LMFDB - Embedded newform 8024.2.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.54005 q^{3} +2.36838 q^{5} +1.43029 q^{7} +3.45187 q^{9} +O(q^{10})$	$q-2.54005 q^{3} +2.36838 q^{5} +1.43029 q^{7} +3.45187 q^{9} +0.0740282 q^{11} -0.356078 q^{13} -6.01582 q^{15} -1.00000 q^{17} +5.87699 q^{19} -3.63301 q^{21} +2.37381 q^{23} +0.609244 q^{25} -1.14778 q^{27} -7.93401 q^{29} -5.12619 q^{31} -0.188036 q^{33} +3.38747 q^{35} -4.00476 q^{37} +0.904458 q^{39} -3.80620 q^{41} +3.55903 q^{43} +8.17536 q^{45} -1.51319 q^{47} -4.95428 q^{49} +2.54005 q^{51} -7.98460 q^{53} +0.175327 q^{55} -14.9279 q^{57} +1.00000 q^{59} +3.91073 q^{61} +4.93717 q^{63} -0.843330 q^{65} -16.0634 q^{67} -6.02961 q^{69} +12.2742 q^{71} +7.16421 q^{73} -1.54751 q^{75} +0.105882 q^{77} -12.9194 q^{79} -7.44019 q^{81} -3.07834 q^{83} -2.36838 q^{85} +20.1528 q^{87} +5.00229 q^{89} -0.509294 q^{91} +13.0208 q^{93} +13.9190 q^{95} -1.56729 q^{97} +0.255536 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) $ 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9	$ 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9} + 2 q^{11} - 9 q^{13} - 7 q^{15} - 22 q^{17} - 10 q^{19} - 10 q^{21} + 14 q^{23} + 3 q^{25} + 6 q^{29} - 15 q^{31} - 52 q^{33} - 7 q^{35} + 9 q^{37} - 52 q^{41} - 7 q^{43} - 30 q^{45} - 7 q^{47} - 6 q^{49} + 3 q^{51} - 18 q^{53} - 39 q^{55} - 2 q^{57} + 22 q^{59} - 42 q^{61} - 35 q^{65} - 28 q^{67} - 10 q^{69} + 23 q^{71} - 33 q^{73} - 3 q^{75} - 28 q^{77} - 30 q^{79} - 2 q^{81} + 11 q^{83} - 3 q^{85} + 27 q^{87} - 34 q^{89} - 18 q^{91} - 28 q^{93} - 10 q^{95} - 3 q^{97} - 28 q^{99}+O(q^{100}) $ 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9 + 2 * q^11 - 9 * q^13 - 7 * q^15 - 22 * q^17 - 10 * q^19 - 10 * q^21 + 14 * q^23 + 3 * q^25 + 6 * q^29 - 15 * q^31 - 52 * q^33 - 7 * q^35 + 9 * q^37 - 52 * q^41 - 7 * q^43 - 30 * q^45 - 7 * q^47 - 6 * q^49 + 3 * q^51 - 18 * q^53 - 39 * q^55 - 2 * q^57 + 22 * q^59 - 42 * q^61 - 35 * q^65 - 28 * q^67 - 10 * q^69 + 23 * q^71 - 33 * q^73 - 3 * q^75 - 28 * q^77 - 30 * q^79 - 2 * q^81 + 11 * q^83 - 3 * q^85 + 27 * q^87 - 34 * q^89 - 18 * q^91 - 28 * q^93 - 10 * q^95 - 3 * q^97 - 28 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.54005	−1.46650	−0.733250	−	0.679959i	$-0.761998\pi$
$3$	−2.54005	−1.46650	−0.733250	+	0.679959i	$0.761998\pi$
$4$	0	0
$5$	2.36838	1.05917	0.529587	−	0.848256i	$-0.322347\pi$
$5$	2.36838	1.05917	0.529587	+	0.848256i	$0.322347\pi$
$6$	0	0
$7$	1.43029	0.540598	0.270299	−	0.962776i	$-0.412877\pi$
$7$	1.43029	0.540598	0.270299	+	0.962776i	$0.412877\pi$
$8$	0	0
$9$	3.45187	1.15062
$10$	0	0
$11$	0.0740282	0.0223204	0.0111602	−	0.999938i	$-0.496448\pi$
$11$	0.0740282	0.0223204	0.0111602	+	0.999938i	$0.496448\pi$
$12$	0	0
$13$	−0.356078	−0.0987583	−0.0493792	−	0.998780i	$-0.515724\pi$
$13$	−0.356078	−0.0987583	−0.0493792	+	0.998780i	$0.515724\pi$
$14$	0	0
$15$	−6.01582	−1.55328
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	5.87699	1.34827	0.674136	−	0.738607i	$-0.264516\pi$
$19$	5.87699	1.34827	0.674136	+	0.738607i	$0.264516\pi$
$20$	0	0
$21$	−3.63301	−0.792787
$22$	0	0
$23$	2.37381	0.494974	0.247487	−	0.968891i	$-0.420395\pi$
$23$	2.37381	0.494974	0.247487	+	0.968891i	$0.420395\pi$
$24$	0	0
$25$	0.609244	0.121849
$26$	0	0
$27$	−1.14778	−0.220890
$28$	0	0
$29$	−7.93401	−1.47331	−0.736654	−	0.676270i	$-0.763595\pi$
$29$	−7.93401	−1.47331	−0.736654	+	0.676270i	$0.763595\pi$
$30$	0	0
$31$	−5.12619	−0.920691	−0.460346	−	0.887740i	$-0.652275\pi$
$31$	−5.12619	−0.920691	−0.460346	+	0.887740i	$0.652275\pi$
$32$	0	0
$33$	−0.188036	−0.0327328
$34$	0	0
$35$	3.38747	0.572587
$36$	0	0
$37$	−4.00476	−0.658379	−0.329189	−	0.944264i	$-0.606775\pi$
$37$	−4.00476	−0.658379	−0.329189	+	0.944264i	$0.606775\pi$
$38$	0	0
$39$	0.904458	0.144829
$40$	0	0
$41$	−3.80620	−0.594428	−0.297214	−	0.954811i	$-0.596057\pi$
$41$	−3.80620	−0.594428	−0.297214	+	0.954811i	$0.596057\pi$
$42$	0	0
$43$	3.55903	0.542747	0.271374	−	0.962474i	$-0.412522\pi$
$43$	3.55903	0.542747	0.271374	+	0.962474i	$0.412522\pi$
$44$	0	0
$45$	8.17536	1.21871
$46$	0	0
$47$	−1.51319	−0.220721	−0.110361	−	0.993892i	$-0.535201\pi$
$47$	−1.51319	−0.220721	−0.110361	+	0.993892i	$0.535201\pi$
$48$	0	0
$49$	−4.95428	−0.707754
$50$	0	0
$51$	2.54005	0.355679
$52$	0	0
$53$	−7.98460	−1.09677	−0.548384	−	0.836226i	$-0.684757\pi$
$53$	−7.98460	−1.09677	−0.548384	+	0.836226i	$0.684757\pi$
$54$	0	0
$55$	0.175327	0.0236411
$56$	0	0
$57$	−14.9279	−1.97724
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	3.91073	0.500718	0.250359	−	0.968153i	$-0.419451\pi$
$61$	3.91073	0.500718	0.250359	+	0.968153i	$0.419451\pi$
$62$	0	0
$63$	4.93717	0.622025
$64$	0	0
$65$	−0.843330	−0.104602
