LMFDB - Embedded newform 8464.2.a.bw.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:	$\Z[a_1, a_2, a_3]$
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			$-1.68251$ 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)

$f(q)$	$=$	$q-0.372786 q^{3} -3.39788 q^{5} +0.453800 q^{7} -2.86103 q^{9} -3.01732 q^{11} +0.904073 q^{13} +1.26668 q^{15} -6.29602 q^{17} +6.58658 q^{19} -0.169170 q^{21} +6.54557 q^{25} +2.18491 q^{27} -1.55435 q^{29} +7.55271 q^{31} +1.12481 q^{33} -1.54196 q^{35} -1.24173 q^{37} -0.337025 q^{39} +7.68240 q^{41} +6.23295 q^{43} +9.72143 q^{45} +10.0111 q^{47} -6.79407 q^{49} +2.34706 q^{51} +9.36780 q^{53} +10.2525 q^{55} -2.45538 q^{57} +6.77225 q^{59} -3.14442 q^{61} -1.29833 q^{63} -3.07193 q^{65} -7.08842 q^{67} -12.7073 q^{71} -1.41500 q^{73} -2.44009 q^{75} -1.36926 q^{77} -0.944707 q^{79} +7.76859 q^{81} -7.89585 q^{83} +21.3931 q^{85} +0.579438 q^{87} +11.1184 q^{89} +0.410268 q^{91} -2.81554 q^{93} -22.3804 q^{95} -1.46278 q^{97} +8.63265 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{3} - 8 q^{5} + 7 q^{7} - q^{9} + 5 q^{11} - 7 q^{13} - q^{15} - 13 q^{17} + 12 q^{19} - 6 q^{21} + q^{25} + 20 q^{27} - 4 q^{29} - 6 q^{31} - 9 q^{33} - 9 q^{35} - 14 q^{37} + 6 q^{39} + q^{41}+ \cdots - 34 q^{99}+O(q^{100}) $ 5 * q + 2 * q^3 - 8 * q^5 + 7 * q^7 - q^9 + 5 * q^11 - 7 * q^13 - q^15 - 13 * q^17 + 12 * q^19 - 6 * q^21 + q^25 + 20 * q^27 - 4 * q^29 - 6 * q^31 - 9 * q^33 - 9 * q^35 - 14 * q^37 + 6 * q^39 + q^41 + 11 * q^43 - 5 * q^45 - 9 * q^47 - 12 * q^49 - 14 * q^51 - 6 * q^53 + 14 * q^55 + 7 * q^57 - 3 * q^59 - 13 * q^61 - 19 * q^63 + 9 * q^65 + 10 * q^67 - 8 * q^71 - 4 * q^73 - 4 * q^75 + 18 * q^77 - 18 * q^79 + 33 * q^81 + 2 * q^83 + 23 * q^85 - 6 * q^87 - 10 * q^89 - 12 * q^91 - 20 * q^93 - 28 * q^95 - 17 * q^97 - 34 * q^99

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
$3$	−0.372786	−0.215228	−0.107614	−	0.994193i	$-0.534321\pi$
$3$	−0.372786	−0.215228	−0.107614	+	0.994193i	$0.534321\pi$
$4$	0	0
$5$	−3.39788	−1.51958	−0.759788	−	0.650170i	$-0.774698\pi$
$5$	−3.39788	−1.51958	−0.759788	+	0.650170i	$0.774698\pi$
$6$	0	0
$7$	0.453800	0.171520	0.0857601	−	0.996316i	$-0.472668\pi$
$7$	0.453800	0.171520	0.0857601	+	0.996316i	$0.472668\pi$
$8$	0	0
$9$	−2.86103	−0.953677
$10$	0	0
$11$	−3.01732	−0.909756	−0.454878	−	0.890554i	$-0.650317\pi$
$11$	−3.01732	−0.909756	−0.454878	+	0.890554i	$0.650317\pi$
$12$	0	0
$13$	0.904073	0.250745	0.125372	−	0.992110i	$-0.459987\pi$
$13$	0.904073	0.250745	0.125372	+	0.992110i	$0.459987\pi$
$14$	0	0
$15$	1.26668	0.327055
$16$	0	0
