LMFDB - Embedded newform 5600.2.a.bv.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5600,2,Mod(1,5600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5600, 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("5600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5600 = 2^{5} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5600.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:	$44.7162251319$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.504568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 2 $ x^5 - x^4 - 6x^3 + 3x^2 + 7*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.23118$ of defining polynomial
Character	$\chi$	$=$	5600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.746976 q^{3} +1.00000 q^{7} -2.44203 q^{9} +O(q^{10})$	$q-0.746976 q^{3} +1.00000 q^{7} -2.44203 q^{9} -5.90438 q^{11} -3.20933 q^{13} +2.14941 q^{17} +3.56597 q^{19} -0.746976 q^{21} -3.75497 q^{23} +4.06506 q^{27} -6.61177 q^{29} +5.79278 q^{31} +4.41043 q^{33} -0.623035 q^{37} +2.39729 q^{39} -5.43076 q^{41} -12.6768 q^{43} +4.31294 q^{47} +1.00000 q^{49} -1.60556 q^{51} -2.11699 q^{53} -2.66369 q^{57} +7.01926 q^{59} +1.86479 q^{61} -2.44203 q^{63} +6.88405 q^{67} +2.80487 q^{69} +1.81816 q^{71} -6.11699 q^{73} -5.90438 q^{77} +4.35664 q^{79} +4.28958 q^{81} +13.1675 q^{83} +4.93883 q^{87} +12.1708 q^{89} -3.20933 q^{91} -4.32706 q^{93} +9.65935 q^{97} +14.4187 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} + 5 q^{7} + 7 q^{9}+O(q^{10}) $ 5 * q - 2 * q^3 + 5 * q^7 + 7 * q^9	$ 5 q - 2 q^{3} + 5 q^{7} + 7 q^{9} + 4 q^{11} + 4 q^{17} + 12 q^{19} - 2 q^{21} + 8 q^{23} - 14 q^{27} - 12 q^{29} + 12 q^{31} - 8 q^{33} + 12 q^{37} + 32 q^{39} - 2 q^{41} - 8 q^{43} + 14 q^{47} + 5 q^{49} + 12 q^{51} + 8 q^{53} - 8 q^{57} + 16 q^{59} - 10 q^{61} + 7 q^{63} - 4 q^{67} + 4 q^{69} + 4 q^{71} - 12 q^{73} + 4 q^{77} + 32 q^{79} + q^{81} - 24 q^{83} + 2 q^{87} + 2 q^{89} - 20 q^{93} - 12 q^{97} + 40 q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 + 5 * q^7 + 7 * q^9 + 4 * q^11 + 4 * q^17 + 12 * q^19 - 2 * q^21 + 8 * q^23 - 14 * q^27 - 12 * q^29 + 12 * q^31 - 8 * q^33 + 12 * q^37 + 32 * q^39 - 2 * q^41 - 8 * q^43 + 14 * q^47 + 5 * q^49 + 12 * q^51 + 8 * q^53 - 8 * q^57 + 16 * q^59 - 10 * q^61 + 7 * q^63 - 4 * q^67 + 4 * q^69 + 4 * q^71 - 12 * q^73 + 4 * q^77 + 32 * q^79 + q^81 - 24 * q^83 + 2 * q^87 + 2 * q^89 - 20 * q^93 - 12 * q^97 + 40 * 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
$3$	−0.746976	−0.431267	−0.215633	−	0.976474i	$-0.569182\pi$
$3$	−0.746976	−0.431267	−0.215633	+	0.976474i	$0.569182\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.44203	−0.814009
$10$	0	0
$11$	−5.90438	−1.78024	−0.890119	−	0.455728i	$-0.849379\pi$
$11$	−5.90438	−1.78024	−0.890119	+	0.455728i	$0.849379\pi$
$12$	0	0
$13$	−3.20933	−0.890108	−0.445054	−	0.895504i	$-0.646816\pi$
$13$	−3.20933	−0.890108	−0.445054	+	0.895504i	$0.646816\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.14941	0.521309	0.260654	−	0.965432i	$-0.416062\pi$
$17$	2.14941	0.521309	0.260654	+	0.965432i	$0.416062\pi$
$18$	0	0
$19$	3.56597	0.818089	0.409045	−	0.912514i	$-0.365862\pi$
$19$	3.56597	0.818089	0.409045	+	0.912514i	$0.365862\pi$
$20$	0	0
$21$	−0.746976	−0.163003
$22$	0	0
$23$	−3.75497	−0.782965	−0.391483	−	0.920185i	$-0.628038\pi$
$23$	−3.75497	−0.782965	−0.391483	+	0.920185i	$0.628038\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.06506	0.782322
$28$	0	0
$29$	−6.61177	−1.22777	−0.613887	−	0.789394i	$-0.710395\pi$
$29$	−6.61177	−1.22777	−0.613887	+	0.789394i	$0.710395\pi$
$30$	0	0
$31$	5.79278	1.04041	0.520207	−	0.854040i	$-0.325855\pi$
$31$	5.79278	1.04041	0.520207	+	0.854040i	$0.325855\pi$
$32$	0	0
$33$	4.41043	0.767757
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.623035	−0.102426	−0.0512132	−	0.998688i	$-0.516309\pi$
$37$	−0.623035	−0.102426	−0.0512132	+	0.998688i	$0.516309\pi$
$38$	0	0
$39$	2.39729	0.383874
$40$	0	0
$41$	−5.43076	−0.848142	−0.424071	−	0.905629i	$-0.639399\pi$
$41$	−5.43076	−0.848142	−0.424071	+	0.905629i	$0.639399\pi$
$42$	0	0
$43$	−12.6768	−1.93320	−0.966599	−	0.256293i	$-0.917499\pi$
$43$	−12.6768	−1.93320	−0.966599	+	0.256293i	$0.917499\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.31294	0.629107	0.314554	−	0.949240i	$-0.398145\pi$
$47$	4.31294	0.629107	0.314554	+	0.949240i	$0.398145\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.60556	−0.224823
$52$	0	0
$53$	−2.11699	−0.290791	−0.145395	−	0.989374i	$-0.546445\pi$
$53$	−2.11699	−0.290791	−0.145395	+	0.989374i	$0.546445\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.66369	−0.352815
$58$	0	0
$59$	7.01926	0.913830	0.456915	−	0.889510i	$-0.348954\pi$
$59$	7.01926	0.913830	0.456915	+	0.889510i	$0.348954\pi$
$60$	0	0
