LMFDB - Embedded newform 1003.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1003,2,Mod(1,1003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1003, 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("1003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1003 = 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1003.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:	$8.00899532273$
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	$\chi$	$=$	1003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +3.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +3.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +6.00000 q^{9} -1.00000 q^{10} -3.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +3.00000 q^{15} -1.00000 q^{16} +1.00000 q^{17} -6.00000 q^{18} +5.00000 q^{19} -1.00000 q^{20} +3.00000 q^{21} +4.00000 q^{23} +9.00000 q^{24} -4.00000 q^{25} +2.00000 q^{26} +9.00000 q^{27} -1.00000 q^{28} -9.00000 q^{29} -3.00000 q^{30} +10.0000 q^{31} -5.00000 q^{32} -1.00000 q^{34} +1.00000 q^{35} -6.00000 q^{36} -8.00000 q^{37} -5.00000 q^{38} -6.00000 q^{39} +3.00000 q^{40} -3.00000 q^{41} -3.00000 q^{42} +10.0000 q^{43} +6.00000 q^{45} -4.00000 q^{46} +12.0000 q^{47} -3.00000 q^{48} -6.00000 q^{49} +4.00000 q^{50} +3.00000 q^{51} +2.00000 q^{52} +3.00000 q^{53} -9.00000 q^{54} +3.00000 q^{56} +15.0000 q^{57} +9.00000 q^{58} -1.00000 q^{59} -3.00000 q^{60} +8.00000 q^{61} -10.0000 q^{62} +6.00000 q^{63} +7.00000 q^{64} -2.00000 q^{65} -8.00000 q^{67} -1.00000 q^{68} +12.0000 q^{69} -1.00000 q^{70} +8.00000 q^{71} +18.0000 q^{72} -6.00000 q^{73} +8.00000 q^{74} -12.0000 q^{75} -5.00000 q^{76} +6.00000 q^{78} -9.00000 q^{79} -1.00000 q^{80} +9.00000 q^{81} +3.00000 q^{82} -16.0000 q^{83} -3.00000 q^{84} +1.00000 q^{85} -10.0000 q^{86} -27.0000 q^{87} -14.0000 q^{89} -6.00000 q^{90} -2.00000 q^{91} -4.00000 q^{92} +30.0000 q^{93} -12.0000 q^{94} +5.00000 q^{95} -15.0000 q^{96} +2.00000 q^{97} +6.00000 q^{98} +O(q^{100})\)

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−3.00000	−1.22474
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	3.00000	1.06066
$9$	6.00000	2.00000
$10$	−1.00000	−0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−3.00000	−0.866025
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−1.00000	−0.267261
$15$	3.00000	0.774597
$16$	−1.00000	−0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−6.00000	−1.41421
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	−1.00000	−0.223607
$21$	3.00000	0.654654
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	9.00000	1.83712
$25$	−4.00000	−0.800000
$26$	2.00000	0.392232
$27$	9.00000	1.73205
$28$	−1.00000	−0.188982
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	−3.00000	−0.547723
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	−1.00000	−0.171499
$35$	1.00000	0.169031
$36$	−6.00000	−1.00000
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	−5.00000	−0.811107
$39$	−6.00000	−0.960769
$40$	3.00000	0.474342
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	−3.00000	−0.462910
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	0	0
$45$	6.00000	0.894427
$46$	−4.00000	−0.589768
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	−3.00000	−0.433013
$49$	−6.00000	−0.857143
$50$	4.00000	0.565685
$51$	3.00000	0.420084
$52$	2.00000	0.277350
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	−9.00000	−1.22474
$55$	0	0
$56$	3.00000	0.400892
$57$	15.0000	1.98680
$58$	9.00000	1.18176
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	−3.00000	−0.387298
$61$	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
$63$	6.00000	0.755929
$64$	7.00000	0.875000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−1.00000	−0.121268
$69$	12.0000	1.44463
$70$	−1.00000	−0.119523
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	18.0000	2.12132
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	8.00000	0.929981
$75$	−12.0000	−1.38564
$76$	−5.00000	−0.573539
$77$	0	0
$78$	6.00000	0.679366
$79$	−9.00000	−1.01258	−0.506290	−	0.862364i	$-0.668983\pi$
$79$	−9.00000	−1.01258	−0.506290	+	0.862364i	$0.668983\pi$
$80$	−1.00000	−0.111803
$81$	9.00000	1.00000
$82$	3.00000	0.331295
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	−3.00000	−0.327327
$85$	1.00000	0.108465
