LMFDB - Embedded newform 2093.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2093,2,Mod(1,2093)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2093, 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("2093.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2093 = 7 \cdot 13 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2093.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:	$16.7126891431$
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	$\chi$	$=$	2093.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +3.00000 q^{3} +2.00000 q^{4} +3.00000 q^{5} -6.00000 q^{6} -1.00000 q^{7} +6.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +3.00000 q^{3} +2.00000 q^{4} +3.00000 q^{5} -6.00000 q^{6} -1.00000 q^{7} +6.00000 q^{9} -6.00000 q^{10} +3.00000 q^{11} +6.00000 q^{12} -1.00000 q^{13} +2.00000 q^{14} +9.00000 q^{15} -4.00000 q^{16} +4.00000 q^{17} -12.0000 q^{18} +5.00000 q^{19} +6.00000 q^{20} -3.00000 q^{21} -6.00000 q^{22} +1.00000 q^{23} +4.00000 q^{25} +2.00000 q^{26} +9.00000 q^{27} -2.00000 q^{28} -2.00000 q^{29} -18.0000 q^{30} -6.00000 q^{31} +8.00000 q^{32} +9.00000 q^{33} -8.00000 q^{34} -3.00000 q^{35} +12.0000 q^{36} -10.0000 q^{37} -10.0000 q^{38} -3.00000 q^{39} +6.00000 q^{41} +6.00000 q^{42} -4.00000 q^{43} +6.00000 q^{44} +18.0000 q^{45} -2.00000 q^{46} +10.0000 q^{47} -12.0000 q^{48} +1.00000 q^{49} -8.00000 q^{50} +12.0000 q^{51} -2.00000 q^{52} -12.0000 q^{53} -18.0000 q^{54} +9.00000 q^{55} +15.0000 q^{57} +4.00000 q^{58} -10.0000 q^{59} +18.0000 q^{60} +2.00000 q^{61} +12.0000 q^{62} -6.00000 q^{63} -8.00000 q^{64} -3.00000 q^{65} -18.0000 q^{66} +3.00000 q^{67} +8.00000 q^{68} +3.00000 q^{69} +6.00000 q^{70} -2.00000 q^{71} +2.00000 q^{73} +20.0000 q^{74} +12.0000 q^{75} +10.0000 q^{76} -3.00000 q^{77} +6.00000 q^{78} -12.0000 q^{79} -12.0000 q^{80} +9.00000 q^{81} -12.0000 q^{82} +9.00000 q^{83} -6.00000 q^{84} +12.0000 q^{85} +8.00000 q^{86} -6.00000 q^{87} +6.00000 q^{89} -36.0000 q^{90} +1.00000 q^{91} +2.00000 q^{92} -18.0000 q^{93} -20.0000 q^{94} +15.0000 q^{95} +24.0000 q^{96} +13.0000 q^{97} -2.00000 q^{98} +18.0000 q^{99} +O(q^{100})\)

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	$0.750000\pi$
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	2.00000	1.00000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	−6.00000	−2.44949
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	6.00000	2.00000
$10$	−6.00000	−1.89737
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	6.00000	1.73205
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	2.00000	0.534522
$15$	9.00000	2.32379
$16$	−4.00000	−1.00000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−12.0000	−2.82843
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	6.00000	1.34164
$21$	−3.00000	−0.654654
$22$	−6.00000	−1.27920
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	4.00000	0.800000
$26$	2.00000	0.392232
$27$	9.00000	1.73205
$28$	−2.00000	−0.377964
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	−18.0000	−3.28634
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	8.00000	1.41421
$33$	9.00000	1.56670
$34$	−8.00000	−1.37199
$35$	−3.00000	−0.507093
$36$	12.0000	2.00000
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	−10.0000	−1.62221
$39$	−3.00000	−0.480384
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	6.00000	0.925820
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	6.00000	0.904534
$45$	18.0000	2.68328
$46$	−2.00000	−0.294884
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	−12.0000	−1.73205
$49$	1.00000	0.142857
$50$	−8.00000	−1.13137
$51$	12.0000	1.68034
$52$	−2.00000	−0.277350
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	−18.0000	−2.44949
$55$	9.00000	1.21356
$56$	0	0
$57$	15.0000	1.98680
$58$	4.00000	0.525226
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	18.0000	2.32379
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	12.0000	1.52400
$63$	−6.00000	−0.755929
$64$	−8.00000	−1.00000
$65$	−3.00000	−0.372104
$66$	−18.0000	−2.21565
$67$	3.00000	0.366508	0.183254	−	0.983066i	$-0.441337\pi$
$67$	3.00000	0.366508	0.183254	+	0.983066i	$0.441337\pi$
$68$	8.00000	0.970143
$69$	3.00000	0.361158
$70$	6.00000	0.717137
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	20.0000	2.32495
$75$	12.0000	1.38564
$76$	10.0000	1.14708
$77$	−3.00000	−0.341882