$61$	1.86479	0.238762	0.119381	−	0.992849i	$-0.461909\pi$
$61$	1.86479	0.238762	0.119381	+	0.992849i	$0.461909\pi$
$62$	0	0
$63$	−2.44203	−0.307667
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.88405	0.841021	0.420511	−	0.907288i	$-0.361851\pi$
$67$	6.88405	0.841021	0.420511	+	0.907288i	$0.361851\pi$
$68$	0	0
$69$	2.80487	0.337667
$70$	0	0
$71$	1.81816	0.215776	0.107888	−	0.994163i	$-0.465591\pi$
$71$	1.81816	0.215776	0.107888	+	0.994163i	$0.465591\pi$
$72$	0	0
$73$	−6.11699	−0.715939	−0.357970	−	0.933733i	$-0.616531\pi$
$73$	−6.11699	−0.715939	−0.357970	+	0.933733i	$0.616531\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.90438	−0.672867
$78$	0	0
$79$	4.35664	0.490160	0.245080	−	0.969503i	$-0.421186\pi$
$79$	4.35664	0.490160	0.245080	+	0.969503i	$0.421186\pi$
$80$	0	0
$81$	4.28958	0.476620
$82$	0	0
$83$	13.1675	1.44532	0.722661	−	0.691203i	$-0.242919\pi$
$83$	13.1675	1.44532	0.722661	+	0.691203i	$0.242919\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.93883	0.529498
$88$	0	0
$89$	12.1708	1.29010	0.645050	−	0.764140i	$-0.276836\pi$
$89$	12.1708	1.29010	0.645050	+	0.764140i	$0.276836\pi$
$90$	0	0
$91$	−3.20933	−0.336429
$92$	0	0
$93$	−4.32706	−0.448696
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.65935	0.980759	0.490379	−	0.871509i	$-0.336858\pi$
$97$	9.65935	0.980759	0.490379	+	0.871509i	$0.336858\pi$
$98$	0	0
$99$	14.4187	1.44913
$100$	0	0
$101$	−13.2307	−1.31650	−0.658252	−	0.752798i	$-0.728704\pi$
$101$	−13.2307	−1.31650	−0.658252	+	0.752798i	$0.728704\pi$
$102$	0	0
$103$	−3.19700	−0.315010	−0.157505	−	0.987518i	$-0.550345\pi$
$103$	−3.19700	−0.315010	−0.157505	+	0.987518i	$0.550345\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.5220	−1.21055	−0.605276	−	0.796016i	$-0.706937\pi$
$107$	−12.5220	−1.21055	−0.605276	+	0.796016i	$0.706937\pi$
$108$	0	0
$109$	14.1217	1.35261	0.676307	−	0.736620i	$-0.263579\pi$
$109$	14.1217	1.35261	0.676307	+	0.736620i	$0.263579\pi$
$110$	0	0
$111$	0.465392	0.0441731
$112$	0	0
$113$	1.63513	0.153820	0.0769102	−	0.997038i	$-0.475495\pi$
$113$	1.63513	0.153820	0.0769102	+	0.997038i	$0.475495\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	7.83727	0.724556
$118$	0	0
$119$	2.14941	0.197036
$120$	0	0
$121$	23.8617	2.16925
$122$	0	0
$123$	4.05665	0.365775
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1.79563	−0.159336	−0.0796680	−	0.996821i	$-0.525386\pi$
$127$	−1.79563	−0.159336	−0.0796680	+	0.996821i	$0.525386\pi$
$128$	0	0
$129$	9.46928	0.833724
$130$	0	0
$131$	12.3868	1.08224	0.541121	−	0.840945i	$-0.318000\pi$
$131$	12.3868	1.08224	0.541121	+	0.840945i	$0.318000\pi$
$132$	0	0
$133$	3.56597	0.309209
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.3940	−1.22976	−0.614881	−	0.788620i	$-0.710796\pi$
$137$	−14.3940	−1.22976	−0.614881	+	0.788620i	$0.710796\pi$
$138$	0	0
$139$	22.7488	1.92953	0.964766	−	0.263110i	$-0.0847481\pi$
$139$	22.7488	1.92953	0.964766	+	0.263110i	$0.0847481\pi$
$140$	0	0
$141$	−3.22166	−0.271313
$142$	0	0
$143$	18.9491	1.58460
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−0.746976	−0.0616095
$148$	0	0
$149$	−18.7560	−1.53655	−0.768276	−	0.640119i	$-0.778885\pi$
$149$	−18.7560	−1.53655	−0.768276	+	0.640119i	$0.778885\pi$
$150$	0	0
$151$	22.2824	1.81332	0.906658	−	0.421867i	$-0.138625\pi$
$151$	22.2824	1.81332	0.906658	+	0.421867i	$0.138625\pi$
$152$	0	0
$153$	−5.24892	−0.424350
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−16.7110	−1.33369	−0.666843	−	0.745198i	$-0.732355\pi$
$157$	−16.7110	−1.33369	−0.666843	+	0.745198i	$0.732355\pi$
$158$	0	0
$159$	1.58134	0.125408
$160$	0	0
$161$	−3.75497	−0.295933
$162$	0	0
$163$	−16.3555	−1.28106	−0.640530	−	0.767933i	$-0.721285\pi$
$163$	−16.3555	−1.28106	−0.640530	+	0.767933i	$0.721285\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.66452	0.283569	0.141785	−	0.989898i	$-0.454716\pi$
$167$	3.66452	0.283569	0.141785	+	0.989898i	$0.454716\pi$
$168$	0	0
$169$	−2.70019	−0.207707
$170$	0	0
$171$	−8.70819	−0.665932
$172$	0	0
$173$	−8.37622	−0.636832	−0.318416	−	0.947951i	$-0.603151\pi$
$173$	−8.37622	−0.636832	−0.318416	+	0.947951i	$0.603151\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5.24322	−0.394105
$178$	0	0
$179$	1.73613	0.129764	0.0648822	−	0.997893i	$-0.479333\pi$
$179$	1.73613	0.129764	0.0648822	+	0.997893i	$0.479333\pi$
$180$	0	0
$181$	13.4621	1.00063	0.500316	−	0.865843i	$-0.333217\pi$
$181$	13.4621	1.00063	0.500316	+	0.865843i	$0.333217\pi$
$182$	0	0
$183$	−1.39295	−0.102970
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−12.6909	−0.928054
$188$	0	0
$189$	4.06506	0.295690
$190$	0	0
$191$	6.91933	0.500665	0.250333	−	0.968160i	$-0.419460\pi$
$191$	6.91933	0.500665	0.250333	+	0.968160i	$0.419460\pi$
$192$	0	0
$193$	8.32421	0.599190	0.299595	−	0.954066i	$-0.403148\pi$
