LMFDB - Embedded newform 9016.2.a.br.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9016,2,Mod(1,9016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9016, 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("9016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9016 = 2^{3} \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9016.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:	$71.9931224624$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 4 x^{10} - 14 x^{9} + 61 x^{8} + 71 x^{7} - 343 x^{6} - 152 x^{5} + 867 x^{4} + 102 x^{3} + \cdots + 243 $ x^11 - 4x^10 - 14x^9 + 61x^8 + 71x^7 - 343x^6 - 152x^5 + 867x^4 + 102x^3 - 918x^2 + 35x + 243
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1288)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.78911$ of defining polynomial
Character	$\chi$	$=$	9016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.78911 q^{3} -0.157144 q^{5} +0.200921 q^{9} +O(q^{10})$	$q-1.78911 q^{3} -0.157144 q^{5} +0.200921 q^{9} -3.73854 q^{11} -6.06322 q^{13} +0.281148 q^{15} +7.10547 q^{17} -4.70697 q^{19} -1.00000 q^{23} -4.97531 q^{25} +5.00787 q^{27} -5.99404 q^{29} -1.31892 q^{31} +6.68867 q^{33} -3.66737 q^{37} +10.8478 q^{39} -7.95498 q^{41} +11.1910 q^{43} -0.0315735 q^{45} +0.0837838 q^{47} -12.7125 q^{51} -9.65830 q^{53} +0.587489 q^{55} +8.42130 q^{57} -10.6913 q^{59} -1.59569 q^{61} +0.952797 q^{65} -5.89259 q^{67} +1.78911 q^{69} -7.00163 q^{71} -5.23935 q^{73} +8.90138 q^{75} -15.8267 q^{79} -9.56239 q^{81} +16.3052 q^{83} -1.11658 q^{85} +10.7240 q^{87} -3.55531 q^{89} +2.35970 q^{93} +0.739672 q^{95} +14.6341 q^{97} -0.751152 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 4 q^{3} + 3 q^{5} + 11 q^{9}+O(q^{10}) $ 11 * q + 4 * q^3 + 3 * q^5 + 11 * q^9	$ 11 q + 4 q^{3} + 3 q^{5} + 11 q^{9} - 13 q^{13} + 7 q^{17} + 8 q^{19} - 11 q^{23} + 6 q^{25} + 25 q^{27} - 3 q^{29} + 12 q^{31} - 2 q^{33} - q^{37} - 21 q^{39} + 12 q^{41} + 9 q^{43} + 19 q^{45} + 17 q^{47} + 19 q^{51} - 5 q^{53} + 21 q^{55} + 11 q^{57} + 33 q^{59} - 15 q^{61} - 9 q^{65} - 5 q^{67} - 4 q^{69} - 9 q^{71} + 5 q^{73} + 44 q^{75} + 11 q^{79} - 13 q^{81} + 51 q^{83} + 33 q^{85} + 4 q^{87} + 26 q^{89} + 6 q^{93} - 19 q^{95} + 21 q^{97} + 15 q^{99}+O(q^{100}) $ 11 * q + 4 * q^3 + 3 * q^5 + 11 * q^9 - 13 * q^13 + 7 * q^17 + 8 * q^19 - 11 * q^23 + 6 * q^25 + 25 * q^27 - 3 * q^29 + 12 * q^31 - 2 * q^33 - q^37 - 21 * q^39 + 12 * q^41 + 9 * q^43 + 19 * q^45 + 17 * q^47 + 19 * q^51 - 5 * q^53 + 21 * q^55 + 11 * q^57 + 33 * q^59 - 15 * q^61 - 9 * q^65 - 5 * q^67 - 4 * q^69 - 9 * q^71 + 5 * q^73 + 44 * q^75 + 11 * q^79 - 13 * q^81 + 51 * q^83 + 33 * q^85 + 4 * q^87 + 26 * q^89 + 6 * q^93 - 19 * q^95 + 21 * q^97 + 15 * 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
$3$	−1.78911	−1.03294	−0.516472	−	0.856304i	$-0.672755\pi$
$3$	−1.78911	−1.03294	−0.516472	+	0.856304i	$0.672755\pi$
$4$	0	0
$5$	−0.157144	−0.0702769	−0.0351384	−	0.999382i	$-0.511187\pi$
$5$	−0.157144	−0.0702769	−0.0351384	+	0.999382i	$0.511187\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.200921	0.0669737
$10$	0	0
$11$	−3.73854	−1.12721	−0.563606	−	0.826044i	$-0.690586\pi$
$11$	−3.73854	−1.12721	−0.563606	+	0.826044i	$0.690586\pi$
$12$	0	0
$13$	−6.06322	−1.68163	−0.840817	−	0.541320i	$-0.817925\pi$
$13$	−6.06322	−1.68163	−0.840817	+	0.541320i	$0.817925\pi$
$14$	0	0
$15$	0.281148	0.0725921
$16$	0	0
$17$	7.10547	1.72333	0.861664	−	0.507479i	$-0.169422\pi$
$17$	7.10547	1.72333	0.861664	+	0.507479i	$0.169422\pi$
$18$	0	0
$19$	−4.70697	−1.07985	−0.539927	−	0.841712i	$-0.681548\pi$
$19$	−4.70697	−1.07985	−0.539927	+	0.841712i	$0.681548\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.97531	−0.995061
$26$	0	0
$27$	5.00787	0.963764
$28$	0	0
$29$	−5.99404	−1.11307	−0.556533	−	0.830826i	$-0.687869\pi$
$29$	−5.99404	−1.11307	−0.556533	+	0.830826i	$0.687869\pi$
$30$	0	0
$31$	−1.31892	−0.236885	−0.118443	−	0.992961i	$-0.537790\pi$
$31$	−1.31892	−0.236885	−0.118443	+	0.992961i	$0.537790\pi$
$32$	0	0
$33$	6.68867	1.16435
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.66737	−0.602912	−0.301456	−	0.953480i	$-0.597473\pi$
$37$	−3.66737	−0.602912	−0.301456	+	0.953480i	$0.597473\pi$
$38$	0	0
$39$	10.8478	1.73703
$40$	0	0
$41$	−7.95498	−1.24236	−0.621180	−	0.783668i	$-0.713346\pi$
$41$	−7.95498	−1.24236	−0.621180	+	0.783668i	$0.713346\pi$
$42$	0	0
$43$	11.1910	1.70662	0.853309	−	0.521405i	$-0.174592\pi$
$43$	11.1910	1.70662	0.853309	+	0.521405i	$0.174592\pi$
$44$	0	0
$45$	−0.0315735	−0.00470670
$46$	0	0
$47$	0.0837838	0.0122211	0.00611056	−	0.999981i	$-0.498055\pi$
$47$	0.0837838	0.0122211	0.00611056	+	0.999981i	$0.498055\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−12.7125	−1.78010
$52$	0	0
