LMFDB - Embedded newform 5780.2.a.n.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5780,2,Mod(1,5780)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5780, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5780.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5780 = 2^{2} \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5780.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:	$46.1535323683$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.9521152.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} - 8x^{3} + 18x^{2} + 20x + 2 $ x^6 - 10x^4 - 8x^3 + 18x^2 + 20x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 340)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$3.17664$ of defining polynomial
Character	$\chi$	$=$	5780.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.17664 q^{3} +1.00000 q^{5} -2.38045 q^{7} +7.09107 q^{9} +O(q^{10})$	$q+3.17664 q^{3} +1.00000 q^{5} -2.38045 q^{7} +7.09107 q^{9} -4.03562 q^{11} -6.67686 q^{13} +3.17664 q^{15} +1.65043 q^{19} -7.56184 q^{21} -3.93180 q^{23} +1.00000 q^{25} +12.9959 q^{27} +0.169372 q^{29} -9.23751 q^{31} -12.8197 q^{33} -2.38045 q^{35} -4.52660 q^{37} -21.2100 q^{39} -12.0529 q^{41} +0.294449 q^{43} +7.09107 q^{45} -0.832485 q^{47} -1.33346 q^{49} +2.84203 q^{53} -4.03562 q^{55} +5.24282 q^{57} -6.00643 q^{59} -1.10473 q^{61} -16.8799 q^{63} -6.67686 q^{65} +7.21646 q^{67} -12.4899 q^{69} +10.9962 q^{71} +10.9674 q^{73} +3.17664 q^{75} +9.60659 q^{77} +3.51753 q^{79} +20.0101 q^{81} -9.90416 q^{83} +0.538036 q^{87} +12.8899 q^{89} +15.8939 q^{91} -29.3443 q^{93} +1.65043 q^{95} -1.13934 q^{97} -28.6169 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) $ 6 * q + 6 * q^5 - 4 * q^7 + 2 * q^9	$ 6 q + 6 q^{5} - 4 q^{7} + 2 q^{9} - 8 q^{11} - 8 q^{13} + 12 q^{19} - 8 q^{21} - 4 q^{23} + 6 q^{25} + 24 q^{27} - 8 q^{29} - 20 q^{31} - 24 q^{33} - 4 q^{35} - 24 q^{37} - 20 q^{39} - 4 q^{41} - 4 q^{43} + 2 q^{45} + 4 q^{47} + 22 q^{49} - 8 q^{53} - 8 q^{55} + 16 q^{57} - 4 q^{61} - 16 q^{63} - 8 q^{65} + 16 q^{67} - 16 q^{69} - 16 q^{71} - 16 q^{73} - 24 q^{77} - 36 q^{79} + 2 q^{81} + 8 q^{83} + 28 q^{89} + 20 q^{91} - 12 q^{93} + 12 q^{95} - 32 q^{97} - 20 q^{99}+O(q^{100}) $ 6 * q + 6 * q^5 - 4 * q^7 + 2 * q^9 - 8 * q^11 - 8 * q^13 + 12 * q^19 - 8 * q^21 - 4 * q^23 + 6 * q^25 + 24 * q^27 - 8 * q^29 - 20 * q^31 - 24 * q^33 - 4 * q^35 - 24 * q^37 - 20 * q^39 - 4 * q^41 - 4 * q^43 + 2 * q^45 + 4 * q^47 + 22 * q^49 - 8 * q^53 - 8 * q^55 + 16 * q^57 - 4 * q^61 - 16 * q^63 - 8 * q^65 + 16 * q^67 - 16 * q^69 - 16 * q^71 - 16 * q^73 - 24 * q^77 - 36 * q^79 + 2 * q^81 + 8 * q^83 + 28 * q^89 + 20 * q^91 - 12 * q^93 + 12 * q^95 - 32 * q^97 - 20 * 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
$3$	3.17664	1.83404	0.917018	−	0.398845i	$-0.130589\pi$
$3$	3.17664	1.83404	0.917018	+	0.398845i	$0.130589\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.38045	−0.899725	−0.449863	−	0.893098i	$-0.648527\pi$
$7$	−2.38045	−0.899725	−0.449863	+	0.893098i	$0.648527\pi$
$8$	0	0
$9$	7.09107	2.36369
$10$	0	0
$11$	−4.03562	−1.21679	−0.608393	−	0.793636i	$-0.708186\pi$
$11$	−4.03562	−1.21679	−0.608393	+	0.793636i	$0.708186\pi$
$12$	0	0
$13$	−6.67686	−1.85183	−0.925914	−	0.377735i	$-0.876703\pi$
$13$	−6.67686	−1.85183	−0.925914	+	0.377735i	$0.876703\pi$
$14$	0	0
$15$	3.17664	0.820206
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	1.65043	0.378634	0.189317	−	0.981916i	$-0.439373\pi$
$19$	1.65043	0.378634	0.189317	+	0.981916i	$0.439373\pi$
$20$	0	0
$21$	−7.56184	−1.65013
$22$	0	0
$23$	−3.93180	−0.819838	−0.409919	−	0.912122i	$-0.634443\pi$
$23$	−3.93180	−0.819838	−0.409919	+	0.912122i	$0.634443\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	12.9959	2.50106
$28$	0	0
$29$	0.169372	0.0314517	0.0157258	−	0.999876i	$-0.494994\pi$
$29$	0.169372	0.0314517	0.0157258	+	0.999876i	$0.494994\pi$
$30$	0	0
$31$	−9.23751	−1.65911	−0.829553	−	0.558428i	$-0.811405\pi$
$31$	−9.23751	−1.65911	−0.829553	+	0.558428i	$0.811405\pi$
$32$	0	0
$33$	−12.8197	−2.23163
$34$	0	0
$35$	−2.38045	−0.402369
$36$	0	0
$37$	−4.52660	−0.744169	−0.372085	−	0.928199i	$-0.621357\pi$
$37$	−4.52660	−0.744169	−0.372085	+	0.928199i	$0.621357\pi$
$38$	0	0
$39$	−21.2100	−3.39632
$40$	0	0
$41$	−12.0529	−1.88234	−0.941171	−	0.337932i	$-0.890273\pi$
$41$	−12.0529	−1.88234	−0.941171	+	0.337932i	$0.890273\pi$
$42$	0	0
$43$	0.294449	0.0449031	0.0224515	−	0.999748i	$-0.492853\pi$
$43$	0.294449	0.0449031	0.0224515	+	0.999748i	$0.492853\pi$
$44$	0	0
$45$	7.09107	1.05707
$46$	0	0
$47$	−0.832485	−0.121430	−0.0607152	−	0.998155i	$-0.519338\pi$
$47$	−0.832485	−0.121430	−0.0607152	+	0.998155i	$0.519338\pi$
$48$	0	0
$49$	−1.33346	−0.190495
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.84203	0.390383	0.195191	−	0.980765i	$-0.437467\pi$
$53$	2.84203	0.390383	0.195191	+	0.980765i	$0.437467\pi$
$54$	0	0