$17$	−6.29602	−1.52701	−0.763504	−	0.645803i	$-0.776523\pi$
$17$	−6.29602	−1.52701	−0.763504	+	0.645803i	$0.776523\pi$
$18$	0	0
$19$	6.58658	1.51107	0.755533	−	0.655111i	$-0.227378\pi$
$19$	6.58658	1.51107	0.755533	+	0.655111i	$0.227378\pi$
$20$	0	0
$21$	−0.169170	−0.0369159
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	6.54557	1.30911
$26$	0	0
$27$	2.18491	0.420486
$28$	0	0
$29$	−1.55435	−0.288635	−0.144317	−	0.989531i	$-0.546099\pi$
$29$	−1.55435	−0.288635	−0.144317	+	0.989531i	$0.546099\pi$
$30$	0	0
$31$	7.55271	1.35651	0.678254	−	0.734828i	$-0.262737\pi$
$31$	7.55271	1.35651	0.678254	+	0.734828i	$0.262737\pi$
$32$	0	0
$33$	1.12481	0.195805
$34$	0	0
$35$	−1.54196	−0.260638
$36$	0	0
$37$	−1.24173	−0.204139	−0.102069	−	0.994777i	$-0.532546\pi$
$37$	−1.24173	−0.204139	−0.102069	+	0.994777i	$0.532546\pi$
$38$	0	0
$39$	−0.337025	−0.0539672
$40$	0	0
$41$	7.68240	1.19979	0.599895	−	0.800079i	$-0.295209\pi$
$41$	7.68240	1.19979	0.599895	+	0.800079i	$0.295209\pi$
$42$	0	0
$43$	6.23295	0.950516	0.475258	−	0.879846i	$-0.342355\pi$
$43$	6.23295	0.950516	0.475258	+	0.879846i	$0.342355\pi$
$44$	0	0
$45$	9.72143	1.44919
$46$	0	0
$47$	10.0111	1.46027	0.730137	−	0.683301i	$-0.239456\pi$
$47$	10.0111	1.46027	0.730137	+	0.683301i	$0.239456\pi$
$48$	0	0
$49$	−6.79407	−0.970581
$50$	0	0
$51$	2.34706	0.328655
$52$	0	0
$53$	9.36780	1.28677	0.643383	−	0.765544i	$-0.277530\pi$
$53$	9.36780	1.28677	0.643383	+	0.765544i	$0.277530\pi$
$54$	0	0
$55$	10.2525	1.38244
$56$	0	0
$57$	−2.45538	−0.325223
$58$	0	0
$59$	6.77225	0.881671	0.440836	−	0.897588i	$-0.354682\pi$
$59$	6.77225	0.881671	0.440836	+	0.897588i	$0.354682\pi$
$60$	0	0
$61$	−3.14442	−0.402602	−0.201301	−	0.979529i	$-0.564517\pi$
$61$	−3.14442	−0.402602	−0.201301	+	0.979529i	$0.564517\pi$
$62$	0	0
$63$	−1.29833	−0.163575
$64$	0	0
$65$	−3.07193	−0.381026
$66$	0	0
$67$	−7.08842	−0.865989	−0.432994	−	0.901397i	$-0.642543\pi$
$67$	−7.08842	−0.865989	−0.432994	+	0.901397i	$0.642543\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.7073	−1.50808	−0.754042	−	0.656826i	$-0.771899\pi$
$71$	−12.7073	−1.50808	−0.754042	+	0.656826i	$0.771899\pi$
$72$	0	0
$73$	−1.41500	−0.165614	−0.0828068	−	0.996566i	$-0.526388\pi$
$73$	−1.41500	−0.165614	−0.0828068	+	0.996566i	$0.526388\pi$
$74$	0	0
$75$	−2.44009	−0.281758
$76$	0	0
$77$	−1.36926	−0.156042
$78$	0	0
$79$	−0.944707	−0.106288	−0.0531439	−	0.998587i	$-0.516924\pi$
$79$	−0.944707	−0.106288	−0.0531439	+	0.998587i	$0.516924\pi$
$80$	0	0
$81$	7.76859	0.863177
$82$	0	0
$83$	−7.89585	−0.866682	−0.433341	−	0.901230i	$-0.642665\pi$
$83$	−7.89585	−0.866682	−0.433341	+	0.901230i	$0.642665\pi$
