LMFDB - Embedded newform 6020.2.a.g.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6020,2,Mod(1,6020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6020, 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("6020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6020.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:	$48.0699420168$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 16x^{7} + 83x^{5} - 9x^{4} - 160x^{3} + 32x^{2} + 77x - 2 $ x^9 - 16x^7 + 83x^5 - 9x^4 - 160x^3 + 32x^2 + 77x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.10254$ of defining polynomial
Character	$\chi$	$=$	6020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.10254 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.78440 q^{9} +O(q^{10})$	$q+1.10254 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.78440 q^{9} -2.59618 q^{11} +0.469137 q^{13} +1.10254 q^{15} -1.96514 q^{17} +7.18094 q^{19} -1.10254 q^{21} -4.85120 q^{23} +1.00000 q^{25} -5.27501 q^{27} -1.48915 q^{29} +4.92960 q^{31} -2.86240 q^{33} -1.00000 q^{35} +1.57845 q^{37} +0.517244 q^{39} -0.309375 q^{41} -1.00000 q^{43} -1.78440 q^{45} -1.47190 q^{47} +1.00000 q^{49} -2.16666 q^{51} +5.99251 q^{53} -2.59618 q^{55} +7.91729 q^{57} +2.05341 q^{59} -11.8531 q^{61} +1.78440 q^{63} +0.469137 q^{65} -13.5909 q^{67} -5.34866 q^{69} +3.34794 q^{71} -12.3213 q^{73} +1.10254 q^{75} +2.59618 q^{77} -11.9025 q^{79} -0.462726 q^{81} -6.16341 q^{83} -1.96514 q^{85} -1.64186 q^{87} +7.09261 q^{89} -0.469137 q^{91} +5.43510 q^{93} +7.18094 q^{95} -12.9082 q^{97} +4.63261 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{5} - 9 q^{7} + 5 q^{9}+O(q^{10}) $ 9 * q + 9 * q^5 - 9 * q^7 + 5 * q^9	$ 9 q + 9 q^{5} - 9 q^{7} + 5 q^{9} + q^{11} - 14 q^{13} - 11 q^{17} - 2 q^{19} - 6 q^{23} + 9 q^{25} - 6 q^{29} + 6 q^{31} - 14 q^{33} - 9 q^{35} - 20 q^{37} - 6 q^{39} - 6 q^{41} - 9 q^{43} + 5 q^{45} + 9 q^{49} - 8 q^{51} - 31 q^{53} + q^{55} - 16 q^{57} + 2 q^{59} - 13 q^{61} - 5 q^{63} - 14 q^{65} - 10 q^{67} - 18 q^{69} + 12 q^{71} - 32 q^{73} - q^{77} + q^{79} - 27 q^{81} - 10 q^{83} - 11 q^{85} - 5 q^{87} - q^{89} + 14 q^{91} - 49 q^{93} - 2 q^{95} - 28 q^{97} - 14 q^{99}+O(q^{100}) $ 9 * q + 9 * q^5 - 9 * q^7 + 5 * q^9 + q^11 - 14 * q^13 - 11 * q^17 - 2 * q^19 - 6 * q^23 + 9 * q^25 - 6 * q^29 + 6 * q^31 - 14 * q^33 - 9 * q^35 - 20 * q^37 - 6 * q^39 - 6 * q^41 - 9 * q^43 + 5 * q^45 + 9 * q^49 - 8 * q^51 - 31 * q^53 + q^55 - 16 * q^57 + 2 * q^59 - 13 * q^61 - 5 * q^63 - 14 * q^65 - 10 * q^67 - 18 * q^69 + 12 * q^71 - 32 * q^73 - q^77 + q^79 - 27 * q^81 - 10 * q^83 - 11 * q^85 - 5 * q^87 - q^89 + 14 * q^91 - 49 * q^93 - 2 * q^95 - 28 * q^97 - 14 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.10254	0.636554	0.318277	−	0.947998i	$-0.396896\pi$
$3$	1.10254	0.636554	0.318277	+	0.947998i	$0.396896\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.78440	−0.594800
$10$	0	0
$11$	−2.59618	−0.782777	−0.391388	−	0.920226i	$-0.628005\pi$
$11$	−2.59618	−0.782777	−0.391388	+	0.920226i	$0.628005\pi$
$12$	0	0
$13$	0.469137	0.130115	0.0650576	−	0.997882i	$-0.479277\pi$
$13$	0.469137	0.130115	0.0650576	+	0.997882i	$0.479277\pi$
$14$	0	0
$15$	1.10254	0.284675
$16$	0	0
$17$	−1.96514	−0.476617	−0.238309	−	0.971189i	$-0.576593\pi$
$17$	−1.96514	−0.476617	−0.238309	+	0.971189i	$0.576593\pi$
$18$	0	0
$19$	7.18094	1.64742	0.823710	−	0.567011i	$-0.191900\pi$
$19$	7.18094	1.64742	0.823710	+	0.567011i	$0.191900\pi$
$20$	0	0
$21$	−1.10254	−0.240595
$22$	0	0
$23$	−4.85120	−1.01154	−0.505772	−	0.862667i	$-0.668792\pi$
$23$	−4.85120	−1.01154	−0.505772	+	0.862667i	$0.668792\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.27501	−1.01518
$28$	0	0
$29$	−1.48915	−0.276529	−0.138264	−	0.990395i	$-0.544152\pi$
$29$	−1.48915	−0.276529	−0.138264	+	0.990395i	$0.544152\pi$
$30$	0	0
$31$	4.92960	0.885382	0.442691	−	0.896674i	$-0.354024\pi$
$31$	4.92960	0.885382	0.442691	+	0.896674i	$0.354024\pi$
$32$	0	0
$33$	−2.86240	−0.498279
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	1.57845	0.259496	0.129748	−	0.991547i	$-0.458583\pi$
$37$	1.57845	0.259496	0.129748	+	0.991547i	$0.458583\pi$
$38$	0	0
$39$	0.517244	0.0828253
$40$	0	0
$41$	−0.309375	−0.0483163	−0.0241581	−	0.999708i	$-0.507691\pi$
$41$	−0.309375	−0.0483163	−0.0241581	+	0.999708i	$0.507691\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	−1.78440	−0.266002
$46$	0	0
$47$	−1.47190	−0.214699	−0.107350	−	0.994221i	$-0.534236\pi$
$47$	−1.47190	−0.214699	−0.107350	+	0.994221i	$0.534236\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.16666	−0.303392
$52$	0	0
$53$	5.99251	0.823135	0.411567	−	0.911379i	$-0.364981\pi$
$53$	5.99251	0.823135	0.411567	+	0.911379i	$0.364981\pi$
$54$	0	0
$55$	−2.59618	−0.350068
$56$	0	0
