LMFDB - Embedded newform 8550.2.a.bw.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.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:	$68.2720937282$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1710)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{7} +1.00000 q^{8} +3.46410 q^{11} -5.46410 q^{13} -2.00000 q^{14} +1.00000 q^{16} -3.46410 q^{17} +1.00000 q^{19} +3.46410 q^{22} +6.92820 q^{23} -5.46410 q^{26} -2.00000 q^{28} -3.46410 q^{29} -1.46410 q^{31} +1.00000 q^{32} -3.46410 q^{34} +1.46410 q^{37} +1.00000 q^{38} -5.46410 q^{43} +3.46410 q^{44} +6.92820 q^{46} +6.92820 q^{47} -3.00000 q^{49} -5.46410 q^{52} -12.9282 q^{53} -2.00000 q^{56} -3.46410 q^{58} +3.46410 q^{59} +2.00000 q^{61} -1.46410 q^{62} +1.00000 q^{64} -14.9282 q^{67} -3.46410 q^{68} +6.92820 q^{71} +4.92820 q^{73} +1.46410 q^{74} +1.00000 q^{76} -6.92820 q^{77} -1.46410 q^{79} -2.53590 q^{83} -5.46410 q^{86} +3.46410 q^{88} -6.92820 q^{89} +10.9282 q^{91} +6.92820 q^{92} +6.92820 q^{94} -18.3923 q^{97} -3.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 4 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 4 q^{7} + 2 q^{8} - 4 q^{13} - 4 q^{14} + 2 q^{16} + 2 q^{19} - 4 q^{26} - 4 q^{28} + 4 q^{31} + 2 q^{32} - 4 q^{37} + 2 q^{38} - 4 q^{43} - 6 q^{49} - 4 q^{52} - 12 q^{53} - 4 q^{56} + 4 q^{61} + 4 q^{62} + 2 q^{64} - 16 q^{67} - 4 q^{73} - 4 q^{74} + 2 q^{76} + 4 q^{79} - 12 q^{83} - 4 q^{86} + 8 q^{91} - 16 q^{97} - 6 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^7 + 2 * q^8 - 4 * q^13 - 4 * q^14 + 2 * q^16 + 2 * q^19 - 4 * q^26 - 4 * q^28 + 4 * q^31 + 2 * q^32 - 4 * q^37 + 2 * q^38 - 4 * q^43 - 6 * q^49 - 4 * q^52 - 12 * q^53 - 4 * q^56 + 4 * q^61 + 4 * q^62 + 2 * q^64 - 16 * q^67 - 4 * q^73 - 4 * q^74 + 2 * q^76 + 4 * q^79 - 12 * q^83 - 4 * q^86 + 8 * q^91 - 16 * q^97 - 6 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	−5.46410	−1.51547	−0.757735	−	0.652563i	$-0.773694\pi$
$13$	−5.46410	−1.51547	−0.757735	+	0.652563i	$0.773694\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	3.46410	0.738549
$23$	6.92820	1.44463	0.722315	−	0.691564i	$-0.243078\pi$
$23$	6.92820	1.44463	0.722315	+	0.691564i	$0.243078\pi$
$24$	0	0
$25$	0	0
$26$	−5.46410	−1.07160
$27$	0	0
$28$	−2.00000	−0.377964
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	−1.46410	−0.262960	−0.131480	−	0.991319i	$-0.541973\pi$
$31$	−1.46410	−0.262960	−0.131480	+	0.991319i	$0.541973\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.46410	−0.594089
$35$	0	0
$36$	0	0
$37$	1.46410	0.240697	0.120348	−	0.992732i	$-0.461599\pi$
$37$	1.46410	0.240697	0.120348	+	0.992732i	$0.461599\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−5.46410	−0.833268	−0.416634	−	0.909074i	$-0.636790\pi$
$43$	−5.46410	−0.833268	−0.416634	+	0.909074i	$0.636790\pi$
$44$	3.46410	0.522233
$45$	0	0
$46$	6.92820	1.02151
$47$	6.92820	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	−5.46410	−0.757735
$53$	−12.9282	−1.77583	−0.887913	−	0.460012i	$-0.847845\pi$
$53$	−12.9282	−1.77583	−0.887913	+	0.460012i	$0.847845\pi$
$54$	0	0
$55$	0	0
$56$	−2.00000	−0.267261
$57$	0	0
$58$	−3.46410	−0.454859
$59$	3.46410	0.450988	0.225494	−	0.974245i	$-0.427600\pi$
$59$	3.46410	0.450988	0.225494	+	0.974245i	$0.427600\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−1.46410	−0.185941
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−14.9282	−1.82377	−0.911885	−	0.410445i	$-0.865373\pi$
$67$	−14.9282	−1.82377	−0.911885	+	0.410445i	$0.865373\pi$
$68$	−3.46410	−0.420084
$69$	0	0
$70$	0	0
$71$	6.92820	0.822226	0.411113	−	0.911584i	$-0.365140\pi$
$71$	6.92820	0.822226	0.411113	+	0.911584i	$0.365140\pi$
$72$	0	0
$73$	4.92820	0.576803	0.288401	−	0.957510i	$-0.406876\pi$
$73$	4.92820	0.576803	0.288401	+	0.957510i	$0.406876\pi$
$74$	1.46410	0.170198
$75$	0	0
$76$	1.00000	0.114708
$77$	−6.92820	−0.789542
$78$	0	0
$79$	−1.46410	−0.164724	−0.0823622	−	0.996602i	$-0.526246\pi$
$79$	−1.46410	−0.164724	−0.0823622	+	0.996602i	$0.526246\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.53590	−0.278351	−0.139176	−	0.990268i	$-0.544445\pi$
$83$	−2.53590	−0.278351	−0.139176	+	0.990268i	$0.544445\pi$
$84$	0	0
$85$	0	0
$86$	−5.46410	−0.589209
$87$	0	0
$88$	3.46410	0.369274
$89$	−6.92820	−0.734388	−0.367194	−	0.930144i	$-0.619682\pi$
$89$	−6.92820	−0.734388	−0.367194	+	0.930144i	$0.619682\pi$
$90$	0	0