$86$	−10.0000	−1.07833
$87$	−27.0000	−2.89470
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	−6.00000	−0.632456
$91$	−2.00000	−0.209657
$92$	−4.00000	−0.417029
$93$	30.0000	3.11086
$94$	−12.0000	−1.23771
$95$	5.00000	0.512989
$96$	−15.0000	−1.53093
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	6.00000	0.606092
$99$	0	0
$100$	4.00000	0.400000
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	−3.00000	−0.297044
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	−6.00000	−0.588348
$105$	3.00000	0.292770
$106$	−3.00000	−0.291386
$107$	11.0000	1.06341	0.531705	−	0.846930i	$-0.321551\pi$
$107$	11.0000	1.06341	0.531705	+	0.846930i	$0.321551\pi$
$108$	−9.00000	−0.866025
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	−24.0000	−2.27798
$112$	−1.00000	−0.0944911
$113$	16.0000	1.50515	0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000	1.50515	0.752577	+	0.658505i	$0.228811\pi$
$114$	−15.0000	−1.40488
$115$	4.00000	0.373002
$116$	9.00000	0.835629
$117$	−12.0000	−1.10940
$118$	1.00000	0.0920575
$119$	1.00000	0.0916698
$120$	9.00000	0.821584
$121$	−11.0000	−1.00000
$122$	−8.00000	−0.724286
$123$	−9.00000	−0.811503
$124$	−10.0000	−0.898027
$125$	−9.00000	−0.804984
$126$	−6.00000	−0.534522
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	3.00000	0.265165
$129$	30.0000	2.64135
$130$	2.00000	0.175412
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	0	0
$133$	5.00000	0.433555
$134$	8.00000	0.691095
$135$	9.00000	0.774597
$136$	3.00000	0.257248
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	−12.0000	−1.02151
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	−1.00000	−0.0845154
$141$	36.0000	3.03175
$142$	−8.00000	−0.671345
$143$	0	0
$144$	−6.00000	−0.500000
$145$	−9.00000	−0.747409
$146$	6.00000	0.496564
$147$	−18.0000	−1.48461
$148$	8.00000	0.657596
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	12.0000	0.979796
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	15.0000	1.21666
$153$	6.00000	0.485071
$154$	0	0
$155$	10.0000	0.803219
$156$	6.00000	0.480384
$157$	−20.0000	−1.59617	−0.798087	−	0.602542i	$-0.794154\pi$
$157$	−20.0000	−1.59617	−0.798087	+	0.602542i	$0.794154\pi$
$158$	9.00000	0.716002
$159$	9.00000	0.713746
$160$	−5.00000	−0.395285
$161$	4.00000	0.315244
$162$	−9.00000	−0.707107
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	3.00000	0.234261
$165$	0	0
$166$	16.0000	1.24184
$167$	−7.00000	−0.541676	−0.270838	−	0.962625i	$-0.587301\pi$
$167$	−7.00000	−0.541676	−0.270838	+	0.962625i	$0.587301\pi$
$168$	9.00000	0.694365
$169$	−9.00000	−0.692308
$170$	−1.00000	−0.0766965
$171$	30.0000	2.29416
$172$	−10.0000	−0.762493
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	27.0000	2.04686
$175$	−4.00000	−0.302372
$176$	0	0
$177$	−3.00000	−0.225494
$178$	14.0000	1.04934
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	−6.00000	−0.447214
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	2.00000	0.148250
$183$	24.0000	1.77413
$184$	12.0000	0.884652
$185$	−8.00000	−0.588172
$186$	−30.0000	−2.19971
$187$	0	0
$188$	−12.0000	−0.875190
$189$	9.00000	0.654654
$190$	−5.00000	−0.362738
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	21.0000	1.51554
$193$	−19.0000	−1.36765	−0.683825	−	0.729646i	$-0.739685\pi$
$193$	−19.0000	−1.36765	−0.683825	+	0.729646i	$0.739685\pi$
$194$	−2.00000	−0.143592
$195$	−6.00000	−0.429669
$196$	6.00000	0.428571
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	−12.0000	−0.848528
$201$	−24.0000	−1.69283
$202$	−8.00000	−0.562878
$203$	−9.00000	−0.631676
$204$	−3.00000	−0.210042
$205$	−3.00000	−0.209529
$206$	−14.0000	−0.975426
$207$	24.0000	1.66812
$208$	2.00000	0.138675
$209$	0	0
$210$	−3.00000	−0.207020
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	−3.00000	−0.206041
$213$	24.0000	1.64445
$214$	−11.0000	−0.751945
$215$	10.0000	0.681994
$216$	27.0000	1.83712
$217$	10.0000	0.678844
$218$	−6.00000	−0.406371
$219$	−18.0000	−1.21633
$220$	0	0
$221$	−2.00000	−0.134535
$222$	24.0000	1.61077
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	−5.00000	−0.334077
$225$	−24.0000	−1.60000
$226$	−16.0000	−1.06430
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	−15.0000	−0.993399
$229$	18.0000	1.18947	0.594737	−	0.803921i	$-0.297256\pi$
$229$	18.0000	1.18947	0.594737	+	0.803921i	$0.297256\pi$