$53$	−9.65830	−1.32667	−0.663335	−	0.748323i	$-0.730859\pi$
$53$	−9.65830	−1.32667	−0.663335	+	0.748323i	$0.730859\pi$
$54$	0	0
$55$	0.587489	0.0792169
$56$	0	0
$57$	8.42130	1.11543
$58$	0	0
$59$	−10.6913	−1.39189	−0.695945	−	0.718095i	$-0.745014\pi$
$59$	−10.6913	−1.39189	−0.695945	+	0.718095i	$0.745014\pi$
$60$	0	0
$61$	−1.59569	−0.204308	−0.102154	−	0.994769i	$-0.532573\pi$
$61$	−1.59569	−0.204308	−0.102154	+	0.994769i	$0.532573\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.952797	0.118180
$66$	0	0
$67$	−5.89259	−0.719895	−0.359947	−	0.932973i	$-0.617205\pi$
$67$	−5.89259	−0.719895	−0.359947	+	0.932973i	$0.617205\pi$
$68$	0	0
$69$	1.78911	0.215384
$70$	0	0
$71$	−7.00163	−0.830940	−0.415470	−	0.909607i	$-0.636383\pi$
$71$	−7.00163	−0.830940	−0.415470	+	0.909607i	$0.636383\pi$
$72$	0	0
$73$	−5.23935	−0.613219	−0.306610	−	0.951835i	$-0.599195\pi$
$73$	−5.23935	−0.613219	−0.306610	+	0.951835i	$0.599195\pi$
$74$	0	0
$75$	8.90138	1.02784
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−15.8267	−1.78064	−0.890319	−	0.455336i	$-0.849519\pi$
$79$	−15.8267	−1.78064	−0.890319	+	0.455336i	$0.849519\pi$
$80$	0	0
$81$	−9.56239	−1.06249
$82$	0	0
$83$	16.3052	1.78973	0.894863	−	0.446342i	$-0.147273\pi$
$83$	16.3052	1.78973	0.894863	+	0.446342i	$0.147273\pi$
$84$	0	0
$85$	−1.11658	−0.121110
$86$	0	0
$87$	10.7240	1.14974
$88$	0	0
$89$	−3.55531	−0.376862	−0.188431	−	0.982086i	$-0.560340\pi$
$89$	−3.55531	−0.376862	−0.188431	+	0.982086i	$0.560340\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.35970	0.244689
$94$	0	0
$95$	0.739672	0.0758888
$96$	0	0
$97$	14.6341	1.48586	0.742932	−	0.669367i	$-0.233435\pi$
$97$	14.6341	1.48586	0.742932	+	0.669367i	$0.233435\pi$
$98$	0	0
$99$	−0.751152	−0.0754936
$100$	0	0
$101$	0.0682102	0.00678717	0.00339358	−	0.999994i	$-0.498920\pi$
$101$	0.0682102	0.00678717	0.00339358	+	0.999994i	$0.498920\pi$
$102$	0	0
$103$	13.1843	1.29909	0.649543	−	0.760325i	$-0.274960\pi$
$103$	13.1843	1.29909	0.649543	+	0.760325i	$0.274960\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.65444	−0.256614	−0.128307	−	0.991734i	$-0.540954\pi$
$107$	−2.65444	−0.256614	−0.128307	+	0.991734i	$0.540954\pi$
$108$	0	0
$109$	−13.8999	−1.33137	−0.665684	−	0.746234i	$-0.731860\pi$
$109$	−13.8999	−1.33137	−0.665684	+	0.746234i	$0.731860\pi$
$110$	0	0
$111$	6.56133	0.622774
$112$	0	0
$113$	−1.33693	−0.125767	−0.0628837	−	0.998021i	$-0.520030\pi$
$113$	−1.33693	−0.125767	−0.0628837	+	0.998021i	$0.520030\pi$
$114$	0	0
$115$	0.157144	0.0146537
$116$	0	0
$117$	−1.21823	−0.112625
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2.97668	0.270607
$122$	0	0
$123$	14.2324	1.28329
$124$	0	0
$125$	1.56756	0.140207
$126$	0	0
$127$	7.28380	0.646333	0.323167	−	0.946342i	$-0.395253\pi$
$127$	7.28380	0.646333	0.323167	+	0.946342i	$0.395253\pi$
$128$	0	0
$129$	−20.0220	−1.76284
$130$	0	0
$131$	5.80169	0.506896	0.253448	−	0.967349i	$-0.418435\pi$
$131$	5.80169	0.506896	0.253448	+	0.967349i	$0.418435\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.786955	−0.0677303
$136$	0	0
$137$	1.31490	0.112340	0.0561698	−	0.998421i	$-0.482111\pi$
$137$	1.31490	0.112340	0.0561698	+	0.998421i	$0.482111\pi$
$138$	0	0
$139$	7.65324	0.649140	0.324570	−	0.945862i	$-0.394780\pi$
$139$	7.65324	0.649140	0.324570	+	0.945862i	$0.394780\pi$
$140$	0	0
$141$	−0.149899	−0.0126237
$142$	0	0
$143$	22.6676	1.89556
$144$	0	0
$145$	0.941927	0.0782228
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−9.54176	−0.781691	−0.390846	−	0.920456i	$-0.627817\pi$
$149$	−9.54176	−0.781691	−0.390846	+	0.920456i	$0.627817\pi$
$150$	0	0
$151$	7.27069	0.591681	0.295840	−	0.955237i	$-0.404400\pi$
$151$	7.27069	0.591681	0.295840	+	0.955237i	$0.404400\pi$
$152$	0	0
$153$	1.42764	0.115418
$154$	0	0
$155$	0.207261	0.0166476
$156$	0	0
$157$	−19.8822	−1.58677	−0.793385	−	0.608720i	$-0.791683\pi$
$157$	−19.8822	−1.58677	−0.793385	+	0.608720i	$0.791683\pi$
$158$	0	0
$159$	17.2798	1.37038
$160$	0	0
$161$	0	0
$162$	0	0
$163$	13.9720	1.09437	0.547186	−	0.837011i	$-0.315699\pi$
$163$	13.9720	1.09437	0.547186	+	0.837011i	$0.315699\pi$
$164$	0	0
$165$	−1.05108	−0.0818267
$166$	0	0
$167$	16.2328	1.25613	0.628066	−	0.778160i	$-0.283847\pi$
$167$	16.2328	1.25613	0.628066	+	0.778160i	$0.283847\pi$
$168$	0	0
$169$	23.7626	1.82789
$170$	0	0
$171$	−0.945731	−0.0723218
$172$	0	0
$173$	−11.8556	−0.901361	−0.450680	−	0.892685i	$-0.648819\pi$
$173$	−11.8556	−0.901361	−0.450680	+	0.892685i	$0.648819\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	19.1279	1.43774
$178$	0	0
$179$	−22.9170	−1.71290	−0.856450	−	0.516230i	$-0.827335\pi$
$179$	−22.9170	−1.71290	−0.856450	+	0.516230i	$0.827335\pi$
$180$	0	0
$181$	7.87584	0.585407	0.292703	−	0.956203i	$-0.405445\pi$
$181$	7.87584	0.585407	0.292703	+	0.956203i	$0.405445\pi$
