0.801493i	$-0.295961\pi$
$139$	1960.00	1.19601	0.598004	+	0.801493i	$0.295961\pi$
$140$	0	0
$141$	1152.00	0.688056
$142$	0	0
$143$	1440.00	0.842090
$144$	0	0
$145$	−648.000	−0.371127
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3366.00	1.85069	0.925347	−	0.379121i	$-0.123774\pi$
$149$	3366.00	1.85069	0.925347	+	0.379121i	$0.123774\pi$
$150$	0	0
$151$	2736.00	1.47452	0.737260	−	0.675609i	$-0.236119\pi$
$151$	2736.00	1.47452	0.737260	+	0.675609i	$0.236119\pi$
$152$	0	0
$153$	−1480.00	−0.782032
$154$	0	0
$155$	64.0000	0.0331652
$156$	0	0
$157$	−684.000	−0.347702	−0.173851	−	0.984772i	$-0.555621\pi$
$157$	−684.000	−0.347702	−0.173851	+	0.984772i	$0.555621\pi$
$158$	0	0
$159$	3888.00	1.93924
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1656.00	0.795754	0.397877	−	0.917439i	$-0.369747\pi$
$163$	1656.00	0.795754	0.397877	+	0.917439i	$0.369747\pi$
$164$	0	0
$165$	−1280.00	−0.603926
$166$	0	0
$167$	−3600.00	−1.66812	−0.834061	−	0.551672i	$-0.813990\pi$
$167$	−3600.00	−1.66812	−0.834061	+	0.551672i	$0.813990\pi$
$168$	0	0
$169$	−901.000	−0.410105
$170$	0	0
$171$	−2664.00	−1.19135
$172$	0	0
$173$	−3676.00	−1.61550	−0.807749	−	0.589527i	$-0.799314\pi$
$173$	−3676.00	−1.61550	−0.807749	+	0.589527i	$0.799314\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5184.00	−2.20143
$178$	0	0
$179$	2984.00	1.24600	0.623002	−	0.782220i	$-0.285913\pi$
$179$	2984.00	1.24600	0.623002	+	0.782220i	$0.285913\pi$
$180$	0	0
$181$	−2988.00	−1.22705	−0.613526	−	0.789675i	$-0.710249\pi$
$181$	−2988.00	−1.22705	−0.613526	+	0.789675i	$0.710249\pi$
$182$	0	0
$183$	5472.00	2.21039
$184$	0	0
$185$	216.000	0.0858413
$186$	0	0
$187$	−1600.00	−0.625688
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1328.00	−0.503093	−0.251546	−	0.967845i	$-0.580939\pi$
$191$	−1328.00	−0.503093	−0.251546	+	0.967845i	$0.580939\pi$
$192$	0	0
$193$	306.000	0.114126	0.0570631	−	0.998371i	$-0.481826\pi$
$193$	306.000	0.114126	0.0570631	+	0.998371i	$0.481826\pi$
$194$	0	0
$195$	−1152.00	−0.423059
$196$	0	0
$197$	−1674.00	−0.605419	−0.302710	−	0.953083i	$-0.597891\pi$
$197$	−1674.00	−0.605419	−0.302710	+	0.953083i	$0.597891\pi$
$198$	0	0
$199$	2736.00	0.974623	0.487311	−	0.873228i	$-0.337978\pi$
$199$	2736.00	0.974623	0.487311	+	0.873228i	$0.337978\pi$
$200$	0	0
$201$	−1728.00	−0.606387
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1888.00	−0.643237
$206$	0	0
$207$	6512.00	2.18655
$208$	0	0
$209$	−2880.00	−0.953176
$210$	0	0
$211$	4104.00	1.33901	0.669505	−	0.742808i	$-0.266506\pi$
$211$	4104.00	1.33901	0.669505	+	0.742808i	$0.266506\pi$
$212$	0	0
$213$	4864.00	1.56468
$214$	0	0
$215$	−288.000	−0.0913555
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−8064.00	−2.48819
$220$	0	0
$221$	−1440.00	−0.438303
$222$	0	0
$223$	3168.00	0.951323	0.475661	−	0.879628i	$-0.342209\pi$
$223$	3168.00	0.951323	0.475661	+	0.879628i	$0.342209\pi$
$224$	0	0
$225$	−4033.00	−1.19496
$226$	0	0
$227$	−2088.00	−0.610508	−0.305254	−	0.952271i	$-0.598741\pi$
$227$	−2088.00	−0.610508	−0.305254	+	0.952271i	$0.598741\pi$
$228$	0	0
$229$	−3708.00	−1.07001	−0.535003	−	0.844850i	$-0.679690\pi$
$229$	−3708.00	−1.07001	−0.535003	+	0.844850i	$0.679690\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2070.00	0.582018	0.291009	−	0.956720i	$-0.406009\pi$
$233$	2070.00	0.582018	0.291009	+	0.956720i	$0.406009\pi$
$234$	0	0
$235$	−576.000	−0.159890
$236$	0	0
$237$	−8064.00	−2.21018
$238$	0	0
$239$	896.000	0.242500	0.121250	−	0.992622i	$-0.461310\pi$
$239$	896.000	0.242500	0.121250	+	0.992622i	$0.461310\pi$
