LMFDB - Embedded newform 8036.2.a.t.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 30 x^{18} + 128 x^{17} + 348 x^{16} - 1644 x^{15} - 1934 x^{14} + 10948 x^{13} + \cdots + 9 $ x^20 - 4x^19 - 30x^18 + 128x^17 + 348x^16 - 1644x^15 - 1934x^14 + 10948x^13 + 4748x^12 - 40524x^11 - 220x^10 + 82500x^9 - 21992x^8 - 84720x^7 + 37544x^6 + 34656x^5 - 18823x^4 - 2276x^3 + 1130x^2 + 204*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.15199$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.15199 q^{3} -0.0591445 q^{5} +1.63107 q^{9} +O(q^{10})$	$q-2.15199 q^{3} -0.0591445 q^{5} +1.63107 q^{9} -5.19280 q^{11} +1.33719 q^{13} +0.127279 q^{15} -5.25954 q^{17} -6.42758 q^{19} -5.30631 q^{23} -4.99650 q^{25} +2.94592 q^{27} -2.05926 q^{29} -4.21473 q^{31} +11.1749 q^{33} -6.95811 q^{37} -2.87762 q^{39} -1.00000 q^{41} -4.70691 q^{43} -0.0964691 q^{45} +4.09351 q^{47} +11.3185 q^{51} +2.41022 q^{53} +0.307126 q^{55} +13.8321 q^{57} -4.59309 q^{59} -4.59729 q^{61} -0.0790874 q^{65} -9.74977 q^{67} +11.4191 q^{69} +2.36099 q^{71} +16.0737 q^{73} +10.7524 q^{75} -17.6984 q^{79} -11.2328 q^{81} +12.5923 q^{83} +0.311073 q^{85} +4.43152 q^{87} +0.670024 q^{89} +9.07008 q^{93} +0.380156 q^{95} -3.02595 q^{97} -8.46985 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9}+O(q^{10}) $ 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9	$ 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9} - 8 q^{11} + 12 q^{13} + 8 q^{15} + 8 q^{17} + 24 q^{19} + 8 q^{23} + 20 q^{25} + 16 q^{27} - 12 q^{29} + 44 q^{33} + 12 q^{37} + 12 q^{39} - 20 q^{41} + 4 q^{43} + 40 q^{45} + 4 q^{47} + 4 q^{51} - 12 q^{53} - 16 q^{55} + 28 q^{57} + 16 q^{59} + 68 q^{61} - 8 q^{65} + 4 q^{67} + 32 q^{69} + 8 q^{71} + 48 q^{73} + 60 q^{75} - 20 q^{79} + 32 q^{81} - 8 q^{83} - 28 q^{85} + 60 q^{89} - 16 q^{93} + 20 q^{95} + 40 q^{97}+O(q^{100}) $ 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9 - 8 * q^11 + 12 * q^13 + 8 * q^15 + 8 * q^17 + 24 * q^19 + 8 * q^23 + 20 * q^25 + 16 * q^27 - 12 * q^29 + 44 * q^33 + 12 * q^37 + 12 * q^39 - 20 * q^41 + 4 * q^43 + 40 * q^45 + 4 * q^47 + 4 * q^51 - 12 * q^53 - 16 * q^55 + 28 * q^57 + 16 * q^59 + 68 * q^61 - 8 * q^65 + 4 * q^67 + 32 * q^69 + 8 * q^71 + 48 * q^73 + 60 * q^75 - 20 * q^79 + 32 * q^81 - 8 * q^83 - 28 * q^85 + 60 * q^89 - 16 * q^93 + 20 * q^95 + 40 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.15199	−1.24245	−0.621227	−	0.783631i	$-0.713365\pi$
$3$	−2.15199	−1.24245	−0.621227	+	0.783631i	$0.713365\pi$
$4$	0	0
$5$	−0.0591445	−0.0264502	−0.0132251	−	0.999913i	$-0.504210\pi$
$5$	−0.0591445	−0.0264502	−0.0132251	+	0.999913i	$0.504210\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.63107	0.543691
$10$	0	0
$11$	−5.19280	−1.56569	−0.782844	−	0.622218i	$-0.786232\pi$
$11$	−5.19280	−1.56569	−0.782844	+	0.622218i	$0.786232\pi$
$12$	0	0
$13$	1.33719	0.370870	0.185435	−	0.982657i	$-0.440631\pi$
$13$	1.33719	0.370870	0.185435	+	0.982657i	$0.440631\pi$
$14$	0	0
$15$	0.127279	0.0328632
$16$	0	0
$17$	−5.25954	−1.27563	−0.637813	−	0.770191i	$-0.720161\pi$
$17$	−5.25954	−1.27563	−0.637813	+	0.770191i	$0.720161\pi$
$18$	0	0
$19$	−6.42758	−1.47459	−0.737295	−	0.675571i	$-0.763897\pi$
$19$	−6.42758	−1.47459	−0.737295	+	0.675571i	$0.763897\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.30631	−1.10644	−0.553221	−	0.833035i	$-0.686601\pi$
$23$	−5.30631	−1.10644	−0.553221	+	0.833035i	$0.686601\pi$
$24$	0	0
$25$	−4.99650	−0.999300
$26$	0	0
$27$	2.94592	0.566942
$28$	0	0
$29$	−2.05926	−0.382395	−0.191198	−	0.981552i	$-0.561237\pi$
$29$	−2.05926	−0.382395	−0.191198	+	0.981552i	$0.561237\pi$
$30$	0	0
$31$	−4.21473	−0.756989	−0.378494	−	0.925604i	$-0.623558\pi$
$31$	−4.21473	−0.756989	−0.378494	+	0.925604i	$0.623558\pi$
$32$	0	0
$33$	11.1749	1.94530
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.95811	−1.14391	−0.571953	−	0.820287i	$-0.693814\pi$
$37$	−6.95811	−1.14391	−0.571953	+	0.820287i	$0.693814\pi$
$38$	0	0
$39$	−2.87762	−0.460789
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−4.70691	−0.717797	−0.358898	−	0.933377i	$-0.616848\pi$
$43$	−4.70691	−0.717797	−0.358898	+	0.933377i	$0.616848\pi$
$44$	0	0
$45$	−0.0964691	−0.0143808
$46$	0	0
$47$	4.09351	0.597100	0.298550	−	0.954394i	$-0.403497\pi$
$47$	4.09351	0.597100	0.298550	+	0.954394i	$0.403497\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	11.3185	1.58491
$52$	0	0
$53$	2.41022	0.331070	0.165535	−	0.986204i	$-0.447065\pi$
$53$	2.41022	0.331070	0.165535	+	0.986204i	$0.447065\pi$
$54$	0	0
$55$	0.307126	0.0414128
$56$	0	0
$57$	13.8321	1.83211
$58$	0	0
$59$	−4.59309	−0.597969	−0.298984	−	0.954258i	$-0.596648\pi$
$59$	−4.59309	−0.597969	−0.298984	+	0.954258i	$0.596648\pi$
