LMFDB - Embedded newform 5445.2.a.be.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, 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("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.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:	$43.4785439006$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.35567$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.16215 q^{2} -0.649414 q^{4} +1.00000 q^{5} -4.28684 q^{7} +3.07901 q^{8} +O(q^{10})$	$q-1.16215 q^{2} -0.649414 q^{4} +1.00000 q^{5} -4.28684 q^{7} +3.07901 q^{8} -1.16215 q^{10} +5.16724 q^{13} +4.98194 q^{14} -2.27943 q^{16} -5.00000 q^{17} +5.59998 q^{19} -0.649414 q^{20} +0.219819 q^{23} +1.00000 q^{25} -6.00509 q^{26} +2.78393 q^{28} -6.41843 q^{29} -2.83095 q^{31} -3.50898 q^{32} +5.81074 q^{34} -4.28684 q^{35} +3.92802 q^{37} -6.50800 q^{38} +3.07901 q^{40} -5.86100 q^{41} -8.90173 q^{43} -0.255462 q^{46} +0.237878 q^{47} +11.3770 q^{49} -1.16215 q^{50} -3.35567 q^{52} -2.53671 q^{53} -13.1992 q^{56} +7.45917 q^{58} +7.87035 q^{59} -4.85807 q^{61} +3.28999 q^{62} +8.63682 q^{64} +5.16724 q^{65} +12.1280 q^{67} +3.24707 q^{68} +4.98194 q^{70} +9.98194 q^{71} +14.5625 q^{73} -4.56494 q^{74} -3.63670 q^{76} +8.00194 q^{79} -2.27943 q^{80} +6.81134 q^{82} -12.5530 q^{83} -5.00000 q^{85} +10.3451 q^{86} +2.56545 q^{89} -22.1511 q^{91} -0.142753 q^{92} -0.276449 q^{94} +5.59998 q^{95} +2.01199 q^{97} -13.2218 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 5 q^{2} + 9 q^{4} + 4 q^{5} - 2 q^{7} - 15 q^{8}+O(q^{10}) $ 4 * q - 5 * q^2 + 9 * q^4 + 4 * q^5 - 2 * q^7 - 15 * q^8	$ 4 q - 5 q^{2} + 9 q^{4} + 4 q^{5} - 2 q^{7} - 15 q^{8} - 5 q^{10} + 3 q^{13} + 5 q^{14} + 15 q^{16} - 20 q^{17} + 3 q^{19} + 9 q^{20} + 5 q^{23} + 4 q^{25} - 6 q^{26} + 3 q^{28} - 5 q^{29} - q^{31} - 30 q^{32} + 25 q^{34} - 2 q^{35} - 7 q^{37} - q^{38} - 15 q^{40} - 20 q^{41} - 2 q^{43} + 7 q^{46} + 20 q^{47} + 8 q^{49} - 5 q^{50} - 7 q^{52} - 6 q^{53} - 10 q^{56} - 21 q^{58} + 5 q^{59} - 7 q^{61} + 12 q^{62} + 49 q^{64} + 3 q^{65} - 13 q^{67} - 45 q^{68} + 5 q^{70} + 25 q^{71} + 23 q^{73} + 7 q^{74} - 7 q^{76} + 15 q^{80} + 11 q^{82} - 33 q^{83} - 20 q^{85} + 12 q^{86} - 16 q^{89} - 24 q^{91} - 17 q^{94} + 3 q^{95} - 25 q^{98}+O(q^{100}) $ 4 * q - 5 * q^2 + 9 * q^4 + 4 * q^5 - 2 * q^7 - 15 * q^8 - 5 * q^10 + 3 * q^13 + 5 * q^14 + 15 * q^16 - 20 * q^17 + 3 * q^19 + 9 * q^20 + 5 * q^23 + 4 * q^25 - 6 * q^26 + 3 * q^28 - 5 * q^29 - q^31 - 30 * q^32 + 25 * q^34 - 2 * q^35 - 7 * q^37 - q^38 - 15 * q^40 - 20 * q^41 - 2 * q^43 + 7 * q^46 + 20 * q^47 + 8 * q^49 - 5 * q^50 - 7 * q^52 - 6 * q^53 - 10 * q^56 - 21 * q^58 + 5 * q^59 - 7 * q^61 + 12 * q^62 + 49 * q^64 + 3 * q^65 - 13 * q^67 - 45 * q^68 + 5 * q^70 + 25 * q^71 + 23 * q^73 + 7 * q^74 - 7 * q^76 + 15 * q^80 + 11 * q^82 - 33 * q^83 - 20 * q^85 + 12 * q^86 - 16 * q^89 - 24 * q^91 - 17 * q^94 + 3 * q^95 - 25 * 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 $

$2$	−1.16215	−0.821762	−0.410881	−	0.911689i	$-0.634779\pi$
$2$	−1.16215	−0.821762	−0.410881	+	0.911689i	$0.634779\pi$
$3$	0	0
$3$	0	0
$4$	−0.649414	−0.324707
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.28684	−1.62027	−0.810137	−	0.586241i	$-0.800607\pi$
$7$	−4.28684	−1.62027	−0.810137	+	0.586241i	$0.800607\pi$
$8$	3.07901	1.08859
$9$	0	0
$10$	−1.16215	−0.367503
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.16724	1.43313	0.716567	−	0.697519i	$-0.245713\pi$
$13$	5.16724	1.43313	0.716567	+	0.697519i	$0.245713\pi$
$14$	4.98194	1.33148
$15$	0	0
$16$	−2.27943	−0.569858
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	5.59998	1.28472	0.642361	−	0.766402i	$-0.277955\pi$
$19$	5.59998	1.28472	0.642361	+	0.766402i	$0.277955\pi$
$20$	−0.649414	−0.145213
$21$	0	0
$22$	0	0
$23$	0.219819	0.0458354	0.0229177	−	0.999737i	$-0.492704\pi$
$23$	0.219819	0.0458354	0.0229177	+	0.999737i	$0.492704\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−6.00509	−1.17769
$27$	0	0
$28$	2.78393	0.526114
$29$	−6.41843	−1.19187	−0.595937	−	0.803031i	$-0.703219\pi$
$29$	−6.41843	−1.19187	−0.595937	+	0.803031i	$0.703219\pi$
$30$	0	0
$31$	−2.83095	−0.508455	−0.254227	−	0.967145i	$-0.581821\pi$
$31$	−2.83095	−0.508455	−0.254227	+	0.967145i	$0.581821\pi$
$32$	−3.50898	−0.620306
$33$	0	0
$34$	5.81074	0.996533
$35$	−4.28684	−0.724608
$36$	0	0
$37$	3.92802	0.645763	0.322881	−	0.946439i	$-0.395348\pi$
$37$	3.92802	0.645763	0.322881	+	0.946439i	$0.395348\pi$
$38$	−6.50800	−1.05574
$39$	0	0
$40$	3.07901	0.486834
$41$	−5.86100	−0.915334	−0.457667	−	0.889124i	$-0.651315\pi$
$41$	−5.86100	−0.915334	−0.457667	+	0.889124i	$0.651315\pi$
$42$	0	0
$43$	−8.90173	−1.35750	−0.678751	−	0.734369i	$-0.737478\pi$
$43$	−8.90173	−1.35750	−0.678751	+	0.734369i	$0.737478\pi$
$44$	0	0
$45$	0	0
$46$	−0.255462	−0.0376658
$47$	0.237878	0.0346980	0.0173490	−	0.999849i	$-0.494477\pi$
$47$	0.237878	0.0346980	0.0173490	+	0.999849i	$0.494477\pi$
$48$	0	0
$49$	11.3770	1.62529
$50$	−1.16215	−0.164352
$51$	0	0
$52$	−3.35567	−0.465348
$53$	−2.53671	−0.348443	−0.174222	−	0.984706i	$-0.555741\pi$
$53$	−2.53671	−0.348443	−0.174222	+	0.984706i	$0.555741\pi$
$54$	0	0
$55$	0	0
$56$	−13.1992	−1.76382
