LMFDB - Embedded newform 786.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [786,2,Mod(1,786)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(786, 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("786.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 786 = 2 \cdot 3 \cdot 131 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	786.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:	$6.27624159887$
Analytic rank:	$1$
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	$\chi$	$=$	786.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -3.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -5.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +3.00000 q^{21} +1.00000 q^{22} +3.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -3.00000 q^{28} -10.0000 q^{29} +1.00000 q^{30} -2.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -5.00000 q^{34} +3.00000 q^{35} +1.00000 q^{36} +11.0000 q^{37} -4.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} -8.00000 q^{41} +3.00000 q^{42} -4.00000 q^{43} +1.00000 q^{44} -1.00000 q^{45} +3.00000 q^{46} -1.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} -4.00000 q^{50} +5.00000 q^{51} -2.00000 q^{52} -5.00000 q^{53} -1.00000 q^{54} -1.00000 q^{55} -3.00000 q^{56} +4.00000 q^{57} -10.0000 q^{58} +9.00000 q^{59} +1.00000 q^{60} -6.00000 q^{61} -2.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -1.00000 q^{66} +15.0000 q^{67} -5.00000 q^{68} -3.00000 q^{69} +3.00000 q^{70} -1.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} +11.0000 q^{74} +4.00000 q^{75} -4.00000 q^{76} -3.00000 q^{77} +2.00000 q^{78} +16.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -8.00000 q^{82} -6.00000 q^{83} +3.00000 q^{84} +5.00000 q^{85} -4.00000 q^{86} +10.0000 q^{87} +1.00000 q^{88} -14.0000 q^{89} -1.00000 q^{90} +6.00000 q^{91} +3.00000 q^{92} +2.00000 q^{93} -1.00000 q^{94} +4.00000 q^{95} -1.00000 q^{96} +8.00000 q^{97} +2.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	−1.00000	−0.408248
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	−1.00000	−0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−3.00000	−0.801784
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	1.00000	0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−1.00000	−0.223607
$21$	3.00000	0.654654
$22$	1.00000	0.213201
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	−1.00000	−0.204124
$25$	−4.00000	−0.800000
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	−3.00000	−0.566947
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	1.00000	0.182574
$31$	−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
$33$	−1.00000	−0.174078
$34$	−5.00000	−0.857493
$35$	3.00000	0.507093
$36$	1.00000	0.166667
$37$	11.0000	1.80839	0.904194	−	0.427121i	$-0.140472\pi$
$37$	11.0000	1.80839	0.904194	+	0.427121i	$0.140472\pi$
$38$	−4.00000	−0.648886
$39$	2.00000	0.320256
$40$	−1.00000	−0.158114
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	3.00000	0.462910
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	1.00000	0.150756
$45$	−1.00000	−0.149071
$46$	3.00000	0.442326
$47$	−1.00000	−0.145865	−0.0729325	−	0.997337i	$-0.523236\pi$
$47$	−1.00000	−0.145865	−0.0729325	+	0.997337i	$0.523236\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	−4.00000	−0.565685
$51$	5.00000	0.700140
$52$	−2.00000	−0.277350
$53$	−5.00000	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$54$	−1.00000	−0.136083
$55$	−1.00000	−0.134840
$56$	−3.00000	−0.400892
$57$	4.00000	0.529813
$58$	−10.0000	−1.31306
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	1.00000	0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−2.00000	−0.254000
$63$	−3.00000	−0.377964
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	−1.00000	−0.123091
$67$	15.0000	1.83254	0.916271	−	0.400559i	$-0.131184\pi$
$67$	15.0000	1.83254	0.916271	+	0.400559i	$0.131184\pi$
$68$	−5.00000	−0.606339
$69$	−3.00000	−0.361158
$70$	3.00000	0.358569
$71$	−1.00000	−0.118678	−0.0593391	−	0.998238i	$-0.518899\pi$
$71$	−1.00000	−0.118678	−0.0593391	+	0.998238i	$0.518899\pi$
$72$	1.00000	0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	11.0000	1.27872
$75$	4.00000	0.461880
$76$	−4.00000	−0.458831
$77$	−3.00000	−0.341882
$78$	2.00000	0.226455
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−8.00000	−0.883452
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	3.00000	0.327327
$85$	5.00000	0.542326
$86$	−4.00000	−0.431331
$87$	10.0000	1.07211
$88$	1.00000	0.106600
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	−1.00000	−0.105409