$60$	0	0
$61$	−4.59729	−0.588622	−0.294311	−	0.955710i	$-0.595090\pi$
$61$	−4.59729	−0.588622	−0.294311	+	0.955710i	$0.595090\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.0790874	−0.00980959
$66$	0	0
$67$	−9.74977	−1.19112	−0.595562	−	0.803309i	$-0.703071\pi$
$67$	−9.74977	−1.19112	−0.595562	+	0.803309i	$0.703071\pi$
$68$	0	0
$69$	11.4191	1.37470
$70$	0	0
$71$	2.36099	0.280198	0.140099	−	0.990137i	$-0.455258\pi$
$71$	2.36099	0.280198	0.140099	+	0.990137i	$0.455258\pi$
$72$	0	0
$73$	16.0737	1.88129	0.940643	−	0.339397i	$-0.110223\pi$
$73$	16.0737	1.88129	0.940643	+	0.339397i	$0.110223\pi$
$74$	0	0
$75$	10.7524	1.24158
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−17.6984	−1.99122	−0.995611	−	0.0935911i	$-0.970165\pi$
$79$	−17.6984	−1.99122	−0.995611	+	0.0935911i	$0.970165\pi$
$80$	0	0
$81$	−11.2328	−1.24809
$82$	0	0
$83$	12.5923	1.38218	0.691092	−	0.722766i	$-0.257130\pi$
$83$	12.5923	1.38218	0.691092	+	0.722766i	$0.257130\pi$
$84$	0	0
$85$	0.311073	0.0337406
$86$	0	0
$87$	4.43152	0.475109
$88$	0	0
$89$	0.670024	0.0710224	0.0355112	−	0.999369i	$-0.488694\pi$
$89$	0.670024	0.0710224	0.0355112	+	0.999369i	$0.488694\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	9.07008	0.940524
$94$	0	0
$95$	0.380156	0.0390032
$96$	0	0
$97$	−3.02595	−0.307239	−0.153619	−	0.988130i	$-0.549093\pi$
$97$	−3.02595	−0.307239	−0.153619	+	0.988130i	$0.549093\pi$
$98$	0	0
$99$	−8.46985	−0.851252
$100$	0	0
$101$	3.46622	0.344902	0.172451	−	0.985018i	$-0.444831\pi$
$101$	3.46622	0.344902	0.172451	+	0.985018i	$0.444831\pi$
$102$	0	0
$103$	−16.6119	−1.63682	−0.818411	−	0.574634i	$-0.805144\pi$
$103$	−16.6119	−1.63682	−0.818411	+	0.574634i	$0.805144\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.78404	0.462491	0.231246	−	0.972895i	$-0.425720\pi$
$107$	4.78404	0.462491	0.231246	+	0.972895i	$0.425720\pi$
$108$	0	0
$109$	10.1783	0.974905	0.487452	−	0.873150i	$-0.337926\pi$
$109$	10.1783	0.974905	0.487452	+	0.873150i	$0.337926\pi$
$110$	0	0
$111$	14.9738	1.42125
$112$	0	0
$113$	−8.11250	−0.763160	−0.381580	−	0.924336i	$-0.624620\pi$
$113$	−8.11250	−0.763160	−0.381580	+	0.924336i	$0.624620\pi$
$114$	0	0
$115$	0.313839	0.0292656
$116$	0	0
$117$	2.18106	0.201639
$118$	0	0
$119$	0	0
$120$	0	0
$121$	15.9652	1.45138
$122$	0	0
$123$	2.15199	0.194039
$124$	0	0
$125$	0.591238	0.0528820
$126$	0	0
$127$	12.3020	1.09163	0.545815	−	0.837906i	$-0.316220\pi$
$127$	12.3020	1.09163	0.545815	+	0.837906i	$0.316220\pi$
$128$	0	0
$129$	10.1292	0.891829
$130$	0	0
$131$	−13.7437	−1.20079	−0.600397	−	0.799702i	$-0.704991\pi$
$131$	−13.7437	−1.20079	−0.600397	+	0.799702i	$0.704991\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.174235	−0.0149958
$136$	0	0
$137$	−16.3423	−1.39622	−0.698110	−	0.715990i	$-0.745975\pi$
$137$	−16.3423	−1.39622	−0.698110	+	0.715990i	$0.745975\pi$
$138$	0	0
$139$	3.15630	0.267714	0.133857	−	0.991001i	$-0.457264\pi$
$139$	3.15630	0.267714	0.133857	+	0.991001i	$0.457264\pi$
$140$	0	0
$141$	−8.80921	−0.741869
$142$	0	0
$143$	−6.94376	−0.580667
$144$	0	0
$145$	0.121794	0.0101144
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.60110	0.295014	0.147507	−	0.989061i	$-0.452875\pi$
$149$	3.60110	0.295014	0.147507	+	0.989061i	$0.452875\pi$
$150$	0	0
$151$	−20.3106	−1.65286	−0.826428	−	0.563042i	$-0.809631\pi$
$151$	−20.3106	−1.65286	−0.826428	+	0.563042i	$0.809631\pi$
$152$	0	0
$153$	−8.57871	−0.693547
$154$	0	0
$155$	0.249278	0.0200225
$156$	0	0
$157$	−2.62122	−0.209196	−0.104598	−	0.994515i	$-0.533356\pi$
$157$	−2.62122	−0.209196	−0.104598	+	0.994515i	$0.533356\pi$
$158$	0	0
$159$	−5.18678	−0.411339
$160$	0	0
$161$	0	0
$162$	0	0
$163$	17.2802	1.35349	0.676743	−	0.736219i	$-0.263391\pi$
$163$	17.2802	1.35349	0.676743	+	0.736219i	$0.263391\pi$
$164$	0	0
$165$	−0.660932	−0.0514535
$166$	0	0
$167$	−21.7931	−1.68640	−0.843199	−	0.537602i	$-0.819330\pi$
$167$	−21.7931	−1.68640	−0.843199	+	0.537602i	$0.819330\pi$
$168$	0	0
$169$	−11.2119	−0.862456
$170$	0	0
$171$	−10.4839	−0.801722
$172$	0	0
$173$	7.43619	0.565363	0.282682	−	0.959214i	$-0.408776\pi$
$173$	7.43619	0.565363	0.282682	+	0.959214i	$0.408776\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	9.88429	0.742949
$178$	0	0
$179$	−15.6673	−1.17103	−0.585515	−	0.810662i	$-0.699108\pi$
$179$	−15.6673	−1.17103	−0.585515	+	0.810662i	$0.699108\pi$
$180$	0	0
$181$	17.9491	1.33415	0.667075	−	0.744991i	$-0.267546\pi$
$181$	17.9491	1.33415	0.667075	+	0.744991i	$0.267546\pi$
$182$	0	0
$183$	9.89333	0.731336
$184$	0	0
$185$	0.411534	0.0302566
$186$	0	0
$187$	27.3118	1.99723
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.9792	0.794425	0.397213	−	0.917727i	$-0.369978\pi$