$230$	−4.00000	−0.263752
$231$	0	0
$232$	−27.0000	−1.77264
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	12.0000	0.784465
$235$	12.0000	0.782794
$236$	1.00000	0.0650945
$237$	−27.0000	−1.75384
$238$	−1.00000	−0.0648204
$239$	−1.00000	−0.0646846	−0.0323423	−	0.999477i	$-0.510297\pi$
$239$	−1.00000	−0.0646846	−0.0323423	+	0.999477i	$0.510297\pi$
$240$	−3.00000	−0.193649
$241$	3.00000	0.193247	0.0966235	−	0.995321i	$-0.469196\pi$
$241$	3.00000	0.193247	0.0966235	+	0.995321i	$0.469196\pi$
$242$	11.0000	0.707107
$243$	0	0
$244$	−8.00000	−0.512148
$245$	−6.00000	−0.383326
$246$	9.00000	0.573819
$247$	−10.0000	−0.636285
$248$	30.0000	1.90500
$249$	−48.0000	−3.04188
$250$	9.00000	0.569210
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	−6.00000	−0.377964
$253$	0	0
$254$	7.00000	0.439219
$255$	3.00000	0.187867
$256$	−17.0000	−1.06250
$257$	−27.0000	−1.68421	−0.842107	−	0.539311i	$-0.818685\pi$
$257$	−27.0000	−1.68421	−0.842107	+	0.539311i	$0.818685\pi$
$258$	−30.0000	−1.86772
$259$	−8.00000	−0.497096
$260$	2.00000	0.124035
$261$	−54.0000	−3.34252
$262$	−10.0000	−0.617802
$263$	−1.00000	−0.0616626	−0.0308313	−	0.999525i	$-0.509815\pi$
$263$	−1.00000	−0.0616626	−0.0308313	+	0.999525i	$0.509815\pi$
$264$	0	0
$265$	3.00000	0.184289
$266$	−5.00000	−0.306570
$267$	−42.0000	−2.57036
$268$	8.00000	0.488678
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	−9.00000	−0.547723
$271$	−7.00000	−0.425220	−0.212610	−	0.977137i	$-0.568196\pi$
$271$	−7.00000	−0.425220	−0.212610	+	0.977137i	$0.568196\pi$
$272$	−1.00000	−0.0606339
$273$	−6.00000	−0.363137
$274$	9.00000	0.543710
$275$	0	0
$276$	−12.0000	−0.722315
$277$	−7.00000	−0.420589	−0.210295	−	0.977638i	$-0.567442\pi$
$277$	−7.00000	−0.420589	−0.210295	+	0.977638i	$0.567442\pi$
$278$	8.00000	0.479808
$279$	60.0000	3.59211
$280$	3.00000	0.179284
$281$	−33.0000	−1.96861	−0.984307	−	0.176462i	$-0.943535\pi$
$281$	−33.0000	−1.96861	−0.984307	+	0.176462i	$0.943535\pi$
$282$	−36.0000	−2.14377
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	−8.00000	−0.474713
$285$	15.0000	0.888523
$286$	0	0
$287$	−3.00000	−0.177084
$288$	−30.0000	−1.76777
$289$	1.00000	0.0588235
$290$	9.00000	0.528498
$291$	6.00000	0.351726
$292$	6.00000	0.351123
$293$	21.0000	1.22683	0.613417	−	0.789760i	$-0.289795\pi$
$293$	21.0000	1.22683	0.613417	+	0.789760i	$0.289795\pi$
$294$	18.0000	1.04978
$295$	−1.00000	−0.0582223
$296$	−24.0000	−1.39497
$297$	0	0
$298$	4.00000	0.231714
$299$	−8.00000	−0.462652
$300$	12.0000	0.692820
$301$	10.0000	0.576390
$302$	4.00000	0.230174
$303$	24.0000	1.37876
$304$	−5.00000	−0.286770
$305$	8.00000	0.458079
$306$	−6.00000	−0.342997
$307$	3.00000	0.171219	0.0856095	−	0.996329i	$-0.472716\pi$
$307$	3.00000	0.171219	0.0856095	+	0.996329i	$0.472716\pi$
$308$	0	0
$309$	42.0000	2.38930
$310$	−10.0000	−0.567962
$311$	−11.0000	−0.623753	−0.311876	−	0.950123i	$-0.600957\pi$
$311$	−11.0000	−0.623753	−0.311876	+	0.950123i	$0.600957\pi$
$312$	−18.0000	−1.01905
$313$	−28.0000	−1.58265	−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000	−1.58265	−0.791327	+	0.611393i	$0.790609\pi$
$314$	20.0000	1.12867
$315$	6.00000	0.338062
$316$	9.00000	0.506290
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	−9.00000	−0.504695
$319$	0	0
$320$	7.00000	0.391312
$321$	33.0000	1.84188
$322$	−4.00000	−0.222911
$323$	5.00000	0.278207
$324$	−9.00000	−0.500000
$325$	8.00000	0.443760
$326$	12.0000	0.664619
$327$	18.0000	0.995402
$328$	−9.00000	−0.496942
$329$	12.0000	0.661581
$330$	0	0
$331$	13.0000	0.714545	0.357272	−	0.934000i	$-0.383707\pi$
$331$	13.0000	0.714545	0.357272	+	0.934000i	$0.383707\pi$
$332$	16.0000	0.878114
$333$	−48.0000	−2.63038
$334$	7.00000	0.383023
$335$	−8.00000	−0.437087
$336$	−3.00000	−0.163663
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	9.00000	0.489535
$339$	48.0000	2.60700
$340$	−1.00000	−0.0542326
$341$	0	0
$342$	−30.0000	−1.62221
$343$	−13.0000	−0.701934
$344$	30.0000	1.61749
$345$	12.0000	0.646058
$346$	14.0000	0.752645
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	27.0000	1.44735
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	4.00000	0.213809
$351$	−18.0000	−0.960769
$352$	0	0
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	3.00000	0.159448
$355$	8.00000	0.424596
$356$	14.0000	0.741999
