LMFDB - Embedded newform 1682.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1682,2,Mod(1,1682)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1682, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1682.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1682 = 2 \cdot 29^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1682.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:	$13.4308376200$
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	$\chi$	$=$	1682.1

$q$ -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} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -6.00000 q^{11} -2.00000 q^{12} -4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{21} +6.00000 q^{22} +9.00000 q^{23} +2.00000 q^{24} -5.00000 q^{25} +4.00000 q^{26} +4.00000 q^{27} -1.00000 q^{28} +5.00000 q^{31} -1.00000 q^{32} +12.0000 q^{33} +3.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} +4.00000 q^{38} +8.00000 q^{39} -6.00000 q^{41} -2.00000 q^{42} -4.00000 q^{43} -6.00000 q^{44} -9.00000 q^{46} -9.00000 q^{47} -2.00000 q^{48} -6.00000 q^{49} +5.00000 q^{50} +6.00000 q^{51} -4.00000 q^{52} -12.0000 q^{53} -4.00000 q^{54} +1.00000 q^{56} +8.00000 q^{57} +6.00000 q^{59} +2.00000 q^{61} -5.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -12.0000 q^{66} +8.00000 q^{67} -3.00000 q^{68} -18.0000 q^{69} +3.00000 q^{71} -1.00000 q^{72} +11.0000 q^{73} -2.00000 q^{74} +10.0000 q^{75} -4.00000 q^{76} +6.00000 q^{77} -8.00000 q^{78} +11.0000 q^{79} -11.0000 q^{81} +6.00000 q^{82} +2.00000 q^{84} +4.00000 q^{86} +6.00000 q^{88} +3.00000 q^{89} +4.00000 q^{91} +9.00000 q^{92} -10.0000 q^{93} +9.00000 q^{94} +2.00000 q^{96} +17.0000 q^{97} +6.00000 q^{98} -6.00000 q^{99} +O(q^{100})

