LMFDB - Embedded newform 4592.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4592,2,Mod(1,4592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4592, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4592.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4592 = 2^{4} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4592.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:	$36.6673046082$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 574)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	4592.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{3} +2.73205 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-2.00000 q^{3} +2.73205 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.73205 q^{11} -5.46410 q^{13} -5.46410 q^{15} +3.46410 q^{19} +2.00000 q^{21} +2.00000 q^{23} +2.46410 q^{25} +4.00000 q^{27} -7.66025 q^{29} -1.46410 q^{31} -5.46410 q^{33} -2.73205 q^{35} +4.92820 q^{37} +10.9282 q^{39} +1.00000 q^{41} -12.3923 q^{43} +2.73205 q^{45} -0.535898 q^{47} +1.00000 q^{49} +7.66025 q^{53} +7.46410 q^{55} -6.92820 q^{57} -13.1244 q^{59} +6.73205 q^{61} -1.00000 q^{63} -14.9282 q^{65} -0.196152 q^{67} -4.00000 q^{69} -2.53590 q^{71} -8.92820 q^{73} -4.92820 q^{75} -2.73205 q^{77} -11.0000 q^{81} -4.73205 q^{83} +15.3205 q^{87} +12.9282 q^{89} +5.46410 q^{91} +2.92820 q^{93} +9.46410 q^{95} +2.00000 q^{97} +2.73205 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 4 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} - 4 q^{13} - 4 q^{15} + 4 q^{21} + 4 q^{23} - 2 q^{25} + 8 q^{27} + 2 q^{29} + 4 q^{31} - 4 q^{33} - 2 q^{35} - 4 q^{37} + 8 q^{39} + 2 q^{41} - 4 q^{43} + 2 q^{45} - 8 q^{47} + 2 q^{49} - 2 q^{53} + 8 q^{55} - 2 q^{59} + 10 q^{61} - 2 q^{63} - 16 q^{65} + 10 q^{67} - 8 q^{69} - 12 q^{71} - 4 q^{73} + 4 q^{75} - 2 q^{77} - 22 q^{81} - 6 q^{83} - 4 q^{87} + 12 q^{89} + 4 q^{91} - 8 q^{93} + 12 q^{95} + 4 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 4 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 - 4 * q^13 - 4 * q^15 + 4 * q^21 + 4 * q^23 - 2 * q^25 + 8 * q^27 + 2 * q^29 + 4 * q^31 - 4 * q^33 - 2 * q^35 - 4 * q^37 + 8 * q^39 + 2 * q^41 - 4 * q^43 + 2 * q^45 - 8 * q^47 + 2 * q^49 - 2 * q^53 + 8 * q^55 - 2 * q^59 + 10 * q^61 - 2 * q^63 - 16 * q^65 + 10 * q^67 - 8 * q^69 - 12 * q^71 - 4 * q^73 + 4 * q^75 - 2 * q^77 - 22 * q^81 - 6 * q^83 - 4 * q^87 + 12 * q^89 + 4 * q^91 - 8 * q^93 + 12 * q^95 + 4 * 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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	0	0
$5$	2.73205	1.22181	0.610905	−	0.791704i	$-0.290806\pi$
$5$	2.73205	1.22181	0.610905	+	0.791704i	$0.290806\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.73205	0.823744	0.411872	−	0.911242i	$-0.364875\pi$
$11$	2.73205	0.823744	0.411872	+	0.911242i	$0.364875\pi$
$12$	0	0
$13$	−5.46410	−1.51547	−0.757735	−	0.652563i	$-0.773694\pi$
$13$	−5.46410	−1.51547	−0.757735	+	0.652563i	$0.773694\pi$
$14$	0	0
$15$	−5.46410	−1.41082
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	3.46410	0.794719	0.397360	−	0.917663i	$-0.369927\pi$
$19$	3.46410	0.794719	0.397360	+	0.917663i	$0.369927\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	2.46410	0.492820
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	−7.66025	−1.42247	−0.711237	−	0.702953i	$-0.751865\pi$
$29$	−7.66025	−1.42247	−0.711237	+	0.702953i	$0.751865\pi$
$30$	0	0
$31$	−1.46410	−0.262960	−0.131480	−	0.991319i	$-0.541973\pi$
$31$	−1.46410	−0.262960	−0.131480	+	0.991319i	$0.541973\pi$
$32$	0	0
$33$	−5.46410	−0.951178
$34$	0	0
$35$	−2.73205	−0.461801
$36$	0	0
$37$	4.92820	0.810192	0.405096	−	0.914274i	$-0.367238\pi$
$37$	4.92820	0.810192	0.405096	+	0.914274i	$0.367238\pi$
$38$	0	0
$39$	10.9282	1.74991
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−12.3923	−1.88981	−0.944904	−	0.327346i	$-0.893846\pi$
$43$	−12.3923	−1.88981	−0.944904	+	0.327346i	$0.893846\pi$
$44$	0	0
$45$	2.73205	0.407270
$46$	0	0
$47$	−0.535898	−0.0781688	−0.0390844	−	0.999236i	$-0.512444\pi$
$47$	−0.535898	−0.0781688	−0.0390844	+	0.999236i	$0.512444\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.66025	1.05222	0.526108	−	0.850418i	$-0.323651\pi$
$53$	7.66025	1.05222	0.526108	+	0.850418i	$0.323651\pi$
$54$	0	0
$55$	7.46410	1.00646
$56$	0	0
$57$	−6.92820	−0.917663
$58$	0	0
$59$	−13.1244	−1.70865	−0.854323	−	0.519743i	$-0.826028\pi$
$59$	−13.1244	−1.70865	−0.854323	+	0.519743i	$0.826028\pi$
$60$	0	0
$61$	6.73205	0.861951	0.430975	−	0.902364i	$-0.358170\pi$
$61$	6.73205	0.861951	0.430975	+	0.902364i	$0.358170\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−14.9282	−1.85162
