LMFDB - Embedded newform 1050.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,2,Mod(1,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1050.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1050.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:	$8.38429221223$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 210)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1050.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} -6.00000 q^{19} -1.00000 q^{21} +2.00000 q^{22} -8.00000 q^{23} +1.00000 q^{24} +6.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} -2.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} +4.00000 q^{37} -6.00000 q^{38} +6.00000 q^{39} +2.00000 q^{41} -1.00000 q^{42} +4.00000 q^{43} +2.00000 q^{44} -8.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +4.00000 q^{51} +6.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -1.00000 q^{56} -6.00000 q^{57} +6.00000 q^{58} -8.00000 q^{59} -10.0000 q^{61} -2.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} +8.00000 q^{67} +4.00000 q^{68} -8.00000 q^{69} -6.00000 q^{71} +1.00000 q^{72} -14.0000 q^{73} +4.00000 q^{74} -6.00000 q^{76} -2.00000 q^{77} +6.00000 q^{78} -12.0000 q^{79} +1.00000 q^{81} +2.00000 q^{82} -8.00000 q^{83} -1.00000 q^{84} +4.00000 q^{86} +6.00000 q^{87} +2.00000 q^{88} -10.0000 q^{89} -6.00000 q^{91} -8.00000 q^{92} -2.00000 q^{93} +8.00000 q^{94} +1.00000 q^{96} -10.0000 q^{97} +1.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	1.00000	0.288675
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	1.00000	0.235702
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	2.00000	0.426401
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	6.00000	1.17670
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	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
$33$	2.00000	0.348155
$34$	4.00000	0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	−6.00000	−0.973329
$39$	6.00000	0.960769
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−1.00000	−0.154303
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	−8.00000	−1.17954
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	0	0
$51$	4.00000	0.560112
$52$	6.00000	0.832050
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−6.00000	−0.794719
$58$	6.00000	0.787839
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−2.00000	−0.254000
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	4.00000	0.485071
$69$	−8.00000	−0.963087
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	1.00000	0.117851
$73$	−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
$76$	−6.00000	−0.688247
$77$	−2.00000	−0.227921
$78$	6.00000	0.679366
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	4.00000	0.431331
$87$	6.00000	0.643268
$88$	2.00000	0.213201
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	−8.00000	−0.834058
$93$	−2.00000	−0.207390
$94$	8.00000	0.825137
$95$	0	0
$96$	1.00000	0.102062
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	1.00000	0.101015
$99$	2.00000	0.201008
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	4.00000	0.396059
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	6.00000	0.582772
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	1.00000	0.0962250
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	−1.00000	−0.0944911
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−6.00000	−0.561951
$115$	0	0
$116$	6.00000	0.557086
$117$	6.00000	0.554700
$118$	−8.00000	−0.736460
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−10.0000	−0.905357
$123$	2.00000	0.180334
$124$	−2.00000	−0.179605
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	1.00000	0.0883883
$129$	4.00000	0.352180