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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	2.00000	0.816497
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	−2.00000	−0.577350
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	−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	$0.651732\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	6.00000	1.27920
$23$	9.00000	1.87663	0.938315	−	0.345782i	$-0.112386\pi$
$23$	9.00000	1.87663	0.938315	+	0.345782i	$0.112386\pi$
$24$	2.00000	0.408248
$25$	−5.00000	−1.00000
$26$	4.00000	0.784465
$27$	4.00000	0.769800
$28$	−1.00000	−0.188982
$29$	0	0
$29$	0	0
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	−1.00000	−0.176777
$33$	12.0000	2.08893
$34$	3.00000	0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	4.00000	0.648886
$39$	8.00000	1.28103
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−2.00000	−0.308607
$43$	−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
$45$	0	0
$46$	−9.00000	−1.32698
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	−2.00000	−0.288675
$49$	−6.00000	−0.857143
$50$	5.00000	0.707107
$51$	6.00000	0.840168
$52$	−4.00000	−0.554700
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	1.00000	0.133631
$57$	8.00000	1.05963
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−5.00000	−0.635001
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	−12.0000	−1.47710
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−3.00000	−0.363803
$69$	−18.0000	−2.16695
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	−1.00000	−0.117851
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	−2.00000	−0.232495
$75$	10.0000	1.15470
$76$	−4.00000	−0.458831
$77$	6.00000	0.683763
$78$	−8.00000	−0.905822
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	6.00000	0.662589
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	6.00000	0.639602
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	9.00000	0.938315
$93$	−10.0000	−1.03695
$94$	9.00000	0.928279
$95$	0	0
$96$	2.00000	0.204124
$97$	17.0000	1.72609	0.863044	−	0.505128i	$-0.168555\pi$
$97$	17.0000	1.72609	0.863044	+	0.505128i	$0.168555\pi$
$98$	6.00000	0.606092
$99$	−6.00000	−0.603023
$100$	−5.00000	−0.500000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	−6.00000	−0.594089
$103$	5.00000	0.492665	0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000	0.492665	0.246332	+	0.969185i	$0.420775\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	12.0000	1.16554
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	4.00000	0.384900
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	−1.00000	−0.0944911
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	−8.00000	−0.749269
$115$	0	0
$116$	0	0
$117$	−4.00000	−0.369800
$118$	−6.00000	−0.552345
$119$	3.00000	0.275010
$120$	0	0
$121$	25.0000	2.27273
$122$	−2.00000	−0.181071
$123$	12.0000	1.08200
$124$	5.00000	0.449013
$125$	0	0
$126$	1.00000	0.0890871
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000	−0.0883883
$129$	8.00000	0.704361
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	12.0000	1.04447
$133$	4.00000	0.346844
$134$	−8.00000	−0.691095
$135$	0	0
$136$	3.00000	0.257248
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	18.0000	1.53226
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	18.0000	1.51587
$142$	−3.00000	−0.251754
$143$	24.0000	2.00698
$144$	1.00000	0.0833333
$145$	0	0
$146$	−11.0000	−0.910366
$147$	12.0000	0.989743
$148$	2.00000	0.164399
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	−10.0000	−0.816497
$151$	−19.0000	−1.54620	−0.773099	−	0.634285i	$-0.781294\pi$
$151$	−19.0000	−1.54620	−0.773099	+	0.634285i	$0.781294\pi$
$152$	4.00000	0.324443
$153$	−3.00000	−0.242536
$154$	−6.00000	−0.483494
$155$	0	0
$156$	8.00000	0.640513
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	−11.0000	−0.875113
$159$	24.0000	1.90332
$160$	0	0
$161$	−9.00000	−0.709299
$162$	11.0000	0.864242
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	−2.00000	−0.154303
$169$	3.00000	0.230769
$170$	0	0
$171$	−4.00000	−0.305888
