LMFDB - Embedded newform 4560.2.a.bi.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +4.73205 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} +4.73205 q^{7} +1.00000 q^{9} -4.19615 q^{11} -1.26795 q^{13} -1.00000 q^{15} -6.92820 q^{17} -1.00000 q^{19} -4.73205 q^{21} +6.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -10.1962 q^{29} +2.00000 q^{31} +4.19615 q^{33} +4.73205 q^{35} +2.73205 q^{37} +1.26795 q^{39} +11.6603 q^{41} +4.73205 q^{43} +1.00000 q^{45} +10.0000 q^{47} +15.3923 q^{49} +6.92820 q^{51} -4.19615 q^{55} +1.00000 q^{57} +9.46410 q^{59} +5.46410 q^{61} +4.73205 q^{63} -1.26795 q^{65} +14.9282 q^{67} -6.00000 q^{69} -12.3923 q^{71} +0.928203 q^{73} -1.00000 q^{75} -19.8564 q^{77} +8.00000 q^{79} +1.00000 q^{81} -8.53590 q^{83} -6.92820 q^{85} +10.1962 q^{87} -7.66025 q^{89} -6.00000 q^{91} -2.00000 q^{93} -1.00000 q^{95} +1.66025 q^{97} -4.19615 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} + 2 q^{11} - 6 q^{13} - 2 q^{15} - 2 q^{19} - 6 q^{21} + 12 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} + 4 q^{31} - 2 q^{33} + 6 q^{35} + 2 q^{37} + 6 q^{39} + 6 q^{41} + 6 q^{43} + 2 q^{45} + 20 q^{47} + 10 q^{49} + 2 q^{55} + 2 q^{57} + 12 q^{59} + 4 q^{61} + 6 q^{63} - 6 q^{65} + 16 q^{67} - 12 q^{69} - 4 q^{71} - 12 q^{73} - 2 q^{75} - 12 q^{77} + 16 q^{79} + 2 q^{81} - 24 q^{83} + 10 q^{87} + 2 q^{89} - 12 q^{91} - 4 q^{93} - 2 q^{95} - 14 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9 + 2 * q^11 - 6 * q^13 - 2 * q^15 - 2 * q^19 - 6 * q^21 + 12 * q^23 + 2 * q^25 - 2 * q^27 - 10 * q^29 + 4 * q^31 - 2 * q^33 + 6 * q^35 + 2 * q^37 + 6 * q^39 + 6 * q^41 + 6 * q^43 + 2 * q^45 + 20 * q^47 + 10 * q^49 + 2 * q^55 + 2 * q^57 + 12 * q^59 + 4 * q^61 + 6 * q^63 - 6 * q^65 + 16 * q^67 - 12 * q^69 - 4 * q^71 - 12 * q^73 - 2 * q^75 - 12 * q^77 + 16 * q^79 + 2 * q^81 - 24 * q^83 + 10 * q^87 + 2 * q^89 - 12 * q^91 - 4 * q^93 - 2 * q^95 - 14 * q^97 + 2 * 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
$7$	4.73205	1.78855	0.894274	−	0.447521i	$-0.147693\pi$
$7$	4.73205	1.78855	0.894274	+	0.447521i	$0.147693\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.19615	−1.26519	−0.632594	−	0.774484i	$-0.718010\pi$
$11$	−4.19615	−1.26519	−0.632594	+	0.774484i	$0.718010\pi$
$12$	0	0
$13$	−1.26795	−0.351666	−0.175833	−	0.984420i	$-0.556262\pi$
$13$	−1.26795	−0.351666	−0.175833	+	0.984420i	$0.556262\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−6.92820	−1.68034	−0.840168	−	0.542326i	$-0.817544\pi$
$17$	−6.92820	−1.68034	−0.840168	+	0.542326i	$0.817544\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−4.73205	−1.03262
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−10.1962	−1.89338	−0.946689	−	0.322149i	$-0.895595\pi$
$29$	−10.1962	−1.89338	−0.946689	+	0.322149i	$0.895595\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	4.19615	0.730456
$34$	0	0
$35$	4.73205	0.799863
$36$	0	0
$37$	2.73205	0.449146	0.224573	−	0.974457i	$-0.427901\pi$
$37$	2.73205	0.449146	0.224573	+	0.974457i	$0.427901\pi$
$38$	0	0
$39$	1.26795	0.203034
$40$	0	0
$41$	11.6603	1.82103	0.910513	−	0.413481i	$-0.135687\pi$
$41$	11.6603	1.82103	0.910513	+	0.413481i	$0.135687\pi$
$42$	0	0
$43$	4.73205	0.721631	0.360815	−	0.932637i	$-0.382498\pi$
$43$	4.73205	0.721631	0.360815	+	0.932637i	$0.382498\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	0	0
$49$	15.3923	2.19890
$50$	0	0
$51$	6.92820	0.970143
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−4.19615	−0.565809
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	9.46410	1.23212	0.616061	−	0.787699i	$-0.288728\pi$
$59$	9.46410	1.23212	0.616061	+	0.787699i	$0.288728\pi$
$60$	0	0
$61$	5.46410	0.699607	0.349803	−	0.936823i	$-0.386248\pi$
$61$	5.46410	0.699607	0.349803	+	0.936823i	$0.386248\pi$
$62$	0	0
$63$	4.73205	0.596182
$64$	0	0
$65$	−1.26795	−0.157270
$66$	0	0
$67$	14.9282	1.82377	0.911885	−	0.410445i	$-0.134627\pi$
$67$	14.9282	1.82377	0.911885	+	0.410445i	$0.134627\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−12.3923	−1.47070	−0.735348	−	0.677690i	$-0.762981\pi$
$71$	−12.3923	−1.47070	−0.735348	+	0.677690i	$0.762981\pi$
$72$	0	0
$73$	0.928203	0.108638	0.0543190	−	0.998524i	$-0.482701\pi$
$73$	0.928203	0.108638	0.0543190	+	0.998524i	$0.482701\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−19.8564	−2.26285
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.53590	−0.936937	−0.468468	−	0.883480i	$-0.655194\pi$
$83$	−8.53590	−0.936937	−0.468468	+	0.883480i	$0.655194\pi$
