LMFDB - Embedded newform 528.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,4,Mod(1,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, 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("528.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	528.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:	$31.1530084830$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 132)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	528.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} +10.0000 q^{5} -8.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} +10.0000 q^{5} -8.00000 q^{7} +9.00000 q^{9} +11.0000 q^{11} +18.0000 q^{13} -30.0000 q^{15} +46.0000 q^{17} -40.0000 q^{19} +24.0000 q^{21} -44.0000 q^{23} -25.0000 q^{25} -27.0000 q^{27} +186.000 q^{29} +72.0000 q^{31} -33.0000 q^{33} -80.0000 q^{35} -114.000 q^{37} -54.0000 q^{39} +174.000 q^{41} +416.000 q^{43} +90.0000 q^{45} +156.000 q^{47} -279.000 q^{49} -138.000 q^{51} -62.0000 q^{53} +110.000 q^{55} +120.000 q^{57} +348.000 q^{59} -446.000 q^{61} -72.0000 q^{63} +180.000 q^{65} +956.000 q^{67} +132.000 q^{69} +444.000 q^{71} +306.000 q^{73} +75.0000 q^{75} -88.0000 q^{77} +664.000 q^{79} +81.0000 q^{81} +124.000 q^{83} +460.000 q^{85} -558.000 q^{87} +602.000 q^{89} -144.000 q^{91} -216.000 q^{93} -400.000 q^{95} +1522.00 q^{97} +99.0000 q^{99} +O(q^{100})$

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
$5$	10.0000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	10.0000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−8.00000	−0.431959	−0.215980	−	0.976398i	$-0.569295\pi$
$7$	−8.00000	−0.431959	−0.215980	+	0.976398i	$0.569295\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	18.0000	0.384023	0.192012	−	0.981393i	$-0.438499\pi$
$13$	18.0000	0.384023	0.192012	+	0.981393i	$0.438499\pi$
$14$	0	0
$15$	−30.0000	−0.516398
$16$	0	0
$17$	46.0000	0.656273	0.328136	−	0.944630i	$-0.393579\pi$
$17$	46.0000	0.656273	0.328136	+	0.944630i	$0.393579\pi$
$18$	0	0
$19$	−40.0000	−0.482980	−0.241490	−	0.970403i	$-0.577636\pi$
$19$	−40.0000	−0.482980	−0.241490	+	0.970403i	$0.577636\pi$
$20$	0	0
$21$	24.0000	0.249392
$22$	0	0
$23$	−44.0000	−0.398897	−0.199449	−	0.979908i	$-0.563915\pi$
$23$	−44.0000	−0.398897	−0.199449	+	0.979908i	$0.563915\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	186.000	1.19101	0.595506	−	0.803351i	$-0.296952\pi$
$29$	186.000	1.19101	0.595506	+	0.803351i	$0.296952\pi$
$30$	0	0
$31$	72.0000	0.417148	0.208574	−	0.978007i	$-0.433118\pi$
$31$	72.0000	0.417148	0.208574	+	0.978007i	$0.433118\pi$
$32$	0	0
$33$	−33.0000	−0.174078
$34$	0	0
$35$	−80.0000	−0.386356
$36$	0	0
$37$	−114.000	−0.506527	−0.253263	−	0.967397i	$-0.581504\pi$
$37$	−114.000	−0.506527	−0.253263	+	0.967397i	$0.581504\pi$
$38$	0	0
$39$	−54.0000	−0.221716
$40$	0	0
$41$	174.000	0.662786	0.331393	−	0.943493i	$-0.392481\pi$
$41$	174.000	0.662786	0.331393	+	0.943493i	$0.392481\pi$
$42$	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
$45$	90.0000	0.298142
$46$	0	0
$47$	156.000	0.484148	0.242074	−	0.970258i	$-0.422172\pi$
$47$	156.000	0.484148	0.242074	+	0.970258i	$0.422172\pi$
$48$	0	0
$49$	−279.000	−0.813411
$50$	0	0
$51$	−138.000	−0.378899
$52$	0	0
$53$	−62.0000	−0.160686	−0.0803430	−	0.996767i	$-0.525602\pi$
$53$	−62.0000	−0.160686	−0.0803430	+	0.996767i	$0.525602\pi$
$54$	0	0
$55$	110.000	0.269680
$56$	0	0
$57$	120.000	0.278849
$58$	0	0
$59$	348.000	0.767894	0.383947	−	0.923355i	$-0.374565\pi$
$59$	348.000	0.767894	0.383947	+	0.923355i	$0.374565\pi$
$60$	0	0
$61$	−446.000	−0.936138	−0.468069	−	0.883692i	$-0.655050\pi$
$61$	−446.000	−0.936138	−0.468069	+	0.883692i	$0.655050\pi$
$62$	0	0
$63$	−72.0000	−0.143986
$64$	0	0
$65$	180.000	0.343481
$66$	0	0
$67$	956.000	1.74319	0.871597	−	0.490223i	$-0.163085\pi$
$67$	956.000	1.74319	0.871597	+	0.490223i	$0.163085\pi$
$68$	0	0
$69$	132.000	0.230303
$70$	0	0
$71$	444.000	0.742156	0.371078	−	0.928602i	$-0.378988\pi$
$71$	444.000	0.742156	0.371078	+	0.928602i	$0.378988\pi$
$72$	0	0
$73$	306.000	0.490611	0.245305	−	0.969446i	$-0.421112\pi$
$73$	306.000	0.490611	0.245305	+	0.969446i	$0.421112\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	−88.0000	−0.130241
$78$	0	0
$79$	664.000	0.945644	0.472822	−	0.881158i	$-0.343235\pi$
$79$	664.000	0.945644	0.472822	+	0.881158i	$0.343235\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	124.000	0.163985	0.0819926	−	0.996633i	$-0.473872\pi$