$357$	3.00000	0.158777
$358$	−24.0000	−1.26844
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	18.0000	0.948683
$361$	6.00000	0.315789
$362$	−7.00000	−0.367912
$363$	−33.0000	−1.73205
$364$	2.00000	0.104828
$365$	−6.00000	−0.314054
$366$	−24.0000	−1.25450
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	−4.00000	−0.208514
$369$	−18.0000	−0.937043
$370$	8.00000	0.415900
$371$	3.00000	0.155752
$372$	−30.0000	−1.55543
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	−27.0000	−1.39427
$376$	36.0000	1.85656
$377$	18.0000	0.927047
$378$	−9.00000	−0.462910
$379$	25.0000	1.28416	0.642082	−	0.766636i	$-0.278071\pi$
$379$	25.0000	1.28416	0.642082	+	0.766636i	$0.278071\pi$
$380$	−5.00000	−0.256495
$381$	−21.0000	−1.07586
$382$	−8.00000	−0.409316
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	9.00000	0.459279
$385$	0	0
$386$	19.0000	0.967075
$387$	60.0000	3.04997
$388$	−2.00000	−0.101535
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	6.00000	0.303822
$391$	4.00000	0.202289
$392$	−18.0000	−0.909137
$393$	30.0000	1.51330
$394$	−18.0000	−0.906827
$395$	−9.00000	−0.452839
$396$	0	0
$397$	12.0000	0.602263	0.301131	−	0.953583i	$-0.402636\pi$
$397$	12.0000	0.602263	0.301131	+	0.953583i	$0.402636\pi$
$398$	11.0000	0.551380
$399$	15.0000	0.750939
$400$	4.00000	0.200000
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	24.0000	1.19701
$403$	−20.0000	−0.996271
$404$	−8.00000	−0.398015
$405$	9.00000	0.447214
$406$	9.00000	0.446663
$407$	0	0
$408$	9.00000	0.445566
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	3.00000	0.148159
$411$	−27.0000	−1.33181
$412$	−14.0000	−0.689730
$413$	−1.00000	−0.0492068
$414$	−24.0000	−1.17954
$415$	−16.0000	−0.785409
$416$	10.0000	0.490290
$417$	−24.0000	−1.17529
$418$	0	0
$419$	14.0000	0.683945	0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000	0.683945	0.341972	+	0.939710i	$0.388905\pi$
$420$	−3.00000	−0.146385
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	22.0000	1.07094
$423$	72.0000	3.50076
$424$	9.00000	0.437079
$425$	−4.00000	−0.194029
$426$	−24.0000	−1.16280
$427$	8.00000	0.387147
$428$	−11.0000	−0.531705
$429$	0	0
$430$	−10.0000	−0.482243
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	−9.00000	−0.433013
$433$	−25.0000	−1.20142	−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000	−1.20142	−0.600712	+	0.799466i	$0.705116\pi$
$434$	−10.0000	−0.480015
$435$	−27.0000	−1.29455
$436$	−6.00000	−0.287348
$437$	20.0000	0.956730
$438$	18.0000	0.860073
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−36.0000	−1.71429
$442$	2.00000	0.0951303
$443$	22.0000	1.04525	0.522626	−	0.852562i	$-0.324953\pi$
$443$	22.0000	1.04525	0.522626	+	0.852562i	$0.324953\pi$
$444$	24.0000	1.13899
$445$	−14.0000	−0.663664
$446$	12.0000	0.568216
$447$	−12.0000	−0.567581
$448$	7.00000	0.330719
$449$	−5.00000	−0.235965	−0.117982	−	0.993016i	$-0.537643\pi$
$449$	−5.00000	−0.235965	−0.117982	+	0.993016i	$0.537643\pi$
$450$	24.0000	1.13137
$451$	0	0
$452$	−16.0000	−0.752577
$453$	−12.0000	−0.563809
$454$	18.0000	0.844782
$455$	−2.00000	−0.0937614
$456$	45.0000	2.10732
$457$	24.0000	1.12267	0.561336	−	0.827588i	$-0.310287\pi$
$457$	24.0000	1.12267	0.561336	+	0.827588i	$0.310287\pi$
$458$	−18.0000	−0.841085
$459$	9.00000	0.420084
$460$	−4.00000	−0.186501
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	9.00000	0.417815
$465$	30.0000	1.39122
$466$	−12.0000	−0.555889
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	12.0000	0.554700
$469$	−8.00000	−0.369406
$470$	−12.0000	−0.553519
$471$	−60.0000	−2.76465
$472$	−3.00000	−0.138086
$473$	0	0
$474$	27.0000	1.24015
$475$	−20.0000	−0.917663
$476$	−1.00000	−0.0458349
$477$	18.0000	0.824163
$478$	1.00000	0.0457389
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	−15.0000	−0.684653
$481$	16.0000	0.729537
$482$	−3.00000	−0.136646
$483$	12.0000	0.546019
$484$	11.0000	0.500000
$485$	2.00000	0.0908153
$486$	0	0
$487$	17.0000	0.770344	0.385172	−	0.922845i	$-0.374142\pi$
$487$	17.0000	0.770344	0.385172	+	0.922845i	$0.374142\pi$
$488$	24.0000	1.08643
$489$	−36.0000	−1.62798
$490$	6.00000	0.271052
$491$	−13.0000	−0.586682	−0.293341	−	0.956008i	$-0.594767\pi$
$491$	−13.0000	−0.586682	−0.293341	+	0.956008i	$0.594767\pi$
$492$	9.00000	0.405751
$493$	−9.00000	−0.405340
