LMFDB - Embedded newform 10000.2.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10000,2,Mod(1,10000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10000, 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("10000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 10000 = 2^{4} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	10000.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:	$79.8504020213$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.3266578125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 6x^{6} + 23x^{5} + x^{4} - 45x^{3} + 25x^{2} + 10x - 5 $ x^8 - 3x^7 - 6x^6 + 23x^5 + x^4 - 45x^3 + 25x^2 + 10x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 5000)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.36933$ of defining polynomial
Character	$\chi$	$=$	10000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.70759 q^{3} +1.85864 q^{7} -0.0841263 q^{9} +O(q^{10})$	$q-1.70759 q^{3} +1.85864 q^{7} -0.0841263 q^{9} +1.10734 q^{11} +1.23055 q^{13} -3.56647 q^{17} -2.50475 q^{19} -3.17380 q^{21} -6.52968 q^{23} +5.26643 q^{27} -6.38008 q^{29} -0.254911 q^{31} -1.89089 q^{33} -0.617118 q^{37} -2.10129 q^{39} +5.07900 q^{41} +7.17539 q^{43} -0.280635 q^{47} -3.54545 q^{49} +6.09008 q^{51} -4.00515 q^{53} +4.27709 q^{57} -3.39434 q^{59} -13.5349 q^{61} -0.156361 q^{63} +2.15370 q^{67} +11.1500 q^{69} +13.7570 q^{71} +16.6066 q^{73} +2.05815 q^{77} +3.70772 q^{79} -8.74054 q^{81} +12.2394 q^{83} +10.8946 q^{87} +4.73692 q^{89} +2.28716 q^{91} +0.435284 q^{93} -7.90486 q^{97} -0.0931567 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{3} + 2 q^{7} - 3 q^{9}+O(q^{10}) $ 8 * q + 3 * q^3 + 2 * q^7 - 3 * q^9	$ 8 q + 3 q^{3} + 2 q^{7} - 3 q^{9} - 5 q^{11} + 2 q^{13} + 7 q^{17} + 10 q^{19} - 14 q^{21} - 2 q^{23} - 6 q^{27} - 10 q^{29} + 27 q^{31} - 3 q^{33} + 10 q^{37} + 18 q^{39} - 16 q^{41} + 4 q^{43} - 8 q^{47} + 4 q^{49} + 10 q^{51} + 2 q^{53} + 2 q^{57} + 39 q^{59} - 18 q^{61} - 14 q^{63} - 12 q^{67} + 19 q^{69} + 13 q^{71} + 12 q^{73} + 41 q^{77} + 16 q^{79} - 28 q^{81} + 64 q^{83} - 4 q^{87} - 25 q^{89} + 26 q^{91} + 40 q^{93} - 8 q^{97} - 6 q^{99}+O(q^{100}) $ 8 * q + 3 * q^3 + 2 * q^7 - 3 * q^9 - 5 * q^11 + 2 * q^13 + 7 * q^17 + 10 * q^19 - 14 * q^21 - 2 * q^23 - 6 * q^27 - 10 * q^29 + 27 * q^31 - 3 * q^33 + 10 * q^37 + 18 * q^39 - 16 * q^41 + 4 * q^43 - 8 * q^47 + 4 * q^49 + 10 * q^51 + 2 * q^53 + 2 * q^57 + 39 * q^59 - 18 * q^61 - 14 * q^63 - 12 * q^67 + 19 * q^69 + 13 * q^71 + 12 * q^73 + 41 * q^77 + 16 * q^79 - 28 * q^81 + 64 * q^83 - 4 * q^87 - 25 * q^89 + 26 * q^91 + 40 * q^93 - 8 * q^97 - 6 * 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
$3$	−1.70759	−0.985879	−0.492940	−	0.870064i	$-0.664078\pi$
$3$	−1.70759	−0.985879	−0.492940	+	0.870064i	$0.664078\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.85864	0.702500	0.351250	−	0.936282i	$-0.385757\pi$
$7$	1.85864	0.702500	0.351250	+	0.936282i	$0.385757\pi$
$8$	0	0
$9$	−0.0841263	−0.0280421
$10$	0	0
$11$	1.10734	0.333877	0.166938	−	0.985967i	$-0.446612\pi$
$11$	1.10734	0.333877	0.166938	+	0.985967i	$0.446612\pi$
$12$	0	0
$13$	1.23055	0.341294	0.170647	−	0.985332i	$-0.445414\pi$
$13$	1.23055	0.341294	0.170647	+	0.985332i	$0.445414\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.56647	−0.864996	−0.432498	−	0.901635i	$-0.642368\pi$
$17$	−3.56647	−0.864996	−0.432498	+	0.901635i	$0.642368\pi$
$18$	0	0
$19$	−2.50475	−0.574629	−0.287314	−	0.957836i	$-0.592762\pi$
$19$	−2.50475	−0.574629	−0.287314	+	0.957836i	$0.592762\pi$
$20$	0	0
$21$	−3.17380	−0.692580
$22$	0	0
$23$	−6.52968	−1.36153	−0.680766	−	0.732501i	$-0.738353\pi$
$23$	−6.52968	−1.36153	−0.680766	+	0.732501i	$0.738353\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.26643	1.01353
$28$	0	0
$29$	−6.38008	−1.18475	−0.592375	−	0.805662i	$-0.701810\pi$
$29$	−6.38008	−1.18475	−0.592375	+	0.805662i	$0.701810\pi$
$30$	0	0
$31$	−0.254911	−0.0457834	−0.0228917	−	0.999738i	$-0.507287\pi$
$31$	−0.254911	−0.0457834	−0.0228917	+	0.999738i	$0.507287\pi$
$32$	0	0
$33$	−1.89089	−0.329162
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.617118	−0.101454	−0.0507268	−	0.998713i	$-0.516154\pi$
$37$	−0.617118	−0.101454	−0.0507268	+	0.998713i	$0.516154\pi$
$38$	0	0
$39$	−2.10129	−0.336475
$40$	0	0
$41$	5.07900	0.793206	0.396603	−	0.917990i	$-0.370189\pi$
$41$	5.07900	0.793206	0.396603	+	0.917990i	$0.370189\pi$
$42$	0	0
$43$	7.17539	1.09424	0.547118	−	0.837055i	$-0.315725\pi$
$43$	7.17539	1.09424	0.547118	+	0.837055i	$0.315725\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.280635	−0.0409349	−0.0204674	−	0.999791i	$-0.506515\pi$
$47$	−0.280635	−0.0409349	−0.0204674	+	0.999791i	$0.506515\pi$
$48$	0	0
$49$	−3.54545	−0.506493
$50$	0	0