$193$	8.32421	0.599190	0.299595	+	0.954066i	$0.403148\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.6044	1.25426	0.627130	−	0.778914i	$-0.284229\pi$
$197$	17.6044	1.25426	0.627130	+	0.778914i	$0.284229\pi$
$198$	0	0
$199$	4.56924	0.323905	0.161952	−	0.986799i	$-0.448221\pi$
$199$	4.56924	0.323905	0.161952	+	0.986799i	$0.448221\pi$
$200$	0	0
$201$	−5.14222	−0.362704
$202$	0	0
$203$	−6.61177	−0.464055
$204$	0	0
$205$	0	0
$206$	0	0
$207$	9.16974	0.637341
$208$	0	0
$209$	−21.0548	−1.45639
$210$	0	0
$211$	28.2582	1.94537	0.972687	−	0.232120i	$-0.0745662\pi$
$211$	28.2582	1.94537	0.972687	+	0.232120i	$0.0745662\pi$
$212$	0	0
$213$	−1.35812	−0.0930571
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5.79278	0.393239
$218$	0	0
$219$	4.56924	0.308761
$220$	0	0
$221$	−6.89817	−0.464021
$222$	0	0
$223$	25.2130	1.68839	0.844193	−	0.536039i	$-0.180080\pi$
$223$	25.2130	1.68839	0.844193	+	0.536039i	$0.180080\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.5030	−1.16171	−0.580857	−	0.814006i	$-0.697282\pi$
$227$	−17.5030	−1.16171	−0.580857	+	0.814006i	$0.697282\pi$
$228$	0	0
$229$	−0.226808	−0.0149879	−0.00749394	−	0.999972i	$-0.502385\pi$
$229$	−0.226808	−0.0149879	−0.00749394	+	0.999972i	$0.502385\pi$
$230$	0	0
$231$	4.41043	0.290185
$232$	0	0
$233$	−27.3836	−1.79396	−0.896978	−	0.442075i	$-0.854243\pi$
$233$	−27.3836	−1.79396	−0.896978	+	0.442075i	$0.854243\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3.25430	−0.211390
$238$	0	0
$239$	10.2504	0.663044	0.331522	−	0.943448i	$-0.392438\pi$
$239$	10.2504	0.663044	0.331522	+	0.943448i	$0.392438\pi$
$240$	0	0
$241$	−20.9407	−1.34891	−0.674455	−	0.738316i	$-0.735621\pi$
$241$	−20.9407	−1.34891	−0.674455	+	0.738316i	$0.735621\pi$
$242$	0	0
$243$	−15.3994	−0.987872
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−11.4444	−0.728188
$248$	0	0
$249$	−9.83581	−0.623319
$250$	0	0
$251$	4.25670	0.268681	0.134340	−	0.990935i	$-0.457108\pi$
$251$	4.25670	0.268681	0.134340	+	0.990935i	$0.457108\pi$
$252$	0	0
$253$	22.1708	1.39387
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−0.636942	−0.0397314	−0.0198657	−	0.999803i	$-0.506324\pi$
$257$	−0.636942	−0.0397314	−0.0198657	+	0.999803i	$0.506324\pi$
$258$	0	0
$259$	−0.623035	−0.0387135
$260$	0	0
$261$	16.1461	0.999419
$262$	0	0
$263$	24.7560	1.52652	0.763261	−	0.646091i	$-0.223597\pi$
$263$	24.7560	1.52652	0.763261	+	0.646091i	$0.223597\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−9.09128	−0.556377
$268$	0	0
$269$	−0.594054	−0.0362201	−0.0181100	−	0.999836i	$-0.505765\pi$
$269$	−0.594054	−0.0362201	−0.0181100	+	0.999836i	$0.505765\pi$
$270$	0	0
$271$	−28.4774	−1.72988	−0.864939	−	0.501877i	$-0.832643\pi$
$271$	−28.4774	−1.72988	−0.864939	+	0.501877i	$0.832643\pi$
$272$	0	0
$273$	2.39729	0.145091
$274$	0	0
$275$	0	0
$276$	0	0
$277$	13.1049	0.787397	0.393698	−	0.919240i	$-0.371195\pi$
$277$	13.1049	0.787397	0.393698	+	0.919240i	$0.371195\pi$
$278$	0	0
$279$	−14.1461	−0.846906
$280$	0	0
$281$	1.24420	0.0742228	0.0371114	−	0.999311i	$-0.488184\pi$
$281$	1.24420	0.0742228	0.0371114	+	0.999311i	$0.488184\pi$
$282$	0	0
$283$	0.476558	0.0283284	0.0141642	−	0.999900i	$-0.495491\pi$
$283$	0.476558	0.0283284	0.0141642	+	0.999900i	$0.495491\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.43076	−0.320568
$288$	0	0
$289$	−12.3800	−0.728237
$290$	0	0
$291$	−7.21530	−0.422969
$292$	0	0
$293$	32.4190	1.89394	0.946968	−	0.321328i	$-0.104129\pi$
$293$	32.4190	1.89394	0.946968	+	0.321328i	$0.104129\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−24.0017	−1.39272
$298$	0	0
$299$	12.0509	0.696924
$300$	0	0
$301$	−12.6768	−0.730680
$302$	0	0
$303$	9.88301	0.567764
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−0.718737	−0.0410205	−0.0205102	−	0.999790i	$-0.506529\pi$
$307$	−0.718737	−0.0410205	−0.0205102	+	0.999790i	$0.506529\pi$
$308$	0	0
$309$	2.38808	0.135853
$310$	0	0
$311$	19.1237	1.08441	0.542204	−	0.840247i	$-0.317590\pi$
$311$	19.1237	1.08441	0.542204	+	0.840247i	$0.317590\pi$
$312$	0	0
$313$	24.4560	1.38234	0.691168	−	0.722694i	$-0.257096\pi$
$313$	24.4560	1.38234	0.691168	+	0.722694i	$0.257096\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.08203	0.229270	0.114635	−	0.993408i	$-0.463430\pi$
$317$	4.08203	0.229270	0.114635	+	0.993408i	$0.463430\pi$
$318$	0	0
$319$	39.0384	2.18573
$320$	0	0
$321$	9.35366	0.522070
$322$	0	0
$323$	7.66473	0.426477
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−10.5486	−0.583337
$328$	0	0
$329$	4.31294	0.237780
$330$	0	0
$331$	−5.70539	−0.313597	−0.156798	−	0.987631i	$-0.550117\pi$
$331$	−5.70539	−0.313597	−0.156798	+	0.987631i	$0.550117\pi$
$332$	0	0
$333$	1.52147	0.0833760
$334$	0	0
$335$	0	0
$336$	0	0