$494$	10.0000	0.449921
$495$	0	0
$496$	−10.0000	−0.449013
$497$	8.00000	0.358849
$498$	48.0000	2.15093
$499$	35.0000	1.56682	0.783408	−	0.621508i	$-0.213480\pi$
$499$	35.0000	1.56682	0.783408	+	0.621508i	$0.213480\pi$
$500$	9.00000	0.402492
$501$	−21.0000	−0.938211
$502$	−15.0000	−0.669483
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	18.0000	0.801784
$505$	8.00000	0.355995
$506$	0	0
$507$	−27.0000	−1.19911
$508$	7.00000	0.310575
$509$	−8.00000	−0.354594	−0.177297	−	0.984157i	$-0.556735\pi$
$509$	−8.00000	−0.354594	−0.177297	+	0.984157i	$0.556735\pi$
$510$	−3.00000	−0.132842
$511$	−6.00000	−0.265424
$512$	11.0000	0.486136
$513$	45.0000	1.98680
$514$	27.0000	1.19092
$515$	14.0000	0.616914
$516$	−30.0000	−1.32068
$517$	0	0
$518$	8.00000	0.351500
$519$	−42.0000	−1.84360
$520$	−6.00000	−0.263117
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	54.0000	2.36352
$523$	−21.0000	−0.918266	−0.459133	−	0.888368i	$-0.651840\pi$
$523$	−21.0000	−0.918266	−0.459133	+	0.888368i	$0.651840\pi$
$524$	−10.0000	−0.436852
$525$	−12.0000	−0.523723
$526$	1.00000	0.0436021
$527$	10.0000	0.435607
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−3.00000	−0.130312
$531$	−6.00000	−0.260378
$532$	−5.00000	−0.216777
$533$	6.00000	0.259889
$534$	42.0000	1.81752
$535$	11.0000	0.475571
$536$	−24.0000	−1.03664
$537$	72.0000	3.10703
$538$	14.0000	0.603583
$539$	0	0
$540$	−9.00000	−0.387298
$541$	12.0000	0.515920	0.257960	−	0.966156i	$-0.416950\pi$
$541$	12.0000	0.515920	0.257960	+	0.966156i	$0.416950\pi$
$542$	7.00000	0.300676
$543$	21.0000	0.901196
$544$	−5.00000	−0.214373
$545$	6.00000	0.257012
$546$	6.00000	0.256776
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	9.00000	0.384461
$549$	48.0000	2.04859
$550$	0	0
$551$	−45.0000	−1.91706
$552$	36.0000	1.53226
$553$	−9.00000	−0.382719
$554$	7.00000	0.297402
$555$	−24.0000	−1.01874
$556$	8.00000	0.339276
$557$	27.0000	1.14403	0.572013	−	0.820244i	$-0.306163\pi$
$557$	27.0000	1.14403	0.572013	+	0.820244i	$0.306163\pi$
$558$	−60.0000	−2.54000
$559$	−20.0000	−0.845910
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	33.0000	1.39202
$563$	14.0000	0.590030	0.295015	−	0.955493i	$-0.404675\pi$
$563$	14.0000	0.590030	0.295015	+	0.955493i	$0.404675\pi$
$564$	−36.0000	−1.51587
$565$	16.0000	0.673125
$566$	16.0000	0.672530
$567$	9.00000	0.377964
$568$	24.0000	1.00702
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	−15.0000	−0.628281
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	24.0000	1.00261
$574$	3.00000	0.125218
$575$	−16.0000	−0.667246
$576$	42.0000	1.75000
$577$	37.0000	1.54033	0.770165	−	0.637845i	$-0.220174\pi$
$577$	37.0000	1.54033	0.770165	+	0.637845i	$0.220174\pi$
$578$	−1.00000	−0.0415945
$579$	−57.0000	−2.36884
$580$	9.00000	0.373705
$581$	−16.0000	−0.663792
$582$	−6.00000	−0.248708
$583$	0	0
$584$	−18.0000	−0.744845
$585$	−12.0000	−0.496139
$586$	−21.0000	−0.867502
$587$	22.0000	0.908037	0.454019	−	0.890992i	$-0.349990\pi$
$587$	22.0000	0.908037	0.454019	+	0.890992i	$0.349990\pi$
$588$	18.0000	0.742307
$589$	50.0000	2.06021
$590$	1.00000	0.0411693
$591$	54.0000	2.22126
$592$	8.00000	0.328798
$593$	−27.0000	−1.10876	−0.554379	−	0.832265i	$-0.687044\pi$
$593$	−27.0000	−1.10876	−0.554379	+	0.832265i	$0.687044\pi$
$594$	0	0
$595$	1.00000	0.0409960
$596$	4.00000	0.163846
$597$	−33.0000	−1.35060
$598$	8.00000	0.327144
$599$	21.0000	0.858037	0.429018	−	0.903296i	$-0.358860\pi$
$599$	21.0000	0.858037	0.429018	+	0.903296i	$0.358860\pi$
$600$	−36.0000	−1.46969
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	−10.0000	−0.407570
$603$	−48.0000	−1.95471
$604$	4.00000	0.162758
$605$	−11.0000	−0.447214
$606$	−24.0000	−0.974933
$607$	−19.0000	−0.771186	−0.385593	−	0.922669i	$-0.626003\pi$
$607$	−19.0000	−0.771186	−0.385593	+	0.922669i	$0.626003\pi$
$608$	−25.0000	−1.01388
$609$	−27.0000	−1.09410
$610$	−8.00000	−0.323911
$611$	−24.0000	−0.970936
$612$	−6.00000	−0.242536
$613$	34.0000	1.37325	0.686624	−	0.727013i	$-0.259092\pi$
$613$	34.0000	1.37325	0.686624	+	0.727013i	$0.259092\pi$
$614$	−3.00000	−0.121070
$615$	−9.00000	−0.362915
$616$	0	0
$617$	27.0000	1.08698	0.543490	−	0.839416i	$-0.317103\pi$
$617$	27.0000	1.08698	0.543490	+	0.839416i	$0.317103\pi$
$618$	−42.0000	−1.68949
$619$	−3.00000	−0.120580	−0.0602901	−	0.998181i	$-0.519203\pi$