$51$	6.09008	0.852782
$52$	0	0
$53$	−4.00515	−0.550149	−0.275075	−	0.961423i	$-0.588703\pi$
$53$	−4.00515	−0.550149	−0.275075	+	0.961423i	$0.588703\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.27709	0.566515
$58$	0	0
$59$	−3.39434	−0.441905	−0.220953	−	0.975285i	$-0.570917\pi$
$59$	−3.39434	−0.441905	−0.220953	+	0.975285i	$0.570917\pi$
$60$	0	0
$61$	−13.5349	−1.73297	−0.866486	−	0.499201i	$-0.833627\pi$
$61$	−13.5349	−1.73297	−0.866486	+	0.499201i	$0.833627\pi$
$62$	0	0
$63$	−0.156361	−0.0196996
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.15370	0.263116	0.131558	−	0.991308i	$-0.458002\pi$
$67$	2.15370	0.263116	0.131558	+	0.991308i	$0.458002\pi$
$68$	0	0
$69$	11.1500	1.34231
$70$	0	0
$71$	13.7570	1.63265	0.816325	−	0.577592i	$-0.196008\pi$
$71$	13.7570	1.63265	0.816325	+	0.577592i	$0.196008\pi$
$72$	0	0
$73$	16.6066	1.94366	0.971830	−	0.235685i	$-0.0757332\pi$
$73$	16.6066	1.94366	0.971830	+	0.235685i	$0.0757332\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.05815	0.234548
$78$	0	0
$79$	3.70772	0.417151	0.208576	−	0.978006i	$-0.433117\pi$
$79$	3.70772	0.417151	0.208576	+	0.978006i	$0.433117\pi$
$80$	0	0
$81$	−8.74054	−0.971172
$82$	0	0
$83$	12.2394	1.34345	0.671723	−	0.740802i	$-0.265554\pi$
$83$	12.2394	1.34345	0.671723	+	0.740802i	$0.265554\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	10.8946	1.16802
$88$	0	0
$89$	4.73692	0.502113	0.251056	−	0.967972i	$-0.419222\pi$
$89$	4.73692	0.502113	0.251056	+	0.967972i	$0.419222\pi$
$90$	0	0
$91$	2.28716	0.239759
$92$	0	0
$93$	0.435284	0.0451369
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−7.90486	−0.802617	−0.401308	−	0.915943i	$-0.631444\pi$
$97$	−7.90486	−0.802617	−0.401308	+	0.915943i	$0.631444\pi$
$98$	0	0
$99$	−0.0931567	−0.00936260
$100$	0	0
$101$	13.5334	1.34663	0.673313	−	0.739357i	$-0.264871\pi$
$101$	13.5334	1.34663	0.673313	+	0.739357i	$0.264871\pi$
$102$	0	0
$103$	−7.94126	−0.782475	−0.391238	−	0.920290i	$-0.627953\pi$
$103$	−7.94126	−0.782475	−0.391238	+	0.920290i	$0.627953\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.2631	1.76556	0.882780	−	0.469787i	$-0.155669\pi$
$107$	18.2631	1.76556	0.882780	+	0.469787i	$0.155669\pi$
$108$	0	0
$109$	−16.6971	−1.59929	−0.799647	−	0.600471i	$-0.794980\pi$
$109$	−16.6971	−1.59929	−0.799647	+	0.600471i	$0.794980\pi$
$110$	0	0
$111$	1.05379	0.100021
$112$	0	0
$113$	−15.2119	−1.43101	−0.715507	−	0.698605i	$-0.753804\pi$
$113$	−15.2119	−1.43101	−0.715507	+	0.698605i	$0.753804\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.103522	−0.00957061
$118$	0	0
$119$	−6.62879	−0.607660
$120$	0	0
$121$	−9.77379	−0.888526
$122$	0	0
$123$	−8.67286	−0.782005
$124$	0	0
$125$	0	0
$126$	0	0
$127$	5.19718	0.461176	0.230588	−	0.973052i	$-0.425935\pi$
$127$	5.19718	0.461176	0.230588	+	0.973052i	$0.425935\pi$
$128$	0	0
$129$	−12.2526	−1.07879
$130$	0	0
$131$	1.42175	0.124219	0.0621093	−	0.998069i	$-0.480217\pi$
$131$	1.42175	0.124219	0.0621093	+	0.998069i	$0.480217\pi$
$132$	0	0
$133$	−4.65543	−0.403677
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.35100	0.628038	0.314019	−	0.949417i	$-0.398324\pi$
$137$	7.35100	0.628038	0.314019	+	0.949417i	$0.398324\pi$
$138$	0	0
$139$	−18.1163	−1.53660	−0.768301	−	0.640089i	$-0.778897\pi$
$139$	−18.1163	−1.53660	−0.768301	+	0.640089i	$0.778897\pi$
$140$	0	0
$141$	0.479211	0.0403568
$142$	0	0
$143$	1.36265	0.113950
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.05419	0.499341
$148$	0	0
$149$	1.17679	0.0964060	0.0482030	−	0.998838i	$-0.484651\pi$
$149$	1.17679	0.0964060	0.0482030	+	0.998838i	$0.484651\pi$
$150$	0	0
$151$	22.7994	1.85539	0.927693	−	0.373345i	$-0.121789\pi$
$151$	22.7994	1.85539	0.927693	+	0.373345i	$0.121789\pi$
$152$	0	0
$153$	0.300034	0.0242563
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.0645	0.803235	0.401617	−	0.915808i	$-0.368448\pi$
$157$	10.0645	0.803235	0.401617	+	0.915808i	$0.368448\pi$
$158$	0	0
$159$	6.83916	0.542381
$160$	0	0
$161$	−12.1363	−0.956477
$162$	0	0
$163$	3.76191	0.294655	0.147328	−	0.989088i	$-0.452933\pi$
$163$	3.76191	0.294655	0.147328	+	0.989088i	$0.452933\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.78685	−0.370418	−0.185209	−	0.982699i	$-0.559296\pi$
$167$	−4.78685	−0.370418	−0.185209	+	0.982699i	$0.559296\pi$
$168$	0	0
$169$	−11.4857	−0.883518
$170$	0	0
$171$	0.210715	0.0161138
$172$	0	0
$173$	12.0447	0.915740	0.457870	−	0.889019i	$-0.348613\pi$
$173$	12.0447	0.915740	0.457870	+	0.889019i	$0.348613\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.79615	0.435665
$178$	0	0
$179$	12.6428	0.944966	0.472483	−	0.881340i	$-0.343358\pi$
$179$	12.6428	0.944966	0.472483	+	0.881340i	$0.343358\pi$
$180$	0	0
$181$	−7.27939	−0.541073	−0.270536	−	0.962710i	$-0.587201\pi$
