LMFDB - Embedded newform 722.2.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [722,2,Mod(1,722)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("722.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(722, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 722 = 2 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	722.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,-1,1,0,-1,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$5.76519902594$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	722.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +3.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -2.00000 q^{18} +4.00000 q^{21} +3.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} -2.00000 q^{26} +5.00000 q^{27} -4.00000 q^{28} -2.00000 q^{31} +1.00000 q^{32} -3.00000 q^{33} -6.00000 q^{34} -2.00000 q^{36} +10.0000 q^{37} +2.00000 q^{39} -9.00000 q^{41} +4.00000 q^{42} -4.00000 q^{43} +3.00000 q^{44} -6.00000 q^{46} -1.00000 q^{48} +9.00000 q^{49} -5.00000 q^{50} +6.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} +5.00000 q^{54} -4.00000 q^{56} +9.00000 q^{59} -4.00000 q^{61} -2.00000 q^{62} +8.00000 q^{63} +1.00000 q^{64} -3.00000 q^{66} +7.00000 q^{67} -6.00000 q^{68} +6.00000 q^{69} +6.00000 q^{71} -2.00000 q^{72} -1.00000 q^{73} +10.0000 q^{74} +5.00000 q^{75} -12.0000 q^{77} +2.00000 q^{78} +4.00000 q^{79} +1.00000 q^{81} -9.00000 q^{82} +3.00000 q^{83} +4.00000 q^{84} -4.00000 q^{86} +3.00000 q^{88} -6.00000 q^{89} +8.00000 q^{91} -6.00000 q^{92} +2.00000 q^{93} -1.00000 q^{96} -17.0000 q^{97} +9.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−1.00000	−0.408248
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−1.00000	−0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	−2.00000	−0.471405
$19$	0	0
$19$	0	0
$20$	0	0
$21$	4.00000	0.872872
$22$	3.00000	0.639602
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−1.00000	−0.204124
$25$	−5.00000	−1.00000
$26$	−2.00000	−0.392232
$27$	5.00000	0.962250
$28$	−4.00000	−0.755929
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	1.00000	0.176777
$33$	−3.00000	−0.522233
$34$	−6.00000	−1.02899
$35$	0	0
$36$	−2.00000	−0.333333
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	4.00000	0.617213
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	−6.00000	−0.884652
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	9.00000	1.28571
$50$	−5.00000	−0.707107
$51$	6.00000	0.840168
$52$	−2.00000	−0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	5.00000	0.680414
$55$	0	0
$56$	−4.00000	−0.534522
$57$	0	0
$58$	0	0
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	−2.00000	−0.254000
$63$	8.00000	1.00791
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.00000	−0.369274
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	−6.00000	−0.727607
$69$	6.00000	0.722315
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	−2.00000	−0.235702
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	10.0000	1.16248
$75$	5.00000	0.577350
$76$	0	0
$77$	−12.0000	−1.36753
$78$	2.00000	0.226455
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−9.00000	−0.993884
$83$	3.00000	0.329293	0.164646	−	0.986353i	$-0.447352\pi$
$83$	3.00000	0.329293	0.164646	+	0.986353i	$0.447352\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	3.00000	0.319801
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	−6.00000	−0.625543
$93$	2.00000	0.207390
$94$	0	0
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−17.0000	−1.72609	−0.863044	−	0.505128i	$-0.831445\pi$
$97$	−17.0000	−1.72609	−0.863044	+	0.505128i	$0.831445\pi$
$98$	9.00000	0.909137
$99$	−6.00000	−0.603023
$100$	−5.00000	−0.500000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	6.00000	0.594089
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	5.00000	0.481125
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	−4.00000	−0.377964
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.00000	0.369800
$118$	9.00000	0.828517