$55$	−4.03562	−0.544163
$56$	0	0
$57$	5.24282	0.694428
$58$	0	0
$59$	−6.00643	−0.781970	−0.390985	−	0.920397i	$-0.627866\pi$
$59$	−6.00643	−0.781970	−0.390985	+	0.920397i	$0.627866\pi$
$60$	0	0
$61$	−1.10473	−0.141447	−0.0707233	−	0.997496i	$-0.522531\pi$
$61$	−1.10473	−0.141447	−0.0707233	+	0.997496i	$0.522531\pi$
$62$	0	0
$63$	−16.8799	−2.12667
$64$	0	0
$65$	−6.67686	−0.828162
$66$	0	0
$67$	7.21646	0.881631	0.440815	−	0.897598i	$-0.354689\pi$
$67$	7.21646	0.881631	0.440815	+	0.897598i	$0.354689\pi$
$68$	0	0
$69$	−12.4899	−1.50361
$70$	0	0
$71$	10.9962	1.30501	0.652505	−	0.757785i	$-0.273718\pi$
$71$	10.9962	1.30501	0.652505	+	0.757785i	$0.273718\pi$
$72$	0	0
$73$	10.9674	1.28363	0.641817	−	0.766858i	$-0.278181\pi$
$73$	10.9674	1.28363	0.641817	+	0.766858i	$0.278181\pi$
$74$	0	0
$75$	3.17664	0.366807
$76$	0	0
$77$	9.60659	1.09477
$78$	0	0
$79$	3.51753	0.395753	0.197877	−	0.980227i	$-0.436595\pi$
$79$	3.51753	0.395753	0.197877	+	0.980227i	$0.436595\pi$
$80$	0	0
$81$	20.0101	2.22334
$82$	0	0
$83$	−9.90416	−1.08712	−0.543561	−	0.839370i	$-0.682924\pi$
$83$	−9.90416	−1.08712	−0.543561	+	0.839370i	$0.682924\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.538036	0.0576835
$88$	0	0
$89$	12.8899	1.36633	0.683163	−	0.730266i	$-0.260604\pi$
$89$	12.8899	1.36633	0.683163	+	0.730266i	$0.260604\pi$
$90$	0	0
$91$	15.8939	1.66614
$92$	0	0
$93$	−29.3443	−3.04286
$94$	0	0
$95$	1.65043	0.169330
$96$	0	0
$97$	−1.13934	−0.115683	−0.0578413	−	0.998326i	$-0.518422\pi$
$97$	−1.13934	−0.115683	−0.0578413	+	0.998326i	$0.518422\pi$
$98$	0	0
$99$	−28.6169	−2.87611
$100$	0	0
$101$	1.40536	0.139839	0.0699193	−	0.997553i	$-0.477726\pi$
$101$	1.40536	0.139839	0.0699193	+	0.997553i	$0.477726\pi$
$102$	0	0
$103$	−12.6089	−1.24240	−0.621198	−	0.783654i	$-0.713354\pi$
$103$	−12.6089	−1.24240	−0.621198	+	0.783654i	$0.713354\pi$
$104$	0	0
$105$	−7.56184	−0.737960
$106$	0	0
$107$	4.21740	0.407711	0.203856	−	0.979001i	$-0.434653\pi$
$107$	4.21740	0.407711	0.203856	+	0.979001i	$0.434653\pi$
$108$	0	0
$109$	−7.68421	−0.736013	−0.368007	−	0.929823i	$-0.619960\pi$
$109$	−7.68421	−0.736013	−0.368007	+	0.929823i	$0.619960\pi$
$110$	0	0
$111$	−14.3794	−1.36483
$112$	0	0
$113$	−3.95154	−0.371730	−0.185865	−	0.982575i	$-0.559509\pi$
$113$	−3.95154	−0.371730	−0.185865	+	0.982575i	$0.559509\pi$
$114$	0	0
$115$	−3.93180	−0.366643
$116$	0	0
$117$	−47.3461	−4.37715
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.28625	0.480568
$122$	0	0
$123$	−38.2877	−3.45228
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	20.8726	1.85214	0.926072	−	0.377347i	$-0.123164\pi$
$127$	20.8726	1.85214	0.926072	+	0.377347i	$0.123164\pi$
$128$	0	0
$129$	0.935360	0.0823539
$130$	0	0
$131$	10.3335	0.902842	0.451421	−	0.892311i	$-0.350917\pi$
$131$	10.3335	0.902842	0.451421	+	0.892311i	$0.350917\pi$
$132$	0	0
$133$	−3.92876	−0.340666
$134$	0	0
$135$	12.9959	1.11851
$136$	0	0
$137$	−6.67686	−0.570442	−0.285221	−	0.958462i	$-0.592067\pi$
$137$	−6.67686	−0.570442	−0.285221	+	0.958462i	$0.592067\pi$
$138$	0	0
$139$	−2.05045	−0.173917	−0.0869585	−	0.996212i	$-0.527715\pi$
$139$	−2.05045	−0.173917	−0.0869585	+	0.996212i	$0.527715\pi$
$140$	0	0
$141$	−2.64451	−0.222708
$142$	0	0
$143$	26.9453	2.25328
$144$	0	0
$145$	0.169372	0.0140656
$146$	0	0
$147$	−4.23594	−0.349374
$148$	0	0
$149$	−5.60530	−0.459204	−0.229602	−	0.973285i	$-0.573743\pi$
$149$	−5.60530	−0.459204	−0.229602	+	0.973285i	$0.573743\pi$
$150$	0	0
$151$	−16.0846	−1.30894	−0.654471	−	0.756087i	$-0.727109\pi$
$151$	−16.0846	−1.30894	−0.654471	+	0.756087i	$0.727109\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.23751	−0.741975
$156$	0	0
$157$	−10.4004	−0.830045	−0.415023	−	0.909811i	$-0.636226\pi$
$157$	−10.4004	−0.830045	−0.415023	+	0.909811i	$0.636226\pi$
$158$	0	0
$159$	9.02811	0.715976
$160$	0	0
$161$	9.35946	0.737629
$162$	0	0
$163$	20.4333	1.60046	0.800230	−	0.599693i	$-0.204711\pi$
$163$	20.4333	1.60046	0.800230	+	0.599693i	$0.204711\pi$
$164$	0	0
$165$	−12.8197	−0.998015
$166$	0	0
$167$	5.28012	0.408588	0.204294	−	0.978910i	$-0.434510\pi$
$167$	5.28012	0.408588	0.204294	+	0.978910i	$0.434510\pi$
$168$	0	0
$169$	31.5804	2.42926
$170$	0	0
$171$	11.7033	0.894973
$172$	0	0
$173$	−13.9522	−1.06077	−0.530385	−	0.847757i	$-0.677953\pi$
$173$	−13.9522	−1.06077	−0.530385	+	0.847757i	$0.677953\pi$
$174$	0	0
$175$	−2.38045	−0.179945
$176$	0	0
$177$	−19.0803	−1.43416
$178$	0	0
$179$	15.1121	1.12953	0.564766	−	0.825251i	$-0.308966\pi$
$179$	15.1121	1.12953	0.564766	+	0.825251i	$0.308966\pi$
$180$	0	0
$181$	10.7281	0.797412	0.398706	−	0.917079i	$-0.369459\pi$
$181$	10.7281	0.797412	0.398706	+	0.917079i	$0.369459\pi$
$182$	0	0
$183$	−3.50934	−0.259418
$184$	0	0
$185$	−4.52660	−0.332803
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−30.9360	−2.25027
$190$	0	0
$191$	1.34618	0.0974061	0.0487031	−	0.998813i	$-0.484491\pi$