$78$	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	$0.735879\pi$
$80$	−12.0000	−1.34164
$81$	9.00000	1.00000
$82$	−12.0000	−1.32518
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	−6.00000	−0.654654
$85$	12.0000	1.30158
$86$	8.00000	0.862662
$87$	−6.00000	−0.643268
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−36.0000	−3.79473
$91$	1.00000	0.104828
$92$	2.00000	0.208514
$93$	−18.0000	−1.86651
$94$	−20.0000	−2.06284
$95$	15.0000	1.53897
$96$	24.0000	2.44949
$97$	13.0000	1.31995	0.659975	−	0.751288i	$-0.270567\pi$
$97$	13.0000	1.31995	0.659975	+	0.751288i	$0.270567\pi$
$98$	−2.00000	−0.202031
$99$	18.0000	1.80907
$100$	8.00000	0.800000
$101$	7.00000	0.696526	0.348263	−	0.937397i	$-0.386772\pi$
$101$	7.00000	0.696526	0.348263	+	0.937397i	$0.386772\pi$
$102$	−24.0000	−2.37635
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	0	0
$105$	−9.00000	−0.878310
$106$	24.0000	2.33109
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	18.0000	1.73205
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	−18.0000	−1.71623
$111$	−30.0000	−2.84747
$112$	4.00000	0.377964
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−30.0000	−2.80976
$115$	3.00000	0.279751
$116$	−4.00000	−0.371391
$117$	−6.00000	−0.554700
$118$	20.0000	1.84115
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−4.00000	−0.362143
$123$	18.0000	1.62301
$124$	−12.0000	−1.07763
$125$	−3.00000	−0.268328
$126$	12.0000	1.06904
$127$	11.0000	0.976092	0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000	0.976092	0.488046	+	0.872818i	$0.337710\pi$
$128$	0	0
$129$	−12.0000	−1.05654
$130$	6.00000	0.526235
$131$	−13.0000	−1.13582	−0.567908	−	0.823092i	$-0.692247\pi$
$131$	−13.0000	−1.13582	−0.567908	+	0.823092i	$0.692247\pi$
$132$	18.0000	1.56670
$133$	−5.00000	−0.433555
$134$	−6.00000	−0.518321
$135$	27.0000	2.32379
$136$	0	0
$137$	−17.0000	−1.45241	−0.726204	−	0.687479i	$-0.758717\pi$
$137$	−17.0000	−1.45241	−0.726204	+	0.687479i	$0.758717\pi$
$138$	−6.00000	−0.510754
$139$	−9.00000	−0.763370	−0.381685	−	0.924292i	$-0.624656\pi$
$139$	−9.00000	−0.763370	−0.381685	+	0.924292i	$0.624656\pi$
$140$	−6.00000	−0.507093
$141$	30.0000	2.52646
$142$	4.00000	0.335673
$143$	−3.00000	−0.250873
$144$	−24.0000	−2.00000
$145$	−6.00000	−0.498273
$146$	−4.00000	−0.331042
$147$	3.00000	0.247436
$148$	−20.0000	−1.64399
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	−24.0000	−1.95959
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	24.0000	1.94029
$154$	6.00000	0.483494
$155$	−18.0000	−1.44579
$156$	−6.00000	−0.480384
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	24.0000	1.90934
$159$	−36.0000	−2.85499
$160$	24.0000	1.89737
$161$	−1.00000	−0.0788110
$162$	−18.0000	−1.41421
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	12.0000	0.937043
$165$	27.0000	2.10195
$166$	−18.0000	−1.39707
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−24.0000	−1.84072
$171$	30.0000	2.29416
$172$	−8.00000	−0.609994
$173$	1.00000	0.0760286	0.0380143	−	0.999277i	$-0.487897\pi$
$173$	1.00000	0.0760286	0.0380143	+	0.999277i	$0.487897\pi$
$174$	12.0000	0.909718
$175$	−4.00000	−0.302372
$176$	−12.0000	−0.904534
$177$	−30.0000	−2.25494
$178$	−12.0000	−0.899438
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	36.0000	2.68328
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	−2.00000	−0.148250
$183$	6.00000	0.443533
$184$	0	0
$185$	−30.0000	−2.20564
$186$	36.0000	2.63965
$187$	12.0000	0.877527
$188$	20.0000	1.45865
$189$	−9.00000	−0.654654
$190$	−30.0000	−2.17643
$191$	−26.0000	−1.88129	−0.940647	−	0.339387i	$-0.889781\pi$
$191$	−26.0000	−1.88129	−0.940647	+	0.339387i	$0.889781\pi$
$192$	−24.0000	−1.73205
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	−26.0000	−1.86669
$195$	−9.00000	−0.644503
$196$	2.00000	0.142857
$197$	8.00000	0.569976	0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000	0.569976	0.284988	+	0.958531i	$0.408010\pi$
$198$	−36.0000	−2.55841
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	9.00000	0.634811
$202$	−14.0000	−0.985037
$203$	2.00000	0.140372
$204$	24.0000	1.68034
$205$	18.0000	1.25717
$206$	20.0000	1.39347
$207$	6.00000	0.417029
$208$	4.00000	0.277350
$209$	15.0000	1.03757
$210$	18.0000	1.24212
$211$	19.0000	1.30801	0.654007	−	0.756489i	$-0.273087\pi$
$211$	19.0000	1.30801	0.654007	+	0.756489i	$0.273087\pi$
$212$	−24.0000	−1.64833
$213$	−6.00000	−0.411113
$214$	−4.00000	−0.273434