$91$	6.00000	0.628971
$92$	3.00000	0.312772
$93$	2.00000	0.207390
$94$	−1.00000	−0.103142
$95$	4.00000	0.410391
$96$	−1.00000	−0.102062
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	2.00000	0.202031
$99$	1.00000	0.100504
$100$	−4.00000	−0.400000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	5.00000	0.495074
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	−2.00000	−0.196116
$105$	−3.00000	−0.292770
$106$	−5.00000	−0.485643
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	−20.0000	−1.91565	−0.957826	−	0.287348i	$-0.907226\pi$
$109$	−20.0000	−1.91565	−0.957826	+	0.287348i	$0.907226\pi$
$110$	−1.00000	−0.0953463
$111$	−11.0000	−1.04407
$112$	−3.00000	−0.283473
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	4.00000	0.374634
$115$	−3.00000	−0.279751
$116$	−10.0000	−0.928477
$117$	−2.00000	−0.184900
$118$	9.00000	0.828517
$119$	15.0000	1.37505
$120$	1.00000	0.0912871
$121$	−10.0000	−0.909091
$122$	−6.00000	−0.543214
$123$	8.00000	0.721336
$124$	−2.00000	−0.179605
$125$	9.00000	0.804984
$126$	−3.00000	−0.267261
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	1.00000	0.0883883
$129$	4.00000	0.352180
$130$	2.00000	0.175412
$131$	1.00000	0.0873704
$131$	1.00000	0.0873704
$132$	−1.00000	−0.0870388
$133$	12.0000	1.04053
$134$	15.0000	1.29580
$135$	1.00000	0.0860663
$136$	−5.00000	−0.428746
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	−3.00000	−0.255377
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	3.00000	0.253546
$141$	1.00000	0.0842152
$142$	−1.00000	−0.0839181
$143$	−2.00000	−0.167248
$144$	1.00000	0.0833333
$145$	10.0000	0.830455
$146$	10.0000	0.827606
$147$	−2.00000	−0.164957
$148$	11.0000	0.904194
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	4.00000	0.326599
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−4.00000	−0.324443
$153$	−5.00000	−0.404226
$154$	−3.00000	−0.241747
$155$	2.00000	0.160644
$156$	2.00000	0.160128
$157$	3.00000	0.239426	0.119713	−	0.992809i	$-0.461803\pi$
$157$	3.00000	0.239426	0.119713	+	0.992809i	$0.461803\pi$
$158$	16.0000	1.27289
$159$	5.00000	0.396526
$160$	−1.00000	−0.0790569
$161$	−9.00000	−0.709299
$162$	1.00000	0.0785674
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	−8.00000	−0.624695
$165$	1.00000	0.0778499
$166$	−6.00000	−0.465690
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	3.00000	0.231455
$169$	−9.00000	−0.692308
$170$	5.00000	0.383482
$171$	−4.00000	−0.305888
$172$	−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	$0.401639\pi$
$174$	10.0000	0.758098
$175$	12.0000	0.907115
$176$	1.00000	0.0753778
$177$	−9.00000	−0.676481
$178$	−14.0000	−1.04934
$179$	−21.0000	−1.56961	−0.784807	−	0.619740i	$-0.787238\pi$
$179$	−21.0000	−1.56961	−0.784807	+	0.619740i	$0.787238\pi$
$180$	−1.00000	−0.0745356
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	6.00000	0.444750
$183$	6.00000	0.443533
$184$	3.00000	0.221163
$185$	−11.0000	−0.808736
$186$	2.00000	0.146647
$187$	−5.00000	−0.365636
$188$	−1.00000	−0.0729325
$189$	3.00000	0.218218
$190$	4.00000	0.290191
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−1.00000	−0.0721688
$193$	26.0000	1.87152	0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000	1.87152	0.935760	+	0.352636i	$0.114715\pi$
$194$	8.00000	0.574367
$195$	−2.00000	−0.143223
$196$	2.00000	0.142857
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	1.00000	0.0710669
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−4.00000	−0.282843
$201$	−15.0000	−1.05802
$202$	6.00000	0.422159
$203$	30.0000	2.10559
$204$	5.00000	0.350070
$205$	8.00000	0.558744
$206$	−2.00000	−0.139347
$207$	3.00000	0.208514
$208$	−2.00000	−0.138675
$209$	−4.00000	−0.276686
$210$	−3.00000	−0.207020
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	−5.00000	−0.343401
$213$	1.00000	0.0685189
$214$	−12.0000	−0.820303
$215$	4.00000	0.272798
$216$	−1.00000	−0.0680414
$217$	6.00000	0.407307
$218$	−20.0000	−1.35457
$219$	−10.0000	−0.675737
$220$	−1.00000	−0.0674200
$221$	10.0000	0.672673
$222$	−11.0000	−0.738272
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	−3.00000	−0.200446
$225$	−4.00000	−0.266667
$226$	10.0000	0.665190
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	4.00000	0.264906
$229$	5.00000	0.330409	0.165205	−	0.986259i	$-0.447172\pi$
$229$	5.00000	0.330409	0.165205	+	0.986259i	$0.447172\pi$
$230$	−3.00000	−0.197814
$231$	3.00000	0.197386
$232$	−10.0000	−0.656532
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	−2.00000	−0.130744
$235$	1.00000	0.0652328
$236$	9.00000	0.585850
$237$	−16.0000	−1.03931
$238$	15.0000	0.972306