$66$	0	0
$67$	−16.0634	−1.96246	−0.981228	−	0.192850i	$-0.938227\pi$
$67$	−16.0634	−1.96246	−0.981228	+	0.192850i	$0.938227\pi$
$68$	0	0
$69$	−6.02961	−0.725879
$70$	0	0
$71$	12.2742	1.45667	0.728337	−	0.685219i	$-0.240294\pi$
$71$	12.2742	1.45667	0.728337	+	0.685219i	$0.240294\pi$
$72$	0	0
$73$	7.16421	0.838507	0.419254	−	0.907869i	$-0.362292\pi$
$73$	7.16421	0.838507	0.419254	+	0.907869i	$0.362292\pi$
$74$	0	0
$75$	−1.54751	−0.178691
$76$	0	0
$77$	0.105882	0.0120663
$78$	0	0
$79$	−12.9194	−1.45355	−0.726774	−	0.686877i	$-0.758981\pi$
$79$	−12.9194	−1.45355	−0.726774	+	0.686877i	$0.758981\pi$
$80$	0	0
$81$	−7.44019	−0.826688
$82$	0	0
$83$	−3.07834	−0.337892	−0.168946	−	0.985625i	$-0.554036\pi$
$83$	−3.07834	−0.337892	−0.168946	+	0.985625i	$0.554036\pi$
$84$	0	0
$85$	−2.36838	−0.256887
$86$	0	0
$87$	20.1528	2.16061
$88$	0	0
$89$	5.00229	0.530241	0.265121	−	0.964215i	$-0.414588\pi$
$89$	5.00229	0.530241	0.265121	+	0.964215i	$0.414588\pi$
$90$	0	0
$91$	−0.509294	−0.0533885
$92$	0	0
$93$	13.0208	1.35019
$94$	0	0
$95$	13.9190	1.42806
$96$	0	0
$97$	−1.56729	−0.159135	−0.0795673	−	0.996829i	$-0.525354\pi$
$97$	−1.56729	−0.159135	−0.0795673	+	0.996829i	$0.525354\pi$
$98$	0	0
$99$	0.255536	0.0256823
$100$	0	0
$101$	−10.4663	−1.04143	−0.520717	−	0.853729i	$-0.674335\pi$
$101$	−10.4663	−1.04143	−0.520717	+	0.853729i	$0.674335\pi$
$102$	0	0
$103$	4.58129	0.451408	0.225704	−	0.974196i	$-0.427532\pi$
$103$	4.58129	0.451408	0.225704	+	0.974196i	$0.427532\pi$
$104$	0	0
$105$	−8.60436	−0.839699
$106$	0	0
$107$	−0.722190	−0.0698168	−0.0349084	−	0.999391i	$-0.511114\pi$
$107$	−0.722190	−0.0698168	−0.0349084	+	0.999391i	$0.511114\pi$
$108$	0	0
$109$	−11.6914	−1.11984	−0.559918	−	0.828548i	$-0.689167\pi$
$109$	−11.6914	−1.11984	−0.559918	+	0.828548i	$0.689167\pi$
$110$	0	0
$111$	10.1723	0.965513
$112$	0	0
$113$	−14.1812	−1.33406	−0.667028	−	0.745033i	$-0.732434\pi$
$113$	−14.1812	−1.33406	−0.667028	+	0.745033i	$0.732434\pi$
$114$	0	0
$115$	5.62209	0.524263
$116$	0	0
$117$	−1.22914	−0.113634
$118$	0	0
$119$	−1.43029	−0.131114
$120$	0	0
$121$	−10.9945	−0.999502
$122$	0	0
$123$	9.66794	0.871729
$124$	0	0
$125$	−10.3990	−0.930115
$126$	0	0
$127$	−8.78911	−0.779907	−0.389954	−	0.920835i	$-0.627509\pi$
$127$	−8.78911	−0.779907	−0.389954	+	0.920835i	$0.627509\pi$
$128$	0	0
$129$	−9.04013	−0.795939
$130$	0	0
$131$	1.47987	0.129297	0.0646484	−	0.997908i	$-0.479407\pi$
$131$	1.47987	0.129297	0.0646484	+	0.997908i	$0.479407\pi$
$132$	0	0
$133$	8.40578	0.728873
$134$	0	0
$135$	−2.71839	−0.233961
$136$	0	0
$137$	−4.08596	−0.349087	−0.174543	−	0.984649i	$-0.555845\pi$
$137$	−4.08596	−0.349087	−0.174543	+	0.984649i	$0.555845\pi$
$138$	0	0
$139$	21.0812	1.78808	0.894041	−	0.447985i	$-0.147858\pi$
$139$	21.0812	1.78808	0.894041	+	0.447985i	$0.147858\pi$
$140$	0	0
$141$	3.84358	0.323688
$142$	0	0
$143$	−0.0263598	−0.00220432
$144$	0	0
$145$	−18.7908	−1.56049
$146$	0	0
$147$	12.5841	1.03792
$148$	0	0
$149$	−0.662506	−0.0542746	−0.0271373	−	0.999632i	$-0.508639\pi$
$149$	−0.662506	−0.0542746	−0.0271373	+	0.999632i	$0.508639\pi$
$150$	0	0
$151$	7.12716	0.580000	0.290000	−	0.957027i	$-0.406345\pi$
$151$	7.12716	0.580000	0.290000	+	0.957027i	$0.406345\pi$
$152$	0	0
$153$	−3.45187	−0.279067
$154$	0	0
$155$	−12.1408	−0.975172
$156$	0	0
$157$	21.1002	1.68398	0.841991	−	0.539492i	$-0.181384\pi$
$157$	21.1002	1.68398	0.841991	+	0.539492i	$0.181384\pi$
$158$	0	0
$159$	20.2813	1.60841
$160$	0	0
$161$	3.39523	0.267582
$162$	0	0
$163$	−13.7464	−1.07670	−0.538351	−	0.842721i	$-0.680952\pi$
$163$	−13.7464	−1.07670	−0.538351	+	0.842721i	$0.680952\pi$
$164$	0	0
$165$	−0.445341	−0.0346697
$166$	0	0
$167$	−12.4258	−0.961539	−0.480769	−	0.876847i	$-0.659643\pi$
$167$	−12.4258	−0.961539	−0.480769	+	0.876847i	$0.659643\pi$
$168$	0	0
$169$	−12.8732	−0.990247
$170$	0	0
$171$	20.2866	1.55136
$172$	0	0
$173$	6.18872	0.470519	0.235260	−	0.971933i	$-0.424406\pi$
$173$	6.18872	0.470519	0.235260	+	0.971933i	$0.424406\pi$
$174$	0	0
$175$	0.871394	0.0658712
$176$	0	0
$177$	−2.54005	−0.190922
$178$	0	0
$179$	−6.69871	−0.500685	−0.250343	−	0.968157i	$-0.580543\pi$
$179$	−6.69871	−0.500685	−0.250343	+	0.968157i	$0.580543\pi$
$180$	0	0
$181$	−0.474434	−0.0352644	−0.0176322	−	0.999845i	$-0.505613\pi$
$181$	−0.474434	−0.0352644	−0.0176322	+	0.999845i	$0.505613\pi$
$182$	0	0
$183$	−9.93346	−0.734303
$184$	0	0
$185$	−9.48481	−0.697337
$186$	0	0
$187$	−0.0740282	−0.00541348
$188$	0	0
$189$	−1.64166	−0.119413
$190$	0	0
$191$	23.3336	1.68836	0.844180	−	0.536060i	$-0.180088\pi$
$191$	23.3336	1.68836	0.844180	+	0.536060i	$0.180088\pi$
$192$	0	0
$193$	16.3962	1.18022	0.590111	−	0.807322i	$-0.299084\pi$
$193$	16.3962	1.18022	0.590111	+	0.807322i	$0.299084\pi$
