LMFDB - Embedded newform 1078.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1078,2,Mod(1,1078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, 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("1078.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1078 = 2 \cdot 7^{2} \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1078.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$8.60787333789$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1078.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} +2.00000 q^{18} -2.00000 q^{19} +1.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} -1.00000 q^{26} +5.00000 q^{27} +9.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} -6.00000 q^{34} -2.00000 q^{36} +2.00000 q^{37} +2.00000 q^{38} -1.00000 q^{39} +6.00000 q^{41} -4.00000 q^{43} -1.00000 q^{44} +6.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} +5.00000 q^{50} -6.00000 q^{51} +1.00000 q^{52} -5.00000 q^{54} +2.00000 q^{57} -9.00000 q^{58} +3.00000 q^{59} -11.0000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -1.00000 q^{66} +11.0000 q^{67} +6.00000 q^{68} +6.00000 q^{69} +2.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} +5.00000 q^{75} -2.00000 q^{76} +1.00000 q^{78} +5.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +6.00000 q^{83} +4.00000 q^{86} -9.00000 q^{87} +1.00000 q^{88} +18.0000 q^{89} -6.00000 q^{92} -4.00000 q^{93} -6.00000 q^{94} +1.00000 q^{96} +13.0000 q^{97} +2.00000 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
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	2.00000	0.471405
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	1.00000	0.213201
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	1.00000	0.204124
$25$	−5.00000	−1.00000
$26$	−1.00000	−0.196116
$27$	5.00000	0.962250
$28$	0	0
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	1.00000	0.174078
$34$	−6.00000	−1.02899
$35$	0	0
$36$	−2.00000	−0.333333
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	2.00000	0.324443
$39$	−1.00000	−0.160128
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−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
$45$	0	0
$46$	6.00000	0.884652
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	5.00000	0.707107
$51$	−6.00000	−0.840168
$52$	1.00000	0.138675
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	−5.00000	−0.680414
$55$	0	0
$56$	0	0
$57$	2.00000	0.264906
$58$	−9.00000	−1.18176
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	−11.0000	−1.40841	−0.704203	−	0.709999i	$-0.748695\pi$
$61$	−11.0000	−1.40841	−0.704203	+	0.709999i	$0.748695\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	11.0000	1.34386	0.671932	−	0.740613i	$-0.265465\pi$
$67$	11.0000	1.34386	0.671932	+	0.740613i	$0.265465\pi$
$68$	6.00000	0.727607
$69$	6.00000	0.722315
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	2.00000	0.235702
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−2.00000	−0.232495
$75$	5.00000	0.577350
$76$	−2.00000	−0.229416
$77$	0	0
$78$	1.00000	0.113228
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	−9.00000	−0.964901
$88$	1.00000	0.106600
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	0	0
$91$	0	0
$92$	−6.00000	−0.625543
$93$	−4.00000	−0.414781
$94$	−6.00000	−0.618853
$95$	0	0
$96$	1.00000	0.102062
$97$	13.0000	1.31995	0.659975	−	0.751288i	$-0.270567\pi$
$97$	13.0000	1.31995	0.659975	+	0.751288i	$0.270567\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	−5.00000	−0.500000
$101$	15.0000	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$102$	6.00000	0.594089