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	2.00000	0.174078
$133$	6.00000	0.520266
$134$	8.00000	0.691095
$135$	0	0
$136$	4.00000	0.342997
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	−8.00000	−0.681005
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	−6.00000	−0.503509
$143$	12.0000	1.00349
$144$	1.00000	0.0833333
$145$	0	0
$146$	−14.0000	−1.15865
$147$	1.00000	0.0824786
$148$	4.00000	0.328798
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−6.00000	−0.486664
$153$	4.00000	0.323381
$154$	−2.00000	−0.161165
$155$	0	0
$156$	6.00000	0.480384
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	−12.0000	−0.954669
$159$	6.00000	0.475831
$160$	0	0
$161$	8.00000	0.630488
$162$	1.00000	0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	−8.00000	−0.620920
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−1.00000	−0.0771517
$169$	23.0000	1.76923
$170$	0	0
$171$	−6.00000	−0.458831
$172$	4.00000	0.304997
$173$	8.00000	0.608229	0.304114	−	0.952636i	$-0.401639\pi$
$173$	8.00000	0.608229	0.304114	+	0.952636i	$0.401639\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	2.00000	0.150756
$177$	−8.00000	−0.601317
$178$	−10.0000	−0.749532
$179$	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
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−6.00000	−0.444750
$183$	−10.0000	−0.739221
$184$	−8.00000	−0.589768
$185$	0	0
$186$	−2.00000	−0.146647
$187$	8.00000	0.585018
$188$	8.00000	0.583460
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	1.00000	0.0721688
$193$	−8.00000	−0.575853	−0.287926	−	0.957653i	$-0.592966\pi$
$193$	−8.00000	−0.575853	−0.287926	+	0.957653i	$0.592966\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	2.00000	0.142134
$199$	6.00000	0.425329	0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000	0.425329	0.212664	+	0.977125i	$0.431786\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	10.0000	0.703598
$203$	−6.00000	−0.421117
$204$	4.00000	0.280056
$205$	0	0
$206$	8.00000	0.557386
$207$	−8.00000	−0.556038
$208$	6.00000	0.416025
$209$	−12.0000	−0.830057
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	6.00000	0.412082
$213$	−6.00000	−0.411113
$214$	12.0000	0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	2.00000	0.135769
$218$	−14.0000	−0.948200
$219$	−14.0000	−0.946032
$220$	0	0
$221$	24.0000	1.61441
$222$	4.00000	0.268462
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	6.00000	0.399114
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	−6.00000	−0.397360
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	6.00000	0.393919
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	−8.00000	−0.520756
$237$	−12.0000	−0.779484
$238$	−4.00000	−0.259281
$239$	26.0000	1.68180	0.840900	−	0.541190i	$-0.182026\pi$
$239$	26.0000	1.68180	0.840900	+	0.541190i	$0.182026\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	−7.00000	−0.449977
$243$	1.00000	0.0641500
$244$	−10.0000	−0.640184
$245$	0	0
$246$	2.00000	0.127515
$247$	−36.0000	−2.29063
$248$	−2.00000	−0.127000
$249$	−8.00000	−0.506979
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−1.00000	−0.0629941
$253$	−16.0000	−1.00591
$254$	4.00000	0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	4.00000	0.249029
$259$	−4.00000	−0.248548
$260$	0	0
$261$	6.00000	0.371391
$262$	−12.0000	−0.741362
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	2.00000	0.123091
$265$	0	0
$266$	6.00000	0.367884
$267$	−10.0000	−0.611990
$268$	8.00000	0.488678
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	10.0000	0.607457	0.303728	−	0.952759i	$-0.401768\pi$
$271$	10.0000	0.607457	0.303728	+	0.952759i	$0.401768\pi$
$272$	4.00000	0.242536
$273$	−6.00000	−0.363137
$274$	−6.00000	−0.362473