$182$	0	0
$183$	2.85488	0.211039
$184$	0	0
$185$	0.576304	0.0423707
$186$	0	0
$187$	−26.5641	−1.94256
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.13919	−0.0824292	−0.0412146	−	0.999150i	$-0.513123\pi$
$191$	−1.13919	−0.0824292	−0.0412146	+	0.999150i	$0.513123\pi$
$192$	0	0
$193$	21.7181	1.56330	0.781651	−	0.623716i	$-0.214378\pi$
$193$	21.7181	1.56330	0.781651	+	0.623716i	$0.214378\pi$
$194$	0	0
$195$	−1.70466	−0.122073
$196$	0	0
$197$	−14.0408	−1.00037	−0.500183	−	0.865920i	$-0.666734\pi$
$197$	−14.0408	−1.00037	−0.500183	+	0.865920i	$0.666734\pi$
$198$	0	0
$199$	−22.3129	−1.58172	−0.790859	−	0.611998i	$-0.790366\pi$
$199$	−22.3129	−1.58172	−0.790859	+	0.611998i	$0.790366\pi$
$200$	0	0
$201$	10.5425	0.743611
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.25008	0.0873092
$206$	0	0
$207$	−0.200921	−0.0139650
$208$	0	0
$209$	17.5972	1.21722
$210$	0	0
$211$	−28.4780	−1.96050	−0.980252	−	0.197751i	$-0.936636\pi$
$211$	−28.4780	−1.96050	−0.980252	+	0.197751i	$0.936636\pi$
$212$	0	0
$213$	12.5267	0.858315
$214$	0	0
$215$	−1.75860	−0.119936
$216$	0	0
$217$	0	0
$218$	0	0
$219$	9.37378	0.633421
$220$	0	0
$221$	−43.0820	−2.89801
$222$	0	0
$223$	−13.5164	−0.905123	−0.452561	−	0.891733i	$-0.649490\pi$
$223$	−13.5164	−0.905123	−0.452561	+	0.891733i	$0.649490\pi$
$224$	0	0
$225$	−0.999644	−0.0666430
$226$	0	0
$227$	−3.12228	−0.207233	−0.103617	−	0.994617i	$-0.533041\pi$
$227$	−3.12228	−0.207233	−0.103617	+	0.994617i	$0.533041\pi$
$228$	0	0
$229$	−12.2294	−0.808145	−0.404072	−	0.914727i	$-0.632406\pi$
$229$	−12.2294	−0.808145	−0.404072	+	0.914727i	$0.632406\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.9998	−1.37575	−0.687873	−	0.725831i	$-0.741455\pi$
$233$	−20.9998	−1.37575	−0.687873	+	0.725831i	$0.741455\pi$
$234$	0	0
$235$	−0.0131661	−0.000858862 0
$236$	0	0
$237$	28.3157	1.83930
$238$	0	0
$239$	15.0348	0.972519	0.486260	−	0.873814i	$-0.338361\pi$
$239$	15.0348	0.972519	0.486260	+	0.873814i	$0.338361\pi$
$240$	0	0
$241$	11.0061	0.708962	0.354481	−	0.935063i	$-0.384657\pi$
$241$	11.0061	0.708962	0.354481	+	0.935063i	$0.384657\pi$
$242$	0	0
$243$	2.08460	0.133727
$244$	0	0
$245$	0	0
$246$	0	0
$247$	28.5394	1.81592
$248$	0	0
$249$	−29.1718	−1.84869
$250$	0	0
$251$	5.24090	0.330803	0.165401	−	0.986226i	$-0.447108\pi$
$251$	5.24090	0.330803	0.165401	+	0.986226i	$0.447108\pi$
$252$	0	0
$253$	3.73854	0.235040
$254$	0	0
$255$	1.99769	0.125100
$256$	0	0
$257$	23.3282	1.45517	0.727586	−	0.686017i	$-0.240642\pi$
$257$	23.3282	1.45517	0.727586	+	0.686017i	$0.240642\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.20433	−0.0745462
$262$	0	0
$263$	−16.8914	−1.04157	−0.520783	−	0.853689i	$-0.674360\pi$
$263$	−16.8914	−1.04157	−0.520783	+	0.853689i	$0.674360\pi$
$264$	0	0
$265$	1.51774	0.0932342
$266$	0	0
$267$	6.36084	0.389277
$268$	0	0
$269$	25.6185	1.56199	0.780993	−	0.624540i	$-0.214714\pi$
$269$	25.6185	1.56199	0.780993	+	0.624540i	$0.214714\pi$
$270$	0	0
$271$	7.08245	0.430228	0.215114	−	0.976589i	$-0.430988\pi$
$271$	7.08245	0.430228	0.215114	+	0.976589i	$0.430988\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.6004	1.12165
$276$	0	0
$277$	2.88524	0.173357	0.0866786	−	0.996236i	$-0.472375\pi$
$277$	2.88524	0.173357	0.0866786	+	0.996236i	$0.472375\pi$
$278$	0	0
$279$	−0.264999	−0.0158651
$280$	0	0
$281$	6.33558	0.377949	0.188974	−	0.981982i	$-0.439484\pi$
$281$	6.33558	0.377949	0.188974	+	0.981982i	$0.439484\pi$
$282$	0	0
$283$	22.1433	1.31628	0.658142	−	0.752894i	$-0.271343\pi$
$283$	22.1433	1.31628	0.658142	+	0.752894i	$0.271343\pi$
$284$	0	0
$285$	−1.32336	−0.0783889
$286$	0	0
$287$	0	0
$288$	0	0
$289$	33.4877	1.96986
$290$	0	0
$291$	−26.1820	−1.53481
$292$	0	0
$293$	−29.3570	−1.71506	−0.857528	−	0.514438i	$-0.828001\pi$
$293$	−29.3570	−1.71506	−0.857528	+	0.514438i	$0.828001\pi$
$294$	0	0
$295$	1.68007	0.0978176
$296$	0	0
$297$	−18.7221	−1.08637
$298$	0	0
$299$	6.06322	0.350645
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−0.122036	−0.00701077
$304$	0	0
$305$	0.250754	0.0143581
$306$	0	0
$307$	9.24565	0.527677	0.263839	−	0.964567i	$-0.415011\pi$
$307$	9.24565	0.527677	0.263839	+	0.964567i	$0.415011\pi$
$308$	0	0
$309$	−23.5882	−1.34188
$310$	0	0
$311$	4.61229	0.261539	0.130770	−	0.991413i	$-0.458255\pi$
$311$	4.61229	0.261539	0.130770	+	0.991413i	$0.458255\pi$
$312$	0	0
$313$	−12.7187	−0.718902	−0.359451	−	0.933164i	$-0.617036\pi$
$313$	−12.7187	−0.718902	−0.359451	+	0.933164i	$0.617036\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.0292	0.619461	0.309730	−	0.950824i	$-0.399761\pi$
$317$	11.0292	0.619461	0.309730	+	0.950824i	$0.399761\pi$
$318$	0	0
$319$	22.4090	1.25466
$320$	0	0
$321$	4.74908	0.265068
$322$	0	0
$323$	−33.4453	−1.86094
$324$	0	0
$325$	30.1664	1.67333
