LMFDB - Embedded newform 8993.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8993,2,Mod(1,8993)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8993, 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("8993.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8993 = 17 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8993.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:	$71.8094665377$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 17)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8993.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} -4.00000 q^{7} +3.00000 q^{8} -3.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} -4.00000 q^{7} +3.00000 q^{8} -3.00000 q^{9} -2.00000 q^{10} -2.00000 q^{13} +4.00000 q^{14} -1.00000 q^{16} -1.00000 q^{17} +3.00000 q^{18} +4.00000 q^{19} -2.00000 q^{20} -1.00000 q^{25} +2.00000 q^{26} +4.00000 q^{28} +6.00000 q^{29} +4.00000 q^{31} -5.00000 q^{32} +1.00000 q^{34} -8.00000 q^{35} +3.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} +6.00000 q^{40} -6.00000 q^{41} -4.00000 q^{43} -6.00000 q^{45} +9.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} -6.00000 q^{53} -12.0000 q^{56} -6.00000 q^{58} -12.0000 q^{59} +10.0000 q^{61} -4.00000 q^{62} +12.0000 q^{63} +7.00000 q^{64} -4.00000 q^{65} -4.00000 q^{67} +1.00000 q^{68} +8.00000 q^{70} -4.00000 q^{71} -9.00000 q^{72} -6.00000 q^{73} -2.00000 q^{74} -4.00000 q^{76} -12.0000 q^{79} -2.00000 q^{80} +9.00000 q^{81} +6.00000 q^{82} +4.00000 q^{83} -2.00000 q^{85} +4.00000 q^{86} -10.0000 q^{89} +6.00000 q^{90} +8.00000 q^{91} +8.00000 q^{95} -2.00000 q^{97} -9.00000 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	−1.00000	−0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	3.00000	1.06066
$9$	−3.00000	−1.00000
$10$	−2.00000	−0.632456
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	3.00000	0.707107
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−1.00000	−0.200000
$26$	2.00000	0.392232
$27$	0	0
$28$	4.00000	0.755929
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	1.00000	0.171499
$35$	−8.00000	−1.35225
$36$	3.00000	0.500000
$37$	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
$39$	0	0
$40$	6.00000	0.948683
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	−6.00000	−0.894427
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	−12.0000	−1.60357
$57$	0	0
$58$	−6.00000	−0.787839
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−4.00000	−0.508001
$63$	12.0000	1.51186
$64$	7.00000	0.875000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	8.00000	0.956183
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	−9.00000	−1.06066
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−4.00000	−0.458831
$77$	0	0
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	−2.00000	−0.223607
$81$	9.00000	1.00000
$82$	6.00000	0.662589
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	4.00000	0.431331
$87$	0	0
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	6.00000	0.632456
$91$	8.00000	0.838628
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−9.00000	−0.909137
$99$	0	0
$100$	1.00000	0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	6.00000	0.582772
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	0	0
$112$	4.00000	0.377964
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	6.00000	0.554700
$118$	12.0000	1.10469
$119$	4.00000	0.366679
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−10.0000	−0.905357
$123$	0	0
$124$	−4.00000	−0.359211
$125$	−12.0000	−1.07331
$126$	−12.0000	−1.06904
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	4.00000	0.350823
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	−16.0000	−1.38738
$134$	4.00000	0.345547
$135$	0	0
$136$	−3.00000	−0.257248