$275$	0	0
$276$	−8.00000	−0.481543
$277$	28.0000	1.68236	0.841178	−	0.540758i	$-0.181862\pi$
$277$	28.0000	1.68236	0.841178	+	0.540758i	$0.181862\pi$
$278$	−14.0000	−0.839664
$279$	−2.00000	−0.119737
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	8.00000	0.476393
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	12.0000	0.709575
$287$	−2.00000	−0.118056
$288$	1.00000	0.0589256
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−10.0000	−0.586210
$292$	−14.0000	−0.819288
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	1.00000	0.0583212
$295$	0	0
$296$	4.00000	0.232495
$297$	2.00000	0.116052
$298$	10.0000	0.579284
$299$	−48.0000	−2.77591
$300$	0	0
$301$	−4.00000	−0.230556
$302$	−8.00000	−0.460348
$303$	10.0000	0.574485
$304$	−6.00000	−0.344124
$305$	0	0
$306$	4.00000	0.228665
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	−2.00000	−0.113961
$309$	8.00000	0.455104
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	6.00000	0.339683
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	−22.0000	−1.24153
$315$	0	0
$316$	−12.0000	−0.675053
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	6.00000	0.336463
$319$	12.0000	0.671871
$320$	0	0
$321$	12.0000	0.669775
$322$	8.00000	0.445823
$323$	−24.0000	−1.33540
$324$	1.00000	0.0555556
$325$	0	0
$326$	−4.00000	−0.221540
$327$	−14.0000	−0.774202
$328$	2.00000	0.110432
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	−8.00000	−0.439057
$333$	4.00000	0.219199
$334$	−12.0000	−0.656611
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	23.0000	1.25104
$339$	6.00000	0.325875
$340$	0	0
$341$	−4.00000	−0.216612
$342$	−6.00000	−0.324443
$343$	−1.00000	−0.0539949
$344$	4.00000	0.215666
$345$	0	0
$346$	8.00000	0.430083
$347$	36.0000	1.93258	0.966291	−	0.257454i	$-0.0828835\pi$
$347$	36.0000	1.93258	0.966291	+	0.257454i	$0.0828835\pi$
$348$	6.00000	0.321634
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	2.00000	0.106600
$353$	20.0000	1.06449	0.532246	−	0.846590i	$-0.321348\pi$
$353$	20.0000	1.06449	0.532246	+	0.846590i	$0.321348\pi$
$354$	−8.00000	−0.425195
$355$	0	0
$356$	−10.0000	−0.529999
$357$	−4.00000	−0.211702
$358$	10.0000	0.528516
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	2.00000	0.105118
$363$	−7.00000	−0.367405
$364$	−6.00000	−0.314485
$365$	0	0
$366$	−10.0000	−0.522708
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	−8.00000	−0.417029
$369$	2.00000	0.104116
$370$	0	0
$371$	−6.00000	−0.311504
$372$	−2.00000	−0.103695
$373$	−24.0000	−1.24267	−0.621336	−	0.783544i	$-0.713410\pi$
$373$	−24.0000	−1.24267	−0.621336	+	0.783544i	$0.713410\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	8.00000	0.412568
$377$	36.0000	1.85409
$378$	−1.00000	−0.0514344
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	−18.0000	−0.920960
$383$	20.0000	1.02195	0.510976	−	0.859595i	$-0.329284\pi$
$383$	20.0000	1.02195	0.510976	+	0.859595i	$0.329284\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−8.00000	−0.407189
$387$	4.00000	0.203331
$388$	−10.0000	−0.507673
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	0	0
$391$	−32.0000	−1.61831
$392$	1.00000	0.0505076
$393$	−12.0000	−0.605320
$394$	2.00000	0.100759
$395$	0	0
$396$	2.00000	0.100504
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	6.00000	0.300753
$399$	6.00000	0.300376
$400$	0	0
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	8.00000	0.399004
$403$	−12.0000	−0.597763
$404$	10.0000	0.497519
$405$	0	0
$406$	−6.00000	−0.297775
$407$	8.00000	0.396545
$408$	4.00000	0.198030
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	8.00000	0.394132
$413$	8.00000	0.393654
$414$	−8.00000	−0.393179
$415$	0	0
$416$	6.00000	0.294174