$191$	10.9792	0.794425	0.397213	+	0.917727i	$0.369978\pi$
$192$	0	0
$193$	−8.35472	−0.601386	−0.300693	−	0.953721i	$-0.597218\pi$
$193$	−8.35472	−0.601386	−0.300693	+	0.953721i	$0.597218\pi$
$194$	0	0
$195$	0.170196	0.0121880
$196$	0	0
$197$	−8.62720	−0.614663	−0.307331	−	0.951603i	$-0.599436\pi$
$197$	−8.62720	−0.614663	−0.307331	+	0.951603i	$0.599436\pi$
$198$	0	0
$199$	15.1574	1.07448	0.537240	−	0.843430i	$-0.319467\pi$
$199$	15.1574	1.07448	0.537240	+	0.843430i	$0.319467\pi$
$200$	0	0
$201$	20.9814	1.47992
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0.0591445	0.00413083
$206$	0	0
$207$	−8.65498	−0.601563
$208$	0	0
$209$	33.3772	2.30875
$210$	0	0
$211$	−21.4024	−1.47340	−0.736701	−	0.676218i	$-0.763618\pi$
$211$	−21.4024	−1.47340	−0.736701	+	0.676218i	$0.763618\pi$
$212$	0	0
$213$	−5.08084	−0.348133
$214$	0	0
$215$	0.278388	0.0189859
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−34.5905	−2.33741
$220$	0	0
$221$	−7.03301	−0.473091
$222$	0	0
$223$	−13.4975	−0.903859	−0.451929	−	0.892054i	$-0.649264\pi$
$223$	−13.4975	−0.903859	−0.451929	+	0.892054i	$0.649264\pi$
$224$	0	0
$225$	−8.14967	−0.543311
$226$	0	0
$227$	5.90820	0.392141	0.196071	−	0.980590i	$-0.437182\pi$
$227$	5.90820	0.392141	0.196071	+	0.980590i	$0.437182\pi$
$228$	0	0
$229$	26.5285	1.75305	0.876525	−	0.481356i	$-0.159855\pi$
$229$	26.5285	1.75305	0.876525	+	0.481356i	$0.159855\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.34879	−0.415923	−0.207962	−	0.978137i	$-0.566683\pi$
$233$	−6.34879	−0.415923	−0.207962	+	0.978137i	$0.566683\pi$
$234$	0	0
$235$	−0.242109	−0.0157934
$236$	0	0
$237$	38.0868	2.47400
$238$	0	0
$239$	−26.2653	−1.69896	−0.849482	−	0.527618i	$-0.823085\pi$
$239$	−26.2653	−1.69896	−0.849482	+	0.527618i	$0.823085\pi$
$240$	0	0
$241$	17.5763	1.13219	0.566096	−	0.824340i	$-0.308453\pi$
$241$	17.5763	1.13219	0.566096	+	0.824340i	$0.308453\pi$
$242$	0	0
$243$	15.3352	0.983753
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.59490	−0.546881
$248$	0	0
$249$	−27.0985	−1.71730
$250$	0	0
$251$	−28.5313	−1.80088	−0.900439	−	0.434983i	$-0.856754\pi$
$251$	−28.5313	−1.80088	−0.900439	+	0.434983i	$0.856754\pi$
$252$	0	0
$253$	27.5546	1.73234
$254$	0	0
$255$	−0.669427	−0.0419212
$256$	0	0
$257$	−20.5221	−1.28013	−0.640066	−	0.768320i	$-0.721093\pi$
$257$	−20.5221	−1.28013	−0.640066	+	0.768320i	$0.721093\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3.35881	−0.207905
$262$	0	0
$263$	7.82507	0.482514	0.241257	−	0.970461i	$-0.422440\pi$
$263$	7.82507	0.482514	0.241257	+	0.970461i	$0.422440\pi$
$264$	0	0
$265$	−0.142551	−0.00875687
$266$	0	0
$267$	−1.44189	−0.0882421
$268$	0	0
$269$	−23.4164	−1.42773	−0.713863	−	0.700286i	$-0.753056\pi$
$269$	−23.4164	−1.42773	−0.713863	+	0.700286i	$0.753056\pi$
$270$	0	0
$271$	−2.96639	−0.180195	−0.0900976	−	0.995933i	$-0.528718\pi$
$271$	−2.96639	−0.180195	−0.0900976	+	0.995933i	$0.528718\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	25.9458	1.56459
$276$	0	0
$277$	−14.9330	−0.897236	−0.448618	−	0.893723i	$-0.648084\pi$
$277$	−14.9330	−0.897236	−0.448618	+	0.893723i	$0.648084\pi$
$278$	0	0
$279$	−6.87455	−0.411568
$280$	0	0
$281$	−3.65734	−0.218178	−0.109089	−	0.994032i	$-0.534793\pi$
$281$	−3.65734	−0.218178	−0.109089	+	0.994032i	$0.534793\pi$
$282$	0	0
$283$	19.6498	1.16806	0.584029	−	0.811733i	$-0.301475\pi$
$283$	19.6498	1.16806	0.584029	+	0.811733i	$0.301475\pi$
$284$	0	0
$285$	−0.818094	−0.0484597
$286$	0	0
$287$	0	0
$288$	0	0
$289$	10.6628	0.627223
$290$	0	0
$291$	6.51183	0.381730
$292$	0	0
$293$	−12.9213	−0.754872	−0.377436	−	0.926036i	$-0.623194\pi$
$293$	−12.9213	−0.754872	−0.377436	+	0.926036i	$0.623194\pi$
$294$	0	0
$295$	0.271656	0.0158164
$296$	0	0
$297$	−15.2976	−0.887655
$298$	0	0
$299$	−7.09554	−0.410346
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−7.45929	−0.428525
$304$	0	0
$305$	0.271904	0.0155692
$306$	0	0
$307$	6.33218	0.361397	0.180698	−	0.983539i	$-0.442164\pi$
$307$	6.33218	0.361397	0.180698	+	0.983539i	$0.442164\pi$
$308$	0	0
$309$	35.7487	2.03367
$310$	0	0
$311$	9.54819	0.541428	0.270714	−	0.962660i	$-0.412740\pi$
$311$	9.54819	0.541428	0.270714	+	0.962660i	$0.412740\pi$
$312$	0	0
$313$	15.6169	0.882721	0.441360	−	0.897330i	$-0.354496\pi$
$313$	15.6169	0.882721	0.441360	+	0.897330i	$0.354496\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	25.6886	1.44282	0.721408	−	0.692510i	$-0.243495\pi$
$317$	25.6886	1.44282	0.721408	+	0.692510i	$0.243495\pi$
$318$	0	0
$319$	10.6933	0.598712
$320$	0	0
$321$	−10.2952	−0.574624
$322$	0	0
$323$	33.8062	1.88102
$324$	0	0
$325$	−6.68127	−0.370610
$326$	0	0
$327$	−21.9036	−1.21127
$328$	0	0
$329$	0	0
$330$	0	0
$331$	10.1815	0.559625	0.279812	−	0.960055i	$-0.409728\pi$