$194$	0	0
$195$	2.14210	0.153399
$196$	0	0
$197$	23.3752	1.66542	0.832708	−	0.553713i	$-0.186789\pi$
$197$	23.3752	1.66542	0.832708	+	0.553713i	$0.186789\pi$
$198$	0	0
$199$	21.9174	1.55369	0.776843	−	0.629695i	$-0.216820\pi$
$199$	21.9174	1.55369	0.776843	+	0.629695i	$0.216820\pi$
$200$	0	0
$201$	40.8019	2.87794
$202$	0	0
$203$	−11.3479	−0.796467
$204$	0	0
$205$	−9.01453	−0.629602
$206$	0	0
$207$	8.19409	0.569529
$208$	0	0
$209$	0.435063	0.0300939
$210$	0	0
$211$	−25.3057	−1.74212	−0.871058	−	0.491180i	$-0.836566\pi$
$211$	−25.3057	−1.74212	−0.871058	+	0.491180i	$0.836566\pi$
$212$	0	0
$213$	−31.1770	−2.13621
$214$	0	0
$215$	8.42915	0.574863
$216$	0	0
$217$	−7.33193	−0.497724
$218$	0	0
$219$	−18.1975	−1.22967
$220$	0	0
$221$	0.356078	0.0239524
$222$	0	0
$223$	−6.66196	−0.446118	−0.223059	−	0.974805i	$-0.571604\pi$
$223$	−6.66196	−0.446118	−0.223059	+	0.974805i	$0.571604\pi$
$224$	0	0
$225$	2.10303	0.140202
$226$	0	0
$227$	20.7778	1.37907	0.689536	−	0.724251i	$-0.257814\pi$
$227$	20.7778	1.37907	0.689536	+	0.724251i	$0.257814\pi$
$228$	0	0
$229$	−8.85464	−0.585131	−0.292566	−	0.956245i	$-0.594509\pi$
$229$	−8.85464	−0.585131	−0.292566	+	0.956245i	$0.594509\pi$
$230$	0	0
$231$	−0.268945	−0.0176953
$232$	0	0
$233$	5.91059	0.387215	0.193608	−	0.981079i	$-0.437981\pi$
$233$	5.91059	0.387215	0.193608	+	0.981079i	$0.437981\pi$
$234$	0	0
$235$	−3.58381	−0.233782
$236$	0	0
$237$	32.8160	2.13163
$238$	0	0
$239$	5.25552	0.339951	0.169976	−	0.985448i	$-0.445631\pi$
$239$	5.25552	0.339951	0.169976	+	0.985448i	$0.445631\pi$
$240$	0	0
$241$	−17.3229	−1.11586	−0.557932	−	0.829887i	$-0.688405\pi$
$241$	−17.3229	−1.11586	−0.557932	+	0.829887i	$0.688405\pi$
$242$	0	0
$243$	22.3418	1.43323
$244$	0	0
$245$	−11.7336	−0.749634
$246$	0	0
$247$	−2.09267	−0.133153
$248$	0	0
$249$	7.81915	0.495518
$250$	0	0
$251$	23.7891	1.50155	0.750776	−	0.660557i	$-0.229680\pi$
$251$	23.7891	1.50155	0.750776	+	0.660557i	$0.229680\pi$
$252$	0	0
$253$	0.175729	0.0110480
$254$	0	0
$255$	6.01582	0.376725
$256$	0	0
$257$	−25.6302	−1.59877	−0.799384	−	0.600821i	$-0.794841\pi$
$257$	−25.6302	−1.59877	−0.799384	+	0.600821i	$0.794841\pi$
$258$	0	0
$259$	−5.72796	−0.355918
$260$	0	0
$261$	−27.3872	−1.69522
$262$	0	0
$263$	9.54859	0.588791	0.294396	−	0.955684i	$-0.404882\pi$
$263$	9.54859	0.588791	0.294396	+	0.955684i	$0.404882\pi$
$264$	0	0
$265$	−18.9106	−1.16167
$266$	0	0
$267$	−12.7061	−0.777599
$268$	0	0
$269$	−29.6278	−1.80644	−0.903219	−	0.429180i	$-0.858802\pi$
$269$	−29.6278	−1.80644	−0.903219	+	0.429180i	$0.858802\pi$
$270$	0	0
$271$	−27.1978	−1.65215	−0.826075	−	0.563560i	$-0.809431\pi$
$271$	−27.1978	−1.65215	−0.826075	+	0.563560i	$0.809431\pi$
$272$	0	0
$273$	1.29363	0.0782943
$274$	0	0
$275$	0.0451012	0.00271971
$276$	0	0
$277$	5.02231	0.301761	0.150881	−	0.988552i	$-0.451789\pi$
$277$	5.02231	0.301761	0.150881	+	0.988552i	$0.451789\pi$
$278$	0	0
$279$	−17.6950	−1.05937
$280$	0	0
$281$	21.8771	1.30508	0.652540	−	0.757755i	$-0.273704\pi$
$281$	21.8771	1.30508	0.652540	+	0.757755i	$0.273704\pi$
$282$	0	0
$283$	11.9592	0.710900	0.355450	−	0.934695i	$-0.384328\pi$
$283$	11.9592	0.710900	0.355450	+	0.934695i	$0.384328\pi$
$284$	0	0
$285$	−35.3549	−2.09424
$286$	0	0
$287$	−5.44395	−0.321346
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	3.98101	0.233371
$292$	0	0
$293$	24.9387	1.45694	0.728468	−	0.685080i	$-0.240233\pi$
$293$	24.9387	1.45694	0.728468	+	0.685080i	$0.240233\pi$
$294$	0	0
$295$	2.36838	0.137893
$296$	0	0
$297$	−0.0849682	−0.00493035
$298$	0	0
$299$	−0.845262	−0.0488828
$300$	0	0
$301$	5.09044	0.293408
$302$	0	0
$303$	26.5849	1.52726
$304$	0	0
$305$	9.26211	0.530347
$306$	0	0
$307$	−19.6691	−1.12257	−0.561286	−	0.827622i	$-0.689693\pi$
$307$	−19.6691	−1.12257	−0.561286	+	0.827622i	$0.689693\pi$
$308$	0	0
$309$	−11.6367	−0.661991
$310$	0	0
$311$	6.08733	0.345181	0.172590	−	0.984994i	$-0.444786\pi$
$311$	6.08733	0.345181	0.172590	+	0.984994i	$0.444786\pi$
$312$	0	0
$313$	0.618473	0.0349582	0.0174791	−	0.999847i	$-0.494436\pi$
$313$	0.618473	0.0349582	0.0174791	+	0.999847i	$0.494436\pi$
$314$	0	0
$315$	11.6931	0.658832
$316$	0	0
$317$	27.4938	1.54421	0.772103	−	0.635498i	$-0.219205\pi$
$317$	27.4938	1.54421	0.772103	+	0.635498i	$0.219205\pi$
$318$	0	0
$319$	−0.587341	−0.0328848
$320$	0	0
$321$	1.83440	0.102386
$322$	0	0
$323$	−5.87699	−0.327004
$324$	0	0
$325$	−0.216938	−0.0120336
$326$	0	0
$327$	29.6968	1.64224
$328$	0	0
$329$	−2.16429	−0.119321
$330$	0	0
$331$	13.1137	0.720795	0.360398	−	0.932799i	$-0.382641\pi$
$331$	13.1137	0.720795	0.360398	+	0.932799i	$0.382641\pi$
$332$	0	0
$333$	−13.8239	−0.757546
$334$	0	0
$335$	−38.0443	−2.07858
$336$	0	0
$337$	−33.1375	−1.80511	−0.902557	−	0.430571i	$-0.858312\pi$