$191$	1.34618	0.0974061	0.0487031	+	0.998813i	$0.484491\pi$
$192$	0	0
$193$	−18.4663	−1.32923	−0.664616	−	0.747185i	$-0.731405\pi$
$193$	−18.4663	−1.32923	−0.664616	+	0.747185i	$0.731405\pi$
$194$	0	0
$195$	−21.2100	−1.51888
$196$	0	0
$197$	−26.0297	−1.85454	−0.927271	−	0.374391i	$-0.877852\pi$
$197$	−26.0297	−1.85454	−0.927271	+	0.374391i	$0.877852\pi$
$198$	0	0
$199$	−9.20017	−0.652183	−0.326091	−	0.945338i	$-0.605732\pi$
$199$	−9.20017	−0.652183	−0.326091	+	0.945338i	$0.605732\pi$
$200$	0	0
$201$	22.9241	1.61694
$202$	0	0
$203$	−0.403183	−0.0282979
$204$	0	0
$205$	−12.0529	−0.841809
$206$	0	0
$207$	−27.8807	−1.93784
$208$	0	0
$209$	−6.66050	−0.460716
$210$	0	0
$211$	8.77482	0.604084	0.302042	−	0.953295i	$-0.402332\pi$
$211$	8.77482	0.604084	0.302042	+	0.953295i	$0.402332\pi$
$212$	0	0
$213$	34.9310	2.39344
$214$	0	0
$215$	0.294449	0.0200813
$216$	0	0
$217$	21.9894	1.49274
$218$	0	0
$219$	34.8394	2.35423
$220$	0	0
$221$	0	0
$222$	0	0
$223$	17.9674	1.20319	0.601595	−	0.798802i	$-0.294532\pi$
$223$	17.9674	1.20319	0.601595	+	0.798802i	$0.294532\pi$
$224$	0	0
$225$	7.09107	0.472738
$226$	0	0
$227$	−12.5226	−0.831153	−0.415577	−	0.909558i	$-0.636420\pi$
$227$	−12.5226	−0.831153	−0.415577	+	0.909558i	$0.636420\pi$
$228$	0	0
$229$	0.386906	0.0255675	0.0127837	−	0.999918i	$-0.495931\pi$
$229$	0.386906	0.0255675	0.0127837	+	0.999918i	$0.495931\pi$
$230$	0	0
$231$	30.5167	2.00785
$232$	0	0
$233$	−14.7746	−0.967915	−0.483958	−	0.875091i	$-0.660801\pi$
$233$	−14.7746	−0.967915	−0.483958	+	0.875091i	$0.660801\pi$
$234$	0	0
$235$	−0.832485	−0.0543054
$236$	0	0
$237$	11.1739	0.725826
$238$	0	0
$239$	−11.7274	−0.758584	−0.379292	−	0.925277i	$-0.623832\pi$
$239$	−11.7274	−0.758584	−0.379292	+	0.925277i	$0.623832\pi$
$240$	0	0
$241$	3.20026	0.206147	0.103074	−	0.994674i	$-0.467132\pi$
$241$	3.20026	0.206147	0.103074	+	0.994674i	$0.467132\pi$
$242$	0	0
$243$	24.5773	1.57663
$244$	0	0
$245$	−1.33346	−0.0851918
$246$	0	0
$247$	−11.0197	−0.701164
$248$	0	0
$249$	−31.4620	−1.99382
$250$	0	0
$251$	3.37527	0.213045	0.106523	−	0.994310i	$-0.466028\pi$
$251$	3.37527	0.213045	0.106523	+	0.994310i	$0.466028\pi$
$252$	0	0
$253$	15.8673	0.997567
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.8179	−0.674805	−0.337402	−	0.941361i	$-0.609548\pi$
$257$	−10.8179	−0.674805	−0.337402	+	0.941361i	$0.609548\pi$
$258$	0	0
$259$	10.7753	0.669548
$260$	0	0
$261$	1.20103	0.0743420
$262$	0	0
$263$	0.208039	0.0128283	0.00641413	−	0.999979i	$-0.497958\pi$
$263$	0.208039	0.0128283	0.00641413	+	0.999979i	$0.497958\pi$
$264$	0	0
$265$	2.84203	0.174584
$266$	0	0
$267$	40.9466	2.50589
$268$	0	0
$269$	3.97891	0.242598	0.121299	−	0.992616i	$-0.461294\pi$
$269$	3.97891	0.242598	0.121299	+	0.992616i	$0.461294\pi$
$270$	0	0
$271$	18.3931	1.11730	0.558652	−	0.829402i	$-0.311319\pi$
$271$	18.3931	1.11730	0.558652	+	0.829402i	$0.311319\pi$
$272$	0	0
$273$	50.4893	3.05575
$274$	0	0
$275$	−4.03562	−0.243357
$276$	0	0
$277$	0.275331	0.0165431	0.00827153	−	0.999966i	$-0.497367\pi$
$277$	0.275331	0.0165431	0.00827153	+	0.999966i	$0.497367\pi$
$278$	0	0
$279$	−65.5038	−3.92161
$280$	0	0
$281$	18.2766	1.09029	0.545144	−	0.838342i	$-0.316475\pi$
$281$	18.2766	1.09029	0.545144	+	0.838342i	$0.316475\pi$
$282$	0	0
$283$	−13.6950	−0.814084	−0.407042	−	0.913410i	$-0.633440\pi$
$283$	−13.6950	−0.814084	−0.407042	+	0.913410i	$0.633440\pi$
$284$	0	0
$285$	5.24282	0.310558
$286$	0	0
$287$	28.6912	1.69359
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−3.61928	−0.212166
$292$	0	0
$293$	−4.81005	−0.281006	−0.140503	−	0.990080i	$-0.544872\pi$
$293$	−4.81005	−0.281006	−0.140503	+	0.990080i	$0.544872\pi$
$294$	0	0
$295$	−6.00643	−0.349708
$296$	0	0
$297$	−52.4465	−3.04325
$298$	0	0
$299$	26.2521	1.51820
$300$	0	0
$301$	−0.700921	−0.0404004
$302$	0	0
$303$	4.46433	0.256469
$304$	0	0
$305$	−1.10473	−0.0632568
$306$	0	0
$307$	9.83568	0.561352	0.280676	−	0.959803i	$-0.409441\pi$
$307$	9.83568	0.561352	0.280676	+	0.959803i	$0.409441\pi$
$308$	0	0
$309$	−40.0541	−2.27860
$310$	0	0
$311$	−12.2861	−0.696683	−0.348342	−	0.937368i	$-0.613255\pi$
$311$	−12.2861	−0.696683	−0.348342	+	0.937368i	$0.613255\pi$
$312$	0	0
$313$	29.1313	1.64660	0.823299	−	0.567609i	$-0.192131\pi$
$313$	29.1313	1.64660	0.823299	+	0.567609i	$0.192131\pi$
$314$	0	0
$315$	−16.8799	−0.951077
$316$	0	0
$317$	−12.3614	−0.694286	−0.347143	−	0.937812i	$-0.612848\pi$
$317$	−12.3614	−0.694286	−0.347143	+	0.937812i	$0.612848\pi$
$318$	0	0
$319$	−0.683523	−0.0382700
$320$	0	0
$321$	13.3972	0.747758
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−6.67686	−0.370365
$326$	0	0
$327$	−24.4100	−1.34988
$328$	0	0
$329$	1.98169	0.109254
$330$	0	0
$331$	−20.9885	−1.15363	−0.576815	−	0.816874i	$-0.695705\pi$
$331$	−20.9885	−1.15363	−0.576815	+	0.816874i	$0.695705\pi$
$332$	0	0
$333$	−32.0985	−1.75899
$334$	0	0
$335$	7.21646	0.394277
$336$	0	0
