LMFDB - Embedded newform 5456.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5456,2,Mod(1,5456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5456, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5456.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5456 = 2^{4} \cdot 11 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5456.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:	$43.5663793428$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 341)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	5456.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.61803 q^{5} +2.85410 q^{7} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.61803 q^{5} +2.85410 q^{7} -2.00000 q^{9} -1.00000 q^{11} +3.47214 q^{13} -2.61803 q^{15} +0.236068 q^{17} +2.76393 q^{19} +2.85410 q^{21} -1.23607 q^{23} +1.85410 q^{25} -5.00000 q^{27} -1.00000 q^{31} -1.00000 q^{33} -7.47214 q^{35} -4.23607 q^{37} +3.47214 q^{39} -1.09017 q^{41} +9.61803 q^{43} +5.23607 q^{45} +10.0902 q^{47} +1.14590 q^{49} +0.236068 q^{51} +7.09017 q^{53} +2.61803 q^{55} +2.76393 q^{57} +2.23607 q^{59} +0.0901699 q^{61} -5.70820 q^{63} -9.09017 q^{65} +7.00000 q^{67} -1.23607 q^{69} -0.0901699 q^{71} -2.70820 q^{73} +1.85410 q^{75} -2.85410 q^{77} +5.00000 q^{79} +1.00000 q^{81} -12.6180 q^{83} -0.618034 q^{85} +5.32624 q^{89} +9.90983 q^{91} -1.00000 q^{93} -7.23607 q^{95} +16.9443 q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 3 q^{5} - q^{7} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 3 * q^5 - q^7 - 4 * q^9	$ 2 q + 2 q^{3} - 3 q^{5} - q^{7} - 4 q^{9} - 2 q^{11} - 2 q^{13} - 3 q^{15} - 4 q^{17} + 10 q^{19} - q^{21} + 2 q^{23} - 3 q^{25} - 10 q^{27} - 2 q^{31} - 2 q^{33} - 6 q^{35} - 4 q^{37} - 2 q^{39} + 9 q^{41} + 17 q^{43} + 6 q^{45} + 9 q^{47} + 9 q^{49} - 4 q^{51} + 3 q^{53} + 3 q^{55} + 10 q^{57} - 11 q^{61} + 2 q^{63} - 7 q^{65} + 14 q^{67} + 2 q^{69} + 11 q^{71} + 8 q^{73} - 3 q^{75} + q^{77} + 10 q^{79} + 2 q^{81} - 23 q^{83} + q^{85} - 5 q^{89} + 31 q^{91} - 2 q^{93} - 10 q^{95} + 16 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 3 * q^5 - q^7 - 4 * q^9 - 2 * q^11 - 2 * q^13 - 3 * q^15 - 4 * q^17 + 10 * q^19 - q^21 + 2 * q^23 - 3 * q^25 - 10 * q^27 - 2 * q^31 - 2 * q^33 - 6 * q^35 - 4 * q^37 - 2 * q^39 + 9 * q^41 + 17 * q^43 + 6 * q^45 + 9 * q^47 + 9 * q^49 - 4 * q^51 + 3 * q^53 + 3 * q^55 + 10 * q^57 - 11 * q^61 + 2 * q^63 - 7 * q^65 + 14 * q^67 + 2 * q^69 + 11 * q^71 + 8 * q^73 - 3 * q^75 + q^77 + 10 * q^79 + 2 * q^81 - 23 * q^83 + q^85 - 5 * q^89 + 31 * q^91 - 2 * q^93 - 10 * q^95 + 16 * q^97 + 4 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	−2.61803	−1.17082	−0.585410	−	0.810737i	$-0.699067\pi$
$5$	−2.61803	−1.17082	−0.585410	+	0.810737i	$0.699067\pi$
$6$	0	0
$7$	2.85410	1.07875	0.539375	−	0.842066i	$-0.318661\pi$
$7$	2.85410	1.07875	0.539375	+	0.842066i	$0.318661\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.47214	0.962997	0.481499	−	0.876447i	$-0.340093\pi$
$13$	3.47214	0.962997	0.481499	+	0.876447i	$0.340093\pi$
$14$	0	0
$15$	−2.61803	−0.675973
$16$	0	0
$17$	0.236068	0.0572549	0.0286274	−	0.999590i	$-0.490886\pi$
$17$	0.236068	0.0572549	0.0286274	+	0.999590i	$0.490886\pi$
$18$	0	0
$19$	2.76393	0.634089	0.317045	−	0.948411i	$-0.397309\pi$
$19$	2.76393	0.634089	0.317045	+	0.948411i	$0.397309\pi$
$20$	0	0
$21$	2.85410	0.622816
$22$	0	0
$23$	−1.23607	−0.257738	−0.128869	−	0.991662i	$-0.541135\pi$
$23$	−1.23607	−0.257738	−0.128869	+	0.991662i	$0.541135\pi$
$24$	0	0
$25$	1.85410	0.370820
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−7.47214	−1.26302
$36$	0	0
$37$	−4.23607	−0.696405	−0.348203	−	0.937419i	$-0.613208\pi$
$37$	−4.23607	−0.696405	−0.348203	+	0.937419i	$0.613208\pi$
$38$	0	0
$39$	3.47214	0.555987
$40$	0	0
$41$	−1.09017	−0.170256	−0.0851280	−	0.996370i	$-0.527130\pi$
$41$	−1.09017	−0.170256	−0.0851280	+	0.996370i	$0.527130\pi$
$42$	0	0
$43$	9.61803	1.46674	0.733368	−	0.679832i	$-0.237947\pi$
$43$	9.61803	1.46674	0.733368	+	0.679832i	$0.237947\pi$
$44$	0	0
$45$	5.23607	0.780547
$46$	0	0
$47$	10.0902	1.47180	0.735901	−	0.677089i	$-0.236759\pi$
$47$	10.0902	1.47180	0.735901	+	0.677089i	$0.236759\pi$
$48$	0	0
$49$	1.14590	0.163700
$50$	0	0
$51$	0.236068	0.0330561
$52$	0	0
$53$	7.09017	0.973910	0.486955	−	0.873427i	$-0.338108\pi$
$53$	7.09017	0.973910	0.486955	+	0.873427i	$0.338108\pi$
$54$	0	0
$55$	2.61803	0.353016
$56$	0	0
$57$	2.76393	0.366092
$58$	0	0
$59$	2.23607	0.291111	0.145556	−	0.989350i	$-0.453503\pi$
$59$	2.23607	0.291111	0.145556	+	0.989350i	$0.453503\pi$
$60$	0	0
$61$	0.0901699	0.0115451	0.00577254	−	0.999983i	$-0.498163\pi$
$61$	0.0901699	0.0115451	0.00577254	+	0.999983i	$0.498163\pi$