$239$	−4.00000	−0.258738	−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000	−0.258738	−0.129369	+	0.991596i	$0.541295\pi$
$240$	1.00000	0.0645497
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	−10.0000	−0.642824
$243$	−1.00000	−0.0641500
$244$	−6.00000	−0.384111
$245$	−2.00000	−0.127775
$246$	8.00000	0.510061
$247$	8.00000	0.509028
$248$	−2.00000	−0.127000
$249$	6.00000	0.380235
$250$	9.00000	0.569210
$251$	−14.0000	−0.883672	−0.441836	−	0.897096i	$-0.645673\pi$
$251$	−14.0000	−0.883672	−0.441836	+	0.897096i	$0.645673\pi$
$252$	−3.00000	−0.188982
$253$	3.00000	0.188608
$254$	14.0000	0.878438
$255$	−5.00000	−0.313112
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	4.00000	0.249029
$259$	−33.0000	−2.05052
$260$	2.00000	0.124035
$261$	−10.0000	−0.618984
$262$	1.00000	0.0617802
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	−1.00000	−0.0615457
$265$	5.00000	0.307148
$266$	12.0000	0.735767
$267$	14.0000	0.856786
$268$	15.0000	0.916271
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	1.00000	0.0608581
$271$	5.00000	0.303728	0.151864	−	0.988401i	$-0.451472\pi$
$271$	5.00000	0.303728	0.151864	+	0.988401i	$0.451472\pi$
$272$	−5.00000	−0.303170
$273$	−6.00000	−0.363137
$274$	9.00000	0.543710
$275$	−4.00000	−0.241209
$276$	−3.00000	−0.180579
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	−13.0000	−0.779688
$279$	−2.00000	−0.119737
$280$	3.00000	0.179284
$281$	−7.00000	−0.417585	−0.208792	−	0.977960i	$-0.566953\pi$
$281$	−7.00000	−0.417585	−0.208792	+	0.977960i	$0.566953\pi$
$282$	1.00000	0.0595491
$283$	−10.0000	−0.594438	−0.297219	−	0.954809i	$-0.596059\pi$
$283$	−10.0000	−0.594438	−0.297219	+	0.954809i	$0.596059\pi$
$284$	−1.00000	−0.0593391
$285$	−4.00000	−0.236940
$286$	−2.00000	−0.118262
$287$	24.0000	1.41668
$288$	1.00000	0.0589256
$289$	8.00000	0.470588
$290$	10.0000	0.587220
$291$	−8.00000	−0.468968
$292$	10.0000	0.585206
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	−2.00000	−0.116642
$295$	−9.00000	−0.524000
$296$	11.0000	0.639362
$297$	−1.00000	−0.0580259
$298$	−18.0000	−1.04271
$299$	−6.00000	−0.346989
$300$	4.00000	0.230940
$301$	12.0000	0.691669
$302$	0	0
$303$	−6.00000	−0.344691
$304$	−4.00000	−0.229416
$305$	6.00000	0.343559
$306$	−5.00000	−0.285831
$307$	−22.0000	−1.25561	−0.627803	−	0.778372i	$-0.716046\pi$
$307$	−22.0000	−1.25561	−0.627803	+	0.778372i	$0.716046\pi$
$308$	−3.00000	−0.170941
$309$	2.00000	0.113776
$310$	2.00000	0.113592
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	2.00000	0.113228
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	3.00000	0.169300
$315$	3.00000	0.169031
$316$	16.0000	0.900070
$317$	−7.00000	−0.393159	−0.196580	−	0.980488i	$-0.562983\pi$
$317$	−7.00000	−0.393159	−0.196580	+	0.980488i	$0.562983\pi$
$318$	5.00000	0.280386
$319$	−10.0000	−0.559893
$320$	−1.00000	−0.0559017
$321$	12.0000	0.669775
$322$	−9.00000	−0.501550
$323$	20.0000	1.11283
$324$	1.00000	0.0555556
$325$	8.00000	0.443760
$326$	8.00000	0.443079
$327$	20.0000	1.10600
$328$	−8.00000	−0.441726
$329$	3.00000	0.165395
$330$	1.00000	0.0550482
$331$	11.0000	0.604615	0.302307	−	0.953211i	$-0.402243\pi$
$331$	11.0000	0.604615	0.302307	+	0.953211i	$0.402243\pi$
$332$	−6.00000	−0.329293
$333$	11.0000	0.602796
$334$	6.00000	0.328305
$335$	−15.0000	−0.819538
$336$	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
$339$	−10.0000	−0.543125
$340$	5.00000	0.271163
$341$	−2.00000	−0.108306
$342$	−4.00000	−0.216295
$343$	15.0000	0.809924
$344$	−4.00000	−0.215666
$345$	3.00000	0.161515
$346$	8.00000	0.430083
$347$	10.0000	0.536828	0.268414	−	0.963304i	$-0.413500\pi$
$347$	10.0000	0.536828	0.268414	+	0.963304i	$0.413500\pi$
$348$	10.0000	0.536056
$349$	7.00000	0.374701	0.187351	−	0.982293i	$-0.440010\pi$
$349$	7.00000	0.374701	0.187351	+	0.982293i	$0.440010\pi$
$350$	12.0000	0.641427
$351$	2.00000	0.106752
$352$	1.00000	0.0533002
$353$	−12.0000	−0.638696	−0.319348	−	0.947638i	$-0.603464\pi$
$353$	−12.0000	−0.638696	−0.319348	+	0.947638i	$0.603464\pi$
$354$	−9.00000	−0.478345
$355$	1.00000	0.0530745
$356$	−14.0000	−0.741999
$357$	−15.0000	−0.793884
$358$	−21.0000	−1.10988
$359$	−29.0000	−1.53056	−0.765281	−	0.643697i	$-0.777400\pi$
$359$	−29.0000	−1.53056	−0.765281	+	0.643697i	$0.777400\pi$
$360$	−1.00000	−0.0527046
$361$	−3.00000	−0.157895
$362$	18.0000	0.946059
$363$	10.0000	0.524864
$364$	6.00000	0.314485
$365$	−10.0000	−0.523424
$366$	6.00000	0.313625
$367$	−7.00000	−0.365397	−0.182699	−	0.983169i	$-0.558483\pi$
$367$	−7.00000	−0.365397	−0.182699	+	0.983169i	$0.558483\pi$
