LMFDB - Embedded newform 8464.2.a.bv.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8464,2,Mod(1,8464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8464, 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("8464.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.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:	$67.5853802708$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 46)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.830830$ of defining polynomial
Character	$\chi$	$=$	8464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.609264 q^{3} -0.372786 q^{5} +2.05954 q^{7} -2.62880 q^{9} +O(q^{10})$	$q+0.609264 q^{3} -0.372786 q^{5} +2.05954 q^{7} -2.62880 q^{9} +2.46519 q^{11} -3.73269 q^{13} -0.227125 q^{15} +6.30909 q^{17} -3.28887 q^{19} +1.25480 q^{21} -4.86103 q^{25} -3.42943 q^{27} +7.13278 q^{29} -5.84511 q^{31} +1.50195 q^{33} -0.767766 q^{35} +4.67961 q^{37} -2.27420 q^{39} -4.78084 q^{41} -11.1379 q^{43} +0.979978 q^{45} -2.44502 q^{47} -2.75831 q^{49} +3.84391 q^{51} -6.04752 q^{53} -0.918986 q^{55} -2.00379 q^{57} +11.8100 q^{59} -1.68675 q^{61} -5.41411 q^{63} +1.39149 q^{65} -14.0947 q^{67} -4.03869 q^{71} +8.69002 q^{73} -2.96165 q^{75} +5.07714 q^{77} -2.38398 q^{79} +5.79696 q^{81} +4.23705 q^{83} -2.35194 q^{85} +4.34575 q^{87} +9.03115 q^{89} -7.68762 q^{91} -3.56122 q^{93} +1.22605 q^{95} +11.3067 q^{97} -6.48047 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{5} - 9 q^{7} + 7 q^{9}+O(q^{10}) $ 5 * q + 2 * q^5 - 9 * q^7 + 7 * q^9	$ 5 q + 2 q^{5} - 9 q^{7} + 7 q^{9} - q^{11} + q^{13} - 11 q^{15} - q^{17} - 10 q^{19} - 11 q^{25} + 10 q^{29} + 11 q^{33} + 3 q^{35} + 20 q^{37} + 22 q^{39} - 3 q^{41} - 27 q^{43} + 5 q^{45} + 11 q^{47} + 12 q^{49} - 2 q^{53} + 4 q^{55} + 11 q^{57} + 17 q^{59} + 7 q^{61} - 39 q^{63} - 15 q^{65} - 28 q^{67} + 8 q^{73} - 22 q^{75} + 26 q^{77} - 16 q^{79} + q^{81} - 18 q^{83} - 7 q^{85} - 22 q^{87} - 14 q^{89} - 26 q^{91} - 44 q^{93} - 26 q^{95} + 47 q^{97} - 8 q^{99}+O(q^{100}) $ 5 * q + 2 * q^5 - 9 * q^7 + 7 * q^9 - q^11 + q^13 - 11 * q^15 - q^17 - 10 * q^19 - 11 * q^25 + 10 * q^29 + 11 * q^33 + 3 * q^35 + 20 * q^37 + 22 * q^39 - 3 * q^41 - 27 * q^43 + 5 * q^45 + 11 * q^47 + 12 * q^49 - 2 * q^53 + 4 * q^55 + 11 * q^57 + 17 * q^59 + 7 * q^61 - 39 * q^63 - 15 * q^65 - 28 * q^67 + 8 * q^73 - 22 * q^75 + 26 * q^77 - 16 * q^79 + q^81 - 18 * q^83 - 7 * q^85 - 22 * q^87 - 14 * q^89 - 26 * q^91 - 44 * q^93 - 26 * q^95 + 47 * q^97 - 8 * 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
$3$	0.609264	0.351759	0.175880	−	0.984412i	$-0.443723\pi$
$3$	0.609264	0.351759	0.175880	+	0.984412i	$0.443723\pi$
$4$	0	0
$5$	−0.372786	−0.166715	−0.0833574	−	0.996520i	$-0.526564\pi$
$5$	−0.372786	−0.166715	−0.0833574	+	0.996520i	$0.526564\pi$
$6$	0	0
$7$	2.05954	0.778432	0.389216	−	0.921147i	$-0.372746\pi$
$7$	2.05954	0.778432	0.389216	+	0.921147i	$0.372746\pi$
$8$	0	0
$9$	−2.62880	−0.876266
$10$	0	0
$11$	2.46519	0.743282	0.371641	−	0.928377i	$-0.378795\pi$
$11$	2.46519	0.743282	0.371641	+	0.928377i	$0.378795\pi$
$12$	0	0
$13$	−3.73269	−1.03526	−0.517631	−	0.855604i	$-0.673186\pi$
$13$	−3.73269	−1.03526	−0.517631	+	0.855604i	$0.673186\pi$
$14$	0	0
$15$	−0.227125	−0.0586434
$16$	0	0
$17$	6.30909	1.53018	0.765090	−	0.643924i	$-0.222695\pi$
$17$	6.30909	1.53018	0.765090	+	0.643924i	$0.222695\pi$
$18$	0	0
$19$	−3.28887	−0.754520	−0.377260	−	0.926107i	$-0.623134\pi$
$19$	−3.28887	−0.754520	−0.377260	+	0.926107i	$0.623134\pi$
$20$	0	0
$21$	1.25480	0.273820
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−4.86103	−0.972206
$26$	0	0
$27$	−3.42943	−0.659993
$28$	0	0
$29$	7.13278	1.32452	0.662262	−	0.749272i	$-0.269597\pi$
$29$	7.13278	1.32452	0.662262	+	0.749272i	$0.269597\pi$
$30$	0	0
$31$	−5.84511	−1.04981	−0.524907	−	0.851160i	$-0.675900\pi$
$31$	−5.84511	−1.04981	−0.524907	+	0.851160i	$0.675900\pi$
$32$	0	0
$33$	1.50195	0.261456
$34$	0	0
$35$	−0.767766	−0.129776
$36$	0	0
$37$	4.67961	0.769323	0.384662	−	0.923058i	$-0.374318\pi$
$37$	4.67961	0.769323	0.384662	+	0.923058i	$0.374318\pi$
$38$	0	0
$39$	−2.27420	−0.364163
$40$	0	0
$41$	−4.78084	−0.746642	−0.373321	−	0.927702i	$-0.621781\pi$
$41$	−4.78084	−0.746642	−0.373321	+	0.927702i	$0.621781\pi$
$42$	0	0
$43$	−11.1379	−1.69851	−0.849256	−	0.527981i	$-0.822949\pi$
$43$	−11.1379	−1.69851	−0.849256	+	0.527981i	$0.822949\pi$
$44$	0	0
$45$	0.979978	0.146086
$46$	0	0
$47$	−2.44502	−0.356643	−0.178322	−	0.983972i	$-0.557067\pi$
$47$	−2.44502	−0.356643	−0.178322	+	0.983972i	$0.557067\pi$
$48$	0	0
$49$	−2.75831	−0.394044
$50$	0	0
$51$	3.84391	0.538254
$52$	0	0
$53$	−6.04752	−0.830691	−0.415345	−	0.909664i	$-0.636339\pi$
$53$	−6.04752	−0.830691	−0.415345	+	0.909664i	$0.636339\pi$
$54$	0	0
$55$	−0.918986	−0.123916
$56$	0	0
$57$	−2.00379	−0.265409
$58$	0	0