$91$	10.9282	1.14559
$92$	6.92820	0.722315
$93$	0	0
$94$	6.92820	0.714590
$95$	0	0
$96$	0	0
$97$	−18.3923	−1.86746	−0.933728	−	0.357984i	$-0.883464\pi$
$97$	−18.3923	−1.86746	−0.933728	+	0.357984i	$0.883464\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	4.92820	0.485590	0.242795	−	0.970078i	$-0.421936\pi$
$103$	4.92820	0.485590	0.242795	+	0.970078i	$0.421936\pi$
$104$	−5.46410	−0.535799
$105$	0	0
$106$	−12.9282	−1.25570
$107$	6.92820	0.669775	0.334887	−	0.942258i	$-0.391302\pi$
$107$	6.92820	0.669775	0.334887	+	0.942258i	$0.391302\pi$
$108$	0	0
$109$	−14.3923	−1.37853	−0.689266	−	0.724508i	$-0.742067\pi$
$109$	−14.3923	−1.37853	−0.689266	+	0.724508i	$0.742067\pi$
$110$	0	0
$111$	0	0
$112$	−2.00000	−0.188982
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	−3.46410	−0.321634
$117$	0	0
$118$	3.46410	0.318896
$119$	6.92820	0.635107
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.00000	0.181071
$123$	0	0
$124$	−1.46410	−0.131480
$125$	0	0
$126$	0	0
$127$	−14.0000	−1.24230	−0.621150	−	0.783692i	$-0.713334\pi$
$127$	−14.0000	−1.24230	−0.621150	+	0.783692i	$0.713334\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−3.46410	−0.302660	−0.151330	−	0.988483i	$-0.548356\pi$
$131$	−3.46410	−0.302660	−0.151330	+	0.988483i	$0.548356\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	−14.9282	−1.28960
$135$	0	0
$136$	−3.46410	−0.297044
$137$	−15.4641	−1.32119	−0.660594	−	0.750744i	$-0.729695\pi$
$137$	−15.4641	−1.32119	−0.660594	+	0.750744i	$0.729695\pi$
$138$	0	0
$139$	9.85641	0.836009	0.418005	−	0.908445i	$-0.362730\pi$
$139$	9.85641	0.836009	0.418005	+	0.908445i	$0.362730\pi$
$140$	0	0
$141$	0	0
$142$	6.92820	0.581402
$143$	−18.9282	−1.58286
$144$	0	0
$145$	0	0
$146$	4.92820	0.407861
$147$	0	0
$148$	1.46410	0.120348
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−8.39230	−0.682956	−0.341478	−	0.939890i	$-0.610927\pi$
$151$	−8.39230	−0.682956	−0.341478	+	0.939890i	$0.610927\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	−6.92820	−0.558291
$155$	0	0
$156$	0	0
$157$	15.3205	1.22271	0.611355	−	0.791357i	$-0.290625\pi$
$157$	15.3205	1.22271	0.611355	+	0.791357i	$0.290625\pi$
$158$	−1.46410	−0.116478
$159$	0	0
$160$	0	0
$161$	−13.8564	−1.09204
$162$	0	0
$163$	6.53590	0.511931	0.255966	−	0.966686i	$-0.417607\pi$
$163$	6.53590	0.511931	0.255966	+	0.966686i	$0.417607\pi$
$164$	0	0
$165$	0	0
$166$	−2.53590	−0.196824
$167$	−18.9282	−1.46471	−0.732354	−	0.680924i	$-0.761578\pi$
$167$	−18.9282	−1.46471	−0.732354	+	0.680924i	$0.761578\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	0	0
$171$	0	0
$172$	−5.46410	−0.416634
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	0	0
$176$	3.46410	0.261116
$177$	0	0
$178$	−6.92820	−0.519291
$179$	−1.60770	−0.120165	−0.0600824	−	0.998193i	$-0.519136\pi$
$179$	−1.60770	−0.120165	−0.0600824	+	0.998193i	$0.519136\pi$
$180$	0	0
$181$	18.3923	1.36709	0.683545	−	0.729909i	$-0.260437\pi$
$181$	18.3923	1.36709	0.683545	+	0.729909i	$0.260437\pi$
$182$	10.9282	0.810052
$183$	0	0
$184$	6.92820	0.510754
$185$	0	0
$186$	0	0
$187$	−12.0000	−0.877527
$188$	6.92820	0.505291
$189$	0	0
$190$	0	0
$191$	−11.3205	−0.819123	−0.409562	−	0.912282i	$-0.634318\pi$
$191$	−11.3205	−0.819123	−0.409562	+	0.912282i	$0.634318\pi$
$192$	0	0
$193$	−6.39230	−0.460128	−0.230064	−	0.973175i	$-0.573894\pi$
$193$	−6.39230	−0.460128	−0.230064	+	0.973175i	$0.573894\pi$
$194$	−18.3923	−1.32049
$195$	0	0
$196$	−3.00000	−0.214286
$197$	0.928203	0.0661317	0.0330659	−	0.999453i	$-0.489473\pi$
$197$	0.928203	0.0661317	0.0330659	+	0.999453i	$0.489473\pi$
$198$	0	0
$199$	2.92820	0.207575	0.103787	−	0.994600i	$-0.466904\pi$
$199$	2.92820	0.207575	0.103787	+	0.994600i	$0.466904\pi$
$200$	0	0
$201$	0	0
$202$	6.00000	0.422159
$203$	6.92820	0.486265
$204$	0	0
$205$	0	0
$206$	4.92820	0.343364
$207$	0	0
$208$	−5.46410	−0.378867
$209$	3.46410	0.239617
$210$	0	0
$211$	−17.8564	−1.22929	−0.614643	−	0.788806i	$-0.710700\pi$
$211$	−17.8564	−1.22929	−0.614643	+	0.788806i	$0.710700\pi$
$212$	−12.9282	−0.887913
$213$	0	0
$214$	6.92820	0.473602
$215$	0	0
$216$	0	0
$217$	2.92820	0.198779
$218$	−14.3923	−0.974770
$219$	0	0
$220$	0	0
$221$	18.9282	1.27325
$222$	0	0
$223$	23.8564	1.59754	0.798772	−	0.601634i	$-0.205484\pi$
$223$	23.8564	1.59754	0.798772	+	0.601634i	$0.205484\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	−6.00000	−0.399114