$337$	−27.6103	−1.50403	−0.752015	−	0.659146i	$-0.770918\pi$
$337$	−27.6103	−1.50403	−0.752015	+	0.659146i	$0.770918\pi$
$338$	0	0
$339$	−12.5526	−0.681766
$340$	0	0
$341$	37.2791	2.01878
$342$	0	0
$343$	19.8374	1.07112
$344$	0	0
$345$	−12.4899	−0.672436
$346$	0	0
$347$	21.5835	1.15866	0.579332	−	0.815092i	$-0.303314\pi$
$347$	21.5835	1.15866	0.579332	+	0.815092i	$0.303314\pi$
$348$	0	0
$349$	−12.2442	−0.655417	−0.327709	−	0.944779i	$-0.606276\pi$
$349$	−12.2442	−0.655417	−0.327709	+	0.944779i	$0.606276\pi$
$350$	0	0
$351$	−86.7717	−4.63153
$352$	0	0
$353$	−4.83466	−0.257323	−0.128661	−	0.991689i	$-0.541068\pi$
$353$	−4.83466	−0.257323	−0.128661	+	0.991689i	$0.541068\pi$
$354$	0	0
$355$	10.9962	0.583618
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.36465	−0.388691	−0.194346	−	0.980933i	$-0.562258\pi$
$359$	−7.36465	−0.388691	−0.194346	+	0.980933i	$0.562258\pi$
$360$	0	0
$361$	−16.2761	−0.856637
$362$	0	0
$363$	16.7925	0.881379
$364$	0	0
$365$	10.9674	0.574058
$366$	0	0
$367$	−30.3870	−1.58619	−0.793095	−	0.609097i	$-0.791532\pi$
$367$	−30.3870	−1.58619	−0.793095	+	0.609097i	$0.791532\pi$
$368$	0	0
$369$	−85.4677	−4.44927
$370$	0	0
$371$	−6.76530	−0.351237
$372$	0	0
$373$	−10.0984	−0.522877	−0.261439	−	0.965220i	$-0.584197\pi$
$373$	−10.0984	−0.522877	−0.261439	+	0.965220i	$0.584197\pi$
$374$	0	0
$375$	3.17664	0.164041
$376$	0	0
$377$	−1.13088	−0.0582431
$378$	0	0
$379$	−33.1911	−1.70491	−0.852457	−	0.522798i	$-0.824888\pi$
$379$	−33.1911	−1.70491	−0.852457	+	0.522798i	$0.824888\pi$
$380$	0	0
$381$	66.3048	3.39690
$382$	0	0
$383$	1.11550	0.0569996	0.0284998	−	0.999594i	$-0.490927\pi$
$383$	1.11550	0.0569996	0.0284998	+	0.999594i	$0.490927\pi$
$384$	0	0
$385$	9.60659	0.489597
$386$	0	0
$387$	2.08796	0.106137
$388$	0	0
$389$	38.0083	1.92710	0.963548	−	0.267537i	$-0.0862098\pi$
$389$	38.0083	1.92710	0.963548	+	0.267537i	$0.0862098\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	32.8259	1.65585
$394$	0	0
$395$	3.51753	0.176986
$396$	0	0
$397$	−7.82646	−0.392799	−0.196399	−	0.980524i	$-0.562925\pi$
$397$	−7.82646	−0.392799	−0.196399	+	0.980524i	$0.562925\pi$
$398$	0	0
$399$	−12.4803	−0.624794
$400$	0	0
$401$	−29.7586	−1.48607	−0.743036	−	0.669251i	$-0.766615\pi$
$401$	−29.7586	−1.48607	−0.743036	+	0.669251i	$0.766615\pi$
$402$	0	0
$403$	61.6775	3.07238
$404$	0	0
$405$	20.0101	0.994309
$406$	0	0
$407$	18.2677	0.905494
$408$	0	0
$409$	15.3134	0.757198	0.378599	−	0.925561i	$-0.376406\pi$
$409$	15.3134	0.757198	0.378599	+	0.925561i	$0.376406\pi$
$410$	0	0
$411$	−21.2100	−1.04621
$412$	0	0
$413$	14.2980	0.703558
$414$	0	0
$415$	−9.90416	−0.486176
$416$	0	0
$417$	−6.51356	−0.318970
$418$	0	0
$419$	21.9525	1.07245	0.536225	−	0.844075i	$-0.319850\pi$
$419$	21.9525	1.07245	0.536225	+	0.844075i	$0.319850\pi$
$420$	0	0
$421$	29.6919	1.44710	0.723548	−	0.690274i	$-0.242510\pi$
$421$	29.6919	1.44710	0.723548	+	0.690274i	$0.242510\pi$
$422$	0	0
$423$	−5.90321	−0.287024
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.62976	0.127263
$428$	0	0
$429$	85.5956	4.13259
$430$	0	0
$431$	3.44684	0.166028	0.0830142	−	0.996548i	$-0.473545\pi$
$431$	3.44684	0.166028	0.0830142	+	0.996548i	$0.473545\pi$
$432$	0	0
$433$	29.9018	1.43699	0.718494	−	0.695533i	$-0.244832\pi$
$433$	29.9018	1.43699	0.718494	+	0.695533i	$0.244832\pi$
$434$	0	0
$435$	0.538036	0.0257969
$436$	0	0
$437$	−6.48915	−0.310418
$438$	0	0
$439$	5.03962	0.240528	0.120264	−	0.992742i	$-0.461626\pi$
$439$	5.03962	0.240528	0.120264	+	0.992742i	$0.461626\pi$
$440$	0	0
$441$	−9.45568	−0.450270
$442$	0	0
$443$	10.9242	0.519025	0.259512	−	0.965740i	$-0.416438\pi$
$443$	10.9242	0.519025	0.259512	+	0.965740i	$0.416438\pi$
$444$	0	0
$445$	12.8899	0.611040
$446$	0	0
$447$	−17.8061	−0.842198
$448$	0	0
$449$	8.57189	0.404533	0.202266	−	0.979331i	$-0.435169\pi$
$449$	8.57189	0.404533	0.202266	+	0.979331i	$0.435169\pi$
$450$	0	0
$451$	48.6408	2.29041
$452$	0	0
$453$	−51.0949	−2.40065
$454$	0	0
$455$	15.8939	0.745118
$456$	0	0
$457$	6.37994	0.298441	0.149221	−	0.988804i	$-0.452324\pi$
$457$	6.37994	0.298441	0.149221	+	0.988804i	$0.452324\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.53677	0.0715746	0.0357873	−	0.999359i	$-0.488606\pi$
$461$	1.53677	0.0715746	0.0357873	+	0.999359i	$0.488606\pi$
$462$	0	0
$463$	4.29271	0.199499	0.0997496	−	0.995013i	$-0.468196\pi$
$463$	4.29271	0.199499	0.0997496	+	0.995013i	$0.468196\pi$
$464$	0	0
$465$	−29.3443	−1.36081
$466$	0	0
$467$	−25.6584	−1.18733	−0.593665	−	0.804712i	$-0.702320\pi$
$467$	−25.6584	−1.18733	−0.593665	+	0.804712i	$0.702320\pi$
$468$	0	0
$469$	−17.1784	−0.793225
$470$	0	0
$471$	−33.0385	−1.52233
$472$	0	0
$473$	−1.18829	−0.0546374
$474$	0	0
$475$	1.65043	0.0757267
$476$	0	0
$477$	20.1530	0.922744
$478$	0	0
$479$	−25.4317	−1.16200	−0.581002	−	0.813902i	$-0.697339\pi$
$479$	−25.4317	−1.16200	−0.581002	+	0.813902i	$0.697339\pi$
$480$	0	0
$481$	30.2235	1.37807
$482$	0	0