$59$	11.8100	1.53754	0.768768	−	0.639528i	$-0.220870\pi$
$59$	11.8100	1.53754	0.768768	+	0.639528i	$0.220870\pi$
$60$	0	0
$61$	−1.68675	−0.215966	−0.107983	−	0.994153i	$-0.534439\pi$
$61$	−1.68675	−0.215966	−0.107983	+	0.994153i	$0.534439\pi$
$62$	0	0
$63$	−5.41411	−0.682113
$64$	0	0
$65$	1.39149	0.172594
$66$	0	0
$67$	−14.0947	−1.72194	−0.860969	−	0.508658i	$-0.830142\pi$
$67$	−14.0947	−1.72194	−0.860969	+	0.508658i	$0.830142\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.03869	−0.479304	−0.239652	−	0.970859i	$-0.577033\pi$
$71$	−4.03869	−0.479304	−0.239652	+	0.970859i	$0.577033\pi$
$72$	0	0
$73$	8.69002	1.01709	0.508545	−	0.861035i	$-0.330183\pi$
$73$	8.69002	1.01709	0.508545	+	0.861035i	$0.330183\pi$
$74$	0	0
$75$	−2.96165	−0.341982
$76$	0	0
$77$	5.07714	0.578594
$78$	0	0
$79$	−2.38398	−0.268218	−0.134109	−	0.990967i	$-0.542817\pi$
$79$	−2.38398	−0.268218	−0.134109	+	0.990967i	$0.542817\pi$
$80$	0	0
$81$	5.79696	0.644107
$82$	0	0
$83$	4.23705	0.465076	0.232538	−	0.972587i	$-0.425297\pi$
$83$	4.23705	0.465076	0.232538	+	0.972587i	$0.425297\pi$
$84$	0	0
$85$	−2.35194	−0.255104
$86$	0	0
$87$	4.34575	0.465913
$88$	0	0
$89$	9.03115	0.957300	0.478650	−	0.878006i	$-0.341126\pi$
$89$	9.03115	0.957300	0.478650	+	0.878006i	$0.341126\pi$
$90$	0	0
$91$	−7.68762	−0.805881
$92$	0	0
$93$	−3.56122	−0.369281
$94$	0	0
$95$	1.22605	0.125790
$96$	0	0
$97$	11.3067	1.14802	0.574010	−	0.818848i	$-0.305387\pi$
$97$	11.3067	1.14802	0.574010	+	0.818848i	$0.305387\pi$
$98$	0	0
$99$	−6.48047	−0.651312
$100$	0	0
$101$	−8.05177	−0.801181	−0.400590	−	0.916257i	$-0.631195\pi$
$101$	−8.05177	−0.801181	−0.400590	+	0.916257i	$0.631195\pi$
$102$	0	0
$103$	−14.7518	−1.45354	−0.726771	−	0.686879i	$-0.758980\pi$
$103$	−14.7518	−1.45354	−0.726771	+	0.686879i	$0.758980\pi$
$104$	0	0
$105$	−0.467772	−0.0456499
$106$	0	0
$107$	−6.68158	−0.645932	−0.322966	−	0.946411i	$-0.604680\pi$
$107$	−6.68158	−0.645932	−0.322966	+	0.946411i	$0.604680\pi$
$108$	0	0
$109$	4.07970	0.390764	0.195382	−	0.980727i	$-0.437405\pi$
$109$	4.07970	0.390764	0.195382	+	0.980727i	$0.437405\pi$
$110$	0	0
$111$	2.85112	0.270616
$112$	0	0
$113$	−5.58523	−0.525414	−0.262707	−	0.964876i	$-0.584615\pi$
$113$	−5.58523	−0.525414	−0.262707	+	0.964876i	$0.584615\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	9.81249	0.907165
$118$	0	0
$119$	12.9938	1.19114
$120$	0	0
$121$	−4.92286	−0.447532
$122$	0	0
$123$	−2.91280	−0.262638
$124$	0	0
$125$	3.67605	0.328796
$126$	0	0
$127$	7.54884	0.669851	0.334926	−	0.942245i	$-0.391289\pi$
$127$	7.54884	0.669851	0.334926	+	0.942245i	$0.391289\pi$
$128$	0	0
$129$	−6.78592	−0.597467
$130$	0	0
$131$	3.27067	0.285760	0.142880	−	0.989740i	$-0.454364\pi$
$131$	3.27067	0.285760	0.142880	+	0.989740i	$0.454364\pi$
$132$	0	0
$133$	−6.77356	−0.587342
$134$	0	0
$135$	1.27844	0.110031
$136$	0	0
$137$	−15.9297	−1.36096	−0.680482	−	0.732765i	$-0.738229\pi$
$137$	−15.9297	−1.36096	−0.680482	+	0.732765i	$0.738229\pi$
$138$	0	0
$139$	2.85447	0.242113	0.121056	−	0.992646i	$-0.461372\pi$
$139$	2.85447	0.242113	0.121056	+	0.992646i	$0.461372\pi$
$140$	0	0
$141$	−1.48967	−0.125453
$142$	0	0
$143$	−9.20178	−0.769491
$144$	0	0
$145$	−2.65900	−0.220818
$146$	0	0
$147$	−1.68054	−0.138608
$148$	0	0
$149$	2.91873	0.239112	0.119556	−	0.992827i	$-0.461853\pi$
$149$	2.91873	0.239112	0.119556	+	0.992827i	$0.461853\pi$
$150$	0	0
$151$	−24.5622	−1.99884	−0.999421	−	0.0340202i	$-0.989169\pi$
$151$	−24.5622	−1.99884	−0.999421	+	0.0340202i	$0.989169\pi$
$152$	0	0
$153$	−16.5853	−1.34084
$154$	0	0
$155$	2.17897	0.175019
$156$	0	0
$157$	13.1045	1.04585	0.522927	−	0.852378i	$-0.324840\pi$
$157$	13.1045	1.04585	0.522927	+	0.852378i	$0.324840\pi$
$158$	0	0
$159$	−3.68454	−0.292203
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−4.61907	−0.361793	−0.180897	−	0.983502i	$-0.557900\pi$
$163$	−4.61907	−0.361793	−0.180897	+	0.983502i	$0.557900\pi$
$164$	0	0
$165$	−0.559905	−0.0435886
$166$	0	0
$167$	3.93949	0.304847	0.152424	−	0.988315i	$-0.451292\pi$
$167$	3.93949	0.304847	0.152424	+	0.988315i	$0.451292\pi$
$168$	0	0
$169$	0.932979	0.0717677
$170$	0	0
$171$	8.64578	0.661160
$172$	0	0
$173$	19.0007	1.44460	0.722298	−	0.691582i	$-0.243086\pi$
$173$	19.0007	1.44460	0.722298	+	0.691582i	$0.243086\pi$
$174$	0	0
$175$	−10.0115	−0.756796
$176$	0	0
$177$	7.19544	0.540842
$178$	0	0
$179$	−2.39943	−0.179342	−0.0896708	−	0.995971i	$-0.528581\pi$
$179$	−2.39943	−0.179342	−0.0896708	+	0.995971i	$0.528581\pi$
$180$	0	0
$181$	−26.0063	−1.93304	−0.966518	−	0.256600i	$-0.917398\pi$
$181$	−26.0063	−1.93304	−0.966518	+	0.256600i	$0.917398\pi$
$182$	0	0
$183$	−1.02768	−0.0759681
$184$	0	0
$185$	−1.74449	−0.128258
$186$	0	0
$187$	15.5531	1.13735
$188$	0	0
$189$	−7.06303	−0.513760
$190$	0	0
$191$	−10.4691	−0.757521	−0.378760	−	0.925495i	$-0.623650\pi$
$191$	−10.4691	−0.757521	−0.378760	+	0.925495i	$0.623650\pi$