$62$	0	0
$63$	−5.70820	−0.719166
$64$	0	0
$65$	−9.09017	−1.12750
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	0	0
$69$	−1.23607	−0.148805
$70$	0	0
$71$	−0.0901699	−0.0107012	−0.00535060	−	0.999986i	$-0.501703\pi$
$71$	−0.0901699	−0.0107012	−0.00535060	+	0.999986i	$0.501703\pi$
$72$	0	0
$73$	−2.70820	−0.316971	−0.158486	−	0.987361i	$-0.550661\pi$
$73$	−2.70820	−0.316971	−0.158486	+	0.987361i	$0.550661\pi$
$74$	0	0
$75$	1.85410	0.214093
$76$	0	0
$77$	−2.85410	−0.325255
$78$	0	0
$79$	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
$82$	0	0
$83$	−12.6180	−1.38501	−0.692505	−	0.721413i	$-0.743493\pi$
$83$	−12.6180	−1.38501	−0.692505	+	0.721413i	$0.743493\pi$
$84$	0	0
$85$	−0.618034	−0.0670352
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.32624	0.564580	0.282290	−	0.959329i	$-0.408906\pi$
$89$	5.32624	0.564580	0.282290	+	0.959329i	$0.408906\pi$
$90$	0	0
$91$	9.90983	1.03883
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−7.23607	−0.742405
$96$	0	0
$97$	16.9443	1.72043	0.860215	−	0.509931i	$-0.170329\pi$
$97$	16.9443	1.72043	0.860215	+	0.509931i	$0.170329\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	−4.18034	−0.415959	−0.207980	−	0.978133i	$-0.566689\pi$
$101$	−4.18034	−0.415959	−0.207980	+	0.978133i	$0.566689\pi$
$102$	0	0
$103$	−9.00000	−0.886796	−0.443398	−	0.896325i	$-0.646227\pi$
$103$	−9.00000	−0.886796	−0.443398	+	0.896325i	$0.646227\pi$
$104$	0	0
$105$	−7.47214	−0.729206
$106$	0	0
$107$	7.32624	0.708254	0.354127	−	0.935197i	$-0.384778\pi$
$107$	7.32624	0.708254	0.354127	+	0.935197i	$0.384778\pi$
$108$	0	0
$109$	−16.7082	−1.60036	−0.800178	−	0.599763i	$-0.795262\pi$
$109$	−16.7082	−1.60036	−0.800178	+	0.599763i	$0.795262\pi$
$110$	0	0
$111$	−4.23607	−0.402070
$112$	0	0
$113$	−4.94427	−0.465118	−0.232559	−	0.972582i	$-0.574710\pi$
$113$	−4.94427	−0.465118	−0.232559	+	0.972582i	$0.574710\pi$
$114$	0	0
$115$	3.23607	0.301765
$116$	0	0
$117$	−6.94427	−0.641998
$118$	0	0
$119$	0.673762	0.0617637
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.09017	−0.0982973
$124$	0	0
$125$	8.23607	0.736656
$126$	0	0
$127$	18.7082	1.66008	0.830042	−	0.557700i	$-0.188316\pi$
$127$	18.7082	1.66008	0.830042	+	0.557700i	$0.188316\pi$
$128$	0	0
$129$	9.61803	0.846821
$130$	0	0
$131$	4.90983	0.428974	0.214487	−	0.976727i	$-0.431192\pi$
$131$	4.90983	0.428974	0.214487	+	0.976727i	$0.431192\pi$
$132$	0	0
$133$	7.88854	0.684023
$134$	0	0
$135$	13.0902	1.12662
$136$	0	0
$137$	−3.70820	−0.316813	−0.158407	−	0.987374i	$-0.550636\pi$
$137$	−3.70820	−0.316813	−0.158407	+	0.987374i	$0.550636\pi$
$138$	0	0
$139$	18.4164	1.56206	0.781030	−	0.624494i	$-0.214695\pi$
$139$	18.4164	1.56206	0.781030	+	0.624494i	$0.214695\pi$
$140$	0	0
$141$	10.0902	0.849746
$142$	0	0
$143$	−3.47214	−0.290355
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.14590	0.0945121
$148$	0	0
$149$	−2.23607	−0.183186	−0.0915929	−	0.995797i	$-0.529196\pi$
$149$	−2.23607	−0.183186	−0.0915929	+	0.995797i	$0.529196\pi$
$150$	0	0
$151$	−16.2705	−1.32408	−0.662038	−	0.749471i	$-0.730308\pi$
$151$	−16.2705	−1.32408	−0.662038	+	0.749471i	$0.730308\pi$
$152$	0	0
$153$	−0.472136	−0.0381699
$154$	0	0
$155$	2.61803	0.210286
$156$	0	0
$157$	10.5623	0.842964	0.421482	−	0.906837i	$-0.361510\pi$
$157$	10.5623	0.842964	0.421482	+	0.906837i	$0.361510\pi$
$158$	0	0
$159$	7.09017	0.562287
$160$	0	0
$161$	−3.52786	−0.278035
$162$	0	0
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	0	0
$165$	2.61803	0.203814
$166$	0	0
$167$	−11.6180	−0.899030	−0.449515	−	0.893273i	$-0.648403\pi$
$167$	−11.6180	−0.899030	−0.449515	+	0.893273i	$0.648403\pi$
$168$	0	0
$169$	−0.944272	−0.0726363
$170$	0	0
$171$	−5.52786	−0.422726
$172$	0	0
$173$	22.0902	1.67948	0.839742	−	0.542985i	$-0.182706\pi$
$173$	22.0902	1.67948	0.839742	+	0.542985i	$0.182706\pi$
$174$	0	0
$175$	5.29180	0.400022
$176$	0	0
$177$	2.23607	0.168073
$178$	0	0
$179$	22.2361	1.66200	0.831001	−	0.556271i	$-0.187768\pi$
$179$	22.2361	1.66200	0.831001	+	0.556271i	$0.187768\pi$
$180$	0	0
$181$	5.09017	0.378349	0.189175	−	0.981943i	$-0.439419\pi$
$181$	5.09017	0.378349	0.189175	+	0.981943i	$0.439419\pi$
$182$	0	0
$183$	0.0901699	0.00666555
$184$	0	0
$185$	11.0902	0.815366
$186$	0	0
$187$	−0.236068	−0.0172630
$188$	0	0
$189$	−14.2705	−1.03803
$190$	0	0
$191$	14.1803	1.02605	0.513027	−	0.858373i	$-0.328524\pi$
$191$	14.1803	1.02605	0.513027	+	0.858373i	$0.328524\pi$
$192$	0	0
$193$	25.1803	1.81252	0.906260	−	0.422720i	$-0.138925\pi$
$193$	25.1803	1.81252	0.906260	+	0.422720i	$0.138925\pi$
$194$	0	0
$195$	−9.09017	−0.650961
$196$	0	0
$197$	−3.70820	−0.264199	−0.132099	−	0.991236i	$-0.542172\pi$
$197$	−3.70820	−0.264199	−0.132099	+	0.991236i	$0.542172\pi$