$326$	0	0
$327$	24.8685	1.37523
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−14.3450	−0.788474	−0.394237	−	0.919009i	$-0.628991\pi$
$331$	−14.3450	−0.788474	−0.394237	+	0.919009i	$0.628991\pi$
$332$	0	0
$333$	−0.736852	−0.0403792
$334$	0	0
$335$	0.925984	0.0505919
$336$	0	0
$337$	−3.59148	−0.195640	−0.0978201	−	0.995204i	$-0.531187\pi$
$337$	−3.59148	−0.195640	−0.0978201	+	0.995204i	$0.531187\pi$
$338$	0	0
$339$	2.39191	0.129911
$340$	0	0
$341$	4.93084	0.267020
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.281148	−0.0151365
$346$	0	0
$347$	19.6213	1.05333	0.526664	−	0.850073i	$-0.323443\pi$
$347$	19.6213	1.05333	0.526664	+	0.850073i	$0.323443\pi$
$348$	0	0
$349$	34.1822	1.82973	0.914866	−	0.403757i	$-0.132296\pi$
$349$	34.1822	1.82973	0.914866	+	0.403757i	$0.132296\pi$
$350$	0	0
$351$	−30.3638	−1.62070
$352$	0	0
$353$	−6.40882	−0.341107	−0.170553	−	0.985348i	$-0.554556\pi$
$353$	−6.40882	−0.341107	−0.170553	+	0.985348i	$0.554556\pi$
$354$	0	0
$355$	1.10026	0.0583959
$356$	0	0
$357$	0	0
$358$	0	0
$359$	24.2541	1.28008	0.640040	−	0.768341i	$-0.278918\pi$
$359$	24.2541	1.28008	0.640040	+	0.768341i	$0.278918\pi$
$360$	0	0
$361$	3.15561	0.166085
$362$	0	0
$363$	−5.32562	−0.279522
$364$	0	0
$365$	0.823331	0.0430951
$366$	0	0
$367$	16.3835	0.855214	0.427607	−	0.903965i	$-0.359357\pi$
$367$	16.3835	0.855214	0.427607	+	0.903965i	$0.359357\pi$
$368$	0	0
$369$	−1.59832	−0.0832055
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−30.7718	−1.59331	−0.796653	−	0.604437i	$-0.793398\pi$
$373$	−30.7718	−1.59331	−0.796653	+	0.604437i	$0.793398\pi$
$374$	0	0
$375$	−2.80454	−0.144826
$376$	0	0
$377$	36.3432	1.87177
$378$	0	0
$379$	18.1017	0.929822	0.464911	−	0.885358i	$-0.346086\pi$
$379$	18.1017	0.929822	0.464911	+	0.885358i	$0.346086\pi$
$380$	0	0
$381$	−13.0315	−0.667626
$382$	0	0
$383$	−14.7212	−0.752218	−0.376109	−	0.926575i	$-0.622738\pi$
$383$	−14.7212	−0.752218	−0.376109	+	0.926575i	$0.622738\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.24852	0.114299
$388$	0	0
$389$	17.4293	0.883699	0.441850	−	0.897089i	$-0.354323\pi$
$389$	17.4293	0.883699	0.441850	+	0.897089i	$0.354323\pi$
$390$	0	0
$391$	−7.10547	−0.359339
$392$	0	0
$393$	−10.3799	−0.523596
$394$	0	0
$395$	2.48706	0.125138
$396$	0	0
$397$	8.43436	0.423308	0.211654	−	0.977345i	$-0.432115\pi$
$397$	8.43436	0.423308	0.211654	+	0.977345i	$0.432115\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−27.1008	−1.35335	−0.676674	−	0.736283i	$-0.736579\pi$
$401$	−27.1008	−1.35335	−0.676674	+	0.736283i	$0.736579\pi$
$402$	0	0
$403$	7.99691	0.398354
$404$	0	0
$405$	1.50267	0.0746684
$406$	0	0
$407$	13.7106	0.679609
$408$	0	0
$409$	−19.8788	−0.982942	−0.491471	−	0.870894i	$-0.663541\pi$
$409$	−19.8788	−0.982942	−0.491471	+	0.870894i	$0.663541\pi$
$410$	0	0
$411$	−2.35251	−0.116041
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−2.56226	−0.125776
$416$	0	0
$417$	−13.6925	−0.670525
$418$	0	0
$419$	32.9383	1.60914	0.804570	−	0.593858i	$-0.202396\pi$
$419$	32.9383	1.60914	0.804570	+	0.593858i	$0.202396\pi$
$420$	0	0
$421$	18.4653	0.899944	0.449972	−	0.893043i	$-0.351434\pi$
$421$	18.4653	0.899944	0.449972	+	0.893043i	$0.351434\pi$
$422$	0	0
$423$	0.0168339	0.000818494 0
$424$	0	0
$425$	−35.3519	−1.71482
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−40.5548	−1.95801
$430$	0	0
$431$	−2.78933	−0.134357	−0.0671787	−	0.997741i	$-0.521400\pi$
$431$	−2.78933	−0.134357	−0.0671787	+	0.997741i	$0.521400\pi$
$432$	0	0
$433$	−23.5930	−1.13381	−0.566904	−	0.823784i	$-0.691859\pi$
$433$	−23.5930	−1.13381	−0.566904	+	0.823784i	$0.691859\pi$
$434$	0	0
$435$	−1.68521	−0.0807998
$436$	0	0
$437$	4.70697	0.225165
$438$	0	0
$439$	1.45768	0.0695712	0.0347856	−	0.999395i	$-0.488925\pi$
$439$	1.45768	0.0695712	0.0347856	+	0.999395i	$0.488925\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.1104	0.575385	0.287692	−	0.957723i	$-0.407112\pi$
$443$	12.1104	0.575385	0.287692	+	0.957723i	$0.407112\pi$
$444$	0	0
$445$	0.558695	0.0264847
$446$	0	0
$447$	17.0713	0.807444
$448$	0	0
$449$	33.2562	1.56946	0.784729	−	0.619838i	$-0.212802\pi$
$449$	33.2562	1.56946	0.784729	+	0.619838i	$0.212802\pi$
$450$	0	0
$451$	29.7400	1.40040
$452$	0	0
$453$	−13.0081	−0.611173
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.3856	1.32782	0.663911	−	0.747812i	$-0.268896\pi$
$457$	28.3856	1.32782	0.663911	+	0.747812i	$0.268896\pi$
$458$	0	0
$459$	35.5832	1.66088
$460$	0	0
$461$	−35.1572	−1.63744	−0.818718	−	0.574196i	$-0.805315\pi$
$461$	−35.1572	−1.63744	−0.818718	+	0.574196i	$0.805315\pi$
$462$	0	0
$463$	7.69153	0.357456	0.178728	−	0.983899i	$-0.442802\pi$
$463$	7.69153	0.357456	0.178728	+	0.983899i	$0.442802\pi$
$464$	0	0
$465$	−0.370812	−0.0171960
$466$	0	0
$467$	−34.7129	−1.60632	−0.803160	−	0.595764i	$-0.796849\pi$