$181$	−7.27939	−0.541073	−0.270536	+	0.962710i	$0.587201\pi$
$182$	0	0
$183$	23.1122	1.70850
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.94931	−0.288802
$188$	0	0
$189$	9.78841	0.712002
$190$	0	0
$191$	14.6309	1.05866	0.529328	−	0.848417i	$-0.322444\pi$
$191$	14.6309	1.05866	0.529328	+	0.848417i	$0.322444\pi$
$192$	0	0
$193$	−2.62660	−0.189067	−0.0945333	−	0.995522i	$-0.530136\pi$
$193$	−2.62660	−0.189067	−0.0945333	+	0.995522i	$0.530136\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.85779	0.417350	0.208675	−	0.977985i	$-0.433085\pi$
$197$	5.85779	0.417350	0.208675	+	0.977985i	$0.433085\pi$
$198$	0	0
$199$	1.15154	0.0816308	0.0408154	−	0.999167i	$-0.487004\pi$
$199$	1.15154	0.0816308	0.0408154	+	0.999167i	$0.487004\pi$
$200$	0	0
$201$	−3.67764	−0.259401
$202$	0	0
$203$	−11.8583	−0.832288
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.549318	0.0381802
$208$	0	0
$209$	−2.77362	−0.191855
$210$	0	0
$211$	1.23590	0.0850827	0.0425414	−	0.999095i	$-0.486455\pi$
$211$	1.23590	0.0850827	0.0425414	+	0.999095i	$0.486455\pi$
$212$	0	0
$213$	−23.4913	−1.60960
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−0.473788	−0.0321628
$218$	0	0
$219$	−28.3574	−1.91621
$220$	0	0
$221$	−4.38874	−0.295218
$222$	0	0
$223$	−9.57344	−0.641085	−0.320542	−	0.947234i	$-0.603865\pi$
$223$	−9.57344	−0.641085	−0.320542	+	0.947234i	$0.603865\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.00120	−0.132824	−0.0664121	−	0.997792i	$-0.521155\pi$
$227$	−2.00120	−0.132824	−0.0664121	+	0.997792i	$0.521155\pi$
$228$	0	0
$229$	−18.5817	−1.22792	−0.613958	−	0.789339i	$-0.710424\pi$
$229$	−18.5817	−1.22792	−0.613958	+	0.789339i	$0.710424\pi$
$230$	0	0
$231$	−3.51449	−0.231236
$232$	0	0
$233$	14.6878	0.962231	0.481116	−	0.876657i	$-0.340232\pi$
$233$	14.6878	0.962231	0.481116	+	0.876657i	$0.340232\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−6.33128	−0.411261
$238$	0	0
$239$	−21.7241	−1.40522	−0.702608	−	0.711577i	$-0.747981\pi$
$239$	−21.7241	−1.40522	−0.702608	+	0.711577i	$0.747981\pi$
$240$	0	0
$241$	−15.2726	−0.983793	−0.491897	−	0.870654i	$-0.663696\pi$
$241$	−15.2726	−0.983793	−0.491897	+	0.870654i	$0.663696\pi$
$242$	0	0
$243$	−0.874006	−0.0560675
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.08223	−0.196118
$248$	0	0
$249$	−20.8999	−1.32448
$250$	0	0
$251$	20.8910	1.31863	0.659314	−	0.751868i	$-0.270847\pi$
$251$	20.8910	1.31863	0.659314	+	0.751868i	$0.270847\pi$
$252$	0	0
$253$	−7.23060	−0.454584
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.21677	0.574926	0.287463	−	0.957792i	$-0.407188\pi$
$257$	9.21677	0.574926	0.287463	+	0.957792i	$0.407188\pi$
$258$	0	0
$259$	−1.14700	−0.0712712
$260$	0	0
$261$	0.536733	0.0332229
$262$	0	0
$263$	−1.99868	−0.123244	−0.0616220	−	0.998100i	$-0.519627\pi$
$263$	−1.99868	−0.123244	−0.0616220	+	0.998100i	$0.519627\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−8.08873	−0.495022
$268$	0	0
$269$	24.0953	1.46912	0.734558	−	0.678546i	$-0.237390\pi$
$269$	24.0953	1.46912	0.734558	+	0.678546i	$0.237390\pi$
$270$	0	0
$271$	−27.5513	−1.67362	−0.836810	−	0.547494i	$-0.815582\pi$
$271$	−27.5513	−1.67362	−0.836810	+	0.547494i	$0.815582\pi$
$272$	0	0
$273$	−3.90554	−0.236374
$274$	0	0
$275$	0	0
$276$	0	0
$277$	24.3869	1.46526	0.732632	−	0.680625i	$-0.238292\pi$
$277$	24.3869	1.46526	0.732632	+	0.680625i	$0.238292\pi$
$278$	0	0
$279$	0.0214447	0.00128386
$280$	0	0
$281$	−26.4306	−1.57672	−0.788359	−	0.615215i	$-0.789069\pi$
$281$	−26.4306	−1.57672	−0.788359	+	0.615215i	$0.789069\pi$
$282$	0	0
$283$	16.5693	0.984941	0.492471	−	0.870329i	$-0.336094\pi$
$283$	16.5693	0.984941	0.492471	+	0.870329i	$0.336094\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.44003	0.557228
$288$	0	0
$289$	−4.28029	−0.251782
$290$	0	0
$291$	13.4983	0.791283
$292$	0	0
$293$	30.9928	1.81062	0.905310	−	0.424752i	$-0.139639\pi$
$293$	30.9928	1.81062	0.905310	+	0.424752i	$0.139639\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.83175	0.338392
$298$	0	0
$299$	−8.03513	−0.464683
$300$	0	0
$301$	13.3365	0.768702
$302$	0	0
$303$	−23.1096	−1.32761
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.12128	−0.0639947	−0.0319973	−	0.999488i	$-0.510187\pi$
$307$	−1.12128	−0.0639947	−0.0319973	+	0.999488i	$0.510187\pi$
$308$	0	0
$309$	13.5604	0.771426
$310$	0	0
$311$	−14.7068	−0.833946	−0.416973	−	0.908919i	$-0.636909\pi$
$311$	−14.7068	−0.833946	−0.416973	+	0.908919i	$0.636909\pi$
$312$	0	0
$313$	−34.8295	−1.96868	−0.984341	−	0.176273i	$-0.943596\pi$
$313$	−34.8295	−1.96868	−0.984341	+	0.176273i	$0.943596\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.47691	0.251448	0.125724	−	0.992065i	$-0.459875\pi$
$317$	4.47691	0.251448	0.125724	+	0.992065i	$0.459875\pi$
$318$	0	0