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	8.00000	0.676123
$141$	0	0
$142$	4.00000	0.335673
$143$	0	0
$144$	3.00000	0.250000
$145$	12.0000	0.996546
$146$	6.00000	0.496564
$147$	0	0
$148$	−2.00000	−0.164399
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	12.0000	0.973329
$153$	3.00000	0.242536
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	12.0000	0.954669
$159$	0	0
$160$	−10.0000	−0.790569
$161$	0	0
$162$	−9.00000	−0.707107
$163$	24.0000	1.87983	0.939913	−	0.341415i	$-0.110906\pi$
$163$	24.0000	1.87983	0.939913	+	0.341415i	$0.110906\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−4.00000	−0.309529	−0.154765	−	0.987951i	$-0.549462\pi$
$167$	−4.00000	−0.309529	−0.154765	+	0.987951i	$0.549462\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	2.00000	0.153393
$171$	−12.0000	−0.917663
$172$	4.00000	0.304997
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	0	0
$178$	10.0000	0.749532
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	6.00000	0.447214
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−8.00000	−0.592999
$183$	0	0
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	−8.00000	−0.580381
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−9.00000	−0.642857
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	−3.00000	−0.212132
$201$	0	0
$202$	10.0000	0.703598
$203$	−24.0000	−1.68447
$204$	0	0
$205$	−12.0000	−0.838116
$206$	8.00000	0.557386
$207$	0	0
$208$	2.00000	0.138675
$209$	0	0
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	8.00000	0.546869
$215$	−8.00000	−0.545595
$216$	0	0
$217$	−16.0000	−1.08615
$218$	6.00000	0.406371
$219$	0	0
$220$	0	0
$221$	2.00000	0.134535
$222$	0	0
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	20.0000	1.33631
$225$	3.00000	0.200000
$226$	−14.0000	−0.931266
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	18.0000	1.18176
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	12.0000	0.781133
$237$	0	0
$238$	−4.00000	−0.259281
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	11.0000	0.707107
$243$	0	0
$244$	−10.0000	−0.640184
$245$	18.0000	1.14998
$246$	0	0
$247$	−8.00000	−0.509028
$248$	12.0000	0.762001
$249$	0	0
$250$	12.0000	0.758947
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	−12.0000	−0.755929
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	4.00000	0.248069
$261$	−18.0000	−1.11417
$262$	−16.0000	−0.988483
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	16.0000	0.981023
$267$	0	0
$268$	4.00000	0.244339
$269$	22.0000	1.34136	0.670682	−	0.741745i	$-0.266002\pi$
$269$	22.0000	1.34136	0.670682	+	0.741745i	$0.266002\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	8.00000	0.479808
$279$	−12.0000	−0.718421
$280$	−24.0000	−1.43427
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	4.00000	0.237356
$285$	0	0
$286$	0	0
$287$	24.0000	1.41668
$288$	15.0000	0.883883
$289$	1.00000	0.0588235
$290$	−12.0000	−0.704664
$291$	0	0
$292$	6.00000	0.351123
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	6.00000	0.348743
$297$	0	0
$298$	−10.0000	−0.579284
$299$	0	0
$300$	0	0
$301$	16.0000	0.922225
$302$	16.0000	0.920697
$303$	0	0
$304$	−4.00000	−0.229416
$305$	20.0000	1.14520
$306$	−3.00000	−0.171499
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	0	0
$310$	−8.00000	−0.454369
$311$	28.0000	1.58773	0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000	1.58773	0.793867	+	0.608091i	$0.208065\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	−2.00000	−0.112867
$315$	24.0000	1.35225
$316$	12.0000	0.675053
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	0	0
$319$	0	0
$320$	14.0000	0.782624
$321$	0	0
$322$	0	0
$323$	−4.00000	−0.222566
$324$	−9.00000	−0.500000
$325$	2.00000	0.110940
$326$	−24.0000	−1.32924
$327$	0	0
$328$	−18.0000	−0.993884
$329$	0	0