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	5.00000	0.481125
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	−2.00000	−0.187317
$115$	0	0
$116$	9.00000	0.835629
$117$	−2.00000	−0.184900
$118$	−3.00000	−0.276172
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	11.0000	0.995893
$123$	−6.00000	−0.541002
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	−1.00000	−0.0883883
$129$	4.00000	0.352180
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	−11.0000	−0.950255
$135$	0	0
$136$	−6.00000	−0.514496
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	−6.00000	−0.510754
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	−2.00000	−0.166667
$145$	0	0
$146$	2.00000	0.165521
$147$	0	0
$148$	2.00000	0.164399
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−5.00000	−0.408248
$151$	−19.0000	−1.54620	−0.773099	−	0.634285i	$-0.781294\pi$
$151$	−19.0000	−1.54620	−0.773099	+	0.634285i	$0.781294\pi$
$152$	2.00000	0.162221
$153$	−12.0000	−0.970143
$154$	0	0
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	−5.00000	−0.397779
$159$	0	0
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	17.0000	1.33154	0.665771	−	0.746156i	$-0.268103\pi$
$163$	17.0000	1.33154	0.665771	+	0.746156i	$0.268103\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−6.00000	−0.465690
$167$	−3.00000	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	4.00000	0.305888
$172$	−4.00000	−0.304997
$173$	21.0000	1.59660	0.798300	−	0.602260i	$-0.205733\pi$
$173$	21.0000	1.59660	0.798300	+	0.602260i	$0.205733\pi$
$174$	9.00000	0.682288
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−3.00000	−0.225494
$178$	−18.0000	−1.34916
$179$	−15.0000	−1.12115	−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000	−1.12115	−0.560576	+	0.828103i	$0.689420\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	11.0000	0.813143
$184$	6.00000	0.442326
$185$	0	0
$186$	4.00000	0.293294
$187$	−6.00000	−0.438763
$188$	6.00000	0.437595
$189$	0	0
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	−1.00000	−0.0721688
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−13.0000	−0.933346
$195$	0	0
$196$	0	0
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	−2.00000	−0.142134
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	5.00000	0.353553
$201$	−11.0000	−0.775880
$202$	−15.0000	−1.05540
$203$	0	0
$204$	−6.00000	−0.420084
$205$	0	0
$206$	−16.0000	−1.11477
$207$	12.0000	0.834058
$208$	1.00000	0.0693375
$209$	2.00000	0.138343
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	0	0
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	−5.00000	−0.340207
$217$	0	0
$218$	−2.00000	−0.135457
$219$	2.00000	0.135147
$220$	0	0
$221$	6.00000	0.403604
$222$	2.00000	0.134231
$223$	−26.0000	−1.74109	−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000	−1.74109	−0.870544	+	0.492090i	$0.836233\pi$
$224$	0	0
$225$	10.0000	0.666667
$226$	−9.00000	−0.598671
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	2.00000	0.132453
$229$	16.0000	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$230$	0	0
$231$	0	0
$232$	−9.00000	−0.590879
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	3.00000	0.195283
$237$	−5.00000	−0.324785
$238$	0	0
$239$	−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
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	−1.00000	−0.0642824
$243$	−16.0000	−1.02640
$244$	−11.0000	−0.704203
$245$	0	0
$246$	6.00000	0.382546
$247$	−2.00000	−0.127257
$248$	−4.00000	−0.254000
$249$	−6.00000	−0.380235
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	7.00000	0.439219
