LMFDB - Embedded newform 4560.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4560,2,Mod(1,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4560.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.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:	$36.4117833217$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	4560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +1.41421 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +1.41421 q^{7} +1.00000 q^{9} +2.24264 q^{11} -3.41421 q^{13} +1.00000 q^{15} +1.17157 q^{17} +1.00000 q^{19} -1.41421 q^{21} -7.65685 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.41421 q^{29} +3.17157 q^{31} -2.24264 q^{33} -1.41421 q^{35} -3.41421 q^{37} +3.41421 q^{39} -0.242641 q^{41} -12.2426 q^{43} -1.00000 q^{45} +7.65685 q^{47} -5.00000 q^{49} -1.17157 q^{51} +8.00000 q^{53} -2.24264 q^{55} -1.00000 q^{57} -12.4853 q^{59} +7.31371 q^{61} +1.41421 q^{63} +3.41421 q^{65} +9.65685 q^{67} +7.65685 q^{69} +10.8284 q^{71} -7.65685 q^{73} -1.00000 q^{75} +3.17157 q^{77} +1.00000 q^{81} -12.8284 q^{83} -1.17157 q^{85} -1.41421 q^{87} +5.89949 q^{89} -4.82843 q^{91} -3.17157 q^{93} -1.00000 q^{95} -9.75736 q^{97} +2.24264 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{11} - 4 q^{13} + 2 q^{15} + 8 q^{17} + 2 q^{19} - 4 q^{23} + 2 q^{25} - 2 q^{27} + 12 q^{31} + 4 q^{33} - 4 q^{37} + 4 q^{39} + 8 q^{41} - 16 q^{43} - 2 q^{45} + 4 q^{47} - 10 q^{49} - 8 q^{51} + 16 q^{53} + 4 q^{55} - 2 q^{57} - 8 q^{59} - 8 q^{61} + 4 q^{65} + 8 q^{67} + 4 q^{69} + 16 q^{71} - 4 q^{73} - 2 q^{75} + 12 q^{77} + 2 q^{81} - 20 q^{83} - 8 q^{85} - 8 q^{89} - 4 q^{91} - 12 q^{93} - 2 q^{95} - 28 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 - 4 * q^11 - 4 * q^13 + 2 * q^15 + 8 * q^17 + 2 * q^19 - 4 * q^23 + 2 * q^25 - 2 * q^27 + 12 * q^31 + 4 * q^33 - 4 * q^37 + 4 * q^39 + 8 * q^41 - 16 * q^43 - 2 * q^45 + 4 * q^47 - 10 * q^49 - 8 * q^51 + 16 * q^53 + 4 * q^55 - 2 * q^57 - 8 * q^59 - 8 * q^61 + 4 * q^65 + 8 * q^67 + 4 * q^69 + 16 * q^71 - 4 * q^73 - 2 * q^75 + 12 * q^77 + 2 * q^81 - 20 * q^83 - 8 * q^85 - 8 * q^89 - 4 * q^91 - 12 * q^93 - 2 * q^95 - 28 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.41421	0.534522	0.267261	−	0.963624i	$-0.413881\pi$
$7$	1.41421	0.534522	0.267261	+	0.963624i	$0.413881\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.24264	0.676182	0.338091	−	0.941113i	$-0.390219\pi$
$11$	2.24264	0.676182	0.338091	+	0.941113i	$0.390219\pi$
$12$	0	0
$13$	−3.41421	−0.946932	−0.473466	−	0.880812i	$-0.656997\pi$
$13$	−3.41421	−0.946932	−0.473466	+	0.880812i	$0.656997\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	1.17157	0.284148	0.142074	−	0.989856i	$-0.454623\pi$
$17$	1.17157	0.284148	0.142074	+	0.989856i	$0.454623\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.41421	−0.308607
$22$	0	0
$23$	−7.65685	−1.59656	−0.798282	−	0.602284i	$-0.794258\pi$
$23$	−7.65685	−1.59656	−0.798282	+	0.602284i	$0.794258\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.41421	0.262613	0.131306	−	0.991342i	$-0.458083\pi$
$29$	1.41421	0.262613	0.131306	+	0.991342i	$0.458083\pi$
$30$	0	0
$31$	3.17157	0.569631	0.284816	−	0.958582i	$-0.408068\pi$
$31$	3.17157	0.569631	0.284816	+	0.958582i	$0.408068\pi$
$32$	0	0
$33$	−2.24264	−0.390394
$34$	0	0
$35$	−1.41421	−0.239046
$36$	0	0
$37$	−3.41421	−0.561293	−0.280647	−	0.959811i	$-0.590549\pi$
$37$	−3.41421	−0.561293	−0.280647	+	0.959811i	$0.590549\pi$
$38$	0	0
$39$	3.41421	0.546712
$40$	0	0
$41$	−0.242641	−0.0378941	−0.0189471	−	0.999820i	$-0.506031\pi$
$41$	−0.242641	−0.0378941	−0.0189471	+	0.999820i	$0.506031\pi$
$42$	0	0
$43$	−12.2426	−1.86699	−0.933493	−	0.358597i	$-0.883255\pi$
$43$	−12.2426	−1.86699	−0.933493	+	0.358597i	$0.883255\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	7.65685	1.11687	0.558433	−	0.829549i	$-0.311403\pi$
$47$	7.65685	1.11687	0.558433	+	0.829549i	$0.311403\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	−1.17157	−0.164053
$52$	0	0
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	0	0
$55$	−2.24264	−0.302398
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−12.4853	−1.62545	−0.812723	−	0.582651i	$-0.802016\pi$
$59$	−12.4853	−1.62545	−0.812723	+	0.582651i	$0.802016\pi$
$60$	0	0
$61$	7.31371	0.936424	0.468212	−	0.883616i	$-0.344898\pi$
$61$	7.31371	0.936424	0.468212	+	0.883616i	$0.344898\pi$
$62$	0	0
$63$	1.41421	0.178174
$64$	0	0
$65$	3.41421	0.423481
$66$	0	0
$67$	9.65685	1.17977	0.589886	−	0.807486i	$-0.299173\pi$
$67$	9.65685	1.17977	0.589886	+	0.807486i	$0.299173\pi$
$68$	0	0
$69$	7.65685	0.921777