$467$	−34.7129	−1.60632	−0.803160	+	0.595764i	$0.796849\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	35.5714	1.63905
$472$	0	0
$473$	−41.8382	−1.92372
$474$	0	0
$475$	23.4186	1.07452
$476$	0	0
$477$	−1.94056	−0.0888520
$478$	0	0
$479$	−0.962923	−0.0439971	−0.0219985	−	0.999758i	$-0.507003\pi$
$479$	−0.962923	−0.0439971	−0.0219985	+	0.999758i	$0.507003\pi$
$480$	0	0
$481$	22.2360	1.01388
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.29965	−0.104422
$486$	0	0
$487$	10.4093	0.471691	0.235845	−	0.971791i	$-0.424214\pi$
$487$	10.4093	0.471691	0.235845	+	0.971791i	$0.424214\pi$
$488$	0	0
$489$	−24.9975	−1.13042
$490$	0	0
$491$	22.3508	1.00868	0.504338	−	0.863506i	$-0.331736\pi$
$491$	22.3508	1.00868	0.504338	+	0.863506i	$0.331736\pi$
$492$	0	0
$493$	−42.5905	−1.91818
$494$	0	0
$495$	0.118039	0.00530545
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.95539	0.221834	0.110917	−	0.993830i	$-0.464621\pi$
$499$	4.95539	0.221834	0.110917	+	0.993830i	$0.464621\pi$
$500$	0	0
$501$	−29.0423	−1.29751
$502$	0	0
$503$	25.6544	1.14387	0.571936	−	0.820298i	$-0.306192\pi$
$503$	25.6544	1.14387	0.571936	+	0.820298i	$0.306192\pi$
$504$	0	0
$505$	−0.0107188	−0.000476981 0
$506$	0	0
$507$	−42.5139	−1.88811
$508$	0	0
$509$	2.93131	0.129928	0.0649641	−	0.997888i	$-0.479307\pi$
$509$	2.93131	0.129928	0.0649641	+	0.997888i	$0.479307\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−23.5719	−1.04072
$514$	0	0
$515$	−2.07183	−0.0912957
$516$	0	0
$517$	−0.313229	−0.0137758
$518$	0	0
$519$	21.2109	0.931056
$520$	0	0
$521$	−7.08152	−0.310247	−0.155123	−	0.987895i	$-0.549578\pi$
$521$	−7.08152	−0.310247	−0.155123	+	0.987895i	$0.549578\pi$
$522$	0	0
$523$	37.1377	1.62392	0.811959	−	0.583715i	$-0.198401\pi$
$523$	37.1377	1.62392	0.811959	+	0.583715i	$0.198401\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.37156	−0.408231
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−2.14811	−0.0932200
$532$	0	0
$533$	48.2328	2.08919
$534$	0	0
$535$	0.417128	0.0180340
$536$	0	0
$537$	41.0011	1.76933
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−22.7670	−0.978829	−0.489415	−	0.872051i	$-0.662790\pi$
$541$	−22.7670	−0.978829	−0.489415	+	0.872051i	$0.662790\pi$
$542$	0	0
$543$	−14.0908	−0.604692
$544$	0	0
$545$	2.18428	0.0935644
$546$	0	0
$547$	−10.6171	−0.453956	−0.226978	−	0.973900i	$-0.572885\pi$
$547$	−10.6171	−0.453956	−0.226978	+	0.973900i	$0.572885\pi$
$548$	0	0
$549$	−0.320609	−0.0136833
$550$	0	0
$551$	28.2138	1.20195
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.03107	−0.0437666
$556$	0	0
$557$	−9.91369	−0.420057	−0.210028	−	0.977695i	$-0.567356\pi$
$557$	−9.91369	−0.420057	−0.210028	+	0.977695i	$0.567356\pi$
$558$	0	0
$559$	−67.8537	−2.86991
$560$	0	0
$561$	47.5261	2.00655
$562$	0	0
$563$	6.66672	0.280969	0.140484	−	0.990083i	$-0.455134\pi$
$563$	6.66672	0.280969	0.140484	+	0.990083i	$0.455134\pi$
$564$	0	0
$565$	0.210090	0.00883854
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−0.898819	−0.0376804	−0.0188402	−	0.999823i	$-0.505997\pi$
$569$	−0.898819	−0.0376804	−0.0188402	+	0.999823i	$0.505997\pi$
$570$	0	0
$571$	−33.8888	−1.41820	−0.709100	−	0.705108i	$-0.750899\pi$
$571$	−33.8888	−1.41820	−0.709100	+	0.705108i	$0.750899\pi$
$572$	0	0
$573$	2.03814	0.0851447
$574$	0	0
$575$	4.97531	0.207485
$576$	0	0
$577$	−29.3431	−1.22157	−0.610785	−	0.791796i	$-0.709146\pi$
$577$	−29.3431	−1.22157	−0.610785	+	0.791796i	$0.709146\pi$
$578$	0	0
$579$	−38.8561	−1.61480
$580$	0	0
$581$	0	0
$582$	0	0
$583$	36.1079	1.49544
$584$	0	0
$585$	0.191437	0.00791495
$586$	0	0
$587$	31.3187	1.29266	0.646330	−	0.763058i	$-0.276303\pi$
$587$	31.3187	1.29266	0.646330	+	0.763058i	$0.276303\pi$
$588$	0	0
$589$	6.20813	0.255802
$590$	0	0
$591$	25.1206	1.03332
$592$	0	0
$593$	−8.16258	−0.335197	−0.167598	−	0.985855i	$-0.553601\pi$
$593$	−8.16258	−0.335197	−0.167598	+	0.985855i	$0.553601\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	39.9202	1.63383
$598$	0	0
$599$	11.5153	0.470502	0.235251	−	0.971935i	$-0.424409\pi$
$599$	11.5153	0.470502	0.235251	+	0.971935i	$0.424409\pi$
$600$	0	0
$601$	18.6852	0.762183	0.381092	−	0.924537i	$-0.375548\pi$
$601$	18.6852	0.762183	0.381092	+	0.924537i	$0.375548\pi$
$602$	0	0
$603$	−1.18395	−0.0482140
$604$	0	0
$605$	−0.467767	−0.0190174
$606$	0	0
$607$	1.24980	0.0507277	0.0253638	−	0.999678i	$-0.491926\pi$
$607$	1.24980	0.0507277	0.0253638	+	0.999678i	$0.491926\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.507999	−0.0205514
$612$	0	0
$613$	42.4008	1.71255	0.856277	−	0.516517i	$-0.172772\pi$
$613$	42.4008	1.71255	0.856277	+	0.516517i	$0.172772\pi$
$614$	0	0
$615$	−2.23653	−0.0901855
$616$	0	0
$617$	−45.0066	−1.81190	−0.905949	−	0.423388i	$-0.860841\pi$