$198$	0	0
$199$	12.8885	0.913645	0.456822	−	0.889558i	$-0.348988\pi$
$199$	12.8885	0.913645	0.456822	+	0.889558i	$0.348988\pi$
$200$	0	0
$201$	7.00000	0.493742
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.85410	0.199339
$206$	0	0
$207$	2.47214	0.171825
$208$	0	0
$209$	−2.76393	−0.191185
$210$	0	0
$211$	18.0000	1.23917	0.619586	−	0.784929i	$-0.287301\pi$
$211$	18.0000	1.23917	0.619586	+	0.784929i	$0.287301\pi$
$212$	0	0
$213$	−0.0901699	−0.00617834
$214$	0	0
$215$	−25.1803	−1.71728
$216$	0	0
$217$	−2.85410	−0.193749
$218$	0	0
$219$	−2.70820	−0.183003
$220$	0	0
$221$	0.819660	0.0551363
$222$	0	0
$223$	22.7082	1.52065	0.760327	−	0.649541i	$-0.225039\pi$
$223$	22.7082	1.52065	0.760327	+	0.649541i	$0.225039\pi$
$224$	0	0
$225$	−3.70820	−0.247214
$226$	0	0
$227$	−15.3607	−1.01952	−0.509762	−	0.860315i	$-0.670267\pi$
$227$	−15.3607	−1.01952	−0.509762	+	0.860315i	$0.670267\pi$
$228$	0	0
$229$	−19.1459	−1.26520	−0.632598	−	0.774480i	$-0.718012\pi$
$229$	−19.1459	−1.26520	−0.632598	+	0.774480i	$0.718012\pi$
$230$	0	0
$231$	−2.85410	−0.187786
$232$	0	0
$233$	−11.5279	−0.755215	−0.377608	−	0.925966i	$-0.623253\pi$
$233$	−11.5279	−0.755215	−0.377608	+	0.925966i	$0.623253\pi$
$234$	0	0
$235$	−26.4164	−1.72322
$236$	0	0
$237$	5.00000	0.324785
$238$	0	0
$239$	−16.3820	−1.05966	−0.529831	−	0.848103i	$-0.677745\pi$
$239$	−16.3820	−1.05966	−0.529831	+	0.848103i	$0.677745\pi$
$240$	0	0
$241$	15.0902	0.972043	0.486022	−	0.873947i	$-0.338448\pi$
$241$	15.0902	0.972043	0.486022	+	0.873947i	$0.338448\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	9.59675	0.610626
$248$	0	0
$249$	−12.6180	−0.799635
$250$	0	0
$251$	−3.90983	−0.246786	−0.123393	−	0.992358i	$-0.539378\pi$
$251$	−3.90983	−0.246786	−0.123393	+	0.992358i	$0.539378\pi$
$252$	0	0
$253$	1.23607	0.0777109
$254$	0	0
$255$	−0.618034	−0.0387028
$256$	0	0
$257$	−31.0689	−1.93802	−0.969012	−	0.247014i	$-0.920551\pi$
$257$	−31.0689	−1.93802	−0.969012	+	0.247014i	$0.920551\pi$
$258$	0	0
$259$	−12.0902	−0.751247
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−22.6180	−1.39469	−0.697344	−	0.716737i	$-0.745635\pi$
$263$	−22.6180	−1.39469	−0.697344	+	0.716737i	$0.745635\pi$
$264$	0	0
$265$	−18.5623	−1.14027
$266$	0	0
$267$	5.32624	0.325960
$268$	0	0
$269$	28.7426	1.75247	0.876235	−	0.481884i	$-0.160047\pi$
$269$	28.7426	1.75247	0.876235	+	0.481884i	$0.160047\pi$
$270$	0	0
$271$	−11.2705	−0.684635	−0.342317	−	0.939584i	$-0.611212\pi$
$271$	−11.2705	−0.684635	−0.342317	+	0.939584i	$0.611212\pi$
$272$	0	0
$273$	9.90983	0.599770
$274$	0	0
$275$	−1.85410	−0.111807
$276$	0	0
$277$	−0.291796	−0.0175323	−0.00876616	−	0.999962i	$-0.502790\pi$
$277$	−0.291796	−0.0175323	−0.00876616	+	0.999962i	$0.502790\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	0	0
$283$	29.2148	1.73664	0.868319	−	0.496006i	$-0.165201\pi$
$283$	29.2148	1.73664	0.868319	+	0.496006i	$0.165201\pi$
$284$	0	0
$285$	−7.23607	−0.428628
$286$	0	0
$287$	−3.11146	−0.183663
$288$	0	0
$289$	−16.9443	−0.996722
$290$	0	0
$291$	16.9443	0.993291
$292$	0	0
$293$	17.9443	1.04832	0.524158	−	0.851621i	$-0.324380\pi$
$293$	17.9443	1.04832	0.524158	+	0.851621i	$0.324380\pi$
$294$	0	0
$295$	−5.85410	−0.340839
$296$	0	0
$297$	5.00000	0.290129
$298$	0	0
$299$	−4.29180	−0.248201
$300$	0	0
$301$	27.4508	1.58224
$302$	0	0
$303$	−4.18034	−0.240154
$304$	0	0
$305$	−0.236068	−0.0135172
$306$	0	0
$307$	34.6869	1.97969	0.989843	−	0.142161i	$-0.0454052\pi$
$307$	34.6869	1.97969	0.989843	+	0.142161i	$0.0454052\pi$
$308$	0	0
$309$	−9.00000	−0.511992
$310$	0	0
$311$	−27.4508	−1.55659	−0.778297	−	0.627896i	$-0.783916\pi$
$311$	−27.4508	−1.55659	−0.778297	+	0.627896i	$0.783916\pi$
$312$	0	0
$313$	29.1246	1.64622	0.823110	−	0.567882i	$-0.192237\pi$
$313$	29.1246	1.64622	0.823110	+	0.567882i	$0.192237\pi$
$314$	0	0
$315$	14.9443	0.842014
$316$	0	0
$317$	23.1246	1.29881	0.649404	−	0.760444i	$-0.275019\pi$
$317$	23.1246	1.29881	0.649404	+	0.760444i	$0.275019\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	7.32624	0.408911
$322$	0	0
$323$	0.652476	0.0363047
$324$	0	0
$325$	6.43769	0.357099
$326$	0	0
$327$	−16.7082	−0.923966
$328$	0	0
$329$	28.7984	1.58771
$330$	0	0
$331$	34.1803	1.87872	0.939361	−	0.342931i	$-0.111420\pi$
$331$	34.1803	1.87872	0.939361	+	0.342931i	$0.111420\pi$
$332$	0	0
$333$	8.47214	0.464270
$334$	0	0
$335$	−18.3262	−1.00127
$336$	0	0
$337$	22.1459	1.20636	0.603182	−	0.797604i	$-0.293899\pi$
$337$	22.1459	1.20636	0.603182	+	0.797604i	$0.293899\pi$
$338$	0	0
$339$	−4.94427	−0.268536
$340$	0	0
$341$	1.00000	0.0541530
$342$	0	0
$343$	−16.7082	−0.902158
$344$	0	0
$345$	3.23607	0.174224
$346$	0	0
$347$	−30.8885	−1.65818	−0.829092	−	0.559112i	$-0.811142\pi$
