LMFDB - Embedded newform 1425.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1425,4,Mod(1,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, 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("1425.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1425 = 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1425.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:	$84.0777217582$
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	$\chi$	$=$	1425.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -3.00000 q^{3} -4.00000 q^{4} -6.00000 q^{6} -9.00000 q^{7} -24.0000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} -3.00000 q^{3} -4.00000 q^{4} -6.00000 q^{6} -9.00000 q^{7} -24.0000 q^{8} +9.00000 q^{9} -62.0000 q^{11} +12.0000 q^{12} +38.0000 q^{13} -18.0000 q^{14} -16.0000 q^{16} -76.0000 q^{17} +18.0000 q^{18} -19.0000 q^{19} +27.0000 q^{21} -124.000 q^{22} -42.0000 q^{23} +72.0000 q^{24} +76.0000 q^{26} -27.0000 q^{27} +36.0000 q^{28} -259.000 q^{29} -120.000 q^{31} +160.000 q^{32} +186.000 q^{33} -152.000 q^{34} -36.0000 q^{36} -230.000 q^{37} -38.0000 q^{38} -114.000 q^{39} +455.000 q^{41} +54.0000 q^{42} -340.000 q^{43} +248.000 q^{44} -84.0000 q^{46} +224.000 q^{47} +48.0000 q^{48} -262.000 q^{49} +228.000 q^{51} -152.000 q^{52} -61.0000 q^{53} -54.0000 q^{54} +216.000 q^{56} +57.0000 q^{57} -518.000 q^{58} -119.000 q^{59} -113.000 q^{61} -240.000 q^{62} -81.0000 q^{63} +448.000 q^{64} +372.000 q^{66} +468.000 q^{67} +304.000 q^{68} +126.000 q^{69} +995.000 q^{71} -216.000 q^{72} -271.000 q^{73} -460.000 q^{74} +76.0000 q^{76} +558.000 q^{77} -228.000 q^{78} +318.000 q^{79} +81.0000 q^{81} +910.000 q^{82} -336.000 q^{83} -108.000 q^{84} -680.000 q^{86} +777.000 q^{87} +1488.00 q^{88} -945.000 q^{89} -342.000 q^{91} +168.000 q^{92} +360.000 q^{93} +448.000 q^{94} -480.000 q^{96} -872.000 q^{97} -524.000 q^{98} -558.000 q^{99} +O(q^{100})\)

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 $

$2$	2.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	2.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−6.00000	−0.408248
$7$	−9.00000	−0.485954	−0.242977	−	0.970032i	$-0.578124\pi$
$7$	−9.00000	−0.485954	−0.242977	+	0.970032i	$0.578124\pi$
$8$	−24.0000	−1.06066
$9$	9.00000	0.333333
$10$	0	0
$11$	−62.0000	−1.69943	−0.849714	−	0.527244i	$-0.823225\pi$
$11$	−62.0000	−1.69943	−0.849714	+	0.527244i	$0.823225\pi$
$12$	12.0000	0.288675
$13$	38.0000	0.810716	0.405358	−	0.914158i	$-0.367147\pi$
$13$	38.0000	0.810716	0.405358	+	0.914158i	$0.367147\pi$
$14$	−18.0000	−0.343622
$15$	0	0
$16$	−16.0000	−0.250000
$17$	−76.0000	−1.08428	−0.542138	−	0.840289i	$-0.682385\pi$
$17$	−76.0000	−1.08428	−0.542138	+	0.840289i	$0.682385\pi$
$18$	18.0000	0.235702
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	27.0000	0.280566
$22$	−124.000	−1.20168
$23$	−42.0000	−0.380765	−0.190383	−	0.981710i	$-0.560973\pi$
$23$	−42.0000	−0.380765	−0.190383	+	0.981710i	$0.560973\pi$
$24$	72.0000	0.612372
$25$	0	0
$26$	76.0000	0.573263
$27$	−27.0000	−0.192450
$28$	36.0000	0.242977
$29$	−259.000	−1.65845	−0.829226	−	0.558914i	$-0.811218\pi$
$29$	−259.000	−1.65845	−0.829226	+	0.558914i	$0.811218\pi$
$30$	0	0
$31$	−120.000	−0.695246	−0.347623	−	0.937634i	$-0.613011\pi$
$31$	−120.000	−0.695246	−0.347623	+	0.937634i	$0.613011\pi$
$32$	160.000	0.883883
$33$	186.000	0.981165
$34$	−152.000	−0.766700
$35$	0	0
$36$	−36.0000	−0.166667
$37$	−230.000	−1.02194	−0.510970	−	0.859599i	$-0.670714\pi$
$37$	−230.000	−1.02194	−0.510970	+	0.859599i	$0.670714\pi$
$38$	−38.0000	−0.162221
$39$	−114.000	−0.468067
$40$	0	0
$41$	455.000	1.73315	0.866574	−	0.499049i	$-0.166317\pi$
$41$	455.000	1.73315	0.866574	+	0.499049i	$0.166317\pi$
$42$	54.0000	0.198390
$43$	−340.000	−1.20580	−0.602901	−	0.797816i	$-0.705989\pi$
$43$	−340.000	−1.20580	−0.602901	+	0.797816i	$0.705989\pi$
$44$	248.000	0.849714
$45$	0	0
$46$	−84.0000	−0.269242
$47$	224.000	0.695186	0.347593	−	0.937645i	$-0.386999\pi$
$47$	224.000	0.695186	0.347593	+	0.937645i	$0.386999\pi$
$48$	48.0000	0.144338
$49$	−262.000	−0.763848
$50$	0	0
$51$	228.000	0.626008
$52$	−152.000	−0.405358
$53$	−61.0000	−0.158094	−0.0790471	−	0.996871i	$-0.525188\pi$
$53$	−61.0000	−0.158094	−0.0790471	+	0.996871i	$0.525188\pi$
$54$	−54.0000	−0.136083
$55$	0	0
$56$	216.000	0.515432
$57$	57.0000	0.132453
$58$	−518.000	−1.17270
$59$	−119.000	−0.262584	−0.131292	−	0.991344i	$-0.541913\pi$
$59$	−119.000	−0.262584	−0.131292	+	0.991344i	$0.541913\pi$
$60$	0	0
$61$	−113.000	−0.237183	−0.118592	−	0.992943i	$-0.537838\pi$
$61$	−113.000	−0.237183	−0.118592	+	0.992943i	$0.537838\pi$
$62$	−240.000	−0.491613
$63$	−81.0000	−0.161985
$64$	448.000	0.875000
$65$	0	0
$66$	372.000	0.693788
$67$	468.000	0.853363	0.426681	−	0.904402i	$-0.359683\pi$
$67$	468.000	0.853363	0.426681	+	0.904402i	$0.359683\pi$
$68$	304.000	0.542138
$69$	126.000	0.219835
$70$	0	0
$71$	995.000	1.66317	0.831583	−	0.555401i	$-0.187435\pi$