$192$	0	0
$193$	0.771242	0.0555152	0.0277576	−	0.999615i	$-0.491163\pi$
$193$	0.771242	0.0555152	0.0277576	+	0.999615i	$0.491163\pi$
$194$	0	0
$195$	0.847787	0.0607113
$196$	0	0
$197$	5.20731	0.371005	0.185503	−	0.982644i	$-0.440609\pi$
$197$	5.20731	0.371005	0.185503	+	0.982644i	$0.440609\pi$
$198$	0	0
$199$	−23.3427	−1.65472	−0.827361	−	0.561670i	$-0.810159\pi$
$199$	−23.3427	−1.65472	−0.827361	+	0.561670i	$0.810159\pi$
$200$	0	0
$201$	−8.58738	−0.605707
$202$	0	0
$203$	14.6902	1.03105
$204$	0	0
$205$	1.78223	0.124476
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8.10769	−0.560821
$210$	0	0
$211$	8.69603	0.598659	0.299330	−	0.954150i	$-0.403237\pi$
$211$	8.69603	0.598659	0.299330	+	0.954150i	$0.403237\pi$
$212$	0	0
$213$	−2.46063	−0.168600
$214$	0	0
$215$	4.15204	0.283167
$216$	0	0
$217$	−12.0382	−0.817208
$218$	0	0
$219$	5.29452	0.357771
$220$	0	0
$221$	−23.5499	−1.58414
$222$	0	0
$223$	3.84115	0.257223	0.128611	−	0.991695i	$-0.458948\pi$
$223$	3.84115	0.257223	0.128611	+	0.991695i	$0.458948\pi$
$224$	0	0
$225$	12.7787	0.851911
$226$	0	0
$227$	2.03403	0.135003	0.0675016	−	0.997719i	$-0.478497\pi$
$227$	2.03403	0.135003	0.0675016	+	0.997719i	$0.478497\pi$
$228$	0	0
$229$	−27.4406	−1.81333	−0.906664	−	0.421853i	$-0.861380\pi$
$229$	−27.4406	−1.81333	−0.906664	+	0.421853i	$0.861380\pi$
$230$	0	0
$231$	3.09332	0.203526
$232$	0	0
$233$	7.04480	0.461520	0.230760	−	0.973011i	$-0.425879\pi$
$233$	7.04480	0.461520	0.230760	+	0.973011i	$0.425879\pi$
$234$	0	0
$235$	0.911470	0.0594577
$236$	0	0
$237$	−1.45247	−0.0943482
$238$	0	0
$239$	−26.0807	−1.68702	−0.843510	−	0.537113i	$-0.819515\pi$
$239$	−26.0807	−1.68702	−0.843510	+	0.537113i	$0.819515\pi$
$240$	0	0
$241$	−1.39021	−0.0895515	−0.0447757	−	0.998997i	$-0.514257\pi$
$241$	−1.39021	−0.0895515	−0.0447757	+	0.998997i	$0.514257\pi$
$242$	0	0
$243$	13.8202	0.886564
$244$	0	0
$245$	1.02826	0.0656929
$246$	0	0
$247$	12.2764	0.781126
$248$	0	0
$249$	2.58148	0.163595
$250$	0	0
$251$	26.0912	1.64686	0.823431	−	0.567416i	$-0.192057\pi$
$251$	26.0912	1.64686	0.823431	+	0.567416i	$0.192057\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−1.43295	−0.0897350
$256$	0	0
$257$	−27.9953	−1.74630	−0.873149	−	0.487453i	$-0.837926\pi$
$257$	−27.9953	−1.74630	−0.873149	+	0.487453i	$0.837926\pi$
$258$	0	0
$259$	9.63783	0.598866
$260$	0	0
$261$	−18.7506	−1.16063
$262$	0	0
$263$	9.43346	0.581692	0.290846	−	0.956770i	$-0.406063\pi$
$263$	9.43346	0.581692	0.290846	+	0.956770i	$0.406063\pi$
$264$	0	0
$265$	2.25443	0.138488
$266$	0	0
$267$	5.50236	0.336739
$268$	0	0
$269$	17.3711	1.05914	0.529568	−	0.848268i	$-0.322354\pi$
$269$	17.3711	1.05914	0.529568	+	0.848268i	$0.322354\pi$
$270$	0	0
$271$	−17.9046	−1.08763	−0.543813	−	0.839206i	$-0.683020\pi$
$271$	−17.9046	−1.08763	−0.543813	+	0.839206i	$0.683020\pi$
$272$	0	0
$273$	−4.68379	−0.283476
$274$	0	0
$275$	−11.9833	−0.722623
$276$	0	0
$277$	−15.2821	−0.918212	−0.459106	−	0.888382i	$-0.651830\pi$
$277$	−15.2821	−0.918212	−0.459106	+	0.888382i	$0.651830\pi$
$278$	0	0
$279$	15.3656	0.919915
$280$	0	0
$281$	−5.92708	−0.353580	−0.176790	−	0.984249i	$-0.556571\pi$
$281$	−5.92708	−0.353580	−0.176790	+	0.984249i	$0.556571\pi$
$282$	0	0
$283$	1.56356	0.0929440	0.0464720	−	0.998920i	$-0.485202\pi$
$283$	1.56356	0.0929440	0.0464720	+	0.998920i	$0.485202\pi$
$284$	0	0
$285$	0.746986	0.0442476
$286$	0	0
$287$	−9.84632	−0.581210
$288$	0	0
$289$	22.8046	1.34145
$290$	0	0
$291$	6.88876	0.403826
$292$	0	0
$293$	−2.05817	−0.120239	−0.0601197	−	0.998191i	$-0.519148\pi$
$293$	−2.05817	−0.120239	−0.0601197	+	0.998191i	$0.519148\pi$
$294$	0	0
$295$	−4.40261	−0.256330
$296$	0	0
$297$	−8.45417	−0.490561
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−22.9389	−1.32218
$302$	0	0
$303$	−4.90566	−0.281823
$304$	0	0
$305$	0.628797	0.0360048
$306$	0	0
$307$	−10.8341	−0.618333	−0.309167	−	0.951008i	$-0.600050\pi$
$307$	−10.8341	−0.618333	−0.309167	+	0.951008i	$0.600050\pi$
$308$	0	0
$309$	−8.98778	−0.511297
$310$	0	0
$311$	22.8972	1.29838	0.649191	−	0.760625i	$-0.275107\pi$
$311$	22.8972	1.29838	0.649191	+	0.760625i	$0.275107\pi$
$312$	0	0
$313$	23.3481	1.31971	0.659856	−	0.751392i	$-0.270617\pi$
$313$	23.3481	1.31971	0.659856	+	0.751392i	$0.270617\pi$
$314$	0	0
$315$	2.01830	0.113718
$316$	0	0
$317$	0.503430	0.0282754	0.0141377	−	0.999900i	$-0.495500\pi$
$317$	0.503430	0.0282754	0.0141377	+	0.999900i	$0.495500\pi$
$318$	0	0
$319$	17.5836	0.984494
$320$	0	0
$321$	−4.07085	−0.227213
$322$	0	0
$323$	−20.7498	−1.15455
$324$	0	0
$325$	18.1447	1.00649
$326$	0	0
$327$	2.48562	0.137455
$328$	0	0
$329$	−5.03562	−0.277623
$330$	0	0
$331$	−20.2032	−1.11047	−0.555235	−	0.831694i	$-0.687372\pi$
$331$	−20.2032	−1.11047	−0.555235	+	0.831694i	$0.687372\pi$
$332$	0	0
$333$	−12.3017	−0.674131
$334$	0	0
$335$	5.25429	0.287072