$483$	29.7317	1.35284
$484$	0	0
$485$	−1.13934	−0.0517348
$486$	0	0
$487$	−25.1146	−1.13805	−0.569025	−	0.822320i	$-0.692679\pi$
$487$	−25.1146	−1.13805	−0.569025	+	0.822320i	$0.692679\pi$
$488$	0	0
$489$	64.9094	2.93530
$490$	0	0
$491$	−7.58321	−0.342225	−0.171113	−	0.985251i	$-0.554736\pi$
$491$	−7.58321	−0.342225	−0.171113	+	0.985251i	$0.554736\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−28.6169	−1.28623
$496$	0	0
$497$	−26.1759	−1.17415
$498$	0	0
$499$	28.1405	1.25974	0.629872	−	0.776699i	$-0.283107\pi$
$499$	28.1405	1.25974	0.629872	+	0.776699i	$0.283107\pi$
$500$	0	0
$501$	16.7731	0.749365
$502$	0	0
$503$	−8.86961	−0.395476	−0.197738	−	0.980255i	$-0.563360\pi$
$503$	−8.86961	−0.395476	−0.197738	+	0.980255i	$0.563360\pi$
$504$	0	0
$505$	1.40536	0.0625377
$506$	0	0
$507$	100.320	4.45536
$508$	0	0
$509$	23.6529	1.04840	0.524198	−	0.851596i	$-0.324365\pi$
$509$	23.6529	1.04840	0.524198	+	0.851596i	$0.324365\pi$
$510$	0	0
$511$	−26.1073	−1.15492
$512$	0	0
$513$	21.4487	0.946985
$514$	0	0
$515$	−12.6089	−0.555616
$516$	0	0
$517$	3.35960	0.147755
$518$	0	0
$519$	−44.3213	−1.94549
$520$	0	0
$521$	35.5738	1.55852	0.779258	−	0.626703i	$-0.215596\pi$
$521$	35.5738	1.55852	0.779258	+	0.626703i	$0.215596\pi$
$522$	0	0
$523$	1.25121	0.0547117	0.0273558	−	0.999626i	$-0.491291\pi$
$523$	1.25121	0.0547117	0.0273558	+	0.999626i	$0.491291\pi$
$524$	0	0
$525$	−7.56184	−0.330026
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.54092	−0.327866
$530$	0	0
$531$	−42.5920	−1.84834
$532$	0	0
$533$	80.4753	3.48577
$534$	0	0
$535$	4.21740	0.182334
$536$	0	0
$537$	48.0058	2.07160
$538$	0	0
$539$	5.38135	0.231791
$540$	0	0
$541$	−34.9047	−1.50067	−0.750336	−	0.661057i	$-0.770108\pi$
$541$	−34.9047	−1.50067	−0.750336	+	0.661057i	$0.770108\pi$
$542$	0	0
$543$	34.0793	1.46248
$544$	0	0
$545$	−7.68421	−0.329155
$546$	0	0
$547$	−21.2710	−0.909483	−0.454742	−	0.890623i	$-0.650268\pi$
$547$	−21.2710	−0.909483	−0.454742	+	0.890623i	$0.650268\pi$
$548$	0	0
$549$	−7.83374	−0.334336
$550$	0	0
$551$	0.279537	0.0119087
$552$	0	0
$553$	−8.37330	−0.356069
$554$	0	0
$555$	−14.3794	−0.610372
$556$	0	0
$557$	−44.2856	−1.87644	−0.938220	−	0.346039i	$-0.887527\pi$
$557$	−44.2856	−1.87644	−0.938220	+	0.346039i	$0.887527\pi$
$558$	0	0
$559$	−1.96600	−0.0831527
$560$	0	0
$561$	0	0
$562$	0	0
$563$	28.0288	1.18127	0.590637	−	0.806937i	$-0.298876\pi$
$563$	28.0288	1.18127	0.590637	+	0.806937i	$0.298876\pi$
$564$	0	0
$565$	−3.95154	−0.166243
$566$	0	0
$567$	−47.6330	−2.00040
$568$	0	0
$569$	−34.0926	−1.42924	−0.714618	−	0.699515i	$-0.753399\pi$
$569$	−34.0926	−1.42924	−0.714618	+	0.699515i	$0.753399\pi$
$570$	0	0
$571$	−7.01735	−0.293667	−0.146833	−	0.989161i	$-0.546908\pi$
$571$	−7.01735	−0.293667	−0.146833	+	0.989161i	$0.546908\pi$
$572$	0	0
$573$	4.27633	0.178646
$574$	0	0
$575$	−3.93180	−0.163968
$576$	0	0
$577$	28.8945	1.20289	0.601446	−	0.798913i	$-0.294591\pi$
$577$	28.8945	1.20289	0.601446	+	0.798913i	$0.294591\pi$
$578$	0	0
$579$	−58.6608	−2.43786
$580$	0	0
$581$	23.5763	0.978111
$582$	0	0
$583$	−11.4694	−0.475012
$584$	0	0
$585$	−47.3461	−1.95752
$586$	0	0
$587$	−11.5333	−0.476031	−0.238016	−	0.971261i	$-0.576497\pi$
$587$	−11.5333	−0.476031	−0.238016	+	0.971261i	$0.576497\pi$
$588$	0	0
$589$	−15.2458	−0.628193
$590$	0	0
$591$	−82.6872	−3.40130
$592$	0	0
$593$	−44.4853	−1.82679	−0.913396	−	0.407073i	$-0.866549\pi$
$593$	−44.4853	−1.82679	−0.913396	+	0.407073i	$0.866549\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−29.2257	−1.19613
$598$	0	0
$599$	31.9610	1.30589	0.652945	−	0.757406i	$-0.273533\pi$
$599$	31.9610	1.30589	0.652945	+	0.757406i	$0.273533\pi$
$600$	0	0
$601$	−32.3861	−1.32106	−0.660528	−	0.750801i	$-0.729668\pi$
$601$	−32.3861	−1.32106	−0.660528	+	0.750801i	$0.729668\pi$
$602$	0	0
$603$	51.1724	2.08390
$604$	0	0
$605$	5.28625	0.214917
$606$	0	0
$607$	25.4596	1.03337	0.516686	−	0.856175i	$-0.327165\pi$
$607$	25.4596	1.03337	0.516686	+	0.856175i	$0.327165\pi$
$608$	0	0
$609$	−1.28077	−0.0518993
$610$	0	0
$611$	5.55839	0.224868
$612$	0	0
$613$	15.5210	0.626888	0.313444	−	0.949607i	$-0.398517\pi$
$613$	15.5210	0.626888	0.313444	+	0.949607i	$0.398517\pi$
$614$	0	0
$615$	−38.2877	−1.54391
$616$	0	0
$617$	16.6230	0.669216	0.334608	−	0.942357i	$-0.391396\pi$
$617$	16.6230	0.669216	0.334608	+	0.942357i	$0.391396\pi$
$618$	0	0
$619$	−3.56495	−0.143288	−0.0716438	−	0.997430i	$-0.522824\pi$
$619$	−3.56495	−0.143288	−0.0716438	+	0.997430i	$0.522824\pi$
$620$	0	0
$621$	−51.0973	−2.05046
$622$	0	0
$623$	−30.6837	−1.22932
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−21.1580	−0.844970
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−9.38997	−0.373809	−0.186904	−	0.982378i	$-0.559845\pi$
$631$	−9.38997	−0.373809	−0.186904	+	0.982378i	$0.559845\pi$
$632$	0	0
$633$	27.8745	1.10791
$634$	0	0
$635$	20.8726	0.828304
$636$	0	0
$637$	8.90334	0.352763