$66$	0	0
$67$	−0.196152	−0.0239638	−0.0119819	−	0.999928i	$-0.503814\pi$
$67$	−0.196152	−0.0239638	−0.0119819	+	0.999928i	$0.503814\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−2.53590	−0.300956	−0.150478	−	0.988613i	$-0.548081\pi$
$71$	−2.53590	−0.300956	−0.150478	+	0.988613i	$0.548081\pi$
$72$	0	0
$73$	−8.92820	−1.04497	−0.522484	−	0.852649i	$-0.674994\pi$
$73$	−8.92820	−1.04497	−0.522484	+	0.852649i	$0.674994\pi$
$74$	0	0
$75$	−4.92820	−0.569060
$76$	0	0
$77$	−2.73205	−0.311346
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−4.73205	−0.519410	−0.259705	−	0.965688i	$-0.583625\pi$
$83$	−4.73205	−0.519410	−0.259705	+	0.965688i	$0.583625\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	15.3205	1.64253
$88$	0	0
$89$	12.9282	1.37039	0.685193	−	0.728361i	$-0.259718\pi$
$89$	12.9282	1.37039	0.685193	+	0.728361i	$0.259718\pi$
$90$	0	0
$91$	5.46410	0.572793
$92$	0	0
$93$	2.92820	0.303641
$94$	0	0
$95$	9.46410	0.970996
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	2.73205	0.274581
$100$	0	0
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	0	0
$103$	−5.46410	−0.538394	−0.269197	−	0.963085i	$-0.586758\pi$
$103$	−5.46410	−0.538394	−0.269197	+	0.963085i	$0.586758\pi$
$104$	0	0
$105$	5.46410	0.533242
$106$	0	0
$107$	2.92820	0.283080	0.141540	−	0.989933i	$-0.454795\pi$
$107$	2.92820	0.283080	0.141540	+	0.989933i	$0.454795\pi$
$108$	0	0
$109$	−15.6603	−1.49998	−0.749990	−	0.661449i	$-0.769942\pi$
$109$	−15.6603	−1.49998	−0.749990	+	0.661449i	$0.769942\pi$
$110$	0	0
$111$	−9.85641	−0.935529
$112$	0	0
$113$	−9.46410	−0.890308	−0.445154	−	0.895454i	$-0.646851\pi$
$113$	−9.46410	−0.890308	−0.445154	+	0.895454i	$0.646851\pi$
$114$	0	0
$115$	5.46410	0.509530
$116$	0	0
$117$	−5.46410	−0.505156
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3.53590	−0.321445
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	−6.92820	−0.619677
$126$	0	0
$127$	−10.0000	−0.887357	−0.443678	−	0.896186i	$-0.646327\pi$
$127$	−10.0000	−0.887357	−0.443678	+	0.896186i	$0.646327\pi$
$128$	0	0
$129$	24.7846	2.18216
$130$	0	0
$131$	9.12436	0.797199	0.398599	−	0.917125i	$-0.369496\pi$
$131$	9.12436	0.797199	0.398599	+	0.917125i	$0.369496\pi$
$132$	0	0
$133$	−3.46410	−0.300376
$134$	0	0
$135$	10.9282	0.940550
$136$	0	0
$137$	−3.46410	−0.295958	−0.147979	−	0.988990i	$-0.547277\pi$
$137$	−3.46410	−0.295958	−0.147979	+	0.988990i	$0.547277\pi$
$138$	0	0
$139$	−6.19615	−0.525551	−0.262775	−	0.964857i	$-0.584638\pi$
$139$	−6.19615	−0.525551	−0.262775	+	0.964857i	$0.584638\pi$
$140$	0	0
$141$	1.07180	0.0902616
$142$	0	0
$143$	−14.9282	−1.24836
$144$	0	0
$145$	−20.9282	−1.73799
$146$	0	0
$147$	−2.00000	−0.164957
$148$	0	0
$149$	−19.6603	−1.61063	−0.805315	−	0.592847i	$-0.798004\pi$
$149$	−19.6603	−1.61063	−0.805315	+	0.592847i	$0.798004\pi$
$150$	0	0
$151$	17.4641	1.42121	0.710604	−	0.703592i	$-0.248422\pi$
$151$	17.4641	1.42121	0.710604	+	0.703592i	$0.248422\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	21.4641	1.71302	0.856511	−	0.516129i	$-0.172627\pi$
$157$	21.4641	1.71302	0.856511	+	0.516129i	$0.172627\pi$
$158$	0	0
$159$	−15.3205	−1.21500
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	17.4641	1.36789	0.683947	−	0.729532i	$-0.260262\pi$
$163$	17.4641	1.36789	0.683947	+	0.729532i	$0.260262\pi$
$164$	0	0
$165$	−14.9282	−1.16216
$166$	0	0
$167$	−14.9282	−1.15518	−0.577590	−	0.816327i	$-0.696007\pi$
$167$	−14.9282	−1.15518	−0.577590	+	0.816327i	$0.696007\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	0	0
$171$	3.46410	0.264906
$172$	0	0
$173$	−10.7321	−0.815943	−0.407971	−	0.912995i	$-0.633764\pi$
$173$	−10.7321	−0.815943	−0.407971	+	0.912995i	$0.633764\pi$
$174$	0	0
$175$	−2.46410	−0.186269
$176$	0	0
$177$	26.2487	1.97297
$178$	0	0
$179$	5.26795	0.393745	0.196873	−	0.980429i	$-0.436921\pi$
$179$	5.26795	0.393745	0.196873	+	0.980429i	$0.436921\pi$
$180$	0	0
$181$	14.9282	1.10960	0.554802	−	0.831982i	$-0.312794\pi$
$181$	14.9282	1.10960	0.554802	+	0.831982i	$0.312794\pi$
$182$	0	0
$183$	−13.4641	−0.995295
$184$	0	0
$185$	13.4641	0.989900
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	−12.3923	−0.896676	−0.448338	−	0.893864i	$-0.647984\pi$
$191$	−12.3923	−0.896676	−0.448338	+	0.893864i	$0.647984\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	0	0
$195$	29.8564	2.13806
$196$	0	0
$197$	2.39230	0.170445	0.0852223	−	0.996362i	$-0.472840\pi$
$197$	2.39230	0.170445	0.0852223	+	0.996362i	$0.472840\pi$
$198$	0	0
$199$	24.2487	1.71895	0.859473	−	0.511182i	$-0.170792\pi$
$199$	24.2487	1.71895	0.859473	+	0.511182i	$0.170792\pi$
$200$	0	0
$201$	0.392305	0.0276711