$336$	0	0
$337$	−4.10576	−0.223655	−0.111827	−	0.993728i	$-0.535670\pi$
$337$	−4.10576	−0.223655	−0.111827	+	0.993728i	$0.535670\pi$
$338$	0	0
$339$	−3.40288	−0.184819
$340$	0	0
$341$	−14.4093	−0.780307
$342$	0	0
$343$	−20.0976	−1.08517
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−24.4155	−1.31069	−0.655347	−	0.755328i	$-0.727478\pi$
$347$	−24.4155	−1.31069	−0.655347	+	0.755328i	$0.727478\pi$
$348$	0	0
$349$	17.2951	0.925785	0.462893	−	0.886414i	$-0.346812\pi$
$349$	17.2951	0.925785	0.462893	+	0.886414i	$0.346812\pi$
$350$	0	0
$351$	12.8010	0.683266
$352$	0	0
$353$	−9.57877	−0.509827	−0.254913	−	0.966964i	$-0.582047\pi$
$353$	−9.57877	−0.509827	−0.254913	+	0.966964i	$0.582047\pi$
$354$	0	0
$355$	1.50557	0.0799071
$356$	0	0
$357$	7.91667	0.418994
$358$	0	0
$359$	0.125276	0.00661183	0.00330591	−	0.999995i	$-0.498948\pi$
$359$	0.125276	0.00661183	0.00330591	+	0.999995i	$0.498948\pi$
$360$	0	0
$361$	−8.18330	−0.430700
$362$	0	0
$363$	−2.99932	−0.157424
$364$	0	0
$365$	−3.23952	−0.169564
$366$	0	0
$367$	−4.23470	−0.221050	−0.110525	−	0.993873i	$-0.535253\pi$
$367$	−4.23470	−0.221050	−0.110525	+	0.993873i	$0.535253\pi$
$368$	0	0
$369$	12.5679	0.654257
$370$	0	0
$371$	−12.4551	−0.646636
$372$	0	0
$373$	0.626453	0.0324365	0.0162183	−	0.999868i	$-0.494837\pi$
$373$	0.626453	0.0324365	0.0162183	+	0.999868i	$0.494837\pi$
$374$	0	0
$375$	2.23969	0.115657
$376$	0	0
$377$	−26.6245	−1.37123
$378$	0	0
$379$	−10.5210	−0.540427	−0.270214	−	0.962800i	$-0.587094\pi$
$379$	−10.5210	−0.540427	−0.270214	+	0.962800i	$0.587094\pi$
$380$	0	0
$381$	4.59924	0.235626
$382$	0	0
$383$	−15.3699	−0.785364	−0.392682	−	0.919674i	$-0.628453\pi$
$383$	−15.3699	−0.785364	−0.392682	+	0.919674i	$0.628453\pi$
$384$	0	0
$385$	−1.89269	−0.0964602
$386$	0	0
$387$	29.2792	1.48835
$388$	0	0
$389$	−21.7347	−1.10200	−0.550998	−	0.834507i	$-0.685753\pi$
$389$	−21.7347	−1.10200	−0.550998	+	0.834507i	$0.685753\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	1.99270	0.100519
$394$	0	0
$395$	0.888712	0.0447160
$396$	0	0
$397$	10.1590	0.509866	0.254933	−	0.966959i	$-0.417947\pi$
$397$	10.1590	0.509866	0.254933	+	0.966959i	$0.417947\pi$
$398$	0	0
$399$	−4.12689	−0.206603
$400$	0	0
$401$	31.8980	1.59291	0.796456	−	0.604696i	$-0.206706\pi$
$401$	31.8980	1.59291	0.796456	+	0.604696i	$0.206706\pi$
$402$	0	0
$403$	21.8180	1.08683
$404$	0	0
$405$	−2.16102	−0.107382
$406$	0	0
$407$	11.5361	0.571824
$408$	0	0
$409$	−13.2147	−0.653424	−0.326712	−	0.945124i	$-0.605941\pi$
$409$	−13.2147	−0.653424	−0.326712	+	0.945124i	$0.605941\pi$
$410$	0	0
$411$	−9.70538	−0.478731
$412$	0	0
$413$	24.3232	1.19687
$414$	0	0
$415$	−1.57951	−0.0775351
$416$	0	0
$417$	1.73913	0.0851654
$418$	0	0
$419$	4.20296	0.205328	0.102664	−	0.994716i	$-0.467263\pi$
$419$	4.20296	0.205328	0.102664	+	0.994716i	$0.467263\pi$
$420$	0	0
$421$	−7.03533	−0.342881	−0.171440	−	0.985194i	$-0.554842\pi$
$421$	−7.03533	−0.342881	−0.171440	+	0.985194i	$0.554842\pi$
$422$	0	0
$423$	6.42747	0.312514
$424$	0	0
$425$	−30.6687	−1.48765
$426$	0	0
$427$	−3.47393	−0.168115
$428$	0	0
$429$	−5.60632	−0.270676
$430$	0	0
$431$	36.8195	1.77353	0.886766	−	0.462219i	$-0.152947\pi$
$431$	36.8195	1.77353	0.886766	+	0.462219i	$0.152947\pi$
$432$	0	0
$433$	27.2275	1.30847	0.654235	−	0.756291i	$-0.272991\pi$
$433$	27.2275	1.30847	0.654235	+	0.756291i	$0.272991\pi$
$434$	0	0
$435$	−1.62003	−0.0776746
$436$	0	0
$437$	0	0
$438$	0	0
$439$	24.1877	1.15441	0.577207	−	0.816598i	$-0.304143\pi$
$439$	24.1877	1.15441	0.577207	+	0.816598i	$0.304143\pi$
$440$	0	0
$441$	7.25102	0.345287
$442$	0	0
$443$	−15.4275	−0.732982	−0.366491	−	0.930422i	$-0.619441\pi$
$443$	−15.4275	−0.732982	−0.366491	+	0.930422i	$0.619441\pi$
$444$	0	0
$445$	−3.36668	−0.159596
$446$	0	0
$447$	1.77828	0.0841097
$448$	0	0
$449$	17.3241	0.817574	0.408787	−	0.912630i	$-0.365952\pi$
$449$	17.3241	0.817574	0.408787	+	0.912630i	$0.365952\pi$
$450$	0	0
$451$	−11.7857	−0.554965
$452$	0	0
$453$	−14.9649	−0.703111
$454$	0	0
$455$	2.86583	0.134352
$456$	0	0
$457$	6.86896	0.321317	0.160658	−	0.987010i	$-0.448638\pi$
$457$	6.86896	0.321317	0.160658	+	0.987010i	$0.448638\pi$
$458$	0	0
$459$	−21.6366	−1.00991
$460$	0	0
$461$	−0.341091	−0.0158862	−0.00794309	−	0.999968i	$-0.502528\pi$
$461$	−0.341091	−0.0158862	−0.00794309	+	0.999968i	$0.502528\pi$
$462$	0	0
$463$	−16.3184	−0.758378	−0.379189	−	0.925319i	$-0.623797\pi$
$463$	−16.3184	−0.758378	−0.379189	+	0.925319i	$0.623797\pi$
$464$	0	0
$465$	1.32757	0.0615647
$466$	0	0
$467$	−36.2344	−1.67673	−0.838365	−	0.545109i	$-0.816488\pi$
$467$	−36.2344	−1.67673	−0.838365	+	0.545109i	$0.816488\pi$
$468$	0	0
$469$	−29.0285	−1.34041
$470$	0	0
$471$	7.98411	0.367888
$472$	0	0
$473$	−27.4570	−1.26247
$474$	0	0
$475$	15.9873	0.733549
$476$	0	0
$477$	15.8977	0.727906
$478$	0	0
$479$	−12.7557	−0.582823	−0.291411	−	0.956598i	$-0.594125\pi$