$619$	−3.00000	−0.120580	−0.0602901	+	0.998181i	$0.519203\pi$
$620$	−10.0000	−0.401610
$621$	36.0000	1.44463
$622$	11.0000	0.441060
$623$	−14.0000	−0.560898
$624$	6.00000	0.240192
$625$	11.0000	0.440000
$626$	28.0000	1.11911
$627$	0	0
$628$	20.0000	0.798087
$629$	−8.00000	−0.318981
$630$	−6.00000	−0.239046
$631$	28.0000	1.11466	0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000	1.11466	0.557331	+	0.830290i	$0.311825\pi$
$632$	−27.0000	−1.07400
$633$	−66.0000	−2.62326
$634$	−2.00000	−0.0794301
$635$	−7.00000	−0.277787
$636$	−9.00000	−0.356873
$637$	12.0000	0.475457
$638$	0	0
$639$	48.0000	1.89885
$640$	3.00000	0.118585
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	−33.0000	−1.30241
$643$	35.0000	1.38027	0.690133	−	0.723683i	$-0.257552\pi$
$643$	35.0000	1.38027	0.690133	+	0.723683i	$0.257552\pi$
$644$	−4.00000	−0.157622
$645$	30.0000	1.18125
$646$	−5.00000	−0.196722
$647$	23.0000	0.904223	0.452112	−	0.891961i	$-0.350671\pi$
$647$	23.0000	0.904223	0.452112	+	0.891961i	$0.350671\pi$
$648$	27.0000	1.06066
$649$	0	0
$650$	−8.00000	−0.313786
$651$	30.0000	1.17579
$652$	12.0000	0.469956
$653$	33.0000	1.29139	0.645695	−	0.763596i	$-0.276568\pi$
$653$	33.0000	1.29139	0.645695	+	0.763596i	$0.276568\pi$
$654$	−18.0000	−0.703856
$655$	10.0000	0.390732
$656$	3.00000	0.117130
$657$	−36.0000	−1.40449
$658$	−12.0000	−0.467809
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	0	0
$661$	35.0000	1.36134	0.680671	−	0.732589i	$-0.261688\pi$
$661$	35.0000	1.36134	0.680671	+	0.732589i	$0.261688\pi$
$662$	−13.0000	−0.505259
$663$	−6.00000	−0.233021
$664$	−48.0000	−1.86276
$665$	5.00000	0.193892
$666$	48.0000	1.85996
$667$	−36.0000	−1.39393
$668$	7.00000	0.270838
$669$	−36.0000	−1.39184
$670$	8.00000	0.309067
$671$	0	0
$672$	−15.0000	−0.578638
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	14.0000	0.539260
$675$	−36.0000	−1.38564
$676$	9.00000	0.346154
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	−48.0000	−1.84343
$679$	2.00000	0.0767530
$680$	3.00000	0.115045
$681$	−54.0000	−2.06928
$682$	0	0
$683$	14.0000	0.535695	0.267848	−	0.963461i	$-0.413688\pi$
$683$	14.0000	0.535695	0.267848	+	0.963461i	$0.413688\pi$
$684$	−30.0000	−1.14708
$685$	−9.00000	−0.343872
$686$	13.0000	0.496342
$687$	54.0000	2.06023
$688$	−10.0000	−0.381246
$689$	−6.00000	−0.228582
$690$	−12.0000	−0.456832
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	14.0000	0.532200
$693$	0	0
$694$	28.0000	1.06287
$695$	−8.00000	−0.303457
$696$	−81.0000	−3.07030
$697$	−3.00000	−0.113633
$698$	6.00000	0.227103
$699$	36.0000	1.36165
$700$	4.00000	0.151186
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	18.0000	0.679366
$703$	−40.0000	−1.50863
$704$	0	0
$705$	36.0000	1.35584
$706$	−24.0000	−0.903252
$707$	8.00000	0.300871
$708$	3.00000	0.112747
$709$	−13.0000	−0.488225	−0.244113	−	0.969747i	$-0.578497\pi$
$709$	−13.0000	−0.488225	−0.244113	+	0.969747i	$0.578497\pi$
$710$	−8.00000	−0.300235
$711$	−54.0000	−2.02516
$712$	−42.0000	−1.57402
$713$	40.0000	1.49801
$714$	−3.00000	−0.112272
$715$	0	0
$716$	−24.0000	−0.896922
$717$	−3.00000	−0.112037
$718$	15.0000	0.559795
$719$	−14.0000	−0.522112	−0.261056	−	0.965324i	$-0.584071\pi$
$719$	−14.0000	−0.522112	−0.261056	+	0.965324i	$0.584071\pi$
$720$	−6.00000	−0.223607
$721$	14.0000	0.521387
$722$	−6.00000	−0.223297
$723$	9.00000	0.334714
$724$	−7.00000	−0.260153
$725$	36.0000	1.33701
$726$	33.0000	1.22474
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	−6.00000	−0.222375
$729$	−27.0000	−1.00000
$730$	6.00000	0.222070
$731$	10.0000	0.369863
$732$	−24.0000	−0.887066
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	32.0000	1.18114
$735$	−18.0000	−0.663940
$736$	−20.0000	−0.737210
$737$	0	0
$738$	18.0000	0.662589
$739$	−38.0000	−1.39785	−0.698926	−	0.715194i	$-0.746338\pi$
$739$	−38.0000	−1.39785	−0.698926	+	0.715194i	$0.746338\pi$
$740$	8.00000	0.294086
$741$	−30.0000	−1.10208
$742$	−3.00000	−0.110133
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	90.0000	3.29956
$745$	−4.00000	−0.146549
$746$	−6.00000	−0.219676
$747$	−96.0000	−3.51246
$748$	0	0
$749$	11.0000	0.401931
$750$	27.0000	0.985901
$751$	−46.0000	−1.67856	−0.839282	−	0.543696i	$-0.817024\pi$
$751$	−46.0000	−1.67856	−0.839282	+	0.543696i	$0.817024\pi$
$752$	−12.0000	−0.437595