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	0	0
$261$	−18.0000	−1.11417
$262$	18.0000	1.11204
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	−1.00000	−0.0615457
$265$	0	0
$266$	0	0
$267$	−18.0000	−1.10158
$268$	11.0000	0.671932
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	0	0
$271$	−29.0000	−1.76162	−0.880812	−	0.473466i	$-0.843003\pi$
$271$	−29.0000	−1.76162	−0.880812	+	0.473466i	$0.843003\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	9.00000	0.543710
$275$	5.00000	0.301511
$276$	6.00000	0.361158
$277$	29.0000	1.74244	0.871221	−	0.490892i	$-0.163329\pi$
$277$	29.0000	1.74244	0.871221	+	0.490892i	$0.163329\pi$
$278$	8.00000	0.479808
$279$	−8.00000	−0.478947
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	6.00000	0.357295
$283$	10.0000	0.594438	0.297219	−	0.954809i	$-0.403941\pi$
$283$	10.0000	0.594438	0.297219	+	0.954809i	$0.403941\pi$
$284$	0	0
$285$	0	0
$286$	1.00000	0.0591312
$287$	0	0
$288$	2.00000	0.117851
$289$	19.0000	1.11765
$290$	0	0
$291$	−13.0000	−0.762073
$292$	−2.00000	−0.117041
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	0	0
$295$	0	0
$296$	−2.00000	−0.116248
$297$	−5.00000	−0.290129
$298$	−6.00000	−0.347571
$299$	−6.00000	−0.346989
$300$	5.00000	0.288675
$301$	0	0
$302$	19.0000	1.09333
$303$	−15.0000	−0.861727
$304$	−2.00000	−0.114708
$305$	0	0
$306$	12.0000	0.685994
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	1.00000	0.0566139
$313$	−17.0000	−0.960897	−0.480448	−	0.877023i	$-0.659526\pi$
$313$	−17.0000	−0.960897	−0.480448	+	0.877023i	$0.659526\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	5.00000	0.281272
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	0	0
$319$	−9.00000	−0.503903
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	−12.0000	−0.667698
$324$	1.00000	0.0555556
$325$	−5.00000	−0.277350
$326$	−17.0000	−0.941543
$327$	−2.00000	−0.110600
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	35.0000	1.92377	0.961887	−	0.273447i	$-0.0881639\pi$
$331$	35.0000	1.92377	0.961887	+	0.273447i	$0.0881639\pi$
$332$	6.00000	0.329293
$333$	−4.00000	−0.219199
$334$	3.00000	0.164153
$335$	0	0
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	12.0000	0.652714
$339$	−9.00000	−0.488813
$340$	0	0
$341$	−4.00000	−0.216612
$342$	−4.00000	−0.216295
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	−21.0000	−1.12897
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	−9.00000	−0.482451
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	1.00000	0.0533002
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	3.00000	0.159448
$355$	0	0
$356$	18.0000	0.953998
$357$	0	0
$358$	15.0000	0.792775
$359$	−3.00000	−0.158334	−0.0791670	−	0.996861i	$-0.525226\pi$
$359$	−3.00000	−0.158334	−0.0791670	+	0.996861i	$0.525226\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	2.00000	0.105118
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	0	0
$366$	−11.0000	−0.574979
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	−6.00000	−0.312772
$369$	−12.0000	−0.624695
$370$	0	0
$371$	0	0
$372$	−4.00000	−0.207390
$373$	−31.0000	−1.60512	−0.802560	−	0.596572i	$-0.796529\pi$
$373$	−31.0000	−1.60512	−0.802560	+	0.596572i	$0.796529\pi$
$374$	6.00000	0.310253
$375$	0	0
$376$	−6.00000	−0.309426
$377$	9.00000	0.463524
$378$	0	0
$379$	23.0000	1.18143	0.590715	−	0.806880i	$-0.298846\pi$
$379$	23.0000	1.18143	0.590715	+	0.806880i	$0.298846\pi$
$380$	0	0
$381$	7.00000	0.358621
$382$	−6.00000	−0.306987
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	8.00000	0.406663
