LMFDB - Embedded newform 1075.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1075,2,Mod(1,1075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1075, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1075 = 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1075.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:	$8.58391821729$
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	$\chi$	$=$	1075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +2.00000 q^{3} +2.00000 q^{4} -4.00000 q^{6} -2.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +2.00000 q^{3} +2.00000 q^{4} -4.00000 q^{6} -2.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} +4.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} -4.00000 q^{16} -3.00000 q^{17} -2.00000 q^{18} +6.00000 q^{19} -4.00000 q^{21} -8.00000 q^{22} -3.00000 q^{23} -4.00000 q^{26} -4.00000 q^{27} -4.00000 q^{28} +4.00000 q^{29} +5.00000 q^{31} +8.00000 q^{32} +8.00000 q^{33} +6.00000 q^{34} +2.00000 q^{36} +2.00000 q^{37} -12.0000 q^{38} +4.00000 q^{39} +5.00000 q^{41} +8.00000 q^{42} -1.00000 q^{43} +8.00000 q^{44} +6.00000 q^{46} +9.00000 q^{47} -8.00000 q^{48} -3.00000 q^{49} -6.00000 q^{51} +4.00000 q^{52} +3.00000 q^{53} +8.00000 q^{54} +12.0000 q^{57} -8.00000 q^{58} +3.00000 q^{59} +8.00000 q^{61} -10.0000 q^{62} -2.00000 q^{63} -8.00000 q^{64} -16.0000 q^{66} -3.00000 q^{67} -6.00000 q^{68} -6.00000 q^{69} +4.00000 q^{71} -14.0000 q^{73} -4.00000 q^{74} +12.0000 q^{76} -8.00000 q^{77} -8.00000 q^{78} -1.00000 q^{79} -11.0000 q^{81} -10.0000 q^{82} +12.0000 q^{83} -8.00000 q^{84} +2.00000 q^{86} +8.00000 q^{87} +2.00000 q^{89} -4.00000 q^{91} -6.00000 q^{92} +10.0000 q^{93} -18.0000 q^{94} +16.0000 q^{96} -9.00000 q^{97} +6.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	2.00000	1.00000
$5$	0	0
$5$	0	0
$6$	−4.00000	−1.63299
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	4.00000	1.15470
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	−2.00000	−0.471405
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	−8.00000	−1.70561
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	0	0
$26$	−4.00000	−0.784465
$27$	−4.00000	−0.769800
$28$	−4.00000	−0.755929
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	8.00000	1.41421
$33$	8.00000	1.39262
$34$	6.00000	1.02899
$35$	0	0
$36$	2.00000	0.333333
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−12.0000	−1.94666
$39$	4.00000	0.640513
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	8.00000	1.23443
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	8.00000	1.20605
$45$	0	0
$46$	6.00000	0.884652
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	−8.00000	−1.15470
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−6.00000	−0.840168
$52$	4.00000	0.554700
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	8.00000	1.08866
$55$	0	0
$56$	0	0
$57$	12.0000	1.58944
$58$	−8.00000	−1.05045
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	−10.0000	−1.27000
$63$	−2.00000	−0.251976
$64$	−8.00000	−1.00000
$65$	0	0
$66$	−16.0000	−1.96946
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	−6.00000	−0.727607
$69$	−6.00000	−0.722315
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	12.0000	1.37649
$77$	−8.00000	−0.911685
$78$	−8.00000	−0.905822
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	−10.0000	−1.10432
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	−8.00000	−0.872872
$85$	0	0
$86$	2.00000	0.215666
$87$	8.00000	0.857690
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	−6.00000	−0.625543
$93$	10.0000	1.03695
$94$	−18.0000	−1.85656
$95$	0	0
$96$	16.0000	1.63299
$97$	−9.00000	−0.913812	−0.456906	−	0.889515i	$-0.651042\pi$
$97$	−9.00000	−0.913812	−0.456906	+	0.889515i	$0.651042\pi$
$98$	6.00000	0.606092
$99$	4.00000	0.402015
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	12.0000	1.18818
$103$	13.0000	1.28093	0.640464	−	0.767988i	$-0.278742\pi$