$119$	24.0000	2.20008
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−4.00000	−0.362143
$123$	9.00000	0.811503
$124$	−2.00000	−0.179605
$125$	0	0
$126$	8.00000	0.712697
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	1.00000	0.0883883
$129$	4.00000	0.352180
$130$	0	0
$131$	9.00000	0.786334	0.393167	−	0.919467i	$-0.371379\pi$
$131$	9.00000	0.786334	0.393167	+	0.919467i	$0.371379\pi$
$132$	−3.00000	−0.261116
$133$	0	0
$134$	7.00000	0.604708
$135$	0	0
$136$	−6.00000	−0.514496
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	6.00000	0.510754
$139$	11.0000	0.933008	0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000	0.933008	0.466504	+	0.884519i	$0.345513\pi$
$140$	0	0
$141$	0	0
$142$	6.00000	0.503509
$143$	−6.00000	−0.501745
$144$	−2.00000	−0.166667
$145$	0	0
$146$	−1.00000	−0.0827606
$147$	−9.00000	−0.742307
$148$	10.0000	0.821995
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	5.00000	0.408248
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	0	0
$153$	12.0000	0.970143
$154$	−12.0000	−0.966988
$155$	0	0
$156$	2.00000	0.160128
$157$	−16.0000	−1.27694	−0.638470	−	0.769647i	$-0.720432\pi$
$157$	−16.0000	−1.27694	−0.638470	+	0.769647i	$0.720432\pi$
$158$	4.00000	0.318223
$159$	6.00000	0.475831
$160$	0	0
$161$	24.0000	1.89146
$162$	1.00000	0.0785674
$163$	−19.0000	−1.48819	−0.744097	−	0.668071i	$-0.767120\pi$
$163$	−19.0000	−1.48819	−0.744097	+	0.668071i	$0.767120\pi$
$164$	−9.00000	−0.702782
$165$	0	0
$166$	3.00000	0.232845
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	4.00000	0.308607
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	20.0000	1.51186
$176$	3.00000	0.226134
$177$	−9.00000	−0.676481
$178$	−6.00000	−0.449719
$179$	−9.00000	−0.672692	−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000	−0.672692	−0.336346	+	0.941739i	$0.609191\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	8.00000	0.592999
$183$	4.00000	0.295689
$184$	−6.00000	−0.442326
$185$	0	0
$186$	2.00000	0.146647
$187$	−18.0000	−1.31629
$188$	0	0
$189$	−20.0000	−1.45479
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−1.00000	−0.0721688
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−17.0000	−1.22053
$195$	0	0
$196$	9.00000	0.642857
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	−6.00000	−0.426401
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	−5.00000	−0.353553
$201$	−7.00000	−0.493742
$202$	0	0
$203$	0	0
$204$	6.00000	0.420084
$205$	0	0
$206$	−2.00000	−0.139347
$207$	12.0000	0.834058
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−6.00000	−0.412082
$213$	−6.00000	−0.411113
$214$	0	0
$215$	0	0
$216$	5.00000	0.340207
$217$	8.00000	0.543075
$218$	16.0000	1.08366
$219$	1.00000	0.0675737
$220$	0	0
$221$	12.0000	0.807207
$222$	−10.0000	−0.671156
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	−4.00000	−0.267261
$225$	10.0000	0.666667
$226$	−15.0000	−0.997785
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	0	0
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	0	0
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	9.00000	0.585850
$237$	−4.00000	−0.259828
$238$	24.0000	1.55569
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−5.00000	−0.322078	−0.161039	−	0.986948i	$-0.551485\pi$
$241$	−5.00000	−0.322078	−0.161039	+	0.986948i	$0.551485\pi$
$242$	−2.00000	−0.128565
$243$	−16.0000	−1.02640
$244$	−4.00000	−0.256074
$245$	0	0
$246$	9.00000	0.573819
$247$	0	0
$248$	−2.00000	−0.127000
$249$	−3.00000	−0.190117
$250$	0	0
$251$	−3.00000	−0.189358	−0.0946792	−	0.995508i	$-0.530183\pi$
$251$	−3.00000	−0.189358	−0.0946792	+	0.995508i	$0.530183\pi$
$252$	8.00000	0.503953
$253$	−18.0000	−1.13165
$254$	−2.00000	−0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	4.00000	0.249029
$259$	−40.0000	−2.48548
$260$	0	0
$261$	0	0
$262$	9.00000	0.556022
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	0	0
$267$	6.00000	0.367194
$268$	7.00000	0.427593
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	−6.00000	−0.363803
