LMFDB - Embedded newform 5445.2.a.m.1.1

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
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			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} -0.828427 q^{7} -4.41421 q^{8} +O(q^{10})$	$q-2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} -0.828427 q^{7} -4.41421 q^{8} -2.41421 q^{10} +5.65685 q^{13} +2.00000 q^{14} +3.00000 q^{16} -1.17157 q^{17} +6.82843 q^{19} +3.82843 q^{20} +4.00000 q^{23} +1.00000 q^{25} -13.6569 q^{26} -3.17157 q^{28} -4.82843 q^{29} +1.58579 q^{32} +2.82843 q^{34} -0.828427 q^{35} +11.6569 q^{37} -16.4853 q^{38} -4.41421 q^{40} +4.82843 q^{41} +8.82843 q^{43} -9.65685 q^{46} +4.00000 q^{47} -6.31371 q^{49} -2.41421 q^{50} +21.6569 q^{52} -9.31371 q^{53} +3.65685 q^{56} +11.6569 q^{58} +4.00000 q^{59} +11.6569 q^{61} -9.82843 q^{64} +5.65685 q^{65} -5.65685 q^{67} -4.48528 q^{68} +2.00000 q^{70} -2.34315 q^{71} -11.3137 q^{73} -28.1421 q^{74} +26.1421 q^{76} -8.48528 q^{79} +3.00000 q^{80} -11.6569 q^{82} -10.0000 q^{83} -1.17157 q^{85} -21.3137 q^{86} -3.65685 q^{89} -4.68629 q^{91} +15.3137 q^{92} -9.65685 q^{94} +6.82843 q^{95} +11.6569 q^{97} +15.2426 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 + 4 * q^7 - 6 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} - 6 q^{8} - 2 q^{10} + 4 q^{14} + 6 q^{16} - 8 q^{17} + 8 q^{19} + 2 q^{20} + 8 q^{23} + 2 q^{25} - 16 q^{26} - 12 q^{28} - 4 q^{29} + 6 q^{32} + 4 q^{35} + 12 q^{37} - 16 q^{38} - 6 q^{40} + 4 q^{41} + 12 q^{43} - 8 q^{46} + 8 q^{47} + 10 q^{49} - 2 q^{50} + 32 q^{52} + 4 q^{53} - 4 q^{56} + 12 q^{58} + 8 q^{59} + 12 q^{61} - 14 q^{64} + 8 q^{68} + 4 q^{70} - 16 q^{71} - 28 q^{74} + 24 q^{76} + 6 q^{80} - 12 q^{82} - 20 q^{83} - 8 q^{85} - 20 q^{86} + 4 q^{89} - 32 q^{91} + 8 q^{92} - 8 q^{94} + 8 q^{95} + 12 q^{97} + 22 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 + 4 * q^7 - 6 * q^8 - 2 * q^10 + 4 * q^14 + 6 * q^16 - 8 * q^17 + 8 * q^19 + 2 * q^20 + 8 * q^23 + 2 * q^25 - 16 * q^26 - 12 * q^28 - 4 * q^29 + 6 * q^32 + 4 * q^35 + 12 * q^37 - 16 * q^38 - 6 * q^40 + 4 * q^41 + 12 * q^43 - 8 * q^46 + 8 * q^47 + 10 * q^49 - 2 * q^50 + 32 * q^52 + 4 * q^53 - 4 * q^56 + 12 * q^58 + 8 * q^59 + 12 * q^61 - 14 * q^64 + 8 * q^68 + 4 * q^70 - 16 * q^71 - 28 * q^74 + 24 * q^76 + 6 * q^80 - 12 * q^82 - 20 * q^83 - 8 * q^85 - 20 * q^86 + 4 * q^89 - 32 * q^91 + 8 * q^92 - 8 * q^94 + 8 * q^95 + 12 * q^97 + 22 * 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 $

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.828427	−0.313116	−0.156558	−	0.987669i	$-0.550040\pi$
$7$	−0.828427	−0.313116	−0.156558	+	0.987669i	$0.550040\pi$
$8$	−4.41421	−1.56066
$9$	0	0
$10$	−2.41421	−0.763441
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.65685	1.56893	0.784465	−	0.620174i	$-0.212938\pi$
$13$	5.65685	1.56893	0.784465	+	0.620174i	$0.212938\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	3.00000	0.750000
$17$	−1.17157	−0.284148	−0.142074	−	0.989856i	$-0.545377\pi$
$17$	−1.17157	−0.284148	−0.142074	+	0.989856i	$0.545377\pi$
$18$	0	0
$19$	6.82843	1.56655	0.783274	−	0.621676i	$-0.213548\pi$
$19$	6.82843	1.56655	0.783274	+	0.621676i	$0.213548\pi$
$20$	3.82843	0.856062
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−13.6569	−2.67833
$27$	0	0
$28$	−3.17157	−0.599371
$29$	−4.82843	−0.896616	−0.448308	−	0.893879i	$-0.647973\pi$
$29$	−4.82843	−0.896616	−0.448308	+	0.893879i	$0.647973\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.58579	0.280330
$33$	0	0
$34$	2.82843	0.485071
$35$	−0.828427	−0.140030
$36$	0	0
$37$	11.6569	1.91638	0.958188	−	0.286141i	$-0.0923726\pi$
$37$	11.6569	1.91638	0.958188	+	0.286141i	$0.0923726\pi$
$38$	−16.4853	−2.67427
$39$	0	0
$40$	−4.41421	−0.697948
$41$	4.82843	0.754074	0.377037	−	0.926198i	$-0.376943\pi$
$41$	4.82843	0.754074	0.377037	+	0.926198i	$0.376943\pi$
$42$	0	0
$43$	8.82843	1.34632	0.673161	−	0.739496i	$-0.264936\pi$
$43$	8.82843	1.34632	0.673161	+	0.739496i	$0.264936\pi$
$44$	0	0
$45$	0	0
$46$	−9.65685	−1.42383
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	−6.31371	−0.901958
$50$	−2.41421	−0.341421
$51$	0	0
$52$	21.6569	3.00327
$53$	−9.31371	−1.27934	−0.639668	−	0.768651i	$-0.720928\pi$
$53$	−9.31371	−1.27934	−0.639668	+	0.768651i	$0.720928\pi$
$54$	0	0
$55$	0	0
$56$	3.65685	0.488668
$57$	0	0
$58$	11.6569	1.53062
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	11.6569	1.49251	0.746254	−	0.665662i	$-0.231851\pi$
$61$	11.6569	1.49251	0.746254	+	0.665662i	$0.231851\pi$
$62$	0	0
$63$	0	0
$64$	−9.82843	−1.22855
$65$	5.65685	0.701646
$66$	0	0
$67$	−5.65685	−0.691095	−0.345547	−	0.938401i	$-0.612307\pi$
$67$	−5.65685	−0.691095	−0.345547	+	0.938401i	$0.612307\pi$
$68$	−4.48528	−0.543920
$69$	0	0
$70$	2.00000	0.239046
$71$	−2.34315	−0.278080	−0.139040	−	0.990287i	$-0.544402\pi$