$57$	7.91729	1.04867
$58$	0	0
$59$	2.05341	0.267331	0.133666	−	0.991026i	$-0.457325\pi$
$59$	2.05341	0.267331	0.133666	+	0.991026i	$0.457325\pi$
$60$	0	0
$61$	−11.8531	−1.51763	−0.758816	−	0.651305i	$-0.774222\pi$
$61$	−11.8531	−1.51763	−0.758816	+	0.651305i	$0.774222\pi$
$62$	0	0
$63$	1.78440	0.224813
$64$	0	0
$65$	0.469137	0.0581893
$66$	0	0
$67$	−13.5909	−1.66039	−0.830194	−	0.557474i	$-0.811771\pi$
$67$	−13.5909	−1.66039	−0.830194	+	0.557474i	$0.811771\pi$
$68$	0	0
$69$	−5.34866	−0.643903
$70$	0	0
$71$	3.34794	0.397327	0.198664	−	0.980068i	$-0.436340\pi$
$71$	3.34794	0.397327	0.198664	+	0.980068i	$0.436340\pi$
$72$	0	0
$73$	−12.3213	−1.44210	−0.721048	−	0.692885i	$-0.756339\pi$
$73$	−12.3213	−1.44210	−0.721048	+	0.692885i	$0.756339\pi$
$74$	0	0
$75$	1.10254	0.127311
$76$	0	0
$77$	2.59618	0.295862
$78$	0	0
$79$	−11.9025	−1.33914	−0.669568	−	0.742751i	$-0.733521\pi$
$79$	−11.9025	−1.33914	−0.669568	+	0.742751i	$0.733521\pi$
$80$	0	0
$81$	−0.462726	−0.0514140
$82$	0	0
$83$	−6.16341	−0.676522	−0.338261	−	0.941052i	$-0.609839\pi$
$83$	−6.16341	−0.676522	−0.338261	+	0.941052i	$0.609839\pi$
$84$	0	0
$85$	−1.96514	−0.213150
$86$	0	0
$87$	−1.64186	−0.176025
$88$	0	0
$89$	7.09261	0.751816	0.375908	−	0.926657i	$-0.377331\pi$
$89$	7.09261	0.751816	0.375908	+	0.926657i	$0.377331\pi$
$90$	0	0
$91$	−0.469137	−0.0491789
$92$	0	0
$93$	5.43510	0.563593
$94$	0	0
$95$	7.18094	0.736749
$96$	0	0
$97$	−12.9082	−1.31062	−0.655312	−	0.755358i	$-0.727463\pi$
$97$	−12.9082	−1.31062	−0.655312	+	0.755358i	$0.727463\pi$
$98$	0	0
$99$	4.63261	0.465595
$100$	0	0
$101$	−10.0096	−0.995990	−0.497995	−	0.867180i	$-0.665930\pi$
$101$	−10.0096	−0.995990	−0.497995	+	0.867180i	$0.665930\pi$
$102$	0	0
$103$	−11.0019	−1.08405	−0.542026	−	0.840362i	$-0.682343\pi$
$103$	−11.0019	−1.08405	−0.542026	+	0.840362i	$0.682343\pi$
$104$	0	0
$105$	−1.10254	−0.107597
$106$	0	0
$107$	3.80036	0.367395	0.183697	−	0.982983i	$-0.441193\pi$
$107$	3.80036	0.367395	0.183697	+	0.982983i	$0.441193\pi$
$108$	0	0
$109$	6.53836	0.626262	0.313131	−	0.949710i	$-0.398622\pi$
$109$	6.53836	0.626262	0.313131	+	0.949710i	$0.398622\pi$
$110$	0	0
$111$	1.74031	0.165183
$112$	0	0
$113$	2.29421	0.215821	0.107911	−	0.994161i	$-0.465584\pi$
$113$	2.29421	0.215821	0.107911	+	0.994161i	$0.465584\pi$
$114$	0	0
$115$	−4.85120	−0.452377
$116$	0	0
$117$	−0.837127	−0.0773925
$118$	0	0
$119$	1.96514	0.180144
$120$	0	0
$121$	−4.25987	−0.387261
$122$	0	0
$123$	−0.341099	−0.0307559
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.120671	0.0107078	0.00535392	−	0.999986i	$-0.498296\pi$
$127$	0.120671	0.0107078	0.00535392	+	0.999986i	$0.498296\pi$
$128$	0	0
$129$	−1.10254	−0.0970735
$130$	0	0
$131$	12.5472	1.09625	0.548127	−	0.836395i	$-0.315341\pi$
$131$	12.5472	1.09625	0.548127	+	0.836395i	$0.315341\pi$
$132$	0	0
$133$	−7.18094	−0.622666
$134$	0	0
$135$	−5.27501	−0.454000
$136$	0	0
$137$	−4.99506	−0.426757	−0.213379	−	0.976970i	$-0.568447\pi$
$137$	−4.99506	−0.426757	−0.213379	+	0.976970i	$0.568447\pi$
$138$	0	0
$139$	−13.7101	−1.16287	−0.581437	−	0.813591i	$-0.697509\pi$
$139$	−13.7101	−1.16287	−0.581437	+	0.813591i	$0.697509\pi$
$140$	0	0
$141$	−1.62284	−0.136668
$142$	0	0
$143$	−1.21796	−0.101851
$144$	0	0
$145$	−1.48915	−0.123667
$146$	0	0
$147$	1.10254	0.0909362
$148$	0	0
$149$	−0.413647	−0.0338873	−0.0169436	−	0.999856i	$-0.505394\pi$
$149$	−0.413647	−0.0338873	−0.0169436	+	0.999856i	$0.505394\pi$
$150$	0	0
$151$	−22.0194	−1.79191	−0.895956	−	0.444143i	$-0.853508\pi$
$151$	−22.0194	−1.79191	−0.895956	+	0.444143i	$0.853508\pi$
$152$	0	0
$153$	3.50660	0.283492
$154$	0	0
$155$	4.92960	0.395955
$156$	0	0
$157$	−1.44739	−0.115514	−0.0577571	−	0.998331i	$-0.518395\pi$
$157$	−1.44739	−0.115514	−0.0577571	+	0.998331i	$0.518395\pi$
$158$	0	0
$159$	6.60700	0.523970
$160$	0	0
$161$	4.85120	0.382328
$162$	0	0
$163$	−0.893537	−0.0699872	−0.0349936	−	0.999388i	$-0.511141\pi$
$163$	−0.893537	−0.0699872	−0.0349936	+	0.999388i	$0.511141\pi$
$164$	0	0
$165$	−2.86240	−0.222837
$166$	0	0
$167$	−9.00700	−0.696983	−0.348491	−	0.937312i	$-0.613306\pi$
$167$	−9.00700	−0.696983	−0.348491	+	0.937312i	$0.613306\pi$
$168$	0	0
$169$	−12.7799	−0.983070
$170$	0	0
$171$	−12.8137	−0.979885
$172$	0	0
$173$	−26.0495	−1.98051	−0.990253	−	0.139277i	$-0.955522\pi$
$173$	−26.0495	−1.98051	−0.990253	+	0.139277i	$0.955522\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	2.26397	0.170171
$178$	0	0
$179$	−1.84193	−0.137672	−0.0688362	−	0.997628i	$-0.521929\pi$
$179$	−1.84193	−0.137672	−0.0688362	+	0.997628i	$0.521929\pi$
$180$	0	0
$181$	−22.0050	−1.63562	−0.817808	−	0.575491i	$-0.804811\pi$
$181$	−22.0050	−1.63562	−0.817808	+	0.575491i	$0.804811\pi$
$182$	0	0
$183$	−13.0685	−0.966054
$184$	0	0
$185$	1.57845	0.116050
$186$	0	0