$319$	−7.06494	−0.395561
$320$	0	0
$321$	−31.1859	−1.74063
$322$	0	0
$323$	8.93311	0.497052
$324$	0	0
$325$	0	0
$326$	0	0
$327$	28.5119	1.57671
$328$	0	0
$329$	−0.521600	−0.0287568
$330$	0	0
$331$	−5.15648	−0.283426	−0.141713	−	0.989908i	$-0.545261\pi$
$331$	−5.15648	−0.283426	−0.141713	+	0.989908i	$0.545261\pi$
$332$	0	0
$333$	0.0519159	0.00284497
$334$	0	0
$335$	0	0
$336$	0	0
$337$	32.2062	1.75438	0.877192	−	0.480140i	$-0.159414\pi$
$337$	32.2062	1.75438	0.877192	+	0.480140i	$0.159414\pi$
$338$	0	0
$339$	25.9757	1.41081
$340$	0	0
$341$	−0.282274	−0.0152860
$342$	0	0
$343$	−19.6002	−1.05831
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.83337	−0.313152	−0.156576	−	0.987666i	$-0.550046\pi$
$347$	−5.83337	−0.313152	−0.156576	+	0.987666i	$0.550046\pi$
$348$	0	0
$349$	0.910922	0.0487606	0.0243803	−	0.999703i	$-0.492239\pi$
$349$	0.910922	0.0487606	0.0243803	+	0.999703i	$0.492239\pi$
$350$	0	0
$351$	6.48063	0.345910
$352$	0	0
$353$	28.8303	1.53448	0.767240	−	0.641360i	$-0.221630\pi$
$353$	28.8303	1.53448	0.767240	+	0.641360i	$0.221630\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	11.3193	0.599079
$358$	0	0
$359$	22.8546	1.20622	0.603109	−	0.797659i	$-0.293929\pi$
$359$	22.8546	1.20622	0.603109	+	0.797659i	$0.293929\pi$
$360$	0	0
$361$	−12.7262	−0.669802
$362$	0	0
$363$	16.6897	0.875980
$364$	0	0
$365$	0	0
$366$	0	0
$367$	20.0952	1.04896	0.524479	−	0.851423i	$-0.324260\pi$
$367$	20.0952	1.04896	0.524479	+	0.851423i	$0.324260\pi$
$368$	0	0
$369$	−0.427277	−0.0222432
$370$	0	0
$371$	−7.44413	−0.386480
$372$	0	0
$373$	15.3155	0.793007	0.396504	−	0.918033i	$-0.370223\pi$
$373$	15.3155	0.793007	0.396504	+	0.918033i	$0.370223\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.85103	−0.404349
$378$	0	0
$379$	36.3005	1.86463	0.932316	−	0.361645i	$-0.117785\pi$
$379$	36.3005	1.86463	0.932316	+	0.361645i	$0.117785\pi$
$380$	0	0
$381$	−8.87467	−0.454663
$382$	0	0
$383$	5.20909	0.266172	0.133086	−	0.991104i	$-0.457511\pi$
$383$	5.20909	0.266172	0.133086	+	0.991104i	$0.457511\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.603639	−0.0306847
$388$	0	0
$389$	17.6241	0.893579	0.446789	−	0.894639i	$-0.352567\pi$
$389$	17.6241	0.893579	0.446789	+	0.894639i	$0.352567\pi$
$390$	0	0
$391$	23.2879	1.17772
$392$	0	0
$393$	−2.42776	−0.122464
$394$	0	0
$395$	0	0
$396$	0	0
$397$	20.5429	1.03102	0.515510	−	0.856883i	$-0.327602\pi$
$397$	20.5429	1.03102	0.515510	+	0.856883i	$0.327602\pi$
$398$	0	0
$399$	7.94958	0.397977
$400$	0	0
$401$	−19.4665	−0.972113	−0.486057	−	0.873927i	$-0.661565\pi$
$401$	−19.4665	−0.972113	−0.486057	+	0.873927i	$0.661565\pi$
$402$	0	0
$403$	−0.313682	−0.0156256
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.683361	−0.0338730
$408$	0	0
$409$	−1.24227	−0.0614263	−0.0307132	−	0.999528i	$-0.509778\pi$
$409$	−1.24227	−0.0614263	−0.0307132	+	0.999528i	$0.509778\pi$
$410$	0	0
$411$	−12.5525	−0.619170
$412$	0	0
$413$	−6.30886	−0.310439
$414$	0	0
$415$	0	0
$416$	0	0
$417$	30.9352	1.51490
$418$	0	0
$419$	16.6936	0.815538	0.407769	−	0.913085i	$-0.366307\pi$
$419$	16.6936	0.815538	0.407769	+	0.913085i	$0.366307\pi$
$420$	0	0
$421$	16.1512	0.787161	0.393581	−	0.919290i	$-0.371236\pi$
$421$	16.1512	0.787161	0.393581	+	0.919290i	$0.371236\pi$
$422$	0	0
$423$	0.0236088	0.00114790
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−25.1566	−1.21741
$428$	0	0
$429$	−2.32684	−0.112341
$430$	0	0
$431$	−3.83273	−0.184616	−0.0923081	−	0.995730i	$-0.529424\pi$
$431$	−3.83273	−0.184616	−0.0923081	+	0.995730i	$0.529424\pi$
$432$	0	0
$433$	26.4777	1.27244	0.636219	−	0.771509i	$-0.280498\pi$
$433$	26.4777	1.27244	0.636219	+	0.771509i	$0.280498\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	16.3552	0.782376
$438$	0	0
$439$	−7.68086	−0.366587	−0.183294	−	0.983058i	$-0.558676\pi$
$439$	−7.68086	−0.366587	−0.183294	+	0.983058i	$0.558676\pi$
$440$	0	0
$441$	0.298266	0.0142031
$442$	0	0
$443$	−5.67017	−0.269398	−0.134699	−	0.990887i	$-0.543007\pi$
$443$	−5.67017	−0.269398	−0.134699	+	0.990887i	$0.543007\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−2.00947	−0.0950447
$448$	0	0
$449$	19.7768	0.933325	0.466663	−	0.884435i	$-0.345456\pi$
$449$	19.7768	0.933325	0.466663	+	0.884435i	$0.345456\pi$
$450$	0	0
$451$	5.62419	0.264833
$452$	0	0
$453$	−38.9320	−1.82919
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.9232	0.651302	0.325651	−	0.945490i	$-0.394417\pi$
$457$	13.9232	0.651302	0.325651	+	0.945490i	$0.394417\pi$
$458$	0	0
$459$	−18.7826	−0.876696
$460$	0	0
$461$	14.1517	0.659110	0.329555	−	0.944136i	$-0.393101\pi$
$461$	14.1517	0.659110	0.329555	+	0.944136i	$0.393101\pi$
$462$	0	0
$463$	−26.9829	−1.25400	−0.627000	−	0.779019i	$-0.715717\pi$