$103$	13.0000	1.28093	0.640464	+	0.767988i	$0.278742\pi$
$104$	0	0
$105$	0	0
$106$	−6.00000	−0.582772
$107$	13.0000	1.25676	0.628379	−	0.777908i	$-0.283719\pi$
$107$	13.0000	1.25676	0.628379	+	0.777908i	$0.283719\pi$
$108$	−8.00000	−0.769800
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	8.00000	0.755929
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	−24.0000	−2.24781
$115$	0	0
$116$	8.00000	0.742781
$117$	2.00000	0.184900
$118$	−6.00000	−0.552345
$119$	6.00000	0.550019
$120$	0	0
$121$	5.00000	0.454545
$122$	−16.0000	−1.44857
$123$	10.0000	0.901670
$124$	10.0000	0.898027
$125$	0	0
$126$	4.00000	0.356348
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	16.0000	1.39262
$133$	−12.0000	−1.04053
$134$	6.00000	0.518321
$135$	0	0
$136$	0	0
$137$	20.0000	1.70872	0.854358	−	0.519685i	$-0.173951\pi$
$137$	20.0000	1.70872	0.854358	+	0.519685i	$0.173951\pi$
$138$	12.0000	1.02151
$139$	−11.0000	−0.933008	−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000	−0.933008	−0.466504	+	0.884519i	$0.654487\pi$
$140$	0	0
$141$	18.0000	1.51587
$142$	−8.00000	−0.671345
$143$	8.00000	0.668994
$144$	−4.00000	−0.333333
$145$	0	0
$146$	28.0000	2.31730
$147$	−6.00000	−0.494872
$148$	4.00000	0.328798
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	0	0
$151$	−22.0000	−1.79033	−0.895167	−	0.445730i	$-0.852944\pi$
$151$	−22.0000	−1.79033	−0.895167	+	0.445730i	$0.852944\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	16.0000	1.28932
$155$	0	0
$156$	8.00000	0.640513
$157$	16.0000	1.27694	0.638470	−	0.769647i	$-0.279568\pi$
$157$	16.0000	1.27694	0.638470	+	0.769647i	$0.279568\pi$
$158$	2.00000	0.159111
$159$	6.00000	0.475831
$160$	0	0
$161$	6.00000	0.472866
$162$	22.0000	1.72848
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	−24.0000	−1.86276
$167$	−21.0000	−1.62503	−0.812514	−	0.582941i	$-0.801902\pi$
$167$	−21.0000	−1.62503	−0.812514	+	0.582941i	$0.801902\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	6.00000	0.458831
$172$	−2.00000	−0.152499
$173$	1.00000	0.0760286	0.0380143	−	0.999277i	$-0.487897\pi$
$173$	1.00000	0.0760286	0.0380143	+	0.999277i	$0.487897\pi$
$174$	−16.0000	−1.21296
$175$	0	0
$176$	−16.0000	−1.20605
$177$	6.00000	0.450988
$178$	−4.00000	−0.299813
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	25.0000	1.85824	0.929118	−	0.369784i	$-0.120568\pi$
$181$	25.0000	1.85824	0.929118	+	0.369784i	$0.120568\pi$
$182$	8.00000	0.592999
$183$	16.0000	1.18275
$184$	0	0
$185$	0	0
$186$	−20.0000	−1.46647
$187$	−12.0000	−0.877527
$188$	18.0000	1.31278
$189$	8.00000	0.581914
$190$	0	0
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	−16.0000	−1.15470
$193$	−17.0000	−1.22369	−0.611843	−	0.790979i	$-0.709572\pi$
$193$	−17.0000	−1.22369	−0.611843	+	0.790979i	$0.709572\pi$
$194$	18.0000	1.29232
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	−8.00000	−0.568535
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	−6.00000	−0.423207
$202$	−12.0000	−0.844317
$203$	−8.00000	−0.561490
$204$	−12.0000	−0.840168
$205$	0	0
$206$	−26.0000	−1.81151
$207$	−3.00000	−0.208514
$208$	−8.00000	−0.554700
$209$	24.0000	1.66011
$210$	0	0
$211$	−26.0000	−1.78991	−0.894957	−	0.446153i	$-0.852794\pi$
$211$	−26.0000	−1.78991	−0.894957	+	0.446153i	$0.852794\pi$
$212$	6.00000	0.412082
$213$	8.00000	0.548151
$214$	−26.0000	−1.77732
$215$	0	0
$216$	0	0
$217$	−10.0000	−0.678844
$218$	10.0000	0.677285
$219$	−28.0000	−1.89206
$220$	0	0
$221$	−6.00000	−0.403604
$222$	−8.00000	−0.536925
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	−16.0000	−1.06904
$225$	0	0
$226$	−24.0000	−1.59646
$227$	−26.0000	−1.72568	−0.862840	−	0.505477i	$-0.831317\pi$
$227$	−26.0000	−1.72568	−0.862840	+	0.505477i	$0.831317\pi$
$228$	24.0000	1.58944
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	−16.0000	−1.05272
$232$	0	0
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	6.00000	0.390567
$237$	−2.00000	−0.129914
$238$	−12.0000	−0.777844