$83$	124.000	0.163985	0.0819926	+	0.996633i	$0.473872\pi$
$84$	0	0
$85$	460.000	0.586988
$86$	0	0
$87$	−558.000	−0.687631
$88$	0	0
$89$	602.000	0.716987	0.358494	−	0.933532i	$-0.383290\pi$
$89$	602.000	0.716987	0.358494	+	0.933532i	$0.383290\pi$
$90$	0	0
$91$	−144.000	−0.165882
$92$	0	0
$93$	−216.000	−0.240840
$94$	0	0
$95$	−400.000	−0.431991
$96$	0	0
$97$	1522.00	1.59315	0.796576	−	0.604539i	$-0.206643\pi$
$97$	1522.00	1.59315	0.796576	+	0.604539i	$0.206643\pi$
$98$	0	0
$99$	99.0000	0.100504
$100$	0	0
$101$	1090.00	1.07385	0.536926	−	0.843629i	$-0.319585\pi$
$101$	1090.00	1.07385	0.536926	+	0.843629i	$0.319585\pi$
$102$	0	0
$103$	1392.00	1.33163	0.665815	−	0.746117i	$-0.268084\pi$
$103$	1392.00	1.33163	0.665815	+	0.746117i	$0.268084\pi$
$104$	0	0
$105$	240.000	0.223063
$106$	0	0
$107$	−308.000	−0.278276	−0.139138	−	0.990273i	$-0.544433\pi$
$107$	−308.000	−0.278276	−0.139138	+	0.990273i	$0.544433\pi$
$108$	0	0
$109$	978.000	0.859407	0.429704	−	0.902970i	$-0.358618\pi$
$109$	978.000	0.859407	0.429704	+	0.902970i	$0.358618\pi$
$110$	0	0
$111$	342.000	0.292443
$112$	0	0
$113$	1162.00	0.967361	0.483680	−	0.875245i	$-0.339300\pi$
$113$	1162.00	0.967361	0.483680	+	0.875245i	$0.339300\pi$
$114$	0	0
$115$	−440.000	−0.356784
$116$	0	0
$117$	162.000	0.128008
$118$	0	0
$119$	−368.000	−0.283483
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−522.000	−0.382660
$124$	0	0
$125$	−1500.00	−1.07331
$126$	0	0
$127$	984.000	0.687527	0.343763	−	0.939056i	$-0.388298\pi$
$127$	984.000	0.687527	0.343763	+	0.939056i	$0.388298\pi$
$128$	0	0
$129$	−1248.00	−0.851785
$130$	0	0
$131$	−2012.00	−1.34190	−0.670951	−	0.741501i	$-0.734114\pi$
$131$	−2012.00	−1.34190	−0.670951	+	0.741501i	$0.734114\pi$
$132$	0	0
$133$	320.000	0.208628
$134$	0	0
$135$	−270.000	−0.172133
$136$	0	0
$137$	−1286.00	−0.801974	−0.400987	−	0.916084i	$-0.631333\pi$
$137$	−1286.00	−0.801974	−0.400987	+	0.916084i	$0.631333\pi$
$138$	0	0
$139$	−464.000	−0.283136	−0.141568	−	0.989929i	$-0.545214\pi$
$139$	−464.000	−0.283136	−0.141568	+	0.989929i	$0.545214\pi$
$140$	0	0
$141$	−468.000	−0.279523
$142$	0	0
$143$	198.000	0.115787
$144$	0	0
$145$	1860.00	1.06527
$146$	0	0
$147$	837.000	0.469623
$148$	0	0
$149$	−2894.00	−1.59118	−0.795590	−	0.605836i	$-0.792839\pi$
$149$	−2894.00	−1.59118	−0.795590	+	0.605836i	$0.792839\pi$
$150$	0	0
$151$	−976.000	−0.525998	−0.262999	−	0.964796i	$-0.584712\pi$
$151$	−976.000	−0.525998	−0.262999	+	0.964796i	$0.584712\pi$
$152$	0	0
$153$	414.000	0.218758
$154$	0	0
$155$	720.000	0.373108
$156$	0	0
$157$	1646.00	0.836720	0.418360	−	0.908281i	$-0.362605\pi$
$157$	1646.00	0.836720	0.418360	+	0.908281i	$0.362605\pi$
$158$	0	0
$159$	186.000	0.0927721
$160$	0	0
$161$	352.000	0.172307
$162$	0	0
$163$	−3268.00	−1.57037	−0.785183	−	0.619264i	$-0.787431\pi$
$163$	−3268.00	−1.57037	−0.785183	+	0.619264i	$0.787431\pi$
$164$	0	0
$165$	−330.000	−0.155700
$166$	0	0
$167$	−1608.00	−0.745094	−0.372547	−	0.928013i	$-0.621516\pi$
$167$	−1608.00	−0.745094	−0.372547	+	0.928013i	$0.621516\pi$
$168$	0	0
$169$	−1873.00	−0.852526
$170$	0	0
$171$	−360.000	−0.160993
$172$	0	0
$173$	−4070.00	−1.78865	−0.894325	−	0.447418i	$-0.852343\pi$
$173$	−4070.00	−1.78865	−0.894325	+	0.447418i	$0.852343\pi$
$174$	0	0
$175$	200.000	0.0863919
$176$	0	0
$177$	−1044.00	−0.443344
$178$	0	0
$179$	−52.0000	−0.0217132	−0.0108566	−	0.999941i	$-0.503456\pi$
$179$	−52.0000	−0.0217132	−0.0108566	+	0.999941i	$0.503456\pi$
$180$	0	0
$181$	1798.00	0.738366	0.369183	−	0.929357i	$-0.379638\pi$
$181$	1798.00	0.738366	0.369183	+	0.929357i	$0.379638\pi$
$182$	0	0
$183$	1338.00	0.540480
$184$	0	0
$185$	−1140.00	−0.453051
$186$	0	0
$187$	506.000	0.197874
$188$	0	0
$189$	216.000	0.0831306
$190$	0	0
$191$	3852.00	1.45927	0.729636	−	0.683836i	$-0.239690\pi$
$191$	3852.00	1.45927	0.729636	+	0.683836i	$0.239690\pi$
$192$	0	0
$193$	−1958.00	−0.730259	−0.365129	−	0.930957i	$-0.618975\pi$
$193$	−1958.00	−0.730259	−0.365129	+	0.930957i	$0.618975\pi$
$194$	0	0
$195$	−540.000	−0.198309
$196$	0	0
$197$	−1630.00	−0.589506	−0.294753	−	0.955573i	$-0.595237\pi$
$197$	−1630.00	−0.589506	−0.294753	+	0.955573i	$0.595237\pi$
$198$	0	0
$199$	4504.00	1.60442	0.802211	−	0.597040i	$-0.203657\pi$
$199$	4504.00	1.60442	0.802211	+	0.597040i	$0.203657\pi$
$200$	0	0
$201$	−2868.00	−1.00643
$202$	0	0