$57$	0	0
$58$	7.45917	0.979436
$59$	7.87035	1.02463	0.512316	−	0.858797i	$-0.328788\pi$
$59$	7.87035	1.02463	0.512316	+	0.858797i	$0.328788\pi$
$60$	0	0
$61$	−4.85807	−0.622012	−0.311006	−	0.950408i	$-0.600666\pi$
$61$	−4.85807	−0.622012	−0.311006	+	0.950408i	$0.600666\pi$
$62$	3.28999	0.417829
$63$	0	0
$64$	8.63682	1.07960
$65$	5.16724	0.640917
$66$	0	0
$67$	12.1280	1.48167	0.740834	−	0.671688i	$-0.234431\pi$
$67$	12.1280	1.48167	0.740834	+	0.671688i	$0.234431\pi$
$68$	3.24707	0.393765
$69$	0	0
$70$	4.98194	0.595456
$71$	9.98194	1.18464	0.592319	−	0.805703i	$-0.298212\pi$
$71$	9.98194	1.18464	0.592319	+	0.805703i	$0.298212\pi$
$72$	0	0
$73$	14.5625	1.70441	0.852207	−	0.523205i	$-0.175264\pi$
$73$	14.5625	1.70441	0.852207	+	0.523205i	$0.175264\pi$
$74$	−4.56494	−0.530664
$75$	0	0
$76$	−3.63670	−0.417158
$77$	0	0
$78$	0	0
$79$	8.00194	0.900289	0.450144	−	0.892956i	$-0.351372\pi$
$79$	8.00194	0.900289	0.450144	+	0.892956i	$0.351372\pi$
$80$	−2.27943	−0.254848
$81$	0	0
$82$	6.81134	0.752187
$83$	−12.5530	−1.37787	−0.688933	−	0.724825i	$-0.741921\pi$
$83$	−12.5530	−1.37787	−0.688933	+	0.724825i	$0.741921\pi$
$84$	0	0
$85$	−5.00000	−0.542326
$86$	10.3451	1.11554
$87$	0	0
$88$	0	0
$89$	2.56545	0.271937	0.135969	−	0.990713i	$-0.456585\pi$
$89$	2.56545	0.271937	0.135969	+	0.990713i	$0.456585\pi$
$90$	0	0
$91$	−22.1511	−2.32207
$92$	−0.142753	−0.0148831
$93$	0	0
$94$	−0.276449	−0.0285135
$95$	5.59998	0.574545
$96$	0	0
$97$	2.01199	0.204286	0.102143	−	0.994770i	$-0.467430\pi$
$97$	2.01199	0.204286	0.102143	+	0.994770i	$0.467430\pi$
$98$	−13.2218	−1.33560
$99$	0	0
$100$	−0.649414	−0.0649414
$101$	7.58484	0.754720	0.377360	−	0.926067i	$-0.376832\pi$
$101$	7.58484	0.754720	0.377360	+	0.926067i	$0.376832\pi$
$102$	0	0
$103$	−16.8685	−1.66211	−0.831053	−	0.556193i	$-0.812262\pi$
$103$	−16.8685	−1.66211	−0.831053	+	0.556193i	$0.812262\pi$
$104$	15.9100	1.56010
$105$	0	0
$106$	2.94803	0.286338
$107$	−7.76161	−0.750343	−0.375172	−	0.926955i	$-0.622416\pi$
$107$	−7.76161	−0.750343	−0.375172	+	0.926955i	$0.622416\pi$
$108$	0	0
$109$	4.45671	0.426876	0.213438	−	0.976957i	$-0.431534\pi$
$109$	4.45671	0.426876	0.213438	+	0.976957i	$0.431534\pi$
$110$	0	0
$111$	0	0
$112$	9.77157	0.923327
$113$	7.71247	0.725528	0.362764	−	0.931881i	$-0.381833\pi$
$113$	7.71247	0.725528	0.362764	+	0.931881i	$0.381833\pi$
$114$	0	0
$115$	0.219819	0.0204982
$116$	4.16822	0.387010
$117$	0	0
$118$	−9.14651	−0.842004
$119$	21.4342	1.96487
$120$	0	0
$121$	0	0
$122$	5.64579	0.511146
$123$	0	0
$124$	1.83846	0.165099
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.53114	0.135867	0.0679335	−	0.997690i	$-0.478359\pi$
$127$	1.53114	0.135867	0.0679335	+	0.997690i	$0.478359\pi$
$128$	−3.01930	−0.266871
$129$	0	0
$130$	−6.00509	−0.526681
$131$	−2.66108	−0.232500	−0.116250	−	0.993220i	$-0.537087\pi$
$131$	−2.66108	−0.232500	−0.116250	+	0.993220i	$0.537087\pi$
$132$	0	0
$133$	−24.0062	−2.08160
$134$	−14.0945	−1.21758
$135$	0	0
$136$	−15.3950	−1.32011
$137$	−10.2555	−0.876183	−0.438092	−	0.898930i	$-0.644345\pi$
$137$	−10.2555	−0.876183	−0.438092	+	0.898930i	$0.644345\pi$
$138$	0	0
$139$	−5.29507	−0.449122	−0.224561	−	0.974460i	$-0.572095\pi$
$139$	−5.29507	−0.449122	−0.224561	+	0.974460i	$0.572095\pi$
$140$	2.78393	0.235285
$141$	0	0
$142$	−11.6005	−0.973491
$143$	0	0
$144$	0	0
$145$	−6.41843	−0.533022
$146$	−16.9238	−1.40062
$147$	0	0
$148$	−2.55091	−0.209684
$149$	−3.78841	−0.310359	−0.155179	−	0.987886i	$-0.549596\pi$
$149$	−3.78841	−0.310359	−0.155179	+	0.987886i	$0.549596\pi$
$150$	0	0
$151$	−5.92001	−0.481763	−0.240882	−	0.970554i	$-0.577437\pi$
$151$	−5.92001	−0.481763	−0.240882	+	0.970554i	$0.577437\pi$
$152$	17.2424	1.39854
$153$	0	0
$154$	0	0
$155$	−2.83095	−0.227388
$156$	0	0
$157$	−7.42909	−0.592906	−0.296453	−	0.955048i	$-0.595804\pi$
$157$	−7.42909	−0.592906	−0.296453	+	0.955048i	$0.595804\pi$
$158$	−9.29944	−0.739823
$159$	0	0
$160$	−3.50898	−0.277409
$161$	−0.942328	−0.0742659
$162$	0	0
$163$	5.32086	0.416762	0.208381	−	0.978048i	$-0.433181\pi$
$163$	5.32086	0.416762	0.208381	+	0.978048i	$0.433181\pi$
$164$	3.80621	0.297215
$165$	0	0
$166$	14.5884	1.13228
$167$	1.97139	0.152551	0.0762753	−	0.997087i	$-0.475697\pi$
$167$	1.97139	0.152551	0.0762753	+	0.997087i	$0.475697\pi$
$168$	0	0
$169$	13.7003	1.05387
$170$	5.81074	0.445663
$171$	0	0
$172$	5.78091	0.440790
$173$	1.97504	0.150160	0.0750799	−	0.997178i	$-0.476079\pi$
$173$	1.97504	0.150160	0.0750799	+	0.997178i	$0.476079\pi$
$174$	0	0
$175$	−4.28684	−0.324055
$176$	0	0
$177$	0	0
$178$	−2.98143	−0.223468
$179$	2.02315	0.151217	0.0756086	−	0.997138i	$-0.475910\pi$
$179$	2.02315	0.151217	0.0756086	+	0.997138i	$0.475910\pi$
$180$	0	0
$181$	20.1380	1.49685	0.748423	−	0.663221i	$-0.230811\pi$
$181$	20.1380	1.49685	0.748423	+	0.663221i	$0.230811\pi$
$182$	25.7429	1.90819
$183$	0	0
$184$	0.676824	0.0498961
$185$	3.92802	0.288794
$186$	0	0
$187$	0	0
$188$	−0.154481	−0.0112667
$189$	0	0
$190$	−6.50800	−0.472140
$191$	−12.6652	−0.916418	−0.458209	−	0.888844i	$-0.651509\pi$
$191$	−12.6652	−0.916418	−0.458209	+	0.888844i	$0.651509\pi$
$192$	0	0
$193$	−13.9321	−1.00286	−0.501428	−	0.865199i	$-0.667192\pi$