$71$	−2.34315	−0.278080	−0.139040	+	0.990287i	$0.544402\pi$
$72$	0	0
$73$	−11.3137	−1.32417	−0.662085	−	0.749429i	$-0.730328\pi$
$73$	−11.3137	−1.32417	−0.662085	+	0.749429i	$0.730328\pi$
$74$	−28.1421	−3.27146
$75$	0	0
$76$	26.1421	2.99871
$77$	0	0
$78$	0	0
$79$	−8.48528	−0.954669	−0.477334	−	0.878722i	$-0.658397\pi$
$79$	−8.48528	−0.954669	−0.477334	+	0.878722i	$0.658397\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	−11.6569	−1.28728
$83$	−10.0000	−1.09764	−0.548821	−	0.835940i	$-0.684923\pi$
$83$	−10.0000	−1.09764	−0.548821	+	0.835940i	$0.684923\pi$
$84$	0	0
$85$	−1.17157	−0.127075
$86$	−21.3137	−2.29832
$87$	0	0
$88$	0	0
$89$	−3.65685	−0.387626	−0.193813	−	0.981039i	$-0.562085\pi$
$89$	−3.65685	−0.387626	−0.193813	+	0.981039i	$0.562085\pi$
$90$	0	0
$91$	−4.68629	−0.491257
$92$	15.3137	1.59656
$93$	0	0
$94$	−9.65685	−0.996028
$95$	6.82843	0.700582
$96$	0	0
$97$	11.6569	1.18357	0.591787	−	0.806094i	$-0.298423\pi$
$97$	11.6569	1.18357	0.591787	+	0.806094i	$0.298423\pi$
$98$	15.2426	1.53974
$99$	0	0
$100$	3.82843	0.382843
$101$	−0.828427	−0.0824316	−0.0412158	−	0.999150i	$-0.513123\pi$
$101$	−0.828427	−0.0824316	−0.0412158	+	0.999150i	$0.513123\pi$
$102$	0	0
$103$	−3.31371	−0.326509	−0.163255	−	0.986584i	$-0.552199\pi$
$103$	−3.31371	−0.326509	−0.163255	+	0.986584i	$0.552199\pi$
$104$	−24.9706	−2.44857
$105$	0	0
$106$	22.4853	2.18396
$107$	17.3137	1.67378	0.836890	−	0.547372i	$-0.184372\pi$
$107$	17.3137	1.67378	0.836890	+	0.547372i	$0.184372\pi$
$108$	0	0
$109$	−17.3137	−1.65835	−0.829176	−	0.558987i	$-0.811190\pi$
$109$	−17.3137	−1.65835	−0.829176	+	0.558987i	$0.811190\pi$
$110$	0	0
$111$	0	0
$112$	−2.48528	−0.234837
$113$	18.9706	1.78460	0.892300	−	0.451442i	$-0.149090\pi$
$113$	18.9706	1.78460	0.892300	+	0.451442i	$0.149090\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	−18.4853	−1.71632
$117$	0	0
$118$	−9.65685	−0.888985
$119$	0.970563	0.0889713
$120$	0	0
$121$	0	0
$122$	−28.1421	−2.54787
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	14.4853	1.28536	0.642680	−	0.766134i	$-0.277822\pi$
$127$	14.4853	1.28536	0.642680	+	0.766134i	$0.277822\pi$
$128$	20.5563	1.81694
$129$	0	0
$130$	−13.6569	−1.19779
$131$	3.31371	0.289520	0.144760	−	0.989467i	$-0.453759\pi$
$131$	3.31371	0.289520	0.144760	+	0.989467i	$0.453759\pi$
$132$	0	0
$133$	−5.65685	−0.490511
$134$	13.6569	1.17977
$135$	0	0
$136$	5.17157	0.443459
$137$	13.3137	1.13747	0.568733	−	0.822522i	$-0.307434\pi$
$137$	13.3137	1.13747	0.568733	+	0.822522i	$0.307434\pi$
$138$	0	0
$139$	−0.485281	−0.0411610	−0.0205805	−	0.999788i	$-0.506551\pi$
$139$	−0.485281	−0.0411610	−0.0205805	+	0.999788i	$0.506551\pi$
$140$	−3.17157	−0.268047
$141$	0	0
$142$	5.65685	0.474713
$143$	0	0
$144$	0	0
$145$	−4.82843	−0.400979
$146$	27.3137	2.26050
$147$	0	0
$148$	44.6274	3.66835
$149$	1.51472	0.124091	0.0620453	−	0.998073i	$-0.480238\pi$
$149$	1.51472	0.124091	0.0620453	+	0.998073i	$0.480238\pi$
$150$	0	0
$151$	−16.4853	−1.34155	−0.670777	−	0.741659i	$-0.734039\pi$
$151$	−16.4853	−1.34155	−0.670777	+	0.741659i	$0.734039\pi$
$152$	−30.1421	−2.44485
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	20.4853	1.62972
$159$	0	0
$160$	1.58579	0.125367
$161$	−3.31371	−0.261157
$162$	0	0
$163$	−7.31371	−0.572854	−0.286427	−	0.958102i	$-0.592468\pi$
$163$	−7.31371	−0.572854	−0.286427	+	0.958102i	$0.592468\pi$
$164$	18.4853	1.44346
$165$	0	0
$166$	24.1421	1.87379
$167$	−13.3137	−1.03025	−0.515123	−	0.857116i	$-0.672254\pi$
$167$	−13.3137	−1.03025	−0.515123	+	0.857116i	$0.672254\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	2.82843	0.216930
$171$	0	0
$172$	33.7990	2.57715
$173$	2.82843	0.215041	0.107521	−	0.994203i	$-0.465709\pi$
$173$	2.82843	0.215041	0.107521	+	0.994203i	$0.465709\pi$
$174$	0	0
$175$	−0.828427	−0.0626232
$176$	0	0
$177$	0	0
$178$	8.82843	0.661719
$179$	17.6569	1.31974	0.659868	−	0.751382i	$-0.270612\pi$
$179$	17.6569	1.31974	0.659868	+	0.751382i	$0.270612\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	11.3137	0.838628
$183$	0	0
$184$	−17.6569	−1.30168
$185$	11.6569	0.857029
$186$	0	0
$187$	0	0
$188$	15.3137	1.11687
$189$	0	0
$190$	−16.4853	−1.19597
$191$	−5.65685	−0.409316	−0.204658	−	0.978834i	$-0.565608\pi$
$191$	−5.65685	−0.409316	−0.204658	+	0.978834i	$0.565608\pi$
$192$	0	0
$193$	−13.6569	−0.983042	−0.491521	−	0.870866i	$-0.663559\pi$
$193$	−13.6569	−0.983042	−0.491521	+	0.870866i	$0.663559\pi$
$194$	−28.1421	−2.02049
$195$	0	0
$196$	−24.1716	−1.72654
$197$	−8.48528	−0.604551	−0.302276	−	0.953221i	$-0.597746\pi$
$197$	−8.48528	−0.604551	−0.302276	+	0.953221i	$0.597746\pi$
$198$	0	0
$199$	−21.6569	−1.53521	−0.767607	−	0.640921i	$-0.778553\pi$
$199$	−21.6569	−1.53521	−0.767607	+	0.640921i	$0.778553\pi$
$200$	−4.41421	−0.312132
$201$	0	0
$202$	2.00000	0.140720