$84$	0	0
$85$	21.3931	2.32041
$86$	0	0
$87$	0.579438	0.0621223
$88$	0	0
$89$	11.1184	1.17855	0.589276	−	0.807931i	$-0.299413\pi$
$89$	11.1184	1.17855	0.589276	+	0.807931i	$0.299413\pi$
$90$	0	0
$91$	0.410268	0.0430078
$92$	0	0
$93$	−2.81554	−0.291958
$94$	0	0
$95$	−22.3804	−2.29618
$96$	0	0
$97$	−1.46278	−0.148523	−0.0742614	−	0.997239i	$-0.523660\pi$
$97$	−1.46278	−0.148523	−0.0742614	+	0.997239i	$0.523660\pi$
$98$	0	0
$99$	8.63265	0.867614
$100$	0	0
$101$	2.39300	0.238113	0.119056	−	0.992887i	$-0.462013\pi$
$101$	2.39300	0.238113	0.119056	+	0.992887i	$0.462013\pi$
$102$	0	0
$103$	−11.3298	−1.11636	−0.558181	−	0.829719i	$-0.688500\pi$
$103$	−11.3298	−1.11636	−0.558181	+	0.829719i	$0.688500\pi$
$104$	0	0
$105$	0.574819	0.0560966
$106$	0	0
$107$	−3.14915	−0.304440	−0.152220	−	0.988347i	$-0.548642\pi$
$107$	−3.14915	−0.304440	−0.152220	+	0.988347i	$0.548642\pi$
$108$	0	0
$109$	−4.97465	−0.476485	−0.238243	−	0.971206i	$-0.576571\pi$
$109$	−4.97465	−0.476485	−0.238243	+	0.971206i	$0.576571\pi$
$110$	0	0
$111$	0.462898	0.0439363
$112$	0	0
$113$	−9.32983	−0.877677	−0.438838	−	0.898566i	$-0.644610\pi$
$113$	−9.32983	−0.877677	−0.438838	+	0.898566i	$0.644610\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.58658	−0.239129
$118$	0	0
$119$	−2.85713	−0.261913
$120$	0	0
$121$	−1.89578	−0.172343
$122$	0	0
$123$	−2.86389	−0.258228
$124$	0	0
$125$	−5.25166	−0.469723
$126$	0	0
$127$	2.81554	0.249839	0.124919	−	0.992167i	$-0.460133\pi$
$127$	2.81554	0.249839	0.124919	+	0.992167i	$0.460133\pi$
$128$	0	0
$129$	−2.32355	−0.204578
$130$	0	0
$131$	4.44047	0.387966	0.193983	−	0.981005i	$-0.437859\pi$
$131$	4.44047	0.387966	0.193983	+	0.981005i	$0.437859\pi$
$132$	0	0
$133$	2.98899	0.259178
$134$	0	0
$135$	−7.42405	−0.638960
$136$	0	0
$137$	−8.55865	−0.731215	−0.365607	−	0.930769i	$-0.619139\pi$
$137$	−8.55865	−0.731215	−0.365607	+	0.930769i	$0.619139\pi$
$138$	0	0
$139$	−1.44220	−0.122326	−0.0611629	−	0.998128i	$-0.519481\pi$
$139$	−1.44220	−0.122326	−0.0611629	+	0.998128i	$0.519481\pi$
$140$	0	0
$141$	−3.73201	−0.314292
$142$	0	0
$143$	−2.72788	−0.228117
$144$	0	0
$145$	5.28148	0.438603
$146$	0	0
$147$	2.53273	0.208896
$148$	0	0
$149$	−2.48672	−0.203720	−0.101860	−	0.994799i	$-0.532479\pi$
$149$	−2.48672	−0.203720	−0.101860	+	0.994799i	$0.532479\pi$
$150$	0	0
$151$	−2.57153	−0.209268	−0.104634	−	0.994511i	$-0.533367\pi$
$151$	−2.57153	−0.209268	−0.104634	+	0.994511i	$0.533367\pi$
$152$	0	0
$153$	18.0131	1.45627
$154$	0	0
$155$	−25.6632	−2.06132
$156$	0	0
$157$	4.78622	0.381982	0.190991	−	0.981592i	$-0.438830\pi$
$157$	4.78622	0.381982	0.190991	+	0.981592i	$0.438830\pi$