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−4.00000	−0.219529
$333$	−6.00000	−0.328798
$334$	4.00000	0.218870
$335$	−8.00000	−0.437087
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	2.00000	0.108465
$341$	0	0
$342$	12.0000	0.648886
$343$	−8.00000	−0.431959
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−22.0000	−1.18273
$347$	32.0000	1.71785	0.858925	−	0.512101i	$-0.171133\pi$
$347$	32.0000	1.71785	0.858925	+	0.512101i	$0.171133\pi$
$348$	0	0
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	0	0
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	10.0000	0.529999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	−18.0000	−0.948683
$361$	−3.00000	−0.157895
$362$	−2.00000	−0.105118
$363$	0	0
$364$	−8.00000	−0.419314
$365$	−12.0000	−0.628109
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	18.0000	0.937043
$370$	−4.00000	−0.207950
$371$	24.0000	1.24602
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	−8.00000	−0.410391
$381$	0	0
$382$	−16.0000	−0.818631
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	0	0
$386$	−2.00000	−0.101797
$387$	12.0000	0.609994
$388$	2.00000	0.101535
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	0	0
$392$	27.0000	1.36371
$393$	0	0
$394$	18.0000	0.906827
$395$	−24.0000	−1.20757
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	−20.0000	−1.00251
$399$	0	0
$400$	1.00000	0.0500000
$401$	14.0000	0.699127	0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000	0.699127	0.349563	+	0.936913i	$0.386330\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	10.0000	0.497519
$405$	18.0000	0.894427
$406$	24.0000	1.19110
$407$	0	0
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	12.0000	0.592638
$411$	0	0
$412$	8.00000	0.394132
$413$	48.0000	2.36193
$414$	0	0
$415$	8.00000	0.392705
$416$	10.0000	0.490290
$417$	0	0
$418$	0	0
$419$	−8.00000	−0.390826	−0.195413	−	0.980721i	$-0.562605\pi$
$419$	−8.00000	−0.390826	−0.195413	+	0.980721i	$0.562605\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	−8.00000	−0.389434
$423$	0	0
$424$	−18.0000	−0.874157
$425$	1.00000	0.0485071
$426$	0	0
$427$	−40.0000	−1.93574
$428$	8.00000	0.386695
$429$	0	0
$430$	8.00000	0.385794
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	16.0000	0.768025
$435$	0	0
$436$	6.00000	0.287348
$437$	0	0
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	−27.0000	−1.28571
$442$	−2.00000	−0.0951303
$443$	28.0000	1.33032	0.665160	−	0.746701i	$-0.268363\pi$
$443$	28.0000	1.33032	0.665160	+	0.746701i	$0.268363\pi$
$444$	0	0
$445$	−20.0000	−0.948091
$446$	−24.0000	−1.13643
$447$	0	0
$448$	−28.0000	−1.32288
$449$	34.0000	1.60456	0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000	1.60456	0.802280	+	0.596948i	$0.203620\pi$
$450$	−3.00000	−0.141421
$451$	0	0
$452$	−14.0000	−0.658505
$453$	0	0
$454$	−24.0000	−1.12638
$455$	16.0000	0.750092
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	6.00000	0.280362
$459$	0	0
$460$	0	0
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	0	0
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	−6.00000	−0.278543
$465$	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	$0.589555\pi$
$468$	−6.00000	−0.277350
$469$	16.0000	0.738811
$470$	0	0
$471$	0	0
$472$	−36.0000	−1.65703
$473$	0	0
$474$	0	0
$475$	−4.00000	−0.183533
$476$	−4.00000	−0.183340
$477$	18.0000	0.824163
$478$	16.0000	0.731823
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	18.0000	0.819878
$483$	0	0
$484$	11.0000	0.500000
$485$	−4.00000	−0.181631
$486$	0	0
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	30.0000	1.35804
$489$	0	0
$490$	−18.0000	−0.813157
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	−6.00000	−0.270226
$494$	8.00000	0.359937
$495$	0	0
$496$	−4.00000	−0.179605
$497$	16.0000	0.717698
$498$	0	0
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	12.0000	0.535586