$71$	995.000	1.66317	0.831583	+	0.555401i	$0.187435\pi$
$72$	−216.000	−0.353553
$73$	−271.000	−0.434495	−0.217248	−	0.976117i	$-0.569708\pi$
$73$	−271.000	−0.434495	−0.217248	+	0.976117i	$0.569708\pi$
$74$	−460.000	−0.722620
$75$	0	0
$76$	76.0000	0.114708
$77$	558.000	0.825844
$78$	−228.000	−0.330973
$79$	318.000	0.452883	0.226442	−	0.974025i	$-0.427291\pi$
$79$	318.000	0.452883	0.226442	+	0.974025i	$0.427291\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	910.000	1.22552
$83$	−336.000	−0.444347	−0.222173	−	0.975007i	$-0.571315\pi$
$83$	−336.000	−0.444347	−0.222173	+	0.975007i	$0.571315\pi$
$84$	−108.000	−0.140283
$85$	0	0
$86$	−680.000	−0.852631
$87$	777.000	0.957507
$88$	1488.00	1.80252
$89$	−945.000	−1.12550	−0.562752	−	0.826626i	$-0.690257\pi$
$89$	−945.000	−1.12550	−0.562752	+	0.826626i	$0.690257\pi$
$90$	0	0
$91$	−342.000	−0.393971
$92$	168.000	0.190383
$93$	360.000	0.401401
$94$	448.000	0.491571
$95$	0	0
$96$	−480.000	−0.510310
$97$	−872.000	−0.912765	−0.456382	−	0.889784i	$-0.650855\pi$
$97$	−872.000	−0.912765	−0.456382	+	0.889784i	$0.650855\pi$
$98$	−524.000	−0.540122
$99$	−558.000	−0.566476
$100$	0	0
$101$	954.000	0.939867	0.469933	−	0.882702i	$-0.344278\pi$
$101$	954.000	0.939867	0.469933	+	0.882702i	$0.344278\pi$
$102$	456.000	0.442654
$103$	1246.00	1.19196	0.595981	−	0.802999i	$-0.296763\pi$
$103$	1246.00	1.19196	0.595981	+	0.802999i	$0.296763\pi$
$104$	−912.000	−0.859894
$105$	0	0
$106$	−122.000	−0.111790
$107$	1581.00	1.42842	0.714210	−	0.699931i	$-0.246786\pi$
$107$	1581.00	1.42842	0.714210	+	0.699931i	$0.246786\pi$
$108$	108.000	0.0962250
$109$	−790.000	−0.694204	−0.347102	−	0.937827i	$-0.612834\pi$
$109$	−790.000	−0.694204	−0.347102	+	0.937827i	$0.612834\pi$
$110$	0	0
$111$	690.000	0.590017
$112$	144.000	0.121489
$113$	1922.00	1.60006	0.800029	−	0.599961i	$-0.204817\pi$
$113$	1922.00	1.60006	0.800029	+	0.599961i	$0.204817\pi$
$114$	114.000	0.0936586
$115$	0	0
$116$	1036.00	0.829226
$117$	342.000	0.270239
$118$	−238.000	−0.185675
$119$	684.000	0.526909
$120$	0	0
$121$	2513.00	1.88805
$122$	−226.000	−0.167714
$123$	−1365.00	−1.00063
$124$	480.000	0.347623
$125$	0	0
$126$	−162.000	−0.114541
$127$	−1972.00	−1.37785	−0.688924	−	0.724834i	$-0.741917\pi$
$127$	−1972.00	−1.37785	−0.688924	+	0.724834i	$0.741917\pi$
$128$	−384.000	−0.265165
$129$	1020.00	0.696170
$130$	0	0
$131$	1086.00	0.724307	0.362154	−	0.932118i	$-0.382042\pi$
$131$	1086.00	0.724307	0.362154	+	0.932118i	$0.382042\pi$
$132$	−744.000	−0.490582
$133$	171.000	0.111486
$134$	936.000	0.603419
$135$	0	0
$136$	1824.00	1.15005
$137$	−2568.00	−1.60145	−0.800726	−	0.599030i	$-0.795553\pi$
$137$	−2568.00	−1.60145	−0.800726	+	0.599030i	$0.795553\pi$
$138$	252.000	0.155447
$139$	1547.00	0.943992	0.471996	−	0.881601i	$-0.343534\pi$
$139$	1547.00	0.943992	0.471996	+	0.881601i	$0.343534\pi$
$140$	0	0
$141$	−672.000	−0.401366
$142$	1990.00	1.17604
$143$	−2356.00	−1.37775
$144$	−144.000	−0.0833333
$145$	0	0
$146$	−542.000	−0.307235
$147$	786.000	0.441008
$148$	920.000	0.510970
$149$	−1102.00	−0.605902	−0.302951	−	0.953006i	$-0.597972\pi$
$149$	−1102.00	−0.605902	−0.302951	+	0.953006i	$0.597972\pi$
$150$	0	0
$151$	2524.00	1.36027	0.680133	−	0.733089i	$-0.261922\pi$
$151$	2524.00	1.36027	0.680133	+	0.733089i	$0.261922\pi$
$152$	456.000	0.243332
$153$	−684.000	−0.361426
$154$	1116.00	0.583960
$155$	0	0
$156$	456.000	0.234033
$157$	−2381.00	−1.21035	−0.605174	−	0.796094i	$-0.706896\pi$
$157$	−2381.00	−1.21035	−0.605174	+	0.796094i	$0.706896\pi$
$158$	636.000	0.320237
$159$	183.000	0.0912757
$160$	0	0
$161$	378.000	0.185035
$162$	162.000	0.0785674
$163$	−953.000	−0.457943	−0.228972	−	0.973433i	$-0.573536\pi$
$163$	−953.000	−0.457943	−0.228972	+	0.973433i	$0.573536\pi$
$164$	−1820.00	−0.866574
$165$	0	0
$166$	−672.000	−0.314201
$167$	2889.00	1.33867	0.669334	−	0.742962i	$-0.266580\pi$
$167$	2889.00	1.33867	0.669334	+	0.742962i	$0.266580\pi$
$168$	−648.000	−0.297585
$169$	−753.000	−0.342740
$170$	0	0
$171$	−171.000	−0.0764719
$172$	1360.00	0.602901
$173$	−1851.00	−0.813462	−0.406731	−	0.913548i	$-0.633331\pi$
$173$	−1851.00	−0.813462	−0.406731	+	0.913548i	$0.633331\pi$
$174$	1554.00	0.677060
$175$	0	0
$176$	992.000	0.424857
$177$	357.000	0.151603
$178$	−1890.00	−0.795851
$179$	2407.00	1.00507	0.502535	−	0.864557i	$-0.332401\pi$
$179$	2407.00	1.00507	0.502535	+	0.864557i	$0.332401\pi$
$180$	0	0
$181$	−340.000	−0.139624	−0.0698122	−	0.997560i	$-0.522240\pi$
$181$	−340.000	−0.139624	−0.0698122	+	0.997560i	$0.522240\pi$
$182$	−684.000	−0.278579
$183$	339.000	0.136938
$184$	1008.00	0.403863
$185$	0	0
$186$	720.000	0.283833
$187$	4712.00	1.84265
$188$	−896.000	−0.347593
$189$	243.000	0.0935220
$190$	0	0
$191$	2552.00	0.966787	0.483393	−	0.875403i	$-0.339404\pi$
$191$	2552.00	0.966787	0.483393	+	0.875403i	$0.339404\pi$
$192$	−1344.00	−0.505181
$193$	3442.00	1.28373	0.641867	−	0.766816i	$-0.278160\pi$
$193$	3442.00	1.28373	0.641867	+	0.766816i	$0.278160\pi$