$158$	0	0
$159$	−3.49218	−0.276948
$160$	0	0
$161$	0	0
$162$	0	0
$163$	15.8815	1.24393	0.621967	−	0.783044i	$-0.286334\pi$
$163$	15.8815	1.24393	0.621967	+	0.783044i	$0.286334\pi$
$164$	0	0
$165$	−3.82198	−0.297541
$166$	0	0
$167$	5.26720	0.407588	0.203794	−	0.979014i	$-0.434673\pi$
$167$	5.26720	0.407588	0.203794	+	0.979014i	$0.434673\pi$
$168$	0	0
$169$	−12.1827	−0.937127
$170$	0	0
$171$	−18.8444	−1.44107
$172$	0	0
$173$	−1.69121	−0.128580	−0.0642901	−	0.997931i	$-0.520478\pi$
$173$	−1.69121	−0.128580	−0.0642901	+	0.997931i	$0.520478\pi$
$174$	0	0
$175$	2.97038	0.224539
$176$	0	0
$177$	−2.52460	−0.189760
$178$	0	0
$179$	−11.0822	−0.828320	−0.414160	−	0.910204i	$-0.635925\pi$
$179$	−11.0822	−0.828320	−0.414160	+	0.910204i	$0.635925\pi$
$180$	0	0
$181$	−2.59329	−0.192758	−0.0963789	−	0.995345i	$-0.530726\pi$
$181$	−2.59329	−0.192758	−0.0963789	+	0.995345i	$0.530726\pi$
$182$	0	0
$183$	1.17220	0.0866512
$184$	0	0
$185$	4.21924	0.310205
$186$	0	0
$187$	18.9971	1.38921
$188$	0	0
$189$	0.991510	0.0721218
$190$	0	0
$191$	−14.3769	−1.04028	−0.520138	−	0.854082i	$-0.674119\pi$
$191$	−14.3769	−1.04028	−0.520138	+	0.854082i	$0.674119\pi$
$192$	0	0
$193$	13.7505	0.989780	0.494890	−	0.868956i	$-0.335208\pi$
$193$	13.7505	0.989780	0.494890	+	0.868956i	$0.335208\pi$
$194$	0	0
$195$	1.14517	0.0820074
$196$	0	0
$197$	−3.22770	−0.229964	−0.114982	−	0.993368i	$-0.536681\pi$
$197$	−3.22770	−0.229964	−0.114982	+	0.993368i	$0.536681\pi$
$198$	0	0
$199$	9.42764	0.668308	0.334154	−	0.942519i	$-0.391549\pi$
$199$	9.42764	0.668308	0.334154	+	0.942519i	$0.391549\pi$
$200$	0	0
$201$	2.64246	0.186385
$202$	0	0
$203$	−0.705362	−0.0495067
$204$	0	0
$205$	−26.1039	−1.82317
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−19.8738	−1.37470
$210$	0	0
$211$	26.6649	1.83568	0.917842	−	0.396946i	$-0.129930\pi$
$211$	26.6649	1.83568	0.917842	+	0.396946i	$0.129930\pi$
$212$	0	0
$213$	4.73711	0.324582
$214$	0	0
$215$	−21.1788	−1.44438
$216$	0	0
$217$	3.42742	0.232668
$218$	0	0
$219$	0.527493	0.0356446
$220$	0	0
$221$	−5.69206	−0.382889
$222$	0	0
$223$	26.3203	1.76254	0.881269	−	0.472616i	$-0.156690\pi$
$223$	26.3203	1.76254	0.881269	+	0.472616i	$0.156690\pi$
$224$	0	0
$225$	−18.7271	−1.24847
$226$	0	0
$227$	3.84884	0.255456	0.127728	−	0.991809i	$-0.459232\pi$
$227$	3.84884	0.255456	0.127728	+	0.991809i	$0.459232\pi$
$228$	0	0
$229$	−21.9552	−1.45084	−0.725422	−	0.688305i	$-0.758355\pi$
$229$	−21.9552	−1.45084	−0.725422	+	0.688305i	$0.758355\pi$
$230$	0	0
$231$	0.510440	0.0335845
$232$	0	0
$233$	−12.5694	−0.823447	−0.411724	−	0.911309i	$-0.635073\pi$