$215$	−12.0000	−0.818393
$216$	0	0
$217$	6.00000	0.407307
$218$	10.0000	0.677285
$219$	6.00000	0.405442
$220$	18.0000	1.21356
$221$	−4.00000	−0.269069
$222$	60.0000	4.02694
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	−8.00000	−0.534522
$225$	24.0000	1.60000
$226$	12.0000	0.798228
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	30.0000	1.98680
$229$	11.0000	0.726900	0.363450	−	0.931614i	$-0.381599\pi$
$229$	11.0000	0.726900	0.363450	+	0.931614i	$0.381599\pi$
$230$	−6.00000	−0.395628
$231$	−9.00000	−0.592157
$232$	0	0
$233$	−15.0000	−0.982683	−0.491341	−	0.870967i	$-0.663493\pi$
$233$	−15.0000	−0.982683	−0.491341	+	0.870967i	$0.663493\pi$
$234$	12.0000	0.784465
$235$	30.0000	1.95698
$236$	−20.0000	−1.30189
$237$	−36.0000	−2.33845
$238$	8.00000	0.518563
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	−36.0000	−2.32379
$241$	19.0000	1.22390	0.611949	−	0.790897i	$-0.290386\pi$
$241$	19.0000	1.22390	0.611949	+	0.790897i	$0.290386\pi$
$242$	4.00000	0.257130
$243$	0	0
$244$	4.00000	0.256074
$245$	3.00000	0.191663
$246$	−36.0000	−2.29528
$247$	−5.00000	−0.318142
$248$	0	0
$249$	27.0000	1.71106
$250$	6.00000	0.379473
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	−12.0000	−0.755929
$253$	3.00000	0.188608
$254$	−22.0000	−1.38040
$255$	36.0000	2.25441
$256$	16.0000	1.00000
$257$	19.0000	1.18519	0.592594	−	0.805502i	$-0.298104\pi$
$257$	19.0000	1.18519	0.592594	+	0.805502i	$0.298104\pi$
$258$	24.0000	1.49417
$259$	10.0000	0.621370
$260$	−6.00000	−0.372104
$261$	−12.0000	−0.742781
$262$	26.0000	1.60629
$263$	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
$265$	−36.0000	−2.21146
$266$	10.0000	0.613139
$267$	18.0000	1.10158
$268$	6.00000	0.366508
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	−54.0000	−3.28634
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−16.0000	−0.970143
$273$	3.00000	0.181568
$274$	34.0000	2.05402
$275$	12.0000	0.723627
$276$	6.00000	0.361158
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	18.0000	1.07957
$279$	−36.0000	−2.15526
$280$	0	0
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	−60.0000	−3.57295
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	−4.00000	−0.237356
$285$	45.0000	2.66557
$286$	6.00000	0.354787
$287$	−6.00000	−0.354169
$288$	48.0000	2.82843
$289$	−1.00000	−0.0588235
$290$	12.0000	0.704664
$291$	39.0000	2.28622
$292$	4.00000	0.234082
$293$	−1.00000	−0.0584206	−0.0292103	−	0.999573i	$-0.509299\pi$
$293$	−1.00000	−0.0584206	−0.0292103	+	0.999573i	$0.509299\pi$
$294$	−6.00000	−0.349927
$295$	−30.0000	−1.74667
$296$	0	0
$297$	27.0000	1.56670
$298$	−36.0000	−2.08542
$299$	−1.00000	−0.0578315
$300$	24.0000	1.38564
$301$	4.00000	0.230556
$302$	0	0
$303$	21.0000	1.20642
$304$	−20.0000	−1.14708
$305$	6.00000	0.343559
$306$	−48.0000	−2.74398
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	−6.00000	−0.341882
$309$	−30.0000	−1.70664
$310$	36.0000	2.04466
$311$	9.00000	0.510343	0.255172	−	0.966896i	$-0.417868\pi$
$311$	9.00000	0.510343	0.255172	+	0.966896i	$0.417868\pi$
$312$	0	0
$313$	−20.0000	−1.13047	−0.565233	−	0.824931i	$-0.691214\pi$
$313$	−20.0000	−1.13047	−0.565233	+	0.824931i	$0.691214\pi$
$314$	4.00000	0.225733
$315$	−18.0000	−1.01419
$316$	−24.0000	−1.35011
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	72.0000	4.03756
$319$	−6.00000	−0.335936
$320$	−24.0000	−1.34164
$321$	6.00000	0.334887
$322$	2.00000	0.111456
$323$	20.0000	1.11283
$324$	18.0000	1.00000
$325$	−4.00000	−0.221880
$326$	8.00000	0.443079
$327$	−15.0000	−0.829502
$328$	0	0
$329$	−10.0000	−0.551318
$330$	−54.0000	−2.97260
$331$	16.0000	0.879440	0.439720	−	0.898135i	$-0.355078\pi$
$331$	16.0000	0.879440	0.439720	+	0.898135i	$0.355078\pi$
$332$	18.0000	0.987878
$333$	−60.0000	−3.28798
$334$	32.0000	1.75096
$335$	9.00000	0.491723
$336$	12.0000	0.654654
$337$	−34.0000	−1.85210	−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−34.0000	−1.85210	−0.926049	+	0.377403i	$0.876817\pi$
$338$	−2.00000	−0.108786
$339$	−18.0000	−0.977626
$340$	24.0000	1.30158
$341$	−18.0000	−0.974755
$342$	−60.0000	−3.24443
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	9.00000	0.484544
$346$	−2.00000	−0.107521
$347$	7.00000	0.375780	0.187890	−	0.982190i	$-0.439835\pi$
$347$	7.00000	0.375780	0.187890	+	0.982190i	$0.439835\pi$
$348$	−12.0000	−0.643268
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	8.00000	0.427618