$347$	−30.8885	−1.65818	−0.829092	+	0.559112i	$0.811142\pi$
$348$	0	0
$349$	−20.9787	−1.12296	−0.561482	−	0.827489i	$-0.689769\pi$
$349$	−20.9787	−1.12296	−0.561482	+	0.827489i	$0.689769\pi$
$350$	0	0
$351$	−17.3607	−0.926645
$352$	0	0
$353$	12.7426	0.678223	0.339111	−	0.940746i	$-0.389874\pi$
$353$	12.7426	0.678223	0.339111	+	0.940746i	$0.389874\pi$
$354$	0	0
$355$	0.236068	0.0125292
$356$	0	0
$357$	0.673762	0.0356593
$358$	0	0
$359$	−20.1246	−1.06214	−0.531068	−	0.847329i	$-0.678209\pi$
$359$	−20.1246	−1.06214	−0.531068	+	0.847329i	$0.678209\pi$
$360$	0	0
$361$	−11.3607	−0.597931
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	7.09017	0.371116
$366$	0	0
$367$	−13.8541	−0.723178	−0.361589	−	0.932338i	$-0.617766\pi$
$367$	−13.8541	−0.723178	−0.361589	+	0.932338i	$0.617766\pi$
$368$	0	0
$369$	2.18034	0.113504
$370$	0	0
$371$	20.2361	1.05060
$372$	0	0
$373$	5.90983	0.305999	0.153000	−	0.988226i	$-0.451107\pi$
$373$	5.90983	0.305999	0.153000	+	0.988226i	$0.451107\pi$
$374$	0	0
$375$	8.23607	0.425309
$376$	0	0
$377$	0	0
$378$	0	0
$379$	19.2705	0.989860	0.494930	−	0.868933i	$-0.335194\pi$
$379$	19.2705	0.989860	0.494930	+	0.868933i	$0.335194\pi$
$380$	0	0
$381$	18.7082	0.958450
$382$	0	0
$383$	15.4721	0.790589	0.395295	−	0.918554i	$-0.370643\pi$
$383$	15.4721	0.790589	0.395295	+	0.918554i	$0.370643\pi$
$384$	0	0
$385$	7.47214	0.380815
$386$	0	0
$387$	−19.2361	−0.977824
$388$	0	0
$389$	−27.7639	−1.40769	−0.703844	−	0.710355i	$-0.748534\pi$
$389$	−27.7639	−1.40769	−0.703844	+	0.710355i	$0.748534\pi$
$390$	0	0
$391$	−0.291796	−0.0147568
$392$	0	0
$393$	4.90983	0.247668
$394$	0	0
$395$	−13.0902	−0.658638
$396$	0	0
$397$	13.0000	0.652451	0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000	0.652451	0.326226	+	0.945292i	$0.394223\pi$
$398$	0	0
$399$	7.88854	0.394921
$400$	0	0
$401$	−6.81966	−0.340558	−0.170279	−	0.985396i	$-0.554467\pi$
$401$	−6.81966	−0.340558	−0.170279	+	0.985396i	$0.554467\pi$
$402$	0	0
$403$	−3.47214	−0.172959
$404$	0	0
$405$	−2.61803	−0.130091
$406$	0	0
$407$	4.23607	0.209974
$408$	0	0
$409$	−10.1246	−0.500630	−0.250315	−	0.968164i	$-0.580534\pi$
$409$	−10.1246	−0.500630	−0.250315	+	0.968164i	$0.580534\pi$
$410$	0	0
$411$	−3.70820	−0.182912
$412$	0	0
$413$	6.38197	0.314036
$414$	0	0
$415$	33.0344	1.62160
$416$	0	0
$417$	18.4164	0.901855
$418$	0	0
$419$	5.52786	0.270054	0.135027	−	0.990842i	$-0.456888\pi$
$419$	5.52786	0.270054	0.135027	+	0.990842i	$0.456888\pi$
$420$	0	0
$421$	−3.00000	−0.146211	−0.0731055	−	0.997324i	$-0.523291\pi$
$421$	−3.00000	−0.146211	−0.0731055	+	0.997324i	$0.523291\pi$
$422$	0	0
$423$	−20.1803	−0.981202
$424$	0	0
$425$	0.437694	0.0212313
$426$	0	0
$427$	0.257354	0.0124542
$428$	0	0
$429$	−3.47214	−0.167636
$430$	0	0
$431$	14.9098	0.718181	0.359091	−	0.933303i	$-0.383087\pi$
$431$	14.9098	0.718181	0.359091	+	0.933303i	$0.383087\pi$
$432$	0	0
$433$	−28.5623	−1.37262	−0.686308	−	0.727311i	$-0.740770\pi$
$433$	−28.5623	−1.37262	−0.686308	+	0.727311i	$0.740770\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.41641	−0.163429
$438$	0	0
$439$	16.1803	0.772245	0.386123	−	0.922447i	$-0.373814\pi$
$439$	16.1803	0.772245	0.386123	+	0.922447i	$0.373814\pi$
$440$	0	0
$441$	−2.29180	−0.109133
$442$	0	0
$443$	32.8328	1.55993	0.779967	−	0.625821i	$-0.215236\pi$
$443$	32.8328	1.55993	0.779967	+	0.625821i	$0.215236\pi$
$444$	0	0
$445$	−13.9443	−0.661022
$446$	0	0
$447$	−2.23607	−0.105762
$448$	0	0
$449$	4.79837	0.226449	0.113225	−	0.993569i	$-0.463882\pi$
$449$	4.79837	0.226449	0.113225	+	0.993569i	$0.463882\pi$
$450$	0	0
$451$	1.09017	0.0513341
$452$	0	0
$453$	−16.2705	−0.764455
$454$	0	0
$455$	−25.9443	−1.21629
$456$	0	0
$457$	9.50658	0.444699	0.222349	−	0.974967i	$-0.428627\pi$
$457$	9.50658	0.444699	0.222349	+	0.974967i	$0.428627\pi$
$458$	0	0
$459$	−1.18034	−0.0550935
$460$	0	0
$461$	8.18034	0.380996	0.190498	−	0.981688i	$-0.438990\pi$
$461$	8.18034	0.380996	0.190498	+	0.981688i	$0.438990\pi$
$462$	0	0
$463$	−8.47214	−0.393734	−0.196867	−	0.980430i	$-0.563077\pi$
$463$	−8.47214	−0.393734	−0.196867	+	0.980430i	$0.563077\pi$
$464$	0	0
$465$	2.61803	0.121408
$466$	0	0
$467$	1.14590	0.0530258	0.0265129	−	0.999648i	$-0.491560\pi$
$467$	1.14590	0.0530258	0.0265129	+	0.999648i	$0.491560\pi$
$468$	0	0
$469$	19.9787	0.922531
$470$	0	0
$471$	10.5623	0.486685
$472$	0	0
$473$	−9.61803	−0.442238
$474$	0	0
$475$	5.12461	0.235133
$476$	0	0
$477$	−14.1803	−0.649273
$478$	0	0
$479$	−8.61803	−0.393768	−0.196884	−	0.980427i	$-0.563082\pi$
$479$	−8.61803	−0.393768	−0.196884	+	0.980427i	$0.563082\pi$
$480$	0	0
$481$	−14.7082	−0.670636
$482$	0	0
$483$	−3.52786	−0.160523
$484$	0	0
$485$	−44.3607	−2.01431
$486$	0	0