$202$	0	0
$203$	7.66025	0.537644
$204$	0	0
$205$	2.73205	0.190815
$206$	0	0
$207$	2.00000	0.139010
$208$	0	0
$209$	9.46410	0.654646
$210$	0	0
$211$	10.0526	0.692047	0.346023	−	0.938226i	$-0.387532\pi$
$211$	10.0526	0.692047	0.346023	+	0.938226i	$0.387532\pi$
$212$	0	0
$213$	5.07180	0.347514
$214$	0	0
$215$	−33.8564	−2.30899
$216$	0	0
$217$	1.46410	0.0993897
$218$	0	0
$219$	17.8564	1.20662
$220$	0	0
$221$	0	0
$222$	0	0
$223$	12.7846	0.856121	0.428060	−	0.903750i	$-0.359197\pi$
$223$	12.7846	0.856121	0.428060	+	0.903750i	$0.359197\pi$
$224$	0	0
$225$	2.46410	0.164273
$226$	0	0
$227$	−15.4641	−1.02639	−0.513194	−	0.858272i	$-0.671538\pi$
$227$	−15.4641	−1.02639	−0.513194	+	0.858272i	$0.671538\pi$
$228$	0	0
$229$	−18.9282	−1.25081	−0.625405	−	0.780300i	$-0.715066\pi$
$229$	−18.9282	−1.25081	−0.625405	+	0.780300i	$0.715066\pi$
$230$	0	0
$231$	5.46410	0.359511
$232$	0	0
$233$	−28.9282	−1.89515	−0.947575	−	0.319534i	$-0.896474\pi$
$233$	−28.9282	−1.89515	−0.947575	+	0.319534i	$0.896474\pi$
$234$	0	0
$235$	−1.46410	−0.0955075
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−25.4641	−1.64714	−0.823568	−	0.567218i	$-0.808020\pi$
$239$	−25.4641	−1.64714	−0.823568	+	0.567218i	$0.808020\pi$
$240$	0	0
$241$	−20.9282	−1.34810	−0.674052	−	0.738684i	$-0.735448\pi$
$241$	−20.9282	−1.34810	−0.674052	+	0.738684i	$0.735448\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	0	0
$245$	2.73205	0.174544
$246$	0	0
$247$	−18.9282	−1.20437
$248$	0	0
$249$	9.46410	0.599763
$250$	0	0
$251$	−16.7321	−1.05612	−0.528059	−	0.849208i	$-0.677080\pi$
$251$	−16.7321	−1.05612	−0.528059	+	0.849208i	$0.677080\pi$
$252$	0	0
$253$	5.46410	0.343525
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−22.7846	−1.42126	−0.710632	−	0.703563i	$-0.751591\pi$
$257$	−22.7846	−1.42126	−0.710632	+	0.703563i	$0.751591\pi$
$258$	0	0
$259$	−4.92820	−0.306224
$260$	0	0
$261$	−7.66025	−0.474158
$262$	0	0
$263$	−26.9282	−1.66046	−0.830232	−	0.557418i	$-0.811792\pi$
$263$	−26.9282	−1.66046	−0.830232	+	0.557418i	$0.811792\pi$
$264$	0	0
$265$	20.9282	1.28561
$266$	0	0
$267$	−25.8564	−1.58239
$268$	0	0
$269$	7.80385	0.475809	0.237904	−	0.971289i	$-0.423539\pi$
$269$	7.80385	0.475809	0.237904	+	0.971289i	$0.423539\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	−10.9282	−0.661405
$274$	0	0
$275$	6.73205	0.405958
$276$	0	0
$277$	−0.535898	−0.0321990	−0.0160995	−	0.999870i	$-0.505125\pi$
$277$	−0.535898	−0.0321990	−0.0160995	+	0.999870i	$0.505125\pi$
$278$	0	0
$279$	−1.46410	−0.0876535
$280$	0	0
$281$	20.9282	1.24847	0.624236	−	0.781236i	$-0.285410\pi$
$281$	20.9282	1.24847	0.624236	+	0.781236i	$0.285410\pi$
$282$	0	0
$283$	−7.66025	−0.455355	−0.227677	−	0.973737i	$-0.573113\pi$
$283$	−7.66025	−0.455355	−0.227677	+	0.973737i	$0.573113\pi$
$284$	0	0
$285$	−18.9282	−1.12121
$286$	0	0
$287$	−1.00000	−0.0590281
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−4.00000	−0.234484
$292$	0	0
$293$	21.4641	1.25395	0.626973	−	0.779041i	$-0.284294\pi$
$293$	21.4641	1.25395	0.626973	+	0.779041i	$0.284294\pi$
$294$	0	0
$295$	−35.8564	−2.08764
$296$	0	0
$297$	10.9282	0.634119
$298$	0	0
$299$	−10.9282	−0.631994
$300$	0	0
$301$	12.3923	0.714281
$302$	0	0
$303$	−16.0000	−0.919176
$304$	0	0
$305$	18.3923	1.05314
$306$	0	0
$307$	−9.80385	−0.559535	−0.279768	−	0.960068i	$-0.590257\pi$
$307$	−9.80385	−0.559535	−0.279768	+	0.960068i	$0.590257\pi$
$308$	0	0
$309$	10.9282	0.621684
$310$	0	0
$311$	−7.46410	−0.423250	−0.211625	−	0.977351i	$-0.567876\pi$
$311$	−7.46410	−0.423250	−0.211625	+	0.977351i	$0.567876\pi$
$312$	0	0
$313$	−20.0000	−1.13047	−0.565233	−	0.824931i	$-0.691214\pi$
$313$	−20.0000	−1.13047	−0.565233	+	0.824931i	$0.691214\pi$
$314$	0	0
$315$	−2.73205	−0.153934
$316$	0	0
$317$	4.33975	0.243744	0.121872	−	0.992546i	$-0.461110\pi$
$317$	4.33975	0.243744	0.121872	+	0.992546i	$0.461110\pi$
$318$	0	0
$319$	−20.9282	−1.17175
$320$	0	0
$321$	−5.85641	−0.326873
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−13.4641	−0.746854
$326$	0	0
$327$	31.3205	1.73203
$328$	0	0
$329$	0.535898	0.0295450
$330$	0	0
$331$	12.5885	0.691924	0.345962	−	0.938248i	$-0.387553\pi$
$331$	12.5885	0.691924	0.345962	+	0.938248i	$0.387553\pi$
$332$	0	0
$333$	4.92820	0.270064
$334$	0	0
$335$	−0.535898	−0.0292793
$336$	0	0
$337$	−22.7846	−1.24116	−0.620578	−	0.784144i	$-0.713102\pi$
$337$	−22.7846	−1.24116	−0.620578	+	0.784144i	$0.713102\pi$
$338$	0	0
$339$	18.9282	1.02804
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−10.9282	−0.588355
$346$	0	0
$347$	−32.1962	−1.72838	−0.864190	−	0.503166i	$-0.832169\pi$
$347$	−32.1962	−1.72838	−0.864190	+	0.503166i	$0.832169\pi$