$273$	−8.00000	−0.484182
$274$	9.00000	0.543710
$275$	−15.0000	−0.904534
$276$	6.00000	0.361158
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	11.0000	0.659736
$279$	4.00000	0.239474
$280$	0	0
$281$	−27.0000	−1.61068	−0.805342	−	0.592810i	$-0.798019\pi$
$281$	−27.0000	−1.61068	−0.805342	+	0.592810i	$0.798019\pi$
$282$	0	0
$283$	5.00000	0.297219	0.148610	−	0.988896i	$-0.452520\pi$
$283$	5.00000	0.297219	0.148610	+	0.988896i	$0.452520\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	−6.00000	−0.354787
$287$	36.0000	2.12501
$288$	−2.00000	−0.117851
$289$	19.0000	1.11765
$290$	0	0
$291$	17.0000	0.996558
$292$	−1.00000	−0.0585206
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	−9.00000	−0.524891
$295$	0	0
$296$	10.0000	0.581238
$297$	15.0000	0.870388
$298$	18.0000	1.04271
$299$	12.0000	0.693978
$300$	5.00000	0.288675
$301$	16.0000	0.922225
$302$	10.0000	0.575435
$303$	0	0
$304$	0	0
$305$	0	0
$306$	12.0000	0.685994
$307$	7.00000	0.399511	0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000	0.399511	0.199756	+	0.979846i	$0.435985\pi$
$308$	−12.0000	−0.683763
$309$	2.00000	0.113776
$310$	0	0
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	2.00000	0.113228
$313$	−19.0000	−1.07394	−0.536972	−	0.843600i	$-0.680432\pi$
$313$	−19.0000	−1.07394	−0.536972	+	0.843600i	$0.680432\pi$
$314$	−16.0000	−0.902932
$315$	0	0
$316$	4.00000	0.225018
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	6.00000	0.336463
$319$	0	0
$320$	0	0
$321$	0	0
$322$	24.0000	1.33747
$323$	0	0
$324$	1.00000	0.0555556
$325$	10.0000	0.554700
$326$	−19.0000	−1.05231
$327$	−16.0000	−0.884802
$328$	−9.00000	−0.496942
$329$	0	0
$330$	0	0
$331$	−5.00000	−0.274825	−0.137412	−	0.990514i	$-0.543879\pi$
$331$	−5.00000	−0.274825	−0.137412	+	0.990514i	$0.543879\pi$
$332$	3.00000	0.164646
$333$	−20.0000	−1.09599
$334$	24.0000	1.31322
$335$	0	0
$336$	4.00000	0.218218
$337$	−11.0000	−0.599208	−0.299604	−	0.954064i	$-0.596855\pi$
$337$	−11.0000	−0.599208	−0.299604	+	0.954064i	$0.596855\pi$
$338$	−9.00000	−0.489535
$339$	15.0000	0.814688
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	−8.00000	−0.431959
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−9.00000	−0.483145	−0.241573	−	0.970383i	$-0.577663\pi$
$347$	−9.00000	−0.483145	−0.241573	+	0.970383i	$0.577663\pi$
$348$	0	0
$349$	−4.00000	−0.214115	−0.107058	−	0.994253i	$-0.534143\pi$
$349$	−4.00000	−0.214115	−0.107058	+	0.994253i	$0.534143\pi$
$350$	20.0000	1.06904
$351$	−10.0000	−0.533761
$352$	3.00000	0.159901
$353$	3.00000	0.159674	0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000	0.159674	0.0798369	+	0.996808i	$0.474560\pi$
$354$	−9.00000	−0.478345
$355$	0	0
$356$	−6.00000	−0.317999
$357$	−24.0000	−1.27021
$358$	−9.00000	−0.475665
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	0	0
$362$	−2.00000	−0.105118
$363$	2.00000	0.104973
$364$	8.00000	0.419314
$365$	0	0
$366$	4.00000	0.209083
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	−6.00000	−0.312772
$369$	18.0000	0.937043
$370$	0	0
$371$	24.0000	1.24602
$372$	2.00000	0.103695
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	−18.0000	−0.930758
$375$	0	0
$376$	0	0
$377$	0	0
$378$	−20.0000	−1.02869
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	12.0000	0.613973
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−2.00000	−0.101797
$387$	8.00000	0.406663
$388$	−17.0000	−0.863044
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	0	0
$391$	36.0000	1.82060
$392$	9.00000	0.454569
$393$	−9.00000	−0.453990
$394$	18.0000	0.906827
$395$	0	0
$396$	−6.00000	−0.301511
$397$	−10.0000	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$398$	−10.0000	−0.501255
$399$	0	0
$400$	−5.00000	−0.250000
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	−7.00000	−0.349128
$403$	4.00000	0.199254
$404$	0	0
$405$	0	0
$406$	0	0
$407$	30.0000	1.48704
$408$	6.00000	0.297044
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	0	0
$411$	−9.00000	−0.443937