$417$	−14.0000	−0.685583
$418$	−12.0000	−0.586939
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	6.00000	0.291386
$425$	0	0
$426$	−6.00000	−0.290701
$427$	10.0000	0.483934
$428$	12.0000	0.580042
$429$	12.0000	0.579365
$430$	0	0
$431$	2.00000	0.0963366	0.0481683	−	0.998839i	$-0.484662\pi$
$431$	2.00000	0.0963366	0.0481683	+	0.998839i	$0.484662\pi$
$432$	1.00000	0.0481125
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	2.00000	0.0960031
$435$	0	0
$436$	−14.0000	−0.670478
$437$	48.0000	2.29615
$438$	−14.0000	−0.668946
$439$	−18.0000	−0.859093	−0.429547	−	0.903045i	$-0.641327\pi$
$439$	−18.0000	−0.859093	−0.429547	+	0.903045i	$0.641327\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	24.0000	1.14156
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	−8.00000	−0.378811
$447$	10.0000	0.472984
$448$	−1.00000	−0.0472456
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	6.00000	0.282216
$453$	−8.00000	−0.375873
$454$	−8.00000	−0.375459
$455$	0	0
$456$	−6.00000	−0.280976
$457$	28.0000	1.30978	0.654892	−	0.755722i	$-0.272714\pi$
$457$	28.0000	1.30978	0.654892	+	0.755722i	$0.272714\pi$
$458$	14.0000	0.654177
$459$	4.00000	0.186704
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	−2.00000	−0.0930484
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−10.0000	−0.463241
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	6.00000	0.277350
$469$	−8.00000	−0.369406
$470$	0	0
$471$	−22.0000	−1.01371
$472$	−8.00000	−0.368230
$473$	8.00000	0.367840
$474$	−12.0000	−0.551178
$475$	0	0
$476$	−4.00000	−0.183340
$477$	6.00000	0.274721
$478$	26.0000	1.18921
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	24.0000	1.09431
$482$	−26.0000	−1.18427
$483$	8.00000	0.364013
$484$	−7.00000	−0.318182
$485$	0	0
$486$	1.00000	0.0453609
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	−10.0000	−0.452679
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	2.00000	0.0901670
$493$	24.0000	1.08091
$494$	−36.0000	−1.61972
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	6.00000	0.269137
$498$	−8.00000	−0.358489
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	0	0
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	−16.0000	−0.711287
$507$	23.0000	1.02147
$508$	4.00000	0.177471
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	14.0000	0.619324
$512$	1.00000	0.0441942
$513$	−6.00000	−0.264906
$514$	0	0
$515$	0	0
$516$	4.00000	0.176090
$517$	16.0000	0.703679
$518$	−4.00000	−0.175750
$519$	8.00000	0.351161
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	6.00000	0.262613
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	16.0000	0.697633
$527$	−8.00000	−0.348485
$528$	2.00000	0.0870388
$529$	41.0000	1.78261
$530$	0	0
$531$	−8.00000	−0.347170
$532$	6.00000	0.260133
$533$	12.0000	0.519778
$534$	−10.0000	−0.432742
$535$	0	0
$536$	8.00000	0.345547
$537$	10.0000	0.431532
$538$	18.0000	0.776035
$539$	2.00000	0.0861461
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	10.0000	0.429537
$543$	2.00000	0.0858282
$544$	4.00000	0.171499
$545$	0	0
$546$	−6.00000	−0.256776
$547$	40.0000	1.71028	0.855138	−	0.518400i	$-0.173472\pi$
$547$	40.0000	1.71028	0.855138	+	0.518400i	$0.173472\pi$
$548$	−6.00000	−0.256307
$549$	−10.0000	−0.426790
$550$	0	0
$551$	−36.0000	−1.53365
$552$	−8.00000	−0.340503
$553$	12.0000	0.510292
$554$	28.0000	1.18961
$555$	0	0
$556$	−14.0000	−0.593732
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	−2.00000	−0.0846668
$559$	24.0000	1.01509
$560$	0	0
$561$	8.00000	0.337760
$562$	30.0000	1.26547
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	8.00000	0.336861
$565$	0	0
$566$	20.0000	0.840663
$567$	−1.00000	−0.0419961
$568$	−6.00000	−0.251754