$388$	13.0000	0.659975
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	−36.0000	−1.82060
$392$	0	0
$393$	18.0000	0.907980
$394$	3.00000	0.151138
$395$	0	0
$396$	2.00000	0.100504
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	14.0000	0.701757
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−33.0000	−1.64794	−0.823971	−	0.566632i	$-0.808246\pi$
$401$	−33.0000	−1.64794	−0.823971	+	0.566632i	$0.808246\pi$
$402$	11.0000	0.548630
$403$	4.00000	0.199254
$404$	15.0000	0.746278
$405$	0	0
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	6.00000	0.297044
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	9.00000	0.443937
$412$	16.0000	0.788263
$413$	0	0
$414$	−12.0000	−0.589768
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	8.00000	0.391762
$418$	−2.00000	−0.0978232
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	10.0000	0.486792
$423$	−12.0000	−0.583460
$424$	0	0
$425$	−30.0000	−1.45521
$426$	0	0
$427$	0	0
$428$	12.0000	0.580042
$429$	1.00000	0.0482805
$430$	0	0
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	5.00000	0.240563
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	12.0000	0.574038
$438$	−2.00000	−0.0955637
$439$	1.00000	0.0477274	0.0238637	−	0.999715i	$-0.492403\pi$
$439$	1.00000	0.0477274	0.0238637	+	0.999715i	$0.492403\pi$
$440$	0	0
$441$	0	0
$442$	−6.00000	−0.285391
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	26.0000	1.23114
$447$	−6.00000	−0.283790
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	−10.0000	−0.471405
$451$	−6.00000	−0.282529
$452$	9.00000	0.423324
$453$	19.0000	0.892698
$454$	18.0000	0.844782
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−16.0000	−0.747631
$459$	30.0000	1.40028
$460$	0	0
$461$	−21.0000	−0.978068	−0.489034	−	0.872265i	$-0.662651\pi$
$461$	−21.0000	−0.978068	−0.489034	+	0.872265i	$0.662651\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	9.00000	0.417815
$465$	0	0
$466$	6.00000	0.277945
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	0	0
$471$	−4.00000	−0.184310
$472$	−3.00000	−0.138086
$473$	4.00000	0.183920
$474$	5.00000	0.229658
$475$	10.0000	0.458831
$476$	0	0
$477$	0	0
$478$	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
$481$	2.00000	0.0911922
$482$	26.0000	1.18427
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	16.0000	0.725775
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	11.0000	0.497947
$489$	−17.0000	−0.768767
$490$	0	0
$491$	42.0000	1.89543	0.947717	−	0.319113i	$-0.103385\pi$
$491$	42.0000	1.89543	0.947717	+	0.319113i	$0.103385\pi$
$492$	−6.00000	−0.270501
$493$	54.0000	2.43204
$494$	2.00000	0.0899843
$495$	0	0
$496$	4.00000	0.179605
$497$	0	0
$498$	6.00000	0.268866
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	0	0
$501$	3.00000	0.134030
$502$	−12.0000	−0.535586
$503$	33.0000	1.47140	0.735699	−	0.677309i	$-0.236854\pi$
$503$	33.0000	1.47140	0.735699	+	0.677309i	$0.236854\pi$
$504$	0	0
$505$	0	0
$506$	−6.00000	−0.266733
$507$	12.0000	0.532939
$508$	−7.00000	−0.310575
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−10.0000	−0.441511
$514$	3.00000	0.132324
$515$	0	0
$516$	4.00000	0.176090
$517$	−6.00000	−0.263880
$518$	0	0
$519$	−21.0000	−0.921798
$520$	0	0
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	18.0000	0.787839
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	−18.0000	−0.786334
$525$	0	0
$526$	9.00000	0.392419
$527$	24.0000	1.04546
$528$	1.00000	0.0435194
$529$	13.0000	0.565217
$530$	0	0
$531$	−6.00000	−0.260378
$532$	0	0
$533$	6.00000	0.259889
$534$	18.0000	0.778936
$535$	0	0
$536$	−11.0000	−0.475128