$487$	−15.0344	−0.681276	−0.340638	−	0.940195i	$-0.610643\pi$
$487$	−15.0344	−0.681276	−0.340638	+	0.940195i	$0.610643\pi$
$488$	0	0
$489$	11.0000	0.497437
$490$	0	0
$491$	18.0000	0.812329	0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000	0.812329	0.406164	+	0.913800i	$0.366866\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−5.23607	−0.235344
$496$	0	0
$497$	−0.257354	−0.0115439
$498$	0	0
$499$	−34.3951	−1.53974	−0.769869	−	0.638202i	$-0.779678\pi$
$499$	−34.3951	−1.53974	−0.769869	+	0.638202i	$0.779678\pi$
$500$	0	0
$501$	−11.6180	−0.519055
$502$	0	0
$503$	−11.2361	−0.500992	−0.250496	−	0.968118i	$-0.580594\pi$
$503$	−11.2361	−0.500992	−0.250496	+	0.968118i	$0.580594\pi$
$504$	0	0
$505$	10.9443	0.487014
$506$	0	0
$507$	−0.944272	−0.0419366
$508$	0	0
$509$	−26.1803	−1.16042	−0.580212	−	0.814466i	$-0.697030\pi$
$509$	−26.1803	−1.16042	−0.580212	+	0.814466i	$0.697030\pi$
$510$	0	0
$511$	−7.72949	−0.341933
$512$	0	0
$513$	−13.8197	−0.610153
$514$	0	0
$515$	23.5623	1.03828
$516$	0	0
$517$	−10.0902	−0.443765
$518$	0	0
$519$	22.0902	0.969651
$520$	0	0
$521$	−38.4508	−1.68456	−0.842281	−	0.539038i	$-0.818788\pi$
$521$	−38.4508	−1.68456	−0.842281	+	0.539038i	$0.818788\pi$
$522$	0	0
$523$	15.2705	0.667733	0.333866	−	0.942620i	$-0.391647\pi$
$523$	15.2705	0.667733	0.333866	+	0.942620i	$0.391647\pi$
$524$	0	0
$525$	5.29180	0.230953
$526$	0	0
$527$	−0.236068	−0.0102833
$528$	0	0
$529$	−21.4721	−0.933571
$530$	0	0
$531$	−4.47214	−0.194074
$532$	0	0
$533$	−3.78522	−0.163956
$534$	0	0
$535$	−19.1803	−0.829238
$536$	0	0
$537$	22.2361	0.959557
$538$	0	0
$539$	−1.14590	−0.0493573
$540$	0	0
$541$	−5.36068	−0.230474	−0.115237	−	0.993338i	$-0.536763\pi$
$541$	−5.36068	−0.230474	−0.115237	+	0.993338i	$0.536763\pi$
$542$	0	0
$543$	5.09017	0.218440
$544$	0	0
$545$	43.7426	1.87373
$546$	0	0
$547$	−6.61803	−0.282967	−0.141483	−	0.989941i	$-0.545187\pi$
$547$	−6.61803	−0.282967	−0.141483	+	0.989941i	$0.545187\pi$
$548$	0	0
$549$	−0.180340	−0.00769672
$550$	0	0
$551$	0	0
$552$	0	0
$553$	14.2705	0.606844
$554$	0	0
$555$	11.0902	0.470751
$556$	0	0
$557$	−25.4164	−1.07693	−0.538464	−	0.842649i	$-0.680995\pi$
$557$	−25.4164	−1.07693	−0.538464	+	0.842649i	$0.680995\pi$
$558$	0	0
$559$	33.3951	1.41246
$560$	0	0
$561$	−0.236068	−0.00996680
$562$	0	0
$563$	0.875388	0.0368932	0.0184466	−	0.999830i	$-0.494128\pi$
$563$	0.875388	0.0368932	0.0184466	+	0.999830i	$0.494128\pi$
$564$	0	0
$565$	12.9443	0.544570
$566$	0	0
$567$	2.85410	0.119861
$568$	0	0
$569$	1.58359	0.0663876	0.0331938	−	0.999449i	$-0.489432\pi$
$569$	1.58359	0.0663876	0.0331938	+	0.999449i	$0.489432\pi$
$570$	0	0
$571$	−21.2705	−0.890143	−0.445072	−	0.895495i	$-0.646822\pi$
$571$	−21.2705	−0.890143	−0.445072	+	0.895495i	$0.646822\pi$
$572$	0	0
$573$	14.1803	0.592392
$574$	0	0
$575$	−2.29180	−0.0955745
$576$	0	0
$577$	5.03444	0.209587	0.104793	−	0.994494i	$-0.466582\pi$
$577$	5.03444	0.209587	0.104793	+	0.994494i	$0.466582\pi$
$578$	0	0
$579$	25.1803	1.04646
$580$	0	0
$581$	−36.0132	−1.49408
$582$	0	0
$583$	−7.09017	−0.293645
$584$	0	0
$585$	18.1803	0.751665
$586$	0	0
$587$	30.0902	1.24195	0.620977	−	0.783829i	$-0.286736\pi$
$587$	30.0902	1.24195	0.620977	+	0.783829i	$0.286736\pi$
$588$	0	0
$589$	−2.76393	−0.113886
$590$	0	0
$591$	−3.70820	−0.152535
$592$	0	0
$593$	−12.5066	−0.513584	−0.256792	−	0.966467i	$-0.582665\pi$
$593$	−12.5066	−0.513584	−0.256792	+	0.966467i	$0.582665\pi$
$594$	0	0
$595$	−1.76393	−0.0723142
$596$	0	0
$597$	12.8885	0.527493
$598$	0	0
$599$	−15.6525	−0.639543	−0.319771	−	0.947495i	$-0.603606\pi$
$599$	−15.6525	−0.639543	−0.319771	+	0.947495i	$0.603606\pi$
$600$	0	0
$601$	−18.0000	−0.734235	−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000	−0.734235	−0.367118	+	0.930175i	$0.619655\pi$
$602$	0	0
$603$	−14.0000	−0.570124
$604$	0	0
$605$	−2.61803	−0.106438
$606$	0	0
$607$	7.12461	0.289179	0.144590	−	0.989492i	$-0.453814\pi$
$607$	7.12461	0.289179	0.144590	+	0.989492i	$0.453814\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	35.0344	1.41734
$612$	0	0
$613$	−23.7639	−0.959816	−0.479908	−	0.877319i	$-0.659330\pi$
$613$	−23.7639	−0.959816	−0.479908	+	0.877319i	$0.659330\pi$
$614$	0	0
$615$	2.85410	0.115088
$616$	0	0
$617$	29.3050	1.17977	0.589886	−	0.807486i	$-0.299172\pi$
$617$	29.3050	1.17977	0.589886	+	0.807486i	$0.299172\pi$
$618$	0	0
$619$	11.3820	0.457480	0.228740	−	0.973488i	$-0.426539\pi$
$619$	11.3820	0.457480	0.228740	+	0.973488i	$0.426539\pi$
$620$	0	0
$621$	6.18034	0.248008
$622$	0	0
$623$	15.2016	0.609040
$624$	0	0
$625$	−30.8328	−1.23331
$626$	0	0
$627$	−2.76393	−0.110381
$628$	0	0
$629$	−1.00000	−0.0398726
$630$	0	0
$631$	−6.27051	−0.249625	−0.124813	−	0.992180i	$-0.539833\pi$