$240$	0	0
$241$	3240.00	0.866003	0.433002	−	0.901393i	$-0.357454\pi$
$241$	3240.00	0.866003	0.433002	+	0.901393i	$0.357454\pi$
$242$	0	0
$243$	−5032.00	−1.32841
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2592.00	−0.667713
$248$	0	0
$249$	1728.00	0.439789
$250$	0	0
$251$	3816.00	0.959617	0.479808	−	0.877373i	$-0.340706\pi$
$251$	3816.00	0.959617	0.479808	+	0.877373i	$0.340706\pi$
$252$	0	0
$253$	7040.00	1.74941
$254$	0	0
$255$	1280.00	0.314340
$256$	0	0
$257$	−4064.00	−0.986402	−0.493201	−	0.869915i	$-0.664173\pi$
$257$	−4064.00	−0.986402	−0.493201	+	0.869915i	$0.664173\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	5994.00	1.42153
$262$	0	0
$263$	256.000	0.0600214	0.0300107	−	0.999550i	$-0.490446\pi$
$263$	256.000	0.0600214	0.0300107	+	0.999550i	$0.490446\pi$
$264$	0	0
$265$	−1944.00	−0.450638
$266$	0	0
$267$	8320.00	1.90703
$268$	0	0
$269$	−1868.00	−0.423398	−0.211699	−	0.977335i	$-0.567900\pi$
$269$	−1868.00	−0.423398	−0.211699	+	0.977335i	$0.567900\pi$
$270$	0	0
$271$	2432.00	0.545142	0.272571	−	0.962136i	$-0.412126\pi$
$271$	2432.00	0.545142	0.272571	+	0.962136i	$0.412126\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4360.00	−0.956065
$276$	0	0
$277$	1926.00	0.417769	0.208885	−	0.977940i	$-0.433017\pi$
$277$	1926.00	0.417769	0.208885	+	0.977940i	$0.433017\pi$
$278$	0	0
$279$	−592.000	−0.127033
$280$	0	0
$281$	−6714.00	−1.42535	−0.712676	−	0.701494i	$-0.752517\pi$
$281$	−6714.00	−1.42535	−0.712676	+	0.701494i	$0.752517\pi$
$282$	0	0
$283$	−72.0000	−0.0151235	−0.00756176	−	0.999971i	$-0.502407\pi$
$283$	−72.0000	−0.0151235	−0.00756176	+	0.999971i	$0.502407\pi$
$284$	0	0
$285$	2304.00	0.478867
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−3313.00	−0.674333
$290$	0	0
$291$	7488.00	1.50843
$292$	0	0
$293$	−2308.00	−0.460187	−0.230094	−	0.973169i	$-0.573903\pi$
$293$	−2308.00	−0.460187	−0.230094	+	0.973169i	$0.573903\pi$
$294$	0	0
$295$	2592.00	0.511566
$296$	0	0
$297$	3200.00	0.625195
$298$	0	0
$299$	6336.00	1.22549
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1184.00	−0.224485
$304$	0	0
$305$	−2736.00	−0.513649
$306$	0	0
$307$	−3816.00	−0.709416	−0.354708	−	0.934977i	$-0.615420\pi$
$307$	−3816.00	−0.709416	−0.354708	+	0.934977i	$0.615420\pi$
$308$	0	0
$309$	13952.0	2.56861
$310$	0	0
$311$	−4032.00	−0.735157	−0.367578	−	0.929993i	$-0.619813\pi$
$311$	−4032.00	−0.735157	−0.367578	+	0.929993i	$0.619813\pi$
$312$	0	0
$313$	−8712.00	−1.57326	−0.786632	−	0.617423i	$-0.788177\pi$
$313$	−8712.00	−1.57326	−0.786632	+	0.617423i	$0.788177\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1854.00	0.328489	0.164245	−	0.986420i	$-0.447481\pi$
$317$	1854.00	0.328489	0.164245	+	0.986420i	$0.447481\pi$
$318$	0	0
$319$	6480.00	1.13734
$320$	0	0
$321$	−2624.00	−0.456254
$322$	0	0
$323$	2880.00	0.496122
$324$	0	0
$325$	−3924.00	−0.669736
$326$	0	0
$327$	−10352.0	−1.75066
$328$	0	0
$329$	0	0
$330$	0	0
$331$	6696.00	1.11192	0.555960	−	0.831209i	$-0.312351\pi$
$331$	6696.00	1.11192	0.555960	+	0.831209i	$0.312351\pi$
$332$	0	0
$333$	−1998.00	−0.328798
$334$	0	0
$335$	864.000	0.140912
$336$	0	0
$337$	−3634.00	−0.587408	−0.293704	−	0.955896i	$-0.594888\pi$
$337$	−3634.00	−0.587408	−0.293704	+	0.955896i	$0.594888\pi$
$338$	0	0
$339$	4752.00	0.761337
$340$	0	0
$341$	−640.000	−0.101636
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−5632.00	−0.878889
$346$	0	0
$347$	7016.00	1.08541	0.542707	−	0.839922i	$-0.317399\pi$
$347$	7016.00	1.08541	0.542707	+	0.839922i	$0.317399\pi$
$348$	0	0