$172$	−4.00000	−0.304997
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	5.00000	0.377964
$176$	−6.00000	−0.452267
$177$	−12.0000	−0.901975
$178$	−3.00000	−0.224860
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	−4.00000	−0.296500
$183$	−4.00000	−0.295689
$184$	−9.00000	−0.663489
$185$	0	0
$186$	10.0000	0.733236
$187$	18.0000	1.31629
$188$	−9.00000	−0.656392
$189$	−4.00000	−0.290957
$190$	0	0
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	−2.00000	−0.144338
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−17.0000	−1.22053
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	6.00000	0.426401
$199$	−7.00000	−0.496217	−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000	−0.496217	−0.248108	+	0.968732i	$0.579809\pi$
$200$	5.00000	0.353553
$201$	−16.0000	−1.12855
$202$	6.00000	0.422159
$203$	0	0
$204$	6.00000	0.420084
$205$	0	0
$206$	−5.00000	−0.348367
$207$	9.00000	0.625543
$208$	−4.00000	−0.277350
$209$	24.0000	1.66011
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	−12.0000	−0.824163
$213$	−6.00000	−0.411113
$214$	0	0
$215$	0	0
$216$	−4.00000	−0.272166
$217$	−5.00000	−0.339422
$218$	10.0000	0.677285
$219$	−22.0000	−1.48662
$220$	0	0
$221$	12.0000	0.807207
$222$	4.00000	0.268462
$223$	23.0000	1.54019	0.770097	−	0.637927i	$-0.220208\pi$
$223$	23.0000	1.54019	0.770097	+	0.637927i	$0.220208\pi$
$224$	1.00000	0.0668153
$225$	−5.00000	−0.333333
$226$	9.00000	0.598671
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	8.00000	0.529813
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	0	0
$231$	−12.0000	−0.789542
$232$	0	0
$233$	27.0000	1.76883	0.884414	−	0.466702i	$-0.154558\pi$
$233$	27.0000	1.76883	0.884414	+	0.466702i	$0.154558\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	6.00000	0.390567
$237$	−22.0000	−1.42905
$238$	−3.00000	−0.194461
$239$	−9.00000	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$240$	0	0
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	−25.0000	−1.60706
$243$	10.0000	0.641500
$244$	2.00000	0.128037
$245$	0	0
$246$	−12.0000	−0.765092
$247$	16.0000	1.01806
$248$	−5.00000	−0.317500
$249$	0	0
$250$	0	0
$251$	−30.0000	−1.89358	−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000	−1.89358	−0.946792	+	0.321847i	$0.895696\pi$
$252$	−1.00000	−0.0629941
$253$	−54.0000	−3.39495
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	15.0000	0.935674	0.467837	−	0.883815i	$-0.345033\pi$
$257$	15.0000	0.935674	0.467837	+	0.883815i	$0.345033\pi$
$258$	−8.00000	−0.498058
$259$	−2.00000	−0.124274
$260$	0	0
$261$	0	0
$262$	−6.00000	−0.370681
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	−12.0000	−0.738549
$265$	0	0
$266$	−4.00000	−0.245256
$267$	−6.00000	−0.367194
$268$	8.00000	0.488678
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	−25.0000	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$272$	−3.00000	−0.181902
$273$	−8.00000	−0.484182
$274$	−9.00000	−0.543710
$275$	30.0000	1.80907
$276$	−18.0000	−1.08347
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	−14.0000	−0.839664
$279$	5.00000	0.299342
$280$	0	0
$281$	−9.00000	−0.536895	−0.268447	−	0.963294i	$-0.586511\pi$
$281$	−9.00000	−0.536895	−0.268447	+	0.963294i	$0.586511\pi$
$282$	−18.0000	−1.07188
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	3.00000	0.178017
$285$	0	0
$286$	−24.0000	−1.41915
$287$	6.00000	0.354169
$288$	−1.00000	−0.0589256
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−34.0000	−1.99312
$292$	11.0000	0.643726
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	−12.0000	−0.699854
$295$	0	0
$296$	−2.00000	−0.116248
$297$	−24.0000	−1.39262
$298$	18.0000	1.04271
$299$	−36.0000	−2.08193
$300$	10.0000	0.577350
$301$	4.00000	0.230556
$302$	19.0000	1.09333
$303$	12.0000	0.689382
$304$	−4.00000	−0.229416
$305$	0	0
$306$	3.00000	0.171499
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	6.00000	0.341882
$309$	−10.0000	−0.568880
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−8.00000	−0.452911
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	11.0000	0.618798
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	−24.0000	−1.34585