$412$	−2.00000	−0.0985329
$413$	−36.0000	−1.77144
$414$	12.0000	0.589768
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	−11.0000	−0.538672
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	−20.0000	−0.973585
$423$	0	0
$424$	−6.00000	−0.291386
$425$	30.0000	1.45521
$426$	−6.00000	−0.290701
$427$	16.0000	0.774294
$428$	0	0
$429$	6.00000	0.289683
$430$	0	0
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	5.00000	0.240563
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	16.0000	0.766261
$437$	0	0
$438$	1.00000	0.0477818
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	0	0
$441$	−18.0000	−0.857143
$442$	12.0000	0.570782
$443$	9.00000	0.427603	0.213801	−	0.976877i	$-0.431415\pi$
$443$	9.00000	0.427603	0.213801	+	0.976877i	$0.431415\pi$
$444$	−10.0000	−0.474579
$445$	0	0
$446$	−14.0000	−0.662919
$447$	−18.0000	−0.851371
$448$	−4.00000	−0.188982
$449$	−9.00000	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$450$	10.0000	0.471405
$451$	−27.0000	−1.27138
$452$	−15.0000	−0.705541
$453$	−10.0000	−0.469841
$454$	−3.00000	−0.140797
$455$	0	0
$456$	0	0
$457$	5.00000	0.233890	0.116945	−	0.993138i	$-0.462690\pi$
$457$	5.00000	0.233890	0.116945	+	0.993138i	$0.462690\pi$
$458$	−16.0000	−0.747631
$459$	−30.0000	−1.40028
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	12.0000	0.558291
$463$	−34.0000	−1.58011	−0.790057	−	0.613033i	$-0.789949\pi$
$463$	−34.0000	−1.58011	−0.790057	+	0.613033i	$0.789949\pi$
$464$	0	0
$465$	0	0
$466$	3.00000	0.138972
$467$	−27.0000	−1.24941	−0.624705	−	0.780860i	$-0.714781\pi$
$467$	−27.0000	−1.24941	−0.624705	+	0.780860i	$0.714781\pi$
$468$	4.00000	0.184900
$469$	−28.0000	−1.29292
$470$	0	0
$471$	16.0000	0.737241
$472$	9.00000	0.414259
$473$	−12.0000	−0.551761
$474$	−4.00000	−0.183726
$475$	0	0
$476$	24.0000	1.10004
$477$	12.0000	0.549442
$478$	−12.0000	−0.548867
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	−5.00000	−0.227744
$483$	−24.0000	−1.09204
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	−16.0000	−0.725775
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	−4.00000	−0.181071
$489$	19.0000	0.859210
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	9.00000	0.405751
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	−24.0000	−1.07655
$498$	−3.00000	−0.134433
$499$	−25.0000	−1.11915	−0.559577	−	0.828778i	$-0.689036\pi$
$499$	−25.0000	−1.11915	−0.559577	+	0.828778i	$0.689036\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	−3.00000	−0.133897
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	8.00000	0.356348
$505$	0	0
$506$	−18.0000	−0.800198
$507$	9.00000	0.399704
$508$	−2.00000	−0.0887357
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	1.00000	0.0441942
$513$	0	0
$514$	−3.00000	−0.132324
$515$	0	0
$516$	4.00000	0.176090
$517$	0	0
$518$	−40.0000	−1.75750
$519$	6.00000	0.263371
$520$	0	0
$521$	−9.00000	−0.394297	−0.197149	−	0.980374i	$-0.563168\pi$
$521$	−9.00000	−0.394297	−0.197149	+	0.980374i	$0.563168\pi$
$522$	0	0
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	9.00000	0.393167
$525$	−20.0000	−0.872872
$526$	−12.0000	−0.523225
$527$	12.0000	0.522728
$528$	−3.00000	−0.130558
$529$	13.0000	0.565217
$530$	0	0
$531$	−18.0000	−0.781133
$532$	0	0
$533$	18.0000	0.779667
$534$	6.00000	0.259645
$535$	0	0
$536$	7.00000	0.302354
$537$	9.00000	0.388379
$538$	−12.0000	−0.517357
$539$	27.0000	1.16297
$540$	0	0
$541$	44.0000	1.89171	0.945854	−	0.324593i	$-0.105227\pi$
$541$	44.0000	1.89171	0.945854	+	0.324593i	$0.105227\pi$
$542$	−16.0000	−0.687259
$543$	2.00000	0.0858282
$544$	−6.00000	−0.257248
$545$	0	0
$546$	−8.00000	−0.342368
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	9.00000	0.384461
$549$	8.00000	0.341432
$550$	−15.0000	−0.639602
$551$	0	0
$552$	6.00000	0.255377
$553$	−16.0000	−0.680389
$554$	8.00000	0.339887
$555$	0	0
$556$	11.0000	0.466504
$557$	24.0000	1.01691	0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000	1.01691	0.508456	+	0.861088i	$0.330216\pi$
$558$	4.00000	0.169334