$638$	0	0
$639$	77.9749	3.08464
$640$	0	0
$641$	32.4403	1.28131	0.640657	−	0.767827i	$-0.278662\pi$
$641$	32.4403	1.28131	0.640657	+	0.767827i	$0.278662\pi$
$642$	0	0
$643$	−26.7459	−1.05476	−0.527378	−	0.849631i	$-0.676825\pi$
$643$	−26.7459	−1.05476	−0.527378	+	0.849631i	$0.676825\pi$
$644$	0	0
$645$	0.935360	0.0368298
$646$	0	0
$647$	5.73115	0.225315	0.112657	−	0.993634i	$-0.464064\pi$
$647$	5.73115	0.225315	0.112657	+	0.993634i	$0.464064\pi$
$648$	0	0
$649$	24.2397	0.951490
$650$	0	0
$651$	69.8526	2.73774
$652$	0	0
$653$	9.08972	0.355708	0.177854	−	0.984057i	$-0.443084\pi$
$653$	9.08972	0.355708	0.177854	+	0.984057i	$0.443084\pi$
$654$	0	0
$655$	10.3335	0.403763
$656$	0	0
$657$	77.7704	3.03411
$658$	0	0
$659$	−39.8600	−1.55272	−0.776362	−	0.630287i	$-0.782937\pi$
$659$	−39.8600	−1.55272	−0.776362	+	0.630287i	$0.782937\pi$
$660$	0	0
$661$	28.9040	1.12423	0.562117	−	0.827057i	$-0.309987\pi$
$661$	28.9040	1.12423	0.562117	+	0.827057i	$0.309987\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.92876	−0.152351
$666$	0	0
$667$	−0.665939	−0.0257853
$668$	0	0
$669$	57.0762	2.20669
$670$	0	0
$671$	4.45828	0.172110
$672$	0	0
$673$	−12.3250	−0.475092	−0.237546	−	0.971376i	$-0.576343\pi$
$673$	−12.3250	−0.475092	−0.237546	+	0.971376i	$0.576343\pi$
$674$	0	0
$675$	12.9959	0.500212
$676$	0	0
$677$	−33.2229	−1.27686	−0.638431	−	0.769679i	$-0.720416\pi$
$677$	−33.2229	−1.27686	−0.638431	+	0.769679i	$0.720416\pi$
$678$	0	0
$679$	2.71214	0.104083
$680$	0	0
$681$	−39.7798	−1.52437
$682$	0	0
$683$	−3.90250	−0.149325	−0.0746626	−	0.997209i	$-0.523788\pi$
$683$	−3.90250	−0.149325	−0.0746626	+	0.997209i	$0.523788\pi$
$684$	0	0
$685$	−6.67686	−0.255110
$686$	0	0
$687$	1.22906	0.0468917
$688$	0	0
$689$	−18.9758	−0.722921
$690$	0	0
$691$	6.47028	0.246141	0.123071	−	0.992398i	$-0.460726\pi$
$691$	6.47028	0.246141	0.123071	+	0.992398i	$0.460726\pi$
$692$	0	0
$693$	68.1210	2.58770
$694$	0	0
$695$	−2.05045	−0.0777781
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−46.9336	−1.77519
$700$	0	0
$701$	42.8631	1.61892	0.809458	−	0.587178i	$-0.199761\pi$
$701$	42.8631	1.61892	0.809458	+	0.587178i	$0.199761\pi$
$702$	0	0
$703$	−7.47082	−0.281767
$704$	0	0
$705$	−2.64451	−0.0995980
$706$	0	0
$707$	−3.34539	−0.125816
$708$	0	0
$709$	−34.2011	−1.28445	−0.642224	−	0.766517i	$-0.721988\pi$
$709$	−34.2011	−1.28445	−0.642224	+	0.766517i	$0.721988\pi$
$710$	0	0
$711$	24.9431	0.935438
$712$	0	0
$713$	36.3201	1.36020
$714$	0	0
$715$	26.9453	1.00770
$716$	0	0
$717$	−37.2539	−1.39127
$718$	0	0
$719$	32.5582	1.21422	0.607109	−	0.794619i	$-0.292329\pi$
$719$	32.5582	1.21422	0.607109	+	0.794619i	$0.292329\pi$
$720$	0	0
$721$	30.0149	1.11781
$722$	0	0
$723$	10.1661	0.378081
$724$	0	0
$725$	0.169372	0.00629034
$726$	0	0
$727$	−2.64319	−0.0980304	−0.0490152	−	0.998798i	$-0.515608\pi$
$727$	−2.64319	−0.0980304	−0.0490152	+	0.998798i	$0.515608\pi$
$728$	0	0
$729$	18.0430	0.668261
$730$	0	0
$731$	0	0
$732$	0	0
$733$	5.11130	0.188790	0.0943950	−	0.995535i	$-0.469908\pi$
$733$	5.11130	0.188790	0.0943950	+	0.995535i	$0.469908\pi$
$734$	0	0
$735$	−4.23594	−0.156245
$736$	0	0
$737$	−29.1229	−1.07276
$738$	0	0
$739$	33.0337	1.21516	0.607582	−	0.794257i	$-0.292140\pi$
$739$	33.0337	1.21516	0.607582	+	0.794257i	$0.292140\pi$
$740$	0	0
$741$	−35.0055	−1.28596
$742$	0	0
$743$	1.70637	0.0626008	0.0313004	−	0.999510i	$-0.490035\pi$
$743$	1.70637	0.0626008	0.0313004	+	0.999510i	$0.490035\pi$
$744$	0	0
$745$	−5.60530	−0.205362
$746$	0	0
$747$	−70.2311	−2.56962
$748$	0	0
$749$	−10.0393	−0.366828
$750$	0	0
$751$	−40.4315	−1.47536	−0.737682	−	0.675148i	$-0.764080\pi$
$751$	−40.4315	−1.47536	−0.737682	+	0.675148i	$0.764080\pi$
$752$	0	0
$753$	10.7220	0.390732
$754$	0	0
$755$	−16.0846	−0.585377
$756$	0	0
$757$	38.2924	1.39176	0.695880	−	0.718158i	$-0.255015\pi$
$757$	38.2924	1.39176	0.695880	+	0.718158i	$0.255015\pi$
$758$	0	0
$759$	50.4047	1.82957
$760$	0	0
$761$	−19.7835	−0.717153	−0.358576	−	0.933500i	$-0.616738\pi$
$761$	−19.7835	−0.717153	−0.358576	+	0.933500i	$0.616738\pi$
$762$	0	0
$763$	18.2919	0.662210
$764$	0	0
$765$	0	0
$766$	0	0
$767$	40.1041	1.44807
$768$	0	0
$769$	−19.1039	−0.688905	−0.344453	−	0.938804i	$-0.611935\pi$
$769$	−19.1039	−0.688905	−0.344453	+	0.938804i	$0.611935\pi$
$770$	0	0
$771$	−34.3648	−1.23762
$772$	0	0
$773$	13.4065	0.482199	0.241099	−	0.970500i	$-0.422492\pi$
$773$	13.4065	0.482199	0.241099	+	0.970500i	$0.422492\pi$
$774$	0	0
$775$	−9.23751	−0.331821
$776$	0	0
$777$	34.2295	1.22797
$778$	0	0
$779$	−19.8924	−0.712718
$780$	0	0
$781$	−44.3765	−1.58792
$782$	0	0
$783$	2.20114	0.0786625
$784$	0	0
$785$	−10.4004	−0.371207
$786$	0	0
$787$	21.5832	0.769358	0.384679	−	0.923050i	$-0.374312\pi$
$787$	21.5832	0.769358	0.384679	+	0.923050i	$0.374312\pi$
$788$	0	0
$789$	0.660867	0.0235275
$790$	0	0
$791$	9.40644	0.334455
$792$	0	0
$793$	7.37614	0.261935
$794$	0	0