$239$	21.0000	1.35838	0.679189	−	0.733964i	$-0.262332\pi$
$239$	21.0000	1.35838	0.679189	+	0.733964i	$0.262332\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	−10.0000	−0.642824
$243$	−10.0000	−0.641500
$244$	16.0000	1.02430
$245$	0	0
$246$	−20.0000	−1.27515
$247$	12.0000	0.763542
$248$	0	0
$249$	24.0000	1.52094
$250$	0	0
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	−4.00000	−0.251976
$253$	−12.0000	−0.754434
$254$	−16.0000	−1.00393
$255$	0	0
$256$	16.0000	1.00000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	4.00000	0.249029
$259$	−4.00000	−0.248548
$260$	0	0
$261$	4.00000	0.247594
$262$	20.0000	1.23560
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	0	0
$266$	24.0000	1.47153
$267$	4.00000	0.244796
$268$	−6.00000	−0.366508
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	0	0
$271$	27.0000	1.64013	0.820067	−	0.572268i	$-0.193936\pi$
$271$	27.0000	1.64013	0.820067	+	0.572268i	$0.193936\pi$
$272$	12.0000	0.727607
$273$	−8.00000	−0.484182
$274$	−40.0000	−2.41649
$275$	0	0
$276$	−12.0000	−0.722315
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	22.0000	1.31947
$279$	5.00000	0.299342
$280$	0	0
$281$	−21.0000	−1.25275	−0.626377	−	0.779520i	$-0.715463\pi$
$281$	−21.0000	−1.25275	−0.626377	+	0.779520i	$0.715463\pi$
$282$	−36.0000	−2.14377
$283$	−25.0000	−1.48610	−0.743048	−	0.669238i	$-0.766621\pi$
$283$	−25.0000	−1.48610	−0.743048	+	0.669238i	$0.766621\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−16.0000	−0.946100
$287$	−10.0000	−0.590281
$288$	8.00000	0.471405
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−18.0000	−1.05518
$292$	−28.0000	−1.63858
$293$	−25.0000	−1.46052	−0.730258	−	0.683172i	$-0.760600\pi$
$293$	−25.0000	−1.46052	−0.730258	+	0.683172i	$0.760600\pi$
$294$	12.0000	0.699854
$295$	0	0
$296$	0	0
$297$	−16.0000	−0.928414
$298$	−16.0000	−0.926855
$299$	−6.00000	−0.346989
$300$	0	0
$301$	2.00000	0.115278
$302$	44.0000	2.53192
$303$	12.0000	0.689382
$304$	−24.0000	−1.37649
$305$	0	0
$306$	6.00000	0.342997
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	−16.0000	−0.911685
$309$	26.0000	1.47909
$310$	0	0
$311$	−9.00000	−0.510343	−0.255172	−	0.966896i	$-0.582132\pi$
$311$	−9.00000	−0.510343	−0.255172	+	0.966896i	$0.582132\pi$
$312$	0	0
$313$	−20.0000	−1.13047	−0.565233	−	0.824931i	$-0.691214\pi$
$313$	−20.0000	−1.13047	−0.565233	+	0.824931i	$0.691214\pi$
$314$	−32.0000	−1.80586
$315$	0	0
$316$	−2.00000	−0.112509
$317$	−15.0000	−0.842484	−0.421242	−	0.906948i	$-0.638406\pi$
$317$	−15.0000	−0.842484	−0.421242	+	0.906948i	$0.638406\pi$
$318$	−12.0000	−0.672927
$319$	16.0000	0.895828
$320$	0	0
$321$	26.0000	1.45118
$322$	−12.0000	−0.668734
$323$	−18.0000	−1.00155
$324$	−22.0000	−1.22222
$325$	0	0
$326$	−28.0000	−1.55078
$327$	−10.0000	−0.553001
$328$	0	0
$329$	−18.0000	−0.992372
$330$	0	0
$331$	16.0000	0.879440	0.439720	−	0.898135i	$-0.355078\pi$
$331$	16.0000	0.879440	0.439720	+	0.898135i	$0.355078\pi$
$332$	24.0000	1.31717
$333$	2.00000	0.109599
$334$	42.0000	2.29814
$335$	0	0
$336$	16.0000	0.872872
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	18.0000	0.979071
$339$	24.0000	1.30350
$340$	0	0
$341$	20.0000	1.08306
$342$	−12.0000	−0.648886
$343$	20.0000	1.07990
$344$	0	0
$345$	0	0
$346$	−2.00000	−0.107521
$347$	−14.0000	−0.751559	−0.375780	−	0.926709i	$-0.622625\pi$
$347$	−14.0000	−0.751559	−0.375780	+	0.926709i	$0.622625\pi$
$348$	16.0000	0.857690
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	−8.00000	−0.427008
$352$	32.0000	1.70561
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	4.00000	0.212000
$357$	12.0000	0.635107
$358$	−20.0000	−1.05703
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−50.0000	−2.62794
$363$	10.0000	0.524864
$364$	−8.00000	−0.419314
$365$	0	0
$366$	−32.0000	−1.67267
$367$	−13.0000	−0.678594	−0.339297	−	0.940679i	$-0.610189\pi$
$367$	−13.0000	−0.678594	−0.339297	+	0.940679i	$0.610189\pi$
$368$	12.0000	0.625543
$369$	5.00000	0.260290
$370$	0	0