$537$	15.0000	0.647298
$538$	12.0000	0.517357
$539$	0	0
$540$	0	0
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	29.0000	1.24566
$543$	2.00000	0.0858282
$544$	−6.00000	−0.257248
$545$	0	0
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	−9.00000	−0.384461
$549$	22.0000	0.938937
$550$	−5.00000	−0.213201
$551$	−18.0000	−0.766826
$552$	−6.00000	−0.255377
$553$	0	0
$554$	−29.0000	−1.23209
$555$	0	0
$556$	−8.00000	−0.339276
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	8.00000	0.338667
$559$	−4.00000	−0.169182
$560$	0	0
$561$	6.00000	0.253320
$562$	−18.0000	−0.759284
$563$	18.0000	0.758610	0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000	0.758610	0.379305	+	0.925272i	$0.376163\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	−10.0000	−0.420331
$567$	0	0
$568$	0	0
$569$	−36.0000	−1.50920	−0.754599	−	0.656186i	$-0.772169\pi$
$569$	−36.0000	−1.50920	−0.754599	+	0.656186i	$0.772169\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	−1.00000	−0.0418121
$573$	−6.00000	−0.250654
$574$	0	0
$575$	30.0000	1.25109
$576$	−2.00000	−0.0833333
$577$	7.00000	0.291414	0.145707	−	0.989328i	$-0.453454\pi$
$577$	7.00000	0.291414	0.145707	+	0.989328i	$0.453454\pi$
$578$	−19.0000	−0.790296
$579$	−14.0000	−0.581820
$580$	0	0
$581$	0	0
$582$	13.0000	0.538867
$583$	0	0
$584$	2.00000	0.0827606
$585$	0	0
$586$	−30.0000	−1.23929
$587$	−9.00000	−0.371470	−0.185735	−	0.982600i	$-0.559467\pi$
$587$	−9.00000	−0.371470	−0.185735	+	0.982600i	$0.559467\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	3.00000	0.123404
$592$	2.00000	0.0821995
$593$	−36.0000	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	5.00000	0.205152
$595$	0	0
$596$	6.00000	0.245770
$597$	14.0000	0.572982
$598$	6.00000	0.245358
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	−5.00000	−0.204124
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	−22.0000	−0.895909
$604$	−19.0000	−0.773099
$605$	0	0
$606$	15.0000	0.609333
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	0	0
$611$	6.00000	0.242734
$612$	−12.0000	−0.485071
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	0	0
$617$	21.0000	0.845428	0.422714	−	0.906263i	$-0.361077\pi$
$617$	21.0000	0.845428	0.422714	+	0.906263i	$0.361077\pi$
$618$	16.0000	0.643614
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	−30.0000	−1.20386
$622$	24.0000	0.962312
$623$	0	0
$624$	−1.00000	−0.0400320
$625$	25.0000	1.00000
$626$	17.0000	0.679457
$627$	−2.00000	−0.0798723
$628$	4.00000	0.159617
$629$	12.0000	0.478471
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	−5.00000	−0.198889
$633$	10.0000	0.397464
$634$	−12.0000	−0.476581
$635$	0	0
$636$	0	0
$637$	0	0
$638$	9.00000	0.356313
$639$	0	0
$640$	0	0
$641$	9.00000	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$642$	12.0000	0.473602
$643$	−5.00000	−0.197181	−0.0985904	−	0.995128i	$-0.531433\pi$
$643$	−5.00000	−0.197181	−0.0985904	+	0.995128i	$0.531433\pi$
$644$	0	0
$645$	0	0
$646$	12.0000	0.472134
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	−1.00000	−0.0392837
$649$	−3.00000	−0.117760
$650$	5.00000	0.196116
$651$	0	0
$652$	17.0000	0.665771
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	2.00000	0.0782062
$655$	0	0
$656$	6.00000	0.234261
$657$	4.00000	0.156055
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	−35.0000	−1.36031
$663$	−6.00000	−0.233021
$664$	−6.00000	−0.232845
$665$	0	0
$666$	4.00000	0.154997
$667$	−54.0000	−2.09089
$668$	−3.00000	−0.116073
$669$	26.0000	1.00522
$670$	0	0
$671$	11.0000	0.424650
$672$	0	0
$673$	−28.0000	−1.07932	−0.539660	−	0.841883i	$-0.681447\pi$