$331$	10.1815	0.559625	0.279812	+	0.960055i	$0.409728\pi$
$332$	0	0
$333$	−11.3492	−0.621932
$334$	0	0
$335$	0.576646	0.0315055
$336$	0	0
$337$	26.1130	1.42246	0.711232	−	0.702958i	$-0.248138\pi$
$337$	26.1130	1.42246	0.711232	+	0.702958i	$0.248138\pi$
$338$	0	0
$339$	17.4580	0.948191
$340$	0	0
$341$	21.8863	1.18521
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.675379	−0.0363612
$346$	0	0
$347$	−29.2380	−1.56958	−0.784788	−	0.619764i	$-0.787228\pi$
$347$	−29.2380	−1.56958	−0.784788	+	0.619764i	$0.787228\pi$
$348$	0	0
$349$	30.6815	1.64234	0.821172	−	0.570681i	$-0.193321\pi$
$349$	30.6815	1.64234	0.821172	+	0.570681i	$0.193321\pi$
$350$	0	0
$351$	3.93925	0.210262
$352$	0	0
$353$	−7.83832	−0.417191	−0.208596	−	0.978002i	$-0.566889\pi$
$353$	−7.83832	−0.417191	−0.208596	+	0.978002i	$0.566889\pi$
$354$	0	0
$355$	−0.139640	−0.00741131
$356$	0	0
$357$	0	0
$358$	0	0
$359$	25.2494	1.33261	0.666305	−	0.745679i	$-0.267875\pi$
$359$	25.2494	1.33261	0.666305	+	0.745679i	$0.267875\pi$
$360$	0	0
$361$	22.3138	1.17441
$362$	0	0
$363$	−34.3570	−1.80327
$364$	0	0
$365$	−0.950672	−0.0497605
$366$	0	0
$367$	34.5939	1.80579	0.902893	−	0.429866i	$-0.141439\pi$
$367$	34.5939	1.80579	0.902893	+	0.429866i	$0.141439\pi$
$368$	0	0
$369$	−1.63107	−0.0849103
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−5.08439	−0.263260	−0.131630	−	0.991299i	$-0.542021\pi$
$373$	−5.08439	−0.263260	−0.131630	+	0.991299i	$0.542021\pi$
$374$	0	0
$375$	−1.27234	−0.0657034
$376$	0	0
$377$	−2.75362	−0.141819
$378$	0	0
$379$	−31.8842	−1.63778	−0.818891	−	0.573950i	$-0.805411\pi$
$379$	−31.8842	−1.63778	−0.818891	+	0.573950i	$0.805411\pi$
$380$	0	0
$381$	−26.4739	−1.35630
$382$	0	0
$383$	−3.82339	−0.195366	−0.0976831	−	0.995218i	$-0.531143\pi$
$383$	−3.82339	−0.195366	−0.0976831	+	0.995218i	$0.531143\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.67732	−0.390260
$388$	0	0
$389$	6.74436	0.341953	0.170976	−	0.985275i	$-0.445308\pi$
$389$	6.74436	0.341953	0.170976	+	0.985275i	$0.445308\pi$
$390$	0	0
$391$	27.9088	1.41141
$392$	0	0
$393$	29.5764	1.49193
$394$	0	0
$395$	1.04676	0.0526683
$396$	0	0
$397$	−8.07782	−0.405414	−0.202707	−	0.979239i	$-0.564974\pi$
$397$	−8.07782	−0.405414	−0.202707	+	0.979239i	$0.564974\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−37.6125	−1.87828	−0.939138	−	0.343539i	$-0.888374\pi$
$401$	−37.6125	−1.87828	−0.939138	+	0.343539i	$0.888374\pi$
$402$	0	0
$403$	−5.63590	−0.280744
$404$	0	0
$405$	0.664360	0.0330123
$406$	0	0
$407$	36.1321	1.79100
$408$	0	0
$409$	−4.69310	−0.232059	−0.116029	−	0.993246i	$-0.537017\pi$
$409$	−4.69310	−0.232059	−0.116029	+	0.993246i	$0.537017\pi$
$410$	0	0
$411$	35.1686	1.73474
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−0.744766	−0.0365591
$416$	0	0
$417$	−6.79234	−0.332622
$418$	0	0
$419$	−10.3430	−0.505287	−0.252643	−	0.967559i	$-0.581300\pi$
$419$	−10.3430	−0.505287	−0.252643	+	0.967559i	$0.581300\pi$
$420$	0	0
$421$	30.5165	1.48728	0.743641	−	0.668579i	$-0.233097\pi$
$421$	30.5165	1.48728	0.743641	+	0.668579i	$0.233097\pi$
$422$	0	0
$423$	6.67682	0.324638
$424$	0	0
$425$	26.2793	1.27473
$426$	0	0
$427$	0	0
$428$	0	0
$429$	14.9429	0.721451
$430$	0	0
$431$	−19.6284	−0.945467	−0.472733	−	0.881205i	$-0.656733\pi$
$431$	−19.6284	−0.945467	−0.472733	+	0.881205i	$0.656733\pi$
$432$	0	0
$433$	−11.7482	−0.564583	−0.282291	−	0.959329i	$-0.591095\pi$
$433$	−11.7482	−0.564583	−0.282291	+	0.959329i	$0.591095\pi$
$434$	0	0
$435$	−0.262100	−0.0125667
$436$	0	0
$437$	34.1067	1.63155
$438$	0	0
$439$	−11.6849	−0.557688	−0.278844	−	0.960336i	$-0.589951\pi$
$439$	−11.6849	−0.557688	−0.278844	+	0.960336i	$0.589951\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.09105	0.0993490	0.0496745	−	0.998765i	$-0.484182\pi$
$443$	2.09105	0.0993490	0.0496745	+	0.998765i	$0.484182\pi$
$444$	0	0
$445$	−0.0396282	−0.00187856
$446$	0	0
$447$	−7.74955	−0.366541
$448$	0	0
$449$	11.2663	0.531691	0.265845	−	0.964016i	$-0.414349\pi$
$449$	11.2663	0.531691	0.265845	+	0.964016i	$0.414349\pi$
$450$	0	0
$451$	5.19280	0.244519
$452$	0	0
$453$	43.7084	2.05360
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.8612	0.835513	0.417757	−	0.908559i	$-0.362817\pi$
$457$	17.8612	0.835513	0.417757	+	0.908559i	$0.362817\pi$
$458$	0	0
$459$	−15.4942	−0.723207
$460$	0	0
$461$	3.76709	0.175451	0.0877254	−	0.996145i	$-0.472040\pi$
$461$	3.76709	0.175451	0.0877254	+	0.996145i	$0.472040\pi$
$462$	0	0
$463$	25.5750	1.18857	0.594286	−	0.804254i	$-0.297435\pi$
$463$	25.5750	1.18857	0.594286	+	0.804254i	$0.297435\pi$
$464$	0	0
$465$	−0.536445	−0.0248771
$466$	0	0
$467$	0.506752	0.0234497	0.0117249	−	0.999931i	$-0.496268\pi$