$559$	8.00000	0.338364
$560$	0	0
$561$	18.0000	0.759961
$562$	−27.0000	−1.13893
$563$	21.0000	0.885044	0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000	0.885044	0.442522	+	0.896758i	$0.354084\pi$
$564$	0	0
$565$	0	0
$566$	5.00000	0.210166
$567$	−4.00000	−0.167984
$568$	6.00000	0.251754
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−7.00000	−0.292941	−0.146470	−	0.989215i	$-0.546791\pi$
$571$	−7.00000	−0.292941	−0.146470	+	0.989215i	$0.546791\pi$
$572$	−6.00000	−0.250873
$573$	−12.0000	−0.501307
$574$	36.0000	1.50261
$575$	30.0000	1.25109
$576$	−2.00000	−0.0833333
$577$	11.0000	0.457936	0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000	0.457936	0.228968	+	0.973434i	$0.426465\pi$
$578$	19.0000	0.790296
$579$	2.00000	0.0831172
$580$	0	0
$581$	−12.0000	−0.497844
$582$	17.0000	0.704673
$583$	−18.0000	−0.745484
$584$	−1.00000	−0.0413803
$585$	0	0
$586$	−24.0000	−0.991431
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	−9.00000	−0.371154
$589$	0	0
$590$	0	0
$591$	−18.0000	−0.740421
$592$	10.0000	0.410997
$593$	−21.0000	−0.862367	−0.431183	−	0.902264i	$-0.641904\pi$
$593$	−21.0000	−0.862367	−0.431183	+	0.902264i	$0.641904\pi$
$594$	15.0000	0.615457
$595$	0	0
$596$	18.0000	0.737309
$597$	10.0000	0.409273
$598$	12.0000	0.490716
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	5.00000	0.204124
$601$	13.0000	0.530281	0.265141	−	0.964210i	$-0.414582\pi$
$601$	13.0000	0.530281	0.265141	+	0.964210i	$0.414582\pi$
$602$	16.0000	0.652111
$603$	−14.0000	−0.570124
$604$	10.0000	0.406894
$605$	0	0
$606$	0	0
$607$	−20.0000	−0.811775	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000	−0.811775	−0.405887	+	0.913923i	$0.633038\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	12.0000	0.485071
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	7.00000	0.282497
$615$	0	0
$616$	−12.0000	−0.483494
$617$	3.00000	0.120775	0.0603877	−	0.998175i	$-0.480766\pi$
$617$	3.00000	0.120775	0.0603877	+	0.998175i	$0.480766\pi$
$618$	2.00000	0.0804518
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	−30.0000	−1.20386
$622$	−30.0000	−1.20289
$623$	24.0000	0.961540
$624$	2.00000	0.0800641
$625$	25.0000	1.00000
$626$	−19.0000	−0.759393
$627$	0	0
$628$	−16.0000	−0.638470
$629$	−60.0000	−2.39236
$630$	0	0
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	4.00000	0.159111
$633$	20.0000	0.794929
$634$	18.0000	0.714871
$635$	0	0
$636$	6.00000	0.237915
$637$	−18.0000	−0.713186
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	39.0000	1.54041	0.770204	−	0.637798i	$-0.220155\pi$
$641$	39.0000	1.54041	0.770204	+	0.637798i	$0.220155\pi$
$642$	0	0
$643$	−43.0000	−1.69575	−0.847877	−	0.530193i	$-0.822120\pi$
$643$	−43.0000	−1.69575	−0.847877	+	0.530193i	$0.822120\pi$
$644$	24.0000	0.945732
$645$	0	0
$646$	0	0
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	1.00000	0.0392837
$649$	27.0000	1.05984
$650$	10.0000	0.392232
$651$	−8.00000	−0.313545
$652$	−19.0000	−0.744097
$653$	−12.0000	−0.469596	−0.234798	−	0.972044i	$-0.575443\pi$
$653$	−12.0000	−0.469596	−0.234798	+	0.972044i	$0.575443\pi$
$654$	−16.0000	−0.625650
$655$	0	0
$656$	−9.00000	−0.351391
$657$	2.00000	0.0780274
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	40.0000	1.55582	0.777910	−	0.628376i	$-0.216280\pi$
$661$	40.0000	1.55582	0.777910	+	0.628376i	$0.216280\pi$
$662$	−5.00000	−0.194331
$663$	−12.0000	−0.466041
$664$	3.00000	0.116423
$665$	0	0
$666$	−20.0000	−0.774984
$667$	0	0
$668$	24.0000	0.928588
$669$	14.0000	0.541271
$670$	0	0
$671$	−12.0000	−0.463255
$672$	4.00000	0.154303
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	−11.0000	−0.423704
$675$	−25.0000	−0.962250
$676$	−9.00000	−0.346154
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	15.0000	0.576072
$679$	68.0000	2.60960
$680$	0	0
$681$	3.00000	0.114960
$682$	−6.00000	−0.229752
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	0	0
$686$	−8.00000	−0.305441
$687$	16.0000	0.610438
$688$	−4.00000	−0.152499