$84$	0	0
$85$	−6.92820	−0.751469
$86$	0	0
$87$	10.1962	1.09314
$88$	0	0
$89$	−7.66025	−0.811985	−0.405993	−	0.913876i	$-0.633074\pi$
$89$	−7.66025	−0.811985	−0.405993	+	0.913876i	$0.633074\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	1.66025	0.168573	0.0842866	−	0.996442i	$-0.473139\pi$
$97$	1.66025	0.168573	0.0842866	+	0.996442i	$0.473139\pi$
$98$	0	0
$99$	−4.19615	−0.421729
$100$	0	0
$101$	−3.46410	−0.344691	−0.172345	−	0.985037i	$-0.555135\pi$
$101$	−3.46410	−0.344691	−0.172345	+	0.985037i	$0.555135\pi$
$102$	0	0
$103$	13.8564	1.36531	0.682656	−	0.730740i	$-0.260825\pi$
$103$	13.8564	1.36531	0.682656	+	0.730740i	$0.260825\pi$
$104$	0	0
$105$	−4.73205	−0.461801
$106$	0	0
$107$	10.9282	1.05647	0.528235	−	0.849098i	$-0.322854\pi$
$107$	10.9282	1.05647	0.528235	+	0.849098i	$0.322854\pi$
$108$	0	0
$109$	−10.3923	−0.995402	−0.497701	−	0.867349i	$-0.665822\pi$
$109$	−10.3923	−0.995402	−0.497701	+	0.867349i	$0.665822\pi$
$110$	0	0
$111$	−2.73205	−0.259315
$112$	0	0
$113$	2.53590	0.238557	0.119279	−	0.992861i	$-0.461942\pi$
$113$	2.53590	0.238557	0.119279	+	0.992861i	$0.461942\pi$
$114$	0	0
$115$	6.00000	0.559503
$116$	0	0
$117$	−1.26795	−0.117222
$118$	0	0
$119$	−32.7846	−3.00536
$120$	0	0
$121$	6.60770	0.600700
$122$	0	0
$123$	−11.6603	−1.05137
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.07180	0.0951066	0.0475533	−	0.998869i	$-0.484858\pi$
$127$	1.07180	0.0951066	0.0475533	+	0.998869i	$0.484858\pi$
$128$	0	0
$129$	−4.73205	−0.416634
$130$	0	0
$131$	20.5885	1.79882	0.899411	−	0.437104i	$-0.143996\pi$
$131$	20.5885	1.79882	0.899411	+	0.437104i	$0.143996\pi$
$132$	0	0
$133$	−4.73205	−0.410321
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	22.7846	1.94662	0.973310	−	0.229493i	$-0.0737068\pi$
$137$	22.7846	1.94662	0.973310	+	0.229493i	$0.0737068\pi$
$138$	0	0
$139$	17.4641	1.48129	0.740643	−	0.671899i	$-0.234521\pi$
$139$	17.4641	1.48129	0.740643	+	0.671899i	$0.234521\pi$
$140$	0	0
$141$	−10.0000	−0.842152
$142$	0	0
$143$	5.32051	0.444923
$144$	0	0
$145$	−10.1962	−0.846744
$146$	0	0
$147$	−15.3923	−1.26954
$148$	0	0
$149$	8.92820	0.731427	0.365713	−	0.930727i	$-0.380825\pi$
$149$	8.92820	0.731427	0.365713	+	0.930727i	$0.380825\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	−6.92820	−0.560112
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	−3.46410	−0.276465	−0.138233	−	0.990400i	$-0.544142\pi$
$157$	−3.46410	−0.276465	−0.138233	+	0.990400i	$0.544142\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	28.3923	2.23763
$162$	0	0
$163$	−25.1244	−1.96789	−0.983946	−	0.178468i	$-0.942886\pi$
$163$	−25.1244	−1.96789	−0.983946	+	0.178468i	$0.942886\pi$
$164$	0	0
$165$	4.19615	0.326670
$166$	0	0
$167$	−14.3923	−1.11371	−0.556855	−	0.830610i	$-0.687992\pi$
$167$	−14.3923	−1.11371	−0.556855	+	0.830610i	$0.687992\pi$
$168$	0	0
$169$	−11.3923	−0.876331
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	12.3923	0.942169	0.471085	−	0.882088i	$-0.343863\pi$
$173$	12.3923	0.942169	0.471085	+	0.882088i	$0.343863\pi$
$174$	0	0
$175$	4.73205	0.357709
$176$	0	0
$177$	−9.46410	−0.711365
$178$	0	0
$179$	−21.4641	−1.60430	−0.802151	−	0.597121i	$-0.796311\pi$
$179$	−21.4641	−1.60430	−0.802151	+	0.597121i	$0.796311\pi$
$180$	0	0
$181$	4.53590	0.337151	0.168575	−	0.985689i	$-0.446083\pi$
$181$	4.53590	0.337151	0.168575	+	0.985689i	$0.446083\pi$
$182$	0	0
$183$	−5.46410	−0.403918
$184$	0	0
$185$	2.73205	0.200864
$186$	0	0
$187$	29.0718	2.12594
$188$	0	0
$189$	−4.73205	−0.344206
$190$	0	0
$191$	13.2679	0.960035	0.480018	−	0.877259i	$-0.340630\pi$
$191$	13.2679	0.960035	0.480018	+	0.877259i	$0.340630\pi$
$192$	0	0
$193$	−11.8038	−0.849660	−0.424830	−	0.905273i	$-0.639666\pi$
$193$	−11.8038	−0.849660	−0.424830	+	0.905273i	$0.639666\pi$
$194$	0	0
$195$	1.26795	0.0907997
$196$	0	0
$197$	1.07180	0.0763624	0.0381812	−	0.999271i	$-0.487844\pi$
$197$	1.07180	0.0763624	0.0381812	+	0.999271i	$0.487844\pi$
$198$	0	0
$199$	2.53590	0.179765	0.0898825	−	0.995952i	$-0.471351\pi$
$199$	2.53590	0.179765	0.0898825	+	0.995952i	$0.471351\pi$
$200$	0	0
$201$	−14.9282	−1.05295
$202$	0	0
$203$	−48.2487	−3.38640
$204$	0	0
$205$	11.6603	0.814387
$206$	0	0
$207$	6.00000	0.417029
$208$	0	0
$209$	4.19615	0.290254
$210$	0	0
$211$	6.92820	0.476957	0.238479	−	0.971148i	$-0.423351\pi$
$211$	6.92820	0.476957	0.238479	+	0.971148i	$0.423351\pi$
$212$	0	0
$213$	12.3923	0.849107
$214$	0	0
$215$	4.73205	0.322723
$216$	0	0
$217$	9.46410	0.642465
$218$	0	0
$219$	−0.928203	−0.0627222
$220$	0	0