$194$	−1744.00	−0.645422
$195$	0	0
$196$	1048.00	0.381924
$197$	456.000	0.164917	0.0824585	−	0.996594i	$-0.473723\pi$
$197$	456.000	0.164917	0.0824585	+	0.996594i	$0.473723\pi$
$198$	−1116.00	−0.400559
$199$	1363.00	0.485530	0.242765	−	0.970085i	$-0.421946\pi$
$199$	1363.00	0.485530	0.242765	+	0.970085i	$0.421946\pi$
$200$	0	0
$201$	−1404.00	−0.492689
$202$	1908.00	0.664586
$203$	2331.00	0.805932
$204$	−912.000	−0.313004
$205$	0	0
$206$	2492.00	0.842844
$207$	−378.000	−0.126922
$208$	−608.000	−0.202679
$209$	1178.00	0.389875
$210$	0	0
$211$	1988.00	0.648624	0.324312	−	0.945950i	$-0.394867\pi$
$211$	1988.00	0.648624	0.324312	+	0.945950i	$0.394867\pi$
$212$	244.000	0.0790471
$213$	−2985.00	−0.960229
$214$	3162.00	1.01005
$215$	0	0
$216$	648.000	0.204124
$217$	1080.00	0.337858
$218$	−1580.00	−0.490877
$219$	813.000	0.250856
$220$	0	0
$221$	−2888.00	−0.879040
$222$	1380.00	0.417205
$223$	−5168.00	−1.55191	−0.775953	−	0.630791i	$-0.782730\pi$
$223$	−5168.00	−1.55191	−0.775953	+	0.630791i	$0.782730\pi$
$224$	−1440.00	−0.429527
$225$	0	0
$226$	3844.00	1.13141
$227$	−5171.00	−1.51194	−0.755972	−	0.654604i	$-0.772835\pi$
$227$	−5171.00	−1.51194	−0.755972	+	0.654604i	$0.772835\pi$
$228$	−228.000	−0.0662266
$229$	2874.00	0.829342	0.414671	−	0.909971i	$-0.363897\pi$
$229$	2874.00	0.829342	0.414671	+	0.909971i	$0.363897\pi$
$230$	0	0
$231$	−1674.00	−0.476801
$232$	6216.00	1.75905
$233$	−2484.00	−0.698422	−0.349211	−	0.937044i	$-0.613550\pi$
$233$	−2484.00	−0.698422	−0.349211	+	0.937044i	$0.613550\pi$
$234$	684.000	0.191088
$235$	0	0
$236$	476.000	0.131292
$237$	−954.000	−0.261472
$238$	1368.00	0.372581
$239$	4066.00	1.10045	0.550225	−	0.835016i	$-0.314542\pi$
$239$	4066.00	1.10045	0.550225	+	0.835016i	$0.314542\pi$
$240$	0	0
$241$	1246.00	0.333037	0.166518	−	0.986038i	$-0.446747\pi$
$241$	1246.00	0.333037	0.166518	+	0.986038i	$0.446747\pi$
$242$	5026.00	1.33506
$243$	−243.000	−0.0641500
$244$	452.000	0.118592
$245$	0	0
$246$	−2730.00	−0.707555
$247$	−722.000	−0.185991
$248$	2880.00	0.737420
$249$	1008.00	0.256544
$250$	0	0
$251$	−3288.00	−0.826840	−0.413420	−	0.910541i	$-0.635666\pi$
$251$	−3288.00	−0.826840	−0.413420	+	0.910541i	$0.635666\pi$
$252$	324.000	0.0809924
$253$	2604.00	0.647083
$254$	−3944.00	−0.974286
$255$	0	0
$256$	−4352.00	−1.06250
$257$	6879.00	1.66965	0.834825	−	0.550515i	$-0.185569\pi$
$257$	6879.00	1.66965	0.834825	+	0.550515i	$0.185569\pi$
$258$	2040.00	0.492267
$259$	2070.00	0.496616
$260$	0	0
$261$	−2331.00	−0.552817
$262$	2172.00	0.512163
$263$	4492.00	1.05319	0.526594	−	0.850117i	$-0.323469\pi$
$263$	4492.00	1.05319	0.526594	+	0.850117i	$0.323469\pi$
$264$	−4464.00	−1.04068
$265$	0	0
$266$	342.000	0.0788322
$267$	2835.00	0.649810
$268$	−1872.00	−0.426681
$269$	7846.00	1.77836	0.889180	−	0.457557i	$-0.151275\pi$
$269$	7846.00	1.77836	0.889180	+	0.457557i	$0.151275\pi$
$270$	0	0
$271$	−203.000	−0.0455032	−0.0227516	−	0.999741i	$-0.507243\pi$
$271$	−203.000	−0.0455032	−0.0227516	+	0.999741i	$0.507243\pi$
$272$	1216.00	0.271069
$273$	1026.00	0.227459
$274$	−5136.00	−1.13240
$275$	0	0
$276$	−504.000	−0.109918
$277$	−8331.00	−1.80708	−0.903540	−	0.428503i	$-0.859041\pi$
$277$	−8331.00	−1.80708	−0.903540	+	0.428503i	$0.859041\pi$
$278$	3094.00	0.667503
$279$	−1080.00	−0.231749
$280$	0	0
$281$	4578.00	0.971888	0.485944	−	0.873990i	$-0.338476\pi$
$281$	4578.00	0.971888	0.485944	+	0.873990i	$0.338476\pi$
$282$	−1344.00	−0.283809
$283$	−5412.00	−1.13678	−0.568392	−	0.822758i	$-0.692434\pi$
$283$	−5412.00	−1.13678	−0.568392	+	0.822758i	$0.692434\pi$
$284$	−3980.00	−0.831583
$285$	0	0
$286$	−4712.00	−0.974218
$287$	−4095.00	−0.842231
$288$	1440.00	0.294628
$289$	863.000	0.175656
$290$	0	0
$291$	2616.00	0.526985
$292$	1084.00	0.217248
$293$	−3282.00	−0.654391	−0.327195	−	0.944957i	$-0.606104\pi$
$293$	−3282.00	−0.654391	−0.327195	+	0.944957i	$0.606104\pi$
$294$	1572.00	0.311840
$295$	0	0
$296$	5520.00	1.08393
$297$	1674.00	0.327055
$298$	−2204.00	−0.428437
$299$	−1596.00	−0.308693
$300$	0	0
$301$	3060.00	0.585965
$302$	5048.00	0.961854
$303$	−2862.00	−0.542632
$304$	304.000	0.0573539
$305$	0	0
$306$	−1368.00	−0.255567
$307$	−7030.00	−1.30692	−0.653458	−	0.756963i	$-0.726682\pi$
$307$	−7030.00	−1.30692	−0.653458	+	0.756963i	$0.726682\pi$
$308$	−2232.00	−0.412922
$309$	−3738.00	−0.688179
$310$	0	0
$311$	552.000	0.100646	0.0503232	−	0.998733i	$-0.483975\pi$
$311$	552.000	0.100646	0.0503232	+	0.998733i	$0.483975\pi$
$312$	2736.00	0.496460
$313$	−9182.00	−1.65814	−0.829069	−	0.559146i	$-0.811129\pi$
$313$	−9182.00	−1.65814	−0.829069	+	0.559146i	$0.811129\pi$
$314$	−4762.00	−0.855845
$315$	0	0
$316$	−1272.00	−0.226442
$317$	3733.00	0.661407	0.330704	−	0.943735i	$-0.392714\pi$
$317$	3733.00	0.661407	0.330704	+	0.943735i	$0.392714\pi$
$318$	366.000	0.0645417
$319$	16058.0	2.81842
$320$	0	0
$321$	−4743.00	−0.824699
$322$	756.000	0.130839
$323$	1444.00	0.248750
$324$	−324.000	−0.0555556
$325$	0	0
$326$	−1906.00	−0.323815
