LMFDB - Embedded newform 8550.2.a.cu.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:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1710)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.517638$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.03528 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.03528 q^{7} -1.00000 q^{8} -3.86370 q^{11} +1.03528 q^{13} +1.03528 q^{14} +1.00000 q^{16} +7.46410 q^{17} -1.00000 q^{19} +3.86370 q^{22} -1.46410 q^{23} -1.03528 q^{26} -1.03528 q^{28} +9.52056 q^{29} +2.92820 q^{31} -1.00000 q^{32} -7.46410 q^{34} -6.69213 q^{37} +1.00000 q^{38} +6.69213 q^{41} -1.79315 q^{43} -3.86370 q^{44} +1.46410 q^{46} +9.46410 q^{47} -5.92820 q^{49} +1.03528 q^{52} +6.00000 q^{53} +1.03528 q^{56} -9.52056 q^{58} -12.6264 q^{59} -2.00000 q^{61} -2.92820 q^{62} +1.00000 q^{64} +3.58630 q^{67} +7.46410 q^{68} -15.4548 q^{71} -13.3843 q^{73} +6.69213 q^{74} -1.00000 q^{76} +4.00000 q^{77} -4.00000 q^{79} -6.69213 q^{82} -4.39230 q^{83} +1.79315 q^{86} +3.86370 q^{88} +1.03528 q^{89} -1.07180 q^{91} -1.46410 q^{92} -9.46410 q^{94} +17.2480 q^{97} +5.92820 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{16} + 16 q^{17} - 4 q^{19} + 8 q^{23} - 16 q^{31} - 4 q^{32} - 16 q^{34} + 4 q^{38} - 8 q^{46} + 24 q^{47} + 4 q^{49} + 24 q^{53} - 8 q^{61} + 16 q^{62} + 4 q^{64} + 16 q^{68} - 4 q^{76} + 16 q^{77} - 16 q^{79} + 24 q^{83} - 32 q^{91} + 8 q^{92} - 24 q^{94} - 4 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 4 * q^16 + 16 * q^17 - 4 * q^19 + 8 * q^23 - 16 * q^31 - 4 * q^32 - 16 * q^34 + 4 * q^38 - 8 * q^46 + 24 * q^47 + 4 * q^49 + 24 * q^53 - 8 * q^61 + 16 * q^62 + 4 * q^64 + 16 * q^68 - 4 * q^76 + 16 * q^77 - 16 * q^79 + 24 * q^83 - 32 * q^91 + 8 * q^92 - 24 * q^94 - 4 * 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
$7$	−1.03528	−0.391298	−0.195649	−	0.980674i	$-0.562681\pi$
$7$	−1.03528	−0.391298	−0.195649	+	0.980674i	$0.562681\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−3.86370	−1.16495	−0.582475	−	0.812848i	$-0.697916\pi$
$11$	−3.86370	−1.16495	−0.582475	+	0.812848i	$0.697916\pi$
$12$	0	0
$13$	1.03528	0.287134	0.143567	−	0.989641i	$-0.454143\pi$
$13$	1.03528	0.287134	0.143567	+	0.989641i	$0.454143\pi$
$14$	1.03528	0.276689
$15$	0	0
$16$	1.00000	0.250000
$17$	7.46410	1.81031	0.905155	−	0.425081i	$-0.139754\pi$
$17$	7.46410	1.81031	0.905155	+	0.425081i	$0.139754\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	3.86370	0.823744
$23$	−1.46410	−0.305286	−0.152643	−	0.988281i	$-0.548779\pi$
$23$	−1.46410	−0.305286	−0.152643	+	0.988281i	$0.548779\pi$
$24$	0	0
$25$	0	0
$26$	−1.03528	−0.203034
$27$	0	0
$28$	−1.03528	−0.195649
$29$	9.52056	1.76792	0.883962	−	0.467560i	$-0.154867\pi$
$29$	9.52056	1.76792	0.883962	+	0.467560i	$0.154867\pi$
$30$	0	0
$31$	2.92820	0.525921	0.262960	−	0.964807i	$-0.415301\pi$
$31$	2.92820	0.525921	0.262960	+	0.964807i	$0.415301\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−7.46410	−1.28008
$35$	0	0
$36$	0	0
$37$	−6.69213	−1.10018	−0.550090	−	0.835106i	$-0.685406\pi$
$37$	−6.69213	−1.10018	−0.550090	+	0.835106i	$0.685406\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	6.69213	1.04514	0.522568	−	0.852598i	$-0.324974\pi$
$41$	6.69213	1.04514	0.522568	+	0.852598i	$0.324974\pi$
$42$	0	0
$43$	−1.79315	−0.273453	−0.136726	−	0.990609i	$-0.543658\pi$
$43$	−1.79315	−0.273453	−0.136726	+	0.990609i	$0.543658\pi$
$44$	−3.86370	−0.582475
$45$	0	0
$46$	1.46410	0.215870
$47$	9.46410	1.38048	0.690241	−	0.723580i	$-0.257505\pi$
$47$	9.46410	1.38048	0.690241	+	0.723580i	$0.257505\pi$
$48$	0	0
$49$	−5.92820	−0.846886
$50$	0	0
$51$	0	0
$52$	1.03528	0.143567
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	1.03528	0.138345
$57$	0	0
$58$	−9.52056	−1.25011
$59$	−12.6264	−1.64382	−0.821908	−	0.569621i	$-0.807090\pi$
$59$	−12.6264	−1.64382	−0.821908	+	0.569621i	$0.807090\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−2.92820	−0.371882
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	3.58630	0.438137	0.219068	−	0.975710i	$-0.429698\pi$
$67$	3.58630	0.438137	0.219068	+	0.975710i	$0.429698\pi$
$68$	7.46410	0.905155
$69$	0	0
$70$	0	0
$71$	−15.4548	−1.83415	−0.917074	−	0.398716i	$-0.869456\pi$
$71$	−15.4548	−1.83415	−0.917074	+	0.398716i	$0.869456\pi$
$72$	0	0
$73$	−13.3843	−1.56651	−0.783255	−	0.621701i	$-0.786442\pi$
$73$	−13.3843	−1.56651	−0.783255	+	0.621701i	$0.786442\pi$
$74$	6.69213	0.777944
$75$	0	0
$76$	−1.00000	−0.114708
$77$	4.00000	0.455842
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	0	0
$82$	−6.69213	−0.739022