$221$	8.78461	0.590917
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−23.4641	−1.55737	−0.778684	−	0.627417i	$-0.784112\pi$
$227$	−23.4641	−1.55737	−0.778684	+	0.627417i	$0.784112\pi$
$228$	0	0
$229$	17.4641	1.15406	0.577030	−	0.816723i	$-0.304212\pi$
$229$	17.4641	1.15406	0.577030	+	0.816723i	$0.304212\pi$
$230$	0	0
$231$	19.8564	1.30646
$232$	0	0
$233$	0.143594	0.00940713	0.00470356	−	0.999989i	$-0.498503\pi$
$233$	0.143594	0.00940713	0.00470356	+	0.999989i	$0.498503\pi$
$234$	0	0
$235$	10.0000	0.652328
$236$	0	0
$237$	−8.00000	−0.519656
$238$	0	0
$239$	22.0526	1.42646	0.713231	−	0.700929i	$-0.247231\pi$
$239$	22.0526	1.42646	0.713231	+	0.700929i	$0.247231\pi$
$240$	0	0
$241$	−8.92820	−0.575116	−0.287558	−	0.957763i	$-0.592843\pi$
$241$	−8.92820	−0.575116	−0.287558	+	0.957763i	$0.592843\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	15.3923	0.983378
$246$	0	0
$247$	1.26795	0.0806777
$248$	0	0
$249$	8.53590	0.540941
$250$	0	0
$251$	28.1962	1.77973	0.889863	−	0.456228i	$-0.150800\pi$
$251$	28.1962	1.77973	0.889863	+	0.456228i	$0.150800\pi$
$252$	0	0
$253$	−25.1769	−1.58286
$254$	0	0
$255$	6.92820	0.433861
$256$	0	0
$257$	27.3205	1.70421	0.852103	−	0.523374i	$-0.175327\pi$
$257$	27.3205	1.70421	0.852103	+	0.523374i	$0.175327\pi$
$258$	0	0
$259$	12.9282	0.803319
$260$	0	0
$261$	−10.1962	−0.631126
$262$	0	0
$263$	−3.46410	−0.213606	−0.106803	−	0.994280i	$-0.534061\pi$
$263$	−3.46410	−0.213606	−0.106803	+	0.994280i	$0.534061\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	7.66025	0.468800
$268$	0	0
$269$	−25.5167	−1.55578	−0.777889	−	0.628402i	$-0.783709\pi$
$269$	−25.5167	−1.55578	−0.777889	+	0.628402i	$0.783709\pi$
$270$	0	0
$271$	−23.3205	−1.41662	−0.708310	−	0.705902i	$-0.750542\pi$
$271$	−23.3205	−1.41662	−0.708310	+	0.705902i	$0.750542\pi$
$272$	0	0
$273$	6.00000	0.363137
$274$	0	0
$275$	−4.19615	−0.253038
$276$	0	0
$277$	11.8564	0.712382	0.356191	−	0.934413i	$-0.384075\pi$
$277$	11.8564	0.712382	0.356191	+	0.934413i	$0.384075\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	−8.05256	−0.480375	−0.240188	−	0.970726i	$-0.577209\pi$
$281$	−8.05256	−0.480375	−0.240188	+	0.970726i	$0.577209\pi$
$282$	0	0
$283$	13.8038	0.820554	0.410277	−	0.911961i	$-0.365432\pi$
$283$	13.8038	0.820554	0.410277	+	0.911961i	$0.365432\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	0	0
$287$	55.1769	3.25699
$288$	0	0
$289$	31.0000	1.82353
$290$	0	0
$291$	−1.66025	−0.0973258
$292$	0	0
$293$	−24.3923	−1.42501	−0.712507	−	0.701665i	$-0.752440\pi$
$293$	−24.3923	−1.42501	−0.712507	+	0.701665i	$0.752440\pi$
$294$	0	0
$295$	9.46410	0.551021
$296$	0	0
$297$	4.19615	0.243485
$298$	0	0
$299$	−7.60770	−0.439964
$300$	0	0
$301$	22.3923	1.29067
$302$	0	0
$303$	3.46410	0.199007
$304$	0	0
$305$	5.46410	0.312874
$306$	0	0
$307$	−7.32051	−0.417803	−0.208902	−	0.977937i	$-0.566989\pi$
$307$	−7.32051	−0.417803	−0.208902	+	0.977937i	$0.566989\pi$
$308$	0	0
$309$	−13.8564	−0.788263
$310$	0	0
$311$	−21.2679	−1.20599	−0.602997	−	0.797743i	$-0.706027\pi$
$311$	−21.2679	−1.20599	−0.602997	+	0.797743i	$0.706027\pi$
$312$	0	0
$313$	−13.3205	−0.752920	−0.376460	−	0.926433i	$-0.622859\pi$
$313$	−13.3205	−0.752920	−0.376460	+	0.926433i	$0.622859\pi$
$314$	0	0
$315$	4.73205	0.266621
$316$	0	0
$317$	1.07180	0.0601981	0.0300991	−	0.999547i	$-0.490418\pi$
$317$	1.07180	0.0601981	0.0300991	+	0.999547i	$0.490418\pi$
$318$	0	0
$319$	42.7846	2.39548
$320$	0	0
$321$	−10.9282	−0.609953
$322$	0	0
$323$	6.92820	0.385496
$324$	0	0
$325$	−1.26795	−0.0703332
$326$	0	0
$327$	10.3923	0.574696
$328$	0	0
$329$	47.3205	2.60886
$330$	0	0
$331$	16.9282	0.930458	0.465229	−	0.885190i	$-0.345972\pi$
$331$	16.9282	0.930458	0.465229	+	0.885190i	$0.345972\pi$
$332$	0	0
$333$	2.73205	0.149715
$334$	0	0
$335$	14.9282	0.815615
$336$	0	0
$337$	−2.73205	−0.148824	−0.0744121	−	0.997228i	$-0.523708\pi$
$337$	−2.73205	−0.148824	−0.0744121	+	0.997228i	$0.523708\pi$
$338$	0	0
$339$	−2.53590	−0.137731
$340$	0	0
$341$	−8.39230	−0.454469
$342$	0	0
$343$	39.7128	2.14429
$344$	0	0
$345$	−6.00000	−0.323029
$346$	0	0
$347$	3.46410	0.185963	0.0929814	−	0.995668i	$-0.470360\pi$
$347$	3.46410	0.185963	0.0929814	+	0.995668i	$0.470360\pi$
$348$	0	0
$349$	29.7128	1.59049	0.795245	−	0.606288i	$-0.207342\pi$
$349$	29.7128	1.59049	0.795245	+	0.606288i	$0.207342\pi$
$350$	0	0
$351$	1.26795	0.0676781
$352$	0	0
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	0	0
$355$	−12.3923	−0.657715
$356$	0	0
$357$	32.7846	1.73515
$358$	0	0