$193$	−13.9321	−1.00286	−0.501428	+	0.865199i	$0.667192\pi$
$194$	−2.33822	−0.167875
$195$	0	0
$196$	−7.38839	−0.527742
$197$	−14.6060	−1.04064	−0.520319	−	0.853972i	$-0.674187\pi$
$197$	−14.6060	−1.04064	−0.520319	+	0.853972i	$0.674187\pi$
$198$	0	0
$199$	−11.8748	−0.841784	−0.420892	−	0.907111i	$-0.638283\pi$
$199$	−11.8748	−0.841784	−0.420892	+	0.907111i	$0.638283\pi$
$200$	3.07901	0.217719
$201$	0	0
$202$	−8.81471	−0.620201
$203$	27.5148	1.93116
$204$	0	0
$205$	−5.86100	−0.409350
$206$	19.6037	1.36586
$207$	0	0
$208$	−11.7784	−0.816683
$209$	0	0
$210$	0	0
$211$	−22.3518	−1.53876	−0.769382	−	0.638789i	$-0.779436\pi$
$211$	−22.3518	−1.53876	−0.769382	+	0.638789i	$0.779436\pi$
$212$	1.64737	0.113142
$213$	0	0
$214$	9.02014	0.616604
$215$	−8.90173	−0.607093
$216$	0	0
$217$	12.1359	0.823836
$218$	−5.17936	−0.350790
$219$	0	0
$220$	0	0
$221$	−25.8362	−1.73793
$222$	0	0
$223$	−19.1583	−1.28294	−0.641468	−	0.767150i	$-0.721674\pi$
$223$	−19.1583	−1.28294	−0.641468	+	0.767150i	$0.721674\pi$
$224$	15.0424	1.00507
$225$	0	0
$226$	−8.96302	−0.596211
$227$	−15.3759	−1.02053	−0.510267	−	0.860016i	$-0.670453\pi$
$227$	−15.3759	−1.02053	−0.510267	+	0.860016i	$0.670453\pi$
$228$	0	0
$229$	8.90126	0.588212	0.294106	−	0.955773i	$-0.404978\pi$
$229$	8.90126	0.588212	0.294106	+	0.955773i	$0.404978\pi$
$230$	−0.255462	−0.0168447
$231$	0	0
$232$	−19.7624	−1.29747
$233$	−22.8362	−1.49605	−0.748024	−	0.663672i	$-0.768997\pi$
$233$	−22.8362	−1.49605	−0.748024	+	0.663672i	$0.768997\pi$
$234$	0	0
$235$	0.237878	0.0155174
$236$	−5.11112	−0.332705
$237$	0	0
$238$	−24.9097	−1.61466
$239$	−16.8605	−1.09062	−0.545308	−	0.838236i	$-0.683587\pi$
$239$	−16.8605	−1.09062	−0.545308	+	0.838236i	$0.683587\pi$
$240$	0	0
$241$	−11.9607	−0.770459	−0.385229	−	0.922821i	$-0.625878\pi$
$241$	−11.9607	−0.770459	−0.385229	+	0.922821i	$0.625878\pi$
$242$	0	0
$243$	0	0
$244$	3.15490	0.201972
$245$	11.3770	0.726851
$246$	0	0
$247$	28.9364	1.84118
$248$	−8.71654	−0.553501
$249$	0	0
$250$	−1.16215	−0.0735006
$251$	−16.0034	−1.01013	−0.505063	−	0.863082i	$-0.668531\pi$
$251$	−16.0034	−1.01013	−0.505063	+	0.863082i	$0.668531\pi$
$252$	0	0
$253$	0	0
$254$	−1.77941	−0.111650
$255$	0	0
$256$	−13.7648	−0.860298
$257$	−5.47446	−0.341487	−0.170744	−	0.985315i	$-0.554617\pi$
$257$	−5.47446	−0.341487	−0.170744	+	0.985315i	$0.554617\pi$
$258$	0	0
$259$	−16.8388	−1.04631
$260$	−3.35567	−0.208110
$261$	0	0
$262$	3.09257	0.191060
$263$	−11.9841	−0.738971	−0.369486	−	0.929236i	$-0.620466\pi$
$263$	−11.9841	−0.738971	−0.369486	+	0.929236i	$0.620466\pi$
$264$	0	0
$265$	−2.53671	−0.155829
$266$	27.8987	1.71058
$267$	0	0
$268$	−7.87607	−0.481108
$269$	−13.6465	−0.832041	−0.416020	−	0.909355i	$-0.636576\pi$
$269$	−13.6465	−0.832041	−0.416020	+	0.909355i	$0.636576\pi$
$270$	0	0
$271$	11.8379	0.719098	0.359549	−	0.933126i	$-0.382930\pi$
$271$	11.8379	0.719098	0.359549	+	0.933126i	$0.382930\pi$
$272$	11.3972	0.691055
$273$	0	0
$274$	11.9184	0.720014
$275$	0	0
$276$	0	0
$277$	10.2638	0.616694	0.308347	−	0.951274i	$-0.400224\pi$
$277$	10.2638	0.616694	0.308347	+	0.951274i	$0.400224\pi$
$278$	6.15366	0.369072
$279$	0	0
$280$	−13.1992	−0.788804
$281$	2.93889	0.175319	0.0876597	−	0.996150i	$-0.472061\pi$
$281$	2.93889	0.175319	0.0876597	+	0.996150i	$0.472061\pi$
$282$	0	0
$283$	21.8279	1.29754	0.648768	−	0.760986i	$-0.275284\pi$
$283$	21.8279	1.29754	0.648768	+	0.760986i	$0.275284\pi$
$284$	−6.48241	−0.384660
$285$	0	0
$286$	0	0
$287$	25.1252	1.48309
$288$	0	0
$289$	8.00000	0.470588
$290$	7.45917	0.438017
$291$	0	0
$292$	−9.45710	−0.553435
$293$	24.7665	1.44688	0.723438	−	0.690390i	$-0.242561\pi$
$293$	24.7665	1.44688	0.723438	+	0.690390i	$0.242561\pi$
$294$	0	0
$295$	7.87035	0.458230
$296$	12.0944	0.702974
$297$	0	0
$298$	4.40269	0.255041
$299$	1.13586	0.0656882
$300$	0	0
$301$	38.1603	2.19952
$302$	6.87992	0.395895
$303$	0	0
$304$	−12.7648	−0.732110
$305$	−4.85807	−0.278172
$306$	0	0
$307$	−4.94023	−0.281954	−0.140977	−	0.990013i	$-0.545024\pi$
$307$	−4.94023	−0.281954	−0.140977	+	0.990013i	$0.545024\pi$
$308$	0	0
$309$	0	0
$310$	3.28999	0.186859
$311$	13.8096	0.783070	0.391535	−	0.920163i	$-0.371944\pi$
$311$	13.8096	0.783070	0.391535	+	0.920163i	$0.371944\pi$
$312$	0	0
$313$	−4.06934	−0.230013	−0.115006	−	0.993365i	$-0.536689\pi$
$313$	−4.06934	−0.230013	−0.115006	+	0.993365i	$0.536689\pi$
$314$	8.63369	0.487227
$315$	0	0
$316$	−5.19657	−0.292330
$317$	−25.8199	−1.45019	−0.725096	−	0.688648i	$-0.758205\pi$
$317$	−25.8199	−1.45019	−0.725096	+	0.688648i	$0.758205\pi$
$318$	0	0
$319$	0	0
$320$	8.63682	0.482813
$321$	0	0
$322$	1.09512	0.0610289
$323$	−27.9999	−1.55795
$324$	0	0
$325$	5.16724	0.286627
$326$	−6.18362	−0.342479
$327$	0	0
$328$	−18.0461	−0.996428
$329$	−1.01974	−0.0562203
$330$	0	0
$331$	−3.35008	−0.184137	−0.0920684	−	0.995753i	$-0.529348\pi$
$331$	−3.35008	−0.184137	−0.0920684	+	0.995753i	$0.529348\pi$
$332$	8.15206	0.447403
$333$	0	0
$334$	−2.29104	−0.125360
$335$	12.1280	0.662622
$336$	0	0
$337$	−27.3998	−1.49256	−0.746282	−	0.665630i	$-0.768163\pi$
$337$	−27.3998	−1.49256	−0.746282	+	0.665630i	$0.768163\pi$
$338$	−15.9218	−0.866031
$339$	0	0
$340$	3.24707	0.176097
$341$	0	0