$70$	0	0
$71$	10.8284	1.28510	0.642549	−	0.766245i	$-0.277877\pi$
$71$	10.8284	1.28510	0.642549	+	0.766245i	$0.277877\pi$
$72$	0	0
$73$	−7.65685	−0.896167	−0.448084	−	0.893992i	$-0.647893\pi$
$73$	−7.65685	−0.896167	−0.448084	+	0.893992i	$0.647893\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	3.17157	0.361434
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.8284	−1.40810	−0.704051	−	0.710149i	$-0.748628\pi$
$83$	−12.8284	−1.40810	−0.704051	+	0.710149i	$0.748628\pi$
$84$	0	0
$85$	−1.17157	−0.127075
$86$	0	0
$87$	−1.41421	−0.151620
$88$	0	0
$89$	5.89949	0.625345	0.312673	−	0.949861i	$-0.398776\pi$
$89$	5.89949	0.625345	0.312673	+	0.949861i	$0.398776\pi$
$90$	0	0
$91$	−4.82843	−0.506157
$92$	0	0
$93$	−3.17157	−0.328877
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−9.75736	−0.990710	−0.495355	−	0.868691i	$-0.664962\pi$
$97$	−9.75736	−0.990710	−0.495355	+	0.868691i	$0.664962\pi$
$98$	0	0
$99$	2.24264	0.225394
$100$	0	0
$101$	−3.17157	−0.315583	−0.157792	−	0.987472i	$-0.550437\pi$
$101$	−3.17157	−0.315583	−0.157792	+	0.987472i	$0.550437\pi$
$102$	0	0
$103$	7.31371	0.720641	0.360321	−	0.932829i	$-0.382667\pi$
$103$	7.31371	0.720641	0.360321	+	0.932829i	$0.382667\pi$
$104$	0	0
$105$	1.41421	0.138013
$106$	0	0
$107$	−19.3137	−1.86713	−0.933563	−	0.358412i	$-0.883318\pi$
$107$	−19.3137	−1.86713	−0.933563	+	0.358412i	$0.883318\pi$
$108$	0	0
$109$	−6.48528	−0.621177	−0.310589	−	0.950544i	$-0.600526\pi$
$109$	−6.48528	−0.621177	−0.310589	+	0.950544i	$0.600526\pi$
$110$	0	0
$111$	3.41421	0.324063
$112$	0	0
$113$	10.1421	0.954092	0.477046	−	0.878878i	$-0.341708\pi$
$113$	10.1421	0.954092	0.477046	+	0.878878i	$0.341708\pi$
$114$	0	0
$115$	7.65685	0.714005
$116$	0	0
$117$	−3.41421	−0.315644
$118$	0	0
$119$	1.65685	0.151884
$120$	0	0
$121$	−5.97056	−0.542778
$122$	0	0
$123$	0.242641	0.0218782
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−19.3137	−1.71381	−0.856907	−	0.515471i	$-0.827617\pi$
$127$	−19.3137	−1.71381	−0.856907	+	0.515471i	$0.827617\pi$
$128$	0	0
$129$	12.2426	1.07790
$130$	0	0
$131$	8.58579	0.750144	0.375072	−	0.926996i	$-0.377618\pi$
$131$	8.58579	0.750144	0.375072	+	0.926996i	$0.377618\pi$
$132$	0	0
$133$	1.41421	0.122628
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	14.1421	1.19952	0.599760	−	0.800180i	$-0.295263\pi$
$139$	14.1421	1.19952	0.599760	+	0.800180i	$0.295263\pi$
$140$	0	0
$141$	−7.65685	−0.644823
$142$	0	0
$143$	−7.65685	−0.640298
$144$	0	0
$145$	−1.41421	−0.117444
$146$	0	0
$147$	5.00000	0.412393
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	6.48528	0.527765	0.263882	−	0.964555i	$-0.414997\pi$
$151$	6.48528	0.527765	0.263882	+	0.964555i	$0.414997\pi$
$152$	0	0
$153$	1.17157	0.0947161
$154$	0	0
$155$	−3.17157	−0.254747
$156$	0	0
$157$	10.4853	0.836817	0.418408	−	0.908259i	$-0.362588\pi$
$157$	10.4853	0.836817	0.418408	+	0.908259i	$0.362588\pi$
$158$	0	0
$159$	−8.00000	−0.634441
$160$	0	0
$161$	−10.8284	−0.853400
$162$	0	0
$163$	−21.8995	−1.71530	−0.857650	−	0.514233i	$-0.828077\pi$
$163$	−21.8995	−1.71530	−0.857650	+	0.514233i	$0.828077\pi$
$164$	0	0
$165$	2.24264	0.174589
$166$	0	0
$167$	17.3137	1.33977	0.669887	−	0.742463i	$-0.266342\pi$
$167$	17.3137	1.33977	0.669887	+	0.742463i	$0.266342\pi$
$168$	0	0
$169$	−1.34315	−0.103319
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−19.7990	−1.50529	−0.752645	−	0.658427i	$-0.771222\pi$
$173$	−19.7990	−1.50529	−0.752645	+	0.658427i	$0.771222\pi$
$174$	0	0
$175$	1.41421	0.106904
$176$	0	0
$177$	12.4853	0.938451
$178$	0	0
$179$	−16.4853	−1.23217	−0.616084	−	0.787681i	$-0.711282\pi$
$179$	−16.4853	−1.23217	−0.616084	+	0.787681i	$0.711282\pi$
$180$	0	0
$181$	−20.8284	−1.54816	−0.774082	−	0.633085i	$-0.781788\pi$
$181$	−20.8284	−1.54816	−0.774082	+	0.633085i	$0.781788\pi$
$182$	0	0
$183$	−7.31371	−0.540645
$184$	0	0
$185$	3.41421	0.251018
$186$	0	0
$187$	2.62742	0.192136
$188$	0	0
$189$	−1.41421	−0.102869
$190$	0	0
$191$	−10.2426	−0.741131	−0.370566	−	0.928806i	$-0.620836\pi$
$191$	−10.2426	−0.741131	−0.370566	+	0.928806i	$0.620836\pi$
$192$	0	0
$193$	5.07107	0.365023	0.182512	−	0.983204i	$-0.441577\pi$
$193$	5.07107	0.365023	0.182512	+	0.983204i	$0.441577\pi$
$194$	0	0
$195$	−3.41421	−0.244497
$196$	0	0
$197$	6.82843	0.486505	0.243253	−	0.969963i	$-0.421786\pi$
$197$	6.82843	0.486505	0.243253	+	0.969963i	$0.421786\pi$
$198$	0	0
$199$	−18.1421	−1.28606	−0.643031	−	0.765840i	$-0.722323\pi$
$199$	−18.1421	−1.28606	−0.643031	+	0.765840i	$0.722323\pi$
$200$	0	0
$201$	−9.65685	−0.681142
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	0.242641	0.0169468