$348$	0	0
$349$	7.80385	0.417730	0.208865	−	0.977944i	$-0.433023\pi$
$349$	7.80385	0.417730	0.208865	+	0.977944i	$0.433023\pi$
$350$	0	0
$351$	−21.8564	−1.16661
$352$	0	0
$353$	−29.3205	−1.56057	−0.780287	−	0.625422i	$-0.784927\pi$
$353$	−29.3205	−1.56057	−0.780287	+	0.625422i	$0.784927\pi$
$354$	0	0
$355$	−6.92820	−0.367711
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.92820	−0.154545	−0.0772723	−	0.997010i	$-0.524621\pi$
$359$	−2.92820	−0.154545	−0.0772723	+	0.997010i	$0.524621\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	0	0
$363$	7.07180	0.371173
$364$	0	0
$365$	−24.3923	−1.27675
$366$	0	0
$367$	−25.8564	−1.34969	−0.674847	−	0.737958i	$-0.735790\pi$
$367$	−25.8564	−1.34969	−0.674847	+	0.737958i	$0.735790\pi$
$368$	0	0
$369$	1.00000	0.0520579
$370$	0	0
$371$	−7.66025	−0.397701
$372$	0	0
$373$	−32.9282	−1.70496	−0.852479	−	0.522762i	$-0.824902\pi$
$373$	−32.9282	−1.70496	−0.852479	+	0.522762i	$0.824902\pi$
$374$	0	0
$375$	13.8564	0.715542
$376$	0	0
$377$	41.8564	2.15571
$378$	0	0
$379$	16.7846	0.862167	0.431084	−	0.902312i	$-0.358131\pi$
$379$	16.7846	0.862167	0.431084	+	0.902312i	$0.358131\pi$
$380$	0	0
$381$	20.0000	1.02463
$382$	0	0
$383$	−25.8564	−1.32120	−0.660600	−	0.750738i	$-0.729698\pi$
$383$	−25.8564	−1.32120	−0.660600	+	0.750738i	$0.729698\pi$
$384$	0	0
$385$	−7.46410	−0.380406
$386$	0	0
$387$	−12.3923	−0.629936
$388$	0	0
$389$	−26.7846	−1.35803	−0.679017	−	0.734123i	$-0.737594\pi$
$389$	−26.7846	−1.35803	−0.679017	+	0.734123i	$0.737594\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−18.2487	−0.920526
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.78461	0.240133	0.120066	−	0.992766i	$-0.461689\pi$
$397$	4.78461	0.240133	0.120066	+	0.992766i	$0.461689\pi$
$398$	0	0
$399$	6.92820	0.346844
$400$	0	0
$401$	27.3205	1.36432	0.682161	−	0.731202i	$-0.261041\pi$
$401$	27.3205	1.36432	0.682161	+	0.731202i	$0.261041\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	0	0
$405$	−30.0526	−1.49332
$406$	0	0
$407$	13.4641	0.667391
$408$	0	0
$409$	−15.0718	−0.745252	−0.372626	−	0.927982i	$-0.621543\pi$
$409$	−15.0718	−0.745252	−0.372626	+	0.927982i	$0.621543\pi$
$410$	0	0
$411$	6.92820	0.341743
$412$	0	0
$413$	13.1244	0.645807
$414$	0	0
$415$	−12.9282	−0.634621
$416$	0	0
$417$	12.3923	0.606854
$418$	0	0
$419$	−20.4449	−0.998797	−0.499398	−	0.866372i	$-0.666446\pi$
$419$	−20.4449	−0.998797	−0.499398	+	0.866372i	$0.666446\pi$
$420$	0	0
$421$	18.5885	0.905946	0.452973	−	0.891524i	$-0.350363\pi$
$421$	18.5885	0.905946	0.452973	+	0.891524i	$0.350363\pi$
$422$	0	0
$423$	−0.535898	−0.0260563
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.73205	−0.325787
$428$	0	0
$429$	29.8564	1.44148
$430$	0	0
$431$	25.7128	1.23854	0.619271	−	0.785177i	$-0.287428\pi$
$431$	25.7128	1.23854	0.619271	+	0.785177i	$0.287428\pi$
$432$	0	0
$433$	29.7128	1.42791	0.713953	−	0.700193i	$-0.246903\pi$
$433$	29.7128	1.42791	0.713953	+	0.700193i	$0.246903\pi$
$434$	0	0
$435$	41.8564	2.00686
$436$	0	0
$437$	6.92820	0.331421
$438$	0	0
$439$	31.4641	1.50170	0.750850	−	0.660473i	$-0.229644\pi$
$439$	31.4641	1.50170	0.750850	+	0.660473i	$0.229644\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−21.4641	−1.01979	−0.509895	−	0.860237i	$-0.670316\pi$
$443$	−21.4641	−1.01979	−0.509895	+	0.860237i	$0.670316\pi$
$444$	0	0
$445$	35.3205	1.67435
$446$	0	0
$447$	39.3205	1.85980
$448$	0	0
$449$	−4.92820	−0.232576	−0.116288	−	0.993216i	$-0.537100\pi$
$449$	−4.92820	−0.232576	−0.116288	+	0.993216i	$0.537100\pi$
$450$	0	0
$451$	2.73205	0.128647
$452$	0	0
$453$	−34.9282	−1.64107
$454$	0	0
$455$	14.9282	0.699845
$456$	0	0
$457$	−11.8564	−0.554619	−0.277310	−	0.960781i	$-0.589443\pi$
$457$	−11.8564	−0.554619	−0.277310	+	0.960781i	$0.589443\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.875644	0.0407828	0.0203914	−	0.999792i	$-0.493509\pi$
$461$	0.875644	0.0407828	0.0203914	+	0.999792i	$0.493509\pi$
$462$	0	0
$463$	−13.0718	−0.607498	−0.303749	−	0.952752i	$-0.598238\pi$
$463$	−13.0718	−0.607498	−0.303749	+	0.952752i	$0.598238\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	0	0
$467$	20.4449	0.946075	0.473038	−	0.881042i	$-0.343157\pi$
$467$	20.4449	0.946075	0.473038	+	0.881042i	$0.343157\pi$
$468$	0	0
$469$	0.196152	0.00905748
$470$	0	0
$471$	−42.9282	−1.97803
$472$	0	0
$473$	−33.8564	−1.55672
$474$	0	0
$475$	8.53590	0.391654
$476$	0	0
$477$	7.66025	0.350739
$478$	0	0
$479$	33.3205	1.52245	0.761226	−	0.648486i	$-0.224598\pi$
$479$	33.3205	1.52245	0.761226	+	0.648486i	$0.224598\pi$
$480$	0	0
$481$	−26.9282	−1.22782
$482$	0	0
$483$	4.00000	0.182006