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	36.0000	1.60357
$505$	−20.0000	−0.889988
$506$	0	0
$507$	0	0
$508$	−8.00000	−0.354943
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	11.0000	0.486136
$513$	0	0
$514$	−18.0000	−0.793946
$515$	−16.0000	−0.705044
$516$	0	0
$517$	0	0
$518$	8.00000	0.351500
$519$	0	0
$520$	−12.0000	−0.526235
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	18.0000	0.787839
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	−16.0000	−0.698963
$525$	0	0
$526$	−16.0000	−0.697633
$527$	−4.00000	−0.174243
$528$	0	0
$529$	0	0
$530$	12.0000	0.521247
$531$	36.0000	1.56227
$532$	16.0000	0.693688
$533$	12.0000	0.519778
$534$	0	0
$535$	−16.0000	−0.691740
$536$	−12.0000	−0.518321
$537$	0	0
$538$	−22.0000	−0.948487
$539$	0	0
$540$	0	0
$541$	6.00000	0.257960	0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000	0.257960	0.128980	+	0.991647i	$0.458830\pi$
$542$	16.0000	0.687259
$543$	0	0
$544$	5.00000	0.214373
$545$	−12.0000	−0.514024
$546$	0	0
$547$	−32.0000	−1.36822	−0.684111	−	0.729378i	$-0.739809\pi$
$547$	−32.0000	−1.36822	−0.684111	+	0.729378i	$0.739809\pi$
$548$	−6.00000	−0.256307
$549$	−30.0000	−1.28037
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	48.0000	2.04117
$554$	−14.0000	−0.594803
$555$	0	0
$556$	8.00000	0.339276
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	12.0000	0.508001
$559$	8.00000	0.338364
$560$	8.00000	0.338062
$561$	0	0
$562$	−6.00000	−0.253095
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	28.0000	1.17797
$566$	−16.0000	−0.672530
$567$	−36.0000	−1.51186
$568$	−12.0000	−0.503509
$569$	38.0000	1.59304	0.796521	−	0.604610i	$-0.206671\pi$
$569$	38.0000	1.59304	0.796521	+	0.604610i	$0.206671\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	0	0
$574$	−24.0000	−1.00174
$575$	0	0
$576$	−21.0000	−0.875000
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	−1.00000	−0.0415945
$579$	0	0
$580$	−12.0000	−0.498273
$581$	−16.0000	−0.663792
$582$	0	0
$583$	0	0
$584$	−18.0000	−0.744845
$585$	12.0000	0.496139
$586$	6.00000	0.247858
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	24.0000	0.988064
$591$	0	0
$592$	−2.00000	−0.0821995
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	8.00000	0.327968
$596$	−10.0000	−0.409616
$597$	0	0
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	−16.0000	−0.652111
$603$	12.0000	0.488678
$604$	16.0000	0.651031
$605$	−22.0000	−0.894427
$606$	0	0
$607$	20.0000	0.811775	0.405887	−	0.913923i	$-0.366962\pi$
$607$	20.0000	0.811775	0.405887	+	0.913923i	$0.366962\pi$
$608$	−20.0000	−0.811107
$609$	0	0
$610$	−20.0000	−0.809776
$611$	0	0
$612$	−3.00000	−0.121268
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	48.0000	1.92928	0.964641	−	0.263566i	$-0.0848986\pi$
$619$	48.0000	1.92928	0.964641	+	0.263566i	$0.0848986\pi$
$620$	−8.00000	−0.321288
$621$	0	0
$622$	−28.0000	−1.12270
$623$	40.0000	1.60257
$624$	0	0
$625$	−19.0000	−0.760000
$626$	−22.0000	−0.879297
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	−2.00000	−0.0797452
$630$	−24.0000	−0.956183
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	−36.0000	−1.43200
$633$	0	0
$634$	10.0000	0.397151
$635$	16.0000	0.634941
$636$	0	0
$637$	−18.0000	−0.713186
$638$	0	0
$639$	12.0000	0.474713
$640$	6.00000	0.237171
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	−32.0000	−1.26196	−0.630978	−	0.775800i	$-0.717346\pi$
$643$	−32.0000	−1.26196	−0.630978	+	0.775800i	$0.717346\pi$
$644$	0	0
$645$	0	0
$646$	4.00000	0.157378
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	27.0000	1.06066
$649$	0	0
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	−24.0000	−0.939913
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	32.0000	1.25034
$656$	6.00000	0.234261
$657$	18.0000	0.702247