$327$	2370.00	0.400799
$328$	−10920.0	−1.83828
$329$	−2016.00	−0.337829
$330$	0	0
$331$	10962.0	1.82032	0.910160	−	0.414257i	$-0.135958\pi$
$331$	10962.0	1.82032	0.910160	+	0.414257i	$0.135958\pi$
$332$	1344.00	0.222173
$333$	−2070.00	−0.340647
$334$	5778.00	0.946581
$335$	0	0
$336$	−432.000	−0.0701415
$337$	−3838.00	−0.620383	−0.310192	−	0.950674i	$-0.600393\pi$
$337$	−3838.00	−0.620383	−0.310192	+	0.950674i	$0.600393\pi$
$338$	−1506.00	−0.242354
$339$	−5766.00	−0.923794
$340$	0	0
$341$	7440.00	1.18152
$342$	−342.000	−0.0540738
$343$	5445.00	0.857150
$344$	8160.00	1.27895
$345$	0	0
$346$	−3702.00	−0.575204
$347$	−316.000	−0.0488869	−0.0244435	−	0.999701i	$-0.507781\pi$
$347$	−316.000	−0.0488869	−0.0244435	+	0.999701i	$0.507781\pi$
$348$	−3108.00	−0.478754
$349$	−5027.00	−0.771029	−0.385515	−	0.922702i	$-0.625976\pi$
$349$	−5027.00	−0.771029	−0.385515	+	0.922702i	$0.625976\pi$
$350$	0	0
$351$	−1026.00	−0.156022
$352$	−9920.00	−1.50210
$353$	−2256.00	−0.340155	−0.170078	−	0.985431i	$-0.554402\pi$
$353$	−2256.00	−0.340155	−0.170078	+	0.985431i	$0.554402\pi$
$354$	714.000	0.107200
$355$	0	0
$356$	3780.00	0.562752
$357$	−2052.00	−0.304211
$358$	4814.00	0.710692
$359$	−7698.00	−1.13171	−0.565856	−	0.824504i	$-0.691454\pi$
$359$	−7698.00	−1.13171	−0.565856	+	0.824504i	$0.691454\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	−680.000	−0.0987293
$363$	−7539.00	−1.09007
$364$	1368.00	0.196985
$365$	0	0
$366$	678.000	0.0968296
$367$	−2880.00	−0.409632	−0.204816	−	0.978801i	$-0.565660\pi$
$367$	−2880.00	−0.409632	−0.204816	+	0.978801i	$0.565660\pi$
$368$	672.000	0.0951914
$369$	4095.00	0.577716
$370$	0	0
$371$	549.000	0.0768266
$372$	−1440.00	−0.200700
$373$	−7258.00	−1.00752	−0.503760	−	0.863844i	$-0.668050\pi$
$373$	−7258.00	−1.00752	−0.503760	+	0.863844i	$0.668050\pi$
$374$	9424.00	1.30295
$375$	0	0
$376$	−5376.00	−0.737356
$377$	−9842.00	−1.34453
$378$	486.000	0.0661300
$379$	5830.00	0.790150	0.395075	−	0.918649i	$-0.370719\pi$
$379$	5830.00	0.790150	0.395075	+	0.918649i	$0.370719\pi$
$380$	0	0
$381$	5916.00	0.795501
$382$	5104.00	0.683621
$383$	1869.00	0.249351	0.124676	−	0.992198i	$-0.460211\pi$
$383$	1869.00	0.249351	0.124676	+	0.992198i	$0.460211\pi$
$384$	1152.00	0.153093
$385$	0	0
$386$	6884.00	0.907737
$387$	−3060.00	−0.401934
$388$	3488.00	0.456382
$389$	−14970.0	−1.95118	−0.975590	−	0.219599i	$-0.929525\pi$
$389$	−14970.0	−1.95118	−0.975590	+	0.219599i	$0.929525\pi$
$390$	0	0
$391$	3192.00	0.412855
$392$	6288.00	0.810184
$393$	−3258.00	−0.418179
$394$	912.000	0.116614
$395$	0	0
$396$	2232.00	0.283238
$397$	−934.000	−0.118076	−0.0590379	−	0.998256i	$-0.518803\pi$
$397$	−934.000	−0.118076	−0.0590379	+	0.998256i	$0.518803\pi$
$398$	2726.00	0.343322
$399$	−513.000	−0.0643662
$400$	0	0
$401$	−14778.0	−1.84034	−0.920172	−	0.391514i	$-0.871951\pi$
$401$	−14778.0	−1.84034	−0.920172	+	0.391514i	$0.871951\pi$
$402$	−2808.00	−0.348384
$403$	−4560.00	−0.563647
$404$	−3816.00	−0.469933
$405$	0	0
$406$	4662.00	0.569880
$407$	14260.0	1.73671
$408$	−5472.00	−0.663981
$409$	−4204.00	−0.508250	−0.254125	−	0.967171i	$-0.581788\pi$
$409$	−4204.00	−0.508250	−0.254125	+	0.967171i	$0.581788\pi$
$410$	0	0
$411$	7704.00	0.924599
$412$	−4984.00	−0.595981
$413$	1071.00	0.127604
$414$	−756.000	−0.0897473
$415$	0	0
$416$	6080.00	0.716578
$417$	−4641.00	−0.545014
$418$	2356.00	0.275684
$419$	−13744.0	−1.60248	−0.801239	−	0.598344i	$-0.795825\pi$
$419$	−13744.0	−1.60248	−0.801239	+	0.598344i	$0.795825\pi$
$420$	0	0
$421$	−3284.00	−0.380172	−0.190086	−	0.981767i	$-0.560877\pi$
$421$	−3284.00	−0.380172	−0.190086	+	0.981767i	$0.560877\pi$
$422$	3976.00	0.458646
$423$	2016.00	0.231729
$424$	1464.00	0.167684
$425$	0	0
$426$	−5970.00	−0.678985
$427$	1017.00	0.115260
$428$	−6324.00	−0.714210
$429$	7068.00	0.795446
$430$	0	0
$431$	14541.0	1.62509	0.812547	−	0.582896i	$-0.198080\pi$
$431$	14541.0	1.62509	0.812547	+	0.582896i	$0.198080\pi$
$432$	432.000	0.0481125
$433$	−1588.00	−0.176246	−0.0881229	−	0.996110i	$-0.528087\pi$
$433$	−1588.00	−0.176246	−0.0881229	+	0.996110i	$0.528087\pi$
$434$	2160.00	0.238902
$435$	0	0
$436$	3160.00	0.347102
$437$	798.000	0.0873536
$438$	1626.00	0.177382
$439$	18070.0	1.96454	0.982271	−	0.187466i	$-0.0600274\pi$
$439$	18070.0	1.96454	0.982271	+	0.187466i	$0.0600274\pi$
$440$	0	0
$441$	−2358.00	−0.254616
$442$	−5776.00	−0.621575
$443$	6378.00	0.684036	0.342018	−	0.939693i	$-0.388890\pi$
$443$	6378.00	0.684036	0.342018	+	0.939693i	$0.388890\pi$
$444$	−2760.00	−0.295009
$445$	0	0
$446$	−10336.0	−1.09736
$447$	3306.00	0.349818
$448$	−4032.00	−0.425210
$449$	−5075.00	−0.533417	−0.266708	−	0.963777i	$-0.585936\pi$
$449$	−5075.00	−0.533417	−0.266708	+	0.963777i	$0.585936\pi$
$450$	0	0
$451$	−28210.0	−2.94536
$452$	−7688.00	−0.800029
$453$	−7572.00	−0.785350
$454$	−10342.0	−1.06911
$455$	0	0
$456$	−1368.00	−0.140488
$457$	1127.00	0.115359	0.0576793	−	0.998335i	$-0.481630\pi$