$467$	0.506752	0.0234497	0.0117249	+	0.999931i	$0.496268\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	5.64085	0.259917
$472$	0	0
$473$	24.4420	1.12385
$474$	0	0
$475$	32.1154	1.47356
$476$	0	0
$477$	3.93125	0.180000
$478$	0	0
$479$	−0.838081	−0.0382929	−0.0191465	−	0.999817i	$-0.506095\pi$
$479$	−0.838081	−0.0382929	−0.0191465	+	0.999817i	$0.506095\pi$
$480$	0	0
$481$	−9.30431	−0.424240
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.178968	0.00812654
$486$	0	0
$487$	10.3497	0.468992	0.234496	−	0.972117i	$-0.424656\pi$
$487$	10.3497	0.468992	0.234496	+	0.972117i	$0.424656\pi$
$488$	0	0
$489$	−37.1868	−1.68164
$490$	0	0
$491$	15.1429	0.683391	0.341695	−	0.939811i	$-0.388999\pi$
$491$	15.1429	0.683391	0.341695	+	0.939811i	$0.388999\pi$
$492$	0	0
$493$	10.8308	0.487794
$494$	0	0
$495$	0.500945	0.0225158
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−7.63088	−0.341605	−0.170803	−	0.985305i	$-0.554636\pi$
$499$	−7.63088	−0.341605	−0.170803	+	0.985305i	$0.554636\pi$
$500$	0	0
$501$	46.8985	2.09527
$502$	0	0
$503$	−36.2600	−1.61675	−0.808377	−	0.588665i	$-0.799654\pi$
$503$	−36.2600	−1.61675	−0.808377	+	0.588665i	$0.799654\pi$
$504$	0	0
$505$	−0.205008	−0.00912274
$506$	0	0
$507$	24.1280	1.07156
$508$	0	0
$509$	7.10944	0.315120	0.157560	−	0.987509i	$-0.449637\pi$
$509$	7.10944	0.315120	0.157560	+	0.987509i	$0.449637\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−18.9351	−0.836007
$514$	0	0
$515$	0.982504	0.0432943
$516$	0	0
$517$	−21.2568	−0.934872
$518$	0	0
$519$	−16.0026	−0.702437
$520$	0	0
$521$	−12.0964	−0.529954	−0.264977	−	0.964255i	$-0.585364\pi$
$521$	−12.0964	−0.529954	−0.264977	+	0.964255i	$0.585364\pi$
$522$	0	0
$523$	−14.5419	−0.635875	−0.317938	−	0.948112i	$-0.602990\pi$
$523$	−14.5419	−0.635875	−0.317938	+	0.948112i	$0.602990\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22.1676	0.965635
$528$	0	0
$529$	5.15690	0.224213
$530$	0	0
$531$	−7.49166	−0.325111
$532$	0	0
$533$	−1.33719	−0.0579201
$534$	0	0
$535$	−0.282950	−0.0122330
$536$	0	0
$537$	33.7159	1.45495
$538$	0	0
$539$	0	0
$540$	0	0
$541$	10.8069	0.464624	0.232312	−	0.972641i	$-0.425371\pi$
$541$	10.8069	0.464624	0.232312	+	0.972641i	$0.425371\pi$
$542$	0	0
$543$	−38.6264	−1.65762
$544$	0	0
$545$	−0.601991	−0.0257865
$546$	0	0
$547$	11.6640	0.498716	0.249358	−	0.968411i	$-0.419780\pi$
$547$	11.6640	0.498716	0.249358	+	0.968411i	$0.419780\pi$
$548$	0	0
$549$	−7.49852	−0.320029
$550$	0	0
$551$	13.2361	0.563876
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−0.885618	−0.0375924
$556$	0	0
$557$	13.7853	0.584100	0.292050	−	0.956403i	$-0.405663\pi$
$557$	13.7853	0.584100	0.292050	+	0.956403i	$0.405663\pi$
$558$	0	0
$559$	−6.29403	−0.266209
$560$	0	0
$561$	−58.7747	−2.48147
$562$	0	0
$563$	28.4480	1.19894	0.599470	−	0.800397i	$-0.295378\pi$
$563$	28.4480	1.19894	0.599470	+	0.800397i	$0.295378\pi$
$564$	0	0
$565$	0.479810	0.0201858
$566$	0	0
$567$	0	0
$568$	0	0
$569$	21.8115	0.914385	0.457193	−	0.889368i	$-0.348855\pi$
$569$	21.8115	0.914385	0.457193	+	0.889368i	$0.348855\pi$
$570$	0	0
$571$	22.5978	0.945687	0.472843	−	0.881147i	$-0.343228\pi$
$571$	22.5978	0.945687	0.472843	+	0.881147i	$0.343228\pi$
$572$	0	0
$573$	−23.6271	−0.987037
$574$	0	0
$575$	26.5130	1.10567
$576$	0	0
$577$	−34.7359	−1.44607	−0.723036	−	0.690810i	$-0.757254\pi$
$577$	−34.7359	−1.44607	−0.723036	+	0.690810i	$0.757254\pi$
$578$	0	0
$579$	17.9793	0.747195
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−12.5158	−0.518352
$584$	0	0
$585$	−0.128998	−0.00533339
$586$	0	0
$587$	−45.3646	−1.87240	−0.936198	−	0.351473i	$-0.885681\pi$
$587$	−45.3646	−1.87240	−0.936198	+	0.351473i	$0.885681\pi$
$588$	0	0
$589$	27.0906	1.11625
$590$	0	0
$591$	18.5657	0.763690
$592$	0	0
$593$	−16.5862	−0.681112	−0.340556	−	0.940224i	$-0.610615\pi$
$593$	−16.5862	−0.681112	−0.340556	+	0.940224i	$0.610615\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−32.6186	−1.33499
$598$	0	0
$599$	17.7376	0.724738	0.362369	−	0.932035i	$-0.381968\pi$
$599$	17.7376	0.724738	0.362369	+	0.932035i	$0.381968\pi$
$600$	0	0
$601$	−9.18453	−0.374645	−0.187322	−	0.982298i	$-0.559981\pi$
$601$	−9.18453	−0.374645	−0.187322	+	0.982298i	$0.559981\pi$
$602$	0	0
$603$	−15.9026	−0.647604
$604$	0	0
$605$	−0.944253	−0.0383894
$606$	0	0
$607$	27.8378	1.12990	0.564951	−	0.825124i	$-0.308895\pi$
$607$	27.8378	1.12990	0.564951	+	0.825124i	$0.308895\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.47380	0.221446
$612$	0	0
$613$	25.9148	1.04669	0.523345	−	0.852121i	$-0.324684\pi$
$613$	25.9148	1.04669	0.523345	+	0.852121i	$0.324684\pi$
$614$	0	0
$615$	−0.127279	−0.00513237
$616$	0	0
$617$	−45.0746	−1.81463	−0.907317	−	0.420448i	$-0.861873\pi$