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	12.0000	0.465690
$665$	−32.0000	−1.24091
$666$	6.00000	0.232495
$667$	0	0
$668$	4.00000	0.154765
$669$	0	0
$670$	8.00000	0.309067
$671$	0	0
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	9.00000	0.346154
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	−6.00000	−0.230089
$681$	0	0
$682$	0	0
$683$	−40.0000	−1.53056	−0.765279	−	0.643699i	$-0.777399\pi$
$683$	−40.0000	−1.53056	−0.765279	+	0.643699i	$0.777399\pi$
$684$	12.0000	0.458831
$685$	12.0000	0.458496
$686$	8.00000	0.305441
$687$	0	0
$688$	4.00000	0.152499
$689$	12.0000	0.457164
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	−22.0000	−0.836315
$693$	0	0
$694$	−32.0000	−1.21470
$695$	−16.0000	−0.606915
$696$	0	0
$697$	6.00000	0.227266
$698$	18.0000	0.681310
$699$	0	0
$700$	−4.00000	−0.151186
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	0	0
$706$	30.0000	1.12906
$707$	40.0000	1.50435
$708$	0	0
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	8.00000	0.300235
$711$	36.0000	1.35011
$712$	−30.0000	−1.12430
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	0	0
$719$	4.00000	0.149175	0.0745874	−	0.997214i	$-0.476236\pi$
$719$	4.00000	0.149175	0.0745874	+	0.997214i	$0.476236\pi$
$720$	6.00000	0.223607
$721$	32.0000	1.19174
$722$	3.00000	0.111648
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	−6.00000	−0.222834
$726$	0	0
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	24.0000	0.889499
$729$	−27.0000	−1.00000
$730$	12.0000	0.444140
$731$	4.00000	0.147945
$732$	0	0
$733$	50.0000	1.84679	0.923396	−	0.383849i	$-0.125402\pi$
$733$	50.0000	1.84679	0.923396	+	0.383849i	$0.125402\pi$
$734$	28.0000	1.03350
$735$	0	0
$736$	0	0
$737$	0	0
$738$	−18.0000	−0.662589
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	−24.0000	−0.881068
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	0	0
$745$	20.0000	0.732743
$746$	6.00000	0.219676
$747$	−12.0000	−0.439057
$748$	0	0
$749$	32.0000	1.16925
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	0	0
$753$	0	0
$754$	12.0000	0.437014
$755$	−32.0000	−1.16460
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	−8.00000	−0.290573
$759$	0	0
$760$	24.0000	0.870572
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	0	0
$763$	24.0000	0.868858
$764$	−16.0000	−0.578860
$765$	6.00000	0.216930
$766$	−24.0000	−0.867155
$767$	24.0000	0.866590
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	0	0
$772$	−2.00000	−0.0719816
$773$	26.0000	0.935155	0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000	0.935155	0.467578	+	0.883952i	$0.345127\pi$
$774$	−12.0000	−0.431331
$775$	−4.00000	−0.143684
$776$	−6.00000	−0.215387
$777$	0	0
$778$	6.00000	0.215110
$779$	−24.0000	−0.859889
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−9.00000	−0.321429
$785$	4.00000	0.142766
$786$	0	0
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	24.0000	0.853882
$791$	−56.0000	−1.99113
$792$	0	0
$793$	−20.0000	−0.710221
$794$	−6.00000	−0.212932
$795$	0	0
$796$	−20.0000	−0.708881
$797$	50.0000	1.77109	0.885545	−	0.464553i	$-0.153785\pi$
$797$	50.0000	1.77109	0.885545	+	0.464553i	$0.153785\pi$
$798$	0	0
$799$	0	0
$800$	5.00000	0.176777
$801$	30.0000	1.06000
$802$	−14.0000	−0.494357
$803$	0	0
$804$	0	0
$805$	0	0
$806$	8.00000	0.281788
$807$	0	0
$808$	−30.0000	−1.05540
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	−18.0000	−0.632456
$811$	40.0000	1.40459	0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000	1.40459	0.702295	+	0.711886i	$0.252159\pi$
$812$	24.0000	0.842235
$813$	0	0
$814$	0	0
$815$	48.0000	1.68137
$816$	0	0
$817$	−16.0000	−0.559769
$818$	−26.0000	−0.909069
$819$	−24.0000	−0.838628
$820$	12.0000	0.419058
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	20.0000	0.697156	0.348578	−	0.937280i	$-0.386665\pi$