$631$	−6.27051	−0.249625	−0.124813	+	0.992180i	$0.539833\pi$
$632$	0	0
$633$	18.0000	0.715436
$634$	0	0
$635$	−48.9787	−1.94366
$636$	0	0
$637$	3.97871	0.157642
$638$	0	0
$639$	0.180340	0.00713414
$640$	0	0
$641$	25.0902	0.991002	0.495501	−	0.868607i	$-0.334984\pi$
$641$	25.0902	0.991002	0.495501	+	0.868607i	$0.334984\pi$
$642$	0	0
$643$	8.43769	0.332750	0.166375	−	0.986063i	$-0.446794\pi$
$643$	8.43769	0.332750	0.166375	+	0.986063i	$0.446794\pi$
$644$	0	0
$645$	−25.1803	−0.991475
$646$	0	0
$647$	3.50658	0.137858	0.0689289	−	0.997622i	$-0.478042\pi$
$647$	3.50658	0.137858	0.0689289	+	0.997622i	$0.478042\pi$
$648$	0	0
$649$	−2.23607	−0.0877733
$650$	0	0
$651$	−2.85410	−0.111861
$652$	0	0
$653$	−36.4508	−1.42643	−0.713216	−	0.700944i	$-0.752762\pi$
$653$	−36.4508	−1.42643	−0.713216	+	0.700944i	$0.752762\pi$
$654$	0	0
$655$	−12.8541	−0.502251
$656$	0	0
$657$	5.41641	0.211314
$658$	0	0
$659$	−35.1246	−1.36826	−0.684130	−	0.729360i	$-0.739818\pi$
$659$	−35.1246	−1.36826	−0.684130	+	0.729360i	$0.739818\pi$
$660$	0	0
$661$	−46.5410	−1.81024	−0.905118	−	0.425161i	$-0.860218\pi$
$661$	−46.5410	−1.81024	−0.905118	+	0.425161i	$0.860218\pi$
$662$	0	0
$663$	0.819660	0.0318330
$664$	0	0
$665$	−20.6525	−0.800869
$666$	0	0
$667$	0	0
$668$	0	0
$669$	22.7082	0.877950
$670$	0	0
$671$	−0.0901699	−0.00348097
$672$	0	0
$673$	41.3607	1.59434	0.797169	−	0.603757i	$-0.206330\pi$
$673$	41.3607	1.59434	0.797169	+	0.603757i	$0.206330\pi$
$674$	0	0
$675$	−9.27051	−0.356822
$676$	0	0
$677$	−40.7426	−1.56587	−0.782934	−	0.622105i	$-0.786278\pi$
$677$	−40.7426	−1.56587	−0.782934	+	0.622105i	$0.786278\pi$
$678$	0	0
$679$	48.3607	1.85591
$680$	0	0
$681$	−15.3607	−0.588623
$682$	0	0
$683$	13.3607	0.511232	0.255616	−	0.966778i	$-0.417722\pi$
$683$	13.3607	0.511232	0.255616	+	0.966778i	$0.417722\pi$
$684$	0	0
$685$	9.70820	0.370931
$686$	0	0
$687$	−19.1459	−0.730462
$688$	0	0
$689$	24.6180	0.937872
$690$	0	0
$691$	11.8197	0.449641	0.224821	−	0.974400i	$-0.427820\pi$
$691$	11.8197	0.449641	0.224821	+	0.974400i	$0.427820\pi$
$692$	0	0
$693$	5.70820	0.216837
$694$	0	0
$695$	−48.2148	−1.82889
$696$	0	0
$697$	−0.257354	−0.00974799
$698$	0	0
$699$	−11.5279	−0.436024
$700$	0	0
$701$	−19.9098	−0.751984	−0.375992	−	0.926623i	$-0.622698\pi$
$701$	−19.9098	−0.751984	−0.375992	+	0.926623i	$0.622698\pi$
$702$	0	0
$703$	−11.7082	−0.441583
$704$	0	0
$705$	−26.4164	−0.994899
$706$	0	0
$707$	−11.9311	−0.448716
$708$	0	0
$709$	−33.5410	−1.25966	−0.629830	−	0.776733i	$-0.716875\pi$
$709$	−33.5410	−1.25966	−0.629830	+	0.776733i	$0.716875\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	0	0
$713$	1.23607	0.0462911
$714$	0	0
$715$	9.09017	0.339953
$716$	0	0
$717$	−16.3820	−0.611796
$718$	0	0
$719$	12.8885	0.480662	0.240331	−	0.970691i	$-0.422744\pi$
$719$	12.8885	0.480662	0.240331	+	0.970691i	$0.422744\pi$
$720$	0	0
$721$	−25.6869	−0.956631
$722$	0	0
$723$	15.0902	0.561209
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−24.7082	−0.916377	−0.458188	−	0.888855i	$-0.651501\pi$
$727$	−24.7082	−0.916377	−0.458188	+	0.888855i	$0.651501\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	2.27051	0.0839778
$732$	0	0
$733$	25.8328	0.954157	0.477078	−	0.878861i	$-0.341696\pi$
$733$	25.8328	0.954157	0.477078	+	0.878861i	$0.341696\pi$
$734$	0	0
$735$	−3.00000	−0.110657
$736$	0	0
$737$	−7.00000	−0.257848
$738$	0	0
$739$	1.58359	0.0582534	0.0291267	−	0.999576i	$-0.490727\pi$
$739$	1.58359	0.0582534	0.0291267	+	0.999576i	$0.490727\pi$
$740$	0	0
$741$	9.59675	0.352545
$742$	0	0
$743$	1.85410	0.0680204	0.0340102	−	0.999421i	$-0.489172\pi$
$743$	1.85410	0.0680204	0.0340102	+	0.999421i	$0.489172\pi$
$744$	0	0
$745$	5.85410	0.214478
$746$	0	0
$747$	25.2361	0.923339
$748$	0	0
$749$	20.9098	0.764029
$750$	0	0
$751$	26.5410	0.968496	0.484248	−	0.874931i	$-0.339093\pi$
$751$	26.5410	0.968496	0.484248	+	0.874931i	$0.339093\pi$
$752$	0	0
$753$	−3.90983	−0.142482
$754$	0	0
$755$	42.5967	1.55025
$756$	0	0
$757$	46.4164	1.68703	0.843517	−	0.537103i	$-0.180481\pi$
$757$	46.4164	1.68703	0.843517	+	0.537103i	$0.180481\pi$
$758$	0	0
$759$	1.23607	0.0448664
$760$	0	0
$761$	−31.0902	−1.12702	−0.563509	−	0.826110i	$-0.690549\pi$
$761$	−31.0902	−1.12702	−0.563509	+	0.826110i	$0.690549\pi$
$762$	0	0
$763$	−47.6869	−1.72638
$764$	0	0
$765$	1.23607	0.0446901
$766$	0	0
$767$	7.76393	0.280339
$768$	0	0
$769$	45.0000	1.62274	0.811371	−	0.584532i	$-0.198722\pi$
$769$	45.0000	1.62274	0.811371	+	0.584532i	$0.198722\pi$
$770$	0	0
$771$	−31.0689	−1.11892
$772$	0	0
$773$	47.9443	1.72444	0.862218	−	0.506538i	$-0.169075\pi$
$773$	47.9443	1.72444	0.862218	+	0.506538i	$0.169075\pi$
$774$	0	0
$775$	−1.85410	−0.0666013
$776$	0	0
$777$	−12.0902	−0.433732