$349$	11268.0	1.72826	0.864129	−	0.503270i	$-0.167870\pi$
$349$	11268.0	1.72826	0.864129	+	0.503270i	$0.167870\pi$
$350$	0	0
$351$	2880.00	0.437957
$352$	0	0
$353$	−2768.00	−0.417353	−0.208677	−	0.977985i	$-0.566916\pi$
$353$	−2768.00	−0.417353	−0.208677	+	0.977985i	$0.566916\pi$
$354$	0	0
$355$	−2432.00	−0.363598
$356$	0	0
$357$	0	0
$358$	0	0
$359$	176.000	0.0258744	0.0129372	−	0.999916i	$-0.495882\pi$
$359$	176.000	0.0258744	0.0129372	+	0.999916i	$0.495882\pi$
$360$	0	0
$361$	−1675.00	−0.244205
$362$	0	0
$363$	2152.00	0.311159
$364$	0	0
$365$	4032.00	0.578204
$366$	0	0
$367$	−10656.0	−1.51564	−0.757818	−	0.652466i	$-0.773735\pi$
$367$	−10656.0	−1.51564	−0.757818	+	0.652466i	$0.773735\pi$
$368$	0	0
$369$	17464.0	2.46379
$370$	0	0
$371$	0	0
$372$	0	0
$373$	5222.00	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	5222.00	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	7488.00	1.03114
$376$	0	0
$377$	5832.00	0.796720
$378$	0	0
$379$	−7992.00	−1.08317	−0.541585	−	0.840646i	$-0.682176\pi$
$379$	−7992.00	−1.08317	−0.541585	+	0.840646i	$0.682176\pi$
$380$	0	0
$381$	4608.00	0.619619
$382$	0	0
$383$	4176.00	0.557137	0.278569	−	0.960416i	$-0.410140\pi$
$383$	4176.00	0.557137	0.278569	+	0.960416i	$0.410140\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2664.00	0.349919
$388$	0	0
$389$	−1494.00	−0.194727	−0.0973635	−	0.995249i	$-0.531041\pi$
$389$	−1494.00	−0.194727	−0.0973635	+	0.995249i	$0.531041\pi$
$390$	0	0
$391$	−7040.00	−0.910558
$392$	0	0
$393$	−16704.0	−2.14403
$394$	0	0
$395$	4032.00	0.513600
$396$	0	0
$397$	3348.00	0.423253	0.211626	−	0.977351i	$-0.432124\pi$
$397$	3348.00	0.423253	0.211626	+	0.977351i	$0.432124\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7650.00	−0.952675	−0.476338	−	0.879262i	$-0.658036\pi$
$401$	−7650.00	−0.952675	−0.476338	+	0.879262i	$0.658036\pi$
$402$	0	0
$403$	−576.000	−0.0711975
$404$	0	0
$405$	1436.00	0.176186
$406$	0	0
$407$	−2160.00	−0.263064
$408$	0	0
$409$	−4680.00	−0.565797	−0.282899	−	0.959150i	$-0.591296\pi$
$409$	−4680.00	−0.565797	−0.282899	+	0.959150i	$0.591296\pi$
$410$	0	0
$411$	20016.0	2.40223
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−864.000	−0.102198
$416$	0	0
$417$	15680.0	1.84137
$418$	0	0
$419$	−13176.0	−1.53625	−0.768126	−	0.640299i	$-0.778811\pi$
$419$	−13176.0	−1.53625	−0.768126	+	0.640299i	$0.778811\pi$
$420$	0	0
$421$	−8874.00	−1.02730	−0.513649	−	0.858001i	$-0.671707\pi$
$421$	−8874.00	−1.02730	−0.513649	+	0.858001i	$0.671707\pi$
$422$	0	0
$423$	5328.00	0.612426
$424$	0	0
$425$	4360.00	0.497626
$426$	0	0
$427$	0	0
$428$	0	0
$429$	11520.0	1.29648
$430$	0	0
$431$	−8096.00	−0.904804	−0.452402	−	0.891814i	$-0.649433\pi$
$431$	−8096.00	−0.904804	−0.452402	+	0.891814i	$0.649433\pi$
$432$	0	0
$433$	11016.0	1.22262	0.611311	−	0.791391i	$-0.290643\pi$
$433$	11016.0	1.22262	0.611311	+	0.791391i	$0.290643\pi$
$434$	0	0
$435$	−5184.00	−0.571388
$436$	0	0
$437$	−12672.0	−1.38715
$438$	0	0
$439$	−16128.0	−1.75341	−0.876706	−	0.481027i	$-0.840264\pi$
$439$	−16128.0	−1.75341	−0.876706	+	0.481027i	$0.840264\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5368.00	0.575714	0.287857	−	0.957673i	$-0.407057\pi$
$443$	5368.00	0.575714	0.287857	+	0.957673i	$0.407057\pi$
$444$	0	0
$445$	−4160.00	−0.443153
$446$	0	0
$447$	26928.0	2.84933
$448$	0	0
$449$	306.000	0.0321627	0.0160813	−	0.999871i	$-0.494881\pi$
$449$	306.000	0.0321627	0.0160813	+	0.999871i	$0.494881\pi$
$450$	0	0
$451$	18880.0	1.97123
$452$	0	0
$453$	21888.0	2.27017