$569$	−46.0000	−1.92842	−0.964210	−	0.265139i	$-0.914582\pi$
$569$	−46.0000	−1.92842	−0.964210	+	0.265139i	$0.914582\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	12.0000	0.501745
$573$	−18.0000	−0.751961
$574$	−2.00000	−0.0834784
$575$	0	0
$576$	1.00000	0.0416667
$577$	−26.0000	−1.08239	−0.541197	−	0.840896i	$-0.682029\pi$
$577$	−26.0000	−1.08239	−0.541197	+	0.840896i	$0.682029\pi$
$578$	−1.00000	−0.0415945
$579$	−8.00000	−0.332469
$580$	0	0
$581$	8.00000	0.331896
$582$	−10.0000	−0.414513
$583$	12.0000	0.496989
$584$	−14.0000	−0.579324
$585$	0	0
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	1.00000	0.0412393
$589$	12.0000	0.494451
$590$	0	0
$591$	2.00000	0.0822690
$592$	4.00000	0.164399
$593$	12.0000	0.492781	0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000	0.492781	0.246390	+	0.969171i	$0.420755\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	10.0000	0.409616
$597$	6.00000	0.245564
$598$	−48.0000	−1.96287
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	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
$603$	8.00000	0.325785
$604$	−8.00000	−0.325515
$605$	0	0
$606$	10.0000	0.406222
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	−6.00000	−0.243332
$609$	−6.00000	−0.243132
$610$	0	0
$611$	48.0000	1.94187
$612$	4.00000	0.161690
$613$	28.0000	1.13091	0.565455	−	0.824779i	$-0.308701\pi$
$613$	28.0000	1.13091	0.565455	+	0.824779i	$0.308701\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	−2.00000	−0.0805823
$617$	−26.0000	−1.04672	−0.523360	−	0.852111i	$-0.675322\pi$
$617$	−26.0000	−1.04672	−0.523360	+	0.852111i	$0.675322\pi$
$618$	8.00000	0.321807
$619$	−22.0000	−0.884255	−0.442127	−	0.896952i	$-0.645776\pi$
$619$	−22.0000	−0.884255	−0.442127	+	0.896952i	$0.645776\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	24.0000	0.962312
$623$	10.0000	0.400642
$624$	6.00000	0.240192
$625$	0	0
$626$	−2.00000	−0.0799361
$627$	−12.0000	−0.479234
$628$	−22.0000	−0.877896
$629$	16.0000	0.637962
$630$	0	0
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	−12.0000	−0.477334
$633$	0	0
$634$	−2.00000	−0.0794301
$635$	0	0
$636$	6.00000	0.237915
$637$	6.00000	0.237729
$638$	12.0000	0.475085
$639$	−6.00000	−0.237356
$640$	0	0
$641$	14.0000	0.552967	0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000	0.552967	0.276483	+	0.961019i	$0.410831\pi$
$642$	12.0000	0.473602
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	−24.0000	−0.944267
$647$	36.0000	1.41531	0.707653	−	0.706560i	$-0.249754\pi$
$647$	36.0000	1.41531	0.707653	+	0.706560i	$0.249754\pi$
$648$	1.00000	0.0392837
$649$	−16.0000	−0.628055
$650$	0	0
$651$	2.00000	0.0783862
$652$	−4.00000	−0.156652
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	−14.0000	−0.547443
$655$	0	0
$656$	2.00000	0.0780869
$657$	−14.0000	−0.546192
$658$	−8.00000	−0.311872
$659$	−26.0000	−1.01282	−0.506408	−	0.862294i	$-0.669027\pi$
$659$	−26.0000	−1.01282	−0.506408	+	0.862294i	$0.669027\pi$
$660$	0	0
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	−8.00000	−0.310929
$663$	24.0000	0.932083
$664$	−8.00000	−0.310460
$665$	0	0
$666$	4.00000	0.154997
$667$	−48.0000	−1.85857
$668$	−12.0000	−0.464294
$669$	−8.00000	−0.309298
$670$	0	0
$671$	−20.0000	−0.772091
$672$	−1.00000	−0.0385758
$673$	12.0000	0.462566	0.231283	−	0.972887i	$-0.425708\pi$
$673$	12.0000	0.462566	0.231283	+	0.972887i	$0.425708\pi$
$674$	−8.00000	−0.308148
$675$	0	0
$676$	23.0000	0.884615
$677$	40.0000	1.53732	0.768662	−	0.639655i	$-0.220923\pi$
$677$	40.0000	1.53732	0.768662	+	0.639655i	$0.220923\pi$
$678$	6.00000	0.230429
$679$	10.0000	0.383765
$680$	0	0
$681$	−8.00000	−0.306561
$682$	−4.00000	−0.153168
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	−6.00000	−0.229416