$337$	−33.1375	−1.80511	−0.902557	+	0.430571i	$0.858312\pi$
$338$	0	0
$339$	36.0210	1.95639
$340$	0	0
$341$	−0.379483	−0.0205502
$342$	0	0
$343$	−17.0981	−0.923208
$344$	0	0
$345$	−14.2804	−0.768832
$346$	0	0
$347$	−13.5021	−0.724830	−0.362415	−	0.932017i	$-0.618048\pi$
$347$	−13.5021	−0.724830	−0.362415	+	0.932017i	$0.618048\pi$
$348$	0	0
$349$	25.4079	1.36005	0.680027	−	0.733187i	$-0.261968\pi$
$349$	25.4079	1.36005	0.680027	+	0.733187i	$0.261968\pi$
$350$	0	0
$351$	0.408700	0.0218148
$352$	0	0
$353$	−10.9965	−0.585283	−0.292642	−	0.956222i	$-0.594534\pi$
$353$	−10.9965	−0.585283	−0.292642	+	0.956222i	$0.594534\pi$
$354$	0	0
$355$	29.0699	1.54287
$356$	0	0
$357$	3.63301	0.192279
$358$	0	0
$359$	−18.6527	−0.984453	−0.492226	−	0.870467i	$-0.663817\pi$
$359$	−18.6527	−0.984453	−0.492226	+	0.870467i	$0.663817\pi$
$360$	0	0
$361$	15.5390	0.817840
$362$	0	0
$363$	27.9267	1.46577
$364$	0	0
$365$	16.9676	0.888125
$366$	0	0
$367$	−7.38469	−0.385478	−0.192739	−	0.981250i	$-0.561737\pi$
$367$	−7.38469	−0.385478	−0.192739	+	0.981250i	$0.561737\pi$
$368$	0	0
$369$	−13.1385	−0.683963
$370$	0	0
$371$	−11.4203	−0.592911
$372$	0	0
$373$	−16.2154	−0.839601	−0.419800	−	0.907616i	$-0.637900\pi$
$373$	−16.2154	−0.839601	−0.419800	+	0.907616i	$0.637900\pi$
$374$	0	0
$375$	26.4140	1.36401
$376$	0	0
$377$	2.82513	0.145501
$378$	0	0
$379$	−5.86295	−0.301160	−0.150580	−	0.988598i	$-0.548114\pi$
$379$	−5.86295	−0.301160	−0.150580	+	0.988598i	$0.548114\pi$
$380$	0	0
$381$	22.3248	1.14373
$382$	0	0
$383$	−22.1348	−1.13103	−0.565517	−	0.824737i	$-0.691323\pi$
$383$	−22.1348	−1.13103	−0.565517	+	0.824737i	$0.691323\pi$
$384$	0	0
$385$	0.250768	0.0127803
$386$	0	0
$387$	12.2853	0.624498
$388$	0	0
$389$	2.95835	0.149994	0.0749970	−	0.997184i	$-0.476105\pi$
$389$	2.95835	0.149994	0.0749970	+	0.997184i	$0.476105\pi$
$390$	0	0
$391$	−2.37381	−0.120049
$392$	0	0
$393$	−3.75895	−0.189614
$394$	0	0
$395$	−30.5981	−1.53956
$396$	0	0
$397$	21.1692	1.06245	0.531225	−	0.847231i	$-0.321732\pi$
$397$	21.1692	1.06245	0.531225	+	0.847231i	$0.321732\pi$
$398$	0	0
$399$	−21.3511	−1.06889
$400$	0	0
$401$	−9.90577	−0.494671	−0.247335	−	0.968930i	$-0.579555\pi$
$401$	−9.90577	−0.494671	−0.247335	+	0.968930i	$0.579555\pi$
$402$	0	0
$403$	1.82533	0.0909259
$404$	0	0
$405$	−17.6212	−0.875606
$406$	0	0
$407$	−0.296465	−0.0146952
$408$	0	0
$409$	38.8031	1.91869	0.959345	−	0.282238i	$-0.0910767\pi$
$409$	38.8031	1.91869	0.959345	+	0.282238i	$0.0910767\pi$
$410$	0	0
$411$	10.3785	0.511936
$412$	0	0
$413$	1.43029	0.0703798
$414$	0	0
$415$	−7.29069	−0.357886
$416$	0	0
$417$	−53.5473	−2.62222
$418$	0	0
$419$	−28.3517	−1.38507	−0.692535	−	0.721384i	$-0.743506\pi$
$419$	−28.3517	−1.38507	−0.692535	+	0.721384i	$0.743506\pi$
$420$	0	0
$421$	−18.2736	−0.890602	−0.445301	−	0.895381i	$-0.646903\pi$
$421$	−18.2736	−0.890602	−0.445301	+	0.895381i	$0.646903\pi$
$422$	0	0
$423$	−5.22333	−0.253967
$424$	0	0
$425$	−0.609244	−0.0295527
$426$	0	0
$427$	5.59347	0.270687
$428$	0	0
$429$	0.0669554	0.00323264
$430$	0	0
$431$	−14.1801	−0.683033	−0.341516	−	0.939876i	$-0.610940\pi$
$431$	−14.1801	−0.683033	−0.341516	+	0.939876i	$0.610940\pi$
$432$	0	0
$433$	−33.8505	−1.62675	−0.813375	−	0.581740i	$-0.802372\pi$
$433$	−33.8505	−1.62675	−0.813375	+	0.581740i	$0.802372\pi$
$434$	0	0
$435$	47.7296	2.28846
$436$	0	0
$437$	13.9508	0.667360
$438$	0	0
$439$	−18.6851	−0.891794	−0.445897	−	0.895084i	$-0.647115\pi$
$439$	−18.6851	−0.891794	−0.445897	+	0.895084i	$0.647115\pi$
$440$	0	0
$441$	−17.1015	−0.814359
$442$	0	0
$443$	14.4207	0.685149	0.342575	−	0.939491i	$-0.388701\pi$
$443$	14.4207	0.685149	0.342575	+	0.939491i	$0.388701\pi$
$444$	0	0
$445$	11.8473	0.561618
$446$	0	0
$447$	1.68280	0.0795937
$448$	0	0
$449$	−3.96847	−0.187284	−0.0936418	−	0.995606i	$-0.529851\pi$
$449$	−3.96847	−0.187284	−0.0936418	+	0.995606i	$0.529851\pi$
$450$	0	0
$451$	−0.281766	−0.0132678
$452$	0	0
$453$	−18.1034	−0.850570
$454$	0	0
$455$	−1.20620	−0.0565477
$456$	0	0
$457$	16.9654	0.793609	0.396804	−	0.917903i	$-0.370119\pi$
$457$	16.9654	0.793609	0.396804	+	0.917903i	$0.370119\pi$
$458$	0	0
$459$	1.14778	0.0535738
$460$	0	0
$461$	−28.6102	−1.33251	−0.666255	−	0.745724i	$-0.732104\pi$
$461$	−28.6102	−1.33251	−0.666255	+	0.745724i	$0.732104\pi$
$462$	0	0
$463$	−37.5642	−1.74575	−0.872877	−	0.487940i	$-0.837749\pi$
$463$	−37.5642	−1.74575	−0.872877	+	0.487940i	$0.837749\pi$
$464$	0	0
$465$	30.8383	1.43009
$466$	0	0
$467$	−12.9155	−0.597659	−0.298829	−	0.954307i	$-0.596596\pi$
$467$	−12.9155	−0.597659	−0.298829	+	0.954307i	$0.596596\pi$
$468$	0	0
$469$	−22.9753	−1.06090
$470$	0	0
$471$	−53.5957	−2.46956
$472$	0	0
$473$	0.263469	0.0121143
$474$	0	0