$187$	5.10186	0.373085
$188$	0	0
$189$	5.27501	0.383700
$190$	0	0
$191$	10.5631	0.764319	0.382160	−	0.924096i	$-0.375180\pi$
$191$	10.5631	0.764319	0.382160	+	0.924096i	$0.375180\pi$
$192$	0	0
$193$	16.2250	1.16790	0.583952	−	0.811788i	$-0.301506\pi$
$193$	16.2250	1.16790	0.583952	+	0.811788i	$0.301506\pi$
$194$	0	0
$195$	0.517244	0.0370406
$196$	0	0
$197$	8.45134	0.602133	0.301067	−	0.953603i	$-0.402657\pi$
$197$	8.45134	0.602133	0.301067	+	0.953603i	$0.402657\pi$
$198$	0	0
$199$	7.73797	0.548530	0.274265	−	0.961654i	$-0.411565\pi$
$199$	7.73797	0.548530	0.274265	+	0.961654i	$0.411565\pi$
$200$	0	0
$201$	−14.9845	−1.05693
$202$	0	0
$203$	1.48915	0.104518
$204$	0	0
$205$	−0.309375	−0.0216077
$206$	0	0
$207$	8.65647	0.601666
$208$	0	0
$209$	−18.6430	−1.28956
$210$	0	0
$211$	2.87990	0.198260	0.0991302	−	0.995074i	$-0.468394\pi$
$211$	2.87990	0.198260	0.0991302	+	0.995074i	$0.468394\pi$
$212$	0	0
$213$	3.69125	0.252920
$214$	0	0
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	−4.92960	−0.334643
$218$	0	0
$219$	−13.5847	−0.917972
$220$	0	0
$221$	−0.921921	−0.0620151
$222$	0	0
$223$	−25.2999	−1.69421	−0.847104	−	0.531427i	$-0.821656\pi$
$223$	−25.2999	−1.69421	−0.847104	+	0.531427i	$0.821656\pi$
$224$	0	0
$225$	−1.78440	−0.118960
$226$	0	0
$227$	10.3014	0.683730	0.341865	−	0.939749i	$-0.388941\pi$
$227$	10.3014	0.683730	0.341865	+	0.939749i	$0.388941\pi$
$228$	0	0
$229$	−5.43377	−0.359073	−0.179537	−	0.983751i	$-0.557460\pi$
$229$	−5.43377	−0.359073	−0.179537	+	0.983751i	$0.557460\pi$
$230$	0	0
$231$	2.86240	0.188332
$232$	0	0
$233$	−1.09840	−0.0719584	−0.0359792	−	0.999353i	$-0.511455\pi$
$233$	−1.09840	−0.0719584	−0.0359792	+	0.999353i	$0.511455\pi$
$234$	0	0
$235$	−1.47190	−0.0960165
$236$	0	0
$237$	−13.1230	−0.852432
$238$	0	0
$239$	12.2795	0.794294	0.397147	−	0.917755i	$-0.370000\pi$
$239$	12.2795	0.794294	0.397147	+	0.917755i	$0.370000\pi$
$240$	0	0
$241$	−15.8434	−1.02056	−0.510280	−	0.860008i	$-0.670458\pi$
$241$	−15.8434	−1.02056	−0.510280	+	0.860008i	$0.670458\pi$
$242$	0	0
$243$	15.3148	0.982448
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	3.36884	0.214354
$248$	0	0
$249$	−6.79543	−0.430643
$250$	0	0
$251$	12.3318	0.778373	0.389187	−	0.921159i	$-0.372756\pi$
$251$	12.3318	0.778373	0.389187	+	0.921159i	$0.372756\pi$
$252$	0	0
$253$	12.5946	0.791814
$254$	0	0
$255$	−2.16666	−0.135681
$256$	0	0
$257$	11.3984	0.711015	0.355507	−	0.934673i	$-0.384308\pi$
$257$	11.3984	0.711015	0.355507	+	0.934673i	$0.384308\pi$
$258$	0	0
$259$	−1.57845	−0.0980802
$260$	0	0
$261$	2.65724	0.164479
$262$	0	0
$263$	6.27777	0.387104	0.193552	−	0.981090i	$-0.437999\pi$
$263$	6.27777	0.387104	0.193552	+	0.981090i	$0.437999\pi$
$264$	0	0
$265$	5.99251	0.368117
$266$	0	0
$267$	7.81991	0.478571
$268$	0	0
$269$	14.4173	0.879036	0.439518	−	0.898234i	$-0.355149\pi$
$269$	14.4173	0.879036	0.439518	+	0.898234i	$0.355149\pi$
$270$	0	0
$271$	16.9507	1.02968	0.514840	−	0.857286i	$-0.327851\pi$
$271$	16.9507	1.02968	0.514840	+	0.857286i	$0.327851\pi$
$272$	0	0
$273$	−0.517244	−0.0313050
$274$	0	0
$275$	−2.59618	−0.156555
$276$	0	0
$277$	16.8443	1.01208	0.506038	−	0.862511i	$-0.331110\pi$
$277$	16.8443	1.01208	0.506038	+	0.862511i	$0.331110\pi$
$278$	0	0
$279$	−8.79637	−0.526625
$280$	0	0
$281$	−30.4281	−1.81519	−0.907594	−	0.419848i	$-0.862083\pi$
$281$	−30.4281	−1.81519	−0.907594	+	0.419848i	$0.862083\pi$
$282$	0	0
$283$	−5.20987	−0.309694	−0.154847	−	0.987938i	$-0.549489\pi$
$283$	−5.20987	−0.309694	−0.154847	+	0.987938i	$0.549489\pi$
$284$	0	0
$285$	7.91729	0.468980
$286$	0	0
$287$	0.309375	0.0182618
$288$	0	0
$289$	−13.1382	−0.772836
$290$	0	0
$291$	−14.2318	−0.834283
$292$	0	0
$293$	20.6168	1.20444	0.602222	−	0.798329i	$-0.294282\pi$
$293$	20.6168	1.20444	0.602222	+	0.798329i	$0.294282\pi$
$294$	0	0
$295$	2.05341	0.119554
$296$	0	0
$297$	13.6948	0.794656
$298$	0	0
$299$	−2.27588	−0.131617
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	−11.0360	−0.634001
$304$	0	0
$305$	−11.8531	−0.678706
$306$	0	0
$307$	23.5926	1.34650	0.673250	−	0.739415i	$-0.264898\pi$
$307$	23.5926	1.34650	0.673250	+	0.739415i	$0.264898\pi$
$308$	0	0
$309$	−12.1301	−0.690057
$310$	0	0
$311$	11.3815	0.645388	0.322694	−	0.946503i	$-0.395412\pi$
$311$	11.3815	0.645388	0.322694	+	0.946503i	$0.395412\pi$
$312$	0	0
$313$	6.60746	0.373476	0.186738	−	0.982410i	$-0.440208\pi$
$313$	6.60746	0.373476	0.186738	+	0.982410i	$0.440208\pi$
$314$	0	0
$315$	1.78440	0.100539
$316$	0	0
$317$	−26.0656	−1.46399	−0.731995	−	0.681310i	$-0.761411\pi$
$317$	−26.0656	−1.46399	−0.731995	+	0.681310i	$0.761411\pi$
$318$	0	0
$319$	3.86610	0.216460
$320$	0	0
$321$	4.19006	0.233867
$322$	0	0
$323$	−14.1116	−0.785189
$324$	0	0
$325$	0.469137	0.0260230
$326$	0	0
$327$	7.20883	0.398649
$328$	0	0