$233$	−12.5694	−0.823447	−0.411724	+	0.911309i	$0.635073\pi$
$234$	0	0
$235$	−34.0166	−2.21900
$236$	0	0
$237$	0.352173	0.0228761
$238$	0	0
$239$	−24.4477	−1.58139	−0.790695	−	0.612210i	$-0.790281\pi$
$239$	−24.4477	−1.58139	−0.790695	+	0.612210i	$0.790281\pi$
$240$	0	0
$241$	14.3863	0.926705	0.463353	−	0.886174i	$-0.346646\pi$
$241$	14.3863	0.926705	0.463353	+	0.886174i	$0.346646\pi$
$242$	0	0
$243$	−9.45074	−0.606265
$244$	0	0
$245$	23.0854	1.47487
$246$	0	0
$247$	5.95475	0.378891
$248$	0	0
$249$	2.94346	0.186534
$250$	0	0
$251$	−5.90345	−0.372623	−0.186311	−	0.982491i	$-0.559653\pi$
$251$	−5.90345	−0.372623	−0.186311	+	0.982491i	$0.559653\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−7.97504	−0.499416
$256$	0	0
$257$	0.890964	0.0555768	0.0277884	−	0.999614i	$-0.491154\pi$
$257$	0.890964	0.0555768	0.0277884	+	0.999614i	$0.491154\pi$
$258$	0	0
$259$	−0.563495	−0.0350139
$260$	0	0
$261$	4.44703	0.275264
$262$	0	0
$263$	−5.81974	−0.358861	−0.179430	−	0.983771i	$-0.557425\pi$
$263$	−5.81974	−0.358861	−0.179430	+	0.983771i	$0.557425\pi$
$264$	0	0
$265$	−31.8307	−1.95534
$266$	0	0
$267$	−4.14480	−0.253657
$268$	0	0
$269$	−12.1895	−0.743210	−0.371605	−	0.928391i	$-0.621192\pi$
$269$	−12.1895	−0.743210	−0.371605	+	0.928391i	$0.621192\pi$
$270$	0	0
$271$	−15.7102	−0.954327	−0.477164	−	0.878814i	$-0.658335\pi$
$271$	−15.7102	−0.954327	−0.477164	+	0.878814i	$0.658335\pi$
$272$	0	0
$273$	−0.152942	−0.00925647
$274$	0	0
$275$	−19.7501	−1.19097
$276$	0	0
$277$	22.1047	1.32814	0.664070	−	0.747670i	$-0.268828\pi$
$277$	22.1047	1.32814	0.664070	+	0.747670i	$0.268828\pi$
$278$	0	0
$279$	−21.6085	−1.29367
$280$	0	0
$281$	−32.2014	−1.92097	−0.960486	−	0.278328i	$-0.910220\pi$
$281$	−32.2014	−1.92097	−0.960486	+	0.278328i	$0.910220\pi$
$282$	0	0
$283$	25.0460	1.48883	0.744415	−	0.667717i	$-0.232729\pi$
$283$	25.0460	1.48883	0.744415	+	0.667717i	$0.232729\pi$
$284$	0	0
$285$	8.34309	0.494202
$286$	0	0
$287$	3.48627	0.205788
$288$	0	0
$289$	22.6398	1.33175
$290$	0	0
$291$	0.545303	0.0319662
$292$	0	0
$293$	9.35234	0.546369	0.273185	−	0.961962i	$-0.411923\pi$
$293$	9.35234	0.546369	0.273185	+	0.961962i	$0.411923\pi$
$294$	0	0
$295$	−23.0113	−1.33977
$296$	0	0
$297$	−6.59257	−0.382540
$298$	0	0
$299$	0	0
$300$	0	0
$301$	2.82851	0.163033
$302$	0	0
$303$	−0.892077	−0.0512485
$304$	0	0
$305$	10.6844	0.611785
$306$	0	0
$307$	−24.3747	−1.39114	−0.695568	−	0.718460i	$-0.744847\pi$
$307$	−24.3747	−1.39114	−0.695568	+	0.718460i	$0.744847\pi$
$308$	0	0
$309$	4.22360	0.240272
$310$	0	0
$311$	−12.5420	−0.711189	−0.355595	−	0.934640i	$-0.615722\pi$