$673$	−28.0000	−1.07932	−0.539660	+	0.841883i	$0.681447\pi$
$674$	−14.0000	−0.539260
$675$	−25.0000	−0.962250
$676$	−12.0000	−0.461538
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	9.00000	0.345643
$679$	0	0
$680$	0	0
$681$	18.0000	0.689761
$682$	4.00000	0.153168
$683$	−21.0000	−0.803543	−0.401771	−	0.915740i	$-0.631605\pi$
$683$	−21.0000	−0.803543	−0.401771	+	0.915740i	$0.631605\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	0	0
$687$	−16.0000	−0.610438
$688$	−4.00000	−0.152499
$689$	0	0
$690$	0	0
$691$	−11.0000	−0.418460	−0.209230	−	0.977866i	$-0.567096\pi$
$691$	−11.0000	−0.418460	−0.209230	+	0.977866i	$0.567096\pi$
$692$	21.0000	0.798300
$693$	0	0
$694$	0	0
$695$	0	0
$696$	9.00000	0.341144
$697$	36.0000	1.36360
$698$	2.00000	0.0757011
$699$	6.00000	0.226941
$700$	0	0
$701$	39.0000	1.47301	0.736505	−	0.676432i	$-0.236475\pi$
$701$	39.0000	1.47301	0.736505	+	0.676432i	$0.236475\pi$
$702$	−5.00000	−0.188713
$703$	−4.00000	−0.150863
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−18.0000	−0.677439
$707$	0	0
$708$	−3.00000	−0.112747
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	−18.0000	−0.674579
$713$	−24.0000	−0.898807
$714$	0	0
$715$	0	0
$716$	−15.0000	−0.560576
$717$	9.00000	0.336111
$718$	3.00000	0.111959
$719$	42.0000	1.56634	0.783168	−	0.621810i	$-0.213603\pi$
$719$	42.0000	1.56634	0.783168	+	0.621810i	$0.213603\pi$
$720$	0	0
$721$	0	0
$722$	15.0000	0.558242
$723$	26.0000	0.966950
$724$	−2.00000	−0.0743294
$725$	−45.0000	−1.67126
$726$	1.00000	0.0371135
$727$	−14.0000	−0.519231	−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000	−0.519231	−0.259616	+	0.965712i	$0.583596\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−24.0000	−0.887672
$732$	11.0000	0.406572
$733$	25.0000	0.923396	0.461698	−	0.887037i	$-0.347240\pi$
$733$	25.0000	0.923396	0.461698	+	0.887037i	$0.347240\pi$
$734$	−10.0000	−0.369107
$735$	0	0
$736$	6.00000	0.221163
$737$	−11.0000	−0.405190
$738$	12.0000	0.441726
$739$	50.0000	1.83928	0.919640	−	0.392763i	$-0.128481\pi$
$739$	50.0000	1.83928	0.919640	+	0.392763i	$0.128481\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	4.00000	0.146647
$745$	0	0
$746$	31.0000	1.13499
$747$	−12.0000	−0.439057
$748$	−6.00000	−0.219382
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	6.00000	0.218797
$753$	−12.0000	−0.437304
$754$	−9.00000	−0.327761
$755$	0	0
$756$	0	0
$757$	−46.0000	−1.67190	−0.835949	−	0.548807i	$-0.815082\pi$
$757$	−46.0000	−1.67190	−0.835949	+	0.548807i	$0.815082\pi$
$758$	−23.0000	−0.835398
$759$	−6.00000	−0.217786
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	−7.00000	−0.253583
$763$	0	0
$764$	6.00000	0.217072
$765$	0	0
$766$	−36.0000	−1.30073
$767$	3.00000	0.108324
$768$	−1.00000	−0.0360844
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	3.00000	0.108042
$772$	14.0000	0.503871
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	−8.00000	−0.287554
$775$	−20.0000	−0.718421
$776$	−13.0000	−0.466673
$777$	0	0
$778$	−12.0000	−0.430221
$779$	−12.0000	−0.429945
$780$	0	0
$781$	0	0
$782$	36.0000	1.28736
$783$	45.0000	1.60817
$784$	0	0
$785$	0	0
$786$	−18.0000	−0.642039
$787$	−14.0000	−0.499046	−0.249523	−	0.968369i	$-0.580274\pi$
$787$	−14.0000	−0.499046	−0.249523	+	0.968369i	$0.580274\pi$
$788$	−3.00000	−0.106871
$789$	9.00000	0.320408
$790$	0	0
$791$	0	0
$792$	−2.00000	−0.0710669
$793$	−11.0000	−0.390621
$794$	−22.0000	−0.780751
$795$	0	0
$796$	−14.0000	−0.496217
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	36.0000	1.27359
$800$	5.00000	0.176777