$484$	0	0
$485$	5.46410	0.248112
$486$	0	0
$487$	−10.9282	−0.495204	−0.247602	−	0.968862i	$-0.579643\pi$
$487$	−10.9282	−0.495204	−0.247602	+	0.968862i	$0.579643\pi$
$488$	0	0
$489$	−34.9282	−1.57951
$490$	0	0
$491$	11.6077	0.523848	0.261924	−	0.965089i	$-0.415643\pi$
$491$	11.6077	0.523848	0.261924	+	0.965089i	$0.415643\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	7.46410	0.335486
$496$	0	0
$497$	2.53590	0.113751
$498$	0	0
$499$	−5.66025	−0.253388	−0.126694	−	0.991942i	$-0.540437\pi$
$499$	−5.66025	−0.253388	−0.126694	+	0.991942i	$0.540437\pi$
$500$	0	0
$501$	29.8564	1.33389
$502$	0	0
$503$	2.14359	0.0955781	0.0477891	−	0.998857i	$-0.484782\pi$
$503$	2.14359	0.0955781	0.0477891	+	0.998857i	$0.484782\pi$
$504$	0	0
$505$	21.8564	0.972597
$506$	0	0
$507$	−33.7128	−1.49724
$508$	0	0
$509$	16.0000	0.709188	0.354594	−	0.935020i	$-0.384619\pi$
$509$	16.0000	0.709188	0.354594	+	0.935020i	$0.384619\pi$
$510$	0	0
$511$	8.92820	0.394960
$512$	0	0
$513$	13.8564	0.611775
$514$	0	0
$515$	−14.9282	−0.657815
$516$	0	0
$517$	−1.46410	−0.0643911
$518$	0	0
$519$	21.4641	0.942169
$520$	0	0
$521$	8.78461	0.384861	0.192430	−	0.981311i	$-0.438363\pi$
$521$	8.78461	0.384861	0.192430	+	0.981311i	$0.438363\pi$
$522$	0	0
$523$	14.1962	0.620754	0.310377	−	0.950613i	$-0.399545\pi$
$523$	14.1962	0.620754	0.310377	+	0.950613i	$0.399545\pi$
$524$	0	0
$525$	4.92820	0.215084
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−13.1244	−0.569549
$532$	0	0
$533$	−5.46410	−0.236677
$534$	0	0
$535$	8.00000	0.345870
$536$	0	0
$537$	−10.5359	−0.454658
$538$	0	0
$539$	2.73205	0.117678
$540$	0	0
$541$	−18.7846	−0.807613	−0.403807	−	0.914844i	$-0.632313\pi$
$541$	−18.7846	−0.807613	−0.403807	+	0.914844i	$0.632313\pi$
$542$	0	0
$543$	−29.8564	−1.28126
$544$	0	0
$545$	−42.7846	−1.83269
$546$	0	0
$547$	19.1244	0.817698	0.408849	−	0.912602i	$-0.365930\pi$
$547$	19.1244	0.817698	0.408849	+	0.912602i	$0.365930\pi$
$548$	0	0
$549$	6.73205	0.287317
$550$	0	0
$551$	−26.5359	−1.13047
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−26.9282	−1.14304
$556$	0	0
$557$	7.26795	0.307953	0.153976	−	0.988075i	$-0.450792\pi$
$557$	7.26795	0.307953	0.153976	+	0.988075i	$0.450792\pi$
$558$	0	0
$559$	67.7128	2.86395
$560$	0	0
$561$	0	0
$562$	0	0
$563$	28.5359	1.20264	0.601322	−	0.799007i	$-0.294641\pi$
$563$	28.5359	1.20264	0.601322	+	0.799007i	$0.294641\pi$
$564$	0	0
$565$	−25.8564	−1.08779
$566$	0	0
$567$	11.0000	0.461957
$568$	0	0
$569$	11.3205	0.474580	0.237290	−	0.971439i	$-0.423741\pi$
$569$	11.3205	0.474580	0.237290	+	0.971439i	$0.423741\pi$
$570$	0	0
$571$	22.7321	0.951307	0.475653	−	0.879633i	$-0.342212\pi$
$571$	22.7321	0.951307	0.475653	+	0.879633i	$0.342212\pi$
$572$	0	0
$573$	24.7846	1.03539
$574$	0	0
$575$	4.92820	0.205520
$576$	0	0
$577$	−25.8564	−1.07642	−0.538208	−	0.842812i	$-0.680899\pi$
$577$	−25.8564	−1.07642	−0.538208	+	0.842812i	$0.680899\pi$
$578$	0	0
$579$	20.0000	0.831172
$580$	0	0
$581$	4.73205	0.196319
$582$	0	0
$583$	20.9282	0.866758
$584$	0	0
$585$	−14.9282	−0.617205
$586$	0	0
$587$	36.9282	1.52419	0.762095	−	0.647465i	$-0.224171\pi$
$587$	36.9282	1.52419	0.762095	+	0.647465i	$0.224171\pi$
$588$	0	0
$589$	−5.07180	−0.208980
$590$	0	0
$591$	−4.78461	−0.196813
$592$	0	0
$593$	6.78461	0.278611	0.139305	−	0.990249i	$-0.455513\pi$
$593$	6.78461	0.278611	0.139305	+	0.990249i	$0.455513\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−48.4974	−1.98487
$598$	0	0
$599$	−32.7846	−1.33954	−0.669771	−	0.742567i	$-0.733608\pi$
$599$	−32.7846	−1.33954	−0.669771	+	0.742567i	$0.733608\pi$
$600$	0	0
$601$	14.9282	0.608934	0.304467	−	0.952523i	$-0.401522\pi$
$601$	14.9282	0.608934	0.304467	+	0.952523i	$0.401522\pi$
$602$	0	0
$603$	−0.196152	−0.00798794
$604$	0	0
$605$	−9.66025	−0.392745
$606$	0	0
$607$	−35.7128	−1.44954	−0.724769	−	0.688992i	$-0.758054\pi$
$607$	−35.7128	−1.44954	−0.724769	+	0.688992i	$0.758054\pi$
$608$	0	0
$609$	−15.3205	−0.620818
$610$	0	0
$611$	2.92820	0.118462
$612$	0	0
$613$	−13.6077	−0.549610	−0.274805	−	0.961500i	$-0.588613\pi$
$613$	−13.6077	−0.549610	−0.274805	+	0.961500i	$0.588613\pi$
$614$	0	0
$615$	−5.46410	−0.220334
$616$	0	0
$617$	−26.5359	−1.06830	−0.534148	−	0.845391i	$-0.679367\pi$
$617$	−26.5359	−1.06830	−0.534148	+	0.845391i	$0.679367\pi$
$618$	0	0
$619$	33.5167	1.34715	0.673574	−	0.739120i	$-0.264758\pi$
$619$	33.5167	1.34715	0.673574	+	0.739120i	$0.264758\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	0	0
$623$	−12.9282	−0.517958
$624$	0	0
$625$	−31.2487	−1.24995
$626$	0	0
$627$	−18.9282	−0.755920
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−28.6410	−1.14018	−0.570090	−	0.821582i	$-0.693092\pi$