$206$	0	0
$207$	−7.65685	−0.532188
$208$	0	0
$209$	2.24264	0.155127
$210$	0	0
$211$	−7.31371	−0.503496	−0.251748	−	0.967793i	$-0.581005\pi$
$211$	−7.31371	−0.503496	−0.251748	+	0.967793i	$0.581005\pi$
$212$	0	0
$213$	−10.8284	−0.741952
$214$	0	0
$215$	12.2426	0.834941
$216$	0	0
$217$	4.48528	0.304481
$218$	0	0
$219$	7.65685	0.517402
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−18.6274	−1.24738	−0.623692	−	0.781670i	$-0.714368\pi$
$223$	−18.6274	−1.24738	−0.623692	+	0.781670i	$0.714368\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−14.9706	−0.993631	−0.496816	−	0.867856i	$-0.665497\pi$
$227$	−14.9706	−0.993631	−0.496816	+	0.867856i	$0.665497\pi$
$228$	0	0
$229$	−22.6274	−1.49526	−0.747631	−	0.664114i	$-0.768809\pi$
$229$	−22.6274	−1.49526	−0.747631	+	0.664114i	$0.768809\pi$
$230$	0	0
$231$	−3.17157	−0.208674
$232$	0	0
$233$	−0.343146	−0.0224802	−0.0112401	−	0.999937i	$-0.503578\pi$
$233$	−0.343146	−0.0224802	−0.0112401	+	0.999937i	$0.503578\pi$
$234$	0	0
$235$	−7.65685	−0.499478
$236$	0	0
$237$	0	0
$238$	0	0
$239$	26.7279	1.72889	0.864443	−	0.502731i	$-0.167671\pi$
$239$	26.7279	1.72889	0.864443	+	0.502731i	$0.167671\pi$
$240$	0	0
$241$	−19.6569	−1.26621	−0.633105	−	0.774066i	$-0.718220\pi$
$241$	−19.6569	−1.26621	−0.633105	+	0.774066i	$0.718220\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	5.00000	0.319438
$246$	0	0
$247$	−3.41421	−0.217241
$248$	0	0
$249$	12.8284	0.812969
$250$	0	0
$251$	1.75736	0.110924	0.0554618	−	0.998461i	$-0.482337\pi$
$251$	1.75736	0.110924	0.0554618	+	0.998461i	$0.482337\pi$
$252$	0	0
$253$	−17.1716	−1.07957
$254$	0	0
$255$	1.17157	0.0733667
$256$	0	0
$257$	−4.48528	−0.279784	−0.139892	−	0.990167i	$-0.544676\pi$
$257$	−4.48528	−0.279784	−0.139892	+	0.990167i	$0.544676\pi$
$258$	0	0
$259$	−4.82843	−0.300024
$260$	0	0
$261$	1.41421	0.0875376
$262$	0	0
$263$	23.4558	1.44635	0.723175	−	0.690665i	$-0.242682\pi$
$263$	23.4558	1.44635	0.723175	+	0.690665i	$0.242682\pi$
$264$	0	0
$265$	−8.00000	−0.491436
$266$	0	0
$267$	−5.89949	−0.361043
$268$	0	0
$269$	−3.07107	−0.187246	−0.0936232	−	0.995608i	$-0.529845\pi$
$269$	−3.07107	−0.187246	−0.0936232	+	0.995608i	$0.529845\pi$
$270$	0	0
$271$	10.8284	0.657780	0.328890	−	0.944368i	$-0.393325\pi$
$271$	10.8284	0.657780	0.328890	+	0.944368i	$0.393325\pi$
$272$	0	0
$273$	4.82843	0.292230
$274$	0	0
$275$	2.24264	0.135236
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	0	0
$279$	3.17157	0.189877
$280$	0	0
$281$	−14.5858	−0.870115	−0.435058	−	0.900403i	$-0.643272\pi$
$281$	−14.5858	−0.870115	−0.435058	+	0.900403i	$0.643272\pi$
$282$	0	0
$283$	10.3848	0.617311	0.308655	−	0.951174i	$-0.400121\pi$
$283$	10.3848	0.617311	0.308655	+	0.951174i	$0.400121\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	0	0
$287$	−0.343146	−0.0202553
$288$	0	0
$289$	−15.6274	−0.919260
$290$	0	0
$291$	9.75736	0.571987
$292$	0	0
$293$	−11.5147	−0.672697	−0.336349	−	0.941738i	$-0.609192\pi$
$293$	−11.5147	−0.672697	−0.336349	+	0.941738i	$0.609192\pi$
$294$	0	0
$295$	12.4853	0.726921
$296$	0	0
$297$	−2.24264	−0.130131
$298$	0	0
$299$	26.1421	1.51184
$300$	0	0
$301$	−17.3137	−0.997946
$302$	0	0
$303$	3.17157	0.182202
$304$	0	0
$305$	−7.31371	−0.418782
$306$	0	0
$307$	−21.1716	−1.20833	−0.604163	−	0.796861i	$-0.706492\pi$
$307$	−21.1716	−1.20833	−0.604163	+	0.796861i	$0.706492\pi$
$308$	0	0
$309$	−7.31371	−0.416062
$310$	0	0
$311$	6.24264	0.353988	0.176994	−	0.984212i	$-0.443363\pi$
$311$	6.24264	0.353988	0.176994	+	0.984212i	$0.443363\pi$
$312$	0	0
$313$	5.79899	0.327778	0.163889	−	0.986479i	$-0.447596\pi$
$313$	5.79899	0.327778	0.163889	+	0.986479i	$0.447596\pi$
$314$	0	0
$315$	−1.41421	−0.0796819
$316$	0	0
$317$	−26.6274	−1.49554	−0.747772	−	0.663955i	$-0.768877\pi$
$317$	−26.6274	−1.49554	−0.747772	+	0.663955i	$0.768877\pi$
$318$	0	0
$319$	3.17157	0.177574
$320$	0	0
$321$	19.3137	1.07799
$322$	0	0
$323$	1.17157	0.0651881
$324$	0	0
$325$	−3.41421	−0.189386
$326$	0	0
$327$	6.48528	0.358637
$328$	0	0
$329$	10.8284	0.596991
$330$	0	0
$331$	−28.1421	−1.54683	−0.773416	−	0.633899i	$-0.781453\pi$
$331$	−28.1421	−1.54683	−0.773416	+	0.633899i	$0.781453\pi$
$332$	0	0
$333$	−3.41421	−0.187098
$334$	0	0
$335$	−9.65685	−0.527610
$336$	0	0
$337$	−24.5858	−1.33927	−0.669637	−	0.742689i	$-0.733550\pi$
$337$	−24.5858	−1.33927	−0.669637	+	0.742689i	$0.733550\pi$
$338$	0	0
$339$	−10.1421	−0.550845
$340$	0	0
$341$	7.11270	0.385174
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	−7.65685	−0.412231
$346$	0	0
$347$	27.4558	1.47391	0.736953	−	0.675943i	$-0.236264\pi$