$311$	−12.5420	−0.711189	−0.355595	+	0.934640i	$0.615722\pi$
$312$	0	0
$313$	−10.0373	−0.567344	−0.283672	−	0.958921i	$-0.591553\pi$
$313$	−10.0373	−0.567344	−0.283672	+	0.958921i	$0.591553\pi$
$314$	0	0
$315$	4.41158	0.248565
$316$	0	0
$317$	−31.9320	−1.79348	−0.896739	−	0.442560i	$-0.854071\pi$
$317$	−31.9320	−1.79348	−0.896739	+	0.442560i	$0.854071\pi$
$318$	0	0
$319$	4.68996	0.262587
$320$	0	0
$321$	1.17396	0.0655239
$322$	0	0
$323$	−41.4692	−2.30741
$324$	0	0
$325$	5.91767	0.328253
$326$	0	0
$327$	1.85448	0.102553
$328$	0	0
$329$	4.54305	0.250466
$330$	0	0
$331$	30.7148	1.68824	0.844120	−	0.536154i	$-0.180123\pi$
$331$	30.7148	1.68824	0.844120	+	0.536154i	$0.180123\pi$
$332$	0	0
$333$	3.55262	0.194682
$334$	0	0
$335$	24.0856	1.31594
$336$	0	0
$337$	−27.9610	−1.52313	−0.761566	−	0.648088i	$-0.775569\pi$
$337$	−27.9610	−1.52313	−0.761566	+	0.648088i	$0.775569\pi$
$338$	0	0
$339$	3.47803	0.188901
$340$	0	0
$341$	−22.7890	−1.23409
$342$	0	0
$343$	−6.25974	−0.337994
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.0252	0.806596	0.403298	−	0.915069i	$-0.367864\pi$
$347$	15.0252	0.806596	0.403298	+	0.915069i	$0.367864\pi$
$348$	0	0
$349$	28.0511	1.50154	0.750770	−	0.660564i	$-0.229683\pi$
$349$	28.0511	1.50154	0.750770	+	0.660564i	$0.229683\pi$
$350$	0	0
$351$	1.97532	0.105435
$352$	0	0
$353$	0.637776	0.0339454	0.0169727	−	0.999856i	$-0.494597\pi$
$353$	0.637776	0.0339454	0.0169727	+	0.999856i	$0.494597\pi$
$354$	0	0
$355$	43.1780	2.29165
$356$	0	0
$357$	1.06510	0.0563709
$358$	0	0
$359$	−9.91391	−0.523236	−0.261618	−	0.965171i	$-0.584256\pi$
$359$	−9.91391	−0.523236	−0.261618	+	0.965171i	$0.584256\pi$
$360$	0	0
$361$	24.3830	1.28332
$362$	0	0
$363$	0.706719	0.0370931
$364$	0	0
$365$	4.80801	0.251663
$366$	0	0
$367$	7.90826	0.412808	0.206404	−	0.978467i	$-0.433824\pi$
$367$	7.90826	0.412808	0.206404	+	0.978467i	$0.433824\pi$
$368$	0	0
$369$	−21.9796	−1.14421
$370$	0	0
$371$	4.25111	0.220706
$372$	0	0
$373$	17.1900	0.890063	0.445032	−	0.895515i	$-0.353192\pi$
$373$	17.1900	0.890063	0.445032	+	0.895515i	$0.353192\pi$
$374$	0	0
$375$	1.95774	0.101097
$376$	0	0
$377$	−1.40524	−0.0723736
$378$	0	0
$379$	11.2191	0.576287	0.288144	−	0.957587i	$-0.406962\pi$
$379$	11.2191	0.576287	0.288144	+	0.957587i	$0.406962\pi$
$380$	0	0
$381$	−1.04959	−0.0537723
$382$	0	0
$383$	−5.48600	−0.280322	−0.140161	−	0.990129i	$-0.544762\pi$
$383$	−5.48600	−0.280322	−0.140161	+	0.990129i	$0.544762\pi$
$384$	0	0
$385$	4.65257	0.237117
$386$	0	0
$387$	−17.8327	−0.906485
$388$	0	0
$389$	18.9648	0.961553	0.480776	−	0.876843i	$-0.340355\pi$