$203$	−1488.00	−0.514469
$204$	0	0
$205$	1740.00	0.592814
$206$	0	0
$207$	−396.000	−0.132966
$208$	0	0
$209$	−440.000	−0.145624
$210$	0	0
$211$	−3776.00	−1.23199	−0.615997	−	0.787749i	$-0.711246\pi$
$211$	−3776.00	−1.23199	−0.615997	+	0.787749i	$0.711246\pi$
$212$	0	0
$213$	−1332.00	−0.428484
$214$	0	0
$215$	4160.00	1.31958
$216$	0	0
$217$	−576.000	−0.180191
$218$	0	0
$219$	−918.000	−0.283254
$220$	0	0
$221$	828.000	0.252024
$222$	0	0
$223$	1728.00	0.518903	0.259452	−	0.965756i	$-0.416458\pi$
$223$	1728.00	0.518903	0.259452	+	0.965756i	$0.416458\pi$
$224$	0	0
$225$	−225.000	−0.0666667
$226$	0	0
$227$	−4716.00	−1.37891	−0.689454	−	0.724330i	$-0.742149\pi$
$227$	−4716.00	−1.37891	−0.689454	+	0.724330i	$0.742149\pi$
$228$	0	0
$229$	1766.00	0.509609	0.254805	−	0.966993i	$-0.417989\pi$
$229$	1766.00	0.509609	0.254805	+	0.966993i	$0.417989\pi$
$230$	0	0
$231$	264.000	0.0751945
$232$	0	0
$233$	4382.00	1.23208	0.616039	−	0.787715i	$-0.288736\pi$
$233$	4382.00	1.23208	0.616039	+	0.787715i	$0.288736\pi$
$234$	0	0
$235$	1560.00	0.433035
$236$	0	0
$237$	−1992.00	−0.545968
$238$	0	0
$239$	−1264.00	−0.342098	−0.171049	−	0.985263i	$-0.554716\pi$
$239$	−1264.00	−0.342098	−0.171049	+	0.985263i	$0.554716\pi$
$240$	0	0
$241$	3010.00	0.804528	0.402264	−	0.915524i	$-0.368223\pi$
$241$	3010.00	0.804528	0.402264	+	0.915524i	$0.368223\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−2790.00	−0.727537
$246$	0	0
$247$	−720.000	−0.185476
$248$	0	0
$249$	−372.000	−0.0946769
$250$	0	0
$251$	−5404.00	−1.35895	−0.679477	−	0.733697i	$-0.737793\pi$
$251$	−5404.00	−1.35895	−0.679477	+	0.733697i	$0.737793\pi$
$252$	0	0
$253$	−484.000	−0.120272
$254$	0	0
$255$	−1380.00	−0.338898
$256$	0	0
$257$	1618.00	0.392716	0.196358	−	0.980532i	$-0.437088\pi$
$257$	1618.00	0.392716	0.196358	+	0.980532i	$0.437088\pi$
$258$	0	0
$259$	912.000	0.218799
$260$	0	0
$261$	1674.00	0.397004
$262$	0	0
$263$	−48.0000	−0.0112540	−0.00562701	−	0.999984i	$-0.501791\pi$
$263$	−48.0000	−0.0112540	−0.00562701	+	0.999984i	$0.501791\pi$
$264$	0	0
$265$	−620.000	−0.143722
$266$	0	0
$267$	−1806.00	−0.413953
$268$	0	0
$269$	−1814.00	−0.411158	−0.205579	−	0.978641i	$-0.565908\pi$
$269$	−1814.00	−0.411158	−0.205579	+	0.978641i	$0.565908\pi$
$270$	0	0
$271$	−1016.00	−0.227740	−0.113870	−	0.993496i	$-0.536325\pi$
$271$	−1016.00	−0.227740	−0.113870	+	0.993496i	$0.536325\pi$
$272$	0	0
$273$	432.000	0.0957723
$274$	0	0
$275$	−275.000	−0.0603023
$276$	0	0
$277$	6914.00	1.49972	0.749859	−	0.661597i	$-0.230121\pi$
$277$	6914.00	1.49972	0.749859	+	0.661597i	$0.230121\pi$
$278$	0	0
$279$	648.000	0.139049
$280$	0	0
$281$	−6098.00	−1.29458	−0.647289	−	0.762245i	$-0.724097\pi$
$281$	−6098.00	−1.29458	−0.647289	+	0.762245i	$0.724097\pi$
$282$	0	0
$283$	−4192.00	−0.880525	−0.440262	−	0.897869i	$-0.645115\pi$
$283$	−4192.00	−0.880525	−0.440262	+	0.897869i	$0.645115\pi$
$284$	0	0
$285$	1200.00	0.249410
$286$	0	0
$287$	−1392.00	−0.286297
$288$	0	0
$289$	−2797.00	−0.569306
$290$	0	0
$291$	−4566.00	−0.919806
$292$	0	0
$293$	9314.00	1.85710	0.928549	−	0.371210i	$-0.121057\pi$
$293$	9314.00	1.85710	0.928549	+	0.371210i	$0.121057\pi$
$294$	0	0
$295$	3480.00	0.686825
$296$	0	0
$297$	−297.000	−0.0580259
$298$	0	0
$299$	−792.000	−0.153186
$300$	0	0
$301$	−3328.00	−0.637285
$302$	0	0
$303$	−3270.00	−0.619989
$304$	0	0
$305$	−4460.00	−0.837308
$306$	0	0
$307$	−1384.00	−0.257293	−0.128647	−	0.991690i	$-0.541063\pi$
$307$	−1384.00	−0.257293	−0.128647	+	0.991690i	$0.541063\pi$
$308$	0	0
$309$	−4176.00	−0.768817
$310$	0	0
$311$	−7916.00	−1.44333	−0.721664	−	0.692243i	$-0.756623\pi$
$311$	−7916.00	−1.44333	−0.721664	+	0.692243i	$0.756623\pi$
$312$	0	0
$313$	218.000	0.0393677	0.0196838	−	0.999806i	$-0.493734\pi$
$313$	218.000	0.0393677	0.0196838	+	0.999806i	$0.493734\pi$
$314$	0	0
$315$	−720.000	−0.128785
$316$	0	0
$317$	666.000	0.118001	0.0590005	−	0.998258i	$-0.481209\pi$
$317$	666.000	0.118001	0.0590005	+	0.998258i	$0.481209\pi$
$318$	0	0
$319$	2046.00	0.359103
$320$	0	0
$321$	924.000	0.160662
$322$	0	0
$323$	−1840.00	−0.316967
$324$	0	0
$325$	−450.000	−0.0768046
$326$	0	0
$327$	−2934.00	−0.496179
$328$	0	0
$329$	−1248.00	−0.209132
$330$	0	0
$331$	2500.00	0.415143	0.207572	−	0.978220i	$-0.433444\pi$
$331$	2500.00	0.415143	0.207572	+	0.978220i	$0.433444\pi$
$332$	0	0
$333$	−1026.00	−0.168842
$334$	0	0
$335$	9560.00	1.55916
$336$	0	0