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−28.9282	−1.91163	−0.955815	−	0.293970i	$-0.905024\pi$
$229$	−28.9282	−1.91163	−0.955815	+	0.293970i	$0.905024\pi$
$230$	0	0
$231$	0	0
$232$	−3.46410	−0.227429
$233$	−8.53590	−0.559205	−0.279603	−	0.960116i	$-0.590203\pi$
$233$	−8.53590	−0.559205	−0.279603	+	0.960116i	$0.590203\pi$
$234$	0	0
$235$	0	0
$236$	3.46410	0.225494
$237$	0	0
$238$	6.92820	0.449089
$239$	−2.53590	−0.164034	−0.0820168	−	0.996631i	$-0.526136\pi$
$239$	−2.53590	−0.164034	−0.0820168	+	0.996631i	$0.526136\pi$
$240$	0	0
$241$	0.143594	0.00924967	0.00462484	−	0.999989i	$-0.498528\pi$
$241$	0.143594	0.00924967	0.00462484	+	0.999989i	$0.498528\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	−5.46410	−0.347672
$248$	−1.46410	−0.0929705
$249$	0	0
$250$	0	0
$251$	10.3923	0.655956	0.327978	−	0.944685i	$-0.393633\pi$
$251$	10.3923	0.655956	0.327978	+	0.944685i	$0.393633\pi$
$252$	0	0
$253$	24.0000	1.50887
$254$	−14.0000	−0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	−24.9282	−1.55498	−0.777489	−	0.628896i	$-0.783507\pi$
$257$	−24.9282	−1.55498	−0.777489	+	0.628896i	$0.783507\pi$
$258$	0	0
$259$	−2.92820	−0.181950
$260$	0	0
$261$	0	0
$262$	−3.46410	−0.214013
$263$	−25.8564	−1.59437	−0.797187	−	0.603732i	$-0.793680\pi$
$263$	−25.8564	−1.59437	−0.797187	+	0.603732i	$0.793680\pi$
$264$	0	0
$265$	0	0
$266$	−2.00000	−0.122628
$267$	0	0
$268$	−14.9282	−0.911885
$269$	−22.3923	−1.36528	−0.682641	−	0.730753i	$-0.739169\pi$
$269$	−22.3923	−1.36528	−0.682641	+	0.730753i	$0.739169\pi$
$270$	0	0
$271$	16.7846	1.01959	0.509796	−	0.860295i	$-0.329721\pi$
$271$	16.7846	1.01959	0.509796	+	0.860295i	$0.329721\pi$
$272$	−3.46410	−0.210042
$273$	0	0
$274$	−15.4641	−0.934221
$275$	0	0
$276$	0	0
$277$	−0.392305	−0.0235713	−0.0117857	−	0.999931i	$-0.503752\pi$
$277$	−0.392305	−0.0235713	−0.0117857	+	0.999931i	$0.503752\pi$
$278$	9.85641	0.591148
$279$	0	0
$280$	0	0
$281$	17.0718	1.01842	0.509209	−	0.860643i	$-0.329938\pi$
$281$	17.0718	1.01842	0.509209	+	0.860643i	$0.329938\pi$
$282$	0	0
$283$	20.3923	1.21220	0.606098	−	0.795390i	$-0.292734\pi$
$283$	20.3923	1.21220	0.606098	+	0.795390i	$0.292734\pi$
$284$	6.92820	0.411113
$285$	0	0
$286$	−18.9282	−1.11925
$287$	0	0
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	0	0
$292$	4.92820	0.288401
$293$	12.9282	0.755274	0.377637	−	0.925954i	$-0.376737\pi$
$293$	12.9282	0.755274	0.377637	+	0.925954i	$0.376737\pi$
$294$	0	0
$295$	0	0
$296$	1.46410	0.0850992
$297$	0	0
$298$	−6.00000	−0.347571
$299$	−37.8564	−2.18929
$300$	0	0
$301$	10.9282	0.629891
$302$	−8.39230	−0.482923
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	24.7846	1.41453	0.707266	−	0.706947i	$-0.249928\pi$
$307$	24.7846	1.41453	0.707266	+	0.706947i	$0.249928\pi$
$308$	−6.92820	−0.394771
$309$	0	0
$310$	0	0
$311$	16.3923	0.929522	0.464761	−	0.885436i	$-0.346140\pi$
$311$	16.3923	0.929522	0.464761	+	0.885436i	$0.346140\pi$
$312$	0	0
$313$	−22.7846	−1.28786	−0.643931	−	0.765083i	$-0.722698\pi$
$313$	−22.7846	−1.28786	−0.643931	+	0.765083i	$0.722698\pi$
$314$	15.3205	0.864586
$315$	0	0
$316$	−1.46410	−0.0823622
$317$	−19.8564	−1.11525	−0.557623	−	0.830094i	$-0.688287\pi$
$317$	−19.8564	−1.11525	−0.557623	+	0.830094i	$0.688287\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	0	0
$322$	−13.8564	−0.772187
$323$	−3.46410	−0.192748
$324$	0	0
$325$	0	0
$326$	6.53590	0.361990
$327$	0	0
$328$	0	0
$329$	−13.8564	−0.763928
$330$	0	0
$331$	14.9282	0.820528	0.410264	−	0.911967i	$-0.365437\pi$
$331$	14.9282	0.820528	0.410264	+	0.911967i	$0.365437\pi$
$332$	−2.53590	−0.139176
$333$	0	0
$334$	−18.9282	−1.03571
$335$	0	0
$336$	0	0
$337$	−1.32051	−0.0719327	−0.0359663	−	0.999353i	$-0.511451\pi$
$337$	−1.32051	−0.0719327	−0.0359663	+	0.999353i	$0.511451\pi$
$338$	16.8564	0.916868
$339$	0	0
$340$	0	0
$341$	−5.07180	−0.274653
$342$	0	0
$343$	20.0000	1.07990
$344$	−5.46410	−0.294605
$345$	0	0
$346$	−18.0000	−0.967686
$347$	37.1769	1.99576	0.997881	−	0.0650705i	$-0.0207272\pi$
$347$	37.1769	1.99576	0.997881	+	0.0650705i	$0.0207272\pi$
$348$	0	0
$349$	−11.8564	−0.634659	−0.317329	−	0.948315i	$-0.602786\pi$
$349$	−11.8564	−0.634659	−0.317329	+	0.948315i	$0.602786\pi$
$350$	0	0
$351$	0	0
$352$	3.46410	0.184637
$353$	−13.6077	−0.724265	−0.362132	−	0.932127i	$-0.617951\pi$
$353$	−13.6077	−0.724265	−0.362132	+	0.932127i	$0.617951\pi$
$354$	0	0
$355$	0	0