$347$	27.4558	1.47391	0.736953	+	0.675943i	$0.236264\pi$
$348$	0	0
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	0	0
$351$	3.41421	0.182237
$352$	0	0
$353$	3.65685	0.194635	0.0973174	−	0.995253i	$-0.468974\pi$
$353$	3.65685	0.194635	0.0973174	+	0.995253i	$0.468974\pi$
$354$	0	0
$355$	−10.8284	−0.574713
$356$	0	0
$357$	−1.65685	−0.0876900
$358$	0	0
$359$	−24.8701	−1.31259	−0.656296	−	0.754504i	$-0.727878\pi$
$359$	−24.8701	−1.31259	−0.656296	+	0.754504i	$0.727878\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	5.97056	0.313373
$364$	0	0
$365$	7.65685	0.400778
$366$	0	0
$367$	14.3848	0.750879	0.375440	−	0.926847i	$-0.377492\pi$
$367$	14.3848	0.750879	0.375440	+	0.926847i	$0.377492\pi$
$368$	0	0
$369$	−0.242641	−0.0126314
$370$	0	0
$371$	11.3137	0.587378
$372$	0	0
$373$	0.585786	0.0303309	0.0151654	−	0.999885i	$-0.495173\pi$
$373$	0.585786	0.0303309	0.0151654	+	0.999885i	$0.495173\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−4.82843	−0.248677
$378$	0	0
$379$	−3.17157	−0.162913	−0.0814564	−	0.996677i	$-0.525957\pi$
$379$	−3.17157	−0.162913	−0.0814564	+	0.996677i	$0.525957\pi$
$380$	0	0
$381$	19.3137	0.989471
$382$	0	0
$383$	−28.0000	−1.43073	−0.715367	−	0.698749i	$-0.753740\pi$
$383$	−28.0000	−1.43073	−0.715367	+	0.698749i	$0.753740\pi$
$384$	0	0
$385$	−3.17157	−0.161638
$386$	0	0
$387$	−12.2426	−0.622328
$388$	0	0
$389$	30.9706	1.57027	0.785135	−	0.619325i	$-0.212594\pi$
$389$	30.9706	1.57027	0.785135	+	0.619325i	$0.212594\pi$
$390$	0	0
$391$	−8.97056	−0.453661
$392$	0	0
$393$	−8.58579	−0.433096
$394$	0	0
$395$	0	0
$396$	0	0
$397$	28.6274	1.43677	0.718384	−	0.695646i	$-0.244882\pi$
$397$	28.6274	1.43677	0.718384	+	0.695646i	$0.244882\pi$
$398$	0	0
$399$	−1.41421	−0.0707992
$400$	0	0
$401$	−7.75736	−0.387384	−0.193692	−	0.981062i	$-0.562046\pi$
$401$	−7.75736	−0.387384	−0.193692	+	0.981062i	$0.562046\pi$
$402$	0	0
$403$	−10.8284	−0.539402
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−7.65685	−0.379536
$408$	0	0
$409$	−12.8284	−0.634325	−0.317162	−	0.948371i	$-0.602730\pi$
$409$	−12.8284	−0.634325	−0.317162	+	0.948371i	$0.602730\pi$
$410$	0	0
$411$	−10.0000	−0.493264
$412$	0	0
$413$	−17.6569	−0.868837
$414$	0	0
$415$	12.8284	0.629723
$416$	0	0
$417$	−14.1421	−0.692543
$418$	0	0
$419$	−3.41421	−0.166795	−0.0833976	−	0.996516i	$-0.526577\pi$
$419$	−3.41421	−0.166795	−0.0833976	+	0.996516i	$0.526577\pi$
$420$	0	0
$421$	9.31371	0.453922	0.226961	−	0.973904i	$-0.427121\pi$
$421$	9.31371	0.453922	0.226961	+	0.973904i	$0.427121\pi$
$422$	0	0
$423$	7.65685	0.372289
$424$	0	0
$425$	1.17157	0.0568296
$426$	0	0
$427$	10.3431	0.500540
$428$	0	0
$429$	7.65685	0.369676
$430$	0	0
$431$	31.1127	1.49865	0.749323	−	0.662205i	$-0.230379\pi$
$431$	31.1127	1.49865	0.749323	+	0.662205i	$0.230379\pi$
$432$	0	0
$433$	−10.9289	−0.525211	−0.262605	−	0.964903i	$-0.584582\pi$
$433$	−10.9289	−0.525211	−0.262605	+	0.964903i	$0.584582\pi$
$434$	0	0
$435$	1.41421	0.0678064
$436$	0	0
$437$	−7.65685	−0.366277
$438$	0	0
$439$	21.6569	1.03363	0.516813	−	0.856099i	$-0.327118\pi$
$439$	21.6569	1.03363	0.516813	+	0.856099i	$0.327118\pi$
$440$	0	0
$441$	−5.00000	−0.238095
$442$	0	0
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	0	0
$445$	−5.89949	−0.279663
$446$	0	0
$447$	6.00000	0.283790
$448$	0	0
$449$	−26.8701	−1.26808	−0.634038	−	0.773302i	$-0.718604\pi$
$449$	−26.8701	−1.26808	−0.634038	+	0.773302i	$0.718604\pi$
$450$	0	0
$451$	−0.544156	−0.0256233
$452$	0	0
$453$	−6.48528	−0.304705
$454$	0	0
$455$	4.82843	0.226360
$456$	0	0
$457$	32.8284	1.53565	0.767825	−	0.640660i	$-0.221339\pi$
$457$	32.8284	1.53565	0.767825	+	0.640660i	$0.221339\pi$
$458$	0	0
$459$	−1.17157	−0.0546843
$460$	0	0
$461$	19.6569	0.915511	0.457755	−	0.889078i	$-0.348654\pi$
$461$	19.6569	0.915511	0.457755	+	0.889078i	$0.348654\pi$
$462$	0	0
$463$	−24.2426	−1.12665	−0.563326	−	0.826235i	$-0.690478\pi$
$463$	−24.2426	−1.12665	−0.563326	+	0.826235i	$0.690478\pi$
$464$	0	0
$465$	3.17157	0.147078
$466$	0	0
$467$	−35.6569	−1.65000	−0.825001	−	0.565131i	$-0.808826\pi$
$467$	−35.6569	−1.65000	−0.825001	+	0.565131i	$0.808826\pi$
$468$	0	0
$469$	13.6569	0.630615
$470$	0	0
$471$	−10.4853	−0.483136
$472$	0	0
$473$	−27.4558	−1.26242
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	8.00000	0.366295
$478$	0	0
$479$	−17.0711	−0.779997	−0.389998	−	0.920815i	$-0.627524\pi$
$479$	−17.0711	−0.779997	−0.389998	+	0.920815i	$0.627524\pi$
$480$	0	0
$481$	11.6569	0.531507
$482$	0	0
$483$	10.8284	0.492710
$484$	0	0
$485$	9.75736	0.443059
$486$	0	0
$487$	−20.4853	−0.928277	−0.464138	−	0.885763i	$-0.653636\pi$