$351$	−9.00000	−0.480384
$352$	24.0000	1.27920
$353$	20.0000	1.06449	0.532246	−	0.846590i	$-0.321348\pi$
$353$	20.0000	1.06449	0.532246	+	0.846590i	$0.321348\pi$
$354$	60.0000	3.18896
$355$	−6.00000	−0.318447
$356$	12.0000	0.635999
$357$	−12.0000	−0.635107
$358$	8.00000	0.422813
$359$	−19.0000	−1.00278	−0.501391	−	0.865221i	$-0.667178\pi$
$359$	−19.0000	−1.00278	−0.501391	+	0.865221i	$0.667178\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	44.0000	2.31259
$363$	−6.00000	−0.314918
$364$	2.00000	0.104828
$365$	6.00000	0.314054
$366$	−12.0000	−0.627250
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	−4.00000	−0.208514
$369$	36.0000	1.87409
$370$	60.0000	3.11925
$371$	12.0000	0.623009
$372$	−36.0000	−1.86651
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	−24.0000	−1.24101
$375$	−9.00000	−0.464758
$376$	0	0
$377$	2.00000	0.103005
$378$	18.0000	0.925820
$379$	−33.0000	−1.69510	−0.847548	−	0.530719i	$-0.821922\pi$
$379$	−33.0000	−1.69510	−0.847548	+	0.530719i	$0.821922\pi$
$380$	30.0000	1.53897
$381$	33.0000	1.69064
$382$	52.0000	2.66055
$383$	21.0000	1.07305	0.536525	−	0.843884i	$-0.319737\pi$
$383$	21.0000	1.07305	0.536525	+	0.843884i	$0.319737\pi$
$384$	0	0
$385$	−9.00000	−0.458682
$386$	−44.0000	−2.23954
$387$	−24.0000	−1.21999
$388$	26.0000	1.31995
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	18.0000	0.911465
$391$	4.00000	0.202289
$392$	0	0
$393$	−39.0000	−1.96729
$394$	−16.0000	−0.806068
$395$	−36.0000	−1.81136
$396$	36.0000	1.80907
$397$	20.0000	1.00377	0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000	1.00377	0.501886	+	0.864934i	$0.332640\pi$
$398$	−8.00000	−0.401004
$399$	−15.0000	−0.750939
$400$	−16.0000	−0.800000
$401$	−29.0000	−1.44819	−0.724095	−	0.689700i	$-0.757743\pi$
$401$	−29.0000	−1.44819	−0.724095	+	0.689700i	$0.757743\pi$
$402$	−18.0000	−0.897758
$403$	6.00000	0.298881
$404$	14.0000	0.696526
$405$	27.0000	1.34164
$406$	−4.00000	−0.198517
$407$	−30.0000	−1.48704
$408$	0	0
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	−36.0000	−1.77791
$411$	−51.0000	−2.51564
$412$	−20.0000	−0.985329
$413$	10.0000	0.492068
$414$	−12.0000	−0.589768
$415$	27.0000	1.32538
$416$	−8.00000	−0.392232
$417$	−27.0000	−1.32220
$418$	−30.0000	−1.46735
$419$	14.0000	0.683945	0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000	0.683945	0.341972	+	0.939710i	$0.388905\pi$
$420$	−18.0000	−0.878310
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	−38.0000	−1.84981
$423$	60.0000	2.91730
$424$	0	0
$425$	16.0000	0.776114
$426$	12.0000	0.581402
$427$	−2.00000	−0.0967868
$428$	4.00000	0.193347
$429$	−9.00000	−0.434524
$430$	24.0000	1.15738
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	−36.0000	−1.73205
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	−12.0000	−0.576018
$435$	−18.0000	−0.863034
$436$	−10.0000	−0.478913
$437$	5.00000	0.239182
$438$	−12.0000	−0.573382
$439$	5.00000	0.238637	0.119318	−	0.992856i	$-0.461929\pi$
$439$	5.00000	0.238637	0.119318	+	0.992856i	$0.461929\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	8.00000	0.380521
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	−60.0000	−2.84747
$445$	18.0000	0.853282
$446$	−48.0000	−2.27287
$447$	54.0000	2.55411
$448$	8.00000	0.377964
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	−48.0000	−2.26274
$451$	18.0000	0.847587
$452$	−12.0000	−0.564433
$453$	0	0
$454$	−16.0000	−0.750917
$455$	3.00000	0.140642
$456$	0	0
$457$	27.0000	1.26301	0.631503	−	0.775373i	$-0.282438\pi$
$457$	27.0000	1.26301	0.631503	+	0.775373i	$0.282438\pi$
$458$	−22.0000	−1.02799
$459$	36.0000	1.68034
$460$	6.00000	0.279751
$461$	8.00000	0.372597	0.186299	−	0.982493i	$-0.440351\pi$
$461$	8.00000	0.372597	0.186299	+	0.982493i	$0.440351\pi$
$462$	18.0000	0.837436
$463$	−14.0000	−0.650635	−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000	−0.650635	−0.325318	+	0.945605i	$0.605471\pi$
$464$	8.00000	0.371391
$465$	−54.0000	−2.50419
$466$	30.0000	1.38972
$467$	−10.0000	−0.462745	−0.231372	−	0.972865i	$-0.574322\pi$
$467$	−10.0000	−0.462745	−0.231372	+	0.972865i	$0.574322\pi$
$468$	−12.0000	−0.554700
$469$	−3.00000	−0.138527
$470$	−60.0000	−2.76759
$471$	−6.00000	−0.276465
$472$	0	0
$473$	−12.0000	−0.551761
$474$	72.0000	3.30707
$475$	20.0000	0.917663
$476$	−8.00000	−0.366679
$477$	−72.0000	−3.29665
$478$	32.0000	1.46365
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	72.0000	3.28634