$337$	19.5071	1.06262	0.531309	−	0.847178i	$-0.321700\pi$
$337$	19.5071	1.06262	0.531309	+	0.847178i	$0.321700\pi$
$338$	0	0
$339$	−1.22140	−0.0663376
$340$	0	0
$341$	−34.2028	−1.85218
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−30.5488	−1.63994	−0.819972	−	0.572403i	$-0.806011\pi$
$347$	−30.5488	−1.63994	−0.819972	+	0.572403i	$0.806011\pi$
$348$	0	0
$349$	−21.1043	−1.12969	−0.564844	−	0.825198i	$-0.691064\pi$
$349$	−21.1043	−1.12969	−0.564844	+	0.825198i	$0.691064\pi$
$350$	0	0
$351$	−13.0461	−0.696351
$352$	0	0
$353$	−25.4989	−1.35717	−0.678584	−	0.734523i	$-0.737406\pi$
$353$	−25.4989	−1.35717	−0.678584	+	0.734523i	$0.737406\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.60556	−0.0849752
$358$	0	0
$359$	22.3658	1.18042	0.590210	−	0.807250i	$-0.299045\pi$
$359$	22.3658	1.18042	0.590210	+	0.807250i	$0.299045\pi$
$360$	0	0
$361$	−6.28388	−0.330730
$362$	0	0
$363$	−17.8241	−0.935524
$364$	0	0
$365$	0	0
$366$	0	0
$367$	18.0126	0.940252	0.470126	−	0.882599i	$-0.344209\pi$
$367$	18.0126	0.940252	0.470126	+	0.882599i	$0.344209\pi$
$368$	0	0
$369$	13.2621	0.690395
$370$	0	0
$371$	−2.11699	−0.109908
$372$	0	0
$373$	−22.8397	−1.18259	−0.591297	−	0.806454i	$-0.701384\pi$
$373$	−22.8397	−1.18259	−0.591297	+	0.806454i	$0.701384\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	21.2193	1.09285
$378$	0	0
$379$	−5.98775	−0.307570	−0.153785	−	0.988104i	$-0.549146\pi$
$379$	−5.98775	−0.307570	−0.153785	+	0.988104i	$0.549146\pi$
$380$	0	0
$381$	1.34129	0.0687163
$382$	0	0
$383$	25.6298	1.30962	0.654810	−	0.755793i	$-0.272749\pi$
$383$	25.6298	1.30962	0.654810	+	0.755793i	$0.272749\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	30.9572	1.57364
$388$	0	0
$389$	24.3043	1.23228	0.616138	−	0.787639i	$-0.288697\pi$
$389$	24.3043	1.23228	0.616138	+	0.787639i	$0.288697\pi$
$390$	0	0
$391$	−8.07098	−0.408167
$392$	0	0
$393$	−9.25266	−0.466735
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3.65219	−0.183298	−0.0916490	−	0.995791i	$-0.529214\pi$
$397$	−3.65219	−0.183298	−0.0916490	+	0.995791i	$0.529214\pi$
$398$	0	0
$399$	−2.66369	−0.133351
$400$	0	0
$401$	16.3543	0.816696	0.408348	−	0.912826i	$-0.366105\pi$
$401$	16.3543	0.816696	0.408348	+	0.912826i	$0.366105\pi$
$402$	0	0
$403$	−18.5909	−0.926080
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.67864	0.182343
$408$	0	0
$409$	35.8675	1.77353	0.886767	−	0.462217i	$-0.152946\pi$
$409$	35.8675	1.77353	0.886767	+	0.462217i	$0.152946\pi$
$410$	0	0
$411$	10.7520	0.530355
$412$	0	0
$413$	7.01926	0.345395
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−16.9928	−0.832143
$418$	0	0
$419$	−22.5741	−1.10282	−0.551408	−	0.834236i	$-0.685909\pi$
$419$	−22.5741	−1.10282	−0.551408	+	0.834236i	$0.685909\pi$
$420$	0	0
$421$	19.5931	0.954910	0.477455	−	0.878656i	$-0.341559\pi$
$421$	19.5931	0.954910	0.477455	+	0.878656i	$0.341559\pi$
$422$	0	0
$423$	−10.5323	−0.512099
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.86479	0.0902436
$428$	0	0
$429$	−14.1545	−0.683387
$430$	0	0
$431$	4.32304	0.208234	0.104117	−	0.994565i	$-0.466798\pi$
$431$	4.32304	0.208234	0.104117	+	0.994565i	$0.466798\pi$
$432$	0	0
$433$	3.71479	0.178521	0.0892606	−	0.996008i	$-0.471550\pi$
$433$	3.71479	0.178521	0.0892606	+	0.996008i	$0.471550\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−13.3901	−0.640535
$438$	0	0
$439$	11.0528	0.527519	0.263760	−	0.964588i	$-0.415037\pi$
$439$	11.0528	0.527519	0.263760	+	0.964588i	$0.415037\pi$
$440$	0	0
$441$	−2.44203	−0.116287
$442$	0	0
$443$	0.538500	0.0255849	0.0127925	−	0.999918i	$-0.495928\pi$
$443$	0.538500	0.0255849	0.0127925	+	0.999918i	$0.495928\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	14.0103	0.662664
$448$	0	0
$449$	−9.38003	−0.442671	−0.221335	−	0.975198i	$-0.571042\pi$
$449$	−9.38003	−0.442671	−0.221335	+	0.975198i	$0.571042\pi$
$450$	0	0
$451$	32.0653	1.50989
$452$	0	0
$453$	−16.6444	−0.782022
$454$	0	0
$455$	0	0
$456$	0	0
$457$	23.7242	1.10977	0.554886	−	0.831926i	$-0.312762\pi$
$457$	23.7242	1.10977	0.554886	+	0.831926i	$0.312762\pi$
$458$	0	0
$459$	8.73749	0.407831
$460$	0	0
$461$	36.8198	1.71487	0.857435	−	0.514592i	$-0.172057\pi$
$461$	36.8198	1.71487	0.857435	+	0.514592i	$0.172057\pi$
$462$	0	0
$463$	14.3287	0.665912	0.332956	−	0.942942i	$-0.391954\pi$
$463$	14.3287	0.665912	0.332956	+	0.942942i	$0.391954\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.3887	0.850927	0.425464	−	0.904975i	$-0.360111\pi$
$467$	18.3887	0.850927	0.425464	+	0.904975i	$0.360111\pi$
$468$	0	0
$469$	6.88405	0.317876
$470$	0	0
$471$	12.4827	0.575174
$472$	0	0
$473$	74.8488	3.44155
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.16974	0.236706
$478$	0	0
$479$	26.1400	1.19437	0.597184	−	0.802104i	$-0.296286\pi$
$479$	26.1400	1.19437	0.597184	+	0.802104i	$0.296286\pi$