$479$	−12.7557	−0.582823	−0.291411	+	0.956598i	$0.594125\pi$
$480$	0	0
$481$	−17.4675	−0.796451
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.21497	−0.191392
$486$	0	0
$487$	−33.5403	−1.51985	−0.759927	−	0.650008i	$-0.774765\pi$
$487$	−33.5403	−1.51985	−0.759927	+	0.650008i	$0.774765\pi$
$488$	0	0
$489$	−2.81423	−0.127264
$490$	0	0
$491$	36.8776	1.66426	0.832132	−	0.554577i	$-0.187120\pi$
$491$	36.8776	1.66426	0.832132	+	0.554577i	$0.187120\pi$
$492$	0	0
$493$	45.0014	2.02676
$494$	0	0
$495$	2.41583	0.108583
$496$	0	0
$497$	−8.31783	−0.373106
$498$	0	0
$499$	−1.81660	−0.0813223	−0.0406611	−	0.999173i	$-0.512946\pi$
$499$	−1.81660	−0.0813223	−0.0406611	+	0.999173i	$0.512946\pi$
$500$	0	0
$501$	2.40019	0.107233
$502$	0	0
$503$	−43.2260	−1.92735	−0.963676	−	0.267076i	$-0.913943\pi$
$503$	−43.2260	−1.92735	−0.963676	+	0.267076i	$0.913943\pi$
$504$	0	0
$505$	3.00158	0.133569
$506$	0	0
$507$	0.568431	0.0252449
$508$	0	0
$509$	23.6091	1.04646	0.523228	−	0.852193i	$-0.324728\pi$
$509$	23.6091	1.04646	0.523228	+	0.852193i	$0.324728\pi$
$510$	0	0
$511$	17.8974	0.791736
$512$	0	0
$513$	11.2790	0.497978
$514$	0	0
$515$	5.49928	0.242327
$516$	0	0
$517$	−6.02744	−0.265087
$518$	0	0
$519$	11.5765	0.508150
$520$	0	0
$521$	−13.2531	−0.580627	−0.290314	−	0.956932i	$-0.593760\pi$
$521$	−13.2531	−0.580627	−0.290314	+	0.956932i	$0.593760\pi$
$522$	0	0
$523$	−16.5018	−0.721572	−0.360786	−	0.932649i	$-0.617491\pi$
$523$	−16.5018	−0.721572	−0.360786	+	0.932649i	$0.617491\pi$
$524$	0	0
$525$	−6.09964	−0.266210
$526$	0	0
$527$	−36.8774	−1.60640
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−31.0462	−1.34729
$532$	0	0
$533$	17.8454	0.772970
$534$	0	0
$535$	2.49080	0.107686
$536$	0	0
$537$	−1.46189	−0.0630850
$538$	0	0
$539$	−6.79974	−0.292885
$540$	0	0
$541$	−17.5752	−0.755615	−0.377808	−	0.925884i	$-0.623322\pi$
$541$	−17.5752	−0.755615	−0.377808	+	0.925884i	$0.623322\pi$
$542$	0	0
$543$	−15.8447	−0.679963
$544$	0	0
$545$	−1.52085	−0.0651462
$546$	0	0
$547$	16.7827	0.717576	0.358788	−	0.933419i	$-0.383190\pi$
$547$	16.7827	0.717576	0.358788	+	0.933419i	$0.383190\pi$
$548$	0	0
$549$	4.43413	0.189244
$550$	0	0
$551$	−23.4588	−0.999379
$552$	0	0
$553$	−4.90989	−0.208790
$554$	0	0
$555$	−1.06286	−0.0451157
$556$	0	0
$557$	7.07870	0.299934	0.149967	−	0.988691i	$-0.452083\pi$
$557$	7.07870	0.299934	0.149967	+	0.988691i	$0.452083\pi$
$558$	0	0
$559$	41.5743	1.75841
$560$	0	0
$561$	9.47594	0.400075
$562$	0	0
$563$	21.1902	0.893061	0.446530	−	0.894769i	$-0.352660\pi$
$563$	21.1902	0.893061	0.446530	+	0.894769i	$0.352660\pi$
$564$	0	0
$565$	2.08209	0.0875944
$566$	0	0
$567$	11.9391	0.501393
$568$	0	0
$569$	13.2182	0.554134	0.277067	−	0.960851i	$-0.410638\pi$
$569$	13.2182	0.554134	0.277067	+	0.960851i	$0.410638\pi$
$570$	0	0
$571$	22.5366	0.943126	0.471563	−	0.881832i	$-0.343690\pi$
$571$	22.5366	0.943126	0.471563	+	0.881832i	$0.343690\pi$
$572$	0	0
$573$	−6.37848	−0.266465
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−14.7870	−0.615592	−0.307796	−	0.951452i	$-0.599591\pi$
$577$	−14.7870	−0.615592	−0.307796	+	0.951452i	$0.599591\pi$
$578$	0	0
$579$	0.469890	0.0195280
$580$	0	0
$581$	8.72636	0.362030
$582$	0	0
$583$	−14.9083	−0.617437
$584$	0	0
$585$	−3.65795	−0.151238
$586$	0	0
$587$	−15.2561	−0.629689	−0.314844	−	0.949143i	$-0.601952\pi$
$587$	−15.2561	−0.629689	−0.314844	+	0.949143i	$0.601952\pi$
$588$	0	0
$589$	19.2238	0.792105
$590$	0	0
$591$	3.17263	0.130504
$592$	0	0
$593$	−3.71324	−0.152484	−0.0762422	−	0.997089i	$-0.524292\pi$
$593$	−3.71324	−0.152484	−0.0762422	+	0.997089i	$0.524292\pi$
$594$	0	0
$595$	−4.84391	−0.198581
$596$	0	0
$597$	−14.2219	−0.582063
$598$	0	0
$599$	8.17752	0.334124	0.167062	−	0.985946i	$-0.446572\pi$
$599$	8.17752	0.334124	0.167062	+	0.985946i	$0.446572\pi$
$600$	0	0
$601$	34.4600	1.40565	0.702825	−	0.711362i	$-0.251921\pi$
$601$	34.4600	1.40565	0.702825	+	0.711362i	$0.251921\pi$
$602$	0	0
$603$	37.0520	1.50887
$604$	0	0
$605$	1.83517	0.0746103
$606$	0	0
$607$	−19.0695	−0.774009	−0.387005	−	0.922078i	$-0.626490\pi$
$607$	−19.0695	−0.774009	−0.387005	+	0.922078i	$0.626490\pi$
$608$	0	0
$609$	8.95023	0.362682
$610$	0	0
$611$	9.12652	0.369219
$612$	0	0
$613$	−11.8822	−0.479916	−0.239958	−	0.970783i	$-0.577134\pi$
$613$	−11.8822	−0.479916	−0.239958	+	0.970783i	$0.577134\pi$
$614$	0	0
$615$	1.08585	0.0437856
$616$	0	0
$617$	10.7162	0.431418	0.215709	−	0.976458i	$-0.430794\pi$
$617$	10.7162	0.431418	0.215709	+	0.976458i	$0.430794\pi$
$618$	0	0
$619$	−5.74573	−0.230940	−0.115470	−	0.993311i	$-0.536837\pi$
$619$	−5.74573	−0.230940	−0.115470	+	0.993311i	$0.536837\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	18.6000	0.745193
$624$	0	0
$625$	22.9348	0.917391
$626$	0	0
$627$	−4.93973	−0.197274
$628$	0	0
$629$	29.5241	1.17720
$630$	0	0