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	14.0000	0.534133
$688$	4.00000	0.152499
$689$	36.0000	1.37149
$690$	0	0
$691$	−2.00000	−0.0760836	−0.0380418	−	0.999276i	$-0.512112\pi$
$691$	−2.00000	−0.0760836	−0.0380418	+	0.999276i	$0.512112\pi$
$692$	8.00000	0.304114
$693$	−2.00000	−0.0759737
$694$	36.0000	1.36654
$695$	0	0
$696$	6.00000	0.227429
$697$	8.00000	0.303022
$698$	−26.0000	−0.984115
$699$	−10.0000	−0.378235
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	6.00000	0.226455
$703$	−24.0000	−0.905177
$704$	2.00000	0.0753778
$705$	0	0
$706$	20.0000	0.752710
$707$	−10.0000	−0.376089
$708$	−8.00000	−0.300658
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	−12.0000	−0.450035
$712$	−10.0000	−0.374766
$713$	16.0000	0.599205
$714$	−4.00000	−0.149696
$715$	0	0
$716$	10.0000	0.373718
$717$	26.0000	0.970988
$718$	6.00000	0.223918
$719$	−4.00000	−0.149175	−0.0745874	−	0.997214i	$-0.523764\pi$
$719$	−4.00000	−0.149175	−0.0745874	+	0.997214i	$0.523764\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	17.0000	0.632674
$723$	−26.0000	−0.966950
$724$	2.00000	0.0743294
$725$	0	0
$726$	−7.00000	−0.259794
$727$	24.0000	0.890111	0.445055	−	0.895503i	$-0.353184\pi$
$727$	24.0000	0.890111	0.445055	+	0.895503i	$0.353184\pi$
$728$	−6.00000	−0.222375
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.0000	0.591781
$732$	−10.0000	−0.369611
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	−32.0000	−1.18114
$735$	0	0
$736$	−8.00000	−0.294884
$737$	16.0000	0.589368
$738$	2.00000	0.0736210
$739$	−44.0000	−1.61857	−0.809283	−	0.587419i	$-0.800144\pi$
$739$	−44.0000	−1.61857	−0.809283	+	0.587419i	$0.800144\pi$
$740$	0	0
$741$	−36.0000	−1.32249
$742$	−6.00000	−0.220267
$743$	−48.0000	−1.76095	−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000	−1.76095	−0.880475	+	0.474093i	$0.842776\pi$
$744$	−2.00000	−0.0733236
$745$	0	0
$746$	−24.0000	−0.878702
$747$	−8.00000	−0.292705
$748$	8.00000	0.292509
$749$	−12.0000	−0.438470
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	36.0000	1.31104
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	−32.0000	−1.16306	−0.581530	−	0.813525i	$-0.697546\pi$
$757$	−32.0000	−1.16306	−0.581530	+	0.813525i	$0.697546\pi$
$758$	20.0000	0.726433
$759$	−16.0000	−0.580763
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	4.00000	0.144905
$763$	14.0000	0.506834
$764$	−18.0000	−0.651217
$765$	0	0
$766$	20.0000	0.722629
$767$	−48.0000	−1.73318
$768$	1.00000	0.0360844
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	0	0
$772$	−8.00000	−0.287926
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	−10.0000	−0.358979
$777$	−4.00000	−0.143499
$778$	2.00000	0.0717035
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−12.0000	−0.429394
$782$	−32.0000	−1.14432
$783$	6.00000	0.214423
$784$	1.00000	0.0357143
$785$	0	0
$786$	−12.0000	−0.428026
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	2.00000	0.0712470
$789$	16.0000	0.569615
$790$	0	0
$791$	−6.00000	−0.213335
$792$	2.00000	0.0710669
$793$	−60.0000	−2.13066
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	6.00000	0.212664
$797$	−8.00000	−0.283375	−0.141687	−	0.989911i	$-0.545253\pi$
$797$	−8.00000	−0.283375	−0.141687	+	0.989911i	$0.545253\pi$
$798$	6.00000	0.212398
$799$	32.0000	1.13208
$800$	0	0
$801$	−10.0000	−0.353333
$802$	−14.0000	−0.494357
$803$	−28.0000	−0.988099
$804$	8.00000	0.282138
$805$	0	0
$806$	−12.0000	−0.422682
$807$	18.0000	0.633630
$808$	10.0000	0.351799
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	−14.0000	−0.491606	−0.245803	−	0.969320i	$-0.579052\pi$
$811$	−14.0000	−0.491606	−0.245803	+	0.969320i	$0.579052\pi$
$812$	−6.00000	−0.210559
$813$	10.0000	0.350715