$481$	10.0000	0.455961
$482$	−38.0000	−1.73085
$483$	−3.00000	−0.136505
$484$	−4.00000	−0.181818
$485$	39.0000	1.77090
$486$	0	0
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	0	0
$489$	−12.0000	−0.542659
$490$	−6.00000	−0.271052
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	36.0000	1.62301
$493$	−8.00000	−0.360302
$494$	10.0000	0.449921
$495$	54.0000	2.42712
$496$	24.0000	1.07763
$497$	2.00000	0.0897123
$498$	−54.0000	−2.41980
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−6.00000	−0.268328
$501$	−48.0000	−2.14448
$502$	−56.0000	−2.49940
$503$	38.0000	1.69434	0.847168	−	0.531325i	$-0.178306\pi$
$503$	38.0000	1.69434	0.847168	+	0.531325i	$0.178306\pi$
$504$	0	0
$505$	21.0000	0.934488
$506$	−6.00000	−0.266733
$507$	3.00000	0.133235
$508$	22.0000	0.976092
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	−72.0000	−3.18821
$511$	−2.00000	−0.0884748
$512$	−32.0000	−1.41421
$513$	45.0000	1.98680
$514$	−38.0000	−1.67611
$515$	−30.0000	−1.32196
$516$	−24.0000	−1.05654
$517$	30.0000	1.31940
$518$	−20.0000	−0.878750
$519$	3.00000	0.131685
$520$	0	0
$521$	−20.0000	−0.876216	−0.438108	−	0.898922i	$-0.644351\pi$
$521$	−20.0000	−0.876216	−0.438108	+	0.898922i	$0.644351\pi$
$522$	24.0000	1.05045
$523$	−44.0000	−1.92399	−0.961993	−	0.273075i	$-0.911959\pi$
$523$	−44.0000	−1.92399	−0.961993	+	0.273075i	$0.911959\pi$
$524$	−26.0000	−1.13582
$525$	−12.0000	−0.523723
$526$	−36.0000	−1.56967
$527$	−24.0000	−1.04546
$528$	−36.0000	−1.56670
$529$	1.00000	0.0434783
$530$	72.0000	3.12748
$531$	−60.0000	−2.60378
$532$	−10.0000	−0.433555
$533$	−6.00000	−0.259889
$534$	−36.0000	−1.55787
$535$	6.00000	0.259403
$536$	0	0
$537$	−12.0000	−0.517838
$538$	−18.0000	−0.776035
$539$	3.00000	0.129219
$540$	54.0000	2.32379
$541$	44.0000	1.89171	0.945854	−	0.324593i	$-0.105227\pi$
$541$	44.0000	1.89171	0.945854	+	0.324593i	$0.105227\pi$
$542$	−40.0000	−1.71815
$543$	−66.0000	−2.83233
$544$	32.0000	1.37199
$545$	−15.0000	−0.642529
$546$	−6.00000	−0.256776
$547$	5.00000	0.213785	0.106892	−	0.994271i	$-0.465910\pi$
$547$	5.00000	0.213785	0.106892	+	0.994271i	$0.465910\pi$
$548$	−34.0000	−1.45241
$549$	12.0000	0.512148
$550$	−24.0000	−1.02336
$551$	−10.0000	−0.426014
$552$	0	0
$553$	12.0000	0.510292
$554$	28.0000	1.18961
$555$	−90.0000	−3.82029
$556$	−18.0000	−0.763370
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	72.0000	3.04800
$559$	4.00000	0.169182
$560$	12.0000	0.507093
$561$	36.0000	1.51992
$562$	−6.00000	−0.253095
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	60.0000	2.52646
$565$	−18.0000	−0.757266
$566$	−56.0000	−2.35386
$567$	−9.00000	−0.377964
$568$	0	0
$569$	−32.0000	−1.34151	−0.670755	−	0.741679i	$-0.734030\pi$
$569$	−32.0000	−1.34151	−0.670755	+	0.741679i	$0.734030\pi$
$570$	−90.0000	−3.76969
$571$	22.0000	0.920671	0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000	0.920671	0.460336	+	0.887745i	$0.347729\pi$
$572$	−6.00000	−0.250873
$573$	−78.0000	−3.25850
$574$	12.0000	0.500870
$575$	4.00000	0.166812
$576$	−48.0000	−2.00000
$577$	32.0000	1.33218	0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000	1.33218	0.666089	+	0.745873i	$0.267967\pi$
$578$	2.00000	0.0831890
$579$	66.0000	2.74287
$580$	−12.0000	−0.498273
$581$	−9.00000	−0.373383
$582$	−78.0000	−3.23320
$583$	−36.0000	−1.49097
$584$	0	0
$585$	−18.0000	−0.744208
$586$	2.00000	0.0826192
$587$	−42.0000	−1.73353	−0.866763	−	0.498721i	$-0.833803\pi$
$587$	−42.0000	−1.73353	−0.866763	+	0.498721i	$0.833803\pi$
$588$	6.00000	0.247436
$589$	−30.0000	−1.23613
$590$	60.0000	2.47016
$591$	24.0000	0.987228
$592$	40.0000	1.64399
$593$	−36.0000	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	−54.0000	−2.21565
$595$	−12.0000	−0.491952
$596$	36.0000	1.47462
$597$	12.0000	0.491127
$598$	2.00000	0.0817861
$599$	35.0000	1.43006	0.715031	−	0.699093i	$-0.246413\pi$
$599$	35.0000	1.43006	0.715031	+	0.699093i	$0.246413\pi$
$600$	0	0
$601$	−31.0000	−1.26452	−0.632258	−	0.774758i	$-0.717872\pi$
$601$	−31.0000	−1.26452	−0.632258	+	0.774758i	$0.717872\pi$
$602$	−8.00000	−0.326056
$603$	18.0000	0.733017
$604$	0	0
$605$	−6.00000	−0.243935
$606$	−42.0000	−1.70613
$607$	−25.0000	−1.01472	−0.507359	−	0.861735i	$-0.669378\pi$
$607$	−25.0000	−1.01472	−0.507359	+	0.861735i	$0.669378\pi$
$608$	40.0000	1.62221
$609$	6.00000	0.243132
$610$	−12.0000	−0.485866
$611$	−10.0000	−0.404557
$612$	48.0000	1.94029