$342$	0	0
$343$	−18.7636	−1.01314
$344$	−27.4085	−1.47777
$345$	0	0
$346$	−2.29529	−0.123396
$347$	−15.8265	−0.849610	−0.424805	−	0.905285i	$-0.639657\pi$
$347$	−15.8265	−0.849610	−0.424805	+	0.905285i	$0.639657\pi$
$348$	0	0
$349$	−3.65930	−0.195878	−0.0979388	−	0.995192i	$-0.531225\pi$
$349$	−3.65930	−0.195878	−0.0979388	+	0.995192i	$0.531225\pi$
$350$	4.98194	0.266296
$351$	0	0
$352$	0	0
$353$	27.4937	1.46334	0.731671	−	0.681658i	$-0.238741\pi$
$353$	27.4937	1.46334	0.731671	+	0.681658i	$0.238741\pi$
$354$	0	0
$355$	9.98194	0.529786
$356$	−1.66604	−0.0882999
$357$	0	0
$358$	−2.35119	−0.124265
$359$	−0.478740	−0.0252670	−0.0126335	−	0.999920i	$-0.504021\pi$
$359$	−0.478740	−0.0252670	−0.0126335	+	0.999920i	$0.504021\pi$
$360$	0	0
$361$	12.3597	0.650512
$362$	−23.4033	−1.23005
$363$	0	0
$364$	14.3852	0.753992
$365$	14.5625	0.762237
$366$	0	0
$367$	−25.4582	−1.32891	−0.664453	−	0.747330i	$-0.731335\pi$
$367$	−25.4582	−1.32891	−0.664453	+	0.747330i	$0.731335\pi$
$368$	−0.501062	−0.0261197
$369$	0	0
$370$	−4.56494	−0.237320
$371$	10.8745	0.564574
$372$	0	0
$373$	−2.75967	−0.142890	−0.0714451	−	0.997445i	$-0.522761\pi$
$373$	−2.75967	−0.142890	−0.0714451	+	0.997445i	$0.522761\pi$
$374$	0	0
$375$	0	0
$376$	0.732428	0.0377721
$377$	−33.1656	−1.70811
$378$	0	0
$379$	−21.6535	−1.11227	−0.556133	−	0.831093i	$-0.687716\pi$
$379$	−21.6535	−1.11227	−0.556133	+	0.831093i	$0.687716\pi$
$380$	−3.63670	−0.186559
$381$	0	0
$382$	14.7188	0.753078
$383$	−16.2295	−0.829287	−0.414643	−	0.909984i	$-0.636094\pi$
$383$	−16.2295	−0.829287	−0.414643	+	0.909984i	$0.636094\pi$
$384$	0	0
$385$	0	0
$386$	16.1912	0.824109
$387$	0	0
$388$	−1.30661	−0.0663332
$389$	16.2916	0.826019	0.413009	−	0.910727i	$-0.364478\pi$
$389$	16.2916	0.826019	0.413009	+	0.910727i	$0.364478\pi$
$390$	0	0
$391$	−1.09909	−0.0555836
$392$	35.0299	1.76928
$393$	0	0
$394$	16.9744	0.855157
$395$	8.00194	0.402621
$396$	0	0
$397$	−19.9673	−1.00213	−0.501064	−	0.865410i	$-0.667058\pi$
$397$	−19.9673	−1.00213	−0.501064	+	0.865410i	$0.667058\pi$
$398$	13.8003	0.691747
$399$	0	0
$400$	−2.27943	−0.113972
$401$	27.3269	1.36464	0.682320	−	0.731053i	$-0.260971\pi$
$401$	27.3269	1.36464	0.682320	+	0.731053i	$0.260971\pi$
$402$	0	0
$403$	−14.6282	−0.728683
$404$	−4.92570	−0.245063
$405$	0	0
$406$	−31.9763	−1.58696
$407$	0	0
$408$	0	0
$409$	24.9380	1.23310	0.616552	−	0.787314i	$-0.288529\pi$
$409$	24.9380	1.23310	0.616552	+	0.787314i	$0.288529\pi$
$410$	6.81134	0.336388
$411$	0	0
$412$	10.9547	0.539698
$413$	−33.7389	−1.66019
$414$	0	0
$415$	−12.5530	−0.616200
$416$	−18.1317	−0.888981
$417$	0	0
$418$	0	0
$419$	−16.8256	−0.821986	−0.410993	−	0.911639i	$-0.634818\pi$
$419$	−16.8256	−0.821986	−0.410993	+	0.911639i	$0.634818\pi$
$420$	0	0
$421$	23.6006	1.15022	0.575112	−	0.818074i	$-0.304958\pi$
$421$	23.6006	1.15022	0.575112	+	0.818074i	$0.304958\pi$
$422$	25.9761	1.26450
$423$	0	0
$424$	−7.81054	−0.379313
$425$	−5.00000	−0.242536
$426$	0	0
$427$	20.8258	1.00783
$428$	5.04050	0.243642
$429$	0	0
$430$	10.3451	0.498886
$431$	37.6556	1.81381	0.906903	−	0.421339i	$-0.138440\pi$
$431$	37.6556	1.81381	0.906903	+	0.421339i	$0.138440\pi$
$432$	0	0
$433$	−30.9752	−1.48857	−0.744286	−	0.667861i	$-0.767210\pi$
$433$	−30.9752	−1.48857	−0.744286	+	0.667861i	$0.767210\pi$
$434$	−14.1037	−0.676997
$435$	0	0
$436$	−2.89425	−0.138609
$437$	1.23098	0.0588858
$438$	0	0
$439$	−28.2131	−1.34654	−0.673268	−	0.739399i	$-0.735110\pi$
$439$	−28.2131	−1.34654	−0.673268	+	0.739399i	$0.735110\pi$
$440$	0	0
$441$	0	0
$442$	30.0254	1.42816
$443$	−14.6420	−0.695663	−0.347831	−	0.937557i	$-0.613082\pi$
$443$	−14.6420	−0.695663	−0.347831	+	0.937557i	$0.613082\pi$
$444$	0	0
$445$	2.56545	0.121614
$446$	22.2648	1.05427
$447$	0	0
$448$	−37.0247	−1.74925
$449$	−11.9977	−0.566206	−0.283103	−	0.959090i	$-0.591364\pi$
$449$	−11.9977	−0.566206	−0.283103	+	0.959090i	$0.591364\pi$
$450$	0	0
$451$	0	0
$452$	−5.00858	−0.235584
$453$	0	0
$454$	17.8691	0.838636
$455$	−22.1511	−1.03846
$456$	0	0
$457$	10.0437	0.469824	0.234912	−	0.972017i	$-0.424520\pi$
$457$	10.0437	0.469824	0.234912	+	0.972017i	$0.424520\pi$
$458$	−10.3446	−0.483370
$459$	0	0
$460$	−0.142753	−0.00665591
$461$	13.4491	0.626386	0.313193	−	0.949689i	$-0.398601\pi$
$461$	13.4491	0.626386	0.313193	+	0.949689i	$0.398601\pi$
$462$	0	0
$463$	18.7836	0.872946	0.436473	−	0.899717i	$-0.356227\pi$
$463$	18.7836	0.872946	0.436473	+	0.899717i	$0.356227\pi$
$464$	14.6304	0.679199
$465$	0	0
$466$	26.5390	1.22940
$467$	−26.5510	−1.22863	−0.614316	−	0.789060i	$-0.710568\pi$
$467$	−26.5510	−1.22863	−0.614316	+	0.789060i	$0.710568\pi$
$468$	0	0
$469$	−51.9907	−2.40071
$470$	−0.276449	−0.0127516
$471$	0	0
$472$	24.2329	1.11541
$473$	0	0
$474$	0	0
$475$	5.59998	0.256944
$476$	−13.9197	−0.638007
$477$	0	0
$478$	19.5944	0.896228
$479$	−10.7830	−0.492689	−0.246345	−	0.969182i	$-0.579229\pi$
$479$	−10.7830	−0.492689	−0.246345	+	0.969182i	$0.579229\pi$
$480$	0	0
$481$	20.2970	0.925464
$482$	13.9001	0.633134
$483$	0	0
$484$	0	0
$485$	2.01199	0.0913596
$486$	0	0
$487$	6.81095	0.308634	0.154317	−	0.988021i	$-0.450682\pi$
$487$	6.81095	0.308634	0.154317	+	0.988021i	$0.450682\pi$