$371$	−6.00000	−0.311504
$372$	20.0000	1.03695
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	24.0000	1.24101
$375$	0	0
$376$	0	0
$377$	8.00000	0.412021
$378$	−16.0000	−0.822951
$379$	−29.0000	−1.48963	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000	−1.48963	−0.744815	+	0.667271i	$0.767462\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	−40.0000	−2.04658
$383$	−14.0000	−0.715367	−0.357683	−	0.933843i	$-0.616433\pi$
$383$	−14.0000	−0.715367	−0.357683	+	0.933843i	$0.616433\pi$
$384$	0	0
$385$	0	0
$386$	34.0000	1.73055
$387$	−1.00000	−0.0508329
$388$	−18.0000	−0.913812
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	9.00000	0.455150
$392$	0	0
$393$	−20.0000	−1.00887
$394$	44.0000	2.21669
$395$	0	0
$396$	8.00000	0.402015
$397$	10.0000	0.501886	0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000	0.501886	0.250943	+	0.968002i	$0.419259\pi$
$398$	40.0000	2.00502
$399$	−24.0000	−1.20150
$400$	0	0
$401$	13.0000	0.649189	0.324595	−	0.945853i	$-0.394772\pi$
$401$	13.0000	0.649189	0.324595	+	0.945853i	$0.394772\pi$
$402$	12.0000	0.598506
$403$	10.0000	0.498135
$404$	12.0000	0.597022
$405$	0	0
$406$	16.0000	0.794067
$407$	8.00000	0.396545
$408$	0	0
$409$	−24.0000	−1.18672	−0.593362	−	0.804936i	$-0.702200\pi$
$409$	−24.0000	−1.18672	−0.593362	+	0.804936i	$0.702200\pi$
$410$	0	0
$411$	40.0000	1.97305
$412$	26.0000	1.28093
$413$	−6.00000	−0.295241
$414$	6.00000	0.294884
$415$	0	0
$416$	16.0000	0.784465
$417$	−22.0000	−1.07734
$418$	−48.0000	−2.34776
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	52.0000	2.53132
$423$	9.00000	0.437595
$424$	0	0
$425$	0	0
$426$	−16.0000	−0.775203
$427$	−16.0000	−0.774294
$428$	26.0000	1.25676
$429$	16.0000	0.772487
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	16.0000	0.769800
$433$	28.0000	1.34559	0.672797	−	0.739827i	$-0.265093\pi$
$433$	28.0000	1.34559	0.672797	+	0.739827i	$0.265093\pi$
$434$	20.0000	0.960031
$435$	0	0
$436$	−10.0000	−0.478913
$437$	−18.0000	−0.861057
$438$	56.0000	2.67578
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	12.0000	0.570782
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	8.00000	0.379663
$445$	0	0
$446$	−20.0000	−0.947027
$447$	16.0000	0.756774
$448$	16.0000	0.755929
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	0	0
$451$	20.0000	0.941763
$452$	24.0000	1.12887
$453$	−44.0000	−2.06730
$454$	52.0000	2.44048
$455$	0	0
$456$	0	0
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	4.00000	0.186908
$459$	12.0000	0.560112
$460$	0	0
$461$	−11.0000	−0.512321	−0.256161	−	0.966634i	$-0.582458\pi$
$461$	−11.0000	−0.512321	−0.256161	+	0.966634i	$0.582458\pi$
$462$	32.0000	1.48877
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	−16.0000	−0.742781
$465$	0	0
$466$	−24.0000	−1.11178
$467$	−10.0000	−0.462745	−0.231372	−	0.972865i	$-0.574322\pi$
$467$	−10.0000	−0.462745	−0.231372	+	0.972865i	$0.574322\pi$
$468$	4.00000	0.184900
$469$	6.00000	0.277054
$470$	0	0
$471$	32.0000	1.47448
$472$	0	0
$473$	−4.00000	−0.183920
$474$	4.00000	0.183726
$475$	0	0
$476$	12.0000	0.550019
$477$	3.00000	0.137361
$478$	−42.0000	−1.92104
$479$	−27.0000	−1.23366	−0.616831	−	0.787096i	$-0.711584\pi$
$479$	−27.0000	−1.23366	−0.616831	+	0.787096i	$0.711584\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	12.0000	0.546585
$483$	12.0000	0.546019
$484$	10.0000	0.454545
$485$	0	0
$486$	20.0000	0.907218
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	0	0
$489$	28.0000	1.26620
$490$	0	0
$491$	−44.0000	−1.98569	−0.992846	−	0.119401i	$-0.961903\pi$
$491$	−44.0000	−1.98569	−0.992846	+	0.119401i	$0.961903\pi$
$492$	20.0000	0.901670
$493$	−12.0000	−0.540453
$494$	−24.0000	−1.07981
$495$	0	0
$496$	−20.0000	−0.898027
$497$	−8.00000	−0.358849
$498$	−48.0000	−2.15093
$499$	−30.0000	−1.34298	−0.671492	−	0.741012i	$-0.734346\pi$
$499$	−30.0000	−1.34298	−0.671492	+	0.741012i	$0.734346\pi$
$500$	0	0
$501$	−42.0000	−1.87642
$502$	42.0000	1.87455