$801$	−36.0000	−1.27200
$802$	33.0000	1.16527
$803$	2.00000	0.0705785
$804$	−11.0000	−0.387940
$805$	0	0
$806$	−4.00000	−0.140894
$807$	12.0000	0.422420
$808$	−15.0000	−0.527698
$809$	−48.0000	−1.68759	−0.843795	−	0.536666i	$-0.819684\pi$
$809$	−48.0000	−1.68759	−0.843795	+	0.536666i	$0.819684\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	29.0000	1.01707
$814$	2.00000	0.0701000
$815$	0	0
$816$	−6.00000	−0.210042
$817$	8.00000	0.279885
$818$	−4.00000	−0.139857
$819$	0	0
$820$	0	0
$821$	−15.0000	−0.523504	−0.261752	−	0.965135i	$-0.584300\pi$
$821$	−15.0000	−0.523504	−0.261752	+	0.965135i	$0.584300\pi$
$822$	−9.00000	−0.313911
$823$	−10.0000	−0.348578	−0.174289	−	0.984695i	$-0.555763\pi$
$823$	−10.0000	−0.348578	−0.174289	+	0.984695i	$0.555763\pi$
$824$	−16.0000	−0.557386
$825$	−5.00000	−0.174078
$826$	0	0
$827$	30.0000	1.04320	0.521601	−	0.853189i	$-0.325335\pi$
$827$	30.0000	1.04320	0.521601	+	0.853189i	$0.325335\pi$
$828$	12.0000	0.417029
$829$	52.0000	1.80603	0.903017	−	0.429604i	$-0.141347\pi$
$829$	52.0000	1.80603	0.903017	+	0.429604i	$0.141347\pi$
$830$	0	0
$831$	−29.0000	−1.00600
$832$	1.00000	0.0346688
$833$	0	0
$834$	−8.00000	−0.277017
$835$	0	0
$836$	2.00000	0.0691714
$837$	20.0000	0.691301
$838$	−12.0000	−0.414533
$839$	−6.00000	−0.207143	−0.103572	−	0.994622i	$-0.533027\pi$
$839$	−6.00000	−0.207143	−0.103572	+	0.994622i	$0.533027\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	28.0000	0.964944
$843$	−18.0000	−0.619953
$844$	−10.0000	−0.344214
$845$	0	0
$846$	12.0000	0.412568
$847$	0	0
$848$	0	0
$849$	−10.0000	−0.343199
$850$	30.0000	1.02899
$851$	−12.0000	−0.411355
$852$	0	0
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−36.0000	−1.22974	−0.614868	−	0.788630i	$-0.710791\pi$
$857$	−36.0000	−1.22974	−0.614868	+	0.788630i	$0.710791\pi$
$858$	−1.00000	−0.0341394
$859$	37.0000	1.26242	0.631212	−	0.775610i	$-0.282558\pi$
$859$	37.0000	1.26242	0.631212	+	0.775610i	$0.282558\pi$
$860$	0	0
$861$	0	0
$862$	3.00000	0.102180
$863$	30.0000	1.02121	0.510606	−	0.859815i	$-0.329421\pi$
$863$	30.0000	1.02121	0.510606	+	0.859815i	$0.329421\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	−34.0000	−1.15537
$867$	−19.0000	−0.645274
$868$	0	0
$869$	−5.00000	−0.169613
$870$	0	0
$871$	11.0000	0.372721
$872$	−2.00000	−0.0677285
$873$	−26.0000	−0.879967
$874$	−12.0000	−0.405906
$875$	0	0
$876$	2.00000	0.0675737
$877$	17.0000	0.574049	0.287025	−	0.957923i	$-0.407334\pi$
$877$	17.0000	0.574049	0.287025	+	0.957923i	$0.407334\pi$
$878$	−1.00000	−0.0337484
$879$	−30.0000	−1.01187
$880$	0	0
$881$	−33.0000	−1.11180	−0.555899	−	0.831250i	$-0.687626\pi$
$881$	−33.0000	−1.11180	−0.555899	+	0.831250i	$0.687626\pi$
$882$	0	0
$883$	29.0000	0.975928	0.487964	−	0.872864i	$-0.337740\pi$
$883$	29.0000	0.975928	0.487964	+	0.872864i	$0.337740\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	−12.0000	−0.403148
$887$	−3.00000	−0.100730	−0.0503651	−	0.998731i	$-0.516038\pi$
$887$	−3.00000	−0.100730	−0.0503651	+	0.998731i	$0.516038\pi$
$888$	2.00000	0.0671156
$889$	0	0
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	−26.0000	−0.870544
$893$	−12.0000	−0.401565
$894$	6.00000	0.200670
$895$	0	0
$896$	0	0
$897$	6.00000	0.200334
$898$	18.0000	0.600668
$899$	36.0000	1.20067
$900$	10.0000	0.333333
$901$	0	0
$902$	6.00000	0.199778
$903$	0	0
$904$	−9.00000	−0.299336
$905$	0	0
$906$	−19.0000	−0.631233
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−18.0000	−0.597351
$909$	−30.0000	−0.995037
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	2.00000	0.0662266