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10118.0	−1.03567	−0.517834	−	0.855481i	$-0.673261\pi$
$457$	−10118.0	−1.03567	−0.517834	+	0.855481i	$0.673261\pi$
$458$	0	0
$459$	−3200.00	−0.325410
$460$	0	0
$461$	15412.0	1.55707	0.778534	−	0.627602i	$-0.215964\pi$
$461$	15412.0	1.55707	0.778534	+	0.627602i	$0.215964\pi$
$462$	0	0
$463$	18288.0	1.83567	0.917835	−	0.396962i	$-0.129935\pi$
$463$	18288.0	1.83567	0.917835	+	0.396962i	$0.129935\pi$
$464$	0	0
$465$	512.000	0.0510611
$466$	0	0
$467$	3672.00	0.363854	0.181927	−	0.983312i	$-0.441767\pi$
$467$	3672.00	0.363854	0.181927	+	0.983312i	$0.441767\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−5472.00	−0.535322
$472$	0	0
$473$	2880.00	0.279963
$474$	0	0
$475$	7848.00	0.758086
$476$	0	0
$477$	17982.0	1.72608
$478$	0	0
$479$	−12240.0	−1.16756	−0.583779	−	0.811913i	$-0.698426\pi$
$479$	−12240.0	−1.16756	−0.583779	+	0.811913i	$0.698426\pi$
$480$	0	0
$481$	−1944.00	−0.184280
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3744.00	−0.350528
$486$	0	0
$487$	−12816.0	−1.19250	−0.596251	−	0.802798i	$-0.703344\pi$
$487$	−12816.0	−1.19250	−0.596251	+	0.802798i	$0.703344\pi$
$488$	0	0
$489$	13248.0	1.22514
$490$	0	0
$491$	6520.00	0.599274	0.299637	−	0.954053i	$-0.403134\pi$
$491$	6520.00	0.599274	0.299637	+	0.954053i	$0.403134\pi$
$492$	0	0
$493$	−6480.00	−0.591977
$494$	0	0
$495$	−5920.00	−0.537544
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−15192.0	−1.36290	−0.681450	−	0.731864i	$-0.738650\pi$
$499$	−15192.0	−1.36290	−0.681450	+	0.731864i	$0.738650\pi$
$500$	0	0
$501$	−28800.0	−2.56824
$502$	0	0
$503$	−8928.00	−0.791411	−0.395706	−	0.918377i	$-0.629500\pi$
$503$	−8928.00	−0.791411	−0.395706	+	0.918377i	$0.629500\pi$
$504$	0	0
$505$	592.000	0.0521657
$506$	0	0
$507$	−7208.00	−0.631397
$508$	0	0
$509$	−13388.0	−1.16584	−0.582920	−	0.812529i	$-0.698090\pi$
$509$	−13388.0	−1.16584	−0.582920	+	0.812529i	$0.698090\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−5760.00	−0.495731
$514$	0	0
$515$	−6976.00	−0.596892
$516$	0	0
$517$	5760.00	0.489989
$518$	0	0
$519$	−29408.0	−2.48722
$520$	0	0
$521$	−13496.0	−1.13488	−0.567438	−	0.823416i	$-0.692065\pi$
$521$	−13496.0	−1.13488	−0.567438	+	0.823416i	$0.692065\pi$
$522$	0	0
$523$	13880.0	1.16048	0.580239	−	0.814446i	$-0.302959\pi$
$523$	13880.0	1.16048	0.580239	+	0.814446i	$0.302959\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	640.000	0.0529010
$528$	0	0
$529$	18809.0	1.54590
$530$	0	0
$531$	−23976.0	−1.95945
$532$	0	0
$533$	16992.0	1.38087
$534$	0	0
$535$	1312.00	0.106024
$536$	0	0
$537$	23872.0	1.91835
$538$	0	0
$539$	0	0
$540$	0	0
$541$	23742.0	1.88678	0.943390	−	0.331685i	$-0.107617\pi$
$541$	23742.0	1.88678	0.943390	+	0.331685i	$0.107617\pi$
$542$	0	0
$543$	−23904.0	−1.88917
$544$	0	0
$545$	5176.00	0.406817
$546$	0	0
$547$	−2952.00	−0.230747	−0.115373	−	0.993322i	$-0.536806\pi$
$547$	−2952.00	−0.230747	−0.115373	+	0.993322i	$0.536806\pi$
$548$	0	0
$549$	25308.0	1.96743
$550$	0	0
$551$	−11664.0	−0.901821
$552$	0	0
$553$	0	0
$554$	0	0
$555$	1728.00	0.132161
$556$	0	0
$557$	8622.00	0.655881	0.327941	−	0.944698i	$-0.393645\pi$
$557$	8622.00	0.655881	0.327941	+	0.944698i	$0.393645\pi$
$558$	0	0
$559$	2592.00	0.196118
$560$	0	0
$561$	−12800.0	−0.963309
$562$	0	0
$563$	12744.0	0.953989	0.476994	−	0.878906i	$-0.341726\pi$
$563$	12744.0	0.953989	0.476994	+	0.878906i	$0.341726\pi$
$564$	0	0
$565$	−2376.00	−0.176919
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−14922.0	−1.09941	−0.549704	−	0.835360i	$-0.685259\pi$