$337$	−6678.00	−1.07945	−0.539724	−	0.841842i	$-0.681471\pi$
$337$	−6678.00	−1.07945	−0.539724	+	0.841842i	$0.681471\pi$
$338$	0	0
$339$	−3486.00	−0.558506
$340$	0	0
$341$	792.000	0.125775
$342$	0	0
$343$	4976.00	0.783320
$344$	0	0
$345$	1320.00	0.205990
$346$	0	0
$347$	1892.00	0.292703	0.146351	−	0.989233i	$-0.453247\pi$
$347$	1892.00	0.292703	0.146351	+	0.989233i	$0.453247\pi$
$348$	0	0
$349$	−7350.00	−1.12733	−0.563663	−	0.826005i	$-0.690608\pi$
$349$	−7350.00	−1.12733	−0.563663	+	0.826005i	$0.690608\pi$
$350$	0	0
$351$	−486.000	−0.0739053
$352$	0	0
$353$	2370.00	0.357344	0.178672	−	0.983909i	$-0.442820\pi$
$353$	2370.00	0.357344	0.178672	+	0.983909i	$0.442820\pi$
$354$	0	0
$355$	4440.00	0.663805
$356$	0	0
$357$	1104.00	0.163669
$358$	0	0
$359$	−912.000	−0.134077	−0.0670383	−	0.997750i	$-0.521355\pi$
$359$	−912.000	−0.134077	−0.0670383	+	0.997750i	$0.521355\pi$
$360$	0	0
$361$	−5259.00	−0.766730
$362$	0	0
$363$	−363.000	−0.0524864
$364$	0	0
$365$	3060.00	0.438816
$366$	0	0
$367$	8464.00	1.20386	0.601931	−	0.798548i	$-0.294398\pi$
$367$	8464.00	1.20386	0.601931	+	0.798548i	$0.294398\pi$
$368$	0	0
$369$	1566.00	0.220929
$370$	0	0
$371$	496.000	0.0694098
$372$	0	0
$373$	11890.0	1.65051	0.825256	−	0.564759i	$-0.191031\pi$
$373$	11890.0	1.65051	0.825256	+	0.564759i	$0.191031\pi$
$374$	0	0
$375$	4500.00	0.619677
$376$	0	0
$377$	3348.00	0.457376
$378$	0	0
$379$	9556.00	1.29514	0.647571	−	0.762005i	$-0.275785\pi$
$379$	9556.00	1.29514	0.647571	+	0.762005i	$0.275785\pi$
$380$	0	0
$381$	−2952.00	−0.396944
$382$	0	0
$383$	−5236.00	−0.698556	−0.349278	−	0.937019i	$-0.613573\pi$
$383$	−5236.00	−0.698556	−0.349278	+	0.937019i	$0.613573\pi$
$384$	0	0
$385$	−880.000	−0.116491
$386$	0	0
$387$	3744.00	0.491778
$388$	0	0
$389$	−8262.00	−1.07686	−0.538432	−	0.842669i	$-0.680983\pi$
$389$	−8262.00	−1.07686	−0.538432	+	0.842669i	$0.680983\pi$
$390$	0	0
$391$	−2024.00	−0.261785
$392$	0	0
$393$	6036.00	0.774748
$394$	0	0
$395$	6640.00	0.845809
$396$	0	0
$397$	−2402.00	−0.303660	−0.151830	−	0.988407i	$-0.548517\pi$
$397$	−2402.00	−0.303660	−0.151830	+	0.988407i	$0.548517\pi$
$398$	0	0
$399$	−960.000	−0.120451
$400$	0	0
$401$	6290.00	0.783311	0.391655	−	0.920112i	$-0.371903\pi$
$401$	6290.00	0.783311	0.391655	+	0.920112i	$0.371903\pi$
$402$	0	0
$403$	1296.00	0.160194
$404$	0	0
$405$	810.000	0.0993808
$406$	0	0
$407$	−1254.00	−0.152724
$408$	0	0
$409$	−6950.00	−0.840233	−0.420117	−	0.907470i	$-0.638011\pi$
$409$	−6950.00	−0.840233	−0.420117	+	0.907470i	$0.638011\pi$
$410$	0	0
$411$	3858.00	0.463020
$412$	0	0
$413$	−2784.00	−0.331699
$414$	0	0
$415$	1240.00	0.146673
$416$	0	0
$417$	1392.00	0.163469
$418$	0	0
$419$	−12660.0	−1.47609	−0.738045	−	0.674752i	$-0.764251\pi$
$419$	−12660.0	−1.47609	−0.738045	+	0.674752i	$0.764251\pi$
$420$	0	0
$421$	5342.00	0.618416	0.309208	−	0.950994i	$-0.399936\pi$
$421$	5342.00	0.618416	0.309208	+	0.950994i	$0.399936\pi$
$422$	0	0
$423$	1404.00	0.161383
$424$	0	0
$425$	−1150.00	−0.131255
$426$	0	0
$427$	3568.00	0.404374
$428$	0	0
$429$	−594.000	−0.0668499
$430$	0	0
$431$	13560.0	1.51546	0.757729	−	0.652570i	$-0.226309\pi$
$431$	13560.0	1.51546	0.757729	+	0.652570i	$0.226309\pi$
$432$	0	0
$433$	4658.00	0.516973	0.258486	−	0.966015i	$-0.416776\pi$
$433$	4658.00	0.516973	0.258486	+	0.966015i	$0.416776\pi$
$434$	0	0
$435$	−5580.00	−0.615036
$436$	0	0
$437$	1760.00	0.192660
$438$	0	0
$439$	6392.00	0.694928	0.347464	−	0.937693i	$-0.387043\pi$
$439$	6392.00	0.694928	0.347464	+	0.937693i	$0.387043\pi$
$440$	0	0
$441$	−2511.00	−0.271137
$442$	0	0
$443$	−10772.0	−1.15529	−0.577645	−	0.816288i	$-0.696028\pi$
$443$	−10772.0	−1.15529	−0.577645	+	0.816288i	$0.696028\pi$
$444$	0	0
$445$	6020.00	0.641293
$446$	0	0
$447$	8682.00	0.918668
$448$	0	0
$449$	−5606.00	−0.589228	−0.294614	−	0.955616i	$-0.595191\pi$
$449$	−5606.00	−0.589228	−0.294614	+	0.955616i	$0.595191\pi$
$450$	0	0
$451$	1914.00	0.199838
$452$	0	0
$453$	2928.00	0.303685
$454$	0	0
$455$	−1440.00	−0.148370
$456$	0	0
$457$	17050.0	1.74522	0.872610	−	0.488418i	$-0.162426\pi$
$457$	17050.0	1.74522	0.872610	+	0.488418i	$0.162426\pi$
$458$	0	0
$459$	−1242.00	−0.126300
$460$	0	0
$461$	−8438.00	−0.852488	−0.426244	−	0.904608i	$-0.640163\pi$
$461$	−8438.00	−0.852488	−0.426244	+	0.904608i	$0.640163\pi$
$462$	0	0
$463$	−5064.00	−0.508302	−0.254151	−	0.967164i	$-0.581796\pi$
$463$	−5064.00	−0.508302	−0.254151	+	0.967164i	$0.581796\pi$