$475$	3.58052	0.164285
$476$	0	0
$477$	−27.5618	−1.26197
$478$	0	0
$479$	−24.1554	−1.10369	−0.551845	−	0.833947i	$-0.686076\pi$
$479$	−24.1554	−1.10369	−0.551845	+	0.833947i	$0.686076\pi$
$480$	0	0
$481$	1.42601	0.0650204
$482$	0	0
$483$	−8.62407	−0.392409
$484$	0	0
$485$	−3.71195	−0.168551
$486$	0	0
$487$	−12.0633	−0.546643	−0.273321	−	0.961923i	$-0.588122\pi$
$487$	−12.0633	−0.546643	−0.273321	+	0.961923i	$0.588122\pi$
$488$	0	0
$489$	34.9166	1.57898
$490$	0	0
$491$	−31.9069	−1.43994	−0.719970	−	0.694005i	$-0.755844\pi$
$491$	−31.9069	−1.43994	−0.719970	+	0.694005i	$0.755844\pi$
$492$	0	0
$493$	7.93401	0.357330
$494$	0	0
$495$	0.605207	0.0272021
$496$	0	0
$497$	17.5556	0.787475
$498$	0	0
$499$	35.8061	1.60290	0.801450	−	0.598061i	$-0.204062\pi$
$499$	35.8061	1.60290	0.801450	+	0.598061i	$0.204062\pi$
$500$	0	0
$501$	31.5622	1.41010
$502$	0	0
$503$	−28.1183	−1.25373	−0.626866	−	0.779127i	$-0.715663\pi$
$503$	−28.1183	−1.25373	−0.626866	+	0.779127i	$0.715663\pi$
$504$	0	0
$505$	−24.7882	−1.10306
$506$	0	0
$507$	32.6986	1.45220
$508$	0	0
$509$	−25.0340	−1.10961	−0.554806	−	0.831979i	$-0.687208\pi$
$509$	−25.0340	−1.10961	−0.554806	+	0.831979i	$0.687208\pi$
$510$	0	0
$511$	10.2469	0.453295
$512$	0	0
$513$	−6.74549	−0.297821
$514$	0	0
$515$	10.8503	0.478120
$516$	0	0
$517$	−0.112019	−0.00492657
$518$	0	0
$519$	−15.7197	−0.690017
$520$	0	0
$521$	−25.0993	−1.09962	−0.549810	−	0.835289i	$-0.685300\pi$
$521$	−25.0993	−1.09962	−0.549810	+	0.835289i	$0.685300\pi$
$522$	0	0
$523$	15.9114	0.695756	0.347878	−	0.937540i	$-0.386902\pi$
$523$	15.9114	0.695756	0.347878	+	0.937540i	$0.386902\pi$
$524$	0	0
$525$	−2.21339	−0.0966001
$526$	0	0
$527$	5.12619	0.223300
$528$	0	0
$529$	−17.3650	−0.755001
$530$	0	0
$531$	3.45187	0.149799
$532$	0	0
$533$	1.35530	0.0587047
$534$	0	0
$535$	−1.71042	−0.0739481
$536$	0	0
$537$	17.0151	0.734255
$538$	0	0
$539$	−0.366756	−0.0157973
$540$	0	0
$541$	−26.6341	−1.14509	−0.572546	−	0.819873i	$-0.694044\pi$
$541$	−26.6341	−1.14509	−0.572546	+	0.819873i	$0.694044\pi$
$542$	0	0
$543$	1.20509	0.0517153
$544$	0	0
$545$	−27.6898	−1.18610
$546$	0	0
$547$	25.1972	1.07735	0.538677	−	0.842513i	$-0.318924\pi$
$547$	25.1972	1.07735	0.538677	+	0.842513i	$0.318924\pi$
$548$	0	0
$549$	13.4993	0.576138
$550$	0	0
$551$	−46.6280	−1.98642
$552$	0	0
$553$	−18.4785	−0.785784
$554$	0	0
$555$	24.0919	1.02265
$556$	0	0
$557$	4.90495	0.207829	0.103915	−	0.994586i	$-0.466863\pi$
$557$	4.90495	0.207829	0.103915	+	0.994586i	$0.466863\pi$
$558$	0	0
$559$	−1.26729	−0.0536008
$560$	0	0
$561$	0.188036	0.00793887
$562$	0	0
$563$	−16.3290	−0.688186	−0.344093	−	0.938936i	$-0.611814\pi$
$563$	−16.3290	−0.688186	−0.344093	+	0.938936i	$0.611814\pi$
$564$	0	0
$565$	−33.5865	−1.41300
$566$	0	0
$567$	−10.6416	−0.446906
$568$	0	0
$569$	−20.0255	−0.839514	−0.419757	−	0.907636i	$-0.637885\pi$
$569$	−20.0255	−0.839514	−0.419757	+	0.907636i	$0.637885\pi$
$570$	0	0
$571$	−2.94294	−0.123158	−0.0615791	−	0.998102i	$-0.519614\pi$
$571$	−2.94294	−0.123158	−0.0615791	+	0.998102i	$0.519614\pi$
$572$	0	0
$573$	−59.2686	−2.47598
$574$	0	0
$575$	1.44623	0.0603119
$576$	0	0
$577$	−10.3704	−0.431726	−0.215863	−	0.976424i	$-0.569257\pi$
$577$	−10.3704	−0.431726	−0.215863	+	0.976424i	$0.569257\pi$
$578$	0	0
$579$	−41.6471	−1.73080
$580$	0	0
$581$	−4.40291	−0.182664
$582$	0	0
$583$	−0.591086	−0.0244803
$584$	0	0
$585$	−2.91107	−0.120358
$586$	0	0
$587$	25.7900	1.06447	0.532234	−	0.846597i	$-0.321353\pi$
$587$	25.7900	1.06447	0.532234	+	0.846597i	$0.321353\pi$
$588$	0	0
$589$	−30.1266	−1.24134
$590$	0	0
$591$	−59.3743	−2.44233
$592$	0	0
$593$	−34.5712	−1.41967	−0.709835	−	0.704368i	$-0.751231\pi$
$593$	−34.5712	−1.41967	−0.709835	+	0.704368i	$0.751231\pi$
$594$	0	0
$595$	−3.38747	−0.138873
$596$	0	0
$597$	−55.6715	−2.27848
$598$	0	0
$599$	8.82420	0.360547	0.180274	−	0.983617i	$-0.442302\pi$
$599$	8.82420	0.360547	0.180274	+	0.983617i	$0.442302\pi$
$600$	0	0
$601$	−35.9221	−1.46529	−0.732647	−	0.680608i	$-0.761716\pi$
$601$	−35.9221	−1.46529	−0.732647	+	0.680608i	$0.761716\pi$
$602$	0	0
$603$	−55.4488	−2.25805
$604$	0	0
$605$	−26.0392	−1.05865
$606$	0	0
$607$	37.9106	1.53874	0.769372	−	0.638801i	$-0.220569\pi$
$607$	37.9106	1.53874	0.769372	+	0.638801i	$0.220569\pi$
$608$	0	0
$609$	28.8243	1.16802
$610$	0	0
$611$	0.538813	0.0217980
$612$	0	0
$613$	−12.1939	−0.492507	−0.246254	−	0.969205i	$-0.579200\pi$
$613$	−12.1939	−0.492507	−0.246254	+	0.969205i	$0.579200\pi$
$614$	0	0
$615$	22.8974	0.923312
$616$	0	0
$617$	1.91792	0.0772127	0.0386063	−	0.999254i	$-0.487708\pi$
$617$	1.91792	0.0772127	0.0386063	+	0.999254i	$0.487708\pi$
$618$	0	0
$619$	−12.0382	−0.483856	−0.241928	−	0.970294i	$-0.577780\pi$