$368$	3.00000	0.156386
$369$	−8.00000	−0.416463
$370$	−11.0000	−0.571863
$371$	15.0000	0.778761
$372$	2.00000	0.103695
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	−5.00000	−0.258544
$375$	−9.00000	−0.464758
$376$	−1.00000	−0.0515711
$377$	20.0000	1.03005
$378$	3.00000	0.154303
$379$	−6.00000	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$380$	4.00000	0.205196
$381$	−14.0000	−0.717242
$382$	12.0000	0.613973
$383$	−2.00000	−0.102195	−0.0510976	−	0.998694i	$-0.516272\pi$
$383$	−2.00000	−0.102195	−0.0510976	+	0.998694i	$0.516272\pi$
$384$	−1.00000	−0.0510310
$385$	3.00000	0.152894
$386$	26.0000	1.32337
$387$	−4.00000	−0.203331
$388$	8.00000	0.406138
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−2.00000	−0.101274
$391$	−15.0000	−0.758583
$392$	2.00000	0.101015
$393$	−1.00000	−0.0504433
$394$	0	0
$395$	−16.0000	−0.805047
$396$	1.00000	0.0502519
$397$	−30.0000	−1.50566	−0.752828	−	0.658217i	$-0.771311\pi$
$397$	−30.0000	−1.50566	−0.752828	+	0.658217i	$0.771311\pi$
$398$	0	0
$399$	−12.0000	−0.600751
$400$	−4.00000	−0.200000
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	−15.0000	−0.748132
$403$	4.00000	0.199254
$404$	6.00000	0.298511
$405$	−1.00000	−0.0496904
$406$	30.0000	1.48888
$407$	11.0000	0.545250
$408$	5.00000	0.247537
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	8.00000	0.395092
$411$	−9.00000	−0.443937
$412$	−2.00000	−0.0985329
$413$	−27.0000	−1.32858
$414$	3.00000	0.147442
$415$	6.00000	0.294528
$416$	−2.00000	−0.0980581
$417$	13.0000	0.636613
$418$	−4.00000	−0.195646
$419$	34.0000	1.66101	0.830504	−	0.557012i	$-0.188052\pi$
$419$	34.0000	1.66101	0.830504	+	0.557012i	$0.188052\pi$
$420$	−3.00000	−0.146385
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	10.0000	0.486792
$423$	−1.00000	−0.0486217
$424$	−5.00000	−0.242821
$425$	20.0000	0.970143
$426$	1.00000	0.0484502
$427$	18.0000	0.871081
$428$	−12.0000	−0.580042
$429$	2.00000	0.0965609
$430$	4.00000	0.192897
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	−1.00000	−0.0481125
$433$	−8.00000	−0.384455	−0.192228	−	0.981350i	$-0.561571\pi$
$433$	−8.00000	−0.384455	−0.192228	+	0.981350i	$0.561571\pi$
$434$	6.00000	0.288009
$435$	−10.0000	−0.479463
$436$	−20.0000	−0.957826
$437$	−12.0000	−0.574038
$438$	−10.0000	−0.477818
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	−1.00000	−0.0476731
$441$	2.00000	0.0952381
$442$	10.0000	0.475651
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	−11.0000	−0.522037
$445$	14.0000	0.663664
$446$	10.0000	0.473514
$447$	18.0000	0.851371
$448$	−3.00000	−0.141737
$449$	11.0000	0.519122	0.259561	−	0.965727i	$-0.416422\pi$
$449$	11.0000	0.519122	0.259561	+	0.965727i	$0.416422\pi$
$450$	−4.00000	−0.188562
$451$	−8.00000	−0.376705
$452$	10.0000	0.470360
$453$	0	0
$454$	−18.0000	−0.844782
$455$	−6.00000	−0.281284
$456$	4.00000	0.187317
$457$	−27.0000	−1.26301	−0.631503	−	0.775373i	$-0.717562\pi$
$457$	−27.0000	−1.26301	−0.631503	+	0.775373i	$0.717562\pi$
$458$	5.00000	0.233635
$459$	5.00000	0.233380
$460$	−3.00000	−0.139876
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	3.00000	0.139573
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	−10.0000	−0.464238
$465$	−2.00000	−0.0927478
$466$	−18.0000	−0.833834
$467$	−13.0000	−0.601568	−0.300784	−	0.953692i	$-0.597248\pi$
$467$	−13.0000	−0.601568	−0.300784	+	0.953692i	$0.597248\pi$
$468$	−2.00000	−0.0924500
$469$	−45.0000	−2.07791
$470$	1.00000	0.0461266
$471$	−3.00000	−0.138233
$472$	9.00000	0.414259
$473$	−4.00000	−0.183920
$474$	−16.0000	−0.734904
$475$	16.0000	0.734130
$476$	15.0000	0.687524
$477$	−5.00000	−0.228934
$478$	−4.00000	−0.182956
$479$	15.0000	0.685367	0.342684	−	0.939451i	$-0.388664\pi$
$479$	15.0000	0.685367	0.342684	+	0.939451i	$0.388664\pi$
$480$	1.00000	0.0456435
$481$	−22.0000	−1.00311
$482$	−22.0000	−1.00207
$483$	9.00000	0.409514
$484$	−10.0000	−0.454545
$485$	−8.00000	−0.363261
$486$	−1.00000	−0.0453609
$487$	7.00000	0.317200	0.158600	−	0.987343i	$-0.449302\pi$
$487$	7.00000	0.317200	0.158600	+	0.987343i	$0.449302\pi$
$488$	−6.00000	−0.271607
$489$	−8.00000	−0.361773
$490$	−2.00000	−0.0903508
$491$	18.0000	0.812329	0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000	0.812329	0.406164	+	0.913800i	$0.366866\pi$
$492$	8.00000	0.360668
$493$	50.0000	2.25189
$494$	8.00000	0.359937
$495$	−1.00000	−0.0449467
$496$	−2.00000	−0.0898027
$497$	3.00000	0.134568
$498$	6.00000	0.268866
$499$	−37.0000	−1.65635	−0.828174	−	0.560471i	$-0.810620\pi$
$499$	−37.0000	−1.65635	−0.828174	+	0.560471i	$0.810620\pi$
$500$	9.00000	0.402492
$501$	−6.00000	−0.268060