$83$	−4.39230	−0.482118	−0.241059	−	0.970510i	$-0.577495\pi$
$83$	−4.39230	−0.482118	−0.241059	+	0.970510i	$0.577495\pi$
$84$	0	0
$85$	0	0
$86$	1.79315	0.193360
$87$	0	0
$88$	3.86370	0.411872
$89$	1.03528	0.109739	0.0548695	−	0.998494i	$-0.482526\pi$
$89$	1.03528	0.109739	0.0548695	+	0.998494i	$0.482526\pi$
$90$	0	0
$91$	−1.07180	−0.112355
$92$	−1.46410	−0.152643
$93$	0	0
$94$	−9.46410	−0.976148
$95$	0	0
$96$	0	0
$97$	17.2480	1.75127	0.875633	−	0.482978i	$-0.160445\pi$
$97$	17.2480	1.75127	0.875633	+	0.482978i	$0.160445\pi$
$98$	5.92820	0.598839
$99$	0	0
$100$	0	0
$101$	−0.757875	−0.0754114	−0.0377057	−	0.999289i	$-0.512005\pi$
$101$	−0.757875	−0.0754114	−0.0377057	+	0.999289i	$0.512005\pi$
$102$	0	0
$103$	−4.89898	−0.482711	−0.241355	−	0.970437i	$-0.577592\pi$
$103$	−4.89898	−0.482711	−0.241355	+	0.970437i	$0.577592\pi$
$104$	−1.03528	−0.101517
$105$	0	0
$106$	−6.00000	−0.582772
$107$	10.9282	1.05647	0.528235	−	0.849098i	$-0.322854\pi$
$107$	10.9282	1.05647	0.528235	+	0.849098i	$0.322854\pi$
$108$	0	0
$109$	−4.92820	−0.472036	−0.236018	−	0.971749i	$-0.575842\pi$
$109$	−4.92820	−0.472036	−0.236018	+	0.971749i	$0.575842\pi$
$110$	0	0
$111$	0	0
$112$	−1.03528	−0.0978244
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	9.52056	0.883962
$117$	0	0
$118$	12.6264	1.16235
$119$	−7.72741	−0.708370
$120$	0	0
$121$	3.92820	0.357109
$122$	2.00000	0.181071
$123$	0	0
$124$	2.92820	0.262960
$125$	0	0
$126$	0	0
$127$	8.48528	0.752947	0.376473	−	0.926427i	$-0.377137\pi$
$127$	8.48528	0.752947	0.376473	+	0.926427i	$0.377137\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−11.5911	−1.01272	−0.506360	−	0.862322i	$-0.669009\pi$
$131$	−11.5911	−1.01272	−0.506360	+	0.862322i	$0.669009\pi$
$132$	0	0
$133$	1.03528	0.0897698
$134$	−3.58630	−0.309809
$135$	0	0
$136$	−7.46410	−0.640041
$137$	7.46410	0.637701	0.318851	−	0.947805i	$-0.396703\pi$
$137$	7.46410	0.637701	0.318851	+	0.947805i	$0.396703\pi$
$138$	0	0
$139$	−6.92820	−0.587643	−0.293821	−	0.955860i	$-0.594927\pi$
$139$	−6.92820	−0.587643	−0.293821	+	0.955860i	$0.594927\pi$
$140$	0	0
$141$	0	0
$142$	15.4548	1.29694
$143$	−4.00000	−0.334497
$144$	0	0
$145$	0	0
$146$	13.3843	1.10769
$147$	0	0
$148$	−6.69213	−0.550090
$149$	−9.04008	−0.740593	−0.370296	−	0.928914i	$-0.620744\pi$
$149$	−9.04008	−0.740593	−0.370296	+	0.928914i	$0.620744\pi$
$150$	0	0
$151$	−21.8564	−1.77865	−0.889325	−	0.457277i	$-0.848825\pi$
$151$	−21.8564	−1.77865	−0.889325	+	0.457277i	$0.848825\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	−4.00000	−0.322329
$155$	0	0
$156$	0	0
$157$	14.1421	1.12867	0.564333	−	0.825547i	$-0.309134\pi$
$157$	14.1421	1.12867	0.564333	+	0.825547i	$0.309134\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	0	0
$161$	1.51575	0.119458
$162$	0	0
$163$	18.7637	1.46969	0.734844	−	0.678236i	$-0.237255\pi$
$163$	18.7637	1.46969	0.734844	+	0.678236i	$0.237255\pi$
$164$	6.69213	0.522568
$165$	0	0
$166$	4.39230	0.340909
$167$	−2.92820	−0.226591	−0.113296	−	0.993561i	$-0.536141\pi$
$167$	−2.92820	−0.226591	−0.113296	+	0.993561i	$0.536141\pi$
$168$	0	0
$169$	−11.9282	−0.917554
$170$	0	0
$171$	0	0
$172$	−1.79315	−0.136726
$173$	18.7846	1.42817	0.714084	−	0.700060i	$-0.246844\pi$
$173$	18.7846	1.42817	0.714084	+	0.700060i	$0.246844\pi$
$174$	0	0
$175$	0	0
$176$	−3.86370	−0.291238
$177$	0	0
$178$	−1.03528	−0.0775972
$179$	22.4243	1.67607	0.838037	−	0.545613i	$-0.183703\pi$
$179$	22.4243	1.67607	0.838037	+	0.545613i	$0.183703\pi$
$180$	0	0
$181$	−7.85641	−0.583962	−0.291981	−	0.956424i	$-0.594314\pi$
$181$	−7.85641	−0.583962	−0.291981	+	0.956424i	$0.594314\pi$
$182$	1.07180	0.0794469
$183$	0	0
$184$	1.46410	0.107935
$185$	0	0
$186$	0	0
$187$	−28.8391	−2.10892
$188$	9.46410	0.690241
$189$	0	0
$190$	0	0
$191$	−2.55103	−0.184586	−0.0922929	−	0.995732i	$-0.529420\pi$
$191$	−2.55103	−0.184586	−0.0922929	+	0.995732i	$0.529420\pi$
$192$	0	0
$193$	−5.93426	−0.427157	−0.213579	−	0.976926i	$-0.568512\pi$
$193$	−5.93426	−0.427157	−0.213579	+	0.976926i	$0.568512\pi$
$194$	−17.2480	−1.23833
$195$	0	0
$196$	−5.92820	−0.423443
$197$	16.5359	1.17813	0.589067	−	0.808084i	$-0.299495\pi$
$197$	16.5359	1.17813	0.589067	+	0.808084i	$0.299495\pi$
$198$	0	0
$199$	26.9282	1.90889	0.954445	−	0.298387i	$-0.0964487\pi$
$199$	26.9282	1.90889	0.954445	+	0.298387i	$0.0964487\pi$
$200$	0	0
$201$	0	0
$202$	0.757875	0.0533239
$203$	−9.85641	−0.691784
$204$	0	0
$205$	0	0
$206$	4.89898	0.341328
$207$	0	0
$208$	1.03528	0.0717835
$209$	3.86370	0.267258
$210$	0	0
$211$	9.85641	0.678543	0.339272	−	0.940688i	$-0.389819\pi$
$211$	9.85641	0.678543	0.339272	+	0.940688i	$0.389819\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−10.9282	−0.747037