$487$	−20.4853	−0.928277	−0.464138	+	0.885763i	$0.653636\pi$
$488$	0	0
$489$	21.8995	0.990329
$490$	0	0
$491$	1.75736	0.0793085	0.0396543	−	0.999213i	$-0.487374\pi$
$491$	1.75736	0.0793085	0.0396543	+	0.999213i	$0.487374\pi$
$492$	0	0
$493$	1.65685	0.0746210
$494$	0	0
$495$	−2.24264	−0.100799
$496$	0	0
$497$	15.3137	0.686914
$498$	0	0
$499$	5.17157	0.231511	0.115756	−	0.993278i	$-0.463071\pi$
$499$	5.17157	0.231511	0.115756	+	0.993278i	$0.463071\pi$
$500$	0	0
$501$	−17.3137	−0.773519
$502$	0	0
$503$	−4.82843	−0.215289	−0.107644	−	0.994189i	$-0.534331\pi$
$503$	−4.82843	−0.215289	−0.107644	+	0.994189i	$0.534331\pi$
$504$	0	0
$505$	3.17157	0.141133
$506$	0	0
$507$	1.34315	0.0596512
$508$	0	0
$509$	−0.727922	−0.0322646	−0.0161323	−	0.999870i	$-0.505135\pi$
$509$	−0.727922	−0.0322646	−0.0161323	+	0.999870i	$0.505135\pi$
$510$	0	0
$511$	−10.8284	−0.479021
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	−7.31371	−0.322281
$516$	0	0
$517$	17.1716	0.755205
$518$	0	0
$519$	19.7990	0.869079
$520$	0	0
$521$	7.75736	0.339856	0.169928	−	0.985456i	$-0.445646\pi$
$521$	7.75736	0.339856	0.169928	+	0.985456i	$0.445646\pi$
$522$	0	0
$523$	15.5147	0.678411	0.339206	−	0.940712i	$-0.389842\pi$
$523$	15.5147	0.678411	0.339206	+	0.940712i	$0.389842\pi$
$524$	0	0
$525$	−1.41421	−0.0617213
$526$	0	0
$527$	3.71573	0.161860
$528$	0	0
$529$	35.6274	1.54902
$530$	0	0
$531$	−12.4853	−0.541815
$532$	0	0
$533$	0.828427	0.0358832
$534$	0	0
$535$	19.3137	0.835004
$536$	0	0
$537$	16.4853	0.711392
$538$	0	0
$539$	−11.2132	−0.482987
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	0	0
$543$	20.8284	0.893833
$544$	0	0
$545$	6.48528	0.277799
$546$	0	0
$547$	−24.4853	−1.04692	−0.523458	−	0.852052i	$-0.675358\pi$
$547$	−24.4853	−1.04692	−0.523458	+	0.852052i	$0.675358\pi$
$548$	0	0
$549$	7.31371	0.312141
$550$	0	0
$551$	1.41421	0.0602475
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−3.41421	−0.144925
$556$	0	0
$557$	−25.3137	−1.07258	−0.536288	−	0.844035i	$-0.680174\pi$
$557$	−25.3137	−1.07258	−0.536288	+	0.844035i	$0.680174\pi$
$558$	0	0
$559$	41.7990	1.76791
$560$	0	0
$561$	−2.62742	−0.110930
$562$	0	0
$563$	30.2843	1.27633	0.638165	−	0.769900i	$-0.279694\pi$
$563$	30.2843	1.27633	0.638165	+	0.769900i	$0.279694\pi$
$564$	0	0
$565$	−10.1421	−0.426683
$566$	0	0
$567$	1.41421	0.0593914
$568$	0	0
$569$	31.0711	1.30257	0.651283	−	0.758835i	$-0.274231\pi$
$569$	31.0711	1.30257	0.651283	+	0.758835i	$0.274231\pi$
$570$	0	0
$571$	−9.17157	−0.383818	−0.191909	−	0.981413i	$-0.561468\pi$
$571$	−9.17157	−0.383818	−0.191909	+	0.981413i	$0.561468\pi$
$572$	0	0
$573$	10.2426	0.427892
$574$	0	0
$575$	−7.65685	−0.319313
$576$	0	0
$577$	19.4558	0.809957	0.404979	−	0.914326i	$-0.367279\pi$
$577$	19.4558	0.809957	0.404979	+	0.914326i	$0.367279\pi$
$578$	0	0
$579$	−5.07107	−0.210746
$580$	0	0
$581$	−18.1421	−0.752663
$582$	0	0
$583$	17.9411	0.743045
$584$	0	0
$585$	3.41421	0.141160
$586$	0	0
$587$	12.3431	0.509456	0.254728	−	0.967013i	$-0.418014\pi$
$587$	12.3431	0.509456	0.254728	+	0.967013i	$0.418014\pi$
$588$	0	0
$589$	3.17157	0.130682
$590$	0	0
$591$	−6.82843	−0.280884
$592$	0	0
$593$	36.6274	1.50411	0.752054	−	0.659102i	$-0.229063\pi$
$593$	36.6274	1.50411	0.752054	+	0.659102i	$0.229063\pi$
$594$	0	0
$595$	−1.65685	−0.0679244
$596$	0	0
$597$	18.1421	0.742508
$598$	0	0
$599$	3.02944	0.123779	0.0618897	−	0.998083i	$-0.480287\pi$
$599$	3.02944	0.123779	0.0618897	+	0.998083i	$0.480287\pi$
$600$	0	0
$601$	8.14214	0.332125	0.166062	−	0.986115i	$-0.446895\pi$
$601$	8.14214	0.332125	0.166062	+	0.986115i	$0.446895\pi$
$602$	0	0
$603$	9.65685	0.393258
$604$	0	0
$605$	5.97056	0.242738
$606$	0	0
$607$	−16.4853	−0.669117	−0.334558	−	0.942375i	$-0.608587\pi$
$607$	−16.4853	−0.669117	−0.334558	+	0.942375i	$0.608587\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	−26.1421	−1.05760
$612$	0	0
$613$	−10.4853	−0.423497	−0.211748	−	0.977324i	$-0.567916\pi$
$613$	−10.4853	−0.423497	−0.211748	+	0.977324i	$0.567916\pi$
$614$	0	0
$615$	−0.242641	−0.00978422
$616$	0	0
$617$	28.4853	1.14677	0.573387	−	0.819285i	$-0.305629\pi$
$617$	28.4853	1.14677	0.573387	+	0.819285i	$0.305629\pi$
$618$	0	0
$619$	20.4853	0.823373	0.411686	−	0.911326i	$-0.364940\pi$
$619$	20.4853	0.823373	0.411686	+	0.911326i	$0.364940\pi$
$620$	0	0
$621$	7.65685	0.307259
$622$	0	0
$623$	8.34315	0.334261
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.24264	−0.0895624
$628$	0	0
$629$	−4.00000	−0.159490
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	7.31371	0.290694
$634$	0	0
$635$	19.3137	0.766441
$636$	0	0