$617$	−45.0746	−1.81463	−0.907317	+	0.420448i	$0.861873\pi$
$618$	0	0
$619$	9.98422	0.401300	0.200650	−	0.979663i	$-0.435695\pi$
$619$	9.98422	0.401300	0.200650	+	0.979663i	$0.435695\pi$
$620$	0	0
$621$	−15.6319	−0.627288
$622$	0	0
$623$	0	0
$624$	0	0
$625$	24.9475	0.997902
$626$	0	0
$627$	−71.8274	−2.86851
$628$	0	0
$629$	36.5965	1.45920
$630$	0	0
$631$	−15.2360	−0.606536	−0.303268	−	0.952905i	$-0.598078\pi$
$631$	−15.2360	−0.606536	−0.303268	+	0.952905i	$0.598078\pi$
$632$	0	0
$633$	46.0578	1.83063
$634$	0	0
$635$	−0.727598	−0.0288738
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.85095	0.152341
$640$	0	0
$641$	−18.6951	−0.738413	−0.369207	−	0.929347i	$-0.620371\pi$
$641$	−18.6951	−0.738413	−0.369207	+	0.929347i	$0.620371\pi$
$642$	0	0
$643$	−33.6654	−1.32764	−0.663818	−	0.747895i	$-0.731065\pi$
$643$	−33.6654	−1.32764	−0.663818	+	0.747895i	$0.731065\pi$
$644$	0	0
$645$	−0.599089	−0.0235891
$646$	0	0
$647$	17.7667	0.698482	0.349241	−	0.937033i	$-0.386439\pi$
$647$	17.7667	0.698482	0.349241	+	0.937033i	$0.386439\pi$
$648$	0	0
$649$	23.8510	0.936233
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−14.5760	−0.570403	−0.285201	−	0.958468i	$-0.592060\pi$
$653$	−14.5760	−0.570403	−0.285201	+	0.958468i	$0.592060\pi$
$654$	0	0
$655$	0.812865	0.0317613
$656$	0	0
$657$	26.2174	1.02284
$658$	0	0
$659$	−7.13592	−0.277976	−0.138988	−	0.990294i	$-0.544385\pi$
$659$	−7.13592	−0.277976	−0.138988	+	0.990294i	$0.544385\pi$
$660$	0	0
$661$	−16.9329	−0.658614	−0.329307	−	0.944223i	$-0.606815\pi$
$661$	−16.9329	−0.658614	−0.329307	+	0.944223i	$0.606815\pi$
$662$	0	0
$663$	15.1350	0.587794
$664$	0	0
$665$	0	0
$666$	0	0
$667$	10.9271	0.423098
$668$	0	0
$669$	29.0465	1.12300
$670$	0	0
$671$	23.8728	0.921599
$672$	0	0
$673$	10.7074	0.412738	0.206369	−	0.978474i	$-0.433835\pi$
$673$	10.7074	0.412738	0.206369	+	0.978474i	$0.433835\pi$
$674$	0	0
$675$	−14.7193	−0.566546
$676$	0	0
$677$	37.3668	1.43612	0.718062	−	0.695979i	$-0.245029\pi$
$677$	37.3668	1.43612	0.718062	+	0.695979i	$0.245029\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−12.7144	−0.487217
$682$	0	0
$683$	−49.6732	−1.90069	−0.950346	−	0.311196i	$-0.899270\pi$
$683$	−49.6732	−1.90069	−0.950346	+	0.311196i	$0.899270\pi$
$684$	0	0
$685$	0.966560	0.0369303
$686$	0	0
$687$	−57.0891	−2.17808
$688$	0	0
$689$	3.22293	0.122784
$690$	0	0
$691$	−11.4435	−0.435330	−0.217665	−	0.976024i	$-0.569844\pi$
$691$	−11.4435	−0.435330	−0.217665	+	0.976024i	$0.569844\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.186678	−0.00708110
$696$	0	0
$697$	5.25954	0.199219
$698$	0	0
$699$	13.6626	0.516766
$700$	0	0
$701$	−8.67998	−0.327838	−0.163919	−	0.986474i	$-0.552414\pi$
$701$	−8.67998	−0.327838	−0.163919	+	0.986474i	$0.552414\pi$
$702$	0	0
$703$	44.7238	1.68679
$704$	0	0
$705$	0.521016	0.0196226
$706$	0	0
$707$	0	0
$708$	0	0
$709$	32.8906	1.23523	0.617615	−	0.786480i	$-0.288099\pi$
$709$	32.8906	1.23523	0.617615	+	0.786480i	$0.288099\pi$
$710$	0	0
$711$	−28.8673	−1.08261
$712$	0	0
$713$	22.3647	0.837564
$714$	0	0
$715$	0.410685	0.0153588
$716$	0	0
$717$	56.5228	2.11088
$718$	0	0
$719$	−18.6116	−0.694094	−0.347047	−	0.937848i	$-0.612816\pi$
$719$	−18.6116	−0.694094	−0.347047	+	0.937848i	$0.612816\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−37.8241	−1.40670
$724$	0	0
$725$	10.2891	0.382128
$726$	0	0
$727$	11.7494	0.435760	0.217880	−	0.975976i	$-0.430086\pi$
$727$	11.7494	0.435760	0.217880	+	0.975976i	$0.430086\pi$
$728$	0	0
$729$	0.697225	0.0258231
$730$	0	0
$731$	24.7562	0.915641
$732$	0	0
$733$	−34.0967	−1.25939	−0.629695	−	0.776843i	$-0.716820\pi$
$733$	−34.0967	−1.25939	−0.629695	+	0.776843i	$0.716820\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	50.6286	1.86493
$738$	0	0
$739$	29.0138	1.06729	0.533644	−	0.845709i	$-0.320822\pi$
$739$	29.0138	1.06729	0.533644	+	0.845709i	$0.320822\pi$
$740$	0	0
$741$	18.4962	0.679474
$742$	0	0
$743$	54.4385	1.99715	0.998577	−	0.0533240i	$-0.0169816\pi$
$743$	54.4385	1.99715	0.998577	+	0.0533240i	$0.0169816\pi$
$744$	0	0
$745$	−0.212985	−0.00780318
$746$	0	0
$747$	20.5390	0.751482
$748$	0	0
$749$	0	0
$750$	0	0
$751$	36.4957	1.33175	0.665873	−	0.746065i	$-0.268059\pi$
$751$	36.4957	1.33175	0.665873	+	0.746065i	$0.268059\pi$
$752$	0	0
$753$	61.3991	2.23751
$754$	0	0
$755$	1.20126	0.0437184
$756$	0	0
$757$	−18.4567	−0.670820	−0.335410	−	0.942072i	$-0.608875\pi$
$757$	−18.4567	−0.670820	−0.335410	+	0.942072i	$0.608875\pi$
$758$	0	0
$759$	−59.2973	−2.15236
$760$	0	0
$761$	−4.88895	−0.177224	−0.0886121	−	0.996066i	$-0.528243\pi$
$761$	−4.88895	−0.177224	−0.0886121	+	0.996066i	$0.528243\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0.507383	0.0183445