$215$	0	0
$216$	0	0
$217$	−3.03150	−0.205792
$218$	4.92820	0.333780
$219$	0	0
$220$	0	0
$221$	7.72741	0.519802
$222$	0	0
$223$	−0.757875	−0.0507510	−0.0253755	−	0.999678i	$-0.508078\pi$
$223$	−0.757875	−0.0507510	−0.0253755	+	0.999678i	$0.508078\pi$
$224$	1.03528	0.0691723
$225$	0	0
$226$	−6.00000	−0.399114
$227$	25.8564	1.71615	0.858075	−	0.513524i	$-0.171660\pi$
$227$	25.8564	1.71615	0.858075	+	0.513524i	$0.171660\pi$
$228$	0	0
$229$	23.8564	1.57648	0.788238	−	0.615371i	$-0.210994\pi$
$229$	23.8564	1.57648	0.788238	+	0.615371i	$0.210994\pi$
$230$	0	0
$231$	0	0
$232$	−9.52056	−0.625055
$233$	19.4641	1.27514	0.637568	−	0.770394i	$-0.279941\pi$
$233$	19.4641	1.27514	0.637568	+	0.770394i	$0.279941\pi$
$234$	0	0
$235$	0	0
$236$	−12.6264	−0.821908
$237$	0	0
$238$	7.72741	0.500893
$239$	18.0058	1.16470	0.582350	−	0.812938i	$-0.302133\pi$
$239$	18.0058	1.16470	0.582350	+	0.812938i	$0.302133\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−3.92820	−0.252514
$243$	0	0
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	−1.03528	−0.0658730
$248$	−2.92820	−0.185941
$249$	0	0
$250$	0	0
$251$	25.5302	1.61145	0.805725	−	0.592290i	$-0.201776\pi$
$251$	25.5302	1.61145	0.805725	+	0.592290i	$0.201776\pi$
$252$	0	0
$253$	5.65685	0.355643
$254$	−8.48528	−0.532414
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.7846	−1.17175	−0.585876	−	0.810401i	$-0.699249\pi$
$257$	−18.7846	−1.17175	−0.585876	+	0.810401i	$0.699249\pi$
$258$	0	0
$259$	6.92820	0.430498
$260$	0	0
$261$	0	0
$262$	11.5911	0.716101
$263$	−15.3205	−0.944703	−0.472351	−	0.881410i	$-0.656595\pi$
$263$	−15.3205	−0.944703	−0.472351	+	0.881410i	$0.656595\pi$
$264$	0	0
$265$	0	0
$266$	−1.03528	−0.0634769
$267$	0	0
$268$	3.58630	0.219068
$269$	12.1459	0.740549	0.370275	−	0.928922i	$-0.379264\pi$
$269$	12.1459	0.740549	0.370275	+	0.928922i	$0.379264\pi$
$270$	0	0
$271$	−2.92820	−0.177876	−0.0889378	−	0.996037i	$-0.528347\pi$
$271$	−2.92820	−0.177876	−0.0889378	+	0.996037i	$0.528347\pi$
$272$	7.46410	0.452578
$273$	0	0
$274$	−7.46410	−0.450923
$275$	0	0
$276$	0	0
$277$	1.31268	0.0788712	0.0394356	−	0.999222i	$-0.487444\pi$
$277$	1.31268	0.0788712	0.0394356	+	0.999222i	$0.487444\pi$
$278$	6.92820	0.415526
$279$	0	0
$280$	0	0
$281$	22.7017	1.35427	0.677136	−	0.735858i	$-0.263221\pi$
$281$	22.7017	1.35427	0.677136	+	0.735858i	$0.263221\pi$
$282$	0	0
$283$	−19.3185	−1.14837	−0.574183	−	0.818727i	$-0.694680\pi$
$283$	−19.3185	−1.14837	−0.574183	+	0.818727i	$0.694680\pi$
$284$	−15.4548	−0.917074
$285$	0	0
$286$	4.00000	0.236525
$287$	−6.92820	−0.408959
$288$	0	0
$289$	38.7128	2.27722
$290$	0	0
$291$	0	0
$292$	−13.3843	−0.783255
$293$	7.85641	0.458976	0.229488	−	0.973311i	$-0.426295\pi$
$293$	7.85641	0.458976	0.229488	+	0.973311i	$0.426295\pi$
$294$	0	0
$295$	0	0
$296$	6.69213	0.388972
$297$	0	0
$298$	9.04008	0.523678
$299$	−1.51575	−0.0876581
$300$	0	0
$301$	1.85641	0.107001
$302$	21.8564	1.25769
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	−11.8685	−0.677372	−0.338686	−	0.940900i	$-0.609982\pi$
$307$	−11.8685	−0.677372	−0.338686	+	0.940900i	$0.609982\pi$
$308$	4.00000	0.227921
$309$	0	0
$310$	0	0
$311$	−6.69213	−0.379476	−0.189738	−	0.981835i	$-0.560764\pi$
$311$	−6.69213	−0.379476	−0.189738	+	0.981835i	$0.560764\pi$
$312$	0	0
$313$	19.5959	1.10763	0.553813	−	0.832641i	$-0.313172\pi$
$313$	19.5959	1.10763	0.553813	+	0.832641i	$0.313172\pi$
$314$	−14.1421	−0.798087
$315$	0	0
$316$	−4.00000	−0.225018
$317$	14.0000	0.786318	0.393159	−	0.919470i	$-0.371382\pi$
$317$	14.0000	0.786318	0.393159	+	0.919470i	$0.371382\pi$
$318$	0	0
$319$	−36.7846	−2.05954
$320$	0	0
$321$	0	0
$322$	−1.51575	−0.0844694
$323$	−7.46410	−0.415314
$324$	0	0
$325$	0	0
$326$	−18.7637	−1.03923
$327$	0	0
$328$	−6.69213	−0.369511
$329$	−9.79796	−0.540179
$330$	0	0
$331$	−13.8564	−0.761617	−0.380808	−	0.924654i	$-0.624354\pi$
$331$	−13.8564	−0.761617	−0.380808	+	0.924654i	$0.624354\pi$
$332$	−4.39230	−0.241059
$333$	0	0
$334$	2.92820	0.160224
$335$	0	0
$336$	0	0
$337$	−3.86370	−0.210469	−0.105235	−	0.994447i	$-0.533559\pi$
$337$	−3.86370	−0.210469	−0.105235	+	0.994447i	$0.533559\pi$
$338$	11.9282	0.648809
$339$	0	0
$340$	0	0
$341$	−11.3137	−0.612672
$342$	0	0
$343$	13.3843	0.722682
$344$	1.79315	0.0966802
$345$	0	0
$346$	−18.7846	−1.00987
$347$	25.4641	1.36698	0.683492	−	0.729958i	$-0.260460\pi$
$347$	25.4641	1.36698	0.683492	+	0.729958i	$0.260460\pi$
$348$	0	0
$349$	31.8564	1.70523	0.852617	−	0.522536i	$-0.175014\pi$
$349$	31.8564	1.70523	0.852617	+	0.522536i	$0.175014\pi$
$350$	0	0
$351$	0	0