$778$	0	0
$779$	−3.01316	−0.107958
$780$	0	0
$781$	0.0901699	0.00322653
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−27.6525	−0.986959
$786$	0	0
$787$	−23.1246	−0.824303	−0.412152	−	0.911115i	$-0.635223\pi$
$787$	−23.1246	−0.824303	−0.412152	+	0.911115i	$0.635223\pi$
$788$	0	0
$789$	−22.6180	−0.805223
$790$	0	0
$791$	−14.1115	−0.501746
$792$	0	0
$793$	0.313082	0.0111179
$794$	0	0
$795$	−18.5623	−0.658337
$796$	0	0
$797$	−34.4853	−1.22153	−0.610766	−	0.791811i	$-0.709138\pi$
$797$	−34.4853	−1.22153	−0.610766	+	0.791811i	$0.709138\pi$
$798$	0	0
$799$	2.38197	0.0842679
$800$	0	0
$801$	−10.6525	−0.376387
$802$	0	0
$803$	2.70820	0.0955704
$804$	0	0
$805$	9.23607	0.325529
$806$	0	0
$807$	28.7426	1.01179
$808$	0	0
$809$	24.5967	0.864776	0.432388	−	0.901688i	$-0.357671\pi$
$809$	24.5967	0.864776	0.432388	+	0.901688i	$0.357671\pi$
$810$	0	0
$811$	−39.3607	−1.38214	−0.691070	−	0.722788i	$-0.742860\pi$
$811$	−39.3607	−1.38214	−0.691070	+	0.722788i	$0.742860\pi$
$812$	0	0
$813$	−11.2705	−0.395274
$814$	0	0
$815$	−28.7984	−1.00876
$816$	0	0
$817$	26.5836	0.930042
$818$	0	0
$819$	−19.8197	−0.692555
$820$	0	0
$821$	−21.5410	−0.751787	−0.375893	−	0.926663i	$-0.622664\pi$
$821$	−21.5410	−0.751787	−0.375893	+	0.926663i	$0.622664\pi$
$822$	0	0
$823$	−19.4508	−0.678014	−0.339007	−	0.940784i	$-0.610091\pi$
$823$	−19.4508	−0.678014	−0.339007	+	0.940784i	$0.610091\pi$
$824$	0	0
$825$	−1.85410	−0.0645515
$826$	0	0
$827$	42.6525	1.48317	0.741586	−	0.670858i	$-0.234074\pi$
$827$	42.6525	1.48317	0.741586	+	0.670858i	$0.234074\pi$
$828$	0	0
$829$	−10.1246	−0.351642	−0.175821	−	0.984422i	$-0.556258\pi$
$829$	−10.1246	−0.351642	−0.175821	+	0.984422i	$0.556258\pi$
$830$	0	0
$831$	−0.291796	−0.0101223
$832$	0	0
$833$	0.270510	0.00937261
$834$	0	0
$835$	30.4164	1.05260
$836$	0	0
$837$	5.00000	0.172825
$838$	0	0
$839$	1.58359	0.0546717	0.0273358	−	0.999626i	$-0.491298\pi$
$839$	1.58359	0.0546717	0.0273358	+	0.999626i	$0.491298\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	12.0000	0.413302
$844$	0	0
$845$	2.47214	0.0850441
$846$	0	0
$847$	2.85410	0.0980681
$848$	0	0
$849$	29.2148	1.00265
$850$	0	0
$851$	5.23607	0.179490
$852$	0	0
$853$	12.9443	0.443203	0.221602	−	0.975137i	$-0.428872\pi$
$853$	12.9443	0.443203	0.221602	+	0.975137i	$0.428872\pi$
$854$	0	0
$855$	14.4721	0.494937
$856$	0	0
$857$	−37.2492	−1.27241	−0.636205	−	0.771520i	$-0.719497\pi$
$857$	−37.2492	−1.27241	−0.636205	+	0.771520i	$0.719497\pi$
$858$	0	0
$859$	16.0557	0.547814	0.273907	−	0.961756i	$-0.411684\pi$
$859$	16.0557	0.547814	0.273907	+	0.961756i	$0.411684\pi$
$860$	0	0
$861$	−3.11146	−0.106038
$862$	0	0
$863$	13.0344	0.443698	0.221849	−	0.975081i	$-0.428791\pi$
$863$	13.0344	0.443698	0.221849	+	0.975081i	$0.428791\pi$
$864$	0	0
$865$	−57.8328	−1.96637
$866$	0	0
$867$	−16.9443	−0.575458
$868$	0	0
$869$	−5.00000	−0.169613
$870$	0	0
$871$	24.3050	0.823542
$872$	0	0
$873$	−33.8885	−1.14695
$874$	0	0
$875$	23.5066	0.794667
$876$	0	0
$877$	31.9443	1.07868	0.539341	−	0.842088i	$-0.318674\pi$
$877$	31.9443	1.07868	0.539341	+	0.842088i	$0.318674\pi$
$878$	0	0
$879$	17.9443	0.605245
$880$	0	0
$881$	−14.9098	−0.502325	−0.251162	−	0.967945i	$-0.580813\pi$
$881$	−14.9098	−0.502325	−0.251162	+	0.967945i	$0.580813\pi$
$882$	0	0
$883$	6.00000	0.201916	0.100958	−	0.994891i	$-0.467809\pi$
$883$	6.00000	0.201916	0.100958	+	0.994891i	$0.467809\pi$
$884$	0	0
$885$	−5.85410	−0.196783
$886$	0	0
$887$	−1.09017	−0.0366043	−0.0183022	−	0.999833i	$-0.505826\pi$
$887$	−1.09017	−0.0366043	−0.0183022	+	0.999833i	$0.505826\pi$
$888$	0	0
$889$	53.3951	1.79081
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	27.8885	0.933255
$894$	0	0
$895$	−58.2148	−1.94591
$896$	0	0
$897$	−4.29180	−0.143299
$898$	0	0
$899$	0	0
$900$	0	0
$901$	1.67376	0.0557611
$902$	0	0
$903$	27.4508	0.913507
$904$	0	0
$905$	−13.3262	−0.442979
$906$	0	0
$907$	−26.4164	−0.877142	−0.438571	−	0.898696i	$-0.644515\pi$
$907$	−26.4164	−0.877142	−0.438571	+	0.898696i	$0.644515\pi$
$908$	0	0
$909$	8.36068	0.277306
$910$	0	0
$911$	−33.1803	−1.09931	−0.549657	−	0.835391i	$-0.685242\pi$
$911$	−33.1803	−1.09931	−0.549657	+	0.835391i	$0.685242\pi$
$912$	0	0
$913$	12.6180	0.417596
$914$	0	0
$915$	−0.236068	−0.00780417
$916$	0	0
$917$	14.0132	0.462755
$918$	0	0
$919$	−42.8115	−1.41222	−0.706111	−	0.708101i	$-0.749552\pi$
$919$	−42.8115	−1.41222	−0.706111	+	0.708101i	$0.749552\pi$
$920$	0	0
$921$	34.6869	1.14297
$922$	0	0
$923$	−0.313082	−0.0103052
$924$	0	0
$925$	−7.85410	−0.258241
$926$	0	0
$927$	18.0000	0.591198
$928$	0	0
$929$	−22.9656	−0.753476	−0.376738	−	0.926320i	$-0.622954\pi$
$929$	−22.9656	−0.753476	−0.376738	+	0.926320i	$0.622954\pi$
$930$	0	0