$637$	17.0711	0.676380
$638$	0	0
$639$	10.8284	0.428366
$640$	0	0
$641$	−31.3553	−1.23846	−0.619231	−	0.785209i	$-0.712555\pi$
$641$	−31.3553	−1.23846	−0.619231	+	0.785209i	$0.712555\pi$
$642$	0	0
$643$	−42.3848	−1.67149	−0.835746	−	0.549116i	$-0.814965\pi$
$643$	−42.3848	−1.67149	−0.835746	+	0.549116i	$0.814965\pi$
$644$	0	0
$645$	−12.2426	−0.482054
$646$	0	0
$647$	36.1421	1.42089	0.710447	−	0.703751i	$-0.248493\pi$
$647$	36.1421	1.42089	0.710447	+	0.703751i	$0.248493\pi$
$648$	0	0
$649$	−28.0000	−1.09910
$650$	0	0
$651$	−4.48528	−0.175792
$652$	0	0
$653$	42.8284	1.67601	0.838003	−	0.545666i	$-0.183723\pi$
$653$	42.8284	1.67601	0.838003	+	0.545666i	$0.183723\pi$
$654$	0	0
$655$	−8.58579	−0.335474
$656$	0	0
$657$	−7.65685	−0.298722
$658$	0	0
$659$	18.6274	0.725621	0.362811	−	0.931863i	$-0.381817\pi$
$659$	18.6274	0.725621	0.362811	+	0.931863i	$0.381817\pi$
$660$	0	0
$661$	7.45584	0.289999	0.144999	−	0.989432i	$-0.453682\pi$
$661$	7.45584	0.289999	0.144999	+	0.989432i	$0.453682\pi$
$662$	0	0
$663$	4.00000	0.155347
$664$	0	0
$665$	−1.41421	−0.0548408
$666$	0	0
$667$	−10.8284	−0.419278
$668$	0	0
$669$	18.6274	0.720178
$670$	0	0
$671$	16.4020	0.633193
$672$	0	0
$673$	−44.1838	−1.70316	−0.851580	−	0.524225i	$-0.824355\pi$
$673$	−44.1838	−1.70316	−0.851580	+	0.524225i	$0.824355\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	16.9706	0.652232	0.326116	−	0.945330i	$-0.394260\pi$
$677$	16.9706	0.652232	0.326116	+	0.945330i	$0.394260\pi$
$678$	0	0
$679$	−13.7990	−0.529557
$680$	0	0
$681$	14.9706	0.573673
$682$	0	0
$683$	−18.3431	−0.701881	−0.350940	−	0.936398i	$-0.614138\pi$
$683$	−18.3431	−0.701881	−0.350940	+	0.936398i	$0.614138\pi$
$684$	0	0
$685$	−10.0000	−0.382080
$686$	0	0
$687$	22.6274	0.863290
$688$	0	0
$689$	−27.3137	−1.04057
$690$	0	0
$691$	−46.8284	−1.78144	−0.890719	−	0.454555i	$-0.849798\pi$
$691$	−46.8284	−1.78144	−0.890719	+	0.454555i	$0.849798\pi$
$692$	0	0
$693$	3.17157	0.120478
$694$	0	0
$695$	−14.1421	−0.536442
$696$	0	0
$697$	−0.284271	−0.0107675
$698$	0	0
$699$	0.343146	0.0129790
$700$	0	0
$701$	6.68629	0.252538	0.126269	−	0.991996i	$-0.459700\pi$
$701$	6.68629	0.252538	0.126269	+	0.991996i	$0.459700\pi$
$702$	0	0
$703$	−3.41421	−0.128770
$704$	0	0
$705$	7.65685	0.288374
$706$	0	0
$707$	−4.48528	−0.168686
$708$	0	0
$709$	28.9706	1.08801	0.544006	−	0.839081i	$-0.316907\pi$
$709$	28.9706	1.08801	0.544006	+	0.839081i	$0.316907\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−24.2843	−0.909453
$714$	0	0
$715$	7.65685	0.286350
$716$	0	0
$717$	−26.7279	−0.998173
$718$	0	0
$719$	1.07107	0.0399441	0.0199720	−	0.999801i	$-0.493642\pi$
$719$	1.07107	0.0399441	0.0199720	+	0.999801i	$0.493642\pi$
$720$	0	0
$721$	10.3431	0.385199
$722$	0	0
$723$	19.6569	0.731046
$724$	0	0
$725$	1.41421	0.0525226
$726$	0	0
$727$	−47.3553	−1.75631	−0.878156	−	0.478374i	$-0.841226\pi$
$727$	−47.3553	−1.75631	−0.878156	+	0.478374i	$0.841226\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−14.3431	−0.530500
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	0	0
$735$	−5.00000	−0.184428
$736$	0	0
$737$	21.6569	0.797740
$738$	0	0
$739$	−14.3431	−0.527621	−0.263811	−	0.964575i	$-0.584979\pi$
$739$	−14.3431	−0.527621	−0.263811	+	0.964575i	$0.584979\pi$
$740$	0	0
$741$	3.41421	0.125424
$742$	0	0
$743$	4.00000	0.146746	0.0733729	−	0.997305i	$-0.476624\pi$
$743$	4.00000	0.146746	0.0733729	+	0.997305i	$0.476624\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	0	0
$747$	−12.8284	−0.469368
$748$	0	0
$749$	−27.3137	−0.998021
$750$	0	0
$751$	24.1421	0.880959	0.440480	−	0.897763i	$-0.354808\pi$
$751$	24.1421	0.880959	0.440480	+	0.897763i	$0.354808\pi$
$752$	0	0
$753$	−1.75736	−0.0640417
$754$	0	0
$755$	−6.48528	−0.236024
$756$	0	0
$757$	−32.4264	−1.17856	−0.589279	−	0.807930i	$-0.700588\pi$
$757$	−32.4264	−1.17856	−0.589279	+	0.807930i	$0.700588\pi$
$758$	0	0
$759$	17.1716	0.623289
$760$	0	0
$761$	43.9411	1.59286	0.796432	−	0.604728i	$-0.206718\pi$
$761$	43.9411	1.59286	0.796432	+	0.604728i	$0.206718\pi$
$762$	0	0
$763$	−9.17157	−0.332033
$764$	0	0
$765$	−1.17157	−0.0423583
$766$	0	0
$767$	42.6274	1.53919
$768$	0	0
$769$	42.9706	1.54956	0.774779	−	0.632232i	$-0.217861\pi$
$769$	42.9706	1.54956	0.774779	+	0.632232i	$0.217861\pi$
$770$	0	0
$771$	4.48528	0.161533
$772$	0	0
$773$	−25.6569	−0.922813	−0.461406	−	0.887189i	$-0.652655\pi$
$773$	−25.6569	−0.922813	−0.461406	+	0.887189i	$0.652655\pi$
$774$	0	0
$775$	3.17157	0.113926
$776$	0	0
$777$	4.82843	0.173219
$778$	0	0
$779$	−0.242641	−0.00869350
$780$	0	0
$781$	24.2843	0.868960
$782$	0	0
$783$	−1.41421	−0.0505399