$352$	3.86370	0.205936
$353$	−13.3205	−0.708979	−0.354490	−	0.935060i	$-0.615345\pi$
$353$	−13.3205	−0.708979	−0.354490	+	0.935060i	$0.615345\pi$
$354$	0	0
$355$	0	0
$356$	1.03528	0.0548695
$357$	0	0
$358$	−22.4243	−1.18516
$359$	−9.31749	−0.491758	−0.245879	−	0.969301i	$-0.579077\pi$
$359$	−9.31749	−0.491758	−0.245879	+	0.969301i	$0.579077\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	7.85641	0.412924
$363$	0	0
$364$	−1.07180	−0.0561774
$365$	0	0
$366$	0	0
$367$	−20.0764	−1.04798	−0.523990	−	0.851725i	$-0.675557\pi$
$367$	−20.0764	−1.04798	−0.523990	+	0.851725i	$0.675557\pi$
$368$	−1.46410	−0.0763216
$369$	0	0
$370$	0	0
$371$	−6.21166	−0.322493
$372$	0	0
$373$	−0.480473	−0.0248780	−0.0124390	−	0.999923i	$-0.503960\pi$
$373$	−0.480473	−0.0248780	−0.0124390	+	0.999923i	$0.503960\pi$
$374$	28.8391	1.49123
$375$	0	0
$376$	−9.46410	−0.488074
$377$	9.85641	0.507631
$378$	0	0
$379$	19.7128	1.01258	0.506290	−	0.862364i	$-0.331017\pi$
$379$	19.7128	1.01258	0.506290	+	0.862364i	$0.331017\pi$
$380$	0	0
$381$	0	0
$382$	2.55103	0.130522
$383$	−9.07180	−0.463547	−0.231774	−	0.972770i	$-0.574453\pi$
$383$	−9.07180	−0.463547	−0.231774	+	0.972770i	$0.574453\pi$
$384$	0	0
$385$	0	0
$386$	5.93426	0.302046
$387$	0	0
$388$	17.2480	0.875633
$389$	−19.7990	−1.00385	−0.501924	−	0.864912i	$-0.667374\pi$
$389$	−19.7990	−1.00385	−0.501924	+	0.864912i	$0.667374\pi$
$390$	0	0
$391$	−10.9282	−0.552663
$392$	5.92820	0.299419
$393$	0	0
$394$	−16.5359	−0.833067
$395$	0	0
$396$	0	0
$397$	−1.31268	−0.0658814	−0.0329407	−	0.999457i	$-0.510487\pi$
$397$	−1.31268	−0.0658814	−0.0329407	+	0.999457i	$0.510487\pi$
$398$	−26.9282	−1.34979
$399$	0	0
$400$	0	0
$401$	4.62158	0.230791	0.115395	−	0.993320i	$-0.463187\pi$
$401$	4.62158	0.230791	0.115395	+	0.993320i	$0.463187\pi$
$402$	0	0
$403$	3.03150	0.151010
$404$	−0.757875	−0.0377057
$405$	0	0
$406$	9.85641	0.489165
$407$	25.8564	1.28165
$408$	0	0
$409$	−7.07180	−0.349678	−0.174839	−	0.984597i	$-0.555940\pi$
$409$	−7.07180	−0.349678	−0.174839	+	0.984597i	$0.555940\pi$
$410$	0	0
$411$	0	0
$412$	−4.89898	−0.241355
$413$	13.0718	0.643221
$414$	0	0
$415$	0	0
$416$	−1.03528	−0.0507586
$417$	0	0
$418$	−3.86370	−0.188980
$419$	24.4206	1.19302	0.596511	−	0.802605i	$-0.296553\pi$
$419$	24.4206	1.19302	0.596511	+	0.802605i	$0.296553\pi$
$420$	0	0
$421$	−25.7128	−1.25317	−0.626583	−	0.779355i	$-0.715547\pi$
$421$	−25.7128	−1.25317	−0.626583	+	0.779355i	$0.715547\pi$
$422$	−9.85641	−0.479802
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	2.07055	0.100201
$428$	10.9282	0.528235
$429$	0	0
$430$	0	0
$431$	−36.0117	−1.73462	−0.867311	−	0.497767i	$-0.834153\pi$
$431$	−36.0117	−1.73462	−0.867311	+	0.497767i	$0.834153\pi$
$432$	0	0
$433$	13.6617	0.656538	0.328269	−	0.944584i	$-0.393535\pi$
$433$	13.6617	0.656538	0.328269	+	0.944584i	$0.393535\pi$
$434$	3.03150	0.145517
$435$	0	0
$436$	−4.92820	−0.236018
$437$	1.46410	0.0700375
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	0	0
$442$	−7.72741	−0.367555
$443$	−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
$446$	0.757875	0.0358864
$447$	0	0
$448$	−1.03528	−0.0489122
$449$	3.66063	0.172756	0.0863779	−	0.996262i	$-0.472471\pi$
$449$	3.66063	0.172756	0.0863779	+	0.996262i	$0.472471\pi$
$450$	0	0
$451$	−25.8564	−1.21753
$452$	6.00000	0.282216
$453$	0	0
$454$	−25.8564	−1.21350
$455$	0	0
$456$	0	0
$457$	−33.9411	−1.58770	−0.793849	−	0.608114i	$-0.791926\pi$
$457$	−33.9411	−1.58770	−0.793849	+	0.608114i	$0.791926\pi$
$458$	−23.8564	−1.11474
$459$	0	0
$460$	0	0
$461$	17.7284	0.825696	0.412848	−	0.910800i	$-0.364534\pi$
$461$	17.7284	0.825696	0.412848	+	0.910800i	$0.364534\pi$
$462$	0	0
$463$	−34.0155	−1.58083	−0.790416	−	0.612570i	$-0.790136\pi$
$463$	−34.0155	−1.58083	−0.790416	+	0.612570i	$0.790136\pi$
$464$	9.52056	0.441981
$465$	0	0
$466$	−19.4641	−0.901657
$467$	9.46410	0.437946	0.218973	−	0.975731i	$-0.429729\pi$
$467$	9.46410	0.437946	0.218973	+	0.975731i	$0.429729\pi$
$468$	0	0
$469$	−3.71281	−0.171442
$470$	0	0
$471$	0	0
$472$	12.6264	0.581177
$473$	6.92820	0.318559
$474$	0	0
$475$	0	0
$476$	−7.72741	−0.354185
$477$	0	0
$478$	−18.0058	−0.823568
$479$	31.3901	1.43425	0.717125	−	0.696944i	$-0.245458\pi$
$479$	31.3901	1.43425	0.717125	+	0.696944i	$0.245458\pi$
$480$	0	0
$481$	−6.92820	−0.315899
$482$	−10.0000	−0.455488
$483$	0	0
$484$	3.92820	0.178555
$485$	0	0
$486$	0	0
$487$	26.0106	1.17865	0.589327	−	0.807894i	$-0.299393\pi$
$487$	26.0106	1.17865	0.589327	+	0.807894i	$0.299393\pi$
$488$	2.00000	0.0905357
$489$	0	0
$490$	0	0
$491$	−15.7322	−0.709985	−0.354992	−	0.934869i	$-0.615517\pi$