$631$	−13.7296	−0.546568	−0.273284	−	0.961933i	$-0.588110\pi$
$631$	−13.7296	−0.546568	−0.273284	+	0.961933i	$0.588110\pi$
$632$	0	0
$633$	5.29818	0.210584
$634$	0	0
$635$	−2.81410	−0.111674
$636$	0	0
$637$	10.2959	0.407938
$638$	0	0
$639$	10.6169	0.419998
$640$	0	0
$641$	−36.2879	−1.43328	−0.716642	−	0.697441i	$-0.754322\pi$
$641$	−36.2879	−1.43328	−0.716642	+	0.697441i	$0.754322\pi$
$642$	0	0
$643$	−14.8477	−0.585535	−0.292768	−	0.956184i	$-0.594576\pi$
$643$	−14.8477	−0.585535	−0.292768	+	0.956184i	$0.594576\pi$
$644$	0	0
$645$	2.52969	0.0996066
$646$	0	0
$647$	−1.91936	−0.0754578	−0.0377289	−	0.999288i	$-0.512012\pi$
$647$	−1.91936	−0.0754578	−0.0377289	+	0.999288i	$0.512012\pi$
$648$	0	0
$649$	29.1139	1.14282
$650$	0	0
$651$	−7.33447	−0.287460
$652$	0	0
$653$	−23.8874	−0.934785	−0.467392	−	0.884050i	$-0.654806\pi$
$653$	−23.8874	−0.934785	−0.467392	+	0.884050i	$0.654806\pi$
$654$	0	0
$655$	−1.21926	−0.0476404
$656$	0	0
$657$	−22.8443	−0.891241
$658$	0	0
$659$	−1.74417	−0.0679431	−0.0339716	−	0.999423i	$-0.510816\pi$
$659$	−1.74417	−0.0679431	−0.0339716	+	0.999423i	$0.510816\pi$
$660$	0	0
$661$	−30.7040	−1.19425	−0.597124	−	0.802149i	$-0.703690\pi$
$661$	−30.7040	−1.19425	−0.597124	+	0.802149i	$0.703690\pi$
$662$	0	0
$663$	−14.3481	−0.557234
$664$	0	0
$665$	2.52509	0.0979186
$666$	0	0
$667$	0	0
$668$	0	0
$669$	2.34028	0.0904804
$670$	0	0
$671$	−4.15816	−0.160524
$672$	0	0
$673$	11.7825	0.454181	0.227091	−	0.973874i	$-0.427079\pi$
$673$	11.7825	0.454181	0.227091	+	0.973874i	$0.427079\pi$
$674$	0	0
$675$	16.6705	0.641650
$676$	0	0
$677$	−28.5452	−1.09708	−0.548541	−	0.836124i	$-0.684817\pi$
$677$	−28.5452	−1.09708	−0.548541	+	0.836124i	$0.684817\pi$
$678$	0	0
$679$	23.2865	0.893655
$680$	0	0
$681$	1.23926	0.0474886
$682$	0	0
$683$	−15.0197	−0.574712	−0.287356	−	0.957824i	$-0.592776\pi$
$683$	−15.0197	−0.574712	−0.287356	+	0.957824i	$0.592776\pi$
$684$	0	0
$685$	5.93835	0.226893
$686$	0	0
$687$	−16.7186	−0.637855
$688$	0	0
$689$	22.5735	0.859983
$690$	0	0
$691$	−20.7588	−0.789701	−0.394851	−	0.918745i	$-0.629204\pi$
$691$	−20.7588	−0.789701	−0.394851	+	0.918745i	$0.629204\pi$
$692$	0	0
$693$	−13.3468	−0.507002
$694$	0	0
$695$	−1.06410	−0.0403638
$696$	0	0
$697$	−30.1628	−1.14250
$698$	0	0
$699$	4.29215	0.162344
$700$	0	0
$701$	−23.8262	−0.899902	−0.449951	−	0.893053i	$-0.648559\pi$
$701$	−23.8262	−0.899902	−0.449951	+	0.893053i	$0.648559\pi$
$702$	0	0
$703$	−15.3906	−0.580469
$704$	0	0
$705$	0.555326	0.0209148
$706$	0	0
$707$	−16.5829	−0.623665
$708$	0	0
$709$	18.2057	0.683731	0.341865	−	0.939749i	$-0.388941\pi$
$709$	18.2057	0.683731	0.341865	+	0.939749i	$0.388941\pi$
$710$	0	0
$711$	6.26699	0.235031
$712$	0	0
$713$	0	0
$714$	0	0
$715$	3.43029	0.128286
$716$	0	0
$717$	−15.8900	−0.593425
$718$	0	0
$719$	12.5234	0.467043	0.233521	−	0.972352i	$-0.424975\pi$
$719$	12.5234	0.467043	0.233521	+	0.972352i	$0.424975\pi$
$720$	0	0
$721$	−30.3820	−1.13148
$722$	0	0
$723$	−0.847007	−0.0315005
$724$	0	0
$725$	−34.6727	−1.28771
$726$	0	0
$727$	−12.3982	−0.459823	−0.229911	−	0.973212i	$-0.573844\pi$
$727$	−12.3982	−0.459823	−0.229911	+	0.973212i	$0.573844\pi$
$728$	0	0
$729$	−8.97076	−0.332250
$730$	0	0
$731$	−70.2700	−2.59903
$732$	0	0
$733$	16.4066	0.605992	0.302996	−	0.952992i	$-0.402013\pi$
$733$	16.4066	0.605992	0.302996	+	0.952992i	$0.402013\pi$
$734$	0	0
$735$	0.626480	0.0231081
$736$	0	0
$737$	−34.7460	−1.27988
$738$	0	0
$739$	42.2265	1.55333	0.776663	−	0.629916i	$-0.216911\pi$
$739$	42.2265	1.55333	0.776663	+	0.629916i	$0.216911\pi$
$740$	0	0
$741$	7.47954	0.274768
$742$	0	0
$743$	−14.4754	−0.531052	−0.265526	−	0.964104i	$-0.585546\pi$
$743$	−14.4754	−0.531052	−0.265526	+	0.964104i	$0.585546\pi$
$744$	0	0
$745$	−1.08806	−0.0398635
$746$	0	0
$747$	−11.1383	−0.407530
$748$	0	0
$749$	−13.7610	−0.502814
$750$	0	0
$751$	−29.1984	−1.06547	−0.532733	−	0.846283i	$-0.678835\pi$
$751$	−29.1984	−1.06547	−0.532733	+	0.846283i	$0.678835\pi$
$752$	0	0
$753$	15.8964	0.579299
$754$	0	0
$755$	9.15643	0.333237
$756$	0	0
$757$	16.3495	0.594233	0.297116	−	0.954841i	$-0.403975\pi$
$757$	16.3495	0.594233	0.297116	+	0.954841i	$0.403975\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−5.10898	−0.185201	−0.0926003	−	0.995703i	$-0.529518\pi$
$761$	−5.10898	−0.185201	−0.0926003	+	0.995703i	$0.529518\pi$
$762$	0	0
$763$	8.40229	0.304183
$764$	0	0
$765$	6.18277	0.223538
$766$	0	0
$767$	−44.0832	−1.59175
$768$	0	0
$769$	−28.6372	−1.03269	−0.516343	−	0.856382i	$-0.672707\pi$
$769$	−28.6372	−1.03269	−0.516343	+	0.856382i	$0.672707\pi$
$770$	0	0
$771$	−17.0565	−0.614276
$772$	0	0
$773$	−24.9280	−0.896599	−0.448299	−	0.893884i	$-0.647970\pi$
$773$	−24.9280	−0.896599	−0.448299	+	0.893884i	$0.647970\pi$
$774$	0	0
$775$	28.4133	1.02064
$776$	0	0
$777$	5.87199	0.210656
$778$	0	0
$779$	15.7236	0.563356
$780$	0	0
$781$	−9.95612	−0.356258