$464$	0	0
$465$	−2160.00	−0.215414
$466$	0	0
$467$	14268.0	1.41380	0.706900	−	0.707314i	$-0.250093\pi$
$467$	14268.0	1.41380	0.706900	+	0.707314i	$0.250093\pi$
$468$	0	0
$469$	−7648.00	−0.752989
$470$	0	0
$471$	−4938.00	−0.483081
$472$	0	0
$473$	4576.00	0.444830
$474$	0	0
$475$	1000.00	0.0965961
$476$	0	0
$477$	−558.000	−0.0535620
$478$	0	0
$479$	18312.0	1.74676	0.873379	−	0.487042i	$-0.161924\pi$
$479$	18312.0	1.74676	0.873379	+	0.487042i	$0.161924\pi$
$480$	0	0
$481$	−2052.00	−0.194518
$482$	0	0
$483$	−1056.00	−0.0994817
$484$	0	0
$485$	15220.0	1.42496
$486$	0	0
$487$	−4376.00	−0.407178	−0.203589	−	0.979056i	$-0.565261\pi$
$487$	−4376.00	−0.407178	−0.203589	+	0.979056i	$0.565261\pi$
$488$	0	0
$489$	9804.00	0.906651
$490$	0	0
$491$	−17380.0	−1.59745	−0.798725	−	0.601696i	$-0.794492\pi$
$491$	−17380.0	−1.59745	−0.798725	+	0.601696i	$0.794492\pi$
$492$	0	0
$493$	8556.00	0.781629
$494$	0	0
$495$	990.000	0.0898933
$496$	0	0
$497$	−3552.00	−0.320581
$498$	0	0
$499$	−11324.0	−1.01590	−0.507948	−	0.861388i	$-0.669596\pi$
$499$	−11324.0	−1.01590	−0.507948	+	0.861388i	$0.669596\pi$
$500$	0	0
$501$	4824.00	0.430180
$502$	0	0
$503$	−2392.00	−0.212036	−0.106018	−	0.994364i	$-0.533810\pi$
$503$	−2392.00	−0.212036	−0.106018	+	0.994364i	$0.533810\pi$
$504$	0	0
$505$	10900.0	0.960482
$506$	0	0
$507$	5619.00	0.492206
$508$	0	0
$509$	−1238.00	−0.107806	−0.0539031	−	0.998546i	$-0.517166\pi$
$509$	−1238.00	−0.107806	−0.0539031	+	0.998546i	$0.517166\pi$
$510$	0	0
$511$	−2448.00	−0.211924
$512$	0	0
$513$	1080.00	0.0929496
$514$	0	0
$515$	13920.0	1.19105
$516$	0	0
$517$	1716.00	0.145976
$518$	0	0
$519$	12210.0	1.03268
$520$	0	0
$521$	21738.0	1.82794	0.913972	−	0.405777i	$-0.132999\pi$
$521$	21738.0	1.82794	0.913972	+	0.405777i	$0.132999\pi$
$522$	0	0
$523$	−22016.0	−1.84071	−0.920356	−	0.391081i	$-0.872101\pi$
$523$	−22016.0	−1.84071	−0.920356	+	0.391081i	$0.872101\pi$
$524$	0	0
$525$	−600.000	−0.0498784
$526$	0	0
$527$	3312.00	0.273763
$528$	0	0
$529$	−10231.0	−0.840881
$530$	0	0
$531$	3132.00	0.255965
$532$	0	0
$533$	3132.00	0.254525
$534$	0	0
$535$	−3080.00	−0.248897
$536$	0	0
$537$	156.000	0.0125361
$538$	0	0
$539$	−3069.00	−0.245253
$540$	0	0
$541$	9490.00	0.754172	0.377086	−	0.926178i	$-0.376926\pi$
$541$	9490.00	0.754172	0.377086	+	0.926178i	$0.376926\pi$
$542$	0	0
$543$	−5394.00	−0.426296
$544$	0	0
$545$	9780.00	0.768677
$546$	0	0
$547$	−21632.0	−1.69089	−0.845446	−	0.534061i	$-0.820665\pi$
$547$	−21632.0	−1.69089	−0.845446	+	0.534061i	$0.820665\pi$
$548$	0	0
$549$	−4014.00	−0.312046
$550$	0	0
$551$	−7440.00	−0.575235
$552$	0	0
$553$	−5312.00	−0.408480
$554$	0	0
$555$	3420.00	0.261569
$556$	0	0
$557$	−4854.00	−0.369247	−0.184624	−	0.982809i	$-0.559107\pi$
$557$	−4854.00	−0.369247	−0.184624	+	0.982809i	$0.559107\pi$
$558$	0	0
$559$	7488.00	0.566563
$560$	0	0
$561$	−1518.00	−0.114242
$562$	0	0
$563$	5308.00	0.397346	0.198673	−	0.980066i	$-0.436337\pi$
$563$	5308.00	0.397346	0.198673	+	0.980066i	$0.436337\pi$
$564$	0	0
$565$	11620.0	0.865234
$566$	0	0
$567$	−648.000	−0.0479955
$568$	0	0
$569$	−12490.0	−0.920225	−0.460113	−	0.887861i	$-0.652191\pi$
$569$	−12490.0	−0.920225	−0.460113	+	0.887861i	$0.652191\pi$
$570$	0	0
$571$	7448.00	0.545865	0.272933	−	0.962033i	$-0.412006\pi$
$571$	7448.00	0.545865	0.272933	+	0.962033i	$0.412006\pi$
$572$	0	0
$573$	−11556.0	−0.842511
$574$	0	0
$575$	1100.00	0.0797794
$576$	0	0
$577$	10994.0	0.793217	0.396608	−	0.917988i	$-0.370187\pi$
$577$	10994.0	0.793217	0.396608	+	0.917988i	$0.370187\pi$
$578$	0	0
$579$	5874.00	0.421615
$580$	0	0
$581$	−992.000	−0.0708349
$582$	0	0
$583$	−682.000	−0.0484486
$584$	0	0
$585$	1620.00	0.114494
$586$	0	0
$587$	5148.00	0.361977	0.180989	−	0.983485i	$-0.442070\pi$
$587$	5148.00	0.361977	0.180989	+	0.983485i	$0.442070\pi$
$588$	0	0
$589$	−2880.00	−0.201474
$590$	0	0
$591$	4890.00	0.340351
$592$	0	0
$593$	−22314.0	−1.54524	−0.772619	−	0.634870i	$-0.781054\pi$
$593$	−22314.0	−1.54524	−0.772619	+	0.634870i	$0.781054\pi$
$594$	0	0
$595$	−3680.00	−0.253555
$596$	0	0
$597$	−13512.0	−0.926314
$598$	0	0
$599$	15588.0	1.06329	0.531643	−	0.846968i	$-0.321575\pi$
$599$	15588.0	1.06329	0.531643	+	0.846968i	$0.321575\pi$
$600$	0	0
$601$	−21638.0	−1.46861	−0.734303	−	0.678822i	$-0.762491\pi$
$601$	−21638.0	−1.46861	−0.734303	+	0.678822i	$0.762491\pi$
$602$	0	0
$603$	8604.00	0.581065
$604$	0	0