$491$	−15.7322	−0.709985	−0.354992	+	0.934869i	$0.615517\pi$
$492$	0	0
$493$	71.0624	3.20049
$494$	1.03528	0.0465793
$495$	0	0
$496$	2.92820	0.131480
$497$	16.0000	0.717698
$498$	0	0
$499$	−1.85641	−0.0831042	−0.0415521	−	0.999136i	$-0.513230\pi$
$499$	−1.85641	−0.0831042	−0.0415521	+	0.999136i	$0.513230\pi$
$500$	0	0
$501$	0	0
$502$	−25.5302	−1.13947
$503$	23.3205	1.03981	0.519905	−	0.854224i	$-0.325967\pi$
$503$	23.3205	1.03981	0.519905	+	0.854224i	$0.325967\pi$
$504$	0	0
$505$	0	0
$506$	−5.65685	−0.251478
$507$	0	0
$508$	8.48528	0.376473
$509$	−22.3500	−0.990647	−0.495324	−	0.868709i	$-0.664950\pi$
$509$	−22.3500	−0.990647	−0.495324	+	0.868709i	$0.664950\pi$
$510$	0	0
$511$	13.8564	0.612971
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	18.7846	0.828554
$515$	0	0
$516$	0	0
$517$	−36.5665	−1.60819
$518$	−6.92820	−0.304408
$519$	0	0
$520$	0	0
$521$	4.62158	0.202475	0.101238	−	0.994862i	$-0.467720\pi$
$521$	4.62158	0.202475	0.101238	+	0.994862i	$0.467720\pi$
$522$	0	0
$523$	7.17260	0.313636	0.156818	−	0.987628i	$-0.449876\pi$
$523$	7.17260	0.313636	0.156818	+	0.987628i	$0.449876\pi$
$524$	−11.5911	−0.506360
$525$	0	0
$526$	15.3205	0.668006
$527$	21.8564	0.952080
$528$	0	0
$529$	−20.8564	−0.906800
$530$	0	0
$531$	0	0
$532$	1.03528	0.0448849
$533$	6.92820	0.300094
$534$	0	0
$535$	0	0
$536$	−3.58630	−0.154905
$537$	0	0
$538$	−12.1459	−0.523647
$539$	22.9048	0.986580
$540$	0	0
$541$	26.0000	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$542$	2.92820	0.125777
$543$	0	0
$544$	−7.46410	−0.320021
$545$	0	0
$546$	0	0
$547$	38.6370	1.65200	0.826000	−	0.563670i	$-0.190611\pi$
$547$	38.6370	1.65200	0.826000	+	0.563670i	$0.190611\pi$
$548$	7.46410	0.318851
$549$	0	0
$550$	0	0
$551$	−9.52056	−0.405589
$552$	0	0
$553$	4.14110	0.176098
$554$	−1.31268	−0.0557703
$555$	0	0
$556$	−6.92820	−0.293821
$557$	24.5359	1.03962	0.519810	−	0.854282i	$-0.326003\pi$
$557$	24.5359	1.03962	0.519810	+	0.854282i	$0.326003\pi$
$558$	0	0
$559$	−1.85641	−0.0785176
$560$	0	0
$561$	0	0
$562$	−22.7017	−0.957615
$563$	−1.85641	−0.0782382	−0.0391191	−	0.999235i	$-0.512455\pi$
$563$	−1.85641	−0.0782382	−0.0391191	+	0.999235i	$0.512455\pi$
$564$	0	0
$565$	0	0
$566$	19.3185	0.812018
$567$	0	0
$568$	15.4548	0.648470
$569$	9.72363	0.407636	0.203818	−	0.979009i	$-0.434665\pi$
$569$	9.72363	0.407636	0.203818	+	0.979009i	$0.434665\pi$
$570$	0	0
$571$	1.07180	0.0448533	0.0224266	−	0.999748i	$-0.492861\pi$
$571$	1.07180	0.0448533	0.0224266	+	0.999748i	$0.492861\pi$
$572$	−4.00000	−0.167248
$573$	0	0
$574$	6.92820	0.289178
$575$	0	0
$576$	0	0
$577$	5.65685	0.235498	0.117749	−	0.993043i	$-0.462432\pi$
$577$	5.65685	0.235498	0.117749	+	0.993043i	$0.462432\pi$
$578$	−38.7128	−1.61024
$579$	0	0
$580$	0	0
$581$	4.54725	0.188652
$582$	0	0
$583$	−23.1822	−0.960109
$584$	13.3843	0.553845
$585$	0	0
$586$	−7.85641	−0.324545
$587$	3.60770	0.148906	0.0744528	−	0.997225i	$-0.476279\pi$
$587$	3.60770	0.148906	0.0744528	+	0.997225i	$0.476279\pi$
$588$	0	0
$589$	−2.92820	−0.120655
$590$	0	0
$591$	0	0
$592$	−6.69213	−0.275045
$593$	−11.4641	−0.470774	−0.235387	−	0.971902i	$-0.575636\pi$
$593$	−11.4641	−0.470774	−0.235387	+	0.971902i	$0.575636\pi$
$594$	0	0
$595$	0	0
$596$	−9.04008	−0.370296
$597$	0	0
$598$	1.51575	0.0619836
$599$	−30.3548	−1.24026	−0.620132	−	0.784497i	$-0.712921\pi$
$599$	−30.3548	−1.24026	−0.620132	+	0.784497i	$0.712921\pi$
$600$	0	0
$601$	22.7846	0.929404	0.464702	−	0.885467i	$-0.346162\pi$
$601$	22.7846	0.929404	0.464702	+	0.885467i	$0.346162\pi$
$602$	−1.85641	−0.0756615
$603$	0	0
$604$	−21.8564	−0.889325
$605$	0	0
$606$	0	0
$607$	22.9791	0.932695	0.466347	−	0.884602i	$-0.345570\pi$
$607$	22.9791	0.932695	0.466347	+	0.884602i	$0.345570\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	9.79796	0.396383
$612$	0	0
$613$	−45.0518	−1.81962	−0.909812	−	0.415021i	$-0.863774\pi$
$613$	−45.0518	−1.81962	−0.909812	+	0.415021i	$0.863774\pi$
$614$	11.8685	0.478974
$615$	0	0
$616$	−4.00000	−0.161165
$617$	−20.2487	−0.815182	−0.407591	−	0.913165i	$-0.633631\pi$
$617$	−20.2487	−0.815182	−0.407591	+	0.913165i	$0.633631\pi$
$618$	0	0
$619$	34.6410	1.39234	0.696170	−	0.717877i	$-0.254886\pi$
$619$	34.6410	1.39234	0.696170	+	0.717877i	$0.254886\pi$
$620$	0	0
$621$	0	0
$622$	6.69213	0.268330
$623$	−1.07180	−0.0429406
$624$	0	0
$625$	0	0
$626$	−19.5959	−0.783210
$627$	0	0
$628$	14.1421	0.564333
$629$	−49.9507	−1.99167
$630$	0	0
$631$	−5.85641	−0.233140	−0.116570	−	0.993182i	$-0.537190\pi$
$631$	−5.85641	−0.233140	−0.116570	+	0.993182i	$0.537190\pi$
$632$	4.00000	0.159111
$633$	0	0
$634$	−14.0000	−0.556011