$823$	20.0000	0.697156	0.348578	+	0.937280i	$0.386665\pi$
$824$	−24.0000	−0.836080
$825$	0	0
$826$	−48.0000	−1.67013
$827$	48.0000	1.66912	0.834562	−	0.550914i	$-0.185721\pi$
$827$	48.0000	1.66912	0.834562	+	0.550914i	$0.185721\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	−8.00000	−0.277684
$831$	0	0
$832$	−14.0000	−0.485363
$833$	−9.00000	−0.311832
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	0	0
$838$	8.00000	0.276355
$839$	−20.0000	−0.690477	−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000	−0.690477	−0.345238	+	0.938515i	$0.612202\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	22.0000	0.758170
$843$	0	0
$844$	−8.00000	−0.275371
$845$	−18.0000	−0.619219
$846$	0	0
$847$	44.0000	1.51186
$848$	6.00000	0.206041
$849$	0	0
$850$	−1.00000	−0.0342997
$851$	0	0
$852$	0	0
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	40.0000	1.36877
$855$	−24.0000	−0.820783
$856$	−24.0000	−0.820303
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	0	0
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	12.0000	0.408722
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	0	0
$865$	44.0000	1.49604
$866$	2.00000	0.0679628
$867$	0	0
$868$	16.0000	0.543075
$869$	0	0
$870$	0	0
$871$	8.00000	0.271070
$872$	−18.0000	−0.609557
$873$	6.00000	0.203069
$874$	0	0
$875$	48.0000	1.62270
$876$	0	0
$877$	6.00000	0.202606	0.101303	−	0.994856i	$-0.467699\pi$
$877$	6.00000	0.202606	0.101303	+	0.994856i	$0.467699\pi$
$878$	20.0000	0.674967
$879$	0	0
$880$	0	0
$881$	46.0000	1.54978	0.774890	−	0.632096i	$-0.217805\pi$
$881$	46.0000	1.54978	0.774890	+	0.632096i	$0.217805\pi$
$882$	27.0000	0.909137
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	−28.0000	−0.940678
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	−32.0000	−1.07325
$890$	20.0000	0.670402
$891$	0	0
$892$	−24.0000	−0.803579
$893$	0	0
$894$	0	0
$895$	24.0000	0.802232
$896$	−12.0000	−0.400892
$897$	0	0
$898$	−34.0000	−1.13459
$899$	24.0000	0.800445
$900$	−3.00000	−0.100000
$901$	6.00000	0.199889
$902$	0	0
$903$	0	0
$904$	42.0000	1.39690
$905$	4.00000	0.132964
$906$	0	0
$907$	−32.0000	−1.06254	−0.531271	−	0.847202i	$-0.678286\pi$
$907$	−32.0000	−1.06254	−0.531271	+	0.847202i	$0.678286\pi$
$908$	−24.0000	−0.796468
$909$	30.0000	0.995037
$910$	−16.0000	−0.530395
$911$	4.00000	0.132526	0.0662630	−	0.997802i	$-0.478892\pi$
$911$	4.00000	0.132526	0.0662630	+	0.997802i	$0.478892\pi$
$912$	0	0
$913$	0	0
$914$	−6.00000	−0.198462
$915$	0	0
$916$	6.00000	0.198246
$917$	−64.0000	−2.11347
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	0	0
$922$	2.00000	0.0658665
$923$	8.00000	0.263323
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	−32.0000	−1.05159
$927$	24.0000	0.788263
$928$	−30.0000	−0.984798
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	6.00000	0.196537
$933$	0	0
$934$	12.0000	0.392652
$935$	0	0
$936$	18.0000	0.588348
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	−16.0000	−0.522419
$939$	0	0
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	0	0
$944$	12.0000	0.390567
$945$	0	0
$946$	0	0
$947$	32.0000	1.03986	0.519930	−	0.854209i	$-0.325958\pi$
$947$	32.0000	1.03986	0.519930	+	0.854209i	$0.325958\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	4.00000	0.129777
$951$	0	0
$952$	12.0000	0.388922
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	−18.0000	−0.582772
$955$	32.0000	1.03550
$956$	16.0000	0.517477
$957$	0	0
$958$	36.0000	1.16311
$959$	−24.0000	−0.775000
$960$	0	0
$961$	−15.0000	−0.483871
$962$	4.00000	0.128965
$963$	24.0000	0.773389
$964$	18.0000	0.579741
$965$	4.00000	0.128765
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	−33.0000	−1.06066
$969$	0	0
$970$	4.00000	0.128432
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	32.0000	1.02587
$974$	−20.0000	−0.640841
$975$	0	0