$359$	−0.588457	−0.0310576	−0.0155288	−	0.999879i	$-0.504943\pi$
$359$	−0.588457	−0.0310576	−0.0155288	+	0.999879i	$0.504943\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−6.60770	−0.346814
$364$	0	0
$365$	0.928203	0.0485844
$366$	0	0
$367$	12.7321	0.664608	0.332304	−	0.943172i	$-0.392174\pi$
$367$	12.7321	0.664608	0.332304	+	0.943172i	$0.392174\pi$
$368$	0	0
$369$	11.6603	0.607009
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.5885	0.858918	0.429459	−	0.903086i	$-0.358704\pi$
$373$	16.5885	0.858918	0.429459	+	0.903086i	$0.358704\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	12.9282	0.665836
$378$	0	0
$379$	−23.8564	−1.22542	−0.612711	−	0.790307i	$-0.709921\pi$
$379$	−23.8564	−1.22542	−0.612711	+	0.790307i	$0.709921\pi$
$380$	0	0
$381$	−1.07180	−0.0549098
$382$	0	0
$383$	−34.6410	−1.77007	−0.885037	−	0.465521i	$-0.845867\pi$
$383$	−34.6410	−1.77007	−0.885037	+	0.465521i	$0.845867\pi$
$384$	0	0
$385$	−19.8564	−1.01198
$386$	0	0
$387$	4.73205	0.240544
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	−41.5692	−2.10225
$392$	0	0
$393$	−20.5885	−1.03855
$394$	0	0
$395$	8.00000	0.402524
$396$	0	0
$397$	−16.9282	−0.849602	−0.424801	−	0.905287i	$-0.639656\pi$
$397$	−16.9282	−0.849602	−0.424801	+	0.905287i	$0.639656\pi$
$398$	0	0
$399$	4.73205	0.236899
$400$	0	0
$401$	4.33975	0.216717	0.108358	−	0.994112i	$-0.465441\pi$
$401$	4.33975	0.216717	0.108358	+	0.994112i	$0.465441\pi$
$402$	0	0
$403$	−2.53590	−0.126322
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−11.4641	−0.568254
$408$	0	0
$409$	−0.535898	−0.0264985	−0.0132492	−	0.999912i	$-0.504217\pi$
$409$	−0.535898	−0.0264985	−0.0132492	+	0.999912i	$0.504217\pi$
$410$	0	0
$411$	−22.7846	−1.12388
$412$	0	0
$413$	44.7846	2.20371
$414$	0	0
$415$	−8.53590	−0.419011
$416$	0	0
$417$	−17.4641	−0.855221
$418$	0	0
$419$	−1.26795	−0.0619434	−0.0309717	−	0.999520i	$-0.509860\pi$
$419$	−1.26795	−0.0619434	−0.0309717	+	0.999520i	$0.509860\pi$
$420$	0	0
$421$	−15.0718	−0.734554	−0.367277	−	0.930112i	$-0.619710\pi$
$421$	−15.0718	−0.734554	−0.367277	+	0.930112i	$0.619710\pi$
$422$	0	0
$423$	10.0000	0.486217
$424$	0	0
$425$	−6.92820	−0.336067
$426$	0	0
$427$	25.8564	1.25128
$428$	0	0
$429$	−5.32051	−0.256877
$430$	0	0
$431$	34.2487	1.64970	0.824851	−	0.565350i	$-0.191259\pi$
$431$	34.2487	1.64970	0.824851	+	0.565350i	$0.191259\pi$
$432$	0	0
$433$	10.0526	0.483095	0.241548	−	0.970389i	$-0.422345\pi$
$433$	10.0526	0.483095	0.241548	+	0.970389i	$0.422345\pi$
$434$	0	0
$435$	10.1962	0.488868
$436$	0	0
$437$	−6.00000	−0.287019
$438$	0	0
$439$	−5.85641	−0.279511	−0.139756	−	0.990186i	$-0.544632\pi$
$439$	−5.85641	−0.279511	−0.139756	+	0.990186i	$0.544632\pi$
$440$	0	0
$441$	15.3923	0.732967
$442$	0	0
$443$	−7.07180	−0.335991	−0.167996	−	0.985788i	$-0.553729\pi$
$443$	−7.07180	−0.335991	−0.167996	+	0.985788i	$0.553729\pi$
$444$	0	0
$445$	−7.66025	−0.363131
$446$	0	0
$447$	−8.92820	−0.422290
$448$	0	0
$449$	9.80385	0.462672	0.231336	−	0.972874i	$-0.425690\pi$
$449$	9.80385	0.462672	0.231336	+	0.972874i	$0.425690\pi$
$450$	0	0
$451$	−48.9282	−2.30394
$452$	0	0
$453$	−2.00000	−0.0939682
$454$	0	0
$455$	−6.00000	−0.281284
$456$	0	0
$457$	−19.4641	−0.910492	−0.455246	−	0.890366i	$-0.650449\pi$
$457$	−19.4641	−0.910492	−0.455246	+	0.890366i	$0.650449\pi$
$458$	0	0
$459$	6.92820	0.323381
$460$	0	0
$461$	10.7846	0.502289	0.251145	−	0.967950i	$-0.419193\pi$
$461$	10.7846	0.502289	0.251145	+	0.967950i	$0.419193\pi$
$462$	0	0
$463$	−17.1244	−0.795836	−0.397918	−	0.917421i	$-0.630267\pi$
$463$	−17.1244	−0.795836	−0.397918	+	0.917421i	$0.630267\pi$
$464$	0	0
$465$	−2.00000	−0.0927478
$466$	0	0
$467$	−19.0718	−0.882538	−0.441269	−	0.897375i	$-0.645471\pi$
$467$	−19.0718	−0.882538	−0.441269	+	0.897375i	$0.645471\pi$
$468$	0	0
$469$	70.6410	3.26190
$470$	0	0
$471$	3.46410	0.159617
$472$	0	0
$473$	−19.8564	−0.912999
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.41154	−0.155877	−0.0779387	−	0.996958i	$-0.524834\pi$
$479$	−3.41154	−0.155877	−0.0779387	+	0.996958i	$0.524834\pi$
$480$	0	0
$481$	−3.46410	−0.157949
$482$	0	0
$483$	−28.3923	−1.29189
$484$	0	0
$485$	1.66025	0.0753883
$486$	0	0
$487$	−20.3923	−0.924064	−0.462032	−	0.886863i	$-0.652879\pi$
$487$	−20.3923	−0.924064	−0.462032	+	0.886863i	$0.652879\pi$
$488$	0	0
$489$	25.1244	1.13616
$490$	0	0
$491$	8.19615	0.369887	0.184944	−	0.982749i	$-0.440790\pi$
$491$	8.19615	0.369887	0.184944	+	0.982749i	$0.440790\pi$
$492$	0	0
$493$	70.6410	3.18151
$494$	0	0
$495$	−4.19615	−0.188603
$496$	0	0