$203$	4.00000	0.280745
$204$	0	0
$205$	4.82843	0.337232
$206$	8.00000	0.557386
$207$	0	0
$208$	16.9706	1.17670
$209$	0	0
$210$	0	0
$211$	−1.17157	−0.0806544	−0.0403272	−	0.999187i	$-0.512840\pi$
$211$	−1.17157	−0.0806544	−0.0403272	+	0.999187i	$0.512840\pi$
$212$	−35.6569	−2.44892
$213$	0	0
$214$	−41.7990	−2.85732
$215$	8.82843	0.602094
$216$	0	0
$217$	0	0
$218$	41.7990	2.83098
$219$	0	0
$220$	0	0
$221$	−6.62742	−0.445808
$222$	0	0
$223$	−6.34315	−0.424768	−0.212384	−	0.977186i	$-0.568123\pi$
$223$	−6.34315	−0.424768	−0.212384	+	0.977186i	$0.568123\pi$
$224$	−1.31371	−0.0877758
$225$	0	0
$226$	−45.7990	−3.04650
$227$	−14.0000	−0.929213	−0.464606	−	0.885517i	$-0.653804\pi$
$227$	−14.0000	−0.929213	−0.464606	+	0.885517i	$0.653804\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	−9.65685	−0.636754
$231$	0	0
$232$	21.3137	1.39931
$233$	−18.8284	−1.23349	−0.616746	−	0.787163i	$-0.711549\pi$
$233$	−18.8284	−1.23349	−0.616746	+	0.787163i	$0.711549\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	15.3137	0.996838
$237$	0	0
$238$	−2.34315	−0.151884
$239$	17.6569	1.14213	0.571063	−	0.820906i	$-0.306531\pi$
$239$	17.6569	1.14213	0.571063	+	0.820906i	$0.306531\pi$
$240$	0	0
$241$	12.3431	0.795092	0.397546	−	0.917582i	$-0.369862\pi$
$241$	12.3431	0.795092	0.397546	+	0.917582i	$0.369862\pi$
$242$	0	0
$243$	0	0
$244$	44.6274	2.85698
$245$	−6.31371	−0.403368
$246$	0	0
$247$	38.6274	2.45780
$248$	0	0
$249$	0	0
$250$	−2.41421	−0.152688
$251$	−20.9706	−1.32365	−0.661825	−	0.749658i	$-0.730218\pi$
$251$	−20.9706	−1.32365	−0.661825	+	0.749658i	$0.730218\pi$
$252$	0	0
$253$	0	0
$254$	−34.9706	−2.19425
$255$	0	0
$256$	−29.9706	−1.87316
$257$	16.3431	1.01946	0.509729	−	0.860335i	$-0.329746\pi$
$257$	16.3431	1.01946	0.509729	+	0.860335i	$0.329746\pi$
$258$	0	0
$259$	−9.65685	−0.600048
$260$	21.6569	1.34310
$261$	0	0
$262$	−8.00000	−0.494242
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	−9.31371	−0.572137
$266$	13.6569	0.837355
$267$	0	0
$268$	−21.6569	−1.32290
$269$	−20.6274	−1.25768	−0.628838	−	0.777536i	$-0.716469\pi$
$269$	−20.6274	−1.25768	−0.628838	+	0.777536i	$0.716469\pi$
$270$	0	0
$271$	11.7990	0.716738	0.358369	−	0.933580i	$-0.383333\pi$
$271$	11.7990	0.716738	0.358369	+	0.933580i	$0.383333\pi$
$272$	−3.51472	−0.213111
$273$	0	0
$274$	−32.1421	−1.94178
$275$	0	0
$276$	0	0
$277$	2.34315	0.140786	0.0703930	−	0.997519i	$-0.477575\pi$
$277$	2.34315	0.140786	0.0703930	+	0.997519i	$0.477575\pi$
$278$	1.17157	0.0702663
$279$	0	0
$280$	3.65685	0.218539
$281$	−11.1716	−0.666440	−0.333220	−	0.942849i	$-0.608135\pi$
$281$	−11.1716	−0.666440	−0.333220	+	0.942849i	$0.608135\pi$
$282$	0	0
$283$	8.82843	0.524796	0.262398	−	0.964960i	$-0.415487\pi$
$283$	8.82843	0.524796	0.262398	+	0.964960i	$0.415487\pi$
$284$	−8.97056	−0.532305
$285$	0	0
$286$	0	0
$287$	−4.00000	−0.236113
$288$	0	0
$289$	−15.6274	−0.919260
$290$	11.6569	0.684514
$291$	0	0
$292$	−43.3137	−2.53474
$293$	−6.82843	−0.398921	−0.199460	−	0.979906i	$-0.563919\pi$
$293$	−6.82843	−0.398921	−0.199460	+	0.979906i	$0.563919\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	−51.4558	−2.99081
$297$	0	0
$298$	−3.65685	−0.211836
$299$	22.6274	1.30858
$300$	0	0
$301$	−7.31371	−0.421555
$302$	39.7990	2.29017
$303$	0	0
$304$	20.4853	1.17491
$305$	11.6569	0.667470
$306$	0	0
$307$	3.17157	0.181011	0.0905056	−	0.995896i	$-0.471152\pi$
$307$	3.17157	0.181011	0.0905056	+	0.995896i	$0.471152\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.31371	0.187903	0.0939516	−	0.995577i	$-0.470050\pi$
$311$	3.31371	0.187903	0.0939516	+	0.995577i	$0.470050\pi$
$312$	0	0
$313$	15.6569	0.884978	0.442489	−	0.896774i	$-0.354096\pi$
$313$	15.6569	0.884978	0.442489	+	0.896774i	$0.354096\pi$
$314$	−43.4558	−2.45236
$315$	0	0
$316$	−32.4853	−1.82744
$317$	26.2843	1.47627	0.738136	−	0.674652i	$-0.235706\pi$
$317$	26.2843	1.47627	0.738136	+	0.674652i	$0.235706\pi$
$318$	0	0
$319$	0	0
$320$	−9.82843	−0.549426
$321$	0	0
$322$	8.00000	0.445823
$323$	−8.00000	−0.445132
$324$	0	0
$325$	5.65685	0.313786
$326$	17.6569	0.977923
$327$	0	0
$328$	−21.3137	−1.17685
$329$	−3.31371	−0.182691
$330$	0	0
$331$	6.34315	0.348651	0.174325	−	0.984688i	$-0.444226\pi$
$331$	6.34315	0.348651	0.174325	+	0.984688i	$0.444226\pi$
$332$	−38.2843	−2.10112
$333$	0	0
$334$	32.1421	1.75874
$335$	−5.65685	−0.309067
$336$	0	0
$337$	−3.31371	−0.180509	−0.0902546	−	0.995919i	$-0.528768\pi$
$337$	−3.31371	−0.180509	−0.0902546	+	0.995919i	$0.528768\pi$
$338$	−45.8701	−2.49500
$339$	0	0
$340$	−4.48528	−0.243249
$341$	0	0
$342$	0	0
$343$	11.0294	0.595534
$344$	−38.9706	−2.10115
$345$	0	0
$346$	−6.82843	−0.367099
$347$	−29.3137	−1.57364	−0.786821	−	0.617181i	$-0.788275\pi$
$347$	−29.3137	−1.57364	−0.786821	+	0.617181i	$0.788275\pi$
$348$	0	0