$356$	−6.92820	−0.367194
$357$	0	0
$358$	−1.60770	−0.0849693
$359$	−33.4641	−1.76617	−0.883084	−	0.469215i	$-0.844537\pi$
$359$	−33.4641	−1.76617	−0.883084	+	0.469215i	$0.844537\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	18.3923	0.966678
$363$	0	0
$364$	10.9282	0.572793
$365$	0	0
$366$	0	0
$367$	25.7128	1.34220	0.671099	−	0.741368i	$-0.265823\pi$
$367$	25.7128	1.34220	0.671099	+	0.741368i	$0.265823\pi$
$368$	6.92820	0.361158
$369$	0	0
$370$	0	0
$371$	25.8564	1.34240
$372$	0	0
$373$	−0.392305	−0.0203128	−0.0101564	−	0.999948i	$-0.503233\pi$
$373$	−0.392305	−0.0203128	−0.0101564	+	0.999948i	$0.503233\pi$
$374$	−12.0000	−0.620505
$375$	0	0
$376$	6.92820	0.357295
$377$	18.9282	0.974852
$378$	0	0
$379$	−9.07180	−0.465987	−0.232993	−	0.972478i	$-0.574852\pi$
$379$	−9.07180	−0.465987	−0.232993	+	0.972478i	$0.574852\pi$
$380$	0	0
$381$	0	0
$382$	−11.3205	−0.579208
$383$	18.9282	0.967186	0.483593	−	0.875293i	$-0.339331\pi$
$383$	18.9282	0.967186	0.483593	+	0.875293i	$0.339331\pi$
$384$	0	0
$385$	0	0
$386$	−6.39230	−0.325360
$387$	0	0
$388$	−18.3923	−0.933728
$389$	26.7846	1.35803	0.679017	−	0.734123i	$-0.262406\pi$
$389$	26.7846	1.35803	0.679017	+	0.734123i	$0.262406\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	−3.00000	−0.151523
$393$	0	0
$394$	0.928203	0.0467622
$395$	0	0
$396$	0	0
$397$	−19.3205	−0.969669	−0.484834	−	0.874606i	$-0.661120\pi$
$397$	−19.3205	−0.969669	−0.484834	+	0.874606i	$0.661120\pi$
$398$	2.92820	0.146778
$399$	0	0
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	6.00000	0.298511
$405$	0	0
$406$	6.92820	0.343841
$407$	5.07180	0.251400
$408$	0	0
$409$	−30.7846	−1.52220	−0.761100	−	0.648634i	$-0.775341\pi$
$409$	−30.7846	−1.52220	−0.761100	+	0.648634i	$0.775341\pi$
$410$	0	0
$411$	0	0
$412$	4.92820	0.242795
$413$	−6.92820	−0.340915
$414$	0	0
$415$	0	0
$416$	−5.46410	−0.267900
$417$	0	0
$418$	3.46410	0.169435
$419$	−13.6077	−0.664779	−0.332390	−	0.943142i	$-0.607855\pi$
$419$	−13.6077	−0.664779	−0.332390	+	0.943142i	$0.607855\pi$
$420$	0	0
$421$	39.1769	1.90937	0.954683	−	0.297625i	$-0.0961944\pi$
$421$	39.1769	1.90937	0.954683	+	0.297625i	$0.0961944\pi$
$422$	−17.8564	−0.869236
$423$	0	0
$424$	−12.9282	−0.627849
$425$	0	0
$426$	0	0
$427$	−4.00000	−0.193574
$428$	6.92820	0.334887
$429$	0	0
$430$	0	0
$431$	−18.9282	−0.911739	−0.455870	−	0.890047i	$-0.650672\pi$
$431$	−18.9282	−0.911739	−0.455870	+	0.890047i	$0.650672\pi$
$432$	0	0
$433$	21.3205	1.02460	0.512299	−	0.858807i	$-0.328794\pi$
$433$	21.3205	1.02460	0.512299	+	0.858807i	$0.328794\pi$
$434$	2.92820	0.140558
$435$	0	0
$436$	−14.3923	−0.689266
$437$	6.92820	0.331421
$438$	0	0
$439$	−3.32051	−0.158479	−0.0792396	−	0.996856i	$-0.525249\pi$
$439$	−3.32051	−0.158479	−0.0792396	+	0.996856i	$0.525249\pi$
$440$	0	0
$441$	0	0
$442$	18.9282	0.900323
$443$	−4.39230	−0.208685	−0.104342	−	0.994541i	$-0.533274\pi$
$443$	−4.39230	−0.208685	−0.104342	+	0.994541i	$0.533274\pi$
$444$	0	0
$445$	0	0
$446$	23.8564	1.12963
$447$	0	0
$448$	−2.00000	−0.0944911
$449$	1.85641	0.0876092	0.0438046	−	0.999040i	$-0.486052\pi$
$449$	1.85641	0.0876092	0.0438046	+	0.999040i	$0.486052\pi$
$450$	0	0
$451$	0	0
$452$	−6.00000	−0.282216
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	−28.9282	−1.35173
$459$	0	0
$460$	0	0
$461$	36.9282	1.71992	0.859959	−	0.510363i	$-0.170489\pi$
$461$	36.9282	1.71992	0.859959	+	0.510363i	$0.170489\pi$
$462$	0	0
$463$	15.0718	0.700446	0.350223	−	0.936666i	$-0.386106\pi$
$463$	15.0718	0.700446	0.350223	+	0.936666i	$0.386106\pi$
$464$	−3.46410	−0.160817
$465$	0	0
$466$	−8.53590	−0.395418
$467$	21.4641	0.993240	0.496620	−	0.867968i	$-0.334574\pi$
$467$	21.4641	0.993240	0.496620	+	0.867968i	$0.334574\pi$
$468$	0	0
$469$	29.8564	1.37864
$470$	0	0
$471$	0	0
$472$	3.46410	0.159448
$473$	−18.9282	−0.870320
$474$	0	0
$475$	0	0
$476$	6.92820	0.317554
$477$	0	0
$478$	−2.53590	−0.115989
$479$	−7.60770	−0.347604	−0.173802	−	0.984781i	$-0.555605\pi$
$479$	−7.60770	−0.347604	−0.173802	+	0.984781i	$0.555605\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0.143594	0.00654051
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	0	0
$487$	−34.7846	−1.57624	−0.788121	−	0.615521i	$-0.788946\pi$
$487$	−34.7846	−1.57624	−0.788121	+	0.615521i	$0.788946\pi$
$488$	2.00000	0.0905357
$489$	0	0
$490$	0	0
$491$	−41.3205	−1.86477	−0.932384	−	0.361469i	$-0.882275\pi$