$329$	1.47190	0.0811487
$330$	0	0
$331$	−2.89912	−0.159350	−0.0796750	−	0.996821i	$-0.525388\pi$
$331$	−2.89912	−0.159350	−0.0796750	+	0.996821i	$0.525388\pi$
$332$	0	0
$333$	−2.81659	−0.154348
$334$	0	0
$335$	−13.5909	−0.742548
$336$	0	0
$337$	13.2437	0.721432	0.360716	−	0.932676i	$-0.382532\pi$
$337$	13.2437	0.721432	0.360716	+	0.932676i	$0.382532\pi$
$338$	0	0
$339$	2.52947	0.137382
$340$	0	0
$341$	−12.7981	−0.693057
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−5.34866	−0.287962
$346$	0	0
$347$	23.3362	1.25275	0.626376	−	0.779521i	$-0.284537\pi$
$347$	23.3362	1.25275	0.626376	+	0.779521i	$0.284537\pi$
$348$	0	0
$349$	19.2522	1.03055	0.515273	−	0.857026i	$-0.327691\pi$
$349$	19.2522	1.03055	0.515273	+	0.857026i	$0.327691\pi$
$350$	0	0
$351$	−2.47470	−0.132090
$352$	0	0
$353$	−27.5588	−1.46681	−0.733404	−	0.679793i	$-0.762069\pi$
$353$	−27.5588	−1.46681	−0.733404	+	0.679793i	$0.762069\pi$
$354$	0	0
$355$	3.34794	0.177690
$356$	0	0
$357$	2.16666	0.114672
$358$	0	0
$359$	1.65514	0.0873551	0.0436776	−	0.999046i	$-0.486093\pi$
$359$	1.65514	0.0873551	0.0436776	+	0.999046i	$0.486093\pi$
$360$	0	0
$361$	32.5659	1.71399
$362$	0	0
$363$	−4.69669	−0.246512
$364$	0	0
$365$	−12.3213	−0.644925
$366$	0	0
$367$	32.1942	1.68052	0.840261	−	0.542181i	$-0.182401\pi$
$367$	32.1942	1.68052	0.840261	+	0.542181i	$0.182401\pi$
$368$	0	0
$369$	0.552049	0.0287385
$370$	0	0
$371$	−5.99251	−0.311116
$372$	0	0
$373$	14.1883	0.734644	0.367322	−	0.930094i	$-0.380275\pi$
$373$	14.1883	0.734644	0.367322	+	0.930094i	$0.380275\pi$
$374$	0	0
$375$	1.10254	0.0569351
$376$	0	0
$377$	−0.698617	−0.0359806
$378$	0	0
$379$	3.46249	0.177856	0.0889282	−	0.996038i	$-0.471656\pi$
$379$	3.46249	0.177856	0.0889282	+	0.996038i	$0.471656\pi$
$380$	0	0
$381$	0.133045	0.00681612
$382$	0	0
$383$	−15.4407	−0.788985	−0.394493	−	0.918899i	$-0.629080\pi$
$383$	−15.4407	−0.788985	−0.394493	+	0.918899i	$0.629080\pi$
$384$	0	0
$385$	2.59618	0.132313
$386$	0	0
$387$	1.78440	0.0907061
$388$	0	0
$389$	30.0592	1.52406	0.762031	−	0.647541i	$-0.224203\pi$
$389$	30.0592	1.52406	0.762031	+	0.647541i	$0.224203\pi$
$390$	0	0
$391$	9.53330	0.482120
$392$	0	0
$393$	13.8338	0.697824
$394$	0	0
$395$	−11.9025	−0.598880
$396$	0	0
$397$	5.51386	0.276733	0.138366	−	0.990381i	$-0.455815\pi$
$397$	5.51386	0.276733	0.138366	+	0.990381i	$0.455815\pi$
$398$	0	0
$399$	−7.91729	−0.396360
$400$	0	0
$401$	−1.24454	−0.0621495	−0.0310748	−	0.999517i	$-0.509893\pi$
$401$	−1.24454	−0.0621495	−0.0310748	+	0.999517i	$0.509893\pi$
$402$	0	0
$403$	2.31266	0.115202
$404$	0	0
$405$	−0.462726	−0.0229931
$406$	0	0
$407$	−4.09794	−0.203127
$408$	0	0
$409$	−11.7100	−0.579021	−0.289510	−	0.957175i	$-0.593492\pi$
$409$	−11.7100	−0.579021	−0.289510	+	0.957175i	$0.593492\pi$
$410$	0	0
$411$	−5.50727	−0.271654
$412$	0	0
$413$	−2.05341	−0.101042
$414$	0	0
$415$	−6.16341	−0.302550
$416$	0	0
$417$	−15.1160	−0.740232
$418$	0	0
$419$	−5.57963	−0.272583	−0.136291	−	0.990669i	$-0.543518\pi$
$419$	−5.57963	−0.272583	−0.136291	+	0.990669i	$0.543518\pi$
$420$	0	0
$421$	14.7518	0.718959	0.359479	−	0.933153i	$-0.382954\pi$
$421$	14.7518	0.718959	0.359479	+	0.933153i	$0.382954\pi$
$422$	0	0
$423$	2.62646	0.127703
$424$	0	0
$425$	−1.96514	−0.0953234
$426$	0	0
$427$	11.8531	0.573611
$428$	0	0
$429$	−1.34286	−0.0648337
$430$	0	0
$431$	28.0654	1.35186	0.675931	−	0.736965i	$-0.263742\pi$
$431$	28.0654	1.35186	0.675931	+	0.736965i	$0.263742\pi$
$432$	0	0
$433$	−33.1186	−1.59158	−0.795789	−	0.605574i	$-0.792944\pi$
$433$	−33.1186	−1.59158	−0.795789	+	0.605574i	$0.792944\pi$
$434$	0	0
$435$	−1.64186	−0.0787209
$436$	0	0
$437$	−34.8362	−1.66644
$438$	0	0
$439$	−16.5460	−0.789695	−0.394848	−	0.918747i	$-0.629203\pi$
$439$	−16.5460	−0.789695	−0.394848	+	0.918747i	$0.629203\pi$
$440$	0	0
$441$	−1.78440	−0.0849714
$442$	0	0
$443$	17.4568	0.829395	0.414698	−	0.909959i	$-0.363887\pi$
$443$	17.4568	0.829395	0.414698	+	0.909959i	$0.363887\pi$
$444$	0	0
$445$	7.09261	0.336222
$446$	0	0
$447$	−0.456064	−0.0215711
$448$	0	0
$449$	1.09396	0.0516274	0.0258137	−	0.999667i	$-0.491782\pi$
$449$	1.09396	0.0516274	0.0258137	+	0.999667i	$0.491782\pi$
$450$	0	0
$451$	0.803192	0.0378209
$452$	0	0
$453$	−24.2773	−1.14065
$454$	0	0
$455$	−0.469137	−0.0219935
$456$	0	0
$457$	1.31432	0.0614813	0.0307407	−	0.999527i	$-0.490213\pi$
$457$	1.31432	0.0614813	0.0307407	+	0.999527i	$0.490213\pi$
$458$	0	0
$459$	10.3661	0.483850
$460$	0	0
$461$	35.0581	1.63282	0.816408	−	0.577475i	$-0.195962\pi$
$461$	35.0581	1.63282	0.816408	+	0.577475i	$0.195962\pi$
$462$	0	0
$463$	−9.68092	−0.449910	−0.224955	−	0.974369i	$-0.572224\pi$
$463$	−9.68092	−0.449910	−0.224955	+	0.974369i	$0.572224\pi$
$464$	0	0
$465$	5.43510	0.252047
$466$	0	0
$467$	37.2285	1.72273	0.861366	−	0.507985i	$-0.169610\pi$