$689$	12.0000	0.457164
$690$	0	0
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	−6.00000	−0.228086
$693$	24.0000	0.911685
$694$	−9.00000	−0.341635
$695$	0	0
$696$	0	0
$697$	54.0000	2.04540
$698$	−4.00000	−0.151402
$699$	−3.00000	−0.113470
$700$	20.0000	0.755929
$701$	−24.0000	−0.906467	−0.453234	−	0.891392i	$-0.649730\pi$
$701$	−24.0000	−0.906467	−0.453234	+	0.891392i	$0.649730\pi$
$702$	−10.0000	−0.377426
$703$	0	0
$704$	3.00000	0.113067
$705$	0	0
$706$	3.00000	0.112906
$707$	0	0
$708$	−9.00000	−0.338241
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	−6.00000	−0.224860
$713$	12.0000	0.449404
$714$	−24.0000	−0.898177
$715$	0	0
$716$	−9.00000	−0.336346
$717$	12.0000	0.448148
$718$	−6.00000	−0.223918
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	5.00000	0.185952
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	2.00000	0.0742270
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	8.00000	0.296500
$729$	13.0000	0.481481
$730$	0	0
$731$	24.0000	0.887672
$732$	4.00000	0.147844
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	−22.0000	−0.812035
$735$	0	0
$736$	−6.00000	−0.221163
$737$	21.0000	0.773545
$738$	18.0000	0.662589
$739$	35.0000	1.28750	0.643748	−	0.765238i	$-0.277379\pi$
$739$	35.0000	1.28750	0.643748	+	0.765238i	$0.277379\pi$
$740$	0	0
$741$	0	0
$742$	24.0000	0.881068
$743$	18.0000	0.660356	0.330178	−	0.943919i	$-0.392891\pi$
$743$	18.0000	0.660356	0.330178	+	0.943919i	$0.392891\pi$
$744$	2.00000	0.0733236
$745$	0	0
$746$	4.00000	0.146450
$747$	−6.00000	−0.219529
$748$	−18.0000	−0.658145
$749$	0	0
$750$	0	0
$751$	−38.0000	−1.38664	−0.693320	−	0.720630i	$-0.743853\pi$
$751$	−38.0000	−1.38664	−0.693320	+	0.720630i	$0.743853\pi$
$752$	0	0
$753$	3.00000	0.109326
$754$	0	0
$755$	0	0
$756$	−20.0000	−0.727393
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	28.0000	1.01701
$759$	18.0000	0.653359
$760$	0	0
$761$	−39.0000	−1.41375	−0.706874	−	0.707339i	$-0.749895\pi$
$761$	−39.0000	−1.41375	−0.706874	+	0.707339i	$0.749895\pi$
$762$	2.00000	0.0724524
$763$	−64.0000	−2.31696
$764$	12.0000	0.434145
$765$	0	0
$766$	36.0000	1.30073
$767$	−18.0000	−0.649942
$768$	−1.00000	−0.0360844
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	3.00000	0.108042
$772$	−2.00000	−0.0719816
$773$	48.0000	1.72644	0.863220	−	0.504828i	$-0.168444\pi$
$773$	48.0000	1.72644	0.863220	+	0.504828i	$0.168444\pi$
$774$	8.00000	0.287554
$775$	10.0000	0.359211
$776$	−17.0000	−0.610264
$777$	40.0000	1.43499
$778$	−36.0000	−1.29066
$779$	0	0
$780$	0	0
$781$	18.0000	0.644091
$782$	36.0000	1.28736
$783$	0	0
$784$	9.00000	0.321429
$785$	0	0
$786$	−9.00000	−0.321019
$787$	7.00000	0.249523	0.124762	−	0.992187i	$-0.460183\pi$
$787$	7.00000	0.249523	0.124762	+	0.992187i	$0.460183\pi$
$788$	18.0000	0.641223
$789$	12.0000	0.427211
$790$	0	0
$791$	60.0000	2.13335
$792$	−6.00000	−0.213201
$793$	8.00000	0.284088
$794$	−10.0000	−0.354887
$795$	0	0
$796$	−10.0000	−0.354441
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	0	0
$799$	0	0
$800$	−5.00000	−0.176777
$801$	12.0000	0.423999
$802$	27.0000	0.953403
$803$	−3.00000	−0.105868
$804$	−7.00000	−0.246871
$805$	0	0
$806$	4.00000	0.140894
$807$	12.0000	0.422420
$808$	0	0
$809$	45.0000	1.58212	0.791058	−	0.611741i	$-0.209531\pi$
$809$	45.0000	1.58212	0.791058	+	0.611741i	$0.209531\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	0	0
$813$	16.0000	0.561144
$814$	30.0000	1.05150
$815$	0	0
$816$	6.00000	0.210042
$817$	0	0
$818$	−5.00000	−0.174821
$819$	−16.0000	−0.559085
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	−9.00000	−0.313911
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	−2.00000	−0.0696733
$825$	15.0000	0.522233
$826$	−36.0000	−1.25260
$827$	39.0000	1.35616	0.678081	−	0.734987i	$-0.262812\pi$
$827$	39.0000	1.35616	0.678081	+	0.734987i	$0.262812\pi$
$828$	12.0000	0.417029
$829$	−44.0000	−1.52818	−0.764092	−	0.645108i	$-0.776812\pi$
$829$	−44.0000	−1.52818	−0.764092	+	0.645108i	$0.776812\pi$
$830$	0	0