$488$	−14.9580	−0.677119
$489$	0	0
$490$	−13.2218	−0.597298
$491$	−26.0954	−1.17767	−0.588834	−	0.808254i	$-0.700413\pi$
$491$	−26.0954	−1.17767	−0.588834	+	0.808254i	$0.700413\pi$
$492$	0	0
$493$	32.0922	1.44536
$494$	−33.6283	−1.51301
$495$	0	0
$496$	6.45297	0.289747
$497$	−42.7910	−1.91944
$498$	0	0
$499$	19.3834	0.867719	0.433860	−	0.900981i	$-0.357151\pi$
$499$	19.3834	0.867719	0.433860	+	0.900981i	$0.357151\pi$
$500$	−0.649414	−0.0290427
$501$	0	0
$502$	18.5983	0.830084
$503$	−18.8422	−0.840134	−0.420067	−	0.907493i	$-0.637993\pi$
$503$	−18.8422	−0.840134	−0.420067	+	0.907493i	$0.637993\pi$
$504$	0	0
$505$	7.58484	0.337521
$506$	0	0
$507$	0	0
$508$	−0.994345	−0.0441169
$509$	5.15120	0.228323	0.114162	−	0.993462i	$-0.463582\pi$
$509$	5.15120	0.228323	0.114162	+	0.993462i	$0.463582\pi$
$510$	0	0
$511$	−62.4272	−2.76162
$512$	22.0353	0.973831
$513$	0	0
$514$	6.36212	0.280621
$515$	−16.8685	−0.743317
$516$	0	0
$517$	0	0
$518$	19.5692	0.859820
$519$	0	0
$520$	15.9100	0.697698
$521$	41.5022	1.81824	0.909121	−	0.416532i	$-0.136754\pi$
$521$	41.5022	1.81824	0.909121	+	0.416532i	$0.136754\pi$
$522$	0	0
$523$	−6.92280	−0.302713	−0.151356	−	0.988479i	$-0.548364\pi$
$523$	−6.92280	−0.302713	−0.151356	+	0.988479i	$0.548364\pi$
$524$	1.72815	0.0754944
$525$	0	0
$526$	13.9273	0.607259
$527$	14.1548	0.616592
$528$	0	0
$529$	−22.9517	−0.997899
$530$	2.94803	0.128054
$531$	0	0
$532$	15.5900	0.675911
$533$	−30.2852	−1.31180
$534$	0	0
$535$	−7.76161	−0.335564
$536$	37.3421	1.61293
$537$	0	0
$538$	15.8592	0.683740
$539$	0	0
$540$	0	0
$541$	−28.4183	−1.22180	−0.610899	−	0.791709i	$-0.709192\pi$
$541$	−28.4183	−1.22180	−0.610899	+	0.791709i	$0.709192\pi$
$542$	−13.7573	−0.590928
$543$	0	0
$544$	17.5449	0.752231
$545$	4.45671	0.190905
$546$	0	0
$547$	25.8592	1.10566	0.552829	−	0.833295i	$-0.313548\pi$
$547$	25.8592	1.10566	0.552829	+	0.833295i	$0.313548\pi$
$548$	6.66004	0.284503
$549$	0	0
$550$	0	0
$551$	−35.9431	−1.53123
$552$	0	0
$553$	−34.3031	−1.45871
$554$	−11.9281	−0.506776
$555$	0	0
$556$	3.43869	0.145833
$557$	8.06409	0.341687	0.170843	−	0.985298i	$-0.445351\pi$
$557$	8.06409	0.341687	0.170843	+	0.985298i	$0.445351\pi$
$558$	0	0
$559$	−45.9973	−1.94548
$560$	9.77157	0.412924
$561$	0	0
$562$	−3.41542	−0.144071
$563$	−29.6516	−1.24966	−0.624832	−	0.780759i	$-0.714833\pi$
$563$	−29.6516	−1.24966	−0.624832	+	0.780759i	$0.714833\pi$
$564$	0	0
$565$	7.71247	0.324466
$566$	−25.3673	−1.06627
$567$	0	0
$568$	30.7345	1.28959
$569$	42.1026	1.76503	0.882517	−	0.470280i	$-0.155847\pi$
$569$	42.1026	1.76503	0.882517	+	0.470280i	$0.155847\pi$
$570$	0	0
$571$	−26.6823	−1.11662	−0.558311	−	0.829632i	$-0.688550\pi$
$571$	−26.6823	−1.11662	−0.558311	+	0.829632i	$0.688550\pi$
$572$	0	0
$573$	0	0
$574$	−29.1992	−1.21875
$575$	0.219819	0.00916708
$576$	0	0
$577$	−3.30818	−0.137721	−0.0688606	−	0.997626i	$-0.521936\pi$
$577$	−3.30818	−0.137721	−0.0688606	+	0.997626i	$0.521936\pi$
$578$	−9.29718	−0.386712
$579$	0	0
$580$	4.16822	0.173076
$581$	53.8125	2.23252
$582$	0	0
$583$	0	0
$584$	44.8381	1.85542
$585$	0	0
$586$	−28.7823	−1.18899
$587$	8.52770	0.351976	0.175988	−	0.984392i	$-0.443688\pi$
$587$	8.52770	0.351976	0.175988	+	0.984392i	$0.443688\pi$
$588$	0	0
$589$	−15.8533	−0.653223
$590$	−9.14651	−0.376556
$591$	0	0
$592$	−8.95367	−0.367993
$593$	23.4343	0.962333	0.481167	−	0.876629i	$-0.340213\pi$
$593$	23.4343	0.962333	0.481167	+	0.876629i	$0.340213\pi$
$594$	0	0
$595$	21.4342	0.878717
$596$	2.46025	0.100776
$597$	0	0
$598$	−1.32003	−0.0539801
$599$	−18.8344	−0.769555	−0.384777	−	0.923009i	$-0.625722\pi$
$599$	−18.8344	−0.769555	−0.384777	+	0.923009i	$0.625722\pi$
$600$	0	0
$601$	−37.6585	−1.53612	−0.768062	−	0.640376i	$-0.778779\pi$
$601$	−37.6585	−1.53612	−0.768062	+	0.640376i	$0.778779\pi$
$602$	−44.3479	−1.80749
$603$	0	0
$604$	3.84453	0.156432
$605$	0	0
$606$	0	0
$607$	−29.3680	−1.19201	−0.596004	−	0.802981i	$-0.703246\pi$
$607$	−29.3680	−1.19201	−0.596004	+	0.802981i	$0.703246\pi$
$608$	−19.6502	−0.796921
$609$	0	0
$610$	5.64579	0.228592
$611$	1.22917	0.0497269
$612$	0	0
$613$	−12.9111	−0.521474	−0.260737	−	0.965410i	$-0.583965\pi$
$613$	−12.9111	−0.521474	−0.260737	+	0.965410i	$0.583965\pi$
$614$	5.74127	0.231699
$615$	0	0
$616$	0	0
$617$	−41.6041	−1.67492	−0.837459	−	0.546500i	$-0.815960\pi$
$617$	−41.6041	−1.67492	−0.837459	+	0.546500i	$0.815960\pi$
$618$	0	0
$619$	4.27117	0.171673	0.0858363	−	0.996309i	$-0.472644\pi$
$619$	4.27117	0.171673	0.0858363	+	0.996309i	$0.472644\pi$
$620$	1.83846	0.0738344
$621$	0	0
$622$	−16.0488	−0.643497
$623$	−10.9977	−0.440613
$624$	0	0
$625$	1.00000	0.0400000
$626$	4.72918	0.189016
$627$	0	0
$628$	4.82455	0.192521
$629$	−19.6401	−0.783103
$630$	0	0
$631$	−27.5510	−1.09679	−0.548393	−	0.836221i	$-0.684760\pi$
$631$	−27.5510	−1.09679	−0.548393	+	0.836221i	$0.684760\pi$
$632$	24.6381	0.980049
$633$	0	0
$634$	30.0066	1.19171
$635$	1.53114	0.0607615
$636$	0	0
$637$	58.7877	2.32925
$638$	0	0
$639$	0	0
$640$	−3.01930	−0.119348
$641$	−9.16806	−0.362117	−0.181058	−	0.983472i	$-0.557952\pi$
$641$	−9.16806	−0.362117	−0.181058	+	0.983472i	$0.557952\pi$
$642$	0	0
$643$	−3.40908	−0.134441	−0.0672206	−	0.997738i	$-0.521413\pi$