$814$	8.00000	0.280400
$815$	0	0
$816$	4.00000	0.140028
$817$	−24.0000	−0.839654
$818$	26.0000	0.909069
$819$	−6.00000	−0.209657
$820$	0	0
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	−6.00000	−0.209274
$823$	12.0000	0.418294	0.209147	−	0.977884i	$-0.432931\pi$
$823$	12.0000	0.418294	0.209147	+	0.977884i	$0.432931\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	8.00000	0.278356
$827$	−20.0000	−0.695468	−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000	−0.695468	−0.347734	+	0.937593i	$0.613049\pi$
$828$	−8.00000	−0.278019
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	0	0
$831$	28.0000	0.971309
$832$	6.00000	0.208013
$833$	4.00000	0.138592
$834$	−14.0000	−0.484780
$835$	0	0
$836$	−12.0000	−0.415029
$837$	−2.00000	−0.0691301
$838$	20.0000	0.690889
$839$	−20.0000	−0.690477	−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000	−0.690477	−0.345238	+	0.938515i	$0.612202\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	34.0000	1.17172
$843$	30.0000	1.03325
$844$	0	0
$845$	0	0
$846$	8.00000	0.275046
$847$	7.00000	0.240523
$848$	6.00000	0.206041
$849$	20.0000	0.686398
$850$	0	0
$851$	−32.0000	−1.09695
$852$	−6.00000	−0.205557
$853$	38.0000	1.30110	0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000	1.30110	0.650548	+	0.759465i	$0.274539\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	12.0000	0.410152
$857$	24.0000	0.819824	0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000	0.819824	0.409912	+	0.912125i	$0.365559\pi$
$858$	12.0000	0.409673
$859$	−10.0000	−0.341196	−0.170598	−	0.985341i	$-0.554570\pi$
$859$	−10.0000	−0.341196	−0.170598	+	0.985341i	$0.554570\pi$
$860$	0	0
$861$	−2.00000	−0.0681598
$862$	2.00000	0.0681203
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−26.0000	−0.883516
$867$	−1.00000	−0.0339618
$868$	2.00000	0.0678844
$869$	−24.0000	−0.814144
$870$	0	0
$871$	48.0000	1.62642
$872$	−14.0000	−0.474100
$873$	−10.0000	−0.338449
$874$	48.0000	1.62362
$875$	0	0
$876$	−14.0000	−0.473016
$877$	−28.0000	−0.945493	−0.472746	−	0.881199i	$-0.656737\pi$
$877$	−28.0000	−0.945493	−0.472746	+	0.881199i	$0.656737\pi$
$878$	−18.0000	−0.607471
$879$	0	0
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	1.00000	0.0336718
$883$	24.0000	0.807664	0.403832	−	0.914833i	$-0.367678\pi$
$883$	24.0000	0.807664	0.403832	+	0.914833i	$0.367678\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	−12.0000	−0.403148
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	4.00000	0.134231
$889$	−4.00000	−0.134156
$890$	0	0
$891$	2.00000	0.0670025
$892$	−8.00000	−0.267860
$893$	−48.0000	−1.60626
$894$	10.0000	0.334450
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	−48.0000	−1.60267
$898$	−6.00000	−0.200223
$899$	−12.0000	−0.400222
$900$	0	0
$901$	24.0000	0.799556
$902$	4.00000	0.133185
$903$	−4.00000	−0.133112
$904$	6.00000	0.199557
$905$	0	0
$906$	−8.00000	−0.265782
$907$	24.0000	0.796907	0.398453	−	0.917189i	$-0.369547\pi$
$907$	24.0000	0.796907	0.398453	+	0.917189i	$0.369547\pi$
$908$	−8.00000	−0.265489
$909$	10.0000	0.331679
$910$	0	0
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	−6.00000	−0.198680
$913$	−16.0000	−0.529523
$914$	28.0000	0.926158
$915$	0	0
$916$	14.0000	0.462573
$917$	12.0000	0.396275
$918$	4.00000	0.132020
$919$	60.0000	1.97922	0.989609	−	0.143787i	$-0.0459280\pi$
$919$	60.0000	1.97922	0.989609	+	0.143787i	$0.0459280\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	−18.0000	−0.592798
$923$	−36.0000	−1.18495
$924$	−2.00000	−0.0657952
$925$	0	0
$926$	36.0000	1.18303
$927$	8.00000	0.262754
$928$	6.00000	0.196960
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	−10.0000	−0.327561
$933$	24.0000	0.785725
$934$	−24.0000	−0.785304