$491$	−41.3205	−1.86477	−0.932384	+	0.361469i	$0.882275\pi$
$492$	0	0
$493$	12.0000	0.540453
$494$	−5.46410	−0.245842
$495$	0	0
$496$	−1.46410	−0.0657401
$497$	−13.8564	−0.621545
$498$	0	0
$499$	−22.9282	−1.02641	−0.513204	−	0.858267i	$-0.671541\pi$
$499$	−22.9282	−1.02641	−0.513204	+	0.858267i	$0.671541\pi$
$500$	0	0
$501$	0	0
$502$	10.3923	0.463831
$503$	−6.92820	−0.308913	−0.154457	−	0.988000i	$-0.549363\pi$
$503$	−6.92820	−0.308913	−0.154457	+	0.988000i	$0.549363\pi$
$504$	0	0
$505$	0	0
$506$	24.0000	1.06693
$507$	0	0
$508$	−14.0000	−0.621150
$509$	−36.2487	−1.60670	−0.803348	−	0.595510i	$-0.796950\pi$
$509$	−36.2487	−1.60670	−0.803348	+	0.595510i	$0.796950\pi$
$510$	0	0
$511$	−9.85641	−0.436022
$512$	1.00000	0.0441942
$513$	0	0
$514$	−24.9282	−1.09954
$515$	0	0
$516$	0	0
$517$	24.0000	1.05552
$518$	−2.92820	−0.128658
$519$	0	0
$520$	0	0
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	0	0
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	−3.46410	−0.151330
$525$	0	0
$526$	−25.8564	−1.12739
$527$	5.07180	0.220931
$528$	0	0
$529$	25.0000	1.08696
$530$	0	0
$531$	0	0
$532$	−2.00000	−0.0867110
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−14.9282	−0.644800
$537$	0	0
$538$	−22.3923	−0.965401
$539$	−10.3923	−0.447628
$540$	0	0
$541$	−39.5692	−1.70121	−0.850607	−	0.525802i	$-0.823765\pi$
$541$	−39.5692	−1.70121	−0.850607	+	0.525802i	$0.823765\pi$
$542$	16.7846	0.720961
$543$	0	0
$544$	−3.46410	−0.148522
$545$	0	0
$546$	0	0
$547$	−26.9282	−1.15137	−0.575683	−	0.817673i	$-0.695264\pi$
$547$	−26.9282	−1.15137	−0.575683	+	0.817673i	$0.695264\pi$
$548$	−15.4641	−0.660594
$549$	0	0
$550$	0	0
$551$	−3.46410	−0.147576
$552$	0	0
$553$	2.92820	0.124520
$554$	−0.392305	−0.0166674
$555$	0	0
$556$	9.85641	0.418005
$557$	12.9282	0.547786	0.273893	−	0.961760i	$-0.411689\pi$
$557$	12.9282	0.547786	0.273893	+	0.961760i	$0.411689\pi$
$558$	0	0
$559$	29.8564	1.26279
$560$	0	0
$561$	0	0
$562$	17.0718	0.720130
$563$	−8.78461	−0.370227	−0.185114	−	0.982717i	$-0.559265\pi$
$563$	−8.78461	−0.370227	−0.185114	+	0.982717i	$0.559265\pi$
$564$	0	0
$565$	0	0
$566$	20.3923	0.857153
$567$	0	0
$568$	6.92820	0.290701
$569$	−25.8564	−1.08396	−0.541978	−	0.840392i	$-0.682325\pi$
$569$	−25.8564	−1.08396	−0.541978	+	0.840392i	$0.682325\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	−18.9282	−0.791428
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6.78461	0.282447	0.141223	−	0.989978i	$-0.454896\pi$
$577$	6.78461	0.282447	0.141223	+	0.989978i	$0.454896\pi$
$578$	−5.00000	−0.207973
$579$	0	0
$580$	0	0
$581$	5.07180	0.210414
$582$	0	0
$583$	−44.7846	−1.85479
$584$	4.92820	0.203931
$585$	0	0
$586$	12.9282	0.534059
$587$	40.3923	1.66717	0.833584	−	0.552392i	$-0.186285\pi$
$587$	40.3923	1.66717	0.833584	+	0.552392i	$0.186285\pi$
$588$	0	0
$589$	−1.46410	−0.0603273
$590$	0	0
$591$	0	0
$592$	1.46410	0.0601742
$593$	12.2487	0.502994	0.251497	−	0.967858i	$-0.419077\pi$
$593$	12.2487	0.502994	0.251497	+	0.967858i	$0.419077\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	0	0
$598$	−37.8564	−1.54806
$599$	−8.78461	−0.358929	−0.179465	−	0.983764i	$-0.557437\pi$
$599$	−8.78461	−0.358929	−0.179465	+	0.983764i	$0.557437\pi$
$600$	0	0
$601$	19.0718	0.777955	0.388977	−	0.921247i	$-0.372828\pi$
$601$	19.0718	0.777955	0.388977	+	0.921247i	$0.372828\pi$
$602$	10.9282	0.445400
$603$	0	0
$604$	−8.39230	−0.341478
$605$	0	0
$606$	0	0
$607$	−24.6410	−1.00015	−0.500074	−	0.865983i	$-0.666694\pi$
$607$	−24.6410	−1.00015	−0.500074	+	0.865983i	$0.666694\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	−37.8564	−1.53151
$612$	0	0
$613$	8.39230	0.338962	0.169481	−	0.985533i	$-0.445791\pi$
$613$	8.39230	0.338962	0.169481	+	0.985533i	$0.445791\pi$
$614$	24.7846	1.00023
$615$	0	0
$616$	−6.92820	−0.279145
$617$	−22.3923	−0.901480	−0.450740	−	0.892655i	$-0.648840\pi$
$617$	−22.3923	−0.901480	−0.450740	+	0.892655i	$0.648840\pi$
$618$	0	0
$619$	28.7846	1.15695	0.578476	−	0.815700i	$-0.303648\pi$
$619$	28.7846	1.15695	0.578476	+	0.815700i	$0.303648\pi$
$620$	0	0
$621$	0	0
$622$	16.3923	0.657272
$623$	13.8564	0.555145
$624$	0	0
$625$	0	0
$626$	−22.7846	−0.910656
$627$	0	0
$628$	15.3205	0.611355
$629$	−5.07180	−0.202226
$630$	0	0
$631$	−22.9282	−0.912757	−0.456379	−	0.889786i	$-0.650854\pi$
$631$	−22.9282	−0.912757	−0.456379	+	0.889786i	$0.650854\pi$
$632$	−1.46410	−0.0582388
$633$	0	0
$634$	−19.8564	−0.788599