$569$	−14922.0	−1.09941	−0.549704	+	0.835360i	$0.685259\pi$
$570$	0	0
$571$	−21528.0	−1.57779	−0.788896	−	0.614527i	$-0.789347\pi$
$571$	−21528.0	−1.57779	−0.788896	+	0.614527i	$0.789347\pi$
$572$	0	0
$573$	−10624.0	−0.774562
$574$	0	0
$575$	−19184.0	−1.39135
$576$	0	0
$577$	−19728.0	−1.42338	−0.711688	−	0.702496i	$-0.752069\pi$
$577$	−19728.0	−1.42338	−0.711688	+	0.702496i	$0.752069\pi$
$578$	0	0
$579$	2448.00	0.175709
$580$	0	0
$581$	0	0
$582$	0	0
$583$	19440.0	1.38100
$584$	0	0
$585$	−5328.00	−0.376557
$586$	0	0
$587$	17928.0	1.26059	0.630296	−	0.776355i	$-0.282933\pi$
$587$	17928.0	1.26059	0.630296	+	0.776355i	$0.282933\pi$
$588$	0	0
$589$	1152.00	0.0805897
$590$	0	0
$591$	−13392.0	−0.932104
$592$	0	0
$593$	−8752.00	−0.606073	−0.303037	−	0.952979i	$-0.598000\pi$
$593$	−8752.00	−0.606073	−0.303037	+	0.952979i	$0.598000\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	21888.0	1.50053
$598$	0	0
$599$	16448.0	1.12195	0.560974	−	0.827833i	$-0.310427\pi$
$599$	16448.0	1.12195	0.560974	+	0.827833i	$0.310427\pi$
$600$	0	0
$601$	−11088.0	−0.752561	−0.376280	−	0.926506i	$-0.622797\pi$
$601$	−11088.0	−0.752561	−0.376280	+	0.926506i	$0.622797\pi$
$602$	0	0
$603$	−7992.00	−0.539734
$604$	0	0
$605$	−1076.00	−0.0723068
$606$	0	0
$607$	4032.00	0.269611	0.134805	−	0.990872i	$-0.456959\pi$
$607$	4032.00	0.269611	0.134805	+	0.990872i	$0.456959\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5184.00	0.343244
$612$	0	0
$613$	−8854.00	−0.583376	−0.291688	−	0.956513i	$-0.594217\pi$
$613$	−8854.00	−0.583376	−0.291688	+	0.956513i	$0.594217\pi$
$614$	0	0
$615$	−15104.0	−0.990329
$616$	0	0
$617$	−1962.00	−0.128018	−0.0640090	−	0.997949i	$-0.520389\pi$
$617$	−1962.00	−0.128018	−0.0640090	+	0.997949i	$0.520389\pi$
$618$	0	0
$619$	−1384.00	−0.0898670	−0.0449335	−	0.998990i	$-0.514308\pi$
$619$	−1384.00	−0.0898670	−0.0449335	+	0.998990i	$0.514308\pi$
$620$	0	0
$621$	14080.0	0.909840
$622$	0	0
$623$	0	0
$624$	0	0
$625$	9881.00	0.632384
$626$	0	0
$627$	−23040.0	−1.46751
$628$	0	0
$629$	2160.00	0.136923
$630$	0	0
$631$	8352.00	0.526922	0.263461	−	0.964670i	$-0.415136\pi$
$631$	8352.00	0.526922	0.263461	+	0.964670i	$0.415136\pi$
$632$	0	0
$633$	32832.0	2.06154
$634$	0	0
$635$	−2304.00	−0.143987
$636$	0	0
$637$	0	0
$638$	0	0
$639$	22496.0	1.39269
$640$	0	0
$641$	−4338.00	−0.267302	−0.133651	−	0.991028i	$-0.542670\pi$
$641$	−4338.00	−0.267302	−0.133651	+	0.991028i	$0.542670\pi$
$642$	0	0
$643$	−6264.00	−0.384180	−0.192090	−	0.981377i	$-0.561527\pi$
$643$	−6264.00	−0.384180	−0.192090	+	0.981377i	$0.561527\pi$
$644$	0	0
$645$	−2304.00	−0.140651
$646$	0	0
$647$	−13392.0	−0.813746	−0.406873	−	0.913485i	$-0.633381\pi$
$647$	−13392.0	−0.813746	−0.406873	+	0.913485i	$0.633381\pi$
$648$	0	0
$649$	−25920.0	−1.56772
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−13230.0	−0.792848	−0.396424	−	0.918067i	$-0.629749\pi$
$653$	−13230.0	−0.792848	−0.396424	+	0.918067i	$0.629749\pi$
$654$	0	0
$655$	8352.00	0.498228
$656$	0	0
$657$	−37296.0	−2.21470
$658$	0	0
$659$	−29704.0	−1.75585	−0.877924	−	0.478800i	$-0.841072\pi$
$659$	−29704.0	−1.75585	−0.877924	+	0.478800i	$0.841072\pi$
$660$	0	0
$661$	−10548.0	−0.620680	−0.310340	−	0.950626i	$-0.600443\pi$
$661$	−10548.0	−0.620680	−0.310340	+	0.950626i	$0.600443\pi$
$662$	0	0
$663$	−11520.0	−0.674811
$664$	0	0
$665$	0	0
$666$	0	0
$667$	28512.0	1.65516
$668$	0	0
$669$	25344.0	1.46466
$670$	0	0
$671$	27360.0	1.57410
$672$	0	0
$673$	−20178.0	−1.15573	−0.577864	−	0.816133i	$-0.696114\pi$
$673$	−20178.0	−1.15573	−0.577864	+	0.816133i	$0.696114\pi$