$480$	0	0
$481$	1.99953	0.0911706
$482$	0	0
$483$	2.80487	0.127626
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.27567	0.329692	0.164846	−	0.986319i	$-0.447287\pi$
$487$	7.27567	0.329692	0.164846	+	0.986319i	$0.447287\pi$
$488$	0	0
$489$	12.2171	0.552478
$490$	0	0
$491$	7.73616	0.349128	0.174564	−	0.984646i	$-0.444148\pi$
$491$	7.73616	0.349128	0.174564	+	0.984646i	$0.444148\pi$
$492$	0	0
$493$	−14.2114	−0.640050
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.81816	0.0815558
$498$	0	0
$499$	−8.66694	−0.387986	−0.193993	−	0.981003i	$-0.562144\pi$
$499$	−8.66694	−0.387986	−0.193993	+	0.981003i	$0.562144\pi$
$500$	0	0
$501$	−2.73731	−0.122294
$502$	0	0
$503$	−8.28220	−0.369285	−0.184643	−	0.982806i	$-0.559113\pi$
$503$	−8.28220	−0.369285	−0.184643	+	0.982806i	$0.559113\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.01698	0.0895772
$508$	0	0
$509$	34.9114	1.54742	0.773711	−	0.633539i	$-0.218398\pi$
$509$	34.9114	1.54742	0.773711	+	0.633539i	$0.218398\pi$
$510$	0	0
$511$	−6.11699	−0.270600
$512$	0	0
$513$	14.4959	0.640009
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−25.4653	−1.11996
$518$	0	0
$519$	6.25683	0.274645
$520$	0	0
$521$	12.9125	0.565705	0.282853	−	0.959163i	$-0.408719\pi$
$521$	12.9125	0.565705	0.282853	+	0.959163i	$0.408719\pi$
$522$	0	0
$523$	−24.4885	−1.07081	−0.535405	−	0.844596i	$-0.679841\pi$
$523$	−24.4885	−1.07081	−0.535405	+	0.844596i	$0.679841\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.4511	0.542377
$528$	0	0
$529$	−8.90020	−0.386965
$530$	0	0
$531$	−17.1412	−0.743866
$532$	0	0
$533$	17.4291	0.754938
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.29685	−0.0559631
$538$	0	0
$539$	−5.90438	−0.254320
$540$	0	0
$541$	−10.0318	−0.431299	−0.215650	−	0.976471i	$-0.569187\pi$
$541$	−10.0318	−0.431299	−0.215650	+	0.976471i	$0.569187\pi$
$542$	0	0
$543$	−10.0559	−0.431539
$544$	0	0
$545$	0	0
$546$	0	0
$547$	21.3310	0.912046	0.456023	−	0.889968i	$-0.349274\pi$
$547$	21.3310	0.912046	0.456023	+	0.889968i	$0.349274\pi$
$548$	0	0
$549$	−4.55387	−0.194354
$550$	0	0
$551$	−23.5773	−1.00443
$552$	0	0
$553$	4.35664	0.185263
$554$	0	0
$555$	0	0
$556$	0	0
$557$	32.4179	1.37359	0.686795	−	0.726851i	$-0.259017\pi$
$557$	32.4179	1.37359	0.686795	+	0.726851i	$0.259017\pi$
$558$	0	0
$559$	40.6841	1.72076
$560$	0	0
$561$	9.47983	0.400239
$562$	0	0
$563$	−44.8971	−1.89219	−0.946093	−	0.323894i	$-0.895008\pi$
$563$	−44.8971	−1.89219	−0.946093	+	0.323894i	$0.895008\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.28958	0.180145
$568$	0	0
$569$	−16.1454	−0.676850	−0.338425	−	0.940993i	$-0.609894\pi$
$569$	−16.1454	−0.676850	−0.338425	+	0.940993i	$0.609894\pi$
$570$	0	0
$571$	14.1954	0.594061	0.297031	−	0.954868i	$-0.404004\pi$
$571$	14.1954	0.594061	0.297031	+	0.954868i	$0.404004\pi$
$572$	0	0
$573$	−5.16857	−0.215920
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14.0481	0.584831	0.292415	−	0.956291i	$-0.405541\pi$
$577$	14.0481	0.584831	0.292415	+	0.956291i	$0.405541\pi$
$578$	0	0
$579$	−6.21798	−0.258411
$580$	0	0
$581$	13.1675	0.546280
$582$	0	0
$583$	12.4995	0.517676
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.4742	−0.845061	−0.422531	−	0.906349i	$-0.638858\pi$
$587$	−20.4742	−0.845061	−0.422531	+	0.906349i	$0.638858\pi$
$588$	0	0
$589$	20.6568	0.851151
$590$	0	0
$591$	−13.1501	−0.540921
$592$	0	0
$593$	−16.1372	−0.662674	−0.331337	−	0.943513i	$-0.607500\pi$
$593$	−16.1372	−0.662674	−0.331337	+	0.943513i	$0.607500\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3.41311	−0.139689
$598$	0	0
$599$	−39.5597	−1.61637	−0.808183	−	0.588931i	$-0.799549\pi$
$599$	−39.5597	−1.61637	−0.808183	+	0.588931i	$0.799549\pi$
$600$	0	0
$601$	−29.4061	−1.19950	−0.599750	−	0.800188i	$-0.704733\pi$
$601$	−29.4061	−1.19950	−0.599750	+	0.800188i	$0.704733\pi$
$602$	0	0
$603$	−16.8110	−0.684599
$604$	0	0
$605$	0	0
$606$	0	0
$607$	22.1165	0.897680	0.448840	−	0.893612i	$-0.351837\pi$
$607$	22.1165	0.897680	0.448840	+	0.893612i	$0.351837\pi$
$608$	0	0
$609$	4.93883	0.200131
$610$	0	0
$611$	−13.8417	−0.559974
$612$	0	0
$613$	−27.1411	−1.09622	−0.548109	−	0.836407i	$-0.684652\pi$
$613$	−27.1411	−1.09622	−0.548109	+	0.836407i	$0.684652\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.66337	0.0669648	0.0334824	−	0.999439i	$-0.489340\pi$
$617$	1.66337	0.0669648	0.0334824	+	0.999439i	$0.489340\pi$
$618$	0	0
$619$	19.8283	0.796969	0.398484	−	0.917175i	$-0.369536\pi$
$619$	19.8283	0.796969	0.398484	+	0.917175i	$0.369536\pi$
$620$	0	0
$621$	−15.2642	−0.612531
$622$	0	0
$623$	12.1708	0.487612
$624$	0	0
$625$	0	0
$626$	0	0
$627$	15.7275	0.628094
$628$	0	0
$629$	−1.33916	−0.0533958
$630$	0	0
$631$	20.1113	0.800619	0.400309	−	0.916380i	$-0.368903\pi$