$349$	−10.9706	−0.587241	−0.293620	−	0.955922i	$-0.594860\pi$
$349$	−10.9706	−0.587241	−0.293620	+	0.955922i	$0.594860\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	0	0
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	0	0
$355$	−2.34315	−0.124361
$356$	−14.0000	−0.741999
$357$	0	0
$358$	−42.6274	−2.25293
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	27.6274	1.45407
$362$	33.7990	1.77644
$363$	0	0
$364$	−17.9411	−0.940370
$365$	−11.3137	−0.592187
$366$	0	0
$367$	9.65685	0.504084	0.252042	−	0.967716i	$-0.418898\pi$
$367$	9.65685	0.504084	0.252042	+	0.967716i	$0.418898\pi$
$368$	12.0000	0.625543
$369$	0	0
$370$	−28.1421	−1.46304
$371$	7.71573	0.400581
$372$	0	0
$373$	−10.6274	−0.550267	−0.275133	−	0.961406i	$-0.588722\pi$
$373$	−10.6274	−0.550267	−0.275133	+	0.961406i	$0.588722\pi$
$374$	0	0
$375$	0	0
$376$	−17.6569	−0.910583
$377$	−27.3137	−1.40673
$378$	0	0
$379$	−23.3137	−1.19754	−0.598772	−	0.800919i	$-0.704345\pi$
$379$	−23.3137	−1.19754	−0.598772	+	0.800919i	$0.704345\pi$
$380$	26.1421	1.34106
$381$	0	0
$382$	13.6569	0.698745
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	0	0
$385$	0	0
$386$	32.9706	1.67816
$387$	0	0
$388$	44.6274	2.26561
$389$	23.6569	1.19945	0.599725	−	0.800206i	$-0.295277\pi$
$389$	23.6569	1.19945	0.599725	+	0.800206i	$0.295277\pi$
$390$	0	0
$391$	−4.68629	−0.236996
$392$	27.8701	1.40765
$393$	0	0
$394$	20.4853	1.03203
$395$	−8.48528	−0.426941
$396$	0	0
$397$	−14.9706	−0.751351	−0.375676	−	0.926751i	$-0.622589\pi$
$397$	−14.9706	−0.751351	−0.375676	+	0.926751i	$0.622589\pi$
$398$	52.2843	2.62077
$399$	0	0
$400$	3.00000	0.150000
$401$	6.68629	0.333897	0.166949	−	0.985966i	$-0.446609\pi$
$401$	6.68629	0.333897	0.166949	+	0.985966i	$0.446609\pi$
$402$	0	0
$403$	0	0
$404$	−3.17157	−0.157792
$405$	0	0
$406$	−9.65685	−0.479262
$407$	0	0
$408$	0	0
$409$	19.6569	0.971969	0.485984	−	0.873967i	$-0.338461\pi$
$409$	19.6569	0.971969	0.485984	+	0.873967i	$0.338461\pi$
$410$	−11.6569	−0.575691
$411$	0	0
$412$	−12.6863	−0.625009
$413$	−3.31371	−0.163057
$414$	0	0
$415$	−10.0000	−0.490881
$416$	8.97056	0.439818
$417$	0	0
$418$	0	0
$419$	36.9706	1.80613	0.903065	−	0.429504i	$-0.141311\pi$
$419$	36.9706	1.80613	0.903065	+	0.429504i	$0.141311\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	2.82843	0.137686
$423$	0	0
$424$	41.1127	1.99661
$425$	−1.17157	−0.0568296
$426$	0	0
$427$	−9.65685	−0.467328
$428$	66.2843	3.20397
$429$	0	0
$430$	−21.3137	−1.02784
$431$	21.6569	1.04317	0.521587	−	0.853198i	$-0.325340\pi$
$431$	21.6569	1.04317	0.521587	+	0.853198i	$0.325340\pi$
$432$	0	0
$433$	−15.6569	−0.752420	−0.376210	−	0.926534i	$-0.622773\pi$
$433$	−15.6569	−0.752420	−0.376210	+	0.926534i	$0.622773\pi$
$434$	0	0
$435$	0	0
$436$	−66.2843	−3.17444
$437$	27.3137	1.30659
$438$	0	0
$439$	20.4853	0.977709	0.488855	−	0.872365i	$-0.337415\pi$
$439$	20.4853	0.977709	0.488855	+	0.872365i	$0.337415\pi$
$440$	0	0
$441$	0	0
$442$	16.0000	0.761042
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	−3.65685	−0.173352
$446$	15.3137	0.725125
$447$	0	0
$448$	8.14214	0.384680
$449$	−30.9706	−1.46159	−0.730796	−	0.682596i	$-0.760851\pi$
$449$	−30.9706	−1.46159	−0.730796	+	0.682596i	$0.760851\pi$
$450$	0	0
$451$	0	0
$452$	72.6274	3.41611
$453$	0	0
$454$	33.7990	1.58627
$455$	−4.68629	−0.219697
$456$	0	0
$457$	23.3137	1.09057	0.545285	−	0.838251i	$-0.316422\pi$
$457$	23.3137	1.09057	0.545285	+	0.838251i	$0.316422\pi$
$458$	4.82843	0.225618
$459$	0	0
$460$	15.3137	0.714005
$461$	0.142136	0.00661992	0.00330996	−	0.999995i	$-0.498946\pi$
$461$	0.142136	0.00661992	0.00330996	+	0.999995i	$0.498946\pi$
$462$	0	0
$463$	4.97056	0.231002	0.115501	−	0.993307i	$-0.463153\pi$
$463$	4.97056	0.231002	0.115501	+	0.993307i	$0.463153\pi$
$464$	−14.4853	−0.672462
$465$	0	0
$466$	45.4558	2.10570
$467$	22.6274	1.04707	0.523536	−	0.852004i	$-0.324613\pi$
$467$	22.6274	1.04707	0.523536	+	0.852004i	$0.324613\pi$
$468$	0	0
$469$	4.68629	0.216393
$470$	−9.65685	−0.445437
$471$	0	0
$472$	−17.6569	−0.812723
$473$	0	0
$474$	0	0
$475$	6.82843	0.313310
$476$	3.71573	0.170310
$477$	0	0
$478$	−42.6274	−1.94973
$479$	36.9706	1.68923	0.844614	−	0.535376i	$-0.179830\pi$
$479$	36.9706	1.68923	0.844614	+	0.535376i	$0.179830\pi$
$480$	0	0
$481$	65.9411	3.00666
$482$	−29.7990	−1.35731
$483$	0	0
$484$	0	0
$485$	11.6569	0.529310
$486$	0	0
$487$	−12.9706	−0.587752	−0.293876	−	0.955844i	$-0.594945\pi$
$487$	−12.9706	−0.587752	−0.293876	+	0.955844i	$0.594945\pi$
$488$	−51.4558	−2.32930
$489$	0	0
$490$	15.2426	0.688592
$491$	14.3431	0.647297	0.323649	−	0.946177i	$-0.395090\pi$
$491$	14.3431	0.647297	0.323649	+	0.946177i	$0.395090\pi$
$492$	0	0
$493$	5.65685	0.254772
$494$	−93.2548	−4.19573
$495$	0	0
$496$	0	0
$497$	1.94113	0.0870714
$498$	0	0
$499$	−22.3431	−1.00022	−0.500108	−	0.865963i	$-0.666706\pi$