$319$	0	0
$320$	0	0
$321$	0	0
$322$	9.00000	0.501550
$323$	12.0000	0.667698
$324$	−11.0000	−0.611111
$325$	20.0000	1.10940
$326$	−14.0000	−0.775388
$327$	20.0000	1.10600
$328$	6.00000	0.331295
$329$	9.00000	0.496186
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	0	0
$336$	2.00000	0.109109
$337$	−1.00000	−0.0544735	−0.0272367	−	0.999629i	$-0.508671\pi$
$337$	−1.00000	−0.0544735	−0.0272367	+	0.999629i	$0.508671\pi$
$338$	−3.00000	−0.163178
$339$	18.0000	0.977626
$340$	0	0
$341$	−30.0000	−1.62459
$342$	4.00000	0.216295
$343$	13.0000	0.701934
$344$	4.00000	0.215666
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	0	0
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	−5.00000	−0.267261
$351$	−16.0000	−0.854017
$352$	6.00000	0.319801
$353$	−9.00000	−0.479022	−0.239511	−	0.970894i	$-0.576987\pi$
$353$	−9.00000	−0.479022	−0.239511	+	0.970894i	$0.576987\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	3.00000	0.159000
$357$	−6.00000	−0.317554
$358$	0	0
$359$	15.0000	0.791670	0.395835	−	0.918322i	$-0.370455\pi$
$359$	15.0000	0.791670	0.395835	+	0.918322i	$0.370455\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	16.0000	0.840941
$363$	−50.0000	−2.62432
$364$	4.00000	0.209657
$365$	0	0
$366$	4.00000	0.209083
$367$	−1.00000	−0.0521996	−0.0260998	−	0.999659i	$-0.508309\pi$
$367$	−1.00000	−0.0521996	−0.0260998	+	0.999659i	$0.508309\pi$
$368$	9.00000	0.469157
$369$	−6.00000	−0.312348
$370$	0	0
$371$	12.0000	0.623009
$372$	−10.0000	−0.518476
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	−18.0000	−0.930758
$375$	0	0
$376$	9.00000	0.464140
$377$	0	0
$378$	4.00000	0.205738
$379$	2.00000	0.102733	0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000	0.102733	0.0513665	+	0.998680i	$0.483642\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	−3.00000	−0.153493
$383$	33.0000	1.68622	0.843111	−	0.537740i	$-0.180722\pi$
$383$	33.0000	1.68622	0.843111	+	0.537740i	$0.180722\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	−14.0000	−0.712581
$387$	−4.00000	−0.203331
$388$	17.0000	0.863044
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	0	0
$391$	−27.0000	−1.36545
$392$	6.00000	0.303046
$393$	−12.0000	−0.605320
$394$	12.0000	0.604551
$395$	0	0
$396$	−6.00000	−0.301511
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	7.00000	0.350878
$399$	−8.00000	−0.400501
$400$	−5.00000	−0.250000
$401$	−3.00000	−0.149813	−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000	−0.149813	−0.0749064	+	0.997191i	$0.523866\pi$
$402$	16.0000	0.798007
$403$	−20.0000	−0.996271
$404$	−6.00000	−0.298511
$405$	0	0
$406$	0	0
$407$	−12.0000	−0.594818
$408$	−6.00000	−0.297044
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	5.00000	0.246332
$413$	−6.00000	−0.295241
$414$	−9.00000	−0.442326
$415$	0	0
$416$	4.00000	0.196116
$417$	−28.0000	−1.37117
$418$	−24.0000	−1.17388
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	−8.00000	−0.389434
$423$	−9.00000	−0.437595
$424$	12.0000	0.582772
$425$	15.0000	0.727607
$426$	6.00000	0.290701
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	−48.0000	−2.31746
$430$	0	0
$431$	3.00000	0.144505	0.0722525	−	0.997386i	$-0.476981\pi$
$431$	3.00000	0.144505	0.0722525	+	0.997386i	$0.476981\pi$
$432$	4.00000	0.192450
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	5.00000	0.240008
$435$	0	0
$436$	−10.0000	−0.478913
$437$	−36.0000	−1.72211
$438$	22.0000	1.05120
$439$	−31.0000	−1.47955	−0.739775	−	0.672855i	$-0.765068\pi$
$439$	−31.0000	−1.47955	−0.739775	+	0.672855i	$0.765068\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	−12.0000	−0.570782
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	−23.0000	−1.08908
$447$	36.0000	1.70274
$448$	−1.00000	−0.0472456
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	5.00000	0.235702
$451$	36.0000	1.69517
$452$	−9.00000	−0.423324
$453$	38.0000	1.78540
$454$	0	0
$455$	0	0
$456$	−8.00000	−0.374634
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−20.0000	−0.934539
$459$	−12.0000	−0.560112
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	12.0000	0.558291