$643$	−3.40908	−0.134441	−0.0672206	+	0.997738i	$0.521413\pi$
$644$	0.611961	0.0241146
$645$	0	0
$646$	32.5400	1.28027
$647$	−22.3608	−0.879092	−0.439546	−	0.898220i	$-0.644861\pi$
$647$	−22.3608	−0.879092	−0.439546	+	0.898220i	$0.644861\pi$
$648$	0	0
$649$	0	0
$650$	−6.00509	−0.235539
$651$	0	0
$652$	−3.45544	−0.135325
$653$	14.7957	0.578999	0.289500	−	0.957178i	$-0.406511\pi$
$653$	14.7957	0.578999	0.289500	+	0.957178i	$0.406511\pi$
$654$	0	0
$655$	−2.66108	−0.103977
$656$	13.3598	0.521611
$657$	0	0
$658$	1.18509	0.0461997
$659$	47.4724	1.84926	0.924631	−	0.380864i	$-0.124373\pi$
$659$	47.4724	1.84926	0.924631	+	0.380864i	$0.124373\pi$
$660$	0	0
$661$	−21.6525	−0.842184	−0.421092	−	0.907018i	$-0.638353\pi$
$661$	−21.6525	−0.842184	−0.421092	+	0.907018i	$0.638353\pi$
$662$	3.89328	0.151317
$663$	0	0
$664$	−38.6507	−1.49994
$665$	−24.0062	−0.930921
$666$	0	0
$667$	−1.41089	−0.0546300
$668$	−1.28025	−0.0495342
$669$	0	0
$670$	−14.0945	−0.544518
$671$	0	0
$672$	0	0
$673$	−38.3923	−1.47991	−0.739957	−	0.672654i	$-0.765154\pi$
$673$	−38.3923	−1.47991	−0.739957	+	0.672654i	$0.765154\pi$
$674$	31.8426	1.22653
$675$	0	0
$676$	−8.89718	−0.342199
$677$	22.9333	0.881398	0.440699	−	0.897655i	$-0.354731\pi$
$677$	22.9333	0.881398	0.440699	+	0.897655i	$0.354731\pi$
$678$	0	0
$679$	−8.62507	−0.331000
$680$	−15.3950	−0.590373
$681$	0	0
$682$	0	0
$683$	−25.9084	−0.991359	−0.495680	−	0.868505i	$-0.665081\pi$
$683$	−25.9084	−0.991359	−0.495680	+	0.868505i	$0.665081\pi$
$684$	0	0
$685$	−10.2555	−0.391841
$686$	21.8060	0.832557
$687$	0	0
$688$	20.2909	0.773584
$689$	−13.1078	−0.499366
$690$	0	0
$691$	14.8155	0.563608	0.281804	−	0.959472i	$-0.409067\pi$
$691$	14.8155	0.563608	0.281804	+	0.959472i	$0.409067\pi$
$692$	−1.28262	−0.0487579
$693$	0	0
$694$	18.3927	0.698177
$695$	−5.29507	−0.200854
$696$	0	0
$697$	29.3050	1.11001
$698$	4.25264	0.160965
$699$	0	0
$700$	2.78393	0.105223
$701$	−5.50632	−0.207971	−0.103985	−	0.994579i	$-0.533160\pi$
$701$	−5.50632	−0.207971	−0.103985	+	0.994579i	$0.533160\pi$
$702$	0	0
$703$	21.9968	0.829626
$704$	0	0
$705$	0	0
$706$	−31.9517	−1.20252
$707$	−32.5150	−1.22285
$708$	0	0
$709$	−17.9007	−0.672274	−0.336137	−	0.941813i	$-0.609120\pi$
$709$	−17.9007	−0.672274	−0.336137	+	0.941813i	$0.609120\pi$
$710$	−11.6005	−0.435358
$711$	0	0
$712$	7.89905	0.296029
$713$	−0.622297	−0.0233052
$714$	0	0
$715$	0	0
$716$	−1.31386	−0.0491012
$717$	0	0
$718$	0.556367	0.0207634
$719$	−32.5918	−1.21547	−0.607734	−	0.794141i	$-0.707921\pi$
$719$	−32.5918	−1.21547	−0.607734	+	0.794141i	$0.707921\pi$
$720$	0	0
$721$	72.3128	2.69307
$722$	−14.3638	−0.534566
$723$	0	0
$724$	−13.0779	−0.486037
$725$	−6.41843	−0.238375
$726$	0	0
$727$	−43.0199	−1.59552	−0.797759	−	0.602976i	$-0.793982\pi$
$727$	−43.0199	−1.59552	−0.797759	+	0.602976i	$0.793982\pi$
$728$	−68.2035	−2.52779
$729$	0	0
$730$	−16.9238	−0.626378
$731$	44.5087	1.64621
$732$	0	0
$733$	40.7066	1.50353	0.751766	−	0.659429i	$-0.229202\pi$
$733$	40.7066	1.50353	0.751766	+	0.659429i	$0.229202\pi$
$734$	29.5862	1.09204
$735$	0	0
$736$	−0.771340	−0.0284320
$737$	0	0
$738$	0	0
$739$	38.1240	1.40241	0.701207	−	0.712958i	$-0.252645\pi$
$739$	38.1240	1.40241	0.701207	+	0.712958i	$0.252645\pi$
$740$	−2.55091	−0.0937734
$741$	0	0
$742$	−12.6377	−0.463945
$743$	18.0381	0.661754	0.330877	−	0.943674i	$-0.392655\pi$
$743$	18.0381	0.661754	0.330877	+	0.943674i	$0.392655\pi$
$744$	0	0
$745$	−3.78841	−0.138797
$746$	3.20714	0.117422
$747$	0	0
$748$	0	0
$749$	33.2728	1.21576
$750$	0	0
$751$	−17.5349	−0.639859	−0.319929	−	0.947441i	$-0.603659\pi$
$751$	−17.5349	−0.639859	−0.319929	+	0.947441i	$0.603659\pi$
$752$	−0.542227	−0.0197730
$753$	0	0
$754$	38.5433	1.40366
$755$	−5.92001	−0.215451
$756$	0	0
$757$	−25.9609	−0.943565	−0.471782	−	0.881715i	$-0.656389\pi$
$757$	−25.9609	−0.943565	−0.471782	+	0.881715i	$0.656389\pi$
$758$	25.1646	0.914019
$759$	0	0
$760$	17.2424	0.625447
$761$	−28.0059	−1.01521	−0.507607	−	0.861589i	$-0.669470\pi$
$761$	−28.0059	−1.01521	−0.507607	+	0.861589i	$0.669470\pi$
$762$	0	0
$763$	−19.1052	−0.691655
$764$	8.22493	0.297567
$765$	0	0
$766$	18.8610	0.681477
$767$	40.6680	1.46843
$768$	0	0
$769$	22.9537	0.827732	0.413866	−	0.910338i	$-0.364178\pi$
$769$	22.9537	0.827732	0.413866	+	0.910338i	$0.364178\pi$
$770$	0	0
$771$	0	0
$772$	9.04772	0.325634
$773$	23.5282	0.846251	0.423126	−	0.906071i	$-0.360933\pi$
$773$	23.5282	0.846251	0.423126	+	0.906071i	$0.360933\pi$
$774$	0	0
$775$	−2.83095	−0.101691
$776$	6.19492	0.222385
$777$	0	0
$778$	−18.9333	−0.678791
$779$	−32.8215	−1.17595
$780$	0	0
$781$	0	0
$782$	1.27731	0.0456765
$783$	0	0
$784$	−25.9331	−0.926184
$785$	−7.42909	−0.265155
$786$	0	0
$787$	−25.2725	−0.900869	−0.450434	−	0.892810i	$-0.648731\pi$
$787$	−25.2725	−0.900869	−0.450434	+	0.892810i	$0.648731\pi$
$788$	9.48537	0.337902
$789$	0	0
$790$	−9.29944	−0.330859
$791$	−33.0621	−1.17555
$792$	0	0
$793$	−25.1028	−0.891427
$794$	23.2049	0.823511
$795$	0	0
$796$	7.71168	0.273333
$797$	−8.65645	−0.306627	−0.153314	−	0.988178i	$-0.548994\pi$
$797$	−8.65645	−0.306627	−0.153314	+	0.988178i	$0.548994\pi$
$798$	0	0
$799$	−1.18939	−0.0420775
$800$	−3.50898	−0.124061