$831$	−8.00000	−0.277517
$832$	−2.00000	−0.0693375
$833$	−54.0000	−1.87099
$834$	−11.0000	−0.380899
$835$	0	0
$836$	0	0
$837$	−10.0000	−0.345651
$838$	−12.0000	−0.414533
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	10.0000	0.344623
$843$	27.0000	0.929929
$844$	−20.0000	−0.688428
$845$	0	0
$846$	0	0
$847$	8.00000	0.274883
$848$	−6.00000	−0.206041
$849$	−5.00000	−0.171600
$850$	30.0000	1.02899
$851$	−60.0000	−2.05677
$852$	−6.00000	−0.205557
$853$	−22.0000	−0.753266	−0.376633	−	0.926363i	$-0.622918\pi$
$853$	−22.0000	−0.753266	−0.376633	+	0.926363i	$0.622918\pi$
$854$	16.0000	0.547509
$855$	0	0
$856$	0	0
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	6.00000	0.204837
$859$	−43.0000	−1.46714	−0.733571	−	0.679613i	$-0.762148\pi$
$859$	−43.0000	−1.46714	−0.733571	+	0.679613i	$0.762148\pi$
$860$	0	0
$861$	−36.0000	−1.22688
$862$	30.0000	1.02180
$863$	−18.0000	−0.612727	−0.306364	−	0.951915i	$-0.599112\pi$
$863$	−18.0000	−0.612727	−0.306364	+	0.951915i	$0.599112\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	−26.0000	−0.883516
$867$	−19.0000	−0.645274
$868$	8.00000	0.271538
$869$	12.0000	0.407072
$870$	0	0
$871$	−14.0000	−0.474372
$872$	16.0000	0.541828
$873$	34.0000	1.15073
$874$	0	0
$875$	0	0
$876$	1.00000	0.0337869
$877$	−20.0000	−0.675352	−0.337676	−	0.941262i	$-0.609641\pi$
$877$	−20.0000	−0.675352	−0.337676	+	0.941262i	$0.609641\pi$
$878$	−14.0000	−0.472477
$879$	24.0000	0.809500
$880$	0	0
$881$	−9.00000	−0.303218	−0.151609	−	0.988441i	$-0.548445\pi$
$881$	−9.00000	−0.303218	−0.151609	+	0.988441i	$0.548445\pi$
$882$	−18.0000	−0.606092
$883$	−19.0000	−0.639401	−0.319700	−	0.947519i	$-0.603582\pi$
$883$	−19.0000	−0.639401	−0.319700	+	0.947519i	$0.603582\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	9.00000	0.302361
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	−10.0000	−0.335578
$889$	8.00000	0.268311
$890$	0	0
$891$	3.00000	0.100504
$892$	−14.0000	−0.468755
$893$	0	0
$894$	−18.0000	−0.602010
$895$	0	0
$896$	−4.00000	−0.133631
$897$	−12.0000	−0.400668
$898$	−9.00000	−0.300334
$899$	0	0
$900$	10.0000	0.333333
$901$	36.0000	1.19933
$902$	−27.0000	−0.899002
$903$	−16.0000	−0.532447
$904$	−15.0000	−0.498893
$905$	0	0
$906$	−10.0000	−0.332228
$907$	−17.0000	−0.564476	−0.282238	−	0.959344i	$-0.591077\pi$
$907$	−17.0000	−0.564476	−0.282238	+	0.959344i	$0.591077\pi$
$908$	−3.00000	−0.0995585
$909$	0	0
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	0	0
$913$	9.00000	0.297857
$914$	5.00000	0.165385
$915$	0	0
$916$	−16.0000	−0.528655
$917$	−36.0000	−1.18882
$918$	−30.0000	−0.990148
$919$	−10.0000	−0.329870	−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000	−0.329870	−0.164935	+	0.986304i	$0.552741\pi$
$920$	0	0
$921$	−7.00000	−0.230658
$922$	6.00000	0.197599
$923$	−12.0000	−0.394985
$924$	12.0000	0.394771
$925$	−50.0000	−1.64399
$926$	−34.0000	−1.11731
$927$	4.00000	0.131377
$928$	0	0
$929$	−3.00000	−0.0984268	−0.0492134	−	0.998788i	$-0.515671\pi$
$929$	−3.00000	−0.0984268	−0.0492134	+	0.998788i	$0.515671\pi$
$930$	0	0
$931$	0	0
$932$	3.00000	0.0982683
$933$	30.0000	0.982156
$934$	−27.0000	−0.883467
$935$	0	0
$936$	4.00000	0.130744
$937$	35.0000	1.14340	0.571700	−	0.820463i	$-0.306284\pi$
$937$	35.0000	1.14340	0.571700	+	0.820463i	$0.306284\pi$
$938$	−28.0000	−0.914232
$939$	19.0000	0.620042
$940$	0	0
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	16.0000	0.521308
$943$	54.0000	1.75848
$944$	9.00000	0.292925
$945$	0	0
$946$	−12.0000	−0.390154
$947$	60.0000	1.94974	0.974869	−	0.222779i	$-0.0715128\pi$
$947$	60.0000	1.94974	0.974869	+	0.222779i	$0.0715128\pi$
$948$	−4.00000	−0.129914
$949$	2.00000	0.0649227
$950$	0	0
$951$	−18.0000	−0.583690
$952$	24.0000	0.777844
$953$	15.0000	0.485898	0.242949	−	0.970039i	$-0.421885\pi$
$953$	15.0000	0.485898	0.242949	+	0.970039i	$0.421885\pi$
$954$	12.0000	0.388514
$955$	0	0
$956$	−12.0000	−0.388108
$957$	0	0
$958$	36.0000	1.16311
$959$	−36.0000	−1.16250
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−20.0000	−0.644826