$631$	20.1113	0.800619	0.400309	+	0.916380i	$0.368903\pi$
$632$	0	0
$633$	−21.1082	−0.838975
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−3.20933	−0.127158
$638$	0	0
$639$	−4.44000	−0.175644
$640$	0	0
$641$	−38.5135	−1.52119	−0.760597	−	0.649225i	$-0.775093\pi$
$641$	−38.5135	−1.52119	−0.760597	+	0.649225i	$0.775093\pi$
$642$	0	0
$643$	−39.3844	−1.55317	−0.776584	−	0.630013i	$-0.783049\pi$
$643$	−39.3844	−1.55317	−0.776584	+	0.630013i	$0.783049\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−22.1010	−0.868882	−0.434441	−	0.900700i	$-0.643054\pi$
$647$	−22.1010	−0.868882	−0.434441	+	0.900700i	$0.643054\pi$
$648$	0	0
$649$	−41.4444	−1.62684
$650$	0	0
$651$	−4.32706	−0.169591
$652$	0	0
$653$	−26.5824	−1.04025	−0.520124	−	0.854090i	$-0.674114\pi$
$653$	−26.5824	−1.04025	−0.520124	+	0.854090i	$0.674114\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	14.9378	0.582781
$658$	0	0
$659$	12.6953	0.494541	0.247270	−	0.968947i	$-0.420466\pi$
$659$	12.6953	0.494541	0.247270	+	0.968947i	$0.420466\pi$
$660$	0	0
$661$	2.67371	0.103995	0.0519976	−	0.998647i	$-0.483441\pi$
$661$	2.67371	0.103995	0.0519976	+	0.998647i	$0.483441\pi$
$662$	0	0
$663$	5.15277	0.200117
$664$	0	0
$665$	0	0
$666$	0	0
$667$	24.8270	0.961305
$668$	0	0
$669$	−18.8335	−0.728145
$670$	0	0
$671$	−11.0104	−0.425053
$672$	0	0
$673$	25.2235	0.972296	0.486148	−	0.873877i	$-0.338402\pi$
$673$	25.2235	0.972296	0.486148	+	0.873877i	$0.338402\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.88750	−0.341574	−0.170787	−	0.985308i	$-0.554631\pi$
$677$	−8.88750	−0.341574	−0.170787	+	0.985308i	$0.554631\pi$
$678$	0	0
$679$	9.65935	0.370692
$680$	0	0
$681$	13.0743	0.501009
$682$	0	0
$683$	−36.7217	−1.40512	−0.702558	−	0.711626i	$-0.747959\pi$
$683$	−36.7217	−1.40512	−0.702558	+	0.711626i	$0.747959\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0.169420	0.00646377
$688$	0	0
$689$	6.79411	0.258835
$690$	0	0
$691$	22.9721	0.873898	0.436949	−	0.899486i	$-0.356059\pi$
$691$	22.9721	0.873898	0.436949	+	0.899486i	$0.356059\pi$
$692$	0	0
$693$	14.4187	0.547720
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−11.6729	−0.442144
$698$	0	0
$699$	20.4549	0.773674
$700$	0	0
$701$	−2.86785	−0.108317	−0.0541586	−	0.998532i	$-0.517248\pi$
$701$	−2.86785	−0.108317	−0.0541586	+	0.998532i	$0.517248\pi$
$702$	0	0
$703$	−2.22172	−0.0837939
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−13.2307	−0.497592
$708$	0	0
$709$	−34.9590	−1.31291	−0.656457	−	0.754363i	$-0.727946\pi$
$709$	−34.9590	−1.31291	−0.656457	+	0.754363i	$0.727946\pi$
$710$	0	0
$711$	−10.6390	−0.398995
$712$	0	0
$713$	−21.7517	−0.814608
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−7.65681	−0.285949
$718$	0	0
$719$	−35.5609	−1.32620	−0.663099	−	0.748532i	$-0.730759\pi$
$719$	−35.5609	−1.32620	−0.663099	+	0.748532i	$0.730759\pi$
$720$	0	0
$721$	−3.19700	−0.119062
$722$	0	0
$723$	15.6422	0.581740
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−29.2096	−1.08333	−0.541663	−	0.840596i	$-0.682205\pi$
$727$	−29.2096	−1.08333	−0.541663	+	0.840596i	$0.682205\pi$
$728$	0	0
$729$	−1.36576	−0.0505836
$730$	0	0
$731$	−27.2477	−1.00779
$732$	0	0
$733$	12.5651	0.464103	0.232052	−	0.972703i	$-0.425456\pi$
$733$	12.5651	0.464103	0.232052	+	0.972703i	$0.425456\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−40.6461	−1.49722
$738$	0	0
$739$	−24.5041	−0.901398	−0.450699	−	0.892676i	$-0.648825\pi$
$739$	−24.5041	−0.901398	−0.450699	+	0.892676i	$0.648825\pi$
$740$	0	0
$741$	8.54867	0.314043
$742$	0	0
$743$	22.9661	0.842544	0.421272	−	0.906934i	$-0.361584\pi$
$743$	22.9661	0.842544	0.421272	+	0.906934i	$0.361584\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−32.1554	−1.17650
$748$	0	0
$749$	−12.5220	−0.457545
$750$	0	0
$751$	−8.06696	−0.294368	−0.147184	−	0.989109i	$-0.547021\pi$
$751$	−8.06696	−0.294368	−0.147184	+	0.989109i	$0.547021\pi$
$752$	0	0
$753$	−3.17965	−0.115873
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−43.5310	−1.58216	−0.791081	−	0.611711i	$-0.790481\pi$
$757$	−43.5310	−1.58216	−0.791081	+	0.611711i	$0.790481\pi$
$758$	0	0
$759$	−16.5610	−0.601128
$760$	0	0
$761$	14.5242	0.526501	0.263250	−	0.964728i	$-0.415205\pi$
$761$	14.5242	0.526501	0.263250	+	0.964728i	$0.415205\pi$
$762$	0	0
$763$	14.1217	0.511240
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−22.5271	−0.813408
$768$	0	0
$769$	−12.4577	−0.449235	−0.224617	−	0.974447i	$-0.572113\pi$
$769$	−12.4577	−0.449235	−0.224617	+	0.974447i	$0.572113\pi$
$770$	0	0
$771$	0.475780	0.0171348
$772$	0	0
$773$	−18.8821	−0.679142	−0.339571	−	0.940580i	$-0.610282\pi$
$773$	−18.8821	−0.679142	−0.339571	+	0.940580i	$0.610282\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.465392	0.0166959
$778$	0	0
$779$	−19.3659	−0.693856
$780$	0	0