$457$	1127.00	0.115359	0.0576793	+	0.998335i	$0.481630\pi$
$458$	5748.00	0.586433
$459$	2052.00	0.208669
$460$	0	0
$461$	−6894.00	−0.696498	−0.348249	−	0.937402i	$-0.613224\pi$
$461$	−6894.00	−0.696498	−0.348249	+	0.937402i	$0.613224\pi$
$462$	−3348.00	−0.337149
$463$	−4116.00	−0.413146	−0.206573	−	0.978431i	$-0.566231\pi$
$463$	−4116.00	−0.413146	−0.206573	+	0.978431i	$0.566231\pi$
$464$	4144.00	0.414613
$465$	0	0
$466$	−4968.00	−0.493859
$467$	−578.000	−0.0572733	−0.0286367	−	0.999590i	$-0.509117\pi$
$467$	−578.000	−0.0572733	−0.0286367	+	0.999590i	$0.509117\pi$
$468$	−1368.00	−0.135119
$469$	−4212.00	−0.414695
$470$	0	0
$471$	7143.00	0.698794
$472$	2856.00	0.278513
$473$	21080.0	2.04917
$474$	−1908.00	−0.184889
$475$	0	0
$476$	−2736.00	−0.263455
$477$	−549.000	−0.0526981
$478$	8132.00	0.778136
$479$	−12978.0	−1.23795	−0.618977	−	0.785409i	$-0.712453\pi$
$479$	−12978.0	−1.23795	−0.618977	+	0.785409i	$0.712453\pi$
$480$	0	0
$481$	−8740.00	−0.828502
$482$	2492.00	0.235493
$483$	−1134.00	−0.106830
$484$	−10052.0	−0.944027
$485$	0	0
$486$	−486.000	−0.0453609
$487$	−12614.0	−1.17371	−0.586853	−	0.809693i	$-0.699633\pi$
$487$	−12614.0	−1.17371	−0.586853	+	0.809693i	$0.699633\pi$
$488$	2712.00	0.251571
$489$	2859.00	0.264394
$490$	0	0
$491$	−9136.00	−0.839719	−0.419859	−	0.907589i	$-0.637921\pi$
$491$	−9136.00	−0.839719	−0.419859	+	0.907589i	$0.637921\pi$
$492$	5460.00	0.500317
$493$	19684.0	1.79822
$494$	−1444.00	−0.131515
$495$	0	0
$496$	1920.00	0.173812
$497$	−8955.00	−0.808223
$498$	2016.00	0.181404
$499$	16483.0	1.47872	0.739359	−	0.673311i	$-0.235129\pi$
$499$	16483.0	1.47872	0.739359	+	0.673311i	$0.235129\pi$
$500$	0	0
$501$	−8667.00	−0.772880
$502$	−6576.00	−0.584664
$503$	−10394.0	−0.921363	−0.460681	−	0.887566i	$-0.652395\pi$
$503$	−10394.0	−0.921363	−0.460681	+	0.887566i	$0.652395\pi$
$504$	1944.00	0.171811
$505$	0	0
$506$	5208.00	0.457557
$507$	2259.00	0.197881
$508$	7888.00	0.688924
$509$	1759.00	0.153175	0.0765877	−	0.997063i	$-0.475597\pi$
$509$	1759.00	0.153175	0.0765877	+	0.997063i	$0.475597\pi$
$510$	0	0
$511$	2439.00	0.211145
$512$	−5632.00	−0.486136
$513$	513.000	0.0441511
$514$	13758.0	1.18062
$515$	0	0
$516$	−4080.00	−0.348085
$517$	−13888.0	−1.18142
$518$	4140.00	0.351161
$519$	5553.00	0.469652
$520$	0	0
$521$	4403.00	0.370247	0.185124	−	0.982715i	$-0.440731\pi$
$521$	4403.00	0.370247	0.185124	+	0.982715i	$0.440731\pi$
$522$	−4662.00	−0.390901
$523$	10136.0	0.847450	0.423725	−	0.905791i	$-0.360722\pi$
$523$	10136.0	0.847450	0.423725	+	0.905791i	$0.360722\pi$
$524$	−4344.00	−0.362154
$525$	0	0
$526$	8984.00	0.744717
$527$	9120.00	0.753840
$528$	−2976.00	−0.245291
$529$	−10403.0	−0.855018
$530$	0	0
$531$	−1071.00	−0.0875281
$532$	−684.000	−0.0557428
$533$	17290.0	1.40509
$534$	5670.00	0.459485
$535$	0	0
$536$	−11232.0	−0.905128
$537$	−7221.00	−0.580278
$538$	15692.0	1.25749
$539$	16244.0	1.29811
$540$	0	0
$541$	−1842.00	−0.146384	−0.0731920	−	0.997318i	$-0.523319\pi$
$541$	−1842.00	−0.146384	−0.0731920	+	0.997318i	$0.523319\pi$
$542$	−406.000	−0.0321756
$543$	1020.00	0.0806121
$544$	−12160.0	−0.958374
$545$	0	0
$546$	2052.00	0.160838
$547$	−4738.00	−0.370351	−0.185176	−	0.982705i	$-0.559285\pi$
$547$	−4738.00	−0.370351	−0.185176	+	0.982705i	$0.559285\pi$
$548$	10272.0	0.800726
$549$	−1017.00	−0.0790610
$550$	0	0
$551$	4921.00	0.380475
$552$	−3024.00	−0.233170
$553$	−2862.00	−0.220081
$554$	−16662.0	−1.27780
$555$	0	0
$556$	−6188.00	−0.471996
$557$	−7596.00	−0.577833	−0.288916	−	0.957354i	$-0.593295\pi$
$557$	−7596.00	−0.577833	−0.288916	+	0.957354i	$0.593295\pi$
$558$	−2160.00	−0.163871
$559$	−12920.0	−0.977563
$560$	0	0
$561$	−14136.0	−1.06385
$562$	9156.00	0.687229
$563$	12411.0	0.929061	0.464530	−	0.885557i	$-0.346223\pi$
$563$	12411.0	0.929061	0.464530	+	0.885557i	$0.346223\pi$
$564$	2688.00	0.200683
$565$	0	0
$566$	−10824.0	−0.803828
$567$	−729.000	−0.0539949
$568$	−23880.0	−1.76405
$569$	9345.00	0.688511	0.344256	−	0.938876i	$-0.388131\pi$
$569$	9345.00	0.688511	0.344256	+	0.938876i	$0.388131\pi$
$570$	0	0
$571$	−3319.00	−0.243250	−0.121625	−	0.992576i	$-0.538811\pi$
$571$	−3319.00	−0.243250	−0.121625	+	0.992576i	$0.538811\pi$
$572$	9424.00	0.688876
$573$	−7656.00	−0.558174
$574$	−8190.00	−0.595547
$575$	0	0
$576$	4032.00	0.291667
$577$	−1106.00	−0.0797979	−0.0398989	−	0.999204i	$-0.512704\pi$
$577$	−1106.00	−0.0797979	−0.0398989	+	0.999204i	$0.512704\pi$
$578$	1726.00	0.124208
$579$	−10326.0	−0.741164
$580$	0	0
$581$	3024.00	0.215932
$582$	5232.00	0.372635
$583$	3782.00	0.268670
$584$	6504.00	0.460852
$585$	0	0
$586$	−6564.00	−0.462724
$587$	−864.000	−0.0607514	−0.0303757	−	0.999539i	$-0.509670\pi$
$587$	−864.000	−0.0607514	−0.0303757	+	0.999539i	$0.509670\pi$
$588$	−3144.00	−0.220504
$589$	2280.00	0.159500
$590$	0	0
$591$	−1368.00	−0.0952149
$592$	3680.00	0.255485
$593$	10846.0	0.751082	0.375541	−	0.926806i	$-0.377457\pi$
$593$	10846.0	0.751082	0.375541	+	0.926806i	$0.377457\pi$
$594$	3348.00	0.231263