$499$	−22.3431	−1.00022	−0.500108	+	0.865963i	$0.666706\pi$
$500$	3.82843	0.171212
$501$	0	0
$502$	50.6274	2.25961
$503$	17.3137	0.771980	0.385990	−	0.922503i	$-0.373860\pi$
$503$	17.3137	0.771980	0.385990	+	0.922503i	$0.373860\pi$
$504$	0	0
$505$	−0.828427	−0.0368645
$506$	0	0
$507$	0	0
$508$	55.4558	2.46046
$509$	−18.6863	−0.828255	−0.414128	−	0.910219i	$-0.635913\pi$
$509$	−18.6863	−0.828255	−0.414128	+	0.910219i	$0.635913\pi$
$510$	0	0
$511$	9.37258	0.414619
$512$	31.2426	1.38074
$513$	0	0
$514$	−39.4558	−1.74032
$515$	−3.31371	−0.146019
$516$	0	0
$517$	0	0
$518$	23.3137	1.02435
$519$	0	0
$520$	−24.9706	−1.09503
$521$	32.6274	1.42943	0.714717	−	0.699414i	$-0.246556\pi$
$521$	32.6274	1.42943	0.714717	+	0.699414i	$0.246556\pi$
$522$	0	0
$523$	9.51472	0.416050	0.208025	−	0.978124i	$-0.433297\pi$
$523$	9.51472	0.416050	0.208025	+	0.978124i	$0.433297\pi$
$524$	12.6863	0.554203
$525$	0	0
$526$	−43.4558	−1.89476
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	22.4853	0.976698
$531$	0	0
$532$	−21.6569	−0.938944
$533$	27.3137	1.18309
$534$	0	0
$535$	17.3137	0.748537
$536$	24.9706	1.07856
$537$	0	0
$538$	49.7990	2.14699
$539$	0	0
$540$	0	0
$541$	−17.3137	−0.744374	−0.372187	−	0.928158i	$-0.621392\pi$
$541$	−17.3137	−0.744374	−0.372187	+	0.928158i	$0.621392\pi$
$542$	−28.4853	−1.22355
$543$	0	0
$544$	−1.85786	−0.0796553
$545$	−17.3137	−0.741638
$546$	0	0
$547$	8.14214	0.348133	0.174066	−	0.984734i	$-0.444309\pi$
$547$	8.14214	0.348133	0.174066	+	0.984734i	$0.444309\pi$
$548$	50.9706	2.17735
$549$	0	0
$550$	0	0
$551$	−32.9706	−1.40459
$552$	0	0
$553$	7.02944	0.298922
$554$	−5.65685	−0.240337
$555$	0	0
$556$	−1.85786	−0.0787910
$557$	5.17157	0.219127	0.109563	−	0.993980i	$-0.465055\pi$
$557$	5.17157	0.219127	0.109563	+	0.993980i	$0.465055\pi$
$558$	0	0
$559$	49.9411	2.11228
$560$	−2.48528	−0.105022
$561$	0	0
$562$	26.9706	1.13768
$563$	31.6569	1.33418	0.667089	−	0.744978i	$-0.267540\pi$
$563$	31.6569	1.33418	0.667089	+	0.744978i	$0.267540\pi$
$564$	0	0
$565$	18.9706	0.798098
$566$	−21.3137	−0.895882
$567$	0	0
$568$	10.3431	0.433989
$569$	−35.4558	−1.48639	−0.743193	−	0.669077i	$-0.766690\pi$
$569$	−35.4558	−1.48639	−0.743193	+	0.669077i	$0.766690\pi$
$570$	0	0
$571$	16.4853	0.689888	0.344944	−	0.938623i	$-0.387898\pi$
$571$	16.4853	0.689888	0.344944	+	0.938623i	$0.387898\pi$
$572$	0	0
$573$	0	0
$574$	9.65685	0.403069
$575$	4.00000	0.166812
$576$	0	0
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	37.7279	1.56927
$579$	0	0
$580$	−18.4853	−0.767560
$581$	8.28427	0.343689
$582$	0	0
$583$	0	0
$584$	49.9411	2.06658
$585$	0	0
$586$	16.4853	0.681001
$587$	−14.6274	−0.603738	−0.301869	−	0.953349i	$-0.597611\pi$
$587$	−14.6274	−0.603738	−0.301869	+	0.953349i	$0.597611\pi$
$588$	0	0
$589$	0	0
$590$	−9.65685	−0.397566
$591$	0	0
$592$	34.9706	1.43728
$593$	−22.8284	−0.937451	−0.468726	−	0.883344i	$-0.655287\pi$
$593$	−22.8284	−0.937451	−0.468726	+	0.883344i	$0.655287\pi$
$594$	0	0
$595$	0.970563	0.0397892
$596$	5.79899	0.237536
$597$	0	0
$598$	−54.6274	−2.23388
$599$	−27.3137	−1.11601	−0.558004	−	0.829838i	$-0.688433\pi$
$599$	−27.3137	−1.11601	−0.558004	+	0.829838i	$0.688433\pi$
$600$	0	0
$601$	5.31371	0.216751	0.108375	−	0.994110i	$-0.465435\pi$
$601$	5.31371	0.216751	0.108375	+	0.994110i	$0.465435\pi$
$602$	17.6569	0.719640
$603$	0	0
$604$	−63.1127	−2.56802
$605$	0	0
$606$	0	0
$607$	−1.51472	−0.0614805	−0.0307403	−	0.999527i	$-0.509786\pi$
$607$	−1.51472	−0.0614805	−0.0307403	+	0.999527i	$0.509786\pi$
$608$	10.8284	0.439151
$609$	0	0
$610$	−28.1421	−1.13944
$611$	22.6274	0.915407
$612$	0	0
$613$	45.9411	1.85554	0.927772	−	0.373147i	$-0.121721\pi$
$613$	45.9411	1.85554	0.927772	+	0.373147i	$0.121721\pi$
$614$	−7.65685	−0.309005
$615$	0	0
$616$	0	0
$617$	−0.343146	−0.0138145	−0.00690726	−	0.999976i	$-0.502199\pi$
$617$	−0.343146	−0.0138145	−0.00690726	+	0.999976i	$0.502199\pi$
$618$	0	0
$619$	−14.3431	−0.576500	−0.288250	−	0.957555i	$-0.593073\pi$
$619$	−14.3431	−0.576500	−0.288250	+	0.957555i	$0.593073\pi$
$620$	0	0
$621$	0	0
$622$	−8.00000	−0.320771
$623$	3.02944	0.121372
$624$	0	0
$625$	1.00000	0.0400000
$626$	−37.7990	−1.51075
$627$	0	0
$628$	68.9117	2.74988
$629$	−13.6569	−0.544534
$630$	0	0
$631$	45.6569	1.81757	0.908785	−	0.417264i	$-0.137011\pi$
$631$	45.6569	1.81757	0.908785	+	0.417264i	$0.137011\pi$
$632$	37.4558	1.48991
$633$	0	0
$634$	−63.4558	−2.52015
$635$	14.4853	0.574831
$636$	0	0
$637$	−35.7157	−1.41511
$638$	0	0
$639$	0	0
$640$	20.5563	0.812561
$641$	−6.97056	−0.275321	−0.137660	−	0.990479i	$-0.543958\pi$
$641$	−6.97056	−0.275321	−0.137660	+	0.990479i	$0.543958\pi$
$642$	0	0
$643$	37.9411	1.49625	0.748126	−	0.663557i	$-0.230954\pi$
$643$	37.9411	1.49625	0.748126	+	0.663557i	$0.230954\pi$
$644$	−12.6863	−0.499910
$645$	0	0
$646$	19.3137	0.759888