$631$	−28.6410	−1.14018	−0.570090	+	0.821582i	$0.693092\pi$
$632$	0	0
$633$	−20.1051	−0.799107
$634$	0	0
$635$	−27.3205	−1.08418
$636$	0	0
$637$	−5.46410	−0.216496
$638$	0	0
$639$	−2.53590	−0.100319
$640$	0	0
$641$	31.8564	1.25825	0.629126	−	0.777303i	$-0.283413\pi$
$641$	31.8564	1.25825	0.629126	+	0.777303i	$0.283413\pi$
$642$	0	0
$643$	1.60770	0.0634013	0.0317007	−	0.999497i	$-0.489908\pi$
$643$	1.60770	0.0634013	0.0317007	+	0.999497i	$0.489908\pi$
$644$	0	0
$645$	67.7128	2.66619
$646$	0	0
$647$	5.46410	0.214816	0.107408	−	0.994215i	$-0.465745\pi$
$647$	5.46410	0.214816	0.107408	+	0.994215i	$0.465745\pi$
$648$	0	0
$649$	−35.8564	−1.40749
$650$	0	0
$651$	−2.92820	−0.114765
$652$	0	0
$653$	−9.80385	−0.383654	−0.191827	−	0.981429i	$-0.561441\pi$
$653$	−9.80385	−0.383654	−0.191827	+	0.981429i	$0.561441\pi$
$654$	0	0
$655$	24.9282	0.974025
$656$	0	0
$657$	−8.92820	−0.348322
$658$	0	0
$659$	6.73205	0.262243	0.131122	−	0.991366i	$-0.458142\pi$
$659$	6.73205	0.262243	0.131122	+	0.991366i	$0.458142\pi$
$660$	0	0
$661$	39.1244	1.52176	0.760881	−	0.648892i	$-0.224767\pi$
$661$	39.1244	1.52176	0.760881	+	0.648892i	$0.224767\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−9.46410	−0.367002
$666$	0	0
$667$	−15.3205	−0.593212
$668$	0	0
$669$	−25.5692	−0.988563
$670$	0	0
$671$	18.3923	0.710027
$672$	0	0
$673$	−28.5359	−1.09998	−0.549989	−	0.835172i	$-0.685368\pi$
$673$	−28.5359	−1.09998	−0.549989	+	0.835172i	$0.685368\pi$
$674$	0	0
$675$	9.85641	0.379373
$676$	0	0
$677$	6.33975	0.243656	0.121828	−	0.992551i	$-0.461124\pi$
$677$	6.33975	0.243656	0.121828	+	0.992551i	$0.461124\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	30.9282	1.18517
$682$	0	0
$683$	14.0526	0.537706	0.268853	−	0.963181i	$-0.413355\pi$
$683$	14.0526	0.537706	0.268853	+	0.963181i	$0.413355\pi$
$684$	0	0
$685$	−9.46410	−0.361605
$686$	0	0
$687$	37.8564	1.44431
$688$	0	0
$689$	−41.8564	−1.59460
$690$	0	0
$691$	−36.9282	−1.40482	−0.702408	−	0.711775i	$-0.747892\pi$
$691$	−36.9282	−1.40482	−0.702408	+	0.711775i	$0.747892\pi$
$692$	0	0
$693$	−2.73205	−0.103782
$694$	0	0
$695$	−16.9282	−0.642123
$696$	0	0
$697$	0	0
$698$	0	0
$699$	57.8564	2.18833
$700$	0	0
$701$	3.07180	0.116020	0.0580101	−	0.998316i	$-0.481524\pi$
$701$	3.07180	0.116020	0.0580101	+	0.998316i	$0.481524\pi$
$702$	0	0
$703$	17.0718	0.643875
$704$	0	0
$705$	2.92820	0.110283
$706$	0	0
$707$	−8.00000	−0.300871
$708$	0	0
$709$	47.3731	1.77913	0.889566	−	0.456806i	$-0.151007\pi$
$709$	47.3731	1.77913	0.889566	+	0.456806i	$0.151007\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.92820	−0.109662
$714$	0	0
$715$	−40.7846	−1.52526
$716$	0	0
$717$	50.9282	1.90195
$718$	0	0
$719$	−23.7128	−0.884339	−0.442169	−	0.896932i	$-0.645791\pi$
$719$	−23.7128	−0.884339	−0.442169	+	0.896932i	$0.645791\pi$
$720$	0	0
$721$	5.46410	0.203494
$722$	0	0
$723$	41.8564	1.55666
$724$	0	0
$725$	−18.8756	−0.701024
$726$	0	0
$727$	41.8564	1.55237	0.776184	−	0.630506i	$-0.217153\pi$
$727$	41.8564	1.55237	0.776184	+	0.630506i	$0.217153\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	0	0
$732$	0	0
$733$	13.2679	0.490063	0.245031	−	0.969515i	$-0.421202\pi$
$733$	13.2679	0.490063	0.245031	+	0.969515i	$0.421202\pi$
$734$	0	0
$735$	−5.46410	−0.201546
$736$	0	0
$737$	−0.535898	−0.0197401
$738$	0	0
$739$	−14.5359	−0.534712	−0.267356	−	0.963598i	$-0.586150\pi$
$739$	−14.5359	−0.534712	−0.267356	+	0.963598i	$0.586150\pi$
$740$	0	0
$741$	37.8564	1.39069
$742$	0	0
$743$	27.7128	1.01668	0.508342	−	0.861155i	$-0.330258\pi$
$743$	27.7128	1.01668	0.508342	+	0.861155i	$0.330258\pi$
$744$	0	0
$745$	−53.7128	−1.96789
$746$	0	0
$747$	−4.73205	−0.173137
$748$	0	0
$749$	−2.92820	−0.106994
$750$	0	0
$751$	3.32051	0.121167	0.0605835	−	0.998163i	$-0.480704\pi$
$751$	3.32051	0.121167	0.0605835	+	0.998163i	$0.480704\pi$
$752$	0	0
$753$	33.4641	1.21950
$754$	0	0
$755$	47.7128	1.73645
$756$	0	0
$757$	−35.3731	−1.28566	−0.642828	−	0.766011i	$-0.722239\pi$
$757$	−35.3731	−1.28566	−0.642828	+	0.766011i	$0.722239\pi$
$758$	0	0
$759$	−10.9282	−0.396669
$760$	0	0
$761$	−15.1769	−0.550163	−0.275081	−	0.961421i	$-0.588705\pi$
$761$	−15.1769	−0.550163	−0.275081	+	0.961421i	$0.588705\pi$
$762$	0	0
$763$	15.6603	0.566939
$764$	0	0
$765$	0	0
$766$	0	0
$767$	71.7128	2.58940
$768$	0	0
$769$	15.1769	0.547294	0.273647	−	0.961830i	$-0.411770\pi$
$769$	15.1769	0.547294	0.273647	+	0.961830i	$0.411770\pi$
$770$	0	0
$771$	45.5692	1.64114
$772$	0	0
$773$	41.1769	1.48103	0.740515	−	0.672039i	$-0.234581\pi$
$773$	41.1769	1.48103	0.740515	+	0.672039i	$0.234581\pi$
$774$	0	0