$595$	0	0
$596$	4408.00	0.302951
$597$	−4089.00	−0.280321
$598$	−3192.00	−0.218279
$599$	5264.00	0.359067	0.179534	−	0.983752i	$-0.442541\pi$
$599$	5264.00	0.359067	0.179534	+	0.983752i	$0.442541\pi$
$600$	0	0
$601$	9050.00	0.614238	0.307119	−	0.951671i	$-0.400635\pi$
$601$	9050.00	0.614238	0.307119	+	0.951671i	$0.400635\pi$
$602$	6120.00	0.414340
$603$	4212.00	0.284454
$604$	−10096.0	−0.680133
$605$	0	0
$606$	−5724.00	−0.383699
$607$	12758.0	0.853099	0.426550	−	0.904464i	$-0.359729\pi$
$607$	12758.0	0.853099	0.426550	+	0.904464i	$0.359729\pi$
$608$	−3040.00	−0.202777
$609$	−6993.00	−0.465305
$610$	0	0
$611$	8512.00	0.563598
$612$	2736.00	0.180713
$613$	−1789.00	−0.117874	−0.0589372	−	0.998262i	$-0.518771\pi$
$613$	−1789.00	−0.117874	−0.0589372	+	0.998262i	$0.518771\pi$
$614$	−14060.0	−0.924129
$615$	0	0
$616$	−13392.0	−0.875940
$617$	6288.00	0.410284	0.205142	−	0.978732i	$-0.434234\pi$
$617$	6288.00	0.410284	0.205142	+	0.978732i	$0.434234\pi$
$618$	−7476.00	−0.486616
$619$	−28681.0	−1.86234	−0.931169	−	0.364589i	$-0.881210\pi$
$619$	−28681.0	−1.86234	−0.931169	+	0.364589i	$0.881210\pi$
$620$	0	0
$621$	1134.00	0.0732783
$622$	1104.00	0.0711678
$623$	8505.00	0.546943
$624$	1824.00	0.117017
$625$	0	0
$626$	−18364.0	−1.17248
$627$	−3534.00	−0.225095
$628$	9524.00	0.605174
$629$	17480.0	1.10807
$630$	0	0
$631$	23848.0	1.50455	0.752277	−	0.658847i	$-0.228955\pi$
$631$	23848.0	1.50455	0.752277	+	0.658847i	$0.228955\pi$
$632$	−7632.00	−0.480355
$633$	−5964.00	−0.374483
$634$	7466.00	0.467686
$635$	0	0
$636$	−732.000	−0.0456379
$637$	−9956.00	−0.619264
$638$	32116.0	1.99292
$639$	8955.00	0.554389
$640$	0	0
$641$	23526.0	1.44964	0.724821	−	0.688937i	$-0.241922\pi$
$641$	23526.0	1.44964	0.724821	+	0.688937i	$0.241922\pi$
$642$	−9486.00	−0.583150
$643$	26853.0	1.64693	0.823467	−	0.567364i	$-0.192037\pi$
$643$	26853.0	1.64693	0.823467	+	0.567364i	$0.192037\pi$
$644$	−1512.00	−0.0925173
$645$	0	0
$646$	2888.00	0.175893
$647$	1892.00	0.114965	0.0574824	−	0.998347i	$-0.481693\pi$
$647$	1892.00	0.114965	0.0574824	+	0.998347i	$0.481693\pi$
$648$	−1944.00	−0.117851
$649$	7378.00	0.446243
$650$	0	0
$651$	−3240.00	−0.195062
$652$	3812.00	0.228972
$653$	−3974.00	−0.238154	−0.119077	−	0.992885i	$-0.537994\pi$
$653$	−3974.00	−0.238154	−0.119077	+	0.992885i	$0.537994\pi$
$654$	4740.00	0.283408
$655$	0	0
$656$	−7280.00	−0.433287
$657$	−2439.00	−0.144832
$658$	−4032.00	−0.238881
$659$	−28796.0	−1.70217	−0.851087	−	0.525024i	$-0.824056\pi$
$659$	−28796.0	−1.70217	−0.851087	+	0.525024i	$0.824056\pi$
$660$	0	0
$661$	11788.0	0.693646	0.346823	−	0.937931i	$-0.387260\pi$
$661$	11788.0	0.693646	0.346823	+	0.937931i	$0.387260\pi$
$662$	21924.0	1.28716
$663$	8664.00	0.507514
$664$	8064.00	0.471301
$665$	0	0
$666$	−4140.00	−0.240873
$667$	10878.0	0.631481
$668$	−11556.0	−0.669334
$669$	15504.0	0.895993
$670$	0	0
$671$	7006.00	0.403075
$672$	4320.00	0.247988
$673$	−1518.00	−0.0869459	−0.0434730	−	0.999055i	$-0.513842\pi$
$673$	−1518.00	−0.0869459	−0.0434730	+	0.999055i	$0.513842\pi$
$674$	−7676.00	−0.438677
$675$	0	0
$676$	3012.00	0.171370
$677$	28095.0	1.59495	0.797473	−	0.603355i	$-0.206170\pi$
$677$	28095.0	1.59495	0.797473	+	0.603355i	$0.206170\pi$
$678$	−11532.0	−0.653221
$679$	7848.00	0.443562
$680$	0	0
$681$	15513.0	0.872921
$682$	14880.0	0.835461
$683$	30081.0	1.68524	0.842619	−	0.538510i	$-0.181013\pi$
$683$	30081.0	1.68524	0.842619	+	0.538510i	$0.181013\pi$
$684$	684.000	0.0382360
$685$	0	0
$686$	10890.0	0.606096
$687$	−8622.00	−0.478821
$688$	5440.00	0.301451
$689$	−2318.00	−0.128169
$690$	0	0
$691$	−22976.0	−1.26490	−0.632452	−	0.774600i	$-0.717951\pi$
$691$	−22976.0	−1.26490	−0.632452	+	0.774600i	$0.717951\pi$
$692$	7404.00	0.406731
$693$	5022.00	0.275281
$694$	−632.000	−0.0345683
$695$	0	0
$696$	−18648.0	−1.01559
$697$	−34580.0	−1.87921
$698$	−10054.0	−0.545200
$699$	7452.00	0.403234
$700$	0	0
$701$	−6574.00	−0.354203	−0.177102	−	0.984193i	$-0.556672\pi$
$701$	−6574.00	−0.354203	−0.177102	+	0.984193i	$0.556672\pi$
$702$	−2052.00	−0.110324
$703$	4370.00	0.234449
$704$	−27776.0	−1.48700
$705$	0	0
$706$	−4512.00	−0.240526
$707$	−8586.00	−0.456732
$708$	−1428.00	−0.0758016
$709$	−7373.00	−0.390548	−0.195274	−	0.980749i	$-0.562560\pi$
$709$	−7373.00	−0.390548	−0.195274	+	0.980749i	$0.562560\pi$
$710$	0	0
$711$	2862.00	0.150961
$712$	22680.0	1.19378
$713$	5040.00	0.264726
$714$	−4104.00	−0.215110
$715$	0	0
$716$	−9628.00	−0.502535
$717$	−12198.0	−0.635345
$718$	−15396.0	−0.800242
$719$	−13692.0	−0.710188	−0.355094	−	0.934831i	$-0.615551\pi$
$719$	−13692.0	−0.710188	−0.355094	+	0.934831i	$0.615551\pi$
$720$	0	0
$721$	−11214.0	−0.579239
$722$	722.000	0.0372161
$723$	−3738.00	−0.192279
$724$	1360.00	0.0698122
$725$	0	0
$726$	−15078.0	−0.770795
$727$	−27113.0	−1.38317	−0.691586	−	0.722294i	$-0.743088\pi$
$727$	−27113.0	−1.38317	−0.691586	+	0.722294i	$0.743088\pi$
$728$	8208.00	0.417869
$729$	729.000	0.0370370
$730$	0	0
$731$	25840.0	1.30742