$753$	45.0000	1.63989
$754$	−18.0000	−0.655521
$755$	−4.00000	−0.145575
$756$	−9.00000	−0.327327
$757$	−47.0000	−1.70824	−0.854122	−	0.520073i	$-0.825905\pi$
$757$	−47.0000	−1.70824	−0.854122	+	0.520073i	$0.825905\pi$
$758$	−25.0000	−0.908041
$759$	0	0
$760$	15.0000	0.544107
$761$	−7.00000	−0.253750	−0.126875	−	0.991919i	$-0.540495\pi$
$761$	−7.00000	−0.253750	−0.126875	+	0.991919i	$0.540495\pi$
$762$	21.0000	0.760750
$763$	6.00000	0.217215
$764$	−8.00000	−0.289430
$765$	6.00000	0.216930
$766$	−12.0000	−0.433578
$767$	2.00000	0.0722158
$768$	−51.0000	−1.84030
$769$	4.00000	0.144244	0.0721218	−	0.997396i	$-0.477023\pi$
$769$	4.00000	0.144244	0.0721218	+	0.997396i	$0.477023\pi$
$770$	0	0
$771$	−81.0000	−2.91714
$772$	19.0000	0.683825
$773$	46.0000	1.65451	0.827253	−	0.561830i	$-0.189903\pi$
$773$	46.0000	1.65451	0.827253	+	0.561830i	$0.189903\pi$
$774$	−60.0000	−2.15666
$775$	−40.0000	−1.43684
$776$	6.00000	0.215387
$777$	−24.0000	−0.860995
$778$	−18.0000	−0.645331
$779$	−15.0000	−0.537431
$780$	6.00000	0.214834
$781$	0	0
$782$	−4.00000	−0.143040
$783$	−81.0000	−2.89470
$784$	6.00000	0.214286
$785$	−20.0000	−0.713831
$786$	−30.0000	−1.07006
$787$	52.0000	1.85360	0.926800	−	0.375555i	$-0.122548\pi$
$787$	52.0000	1.85360	0.926800	+	0.375555i	$0.122548\pi$
$788$	−18.0000	−0.641223
$789$	−3.00000	−0.106803
$790$	9.00000	0.320206
$791$	16.0000	0.568895
$792$	0	0
$793$	−16.0000	−0.568177
$794$	−12.0000	−0.425864
$795$	9.00000	0.319197
$796$	11.0000	0.389885
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	−15.0000	−0.530994
$799$	12.0000	0.424529
$800$	20.0000	0.707107
$801$	−84.0000	−2.96799
$802$	14.0000	0.494357
$803$	0	0
$804$	24.0000	0.846415
$805$	4.00000	0.140981
$806$	20.0000	0.704470
$807$	−42.0000	−1.47847
$808$	24.0000	0.844317
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	−9.00000	−0.316228
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	9.00000	0.315838
$813$	−21.0000	−0.736502
$814$	0	0
$815$	−12.0000	−0.420342
$816$	−3.00000	−0.105021
$817$	50.0000	1.74928
$818$	−14.0000	−0.489499
$819$	−12.0000	−0.419314
$820$	3.00000	0.104765
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	27.0000	0.941733
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	42.0000	1.46314
$825$	0	0
$826$	1.00000	0.0347945
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	−24.0000	−0.834058
$829$	−7.00000	−0.243120	−0.121560	−	0.992584i	$-0.538790\pi$
$829$	−7.00000	−0.243120	−0.121560	+	0.992584i	$0.538790\pi$
$830$	16.0000	0.555368
$831$	−21.0000	−0.728482
$832$	−14.0000	−0.485363
$833$	−6.00000	−0.207888
$834$	24.0000	0.831052
$835$	−7.00000	−0.242245
$836$	0	0
$837$	90.0000	3.11086
$838$	−14.0000	−0.483622
$839$	26.0000	0.897620	0.448810	−	0.893627i	$-0.351848\pi$
$839$	26.0000	0.897620	0.448810	+	0.893627i	$0.351848\pi$
$840$	9.00000	0.310530
$841$	52.0000	1.79310
$842$	26.0000	0.896019
$843$	−99.0000	−3.40974
$844$	22.0000	0.757271
$845$	−9.00000	−0.309609
$846$	−72.0000	−2.47541
$847$	−11.0000	−0.377964
$848$	−3.00000	−0.103020
$849$	−48.0000	−1.64736
$850$	4.00000	0.137199
$851$	−32.0000	−1.09695
$852$	−24.0000	−0.822226
$853$	−1.00000	−0.0342393	−0.0171197	−	0.999853i	$-0.505450\pi$
$853$	−1.00000	−0.0342393	−0.0171197	+	0.999853i	$0.505450\pi$
$854$	−8.00000	−0.273754
$855$	30.0000	1.02598
$856$	33.0000	1.12792
$857$	−20.0000	−0.683187	−0.341593	−	0.939848i	$-0.610967\pi$
$857$	−20.0000	−0.683187	−0.341593	+	0.939848i	$0.610967\pi$
$858$	0	0
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	−10.0000	−0.340997
$861$	−9.00000	−0.306719
$862$	−36.0000	−1.22616
$863$	46.0000	1.56586	0.782929	−	0.622111i	$-0.213725\pi$
$863$	46.0000	1.56586	0.782929	+	0.622111i	$0.213725\pi$
$864$	−45.0000	−1.53093
$865$	−14.0000	−0.476014
$866$	25.0000	0.849535
$867$	3.00000	0.101885
$868$	−10.0000	−0.339422
$869$	0	0
$870$	27.0000	0.915386
$871$	16.0000	0.542139
$872$	18.0000	0.609557
$873$	12.0000	0.406138
$874$	−20.0000	−0.676510
$875$	−9.00000	−0.304256
$876$	18.0000	0.608164
$877$	−33.0000	−1.11433	−0.557165	−	0.830402i	$-0.688111\pi$
$877$	−33.0000	−1.11433	−0.557165	+	0.830402i	$0.688111\pi$
$878$	8.00000	0.269987
$879$	63.0000	2.12494
$880$	0	0
$881$	−28.0000	−0.943344	−0.471672	−	0.881774i	$-0.656349\pi$
$881$	−28.0000	−0.943344	−0.471672	+	0.881774i	$0.656349\pi$