$617$	−45.0066	−1.81190	−0.905949	+	0.423388i	$0.860841\pi$
$618$	0	0
$619$	15.3147	0.615551	0.307776	−	0.951459i	$-0.400415\pi$
$619$	15.3147	0.615551	0.307776	+	0.951459i	$0.400415\pi$
$620$	0	0
$621$	−5.00787	−0.200959
$622$	0	0
$623$	0	0
$624$	0	0
$625$	24.6302	0.985208
$626$	0	0
$627$	−31.4834	−1.25733
$628$	0	0
$629$	−26.0584	−1.03901
$630$	0	0
$631$	−6.16116	−0.245272	−0.122636	−	0.992452i	$-0.539135\pi$
$631$	−6.16116	−0.245272	−0.122636	+	0.992452i	$0.539135\pi$
$632$	0	0
$633$	50.9503	2.02509
$634$	0	0
$635$	−1.14461	−0.0454223
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.40678	−0.0556512
$640$	0	0
$641$	19.3289	0.763445	0.381723	−	0.924277i	$-0.375331\pi$
$641$	19.3289	0.763445	0.381723	+	0.924277i	$0.375331\pi$
$642$	0	0
$643$	−11.0711	−0.436603	−0.218302	−	0.975881i	$-0.570052\pi$
$643$	−11.0711	−0.436603	−0.218302	+	0.975881i	$0.570052\pi$
$644$	0	0
$645$	3.14634	0.123887
$646$	0	0
$647$	25.7580	1.01265	0.506326	−	0.862342i	$-0.331003\pi$
$647$	25.7580	1.01265	0.506326	+	0.862342i	$0.331003\pi$
$648$	0	0
$649$	39.9699	1.56895
$650$	0	0
$651$	0	0
$652$	0	0
$653$	21.8116	0.853554	0.426777	−	0.904357i	$-0.359649\pi$
$653$	21.8116	0.853554	0.426777	+	0.904357i	$0.359649\pi$
$654$	0	0
$655$	−0.911701	−0.0356231
$656$	0	0
$657$	−1.05270	−0.0410696
$658$	0	0
$659$	2.87631	0.112045	0.0560225	−	0.998430i	$-0.482158\pi$
$659$	2.87631	0.112045	0.0560225	+	0.998430i	$0.482158\pi$
$660$	0	0
$661$	−19.4422	−0.756214	−0.378107	−	0.925762i	$-0.623425\pi$
$661$	−19.4422	−0.756214	−0.378107	+	0.925762i	$0.623425\pi$
$662$	0	0
$663$	77.0785	2.99348
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.99404	0.232090
$668$	0	0
$669$	24.1823	0.934941
$670$	0	0
$671$	5.96557	0.230298
$672$	0	0
$673$	−12.5891	−0.485276	−0.242638	−	0.970117i	$-0.578013\pi$
$673$	−12.5891	−0.485276	−0.242638	+	0.970117i	$0.578013\pi$
$674$	0	0
$675$	−24.9157	−0.959004
$676$	0	0
$677$	9.92016	0.381263	0.190631	−	0.981662i	$-0.438946\pi$
$677$	9.92016	0.381263	0.190631	+	0.981662i	$0.438946\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	5.58611	0.214060
$682$	0	0
$683$	−30.3140	−1.15993	−0.579967	−	0.814640i	$-0.696935\pi$
$683$	−30.3140	−1.15993	−0.579967	+	0.814640i	$0.696935\pi$
$684$	0	0
$685$	−0.206629	−0.00789487
$686$	0	0
$687$	21.8799	0.834768
$688$	0	0
$689$	58.5603	2.23097
$690$	0	0
$691$	7.90929	0.300884	0.150442	−	0.988619i	$-0.451930\pi$
$691$	7.90929	0.300884	0.150442	+	0.988619i	$0.451930\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.20266	−0.0456195
$696$	0	0
$697$	−56.5239	−2.14099
$698$	0	0
$699$	37.5711	1.42107
$700$	0	0
$701$	−26.7137	−1.00896	−0.504481	−	0.863423i	$-0.668316\pi$
$701$	−26.7137	−1.00896	−0.504481	+	0.863423i	$0.668316\pi$
$702$	0	0
$703$	17.2622	0.651057
$704$	0	0
$705$	0.0235556	0.000887157 0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	5.47434	0.205593	0.102797	−	0.994702i	$-0.467221\pi$
$709$	5.47434	0.205593	0.102797	+	0.994702i	$0.467221\pi$
$710$	0	0
$711$	−3.17991	−0.119256
$712$	0	0
$713$	1.31892	0.0493940
$714$	0	0
$715$	−3.56207	−0.133214
$716$	0	0
$717$	−26.8989	−1.00456
$718$	0	0
$719$	47.4949	1.77126	0.885631	−	0.464391i	$-0.153727\pi$
$719$	47.4949	1.77126	0.885631	+	0.464391i	$0.153727\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−19.6911	−0.732319
$724$	0	0
$725$	29.8222	1.10757
$726$	0	0
$727$	−8.04421	−0.298343	−0.149172	−	0.988811i	$-0.547661\pi$
$727$	−8.04421	−0.298343	−0.149172	+	0.988811i	$0.547661\pi$
$728$	0	0
$729$	24.9576	0.924356
$730$	0	0
$731$	79.5176	2.94106
$732$	0	0
$733$	41.0539	1.51636	0.758179	−	0.652046i	$-0.226089\pi$
$733$	41.0539	1.51636	0.758179	+	0.652046i	$0.226089\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	22.0297	0.811474
$738$	0	0
$739$	12.5093	0.460162	0.230081	−	0.973172i	$-0.426101\pi$
$739$	12.5093	0.460162	0.230081	+	0.973172i	$0.426101\pi$
$740$	0	0
$741$	−51.0602	−1.87574
$742$	0	0
$743$	9.65904	0.354356	0.177178	−	0.984179i	$-0.443303\pi$
$743$	9.65904	0.354356	0.177178	+	0.984179i	$0.443303\pi$
$744$	0	0
$745$	1.49943	0.0549348
$746$	0	0
$747$	3.27605	0.119865
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−7.91262	−0.288736	−0.144368	−	0.989524i	$-0.546115\pi$
$751$	−7.91262	−0.288736	−0.144368	+	0.989524i	$0.546115\pi$
$752$	0	0
$753$	−9.37656	−0.341701
$754$	0	0
$755$	−1.14254	−0.0415815
$756$	0	0
$757$	19.1885	0.697419	0.348709	−	0.937231i	$-0.386620\pi$
$757$	19.1885	0.697419	0.348709	+	0.937231i	$0.386620\pi$
$758$	0	0
$759$	−6.68867	−0.242783
$760$	0	0
$761$	12.9618	0.469864	0.234932	−	0.972012i	$-0.424513\pi$
$761$	12.9618	0.469864	0.234932	+	0.972012i	$0.424513\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−0.224345	−0.00811120
$766$	0	0
$767$	64.8237	2.34065
$768$	0	0