$467$	37.2285	1.72273	0.861366	+	0.507985i	$0.169610\pi$
$468$	0	0
$469$	13.5909	0.627568
$470$	0	0
$471$	−1.59581	−0.0735309
$472$	0	0
$473$	2.59618	0.119372
$474$	0	0
$475$	7.18094	0.329484
$476$	0	0
$477$	−10.6930	−0.489600
$478$	0	0
$479$	8.76771	0.400607	0.200303	−	0.979734i	$-0.435807\pi$
$479$	8.76771	0.400607	0.200303	+	0.979734i	$0.435807\pi$
$480$	0	0
$481$	0.740510	0.0337644
$482$	0	0
$483$	5.34866	0.243372
$484$	0	0
$485$	−12.9082	−0.586129
$486$	0	0
$487$	−12.2662	−0.555836	−0.277918	−	0.960605i	$-0.589644\pi$
$487$	−12.2662	−0.555836	−0.277918	+	0.960605i	$0.589644\pi$
$488$	0	0
$489$	−0.985163	−0.0445506
$490$	0	0
$491$	−20.4503	−0.922911	−0.461456	−	0.887163i	$-0.652673\pi$
$491$	−20.4503	−0.922911	−0.461456	+	0.887163i	$0.652673\pi$
$492$	0	0
$493$	2.92640	0.131798
$494$	0	0
$495$	4.63261	0.208220
$496$	0	0
$497$	−3.34794	−0.150176
$498$	0	0
$499$	−13.6225	−0.609828	−0.304914	−	0.952380i	$-0.598628\pi$
$499$	−13.6225	−0.609828	−0.304914	+	0.952380i	$0.598628\pi$
$500$	0	0
$501$	−9.93061	−0.443667
$502$	0	0
$503$	−31.7270	−1.41464	−0.707319	−	0.706894i	$-0.750096\pi$
$503$	−31.7270	−1.41464	−0.707319	+	0.706894i	$0.750096\pi$
$504$	0	0
$505$	−10.0096	−0.445420
$506$	0	0
$507$	−14.0904	−0.625777
$508$	0	0
$509$	−7.49340	−0.332139	−0.166070	−	0.986114i	$-0.553108\pi$
$509$	−7.49340	−0.332139	−0.166070	+	0.986114i	$0.553108\pi$
$510$	0	0
$511$	12.3213	0.545061
$512$	0	0
$513$	−37.8795	−1.67242
$514$	0	0
$515$	−11.0019	−0.484803
$516$	0	0
$517$	3.82132	0.168062
$518$	0	0
$519$	−28.7207	−1.26070
$520$	0	0
$521$	10.8193	0.474001	0.237000	−	0.971510i	$-0.423836\pi$
$521$	10.8193	0.474001	0.237000	+	0.971510i	$0.423836\pi$
$522$	0	0
$523$	−12.0792	−0.528188	−0.264094	−	0.964497i	$-0.585073\pi$
$523$	−12.0792	−0.528188	−0.264094	+	0.964497i	$0.585073\pi$
$524$	0	0
$525$	−1.10254	−0.0481189
$526$	0	0
$527$	−9.68737	−0.421988
$528$	0	0
$529$	0.534135	0.0232233
$530$	0	0
$531$	−3.66410	−0.159009
$532$	0	0
$533$	−0.145139	−0.00628668
$534$	0	0
$535$	3.80036	0.164304
$536$	0	0
$537$	−2.03081	−0.0876359
$538$	0	0
$539$	−2.59618	−0.111825
$540$	0	0
$541$	−26.3423	−1.13254	−0.566271	−	0.824219i	$-0.691614\pi$
$541$	−26.3423	−1.13254	−0.566271	+	0.824219i	$0.691614\pi$
$542$	0	0
$543$	−24.2614	−1.04116
$544$	0	0
$545$	6.53836	0.280073
$546$	0	0
$547$	10.2573	0.438571	0.219286	−	0.975661i	$-0.429627\pi$
$547$	10.2573	0.438571	0.219286	+	0.975661i	$0.429627\pi$
$548$	0	0
$549$	21.1506	0.902687
$550$	0	0
$551$	−10.6935	−0.455559
$552$	0	0
$553$	11.9025	0.506146
$554$	0	0
$555$	1.74031	0.0738721
$556$	0	0
$557$	−37.7429	−1.59922	−0.799609	−	0.600521i	$-0.794960\pi$
$557$	−37.7429	−1.59922	−0.799609	+	0.600521i	$0.794960\pi$
$558$	0	0
$559$	−0.469137	−0.0198424
$560$	0	0
$561$	5.62502	0.237488
$562$	0	0
$563$	−42.7934	−1.80353	−0.901763	−	0.432230i	$-0.857727\pi$
$563$	−42.7934	−1.80353	−0.901763	+	0.432230i	$0.857727\pi$
$564$	0	0
$565$	2.29421	0.0965183
$566$	0	0
$567$	0.462726	0.0194327
$568$	0	0
$569$	−37.0789	−1.55443	−0.777215	−	0.629235i	$-0.783368\pi$
$569$	−37.0789	−1.55443	−0.777215	+	0.629235i	$0.783368\pi$
$570$	0	0
$571$	13.4810	0.564164	0.282082	−	0.959390i	$-0.408975\pi$
$571$	13.4810	0.564164	0.282082	+	0.959390i	$0.408975\pi$
$572$	0	0
$573$	11.6463	0.486530
$574$	0	0
$575$	−4.85120	−0.202309
$576$	0	0
$577$	−6.12075	−0.254810	−0.127405	−	0.991851i	$-0.540665\pi$
$577$	−6.12075	−0.254810	−0.127405	+	0.991851i	$0.540665\pi$
$578$	0	0
$579$	17.8888	0.743433
$580$	0	0
$581$	6.16341	0.255701
$582$	0	0
$583$	−15.5576	−0.644331
$584$	0	0
$585$	−0.837127	−0.0346110
$586$	0	0
$587$	39.2551	1.62023	0.810116	−	0.586270i	$-0.199404\pi$
$587$	39.2551	1.62023	0.810116	+	0.586270i	$0.199404\pi$
$588$	0	0
$589$	35.3992	1.45860
$590$	0	0
$591$	9.31797	0.383290
$592$	0	0
$593$	−28.7255	−1.17962	−0.589808	−	0.807544i	$-0.700796\pi$
$593$	−28.7255	−1.17962	−0.589808	+	0.807544i	$0.700796\pi$
$594$	0	0
$595$	1.96514	0.0805630
$596$	0	0
$597$	8.53144	0.349169
$598$	0	0
$599$	7.31656	0.298947	0.149473	−	0.988766i	$-0.452242\pi$
$599$	7.31656	0.298947	0.149473	+	0.988766i	$0.452242\pi$
$600$	0	0
$601$	36.5056	1.48910	0.744548	−	0.667569i	$-0.232665\pi$
$601$	36.5056	1.48910	0.744548	+	0.667569i	$0.232665\pi$
$602$	0	0
$603$	24.2515	0.987598
$604$	0	0
$605$	−4.25987	−0.173188
$606$	0	0
$607$	0.797053	0.0323514	0.0161757	−	0.999869i	$-0.494851\pi$
$607$	0.797053	0.0323514	0.0161757	+	0.999869i	$0.494851\pi$
$608$	0	0
$609$	1.64186	0.0665313
$610$	0	0
$611$	−0.690525	−0.0279357
$612$	0	0
$613$	−3.56331	−0.143921	−0.0719604	−	0.997407i	$-0.522926\pi$
$613$	−3.56331	−0.143921	−0.0719604	+	0.997407i	$0.522926\pi$
$614$	0	0
$615$	−0.341099	−0.0137545
$616$	0	0
$617$	3.90186	0.157083	0.0785415	−	0.996911i	$-0.474974\pi$