$463$	−1.00000	−0.0464739	−0.0232370	−	0.999730i	$-0.507397\pi$
$463$	−1.00000	−0.0464739	−0.0232370	+	0.999730i	$0.507397\pi$
$464$	0	0
$465$	0	0
$466$	−27.0000	−1.25075
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	−4.00000	−0.184900
$469$	−8.00000	−0.369406
$470$	0	0
$471$	8.00000	0.368621
$472$	−6.00000	−0.276172
$473$	24.0000	1.10352
$474$	22.0000	1.01049
$475$	20.0000	0.917663
$476$	3.00000	0.137505
$477$	−12.0000	−0.549442
$478$	9.00000	0.411650
$479$	−15.0000	−0.685367	−0.342684	−	0.939451i	$-0.611336\pi$
$479$	−15.0000	−0.685367	−0.342684	+	0.939451i	$0.611336\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	−17.0000	−0.774329
$483$	18.0000	0.819028
$484$	25.0000	1.13636
$485$	0	0
$486$	−10.0000	−0.453609
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	−2.00000	−0.0905357
$489$	−28.0000	−1.26620
$490$	0	0
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	12.0000	0.541002
$493$	0	0
$494$	−16.0000	−0.719874
$495$	0	0
$496$	5.00000	0.224507
$497$	−3.00000	−0.134568
$498$	0	0
$499$	−34.0000	−1.52205	−0.761025	−	0.648723i	$-0.775303\pi$
$499$	−34.0000	−1.52205	−0.761025	+	0.648723i	$0.775303\pi$
$500$	0	0
$501$	0	0
$502$	30.0000	1.33897
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	54.0000	2.40059
$507$	−6.00000	−0.266469
$508$	8.00000	0.354943
$509$	−24.0000	−1.06378	−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000	−1.06378	−0.531891	+	0.846813i	$0.678518\pi$
$510$	0	0
$511$	−11.0000	−0.486611
$512$	−1.00000	−0.0441942
$513$	−16.0000	−0.706417
$514$	−15.0000	−0.661622
$515$	0	0
$516$	8.00000	0.352180
$517$	54.0000	2.37492
$518$	2.00000	0.0878750
$519$	−12.0000	−0.526742
$520$	0	0
$521$	21.0000	0.920027	0.460013	−	0.887912i	$-0.347845\pi$
$521$	21.0000	0.920027	0.460013	+	0.887912i	$0.347845\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	6.00000	0.262111
$525$	−10.0000	−0.436436
$526$	12.0000	0.523225
$527$	−15.0000	−0.653410
$528$	12.0000	0.522233
$529$	58.0000	2.52174
$530$	0	0
$531$	6.00000	0.260378
$532$	4.00000	0.173422
$533$	24.0000	1.03956
$534$	6.00000	0.259645
$535$	0	0
$536$	−8.00000	−0.345547
$537$	0	0
$538$	6.00000	0.258678
$539$	36.0000	1.55063
$540$	0	0
$541$	−16.0000	−0.687894	−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000	−0.687894	−0.343947	+	0.938989i	$0.611764\pi$
$542$	25.0000	1.07384
$543$	32.0000	1.37325
$544$	3.00000	0.128624
$545$	0	0
$546$	8.00000	0.342368
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	9.00000	0.384461
$549$	2.00000	0.0853579
$550$	−30.0000	−1.27920
$551$	0	0
$552$	18.0000	0.766131
$553$	−11.0000	−0.467768
$554$	−14.0000	−0.594803
$555$	0	0
$556$	14.0000	0.593732
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	−5.00000	−0.211667
$559$	16.0000	0.676728
$560$	0	0
$561$	−36.0000	−1.51992
$562$	9.00000	0.379642
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	18.0000	0.757937
$565$	0	0
$566$	−14.0000	−0.588464
$567$	11.0000	0.461957
$568$	−3.00000	−0.125877
$569$	−39.0000	−1.63497	−0.817483	−	0.575953i	$-0.804631\pi$
$569$	−39.0000	−1.63497	−0.817483	+	0.575953i	$0.804631\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	24.0000	1.00349
$573$	−6.00000	−0.250654
$574$	−6.00000	−0.250435
$575$	−45.0000	−1.87663
$576$	1.00000	0.0416667
$577$	11.0000	0.457936	0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000	0.457936	0.228968	+	0.973434i	$0.426465\pi$
$578$	8.00000	0.332756
$579$	−28.0000	−1.16364
$580$	0	0
$581$	0	0
$582$	34.0000	1.40935
$583$	72.0000	2.98194
$584$	−11.0000	−0.455183
$585$	0	0
$586$	−30.0000	−1.23929
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	12.0000	0.494872
$589$	−20.0000	−0.824086
$590$	0	0
$591$	24.0000	0.987228
$592$	2.00000	0.0821995
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	24.0000	0.984732
$595$	0	0
$596$	−18.0000	−0.737309
$597$	14.0000	0.572982
$598$	36.0000	1.47215
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	−10.0000	−0.408248
$601$	23.0000	0.938190	0.469095	−	0.883148i	$-0.344580\pi$
$601$	23.0000	0.938190	0.469095	+	0.883148i	$0.344580\pi$