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	0	0
$615$	54.0000	2.17749
$616$	0	0
$617$	3.00000	0.120775	0.0603877	−	0.998175i	$-0.480766\pi$
$617$	3.00000	0.120775	0.0603877	+	0.998175i	$0.480766\pi$
$618$	60.0000	2.41355
$619$	47.0000	1.88909	0.944545	−	0.328383i	$-0.106504\pi$
$619$	47.0000	1.88909	0.944545	+	0.328383i	$0.106504\pi$
$620$	−36.0000	−1.44579
$621$	9.00000	0.361158
$622$	−18.0000	−0.721734
$623$	−6.00000	−0.240385
$624$	12.0000	0.480384
$625$	−29.0000	−1.16000
$626$	40.0000	1.59872
$627$	45.0000	1.79713
$628$	−4.00000	−0.159617
$629$	−40.0000	−1.59490
$630$	36.0000	1.43427
$631$	25.0000	0.995234	0.497617	−	0.867397i	$-0.334208\pi$
$631$	25.0000	0.995234	0.497617	+	0.867397i	$0.334208\pi$
$632$	0	0
$633$	57.0000	2.26555
$634$	36.0000	1.42974
$635$	33.0000	1.30957
$636$	−72.0000	−2.85499
$637$	−1.00000	−0.0396214
$638$	12.0000	0.475085
$639$	−12.0000	−0.474713
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	−12.0000	−0.473602
$643$	−1.00000	−0.0394362	−0.0197181	−	0.999806i	$-0.506277\pi$
$643$	−1.00000	−0.0394362	−0.0197181	+	0.999806i	$0.506277\pi$
$644$	−2.00000	−0.0788110
$645$	−36.0000	−1.41750
$646$	−40.0000	−1.57378
$647$	−3.00000	−0.117942	−0.0589711	−	0.998260i	$-0.518782\pi$
$647$	−3.00000	−0.117942	−0.0589711	+	0.998260i	$0.518782\pi$
$648$	0	0
$649$	−30.0000	−1.17760
$650$	8.00000	0.313786
$651$	18.0000	0.705476
$652$	−8.00000	−0.313304
$653$	−39.0000	−1.52619	−0.763094	−	0.646288i	$-0.776321\pi$
$653$	−39.0000	−1.52619	−0.763094	+	0.646288i	$0.776321\pi$
$654$	30.0000	1.17309
$655$	−39.0000	−1.52386
$656$	−24.0000	−0.937043
$657$	12.0000	0.468165
$658$	20.0000	0.779681
$659$	14.0000	0.545363	0.272681	−	0.962104i	$-0.412090\pi$
$659$	14.0000	0.545363	0.272681	+	0.962104i	$0.412090\pi$
$660$	54.0000	2.10195
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	−32.0000	−1.24372
$663$	−12.0000	−0.466041
$664$	0	0
$665$	−15.0000	−0.581675
$666$	120.000	4.64991
$667$	−2.00000	−0.0774403
$668$	−32.0000	−1.23812
$669$	72.0000	2.78368
$670$	−18.0000	−0.695401
$671$	6.00000	0.231627
$672$	−24.0000	−0.925820
$673$	−5.00000	−0.192736	−0.0963679	−	0.995346i	$-0.530723\pi$
$673$	−5.00000	−0.192736	−0.0963679	+	0.995346i	$0.530723\pi$
$674$	68.0000	2.61926
$675$	36.0000	1.38564
$676$	2.00000	0.0769231
$677$	−2.00000	−0.0768662	−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	−2.00000	−0.0768662	−0.0384331	+	0.999261i	$0.512237\pi$
$678$	36.0000	1.38257
$679$	−13.0000	−0.498894
$680$	0	0
$681$	24.0000	0.919682
$682$	36.0000	1.37851
$683$	−42.0000	−1.60709	−0.803543	−	0.595247i	$-0.797054\pi$
$683$	−42.0000	−1.60709	−0.803543	+	0.595247i	$0.797054\pi$
$684$	60.0000	2.29416
$685$	−51.0000	−1.94861
$686$	2.00000	0.0763604
$687$	33.0000	1.25903
$688$	16.0000	0.609994
$689$	12.0000	0.457164
$690$	−18.0000	−0.685248
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	2.00000	0.0760286
$693$	−18.0000	−0.683763
$694$	−14.0000	−0.531433
$695$	−27.0000	−1.02417
$696$	0	0
$697$	24.0000	0.909065
$698$	32.0000	1.21122
$699$	−45.0000	−1.70206
$700$	−8.00000	−0.302372
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	18.0000	0.679366
$703$	−50.0000	−1.88579
$704$	−24.0000	−0.904534
$705$	90.0000	3.38960
$706$	−40.0000	−1.50542
$707$	−7.00000	−0.263262
$708$	−60.0000	−2.25494
$709$	11.0000	0.413114	0.206557	−	0.978435i	$-0.433774\pi$
$709$	11.0000	0.413114	0.206557	+	0.978435i	$0.433774\pi$
$710$	12.0000	0.450352
$711$	−72.0000	−2.70021
$712$	0	0
$713$	−6.00000	−0.224702
$714$	24.0000	0.898177
$715$	−9.00000	−0.336581
$716$	−8.00000	−0.298974
$717$	−48.0000	−1.79259
$718$	38.0000	1.41815
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	−72.0000	−2.68328
$721$	10.0000	0.372419
$722$	−12.0000	−0.446594
$723$	57.0000	2.11985
$724$	−44.0000	−1.63525
$725$	−8.00000	−0.297113
$726$	12.0000	0.445362
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	−12.0000	−0.444140
$731$	−16.0000	−0.591781
$732$	12.0000	0.443533
$733$	−27.0000	−0.997268	−0.498634	−	0.866813i	$-0.666165\pi$
$733$	−27.0000	−0.997268	−0.498634	+	0.866813i	$0.666165\pi$
$734$	−40.0000	−1.47643
$735$	9.00000	0.331970
$736$	8.00000	0.294884
$737$	9.00000	0.331519
$738$	−72.0000	−2.65036
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	−60.0000	−2.20564
$741$	−15.0000	−0.551039
$742$	−24.0000	−0.881068
$743$	33.0000	1.21065	0.605326	−	0.795977i	$-0.293043\pi$
$743$	33.0000	1.21065	0.605326	+	0.795977i	$0.293043\pi$