$935$	0	0
$936$	6.00000	0.196116
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	−8.00000	−0.261209
$939$	−2.00000	−0.0652675
$940$	0	0
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	−22.0000	−0.716799
$943$	−16.0000	−0.521032
$944$	−8.00000	−0.260378
$945$	0	0
$946$	8.00000	0.260102
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−12.0000	−0.389742
$949$	−84.0000	−2.72676
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	−4.00000	−0.129641
$953$	−10.0000	−0.323932	−0.161966	−	0.986796i	$-0.551783\pi$
$953$	−10.0000	−0.323932	−0.161966	+	0.986796i	$0.551783\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	26.0000	0.840900
$957$	12.0000	0.387905
$958$	8.00000	0.258468
$959$	6.00000	0.193750
$960$	0	0
$961$	−27.0000	−0.870968
$962$	24.0000	0.773791
$963$	12.0000	0.386695
$964$	−26.0000	−0.837404
$965$	0	0
$966$	8.00000	0.257396
$967$	56.0000	1.80084	0.900419	−	0.435023i	$-0.143260\pi$
$967$	56.0000	1.80084	0.900419	+	0.435023i	$0.143260\pi$
$968$	−7.00000	−0.224989
$969$	−24.0000	−0.770991
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	1.00000	0.0320750
$973$	14.0000	0.448819
$974$	12.0000	0.384505
$975$	0	0
$976$	−10.0000	−0.320092
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	−4.00000	−0.127906
$979$	−20.0000	−0.639203
$980$	0	0
$981$	−14.0000	−0.446986
$982$	−30.0000	−0.957338
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	2.00000	0.0637577
$985$	0	0
$986$	24.0000	0.764316
$987$	−8.00000	−0.254643
$988$	−36.0000	−1.14531
$989$	−32.0000	−1.01754
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	−2.00000	−0.0635001
$993$	−8.00000	−0.253872
$994$	6.00000	0.190308
$995$	0	0
$996$	−8.00000	−0.253490
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	−12.0000	−0.379853
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.2.a.p.1.1		1
3.2	odd	2		3150.2.a.d.1.1		1
4.3	odd	2		8400.2.a.w.1.1		1
5.2	odd	4		210.2.g.b.169.2	yes	2
5.3	odd	4		210.2.g.b.169.1	✓	2
5.4	even	2		1050.2.a.d.1.1		1
7.6	odd	2		7350.2.a.bz.1.1		1
15.2	even	4		630.2.g.c.379.1		2
15.8	even	4		630.2.g.c.379.2		2
15.14	odd	2		3150.2.a.bk.1.1		1
20.3	even	4		1680.2.t.e.1009.1		2
20.7	even	4		1680.2.t.e.1009.2		2
20.19	odd	2		8400.2.a.bp.1.1		1
35.2	odd	12		1470.2.n.b.949.2		4
35.3	even	12		1470.2.n.f.79.2		4
35.12	even	12		1470.2.n.f.949.2		4
35.13	even	4		1470.2.g.b.589.1		2
35.17	even	12		1470.2.n.f.79.1		4
35.18	odd	12		1470.2.n.b.79.2		4
35.23	odd	12		1470.2.n.b.949.1		4
35.27	even	4		1470.2.g.b.589.2		2
35.32	odd	12		1470.2.n.b.79.1		4
35.33	even	12		1470.2.n.f.949.1		4
35.34	odd	2		7350.2.a.bk.1.1		1
60.23	odd	4		5040.2.t.h.1009.1		2
60.47	odd	4		5040.2.t.h.1009.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.g.b.169.1	✓	2	5.3	odd	4
210.2.g.b.169.2	yes	2	5.2	odd	4
630.2.g.c.379.1		2	15.2	even	4
630.2.g.c.379.2		2	15.8	even	4
1050.2.a.d.1.1		1	5.4	even	2
1050.2.a.p.1.1		1	1.1	even	1	trivial
1470.2.g.b.589.1		2	35.13	even	4
1470.2.g.b.589.2		2	35.27	even	4
1470.2.n.b.79.1		4	35.32	odd	12
1470.2.n.b.79.2		4	35.18	odd	12
1470.2.n.b.949.1		4	35.23	odd	12
1470.2.n.b.949.2		4	35.2	odd	12
1470.2.n.f.79.1		4	35.17	even	12
1470.2.n.f.79.2		4	35.3	even	12
1470.2.n.f.949.1		4	35.33	even	12
1470.2.n.f.949.2		4	35.12	even	12
1680.2.t.e.1009.1		2	20.3	even	4
1680.2.t.e.1009.2		2	20.7	even	4
3150.2.a.d.1.1		1	3.2	odd	2
3150.2.a.bk.1.1		1	15.14	odd	2
5040.2.t.h.1009.1		2	60.23	odd	4
5040.2.t.h.1009.2		2	60.47	odd	4
7350.2.a.bk.1.1		1	35.34	odd	2
7350.2.a.bz.1.1		1	7.6	odd	2
8400.2.a.w.1.1		1	4.3	odd	2
8400.2.a.bp.1.1		1	20.19	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:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1050.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more