$602$	−4.00000	−0.163028
$603$	8.00000	0.325785
$604$	−19.0000	−0.773099
$605$	0	0
$606$	−12.0000	−0.487467
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	0	0
$611$	36.0000	1.45640
$612$	−3.00000	−0.121268
$613$	20.0000	0.807792	0.403896	−	0.914805i	$-0.367656\pi$
$613$	20.0000	0.807792	0.403896	+	0.914805i	$0.367656\pi$
$614$	28.0000	1.12999
$615$	0	0
$616$	−6.00000	−0.241747
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	10.0000	0.402259
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	0	0
$621$	36.0000	1.44463
$622$	0	0
$623$	−3.00000	−0.120192
$624$	8.00000	0.320256
$625$	25.0000	1.00000
$626$	−2.00000	−0.0799361
$627$	−48.0000	−1.91694
$628$	−4.00000	−0.159617
$629$	−6.00000	−0.239236
$630$	0	0
$631$	41.0000	1.63218	0.816092	−	0.577922i	$-0.196136\pi$
$631$	41.0000	1.63218	0.816092	+	0.577922i	$0.196136\pi$
$632$	−11.0000	−0.437557
$633$	−16.0000	−0.635943
$634$	6.00000	0.238290
$635$	0	0
$636$	24.0000	0.951662
$637$	24.0000	0.950915
$638$	0	0
$639$	3.00000	0.118678
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	−9.00000	−0.354650
$645$	0	0
$646$	−12.0000	−0.472134
$647$	−21.0000	−0.825595	−0.412798	−	0.910823i	$-0.635448\pi$
$647$	−21.0000	−0.825595	−0.412798	+	0.910823i	$0.635448\pi$
$648$	11.0000	0.432121
$649$	−36.0000	−1.41312
$650$	−20.0000	−0.784465
$651$	10.0000	0.391931
$652$	14.0000	0.548282
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	−20.0000	−0.782062
$655$	0	0
$656$	−6.00000	−0.234261
$657$	11.0000	0.429151
$658$	−9.00000	−0.350857
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−8.00000	−0.310929
$663$	−24.0000	−0.932083
$664$	0	0
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	0	0
$669$	−46.0000	−1.77846
$670$	0	0
$671$	−12.0000	−0.463255
$672$	−2.00000	−0.0771517
$673$	−43.0000	−1.65753	−0.828764	−	0.559598i	$-0.810955\pi$
$673$	−43.0000	−1.65753	−0.828764	+	0.559598i	$0.810955\pi$
$674$	1.00000	0.0385186
$675$	−20.0000	−0.769800
$676$	3.00000	0.115385
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	−18.0000	−0.691286
$679$	−17.0000	−0.652400
$680$	0	0
$681$	0	0
$682$	30.0000	1.14876
$683$	6.00000	0.229584	0.114792	−	0.993390i	$-0.463380\pi$
$683$	6.00000	0.229584	0.114792	+	0.993390i	$0.463380\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	−13.0000	−0.496342
$687$	−40.0000	−1.52610
$688$	−4.00000	−0.152499
$689$	48.0000	1.82865
$690$	0	0
$691$	50.0000	1.90209	0.951045	−	0.309053i	$-0.100012\pi$
$691$	50.0000	1.90209	0.951045	+	0.309053i	$0.100012\pi$
$692$	6.00000	0.228086
$693$	6.00000	0.227921
$694$	24.0000	0.911028
$695$	0	0
$696$	0	0
$697$	18.0000	0.681799
$698$	16.0000	0.605609
$699$	−54.0000	−2.04247
$700$	5.00000	0.188982
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	16.0000	0.603881
$703$	−8.00000	−0.301726
$704$	−6.00000	−0.226134
$705$	0	0
$706$	9.00000	0.338719
$707$	6.00000	0.225653
$708$	−12.0000	−0.450988
$709$	−16.0000	−0.600893	−0.300446	−	0.953799i	$-0.597136\pi$
$709$	−16.0000	−0.600893	−0.300446	+	0.953799i	$0.597136\pi$
$710$	0	0
$711$	11.0000	0.412532
$712$	−3.00000	−0.112430
$713$	45.0000	1.68526
$714$	6.00000	0.224544
$715$	0	0
$716$	0	0
$717$	18.0000	0.672222
$718$	−15.0000	−0.559795
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	−5.00000	−0.186210
$722$	3.00000	0.111648
$723$	−34.0000	−1.26447
$724$	−16.0000	−0.594635
$725$	0	0
$726$	50.0000	1.85567
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	−4.00000	−0.148250
$729$	13.0000	0.481481
$730$	0	0
$731$	12.0000	0.443836
$732$	−4.00000	−0.147844
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	1.00000	0.0369107
$735$	0	0
$736$	−9.00000	−0.331744
$737$	−48.0000	−1.76810
$738$	6.00000	0.220863
$739$	−10.0000	−0.367856	−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000	−0.367856	−0.183928	+	0.982940i	$0.558881\pi$
$740$	0	0
$741$	−32.0000	−1.17555
$742$	−12.0000	−0.440534
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	10.0000	0.366618
$745$	0	0
$746$	10.0000	0.366126
$747$	0	0
$748$	18.0000	0.658145
$749$	0	0
$750$	0	0
$751$	35.0000	1.27717	0.638584	−	0.769552i	$-0.279520\pi$