$882$	36.0000	1.21218
$883$	27.0000	0.908622	0.454311	−	0.890843i	$-0.349885\pi$
$883$	27.0000	0.908622	0.454311	+	0.890843i	$0.349885\pi$
$884$	2.00000	0.0672673
$885$	−3.00000	−0.100844
$886$	−22.0000	−0.739104
$887$	−42.0000	−1.41022	−0.705111	−	0.709097i	$-0.749103\pi$
$887$	−42.0000	−1.41022	−0.705111	+	0.709097i	$0.749103\pi$
$888$	−72.0000	−2.41616
$889$	−7.00000	−0.234772
$890$	14.0000	0.469281
$891$	0	0
$892$	12.0000	0.401790
$893$	60.0000	2.00782
$894$	12.0000	0.401340
$895$	24.0000	0.802232
$896$	3.00000	0.100223
$897$	−24.0000	−0.801337
$898$	5.00000	0.166852
$899$	−90.0000	−3.00167
$900$	24.0000	0.800000
$901$	3.00000	0.0999445
$902$	0	0
$903$	30.0000	0.998337
$904$	48.0000	1.59646
$905$	7.00000	0.232688
$906$	12.0000	0.398673
$907$	23.0000	0.763702	0.381851	−	0.924224i	$-0.375287\pi$
$907$	23.0000	0.763702	0.381851	+	0.924224i	$0.375287\pi$
$908$	18.0000	0.597351
$909$	48.0000	1.59206
$910$	2.00000	0.0662994
$911$	−7.00000	−0.231920	−0.115960	−	0.993254i	$-0.536994\pi$
$911$	−7.00000	−0.231920	−0.115960	+	0.993254i	$0.536994\pi$
$912$	−15.0000	−0.496700
$913$	0	0
$914$	−24.0000	−0.793849
$915$	24.0000	0.793416
$916$	−18.0000	−0.594737
$917$	10.0000	0.330229
$918$	−9.00000	−0.297044
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	12.0000	0.395628
$921$	9.00000	0.296560
$922$	14.0000	0.461065
$923$	−16.0000	−0.526646
$924$	0	0
$925$	32.0000	1.05215
$926$	4.00000	0.131448
$927$	84.0000	2.75892
$928$	45.0000	1.47720
$929$	−12.0000	−0.393707	−0.196854	−	0.980433i	$-0.563072\pi$
$929$	−12.0000	−0.393707	−0.196854	+	0.980433i	$0.563072\pi$
$930$	−30.0000	−0.983739
$931$	−30.0000	−0.983210
$932$	−12.0000	−0.393073
$933$	−33.0000	−1.08037
$934$	−18.0000	−0.588978
$935$	0	0
$936$	−36.0000	−1.17670
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	8.00000	0.261209
$939$	−84.0000	−2.74124
$940$	−12.0000	−0.391397
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	60.0000	1.95491
$943$	−12.0000	−0.390774
$944$	1.00000	0.0325472
$945$	9.00000	0.292770
$946$	0	0
$947$	−37.0000	−1.20234	−0.601169	−	0.799122i	$-0.705298\pi$
$947$	−37.0000	−1.20234	−0.601169	+	0.799122i	$0.705298\pi$
$948$	27.0000	0.876919
$949$	12.0000	0.389536
$950$	20.0000	0.648886
$951$	6.00000	0.194563
$952$	3.00000	0.0972306
$953$	50.0000	1.61966	0.809829	−	0.586665i	$-0.199560\pi$
$953$	50.0000	1.61966	0.809829	+	0.586665i	$0.199560\pi$
$954$	−18.0000	−0.582772
$955$	8.00000	0.258874
$956$	1.00000	0.0323423
$957$	0	0
$958$	−24.0000	−0.775405
$959$	−9.00000	−0.290625
$960$	21.0000	0.677772
$961$	69.0000	2.22581
$962$	−16.0000	−0.515861
$963$	66.0000	2.12682
$964$	−3.00000	−0.0966235
$965$	−19.0000	−0.611632
$966$	−12.0000	−0.386094
$967$	18.0000	0.578841	0.289420	−	0.957202i	$-0.406537\pi$
$967$	18.0000	0.578841	0.289420	+	0.957202i	$0.406537\pi$
$968$	−33.0000	−1.06066
$969$	15.0000	0.481869
$970$	−2.00000	−0.0642161
$971$	−33.0000	−1.05902	−0.529510	−	0.848304i	$-0.677624\pi$
$971$	−33.0000	−1.05902	−0.529510	+	0.848304i	$0.677624\pi$
$972$	0	0
$973$	−8.00000	−0.256468
$974$	−17.0000	−0.544715
$975$	24.0000	0.768615
$976$	−8.00000	−0.256074
$977$	−24.0000	−0.767828	−0.383914	−	0.923369i	$-0.625424\pi$
$977$	−24.0000	−0.767828	−0.383914	+	0.923369i	$0.625424\pi$
$978$	36.0000	1.15115
$979$	0	0
$980$	6.00000	0.191663
$981$	36.0000	1.14939
$982$	13.0000	0.414847
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	−27.0000	−0.860729
$985$	18.0000	0.573528
$986$	9.00000	0.286618
$987$	36.0000	1.14589
$988$	10.0000	0.318142
$989$	40.0000	1.27193
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	−50.0000	−1.58750
$993$	39.0000	1.23763
$994$	−8.00000	−0.253745
$995$	−11.0000	−0.348723
$996$	48.0000	1.52094
$997$	7.00000	0.221692	0.110846	−	0.993838i	$-0.464644\pi$
$997$	7.00000	0.221692	0.110846	+	0.993838i	$0.464644\pi$
$998$	−35.0000	−1.10791
$999$	−72.0000	−2.27798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1003.2.a.a.1.1	✓	1
3.2	odd	2		9027.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1003.2.a.a.1.1	✓	1	1.1	even	1	trivial
9027.2.a.e.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:	\( N \)	\(=\)	\( 1003 = 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1003.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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