$605$	1210.00	0.0813116
$606$	0	0
$607$	7496.00	0.501241	0.250620	−	0.968085i	$-0.419365\pi$
$607$	7496.00	0.501241	0.250620	+	0.968085i	$0.419365\pi$
$608$	0	0
$609$	4464.00	0.297029
$610$	0	0
$611$	2808.00	0.185924
$612$	0	0
$613$	2106.00	0.138761	0.0693805	−	0.997590i	$-0.477898\pi$
$613$	2106.00	0.138761	0.0693805	+	0.997590i	$0.477898\pi$
$614$	0	0
$615$	−5220.00	−0.342261
$616$	0	0
$617$	26962.0	1.75924	0.879619	−	0.475680i	$-0.157798\pi$
$617$	26962.0	1.75924	0.879619	+	0.475680i	$0.157798\pi$
$618$	0	0
$619$	17740.0	1.15191	0.575954	−	0.817482i	$-0.304631\pi$
$619$	17740.0	1.15191	0.575954	+	0.817482i	$0.304631\pi$
$620$	0	0
$621$	1188.00	0.0767678
$622$	0	0
$623$	−4816.00	−0.309709
$624$	0	0
$625$	−11875.0	−0.760000
$626$	0	0
$627$	1320.00	0.0840761
$628$	0	0
$629$	−5244.00	−0.332420
$630$	0	0
$631$	−19360.0	−1.22141	−0.610705	−	0.791858i	$-0.709114\pi$
$631$	−19360.0	−1.22141	−0.610705	+	0.791858i	$0.709114\pi$
$632$	0	0
$633$	11328.0	0.711292
$634$	0	0
$635$	9840.00	0.614943
$636$	0	0
$637$	−5022.00	−0.312369
$638$	0	0
$639$	3996.00	0.247385
$640$	0	0
$641$	−19158.0	−1.18049	−0.590246	−	0.807223i	$-0.700969\pi$
$641$	−19158.0	−1.18049	−0.590246	+	0.807223i	$0.700969\pi$
$642$	0	0
$643$	−11228.0	−0.688630	−0.344315	−	0.938854i	$-0.611889\pi$
$643$	−11228.0	−0.688630	−0.344315	+	0.938854i	$0.611889\pi$
$644$	0	0
$645$	−12480.0	−0.761860
$646$	0	0
$647$	22628.0	1.37496	0.687480	−	0.726204i	$-0.258717\pi$
$647$	22628.0	1.37496	0.687480	+	0.726204i	$0.258717\pi$
$648$	0	0
$649$	3828.00	0.231529
$650$	0	0
$651$	1728.00	0.104033
$652$	0	0
$653$	28338.0	1.69824	0.849121	−	0.528199i	$-0.177132\pi$
$653$	28338.0	1.69824	0.849121	+	0.528199i	$0.177132\pi$
$654$	0	0
$655$	−20120.0	−1.20023
$656$	0	0
$657$	2754.00	0.163537
$658$	0	0
$659$	17052.0	1.00797	0.503985	−	0.863713i	$-0.331867\pi$
$659$	17052.0	1.00797	0.503985	+	0.863713i	$0.331867\pi$
$660$	0	0
$661$	−21354.0	−1.25654	−0.628271	−	0.777995i	$-0.716237\pi$
$661$	−21354.0	−1.25654	−0.628271	+	0.777995i	$0.716237\pi$
$662$	0	0
$663$	−2484.00	−0.145506
$664$	0	0
$665$	3200.00	0.186603
$666$	0	0
$667$	−8184.00	−0.475091
$668$	0	0
$669$	−5184.00	−0.299589
$670$	0	0
$671$	−4906.00	−0.282256
$672$	0	0
$673$	−22198.0	−1.27143	−0.635713	−	0.771925i	$-0.719294\pi$
$673$	−22198.0	−1.27143	−0.635713	+	0.771925i	$0.719294\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	−32974.0	−1.87193	−0.935963	−	0.352099i	$-0.885468\pi$
$677$	−32974.0	−1.87193	−0.935963	+	0.352099i	$0.885468\pi$
$678$	0	0
$679$	−12176.0	−0.688177
$680$	0	0
$681$	14148.0	0.796112
$682$	0	0
$683$	−22572.0	−1.26456	−0.632279	−	0.774740i	$-0.717880\pi$
$683$	−22572.0	−1.26456	−0.632279	+	0.774740i	$0.717880\pi$
$684$	0	0
$685$	−12860.0	−0.717307
$686$	0	0
$687$	−5298.00	−0.294223
$688$	0	0
$689$	−1116.00	−0.0617071
$690$	0	0
$691$	−2700.00	−0.148644	−0.0743219	−	0.997234i	$-0.523679\pi$
$691$	−2700.00	−0.148644	−0.0743219	+	0.997234i	$0.523679\pi$
$692$	0	0
$693$	−792.000	−0.0434136
$694$	0	0
$695$	−4640.00	−0.253245
$696$	0	0
$697$	8004.00	0.434969
$698$	0	0
$699$	−13146.0	−0.711341
$700$	0	0
$701$	−17382.0	−0.936532	−0.468266	−	0.883587i	$-0.655121\pi$
$701$	−17382.0	−0.936532	−0.468266	+	0.883587i	$0.655121\pi$
$702$	0	0
$703$	4560.00	0.244642
$704$	0	0
$705$	−4680.00	−0.250013
$706$	0	0
$707$	−8720.00	−0.463860
$708$	0	0
$709$	20454.0	1.08345	0.541725	−	0.840556i	$-0.317771\pi$
$709$	20454.0	1.08345	0.541725	+	0.840556i	$0.317771\pi$
$710$	0	0
$711$	5976.00	0.315215
$712$	0	0
$713$	−3168.00	−0.166399
$714$	0	0
$715$	1980.00	0.103563
$716$	0	0
$717$	3792.00	0.197510
$718$	0	0
$719$	−10236.0	−0.530930	−0.265465	−	0.964121i	$-0.585525\pi$
$719$	−10236.0	−0.530930	−0.265465	+	0.964121i	$0.585525\pi$
$720$	0	0
$721$	−11136.0	−0.575210
$722$	0	0
$723$	−9030.00	−0.464494
$724$	0	0
$725$	−4650.00	−0.238202
$726$	0	0
$727$	−9672.00	−0.493418	−0.246709	−	0.969090i	$-0.579349\pi$
$727$	−9672.00	−0.493418	−0.246709	+	0.969090i	$0.579349\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	19136.0	0.968222
$732$	0	0
$733$	−28078.0	−1.41485	−0.707425	−	0.706789i	$-0.750143\pi$
$733$	−28078.0	−1.41485	−0.707425	+	0.706789i	$0.750143\pi$
$734$	0	0
$735$	8370.00	0.420044
$736$	0	0
$737$	10516.0	0.525593
$738$	0	0
$739$	26776.0	1.33284	0.666422	−	0.745575i	$-0.267825\pi$
$739$	26776.0	1.33284	0.666422	+	0.745575i	$0.267825\pi$
$740$	0	0