$732$	−1356.00	−0.0684689
$733$	−5273.00	−0.265706	−0.132853	−	0.991136i	$-0.542414\pi$
$733$	−5273.00	−0.265706	−0.132853	+	0.991136i	$0.542414\pi$
$734$	−5760.00	−0.289653
$735$	0	0
$736$	−6720.00	−0.336552
$737$	−29016.0	−1.45023
$738$	8190.00	0.408507
$739$	37261.0	1.85476	0.927380	−	0.374120i	$-0.122055\pi$
$739$	37261.0	1.85476	0.927380	+	0.374120i	$0.122055\pi$
$740$	0	0
$741$	2166.00	0.107382
$742$	1098.00	0.0543246
$743$	8891.00	0.439003	0.219501	−	0.975612i	$-0.429557\pi$
$743$	8891.00	0.439003	0.219501	+	0.975612i	$0.429557\pi$
$744$	−8640.00	−0.425750
$745$	0	0
$746$	−14516.0	−0.712424
$747$	−3024.00	−0.148116
$748$	−18848.0	−0.921325
$749$	−14229.0	−0.694147
$750$	0	0
$751$	−29660.0	−1.44116	−0.720578	−	0.693373i	$-0.756124\pi$
$751$	−29660.0	−1.44116	−0.720578	+	0.693373i	$0.756124\pi$
$752$	−3584.00	−0.173797
$753$	9864.00	0.477376
$754$	−19684.0	−0.950728
$755$	0	0
$756$	−972.000	−0.0467610
$757$	15035.0	0.721871	0.360936	−	0.932591i	$-0.382457\pi$
$757$	15035.0	0.721871	0.360936	+	0.932591i	$0.382457\pi$
$758$	11660.0	0.558721
$759$	−7812.00	−0.373594
$760$	0	0
$761$	−5604.00	−0.266945	−0.133472	−	0.991053i	$-0.542613\pi$
$761$	−5604.00	−0.266945	−0.133472	+	0.991053i	$0.542613\pi$
$762$	11832.0	0.562504
$763$	7110.00	0.337352
$764$	−10208.0	−0.483393
$765$	0	0
$766$	3738.00	0.176318
$767$	−4522.00	−0.212881
$768$	13056.0	0.613435
$769$	21311.0	0.999342	0.499671	−	0.866215i	$-0.333454\pi$
$769$	21311.0	0.999342	0.499671	+	0.866215i	$0.333454\pi$
$770$	0	0
$771$	−20637.0	−0.963973
$772$	−13768.0	−0.641867
$773$	−4923.00	−0.229066	−0.114533	−	0.993419i	$-0.536537\pi$
$773$	−4923.00	−0.229066	−0.114533	+	0.993419i	$0.536537\pi$
$774$	−6120.00	−0.284210
$775$	0	0
$776$	20928.0	0.968133
$777$	−6210.00	−0.286721
$778$	−29940.0	−1.37969
$779$	−8645.00	−0.397611
$780$	0	0
$781$	−61690.0	−2.82643
$782$	6384.00	0.291933
$783$	6993.00	0.319169
$784$	4192.00	0.190962
$785$	0	0
$786$	−6516.00	−0.295697
$787$	−4354.00	−0.197209	−0.0986044	−	0.995127i	$-0.531438\pi$
$787$	−4354.00	−0.197209	−0.0986044	+	0.995127i	$0.531438\pi$
$788$	−1824.00	−0.0824585
$789$	−13476.0	−0.608059
$790$	0	0
$791$	−17298.0	−0.777555
$792$	13392.0	0.600838
$793$	−4294.00	−0.192288
$794$	−1868.00	−0.0834922
$795$	0	0
$796$	−5452.00	−0.242765
$797$	−26063.0	−1.15834	−0.579171	−	0.815206i	$-0.696624\pi$
$797$	−26063.0	−1.15834	−0.579171	+	0.815206i	$0.696624\pi$
$798$	−1026.00	−0.0455138
$799$	−17024.0	−0.753775
$800$	0	0
$801$	−8505.00	−0.375168
$802$	−29556.0	−1.30132
$803$	16802.0	0.738393
$804$	5616.00	0.246345
$805$	0	0
$806$	−9120.00	−0.398559
$807$	−23538.0	−1.02674
$808$	−22896.0	−0.996879
$809$	32768.0	1.42406	0.712028	−	0.702151i	$-0.247777\pi$
$809$	32768.0	1.42406	0.712028	+	0.702151i	$0.247777\pi$
$810$	0	0
$811$	−16896.0	−0.731564	−0.365782	−	0.930700i	$-0.619199\pi$
$811$	−16896.0	−0.731564	−0.365782	+	0.930700i	$0.619199\pi$
$812$	−9324.00	−0.402966
$813$	609.000	0.0262713
$814$	28520.0	1.22804
$815$	0	0
$816$	−3648.00	−0.156502
$817$	6460.00	0.276630
$818$	−8408.00	−0.359387
$819$	−3078.00	−0.131324
$820$	0	0
$821$	4506.00	0.191547	0.0957737	−	0.995403i	$-0.469467\pi$
$821$	4506.00	0.191547	0.0957737	+	0.995403i	$0.469467\pi$
$822$	15408.0	0.653790
$823$	−40667.0	−1.72243	−0.861217	−	0.508238i	$-0.830297\pi$
$823$	−40667.0	−1.72243	−0.861217	+	0.508238i	$0.830297\pi$
$824$	−29904.0	−1.26427
$825$	0	0
$826$	2142.00	0.0902297
$827$	3212.00	0.135057	0.0675285	−	0.997717i	$-0.478489\pi$
$827$	3212.00	0.135057	0.0675285	+	0.997717i	$0.478489\pi$
$828$	1512.00	0.0634609
$829$	10248.0	0.429346	0.214673	−	0.976686i	$-0.431131\pi$
$829$	10248.0	0.429346	0.214673	+	0.976686i	$0.431131\pi$
$830$	0	0
$831$	24993.0	1.04332
$832$	17024.0	0.709376
$833$	19912.0	0.828223
$834$	−9282.00	−0.385383
$835$	0	0
$836$	−4712.00	−0.194938
$837$	3240.00	0.133800
$838$	−27488.0	−1.13312
$839$	−6649.00	−0.273598	−0.136799	−	0.990599i	$-0.543681\pi$
$839$	−6649.00	−0.273598	−0.136799	+	0.990599i	$0.543681\pi$
$840$	0	0
$841$	42692.0	1.75046
$842$	−6568.00	−0.268822
$843$	−13734.0	−0.561120
$844$	−7952.00	−0.324312
$845$	0	0
$846$	4032.00	0.163857
$847$	−22617.0	−0.917508
$848$	976.000	0.0395236
$849$	16236.0	0.656323
$850$	0	0
$851$	9660.00	0.389119
$852$	11940.0	0.480115
$853$	30465.0	1.22286	0.611431	−	0.791298i	$-0.290594\pi$
$853$	30465.0	1.22286	0.611431	+	0.791298i	$0.290594\pi$
$854$	2034.00	0.0815012
$855$	0	0
$856$	−37944.0	−1.51507
$857$	19385.0	0.772671	0.386335	−	0.922358i	$-0.373741\pi$
$857$	19385.0	0.772671	0.386335	+	0.922358i	$0.373741\pi$
$858$	14136.0	0.562465
$859$	−10253.0	−0.407250	−0.203625	−	0.979049i	$-0.565272\pi$
$859$	−10253.0	−0.407250	−0.203625	+	0.979049i	$0.565272\pi$
$860$	0	0
$861$	12285.0	0.486262
$862$	29082.0	1.14911
$863$	32185.0	1.26951	0.634757	−	0.772712i	$-0.281100\pi$
$863$	32185.0	1.26951	0.634757	+	0.772712i	$0.281100\pi$
$864$	−4320.00	−0.170103
$865$	0	0
$866$	−3176.00	−0.124625