$674$	0	0
$675$	−8720.00	−0.497234
$676$	0	0
$677$	−24908.0	−1.41402	−0.707010	−	0.707203i	$-0.749957\pi$
$677$	−24908.0	−1.41402	−0.707010	+	0.707203i	$0.749957\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−16704.0	−0.939939
$682$	0	0
$683$	−8456.00	−0.473733	−0.236867	−	0.971542i	$-0.576120\pi$
$683$	−8456.00	−0.473733	−0.236867	+	0.971542i	$0.576120\pi$
$684$	0	0
$685$	−10008.0	−0.558228
$686$	0	0
$687$	−29664.0	−1.64738
$688$	0	0
$689$	17496.0	0.967409
$690$	0	0
$691$	−24104.0	−1.32700	−0.663502	−	0.748175i	$-0.730930\pi$
$691$	−24104.0	−1.32700	−0.663502	+	0.748175i	$0.730930\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7840.00	−0.427897
$696$	0	0
$697$	−18880.0	−1.02601
$698$	0	0
$699$	16560.0	0.896075
$700$	0	0
$701$	9954.00	0.536316	0.268158	−	0.963375i	$-0.413585\pi$
$701$	9954.00	0.536316	0.268158	+	0.963375i	$0.413585\pi$
$702$	0	0
$703$	3888.00	0.208590
$704$	0	0
$705$	−4608.00	−0.246166
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−28694.0	−1.51992	−0.759962	−	0.649968i	$-0.774782\pi$
$709$	−28694.0	−1.51992	−0.759962	+	0.649968i	$0.774782\pi$
$710$	0	0
$711$	−37296.0	−1.96724
$712$	0	0
$713$	−2816.00	−0.147910
$714$	0	0
$715$	−5760.00	−0.301275
$716$	0	0
$717$	7168.00	0.373353
$718$	0	0
$719$	28080.0	1.45648	0.728239	−	0.685324i	$-0.240339\pi$
$719$	28080.0	1.45648	0.728239	+	0.685324i	$0.240339\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	25920.0	1.33330
$724$	0	0
$725$	−17658.0	−0.904554
$726$	0	0
$727$	−17584.0	−0.897049	−0.448524	−	0.893771i	$-0.648050\pi$
$727$	−17584.0	−0.897049	−0.448524	+	0.893771i	$0.648050\pi$
$728$	0	0
$729$	−30563.0	−1.55276
$730$	0	0
$731$	−2880.00	−0.145719
$732$	0	0
$733$	−28260.0	−1.42402	−0.712010	−	0.702169i	$-0.752215\pi$
$733$	−28260.0	−1.42402	−0.712010	+	0.702169i	$0.752215\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8640.00	−0.431830
$738$	0	0
$739$	−1224.00	−0.0609277	−0.0304638	−	0.999536i	$-0.509698\pi$
$739$	−1224.00	−0.0609277	−0.0304638	+	0.999536i	$0.509698\pi$
$740$	0	0
$741$	−20736.0	−1.02801
$742$	0	0
$743$	−8240.00	−0.406859	−0.203430	−	0.979090i	$-0.565209\pi$
$743$	−8240.00	−0.406859	−0.203430	+	0.979090i	$0.565209\pi$
$744$	0	0
$745$	−13464.0	−0.662125
$746$	0	0
$747$	7992.00	0.391448
$748$	0	0
$749$	0	0
$750$	0	0
$751$	16704.0	0.811635	0.405817	−	0.913954i	$-0.366987\pi$
$751$	16704.0	0.811635	0.405817	+	0.913954i	$0.366987\pi$
$752$	0	0
$753$	30528.0	1.47743
$754$	0	0
$755$	−10944.0	−0.527540
$756$	0	0
$757$	30778.0	1.47774	0.738868	−	0.673851i	$-0.235361\pi$
$757$	30778.0	1.47774	0.738868	+	0.673851i	$0.235361\pi$
$758$	0	0
$759$	56320.0	2.69339
$760$	0	0
$761$	−29624.0	−1.41113	−0.705564	−	0.708646i	$-0.749306\pi$
$761$	−29624.0	−1.41113	−0.705564	+	0.708646i	$0.749306\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	5920.00	0.279788
$766$	0	0
$767$	−23328.0	−1.09821
$768$	0	0
$769$	−11016.0	−0.516576	−0.258288	−	0.966068i	$-0.583158\pi$
$769$	−11016.0	−0.516576	−0.258288	+	0.966068i	$0.583158\pi$
$770$	0	0
$771$	−32512.0	−1.51867
$772$	0	0
$773$	940.000	0.0437380	0.0218690	−	0.999761i	$-0.493038\pi$
$773$	940.000	0.0437380	0.0218690	+	0.999761i	$0.493038\pi$
$774$	0	0
$775$	1744.00	0.0808340
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−33984.0	−1.56303
$780$	0	0
$781$	24320.0	1.11426
$782$	0	0
$783$	12960.0	0.591510
$784$	0	0
$785$	2736.00	0.124397
$786$	0	0
$787$	−1960.00	−0.0887757	−0.0443878	−	0.999014i	$-0.514134\pi$
$787$	−1960.00	−0.0887757	−0.0443878	+	0.999014i	$0.514134\pi$