$782$	0	0
$783$	−24.4613	−0.874177
$784$	0	0
$785$	−4.88517	−0.174359
$786$	0	0
$787$	−5.13189	−0.182932	−0.0914660	−	0.995808i	$-0.529155\pi$
$787$	−5.13189	−0.182932	−0.0914660	+	0.995808i	$0.529155\pi$
$788$	0	0
$789$	5.74747	0.204615
$790$	0	0
$791$	−11.5030	−0.408999
$792$	0	0
$793$	6.29612	0.223582
$794$	0	0
$795$	1.37354	0.0487146
$796$	0	0
$797$	9.52437	0.337371	0.168685	−	0.985670i	$-0.446048\pi$
$797$	9.52437	0.337371	0.168685	+	0.985670i	$0.446048\pi$
$798$	0	0
$799$	−15.4259	−0.545729
$800$	0	0
$801$	−23.7411	−0.838849
$802$	0	0
$803$	21.4225	0.755985
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.5836	0.372560
$808$	0	0
$809$	−10.6104	−0.373042	−0.186521	−	0.982451i	$-0.559721\pi$
$809$	−10.6104	−0.373042	−0.186521	+	0.982451i	$0.559721\pi$
$810$	0	0
$811$	30.8198	1.08223	0.541116	−	0.840948i	$-0.318002\pi$
$811$	30.8198	1.08223	0.541116	+	0.840948i	$0.318002\pi$
$812$	0	0
$813$	−10.9086	−0.382582
$814$	0	0
$815$	1.72192	0.0603163
$816$	0	0
$817$	36.6311	1.28156
$818$	0	0
$819$	20.2092	0.706166
$820$	0	0
$821$	42.0134	1.46628	0.733139	−	0.680079i	$-0.238054\pi$
$821$	42.0134	1.46628	0.733139	+	0.680079i	$0.238054\pi$
$822$	0	0
$823$	28.0616	0.978165	0.489082	−	0.872238i	$-0.337332\pi$
$823$	28.0616	0.978165	0.489082	+	0.872238i	$0.337332\pi$
$824$	0	0
$825$	−7.30103	−0.254189
$826$	0	0
$827$	36.3429	1.26377	0.631883	−	0.775064i	$-0.282282\pi$
$827$	36.3429	1.26377	0.631883	+	0.775064i	$0.282282\pi$
$828$	0	0
$829$	30.0375	1.04325	0.521623	−	0.853176i	$-0.325327\pi$
$829$	30.0375	1.04325	0.521623	+	0.853176i	$0.325327\pi$
$830$	0	0
$831$	−9.31083	−0.322989
$832$	0	0
$833$	−17.4024	−0.602958
$834$	0	0
$835$	−1.46859	−0.0508225
$836$	0	0
$837$	20.0454	0.692870
$838$	0	0
$839$	−33.4245	−1.15394	−0.576971	−	0.816765i	$-0.695765\pi$
$839$	−33.4245	−1.15394	−0.576971	+	0.816765i	$0.695765\pi$
$840$	0	0
$841$	21.8766	0.754364
$842$	0	0
$843$	−3.61116	−0.124375
$844$	0	0
$845$	−0.347801	−0.0119647
$846$	0	0
$847$	−10.1388	−0.348374
$848$	0	0
$849$	0.952622	0.0326939
$850$	0	0
$851$	0	0
$852$	0	0
$853$	5.87395	0.201120	0.100560	−	0.994931i	$-0.467937\pi$
$853$	5.87395	0.201120	0.100560	+	0.994931i	$0.467937\pi$
$854$	0	0
$855$	−3.22302	−0.110225
$856$	0	0
$857$	14.8919	0.508699	0.254349	−	0.967112i	$-0.418139\pi$
$857$	14.8919	0.508699	0.254349	+	0.967112i	$0.418139\pi$
$858$	0	0
$859$	7.91064	0.269908	0.134954	−	0.990852i	$-0.456911\pi$
$859$	7.91064	0.269908	0.134954	+	0.990852i	$0.456911\pi$
$860$	0	0
$861$	−5.99901	−0.204446
$862$	0	0
$863$	4.03424	0.137327	0.0686636	−	0.997640i	$-0.478126\pi$
$863$	4.03424	0.137327	0.0686636	+	0.997640i	$0.478126\pi$
$864$	0	0
$865$	−7.08319	−0.240836
$866$	0	0
$867$	13.8941	0.471867
$868$	0	0
$869$	−5.87695	−0.199362
$870$	0	0
$871$	52.6110	1.78266
$872$	0	0
$873$	−29.7230	−1.00597
$874$	0	0
$875$	7.57096	0.255945
$876$	0	0
$877$	−51.8651	−1.75136	−0.875680	−	0.482892i	$-0.839586\pi$
$877$	−51.8651	−1.75136	−0.875680	+	0.482892i	$0.839586\pi$
$878$	0	0
$879$	−1.25397	−0.0422953
$880$	0	0
$881$	32.4415	1.09298	0.546491	−	0.837465i	$-0.315963\pi$
$881$	32.4415	1.09298	0.546491	+	0.837465i	$0.315963\pi$
$882$	0	0
$883$	6.65684	0.224021	0.112010	−	0.993707i	$-0.464271\pi$
$883$	6.65684	0.224021	0.112010	+	0.993707i	$0.464271\pi$
$884$	0	0
$885$	−2.68236	−0.0901664
$886$	0	0
$887$	−42.6040	−1.43050	−0.715252	−	0.698867i	$-0.753688\pi$
$887$	−42.6040	−1.43050	−0.715252	+	0.698867i	$0.753688\pi$
$888$	0	0
$889$	15.5471	0.521434
$890$	0	0
$891$	14.2906	0.478753
$892$	0	0
$893$	8.04138	0.269094
$894$	0	0
$895$	0.894472	0.0298989
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−41.6919	−1.39050
$900$	0	0
$901$	−38.1544	−1.27111
$902$	0	0
$903$	−13.9759	−0.465087
$904$	0	0
$905$	9.69479	0.322266
$906$	0	0
$907$	34.2518	1.13731	0.568656	−	0.822576i	$-0.307464\pi$
$907$	34.2518	1.13731	0.568656	+	0.822576i	$0.307464\pi$
$908$	0	0
$909$	21.1665	0.702047
$910$	0	0
$911$	−35.2535	−1.16800	−0.584001	−	0.811753i	$-0.698514\pi$
$911$	−35.2535	−1.16800	−0.584001	+	0.811753i	$0.698514\pi$
$912$	0	0
$913$	10.4451	0.345683
$914$	0	0
$915$	0.383104	0.0126650
$916$	0	0
$917$	6.73607	0.222444
$918$	0	0
$919$	34.5723	1.14043	0.570217	−	0.821494i	$-0.306859\pi$
$919$	34.5723	1.14043	0.570217	+	0.821494i	$0.306859\pi$
$920$	0	0
$921$	−6.60081	−0.217504
$922$	0	0
$923$	15.0752	0.496206
$924$	0	0
$925$	−22.7477	−0.747941
$926$	0	0
$927$	38.7796	1.27369
$928$	0	0
$929$	30.8336	1.01162	0.505809	−	0.862646i	$-0.331194\pi$
$929$	30.8336	1.01162	0.505809	+	0.862646i	$0.331194\pi$
$930$	0	0
$931$	9.07172	0.297314
$932$	0	0
$933$	13.9505	0.456718
$934$	0	0
$935$	−5.79797	−0.189614
$936$	0	0
$937$	4.56731	0.149207	0.0746037	−	0.997213i	$-0.476231\pi$
$937$	4.56731	0.149207	0.0746037	+	0.997213i	$0.476231\pi$
$938$	0	0
$939$	14.2252	0.464220
$940$	0	0