$867$	−2589.00	−0.101415
$868$	−4320.00	−0.168929
$869$	−19716.0	−0.769643
$870$	0	0
$871$	17784.0	0.691835
$872$	18960.0	0.736315
$873$	−7848.00	−0.304255
$874$	1596.00	0.0617683
$875$	0	0
$876$	−3252.00	−0.125428
$877$	43494.0	1.67467	0.837336	−	0.546688i	$-0.184112\pi$
$877$	43494.0	1.67467	0.837336	+	0.546688i	$0.184112\pi$
$878$	36140.0	1.38914
$879$	9846.00	0.377813
$880$	0	0
$881$	20770.0	0.794278	0.397139	−	0.917758i	$-0.370003\pi$
$881$	20770.0	0.794278	0.397139	+	0.917758i	$0.370003\pi$
$882$	−4716.00	−0.180041
$883$	15637.0	0.595954	0.297977	−	0.954573i	$-0.403688\pi$
$883$	15637.0	0.595954	0.297977	+	0.954573i	$0.403688\pi$
$884$	11552.0	0.439520
$885$	0	0
$886$	12756.0	0.483686
$887$	12312.0	0.466061	0.233031	−	0.972469i	$-0.425136\pi$
$887$	12312.0	0.466061	0.233031	+	0.972469i	$0.425136\pi$
$888$	−16560.0	−0.625808
$889$	17748.0	0.669571
$890$	0	0
$891$	−5022.00	−0.188825
$892$	20672.0	0.775953
$893$	−4256.00	−0.159487
$894$	6612.00	0.247358
$895$	0	0
$896$	3456.00	0.128858
$897$	4788.00	0.178224
$898$	−10150.0	−0.377183
$899$	31080.0	1.15303
$900$	0	0
$901$	4636.00	0.171418
$902$	−56420.0	−2.08268
$903$	−9180.00	−0.338307
$904$	−46128.0	−1.69712
$905$	0	0
$906$	−15144.0	−0.555326
$907$	23784.0	0.870711	0.435355	−	0.900259i	$-0.356623\pi$
$907$	23784.0	0.870711	0.435355	+	0.900259i	$0.356623\pi$
$908$	20684.0	0.755972
$909$	8586.00	0.313289
$910$	0	0
$911$	−33515.0	−1.21888	−0.609441	−	0.792831i	$-0.708606\pi$
$911$	−33515.0	−1.21888	−0.609441	+	0.792831i	$0.708606\pi$
$912$	−912.000	−0.0331133
$913$	20832.0	0.755135
$914$	2254.00	0.0815708
$915$	0	0
$916$	−11496.0	−0.414671
$917$	−9774.00	−0.351980
$918$	4104.00	0.147551
$919$	−39707.0	−1.42526	−0.712630	−	0.701541i	$-0.752496\pi$
$919$	−39707.0	−1.42526	−0.712630	+	0.701541i	$0.752496\pi$
$920$	0	0
$921$	21090.0	0.754548
$922$	−13788.0	−0.492498
$923$	37810.0	1.34835
$924$	6696.00	0.238401
$925$	0	0
$926$	−8232.00	−0.292139
$927$	11214.0	0.397320
$928$	−41440.0	−1.46588
$929$	49310.0	1.74145	0.870726	−	0.491769i	$-0.163649\pi$
$929$	49310.0	1.74145	0.870726	+	0.491769i	$0.163649\pi$
$930$	0	0
$931$	4978.00	0.175239
$932$	9936.00	0.349211
$933$	−1656.00	−0.0581083
$934$	−1156.00	−0.0404984
$935$	0	0
$936$	−8208.00	−0.286631
$937$	−16269.0	−0.567220	−0.283610	−	0.958940i	$-0.591532\pi$
$937$	−16269.0	−0.567220	−0.283610	+	0.958940i	$0.591532\pi$
$938$	−8424.00	−0.293234
$939$	27546.0	0.957327
$940$	0	0
$941$	24770.0	0.858107	0.429054	−	0.903279i	$-0.358847\pi$
$941$	24770.0	0.858107	0.429054	+	0.903279i	$0.358847\pi$
$942$	14286.0	0.494122
$943$	−19110.0	−0.659923
$944$	1904.00	0.0656461
$945$	0	0
$946$	42160.0	1.44899
$947$	−29364.0	−1.00760	−0.503802	−	0.863819i	$-0.668066\pi$
$947$	−29364.0	−1.00760	−0.503802	+	0.863819i	$0.668066\pi$
$948$	3816.00	0.130736
$949$	−10298.0	−0.352252
$950$	0	0
$951$	−11199.0	−0.381864
$952$	−16416.0	−0.558871
$953$	4763.00	0.161898	0.0809490	−	0.996718i	$-0.474205\pi$
$953$	4763.00	0.161898	0.0809490	+	0.996718i	$0.474205\pi$
$954$	−1098.00	−0.0372632
$955$	0	0
$956$	−16264.0	−0.550225
$957$	−48174.0	−1.62721
$958$	−25956.0	−0.875366
$959$	23112.0	0.778233
$960$	0	0
$961$	−15391.0	−0.516633
$962$	−17480.0	−0.585840
$963$	14229.0	0.476140
$964$	−4984.00	−0.166518
$965$	0	0
$966$	−2268.00	−0.0755401
$967$	−2027.00	−0.0674084	−0.0337042	−	0.999432i	$-0.510730\pi$
$967$	−2027.00	−0.0674084	−0.0337042	+	0.999432i	$0.510730\pi$
$968$	−60312.0	−2.00258
$969$	−4332.00	−0.143616
$970$	0	0
$971$	4581.00	0.151402	0.0757010	−	0.997131i	$-0.475881\pi$
$971$	4581.00	0.151402	0.0757010	+	0.997131i	$0.475881\pi$
$972$	972.000	0.0320750
$973$	−13923.0	−0.458737
$974$	−25228.0	−0.829936
$975$	0	0
$976$	1808.00	0.0592958
$977$	12206.0	0.399698	0.199849	−	0.979827i	$-0.435955\pi$
$977$	12206.0	0.399698	0.199849	+	0.979827i	$0.435955\pi$
$978$	5718.00	0.186954
$979$	58590.0	1.91271
$980$	0	0
$981$	−7110.00	−0.231401
$982$	−18272.0	−0.593771
$983$	−53712.0	−1.74277	−0.871387	−	0.490596i	$-0.836779\pi$
$983$	−53712.0	−1.74277	−0.871387	+	0.490596i	$0.836779\pi$
$984$	32760.0	1.06133
$985$	0	0
$986$	39368.0	1.27153
$987$	6048.00	0.195046
$988$	2888.00	0.0929955
$989$	14280.0	0.459128
$990$	0	0
$991$	2288.00	0.0733408	0.0366704	−	0.999327i	$-0.488325\pi$
$991$	2288.00	0.0733408	0.0366704	+	0.999327i	$0.488325\pi$
$992$	−19200.0	−0.614517
$993$	−32886.0	−1.05096
$994$	−17910.0	−0.571500
$995$	0	0
$996$	−4032.00	−0.128272
$997$	57426.0	1.82417	0.912086	−	0.409999i	$-0.134471\pi$
$997$	57426.0	1.82417	0.912086	+	0.409999i	$0.134471\pi$
$998$	32966.0	1.04561
$999$	6210.00	0.196672

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.4.a.e.1.1	yes	1
5.4	even	2		1425.4.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1425.4.a.a.1.1	✓	1	5.4	even	2
1425.4.a.e.1.1	yes	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1425.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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