$766$	0	0
$767$	−6.14183	−0.221769
$768$	0	0
$769$	5.71646	0.206141	0.103070	−	0.994674i	$-0.467133\pi$
$769$	5.71646	0.206141	0.103070	+	0.994674i	$0.467133\pi$
$770$	0	0
$771$	44.1633	1.59050
$772$	0	0
$773$	−26.7111	−0.960731	−0.480365	−	0.877068i	$-0.659496\pi$
$773$	−26.7111	−0.960731	−0.480365	+	0.877068i	$0.659496\pi$
$774$	0	0
$775$	21.0589	0.756459
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.42758	0.230292
$780$	0	0
$781$	−12.2602	−0.438703
$782$	0	0
$783$	−6.06642	−0.216796
$784$	0	0
$785$	0.155031	0.00553329
$786$	0	0
$787$	10.9270	0.389504	0.194752	−	0.980853i	$-0.437610\pi$
$787$	10.9270	0.389504	0.194752	+	0.980853i	$0.437610\pi$
$788$	0	0
$789$	−16.8395	−0.599502
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−6.14745	−0.218302
$794$	0	0
$795$	0.306770	0.0108800
$796$	0	0
$797$	35.4254	1.25483	0.627416	−	0.778684i	$-0.284112\pi$
$797$	35.4254	1.25483	0.627416	+	0.778684i	$0.284112\pi$
$798$	0	0
$799$	−21.5300	−0.761676
$800$	0	0
$801$	1.09286	0.0386143
$802$	0	0
$803$	−83.4676	−2.94551
$804$	0	0
$805$	0	0
$806$	0	0
$807$	50.3920	1.77388
$808$	0	0
$809$	20.9058	0.735008	0.367504	−	0.930022i	$-0.380212\pi$
$809$	20.9058	0.735008	0.367504	+	0.930022i	$0.380212\pi$
$810$	0	0
$811$	47.1023	1.65398	0.826992	−	0.562214i	$-0.190050\pi$
$811$	47.1023	1.65398	0.826992	+	0.562214i	$0.190050\pi$
$812$	0	0
$813$	6.38364	0.223884
$814$	0	0
$815$	−1.02203	−0.0358000
$816$	0	0
$817$	30.2541	1.05846
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−54.8833	−1.91544	−0.957721	−	0.287700i	$-0.907109\pi$
$821$	−54.8833	−1.91544	−0.957721	+	0.287700i	$0.907109\pi$
$822$	0	0
$823$	13.0764	0.455815	0.227907	−	0.973683i	$-0.426812\pi$
$823$	13.0764	0.455815	0.227907	+	0.973683i	$0.426812\pi$
$824$	0	0
$825$	−55.8353	−1.94393
$826$	0	0
$827$	−37.0855	−1.28959	−0.644794	−	0.764356i	$-0.723057\pi$
$827$	−37.0855	−1.28959	−0.644794	+	0.764356i	$0.723057\pi$
$828$	0	0
$829$	34.3370	1.19257	0.596287	−	0.802771i	$-0.296642\pi$
$829$	34.3370	1.19257	0.596287	+	0.802771i	$0.296642\pi$
$830$	0	0
$831$	32.1357	1.11477
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.28894	0.0446056
$836$	0	0
$837$	−12.4163	−0.429169
$838$	0	0
$839$	−32.9880	−1.13887	−0.569436	−	0.822036i	$-0.692838\pi$
$839$	−32.9880	−1.13887	−0.569436	+	0.822036i	$0.692838\pi$
$840$	0	0
$841$	−24.7594	−0.853774
$842$	0	0
$843$	7.87056	0.271077
$844$	0	0
$845$	0.663124	0.0228121
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−42.2862	−1.45126
$850$	0	0
$851$	36.9218	1.26566
$852$	0	0
$853$	11.8738	0.406551	0.203275	−	0.979122i	$-0.434841\pi$
$853$	11.8738	0.406551	0.203275	+	0.979122i	$0.434841\pi$
$854$	0	0
$855$	0.620063	0.0212057
$856$	0	0
$857$	−17.6802	−0.603944	−0.301972	−	0.953317i	$-0.597645\pi$
$857$	−17.6802	−0.603944	−0.301972	+	0.953317i	$0.597645\pi$
$858$	0	0
$859$	14.0627	0.479812	0.239906	−	0.970796i	$-0.422883\pi$
$859$	14.0627	0.479812	0.239906	+	0.970796i	$0.422883\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−31.8983	−1.08583	−0.542916	−	0.839787i	$-0.682680\pi$
$863$	−31.8983	−1.08583	−0.542916	+	0.839787i	$0.682680\pi$
$864$	0	0
$865$	−0.439810	−0.0149540
$866$	0	0
$867$	−22.9463	−0.779296
$868$	0	0
$869$	91.9041	3.11763
$870$	0	0
$871$	−13.0373	−0.441752
$872$	0	0
$873$	−4.93555	−0.167043
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−27.3189	−0.922494	−0.461247	−	0.887272i	$-0.652598\pi$
$877$	−27.3189	−0.922494	−0.461247	+	0.887272i	$0.652598\pi$
$878$	0	0
$879$	27.8066	0.937893
$880$	0	0
$881$	6.22926	0.209869	0.104935	−	0.994479i	$-0.466537\pi$
$881$	6.22926	0.209869	0.104935	+	0.994479i	$0.466537\pi$
$882$	0	0
$883$	−31.9879	−1.07648	−0.538239	−	0.842792i	$-0.680910\pi$
$883$	−31.9879	−1.07648	−0.538239	+	0.842792i	$0.680910\pi$
$884$	0	0
$885$	−0.584601	−0.0196512
$886$	0	0
$887$	−22.5876	−0.758418	−0.379209	−	0.925311i	$-0.623804\pi$
$887$	−22.5876	−0.758418	−0.379209	+	0.925311i	$0.623804\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	58.3298	1.95412
$892$	0	0
$893$	−26.3114	−0.880477
$894$	0	0
$895$	0.926635	0.0309740
$896$	0	0
$897$	15.2696	0.509836
$898$	0	0
$899$	8.67924	0.289469
$900$	0	0
$901$	−12.6767	−0.422321
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.06159	−0.0352886
$906$	0	0
$907$	27.2201	0.903829	0.451915	−	0.892061i	$-0.350741\pi$
$907$	27.2201	0.903829	0.451915	+	0.892061i	$0.350741\pi$
$908$	0	0
$909$	5.65367	0.187520
$910$	0	0
$911$	26.8820	0.890641	0.445321	−	0.895371i	$-0.353090\pi$
$911$	26.8820	0.890641	0.445321	+	0.895371i	$0.353090\pi$
$912$	0	0
$913$	−65.3893	−2.16407
$914$	0	0
$915$	−0.585136	−0.0193440
$916$	0	0
$917$	0	0
$918$	0	0