$635$	0	0
$636$	0	0
$637$	−6.13733	−0.243170
$638$	36.7846	1.45632
$639$	0	0
$640$	0	0
$641$	2.55103	0.100759	0.0503797	−	0.998730i	$-0.483957\pi$
$641$	2.55103	0.100759	0.0503797	+	0.998730i	$0.483957\pi$
$642$	0	0
$643$	−22.3500	−0.881399	−0.440699	−	0.897655i	$-0.645269\pi$
$643$	−22.3500	−0.881399	−0.440699	+	0.897655i	$0.645269\pi$
$644$	1.51575	0.0597289
$645$	0	0
$646$	7.46410	0.293671
$647$	6.53590	0.256953	0.128476	−	0.991713i	$-0.458991\pi$
$647$	6.53590	0.256953	0.128476	+	0.991713i	$0.458991\pi$
$648$	0	0
$649$	48.7846	1.91496
$650$	0	0
$651$	0	0
$652$	18.7637	0.734844
$653$	24.2487	0.948925	0.474463	−	0.880276i	$-0.342642\pi$
$653$	24.2487	0.948925	0.474463	+	0.880276i	$0.342642\pi$
$654$	0	0
$655$	0	0
$656$	6.69213	0.261284
$657$	0	0
$658$	9.79796	0.381964
$659$	−9.04008	−0.352152	−0.176076	−	0.984377i	$-0.556340\pi$
$659$	−9.04008	−0.352152	−0.176076	+	0.984377i	$0.556340\pi$
$660$	0	0
$661$	30.7846	1.19738	0.598691	−	0.800980i	$-0.295688\pi$
$661$	30.7846	1.19738	0.598691	+	0.800980i	$0.295688\pi$
$662$	13.8564	0.538545
$663$	0	0
$664$	4.39230	0.170454
$665$	0	0
$666$	0	0
$667$	−13.9391	−0.539723
$668$	−2.92820	−0.113296
$669$	0	0
$670$	0	0
$671$	7.72741	0.298313
$672$	0	0
$673$	−32.1480	−1.23921	−0.619607	−	0.784912i	$-0.712708\pi$
$673$	−32.1480	−1.23921	−0.619607	+	0.784912i	$0.712708\pi$
$674$	3.86370	0.148824
$675$	0	0
$676$	−11.9282	−0.458777
$677$	−32.6410	−1.25450	−0.627248	−	0.778820i	$-0.715819\pi$
$677$	−32.6410	−1.25450	−0.627248	+	0.778820i	$0.715819\pi$
$678$	0	0
$679$	−17.8564	−0.685266
$680$	0	0
$681$	0	0
$682$	11.3137	0.433224
$683$	5.07180	0.194067	0.0970335	−	0.995281i	$-0.469065\pi$
$683$	5.07180	0.194067	0.0970335	+	0.995281i	$0.469065\pi$
$684$	0	0
$685$	0	0
$686$	−13.3843	−0.511013
$687$	0	0
$688$	−1.79315	−0.0683632
$689$	6.21166	0.236645
$690$	0	0
$691$	25.0718	0.953776	0.476888	−	0.878964i	$-0.341765\pi$
$691$	25.0718	0.953776	0.476888	+	0.878964i	$0.341765\pi$
$692$	18.7846	0.714084
$693$	0	0
$694$	−25.4641	−0.966604
$695$	0	0
$696$	0	0
$697$	49.9507	1.89202
$698$	−31.8564	−1.20578
$699$	0	0
$700$	0	0
$701$	41.4655	1.56613	0.783064	−	0.621941i	$-0.213655\pi$
$701$	41.4655	1.56613	0.783064	+	0.621941i	$0.213655\pi$
$702$	0	0
$703$	6.69213	0.252398
$704$	−3.86370	−0.145619
$705$	0	0
$706$	13.3205	0.501324
$707$	0.784610	0.0295083
$708$	0	0
$709$	15.8564	0.595500	0.297750	−	0.954644i	$-0.403764\pi$
$709$	15.8564	0.595500	0.297750	+	0.954644i	$0.403764\pi$
$710$	0	0
$711$	0	0
$712$	−1.03528	−0.0387986
$713$	−4.28719	−0.160556
$714$	0	0
$715$	0	0
$716$	22.4243	0.838037
$717$	0	0
$718$	9.31749	0.347725
$719$	−26.8429	−1.00107	−0.500535	−	0.865716i	$-0.666863\pi$
$719$	−26.8429	−1.00107	−0.500535	+	0.865716i	$0.666863\pi$
$720$	0	0
$721$	5.07180	0.188884
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−7.85641	−0.291981
$725$	0	0
$726$	0	0
$727$	−10.2784	−0.381206	−0.190603	−	0.981667i	$-0.561044\pi$
$727$	−10.2784	−0.381206	−0.190603	+	0.981667i	$0.561044\pi$
$728$	1.07180	0.0397234
$729$	0	0
$730$	0	0
$731$	−13.3843	−0.495035
$732$	0	0
$733$	−12.0716	−0.445874	−0.222937	−	0.974833i	$-0.571564\pi$
$733$	−12.0716	−0.445874	−0.222937	+	0.974833i	$0.571564\pi$
$734$	20.0764	0.741033
$735$	0	0
$736$	1.46410	0.0539675
$737$	−13.8564	−0.510407
$738$	0	0
$739$	−31.7128	−1.16657	−0.583287	−	0.812266i	$-0.698234\pi$
$739$	−31.7128	−1.16657	−0.583287	+	0.812266i	$0.698234\pi$
$740$	0	0
$741$	0	0
$742$	6.21166	0.228037
$743$	34.6410	1.27086	0.635428	−	0.772160i	$-0.280824\pi$
$743$	34.6410	1.27086	0.635428	+	0.772160i	$0.280824\pi$
$744$	0	0
$745$	0	0
$746$	0.480473	0.0175914
$747$	0	0
$748$	−28.8391	−1.05446
$749$	−11.3137	−0.413394
$750$	0	0
$751$	−21.8564	−0.797552	−0.398776	−	0.917048i	$-0.630565\pi$
$751$	−21.8564	−0.797552	−0.398776	+	0.917048i	$0.630565\pi$
$752$	9.46410	0.345120
$753$	0	0
$754$	−9.85641	−0.358949
$755$	0	0
$756$	0	0
$757$	50.1538	1.82287	0.911436	−	0.411443i	$-0.134975\pi$
$757$	50.1538	1.82287	0.911436	+	0.411443i	$0.134975\pi$
$758$	−19.7128	−0.716002
$759$	0	0
$760$	0	0
$761$	−6.21166	−0.225172	−0.112586	−	0.993642i	$-0.535913\pi$
$761$	−6.21166	−0.225172	−0.112586	+	0.993642i	$0.535913\pi$
$762$	0	0
$763$	5.10205	0.184707
$764$	−2.55103	−0.0922929
$765$	0	0
$766$	9.07180	0.327777
$767$	−13.0718	−0.471995
$768$	0	0
$769$	8.14359	0.293665	0.146833	−	0.989161i	$-0.453092\pi$
$769$	8.14359	0.293665	0.146833	+	0.989161i	$0.453092\pi$
$770$	0	0
$771$	0	0
$772$	−5.93426	−0.213579
$773$	19.8564	0.714185	0.357093	−	0.934069i	$-0.383768\pi$
$773$	19.8564	0.714185	0.357093	+	0.934069i	$0.383768\pi$
$774$	0	0