$931$	3.16718	0.103800
$932$	0	0
$933$	−27.4508	−0.898700
$934$	0	0
$935$	0.618034	0.0202119
$936$	0	0
$937$	−31.1459	−1.01749	−0.508746	−	0.860917i	$-0.669891\pi$
$937$	−31.1459	−1.01749	−0.508746	+	0.860917i	$0.669891\pi$
$938$	0	0
$939$	29.1246	0.950446
$940$	0	0
$941$	35.0902	1.14391	0.571953	−	0.820286i	$-0.306186\pi$
$941$	35.0902	1.14391	0.571953	+	0.820286i	$0.306186\pi$
$942$	0	0
$943$	1.34752	0.0438814
$944$	0	0
$945$	37.3607	1.21534
$946$	0	0
$947$	−49.4296	−1.60624	−0.803122	−	0.595814i	$-0.796830\pi$
$947$	−49.4296	−1.60624	−0.803122	+	0.595814i	$0.796830\pi$
$948$	0	0
$949$	−9.40325	−0.305242
$950$	0	0
$951$	23.1246	0.749867
$952$	0	0
$953$	−49.8673	−1.61536	−0.807679	−	0.589622i	$-0.799277\pi$
$953$	−49.8673	−1.61536	−0.807679	+	0.589622i	$0.799277\pi$
$954$	0	0
$955$	−37.1246	−1.20132
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.5836	−0.341762
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−14.6525	−0.472169
$964$	0	0
$965$	−65.9230	−2.12214
$966$	0	0
$967$	−13.0000	−0.418052	−0.209026	−	0.977910i	$-0.567029\pi$
$967$	−13.0000	−0.418052	−0.209026	+	0.977910i	$0.567029\pi$
$968$	0	0
$969$	0.652476	0.0209605
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	0	0
$973$	52.5623	1.68507
$974$	0	0
$975$	6.43769	0.206171
$976$	0	0
$977$	38.7771	1.24059	0.620294	−	0.784369i	$-0.287013\pi$
$977$	38.7771	1.24059	0.620294	+	0.784369i	$0.287013\pi$
$978$	0	0
$979$	−5.32624	−0.170227
$980$	0	0
$981$	33.4164	1.06690
$982$	0	0
$983$	21.3262	0.680201	0.340101	−	0.940389i	$-0.389539\pi$
$983$	21.3262	0.680201	0.340101	+	0.940389i	$0.389539\pi$
$984$	0	0
$985$	9.70820	0.309329
$986$	0	0
$987$	28.7984	0.916662
$988$	0	0
$989$	−11.8885	−0.378034
$990$	0	0
$991$	49.1803	1.56226	0.781132	−	0.624365i	$-0.214642\pi$
$991$	49.1803	1.56226	0.781132	+	0.624365i	$0.214642\pi$
$992$	0	0
$993$	34.1803	1.08468
$994$	0	0
$995$	−33.7426	−1.06971
$996$	0	0
$997$	−19.2361	−0.609212	−0.304606	−	0.952478i	$-0.598525\pi$
$997$	−19.2361	−0.609212	−0.304606	+	0.952478i	$0.598525\pi$
$998$	0	0
$999$	21.1803	0.670116

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5456.2.a.r.1.1		2
4.3	odd	2		341.2.a.a.1.2	✓	2
12.11	even	2		3069.2.a.b.1.1		2
20.19	odd	2		8525.2.a.d.1.1		2
44.43	even	2		3751.2.a.a.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
341.2.a.a.1.2	✓	2	4.3	odd	2
3069.2.a.b.1.1		2	12.11	even	2
3751.2.a.a.1.1		2	44.43	even	2
5456.2.a.r.1.1		2	1.1	even	1	trivial
8525.2.a.d.1.1		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5456 = 2^{4} \cdot 11 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5456.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.61803 q^{5} +2.85410 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.61803 q^{5} +2.85410 q^{7} -2.00000 q^{9} -1.00000 q^{11} +3.47214 q^{13} -2.61803 q^{15} +0.236068 q^{17} +2.76393 q^{19} +2.85410 q^{21} -1.23607 q^{23} +1.85410 q^{25} -5.00000 q^{27} -1.00000 q^{31} -1.00000 q^{33} -7.47214 q^{35} -4.23607 q^{37} +3.47214 q^{39} -1.09017 q^{41} +9.61803 q^{43} +5.23607 q^{45} +10.0902 q^{47} +1.14590 q^{49} +0.236068 q^{51} +7.09017 q^{53} +2.61803 q^{55} +2.76393 q^{57} +2.23607 q^{59} +0.0901699 q^{61} -5.70820 q^{63} -9.09017 q^{65} +7.00000 q^{67} -1.23607 q^{69} -0.0901699 q^{71} -2.70820 q^{73} +1.85410 q^{75} -2.85410 q^{77} +5.00000 q^{79} +1.00000 q^{81} -12.6180 q^{83} -0.618034 q^{85} +5.32624 q^{89} +9.90983 q^{91} -1.00000 q^{93} -7.23607 q^{95} +16.9443 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 3 q^{5} - q^{7} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 3 * q^5 - q^7 - 4 * q^9	\( 2 q + 2 q^{3} - 3 q^{5} - q^{7} - 4 q^{9} - 2 q^{11} - 2 q^{13} - 3 q^{15} - 4 q^{17} + 10 q^{19} - q^{21} + 2 q^{23} - 3 q^{25} - 10 q^{27} - 2 q^{31} - 2 q^{33} - 6 q^{35} - 4 q^{37} - 2 q^{39} + 9 q^{41} + 17 q^{43} + 6 q^{45} + 9 q^{47} + 9 q^{49} - 4 q^{51} + 3 q^{53} + 3 q^{55} + 10 q^{57} - 11 q^{61} + 2 q^{63} - 7 q^{65} + 14 q^{67} + 2 q^{69} + 11 q^{71} + 8 q^{73} - 3 q^{75} + q^{77} + 10 q^{79} + 2 q^{81} - 23 q^{83} + q^{85} - 5 q^{89} + 31 q^{91} - 2 q^{93} - 10 q^{95} + 16 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 3 * q^5 - q^7 - 4 * q^9 - 2 * q^11 - 2 * q^13 - 3 * q^15 - 4 * q^17 + 10 * q^19 - q^21 + 2 * q^23 - 3 * q^25 - 10 * q^27 - 2 * q^31 - 2 * q^33 - 6 * q^35 - 4 * q^37 - 2 * q^39 + 9 * q^41 + 17 * q^43 + 6 * q^45 + 9 * q^47 + 9 * q^49 - 4 * q^51 + 3 * q^53 + 3 * q^55 + 10 * q^57 - 11 * q^61 + 2 * q^63 - 7 * q^65 + 14 * q^67 + 2 * q^69 + 11 * q^71 + 8 * q^73 - 3 * q^75 + q^77 + 10 * q^79 + 2 * q^81 - 23 * q^83 + q^85 - 5 * q^89 + 31 * q^91 - 2 * q^93 - 10 * q^95 + 16 * q^97 + 4 * q^99

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