$976$	−10.0000	−0.320092
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	0	0
$980$	−18.0000	−0.574989
$981$	18.0000	0.574696
$982$	−20.0000	−0.638226
$983$	−12.0000	−0.382741	−0.191370	−	0.981518i	$-0.561293\pi$
$983$	−12.0000	−0.382741	−0.191370	+	0.981518i	$0.561293\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	6.00000	0.191079
$987$	0	0
$988$	8.00000	0.254514
$989$	0	0
$990$	0	0
$991$	−12.0000	−0.381193	−0.190596	−	0.981669i	$-0.561042\pi$
$991$	−12.0000	−0.381193	−0.190596	+	0.981669i	$0.561042\pi$
$992$	−20.0000	−0.635001
$993$	0	0
$994$	−16.0000	−0.507489
$995$	40.0000	1.26809
$996$	0	0
$997$	46.0000	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$998$	40.0000	1.26618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8993.2.a.a.1.1		1
23.22	odd	2		17.2.a.a.1.1	✓	1
69.68	even	2		153.2.a.c.1.1		1
92.91	even	2		272.2.a.b.1.1		1
115.22	even	4		425.2.b.b.324.1		2
115.68	even	4		425.2.b.b.324.2		2
115.114	odd	2		425.2.a.d.1.1		1
161.45	even	6		833.2.e.a.324.1		2
161.68	even	6		833.2.e.a.18.1		2
161.114	odd	6		833.2.e.b.18.1		2
161.137	odd	6		833.2.e.b.324.1		2
161.160	even	2		833.2.a.a.1.1		1
184.45	odd	2		1088.2.a.i.1.1		1
184.91	even	2		1088.2.a.h.1.1		1
253.252	even	2		2057.2.a.e.1.1		1
276.275	odd	2		2448.2.a.o.1.1		1
299.298	odd	2		2873.2.a.c.1.1		1
345.344	even	2		3825.2.a.d.1.1		1
391.22	even	16		289.2.d.d.110.2		8
391.45	even	16		289.2.d.d.155.1		8
391.91	even	16		289.2.d.d.155.2		8
391.114	even	16		289.2.d.d.110.1		8
391.160	even	16		289.2.d.d.134.2		8
391.183	odd	4		289.2.b.a.288.1		2
391.206	odd	8		289.2.c.a.38.2		4
391.229	odd	8		289.2.c.a.251.1		4
391.252	even	16		289.2.d.d.179.1		8
391.275	even	16		289.2.d.d.179.2		8
391.298	odd	8		289.2.c.a.251.2		4
391.321	odd	8		289.2.c.a.38.1		4
391.344	odd	4		289.2.b.a.288.2		2
391.367	even	16		289.2.d.d.134.1		8
391.390	odd	2		289.2.a.a.1.1		1
437.436	even	2		6137.2.a.b.1.1		1
460.459	even	2		6800.2.a.n.1.1		1
483.482	odd	2		7497.2.a.l.1.1		1
552.275	odd	2		9792.2.a.i.1.1		1
552.413	even	2		9792.2.a.n.1.1		1
1173.1172	even	2		2601.2.a.g.1.1		1
1564.1563	even	2		4624.2.a.d.1.1		1
1955.1954	odd	2		7225.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.2.a.a.1.1	✓	1	23.22	odd	2
153.2.a.c.1.1		1	69.68	even	2
272.2.a.b.1.1		1	92.91	even	2
289.2.a.a.1.1		1	391.390	odd	2
289.2.b.a.288.1		2	391.183	odd	4
289.2.b.a.288.2		2	391.344	odd	4
289.2.c.a.38.1		4	391.321	odd	8
289.2.c.a.38.2		4	391.206	odd	8
289.2.c.a.251.1		4	391.229	odd	8
289.2.c.a.251.2		4	391.298	odd	8
289.2.d.d.110.1		8	391.114	even	16
289.2.d.d.110.2		8	391.22	even	16
289.2.d.d.134.1		8	391.367	even	16
289.2.d.d.134.2		8	391.160	even	16
289.2.d.d.155.1		8	391.45	even	16
289.2.d.d.155.2		8	391.91	even	16
289.2.d.d.179.1		8	391.252	even	16
289.2.d.d.179.2		8	391.275	even	16
425.2.a.d.1.1		1	115.114	odd	2
425.2.b.b.324.1		2	115.22	even	4
425.2.b.b.324.2		2	115.68	even	4
833.2.a.a.1.1		1	161.160	even	2
833.2.e.a.18.1		2	161.68	even	6
833.2.e.a.324.1		2	161.45	even	6
833.2.e.b.18.1		2	161.114	odd	6
833.2.e.b.324.1		2	161.137	odd	6
1088.2.a.h.1.1		1	184.91	even	2
1088.2.a.i.1.1		1	184.45	odd	2
2057.2.a.e.1.1		1	253.252	even	2
2448.2.a.o.1.1		1	276.275	odd	2
2601.2.a.g.1.1		1	1173.1172	even	2
2873.2.a.c.1.1		1	299.298	odd	2
3825.2.a.d.1.1		1	345.344	even	2
4624.2.a.d.1.1		1	1564.1563	even	2
6137.2.a.b.1.1		1	437.436	even	2
6800.2.a.n.1.1		1	460.459	even	2
7225.2.a.g.1.1		1	1955.1954	odd	2
7497.2.a.l.1.1		1	483.482	odd	2
8993.2.a.a.1.1		1	1.1	even	1	trivial
9792.2.a.i.1.1		1	552.275	odd	2
9792.2.a.n.1.1		1	552.413	even	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:	\( N \)	\(=\)	\( 8993 = 17 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8993.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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