$801$	0	0
$802$	−31.7579	−1.12141
$803$	0	0
$804$	0	0
$805$	−0.942328	−0.0332127
$806$	17.0001	0.598804
$807$	0	0
$808$	23.3538	0.821584
$809$	10.2124	0.359050	0.179525	−	0.983753i	$-0.442544\pi$
$809$	10.2124	0.359050	0.179525	+	0.983753i	$0.442544\pi$
$810$	0	0
$811$	−47.6224	−1.67225	−0.836124	−	0.548540i	$-0.815184\pi$
$811$	−47.6224	−1.67225	−0.836124	+	0.548540i	$0.815184\pi$
$812$	−17.8685	−0.627061
$813$	0	0
$814$	0	0
$815$	5.32086	0.186381
$816$	0	0
$817$	−49.8495	−1.74401
$818$	−28.9816	−1.01332
$819$	0	0
$820$	3.80621	0.132919
$821$	47.5598	1.65985	0.829925	−	0.557875i	$-0.188383\pi$
$821$	47.5598	1.65985	0.829925	+	0.557875i	$0.188383\pi$
$822$	0	0
$823$	−24.7360	−0.862242	−0.431121	−	0.902294i	$-0.641882\pi$
$823$	−24.7360	−0.862242	−0.431121	+	0.902294i	$0.641882\pi$
$824$	−51.9384	−1.80936
$825$	0	0
$826$	39.2096	1.36428
$827$	1.56166	0.0543043	0.0271522	−	0.999631i	$-0.491356\pi$
$827$	1.56166	0.0543043	0.0271522	+	0.999631i	$0.491356\pi$
$828$	0	0
$829$	42.1950	1.46549	0.732747	−	0.680501i	$-0.238238\pi$
$829$	42.1950	1.46549	0.732747	+	0.680501i	$0.238238\pi$
$830$	14.5884	0.506370
$831$	0	0
$832$	44.6285	1.54721
$833$	−56.8851	−1.97095
$834$	0	0
$835$	1.97139	0.0682227
$836$	0	0
$837$	0	0
$838$	19.5539	0.675477
$839$	6.29451	0.217311	0.108655	−	0.994079i	$-0.465346\pi$
$839$	6.29451	0.217311	0.108655	+	0.994079i	$0.465346\pi$
$840$	0	0
$841$	12.1963	0.420562
$842$	−27.4274	−0.945211
$843$	0	0
$844$	14.5156	0.499647
$845$	13.7003	0.471305
$846$	0	0
$847$	0	0
$848$	5.78225	0.198563
$849$	0	0
$850$	5.81074	0.199307
$851$	0.863453	0.0295988
$852$	0	0
$853$	−7.27120	−0.248961	−0.124481	−	0.992222i	$-0.539726\pi$
$853$	−7.27120	−0.248961	−0.124481	+	0.992222i	$0.539726\pi$
$854$	−24.2026	−0.828197
$855$	0	0
$856$	−23.8981	−0.816819
$857$	−39.8590	−1.36156	−0.680779	−	0.732489i	$-0.738359\pi$
$857$	−39.8590	−1.36156	−0.680779	+	0.732489i	$0.738359\pi$
$858$	0	0
$859$	13.5278	0.461564	0.230782	−	0.973006i	$-0.425872\pi$
$859$	13.5278	0.461564	0.230782	+	0.973006i	$0.425872\pi$
$860$	5.78091	0.197127
$861$	0	0
$862$	−43.7613	−1.49052
$863$	−50.3117	−1.71263	−0.856316	−	0.516453i	$-0.827252\pi$
$863$	−50.3117	−1.71263	−0.856316	+	0.516453i	$0.827252\pi$
$864$	0	0
$865$	1.97504	0.0671535
$866$	35.9977	1.22325
$867$	0	0
$868$	−7.88119	−0.267505
$869$	0	0
$870$	0	0
$871$	62.6681	2.12343
$872$	13.7223	0.464694
$873$	0	0
$874$	−1.43058	−0.0483901
$875$	−4.28684	−0.144922
$876$	0	0
$877$	−1.20736	−0.0407696	−0.0203848	−	0.999792i	$-0.506489\pi$
$877$	−1.20736	−0.0407696	−0.0203848	+	0.999792i	$0.506489\pi$
$878$	32.7877	1.10653
$879$	0	0
$880$	0	0
$881$	−1.91816	−0.0646245	−0.0323123	−	0.999478i	$-0.510287\pi$
$881$	−1.91816	−0.0646245	−0.0323123	+	0.999478i	$0.510287\pi$
$882$	0	0
$883$	58.0890	1.95485	0.977425	−	0.211281i	$-0.0677634\pi$
$883$	58.0890	1.95485	0.977425	+	0.211281i	$0.0677634\pi$
$884$	16.7784	0.564318
$885$	0	0
$886$	17.0162	0.571669
$887$	−30.2427	−1.01545	−0.507725	−	0.861519i	$-0.669513\pi$
$887$	−30.2427	−1.01545	−0.507725	+	0.861519i	$0.669513\pi$
$888$	0	0
$889$	−6.56377	−0.220142
$890$	−2.98143	−0.0999378
$891$	0	0
$892$	12.4417	0.416578
$893$	1.33211	0.0445773
$894$	0	0
$895$	2.02315	0.0676263
$896$	12.9432	0.432403
$897$	0	0
$898$	13.9431	0.465286
$899$	18.1703	0.606013
$900$	0	0
$901$	12.6835	0.422550
$902$	0	0
$903$	0	0
$904$	23.7468	0.789805
$905$	20.1380	0.669410
$906$	0	0
$907$	−31.9669	−1.06144	−0.530721	−	0.847547i	$-0.678079\pi$
$907$	−31.9669	−1.06144	−0.530721	+	0.847547i	$0.678079\pi$
$908$	9.98532	0.331374
$909$	0	0
$910$	25.7429	0.853368
$911$	51.8454	1.71771	0.858857	−	0.512216i	$-0.171175\pi$
$911$	51.8454	1.71771	0.858857	+	0.512216i	$0.171175\pi$
$912$	0	0
$913$	0	0
$914$	−11.6722	−0.386083
$915$	0	0
$916$	−5.78060	−0.190996
$917$	11.4076	0.376714
$918$	0	0
$919$	−34.2084	−1.12843	−0.564215	−	0.825628i	$-0.690821\pi$
$919$	−34.2084	−1.12843	−0.564215	+	0.825628i	$0.690821\pi$
$920$	0.676824	0.0223142
$921$	0	0
$922$	−15.6298	−0.514741
$923$	51.5790	1.69774
$924$	0	0
$925$	3.92802	0.129153
$926$	−21.8293	−0.717354
$927$	0	0
$928$	22.5222	0.739326
$929$	−18.3482	−0.601984	−0.300992	−	0.953627i	$-0.597318\pi$
$929$	−18.3482	−0.601984	−0.300992	+	0.953627i	$0.597318\pi$
$930$	0	0
$931$	63.7110	2.08804
$932$	14.8301	0.485777
$933$	0	0
$934$	30.8561	1.00964
$935$	0	0
$936$	0	0
$937$	3.38415	0.110555	0.0552777	−	0.998471i	$-0.482396\pi$
$937$	3.38415	0.110555	0.0552777	+	0.998471i	$0.482396\pi$
$938$	60.4208	1.97281
$939$	0	0
$940$	−0.154481	−0.00503862
$941$	35.1002	1.14423	0.572116	−	0.820172i	$-0.306123\pi$
$941$	35.1002	1.14423	0.572116	+	0.820172i	$0.306123\pi$
$942$	0	0
$943$	−1.28836	−0.0419547
$944$	−17.9399	−0.583895
$945$	0	0
$946$	0	0
$947$	39.0512	1.26899	0.634497	−	0.772925i	$-0.281207\pi$
$947$	39.0512	1.26899	0.634497	+	0.772925i	$0.281207\pi$
$948$	0	0
$949$	75.2480	2.44265
$950$	−6.50800	−0.211147
$951$	0	0
$952$	65.9961	2.13895
$953$	−24.6783	−0.799409	−0.399705	−	0.916644i	$-0.630887\pi$
$953$	−24.6783	−0.799409	−0.399705	+	0.916644i	$0.630887\pi$
$954$	0	0
$955$	−12.6652	−0.409835
$956$	10.9495	0.354131
$957$	0	0
$958$	12.5315	0.404873
$959$	43.9635	1.41966