$617$	3.90186	0.157083	0.0785415	+	0.996911i	$0.474974\pi$
$618$	0	0
$619$	6.03603	0.242609	0.121304	−	0.992615i	$-0.461292\pi$
$619$	6.03603	0.242609	0.121304	+	0.992615i	$0.461292\pi$
$620$	0	0
$621$	25.5901	1.02690
$622$	0	0
$623$	−7.09261	−0.284160
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−20.5547	−0.820875
$628$	0	0
$629$	−3.10188	−0.123680
$630$	0	0
$631$	5.20375	0.207158	0.103579	−	0.994621i	$-0.466971\pi$
$631$	5.20375	0.207158	0.103579	+	0.994621i	$0.466971\pi$
$632$	0	0
$633$	3.17521	0.126203
$634$	0	0
$635$	0.120671	0.00478869
$636$	0	0
$637$	0.469137	0.0185879
$638$	0	0
$639$	−5.97406	−0.236330
$640$	0	0
$641$	20.1475	0.795779	0.397890	−	0.917433i	$-0.369743\pi$
$641$	20.1475	0.795779	0.397890	+	0.917433i	$0.369743\pi$
$642$	0	0
$643$	16.1420	0.636577	0.318289	−	0.947994i	$-0.396892\pi$
$643$	16.1420	0.636577	0.318289	+	0.947994i	$0.396892\pi$
$644$	0	0
$645$	−1.10254	−0.0434126
$646$	0	0
$647$	−6.28895	−0.247244	−0.123622	−	0.992329i	$-0.539451\pi$
$647$	−6.28895	−0.247244	−0.123622	+	0.992329i	$0.539451\pi$
$648$	0	0
$649$	−5.33102	−0.209261
$650$	0	0
$651$	−5.43510	−0.213018
$652$	0	0
$653$	6.55960	0.256697	0.128349	−	0.991729i	$-0.459032\pi$
$653$	6.55960	0.256697	0.128349	+	0.991729i	$0.459032\pi$
$654$	0	0
$655$	12.5472	0.490259
$656$	0	0
$657$	21.9861	0.857758
$658$	0	0
$659$	17.2518	0.672035	0.336018	−	0.941856i	$-0.390920\pi$
$659$	17.2518	0.672035	0.336018	+	0.941856i	$0.390920\pi$
$660$	0	0
$661$	−36.6394	−1.42511	−0.712554	−	0.701617i	$-0.752462\pi$
$661$	−36.6394	−1.42511	−0.712554	+	0.701617i	$0.752462\pi$
$662$	0	0
$663$	−1.01646	−0.0394760
$664$	0	0
$665$	−7.18094	−0.278465
$666$	0	0
$667$	7.22418	0.279721
$668$	0	0
$669$	−27.8942	−1.07845
$670$	0	0
$671$	30.7727	1.18797
$672$	0	0
$673$	−43.6204	−1.68144	−0.840721	−	0.541469i	$-0.817869\pi$
$673$	−43.6204	−1.68144	−0.840721	+	0.541469i	$0.817869\pi$
$674$	0	0
$675$	−5.27501	−0.203035
$676$	0	0
$677$	−5.16917	−0.198667	−0.0993336	−	0.995054i	$-0.531671\pi$
$677$	−5.16917	−0.198667	−0.0993336	+	0.995054i	$0.531671\pi$
$678$	0	0
$679$	12.9082	0.495370
$680$	0	0
$681$	11.3578	0.435231
$682$	0	0
$683$	−23.9948	−0.918135	−0.459067	−	0.888401i	$-0.651816\pi$
$683$	−23.9948	−0.918135	−0.459067	+	0.888401i	$0.651816\pi$
$684$	0	0
$685$	−4.99506	−0.190852
$686$	0	0
$687$	−5.99096	−0.228569
$688$	0	0
$689$	2.81131	0.107102
$690$	0	0
$691$	−23.7654	−0.904077	−0.452038	−	0.891998i	$-0.649303\pi$
$691$	−23.7654	−0.904077	−0.452038	+	0.891998i	$0.649303\pi$
$692$	0	0
$693$	−4.63261	−0.175978
$694$	0	0
$695$	−13.7101	−0.520053
$696$	0	0
$697$	0.607966	0.0230284
$698$	0	0
$699$	−1.21103	−0.0458054
$700$	0	0
$701$	9.95545	0.376012	0.188006	−	0.982168i	$-0.439798\pi$
$701$	9.95545	0.376012	0.188006	+	0.982168i	$0.439798\pi$
$702$	0	0
$703$	11.3348	0.427499
$704$	0	0
$705$	−1.62284	−0.0611196
$706$	0	0
$707$	10.0096	0.376449
$708$	0	0
$709$	21.0892	0.792022	0.396011	−	0.918246i	$-0.370394\pi$
$709$	21.0892	0.792022	0.396011	+	0.918246i	$0.370394\pi$
$710$	0	0
$711$	21.2388	0.796517
$712$	0	0
$713$	−23.9145	−0.895604
$714$	0	0
$715$	−1.21796	−0.0455492
$716$	0	0
$717$	13.5387	0.505610
$718$	0	0
$719$	16.9974	0.633894	0.316947	−	0.948443i	$-0.397342\pi$
$719$	16.9974	0.633894	0.316947	+	0.948443i	$0.397342\pi$
$720$	0	0
$721$	11.0019	0.409733
$722$	0	0
$723$	−17.4680	−0.649641
$724$	0	0
$725$	−1.48915	−0.0553057
$726$	0	0
$727$	−14.7706	−0.547812	−0.273906	−	0.961756i	$-0.588316\pi$
$727$	−14.7706	−0.547812	−0.273906	+	0.961756i	$0.588316\pi$
$728$	0	0
$729$	18.2735	0.676795
$730$	0	0
$731$	1.96514	0.0726834
$732$	0	0
$733$	4.40608	0.162742	0.0813711	−	0.996684i	$-0.474070\pi$
$733$	4.40608	0.162742	0.0813711	+	0.996684i	$0.474070\pi$
$734$	0	0
$735$	1.10254	0.0406679
$736$	0	0
$737$	35.2843	1.29971
$738$	0	0
$739$	−3.19229	−0.117430	−0.0587152	−	0.998275i	$-0.518700\pi$
$739$	−3.19229	−0.117430	−0.0587152	+	0.998275i	$0.518700\pi$
$740$	0	0
$741$	3.71430	0.136448
$742$	0	0
$743$	38.6327	1.41729	0.708647	−	0.705563i	$-0.249306\pi$
$743$	38.6327	1.41729	0.708647	+	0.705563i	$0.249306\pi$
$744$	0	0
$745$	−0.413647	−0.0151549
$746$	0	0
$747$	10.9980	0.402395
$748$	0	0
$749$	−3.80036	−0.138862
$750$	0	0
$751$	−37.4117	−1.36517	−0.682586	−	0.730805i	$-0.739145\pi$
$751$	−37.4117	−1.36517	−0.682586	+	0.730805i	$0.739145\pi$
$752$	0	0
$753$	13.5963	0.495476
$754$	0	0
$755$	−22.0194	−0.801367
$756$	0	0
$757$	−32.5550	−1.18323	−0.591616	−	0.806220i	$-0.701510\pi$
$757$	−32.5550	−1.18323	−0.591616	+	0.806220i	$0.701510\pi$
$758$	0	0
$759$	13.8861	0.504032
$760$	0	0
$761$	26.2904	0.953025	0.476512	−	0.879168i	$-0.341901\pi$
$761$	26.2904	0.953025	0.476512	+	0.879168i	$0.341901\pi$
$762$	0	0
$763$	−6.53836	−0.236705
$764$	0	0
$765$	3.50660	0.126781