$744$	0	0
$745$	54.0000	1.97841
$746$	12.0000	0.439351
$747$	54.0000	1.97576
$748$	24.0000	0.877527
$749$	−2.00000	−0.0730784
$750$	18.0000	0.657267
$751$	−50.0000	−1.82453	−0.912263	−	0.409605i	$-0.865667\pi$
$751$	−50.0000	−1.82453	−0.912263	+	0.409605i	$0.865667\pi$
$752$	−40.0000	−1.45865
$753$	84.0000	3.06113
$754$	−4.00000	−0.145671
$755$	0	0
$756$	−18.0000	−0.654654
$757$	24.0000	0.872295	0.436147	−	0.899875i	$-0.356343\pi$
$757$	24.0000	0.872295	0.436147	+	0.899875i	$0.356343\pi$
$758$	66.0000	2.39723
$759$	9.00000	0.326679
$760$	0	0
$761$	−48.0000	−1.74000	−0.869999	−	0.493053i	$-0.835881\pi$
$761$	−48.0000	−1.74000	−0.869999	+	0.493053i	$0.835881\pi$
$762$	−66.0000	−2.39093
$763$	5.00000	0.181012
$764$	−52.0000	−1.88129
$765$	72.0000	2.60317
$766$	−42.0000	−1.51752
$767$	10.0000	0.361079
$768$	48.0000	1.73205
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	18.0000	0.648675
$771$	57.0000	2.05280
$772$	44.0000	1.58359
$773$	45.0000	1.61854	0.809269	−	0.587439i	$-0.199864\pi$
$773$	45.0000	1.61854	0.809269	+	0.587439i	$0.199864\pi$
$774$	48.0000	1.72532
$775$	−24.0000	−0.862105
$776$	0	0
$777$	30.0000	1.07624
$778$	36.0000	1.29066
$779$	30.0000	1.07486
$780$	−18.0000	−0.644503
$781$	−6.00000	−0.214697
$782$	−8.00000	−0.286079
$783$	−18.0000	−0.643268
$784$	−4.00000	−0.142857
$785$	−6.00000	−0.214149
$786$	78.0000	2.78217
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	16.0000	0.569976
$789$	54.0000	1.92245
$790$	72.0000	2.56165
$791$	6.00000	0.213335
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	−40.0000	−1.41955
$795$	−108.000	−3.83037
$796$	8.00000	0.283552
$797$	54.0000	1.91278	0.956389	−	0.292096i	$-0.0943526\pi$
$797$	54.0000	1.91278	0.956389	+	0.292096i	$0.0943526\pi$
$798$	30.0000	1.06199
$799$	40.0000	1.41510
$800$	32.0000	1.13137
$801$	36.0000	1.27200
$802$	58.0000	2.04805
$803$	6.00000	0.211735
$804$	18.0000	0.634811
$805$	−3.00000	−0.105736
$806$	−12.0000	−0.422682
$807$	27.0000	0.950445
$808$	0	0
$809$	−37.0000	−1.30085	−0.650425	−	0.759570i	$-0.725409\pi$
$809$	−37.0000	−1.30085	−0.650425	+	0.759570i	$0.725409\pi$
$810$	−54.0000	−1.89737
$811$	−22.0000	−0.772524	−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000	−0.772524	−0.386262	+	0.922389i	$0.626234\pi$
$812$	4.00000	0.140372
$813$	60.0000	2.10429
$814$	60.0000	2.10300
$815$	−12.0000	−0.420342
$816$	−48.0000	−1.68034
$817$	−20.0000	−0.699711
$818$	68.0000	2.37756
$819$	6.00000	0.209657
$820$	36.0000	1.25717
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	102.000	3.55766
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	0	0
$825$	36.0000	1.25336
$826$	−20.0000	−0.695889
$827$	−11.0000	−0.382507	−0.191254	−	0.981541i	$-0.561255\pi$
$827$	−11.0000	−0.382507	−0.191254	+	0.981541i	$0.561255\pi$
$828$	12.0000	0.417029
$829$	17.0000	0.590434	0.295217	−	0.955430i	$-0.404608\pi$
$829$	17.0000	0.590434	0.295217	+	0.955430i	$0.404608\pi$
$830$	−54.0000	−1.87437
$831$	−42.0000	−1.45696
$832$	8.00000	0.277350
$833$	4.00000	0.138592
$834$	54.0000	1.86987
$835$	−48.0000	−1.66111
$836$	30.0000	1.03757
$837$	−54.0000	−1.86651
$838$	−28.0000	−0.967244
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−60.0000	−2.06774
$843$	9.00000	0.309976
$844$	38.0000	1.30801
$845$	3.00000	0.103203
$846$	−120.000	−4.12568
$847$	2.00000	0.0687208
$848$	48.0000	1.64833
$849$	84.0000	2.88287
$850$	−32.0000	−1.09759
$851$	−10.0000	−0.342796
$852$	−12.0000	−0.411113
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	4.00000	0.136877
$855$	90.0000	3.07794
$856$	0	0
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	18.0000	0.614510
$859$	13.0000	0.443554	0.221777	−	0.975097i	$-0.428814\pi$
$859$	13.0000	0.443554	0.221777	+	0.975097i	$0.428814\pi$
$860$	−24.0000	−0.818393
$861$	−18.0000	−0.613438
$862$	24.0000	0.817443
$863$	−8.00000	−0.272323	−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000	−0.272323	−0.136162	+	0.990687i	$0.543477\pi$
$864$	72.0000	2.44949
$865$	3.00000	0.102003
$866$	−12.0000	−0.407777
$867$	−3.00000	−0.101885
$868$	12.0000	0.407307
$869$	−36.0000	−1.22122
$870$	36.0000	1.22051
$871$	−3.00000	−0.101651
$872$	0	0
$873$	78.0000	2.63990
$874$	−10.0000	−0.338255
$875$	3.00000	0.101419
$876$	12.0000	0.405442
$877$	−42.0000	−1.41824	−0.709120	−	0.705088i	$-0.750907\pi$
$877$	−42.0000	−1.41824	−0.709120	+	0.705088i	$0.750907\pi$