$784$	0	0
$785$	−10.4853	−0.374236
$786$	0	0
$787$	42.8284	1.52667	0.763334	−	0.646004i	$-0.223561\pi$
$787$	42.8284	1.52667	0.763334	+	0.646004i	$0.223561\pi$
$788$	0	0
$789$	−23.4558	−0.835050
$790$	0	0
$791$	14.3431	0.509984
$792$	0	0
$793$	−24.9706	−0.886731
$794$	0	0
$795$	8.00000	0.283731
$796$	0	0
$797$	−14.1421	−0.500940	−0.250470	−	0.968124i	$-0.580585\pi$
$797$	−14.1421	−0.500940	−0.250470	+	0.968124i	$0.580585\pi$
$798$	0	0
$799$	8.97056	0.317356
$800$	0	0
$801$	5.89949	0.208448
$802$	0	0
$803$	−17.1716	−0.605972
$804$	0	0
$805$	10.8284	0.381652
$806$	0	0
$807$	3.07107	0.108107
$808$	0	0
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	0	0
$811$	−8.68629	−0.305017	−0.152508	−	0.988302i	$-0.548735\pi$
$811$	−8.68629	−0.305017	−0.152508	+	0.988302i	$0.548735\pi$
$812$	0	0
$813$	−10.8284	−0.379770
$814$	0	0
$815$	21.8995	0.767106
$816$	0	0
$817$	−12.2426	−0.428316
$818$	0	0
$819$	−4.82843	−0.168719
$820$	0	0
$821$	14.4853	0.505540	0.252770	−	0.967526i	$-0.418658\pi$
$821$	14.4853	0.505540	0.252770	+	0.967526i	$0.418658\pi$
$822$	0	0
$823$	25.0122	0.871870	0.435935	−	0.899978i	$-0.356418\pi$
$823$	25.0122	0.871870	0.435935	+	0.899978i	$0.356418\pi$
$824$	0	0
$825$	−2.24264	−0.0780787
$826$	0	0
$827$	2.68629	0.0934115	0.0467058	−	0.998909i	$-0.485128\pi$
$827$	2.68629	0.0934115	0.0467058	+	0.998909i	$0.485128\pi$
$828$	0	0
$829$	46.4853	1.61450	0.807250	−	0.590209i	$-0.200955\pi$
$829$	46.4853	1.61450	0.807250	+	0.590209i	$0.200955\pi$
$830$	0	0
$831$	22.0000	0.763172
$832$	0	0
$833$	−5.85786	−0.202963
$834$	0	0
$835$	−17.3137	−0.599166
$836$	0	0
$837$	−3.17157	−0.109626
$838$	0	0
$839$	9.85786	0.340331	0.170166	−	0.985415i	$-0.445570\pi$
$839$	9.85786	0.340331	0.170166	+	0.985415i	$0.445570\pi$
$840$	0	0
$841$	−27.0000	−0.931034
$842$	0	0
$843$	14.5858	0.502361
$844$	0	0
$845$	1.34315	0.0462056
$846$	0	0
$847$	−8.44365	−0.290127
$848$	0	0
$849$	−10.3848	−0.356405
$850$	0	0
$851$	26.1421	0.896141
$852$	0	0
$853$	−16.8284	−0.576194	−0.288097	−	0.957601i	$-0.593023\pi$
$853$	−16.8284	−0.576194	−0.288097	+	0.957601i	$0.593023\pi$
$854$	0	0
$855$	−1.00000	−0.0341993
$856$	0	0
$857$	12.6863	0.433355	0.216678	−	0.976243i	$-0.430478\pi$
$857$	12.6863	0.433355	0.216678	+	0.976243i	$0.430478\pi$
$858$	0	0
$859$	49.9411	1.70397	0.851985	−	0.523567i	$-0.175399\pi$
$859$	49.9411	1.70397	0.851985	+	0.523567i	$0.175399\pi$
$860$	0	0
$861$	0.343146	0.0116944
$862$	0	0
$863$	−8.68629	−0.295685	−0.147842	−	0.989011i	$-0.547233\pi$
$863$	−8.68629	−0.295685	−0.147842	+	0.989011i	$0.547233\pi$
$864$	0	0
$865$	19.7990	0.673186
$866$	0	0
$867$	15.6274	0.530735
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−32.9706	−1.11716
$872$	0	0
$873$	−9.75736	−0.330237
$874$	0	0
$875$	−1.41421	−0.0478091
$876$	0	0
$877$	−34.9289	−1.17947	−0.589733	−	0.807598i	$-0.700767\pi$
$877$	−34.9289	−1.17947	−0.589733	+	0.807598i	$0.700767\pi$
$878$	0	0
$879$	11.5147	0.388382
$880$	0	0
$881$	−5.79899	−0.195373	−0.0976865	−	0.995217i	$-0.531144\pi$
$881$	−5.79899	−0.195373	−0.0976865	+	0.995217i	$0.531144\pi$
$882$	0	0
$883$	12.0416	0.405233	0.202617	−	0.979258i	$-0.435055\pi$
$883$	12.0416	0.405233	0.202617	+	0.979258i	$0.435055\pi$
$884$	0	0
$885$	−12.4853	−0.419688
$886$	0	0
$887$	−25.9411	−0.871018	−0.435509	−	0.900184i	$-0.643432\pi$
$887$	−25.9411	−0.871018	−0.435509	+	0.900184i	$0.643432\pi$
$888$	0	0
$889$	−27.3137	−0.916072
$890$	0	0
$891$	2.24264	0.0751313
$892$	0	0
$893$	7.65685	0.256227
$894$	0	0
$895$	16.4853	0.551042
$896$	0	0
$897$	−26.1421	−0.872861
$898$	0	0
$899$	4.48528	0.149593
$900$	0	0
$901$	9.37258	0.312246
$902$	0	0
$903$	17.3137	0.576164
$904$	0	0
$905$	20.8284	0.692360
$906$	0	0
$907$	18.1421	0.602400	0.301200	−	0.953561i	$-0.402613\pi$
$907$	18.1421	0.602400	0.301200	+	0.953561i	$0.402613\pi$
$908$	0	0
$909$	−3.17157	−0.105194
$910$	0	0
$911$	−8.68629	−0.287790	−0.143895	−	0.989593i	$-0.545963\pi$
$911$	−8.68629	−0.287790	−0.143895	+	0.989593i	$0.545963\pi$
$912$	0	0
$913$	−28.7696	−0.952133
$914$	0	0
$915$	7.31371	0.241784
$916$	0	0
$917$	12.1421	0.400969
$918$	0	0
$919$	12.0000	0.395843	0.197922	−	0.980218i	$-0.436581\pi$
$919$	12.0000	0.395843	0.197922	+	0.980218i	$0.436581\pi$
$920$	0	0
$921$	21.1716	0.697627
$922$	0	0
$923$	−36.9706	−1.21690
$924$	0	0
$925$	−3.41421	−0.112259
$926$	0	0
$927$	7.31371	0.240214
$928$	0	0
$929$	33.1127	1.08639	0.543196	−	0.839606i	$-0.317214\pi$
$929$	33.1127	1.08639	0.543196	+	0.839606i	$0.317214\pi$
$930$	0	0
$931$	−5.00000	−0.163868
$932$	0	0
$933$	−6.24264	−0.204375
$934$	0	0
$935$	−2.62742	−0.0859257