$389$	18.9648	0.961553	0.480776	+	0.876843i	$0.340355\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−1.65534	−0.0835010
$394$	0	0
$395$	3.21000	0.161513
$396$	0	0
$397$	14.3520	0.720307	0.360154	−	0.932893i	$-0.382724\pi$
$397$	14.3520	0.720307	0.360154	+	0.932893i	$0.382724\pi$
$398$	0	0
$399$	−1.11425	−0.0557823
$400$	0	0
$401$	−8.79712	−0.439307	−0.219654	−	0.975578i	$-0.570493\pi$
$401$	−8.79712	−0.439307	−0.219654	+	0.975578i	$0.570493\pi$
$402$	0	0
$403$	6.82820	0.340137
$404$	0	0
$405$	−26.3967	−1.31166
$406$	0	0
$407$	3.74669	0.185717
$408$	0	0
$409$	−8.33825	−0.412300	−0.206150	−	0.978520i	$-0.566093\pi$
$409$	−8.33825	−0.412300	−0.206150	+	0.978520i	$0.566093\pi$
$410$	0	0
$411$	3.19054	0.157378
$412$	0	0
$413$	3.07324	0.151224
$414$	0	0
$415$	26.8291	1.31699
$416$	0	0
$417$	0.537631	0.0263279
$418$	0	0
$419$	−4.16460	−0.203454	−0.101727	−	0.994812i	$-0.532437\pi$
$419$	−4.16460	−0.203454	−0.101727	+	0.994812i	$0.532437\pi$
$420$	0	0
$421$	−1.96849	−0.0959384	−0.0479692	−	0.998849i	$-0.515275\pi$
$421$	−1.96849	−0.0959384	−0.0479692	+	0.998849i	$0.515275\pi$
$422$	0	0
$423$	−28.6422	−1.39263
$424$	0	0
$425$	−41.2110	−1.99903
$426$	0	0
$427$	−1.42694	−0.0690544
$428$	0	0
$429$	1.01691	0.0490970
$430$	0	0
$431$	7.65653	0.368802	0.184401	−	0.982851i	$-0.440965\pi$
$431$	7.65653	0.368802	0.184401	+	0.982851i	$0.440965\pi$
$432$	0	0
$433$	−6.30458	−0.302979	−0.151489	−	0.988459i	$-0.548407\pi$
$433$	−6.30458	−0.302979	−0.151489	+	0.988459i	$0.548407\pi$
$434$	0	0
$435$	−1.96886	−0.0943995
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−11.4992	−0.548827	−0.274413	−	0.961612i	$-0.588484\pi$
$439$	−11.4992	−0.548827	−0.274413	+	0.961612i	$0.588484\pi$
$440$	0	0
$441$	19.4380	0.925621
$442$	0	0
$443$	16.0601	0.763039	0.381520	−	0.924361i	$-0.375401\pi$
$443$	16.0601	0.763039	0.381520	+	0.924361i	$0.375401\pi$
$444$	0	0
$445$	−37.7791	−1.79090
$446$	0	0
$447$	0.927013	0.0438462
$448$	0	0
$449$	20.4873	0.966854	0.483427	−	0.875385i	$-0.339392\pi$
$449$	20.4873	0.966854	0.483427	+	0.875385i	$0.339392\pi$
$450$	0	0
$451$	−23.1803	−1.09152
$452$	0	0
$453$	0.958628	0.0450403
$454$	0	0
$455$	−1.39404	−0.0653536
$456$	0	0
$457$	−41.0150	−1.91860	−0.959299	−	0.282391i	$-0.908873\pi$
$457$	−41.0150	−1.91860	−0.959299	+	0.282391i	$0.908873\pi$
$458$	0	0
$459$	−13.7562	−0.642085
$460$	0	0
$461$	−30.8859	−1.43850	−0.719250	−	0.694751i	$-0.755514\pi$
$461$	−30.8859	−1.43850	−0.719250	+	0.694751i	$0.755514\pi$
$462$	0	0
$463$	−9.33001	−0.433602	−0.216801	−	0.976216i	$-0.569562\pi$
$463$	−9.33001	−0.433602	−0.216801	+	0.976216i	$0.569562\pi$
$464$	0	0
$465$	9.56687	0.443653