$781$	−10.7351	−0.384133
$782$	0	0
$783$	−26.8772	−0.960514
$784$	0	0
$785$	0	0
$786$	0	0
$787$	36.1080	1.28711	0.643555	−	0.765399i	$-0.277459\pi$
$787$	36.1080	1.28711	0.643555	+	0.765399i	$0.277459\pi$
$788$	0	0
$789$	−18.4921	−0.658338
$790$	0	0
$791$	1.63513	0.0581386
$792$	0	0
$793$	−5.98473	−0.212524
$794$	0	0
$795$	0	0
$796$	0	0
$797$	26.4489	0.936866	0.468433	−	0.883499i	$-0.344819\pi$
$797$	26.4489	0.936866	0.468433	+	0.883499i	$0.344819\pi$
$798$	0	0
$799$	9.27029	0.327959
$800$	0	0
$801$	−29.7214	−1.05015
$802$	0	0
$803$	36.1170	1.27454
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0.443744	0.0156205
$808$	0	0
$809$	25.7779	0.906302	0.453151	−	0.891434i	$-0.350300\pi$
$809$	25.7779	0.906302	0.453151	+	0.891434i	$0.350300\pi$
$810$	0	0
$811$	28.3586	0.995805	0.497902	−	0.867233i	$-0.334104\pi$
$811$	28.3586	0.995805	0.497902	+	0.867233i	$0.334104\pi$
$812$	0	0
$813$	21.2719	0.746039
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−45.2052	−1.58153
$818$	0	0
$819$	7.83727	0.273856
$820$	0	0
$821$	−5.00452	−0.174659	−0.0873296	−	0.996179i	$-0.527833\pi$
$821$	−5.00452	−0.174659	−0.0873296	+	0.996179i	$0.527833\pi$
$822$	0	0
$823$	−8.69074	−0.302940	−0.151470	−	0.988462i	$-0.548401\pi$
$823$	−8.69074	−0.302940	−0.151470	+	0.988462i	$0.548401\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−7.27223	−0.252880	−0.126440	−	0.991974i	$-0.540355\pi$
$827$	−7.27223	−0.252880	−0.126440	+	0.991974i	$0.540355\pi$
$828$	0	0
$829$	9.68058	0.336220	0.168110	−	0.985768i	$-0.446234\pi$
$829$	9.68058	0.336220	0.168110	+	0.985768i	$0.446234\pi$
$830$	0	0
$831$	−9.78904	−0.339578
$832$	0	0
$833$	2.14941	0.0744727
$834$	0	0
$835$	0	0
$836$	0	0
$837$	23.5480	0.813938
$838$	0	0
$839$	−37.4160	−1.29174	−0.645872	−	0.763446i	$-0.723506\pi$
$839$	−37.4160	−1.29174	−0.645872	+	0.763446i	$0.723506\pi$
$840$	0	0
$841$	14.7155	0.507430
$842$	0	0
$843$	−0.929388	−0.0320098
$844$	0	0
$845$	0	0
$846$	0	0
$847$	23.8617	0.819899
$848$	0	0
$849$	−0.355977	−0.0122171
$850$	0	0
$851$	2.33948	0.0801963
$852$	0	0
$853$	28.3650	0.971200	0.485600	−	0.874181i	$-0.338601\pi$
$853$	28.3650	0.971200	0.485600	+	0.874181i	$0.338601\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13.3622	0.456444	0.228222	−	0.973609i	$-0.426709\pi$
$857$	13.3622	0.456444	0.228222	+	0.973609i	$0.426709\pi$
$858$	0	0
$859$	−11.6961	−0.399066	−0.199533	−	0.979891i	$-0.563942\pi$
$859$	−11.6961	−0.399066	−0.199533	+	0.979891i	$0.563942\pi$
$860$	0	0
$861$	4.05665	0.138250
$862$	0	0
$863$	18.0895	0.615773	0.307886	−	0.951423i	$-0.400378\pi$
$863$	18.0895	0.615773	0.307886	+	0.951423i	$0.400378\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9.24758	0.314064
$868$	0	0
$869$	−25.7232	−0.872601
$870$	0	0
$871$	−22.0932	−0.748600
$872$	0	0
$873$	−23.5884	−0.798346
$874$	0	0
$875$	0	0
$876$	0	0
$877$	45.9438	1.55141	0.775706	−	0.631094i	$-0.217394\pi$
$877$	45.9438	1.55141	0.775706	+	0.631094i	$0.217394\pi$
$878$	0	0
$879$	−24.2162	−0.816791
$880$	0	0
$881$	−16.1280	−0.543368	−0.271684	−	0.962387i	$-0.587580\pi$
$881$	−16.1280	−0.543368	−0.271684	+	0.962387i	$0.587580\pi$
$882$	0	0
$883$	15.1120	0.508558	0.254279	−	0.967131i	$-0.418162\pi$
$883$	15.1120	0.508558	0.254279	+	0.967131i	$0.418162\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	41.8450	1.40502	0.702508	−	0.711676i	$-0.252063\pi$
$887$	41.8450	1.40502	0.702508	+	0.711676i	$0.252063\pi$
$888$	0	0
$889$	−1.79563	−0.0602234
$890$	0	0
$891$	−25.3273	−0.848497
$892$	0	0
$893$	15.3798	0.514666
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−9.00176	−0.300560
$898$	0	0
$899$	−38.3005	−1.27739
$900$	0	0
$901$	−4.55028	−0.151592
$902$	0	0
$903$	9.46928	0.315118
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−10.1402	−0.336701	−0.168351	−	0.985727i	$-0.553844\pi$
$907$	−10.1402	−0.336701	−0.168351	+	0.985727i	$0.553844\pi$
$908$	0	0
$909$	32.3097	1.07165
$910$	0	0
$911$	34.7638	1.15178	0.575888	−	0.817529i	$-0.304656\pi$
$911$	34.7638	1.15178	0.575888	+	0.817529i	$0.304656\pi$
$912$	0	0
$913$	−77.7460	−2.57302
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.3868	0.409049
$918$	0	0
$919$	−19.8017	−0.653199	−0.326599	−	0.945163i	$-0.605903\pi$
$919$	−19.8017	−0.653199	−0.326599	+	0.945163i	$0.605903\pi$
$920$	0	0
$921$	0.536879	0.0176908
$922$	0	0
$923$	−5.83509	−0.192064
$924$	0	0
$925$	0	0
$926$	0	0
$927$	7.80715	0.256421
$928$	0	0
$929$	16.2758	0.533992	0.266996	−	0.963698i	$-0.413969\pi$
$929$	16.2758	0.533992	0.266996	+	0.963698i	$0.413969\pi$
$930$	0	0
$931$	3.56597	0.116870
$932$	0	0
$933$	−14.2850	−0.467669
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.52172	0.0823809	0.0411904	−	0.999151i	$-0.486885\pi$
$937$	2.52172	0.0823809	0.0411904	+	0.999151i	$0.486885\pi$
$938$	0	0