$497$	−58.6410	−2.63041
$498$	0	0
$499$	7.32051	0.327711	0.163855	−	0.986484i	$-0.447607\pi$
$499$	7.32051	0.327711	0.163855	+	0.986484i	$0.447607\pi$
$500$	0	0
$501$	14.3923	0.643001
$502$	0	0
$503$	−39.1769	−1.74681	−0.873406	−	0.486993i	$-0.838094\pi$
$503$	−39.1769	−1.74681	−0.873406	+	0.486993i	$0.838094\pi$
$504$	0	0
$505$	−3.46410	−0.154150
$506$	0	0
$507$	11.3923	0.505950
$508$	0	0
$509$	6.19615	0.274640	0.137320	−	0.990527i	$-0.456151\pi$
$509$	6.19615	0.274640	0.137320	+	0.990527i	$0.456151\pi$
$510$	0	0
$511$	4.39230	0.194304
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	13.8564	0.610586
$516$	0	0
$517$	−41.9615	−1.84547
$518$	0	0
$519$	−12.3923	−0.543962
$520$	0	0
$521$	14.8756	0.651714	0.325857	−	0.945419i	$-0.394347\pi$
$521$	14.8756	0.651714	0.325857	+	0.945419i	$0.394347\pi$
$522$	0	0
$523$	6.53590	0.285795	0.142897	−	0.989738i	$-0.454358\pi$
$523$	6.53590	0.285795	0.142897	+	0.989738i	$0.454358\pi$
$524$	0	0
$525$	−4.73205	−0.206524
$526$	0	0
$527$	−13.8564	−0.603595
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	9.46410	0.410707
$532$	0	0
$533$	−14.7846	−0.640393
$534$	0	0
$535$	10.9282	0.472467
$536$	0	0
$537$	21.4641	0.926244
$538$	0	0
$539$	−64.5885	−2.78202
$540$	0	0
$541$	−21.7128	−0.933507	−0.466753	−	0.884388i	$-0.654576\pi$
$541$	−21.7128	−0.933507	−0.466753	+	0.884388i	$0.654576\pi$
$542$	0	0
$543$	−4.53590	−0.194654
$544$	0	0
$545$	−10.3923	−0.445157
$546$	0	0
$547$	18.2487	0.780259	0.390129	−	0.920760i	$-0.372430\pi$
$547$	18.2487	0.780259	0.390129	+	0.920760i	$0.372430\pi$
$548$	0	0
$549$	5.46410	0.233202
$550$	0	0
$551$	10.1962	0.434371
$552$	0	0
$553$	37.8564	1.60982
$554$	0	0
$555$	−2.73205	−0.115969
$556$	0	0
$557$	19.8564	0.841343	0.420671	−	0.907213i	$-0.361795\pi$
$557$	19.8564	0.841343	0.420671	+	0.907213i	$0.361795\pi$
$558$	0	0
$559$	−6.00000	−0.253773
$560$	0	0
$561$	−29.0718	−1.22741
$562$	0	0
$563$	−17.6077	−0.742076	−0.371038	−	0.928618i	$-0.620998\pi$
$563$	−17.6077	−0.742076	−0.371038	+	0.928618i	$0.620998\pi$
$564$	0	0
$565$	2.53590	0.106686
$566$	0	0
$567$	4.73205	0.198727
$568$	0	0
$569$	9.80385	0.410999	0.205499	−	0.978657i	$-0.434118\pi$
$569$	9.80385	0.410999	0.205499	+	0.978657i	$0.434118\pi$
$570$	0	0
$571$	−7.60770	−0.318372	−0.159186	−	0.987249i	$-0.550887\pi$
$571$	−7.60770	−0.318372	−0.159186	+	0.987249i	$0.550887\pi$
$572$	0	0
$573$	−13.2679	−0.554277
$574$	0	0
$575$	6.00000	0.250217
$576$	0	0
$577$	25.3205	1.05411	0.527053	−	0.849832i	$-0.323297\pi$
$577$	25.3205	1.05411	0.527053	+	0.849832i	$0.323297\pi$
$578$	0	0
$579$	11.8038	0.490551
$580$	0	0
$581$	−40.3923	−1.67576
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−1.26795	−0.0524232
$586$	0	0
$587$	−22.7846	−0.940421	−0.470211	−	0.882554i	$-0.655822\pi$
$587$	−22.7846	−0.940421	−0.470211	+	0.882554i	$0.655822\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	−1.07180	−0.0440878
$592$	0	0
$593$	2.78461	0.114350	0.0571751	−	0.998364i	$-0.481791\pi$
$593$	2.78461	0.114350	0.0571751	+	0.998364i	$0.481791\pi$
$594$	0	0
$595$	−32.7846	−1.34404
$596$	0	0
$597$	−2.53590	−0.103787
$598$	0	0
$599$	28.0000	1.14405	0.572024	−	0.820237i	$-0.306158\pi$
$599$	28.0000	1.14405	0.572024	+	0.820237i	$0.306158\pi$
$600$	0	0
$601$	−26.3923	−1.07656	−0.538282	−	0.842765i	$-0.680927\pi$
$601$	−26.3923	−1.07656	−0.538282	+	0.842765i	$0.680927\pi$
$602$	0	0
$603$	14.9282	0.607923
$604$	0	0
$605$	6.60770	0.268641
$606$	0	0
$607$	10.5359	0.427639	0.213819	−	0.976873i	$-0.431410\pi$
$607$	10.5359	0.427639	0.213819	+	0.976873i	$0.431410\pi$
$608$	0	0
$609$	48.2487	1.95514
$610$	0	0
$611$	−12.6795	−0.512957
$612$	0	0
$613$	−27.1769	−1.09767	−0.548833	−	0.835932i	$-0.684928\pi$
$613$	−27.1769	−1.09767	−0.548833	+	0.835932i	$0.684928\pi$
$614$	0	0
$615$	−11.6603	−0.470187
$616$	0	0
$617$	4.78461	0.192621	0.0963106	−	0.995351i	$-0.469296\pi$
$617$	4.78461	0.192621	0.0963106	+	0.995351i	$0.469296\pi$
$618$	0	0
$619$	41.1769	1.65504	0.827520	−	0.561436i	$-0.189751\pi$
$619$	41.1769	1.65504	0.827520	+	0.561436i	$0.189751\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	0	0
$623$	−36.2487	−1.45227
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.19615	−0.167578
$628$	0	0
$629$	−18.9282	−0.754717
$630$	0	0
$631$	19.7128	0.784755	0.392377	−	0.919804i	$-0.371653\pi$
$631$	19.7128	0.784755	0.392377	+	0.919804i	$0.371653\pi$
$632$	0	0
$633$	−6.92820	−0.275371
$634$	0	0
$635$	1.07180	0.0425330
$636$	0	0
$637$	−19.5167	−0.773278
$638$	0	0
$639$	−12.3923	−0.490232
$640$	0	0