$647$	−4.68629	−0.184237	−0.0921186	−	0.995748i	$-0.529364\pi$
$647$	−4.68629	−0.184237	−0.0921186	+	0.995748i	$0.529364\pi$
$648$	0	0
$649$	0	0
$650$	−13.6569	−0.535666
$651$	0	0
$652$	−28.0000	−1.09656
$653$	−6.97056	−0.272779	−0.136390	−	0.990655i	$-0.543550\pi$
$653$	−6.97056	−0.272779	−0.136390	+	0.990655i	$0.543550\pi$
$654$	0	0
$655$	3.31371	0.129477
$656$	14.4853	0.565555
$657$	0	0
$658$	8.00000	0.311872
$659$	−15.3137	−0.596537	−0.298269	−	0.954482i	$-0.596409\pi$
$659$	−15.3137	−0.596537	−0.298269	+	0.954482i	$0.596409\pi$
$660$	0	0
$661$	9.31371	0.362261	0.181131	−	0.983459i	$-0.442024\pi$
$661$	9.31371	0.362261	0.181131	+	0.983459i	$0.442024\pi$
$662$	−15.3137	−0.595184
$663$	0	0
$664$	44.1421	1.71305
$665$	−5.65685	−0.219363
$666$	0	0
$667$	−19.3137	−0.747830
$668$	−50.9706	−1.97211
$669$	0	0
$670$	13.6569	0.527610
$671$	0	0
$672$	0	0
$673$	−18.3431	−0.707076	−0.353538	−	0.935420i	$-0.615022\pi$
$673$	−18.3431	−0.707076	−0.353538	+	0.935420i	$0.615022\pi$
$674$	8.00000	0.308148
$675$	0	0
$676$	72.7401	2.79770
$677$	29.4558	1.13208	0.566040	−	0.824378i	$-0.308475\pi$
$677$	29.4558	1.13208	0.566040	+	0.824378i	$0.308475\pi$
$678$	0	0
$679$	−9.65685	−0.370596
$680$	5.17157	0.198321
$681$	0	0
$682$	0	0
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	0	0
$685$	13.3137	0.508691
$686$	−26.6274	−1.01664
$687$	0	0
$688$	26.4853	1.00974
$689$	−52.6863	−2.00719
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	10.8284	0.411635
$693$	0	0
$694$	70.7696	2.68638
$695$	−0.485281	−0.0184078
$696$	0	0
$697$	−5.65685	−0.214269
$698$	26.4853	1.00248
$699$	0	0
$700$	−3.17157	−0.119874
$701$	36.1421	1.36507	0.682535	−	0.730853i	$-0.260878\pi$
$701$	36.1421	1.36507	0.682535	+	0.730853i	$0.260878\pi$
$702$	0	0
$703$	79.5980	3.00209
$704$	0	0
$705$	0	0
$706$	62.7696	2.36236
$707$	0.686292	0.0258106
$708$	0	0
$709$	6.68629	0.251109	0.125554	−	0.992087i	$-0.459929\pi$
$709$	6.68629	0.251109	0.125554	+	0.992087i	$0.459929\pi$
$710$	5.65685	0.212298
$711$	0	0
$712$	16.1421	0.604952
$713$	0	0
$714$	0	0
$715$	0	0
$716$	67.5980	2.52626
$717$	0	0
$718$	−28.9706	−1.08117
$719$	47.5980	1.77511	0.887553	−	0.460706i	$-0.152404\pi$
$719$	47.5980	1.77511	0.887553	+	0.460706i	$0.152404\pi$
$720$	0	0
$721$	2.74517	0.102235
$722$	−66.6985	−2.48226
$723$	0	0
$724$	−53.5980	−1.99195
$725$	−4.82843	−0.179323
$726$	0	0
$727$	33.9411	1.25881	0.629403	−	0.777079i	$-0.283299\pi$
$727$	33.9411	1.25881	0.629403	+	0.777079i	$0.283299\pi$
$728$	20.6863	0.766685
$729$	0	0
$730$	27.3137	1.01093
$731$	−10.3431	−0.382555
$732$	0	0
$733$	6.34315	0.234289	0.117145	−	0.993115i	$-0.462626\pi$
$733$	6.34315	0.234289	0.117145	+	0.993115i	$0.462626\pi$
$734$	−23.3137	−0.860525
$735$	0	0
$736$	6.34315	0.233811
$737$	0	0
$738$	0	0
$739$	15.1127	0.555930	0.277965	−	0.960591i	$-0.410340\pi$
$739$	15.1127	0.555930	0.277965	+	0.960591i	$0.410340\pi$
$740$	44.6274	1.64054
$741$	0	0
$742$	−18.6274	−0.683834
$743$	36.3431	1.33330	0.666650	−	0.745371i	$-0.267727\pi$
$743$	36.3431	1.33330	0.666650	+	0.745371i	$0.267727\pi$
$744$	0	0
$745$	1.51472	0.0554950
$746$	25.6569	0.939364
$747$	0	0
$748$	0	0
$749$	−14.3431	−0.524087
$750$	0	0
$751$	20.2843	0.740184	0.370092	−	0.928995i	$-0.379326\pi$
$751$	20.2843	0.740184	0.370092	+	0.928995i	$0.379326\pi$
$752$	12.0000	0.437595
$753$	0	0
$754$	65.9411	2.40143
$755$	−16.4853	−0.599961
$756$	0	0
$757$	−36.6274	−1.33125	−0.665623	−	0.746288i	$-0.731834\pi$
$757$	−36.6274	−1.33125	−0.665623	+	0.746288i	$0.731834\pi$
$758$	56.2843	2.04434
$759$	0	0
$760$	−30.1421	−1.09337
$761$	−28.8284	−1.04503	−0.522515	−	0.852630i	$-0.675006\pi$
$761$	−28.8284	−1.04503	−0.522515	+	0.852630i	$0.675006\pi$
$762$	0	0
$763$	14.3431	0.519257
$764$	−21.6569	−0.783517
$765$	0	0
$766$	−19.3137	−0.697833
$767$	22.6274	0.817029
$768$	0	0
$769$	−10.6863	−0.385358	−0.192679	−	0.981262i	$-0.561718\pi$
$769$	−10.6863	−0.385358	−0.192679	+	0.981262i	$0.561718\pi$
$770$	0	0
$771$	0	0
$772$	−52.2843	−1.88175
$773$	−3.65685	−0.131528	−0.0657640	−	0.997835i	$-0.520948\pi$
$773$	−3.65685	−0.131528	−0.0657640	+	0.997835i	$0.520948\pi$
$774$	0	0
$775$	0	0
$776$	−51.4558	−1.84716
$777$	0	0
$778$	−57.1127	−2.04759
$779$	32.9706	1.18129
$780$	0	0
$781$	0	0
$782$	11.3137	0.404577
$783$	0	0
$784$	−18.9411	−0.676469
$785$	18.0000	0.642448
$786$	0	0
$787$	20.1421	0.717990	0.358995	−	0.933340i	$-0.383120\pi$
$787$	20.1421	0.717990	0.358995	+	0.933340i	$0.383120\pi$
$788$	−32.4853	−1.15724
$789$	0	0
$790$	20.4853	0.728834
$791$	−15.7157	−0.558787
$792$	0	0
$793$	65.9411	2.34164
$794$	36.1421	1.28264
$795$	0	0
$796$	−82.9117	−2.93873
$797$	−34.9706	−1.23872	−0.619360	−	0.785107i	$-0.712608\pi$
$797$	−34.9706	−1.23872	−0.619360	+	0.785107i	$0.712608\pi$
$798$	0	0
$799$	−4.68629	−0.165789
$800$	1.58579	0.0560660
$801$	0	0