$963$	0	0
$964$	−5.00000	−0.161039
$965$	0	0
$966$	−24.0000	−0.772187
$967$	−34.0000	−1.09337	−0.546683	−	0.837340i	$-0.684110\pi$
$967$	−34.0000	−1.09337	−0.546683	+	0.837340i	$0.684110\pi$
$968$	−2.00000	−0.0642824
$969$	0	0
$970$	0	0
$971$	−21.0000	−0.673922	−0.336961	−	0.941519i	$-0.609399\pi$
$971$	−21.0000	−0.673922	−0.336961	+	0.941519i	$0.609399\pi$
$972$	−16.0000	−0.513200
$973$	−44.0000	−1.41058
$974$	−2.00000	−0.0640841
$975$	−10.0000	−0.320256
$976$	−4.00000	−0.128037
$977$	−33.0000	−1.05576	−0.527882	−	0.849318i	$-0.677014\pi$
$977$	−33.0000	−1.05576	−0.527882	+	0.849318i	$0.677014\pi$
$978$	19.0000	0.607553
$979$	−18.0000	−0.575282
$980$	0	0
$981$	−32.0000	−1.02168
$982$	0	0
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	9.00000	0.286910
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	24.0000	0.763156
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	−2.00000	−0.0635001
$993$	5.00000	0.158670
$994$	−24.0000	−0.761234
$995$	0	0
$996$	−3.00000	−0.0950586
$997$	−4.00000	−0.126681	−0.0633406	−	0.997992i	$-0.520175\pi$
$997$	−4.00000	−0.126681	−0.0633406	+	0.997992i	$0.520175\pi$
$998$	−25.0000	−0.791361
$999$	50.0000	1.58193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	722.2.a.d.1.1		1
3.2	odd	2		6498.2.a.e.1.1		1
4.3	odd	2		5776.2.a.n.1.1		1
19.2	odd	18		722.2.e.j.99.1		6
19.3	odd	18		722.2.e.j.389.1		6
19.4	even	9		722.2.e.i.415.1		6
19.5	even	9		722.2.e.i.595.1		6
19.6	even	9		722.2.e.i.245.1		6
19.7	even	3		722.2.c.b.429.1		2
19.8	odd	6		38.2.c.a.7.1	✓	2
19.9	even	9		722.2.e.i.423.1		6
19.10	odd	18		722.2.e.j.423.1		6
19.11	even	3		722.2.c.b.653.1		2
19.12	odd	6		38.2.c.a.11.1	yes	2
19.13	odd	18		722.2.e.j.245.1		6
19.14	odd	18		722.2.e.j.595.1		6
19.15	odd	18		722.2.e.j.415.1		6
19.16	even	9		722.2.e.i.389.1		6
19.17	even	9		722.2.e.i.99.1		6
19.18	odd	2		722.2.a.c.1.1		1
57.8	even	6		342.2.g.b.235.1		2
57.50	even	6		342.2.g.b.163.1		2
57.56	even	2		6498.2.a.s.1.1		1
76.27	even	6		304.2.i.c.273.1		2
76.31	even	6		304.2.i.c.49.1		2
76.75	even	2		5776.2.a.g.1.1		1
95.8	even	12		950.2.j.e.349.2		4
95.12	even	12		950.2.j.e.49.2		4
95.27	even	12		950.2.j.e.349.1		4
95.69	odd	6		950.2.e.d.201.1		2
95.84	odd	6		950.2.e.d.501.1		2
95.88	even	12		950.2.j.e.49.1		4
152.27	even	6		1216.2.i.d.577.1		2
152.69	odd	6		1216.2.i.h.961.1		2
152.107	even	6		1216.2.i.d.961.1		2
152.141	odd	6		1216.2.i.h.577.1		2
228.107	odd	6		2736.2.s.m.1873.1		2
228.179	odd	6		2736.2.s.m.577.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.2.c.a.7.1	✓	2	19.8	odd	6
38.2.c.a.11.1	yes	2	19.12	odd	6
304.2.i.c.49.1		2	76.31	even	6
304.2.i.c.273.1		2	76.27	even	6
342.2.g.b.163.1		2	57.50	even	6
342.2.g.b.235.1		2	57.8	even	6
722.2.a.c.1.1		1	19.18	odd	2
722.2.a.d.1.1		1	1.1	even	1	trivial
722.2.c.b.429.1		2	19.7	even	3
722.2.c.b.653.1		2	19.11	even	3
722.2.e.i.99.1		6	19.17	even	9
722.2.e.i.245.1		6	19.6	even	9
722.2.e.i.389.1		6	19.16	even	9
722.2.e.i.415.1		6	19.4	even	9
722.2.e.i.423.1		6	19.9	even	9
722.2.e.i.595.1		6	19.5	even	9
722.2.e.j.99.1		6	19.2	odd	18
722.2.e.j.245.1		6	19.13	odd	18
722.2.e.j.389.1		6	19.3	odd	18
722.2.e.j.415.1		6	19.15	odd	18
722.2.e.j.423.1		6	19.10	odd	18
722.2.e.j.595.1		6	19.14	odd	18
950.2.e.d.201.1		2	95.69	odd	6
950.2.e.d.501.1		2	95.84	odd	6
950.2.j.e.49.1		4	95.88	even	12
950.2.j.e.49.2		4	95.12	even	12
950.2.j.e.349.1		4	95.27	even	12
950.2.j.e.349.2		4	95.8	even	12
1216.2.i.d.577.1		2	152.27	even	6
1216.2.i.d.961.1		2	152.107	even	6
1216.2.i.h.577.1		2	152.141	odd	6
1216.2.i.h.961.1		2	152.69	odd	6
2736.2.s.m.577.1		2	228.179	odd	6
2736.2.s.m.1873.1		2	228.107	odd	6
5776.2.a.g.1.1		1	76.75	even	2
5776.2.a.n.1.1		1	4.3	odd	2
6498.2.a.e.1.1		1	3.2	odd	2
6498.2.a.s.1.1		1	57.56	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 722 = 2 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	722.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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