$751$	35.0000	1.27717	0.638584	+	0.769552i	$0.279520\pi$
$752$	−9.00000	−0.328196
$753$	60.0000	2.18652
$754$	0	0
$755$	0	0
$756$	−4.00000	−0.145479
$757$	−52.0000	−1.88997	−0.944986	−	0.327111i	$-0.893925\pi$
$757$	−52.0000	−1.88997	−0.944986	+	0.327111i	$0.893925\pi$
$758$	−2.00000	−0.0726433
$759$	108.000	3.92015
$760$	0	0
$761$	−3.00000	−0.108750	−0.0543750	−	0.998521i	$-0.517317\pi$
$761$	−3.00000	−0.108750	−0.0543750	+	0.998521i	$0.517317\pi$
$762$	16.0000	0.579619
$763$	10.0000	0.362024
$764$	3.00000	0.108536
$765$	0	0
$766$	−33.0000	−1.19234
$767$	−24.0000	−0.866590
$768$	−2.00000	−0.0721688
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	−30.0000	−1.08042
$772$	14.0000	0.503871
$773$	36.0000	1.29483	0.647415	−	0.762138i	$-0.275850\pi$
$773$	36.0000	1.29483	0.647415	+	0.762138i	$0.275850\pi$
$774$	4.00000	0.143777
$775$	−25.0000	−0.898027
$776$	−17.0000	−0.610264
$777$	4.00000	0.143499
$778$	−24.0000	−0.860442
$779$	24.0000	0.859889
$780$	0	0
$781$	−18.0000	−0.644091
$782$	27.0000	0.965518
$783$	0	0
$784$	−6.00000	−0.214286
$785$	0	0
$786$	12.0000	0.428026
$787$	26.0000	0.926800	0.463400	−	0.886149i	$-0.346629\pi$
$787$	26.0000	0.926800	0.463400	+	0.886149i	$0.346629\pi$
$788$	−12.0000	−0.427482
$789$	24.0000	0.854423
$790$	0	0
$791$	9.00000	0.320003
$792$	6.00000	0.213201
$793$	−8.00000	−0.284088
$794$	−26.0000	−0.922705
$795$	0	0
$796$	−7.00000	−0.248108
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	8.00000	0.283197
$799$	27.0000	0.955191
$800$	5.00000	0.176777
$801$	3.00000	0.106000
$802$	3.00000	0.105934
$803$	−66.0000	−2.32909
$804$	−16.0000	−0.564276
$805$	0	0
$806$	20.0000	0.704470
$807$	12.0000	0.422420
$808$	6.00000	0.211079
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	0	0
$813$	50.0000	1.75358
$814$	12.0000	0.420600
$815$	0	0
$816$	6.00000	0.210042
$817$	16.0000	0.559769
$818$	10.0000	0.349642
$819$	4.00000	0.139771
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	18.0000	0.627822
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	−5.00000	−0.174183
$825$	−60.0000	−2.08893
$826$	6.00000	0.208767
$827$	−42.0000	−1.46048	−0.730242	−	0.683189i	$-0.760592\pi$
$827$	−42.0000	−1.46048	−0.730242	+	0.683189i	$0.760592\pi$
$828$	9.00000	0.312772
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	−28.0000	−0.971309
$832$	−4.00000	−0.138675
$833$	18.0000	0.623663
$834$	28.0000	0.969561
$835$	0	0
$836$	24.0000	0.830057
$837$	20.0000	0.691301
$838$	12.0000	0.414533
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	0	0
$842$	−26.0000	−0.896019
$843$	18.0000	0.619953
$844$	8.00000	0.275371
$845$	0	0
$846$	9.00000	0.309426
$847$	−25.0000	−0.859010
$848$	−12.0000	−0.412082
$849$	−28.0000	−0.960958
$850$	−15.0000	−0.514496
$851$	18.0000	0.617032
$852$	−6.00000	−0.205557
$853$	−28.0000	−0.958702	−0.479351	−	0.877623i	$-0.659128\pi$
$853$	−28.0000	−0.958702	−0.479351	+	0.877623i	$0.659128\pi$
$854$	2.00000	0.0684386
$855$	0	0
$856$	0	0
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	48.0000	1.63869
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	−3.00000	−0.102180
$863$	45.0000	1.53182	0.765909	−	0.642949i	$-0.222289\pi$
$863$	45.0000	1.53182	0.765909	+	0.642949i	$0.222289\pi$
$864$	−4.00000	−0.136083
$865$	0	0
$866$	19.0000	0.645646
$867$	16.0000	0.543388
$868$	−5.00000	−0.169711
$869$	−66.0000	−2.23890
$870$	0	0
$871$	−32.0000	−1.08428
$872$	10.0000	0.338643
$873$	17.0000	0.575363
$874$	36.0000	1.21772
$875$	0	0
$876$	−22.0000	−0.743311
$877$	−4.00000	−0.135070	−0.0675352	−	0.997717i	$-0.521513\pi$
$877$	−4.00000	−0.135070	−0.0675352	+	0.997717i	$0.521513\pi$
$878$	31.0000	1.04620
$879$	−60.0000	−2.02375
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	6.00000	0.202031
$883$	−34.0000	−1.14419	−0.572096	−	0.820187i	$-0.693869\pi$
$883$	−34.0000	−1.14419	−0.572096	+	0.820187i	$0.693869\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	6.00000	0.201574
$887$	−45.0000	−1.51095	−0.755476	−	0.655176i	$-0.772594\pi$
$887$	−45.0000	−1.51095	−0.755476	+	0.655176i	$0.772594\pi$
$888$	4.00000	0.134231
$889$	−8.00000	−0.268311