$919$	2.23690	0.0737884	0.0368942	−	0.999319i	$-0.488254\pi$
$919$	2.23690	0.0737884	0.0368942	+	0.999319i	$0.488254\pi$
$920$	0	0
$921$	−13.6268	−0.449019
$922$	0	0
$923$	3.15709	0.103917
$924$	0	0
$925$	34.7662	1.14311
$926$	0	0
$927$	−27.0953	−0.889926
$928$	0	0
$929$	8.47724	0.278129	0.139065	−	0.990283i	$-0.455590\pi$
$929$	8.47724	0.278129	0.139065	+	0.990283i	$0.455590\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−20.5476	−0.672700
$934$	0	0
$935$	−1.61534	−0.0528273
$936$	0	0
$937$	1.35443	0.0442473	0.0221236	−	0.999755i	$-0.492957\pi$
$937$	1.35443	0.0442473	0.0221236	+	0.999755i	$0.492957\pi$
$938$	0	0
$939$	−33.6075	−1.09674
$940$	0	0
$941$	54.7641	1.78526	0.892630	−	0.450790i	$-0.148858\pi$
$941$	54.7641	1.78526	0.892630	+	0.450790i	$0.148858\pi$
$942$	0	0
$943$	5.30631	0.172797
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.23227	−0.137530	−0.0687651	−	0.997633i	$-0.521906\pi$
$947$	−4.23227	−0.137530	−0.0687651	+	0.997633i	$0.521906\pi$
$948$	0	0
$949$	21.4936	0.697712
$950$	0	0
$951$	−55.2817	−1.79263
$952$	0	0
$953$	9.89170	0.320424	0.160212	−	0.987083i	$-0.448782\pi$
$953$	9.89170	0.320424	0.160212	+	0.987083i	$0.448782\pi$
$954$	0	0
$955$	−0.649358	−0.0210127
$956$	0	0
$957$	−23.0120	−0.743872
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−13.2360	−0.426968
$962$	0	0
$963$	7.80313	0.251452
$964$	0	0
$965$	0.494136	0.0159068
$966$	0	0
$967$	2.64614	0.0850942	0.0425471	−	0.999094i	$-0.486453\pi$
$967$	2.64614	0.0850942	0.0425471	+	0.999094i	$0.486453\pi$
$968$	0	0
$969$	−72.7506	−2.33709
$970$	0	0
$971$	−7.20724	−0.231291	−0.115646	−	0.993291i	$-0.536894\pi$
$971$	−7.20724	−0.231291	−0.115646	+	0.993291i	$0.536894\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	14.3781	0.460466
$976$	0	0
$977$	−27.0409	−0.865115	−0.432557	−	0.901606i	$-0.642389\pi$
$977$	−27.0409	−0.865115	−0.432557	+	0.901606i	$0.642389\pi$
$978$	0	0
$979$	−3.47930	−0.111199
$980$	0	0
$981$	16.6016	0.530048
$982$	0	0
$983$	−50.7546	−1.61882	−0.809410	−	0.587244i	$-0.800213\pi$
$983$	−50.7546	−1.61882	−0.809410	+	0.587244i	$0.800213\pi$
$984$	0	0
$985$	0.510252	0.0162580
$986$	0	0
$987$	0	0
$988$	0	0
$989$	24.9763	0.794200
$990$	0	0
$991$	−37.2471	−1.18319	−0.591596	−	0.806234i	$-0.701502\pi$
$991$	−37.2471	−1.18319	−0.591596	+	0.806234i	$0.701502\pi$
$992$	0	0
$993$	−21.9105	−0.695308
$994$	0	0
$995$	−0.896477	−0.0284202
$996$	0	0
$997$	39.2070	1.24170	0.620849	−	0.783930i	$-0.286788\pi$
$997$	39.2070	1.24170	0.620849	+	0.783930i	$0.286788\pi$
$998$	0	0
$999$	−20.4980	−0.648528

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.t.1.4	yes	20
7.6	odd	2		8036.2.a.s.1.17	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.s.1.17	✓	20	7.6	odd	2
8036.2.a.t.1.4	yes	20	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q-2.15199 q^{3} -0.0591445 q^{5} +1.63107 q^{9} +O(q^{10})\)	\(q-2.15199 q^{3} -0.0591445 q^{5} +1.63107 q^{9} -5.19280 q^{11} +1.33719 q^{13} +0.127279 q^{15} -5.25954 q^{17} -6.42758 q^{19} -5.30631 q^{23} -4.99650 q^{25} +2.94592 q^{27} -2.05926 q^{29} -4.21473 q^{31} +11.1749 q^{33} -6.95811 q^{37} -2.87762 q^{39} -1.00000 q^{41} -4.70691 q^{43} -0.0964691 q^{45} +4.09351 q^{47} +11.3185 q^{51} +2.41022 q^{53} +0.307126 q^{55} +13.8321 q^{57} -4.59309 q^{59} -4.59729 q^{61} -0.0790874 q^{65} -9.74977 q^{67} +11.4191 q^{69} +2.36099 q^{71} +16.0737 q^{73} +10.7524 q^{75} -17.6984 q^{79} -11.2328 q^{81} +12.5923 q^{83} +0.311073 q^{85} +4.43152 q^{87} +0.670024 q^{89} +9.07008 q^{93} +0.380156 q^{95} -3.02595 q^{97} -8.46985 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9}+O(q^{10}) \) 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9	\( 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9} - 8 q^{11} + 12 q^{13} + 8 q^{15} + 8 q^{17} + 24 q^{19} + 8 q^{23} + 20 q^{25} + 16 q^{27} - 12 q^{29} + 44 q^{33} + 12 q^{37} + 12 q^{39} - 20 q^{41} + 4 q^{43} + 40 q^{45} + 4 q^{47} + 4 q^{51} - 12 q^{53} - 16 q^{55} + 28 q^{57} + 16 q^{59} + 68 q^{61} - 8 q^{65} + 4 q^{67} + 32 q^{69} + 8 q^{71} + 48 q^{73} + 60 q^{75} - 20 q^{79} + 32 q^{81} - 8 q^{83} - 28 q^{85} + 60 q^{89} - 16 q^{93} + 20 q^{95} + 40 q^{97}+O(q^{100}) \) 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9 - 8 * q^11 + 12 * q^13 + 8 * q^15 + 8 * q^17 + 24 * q^19 + 8 * q^23 + 20 * q^25 + 16 * q^27 - 12 * q^29 + 44 * q^33 + 12 * q^37 + 12 * q^39 - 20 * q^41 + 4 * q^43 + 40 * q^45 + 4 * q^47 + 4 * q^51 - 12 * q^53 - 16 * q^55 + 28 * q^57 + 16 * q^59 + 68 * q^61 - 8 * q^65 + 4 * q^67 + 32 * q^69 + 8 * q^71 + 48 * q^73 + 60 * q^75 - 20 * q^79 + 32 * q^81 - 8 * q^83 - 28 * q^85 + 60 * q^89 - 16 * q^93 + 20 * q^95 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more