$913$	−6.00000	−0.198571
$914$	22.0000	0.727695
$915$	0	0
$916$	16.0000	0.528655
$917$	0	0
$918$	−30.0000	−0.990148
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	21.0000	0.691598
$923$	0	0
$924$	0	0
$925$	−10.0000	−0.328798
$926$	22.0000	0.722965
$927$	−32.0000	−1.05102
$928$	−9.00000	−0.295439
$929$	−39.0000	−1.27955	−0.639774	−	0.768563i	$-0.720972\pi$
$929$	−39.0000	−1.27955	−0.639774	+	0.768563i	$0.720972\pi$
$930$	0	0
$931$	0	0
$932$	−6.00000	−0.196537
$933$	24.0000	0.785725
$934$	12.0000	0.392652
$935$	0	0
$936$	2.00000	0.0653720
$937$	−8.00000	−0.261349	−0.130674	−	0.991425i	$-0.541714\pi$
$937$	−8.00000	−0.261349	−0.130674	+	0.991425i	$0.541714\pi$
$938$	0	0
$939$	17.0000	0.554774
$940$	0	0
$941$	−15.0000	−0.488986	−0.244493	−	0.969651i	$-0.578622\pi$
$941$	−15.0000	−0.488986	−0.244493	+	0.969651i	$0.578622\pi$
$942$	4.00000	0.130327
$943$	−36.0000	−1.17232
$944$	3.00000	0.0976417
$945$	0	0
$946$	−4.00000	−0.130051
$947$	24.0000	0.779895	0.389948	−	0.920837i	$-0.372493\pi$
$947$	24.0000	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$948$	−5.00000	−0.162392
$949$	−2.00000	−0.0649227
$950$	−10.0000	−0.324443
$951$	−12.0000	−0.389127
$952$	0	0
$953$	−36.0000	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$954$	0	0
$955$	0	0
$956$	−9.00000	−0.291081
$957$	9.00000	0.290929
$958$	15.0000	0.484628
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−2.00000	−0.0644826
$963$	−24.0000	−0.773389
$964$	−26.0000	−0.837404
$965$	0	0
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	−1.00000	−0.0321412
$969$	12.0000	0.385496
$970$	0	0
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	−16.0000	−0.513200
$973$	0	0
$974$	−20.0000	−0.640841
$975$	5.00000	0.160128
$976$	−11.0000	−0.352101
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	17.0000	0.543600
$979$	−18.0000	−0.575282
$980$	0	0
$981$	−4.00000	−0.127710
$982$	−42.0000	−1.34027
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	−54.0000	−1.71971
$987$	0	0
$988$	−2.00000	−0.0636285
$989$	24.0000	0.763156
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	−4.00000	−0.127000
$993$	−35.0000	−1.11069
$994$	0	0
$995$	0	0
$996$	−6.00000	−0.190117
$997$	34.0000	1.07679	0.538395	−	0.842692i	$-0.319031\pi$
$997$	34.0000	1.07679	0.538395	+	0.842692i	$0.319031\pi$
$998$	40.0000	1.26618
$999$	10.0000	0.316386

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1078.2.a.c.1.1		1
3.2	odd	2		9702.2.a.bs.1.1		1
4.3	odd	2		8624.2.a.u.1.1		1
7.2	even	3		1078.2.e.k.67.1		2
7.3	odd	6		154.2.e.c.23.1	✓	2
7.4	even	3		1078.2.e.k.177.1		2
7.5	odd	6		154.2.e.c.67.1	yes	2
7.6	odd	2		1078.2.a.e.1.1		1
21.5	even	6		1386.2.k.e.991.1		2
21.17	even	6		1386.2.k.e.793.1		2
21.20	even	2		9702.2.a.br.1.1		1
28.3	even	6		1232.2.q.d.177.1		2
28.19	even	6		1232.2.q.d.529.1		2
28.27	even	2		8624.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.e.c.23.1	✓	2	7.3	odd	6
154.2.e.c.67.1	yes	2	7.5	odd	6
1078.2.a.c.1.1		1	1.1	even	1	trivial
1078.2.a.e.1.1		1	7.6	odd	2
1078.2.e.k.67.1		2	7.2	even	3
1078.2.e.k.177.1		2	7.4	even	3
1232.2.q.d.177.1		2	28.3	even	6
1232.2.q.d.529.1		2	28.19	even	6
1386.2.k.e.793.1		2	21.17	even	6
1386.2.k.e.991.1		2	21.5	even	6
8624.2.a.k.1.1		1	28.27	even	2
8624.2.a.u.1.1		1	4.3	odd	2
9702.2.a.br.1.1		1	21.20	even	2
9702.2.a.bs.1.1		1	3.2	odd	2

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

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	1078.2.a.c.1.1

Level	$1078$
Weight	$2$
Character	1078.1
Self dual	yes
Analytic conductor	$8.608$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$