$802$	−16.1421	−0.569999
$803$	0	0
$804$	0	0
$805$	−3.31371	−0.116793
$806$	0	0
$807$	0	0
$808$	3.65685	0.128648
$809$	28.4264	0.999419	0.499710	−	0.866193i	$-0.333440\pi$
$809$	28.4264	0.999419	0.499710	+	0.866193i	$0.333440\pi$
$810$	0	0
$811$	−0.485281	−0.0170405	−0.00852027	−	0.999964i	$-0.502712\pi$
$811$	−0.485281	−0.0170405	−0.00852027	+	0.999964i	$0.502712\pi$
$812$	15.3137	0.537406
$813$	0	0
$814$	0	0
$815$	−7.31371	−0.256188
$816$	0	0
$817$	60.2843	2.10908
$818$	−47.4558	−1.65925
$819$	0	0
$820$	18.4853	0.645534
$821$	−12.8284	−0.447715	−0.223858	−	0.974622i	$-0.571865\pi$
$821$	−12.8284	−0.447715	−0.223858	+	0.974622i	$0.571865\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	14.6274	0.509570
$825$	0	0
$826$	8.00000	0.278356
$827$	41.3137	1.43662	0.718309	−	0.695724i	$-0.244916\pi$
$827$	41.3137	1.43662	0.718309	+	0.695724i	$0.244916\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	24.1421	0.837986
$831$	0	0
$832$	−55.5980	−1.92751
$833$	7.39697	0.256290
$834$	0	0
$835$	−13.3137	−0.460740
$836$	0	0
$837$	0	0
$838$	−89.2548	−3.08326
$839$	22.6274	0.781185	0.390593	−	0.920564i	$-0.372270\pi$
$839$	22.6274	0.781185	0.390593	+	0.920564i	$0.372270\pi$
$840$	0	0
$841$	−5.68629	−0.196079
$842$	14.4853	0.499196
$843$	0	0
$844$	−4.48528	−0.154390
$845$	19.0000	0.653620
$846$	0	0
$847$	0	0
$848$	−27.9411	−0.959502
$849$	0	0
$850$	2.82843	0.0970143
$851$	46.6274	1.59837
$852$	0	0
$853$	8.68629	0.297413	0.148706	−	0.988881i	$-0.452489\pi$
$853$	8.68629	0.297413	0.148706	+	0.988881i	$0.452489\pi$
$854$	23.3137	0.797779
$855$	0	0
$856$	−76.4264	−2.61220
$857$	−28.4853	−0.973039	−0.486519	−	0.873670i	$-0.661734\pi$
$857$	−28.4853	−0.973039	−0.486519	+	0.873670i	$0.661734\pi$
$858$	0	0
$859$	−52.9706	−1.80733	−0.903666	−	0.428238i	$-0.859135\pi$
$859$	−52.9706	−1.80733	−0.903666	+	0.428238i	$0.859135\pi$
$860$	33.7990	1.15254
$861$	0	0
$862$	−52.2843	−1.78081
$863$	20.6863	0.704170	0.352085	−	0.935968i	$-0.385473\pi$
$863$	20.6863	0.704170	0.352085	+	0.935968i	$0.385473\pi$
$864$	0	0
$865$	2.82843	0.0961694
$866$	37.7990	1.28446
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−32.0000	−1.08428
$872$	76.4264	2.58812
$873$	0	0
$874$	−65.9411	−2.23049
$875$	−0.828427	−0.0280059
$876$	0	0
$877$	−2.62742	−0.0887216	−0.0443608	−	0.999016i	$-0.514125\pi$
$877$	−2.62742	−0.0887216	−0.0443608	+	0.999016i	$0.514125\pi$
$878$	−49.4558	−1.66905
$879$	0	0
$880$	0	0
$881$	46.9706	1.58248	0.791239	−	0.611507i	$-0.209436\pi$
$881$	46.9706	1.58248	0.791239	+	0.611507i	$0.209436\pi$
$882$	0	0
$883$	−5.37258	−0.180802	−0.0904009	−	0.995905i	$-0.528815\pi$
$883$	−5.37258	−0.180802	−0.0904009	+	0.995905i	$0.528815\pi$
$884$	−25.3726	−0.853372
$885$	0	0
$886$	28.9706	0.973285
$887$	−15.6569	−0.525706	−0.262853	−	0.964836i	$-0.584663\pi$
$887$	−15.6569	−0.525706	−0.262853	+	0.964836i	$0.584663\pi$
$888$	0	0
$889$	−12.0000	−0.402467
$890$	8.82843	0.295930
$891$	0	0
$892$	−24.2843	−0.813098
$893$	27.3137	0.914018
$894$	0	0
$895$	17.6569	0.590204
$896$	−17.0294	−0.568914
$897$	0	0
$898$	74.7696	2.49509
$899$	0	0
$900$	0	0
$901$	10.9117	0.363521
$902$	0	0
$903$	0	0
$904$	−83.7401	−2.78515
$905$	−14.0000	−0.465376
$906$	0	0
$907$	40.9706	1.36041	0.680203	−	0.733024i	$-0.261892\pi$
$907$	40.9706	1.36041	0.680203	+	0.733024i	$0.261892\pi$
$908$	−53.5980	−1.77871
$909$	0	0
$910$	11.3137	0.375046
$911$	48.9706	1.62247	0.811234	−	0.584722i	$-0.198797\pi$
$911$	48.9706	1.62247	0.811234	+	0.584722i	$0.198797\pi$
$912$	0	0
$913$	0	0
$914$	−56.2843	−1.86172
$915$	0	0
$916$	−7.65685	−0.252990
$917$	−2.74517	−0.0906534
$918$	0	0
$919$	−11.5147	−0.379836	−0.189918	−	0.981800i	$-0.560822\pi$
$919$	−11.5147	−0.379836	−0.189918	+	0.981800i	$0.560822\pi$
$920$	−17.6569	−0.582129
$921$	0	0
$922$	−0.343146	−0.0113009
$923$	−13.2548	−0.436288
$924$	0	0
$925$	11.6569	0.383275
$926$	−12.0000	−0.394344
$927$	0	0
$928$	−7.65685	−0.251349
$929$	45.5980	1.49602	0.748011	−	0.663687i	$-0.231009\pi$
$929$	45.5980	1.49602	0.748011	+	0.663687i	$0.231009\pi$
$930$	0	0
$931$	−43.1127	−1.41296
$932$	−72.0833	−2.36117
$933$	0	0
$934$	−54.6274	−1.78746
$935$	0	0
$936$	0	0
$937$	−11.0294	−0.360316	−0.180158	−	0.983638i	$-0.557661\pi$
$937$	−11.0294	−0.360316	−0.180158	+	0.983638i	$0.557661\pi$
$938$	−11.3137	−0.369406
$939$	0	0
$940$	15.3137	0.499478
$941$	−34.7696	−1.13346	−0.566728	−	0.823905i	$-0.691791\pi$
$941$	−34.7696	−1.13346	−0.566728	+	0.823905i	$0.691791\pi$
$942$	0	0
$943$	19.3137	0.628941
$944$	12.0000	0.390567
$945$	0	0
$946$	0	0
$947$	−6.62742	−0.215362	−0.107681	−	0.994185i	$-0.534343\pi$
$947$	−6.62742	−0.215362	−0.107681	+	0.994185i	$0.534343\pi$
$948$	0	0
$949$	−64.0000	−2.07753
$950$	−16.4853	−0.534853
$951$	0	0
$952$	−4.28427	−0.138854
$953$	−11.7990	−0.382207	−0.191103	−	0.981570i	$-0.561207\pi$