$502$	−14.0000	−0.624851
$503$	−3.00000	−0.133763	−0.0668817	−	0.997761i	$-0.521305\pi$
$503$	−3.00000	−0.133763	−0.0668817	+	0.997761i	$0.521305\pi$
$504$	−3.00000	−0.133631
$505$	−6.00000	−0.266996
$506$	3.00000	0.133366
$507$	9.00000	0.399704
$508$	14.0000	0.621150
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	−5.00000	−0.221404
$511$	−30.0000	−1.32712
$512$	1.00000	0.0441942
$513$	4.00000	0.176604
$514$	−18.0000	−0.793946
$515$	2.00000	0.0881305
$516$	4.00000	0.176090
$517$	−1.00000	−0.0439799
$518$	−33.0000	−1.44994
$519$	−8.00000	−0.351161
$520$	2.00000	0.0877058
$521$	31.0000	1.35813	0.679067	−	0.734076i	$-0.262384\pi$
$521$	31.0000	1.35813	0.679067	+	0.734076i	$0.262384\pi$
$522$	−10.0000	−0.437688
$523$	−35.0000	−1.53044	−0.765222	−	0.643767i	$-0.777371\pi$
$523$	−35.0000	−1.53044	−0.765222	+	0.643767i	$0.777371\pi$
$524$	1.00000	0.0436852
$525$	−12.0000	−0.523723
$526$	−24.0000	−1.04645
$527$	10.0000	0.435607
$528$	−1.00000	−0.0435194
$529$	−14.0000	−0.608696
$530$	5.00000	0.217186
$531$	9.00000	0.390567
$532$	12.0000	0.520266
$533$	16.0000	0.693037
$534$	14.0000	0.605839
$535$	12.0000	0.518805
$536$	15.0000	0.647901
$537$	21.0000	0.906217
$538$	−15.0000	−0.646696
$539$	2.00000	0.0861461
$540$	1.00000	0.0430331
$541$	−6.00000	−0.257960	−0.128980	−	0.991647i	$-0.541170\pi$
$541$	−6.00000	−0.257960	−0.128980	+	0.991647i	$0.541170\pi$
$542$	5.00000	0.214768
$543$	−18.0000	−0.772454
$544$	−5.00000	−0.214373
$545$	20.0000	0.856706
$546$	−6.00000	−0.256776
$547$	−33.0000	−1.41098	−0.705489	−	0.708721i	$-0.749273\pi$
$547$	−33.0000	−1.41098	−0.705489	+	0.708721i	$0.749273\pi$
$548$	9.00000	0.384461
$549$	−6.00000	−0.256074
$550$	−4.00000	−0.170561
$551$	40.0000	1.70406
$552$	−3.00000	−0.127688
$553$	−48.0000	−2.04117
$554$	−2.00000	−0.0849719
$555$	11.0000	0.466924
$556$	−13.0000	−0.551323
$557$	23.0000	0.974541	0.487271	−	0.873251i	$-0.337993\pi$
$557$	23.0000	0.974541	0.487271	+	0.873251i	$0.337993\pi$
$558$	−2.00000	−0.0846668
$559$	8.00000	0.338364
$560$	3.00000	0.126773
$561$	5.00000	0.211100
$562$	−7.00000	−0.295277
$563$	−37.0000	−1.55936	−0.779682	−	0.626176i	$-0.784619\pi$
$563$	−37.0000	−1.55936	−0.779682	+	0.626176i	$0.784619\pi$
$564$	1.00000	0.0421076
$565$	−10.0000	−0.420703
$566$	−10.0000	−0.420331
$567$	−3.00000	−0.125988
$568$	−1.00000	−0.0419591
$569$	32.0000	1.34151	0.670755	−	0.741679i	$-0.265970\pi$
$569$	32.0000	1.34151	0.670755	+	0.741679i	$0.265970\pi$
$570$	−4.00000	−0.167542
$571$	5.00000	0.209243	0.104622	−	0.994512i	$-0.466637\pi$
$571$	5.00000	0.209243	0.104622	+	0.994512i	$0.466637\pi$
$572$	−2.00000	−0.0836242
$573$	−12.0000	−0.501307
$574$	24.0000	1.00174
$575$	−12.0000	−0.500435
$576$	1.00000	0.0416667
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	8.00000	0.332756
$579$	−26.0000	−1.08052
$580$	10.0000	0.415227
$581$	18.0000	0.746766
$582$	−8.00000	−0.331611
$583$	−5.00000	−0.207079
$584$	10.0000	0.413803
$585$	2.00000	0.0826898
$586$	−12.0000	−0.495715
$587$	39.0000	1.60970	0.804851	−	0.593477i	$-0.202245\pi$
$587$	39.0000	1.60970	0.804851	+	0.593477i	$0.202245\pi$
$588$	−2.00000	−0.0824786
$589$	8.00000	0.329634
$590$	−9.00000	−0.370524
$591$	0	0
$592$	11.0000	0.452097
$593$	−7.00000	−0.287456	−0.143728	−	0.989617i	$-0.545909\pi$
$593$	−7.00000	−0.287456	−0.143728	+	0.989617i	$0.545909\pi$
$594$	−1.00000	−0.0410305
$595$	−15.0000	−0.614940
$596$	−18.0000	−0.737309
$597$	0	0
$598$	−6.00000	−0.245358
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	4.00000	0.163299
$601$	37.0000	1.50926	0.754631	−	0.656150i	$-0.227816\pi$
$601$	37.0000	1.50926	0.754631	+	0.656150i	$0.227816\pi$
$602$	12.0000	0.489083
$603$	15.0000	0.610847
$604$	0	0
$605$	10.0000	0.406558
$606$	−6.00000	−0.243733
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	−4.00000	−0.162221
$609$	−30.0000	−1.21566
$610$	6.00000	0.242933
$611$	2.00000	0.0809113
$612$	−5.00000	−0.202113
$613$	24.0000	0.969351	0.484675	−	0.874694i	$-0.338938\pi$
$613$	24.0000	0.969351	0.484675	+	0.874694i	$0.338938\pi$
$614$	−22.0000	−0.887848
$615$	−8.00000	−0.322591
$616$	−3.00000	−0.120873
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	2.00000	0.0804518
$619$	−13.0000	−0.522514	−0.261257	−	0.965269i	$-0.584137\pi$
$619$	−13.0000	−0.522514	−0.261257	+	0.965269i	$0.584137\pi$
$620$	2.00000	0.0803219
$621$	−3.00000	−0.120386
$622$	24.0000	0.962312
$623$	42.0000	1.68269
$624$	2.00000	0.0800641
$625$	11.0000	0.440000
$626$	−16.0000	−0.639489
$627$	4.00000	0.159745
$628$	3.00000	0.119713
$629$	−55.0000	−2.19299
$630$	3.00000	0.119523