$766$	0	0
$767$	0.963331	0.0347839
$768$	0	0
$769$	−17.8531	−0.643799	−0.321899	−	0.946774i	$-0.604321\pi$
$769$	−17.8531	−0.643799	−0.321899	+	0.946774i	$0.604321\pi$
$770$	0	0
$771$	12.5673	0.452599
$772$	0	0
$773$	30.4647	1.09574	0.547869	−	0.836564i	$-0.315439\pi$
$773$	30.4647	1.09574	0.547869	+	0.836564i	$0.315439\pi$
$774$	0	0
$775$	4.92960	0.177076
$776$	0	0
$777$	−1.74031	−0.0624333
$778$	0	0
$779$	−2.22160	−0.0795972
$780$	0	0
$781$	−8.69184	−0.311019
$782$	0	0
$783$	7.85529	0.280725
$784$	0	0
$785$	−1.44739	−0.0516595
$786$	0	0
$787$	−48.7260	−1.73689	−0.868446	−	0.495783i	$-0.834881\pi$
$787$	−48.7260	−1.73689	−0.868446	+	0.495783i	$0.834881\pi$
$788$	0	0
$789$	6.92151	0.246412
$790$	0	0
$791$	−2.29421	−0.0815729
$792$	0	0
$793$	−5.56072	−0.197467
$794$	0	0
$795$	6.60700	0.234326
$796$	0	0
$797$	3.43682	0.121738	0.0608692	−	0.998146i	$-0.480613\pi$
$797$	3.43682	0.121738	0.0608692	+	0.998146i	$0.480613\pi$
$798$	0	0
$799$	2.89250	0.102329
$800$	0	0
$801$	−12.6561	−0.447180
$802$	0	0
$803$	31.9882	1.12884
$804$	0	0
$805$	4.85120	0.170982
$806$	0	0
$807$	15.8956	0.559553
$808$	0	0
$809$	29.0281	1.02057	0.510286	−	0.860005i	$-0.329540\pi$
$809$	29.0281	1.02057	0.510286	+	0.860005i	$0.329540\pi$
$810$	0	0
$811$	26.9352	0.945822	0.472911	−	0.881110i	$-0.343203\pi$
$811$	26.9352	0.945822	0.472911	+	0.881110i	$0.343203\pi$
$812$	0	0
$813$	18.6888	0.655447
$814$	0	0
$815$	−0.893537	−0.0312992
$816$	0	0
$817$	−7.18094	−0.251229
$818$	0	0
$819$	0.837127	0.0292516
$820$	0	0
$821$	−43.3290	−1.51219	−0.756095	−	0.654461i	$-0.772895\pi$
$821$	−43.3290	−1.51219	−0.756095	+	0.654461i	$0.772895\pi$
$822$	0	0
$823$	36.5000	1.27231	0.636155	−	0.771561i	$-0.280524\pi$
$823$	36.5000	1.27231	0.636155	+	0.771561i	$0.280524\pi$
$824$	0	0
$825$	−2.86240	−0.0996559
$826$	0	0
$827$	12.7580	0.443638	0.221819	−	0.975088i	$-0.428801\pi$
$827$	12.7580	0.443638	0.221819	+	0.975088i	$0.428801\pi$
$828$	0	0
$829$	38.2775	1.32943	0.664716	−	0.747096i	$-0.268553\pi$
$829$	38.2775	1.32943	0.664716	+	0.747096i	$0.268553\pi$
$830$	0	0
$831$	18.5716	0.644241
$832$	0	0
$833$	−1.96514	−0.0680882
$834$	0	0
$835$	−9.00700	−0.311700
$836$	0	0
$837$	−26.0037	−0.898818
$838$	0	0
$839$	−46.2697	−1.59741	−0.798703	−	0.601725i	$-0.794480\pi$
$839$	−46.2697	−1.59741	−0.798703	+	0.601725i	$0.794480\pi$
$840$	0	0
$841$	−26.7824	−0.923532
$842$	0	0
$843$	−33.5483	−1.15547
$844$	0	0
$845$	−12.7799	−0.439642
$846$	0	0
$847$	4.25987	0.146371
$848$	0	0
$849$	−5.74410	−0.197137
$850$	0	0
$851$	−7.65738	−0.262492
$852$	0	0
$853$	−43.1855	−1.47864	−0.739321	−	0.673353i	$-0.764853\pi$
$853$	−43.1855	−1.47864	−0.739321	+	0.673353i	$0.764853\pi$
$854$	0	0
$855$	−12.8137	−0.438218
$856$	0	0
$857$	−24.9507	−0.852299	−0.426150	−	0.904653i	$-0.640130\pi$
$857$	−24.9507	−0.852299	−0.426150	+	0.904653i	$0.640130\pi$
$858$	0	0
$859$	9.27108	0.316325	0.158163	−	0.987413i	$-0.449443\pi$
$859$	9.27108	0.316325	0.158163	+	0.987413i	$0.449443\pi$
$860$	0	0
$861$	0.341099	0.0116246
$862$	0	0
$863$	44.9368	1.52967	0.764833	−	0.644228i	$-0.222821\pi$
$863$	44.9368	1.52967	0.764833	+	0.644228i	$0.222821\pi$
$864$	0	0
$865$	−26.0495	−0.885710
$866$	0	0
$867$	−14.4854	−0.491952
$868$	0	0
$869$	30.9010	1.04824
$870$	0	0
$871$	−6.37598	−0.216042
$872$	0	0
$873$	23.0333	0.779559
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−22.7362	−0.767746	−0.383873	−	0.923386i	$-0.625410\pi$
$877$	−22.7362	−0.767746	−0.383873	+	0.923386i	$0.625410\pi$
$878$	0	0
$879$	22.7309	0.766693
$880$	0	0
$881$	−25.8363	−0.870447	−0.435223	−	0.900323i	$-0.643331\pi$
$881$	−25.8363	−0.870447	−0.435223	+	0.900323i	$0.643331\pi$
$882$	0	0
$883$	−10.2877	−0.346210	−0.173105	−	0.984903i	$-0.555380\pi$
$883$	−10.2877	−0.346210	−0.173105	+	0.984903i	$0.555380\pi$
$884$	0	0
$885$	2.26397	0.0761027
$886$	0	0
$887$	5.51296	0.185107	0.0925535	−	0.995708i	$-0.470497\pi$
$887$	5.51296	0.185107	0.0925535	+	0.995708i	$0.470497\pi$
$888$	0	0
$889$	−0.120671	−0.00404718
$890$	0	0
$891$	1.20132	0.0402457
$892$	0	0
$893$	−10.5697	−0.353700
$894$	0	0
$895$	−1.84193	−0.0615690
$896$	0	0
$897$	−2.50925	−0.0837815
$898$	0	0
$899$	−7.34093	−0.244834
$900$	0	0
$901$	−11.7761	−0.392320
$902$	0	0
$903$	1.10254	0.0366903
$904$	0	0
$905$	−22.0050	−0.731470
$906$	0	0
$907$	9.89739	0.328637	0.164319	−	0.986407i	$-0.447457\pi$
$907$	9.89739	0.328637	0.164319	+	0.986407i	$0.447457\pi$
$908$	0	0
$909$	17.8611	0.592414
$910$	0	0
$911$	59.3812	1.96739	0.983693	−	0.179854i	$-0.0575626\pi$
$911$	59.3812	1.96739	0.983693	+	0.179854i	$0.0575626\pi$
$912$	0	0
$913$	16.0013	0.529566
$914$	0	0
$915$	−13.0685	−0.432032
$916$	0	0
$917$	−12.5472	−0.414345
$918$	0	0
$919$	−9.28833	−0.306394	−0.153197	−	0.988196i	$-0.548957\pi$