$463$	−26.9829	−1.25400	−0.627000	+	0.779019i	$0.715717\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.3500	0.895411	0.447706	−	0.894181i	$-0.352241\pi$
$467$	19.3500	0.895411	0.447706	+	0.894181i	$0.352241\pi$
$468$	0	0
$469$	4.00295	0.184839
$470$	0	0
$471$	−17.1861	−0.791893
$472$	0	0
$473$	7.94562	0.365340
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.336938	0.0154273
$478$	0	0
$479$	−18.7268	−0.855650	−0.427825	−	0.903862i	$-0.640720\pi$
$479$	−18.7268	−0.855650	−0.427825	+	0.903862i	$0.640720\pi$
$480$	0	0
$481$	−0.759397	−0.0346255
$482$	0	0
$483$	20.7239	0.942971
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−5.33189	−0.241611	−0.120805	−	0.992676i	$-0.538548\pi$
$487$	−5.33189	−0.241611	−0.120805	+	0.992676i	$0.538548\pi$
$488$	0	0
$489$	−6.42381	−0.290495
$490$	0	0
$491$	33.5653	1.51478	0.757391	−	0.652962i	$-0.226474\pi$
$491$	33.5653	1.51478	0.757391	+	0.652962i	$0.226474\pi$
$492$	0	0
$493$	22.7544	1.02480
$494$	0	0
$495$	0	0
$496$	0	0
$497$	25.5693	1.14694
$498$	0	0
$499$	3.78538	0.169457	0.0847284	−	0.996404i	$-0.472998\pi$
$499$	3.78538	0.169457	0.0847284	+	0.996404i	$0.472998\pi$
$500$	0	0
$501$	8.17400	0.365187
$502$	0	0
$503$	−16.2154	−0.723010	−0.361505	−	0.932370i	$-0.617737\pi$
$503$	−16.2154	−0.723010	−0.361505	+	0.932370i	$0.617737\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	19.6130	0.871042
$508$	0	0
$509$	−7.07267	−0.313491	−0.156745	−	0.987639i	$-0.550100\pi$
$509$	−7.07267	−0.313491	−0.156745	+	0.987639i	$0.550100\pi$
$510$	0	0
$511$	30.8658	1.36542
$512$	0	0
$513$	−13.1911	−0.582401
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−0.310760	−0.0136672
$518$	0	0
$519$	−20.5674	−0.902809
$520$	0	0
$521$	−17.5413	−0.768499	−0.384249	−	0.923229i	$-0.625540\pi$
$521$	−17.5413	−0.768499	−0.384249	+	0.923229i	$0.625540\pi$
$522$	0	0
$523$	5.43660	0.237726	0.118863	−	0.992911i	$-0.462075\pi$
$523$	5.43660	0.237726	0.118863	+	0.992911i	$0.462075\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.909133	0.0396025
$528$	0	0
$529$	19.6367	0.853771
$530$	0	0
$531$	0.285553	0.0123920
$532$	0	0
$533$	6.24998	0.270717
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−21.5887	−0.931622
$538$	0	0
$539$	−3.92603	−0.169106
$540$	0	0
$541$	27.0924	1.16479	0.582396	−	0.812905i	$-0.302115\pi$
$541$	27.0924	1.16479	0.582396	+	0.812905i	$0.302115\pi$
$542$	0	0
$543$	12.4302	0.533432
$544$	0	0
$545$	0	0
$546$	0	0
$547$	40.2342	1.72029	0.860144	−	0.510051i	$-0.170373\pi$
$547$	40.2342	1.72029	0.860144	+	0.510051i	$0.170373\pi$
$548$	0	0
$549$	1.13865	0.0485962
$550$	0	0
$551$	15.9805	0.680792
$552$	0	0
$553$	6.89132	0.293049
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−17.8135	−0.754782	−0.377391	−	0.926054i	$-0.623179\pi$
$557$	−17.8135	−0.754782	−0.377391	+	0.926054i	$0.623179\pi$
$558$	0	0
$559$	8.82971	0.373457
$560$	0	0
$561$	6.74381	0.284724
$562$	0	0
$563$	−17.4989	−0.737492	−0.368746	−	0.929530i	$-0.620213\pi$
$563$	−17.4989	−0.737492	−0.368746	+	0.929530i	$0.620213\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−16.2455	−0.682248
$568$	0	0
$569$	−9.57047	−0.401215	−0.200608	−	0.979672i	$-0.564292\pi$
$569$	−9.57047	−0.401215	−0.200608	+	0.979672i	$0.564292\pi$
$570$	0	0
$571$	−25.5084	−1.06749	−0.533747	−	0.845644i	$-0.679217\pi$
$571$	−25.5084	−1.06749	−0.533747	+	0.845644i	$0.679217\pi$
$572$	0	0
$573$	−24.9837	−1.04371
$574$	0	0
$575$	0	0
$576$	0	0
$577$	22.0765	0.919058	0.459529	−	0.888163i	$-0.348018\pi$
$577$	22.0765	0.919058	0.459529	+	0.888163i	$0.348018\pi$
$578$	0	0
$579$	4.48516	0.186397
$580$	0	0
$581$	22.7486	0.943772
$582$	0	0
$583$	−4.43507	−0.183682
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.94853	0.410620	0.205310	−	0.978697i	$-0.434180\pi$
$587$	9.94853	0.410620	0.205310	+	0.978697i	$0.434180\pi$
$588$	0	0
$589$	0.638488	0.0263085
$590$	0	0
$591$	−10.0027	−0.411457
$592$	0	0
$593$	−17.4669	−0.717280	−0.358640	−	0.933476i	$-0.616760\pi$
$593$	−17.4669	−0.717280	−0.358640	+	0.933476i	$0.616760\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.96637	−0.0804781
$598$	0	0
$599$	26.8713	1.09793	0.548966	−	0.835844i	$-0.315022\pi$
$599$	26.8713	1.09793	0.548966	+	0.835844i	$0.315022\pi$
$600$	0	0
$601$	19.7272	0.804690	0.402345	−	0.915488i	$-0.368195\pi$
$601$	19.7272	0.804690	0.402345	+	0.915488i	$0.368195\pi$
$602$	0	0
$603$	−0.181183	−0.00737833
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−32.3135	−1.31157	−0.655783	−	0.754949i	$-0.727661\pi$
$607$	−32.3135	−1.31157	−0.655783	+	0.754949i	$0.727661\pi$
$608$	0	0
$609$	20.2491	0.820535
$610$	0	0
$611$	−0.345337	−0.0139708
$612$	0	0
$613$	6.65196	0.268670	0.134335	−	0.990936i	$-0.457110\pi$
$613$	6.65196	0.268670	0.134335	+	0.990936i	$0.457110\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.17755	0.288957	0.144479	−	0.989508i	$-0.453849\pi$