$503$	2.00000	0.0891756	0.0445878	−	0.999005i	$-0.485803\pi$
$503$	2.00000	0.0891756	0.0445878	+	0.999005i	$0.485803\pi$
$504$	0	0
$505$	0	0
$506$	24.0000	1.06693
$507$	−18.0000	−0.799408
$508$	16.0000	0.709885
$509$	21.0000	0.930809	0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000	0.930809	0.465404	+	0.885098i	$0.345909\pi$
$510$	0	0
$511$	28.0000	1.23865
$512$	−32.0000	−1.41421
$513$	−24.0000	−1.05963
$514$	12.0000	0.529297
$515$	0	0
$516$	−4.00000	−0.176090
$517$	36.0000	1.58328
$518$	8.00000	0.351500
$519$	2.00000	0.0877903
$520$	0	0
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	−8.00000	−0.350150
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−20.0000	−0.873704
$525$	0	0
$526$	−36.0000	−1.56967
$527$	−15.0000	−0.653410
$528$	−32.0000	−1.39262
$529$	−14.0000	−0.608696
$530$	0	0
$531$	3.00000	0.130189
$532$	−24.0000	−1.04053
$533$	10.0000	0.433148
$534$	−8.00000	−0.346194
$535$	0	0
$536$	0	0
$537$	20.0000	0.863064
$538$	42.0000	1.81075
$539$	−12.0000	−0.516877
$540$	0	0
$541$	−15.0000	−0.644900	−0.322450	−	0.946586i	$-0.604506\pi$
$541$	−15.0000	−0.644900	−0.322450	+	0.946586i	$0.604506\pi$
$542$	−54.0000	−2.31950
$543$	50.0000	2.14571
$544$	−24.0000	−1.02899
$545$	0	0
$546$	16.0000	0.684737
$547$	17.0000	0.726868	0.363434	−	0.931620i	$-0.381604\pi$
$547$	17.0000	0.726868	0.363434	+	0.931620i	$0.381604\pi$
$548$	40.0000	1.70872
$549$	8.00000	0.341432
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	2.00000	0.0850487
$554$	36.0000	1.52949
$555$	0	0
$556$	−22.0000	−0.933008
$557$	−3.00000	−0.127114	−0.0635570	−	0.997978i	$-0.520244\pi$
$557$	−3.00000	−0.127114	−0.0635570	+	0.997978i	$0.520244\pi$
$558$	−10.0000	−0.423334
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	−24.0000	−1.01328
$562$	42.0000	1.77166
$563$	−21.0000	−0.885044	−0.442522	−	0.896758i	$-0.645916\pi$
$563$	−21.0000	−0.885044	−0.442522	+	0.896758i	$0.645916\pi$
$564$	36.0000	1.51587
$565$	0	0
$566$	50.0000	2.10166
$567$	22.0000	0.923913
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−26.0000	−1.08807	−0.544033	−	0.839064i	$-0.683103\pi$
$571$	−26.0000	−1.08807	−0.544033	+	0.839064i	$0.683103\pi$
$572$	16.0000	0.668994
$573$	40.0000	1.67102
$574$	20.0000	0.834784
$575$	0	0
$576$	−8.00000	−0.333333
$577$	28.0000	1.16566	0.582828	−	0.812596i	$-0.301946\pi$
$577$	28.0000	1.16566	0.582828	+	0.812596i	$0.301946\pi$
$578$	16.0000	0.665512
$579$	−34.0000	−1.41299
$580$	0	0
$581$	−24.0000	−0.995688
$582$	36.0000	1.49225
$583$	12.0000	0.496989
$584$	0	0
$585$	0	0
$586$	50.0000	2.06548
$587$	14.0000	0.577842	0.288921	−	0.957353i	$-0.406704\pi$
$587$	14.0000	0.577842	0.288921	+	0.957353i	$0.406704\pi$
$588$	−12.0000	−0.494872
$589$	30.0000	1.23613
$590$	0	0
$591$	−44.0000	−1.80992
$592$	−8.00000	−0.328798
$593$	−20.0000	−0.821302	−0.410651	−	0.911793i	$-0.634698\pi$
$593$	−20.0000	−0.821302	−0.410651	+	0.911793i	$0.634698\pi$
$594$	32.0000	1.31298
$595$	0	0
$596$	16.0000	0.655386
$597$	−40.0000	−1.63709
$598$	12.0000	0.490716
$599$	−43.0000	−1.75693	−0.878466	−	0.477805i	$-0.841433\pi$
$599$	−43.0000	−1.75693	−0.878466	+	0.477805i	$0.841433\pi$
$600$	0	0
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	−4.00000	−0.163028
$603$	−3.00000	−0.122169
$604$	−44.0000	−1.79033
$605$	0	0
$606$	−24.0000	−0.974933
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	48.0000	1.94666
$609$	−16.0000	−0.648353
$610$	0	0
$611$	18.0000	0.728202
$612$	−6.00000	−0.242536
$613$	−19.0000	−0.767403	−0.383701	−	0.923457i	$-0.625351\pi$
$613$	−19.0000	−0.767403	−0.383701	+	0.923457i	$0.625351\pi$
$614$	24.0000	0.968561
$615$	0	0
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	−52.0000	−2.09175
$619$	−5.00000	−0.200967	−0.100483	−	0.994939i	$-0.532039\pi$
$619$	−5.00000	−0.200967	−0.100483	+	0.994939i	$0.532039\pi$
$620$	0	0
$621$	12.0000	0.481543
$622$	18.0000	0.721734
$623$	−4.00000	−0.160257
$624$	−16.0000	−0.640513
$625$	0	0
$626$	40.0000	1.59872
$627$	48.0000	1.91694