$919$	−9.28833	−0.306394	−0.153197	+	0.988196i	$0.548957\pi$
$920$	0	0
$921$	26.0119	0.857120
$922$	0	0
$923$	1.57064	0.0516983
$924$	0	0
$925$	1.57845	0.0518992
$926$	0	0
$927$	19.6318	0.644794
$928$	0	0
$929$	53.6391	1.75984	0.879922	−	0.475119i	$-0.157595\pi$
$929$	53.6391	1.75984	0.879922	+	0.475119i	$0.157595\pi$
$930$	0	0
$931$	7.18094	0.235346
$932$	0	0
$933$	12.5486	0.410824
$934$	0	0
$935$	5.10186	0.166849
$936$	0	0
$937$	−3.93394	−0.128516	−0.0642582	−	0.997933i	$-0.520468\pi$
$937$	−3.93394	−0.128516	−0.0642582	+	0.997933i	$0.520468\pi$
$938$	0	0
$939$	7.28501	0.237737
$940$	0	0
$941$	44.8418	1.46180	0.730900	−	0.682485i	$-0.239101\pi$
$941$	44.8418	1.46180	0.730900	+	0.682485i	$0.239101\pi$
$942$	0	0
$943$	1.50084	0.0488741
$944$	0	0
$945$	5.27501	0.171596
$946$	0	0
$947$	−35.7643	−1.16218	−0.581092	−	0.813838i	$-0.697374\pi$
$947$	−35.7643	−1.16218	−0.581092	+	0.813838i	$0.697374\pi$
$948$	0	0
$949$	−5.78037	−0.187639
$950$	0	0
$951$	−28.7385	−0.931908
$952$	0	0
$953$	5.10695	0.165430	0.0827152	−	0.996573i	$-0.473641\pi$
$953$	5.10695	0.165430	0.0827152	+	0.996573i	$0.473641\pi$
$954$	0	0
$955$	10.5631	0.341814
$956$	0	0
$957$	4.26255	0.137789
$958$	0	0
$959$	4.99506	0.161299
$960$	0	0
$961$	−6.69903	−0.216098
$962$	0	0
$963$	−6.78136	−0.218526
$964$	0	0
$965$	16.2250	0.522302
$966$	0	0
$967$	−20.3137	−0.653245	−0.326623	−	0.945155i	$-0.605911\pi$
$967$	−20.3137	−0.653245	−0.326623	+	0.945155i	$0.605911\pi$
$968$	0	0
$969$	−15.5586	−0.499815
$970$	0	0
$971$	−21.4964	−0.689851	−0.344925	−	0.938630i	$-0.612096\pi$
$971$	−21.4964	−0.689851	−0.344925	+	0.938630i	$0.612096\pi$
$972$	0	0
$973$	13.7101	0.439525
$974$	0	0
$975$	0.517244	0.0165651
$976$	0	0
$977$	5.87428	0.187935	0.0939674	−	0.995575i	$-0.470045\pi$
$977$	5.87428	0.187935	0.0939674	+	0.995575i	$0.470045\pi$
$978$	0	0
$979$	−18.4137	−0.588504
$980$	0	0
$981$	−11.6670	−0.372500
$982$	0	0
$983$	−24.5061	−0.781624	−0.390812	−	0.920470i	$-0.627806\pi$
$983$	−24.5061	−0.781624	−0.390812	+	0.920470i	$0.627806\pi$
$984$	0	0
$985$	8.45134	0.269282
$986$	0	0
$987$	1.62284	0.0516555
$988$	0	0
$989$	4.85120	0.154259
$990$	0	0
$991$	55.4850	1.76254	0.881270	−	0.472613i	$-0.156689\pi$
$991$	55.4850	1.76254	0.881270	+	0.472613i	$0.156689\pi$
$992$	0	0
$993$	−3.19641	−0.101435
$994$	0	0
$995$	7.73797	0.245310
$996$	0	0
$997$	37.2903	1.18099	0.590497	−	0.807040i	$-0.298932\pi$
$997$	37.2903	1.18099	0.590497	+	0.807040i	$0.298932\pi$
$998$	0	0
$999$	−8.32634	−0.263434

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6020.2.a.g.1.6	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6020.2.a.g.1.6	✓	9	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6020.a (trivial)

\(f(q)\)	\(=\)	\(q+1.10254 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.78440 q^{9} +O(q^{10})\)	\(q+1.10254 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.78440 q^{9} -2.59618 q^{11} +0.469137 q^{13} +1.10254 q^{15} -1.96514 q^{17} +7.18094 q^{19} -1.10254 q^{21} -4.85120 q^{23} +1.00000 q^{25} -5.27501 q^{27} -1.48915 q^{29} +4.92960 q^{31} -2.86240 q^{33} -1.00000 q^{35} +1.57845 q^{37} +0.517244 q^{39} -0.309375 q^{41} -1.00000 q^{43} -1.78440 q^{45} -1.47190 q^{47} +1.00000 q^{49} -2.16666 q^{51} +5.99251 q^{53} -2.59618 q^{55} +7.91729 q^{57} +2.05341 q^{59} -11.8531 q^{61} +1.78440 q^{63} +0.469137 q^{65} -13.5909 q^{67} -5.34866 q^{69} +3.34794 q^{71} -12.3213 q^{73} +1.10254 q^{75} +2.59618 q^{77} -11.9025 q^{79} -0.462726 q^{81} -6.16341 q^{83} -1.96514 q^{85} -1.64186 q^{87} +7.09261 q^{89} -0.469137 q^{91} +5.43510 q^{93} +7.18094 q^{95} -12.9082 q^{97} +4.63261 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{5} - 9 q^{7} + 5 q^{9}+O(q^{10}) \) 9 * q + 9 * q^5 - 9 * q^7 + 5 * q^9	\( 9 q + 9 q^{5} - 9 q^{7} + 5 q^{9} + q^{11} - 14 q^{13} - 11 q^{17} - 2 q^{19} - 6 q^{23} + 9 q^{25} - 6 q^{29} + 6 q^{31} - 14 q^{33} - 9 q^{35} - 20 q^{37} - 6 q^{39} - 6 q^{41} - 9 q^{43} + 5 q^{45} + 9 q^{49} - 8 q^{51} - 31 q^{53} + q^{55} - 16 q^{57} + 2 q^{59} - 13 q^{61} - 5 q^{63} - 14 q^{65} - 10 q^{67} - 18 q^{69} + 12 q^{71} - 32 q^{73} - q^{77} + q^{79} - 27 q^{81} - 10 q^{83} - 11 q^{85} - 5 q^{87} - q^{89} + 14 q^{91} - 49 q^{93} - 2 q^{95} - 28 q^{97} - 14 q^{99}+O(q^{100}) \) 9 * q + 9 * q^5 - 9 * q^7 + 5 * q^9 + q^11 - 14 * q^13 - 11 * q^17 - 2 * q^19 - 6 * q^23 + 9 * q^25 - 6 * q^29 + 6 * q^31 - 14 * q^33 - 9 * q^35 - 20 * q^37 - 6 * q^39 - 6 * q^41 - 9 * q^43 + 5 * q^45 + 9 * q^49 - 8 * q^51 - 31 * q^53 + q^55 - 16 * q^57 + 2 * q^59 - 13 * q^61 - 5 * q^63 - 14 * q^65 - 10 * q^67 - 18 * q^69 + 12 * q^71 - 32 * q^73 - q^77 + q^79 - 27 * q^81 - 10 * q^83 - 11 * q^85 - 5 * q^87 - q^89 + 14 * q^91 - 49 * q^93 - 2 * q^95 - 28 * q^97 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more