$617$	7.17755	0.288957	0.144479	+	0.989508i	$0.453849\pi$
$618$	0	0
$619$	25.9320	1.04230	0.521148	−	0.853466i	$-0.325504\pi$
$619$	25.9320	1.04230	0.521148	+	0.853466i	$0.325504\pi$
$620$	0	0
$621$	−34.3881	−1.37995
$622$	0	0
$623$	8.80424	0.352734
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.73621	0.189146
$628$	0	0
$629$	2.20093	0.0877569
$630$	0	0
$631$	26.8698	1.06967	0.534834	−	0.844957i	$-0.320374\pi$
$631$	26.8698	1.06967	0.534834	+	0.844957i	$0.320374\pi$
$632$	0	0
$633$	−2.11041	−0.0838813
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−4.36287	−0.172863
$638$	0	0
$639$	−1.15732	−0.0457830
$640$	0	0
$641$	28.3190	1.11853	0.559266	−	0.828988i	$-0.311083\pi$
$641$	28.3190	1.11853	0.559266	+	0.828988i	$0.311083\pi$
$642$	0	0
$643$	29.3294	1.15664	0.578319	−	0.815811i	$-0.303709\pi$
$643$	29.3294	1.15664	0.578319	+	0.815811i	$0.303709\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.1363	1.06684	0.533419	−	0.845851i	$-0.320907\pi$
$647$	27.1363	1.06684	0.533419	+	0.845851i	$0.320907\pi$
$648$	0	0
$649$	−3.75870	−0.147542
$650$	0	0
$651$	0.809037	0.0317087
$652$	0	0
$653$	−15.0796	−0.590109	−0.295054	−	0.955480i	$-0.595338\pi$
$653$	−15.0796	−0.590109	−0.295054	+	0.955480i	$0.595338\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.39705	−0.0545043
$658$	0	0
$659$	−9.79959	−0.381738	−0.190869	−	0.981616i	$-0.561131\pi$
$659$	−9.79959	−0.381738	−0.190869	+	0.981616i	$0.561131\pi$
$660$	0	0
$661$	29.8139	1.15962	0.579812	−	0.814750i	$-0.303126\pi$
$661$	29.8139	1.15962	0.579812	+	0.814750i	$0.303126\pi$
$662$	0	0
$663$	7.49417	0.291050
$664$	0	0
$665$	0	0
$666$	0	0
$667$	41.6599	1.61308
$668$	0	0
$669$	16.3475	0.632032
$670$	0	0
$671$	−14.9878	−0.578599
$672$	0	0
$673$	−10.5222	−0.405600	−0.202800	−	0.979220i	$-0.565004\pi$
$673$	−10.5222	−0.405600	−0.202800	+	0.979220i	$0.565004\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−22.6438	−0.870271	−0.435135	−	0.900365i	$-0.643299\pi$
$677$	−22.6438	−0.870271	−0.435135	+	0.900365i	$0.643299\pi$
$678$	0	0
$679$	−14.6923	−0.563838
$680$	0	0
$681$	3.41723	0.130949
$682$	0	0
$683$	21.2769	0.814139	0.407070	−	0.913397i	$-0.366551\pi$
$683$	21.2769	0.814139	0.407070	+	0.913397i	$0.366551\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	31.7301	1.21058
$688$	0	0
$689$	−4.92855	−0.187763
$690$	0	0
$691$	−17.3079	−0.658425	−0.329212	−	0.944256i	$-0.606783\pi$
$691$	−17.3079	−0.658425	−0.329212	+	0.944256i	$0.606783\pi$
$692$	0	0
$693$	−0.173145	−0.00657723
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−18.1141	−0.686120
$698$	0	0
$699$	−25.0808	−0.948644
$700$	0	0
$701$	26.3795	0.996340	0.498170	−	0.867079i	$-0.334005\pi$
$701$	26.3795	0.996340	0.498170	+	0.867079i	$0.334005\pi$
$702$	0	0
$703$	1.54573	0.0582981
$704$	0	0
$705$	0	0
$706$	0	0
$707$	25.1538	0.946006
$708$	0	0
$709$	−45.5677	−1.71133	−0.855666	−	0.517528i	$-0.826852\pi$
$709$	−45.5677	−1.71133	−0.855666	+	0.517528i	$0.826852\pi$
$710$	0	0
$711$	−0.311917	−0.0116978
$712$	0	0
$713$	1.66449	0.0623356
$714$	0	0
$715$	0	0
$716$	0	0
$717$	37.0959	1.38537
$718$	0	0
$719$	49.8091	1.85756	0.928782	−	0.370626i	$-0.120857\pi$
$719$	49.8091	1.85756	0.928782	+	0.370626i	$0.120857\pi$
$720$	0	0
$721$	−14.7600	−0.549689
$722$	0	0
$723$	26.0793	0.969901
$724$	0	0
$725$	0	0
$726$	0	0
$727$	36.1299	1.33998	0.669991	−	0.742369i	$-0.266298\pi$
$727$	36.1299	1.33998	0.669991	+	0.742369i	$0.266298\pi$
$728$	0	0
$729$	27.7141	1.02645
$730$	0	0
$731$	−25.5908	−0.946511
$732$	0	0
$733$	15.2033	0.561547	0.280773	−	0.959774i	$-0.409409\pi$
$733$	15.2033	0.561547	0.280773	+	0.959774i	$0.409409\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.38488	0.0878483
$738$	0	0
$739$	44.7017	1.64438	0.822189	−	0.569214i	$-0.192753\pi$
$739$	44.7017	1.64438	0.822189	+	0.569214i	$0.192753\pi$
$740$	0	0
$741$	5.26319	0.193348
$742$	0	0
$743$	18.6686	0.684883	0.342442	−	0.939539i	$-0.388746\pi$
$743$	18.6686	0.684883	0.342442	+	0.939539i	$0.388746\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.02965	−0.0376731
$748$	0	0
$749$	33.9445	1.24031
$750$	0	0
$751$	−44.2141	−1.61339	−0.806697	−	0.590965i	$-0.798747\pi$
$751$	−44.2141	−1.61339	−0.806697	+	0.590965i	$0.798747\pi$
$752$	0	0
$753$	−35.6733	−1.30001
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−20.6158	−0.749294	−0.374647	−	0.927168i	$-0.622236\pi$
$757$	−20.6158	−0.749294	−0.374647	+	0.927168i	$0.622236\pi$
$758$	0	0
$759$	12.3469	0.448165
$760$	0	0
$761$	19.8479	0.719485	0.359743	−	0.933052i	$-0.382864\pi$
$761$	19.8479	0.719485	0.359743	+	0.933052i	$0.382864\pi$
$762$	0	0
$763$	−31.0339	−1.12350
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.17692	−0.150820