$641$	−37.5167	−1.48182	−0.740909	−	0.671605i	$-0.765605\pi$
$641$	−37.5167	−1.48182	−0.740909	+	0.671605i	$0.765605\pi$
$642$	0	0
$643$	30.9808	1.22176	0.610881	−	0.791722i	$-0.290815\pi$
$643$	30.9808	1.22176	0.610881	+	0.791722i	$0.290815\pi$
$644$	0	0
$645$	−4.73205	−0.186324
$646$	0	0
$647$	6.67949	0.262598	0.131299	−	0.991343i	$-0.458085\pi$
$647$	6.67949	0.262598	0.131299	+	0.991343i	$0.458085\pi$
$648$	0	0
$649$	−39.7128	−1.55886
$650$	0	0
$651$	−9.46410	−0.370927
$652$	0	0
$653$	−7.21539	−0.282360	−0.141180	−	0.989984i	$-0.545090\pi$
$653$	−7.21539	−0.282360	−0.141180	+	0.989984i	$0.545090\pi$
$654$	0	0
$655$	20.5885	0.804458
$656$	0	0
$657$	0.928203	0.0362127
$658$	0	0
$659$	−43.7128	−1.70281	−0.851405	−	0.524509i	$-0.824249\pi$
$659$	−43.7128	−1.70281	−0.851405	+	0.524509i	$0.824249\pi$
$660$	0	0
$661$	34.3923	1.33771	0.668853	−	0.743395i	$-0.266786\pi$
$661$	34.3923	1.33771	0.668853	+	0.743395i	$0.266786\pi$
$662$	0	0
$663$	−8.78461	−0.341166
$664$	0	0
$665$	−4.73205	−0.183501
$666$	0	0
$667$	−61.1769	−2.36878
$668$	0	0
$669$	8.00000	0.309298
$670$	0	0
$671$	−22.9282	−0.885133
$672$	0	0
$673$	−39.5167	−1.52326	−0.761628	−	0.648015i	$-0.775599\pi$
$673$	−39.5167	−1.52326	−0.761628	+	0.648015i	$0.775599\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−40.0000	−1.53732	−0.768662	−	0.639655i	$-0.779077\pi$
$677$	−40.0000	−1.53732	−0.768662	+	0.639655i	$0.779077\pi$
$678$	0	0
$679$	7.85641	0.301501
$680$	0	0
$681$	23.4641	0.899146
$682$	0	0
$683$	8.00000	0.306111	0.153056	−	0.988218i	$-0.451089\pi$
$683$	8.00000	0.306111	0.153056	+	0.988218i	$0.451089\pi$
$684$	0	0
$685$	22.7846	0.870555
$686$	0	0
$687$	−17.4641	−0.666297
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−8.39230	−0.319258	−0.159629	−	0.987177i	$-0.551030\pi$
$691$	−8.39230	−0.319258	−0.159629	+	0.987177i	$0.551030\pi$
$692$	0	0
$693$	−19.8564	−0.754283
$694$	0	0
$695$	17.4641	0.662451
$696$	0	0
$697$	−80.7846	−3.05994
$698$	0	0
$699$	−0.143594	−0.00543121
$700$	0	0
$701$	24.9282	0.941525	0.470763	−	0.882260i	$-0.343979\pi$
$701$	24.9282	0.941525	0.470763	+	0.882260i	$0.343979\pi$
$702$	0	0
$703$	−2.73205	−0.103041
$704$	0	0
$705$	−10.0000	−0.376622
$706$	0	0
$707$	−16.3923	−0.616496
$708$	0	0
$709$	38.2487	1.43646	0.718230	−	0.695806i	$-0.244952\pi$
$709$	38.2487	1.43646	0.718230	+	0.695806i	$0.244952\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	12.0000	0.449404
$714$	0	0
$715$	5.32051	0.198976
$716$	0	0
$717$	−22.0526	−0.823568
$718$	0	0
$719$	28.9808	1.08080	0.540400	−	0.841408i	$-0.318273\pi$
$719$	28.9808	1.08080	0.540400	+	0.841408i	$0.318273\pi$
$720$	0	0
$721$	65.5692	2.44193
$722$	0	0
$723$	8.92820	0.332043
$724$	0	0
$725$	−10.1962	−0.378676
$726$	0	0
$727$	1.41154	0.0523512	0.0261756	−	0.999657i	$-0.491667\pi$
$727$	1.41154	0.0523512	0.0261756	+	0.999657i	$0.491667\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−32.7846	−1.21258
$732$	0	0
$733$	−34.7846	−1.28480	−0.642399	−	0.766370i	$-0.722061\pi$
$733$	−34.7846	−1.28480	−0.642399	+	0.766370i	$0.722061\pi$
$734$	0	0
$735$	−15.3923	−0.567754
$736$	0	0
$737$	−62.6410	−2.30741
$738$	0	0
$739$	22.9282	0.843428	0.421714	−	0.906729i	$-0.361429\pi$
$739$	22.9282	0.843428	0.421714	+	0.906729i	$0.361429\pi$
$740$	0	0
$741$	−1.26795	−0.0465793
$742$	0	0
$743$	26.6410	0.977364	0.488682	−	0.872462i	$-0.337478\pi$
$743$	26.6410	0.977364	0.488682	+	0.872462i	$0.337478\pi$
$744$	0	0
$745$	8.92820	0.327104
$746$	0	0
$747$	−8.53590	−0.312312
$748$	0	0
$749$	51.7128	1.88955
$750$	0	0
$751$	8.14359	0.297164	0.148582	−	0.988900i	$-0.452529\pi$
$751$	8.14359	0.297164	0.148582	+	0.988900i	$0.452529\pi$
$752$	0	0
$753$	−28.1962	−1.02752
$754$	0	0
$755$	2.00000	0.0727875
$756$	0	0
$757$	−11.4641	−0.416670	−0.208335	−	0.978058i	$-0.566804\pi$
$757$	−11.4641	−0.416670	−0.208335	+	0.978058i	$0.566804\pi$
$758$	0	0
$759$	25.1769	0.913864
$760$	0	0
$761$	−27.0718	−0.981352	−0.490676	−	0.871342i	$-0.663250\pi$
$761$	−27.0718	−0.981352	−0.490676	+	0.871342i	$0.663250\pi$
$762$	0	0
$763$	−49.1769	−1.78032
$764$	0	0
$765$	−6.92820	−0.250490
$766$	0	0
$767$	−12.0000	−0.433295
$768$	0	0
$769$	−38.7846	−1.39861	−0.699304	−	0.714824i	$-0.746507\pi$
$769$	−38.7846	−1.39861	−0.699304	+	0.714824i	$0.746507\pi$
$770$	0	0
$771$	−27.3205	−0.983924
$772$	0	0
$773$	1.85641	0.0667703	0.0333851	−	0.999443i	$-0.489371\pi$
$773$	1.85641	0.0667703	0.0333851	+	0.999443i	$0.489371\pi$
$774$	0	0
$775$	2.00000	0.0718421
$776$	0	0
$777$	−12.9282	−0.463797
$778$	0	0