$775$	−3.60770	−0.129592
$776$	0	0
$777$	9.85641	0.353597
$778$	0	0
$779$	3.46410	0.124114
$780$	0	0
$781$	−6.92820	−0.247911
$782$	0	0
$783$	−30.6410	−1.09502
$784$	0	0
$785$	58.6410	2.09299
$786$	0	0
$787$	−18.1962	−0.648623	−0.324311	−	0.945950i	$-0.605133\pi$
$787$	−18.1962	−0.648623	−0.324311	+	0.945950i	$0.605133\pi$
$788$	0	0
$789$	53.8564	1.91734
$790$	0	0
$791$	9.46410	0.336505
$792$	0	0
$793$	−36.7846	−1.30626
$794$	0	0
$795$	−41.8564	−1.48449
$796$	0	0
$797$	44.3013	1.56923	0.784616	−	0.619982i	$-0.212860\pi$
$797$	44.3013	1.56923	0.784616	+	0.619982i	$0.212860\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	12.9282	0.456796
$802$	0	0
$803$	−24.3923	−0.860786
$804$	0	0
$805$	−5.46410	−0.192584
$806$	0	0
$807$	−15.6077	−0.549417
$808$	0	0
$809$	54.7846	1.92612	0.963062	−	0.269279i	$-0.0867854\pi$
$809$	54.7846	1.92612	0.963062	+	0.269279i	$0.0867854\pi$
$810$	0	0
$811$	−46.3013	−1.62586	−0.812929	−	0.582363i	$-0.802128\pi$
$811$	−46.3013	−1.62586	−0.812929	+	0.582363i	$0.802128\pi$
$812$	0	0
$813$	16.0000	0.561144
$814$	0	0
$815$	47.7128	1.67131
$816$	0	0
$817$	−42.9282	−1.50187
$818$	0	0
$819$	5.46410	0.190931
$820$	0	0
$821$	16.5359	0.577107	0.288553	−	0.957464i	$-0.406826\pi$
$821$	16.5359	0.577107	0.288553	+	0.957464i	$0.406826\pi$
$822$	0	0
$823$	21.1769	0.738181	0.369090	−	0.929393i	$-0.379669\pi$
$823$	21.1769	0.738181	0.369090	+	0.929393i	$0.379669\pi$
$824$	0	0
$825$	−13.4641	−0.468760
$826$	0	0
$827$	−0.588457	−0.0204627	−0.0102313	−	0.999948i	$-0.503257\pi$
$827$	−0.588457	−0.0204627	−0.0102313	+	0.999948i	$0.503257\pi$
$828$	0	0
$829$	33.3731	1.15909	0.579547	−	0.814939i	$-0.303229\pi$
$829$	33.3731	1.15909	0.579547	+	0.814939i	$0.303229\pi$
$830$	0	0
$831$	1.07180	0.0371802
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−40.7846	−1.41141
$836$	0	0
$837$	−5.85641	−0.202427
$838$	0	0
$839$	−0.535898	−0.0185013	−0.00925063	−	0.999957i	$-0.502945\pi$
$839$	−0.535898	−0.0185013	−0.00925063	+	0.999957i	$0.502945\pi$
$840$	0	0
$841$	29.6795	1.02343
$842$	0	0
$843$	−41.8564	−1.44161
$844$	0	0
$845$	46.0526	1.58426
$846$	0	0
$847$	3.53590	0.121495
$848$	0	0
$849$	15.3205	0.525798
$850$	0	0
$851$	9.85641	0.337873
$852$	0	0
$853$	−22.4449	−0.768497	−0.384249	−	0.923230i	$-0.625539\pi$
$853$	−22.4449	−0.768497	−0.384249	+	0.923230i	$0.625539\pi$
$854$	0	0
$855$	9.46410	0.323665
$856$	0	0
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	0	0
$859$	24.4449	0.834048	0.417024	−	0.908895i	$-0.363073\pi$
$859$	24.4449	0.834048	0.417024	+	0.908895i	$0.363073\pi$
$860$	0	0
$861$	2.00000	0.0681598
$862$	0	0
$863$	46.6410	1.58768	0.793839	−	0.608128i	$-0.208079\pi$
$863$	46.6410	1.58768	0.793839	+	0.608128i	$0.208079\pi$
$864$	0	0
$865$	−29.3205	−0.996927
$866$	0	0
$867$	34.0000	1.15470
$868$	0	0
$869$	0	0
$870$	0	0
$871$	1.07180	0.0363164
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	6.92820	0.234216
$876$	0	0
$877$	56.2487	1.89938	0.949692	−	0.313185i	$-0.101396\pi$
$877$	56.2487	1.89938	0.949692	+	0.313185i	$0.101396\pi$
$878$	0	0
$879$	−42.9282	−1.44793
$880$	0	0
$881$	21.6077	0.727982	0.363991	−	0.931403i	$-0.381414\pi$
$881$	21.6077	0.727982	0.363991	+	0.931403i	$0.381414\pi$
$882$	0	0
$883$	−50.0526	−1.68440	−0.842201	−	0.539163i	$-0.818741\pi$
$883$	−50.0526	−1.68440	−0.842201	+	0.539163i	$0.818741\pi$
$884$	0	0
$885$	71.7128	2.41060
$886$	0	0
$887$	−35.1769	−1.18113	−0.590563	−	0.806992i	$-0.701094\pi$
$887$	−35.1769	−1.18113	−0.590563	+	0.806992i	$0.701094\pi$
$888$	0	0
$889$	10.0000	0.335389
$890$	0	0
$891$	−30.0526	−1.00680
$892$	0	0
$893$	−1.85641	−0.0621223
$894$	0	0
$895$	14.3923	0.481082
$896$	0	0
$897$	21.8564	0.729764
$898$	0	0
$899$	11.2154	0.374054
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−24.7846	−0.824780
$904$	0	0
$905$	40.7846	1.35573
$906$	0	0
$907$	−10.5359	−0.349839	−0.174919	−	0.984583i	$-0.555966\pi$
$907$	−10.5359	−0.349839	−0.174919	+	0.984583i	$0.555966\pi$
$908$	0	0
$909$	8.00000	0.265343
$910$	0	0
$911$	−4.92820	−0.163279	−0.0816393	−	0.996662i	$-0.526016\pi$
$911$	−4.92820	−0.163279	−0.0816393	+	0.996662i	$0.526016\pi$
$912$	0	0
$913$	−12.9282	−0.427861
$914$	0	0
$915$	−36.7846	−1.21606
$916$	0	0
$917$	−9.12436	−0.301313
$918$	0	0
$919$	−0.392305	−0.0129409	−0.00647047	−	0.999979i	$-0.502060\pi$
$919$	−0.392305	−0.0129409	−0.00647047	+	0.999979i	$0.502060\pi$
$920$	0	0
$921$	19.6077	0.646096
$922$	0	0
$923$	13.8564	0.456089
$924$	0	0
$925$	12.1436	0.399279
$926$	0	0
$927$	−5.46410	−0.179465
$928$	0	0
$929$	8.14359	0.267183	0.133591	−	0.991037i	$-0.457349\pi$
$929$	8.14359	0.267183	0.133591	+	0.991037i	$0.457349\pi$