$768$	0	0
$769$	−19.1429	−0.690310	−0.345155	−	0.938546i	$-0.612174\pi$
$769$	−19.1429	−0.690310	−0.345155	+	0.938546i	$0.612174\pi$
$770$	0	0
$771$	−15.7385	−0.566808
$772$	0	0
$773$	16.6146	0.597584	0.298792	−	0.954318i	$-0.403416\pi$
$773$	16.6146	0.597584	0.298792	+	0.954318i	$0.403416\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.95861	0.0702647
$778$	0	0
$779$	−12.7216	−0.455799
$780$	0	0
$781$	15.2337	0.545104
$782$	0	0
$783$	−33.6003	−1.20077
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−30.1550	−1.07491	−0.537455	−	0.843292i	$-0.680614\pi$
$787$	−30.1550	−1.07491	−0.537455	+	0.843292i	$0.680614\pi$
$788$	0	0
$789$	3.41293	0.121504
$790$	0	0
$791$	−28.2734	−1.00529
$792$	0	0
$793$	−16.6555	−0.591454
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.65214	0.306474	0.153237	−	0.988189i	$-0.451030\pi$
$797$	8.65214	0.306474	0.153237	+	0.988189i	$0.451030\pi$
$798$	0	0
$799$	1.00088	0.0354085
$800$	0	0
$801$	−0.398500	−0.0140803
$802$	0	0
$803$	18.3892	0.648942
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−41.1449	−1.44837
$808$	0	0
$809$	55.1395	1.93860	0.969301	−	0.245878i	$-0.0790764\pi$
$809$	55.1395	1.93860	0.969301	+	0.245878i	$0.0790764\pi$
$810$	0	0
$811$	54.9680	1.93019	0.965093	−	0.261907i	$-0.0843514\pi$
$811$	54.9680	1.93019	0.965093	+	0.261907i	$0.0843514\pi$
$812$	0	0
$813$	47.0463	1.64999
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−17.9726	−0.628780
$818$	0	0
$819$	−0.192410	−0.00672336
$820$	0	0
$821$	−14.1768	−0.494772	−0.247386	−	0.968917i	$-0.579572\pi$
$821$	−14.1768	−0.494772	−0.247386	+	0.968917i	$0.579572\pi$
$822$	0	0
$823$	−26.8408	−0.935612	−0.467806	−	0.883831i	$-0.654955\pi$
$823$	−26.8408	−0.935612	−0.467806	+	0.883831i	$0.654955\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−38.7020	−1.34580	−0.672900	−	0.739733i	$-0.734952\pi$
$827$	−38.7020	−1.34580	−0.672900	+	0.739733i	$0.734952\pi$
$828$	0	0
$829$	−21.4901	−0.746382	−0.373191	−	0.927754i	$-0.621736\pi$
$829$	−21.4901	−0.746382	−0.373191	+	0.927754i	$0.621736\pi$
$830$	0	0
$831$	−41.6428	−1.44457
$832$	0	0
$833$	12.6448	0.438115
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.34247	−0.0464026
$838$	0	0
$839$	−25.6479	−0.885462	−0.442731	−	0.896654i	$-0.645990\pi$
$839$	−25.6479	−0.885462	−0.442731	+	0.896654i	$0.645990\pi$
$840$	0	0
$841$	11.7054	0.403634
$842$	0	0
$843$	45.1327	1.55445
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−18.1660	−0.624190
$848$	0	0
$849$	−28.2936	−0.971033
$850$	0	0
$851$	4.02958	0.138132
$852$	0	0
$853$	−21.9323	−0.750946	−0.375473	−	0.926833i	$-0.622520\pi$
$853$	−21.9323	−0.750946	−0.375473	+	0.926833i	$0.622520\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−49.6943	−1.69753	−0.848763	−	0.528774i	$-0.822652\pi$
$857$	−49.6943	−1.69753	−0.848763	+	0.528774i	$0.822652\pi$
$858$	0	0
$859$	−13.4400	−0.458566	−0.229283	−	0.973360i	$-0.573638\pi$
$859$	−13.4400	−0.458566	−0.229283	+	0.973360i	$0.573638\pi$
$860$	0	0
$861$	−16.1197	−0.549359
$862$	0	0
$863$	−46.1292	−1.57026	−0.785128	−	0.619334i	$-0.787403\pi$
$863$	−46.1292	−1.57026	−0.785128	+	0.619334i	$0.787403\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	7.30899	0.248226
$868$	0	0
$869$	4.10572	0.139277
$870$	0	0
$871$	2.65024	0.0898001
$872$	0	0
$873$	0.665007	0.0225071
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.54440	0.288524	0.144262	−	0.989540i	$-0.453919\pi$
$877$	8.54440	0.288524	0.144262	+	0.989540i	$0.453919\pi$
$878$	0	0
$879$	−52.9231	−1.78505
$880$	0	0
$881$	−0.201941	−0.00680356	−0.00340178	−	0.999994i	$-0.501083\pi$
$881$	−0.201941	−0.00680356	−0.00340178	+	0.999994i	$0.501083\pi$
$882$	0	0
$883$	−2.45751	−0.0827017	−0.0413509	−	0.999145i	$-0.513166\pi$
$883$	−2.45751	−0.0827017	−0.0413509	+	0.999145i	$0.513166\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20.6122	0.692091	0.346046	−	0.938218i	$-0.387524\pi$
$887$	20.6122	0.692091	0.346046	+	0.938218i	$0.387524\pi$
$888$	0	0
$889$	9.65970	0.323976
$890$	0	0
$891$	−9.67878	−0.324251
$892$	0	0
$893$	0.702921	0.0235223
$894$	0	0
$895$	0	0
$896$	0	0
$897$	13.7207	0.458122
$898$	0	0
$899$	1.62635	0.0542419
$900$	0	0
$901$	14.2842	0.475877
$902$	0	0
$903$	−22.7733	−0.757847
$904$	0	0
$905$	0	0
$906$	0	0
$907$	16.9115	0.561538	0.280769	−	0.959775i	$-0.409411\pi$
$907$	16.9115	0.561538	0.280769	+	0.959775i	$0.409411\pi$
$908$	0	0
$909$	−1.13852	−0.0377622
$910$	0	0
$911$	−17.6053	−0.583291	−0.291645	−	0.956527i	$-0.594203\pi$
$911$	−17.6053	−0.583291	−0.291645	+	0.956527i	$0.594203\pi$
$912$	0	0
$913$	13.5532	0.448545
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.64252	0.0872636
$918$	0	0
$919$	49.0185	1.61697	0.808486	−	0.588515i	$-0.200287\pi$
$919$	49.0185	1.61697	0.808486	+	0.588515i	$0.200287\pi$