$619$	−12.0382	−0.483856	−0.241928	+	0.970294i	$0.577780\pi$
$620$	0	0
$621$	−2.72461	−0.109335
$622$	0	0
$623$	7.15471	0.286647
$624$	0	0
$625$	−27.6750	−1.10700
$626$	0	0
$627$	−1.10508	−0.0441328
$628$	0	0
$629$	4.00476	0.159680
$630$	0	0
$631$	−1.79521	−0.0714661	−0.0357330	−	0.999361i	$-0.511377\pi$
$631$	−1.79521	−0.0714661	−0.0357330	+	0.999361i	$0.511377\pi$
$632$	0	0
$633$	64.2779	2.55482
$634$	0	0
$635$	−20.8160	−0.826057
$636$	0	0
$637$	1.76411	0.0698966
$638$	0	0
$639$	42.3688	1.67608
$640$	0	0
$641$	47.2554	1.86647	0.933237	−	0.359260i	$-0.116971\pi$
$641$	47.2554	1.86647	0.933237	+	0.359260i	$0.116971\pi$
$642$	0	0
$643$	12.6156	0.497509	0.248755	−	0.968567i	$-0.419979\pi$
$643$	12.6156	0.497509	0.248755	+	0.968567i	$0.419979\pi$
$644$	0	0
$645$	−21.4105	−0.843038
$646$	0	0
$647$	−12.6790	−0.498464	−0.249232	−	0.968444i	$-0.580178\pi$
$647$	−12.6790	−0.498464	−0.249232	+	0.968444i	$0.580178\pi$
$648$	0	0
$649$	0.0740282	0.00290586
$650$	0	0
$651$	18.6235	0.729912
$652$	0	0
$653$	−26.1031	−1.02149	−0.510746	−	0.859732i	$-0.670631\pi$
$653$	−26.1031	−1.02149	−0.510746	+	0.859732i	$0.670631\pi$
$654$	0	0
$655$	3.50490	0.136948
$656$	0	0
$657$	24.7299	0.964807
$658$	0	0
$659$	−14.3676	−0.559684	−0.279842	−	0.960046i	$-0.590282\pi$
$659$	−14.3676	−0.559684	−0.279842	+	0.960046i	$0.590282\pi$
$660$	0	0
$661$	−15.0811	−0.586585	−0.293293	−	0.956023i	$-0.594751\pi$
$661$	−15.0811	−0.586585	−0.293293	+	0.956023i	$0.594751\pi$
$662$	0	0
$663$	−0.904458	−0.0351262
$664$	0	0
$665$	19.9081	0.772003
$666$	0	0
$667$	−18.8338	−0.729249
$668$	0	0
$669$	16.9217	0.654232
$670$	0	0
$671$	0.289504	0.0111762
$672$	0	0
$673$	15.9530	0.614943	0.307471	−	0.951557i	$-0.400517\pi$
$673$	15.9530	0.614943	0.307471	+	0.951557i	$0.400517\pi$
$674$	0	0
$675$	−0.699278	−0.0269152
$676$	0	0
$677$	45.9438	1.76576	0.882882	−	0.469595i	$-0.155600\pi$
$677$	45.9438	1.76576	0.882882	+	0.469595i	$0.155600\pi$
$678$	0	0
$679$	−2.24168	−0.0860278
$680$	0	0
$681$	−52.7768	−2.02241
$682$	0	0
$683$	12.3888	0.474044	0.237022	−	0.971504i	$-0.423829\pi$
$683$	12.3888	0.474044	0.237022	+	0.971504i	$0.423829\pi$
$684$	0	0
$685$	−9.67711	−0.369743
$686$	0	0
$687$	22.4913	0.858095
$688$	0	0
$689$	2.84314	0.108315
$690$	0	0
$691$	−38.3402	−1.45853	−0.729264	−	0.684232i	$-0.760138\pi$
$691$	−38.3402	−1.45853	−0.729264	+	0.684232i	$0.760138\pi$
$692$	0	0
$693$	0.365490	0.0138838
$694$	0	0
$695$	49.9283	1.89389
$696$	0	0
$697$	3.80620	0.144170
$698$	0	0
$699$	−15.0132	−0.567851
$700$	0	0
$701$	6.94039	0.262135	0.131067	−	0.991373i	$-0.458160\pi$
$701$	6.94039	0.262135	0.131067	+	0.991373i	$0.458160\pi$
$702$	0	0
$703$	−23.5359	−0.887674
$704$	0	0
$705$	9.10307	0.342841
$706$	0	0
$707$	−14.9698	−0.562997
$708$	0	0
$709$	−13.4333	−0.504497	−0.252248	−	0.967663i	$-0.581170\pi$
$709$	−13.4333	−0.504497	−0.252248	+	0.967663i	$0.581170\pi$
$710$	0	0
$711$	−44.5962	−1.67249
$712$	0	0
$713$	−12.1686	−0.455718
$714$	0	0
$715$	−0.0624302	−0.00233476
$716$	0	0
$717$	−13.3493	−0.498539
$718$	0	0
$719$	39.9797	1.49099	0.745496	−	0.666510i	$-0.232213\pi$
$719$	39.9797	1.49099	0.745496	+	0.666510i	$0.232213\pi$
$720$	0	0
$721$	6.55257	0.244030
$722$	0	0
$723$	44.0010	1.63642
$724$	0	0
$725$	−4.83375	−0.179521
$726$	0	0
$727$	−45.9641	−1.70471	−0.852357	−	0.522961i	$-0.824827\pi$
$727$	−45.9641	−1.70471	−0.852357	+	0.522961i	$0.824827\pi$
$728$	0	0
$729$	−34.4289	−1.27514
$730$	0	0
$731$	−3.55903	−0.131636
$732$	0	0
$733$	5.42388	0.200336	0.100168	−	0.994971i	$-0.468062\pi$
$733$	5.42388	0.200336	0.100168	+	0.994971i	$0.468062\pi$
$734$	0	0
$735$	29.8041	1.09934
$736$	0	0
$737$	−1.18914	−0.0438027
$738$	0	0
$739$	42.2672	1.55482	0.777412	−	0.628992i	$-0.216532\pi$
$739$	42.2672	1.55482	0.777412	+	0.628992i	$0.216532\pi$
$740$	0	0
$741$	5.31548	0.195269
$742$	0	0
$743$	−24.4091	−0.895483	−0.447741	−	0.894163i	$-0.647772\pi$
$743$	−24.4091	−0.895483	−0.447741	+	0.894163i	$0.647772\pi$
$744$	0	0
$745$	−1.56907	−0.0574862
$746$	0	0
$747$	−10.6260	−0.388786
$748$	0	0
$749$	−1.03294	−0.0377428
$750$	0	0
$751$	9.70950	0.354305	0.177152	−	0.984183i	$-0.443311\pi$
$751$	9.70950	0.354305	0.177152	+	0.984183i	$0.443311\pi$
$752$	0	0
$753$	−60.4255	−2.20203
$754$	0	0
$755$	16.8798	0.614321
$756$	0	0
$757$	21.7936	0.792102	0.396051	−	0.918228i	$-0.370380\pi$
$757$	21.7936	0.792102	0.396051	+	0.918228i	$0.370380\pi$
$758$	0	0
$759$	−0.446361	−0.0162019
$760$	0	0
$761$	−46.6358	−1.69055	−0.845273	−	0.534335i	$-0.820562\pi$
$761$	−46.6358	−1.69055	−0.845273	+	0.534335i	$0.820562\pi$
$762$	0	0
$763$	−16.7221	−0.605380
$764$	0	0
$765$	−8.17536	−0.295581
$766$	0	0
$767$	−0.356078	−0.0128572
$768$	0	0
$769$	40.9972	1.47840	0.739199	−	0.673487i	$-0.235204\pi$