$779$	−11.6603	−0.417772
$780$	0	0
$781$	52.0000	1.86071
$782$	0	0
$783$	10.1962	0.364381
$784$	0	0
$785$	−3.46410	−0.123639
$786$	0	0
$787$	−1.46410	−0.0521896	−0.0260948	−	0.999659i	$-0.508307\pi$
$787$	−1.46410	−0.0521896	−0.0260948	+	0.999659i	$0.508307\pi$
$788$	0	0
$789$	3.46410	0.123325
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	−6.92820	−0.246028
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.24871	−0.0796534	−0.0398267	−	0.999207i	$-0.512681\pi$
$797$	−2.24871	−0.0796534	−0.0398267	+	0.999207i	$0.512681\pi$
$798$	0	0
$799$	−69.2820	−2.45102
$800$	0	0
$801$	−7.66025	−0.270662
$802$	0	0
$803$	−3.89488	−0.137447
$804$	0	0
$805$	28.3923	1.00070
$806$	0	0
$807$	25.5167	0.898229
$808$	0	0
$809$	42.4974	1.49413	0.747065	−	0.664751i	$-0.231462\pi$
$809$	42.4974	1.49413	0.747065	+	0.664751i	$0.231462\pi$
$810$	0	0
$811$	26.6410	0.935493	0.467746	−	0.883863i	$-0.345066\pi$
$811$	26.6410	0.935493	0.467746	+	0.883863i	$0.345066\pi$
$812$	0	0
$813$	23.3205	0.817886
$814$	0	0
$815$	−25.1244	−0.880068
$816$	0	0
$817$	−4.73205	−0.165554
$818$	0	0
$819$	−6.00000	−0.209657
$820$	0	0
$821$	−21.3205	−0.744091	−0.372045	−	0.928215i	$-0.621343\pi$
$821$	−21.3205	−0.744091	−0.372045	+	0.928215i	$0.621343\pi$
$822$	0	0
$823$	−10.9808	−0.382765	−0.191383	−	0.981516i	$-0.561297\pi$
$823$	−10.9808	−0.382765	−0.191383	+	0.981516i	$0.561297\pi$
$824$	0	0
$825$	4.19615	0.146091
$826$	0	0
$827$	−26.1051	−0.907764	−0.453882	−	0.891062i	$-0.649961\pi$
$827$	−26.1051	−0.907764	−0.453882	+	0.891062i	$0.649961\pi$
$828$	0	0
$829$	17.6077	0.611541	0.305770	−	0.952105i	$-0.401086\pi$
$829$	17.6077	0.611541	0.305770	+	0.952105i	$0.401086\pi$
$830$	0	0
$831$	−11.8564	−0.411294
$832$	0	0
$833$	−106.641	−3.69489
$834$	0	0
$835$	−14.3923	−0.498066
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	0	0
$839$	−25.4641	−0.879118	−0.439559	−	0.898214i	$-0.644865\pi$
$839$	−25.4641	−0.879118	−0.439559	+	0.898214i	$0.644865\pi$
$840$	0	0
$841$	74.9615	2.58488
$842$	0	0
$843$	8.05256	0.277345
$844$	0	0
$845$	−11.3923	−0.391907
$846$	0	0
$847$	31.2679	1.07438
$848$	0	0
$849$	−13.8038	−0.473747
$850$	0	0
$851$	16.3923	0.561921
$852$	0	0
$853$	−24.5359	−0.840093	−0.420047	−	0.907503i	$-0.637986\pi$
$853$	−24.5359	−0.840093	−0.420047	+	0.907503i	$0.637986\pi$
$854$	0	0
$855$	−1.00000	−0.0341993
$856$	0	0
$857$	−21.0718	−0.719799	−0.359899	−	0.932991i	$-0.617189\pi$
$857$	−21.0718	−0.719799	−0.359899	+	0.932991i	$0.617189\pi$
$858$	0	0
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	0	0
$861$	−55.1769	−1.88042
$862$	0	0
$863$	−6.14359	−0.209130	−0.104565	−	0.994518i	$-0.533345\pi$
$863$	−6.14359	−0.209130	−0.104565	+	0.994518i	$0.533345\pi$
$864$	0	0
$865$	12.3923	0.421351
$866$	0	0
$867$	−31.0000	−1.05282
$868$	0	0
$869$	−33.5692	−1.13876
$870$	0	0
$871$	−18.9282	−0.641358
$872$	0	0
$873$	1.66025	0.0561911
$874$	0	0
$875$	4.73205	0.159973
$876$	0	0
$877$	−35.8038	−1.20901	−0.604505	−	0.796601i	$-0.706629\pi$
$877$	−35.8038	−1.20901	−0.604505	+	0.796601i	$0.706629\pi$
$878$	0	0
$879$	24.3923	0.822732
$880$	0	0
$881$	−4.24871	−0.143143	−0.0715714	−	0.997435i	$-0.522801\pi$
$881$	−4.24871	−0.143143	−0.0715714	+	0.997435i	$0.522801\pi$
$882$	0	0
$883$	−35.2679	−1.18686	−0.593430	−	0.804885i	$-0.702227\pi$
$883$	−35.2679	−1.18686	−0.593430	+	0.804885i	$0.702227\pi$
$884$	0	0
$885$	−9.46410	−0.318132
$886$	0	0
$887$	5.07180	0.170294	0.0851471	−	0.996368i	$-0.472864\pi$
$887$	5.07180	0.170294	0.0851471	+	0.996368i	$0.472864\pi$
$888$	0	0
$889$	5.07180	0.170103
$890$	0	0
$891$	−4.19615	−0.140576
$892$	0	0
$893$	−10.0000	−0.334637
$894$	0	0
$895$	−21.4641	−0.717466
$896$	0	0
$897$	7.60770	0.254014
$898$	0	0
$899$	−20.3923	−0.680121
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−22.3923	−0.745169
$904$	0	0
$905$	4.53590	0.150778
$906$	0	0
$907$	47.0333	1.56172	0.780858	−	0.624709i	$-0.214782\pi$
$907$	47.0333	1.56172	0.780858	+	0.624709i	$0.214782\pi$
$908$	0	0
$909$	−3.46410	−0.114897
$910$	0	0
$911$	−17.0718	−0.565614	−0.282807	−	0.959177i	$-0.591266\pi$
$911$	−17.0718	−0.565614	−0.282807	+	0.959177i	$0.591266\pi$
$912$	0	0
$913$	35.8179	1.18540
$914$	0	0
$915$	−5.46410	−0.180638
$916$	0	0
$917$	97.4256	3.21728
$918$	0	0
$919$	22.9282	0.756332	0.378166	−	0.925738i	$-0.376555\pi$
$919$	22.9282	0.756332	0.378166	+	0.925738i	$0.376555\pi$
$920$	0	0
$921$	7.32051	0.241219
$922$	0	0
$923$	15.7128	0.517194
$924$	0	0
$925$	2.73205	0.0898293
$926$	0	0
$927$	13.8564	0.455104
$928$	0	0