$775$	0	0
$776$	−17.2480	−0.619166
$777$	0	0
$778$	19.7990	0.709828
$779$	−6.69213	−0.239770
$780$	0	0
$781$	59.7128	2.13669
$782$	10.9282	0.390792
$783$	0	0
$784$	−5.92820	−0.211722
$785$	0	0
$786$	0	0
$787$	26.7685	0.954195	0.477097	−	0.878850i	$-0.341689\pi$
$787$	26.7685	0.954195	0.477097	+	0.878850i	$0.341689\pi$
$788$	16.5359	0.589067
$789$	0	0
$790$	0	0
$791$	−6.21166	−0.220861
$792$	0	0
$793$	−2.07055	−0.0735275
$794$	1.31268	0.0465852
$795$	0	0
$796$	26.9282	0.954445
$797$	−28.6410	−1.01452	−0.507258	−	0.861794i	$-0.669341\pi$
$797$	−28.6410	−1.01452	−0.507258	+	0.861794i	$0.669341\pi$
$798$	0	0
$799$	70.6410	2.49910
$800$	0	0
$801$	0	0
$802$	−4.62158	−0.163194
$803$	51.7128	1.82491
$804$	0	0
$805$	0	0
$806$	−3.03150	−0.106780
$807$	0	0
$808$	0.757875	0.0266619
$809$	−33.5350	−1.17903	−0.589514	−	0.807758i	$-0.700681\pi$
$809$	−33.5350	−1.17903	−0.589514	+	0.807758i	$0.700681\pi$
$810$	0	0
$811$	56.4974	1.98389	0.991946	−	0.126658i	$-0.0404251\pi$
$811$	56.4974	1.98389	0.991946	+	0.126658i	$0.0404251\pi$
$812$	−9.85641	−0.345892
$813$	0	0
$814$	−25.8564	−0.906267
$815$	0	0
$816$	0	0
$817$	1.79315	0.0627344
$818$	7.07180	0.247260
$819$	0	0
$820$	0	0
$821$	−37.3244	−1.30263	−0.651314	−	0.758808i	$-0.725782\pi$
$821$	−37.3244	−1.30263	−0.651314	+	0.758808i	$0.725782\pi$
$822$	0	0
$823$	55.6819	1.94095	0.970475	−	0.241202i	$-0.0775416\pi$
$823$	55.6819	1.94095	0.970475	+	0.241202i	$0.0775416\pi$
$824$	4.89898	0.170664
$825$	0	0
$826$	−13.0718	−0.454826
$827$	−6.92820	−0.240917	−0.120459	−	0.992718i	$-0.538437\pi$
$827$	−6.92820	−0.240917	−0.120459	+	0.992718i	$0.538437\pi$
$828$	0	0
$829$	−26.7846	−0.930268	−0.465134	−	0.885240i	$-0.653994\pi$
$829$	−26.7846	−0.930268	−0.465134	+	0.885240i	$0.653994\pi$
$830$	0	0
$831$	0	0
$832$	1.03528	0.0358917
$833$	−44.2487	−1.53313
$834$	0	0
$835$	0	0
$836$	3.86370	0.133629
$837$	0	0
$838$	−24.4206	−0.843595
$839$	−28.2843	−0.976481	−0.488241	−	0.872709i	$-0.662361\pi$
$839$	−28.2843	−0.976481	−0.488241	+	0.872709i	$0.662361\pi$
$840$	0	0
$841$	61.6410	2.12555
$842$	25.7128	0.886122
$843$	0	0
$844$	9.85641	0.339272
$845$	0	0
$846$	0	0
$847$	−4.06678	−0.139736
$848$	6.00000	0.206041
$849$	0	0
$850$	0	0
$851$	9.79796	0.335870
$852$	0	0
$853$	14.6969	0.503214	0.251607	−	0.967830i	$-0.419041\pi$
$853$	14.6969	0.503214	0.251607	+	0.967830i	$0.419041\pi$
$854$	−2.07055	−0.0708528
$855$	0	0
$856$	−10.9282	−0.373518
$857$	−52.6410	−1.79818	−0.899091	−	0.437761i	$-0.855772\pi$
$857$	−52.6410	−1.79818	−0.899091	+	0.437761i	$0.855772\pi$
$858$	0	0
$859$	36.7846	1.25507	0.627537	−	0.778586i	$-0.284063\pi$
$859$	36.7846	1.25507	0.627537	+	0.778586i	$0.284063\pi$
$860$	0	0
$861$	0	0
$862$	36.0117	1.22656
$863$	−20.7846	−0.707516	−0.353758	−	0.935337i	$-0.615096\pi$
$863$	−20.7846	−0.707516	−0.353758	+	0.935337i	$0.615096\pi$
$864$	0	0
$865$	0	0
$866$	−13.6617	−0.464242
$867$	0	0
$868$	−3.03150	−0.102896
$869$	15.4548	0.524269
$870$	0	0
$871$	3.71281	0.125804
$872$	4.92820	0.166890
$873$	0	0
$874$	−1.46410	−0.0495240
$875$	0	0
$876$	0	0
$877$	−43.8134	−1.47947	−0.739737	−	0.672896i	$-0.765050\pi$
$877$	−43.8134	−1.47947	−0.739737	+	0.672896i	$0.765050\pi$
$878$	−20.0000	−0.674967
$879$	0	0
$880$	0	0
$881$	10.3528	0.348793	0.174397	−	0.984675i	$-0.444202\pi$
$881$	10.3528	0.348793	0.174397	+	0.984675i	$0.444202\pi$
$882$	0	0
$883$	24.0144	0.808150	0.404075	−	0.914726i	$-0.367594\pi$
$883$	24.0144	0.808150	0.404075	+	0.914726i	$0.367594\pi$
$884$	7.72741	0.259901
$885$	0	0
$886$	4.39230	0.147562
$887$	39.7128	1.33343	0.666713	−	0.745315i	$-0.267701\pi$
$887$	39.7128	1.33343	0.666713	+	0.745315i	$0.267701\pi$
$888$	0	0
$889$	−8.78461	−0.294626
$890$	0	0
$891$	0	0
$892$	−0.757875	−0.0253755
$893$	−9.46410	−0.316704
$894$	0	0
$895$	0	0
$896$	1.03528	0.0345861
$897$	0	0
$898$	−3.66063	−0.122157
$899$	27.8781	0.929788
$900$	0	0
$901$	44.7846	1.49199
$902$	25.8564	0.860924
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	−2.62536	−0.0871735	−0.0435867	−	0.999050i	$-0.513878\pi$
$907$	−2.62536	−0.0871735	−0.0435867	+	0.999050i	$0.513878\pi$
$908$	25.8564	0.858075
$909$	0	0
$910$	0	0
$911$	45.2548	1.49936	0.749680	−	0.661801i	$-0.230208\pi$
$911$	45.2548	1.49936	0.749680	+	0.661801i	$0.230208\pi$
$912$	0	0
$913$	16.9706	0.561644
$914$	33.9411	1.12267
$915$	0	0
$916$	23.8564	0.788238
$917$	12.0000	0.396275
$918$	0	0
$919$	−27.7128	−0.914161	−0.457081	−	0.889425i	$-0.651105\pi$
$919$	−27.7128	−0.914161	−0.457081	+	0.889425i	$0.651105\pi$
$920$	0	0
$921$	0	0
$922$	−17.7284	−0.583855
$923$	−16.0000	−0.526646
$924$	0	0
$925$	0	0