$939$	−18.2681	−0.596155
$940$	0	0
$941$	43.7285	1.42551	0.712755	−	0.701413i	$-0.247447\pi$
$941$	43.7285	1.42551	0.712755	+	0.701413i	$0.247447\pi$
$942$	0	0
$943$	20.3923	0.664066
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−40.9743	−1.33149	−0.665743	−	0.746181i	$-0.731885\pi$
$947$	−40.9743	−1.33149	−0.665743	+	0.746181i	$0.731885\pi$
$948$	0	0
$949$	19.6314	0.637263
$950$	0	0
$951$	−3.04918	−0.0988764
$952$	0	0
$953$	5.09948	0.165188	0.0825942	−	0.996583i	$-0.473679\pi$
$953$	5.09948	0.165188	0.0825942	+	0.996583i	$0.473679\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−29.1607	−0.942633
$958$	0	0
$959$	−14.3940	−0.464806
$960$	0	0
$961$	2.55625	0.0824595
$962$	0	0
$963$	30.5792	0.985400
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−21.7426	−0.699193	−0.349597	−	0.936900i	$-0.613681\pi$
$967$	−21.7426	−0.699193	−0.349597	+	0.936900i	$0.613681\pi$
$968$	0	0
$969$	−5.72537	−0.183925
$970$	0	0
$971$	−57.9688	−1.86031	−0.930153	−	0.367172i	$-0.880326\pi$
$971$	−57.9688	−1.86031	−0.930153	+	0.367172i	$0.880326\pi$
$972$	0	0
$973$	22.7488	0.729295
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.09815	0.131111	0.0655557	−	0.997849i	$-0.479118\pi$
$977$	4.09815	0.131111	0.0655557	+	0.997849i	$0.479118\pi$
$978$	0	0
$979$	−71.8609	−2.29669
$980$	0	0
$981$	−34.4856	−1.10104
$982$	0	0
$983$	21.8461	0.696781	0.348391	−	0.937349i	$-0.386728\pi$
$983$	21.8461	0.696781	0.348391	+	0.937349i	$0.386728\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−3.22166	−0.102547
$988$	0	0
$989$	47.6011	1.51363
$990$	0	0
$991$	−16.4720	−0.523250	−0.261625	−	0.965170i	$-0.584258\pi$
$991$	−16.4720	−0.523250	−0.261625	+	0.965170i	$0.584258\pi$
$992$	0	0
$993$	4.26179	0.135244
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−23.1301	−0.732539	−0.366270	−	0.930509i	$-0.619365\pi$
$997$	−23.1301	−0.732539	−0.366270	+	0.930509i	$0.619365\pi$
$998$	0	0
$999$	−2.53268	−0.0801304

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5600.2.a.bv.1.3		5
4.3	odd	2		5600.2.a.bw.1.3		5
5.2	odd	4		1120.2.g.c.449.6	yes	10
5.3	odd	4		1120.2.g.c.449.5	yes	10
5.4	even	2		5600.2.a.bx.1.3		5
20.3	even	4		1120.2.g.b.449.6	yes	10
20.7	even	4		1120.2.g.b.449.5	✓	10
20.19	odd	2		5600.2.a.bu.1.3		5
40.3	even	4		2240.2.g.o.449.5		10
40.13	odd	4		2240.2.g.n.449.6		10
40.27	even	4		2240.2.g.o.449.6		10
40.37	odd	4		2240.2.g.n.449.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.g.b.449.5	✓	10	20.7	even	4
1120.2.g.b.449.6	yes	10	20.3	even	4
1120.2.g.c.449.5	yes	10	5.3	odd	4
1120.2.g.c.449.6	yes	10	5.2	odd	4
2240.2.g.n.449.5		10	40.37	odd	4
2240.2.g.n.449.6		10	40.13	odd	4
2240.2.g.o.449.5		10	40.3	even	4
2240.2.g.o.449.6		10	40.27	even	4
5600.2.a.bu.1.3		5	20.19	odd	2
5600.2.a.bv.1.3		5	1.1	even	1	trivial
5600.2.a.bw.1.3		5	4.3	odd	2
5600.2.a.bx.1.3		5	5.4	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:	\( N \)	\(=\)	\( 5600 = 2^{5} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5600.a (trivial)

\(f(q)\)	\(=\)	\(q-0.746976 q^{3} +1.00000 q^{7} -2.44203 q^{9} +O(q^{10})\)	\(q-0.746976 q^{3} +1.00000 q^{7} -2.44203 q^{9} -5.90438 q^{11} -3.20933 q^{13} +2.14941 q^{17} +3.56597 q^{19} -0.746976 q^{21} -3.75497 q^{23} +4.06506 q^{27} -6.61177 q^{29} +5.79278 q^{31} +4.41043 q^{33} -0.623035 q^{37} +2.39729 q^{39} -5.43076 q^{41} -12.6768 q^{43} +4.31294 q^{47} +1.00000 q^{49} -1.60556 q^{51} -2.11699 q^{53} -2.66369 q^{57} +7.01926 q^{59} +1.86479 q^{61} -2.44203 q^{63} +6.88405 q^{67} +2.80487 q^{69} +1.81816 q^{71} -6.11699 q^{73} -5.90438 q^{77} +4.35664 q^{79} +4.28958 q^{81} +13.1675 q^{83} +4.93883 q^{87} +12.1708 q^{89} -3.20933 q^{91} -4.32706 q^{93} +9.65935 q^{97} +14.4187 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} + 5 q^{7} + 7 q^{9}+O(q^{10}) \) 5 * q - 2 * q^3 + 5 * q^7 + 7 * q^9	\( 5 q - 2 q^{3} + 5 q^{7} + 7 q^{9} + 4 q^{11} + 4 q^{17} + 12 q^{19} - 2 q^{21} + 8 q^{23} - 14 q^{27} - 12 q^{29} + 12 q^{31} - 8 q^{33} + 12 q^{37} + 32 q^{39} - 2 q^{41} - 8 q^{43} + 14 q^{47} + 5 q^{49} + 12 q^{51} + 8 q^{53} - 8 q^{57} + 16 q^{59} - 10 q^{61} + 7 q^{63} - 4 q^{67} + 4 q^{69} + 4 q^{71} - 12 q^{73} + 4 q^{77} + 32 q^{79} + q^{81} - 24 q^{83} + 2 q^{87} + 2 q^{89} - 20 q^{93} - 12 q^{97} + 40 q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 + 5 * q^7 + 7 * q^9 + 4 * q^11 + 4 * q^17 + 12 * q^19 - 2 * q^21 + 8 * q^23 - 14 * q^27 - 12 * q^29 + 12 * q^31 - 8 * q^33 + 12 * q^37 + 32 * q^39 - 2 * q^41 - 8 * q^43 + 14 * q^47 + 5 * q^49 + 12 * q^51 + 8 * q^53 - 8 * q^57 + 16 * q^59 - 10 * q^61 + 7 * q^63 - 4 * q^67 + 4 * q^69 + 4 * q^71 - 12 * q^73 + 4 * q^77 + 32 * q^79 + q^81 - 24 * q^83 + 2 * q^87 + 2 * q^89 - 20 * q^93 - 12 * q^97 + 40 * q^99

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