$769$	40.9972	1.47840	0.739199	+	0.673487i	$0.235204\pi$
$770$	0	0
$771$	65.1021	2.34459
$772$	0	0
$773$	30.5411	1.09849	0.549244	−	0.835662i	$-0.314916\pi$
$773$	30.5411	1.09849	0.549244	+	0.835662i	$0.314916\pi$
$774$	0	0
$775$	−3.12310	−0.112185
$776$	0	0
$777$	14.5493	0.521954
$778$	0	0
$779$	−22.3690	−0.801451
$780$	0	0
$781$	0.908634	0.0325135
$782$	0	0
$783$	9.10650	0.325440
$784$	0	0
$785$	49.9735	1.78363
$786$	0	0
$787$	53.8001	1.91777	0.958883	−	0.283802i	$-0.0915957\pi$
$787$	53.8001	1.91777	0.958883	+	0.283802i	$0.0915957\pi$
$788$	0	0
$789$	−24.2539	−0.863463
$790$	0	0
$791$	−20.2832	−0.721188
$792$	0	0
$793$	−1.39253	−0.0494500
$794$	0	0
$795$	48.0339	1.70359
$796$	0	0
$797$	10.3435	0.366386	0.183193	−	0.983077i	$-0.441357\pi$
$797$	10.3435	0.366386	0.183193	+	0.983077i	$0.441357\pi$
$798$	0	0
$799$	1.51319	0.0535327
$800$	0	0
$801$	17.2673	0.610108
$802$	0	0
$803$	0.530354	0.0187158
$804$	0	0
$805$	8.04121	0.283415
$806$	0	0
$807$	75.2562	2.64914
$808$	0	0
$809$	12.0992	0.425386	0.212693	−	0.977119i	$-0.431777\pi$
$809$	12.0992	0.425386	0.212693	+	0.977119i	$0.431777\pi$
$810$	0	0
$811$	−23.7915	−0.835431	−0.417716	−	0.908578i	$-0.637169\pi$
$811$	−23.7915	−0.835431	−0.417716	+	0.908578i	$0.637169\pi$
$812$	0	0
$813$	69.0839	2.42288
$814$	0	0
$815$	−32.5568	−1.14041
$816$	0	0
$817$	20.9164	0.731771
$818$	0	0
$819$	−1.75802	−0.0614301
$820$	0	0
$821$	−10.5512	−0.368239	−0.184119	−	0.982904i	$-0.558943\pi$
$821$	−10.5512	−0.368239	−0.184119	+	0.982904i	$0.558943\pi$
$822$	0	0
$823$	7.03466	0.245213	0.122606	−	0.992455i	$-0.460875\pi$
$823$	7.03466	0.245213	0.122606	+	0.992455i	$0.460875\pi$
$824$	0	0
$825$	−0.114560	−0.00398845
$826$	0	0
$827$	13.4401	0.467359	0.233679	−	0.972314i	$-0.424923\pi$
$827$	13.4401	0.467359	0.233679	+	0.972314i	$0.424923\pi$
$828$	0	0
$829$	42.9499	1.49171	0.745856	−	0.666107i	$-0.232041\pi$
$829$	42.9499	1.49171	0.745856	+	0.666107i	$0.232041\pi$
$830$	0	0
$831$	−12.7569	−0.442533
$832$	0	0
$833$	4.95428	0.171656
$834$	0	0
$835$	−29.4291	−1.01844
$836$	0	0
$837$	5.88374	0.203372
$838$	0	0
$839$	−37.4310	−1.29226	−0.646132	−	0.763226i	$-0.723614\pi$
$839$	−37.4310	−1.29226	−0.646132	+	0.763226i	$0.723614\pi$
$840$	0	0
$841$	33.9485	1.17064
$842$	0	0
$843$	−55.5691	−1.91390
$844$	0	0
$845$	−30.4887	−1.04884
$846$	0	0
$847$	−15.7253	−0.540328
$848$	0	0
$849$	−30.3770	−1.04253
$850$	0	0
$851$	−9.50654	−0.325880
$852$	0	0
$853$	−20.8221	−0.712935	−0.356467	−	0.934308i	$-0.616019\pi$
$853$	−20.8221	−0.712935	−0.356467	+	0.934308i	$0.616019\pi$
$854$	0	0
$855$	48.0465	1.64315
$856$	0	0
$857$	52.3656	1.78878	0.894388	−	0.447292i	$-0.147611\pi$
$857$	52.3656	1.78878	0.894388	+	0.447292i	$0.147611\pi$
$858$	0	0
$859$	16.8456	0.574764	0.287382	−	0.957816i	$-0.407215\pi$
$859$	16.8456	0.574764	0.287382	+	0.957816i	$0.407215\pi$
$860$	0	0
$861$	13.8279	0.471255
$862$	0	0
$863$	9.13136	0.310835	0.155418	−	0.987849i	$-0.450328\pi$
$863$	9.13136	0.310835	0.155418	+	0.987849i	$0.450328\pi$
$864$	0	0
$865$	14.6573	0.498362
$866$	0	0
$867$	−2.54005	−0.0862647
$868$	0	0
$869$	−0.956401	−0.0324437
$870$	0	0
$871$	5.71983	0.193809
$872$	0	0
$873$	−5.41010	−0.183104
$874$	0	0
$875$	−14.8736	−0.502818
$876$	0	0
$877$	−41.2465	−1.39279	−0.696397	−	0.717657i	$-0.745215\pi$
$877$	−41.2465	−1.39279	−0.696397	+	0.717657i	$0.745215\pi$
$878$	0	0
$879$	−63.3457	−2.13660
$880$	0	0
$881$	−29.4505	−0.992213	−0.496107	−	0.868262i	$-0.665237\pi$
$881$	−29.4505	−0.992213	−0.496107	+	0.868262i	$0.665237\pi$
$882$	0	0
$883$	−12.5641	−0.422817	−0.211408	−	0.977398i	$-0.567805\pi$
$883$	−12.5641	−0.422817	−0.211408	+	0.977398i	$0.567805\pi$
$884$	0	0
$885$	−6.01582	−0.202220
$886$	0	0
$887$	30.5517	1.02583	0.512914	−	0.858440i	$-0.328566\pi$
$887$	30.5517	1.02583	0.512914	+	0.858440i	$0.328566\pi$
$888$	0	0
$889$	−12.5709	−0.421616
$890$	0	0
$891$	−0.550784	−0.0184520
$892$	0	0
$893$	−8.89298	−0.297592
$894$	0	0
$895$	−15.8651	−0.530313
$896$	0	0
$897$	2.14701	0.0716866
$898$	0	0
$899$	40.6712	1.35646
$900$	0	0
$901$	7.98460	0.266006
$902$	0	0
$903$	−12.9300	−0.430283
$904$	0	0
$905$	−1.12364	−0.0373512
$906$	0	0
$907$	−28.0972	−0.932953	−0.466476	−	0.884534i	$-0.654477\pi$
$907$	−28.0972	−0.932953	−0.466476	+	0.884534i	$0.654477\pi$
$908$	0	0
$909$	−36.1283	−1.19830
$910$	0	0
$911$	41.5745	1.37742	0.688712	−	0.725035i	$-0.258176\pi$
$911$	41.5745	1.37742	0.688712	+	0.725035i	$0.258176\pi$
$912$	0	0
$913$	−0.227884	−0.00754186
$914$	0	0
$915$	−23.5263	−0.777754
$916$	0	0
$917$	2.11664	0.0698976
$918$	0	0
$919$	−17.5280	−0.578194	−0.289097	−	0.957300i	$-0.593355\pi$
$919$	−17.5280	−0.578194	−0.289097	+	0.957300i	$0.593355\pi$
$920$	0	0
$921$	49.9605	1.64625
$922$	0	0