$795$	9.02811	0.320194
$796$	0	0
$797$	−33.8619	−1.19945	−0.599724	−	0.800207i	$-0.704723\pi$
$797$	−33.8619	−1.19945	−0.599724	+	0.800207i	$0.704723\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	91.4032	3.22957
$802$	0	0
$803$	−44.2601	−1.56191
$804$	0	0
$805$	9.35946	0.329878
$806$	0	0
$807$	12.6396	0.444934
$808$	0	0
$809$	−12.2572	−0.430939	−0.215470	−	0.976511i	$-0.569128\pi$
$809$	−12.2572	−0.430939	−0.215470	+	0.976511i	$0.569128\pi$
$810$	0	0
$811$	−15.1937	−0.533523	−0.266762	−	0.963763i	$-0.585954\pi$
$811$	−15.1937	−0.533523	−0.266762	+	0.963763i	$0.585954\pi$
$812$	0	0
$813$	58.4285	2.04918
$814$	0	0
$815$	20.4333	0.715748
$816$	0	0
$817$	0.485967	0.0170018
$818$	0	0
$819$	112.705	3.93823
$820$	0	0
$821$	−48.1232	−1.67951	−0.839756	−	0.542964i	$-0.817302\pi$
$821$	−48.1232	−1.67951	−0.839756	+	0.542964i	$0.817302\pi$
$822$	0	0
$823$	−35.2277	−1.22796	−0.613979	−	0.789322i	$-0.710432\pi$
$823$	−35.2277	−1.22796	−0.613979	+	0.789322i	$0.710432\pi$
$824$	0	0
$825$	−12.8197	−0.446326
$826$	0	0
$827$	40.1107	1.39479	0.697394	−	0.716688i	$-0.254343\pi$
$827$	40.1107	1.39479	0.697394	+	0.716688i	$0.254343\pi$
$828$	0	0
$829$	−25.3864	−0.881706	−0.440853	−	0.897579i	$-0.645324\pi$
$829$	−25.3864	−0.881706	−0.440853	+	0.897579i	$0.645324\pi$
$830$	0	0
$831$	0.874630	0.0303406
$832$	0	0
$833$	0	0
$834$	0	0
$835$	5.28012	0.182726
$836$	0	0
$837$	−120.050	−4.14952
$838$	0	0
$839$	−13.7401	−0.474361	−0.237181	−	0.971466i	$-0.576223\pi$
$839$	−13.7401	−0.474361	−0.237181	+	0.971466i	$0.576223\pi$
$840$	0	0
$841$	−28.9713	−0.999011
$842$	0	0
$843$	58.0582	1.99963
$844$	0	0
$845$	31.5804	1.08640
$846$	0	0
$847$	−12.5836	−0.432379
$848$	0	0
$849$	−43.5042	−1.49306
$850$	0	0
$851$	17.7977	0.610098
$852$	0	0
$853$	14.7610	0.505408	0.252704	−	0.967544i	$-0.418680\pi$
$853$	14.7610	0.505408	0.252704	+	0.967544i	$0.418680\pi$
$854$	0	0
$855$	11.7033	0.400244
$856$	0	0
$857$	19.3513	0.661029	0.330515	−	0.943801i	$-0.392778\pi$
$857$	19.3513	0.661029	0.330515	+	0.943801i	$0.392778\pi$
$858$	0	0
$859$	48.8863	1.66798	0.833990	−	0.551780i	$-0.186051\pi$
$859$	48.8863	1.66798	0.833990	+	0.551780i	$0.186051\pi$
$860$	0	0
$861$	91.1418	3.10611
$862$	0	0
$863$	−15.6077	−0.531292	−0.265646	−	0.964071i	$-0.585585\pi$
$863$	−15.6077	−0.531292	−0.265646	+	0.964071i	$0.585585\pi$
$864$	0	0
$865$	−13.9522	−0.474391
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−14.1954	−0.481547
$870$	0	0
$871$	−48.1833	−1.63263
$872$	0	0
$873$	−8.07915	−0.273438
$874$	0	0
$875$	−2.38045	−0.0804739
$876$	0	0
$877$	−15.4006	−0.520041	−0.260020	−	0.965603i	$-0.583729\pi$
$877$	−15.4006	−0.520041	−0.260020	+	0.965603i	$0.583729\pi$
$878$	0	0
$879$	−15.2798	−0.515375
$880$	0	0
$881$	−2.05316	−0.0691726	−0.0345863	−	0.999402i	$-0.511011\pi$
$881$	−2.05316	−0.0691726	−0.0345863	+	0.999402i	$0.511011\pi$
$882$	0	0
$883$	−55.0584	−1.85286	−0.926431	−	0.376464i	$-0.877140\pi$
$883$	−55.0584	−1.85286	−0.926431	+	0.376464i	$0.877140\pi$
$884$	0	0
$885$	−19.0803	−0.641377
$886$	0	0
$887$	32.7492	1.09961	0.549805	−	0.835293i	$-0.314702\pi$
$887$	32.7492	1.09961	0.549805	+	0.835293i	$0.314702\pi$
$888$	0	0
$889$	−49.6862	−1.66642
$890$	0	0
$891$	−80.7531	−2.70533
$892$	0	0
$893$	−1.37396	−0.0459777
$894$	0	0
$895$	15.1121	0.505142
$896$	0	0
$897$	83.3936	2.78443
$898$	0	0
$899$	−1.56458	−0.0521817
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−2.22658	−0.0740959
$904$	0	0
$905$	10.7281	0.356614
$906$	0	0
$907$	−50.1978	−1.66679	−0.833396	−	0.552676i	$-0.813607\pi$
$907$	−50.1978	−1.66679	−0.833396	+	0.552676i	$0.813607\pi$
$908$	0	0
$909$	9.96551	0.330535
$910$	0	0
$911$	48.4412	1.60493	0.802464	−	0.596700i	$-0.203522\pi$
$911$	48.4412	1.60493	0.802464	+	0.596700i	$0.203522\pi$
$912$	0	0
$913$	39.9694	1.32280
$914$	0	0
$915$	−3.50934	−0.116015
$916$	0	0
$917$	−24.5984	−0.812310
$918$	0	0
$919$	−16.9610	−0.559491	−0.279745	−	0.960074i	$-0.590250\pi$
$919$	−16.9610	−0.559491	−0.279745	+	0.960074i	$0.590250\pi$
$920$	0	0
$921$	31.2445	1.02954
$922$	0	0
$923$	−73.4201	−2.41665
$924$	0	0
$925$	−4.52660	−0.148834
$926$	0	0
$927$	−89.4109	−2.93664
$928$	0	0
$929$	−46.7262	−1.53304	−0.766518	−	0.642223i	$-0.778012\pi$
$929$	−46.7262	−1.53304	−0.766518	+	0.642223i	$0.778012\pi$
$930$	0	0
$931$	−2.20078	−0.0721277
$932$	0	0
$933$	−39.0287	−1.27774
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−15.1370	−0.494506	−0.247253	−	0.968951i	$-0.579528\pi$
$937$	−15.1370	−0.494506	−0.247253	+	0.968951i	$0.579528\pi$
$938$	0	0
$939$	92.5397	3.01992
$940$	0	0
$941$	45.1270	1.47110	0.735549	−	0.677472i	$-0.236924\pi$
$941$	45.1270	1.47110	0.735549	+	0.677472i	$0.236924\pi$
$942$	0	0
$943$	47.3895	1.54321
$944$	0	0
$945$	−30.9360	−1.00635
$946$	0	0
$947$	−37.3190	−1.21270	−0.606352	−	0.795196i	$-0.707368\pi$
$947$	−37.3190	−1.21270	−0.606352	+	0.795196i	$0.707368\pi$
$948$	0	0
$949$	−73.2275	−2.37707