$878$	−10.0000	−0.337484
$879$	−3.00000	−0.101187
$880$	−36.0000	−1.21356
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	−12.0000	−0.404061
$883$	47.0000	1.58168	0.790838	−	0.612026i	$-0.209645\pi$
$883$	47.0000	1.58168	0.790838	+	0.612026i	$0.209645\pi$
$884$	−8.00000	−0.269069
$885$	−90.0000	−3.02532
$886$	−8.00000	−0.268765
$887$	−3.00000	−0.100730	−0.0503651	−	0.998731i	$-0.516038\pi$
$887$	−3.00000	−0.100730	−0.0503651	+	0.998731i	$0.516038\pi$
$888$	0	0
$889$	−11.0000	−0.368928
$890$	−36.0000	−1.20672
$891$	27.0000	0.904534
$892$	48.0000	1.60716
$893$	50.0000	1.67319
$894$	−108.000	−3.61206
$895$	−12.0000	−0.401116
$896$	0	0
$897$	−3.00000	−0.100167
$898$	60.0000	2.00223
$899$	12.0000	0.400222
$900$	48.0000	1.60000
$901$	−48.0000	−1.59911
$902$	−36.0000	−1.19867
$903$	12.0000	0.399335
$904$	0	0
$905$	−66.0000	−2.19391
$906$	0	0
$907$	−26.0000	−0.863316	−0.431658	−	0.902037i	$-0.642071\pi$
$907$	−26.0000	−0.863316	−0.431658	+	0.902037i	$0.642071\pi$
$908$	16.0000	0.530979
$909$	42.0000	1.39305
$910$	−6.00000	−0.198898
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	−60.0000	−1.98680
$913$	27.0000	0.893570
$914$	−54.0000	−1.78616
$915$	18.0000	0.595062
$916$	22.0000	0.726900
$917$	13.0000	0.429298
$918$	−72.0000	−2.37635
$919$	−46.0000	−1.51740	−0.758700	−	0.651440i	$-0.774165\pi$
$919$	−46.0000	−1.51740	−0.758700	+	0.651440i	$0.774165\pi$
$920$	0	0
$921$	0	0
$922$	−16.0000	−0.526932
$923$	2.00000	0.0658308
$924$	−18.0000	−0.592157
$925$	−40.0000	−1.31519
$926$	28.0000	0.920137
$927$	−60.0000	−1.97066
$928$	−16.0000	−0.525226
$929$	−4.00000	−0.131236	−0.0656179	−	0.997845i	$-0.520902\pi$
$929$	−4.00000	−0.131236	−0.0656179	+	0.997845i	$0.520902\pi$
$930$	108.000	3.54146
$931$	5.00000	0.163868
$932$	−30.0000	−0.982683
$933$	27.0000	0.883940
$934$	20.0000	0.654420
$935$	36.0000	1.17733
$936$	0	0
$937$	40.0000	1.30674	0.653372	−	0.757037i	$-0.273354\pi$
$937$	40.0000	1.30674	0.653372	+	0.757037i	$0.273354\pi$
$938$	6.00000	0.195907
$939$	−60.0000	−1.95803
$940$	60.0000	1.95698
$941$	25.0000	0.814977	0.407488	−	0.913210i	$-0.366405\pi$
$941$	25.0000	0.814977	0.407488	+	0.913210i	$0.366405\pi$
$942$	12.0000	0.390981
$943$	6.00000	0.195387
$944$	40.0000	1.30189
$945$	−27.0000	−0.878310
$946$	24.0000	0.780307
$947$	−30.0000	−0.974869	−0.487435	−	0.873160i	$-0.662067\pi$
$947$	−30.0000	−0.974869	−0.487435	+	0.873160i	$0.662067\pi$
$948$	−72.0000	−2.33845
$949$	−2.00000	−0.0649227
$950$	−40.0000	−1.29777
$951$	−54.0000	−1.75107
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	144.000	4.66217
$955$	−78.0000	−2.52402
$956$	−32.0000	−1.03495
$957$	−18.0000	−0.581857
$958$	16.0000	0.516937
$959$	17.0000	0.548959
$960$	−72.0000	−2.32379
$961$	5.00000	0.161290
$962$	−20.0000	−0.644826
$963$	12.0000	0.386695
$964$	38.0000	1.22390
$965$	66.0000	2.12462
$966$	6.00000	0.193047
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	0	0
$969$	60.0000	1.92748
$970$	−78.0000	−2.50443
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	0	0
$973$	9.00000	0.288527
$974$	64.0000	2.05069
$975$	−12.0000	−0.384308
$976$	−8.00000	−0.256074
$977$	−35.0000	−1.11975	−0.559875	−	0.828577i	$-0.689151\pi$
$977$	−35.0000	−1.11975	−0.559875	+	0.828577i	$0.689151\pi$
$978$	24.0000	0.767435
$979$	18.0000	0.575282
$980$	6.00000	0.191663
$981$	−30.0000	−0.957826
$982$	0	0
$983$	59.0000	1.88181	0.940904	−	0.338674i	$-0.109978\pi$
$983$	59.0000	1.88181	0.940904	+	0.338674i	$0.109978\pi$
$984$	0	0
$985$	24.0000	0.764704
$986$	16.0000	0.509544
$987$	−30.0000	−0.954911
$988$	−10.0000	−0.318142
$989$	−4.00000	−0.127193
$990$	−108.000	−3.43247
$991$	25.0000	0.794151	0.397076	−	0.917786i	$-0.370025\pi$
$991$	25.0000	0.794151	0.397076	+	0.917786i	$0.370025\pi$
$992$	−48.0000	−1.52400
$993$	48.0000	1.52323
$994$	−4.00000	−0.126872
$995$	12.0000	0.380426
$996$	54.0000	1.71106
$997$	−1.00000	−0.0316703	−0.0158352	−	0.999875i	$-0.505041\pi$
$997$	−1.00000	−0.0316703	−0.0158352	+	0.999875i	$0.505041\pi$
$998$	8.00000	0.253236
$999$	−90.0000	−2.84747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2093.2.a.c.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2093.2.a.c.1.1	✓	1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2093 = 7 \cdot 13 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2093.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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