$923$	−4.37056	−0.143859
$924$	0	0
$925$	−2.43988	−0.0802226
$926$	0	0
$927$	15.8140	0.519401
$928$	0	0
$929$	−37.5823	−1.23304	−0.616518	−	0.787341i	$-0.711457\pi$
$929$	−37.5823	−1.23304	−0.616518	+	0.787341i	$0.711457\pi$
$930$	0	0
$931$	−29.1162	−0.954246
$932$	0	0
$933$	−15.4622	−0.506208
$934$	0	0
$935$	−0.175327	−0.00573382
$936$	0	0
$937$	−61.1983	−1.99926	−0.999630	−	0.0271965i	$-0.991342\pi$
$937$	−61.1983	−1.99926	−0.999630	+	0.0271965i	$0.991342\pi$
$938$	0	0
$939$	−1.57095	−0.0512662
$940$	0	0
$941$	7.76164	0.253022	0.126511	−	0.991965i	$-0.459622\pi$
$941$	7.76164	0.253022	0.126511	+	0.991965i	$0.459622\pi$
$942$	0	0
$943$	−9.03519	−0.294226
$944$	0	0
$945$	−3.88807	−0.126479
$946$	0	0
$947$	30.8078	1.00112	0.500559	−	0.865702i	$-0.333128\pi$
$947$	30.8078	1.00112	0.500559	+	0.865702i	$0.333128\pi$
$948$	0	0
$949$	−2.55102	−0.0828096
$950$	0	0
$951$	−69.8357	−2.26458
$952$	0	0
$953$	19.2582	0.623836	0.311918	−	0.950109i	$-0.399029\pi$
$953$	19.2582	0.623836	0.311918	+	0.950109i	$0.399029\pi$
$954$	0	0
$955$	55.2629	1.78827
$956$	0	0
$957$	1.49188	0.0482255
$958$	0	0
$959$	−5.84409	−0.188716
$960$	0	0
$961$	−4.72216	−0.152328
$962$	0	0
$963$	−2.49291	−0.0803329
$964$	0	0
$965$	38.8324	1.25006
$966$	0	0
$967$	2.40353	0.0772924	0.0386462	−	0.999253i	$-0.487695\pi$
$967$	2.40353	0.0772924	0.0386462	+	0.999253i	$0.487695\pi$
$968$	0	0
$969$	14.9279	0.479552
$970$	0	0
$971$	11.9868	0.384676	0.192338	−	0.981329i	$-0.438393\pi$
$971$	11.9868	0.384676	0.192338	+	0.981329i	$0.438393\pi$
$972$	0	0
$973$	30.1522	0.966634
$974$	0	0
$975$	0.551035	0.0176473
$976$	0	0
$977$	−58.4078	−1.86863	−0.934316	−	0.356447i	$-0.883988\pi$
$977$	−58.4078	−1.86863	−0.934316	+	0.356447i	$0.883988\pi$
$978$	0	0
$979$	0.370310	0.0118352
$980$	0	0
$981$	−40.3573	−1.28851
$982$	0	0
$983$	33.7637	1.07689	0.538447	−	0.842659i	$-0.319011\pi$
$983$	33.7637	1.07689	0.538447	+	0.842659i	$0.319011\pi$
$984$	0	0
$985$	55.3615	1.76396
$986$	0	0
$987$	5.49742	0.174985
$988$	0	0
$989$	8.44846	0.268646
$990$	0	0
$991$	−60.2151	−1.91279	−0.956397	−	0.292069i	$-0.905656\pi$
$991$	−60.2151	−1.91279	−0.956397	+	0.292069i	$0.905656\pi$
$992$	0	0
$993$	−33.3096	−1.05705
$994$	0	0
$995$	51.9089	1.64562
$996$	0	0
$997$	54.5148	1.72650	0.863251	−	0.504775i	$-0.168425\pi$
$997$	54.5148	1.72650	0.863251	+	0.504775i	$0.168425\pi$
$998$	0	0
$999$	4.59659	0.145430

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.x.1.3	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.x.1.3	✓	22	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.54005 q^{3} +2.36838 q^{5} +1.43029 q^{7} +3.45187 q^{9} +O(q^{10})\)	\(q-2.54005 q^{3} +2.36838 q^{5} +1.43029 q^{7} +3.45187 q^{9} +0.0740282 q^{11} -0.356078 q^{13} -6.01582 q^{15} -1.00000 q^{17} +5.87699 q^{19} -3.63301 q^{21} +2.37381 q^{23} +0.609244 q^{25} -1.14778 q^{27} -7.93401 q^{29} -5.12619 q^{31} -0.188036 q^{33} +3.38747 q^{35} -4.00476 q^{37} +0.904458 q^{39} -3.80620 q^{41} +3.55903 q^{43} +8.17536 q^{45} -1.51319 q^{47} -4.95428 q^{49} +2.54005 q^{51} -7.98460 q^{53} +0.175327 q^{55} -14.9279 q^{57} +1.00000 q^{59} +3.91073 q^{61} +4.93717 q^{63} -0.843330 q^{65} -16.0634 q^{67} -6.02961 q^{69} +12.2742 q^{71} +7.16421 q^{73} -1.54751 q^{75} +0.105882 q^{77} -12.9194 q^{79} -7.44019 q^{81} -3.07834 q^{83} -2.36838 q^{85} +20.1528 q^{87} +5.00229 q^{89} -0.509294 q^{91} +13.0208 q^{93} +13.9190 q^{95} -1.56729 q^{97} +0.255536 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) \) 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9	\( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9} + 2 q^{11} - 9 q^{13} - 7 q^{15} - 22 q^{17} - 10 q^{19} - 10 q^{21} + 14 q^{23} + 3 q^{25} + 6 q^{29} - 15 q^{31} - 52 q^{33} - 7 q^{35} + 9 q^{37} - 52 q^{41} - 7 q^{43} - 30 q^{45} - 7 q^{47} - 6 q^{49} + 3 q^{51} - 18 q^{53} - 39 q^{55} - 2 q^{57} + 22 q^{59} - 42 q^{61} - 35 q^{65} - 28 q^{67} - 10 q^{69} + 23 q^{71} - 33 q^{73} - 3 q^{75} - 28 q^{77} - 30 q^{79} - 2 q^{81} + 11 q^{83} - 3 q^{85} + 27 q^{87} - 34 q^{89} - 18 q^{91} - 28 q^{93} - 10 q^{95} - 3 q^{97} - 28 q^{99}+O(q^{100}) \) 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9 + 2 * q^11 - 9 * q^13 - 7 * q^15 - 22 * q^17 - 10 * q^19 - 10 * q^21 + 14 * q^23 + 3 * q^25 + 6 * q^29 - 15 * q^31 - 52 * q^33 - 7 * q^35 + 9 * q^37 - 52 * q^41 - 7 * q^43 - 30 * q^45 - 7 * q^47 - 6 * q^49 + 3 * q^51 - 18 * q^53 - 39 * q^55 - 2 * q^57 + 22 * q^59 - 42 * q^61 - 35 * q^65 - 28 * q^67 - 10 * q^69 + 23 * q^71 - 33 * q^73 - 3 * q^75 - 28 * q^77 - 30 * q^79 - 2 * q^81 + 11 * q^83 - 3 * q^85 + 27 * q^87 - 34 * q^89 - 18 * q^91 - 28 * q^93 - 10 * q^95 - 3 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more