$769$	34.3174	1.23752	0.618758	−	0.785581i	$-0.287636\pi$
$769$	34.3174	1.23752	0.618758	+	0.785581i	$0.287636\pi$
$770$	0	0
$771$	−41.7367	−1.50311
$772$	0	0
$773$	−28.2844	−1.01732	−0.508660	−	0.860967i	$-0.669859\pi$
$773$	−28.2844	−1.01732	−0.508660	+	0.860967i	$0.669859\pi$
$774$	0	0
$775$	6.56204	0.235715
$776$	0	0
$777$	0	0
$778$	0	0
$779$	37.4439	1.34157
$780$	0	0
$781$	26.1759	0.936646
$782$	0	0
$783$	−30.0174	−1.07273
$784$	0	0
$785$	3.12436	0.111513
$786$	0	0
$787$	−16.5819	−0.591079	−0.295540	−	0.955330i	$-0.595499\pi$
$787$	−16.5819	−0.591079	−0.295540	+	0.955330i	$0.595499\pi$
$788$	0	0
$789$	30.2205	1.07588
$790$	0	0
$791$	0	0
$792$	0	0
$793$	9.67504	0.343571
$794$	0	0
$795$	−2.71541	−0.0963057
$796$	0	0
$797$	28.9430	1.02521	0.512606	−	0.858624i	$-0.328680\pi$
$797$	28.9430	1.02521	0.512606	+	0.858624i	$0.328680\pi$
$798$	0	0
$799$	0.595323	0.0210610
$800$	0	0
$801$	−0.714337	−0.0252398
$802$	0	0
$803$	19.5875	0.691228
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−45.8343	−1.61344
$808$	0	0
$809$	27.2783	0.959054	0.479527	−	0.877527i	$-0.340808\pi$
$809$	27.2783	0.959054	0.479527	+	0.877527i	$0.340808\pi$
$810$	0	0
$811$	7.55589	0.265323	0.132662	−	0.991161i	$-0.457648\pi$
$811$	7.55589	0.265323	0.132662	+	0.991161i	$0.457648\pi$
$812$	0	0
$813$	−12.6713	−0.444402
$814$	0	0
$815$	−2.19561	−0.0769090
$816$	0	0
$817$	−52.6760	−1.84290
$818$	0	0
$819$	0	0
$820$	0	0
$821$	49.2867	1.72012	0.860059	−	0.510194i	$-0.170427\pi$
$821$	49.2867	1.72012	0.860059	+	0.510194i	$0.170427\pi$
$822$	0	0
$823$	−38.9855	−1.35895	−0.679475	−	0.733699i	$-0.737792\pi$
$823$	−38.9855	−1.35895	−0.679475	+	0.733699i	$0.737792\pi$
$824$	0	0
$825$	−33.2782	−1.15860
$826$	0	0
$827$	4.02528	0.139973	0.0699864	−	0.997548i	$-0.477704\pi$
$827$	4.02528	0.139973	0.0699864	+	0.997548i	$0.477704\pi$
$828$	0	0
$829$	−33.1145	−1.15011	−0.575056	−	0.818114i	$-0.695020\pi$
$829$	−33.1145	−1.15011	−0.575056	+	0.818114i	$0.695020\pi$
$830$	0	0
$831$	−5.16202	−0.179068
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−2.55088	−0.0882770
$836$	0	0
$837$	−6.60498	−0.228302
$838$	0	0
$839$	−37.7204	−1.30225	−0.651127	−	0.758969i	$-0.725703\pi$
$839$	−37.7204	−1.30225	−0.651127	+	0.758969i	$0.725703\pi$
$840$	0	0
$841$	6.92856	0.238916
$842$	0	0
$843$	−11.3351	−0.390400
$844$	0	0
$845$	−3.73414	−0.128458
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−39.6169	−1.35965
$850$	0	0
$851$	3.66737	0.125716
$852$	0	0
$853$	−11.0242	−0.377461	−0.188730	−	0.982029i	$-0.560437\pi$
$853$	−11.0242	−0.377461	−0.188730	+	0.982029i	$0.560437\pi$
$854$	0	0
$855$	0.148616	0.00508255
$856$	0	0
$857$	−10.8741	−0.371453	−0.185727	−	0.982601i	$-0.559464\pi$
$857$	−10.8741	−0.371453	−0.185727	+	0.982601i	$0.559464\pi$
$858$	0	0
$859$	−29.4639	−1.00530	−0.502648	−	0.864491i	$-0.667641\pi$
$859$	−29.4639	−1.00530	−0.502648	+	0.864491i	$0.667641\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	41.8112	1.42327	0.711635	−	0.702549i	$-0.247955\pi$
$863$	41.8112	1.42327	0.711635	+	0.702549i	$0.247955\pi$
$864$	0	0
$865$	1.86303	0.0633448
$866$	0	0
$867$	−59.9132	−2.03476
$868$	0	0
$869$	59.1686	2.00716
$870$	0	0
$871$	35.7280	1.21060
$872$	0	0
$873$	2.94029	0.0995139
$874$	0	0
$875$	0	0
$876$	0	0
$877$	28.0049	0.945658	0.472829	−	0.881154i	$-0.343233\pi$
$877$	28.0049	0.945658	0.472829	+	0.881154i	$0.343233\pi$
$878$	0	0
$879$	52.5230	1.77156
$880$	0	0
$881$	−9.20360	−0.310077	−0.155039	−	0.987908i	$-0.549550\pi$
$881$	−9.20360	−0.310077	−0.155039	+	0.987908i	$0.549550\pi$
$882$	0	0
$883$	−41.1547	−1.38496	−0.692482	−	0.721435i	$-0.743483\pi$
$883$	−41.1547	−1.38496	−0.692482	+	0.721435i	$0.743483\pi$
$884$	0	0
$885$	−3.00584	−0.101040
$886$	0	0
$887$	28.1198	0.944169	0.472085	−	0.881553i	$-0.343502\pi$
$887$	28.1198	0.944169	0.472085	+	0.881553i	$0.343502\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	35.7494	1.19765
$892$	0	0
$893$	−0.394368	−0.0131970
$894$	0	0
$895$	3.60127	0.120377
$896$	0	0
$897$	−10.8478	−0.362197
$898$	0	0
$899$	7.90568	0.263669
$900$	0	0
$901$	−68.6267	−2.28629
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.23764	−0.0411405
$906$	0	0
$907$	18.2215	0.605036	0.302518	−	0.953144i	$-0.402173\pi$
$907$	18.2215	0.605036	0.302518	+	0.953144i	$0.402173\pi$
$908$	0	0
$909$	0.0137049	0.000454562 0
$910$	0	0
$911$	−3.82547	−0.126744	−0.0633718	−	0.997990i	$-0.520185\pi$
$911$	−3.82547	−0.126744	−0.0633718	+	0.997990i	$0.520185\pi$
$912$	0	0
$913$	−60.9575	−2.01740
$914$	0	0
$915$	−0.448626	−0.0148311
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−31.0493	−1.02422	−0.512112	−	0.858919i	$-0.671137\pi$
$919$	−31.0493	−1.02422	−0.512112	+	0.858919i	$0.671137\pi$
$920$	0	0
$921$	−16.5415	−0.545061
$922$	0	0
$923$	42.4524	1.39734
$924$	0	0