$635$	0	0
$636$	0	0
$637$	16.3923	0.649487
$638$	−12.0000	−0.475085
$639$	0	0
$640$	0	0
$641$	10.1436	0.400648	0.200324	−	0.979730i	$-0.435801\pi$
$641$	10.1436	0.400648	0.200324	+	0.979730i	$0.435801\pi$
$642$	0	0
$643$	−45.1769	−1.78160	−0.890802	−	0.454392i	$-0.849857\pi$
$643$	−45.1769	−1.78160	−0.890802	+	0.454392i	$0.849857\pi$
$644$	−13.8564	−0.546019
$645$	0	0
$646$	−3.46410	−0.136293
$647$	17.0718	0.671162	0.335581	−	0.942011i	$-0.391067\pi$
$647$	17.0718	0.671162	0.335581	+	0.942011i	$0.391067\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	6.53590	0.255966
$653$	−45.7128	−1.78888	−0.894440	−	0.447187i	$-0.852426\pi$
$653$	−45.7128	−1.78888	−0.894440	+	0.447187i	$0.852426\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	−13.8564	−0.540179
$659$	−22.3923	−0.872280	−0.436140	−	0.899879i	$-0.643655\pi$
$659$	−22.3923	−0.872280	−0.436140	+	0.899879i	$0.643655\pi$
$660$	0	0
$661$	42.3923	1.64887	0.824435	−	0.565957i	$-0.191493\pi$
$661$	42.3923	1.64887	0.824435	+	0.565957i	$0.191493\pi$
$662$	14.9282	0.580201
$663$	0	0
$664$	−2.53590	−0.0984119
$665$	0	0
$666$	0	0
$667$	−24.0000	−0.929284
$668$	−18.9282	−0.732354
$669$	0	0
$670$	0	0
$671$	6.92820	0.267460
$672$	0	0
$673$	−2.67949	−0.103287	−0.0516434	−	0.998666i	$-0.516446\pi$
$673$	−2.67949	−0.103287	−0.0516434	+	0.998666i	$0.516446\pi$
$674$	−1.32051	−0.0508641
$675$	0	0
$676$	16.8564	0.648323
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	36.7846	1.41166
$680$	0	0
$681$	0	0
$682$	−5.07180	−0.194209
$683$	−25.8564	−0.989368	−0.494684	−	0.869073i	$-0.664716\pi$
$683$	−25.8564	−0.989368	−0.494684	+	0.869073i	$0.664716\pi$
$684$	0	0
$685$	0	0
$686$	20.0000	0.763604
$687$	0	0
$688$	−5.46410	−0.208317
$689$	70.6410	2.69121
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	37.1769	1.41122
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−11.8564	−0.448772
$699$	0	0
$700$	0	0
$701$	−23.0718	−0.871410	−0.435705	−	0.900090i	$-0.643501\pi$
$701$	−23.0718	−0.871410	−0.435705	+	0.900090i	$0.643501\pi$
$702$	0	0
$703$	1.46410	0.0552196
$704$	3.46410	0.130558
$705$	0	0
$706$	−13.6077	−0.512132
$707$	−12.0000	−0.451306
$708$	0	0
$709$	46.7846	1.75703	0.878516	−	0.477712i	$-0.158534\pi$
$709$	46.7846	1.75703	0.878516	+	0.477712i	$0.158534\pi$
$710$	0	0
$711$	0	0
$712$	−6.92820	−0.259645
$713$	−10.1436	−0.379881
$714$	0	0
$715$	0	0
$716$	−1.60770	−0.0600824
$717$	0	0
$718$	−33.4641	−1.24887
$719$	−14.5359	−0.542098	−0.271049	−	0.962566i	$-0.587370\pi$
$719$	−14.5359	−0.542098	−0.271049	+	0.962566i	$0.587370\pi$
$720$	0	0
$721$	−9.85641	−0.367072
$722$	1.00000	0.0372161
$723$	0	0
$724$	18.3923	0.683545
$725$	0	0
$726$	0	0
$727$	−0.143594	−0.00532559	−0.00266279	−	0.999996i	$-0.500848\pi$
$727$	−0.143594	−0.00532559	−0.00266279	+	0.999996i	$0.500848\pi$
$728$	10.9282	0.405026
$729$	0	0
$730$	0	0
$731$	18.9282	0.700085
$732$	0	0
$733$	41.1769	1.52090	0.760452	−	0.649394i	$-0.224977\pi$
$733$	41.1769	1.52090	0.760452	+	0.649394i	$0.224977\pi$
$734$	25.7128	0.949077
$735$	0	0
$736$	6.92820	0.255377
$737$	−51.7128	−1.90487
$738$	0	0
$739$	−12.7846	−0.470289	−0.235145	−	0.971960i	$-0.575556\pi$
$739$	−12.7846	−0.470289	−0.235145	+	0.971960i	$0.575556\pi$
$740$	0	0
$741$	0	0
$742$	25.8564	0.949219
$743$	−10.1436	−0.372132	−0.186066	−	0.982537i	$-0.559574\pi$
$743$	−10.1436	−0.372132	−0.186066	+	0.982537i	$0.559574\pi$
$744$	0	0
$745$	0	0
$746$	−0.392305	−0.0143633
$747$	0	0
$748$	−12.0000	−0.438763
$749$	−13.8564	−0.506302
$750$	0	0
$751$	−1.46410	−0.0534258	−0.0267129	−	0.999643i	$-0.508504\pi$
$751$	−1.46410	−0.0534258	−0.0267129	+	0.999643i	$0.508504\pi$
$752$	6.92820	0.252646
$753$	0	0
$754$	18.9282	0.689325
$755$	0	0
$756$	0	0
$757$	41.1769	1.49660	0.748300	−	0.663360i	$-0.230870\pi$
$757$	41.1769	1.49660	0.748300	+	0.663360i	$0.230870\pi$
$758$	−9.07180	−0.329502
$759$	0	0
$760$	0	0
$761$	53.5692	1.94188	0.970941	−	0.239318i	$-0.0769237\pi$
$761$	53.5692	1.94188	0.970941	+	0.239318i	$0.0769237\pi$
$762$	0	0
$763$	28.7846	1.04207
$764$	−11.3205	−0.409562
$765$	0	0
$766$	18.9282	0.683904
$767$	−18.9282	−0.683458
$768$	0	0
$769$	27.8564	1.00453	0.502264	−	0.864714i	$-0.332501\pi$
$769$	27.8564	1.00453	0.502264	+	0.864714i	$0.332501\pi$
$770$	0	0
$771$	0	0
$772$	−6.39230	−0.230064
$773$	7.85641	0.282575	0.141288	−	0.989969i	$-0.454876\pi$
$773$	7.85641	0.282575	0.141288	+	0.989969i	$0.454876\pi$