$930$	0	0
$931$	3.46410	0.113531
$932$	0	0
$933$	14.9282	0.488727
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−12.9282	−0.422346	−0.211173	−	0.977449i	$-0.567728\pi$
$937$	−12.9282	−0.422346	−0.211173	+	0.977449i	$0.567728\pi$
$938$	0	0
$939$	40.0000	1.30535
$940$	0	0
$941$	−28.9808	−0.944746	−0.472373	−	0.881399i	$-0.656602\pi$
$941$	−28.9808	−0.944746	−0.472373	+	0.881399i	$0.656602\pi$
$942$	0	0
$943$	2.00000	0.0651290
$944$	0	0
$945$	−10.9282	−0.355494
$946$	0	0
$947$	−22.6410	−0.735734	−0.367867	−	0.929878i	$-0.619912\pi$
$947$	−22.6410	−0.735734	−0.367867	+	0.929878i	$0.619912\pi$
$948$	0	0
$949$	48.7846	1.58362
$950$	0	0
$951$	−8.67949	−0.281452
$952$	0	0
$953$	18.2487	0.591134	0.295567	−	0.955322i	$-0.404491\pi$
$953$	18.2487	0.591134	0.295567	+	0.955322i	$0.404491\pi$
$954$	0	0
$955$	−33.8564	−1.09557
$956$	0	0
$957$	41.8564	1.35303
$958$	0	0
$959$	3.46410	0.111862
$960$	0	0
$961$	−28.8564	−0.930852
$962$	0	0
$963$	2.92820	0.0943600
$964$	0	0
$965$	−27.3205	−0.879478
$966$	0	0
$967$	−36.7846	−1.18291	−0.591457	−	0.806337i	$-0.701447\pi$
$967$	−36.7846	−1.18291	−0.591457	+	0.806337i	$0.701447\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	30.1051	0.966119	0.483060	−	0.875587i	$-0.339525\pi$
$971$	30.1051	0.966119	0.483060	+	0.875587i	$0.339525\pi$
$972$	0	0
$973$	6.19615	0.198640
$974$	0	0
$975$	26.9282	0.862393
$976$	0	0
$977$	47.1769	1.50932	0.754662	−	0.656114i	$-0.227801\pi$
$977$	47.1769	1.50932	0.754662	+	0.656114i	$0.227801\pi$
$978$	0	0
$979$	35.3205	1.12885
$980$	0	0
$981$	−15.6603	−0.499993
$982$	0	0
$983$	37.4641	1.19492	0.597460	−	0.801899i	$-0.296177\pi$
$983$	37.4641	1.19492	0.597460	+	0.801899i	$0.296177\pi$
$984$	0	0
$985$	6.53590	0.208251
$986$	0	0
$987$	−1.07180	−0.0341157
$988$	0	0
$989$	−24.7846	−0.788105
$990$	0	0
$991$	−7.32051	−0.232544	−0.116272	−	0.993217i	$-0.537094\pi$
$991$	−7.32051	−0.232544	−0.116272	+	0.993217i	$0.537094\pi$
$992$	0	0
$993$	−25.1769	−0.798965
$994$	0	0
$995$	66.2487	2.10023
$996$	0	0
$997$	52.0000	1.64686	0.823428	−	0.567420i	$-0.192059\pi$
$997$	52.0000	1.64686	0.823428	+	0.567420i	$0.192059\pi$
$998$	0	0
$999$	19.7128	0.623686

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4592.2.a.m.1.2		2
4.3	odd	2		574.2.a.k.1.2	✓	2
12.11	even	2		5166.2.a.bo.1.1		2
28.27	even	2		4018.2.a.x.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
574.2.a.k.1.2	✓	2	4.3	odd	2
4018.2.a.x.1.1		2	28.27	even	2
4592.2.a.m.1.2		2	1.1	even	1	trivial
5166.2.a.bo.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:	\( N \)	\(=\)	\( 4592 = 2^{4} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4592.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{3} +2.73205 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{3} +2.73205 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.73205 q^{11} -5.46410 q^{13} -5.46410 q^{15} +3.46410 q^{19} +2.00000 q^{21} +2.00000 q^{23} +2.46410 q^{25} +4.00000 q^{27} -7.66025 q^{29} -1.46410 q^{31} -5.46410 q^{33} -2.73205 q^{35} +4.92820 q^{37} +10.9282 q^{39} +1.00000 q^{41} -12.3923 q^{43} +2.73205 q^{45} -0.535898 q^{47} +1.00000 q^{49} +7.66025 q^{53} +7.46410 q^{55} -6.92820 q^{57} -13.1244 q^{59} +6.73205 q^{61} -1.00000 q^{63} -14.9282 q^{65} -0.196152 q^{67} -4.00000 q^{69} -2.53590 q^{71} -8.92820 q^{73} -4.92820 q^{75} -2.73205 q^{77} -11.0000 q^{81} -4.73205 q^{83} +15.3205 q^{87} +12.9282 q^{89} +5.46410 q^{91} +2.92820 q^{93} +9.46410 q^{95} +2.00000 q^{97} +2.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 4 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} - 4 q^{13} - 4 q^{15} + 4 q^{21} + 4 q^{23} - 2 q^{25} + 8 q^{27} + 2 q^{29} + 4 q^{31} - 4 q^{33} - 2 q^{35} - 4 q^{37} + 8 q^{39} + 2 q^{41} - 4 q^{43} + 2 q^{45} - 8 q^{47} + 2 q^{49} - 2 q^{53} + 8 q^{55} - 2 q^{59} + 10 q^{61} - 2 q^{63} - 16 q^{65} + 10 q^{67} - 8 q^{69} - 12 q^{71} - 4 q^{73} + 4 q^{75} - 2 q^{77} - 22 q^{81} - 6 q^{83} - 4 q^{87} + 12 q^{89} + 4 q^{91} - 8 q^{93} + 12 q^{95} + 4 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 4 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 - 4 * q^13 - 4 * q^15 + 4 * q^21 + 4 * q^23 - 2 * q^25 + 8 * q^27 + 2 * q^29 + 4 * q^31 - 4 * q^33 - 2 * q^35 - 4 * q^37 + 8 * q^39 + 2 * q^41 - 4 * q^43 + 2 * q^45 - 8 * q^47 + 2 * q^49 - 2 * q^53 + 8 * q^55 - 2 * q^59 + 10 * q^61 - 2 * q^63 - 16 * q^65 + 10 * q^67 - 8 * q^69 - 12 * q^71 - 4 * q^73 + 4 * q^75 - 2 * q^77 - 22 * q^81 - 6 * q^83 - 4 * q^87 + 12 * q^89 + 4 * q^91 - 8 * q^93 + 12 * q^95 + 4 * q^97 + 2 * q^99

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