$920$	0	0
$921$	1.91469	0.0630910
$922$	0	0
$923$	16.9287	0.557214
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.668069	0.0219423
$928$	0	0
$929$	40.8140	1.33906	0.669532	−	0.742783i	$-0.266495\pi$
$929$	40.8140	1.33906	0.669532	+	0.742783i	$0.266495\pi$
$930$	0	0
$931$	8.88047	0.291046
$932$	0	0
$933$	25.1132	0.822170
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−8.18526	−0.267401	−0.133700	−	0.991022i	$-0.542686\pi$
$937$	−8.18526	−0.267401	−0.133700	+	0.991022i	$0.542686\pi$
$938$	0	0
$939$	59.4747	1.94088
$940$	0	0
$941$	50.5081	1.64652	0.823259	−	0.567666i	$-0.192154\pi$
$941$	50.5081	1.64652	0.823259	+	0.567666i	$0.192154\pi$
$942$	0	0
$943$	−33.1642	−1.07998
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.4824	1.18552	0.592760	−	0.805380i	$-0.298038\pi$
$947$	36.4824	1.18552	0.592760	+	0.805380i	$0.298038\pi$
$948$	0	0
$949$	20.4354	0.663360
$950$	0	0
$951$	−7.64474	−0.247898
$952$	0	0
$953$	−26.6528	−0.863369	−0.431685	−	0.902025i	$-0.642081\pi$
$953$	−26.6528	−0.863369	−0.431685	+	0.902025i	$0.642081\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	12.0640	0.389975
$958$	0	0
$959$	13.6629	0.441197
$960$	0	0
$961$	−30.9350	−0.997904
$962$	0	0
$963$	−1.53641	−0.0495100
$964$	0	0
$965$	0	0
$966$	0	0
$967$	51.2843	1.64919	0.824596	−	0.565722i	$-0.191402\pi$
$967$	51.2843	1.64919	0.824596	+	0.565722i	$0.191402\pi$
$968$	0	0
$969$	−15.2541	−0.490033
$970$	0	0
$971$	22.2917	0.715376	0.357688	−	0.933841i	$-0.383565\pi$
$971$	22.2917	0.715376	0.357688	+	0.933841i	$0.383565\pi$
$972$	0	0
$973$	−33.6716	−1.07946
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−37.0255	−1.18455	−0.592275	−	0.805736i	$-0.701770\pi$
$977$	−37.0255	−1.18455	−0.592275	+	0.805736i	$0.701770\pi$
$978$	0	0
$979$	5.24540	0.167644
$980$	0	0
$981$	1.40467	0.0448476
$982$	0	0
$983$	−21.1370	−0.674165	−0.337083	−	0.941475i	$-0.609440\pi$
$983$	−21.1370	−0.674165	−0.337083	+	0.941475i	$0.609440\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.890681	0.0283507
$988$	0	0
$989$	−46.8530	−1.48984
$990$	0	0
$991$	18.5433	0.589047	0.294524	−	0.955644i	$-0.404839\pi$
$991$	18.5433	0.589047	0.294524	+	0.955644i	$0.404839\pi$
$992$	0	0
$993$	8.80517	0.279424
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−20.5905	−0.652106	−0.326053	−	0.945351i	$-0.605719\pi$
$997$	−20.5905	−0.652106	−0.326053	+	0.945351i	$0.605719\pi$
$998$	0	0
$999$	−3.25001	−0.102826

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bl.1.2		8
4.3	odd	2		5000.2.a.k.1.7	✓	8
5.4	even	2		10000.2.a.bg.1.7		8
20.19	odd	2		5000.2.a.n.1.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5000.2.a.k.1.7	✓	8	4.3	odd	2
5000.2.a.n.1.2	yes	8	20.19	odd	2
10000.2.a.bg.1.7		8	5.4	even	2
10000.2.a.bl.1.2		8	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:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

\(f(q)\)	\(=\)	\(q-1.70759 q^{3} +1.85864 q^{7} -0.0841263 q^{9} +O(q^{10})\)	\(q-1.70759 q^{3} +1.85864 q^{7} -0.0841263 q^{9} +1.10734 q^{11} +1.23055 q^{13} -3.56647 q^{17} -2.50475 q^{19} -3.17380 q^{21} -6.52968 q^{23} +5.26643 q^{27} -6.38008 q^{29} -0.254911 q^{31} -1.89089 q^{33} -0.617118 q^{37} -2.10129 q^{39} +5.07900 q^{41} +7.17539 q^{43} -0.280635 q^{47} -3.54545 q^{49} +6.09008 q^{51} -4.00515 q^{53} +4.27709 q^{57} -3.39434 q^{59} -13.5349 q^{61} -0.156361 q^{63} +2.15370 q^{67} +11.1500 q^{69} +13.7570 q^{71} +16.6066 q^{73} +2.05815 q^{77} +3.70772 q^{79} -8.74054 q^{81} +12.2394 q^{83} +10.8946 q^{87} +4.73692 q^{89} +2.28716 q^{91} +0.435284 q^{93} -7.90486 q^{97} -0.0931567 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{3} + 2 q^{7} - 3 q^{9}+O(q^{10}) \) 8 * q + 3 * q^3 + 2 * q^7 - 3 * q^9	\( 8 q + 3 q^{3} + 2 q^{7} - 3 q^{9} - 5 q^{11} + 2 q^{13} + 7 q^{17} + 10 q^{19} - 14 q^{21} - 2 q^{23} - 6 q^{27} - 10 q^{29} + 27 q^{31} - 3 q^{33} + 10 q^{37} + 18 q^{39} - 16 q^{41} + 4 q^{43} - 8 q^{47} + 4 q^{49} + 10 q^{51} + 2 q^{53} + 2 q^{57} + 39 q^{59} - 18 q^{61} - 14 q^{63} - 12 q^{67} + 19 q^{69} + 13 q^{71} + 12 q^{73} + 41 q^{77} + 16 q^{79} - 28 q^{81} + 64 q^{83} - 4 q^{87} - 25 q^{89} + 26 q^{91} + 40 q^{93} - 8 q^{97} - 6 q^{99}+O(q^{100}) \) 8 * q + 3 * q^3 + 2 * q^7 - 3 * q^9 - 5 * q^11 + 2 * q^13 + 7 * q^17 + 10 * q^19 - 14 * q^21 - 2 * q^23 - 6 * q^27 - 10 * q^29 + 27 * q^31 - 3 * q^33 + 10 * q^37 + 18 * q^39 - 16 * q^41 + 4 * q^43 - 8 * q^47 + 4 * q^49 + 10 * q^51 + 2 * q^53 + 2 * q^57 + 39 * q^59 - 18 * q^61 - 14 * q^63 - 12 * q^67 + 19 * q^69 + 13 * q^71 + 12 * q^73 + 41 * q^77 + 16 * q^79 - 28 * q^81 + 64 * q^83 - 4 * q^87 - 25 * q^89 + 26 * q^91 + 40 * q^93 - 8 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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