$774$	0	0
$775$	0	0
$776$	−18.3923	−0.660245
$777$	0	0
$778$	26.7846	0.960275
$779$	0	0
$780$	0	0
$781$	24.0000	0.858788
$782$	−24.0000	−0.858238
$783$	0	0
$784$	−3.00000	−0.107143
$785$	0	0
$786$	0	0
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	0.928203	0.0330659
$789$	0	0
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	−10.9282	−0.388072
$794$	−19.3205	−0.685659
$795$	0	0
$796$	2.92820	0.103787
$797$	33.7128	1.19417	0.597085	−	0.802178i	$-0.296326\pi$
$797$	33.7128	1.19417	0.597085	+	0.802178i	$0.296326\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	0	0
$801$	0	0
$802$	12.0000	0.423735
$803$	17.0718	0.602451
$804$	0	0
$805$	0	0
$806$	8.00000	0.281788
$807$	0	0
$808$	6.00000	0.211079
$809$	46.6410	1.63981	0.819905	−	0.572499i	$-0.194026\pi$
$809$	46.6410	1.63981	0.819905	+	0.572499i	$0.194026\pi$
$810$	0	0
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	6.92820	0.243132
$813$	0	0
$814$	5.07180	0.177766
$815$	0	0
$816$	0	0
$817$	−5.46410	−0.191165
$818$	−30.7846	−1.07636
$819$	0	0
$820$	0	0
$821$	−7.85641	−0.274190	−0.137095	−	0.990558i	$-0.543777\pi$
$821$	−7.85641	−0.274190	−0.137095	+	0.990558i	$0.543777\pi$
$822$	0	0
$823$	42.7846	1.49138	0.745689	−	0.666294i	$-0.232121\pi$
$823$	42.7846	1.49138	0.745689	+	0.666294i	$0.232121\pi$
$824$	4.92820	0.171682
$825$	0	0
$826$	−6.92820	−0.241063
$827$	−5.07180	−0.176364	−0.0881818	−	0.996104i	$-0.528106\pi$
$827$	−5.07180	−0.176364	−0.0881818	+	0.996104i	$0.528106\pi$
$828$	0	0
$829$	47.4641	1.64850	0.824248	−	0.566229i	$-0.191598\pi$
$829$	47.4641	1.64850	0.824248	+	0.566229i	$0.191598\pi$
$830$	0	0
$831$	0	0
$832$	−5.46410	−0.189434
$833$	10.3923	0.360072
$834$	0	0
$835$	0	0
$836$	3.46410	0.119808
$837$	0	0
$838$	−13.6077	−0.470070
$839$	13.8564	0.478376	0.239188	−	0.970973i	$-0.423119\pi$
$839$	13.8564	0.478376	0.239188	+	0.970973i	$0.423119\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	39.1769	1.35013
$843$	0	0
$844$	−17.8564	−0.614643
$845$	0	0
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	−12.9282	−0.443956
$849$	0	0
$850$	0	0
$851$	10.1436	0.347718
$852$	0	0
$853$	−19.3205	−0.661522	−0.330761	−	0.943715i	$-0.607305\pi$
$853$	−19.3205	−0.661522	−0.330761	+	0.943715i	$0.607305\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	6.92820	0.236801
$857$	14.7846	0.505033	0.252516	−	0.967593i	$-0.418742\pi$
$857$	14.7846	0.505033	0.252516	+	0.967593i	$0.418742\pi$
$858$	0	0
$859$	−41.8564	−1.42812	−0.714061	−	0.700083i	$-0.753146\pi$
$859$	−41.8564	−1.42812	−0.714061	+	0.700083i	$0.753146\pi$
$860$	0	0
$861$	0	0
$862$	−18.9282	−0.644697
$863$	42.9282	1.46129	0.730647	−	0.682756i	$-0.239219\pi$
$863$	42.9282	1.46129	0.730647	+	0.682756i	$0.239219\pi$
$864$	0	0
$865$	0	0
$866$	21.3205	0.724500
$867$	0	0
$868$	2.92820	0.0993897
$869$	−5.07180	−0.172049
$870$	0	0
$871$	81.5692	2.76387
$872$	−14.3923	−0.487385
$873$	0	0
$874$	6.92820	0.234350
$875$	0	0
$876$	0	0
$877$	23.6077	0.797175	0.398588	−	0.917130i	$-0.369500\pi$
$877$	23.6077	0.797175	0.398588	+	0.917130i	$0.369500\pi$
$878$	−3.32051	−0.112062
$879$	0	0
$880$	0	0
$881$	−6.92820	−0.233417	−0.116709	−	0.993166i	$-0.537234\pi$
$881$	−6.92820	−0.233417	−0.116709	+	0.993166i	$0.537234\pi$
$882$	0	0
$883$	−3.60770	−0.121409	−0.0607043	−	0.998156i	$-0.519335\pi$
$883$	−3.60770	−0.121409	−0.0607043	+	0.998156i	$0.519335\pi$
$884$	18.9282	0.636624
$885$	0	0
$886$	−4.39230	−0.147562
$887$	−41.5692	−1.39576	−0.697879	−	0.716216i	$-0.745873\pi$
$887$	−41.5692	−1.39576	−0.697879	+	0.716216i	$0.745873\pi$
$888$	0	0
$889$	28.0000	0.939090
$890$	0	0
$891$	0	0
$892$	23.8564	0.798772
$893$	6.92820	0.231843
$894$	0	0
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	1.85641	0.0619491
$899$	5.07180	0.169154
$900$	0	0
$901$	44.7846	1.49199
$902$	0	0
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	16.0000	0.531271	0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000	0.531271	0.265636	+	0.964073i	$0.414418\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−10.1436	−0.336072	−0.168036	−	0.985781i	$-0.553743\pi$
$911$	−10.1436	−0.336072	−0.168036	+	0.985781i	$0.553743\pi$
$912$	0	0
$913$	−8.78461	−0.290728
$914$	22.0000	0.727695
$915$	0	0
$916$	−28.9282	−0.955815
$917$	6.92820	0.228789
$918$	0	0
$919$	38.9282	1.28412	0.642061	−	0.766653i	$-0.278079\pi$
$919$	38.9282	1.28412	0.642061	+	0.766653i	$0.278079\pi$