$628$	32.0000	1.27694
$629$	−6.00000	−0.239236
$630$	0	0
$631$	−30.0000	−1.19428	−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000	−1.19428	−0.597141	+	0.802137i	$0.703697\pi$
$632$	0	0
$633$	−52.0000	−2.06681
$634$	30.0000	1.19145
$635$	0	0
$636$	12.0000	0.475831
$637$	−6.00000	−0.237729
$638$	−32.0000	−1.26689
$639$	4.00000	0.158238
$640$	0	0
$641$	22.0000	0.868948	0.434474	−	0.900684i	$-0.356934\pi$
$641$	22.0000	0.868948	0.434474	+	0.900684i	$0.356934\pi$
$642$	−52.0000	−2.05228
$643$	−33.0000	−1.30139	−0.650696	−	0.759338i	$-0.725523\pi$
$643$	−33.0000	−1.30139	−0.650696	+	0.759338i	$0.725523\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	36.0000	1.41640
$647$	42.0000	1.65119	0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000	1.65119	0.825595	+	0.564263i	$0.190840\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	−20.0000	−0.783862
$652$	28.0000	1.09656
$653$	−12.0000	−0.469596	−0.234798	−	0.972044i	$-0.575443\pi$
$653$	−12.0000	−0.469596	−0.234798	+	0.972044i	$0.575443\pi$
$654$	20.0000	0.782062
$655$	0	0
$656$	−20.0000	−0.780869
$657$	−14.0000	−0.546192
$658$	36.0000	1.40343
$659$	31.0000	1.20759	0.603794	−	0.797140i	$-0.293655\pi$
$659$	31.0000	1.20759	0.603794	+	0.797140i	$0.293655\pi$
$660$	0	0
$661$	−45.0000	−1.75030	−0.875149	−	0.483854i	$-0.839236\pi$
$661$	−45.0000	−1.75030	−0.875149	+	0.483854i	$0.839236\pi$
$662$	−32.0000	−1.24372
$663$	−12.0000	−0.466041
$664$	0	0
$665$	0	0
$666$	−4.00000	−0.154997
$667$	−12.0000	−0.464642
$668$	−42.0000	−1.62503
$669$	20.0000	0.773245
$670$	0	0
$671$	32.0000	1.23535
$672$	−32.0000	−1.23443
$673$	12.0000	0.462566	0.231283	−	0.972887i	$-0.425708\pi$
$673$	12.0000	0.462566	0.231283	+	0.972887i	$0.425708\pi$
$674$	−28.0000	−1.07852
$675$	0	0
$676$	−18.0000	−0.692308
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	−48.0000	−1.84343
$679$	18.0000	0.690777
$680$	0	0
$681$	−52.0000	−1.99264
$682$	−40.0000	−1.53168
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	12.0000	0.458831
$685$	0	0
$686$	−40.0000	−1.52721
$687$	−4.00000	−0.152610
$688$	4.00000	0.152499
$689$	6.00000	0.228582
$690$	0	0
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	2.00000	0.0760286
$693$	−8.00000	−0.303895
$694$	28.0000	1.06287
$695$	0	0
$696$	0	0
$697$	−15.0000	−0.568166
$698$	0	0
$699$	24.0000	0.907763
$700$	0	0
$701$	35.0000	1.32193	0.660966	−	0.750416i	$-0.270147\pi$
$701$	35.0000	1.32193	0.660966	+	0.750416i	$0.270147\pi$
$702$	16.0000	0.603881
$703$	12.0000	0.452589
$704$	−32.0000	−1.20605
$705$	0	0
$706$	−20.0000	−0.752710
$707$	−12.0000	−0.451306
$708$	12.0000	0.450988
$709$	−30.0000	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$710$	0	0
$711$	−1.00000	−0.0375029
$712$	0	0
$713$	−15.0000	−0.561754
$714$	−24.0000	−0.898177
$715$	0	0
$716$	20.0000	0.747435
$717$	42.0000	1.56852
$718$	−6.00000	−0.223918
$719$	−15.0000	−0.559406	−0.279703	−	0.960087i	$-0.590236\pi$
$719$	−15.0000	−0.559406	−0.279703	+	0.960087i	$0.590236\pi$
$720$	0	0
$721$	−26.0000	−0.968291
$722$	−34.0000	−1.26535
$723$	−12.0000	−0.446285
$724$	50.0000	1.85824
$725$	0	0
$726$	−20.0000	−0.742270
$727$	−38.0000	−1.40934	−0.704671	−	0.709534i	$-0.748905\pi$
$727$	−38.0000	−1.40934	−0.704671	+	0.709534i	$0.748905\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	3.00000	0.110959
$732$	32.0000	1.18275
$733$	20.0000	0.738717	0.369358	−	0.929287i	$-0.379577\pi$
$733$	20.0000	0.738717	0.369358	+	0.929287i	$0.379577\pi$
$734$	26.0000	0.959678
$735$	0	0
$736$	−24.0000	−0.884652
$737$	−12.0000	−0.442026
$738$	−10.0000	−0.368105
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	24.0000	0.881662
$742$	12.0000	0.440534
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	0	0
$746$	52.0000	1.90386
$747$	12.0000	0.439057
$748$	−24.0000	−0.877527
$749$	−26.0000	−0.950019
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	−36.0000	−1.31278
$753$	−42.0000	−1.53057
$754$	−16.0000	−0.582686
$755$	0	0