$631$	−19.0000	−0.756378	−0.378189	−	0.925728i	$-0.623453\pi$
$631$	−19.0000	−0.756378	−0.378189	+	0.925728i	$0.623453\pi$
$632$	16.0000	0.636446
$633$	−10.0000	−0.397464
$634$	−7.00000	−0.278006
$635$	−14.0000	−0.555573
$636$	5.00000	0.198263
$637$	−4.00000	−0.158486
$638$	−10.0000	−0.395904
$639$	−1.00000	−0.0395594
$640$	−1.00000	−0.0395285
$641$	8.00000	0.315981	0.157991	−	0.987441i	$-0.449498\pi$
$641$	8.00000	0.315981	0.157991	+	0.987441i	$0.449498\pi$
$642$	12.0000	0.473602
$643$	−3.00000	−0.118308	−0.0591542	−	0.998249i	$-0.518840\pi$
$643$	−3.00000	−0.118308	−0.0591542	+	0.998249i	$0.518840\pi$
$644$	−9.00000	−0.354650
$645$	−4.00000	−0.157500
$646$	20.0000	0.786889
$647$	42.0000	1.65119	0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000	1.65119	0.825595	+	0.564263i	$0.190840\pi$
$648$	1.00000	0.0392837
$649$	9.00000	0.353281
$650$	8.00000	0.313786
$651$	−6.00000	−0.235159
$652$	8.00000	0.313304
$653$	10.0000	0.391330	0.195665	−	0.980671i	$-0.437313\pi$
$653$	10.0000	0.391330	0.195665	+	0.980671i	$0.437313\pi$
$654$	20.0000	0.782062
$655$	−1.00000	−0.0390732
$656$	−8.00000	−0.312348
$657$	10.0000	0.390137
$658$	3.00000	0.116952
$659$	−19.0000	−0.740135	−0.370067	−	0.929005i	$-0.620665\pi$
$659$	−19.0000	−0.740135	−0.370067	+	0.929005i	$0.620665\pi$
$660$	1.00000	0.0389249
$661$	29.0000	1.12797	0.563985	−	0.825785i	$-0.309268\pi$
$661$	29.0000	1.12797	0.563985	+	0.825785i	$0.309268\pi$
$662$	11.0000	0.427527
$663$	−10.0000	−0.388368
$664$	−6.00000	−0.232845
$665$	−12.0000	−0.465340
$666$	11.0000	0.426241
$667$	−30.0000	−1.16160
$668$	6.00000	0.232147
$669$	−10.0000	−0.386622
$670$	−15.0000	−0.579501
$671$	−6.00000	−0.231627
$672$	3.00000	0.115728
$673$	16.0000	0.616755	0.308377	−	0.951264i	$-0.400214\pi$
$673$	16.0000	0.616755	0.308377	+	0.951264i	$0.400214\pi$
$674$	−14.0000	−0.539260
$675$	4.00000	0.153960
$676$	−9.00000	−0.346154
$677$	−28.0000	−1.07613	−0.538064	−	0.842904i	$-0.680844\pi$
$677$	−28.0000	−1.07613	−0.538064	+	0.842904i	$0.680844\pi$
$678$	−10.0000	−0.384048
$679$	−24.0000	−0.921035
$680$	5.00000	0.191741
$681$	18.0000	0.689761
$682$	−2.00000	−0.0765840
$683$	−19.0000	−0.727015	−0.363507	−	0.931591i	$-0.618421\pi$
$683$	−19.0000	−0.727015	−0.363507	+	0.931591i	$0.618421\pi$
$684$	−4.00000	−0.152944
$685$	−9.00000	−0.343872
$686$	15.0000	0.572703
$687$	−5.00000	−0.190762
$688$	−4.00000	−0.152499
$689$	10.0000	0.380970
$690$	3.00000	0.114208
$691$	−48.0000	−1.82601	−0.913003	−	0.407953i	$-0.866243\pi$
$691$	−48.0000	−1.82601	−0.913003	+	0.407953i	$0.866243\pi$
$692$	8.00000	0.304114
$693$	−3.00000	−0.113961
$694$	10.0000	0.379595
$695$	13.0000	0.493118
$696$	10.0000	0.379049
$697$	40.0000	1.51511
$698$	7.00000	0.264954
$699$	18.0000	0.680823
$700$	12.0000	0.453557
$701$	47.0000	1.77517	0.887583	−	0.460648i	$-0.152383\pi$
$701$	47.0000	1.77517	0.887583	+	0.460648i	$0.152383\pi$
$702$	2.00000	0.0754851
$703$	−44.0000	−1.65949
$704$	1.00000	0.0376889
$705$	−1.00000	−0.0376622
$706$	−12.0000	−0.451626
$707$	−18.0000	−0.676960
$708$	−9.00000	−0.338241
$709$	17.0000	0.638448	0.319224	−	0.947679i	$-0.396578\pi$
$709$	17.0000	0.638448	0.319224	+	0.947679i	$0.396578\pi$
$710$	1.00000	0.0375293
$711$	16.0000	0.600047
$712$	−14.0000	−0.524672
$713$	−6.00000	−0.224702
$714$	−15.0000	−0.561361
$715$	2.00000	0.0747958
$716$	−21.0000	−0.784807
$717$	4.00000	0.149383
$718$	−29.0000	−1.08227
$719$	−52.0000	−1.93927	−0.969636	−	0.244551i	$-0.921359\pi$
$719$	−52.0000	−1.93927	−0.969636	+	0.244551i	$0.921359\pi$
$720$	−1.00000	−0.0372678
$721$	6.00000	0.223452
$722$	−3.00000	−0.111648
$723$	22.0000	0.818189
$724$	18.0000	0.668965
$725$	40.0000	1.48556
$726$	10.0000	0.371135
$727$	4.00000	0.148352	0.0741759	−	0.997245i	$-0.476367\pi$
$727$	4.00000	0.148352	0.0741759	+	0.997245i	$0.476367\pi$
$728$	6.00000	0.222375
$729$	1.00000	0.0370370
$730$	−10.0000	−0.370117
$731$	20.0000	0.739727
$732$	6.00000	0.221766
$733$	38.0000	1.40356	0.701781	−	0.712393i	$-0.252388\pi$
$733$	38.0000	1.40356	0.701781	+	0.712393i	$0.252388\pi$
$734$	−7.00000	−0.258375
$735$	2.00000	0.0737711
$736$	3.00000	0.110581
$737$	15.0000	0.552532
$738$	−8.00000	−0.294484
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	−11.0000	−0.404368
$741$	−8.00000	−0.293887
$742$	15.0000	0.550667
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	2.00000	0.0733236
$745$	18.0000	0.659469
$746$	−6.00000	−0.219676
$747$	−6.00000	−0.219529
$748$	−5.00000	−0.182818
$749$	36.0000	1.31541
$750$	−9.00000	−0.328634
$751$	10.0000	0.364905	0.182453	−	0.983215i	$-0.441596\pi$