$890$	0	0
$891$	66.0000	2.21108
$892$	23.0000	0.770097
$893$	36.0000	1.20469
$894$	−36.0000	−1.20402
$895$	0	0
$896$	1.00000	0.0334077
$897$	72.0000	2.40401
$898$	30.0000	1.00111
$899$	0	0
$900$	−5.00000	−0.166667
$901$	36.0000	1.19933
$902$	−36.0000	−1.19867
$903$	−8.00000	−0.266223
$904$	9.00000	0.299336
$905$	0	0
$906$	−38.0000	−1.26247
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	15.0000	0.496972	0.248486	−	0.968635i	$-0.420067\pi$
$911$	15.0000	0.496972	0.248486	+	0.968635i	$0.420067\pi$
$912$	8.00000	0.264906
$913$	0	0
$914$	22.0000	0.727695
$915$	0	0
$916$	20.0000	0.660819
$917$	−6.00000	−0.198137
$918$	12.0000	0.396059
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	56.0000	1.84526
$922$	18.0000	0.592798
$923$	−12.0000	−0.394985
$924$	−12.0000	−0.394771
$925$	−10.0000	−0.328798
$926$	1.00000	0.0328620
$927$	5.00000	0.164222
$928$	0	0
$929$	45.0000	1.47640	0.738201	−	0.674581i	$-0.235676\pi$
$929$	45.0000	1.47640	0.738201	+	0.674581i	$0.235676\pi$
$930$	0	0
$931$	24.0000	0.786568
$932$	27.0000	0.884414
$933$	0	0
$934$	−18.0000	−0.588978
$935$	0	0
$936$	4.00000	0.130744
$937$	23.0000	0.751377	0.375689	−	0.926746i	$-0.377406\pi$
$937$	23.0000	0.751377	0.375689	+	0.926746i	$0.377406\pi$
$938$	8.00000	0.261209
$939$	−4.00000	−0.130535
$940$	0	0
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	−8.00000	−0.260654
$943$	−54.0000	−1.75848
$944$	6.00000	0.195283
$945$	0	0
$946$	−24.0000	−0.780307
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	−22.0000	−0.714527
$949$	−44.0000	−1.42830
$950$	−20.0000	−0.648886
$951$	12.0000	0.389127
$952$	−3.00000	−0.0972306
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	12.0000	0.388514
$955$	0	0
$956$	−9.00000	−0.291081
$957$	0	0
$958$	15.0000	0.484628
$959$	−9.00000	−0.290625
$960$	0	0
$961$	−6.00000	−0.193548
$962$	8.00000	0.257930
$963$	0	0
$964$	17.0000	0.547533
$965$	0	0
$966$	−18.0000	−0.579141
$967$	−13.0000	−0.418052	−0.209026	−	0.977910i	$-0.567029\pi$
$967$	−13.0000	−0.418052	−0.209026	+	0.977910i	$0.567029\pi$
$968$	−25.0000	−0.803530
$969$	−24.0000	−0.770991
$970$	0	0
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	10.0000	0.320750
$973$	−14.0000	−0.448819
$974$	−20.0000	−0.640841
$975$	−40.0000	−1.28103
$976$	2.00000	0.0640184
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	28.0000	0.895341
$979$	−18.0000	−0.575282
$980$	0	0
$981$	−10.0000	−0.319275
$982$	−6.00000	−0.191468
$983$	−15.0000	−0.478426	−0.239213	−	0.970967i	$-0.576889\pi$
$983$	−15.0000	−0.478426	−0.239213	+	0.970967i	$0.576889\pi$
$984$	−12.0000	−0.382546
$985$	0	0
$986$	0	0
$987$	−18.0000	−0.572946
$988$	16.0000	0.509028
$989$	−36.0000	−1.14473
$990$	0	0
$991$	56.0000	1.77890	0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000	1.77890	0.889449	+	0.457034i	$0.151088\pi$
$992$	−5.00000	−0.158750
$993$	−16.0000	−0.507745
$994$	3.00000	0.0951542
$995$	0	0
$996$	0	0
$997$	32.0000	1.01345	0.506725	−	0.862108i	$-0.330856\pi$
$997$	32.0000	1.01345	0.506725	+	0.862108i	$0.330856\pi$
$998$	34.0000	1.07625
$999$	8.00000	0.253109

Display

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a_n

instead) (See

a_n

instead) Display

a_n

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n

up to: 50 250 1000 (See only

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a_p

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a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1682.2.a.b.1.1	✓	1
29.12	odd	4		1682.2.b.b.1681.1		2
29.17	odd	4		1682.2.b.b.1681.2		2
29.28	even	2		1682.2.a.i.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1682.2.a.b.1.1	✓	1	1.1	even	1	trivial
1682.2.a.i.1.1	yes	1	29.28	even	2
1682.2.b.b.1681.1		2	29.12	odd	4
1682.2.b.b.1681.2		2	29.17	odd	4

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

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Embedding invariants

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1682.2.a.b.1.1

Level	$1682$
Weight	$2$
Character	1682.1
Self dual	yes
Analytic conductor	$13.431$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$