$950$	0	0
$951$	−39.2678	−1.27335
$952$	0	0
$953$	−15.8268	−0.512679	−0.256340	−	0.966587i	$-0.582517\pi$
$953$	−15.8268	−0.512679	−0.256340	+	0.966587i	$0.582517\pi$
$954$	0	0
$955$	1.34618	0.0435614
$956$	0	0
$957$	−2.17131	−0.0701885
$958$	0	0
$959$	15.8939	0.513241
$960$	0	0
$961$	54.3316	1.75263
$962$	0	0
$963$	29.9059	0.963704
$964$	0	0
$965$	−18.4663	−0.594451
$966$	0	0
$967$	5.34018	0.171729	0.0858643	−	0.996307i	$-0.472635\pi$
$967$	5.34018	0.171729	0.0858643	+	0.996307i	$0.472635\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	33.3942	1.07167	0.535835	−	0.844323i	$-0.319997\pi$
$971$	33.3942	1.07167	0.535835	+	0.844323i	$0.319997\pi$
$972$	0	0
$973$	4.88100	0.156478
$974$	0	0
$975$	−21.2100	−0.679264
$976$	0	0
$977$	−23.1595	−0.740939	−0.370470	−	0.928845i	$-0.620803\pi$
$977$	−23.1595	−0.740939	−0.370470	+	0.928845i	$0.620803\pi$
$978$	0	0
$979$	−52.0187	−1.66253
$980$	0	0
$981$	−54.4892	−1.73971
$982$	0	0
$983$	−2.95585	−0.0942770	−0.0471385	−	0.998888i	$-0.515010\pi$
$983$	−2.95585	−0.0942770	−0.0471385	+	0.998888i	$0.515010\pi$
$984$	0	0
$985$	−26.0297	−0.829376
$986$	0	0
$987$	6.29512	0.200376
$988$	0	0
$989$	−1.15772	−0.0368132
$990$	0	0
$991$	−3.01179	−0.0956728	−0.0478364	−	0.998855i	$-0.515233\pi$
$991$	−3.01179	−0.0956728	−0.0478364	+	0.998855i	$0.515233\pi$
$992$	0	0
$993$	−66.6729	−2.11580
$994$	0	0
$995$	−9.20017	−0.291665
$996$	0	0
$997$	−27.6137	−0.874535	−0.437267	−	0.899332i	$-0.644054\pi$
$997$	−27.6137	−0.874535	−0.437267	+	0.899332i	$0.644054\pi$
$998$	0	0
$999$	−58.8272	−1.86121

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5780.2.a.n.1.6		6
17.2	even	8		340.2.o.a.21.6	✓	12
17.4	even	4		5780.2.c.h.5201.1		12
17.9	even	8		340.2.o.a.81.6	yes	12
17.13	even	4		5780.2.c.h.5201.12		12
17.16	even	2		5780.2.a.m.1.1		6
51.2	odd	8		3060.2.be.b.361.5		12
51.26	odd	8		3060.2.be.b.1441.5		12
68.19	odd	8		1360.2.bt.c.1041.1		12
68.43	odd	8		1360.2.bt.c.81.1		12
85.2	odd	8		1700.2.m.c.1449.1		12
85.9	even	8		1700.2.o.d.1101.1		12
85.19	even	8		1700.2.o.d.701.1		12
85.43	odd	8		1700.2.m.c.149.1		12
85.53	odd	8		1700.2.m.f.1449.6		12
85.77	odd	8		1700.2.m.f.149.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
340.2.o.a.21.6	✓	12	17.2	even	8
340.2.o.a.81.6	yes	12	17.9	even	8
1360.2.bt.c.81.1		12	68.43	odd	8
1360.2.bt.c.1041.1		12	68.19	odd	8
1700.2.m.c.149.1		12	85.43	odd	8
1700.2.m.c.1449.1		12	85.2	odd	8
1700.2.m.f.149.6		12	85.77	odd	8
1700.2.m.f.1449.6		12	85.53	odd	8
1700.2.o.d.701.1		12	85.19	even	8
1700.2.o.d.1101.1		12	85.9	even	8
3060.2.be.b.361.5		12	51.2	odd	8
3060.2.be.b.1441.5		12	51.26	odd	8
5780.2.a.m.1.1		6	17.16	even	2
5780.2.a.n.1.6		6	1.1	even	1	trivial
5780.2.c.h.5201.1		12	17.4	even	4
5780.2.c.h.5201.12		12	17.13	even	4

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:	\( N \)	\(=\)	\( 5780 = 2^{2} \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5780.a (trivial)

\(f(q)\)	\(=\)	\(q+3.17664 q^{3} +1.00000 q^{5} -2.38045 q^{7} +7.09107 q^{9} +O(q^{10})\)	\(q+3.17664 q^{3} +1.00000 q^{5} -2.38045 q^{7} +7.09107 q^{9} -4.03562 q^{11} -6.67686 q^{13} +3.17664 q^{15} +1.65043 q^{19} -7.56184 q^{21} -3.93180 q^{23} +1.00000 q^{25} +12.9959 q^{27} +0.169372 q^{29} -9.23751 q^{31} -12.8197 q^{33} -2.38045 q^{35} -4.52660 q^{37} -21.2100 q^{39} -12.0529 q^{41} +0.294449 q^{43} +7.09107 q^{45} -0.832485 q^{47} -1.33346 q^{49} +2.84203 q^{53} -4.03562 q^{55} +5.24282 q^{57} -6.00643 q^{59} -1.10473 q^{61} -16.8799 q^{63} -6.67686 q^{65} +7.21646 q^{67} -12.4899 q^{69} +10.9962 q^{71} +10.9674 q^{73} +3.17664 q^{75} +9.60659 q^{77} +3.51753 q^{79} +20.0101 q^{81} -9.90416 q^{83} +0.538036 q^{87} +12.8899 q^{89} +15.8939 q^{91} -29.3443 q^{93} +1.65043 q^{95} -1.13934 q^{97} -28.6169 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 6 * q + 6 * q^5 - 4 * q^7 + 2 * q^9	\( 6 q + 6 q^{5} - 4 q^{7} + 2 q^{9} - 8 q^{11} - 8 q^{13} + 12 q^{19} - 8 q^{21} - 4 q^{23} + 6 q^{25} + 24 q^{27} - 8 q^{29} - 20 q^{31} - 24 q^{33} - 4 q^{35} - 24 q^{37} - 20 q^{39} - 4 q^{41} - 4 q^{43} + 2 q^{45} + 4 q^{47} + 22 q^{49} - 8 q^{53} - 8 q^{55} + 16 q^{57} - 4 q^{61} - 16 q^{63} - 8 q^{65} + 16 q^{67} - 16 q^{69} - 16 q^{71} - 16 q^{73} - 24 q^{77} - 36 q^{79} + 2 q^{81} + 8 q^{83} + 28 q^{89} + 20 q^{91} - 12 q^{93} + 12 q^{95} - 32 q^{97} - 20 q^{99}+O(q^{100}) \) 6 * q + 6 * q^5 - 4 * q^7 + 2 * q^9 - 8 * q^11 - 8 * q^13 + 12 * q^19 - 8 * q^21 - 4 * q^23 + 6 * q^25 + 24 * q^27 - 8 * q^29 - 20 * q^31 - 24 * q^33 - 4 * q^35 - 24 * q^37 - 20 * q^39 - 4 * q^41 - 4 * q^43 + 2 * q^45 + 4 * q^47 + 22 * q^49 - 8 * q^53 - 8 * q^55 + 16 * q^57 - 4 * q^61 - 16 * q^63 - 8 * q^65 + 16 * q^67 - 16 * q^69 - 16 * q^71 - 16 * q^73 - 24 * q^77 - 36 * q^79 + 2 * q^81 + 8 * q^83 + 28 * q^89 + 20 * q^91 - 12 * q^93 + 12 * q^95 - 32 * q^97 - 20 * q^99

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