$788$	0	0
$789$	2048.00	0.0924090
$790$	0	0
$791$	0	0
$792$	0	0
$793$	24624.0	1.10268
$794$	0	0
$795$	−15552.0	−0.693802
$796$	0	0
$797$	−37868.0	−1.68300	−0.841501	−	0.540255i	$-0.818328\pi$
$797$	−37868.0	−1.68300	−0.841501	+	0.540255i	$0.818328\pi$
$798$	0	0
$799$	−5760.00	−0.255036
$800$	0	0
$801$	38480.0	1.69741
$802$	0	0
$803$	−40320.0	−1.77193
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−14944.0	−0.651863
$808$	0	0
$809$	−28134.0	−1.22267	−0.611334	−	0.791373i	$-0.709367\pi$
$809$	−28134.0	−1.22267	−0.611334	+	0.791373i	$0.709367\pi$
$810$	0	0
$811$	25112.0	1.08730	0.543651	−	0.839312i	$-0.317042\pi$
$811$	25112.0	1.08730	0.543651	+	0.839312i	$0.317042\pi$
$812$	0	0
$813$	19456.0	0.839301
$814$	0	0
$815$	−6624.00	−0.284698
$816$	0	0
$817$	−5184.00	−0.221989
$818$	0	0
$819$	0	0
$820$	0	0
$821$	43686.0	1.85707	0.928533	−	0.371249i	$-0.121070\pi$
$821$	43686.0	1.85707	0.928533	+	0.371249i	$0.121070\pi$
$822$	0	0
$823$	−43776.0	−1.85411	−0.927057	−	0.374921i	$-0.877670\pi$
$823$	−43776.0	−1.85411	−0.927057	+	0.374921i	$0.877670\pi$
$824$	0	0
$825$	−34880.0	−1.47196
$826$	0	0
$827$	1336.00	0.0561757	0.0280878	−	0.999605i	$-0.491058\pi$
$827$	1336.00	0.0561757	0.0280878	+	0.999605i	$0.491058\pi$
$828$	0	0
$829$	15372.0	0.644019	0.322009	−	0.946736i	$-0.395642\pi$
$829$	15372.0	0.644019	0.322009	+	0.946736i	$0.395642\pi$
$830$	0	0
$831$	15408.0	0.643198
$832$	0	0
$833$	0	0
$834$	0	0
$835$	14400.0	0.596805
$836$	0	0
$837$	−1280.00	−0.0528593
$838$	0	0
$839$	−40752.0	−1.67690	−0.838448	−	0.544982i	$-0.816536\pi$
$839$	−40752.0	−1.67690	−0.838448	+	0.544982i	$0.816536\pi$
$840$	0	0
$841$	1855.00	0.0760589
$842$	0	0
$843$	−53712.0	−2.19447
$844$	0	0
$845$	3604.00	0.146724
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−576.000	−0.0232842
$850$	0	0
$851$	−9504.00	−0.382835
$852$	0	0
$853$	12132.0	0.486977	0.243489	−	0.969904i	$-0.421708\pi$
$853$	12132.0	0.486977	0.243489	+	0.969904i	$0.421708\pi$
$854$	0	0
$855$	10656.0	0.426231
$856$	0	0
$857$	−36392.0	−1.45056	−0.725278	−	0.688456i	$-0.758289\pi$
$857$	−36392.0	−1.45056	−0.725278	+	0.688456i	$0.758289\pi$
$858$	0	0
$859$	10440.0	0.414678	0.207339	−	0.978269i	$-0.433520\pi$
$859$	10440.0	0.414678	0.207339	+	0.978269i	$0.433520\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−13360.0	−0.526975	−0.263488	−	0.964663i	$-0.584873\pi$
$863$	−13360.0	−0.526975	−0.263488	+	0.964663i	$0.584873\pi$
$864$	0	0
$865$	14704.0	0.577978
$866$	0	0
$867$	−26504.0	−1.03820
$868$	0	0
$869$	−40320.0	−1.57395
$870$	0	0
$871$	−7776.00	−0.302503
$872$	0	0
$873$	34632.0	1.34263
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−11086.0	−0.426850	−0.213425	−	0.976959i	$-0.568462\pi$
$877$	−11086.0	−0.426850	−0.213425	+	0.976959i	$0.568462\pi$
$878$	0	0
$879$	−18464.0	−0.708504
$880$	0	0
$881$	36832.0	1.40851	0.704257	−	0.709945i	$-0.251280\pi$
$881$	36832.0	1.40851	0.704257	+	0.709945i	$0.251280\pi$
$882$	0	0
$883$	49608.0	1.89065	0.945324	−	0.326133i	$-0.105746\pi$
$883$	49608.0	1.89065	0.945324	+	0.326133i	$0.105746\pi$
$884$	0	0
$885$	20736.0	0.787608
$886$	0	0
$887$	15696.0	0.594160	0.297080	−	0.954853i	$-0.403987\pi$
$887$	15696.0	0.594160	0.297080	+	0.954853i	$0.403987\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−14360.0	−0.539931
$892$	0	0
$893$	−10368.0	−0.388524
$894$	0	0
$895$	−11936.0	−0.445784
$896$	0	0
$897$	50688.0	1.88676
$898$	0	0
$899$	−2592.00	−0.0961602
$900$	0	0
$901$	−19440.0	−0.718802
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11952.0	0.439003