$953$	−11.7990	−0.382207	−0.191103	+	0.981570i	$0.561207\pi$
$954$	0	0
$955$	−5.65685	−0.183052
$956$	67.5980	2.18627
$957$	0	0
$958$	−89.2548	−2.88369
$959$	−11.0294	−0.356159
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−159.196	−5.13268
$963$	0	0
$964$	47.2548	1.52198
$965$	−13.6569	−0.439630
$966$	0	0
$967$	−11.4558	−0.368395	−0.184198	−	0.982889i	$-0.558969\pi$
$967$	−11.4558	−0.368395	−0.184198	+	0.982889i	$0.558969\pi$
$968$	0	0
$969$	0	0
$970$	−28.1421	−0.903590
$971$	34.6274	1.11125	0.555623	−	0.831434i	$-0.312480\pi$
$971$	34.6274	1.11125	0.555623	+	0.831434i	$0.312480\pi$
$972$	0	0
$973$	0.402020	0.0128882
$974$	31.3137	1.00336
$975$	0	0
$976$	34.9706	1.11938
$977$	−2.68629	−0.0859421	−0.0429710	−	0.999076i	$-0.513682\pi$
$977$	−2.68629	−0.0859421	−0.0429710	+	0.999076i	$0.513682\pi$
$978$	0	0
$979$	0	0
$980$	−24.1716	−0.772133
$981$	0	0
$982$	−34.6274	−1.10501
$983$	30.6274	0.976863	0.488431	−	0.872602i	$-0.337569\pi$
$983$	30.6274	0.976863	0.488431	+	0.872602i	$0.337569\pi$
$984$	0	0
$985$	−8.48528	−0.270364
$986$	−13.6569	−0.434923
$987$	0	0
$988$	147.882	4.70476
$989$	35.3137	1.12291
$990$	0	0
$991$	−30.6274	−0.972912	−0.486456	−	0.873705i	$-0.661711\pi$
$991$	−30.6274	−0.972912	−0.486456	+	0.873705i	$0.661711\pi$
$992$	0	0
$993$	0	0
$994$	−4.68629	−0.148640
$995$	−21.6569	−0.686568
$996$	0	0
$997$	39.3137	1.24508	0.622539	−	0.782589i	$-0.286101\pi$
$997$	39.3137	1.24508	0.622539	+	0.782589i	$0.286101\pi$
$998$	53.9411	1.70748
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.m.1.1		2
3.2	odd	2		1815.2.a.k.1.2		2
11.10	odd	2		495.2.a.d.1.2		2
15.14	odd	2		9075.2.a.v.1.1		2
33.32	even	2		165.2.a.a.1.1	✓	2
44.43	even	2		7920.2.a.cg.1.1		2
55.32	even	4		2475.2.c.m.199.4		4
55.43	even	4		2475.2.c.m.199.1		4
55.54	odd	2		2475.2.a.m.1.1		2
132.131	odd	2		2640.2.a.bb.1.1		2
165.32	odd	4		825.2.c.e.199.1		4
165.98	odd	4		825.2.c.e.199.4		4
165.164	even	2		825.2.a.g.1.2		2
231.230	odd	2		8085.2.a.ba.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.a.1.1	✓	2	33.32	even	2
495.2.a.d.1.2		2	11.10	odd	2
825.2.a.g.1.2		2	165.164	even	2
825.2.c.e.199.1		4	165.32	odd	4
825.2.c.e.199.4		4	165.98	odd	4
1815.2.a.k.1.2		2	3.2	odd	2
2475.2.a.m.1.1		2	55.54	odd	2
2475.2.c.m.199.1		4	55.43	even	4
2475.2.c.m.199.4		4	55.32	even	4
2640.2.a.bb.1.1		2	132.131	odd	2
5445.2.a.m.1.1		2	1.1	even	1	trivial
7920.2.a.cg.1.1		2	44.43	even	2
8085.2.a.ba.1.1		2	231.230	odd	2
9075.2.a.v.1.1		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} -0.828427 q^{7} -4.41421 q^{8} +O(q^{10})\)	\(q-2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} -0.828427 q^{7} -4.41421 q^{8} -2.41421 q^{10} +5.65685 q^{13} +2.00000 q^{14} +3.00000 q^{16} -1.17157 q^{17} +6.82843 q^{19} +3.82843 q^{20} +4.00000 q^{23} +1.00000 q^{25} -13.6569 q^{26} -3.17157 q^{28} -4.82843 q^{29} +1.58579 q^{32} +2.82843 q^{34} -0.828427 q^{35} +11.6569 q^{37} -16.4853 q^{38} -4.41421 q^{40} +4.82843 q^{41} +8.82843 q^{43} -9.65685 q^{46} +4.00000 q^{47} -6.31371 q^{49} -2.41421 q^{50} +21.6569 q^{52} -9.31371 q^{53} +3.65685 q^{56} +11.6569 q^{58} +4.00000 q^{59} +11.6569 q^{61} -9.82843 q^{64} +5.65685 q^{65} -5.65685 q^{67} -4.48528 q^{68} +2.00000 q^{70} -2.34315 q^{71} -11.3137 q^{73} -28.1421 q^{74} +26.1421 q^{76} -8.48528 q^{79} +3.00000 q^{80} -11.6569 q^{82} -10.0000 q^{83} -1.17157 q^{85} -21.3137 q^{86} -3.65685 q^{89} -4.68629 q^{91} +15.3137 q^{92} -9.65685 q^{94} +6.82843 q^{95} +11.6569 q^{97} +15.2426 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 + 4 * q^7 - 6 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} - 6 q^{8} - 2 q^{10} + 4 q^{14} + 6 q^{16} - 8 q^{17} + 8 q^{19} + 2 q^{20} + 8 q^{23} + 2 q^{25} - 16 q^{26} - 12 q^{28} - 4 q^{29} + 6 q^{32} + 4 q^{35} + 12 q^{37} - 16 q^{38} - 6 q^{40} + 4 q^{41} + 12 q^{43} - 8 q^{46} + 8 q^{47} + 10 q^{49} - 2 q^{50} + 32 q^{52} + 4 q^{53} - 4 q^{56} + 12 q^{58} + 8 q^{59} + 12 q^{61} - 14 q^{64} + 8 q^{68} + 4 q^{70} - 16 q^{71} - 28 q^{74} + 24 q^{76} + 6 q^{80} - 12 q^{82} - 20 q^{83} - 8 q^{85} - 20 q^{86} + 4 q^{89} - 32 q^{91} + 8 q^{92} - 8 q^{94} + 8 q^{95} + 12 q^{97} + 22 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 + 4 * q^7 - 6 * q^8 - 2 * q^10 + 4 * q^14 + 6 * q^16 - 8 * q^17 + 8 * q^19 + 2 * q^20 + 8 * q^23 + 2 * q^25 - 16 * q^26 - 12 * q^28 - 4 * q^29 + 6 * q^32 + 4 * q^35 + 12 * q^37 - 16 * q^38 - 6 * q^40 + 4 * q^41 + 12 * q^43 - 8 * q^46 + 8 * q^47 + 10 * q^49 - 2 * q^50 + 32 * q^52 + 4 * q^53 - 4 * q^56 + 12 * q^58 + 8 * q^59 + 12 * q^61 - 14 * q^64 + 8 * q^68 + 4 * q^70 - 16 * q^71 - 28 * q^74 + 24 * q^76 + 6 * q^80 - 12 * q^82 - 20 * q^83 - 8 * q^85 - 20 * q^86 + 4 * q^89 - 32 * q^91 + 8 * q^92 - 8 * q^94 + 8 * q^95 + 12 * q^97 + 22 * q^98

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