$941$	−11.5271	−0.375773	−0.187886	−	0.982191i	$-0.560164\pi$
$941$	−11.5271	−0.375773	−0.187886	+	0.982191i	$0.560164\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	2.63300	0.0856514
$946$	0	0
$947$	60.0335	1.95083	0.975413	−	0.220386i	$-0.0707316\pi$
$947$	60.0335	1.95083	0.975413	+	0.220386i	$0.0707316\pi$
$948$	0	0
$949$	−32.4372	−1.05296
$950$	0	0
$951$	0.306722	0.00994614
$952$	0	0
$953$	9.45086	0.306143	0.153072	−	0.988215i	$-0.451083\pi$
$953$	9.45086	0.306143	0.153072	+	0.988215i	$0.451083\pi$
$954$	0	0
$955$	3.90275	0.126290
$956$	0	0
$957$	10.7131	0.346305
$958$	0	0
$959$	−32.8077	−1.05942
$960$	0	0
$961$	3.16535	0.102108
$962$	0	0
$963$	17.5645	0.566008
$964$	0	0
$965$	−0.287508	−0.00925520
$966$	0	0
$967$	39.0760	1.25660	0.628300	−	0.777971i	$-0.283751\pi$
$967$	39.0760	1.25660	0.628300	+	0.777971i	$0.283751\pi$
$968$	0	0
$969$	−12.6421	−0.406123
$970$	0	0
$971$	−41.4820	−1.33122	−0.665610	−	0.746299i	$-0.731829\pi$
$971$	−41.4820	−1.33122	−0.665610	+	0.746299i	$0.731829\pi$
$972$	0	0
$973$	5.87888	0.188468
$974$	0	0
$975$	11.0549	0.354041
$976$	0	0
$977$	20.7125	0.662651	0.331326	−	0.943516i	$-0.392504\pi$
$977$	20.7125	0.662651	0.331326	+	0.943516i	$0.392504\pi$
$978$	0	0
$979$	22.2635	0.711544
$980$	0	0
$981$	−10.7247	−0.342413
$982$	0	0
$983$	−24.8727	−0.793315	−0.396658	−	0.917967i	$-0.629830\pi$
$983$	−24.8727	−0.793315	−0.396658	+	0.917967i	$0.629830\pi$
$984$	0	0
$985$	−1.94121	−0.0618521
$986$	0	0
$987$	−3.06802	−0.0976563
$988$	0	0
$989$	0	0
$990$	0	0
$991$	9.20609	0.292441	0.146221	−	0.989252i	$-0.453289\pi$
$991$	9.20609	0.292441	0.146221	+	0.989252i	$0.453289\pi$
$992$	0	0
$993$	−12.3091	−0.390618
$994$	0	0
$995$	8.70183	0.275867
$996$	0	0
$997$	42.7206	1.35298	0.676488	−	0.736454i	$-0.263501\pi$
$997$	42.7206	1.35298	0.676488	+	0.736454i	$0.263501\pi$
$998$	0	0
$999$	−16.0484	−0.507748

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.bv.1.3		5
4.3	odd	2		1058.2.a.k.1.3		5
12.11	even	2		9522.2.a.bw.1.3		5
23.17	odd	22		368.2.m.a.289.1		10
23.19	odd	22		368.2.m.a.177.1		10
23.22	odd	2		8464.2.a.bu.1.3		5
92.19	even	22		46.2.c.b.39.1	yes	10
92.63	even	22		46.2.c.b.13.1	✓	10
92.91	even	2		1058.2.a.j.1.3		5
276.155	odd	22		414.2.i.c.289.1		10
276.203	odd	22		414.2.i.c.361.1		10
276.275	odd	2		9522.2.a.bz.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
46.2.c.b.13.1	✓	10	92.63	even	22
46.2.c.b.39.1	yes	10	92.19	even	22
368.2.m.a.177.1		10	23.19	odd	22
368.2.m.a.289.1		10	23.17	odd	22
414.2.i.c.289.1		10	276.155	odd	22
414.2.i.c.361.1		10	276.203	odd	22
1058.2.a.j.1.3		5	92.91	even	2
1058.2.a.k.1.3		5	4.3	odd	2
8464.2.a.bu.1.3		5	23.22	odd	2
8464.2.a.bv.1.3		5	1.1	even	1	trivial
9522.2.a.bw.1.3		5	12.11	even	2
9522.2.a.bz.1.3		5	276.275	odd	2

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:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q+0.609264 q^{3} -0.372786 q^{5} +2.05954 q^{7} -2.62880 q^{9} +O(q^{10})\)	\(q+0.609264 q^{3} -0.372786 q^{5} +2.05954 q^{7} -2.62880 q^{9} +2.46519 q^{11} -3.73269 q^{13} -0.227125 q^{15} +6.30909 q^{17} -3.28887 q^{19} +1.25480 q^{21} -4.86103 q^{25} -3.42943 q^{27} +7.13278 q^{29} -5.84511 q^{31} +1.50195 q^{33} -0.767766 q^{35} +4.67961 q^{37} -2.27420 q^{39} -4.78084 q^{41} -11.1379 q^{43} +0.979978 q^{45} -2.44502 q^{47} -2.75831 q^{49} +3.84391 q^{51} -6.04752 q^{53} -0.918986 q^{55} -2.00379 q^{57} +11.8100 q^{59} -1.68675 q^{61} -5.41411 q^{63} +1.39149 q^{65} -14.0947 q^{67} -4.03869 q^{71} +8.69002 q^{73} -2.96165 q^{75} +5.07714 q^{77} -2.38398 q^{79} +5.79696 q^{81} +4.23705 q^{83} -2.35194 q^{85} +4.34575 q^{87} +9.03115 q^{89} -7.68762 q^{91} -3.56122 q^{93} +1.22605 q^{95} +11.3067 q^{97} -6.48047 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{5} - 9 q^{7} + 7 q^{9}+O(q^{10}) \) 5 * q + 2 * q^5 - 9 * q^7 + 7 * q^9	\( 5 q + 2 q^{5} - 9 q^{7} + 7 q^{9} - q^{11} + q^{13} - 11 q^{15} - q^{17} - 10 q^{19} - 11 q^{25} + 10 q^{29} + 11 q^{33} + 3 q^{35} + 20 q^{37} + 22 q^{39} - 3 q^{41} - 27 q^{43} + 5 q^{45} + 11 q^{47} + 12 q^{49} - 2 q^{53} + 4 q^{55} + 11 q^{57} + 17 q^{59} + 7 q^{61} - 39 q^{63} - 15 q^{65} - 28 q^{67} + 8 q^{73} - 22 q^{75} + 26 q^{77} - 16 q^{79} + q^{81} - 18 q^{83} - 7 q^{85} - 22 q^{87} - 14 q^{89} - 26 q^{91} - 44 q^{93} - 26 q^{95} + 47 q^{97} - 8 q^{99}+O(q^{100}) \) 5 * q + 2 * q^5 - 9 * q^7 + 7 * q^9 - q^11 + q^13 - 11 * q^15 - q^17 - 10 * q^19 - 11 * q^25 + 10 * q^29 + 11 * q^33 + 3 * q^35 + 20 * q^37 + 22 * q^39 - 3 * q^41 - 27 * q^43 + 5 * q^45 + 11 * q^47 + 12 * q^49 - 2 * q^53 + 4 * q^55 + 11 * q^57 + 17 * q^59 + 7 * q^61 - 39 * q^63 - 15 * q^65 - 28 * q^67 + 8 * q^73 - 22 * q^75 + 26 * q^77 - 16 * q^79 + q^81 - 18 * q^83 - 7 * q^85 - 22 * q^87 - 14 * q^89 - 26 * q^91 - 44 * q^93 - 26 * q^95 + 47 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more