$906$	0	0
$907$	−2664.00	−0.0975266	−0.0487633	−	0.998810i	$-0.515528\pi$
$907$	−2664.00	−0.0975266	−0.0487633	+	0.998810i	$0.515528\pi$
$908$	0	0
$909$	−5476.00	−0.199810
$910$	0	0
$911$	−4864.00	−0.176895	−0.0884476	−	0.996081i	$-0.528191\pi$
$911$	−4864.00	−0.176895	−0.0884476	+	0.996081i	$0.528191\pi$
$912$	0	0
$913$	8640.00	0.313190
$914$	0	0
$915$	−21888.0	−0.790814
$916$	0	0
$917$	0	0
$918$	0	0
$919$	12384.0	0.444516	0.222258	−	0.974988i	$-0.428657\pi$
$919$	12384.0	0.444516	0.222258	+	0.974988i	$0.428657\pi$
$920$	0	0
$921$	−30528.0	−1.09222
$922$	0	0
$923$	21888.0	0.780555
$924$	0	0
$925$	5886.00	0.209222
$926$	0	0
$927$	64528.0	2.28628
$928$	0	0
$929$	−33224.0	−1.17335	−0.586676	−	0.809822i	$-0.699564\pi$
$929$	−33224.0	−1.17335	−0.586676	+	0.809822i	$0.699564\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−32256.0	−1.13185
$934$	0	0
$935$	6400.00	0.223853
$936$	0	0
$937$	11088.0	0.386584	0.193292	−	0.981141i	$-0.438084\pi$
$937$	11088.0	0.386584	0.193292	+	0.981141i	$0.438084\pi$
$938$	0	0
$939$	−69696.0	−2.42220
$940$	0	0
$941$	−22324.0	−0.773370	−0.386685	−	0.922212i	$-0.626380\pi$
$941$	−22324.0	−0.773370	−0.386685	+	0.922212i	$0.626380\pi$
$942$	0	0
$943$	83072.0	2.86871
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23800.0	0.816680	0.408340	−	0.912830i	$-0.366108\pi$
$947$	23800.0	0.816680	0.408340	+	0.912830i	$0.366108\pi$
$948$	0	0
$949$	−36288.0	−1.24126
$950$	0	0
$951$	14832.0	0.505742
$952$	0	0
$953$	−2358.00	−0.0801502	−0.0400751	−	0.999197i	$-0.512760\pi$
$953$	−2358.00	−0.0801502	−0.0400751	+	0.999197i	$0.512760\pi$
$954$	0	0
$955$	5312.00	0.179992
$956$	0	0
$957$	51840.0	1.75104
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29535.0	−0.991407
$962$	0	0
$963$	−12136.0	−0.406103
$964$	0	0
$965$	−1224.00	−0.0408310
$966$	0	0
$967$	−8496.00	−0.282537	−0.141268	−	0.989971i	$-0.545118\pi$
$967$	−8496.00	−0.282537	−0.141268	+	0.989971i	$0.545118\pi$
$968$	0	0
$969$	23040.0	0.763830
$970$	0	0
$971$	−16200.0	−0.535410	−0.267705	−	0.963501i	$-0.586265\pi$
$971$	−16200.0	−0.535410	−0.267705	+	0.963501i	$0.586265\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−31392.0	−1.03113
$976$	0	0
$977$	21726.0	0.711439	0.355720	−	0.934593i	$-0.384236\pi$
$977$	21726.0	0.711439	0.355720	+	0.934593i	$0.384236\pi$
$978$	0	0
$979$	41600.0	1.35806
$980$	0	0
$981$	−47878.0	−1.55823
$982$	0	0
$983$	8208.00	0.266322	0.133161	−	0.991094i	$-0.457487\pi$
$983$	8208.00	0.266322	0.133161	+	0.991094i	$0.457487\pi$
$984$	0	0
$985$	6696.00	0.216601
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12672.0	0.407428
$990$	0	0
$991$	18000.0	0.576982	0.288491	−	0.957483i	$-0.406847\pi$
$991$	18000.0	0.576982	0.288491	+	0.957483i	$0.406847\pi$
$992$	0	0
$993$	53568.0	1.71191
$994$	0	0
$995$	−10944.0	−0.348692
$996$	0	0
$997$	−57204.0	−1.81712	−0.908560	−	0.417754i	$-0.862817\pi$
$997$	−57204.0	−1.81712	−0.908560	+	0.417754i	$0.862817\pi$
$998$	0	0
$999$	−4320.00	−0.136816

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.4.a.m.1.1	yes	1
4.3	odd	2		1568.4.a.a.1.1	✓	1
7.6	odd	2		1568.4.a.b.1.1	yes	1
28.27	even	2		1568.4.a.n.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1568.4.a.a.1.1	✓	1	4.3	odd	2
1568.4.a.b.1.1	yes	1	7.6	odd	2
1568.4.a.m.1.1	yes	1	1.1	even	1	trivial
1568.4.a.n.1.1	yes	1	28.27	even	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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1568.a (trivial)

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