$756$	16.0000	0.581914
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	58.0000	2.10665
$759$	−24.0000	−0.871145
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	−32.0000	−1.15924
$763$	10.0000	0.362024
$764$	40.0000	1.44715
$765$	0	0
$766$	28.0000	1.01168
$767$	6.00000	0.216647
$768$	32.0000	1.15470
$769$	27.0000	0.973645	0.486822	−	0.873501i	$-0.338156\pi$
$769$	27.0000	0.973645	0.486822	+	0.873501i	$0.338156\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	−34.0000	−1.22369
$773$	−2.00000	−0.0719350	−0.0359675	−	0.999353i	$-0.511451\pi$
$773$	−2.00000	−0.0719350	−0.0359675	+	0.999353i	$0.511451\pi$
$774$	2.00000	0.0718885
$775$	0	0
$776$	0	0
$777$	−8.00000	−0.286998
$778$	−52.0000	−1.86429
$779$	30.0000	1.07486
$780$	0	0
$781$	16.0000	0.572525
$782$	−18.0000	−0.643679
$783$	−16.0000	−0.571793
$784$	12.0000	0.428571
$785$	0	0
$786$	40.0000	1.42675
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	−44.0000	−1.56744
$789$	36.0000	1.28163
$790$	0	0
$791$	−24.0000	−0.853342
$792$	0	0
$793$	16.0000	0.568177
$794$	−20.0000	−0.709773
$795$	0	0
$796$	−40.0000	−1.41776
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	48.0000	1.69918
$799$	−27.0000	−0.955191
$800$	0	0
$801$	2.00000	0.0706665
$802$	−26.0000	−0.918092
$803$	−56.0000	−1.97620
$804$	−12.0000	−0.423207
$805$	0	0
$806$	−20.0000	−0.704470
$807$	−42.0000	−1.47847
$808$	0	0
$809$	33.0000	1.16022	0.580109	−	0.814539i	$-0.303010\pi$
$809$	33.0000	1.16022	0.580109	+	0.814539i	$0.303010\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	−16.0000	−0.561490
$813$	54.0000	1.89386
$814$	−16.0000	−0.560800
$815$	0	0
$816$	24.0000	0.840168
$817$	−6.00000	−0.209913
$818$	48.0000	1.67828
$819$	−4.00000	−0.139771
$820$	0	0
$821$	57.0000	1.98931	0.994657	−	0.103236i	$-0.0329198\pi$
$821$	57.0000	1.98931	0.994657	+	0.103236i	$0.0329198\pi$
$822$	−80.0000	−2.79032
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	0	0
$825$	0	0
$826$	12.0000	0.417533
$827$	−23.0000	−0.799788	−0.399894	−	0.916561i	$-0.630953\pi$
$827$	−23.0000	−0.799788	−0.399894	+	0.916561i	$0.630953\pi$
$828$	−6.00000	−0.208514
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	0	0
$831$	−36.0000	−1.24883
$832$	−16.0000	−0.554700
$833$	9.00000	0.311832
$834$	44.0000	1.52360
$835$	0	0
$836$	48.0000	1.66011
$837$	−20.0000	−0.691301
$838$	0	0
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−16.0000	−0.551396
$843$	−42.0000	−1.44656
$844$	−52.0000	−1.78991
$845$	0	0
$846$	−18.0000	−0.618853
$847$	−10.0000	−0.343604
$848$	−12.0000	−0.412082
$849$	−50.0000	−1.71600
$850$	0	0
$851$	−6.00000	−0.205677
$852$	16.0000	0.548151
$853$	31.0000	1.06142	0.530710	−	0.847554i	$-0.321925\pi$
$853$	31.0000	1.06142	0.530710	+	0.847554i	$0.321925\pi$
$854$	32.0000	1.09502
$855$	0	0
$856$	0	0
$857$	−1.00000	−0.0341593	−0.0170797	−	0.999854i	$-0.505437\pi$
$857$	−1.00000	−0.0341593	−0.0170797	+	0.999854i	$0.505437\pi$
$858$	−32.0000	−1.09246
$859$	−18.0000	−0.614152	−0.307076	−	0.951685i	$-0.599351\pi$
$859$	−18.0000	−0.614152	−0.307076	+	0.951685i	$0.599351\pi$
$860$	0	0
$861$	−20.0000	−0.681598
$862$	0	0
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	−32.0000	−1.08866
$865$	0	0
$866$	−56.0000	−1.90296
$867$	−16.0000	−0.543388
$868$	−20.0000	−0.678844
$869$	−4.00000	−0.135691
$870$	0	0
$871$	−6.00000	−0.203302
$872$	0	0
$873$	−9.00000	−0.304604
$874$	36.0000	1.21772
$875$	0	0
$876$	−56.0000	−1.89206
$877$	−19.0000	−0.641584	−0.320792	−	0.947150i	$-0.603949\pi$
$877$	−19.0000	−0.641584	−0.320792	+	0.947150i	$0.603949\pi$
$878$	8.00000	0.269987
$879$	−50.0000	−1.68646
$880$	0	0
$881$	−46.0000	−1.54978	−0.774890	−	0.632096i	$-0.782195\pi$
$881$	−46.0000	−1.54978	−0.774890	+	0.632096i	$0.782195\pi$
$882$	6.00000	0.202031
$883$	52.0000	1.74994	0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000	1.74994	0.874970	+	0.484178i	$0.160881\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	8.00000	0.268765