$751$	10.0000	0.364905	0.182453	+	0.983215i	$0.441596\pi$
$752$	−1.00000	−0.0364662
$753$	14.0000	0.510188
$754$	20.0000	0.728357
$755$	0	0
$756$	3.00000	0.109109
$757$	−52.0000	−1.88997	−0.944986	−	0.327111i	$-0.893925\pi$
$757$	−52.0000	−1.88997	−0.944986	+	0.327111i	$0.893925\pi$
$758$	−6.00000	−0.217930
$759$	−3.00000	−0.108893
$760$	4.00000	0.145095
$761$	54.0000	1.95750	0.978749	−	0.205061i	$-0.0657392\pi$
$761$	54.0000	1.95750	0.978749	+	0.205061i	$0.0657392\pi$
$762$	−14.0000	−0.507166
$763$	60.0000	2.17215
$764$	12.0000	0.434145
$765$	5.00000	0.180775
$766$	−2.00000	−0.0722629
$767$	−18.0000	−0.649942
$768$	−1.00000	−0.0360844
$769$	17.0000	0.613036	0.306518	−	0.951865i	$-0.400836\pi$
$769$	17.0000	0.613036	0.306518	+	0.951865i	$0.400836\pi$
$770$	3.00000	0.108112
$771$	18.0000	0.648254
$772$	26.0000	0.935760
$773$	−12.0000	−0.431610	−0.215805	−	0.976436i	$-0.569238\pi$
$773$	−12.0000	−0.431610	−0.215805	+	0.976436i	$0.569238\pi$
$774$	−4.00000	−0.143777
$775$	8.00000	0.287368
$776$	8.00000	0.287183
$777$	33.0000	1.18387
$778$	6.00000	0.215110
$779$	32.0000	1.14652
$780$	−2.00000	−0.0716115
$781$	−1.00000	−0.0357828
$782$	−15.0000	−0.536399
$783$	10.0000	0.357371
$784$	2.00000	0.0714286
$785$	−3.00000	−0.107075
$786$	−1.00000	−0.0356688
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	0	0
$789$	24.0000	0.854423
$790$	−16.0000	−0.569254
$791$	−30.0000	−1.06668
$792$	1.00000	0.0355335
$793$	12.0000	0.426132
$794$	−30.0000	−1.06466
$795$	−5.00000	−0.177332
$796$	0	0
$797$	−3.00000	−0.106265	−0.0531327	−	0.998587i	$-0.516921\pi$
$797$	−3.00000	−0.106265	−0.0531327	+	0.998587i	$0.516921\pi$
$798$	−12.0000	−0.424795
$799$	5.00000	0.176887
$800$	−4.00000	−0.141421
$801$	−14.0000	−0.494666
$802$	3.00000	0.105934
$803$	10.0000	0.352892
$804$	−15.0000	−0.529009
$805$	9.00000	0.317208
$806$	4.00000	0.140894
$807$	15.0000	0.528025
$808$	6.00000	0.211079
$809$	−35.0000	−1.23053	−0.615267	−	0.788319i	$-0.710952\pi$
$809$	−35.0000	−1.23053	−0.615267	+	0.788319i	$0.710952\pi$
$810$	−1.00000	−0.0351364
$811$	40.0000	1.40459	0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000	1.40459	0.702295	+	0.711886i	$0.252159\pi$
$812$	30.0000	1.05279
$813$	−5.00000	−0.175358
$814$	11.0000	0.385550
$815$	−8.00000	−0.280228
$816$	5.00000	0.175035
$817$	16.0000	0.559769
$818$	5.00000	0.174821
$819$	6.00000	0.209657
$820$	8.00000	0.279372
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	−9.00000	−0.313911
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	−2.00000	−0.0696733
$825$	4.00000	0.139262
$826$	−27.0000	−0.939450
$827$	−28.0000	−0.973655	−0.486828	−	0.873498i	$-0.661846\pi$
$827$	−28.0000	−0.973655	−0.486828	+	0.873498i	$0.661846\pi$
$828$	3.00000	0.104257
$829$	−44.0000	−1.52818	−0.764092	−	0.645108i	$-0.776812\pi$
$829$	−44.0000	−1.52818	−0.764092	+	0.645108i	$0.776812\pi$
$830$	6.00000	0.208263
$831$	2.00000	0.0693792
$832$	−2.00000	−0.0693375
$833$	−10.0000	−0.346479
$834$	13.0000	0.450153
$835$	−6.00000	−0.207639
$836$	−4.00000	−0.138343
$837$	2.00000	0.0691301
$838$	34.0000	1.17451
$839$	42.0000	1.45000	0.725001	−	0.688748i	$-0.241839\pi$
$839$	42.0000	1.45000	0.725001	+	0.688748i	$0.241839\pi$
$840$	−3.00000	−0.103510
$841$	71.0000	2.44828
$842$	−10.0000	−0.344623
$843$	7.00000	0.241093
$844$	10.0000	0.344214
$845$	9.00000	0.309609
$846$	−1.00000	−0.0343807
$847$	30.0000	1.03081
$848$	−5.00000	−0.171701
$849$	10.0000	0.343199
$850$	20.0000	0.685994
$851$	33.0000	1.13123
$852$	1.00000	0.0342594
$853$	−27.0000	−0.924462	−0.462231	−	0.886759i	$-0.652951\pi$
$853$	−27.0000	−0.924462	−0.462231	+	0.886759i	$0.652951\pi$
$854$	18.0000	0.615947
$855$	4.00000	0.136797
$856$	−12.0000	−0.410152
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	2.00000	0.0682789
$859$	27.0000	0.921228	0.460614	−	0.887601i	$-0.347629\pi$
$859$	27.0000	0.921228	0.460614	+	0.887601i	$0.347629\pi$
$860$	4.00000	0.136399
$861$	−24.0000	−0.817918
$862$	18.0000	0.613082
$863$	−46.0000	−1.56586	−0.782929	−	0.622111i	$-0.786275\pi$
$863$	−46.0000	−1.56586	−0.782929	+	0.622111i	$0.786275\pi$
$864$	−1.00000	−0.0340207
$865$	−8.00000	−0.272008
$866$	−8.00000	−0.271851
$867$	−8.00000	−0.271694
$868$	6.00000	0.203653
$869$	16.0000	0.542763
$870$	−10.0000	−0.339032
$871$	−30.0000	−1.01651
$872$	−20.0000	−0.677285
$873$	8.00000	0.270759
$874$	−12.0000	−0.405906
$875$	−27.0000	−0.912767
$876$	−10.0000	−0.337869
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	20.0000	0.674967
$879$	12.0000	0.404750
$880$	−1.00000	−0.0337100
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	2.00000	0.0673435