$929$	60.2487	1.97670	0.988348	−	0.152211i	$-0.0486393\pi$
$929$	60.2487	1.97670	0.988348	+	0.152211i	$0.0486393\pi$
$930$	0	0
$931$	−15.3923	−0.504462
$932$	0	0
$933$	21.2679	0.696281
$934$	0	0
$935$	29.0718	0.950749
$936$	0	0
$937$	43.1769	1.41053	0.705264	−	0.708945i	$-0.250828\pi$
$937$	43.1769	1.41053	0.705264	+	0.708945i	$0.250828\pi$
$938$	0	0
$939$	13.3205	0.434698
$940$	0	0
$941$	−15.9474	−0.519872	−0.259936	−	0.965626i	$-0.583701\pi$
$941$	−15.9474	−0.519872	−0.259936	+	0.965626i	$0.583701\pi$
$942$	0	0
$943$	69.9615	2.27826
$944$	0	0
$945$	−4.73205	−0.153934
$946$	0	0
$947$	18.6795	0.607002	0.303501	−	0.952831i	$-0.401844\pi$
$947$	18.6795	0.607002	0.303501	+	0.952831i	$0.401844\pi$
$948$	0	0
$949$	−1.17691	−0.0382043
$950$	0	0
$951$	−1.07180	−0.0347554
$952$	0	0
$953$	17.1769	0.556415	0.278207	−	0.960521i	$-0.410260\pi$
$953$	17.1769	0.556415	0.278207	+	0.960521i	$0.410260\pi$
$954$	0	0
$955$	13.2679	0.429341
$956$	0	0
$957$	−42.7846	−1.38303
$958$	0	0
$959$	107.818	3.48162
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	10.9282	0.352156
$964$	0	0
$965$	−11.8038	−0.379979
$966$	0	0
$967$	46.9808	1.51080	0.755400	−	0.655264i	$-0.227443\pi$
$967$	46.9808	1.51080	0.755400	+	0.655264i	$0.227443\pi$
$968$	0	0
$969$	−6.92820	−0.222566
$970$	0	0
$971$	−60.4974	−1.94145	−0.970727	−	0.240184i	$-0.922792\pi$
$971$	−60.4974	−1.94145	−0.970727	+	0.240184i	$0.922792\pi$
$972$	0	0
$973$	82.6410	2.64935
$974$	0	0
$975$	1.26795	0.0406069
$976$	0	0
$977$	17.8564	0.571277	0.285639	−	0.958337i	$-0.407794\pi$
$977$	17.8564	0.571277	0.285639	+	0.958337i	$0.407794\pi$
$978$	0	0
$979$	32.1436	1.02731
$980$	0	0
$981$	−10.3923	−0.331801
$982$	0	0
$983$	−33.6077	−1.07192	−0.535960	−	0.844244i	$-0.680050\pi$
$983$	−33.6077	−1.07192	−0.535960	+	0.844244i	$0.680050\pi$
$984$	0	0
$985$	1.07180	0.0341503
$986$	0	0
$987$	−47.3205	−1.50623
$988$	0	0
$989$	28.3923	0.902823
$990$	0	0
$991$	16.7846	0.533181	0.266590	−	0.963810i	$-0.414103\pi$
$991$	16.7846	0.533181	0.266590	+	0.963810i	$0.414103\pi$
$992$	0	0
$993$	−16.9282	−0.537200
$994$	0	0
$995$	2.53590	0.0803934
$996$	0	0
$997$	−37.6077	−1.19105	−0.595524	−	0.803338i	$-0.703055\pi$
$997$	−37.6077	−1.19105	−0.595524	+	0.803338i	$0.703055\pi$
$998$	0	0
$999$	−2.73205	−0.0864383

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.a.bi.1.2		2
4.3	odd	2		2280.2.a.q.1.1	✓	2
12.11	even	2		6840.2.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.q.1.1	✓	2	4.3	odd	2
4560.2.a.bi.1.2		2	1.1	even	1	trivial
6840.2.a.v.1.1		2	12.11	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +4.73205 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +4.73205 q^{7} +1.00000 q^{9} -4.19615 q^{11} -1.26795 q^{13} -1.00000 q^{15} -6.92820 q^{17} -1.00000 q^{19} -4.73205 q^{21} +6.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -10.1962 q^{29} +2.00000 q^{31} +4.19615 q^{33} +4.73205 q^{35} +2.73205 q^{37} +1.26795 q^{39} +11.6603 q^{41} +4.73205 q^{43} +1.00000 q^{45} +10.0000 q^{47} +15.3923 q^{49} +6.92820 q^{51} -4.19615 q^{55} +1.00000 q^{57} +9.46410 q^{59} +5.46410 q^{61} +4.73205 q^{63} -1.26795 q^{65} +14.9282 q^{67} -6.00000 q^{69} -12.3923 q^{71} +0.928203 q^{73} -1.00000 q^{75} -19.8564 q^{77} +8.00000 q^{79} +1.00000 q^{81} -8.53590 q^{83} -6.92820 q^{85} +10.1962 q^{87} -7.66025 q^{89} -6.00000 q^{91} -2.00000 q^{93} -1.00000 q^{95} +1.66025 q^{97} -4.19615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} + 2 q^{11} - 6 q^{13} - 2 q^{15} - 2 q^{19} - 6 q^{21} + 12 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} + 4 q^{31} - 2 q^{33} + 6 q^{35} + 2 q^{37} + 6 q^{39} + 6 q^{41} + 6 q^{43} + 2 q^{45} + 20 q^{47} + 10 q^{49} + 2 q^{55} + 2 q^{57} + 12 q^{59} + 4 q^{61} + 6 q^{63} - 6 q^{65} + 16 q^{67} - 12 q^{69} - 4 q^{71} - 12 q^{73} - 2 q^{75} - 12 q^{77} + 16 q^{79} + 2 q^{81} - 24 q^{83} + 10 q^{87} + 2 q^{89} - 12 q^{91} - 4 q^{93} - 2 q^{95} - 14 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9 + 2 * q^11 - 6 * q^13 - 2 * q^15 - 2 * q^19 - 6 * q^21 + 12 * q^23 + 2 * q^25 - 2 * q^27 - 10 * q^29 + 4 * q^31 - 2 * q^33 + 6 * q^35 + 2 * q^37 + 6 * q^39 + 6 * q^41 + 6 * q^43 + 2 * q^45 + 20 * q^47 + 10 * q^49 + 2 * q^55 + 2 * q^57 + 12 * q^59 + 4 * q^61 + 6 * q^63 - 6 * q^65 + 16 * q^67 - 12 * q^69 - 4 * q^71 - 12 * q^73 - 2 * q^75 - 12 * q^77 + 16 * q^79 + 2 * q^81 - 24 * q^83 + 10 * q^87 + 2 * q^89 - 12 * q^91 - 4 * q^93 - 2 * q^95 - 14 * q^97 + 2 * q^99

Introduction

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