$920$	0	0
$921$	0	0
$922$	36.9282	1.21617
$923$	−37.8564	−1.24606
$924$	0	0
$925$	0	0
$926$	15.0718	0.495290
$927$	0	0
$928$	−3.46410	−0.113715
$929$	−12.0000	−0.393707	−0.196854	−	0.980433i	$-0.563072\pi$
$929$	−12.0000	−0.393707	−0.196854	+	0.980433i	$0.563072\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	−8.53590	−0.279603
$933$	0	0
$934$	21.4641	0.702327
$935$	0	0
$936$	0	0
$937$	−43.5692	−1.42334	−0.711672	−	0.702512i	$-0.752062\pi$
$937$	−43.5692	−1.42334	−0.711672	+	0.702512i	$0.752062\pi$
$938$	29.8564	0.974846
$939$	0	0
$940$	0	0
$941$	−48.2487	−1.57286	−0.786432	−	0.617677i	$-0.788074\pi$
$941$	−48.2487	−1.57286	−0.786432	+	0.617677i	$0.788074\pi$
$942$	0	0
$943$	0	0
$944$	3.46410	0.112747
$945$	0	0
$946$	−18.9282	−0.615409
$947$	0.679492	0.0220805	0.0110403	−	0.999939i	$-0.496486\pi$
$947$	0.679492	0.0220805	0.0110403	+	0.999939i	$0.496486\pi$
$948$	0	0
$949$	−26.9282	−0.874126
$950$	0	0
$951$	0	0
$952$	6.92820	0.224544
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	0	0
$955$	0	0
$956$	−2.53590	−0.0820168
$957$	0	0
$958$	−7.60770	−0.245793
$959$	30.9282	0.998724
$960$	0	0
$961$	−28.8564	−0.930852
$962$	−8.00000	−0.257930
$963$	0	0
$964$	0.143594	0.00462484
$965$	0	0
$966$	0	0
$967$	−7.07180	−0.227414	−0.113707	−	0.993514i	$-0.536272\pi$
$967$	−7.07180	−0.227414	−0.113707	+	0.993514i	$0.536272\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	0	0
$971$	−18.6795	−0.599453	−0.299727	−	0.954025i	$-0.596895\pi$
$971$	−18.6795	−0.599453	−0.299727	+	0.954025i	$0.596895\pi$
$972$	0	0
$973$	−19.7128	−0.631964
$974$	−34.7846	−1.11457
$975$	0	0
$976$	2.00000	0.0640184
$977$	23.5692	0.754046	0.377023	−	0.926204i	$-0.376948\pi$
$977$	23.5692	0.754046	0.377023	+	0.926204i	$0.376948\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	0	0
$981$	0	0
$982$	−41.3205	−1.31859
$983$	60.4974	1.92957	0.964784	−	0.263043i	$-0.0847262\pi$
$983$	60.4974	1.92957	0.964784	+	0.263043i	$0.0847262\pi$
$984$	0	0
$985$	0	0
$986$	12.0000	0.382158
$987$	0	0
$988$	−5.46410	−0.173836
$989$	−37.8564	−1.20376
$990$	0	0
$991$	−8.39230	−0.266590	−0.133295	−	0.991076i	$-0.542556\pi$
$991$	−8.39230	−0.266590	−0.133295	+	0.991076i	$0.542556\pi$
$992$	−1.46410	−0.0464853
$993$	0	0
$994$	−13.8564	−0.439499
$995$	0	0
$996$	0	0
$997$	−12.3923	−0.392468	−0.196234	−	0.980557i	$-0.562871\pi$
$997$	−12.3923	−0.392468	−0.196234	+	0.980557i	$0.562871\pi$
$998$	−22.9282	−0.725780
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.bw.1.2		2
3.2	odd	2		8550.2.a.bo.1.1		2
5.4	even	2		1710.2.a.u.1.2	✓	2
15.14	odd	2		1710.2.a.y.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1710.2.a.u.1.2	✓	2	5.4	even	2
1710.2.a.y.1.1	yes	2	15.14	odd	2
8550.2.a.bo.1.1		2	3.2	odd	2
8550.2.a.bw.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{7} +1.00000 q^{8} +3.46410 q^{11} -5.46410 q^{13} -2.00000 q^{14} +1.00000 q^{16} -3.46410 q^{17} +1.00000 q^{19} +3.46410 q^{22} +6.92820 q^{23} -5.46410 q^{26} -2.00000 q^{28} -3.46410 q^{29} -1.46410 q^{31} +1.00000 q^{32} -3.46410 q^{34} +1.46410 q^{37} +1.00000 q^{38} -5.46410 q^{43} +3.46410 q^{44} +6.92820 q^{46} +6.92820 q^{47} -3.00000 q^{49} -5.46410 q^{52} -12.9282 q^{53} -2.00000 q^{56} -3.46410 q^{58} +3.46410 q^{59} +2.00000 q^{61} -1.46410 q^{62} +1.00000 q^{64} -14.9282 q^{67} -3.46410 q^{68} +6.92820 q^{71} +4.92820 q^{73} +1.46410 q^{74} +1.00000 q^{76} -6.92820 q^{77} -1.46410 q^{79} -2.53590 q^{83} -5.46410 q^{86} +3.46410 q^{88} -6.92820 q^{89} +10.9282 q^{91} +6.92820 q^{92} +6.92820 q^{94} -18.3923 q^{97} -3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{7} + 2 q^{8} - 4 q^{13} - 4 q^{14} + 2 q^{16} + 2 q^{19} - 4 q^{26} - 4 q^{28} + 4 q^{31} + 2 q^{32} - 4 q^{37} + 2 q^{38} - 4 q^{43} - 6 q^{49} - 4 q^{52} - 12 q^{53} - 4 q^{56} + 4 q^{61} + 4 q^{62} + 2 q^{64} - 16 q^{67} - 4 q^{73} - 4 q^{74} + 2 q^{76} + 4 q^{79} - 12 q^{83} - 4 q^{86} + 8 q^{91} - 16 q^{97} - 6 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^7 + 2 * q^8 - 4 * q^13 - 4 * q^14 + 2 * q^16 + 2 * q^19 - 4 * q^26 - 4 * q^28 + 4 * q^31 + 2 * q^32 - 4 * q^37 + 2 * q^38 - 4 * q^43 - 6 * q^49 - 4 * q^52 - 12 * q^53 - 4 * q^56 + 4 * q^61 + 4 * q^62 + 2 * q^64 - 16 * q^67 - 4 * q^73 - 4 * q^74 + 2 * q^76 + 4 * q^79 - 12 * q^83 - 4 * q^86 + 8 * q^91 - 16 * q^97 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more