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	10.0000	0.336336
$885$	9.00000	0.302532
$886$	−4.00000	−0.134383
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	−11.0000	−0.369136
$889$	−42.0000	−1.40863
$890$	14.0000	0.469281
$891$	1.00000	0.0335013
$892$	10.0000	0.334825
$893$	4.00000	0.133855
$894$	18.0000	0.602010
$895$	21.0000	0.701953
$896$	−3.00000	−0.100223
$897$	6.00000	0.200334
$898$	11.0000	0.367075
$899$	20.0000	0.667037
$900$	−4.00000	−0.133333
$901$	25.0000	0.832871
$902$	−8.00000	−0.266371
$903$	−12.0000	−0.399335
$904$	10.0000	0.332595
$905$	−18.0000	−0.598340
$906$	0	0
$907$	14.0000	0.464862	0.232431	−	0.972613i	$-0.425332\pi$
$907$	14.0000	0.464862	0.232431	+	0.972613i	$0.425332\pi$
$908$	−18.0000	−0.597351
$909$	6.00000	0.199007
$910$	−6.00000	−0.198898
$911$	52.0000	1.72284	0.861418	−	0.507896i	$-0.169577\pi$
$911$	52.0000	1.72284	0.861418	+	0.507896i	$0.169577\pi$
$912$	4.00000	0.132453
$913$	−6.00000	−0.198571
$914$	−27.0000	−0.893081
$915$	−6.00000	−0.198354
$916$	5.00000	0.165205
$917$	−3.00000	−0.0990687
$918$	5.00000	0.165025
$919$	−46.0000	−1.51740	−0.758700	−	0.651440i	$-0.774165\pi$
$919$	−46.0000	−1.51740	−0.758700	+	0.651440i	$0.774165\pi$
$920$	−3.00000	−0.0989071
$921$	22.0000	0.724925
$922$	−30.0000	−0.987997
$923$	2.00000	0.0658308
$924$	3.00000	0.0986928
$925$	−44.0000	−1.44671
$926$	22.0000	0.722965
$927$	−2.00000	−0.0656886
$928$	−10.0000	−0.328266
$929$	−48.0000	−1.57483	−0.787414	−	0.616424i	$-0.788581\pi$
$929$	−48.0000	−1.57483	−0.787414	+	0.616424i	$0.788581\pi$
$930$	−2.00000	−0.0655826
$931$	−8.00000	−0.262189
$932$	−18.0000	−0.589610
$933$	−24.0000	−0.785725
$934$	−13.0000	−0.425373
$935$	5.00000	0.163517
$936$	−2.00000	−0.0653720
$937$	−3.00000	−0.0980057	−0.0490029	−	0.998799i	$-0.515604\pi$
$937$	−3.00000	−0.0980057	−0.0490029	+	0.998799i	$0.515604\pi$
$938$	−45.0000	−1.46930
$939$	16.0000	0.522140
$940$	1.00000	0.0326164
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	−3.00000	−0.0977453
$943$	−24.0000	−0.781548
$944$	9.00000	0.292925
$945$	−3.00000	−0.0975900
$946$	−4.00000	−0.130051
$947$	6.00000	0.194974	0.0974869	−	0.995237i	$-0.468920\pi$
$947$	6.00000	0.194974	0.0974869	+	0.995237i	$0.468920\pi$
$948$	−16.0000	−0.519656
$949$	−20.0000	−0.649227
$950$	16.0000	0.519109
$951$	7.00000	0.226991
$952$	15.0000	0.486153
$953$	8.00000	0.259145	0.129573	−	0.991570i	$-0.458639\pi$
$953$	8.00000	0.259145	0.129573	+	0.991570i	$0.458639\pi$
$954$	−5.00000	−0.161881
$955$	−12.0000	−0.388311
$956$	−4.00000	−0.129369
$957$	10.0000	0.323254
$958$	15.0000	0.484628
$959$	−27.0000	−0.871875
$960$	1.00000	0.0322749
$961$	−27.0000	−0.870968
$962$	−22.0000	−0.709308
$963$	−12.0000	−0.386695
$964$	−22.0000	−0.708572
$965$	−26.0000	−0.836970
$966$	9.00000	0.289570
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	−10.0000	−0.321412
$969$	−20.0000	−0.642493
$970$	−8.00000	−0.256865
$971$	−2.00000	−0.0641831	−0.0320915	−	0.999485i	$-0.510217\pi$
$971$	−2.00000	−0.0641831	−0.0320915	+	0.999485i	$0.510217\pi$
$972$	−1.00000	−0.0320750
$973$	39.0000	1.25028
$974$	7.00000	0.224294
$975$	−8.00000	−0.256205
$976$	−6.00000	−0.192055
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	−8.00000	−0.255812
$979$	−14.0000	−0.447442
$980$	−2.00000	−0.0638877
$981$	−20.0000	−0.638551
$982$	18.0000	0.574403
$983$	5.00000	0.159475	0.0797376	−	0.996816i	$-0.474592\pi$
$983$	5.00000	0.159475	0.0797376	+	0.996816i	$0.474592\pi$
$984$	8.00000	0.255031
$985$	0	0
$986$	50.0000	1.59232
$987$	−3.00000	−0.0954911
$988$	8.00000	0.254514
$989$	−12.0000	−0.381578
$990$	−1.00000	−0.0317821
$991$	−13.0000	−0.412959	−0.206479	−	0.978451i	$-0.566201\pi$
$991$	−13.0000	−0.412959	−0.206479	+	0.978451i	$0.566201\pi$
$992$	−2.00000	−0.0635001
$993$	−11.0000	−0.349074
$994$	3.00000	0.0951542
$995$	0	0
$996$	6.00000	0.190117
$997$	−20.0000	−0.633406	−0.316703	−	0.948525i	$-0.602576\pi$
$997$	−20.0000	−0.633406	−0.316703	+	0.948525i	$0.602576\pi$
$998$	−37.0000	−1.17121
$999$	−11.0000	−0.348025

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	786.2.a.j.1.1	✓	1
3.2	odd	2		2358.2.a.f.1.1		1
4.3	odd	2		6288.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
786.2.a.j.1.1	✓	1	1.1	even	1	trivial
2358.2.a.f.1.1		1	3.2	odd	2
6288.2.a.k.1.1		1	4.3	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:	\( N \)	\(=\)	\( 786 = 2 \cdot 3 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	786.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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