$741$	2160.00	0.107084
$742$	0	0
$743$	−25280.0	−1.24823	−0.624114	−	0.781333i	$-0.714540\pi$
$743$	−25280.0	−1.24823	−0.624114	+	0.781333i	$0.714540\pi$
$744$	0	0
$745$	−28940.0	−1.42319
$746$	0	0
$747$	1116.00	0.0546617
$748$	0	0
$749$	2464.00	0.120204
$750$	0	0
$751$	−25160.0	−1.22251	−0.611253	−	0.791436i	$-0.709334\pi$
$751$	−25160.0	−1.22251	−0.611253	+	0.791436i	$0.709334\pi$
$752$	0	0
$753$	16212.0	0.784592
$754$	0	0
$755$	−9760.00	−0.470467
$756$	0	0
$757$	28910.0	1.38805	0.694024	−	0.719952i	$-0.255836\pi$
$757$	28910.0	1.38805	0.694024	+	0.719952i	$0.255836\pi$
$758$	0	0
$759$	1452.00	0.0694391
$760$	0	0
$761$	10278.0	0.489589	0.244794	−	0.969575i	$-0.421280\pi$
$761$	10278.0	0.489589	0.244794	+	0.969575i	$0.421280\pi$
$762$	0	0
$763$	−7824.00	−0.371229
$764$	0	0
$765$	4140.00	0.195663
$766$	0	0
$767$	6264.00	0.294889
$768$	0	0
$769$	−29558.0	−1.38607	−0.693036	−	0.720903i	$-0.743727\pi$
$769$	−29558.0	−1.38607	−0.693036	+	0.720903i	$0.743727\pi$
$770$	0	0
$771$	−4854.00	−0.226735
$772$	0	0
$773$	9762.00	0.454223	0.227112	−	0.973869i	$-0.427072\pi$
$773$	9762.00	0.454223	0.227112	+	0.973869i	$0.427072\pi$
$774$	0	0
$775$	−1800.00	−0.0834296
$776$	0	0
$777$	−2736.00	−0.126324
$778$	0	0
$779$	−6960.00	−0.320113
$780$	0	0
$781$	4884.00	0.223769
$782$	0	0
$783$	−5022.00	−0.229210
$784$	0	0
$785$	16460.0	0.748385
$786$	0	0
$787$	−14824.0	−0.671434	−0.335717	−	0.941963i	$-0.608979\pi$
$787$	−14824.0	−0.671434	−0.335717	+	0.941963i	$0.608979\pi$
$788$	0	0
$789$	144.000	0.00649751
$790$	0	0
$791$	−9296.00	−0.417861
$792$	0	0
$793$	−8028.00	−0.359499
$794$	0	0
$795$	1860.00	0.0829779
$796$	0	0
$797$	14946.0	0.664259	0.332130	−	0.943234i	$-0.392233\pi$
$797$	14946.0	0.664259	0.332130	+	0.943234i	$0.392233\pi$
$798$	0	0
$799$	7176.00	0.317733
$800$	0	0
$801$	5418.00	0.238996
$802$	0	0
$803$	3366.00	0.147925
$804$	0	0
$805$	3520.00	0.154116
$806$	0	0
$807$	5442.00	0.237382
$808$	0	0
$809$	14774.0	0.642060	0.321030	−	0.947069i	$-0.395971\pi$
$809$	14774.0	0.642060	0.321030	+	0.947069i	$0.395971\pi$
$810$	0	0
$811$	−22760.0	−0.985464	−0.492732	−	0.870181i	$-0.664002\pi$
$811$	−22760.0	−0.985464	−0.492732	+	0.870181i	$0.664002\pi$
$812$	0	0
$813$	3048.00	0.131486
$814$	0	0
$815$	−32680.0	−1.40458
$816$	0	0
$817$	−16640.0	−0.712558
$818$	0	0
$819$	−1296.00	−0.0552941
$820$	0	0
$821$	16370.0	0.695879	0.347940	−	0.937517i	$-0.386881\pi$
$821$	16370.0	0.695879	0.347940	+	0.937517i	$0.386881\pi$
$822$	0	0
$823$	−1784.00	−0.0755605	−0.0377803	−	0.999286i	$-0.512029\pi$
$823$	−1784.00	−0.0755605	−0.0377803	+	0.999286i	$0.512029\pi$
$824$	0	0
$825$	825.000	0.0348155
$826$	0	0
$827$	22052.0	0.927235	0.463617	−	0.886036i	$-0.346551\pi$
$827$	22052.0	0.927235	0.463617	+	0.886036i	$0.346551\pi$
$828$	0	0
$829$	−29738.0	−1.24589	−0.622945	−	0.782265i	$-0.714064\pi$
$829$	−29738.0	−1.24589	−0.622945	+	0.782265i	$0.714064\pi$
$830$	0	0
$831$	−20742.0	−0.865863
$832$	0	0
$833$	−12834.0	−0.533820
$834$	0	0
$835$	−16080.0	−0.666433
$836$	0	0
$837$	−1944.00	−0.0802801
$838$	0	0
$839$	−43956.0	−1.80874	−0.904368	−	0.426753i	$-0.859657\pi$
$839$	−43956.0	−1.80874	−0.904368	+	0.426753i	$0.859657\pi$
$840$	0	0
$841$	10207.0	0.418508
$842$	0	0
$843$	18294.0	0.747424
$844$	0	0
$845$	−18730.0	−0.762523
$846$	0	0
$847$	−968.000	−0.0392690
$848$	0	0
$849$	12576.0	0.508371
$850$	0	0
$851$	5016.00	0.202052
$852$	0	0
$853$	−6438.00	−0.258421	−0.129210	−	0.991617i	$-0.541244\pi$
$853$	−6438.00	−0.258421	−0.129210	+	0.991617i	$0.541244\pi$
$854$	0	0
$855$	−3600.00	−0.143997
$856$	0	0
$857$	−2282.00	−0.0909587	−0.0454794	−	0.998965i	$-0.514482\pi$
$857$	−2282.00	−0.0909587	−0.0454794	+	0.998965i	$0.514482\pi$
$858$	0	0
$859$	29972.0	1.19049	0.595245	−	0.803544i	$-0.297055\pi$
$859$	29972.0	1.19049	0.595245	+	0.803544i	$0.297055\pi$
$860$	0	0
$861$	4176.00	0.165293
$862$	0	0
$863$	3716.00	0.146575	0.0732874	−	0.997311i	$-0.476651\pi$
$863$	3716.00	0.146575	0.0732874	+	0.997311i	$0.476651\pi$
$864$	0	0
$865$	−40700.0	−1.59982
$866$	0	0
$867$	8391.00	0.328689
$868$	0	0
$869$	7304.00	0.285122
$870$	0	0
$871$	17208.0	0.669427
$872$	0	0
$873$	13698.0	0.531050
$874$	0	0
$875$	12000.0	0.463627
$876$	0	0
$877$	25194.0	0.970058	0.485029	−	0.874498i	$-0.338809\pi$
$877$	25194.0	0.970058	0.485029	+	0.874498i	$0.338809\pi$
$878$	0	0
$879$	−27942.0	−1.07220
$880$	0	0
$881$	27194.0	1.03994	0.519971	−	0.854184i	$-0.325943\pi$