$936$	0	0
$937$	−29.7990	−0.973491	−0.486745	−	0.873544i	$-0.661816\pi$
$937$	−29.7990	−0.973491	−0.486745	+	0.873544i	$0.661816\pi$
$938$	0	0
$939$	−5.79899	−0.189243
$940$	0	0
$941$	22.8701	0.745543	0.372771	−	0.927923i	$-0.378408\pi$
$941$	22.8701	0.745543	0.372771	+	0.927923i	$0.378408\pi$
$942$	0	0
$943$	1.85786	0.0605004
$944$	0	0
$945$	1.41421	0.0460044
$946$	0	0
$947$	47.1716	1.53287	0.766435	−	0.642322i	$-0.222029\pi$
$947$	47.1716	1.53287	0.766435	+	0.642322i	$0.222029\pi$
$948$	0	0
$949$	26.1421	0.848610
$950$	0	0
$951$	26.6274	0.863453
$952$	0	0
$953$	−53.4558	−1.73160	−0.865802	−	0.500386i	$-0.833191\pi$
$953$	−53.4558	−1.73160	−0.865802	+	0.500386i	$0.833191\pi$
$954$	0	0
$955$	10.2426	0.331444
$956$	0	0
$957$	−3.17157	−0.102522
$958$	0	0
$959$	14.1421	0.456673
$960$	0	0
$961$	−20.9411	−0.675520
$962$	0	0
$963$	−19.3137	−0.622376
$964$	0	0
$965$	−5.07107	−0.163243
$966$	0	0
$967$	−8.04163	−0.258601	−0.129301	−	0.991605i	$-0.541273\pi$
$967$	−8.04163	−0.258601	−0.129301	+	0.991605i	$0.541273\pi$
$968$	0	0
$969$	−1.17157	−0.0376363
$970$	0	0
$971$	−22.3431	−0.717026	−0.358513	−	0.933525i	$-0.616716\pi$
$971$	−22.3431	−0.717026	−0.358513	+	0.933525i	$0.616716\pi$
$972$	0	0
$973$	20.0000	0.641171
$974$	0	0
$975$	3.41421	0.109342
$976$	0	0
$977$	32.2843	1.03287	0.516433	−	0.856328i	$-0.327260\pi$
$977$	32.2843	1.03287	0.516433	+	0.856328i	$0.327260\pi$
$978$	0	0
$979$	13.2304	0.422847
$980$	0	0
$981$	−6.48528	−0.207059
$982$	0	0
$983$	24.6274	0.785493	0.392746	−	0.919647i	$-0.371525\pi$
$983$	24.6274	0.785493	0.392746	+	0.919647i	$0.371525\pi$
$984$	0	0
$985$	−6.82843	−0.217572
$986$	0	0
$987$	−10.8284	−0.344673
$988$	0	0
$989$	93.7401	2.98076
$990$	0	0
$991$	−2.34315	−0.0744325	−0.0372162	−	0.999307i	$-0.511849\pi$
$991$	−2.34315	−0.0744325	−0.0372162	+	0.999307i	$0.511849\pi$
$992$	0	0
$993$	28.1421	0.893064
$994$	0	0
$995$	18.1421	0.575144
$996$	0	0
$997$	−38.0833	−1.20611	−0.603054	−	0.797700i	$-0.706050\pi$
$997$	−38.0833	−1.20611	−0.603054	+	0.797700i	$0.706050\pi$
$998$	0	0
$999$	3.41421	0.108021

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.a.bf.1.2		2
4.3	odd	2		285.2.a.g.1.2	✓	2
12.11	even	2		855.2.a.d.1.1		2
20.3	even	4		1425.2.c.l.799.1		4
20.7	even	4		1425.2.c.l.799.4		4
20.19	odd	2		1425.2.a.k.1.1		2
60.59	even	2		4275.2.a.y.1.2		2
76.75	even	2		5415.2.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.g.1.2	✓	2	4.3	odd	2
855.2.a.d.1.1		2	12.11	even	2
1425.2.a.k.1.1		2	20.19	odd	2
1425.2.c.l.799.1		4	20.3	even	4
1425.2.c.l.799.4		4	20.7	even	4
4275.2.a.y.1.2		2	60.59	even	2
4560.2.a.bf.1.2		2	1.1	even	1	trivial
5415.2.a.n.1.1		2	76.75	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:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.41421 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.41421 q^{7} +1.00000 q^{9} +2.24264 q^{11} -3.41421 q^{13} +1.00000 q^{15} +1.17157 q^{17} +1.00000 q^{19} -1.41421 q^{21} -7.65685 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.41421 q^{29} +3.17157 q^{31} -2.24264 q^{33} -1.41421 q^{35} -3.41421 q^{37} +3.41421 q^{39} -0.242641 q^{41} -12.2426 q^{43} -1.00000 q^{45} +7.65685 q^{47} -5.00000 q^{49} -1.17157 q^{51} +8.00000 q^{53} -2.24264 q^{55} -1.00000 q^{57} -12.4853 q^{59} +7.31371 q^{61} +1.41421 q^{63} +3.41421 q^{65} +9.65685 q^{67} +7.65685 q^{69} +10.8284 q^{71} -7.65685 q^{73} -1.00000 q^{75} +3.17157 q^{77} +1.00000 q^{81} -12.8284 q^{83} -1.17157 q^{85} -1.41421 q^{87} +5.89949 q^{89} -4.82843 q^{91} -3.17157 q^{93} -1.00000 q^{95} -9.75736 q^{97} +2.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{11} - 4 q^{13} + 2 q^{15} + 8 q^{17} + 2 q^{19} - 4 q^{23} + 2 q^{25} - 2 q^{27} + 12 q^{31} + 4 q^{33} - 4 q^{37} + 4 q^{39} + 8 q^{41} - 16 q^{43} - 2 q^{45} + 4 q^{47} - 10 q^{49} - 8 q^{51} + 16 q^{53} + 4 q^{55} - 2 q^{57} - 8 q^{59} - 8 q^{61} + 4 q^{65} + 8 q^{67} + 4 q^{69} + 16 q^{71} - 4 q^{73} - 2 q^{75} + 12 q^{77} + 2 q^{81} - 20 q^{83} - 8 q^{85} - 8 q^{89} - 4 q^{91} - 12 q^{93} - 2 q^{95} - 28 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 - 4 * q^11 - 4 * q^13 + 2 * q^15 + 8 * q^17 + 2 * q^19 - 4 * q^23 + 2 * q^25 - 2 * q^27 + 12 * q^31 + 4 * q^33 - 4 * q^37 + 4 * q^39 + 8 * q^41 - 16 * q^43 - 2 * q^45 + 4 * q^47 - 10 * q^49 - 8 * q^51 + 16 * q^53 + 4 * q^55 - 2 * q^57 - 8 * q^59 - 8 * q^61 + 4 * q^65 + 8 * q^67 + 4 * q^69 + 16 * q^71 - 4 * q^73 - 2 * q^75 + 12 * q^77 + 2 * q^81 - 20 * q^83 - 8 * q^85 - 8 * q^89 - 4 * q^91 - 12 * q^93 - 2 * q^95 - 28 * q^97 - 4 * q^99

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