$925$	18.2463	0.599934
$926$	0	0
$927$	2.64900	0.0870046
$928$	0	0
$929$	−21.2949	−0.698664	−0.349332	−	0.936999i	$-0.613591\pi$
$929$	−21.2949	−0.698664	−0.349332	+	0.936999i	$0.613591\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−8.25190	−0.270155
$934$	0	0
$935$	4.17438	0.136517
$936$	0	0
$937$	−9.72057	−0.317557	−0.158779	−	0.987314i	$-0.550756\pi$
$937$	−9.72057	−0.317557	−0.158779	+	0.987314i	$0.550756\pi$
$938$	0	0
$939$	22.7551	0.742586
$940$	0	0
$941$	−0.671851	−0.0219017	−0.0109509	−	0.999940i	$-0.503486\pi$
$941$	−0.671851	−0.0219017	−0.0109509	+	0.999940i	$0.503486\pi$
$942$	0	0
$943$	7.95498	0.259050
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−0.364992	−0.0118607	−0.00593033	−	0.999982i	$-0.501888\pi$
$947$	−0.364992	−0.0118607	−0.00593033	+	0.999982i	$0.501888\pi$
$948$	0	0
$949$	31.7673	1.03121
$950$	0	0
$951$	−19.7325	−0.639869
$952$	0	0
$953$	−12.6450	−0.409612	−0.204806	−	0.978803i	$-0.565656\pi$
$953$	−12.6450	−0.409612	−0.204806	+	0.978803i	$0.565656\pi$
$954$	0	0
$955$	0.179017	0.00579286
$956$	0	0
$957$	−40.0922	−1.29600
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.2604	−0.943885
$962$	0	0
$963$	−0.533333	−0.0171864
$964$	0	0
$965$	−3.41287	−0.109864
$966$	0	0
$967$	19.9186	0.640539	0.320269	−	0.947326i	$-0.396227\pi$
$967$	19.9186	0.640539	0.320269	+	0.947326i	$0.396227\pi$
$968$	0	0
$969$	59.8373	1.92225
$970$	0	0
$971$	11.8342	0.379777	0.189888	−	0.981806i	$-0.439187\pi$
$971$	11.8342	0.379777	0.189888	+	0.981806i	$0.439187\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−53.9710	−1.72845
$976$	0	0
$977$	20.7961	0.665325	0.332663	−	0.943046i	$-0.392053\pi$
$977$	20.7961	0.665325	0.332663	+	0.943046i	$0.392053\pi$
$978$	0	0
$979$	13.2917	0.424803
$980$	0	0
$981$	−2.79278	−0.0891667
$982$	0	0
$983$	−10.2251	−0.326131	−0.163065	−	0.986615i	$-0.552138\pi$
$983$	−10.2251	−0.326131	−0.163065	+	0.986615i	$0.552138\pi$
$984$	0	0
$985$	2.20643	0.0703026
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11.1910	−0.355855
$990$	0	0
$991$	−36.8031	−1.16909	−0.584544	−	0.811362i	$-0.698727\pi$
$991$	−36.8031	−1.16909	−0.584544	+	0.811362i	$0.698727\pi$
$992$	0	0
$993$	25.6649	0.814450
$994$	0	0
$995$	3.50633	0.111158
$996$	0	0
$997$	−6.22140	−0.197034	−0.0985169	−	0.995135i	$-0.531410\pi$
$997$	−6.22140	−0.197034	−0.0985169	+	0.995135i	$0.531410\pi$
$998$	0	0
$999$	−18.3657	−0.581065

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9016.2.a.br.1.3		11
7.3	odd	6		1288.2.q.d.737.3	✓	22
7.5	odd	6		1288.2.q.d.921.3	yes	22
7.6	odd	2		9016.2.a.bk.1.9		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1288.2.q.d.737.3	✓	22	7.3	odd	6
1288.2.q.d.921.3	yes	22	7.5	odd	6
9016.2.a.bk.1.9		11	7.6	odd	2
9016.2.a.br.1.3		11	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:	\( N \)	\(=\)	\( 9016 = 2^{3} \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.78911 q^{3} -0.157144 q^{5} +0.200921 q^{9} +O(q^{10})\)	\(q-1.78911 q^{3} -0.157144 q^{5} +0.200921 q^{9} -3.73854 q^{11} -6.06322 q^{13} +0.281148 q^{15} +7.10547 q^{17} -4.70697 q^{19} -1.00000 q^{23} -4.97531 q^{25} +5.00787 q^{27} -5.99404 q^{29} -1.31892 q^{31} +6.68867 q^{33} -3.66737 q^{37} +10.8478 q^{39} -7.95498 q^{41} +11.1910 q^{43} -0.0315735 q^{45} +0.0837838 q^{47} -12.7125 q^{51} -9.65830 q^{53} +0.587489 q^{55} +8.42130 q^{57} -10.6913 q^{59} -1.59569 q^{61} +0.952797 q^{65} -5.89259 q^{67} +1.78911 q^{69} -7.00163 q^{71} -5.23935 q^{73} +8.90138 q^{75} -15.8267 q^{79} -9.56239 q^{81} +16.3052 q^{83} -1.11658 q^{85} +10.7240 q^{87} -3.55531 q^{89} +2.35970 q^{93} +0.739672 q^{95} +14.6341 q^{97} -0.751152 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 4 q^{3} + 3 q^{5} + 11 q^{9}+O(q^{10}) \) 11 * q + 4 * q^3 + 3 * q^5 + 11 * q^9	\( 11 q + 4 q^{3} + 3 q^{5} + 11 q^{9} - 13 q^{13} + 7 q^{17} + 8 q^{19} - 11 q^{23} + 6 q^{25} + 25 q^{27} - 3 q^{29} + 12 q^{31} - 2 q^{33} - q^{37} - 21 q^{39} + 12 q^{41} + 9 q^{43} + 19 q^{45} + 17 q^{47} + 19 q^{51} - 5 q^{53} + 21 q^{55} + 11 q^{57} + 33 q^{59} - 15 q^{61} - 9 q^{65} - 5 q^{67} - 4 q^{69} - 9 q^{71} + 5 q^{73} + 44 q^{75} + 11 q^{79} - 13 q^{81} + 51 q^{83} + 33 q^{85} + 4 q^{87} + 26 q^{89} + 6 q^{93} - 19 q^{95} + 21 q^{97} + 15 q^{99}+O(q^{100}) \) 11 * q + 4 * q^3 + 3 * q^5 + 11 * q^9 - 13 * q^13 + 7 * q^17 + 8 * q^19 - 11 * q^23 + 6 * q^25 + 25 * q^27 - 3 * q^29 + 12 * q^31 - 2 * q^33 - q^37 - 21 * q^39 + 12 * q^41 + 9 * q^43 + 19 * q^45 + 17 * q^47 + 19 * q^51 - 5 * q^53 + 21 * q^55 + 11 * q^57 + 33 * q^59 - 15 * q^61 - 9 * q^65 - 5 * q^67 - 4 * q^69 - 9 * q^71 + 5 * q^73 + 44 * q^75 + 11 * q^79 - 13 * q^81 + 51 * q^83 + 33 * q^85 + 4 * q^87 + 26 * q^89 + 6 * q^93 - 19 * q^95 + 21 * q^97 + 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more