$881$	27194.0	1.03994	0.519971	+	0.854184i	$0.325943\pi$
$882$	0	0
$883$	14300.0	0.544998	0.272499	−	0.962156i	$-0.412150\pi$
$883$	14300.0	0.544998	0.272499	+	0.962156i	$0.412150\pi$
$884$	0	0
$885$	−10440.0	−0.396539
$886$	0	0
$887$	−4944.00	−0.187151	−0.0935757	−	0.995612i	$-0.529830\pi$
$887$	−4944.00	−0.187151	−0.0935757	+	0.995612i	$0.529830\pi$
$888$	0	0
$889$	−7872.00	−0.296984
$890$	0	0
$891$	891.000	0.0335013
$892$	0	0
$893$	−6240.00	−0.233834
$894$	0	0
$895$	−520.000	−0.0194209
$896$	0	0
$897$	2376.00	0.0884418
$898$	0	0
$899$	13392.0	0.496828
$900$	0	0
$901$	−2852.00	−0.105454
$902$	0	0
$903$	9984.00	0.367937
$904$	0	0
$905$	17980.0	0.660415
$906$	0	0
$907$	24332.0	0.890773	0.445386	−	0.895338i	$-0.353066\pi$
$907$	24332.0	0.890773	0.445386	+	0.895338i	$0.353066\pi$
$908$	0	0
$909$	9810.00	0.357951
$910$	0	0
$911$	−20796.0	−0.756314	−0.378157	−	0.925741i	$-0.623442\pi$
$911$	−20796.0	−0.756314	−0.378157	+	0.925741i	$0.623442\pi$
$912$	0	0
$913$	1364.00	0.0494434
$914$	0	0
$915$	13380.0	0.483420
$916$	0	0
$917$	16096.0	0.579647
$918$	0	0
$919$	−15392.0	−0.552487	−0.276243	−	0.961088i	$-0.589090\pi$
$919$	−15392.0	−0.552487	−0.276243	+	0.961088i	$0.589090\pi$
$920$	0	0
$921$	4152.00	0.148548
$922$	0	0
$923$	7992.00	0.285005
$924$	0	0
$925$	2850.00	0.101305
$926$	0	0
$927$	12528.0	0.443876
$928$	0	0
$929$	−15150.0	−0.535043	−0.267522	−	0.963552i	$-0.586205\pi$
$929$	−15150.0	−0.535043	−0.267522	+	0.963552i	$0.586205\pi$
$930$	0	0
$931$	11160.0	0.392862
$932$	0	0
$933$	23748.0	0.833306
$934$	0	0
$935$	5060.00	0.176984
$936$	0	0
$937$	15610.0	0.544244	0.272122	−	0.962263i	$-0.412275\pi$
$937$	15610.0	0.544244	0.272122	+	0.962263i	$0.412275\pi$
$938$	0	0
$939$	−654.000	−0.0227289
$940$	0	0
$941$	−45222.0	−1.56663	−0.783313	−	0.621627i	$-0.786472\pi$
$941$	−45222.0	−1.56663	−0.783313	+	0.621627i	$0.786472\pi$
$942$	0	0
$943$	−7656.00	−0.264384
$944$	0	0
$945$	2160.00	0.0743543
$946$	0	0
$947$	14820.0	0.508538	0.254269	−	0.967134i	$-0.418165\pi$
$947$	14820.0	0.508538	0.254269	+	0.967134i	$0.418165\pi$
$948$	0	0
$949$	5508.00	0.188406
$950$	0	0
$951$	−1998.00	−0.0681279
$952$	0	0
$953$	5334.00	0.181307	0.0906533	−	0.995883i	$-0.471104\pi$
$953$	5334.00	0.181307	0.0906533	+	0.995883i	$0.471104\pi$
$954$	0	0
$955$	38520.0	1.30521
$956$	0	0
$957$	−6138.00	−0.207328
$958$	0	0
$959$	10288.0	0.346420
$960$	0	0
$961$	−24607.0	−0.825988
$962$	0	0
$963$	−2772.00	−0.0927585
$964$	0	0
$965$	−19580.0	−0.653163
$966$	0	0
$967$	−18400.0	−0.611897	−0.305948	−	0.952048i	$-0.598973\pi$
$967$	−18400.0	−0.611897	−0.305948	+	0.952048i	$0.598973\pi$
$968$	0	0
$969$	5520.00	0.183001
$970$	0	0
$971$	14460.0	0.477903	0.238951	−	0.971032i	$-0.423196\pi$
$971$	14460.0	0.477903	0.238951	+	0.971032i	$0.423196\pi$
$972$	0	0
$973$	3712.00	0.122303
$974$	0	0
$975$	1350.00	0.0443432
$976$	0	0
$977$	−9998.00	−0.327394	−0.163697	−	0.986511i	$-0.552342\pi$
$977$	−9998.00	−0.327394	−0.163697	+	0.986511i	$0.552342\pi$
$978$	0	0
$979$	6622.00	0.216180
$980$	0	0
$981$	8802.00	0.286469
$982$	0	0
$983$	52548.0	1.70501	0.852503	−	0.522722i	$-0.175084\pi$
$983$	52548.0	1.70501	0.852503	+	0.522722i	$0.175084\pi$
$984$	0	0
$985$	−16300.0	−0.527270
$986$	0	0
$987$	3744.00	0.120742
$988$	0	0
$989$	−18304.0	−0.588507
$990$	0	0
$991$	7096.00	0.227459	0.113729	−	0.993512i	$-0.463720\pi$
$991$	7096.00	0.227459	0.113729	+	0.993512i	$0.463720\pi$
$992$	0	0
$993$	−7500.00	−0.239683
$994$	0	0
$995$	45040.0	1.43504
$996$	0	0
$997$	45202.0	1.43587	0.717935	−	0.696111i	$-0.245088\pi$
$997$	45202.0	1.43587	0.717935	+	0.696111i	$0.245088\pi$
$998$	0	0
$999$	3078.00	0.0974811

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.4.a.e.1.1		1
3.2	odd	2		1584.4.a.f.1.1		1
4.3	odd	2		132.4.a.d.1.1	✓	1
8.3	odd	2		2112.4.a.e.1.1		1
8.5	even	2		2112.4.a.q.1.1		1
12.11	even	2		396.4.a.c.1.1		1
44.43	even	2		1452.4.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.4.a.d.1.1	✓	1	4.3	odd	2
396.4.a.c.1.1		1	12.11	even	2
528.4.a.e.1.1		1	1.1	even	1	trivial
1452.4.a.i.1.1		1	44.43	even	2
1584.4.a.f.1.1		1	3.2	odd	2
2112.4.a.e.1.1		1	8.3	odd	2
2112.4.a.q.1.1		1	8.5	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:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	528.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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