$960$	0	0
$961$	−22.9857	−0.741474
$962$	−23.5881	−0.760512
$963$	0	0
$964$	7.76747	0.250173
$965$	−13.9321	−0.448491
$966$	0	0
$967$	1.20724	0.0388222	0.0194111	−	0.999812i	$-0.493821\pi$
$967$	1.20724	0.0388222	0.0194111	+	0.999812i	$0.493821\pi$
$968$	0	0
$969$	0	0
$970$	−2.33822	−0.0750758
$971$	−50.3821	−1.61684	−0.808418	−	0.588608i	$-0.799676\pi$
$971$	−50.3821	−1.61684	−0.808418	+	0.588608i	$0.799676\pi$
$972$	0	0
$973$	22.6991	0.727701
$974$	−7.91533	−0.253624
$975$	0	0
$976$	11.0737	0.354459
$977$	5.88330	0.188223	0.0941117	−	0.995562i	$-0.469999\pi$
$977$	5.88330	0.188223	0.0941117	+	0.995562i	$0.469999\pi$
$978$	0	0
$979$	0	0
$980$	−7.38839	−0.236013
$981$	0	0
$982$	30.3267	0.967763
$983$	−36.4899	−1.16385	−0.581924	−	0.813243i	$-0.697700\pi$
$983$	−36.4899	−1.16385	−0.581924	+	0.813243i	$0.697700\pi$
$984$	0	0
$985$	−14.6060	−0.465387
$986$	−37.2958	−1.18774
$987$	0	0
$988$	−18.7917	−0.597843
$989$	−1.95677	−0.0622216
$990$	0	0
$991$	−36.8404	−1.17027	−0.585137	−	0.810934i	$-0.698959\pi$
$991$	−36.8404	−1.17027	−0.585137	+	0.810934i	$0.698959\pi$
$992$	9.93376	0.315397
$993$	0	0
$994$	49.7294	1.57732
$995$	−11.8748	−0.376457
$996$	0	0
$997$	34.8694	1.10432	0.552162	−	0.833737i	$-0.313803\pi$
$997$	34.8694	1.10432	0.552162	+	0.833737i	$0.313803\pi$
$998$	−22.5263	−0.713059
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.be.1.3		4
3.2	odd	2		1815.2.a.x.1.2		4
11.3	even	5		495.2.n.d.361.2		8
11.4	even	5		495.2.n.d.181.2		8
11.10	odd	2		5445.2.a.bv.1.2		4
15.14	odd	2		9075.2.a.cl.1.3		4
33.14	odd	10		165.2.m.a.31.1	yes	8
33.26	odd	10		165.2.m.a.16.1	✓	8
33.32	even	2		1815.2.a.o.1.3		4
165.14	odd	10		825.2.n.k.526.2		8
165.47	even	20		825.2.bx.h.724.3		16
165.59	odd	10		825.2.n.k.676.2		8
165.92	even	20		825.2.bx.h.49.2		16
165.113	even	20		825.2.bx.h.724.2		16
165.158	even	20		825.2.bx.h.49.3		16
165.164	even	2		9075.2.a.dj.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.m.a.16.1	✓	8	33.26	odd	10
165.2.m.a.31.1	yes	8	33.14	odd	10
495.2.n.d.181.2		8	11.4	even	5
495.2.n.d.361.2		8	11.3	even	5
825.2.n.k.526.2		8	165.14	odd	10
825.2.n.k.676.2		8	165.59	odd	10
825.2.bx.h.49.2		16	165.92	even	20
825.2.bx.h.49.3		16	165.158	even	20
825.2.bx.h.724.2		16	165.113	even	20
825.2.bx.h.724.3		16	165.47	even	20
1815.2.a.o.1.3		4	33.32	even	2
1815.2.a.x.1.2		4	3.2	odd	2
5445.2.a.be.1.3		4	1.1	even	1	trivial
5445.2.a.bv.1.2		4	11.10	odd	2
9075.2.a.cl.1.3		4	15.14	odd	2
9075.2.a.dj.1.2		4	165.164	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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q-1.16215 q^{2} -0.649414 q^{4} +1.00000 q^{5} -4.28684 q^{7} +3.07901 q^{8} +O(q^{10})\)	\(q-1.16215 q^{2} -0.649414 q^{4} +1.00000 q^{5} -4.28684 q^{7} +3.07901 q^{8} -1.16215 q^{10} +5.16724 q^{13} +4.98194 q^{14} -2.27943 q^{16} -5.00000 q^{17} +5.59998 q^{19} -0.649414 q^{20} +0.219819 q^{23} +1.00000 q^{25} -6.00509 q^{26} +2.78393 q^{28} -6.41843 q^{29} -2.83095 q^{31} -3.50898 q^{32} +5.81074 q^{34} -4.28684 q^{35} +3.92802 q^{37} -6.50800 q^{38} +3.07901 q^{40} -5.86100 q^{41} -8.90173 q^{43} -0.255462 q^{46} +0.237878 q^{47} +11.3770 q^{49} -1.16215 q^{50} -3.35567 q^{52} -2.53671 q^{53} -13.1992 q^{56} +7.45917 q^{58} +7.87035 q^{59} -4.85807 q^{61} +3.28999 q^{62} +8.63682 q^{64} +5.16724 q^{65} +12.1280 q^{67} +3.24707 q^{68} +4.98194 q^{70} +9.98194 q^{71} +14.5625 q^{73} -4.56494 q^{74} -3.63670 q^{76} +8.00194 q^{79} -2.27943 q^{80} +6.81134 q^{82} -12.5530 q^{83} -5.00000 q^{85} +10.3451 q^{86} +2.56545 q^{89} -22.1511 q^{91} -0.142753 q^{92} -0.276449 q^{94} +5.59998 q^{95} +2.01199 q^{97} -13.2218 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 5 q^{2} + 9 q^{4} + 4 q^{5} - 2 q^{7} - 15 q^{8}+O(q^{10}) \) 4 * q - 5 * q^2 + 9 * q^4 + 4 * q^5 - 2 * q^7 - 15 * q^8	\( 4 q - 5 q^{2} + 9 q^{4} + 4 q^{5} - 2 q^{7} - 15 q^{8} - 5 q^{10} + 3 q^{13} + 5 q^{14} + 15 q^{16} - 20 q^{17} + 3 q^{19} + 9 q^{20} + 5 q^{23} + 4 q^{25} - 6 q^{26} + 3 q^{28} - 5 q^{29} - q^{31} - 30 q^{32} + 25 q^{34} - 2 q^{35} - 7 q^{37} - q^{38} - 15 q^{40} - 20 q^{41} - 2 q^{43} + 7 q^{46} + 20 q^{47} + 8 q^{49} - 5 q^{50} - 7 q^{52} - 6 q^{53} - 10 q^{56} - 21 q^{58} + 5 q^{59} - 7 q^{61} + 12 q^{62} + 49 q^{64} + 3 q^{65} - 13 q^{67} - 45 q^{68} + 5 q^{70} + 25 q^{71} + 23 q^{73} + 7 q^{74} - 7 q^{76} + 15 q^{80} + 11 q^{82} - 33 q^{83} - 20 q^{85} + 12 q^{86} - 16 q^{89} - 24 q^{91} - 17 q^{94} + 3 q^{95} - 25 q^{98}+O(q^{100}) \) 4 * q - 5 * q^2 + 9 * q^4 + 4 * q^5 - 2 * q^7 - 15 * q^8 - 5 * q^10 + 3 * q^13 + 5 * q^14 + 15 * q^16 - 20 * q^17 + 3 * q^19 + 9 * q^20 + 5 * q^23 + 4 * q^25 - 6 * q^26 + 3 * q^28 - 5 * q^29 - q^31 - 30 * q^32 + 25 * q^34 - 2 * q^35 - 7 * q^37 - q^38 - 15 * q^40 - 20 * q^41 - 2 * q^43 + 7 * q^46 + 20 * q^47 + 8 * q^49 - 5 * q^50 - 7 * q^52 - 6 * q^53 - 10 * q^56 - 21 * q^58 + 5 * q^59 - 7 * q^61 + 12 * q^62 + 49 * q^64 + 3 * q^65 - 13 * q^67 - 45 * q^68 + 5 * q^70 + 25 * q^71 + 23 * q^73 + 7 * q^74 - 7 * q^76 + 15 * q^80 + 11 * q^82 - 33 * q^83 - 20 * q^85 + 12 * q^86 - 16 * q^89 - 24 * q^91 - 17 * q^94 + 3 * q^95 - 25 * q^98

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