$887$	−28.0000	−0.940148	−0.470074	−	0.882627i	$-0.655773\pi$
$887$	−28.0000	−0.940148	−0.470074	+	0.882627i	$0.655773\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	−44.0000	−1.47406
$892$	20.0000	0.669650
$893$	54.0000	1.80704
$894$	−32.0000	−1.07024
$895$	0	0
$896$	0	0
$897$	−12.0000	−0.400668
$898$	68.0000	2.26919
$899$	20.0000	0.667037
$900$	0	0
$901$	−9.00000	−0.299833
$902$	−40.0000	−1.33185
$903$	4.00000	0.133112
$904$	0	0
$905$	0	0
$906$	88.0000	2.92360
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	−52.0000	−1.72568
$909$	6.00000	0.199007
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	−48.0000	−1.58944
$913$	48.0000	1.58857
$914$	−52.0000	−1.72001
$915$	0	0
$916$	−4.00000	−0.132164
$917$	20.0000	0.660458
$918$	−24.0000	−0.792118
$919$	−25.0000	−0.824674	−0.412337	−	0.911031i	$-0.635287\pi$
$919$	−25.0000	−0.824674	−0.412337	+	0.911031i	$0.635287\pi$
$920$	0	0
$921$	−24.0000	−0.790827
$922$	22.0000	0.724531
$923$	8.00000	0.263323
$924$	−32.0000	−1.05272
$925$	0	0
$926$	−28.0000	−0.920137
$927$	13.0000	0.426976
$928$	32.0000	1.05045
$929$	48.0000	1.57483	0.787414	−	0.616424i	$-0.211419\pi$
$929$	48.0000	1.57483	0.787414	+	0.616424i	$0.211419\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	24.0000	0.786146
$933$	−18.0000	−0.589294
$934$	20.0000	0.654420
$935$	0	0
$936$	0	0
$937$	48.0000	1.56809	0.784046	−	0.620703i	$-0.213153\pi$
$937$	48.0000	1.56809	0.784046	+	0.620703i	$0.213153\pi$
$938$	−12.0000	−0.391814
$939$	−40.0000	−1.30535
$940$	0	0
$941$	54.0000	1.76035	0.880175	−	0.474650i	$-0.157425\pi$
$941$	54.0000	1.76035	0.880175	+	0.474650i	$0.157425\pi$
$942$	−64.0000	−2.08523
$943$	−15.0000	−0.488467
$944$	−12.0000	−0.390567
$945$	0	0
$946$	8.00000	0.260102
$947$	24.0000	0.779895	0.389948	−	0.920837i	$-0.372493\pi$
$947$	24.0000	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$948$	−4.00000	−0.129914
$949$	−28.0000	−0.908918
$950$	0	0
$951$	−30.0000	−0.972817
$952$	0	0
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	42.0000	1.35838
$957$	32.0000	1.03441
$958$	54.0000	1.74466
$959$	−40.0000	−1.29167
$960$	0	0
$961$	−6.00000	−0.193548
$962$	−8.00000	−0.257930
$963$	13.0000	0.418919
$964$	−12.0000	−0.386494
$965$	0	0
$966$	−24.0000	−0.772187
$967$	55.0000	1.76868	0.884340	−	0.466843i	$-0.154609\pi$
$967$	55.0000	1.76868	0.884340	+	0.466843i	$0.154609\pi$
$968$	0	0
$969$	−36.0000	−1.15649
$970$	0	0
$971$	3.00000	0.0962746	0.0481373	−	0.998841i	$-0.484672\pi$
$971$	3.00000	0.0962746	0.0481373	+	0.998841i	$0.484672\pi$
$972$	−20.0000	−0.641500
$973$	22.0000	0.705288
$974$	−24.0000	−0.769010
$975$	0	0
$976$	−32.0000	−1.02430
$977$	−13.0000	−0.415907	−0.207953	−	0.978139i	$-0.566680\pi$
$977$	−13.0000	−0.415907	−0.207953	+	0.978139i	$0.566680\pi$
$978$	−56.0000	−1.79068
$979$	8.00000	0.255681
$980$	0	0
$981$	−5.00000	−0.159638
$982$	88.0000	2.80819
$983$	−18.0000	−0.574111	−0.287055	−	0.957914i	$-0.592676\pi$
$983$	−18.0000	−0.574111	−0.287055	+	0.957914i	$0.592676\pi$
$984$	0	0
$985$	0	0
$986$	24.0000	0.764316
$987$	−36.0000	−1.14589
$988$	24.0000	0.763542
$989$	3.00000	0.0953945
$990$	0	0
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	40.0000	1.27000
$993$	32.0000	1.01549
$994$	16.0000	0.507489
$995$	0	0
$996$	48.0000	1.52094
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	60.0000	1.89927
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.2.a.a.1.1	✓	1
3.2	odd	2		9675.2.a.w.1.1		1
5.2	odd	4		1075.2.b.a.474.1		2
5.3	odd	4		1075.2.b.a.474.2		2
5.4	even	2		1075.2.a.g.1.1	yes	1
15.14	odd	2		9675.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1075.2.a.a.1.1	✓	1	1.1	even	1	trivial
1075.2.a.g.1.1	yes	1	5.4	even	2
1075.2.b.a.474.1		2	5.2	odd	4
1075.2.b.a.474.2		2	5.3	odd	4
9675.2.a.c.1.1		1	15.14	odd	2
9675.2.a.w.1.1		1	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1075.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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