$926$	34.0155	1.11782
$927$	0	0
$928$	−9.52056	−0.312528
$929$	11.3137	0.371191	0.185595	−	0.982626i	$-0.440579\pi$
$929$	11.3137	0.371191	0.185595	+	0.982626i	$0.440579\pi$
$930$	0	0
$931$	5.92820	0.194289
$932$	19.4641	0.637568
$933$	0	0
$934$	−9.46410	−0.309675
$935$	0	0
$936$	0	0
$937$	30.3548	0.991649	0.495824	−	0.868423i	$-0.334866\pi$
$937$	30.3548	0.991649	0.495824	+	0.868423i	$0.334866\pi$
$938$	3.71281	0.121228
$939$	0	0
$940$	0	0
$941$	20.8343	0.679178	0.339589	−	0.940574i	$-0.389712\pi$
$941$	20.8343	0.679178	0.339589	+	0.940574i	$0.389712\pi$
$942$	0	0
$943$	−9.79796	−0.319065
$944$	−12.6264	−0.410954
$945$	0	0
$946$	−6.92820	−0.225255
$947$	37.1769	1.20809	0.604044	−	0.796951i	$-0.293555\pi$
$947$	37.1769	1.20809	0.604044	+	0.796951i	$0.293555\pi$
$948$	0	0
$949$	−13.8564	−0.449798
$950$	0	0
$951$	0	0
$952$	7.72741	0.250447
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	0	0
$955$	0	0
$956$	18.0058	0.582350
$957$	0	0
$958$	−31.3901	−1.01417
$959$	−7.72741	−0.249531
$960$	0	0
$961$	−22.4256	−0.723407
$962$	6.92820	0.223374
$963$	0	0
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	2.55103	0.0820355	0.0410177	−	0.999158i	$-0.486940\pi$
$967$	2.55103	0.0820355	0.0410177	+	0.999158i	$0.486940\pi$
$968$	−3.92820	−0.126257
$969$	0	0
$970$	0	0
$971$	24.9010	0.799112	0.399556	−	0.916709i	$-0.369164\pi$
$971$	24.9010	0.799112	0.399556	+	0.916709i	$0.369164\pi$
$972$	0	0
$973$	7.17260	0.229943
$974$	−26.0106	−0.833435
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	−49.7128	−1.59045	−0.795227	−	0.606312i	$-0.792648\pi$
$977$	−49.7128	−1.59045	−0.795227	+	0.606312i	$0.792648\pi$
$978$	0	0
$979$	−4.00000	−0.127841
$980$	0	0
$981$	0	0
$982$	15.7322	0.502035
$983$	48.7846	1.55599	0.777994	−	0.628272i	$-0.216238\pi$
$983$	48.7846	1.55599	0.777994	+	0.628272i	$0.216238\pi$
$984$	0	0
$985$	0	0
$986$	−71.0624	−2.26309
$987$	0	0
$988$	−1.03528	−0.0329365
$989$	2.62536	0.0834814
$990$	0	0
$991$	20.7846	0.660245	0.330122	−	0.943938i	$-0.392910\pi$
$991$	20.7846	0.660245	0.330122	+	0.943938i	$0.392910\pi$
$992$	−2.92820	−0.0929705
$993$	0	0
$994$	−16.0000	−0.507489
$995$	0	0
$996$	0	0
$997$	−1.31268	−0.0415729	−0.0207865	−	0.999784i	$-0.506617\pi$
$997$	−1.31268	−0.0415729	−0.0207865	+	0.999784i	$0.506617\pi$
$998$	1.85641	0.0587635
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.cu.1.2		4
3.2	odd	2		8550.2.a.cv.1.2		4
5.2	odd	4		1710.2.d.g.1369.1	✓	8
5.3	odd	4		1710.2.d.g.1369.6	yes	8
5.4	even	2		8550.2.a.cv.1.3		4
15.2	even	4		1710.2.d.g.1369.8	yes	8
15.8	even	4		1710.2.d.g.1369.3	yes	8
15.14	odd	2	inner	8550.2.a.cu.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1710.2.d.g.1369.1	✓	8	5.2	odd	4
1710.2.d.g.1369.3	yes	8	15.8	even	4
1710.2.d.g.1369.6	yes	8	5.3	odd	4
1710.2.d.g.1369.8	yes	8	15.2	even	4
8550.2.a.cu.1.2		4	1.1	even	1	trivial
8550.2.a.cu.1.3		4	15.14	odd	2	inner
8550.2.a.cv.1.2		4	3.2	odd	2
8550.2.a.cv.1.3		4	5.4	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:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.03528 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.03528 q^{7} -1.00000 q^{8} -3.86370 q^{11} +1.03528 q^{13} +1.03528 q^{14} +1.00000 q^{16} +7.46410 q^{17} -1.00000 q^{19} +3.86370 q^{22} -1.46410 q^{23} -1.03528 q^{26} -1.03528 q^{28} +9.52056 q^{29} +2.92820 q^{31} -1.00000 q^{32} -7.46410 q^{34} -6.69213 q^{37} +1.00000 q^{38} +6.69213 q^{41} -1.79315 q^{43} -3.86370 q^{44} +1.46410 q^{46} +9.46410 q^{47} -5.92820 q^{49} +1.03528 q^{52} +6.00000 q^{53} +1.03528 q^{56} -9.52056 q^{58} -12.6264 q^{59} -2.00000 q^{61} -2.92820 q^{62} +1.00000 q^{64} +3.58630 q^{67} +7.46410 q^{68} -15.4548 q^{71} -13.3843 q^{73} +6.69213 q^{74} -1.00000 q^{76} +4.00000 q^{77} -4.00000 q^{79} -6.69213 q^{82} -4.39230 q^{83} +1.79315 q^{86} +3.86370 q^{88} +1.03528 q^{89} -1.07180 q^{91} -1.46410 q^{92} -9.46410 q^{94} +17.2480 q^{97} +5.92820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{16} + 16 q^{17} - 4 q^{19} + 8 q^{23} - 16 q^{31} - 4 q^{32} - 16 q^{34} + 4 q^{38} - 8 q^{46} + 24 q^{47} + 4 q^{49} + 24 q^{53} - 8 q^{61} + 16 q^{62} + 4 q^{64} + 16 q^{68} - 4 q^{76} + 16 q^{77} - 16 q^{79} + 24 q^{83} - 32 q^{91} + 8 q^{92} - 24 q^{94} - 4 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 4 * q^16 + 16 * q^17 - 4 * q^19 + 8 * q^23 - 16 * q^31 - 4 * q^32 - 16 * q^34 + 4 * q^38 - 8 * q^46 + 24 * q^47 + 4 * q^49 + 24 * q^53 - 8 * q^61 + 16 * q^62 + 4 * q^64 + 16 * q^68 - 4 * q^76 + 16 * q^77 - 16 * q^79 + 24 * q^83 - 32 * q^91 + 8 * q^92 - 24 * q^94 - 4 * q^98

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