$466$	0	0
$467$	28.2709	1.30822	0.654111	−	0.756399i	$-0.273043\pi$
$467$	28.2709	1.30822	0.654111	+	0.756399i	$0.273043\pi$
$468$	0	0
$469$	−3.21672	−0.148535
$470$	0	0
$471$	−1.78423	−0.0822131
$472$	0	0
$473$	−18.8068	−0.864738
$474$	0	0
$475$	43.1129	1.97816
$476$	0	0
$477$	−26.8016	−1.22716
$478$	0	0
$479$	−23.1267	−1.05669	−0.528344	−	0.849030i	$-0.677187\pi$
$479$	−23.1267	−1.05669	−0.528344	+	0.849030i	$0.677187\pi$
$480$	0	0
$481$	−1.12261	−0.0511867
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.97034	0.225692
$486$	0	0
$487$	−37.5582	−1.70193	−0.850963	−	0.525226i	$-0.823981\pi$
$487$	−37.5582	−1.70193	−0.850963	+	0.525226i	$0.823981\pi$
$488$	0	0
$489$	−5.92039	−0.267729
$490$	0	0
$491$	26.5270	1.19715	0.598573	−	0.801069i	$-0.295735\pi$
$491$	26.5270	1.19715	0.598573	+	0.801069i	$0.295735\pi$
$492$	0	0
$493$	9.78619	0.440748
$494$	0	0
$495$	−29.3327	−1.31841
$496$	0	0
$497$	−5.76659	−0.258667
$498$	0	0
$499$	−5.80206	−0.259736	−0.129868	−	0.991531i	$-0.541455\pi$
$499$	−5.80206	−0.259736	−0.129868	+	0.991531i	$0.541455\pi$
$500$	0	0
$501$	−1.96354	−0.0877244
$502$	0	0
$503$	−22.8156	−1.01730	−0.508649	−	0.860974i	$-0.669855\pi$
$503$	−22.8156	−1.01730	−0.508649	+	0.860974i	$0.669855\pi$
$504$	0	0
$505$	−8.13113	−0.361831
$506$	0	0
$507$	4.54152	0.201696
$508$	0	0
$509$	−38.0127	−1.68488	−0.842441	−	0.538789i	$-0.818882\pi$
$509$	−38.0127	−1.68488	−0.842441	+	0.538789i	$0.818882\pi$
$510$	0	0
$511$	−0.642128	−0.0284061
$512$	0	0
$513$	14.3911	0.635381
$514$	0	0
$515$	38.4974	1.69640
$516$	0	0
$517$	−30.2068	−1.32849
$518$	0	0
$519$	0.630459	0.0276741
$520$	0	0
$521$	6.91255	0.302844	0.151422	−	0.988469i	$-0.451615\pi$
$521$	6.91255	0.302844	0.151422	+	0.988469i	$0.451615\pi$
$522$	0	0
$523$	3.04463	0.133132	0.0665662	−	0.997782i	$-0.478796\pi$
$523$	3.04463	0.133132	0.0665662	+	0.997782i	$0.478796\pi$
$524$	0	0
$525$	−1.10731	−0.0483271
$526$	0	0
$527$	−47.5520	−2.07140
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−19.3756	−0.840830
$532$	0	0
$533$	6.94545	0.300841
$534$	0	0
$535$	10.7004	0.462619
$536$	0	0
$537$	4.13127	0.178277
$538$	0	0
$539$	20.4999	0.882992
$540$	0	0
$541$	−8.16110	−0.350873	−0.175436	−	0.984491i	$-0.556134\pi$
$541$	−8.16110	−0.350873	−0.175436	+	0.984491i	$0.556134\pi$
$542$	0	0
$543$	0.966742	0.0414869
$544$	0	0
$545$	16.9033	0.724056
$546$	0	0
$547$	−26.5513	−1.13525	−0.567625	−	0.823287i	$-0.692138\pi$
$547$	−26.5513	−1.13525	−0.567625	+	0.823287i	$0.692138\pi$
$548$	0	0
$549$	8.99629	0.383952
$550$	0	0
$551$	−10.2378	−0.436146
$552$	0	0
$553$	−0.428708	−0.0182305
$554$	0	0
$555$	−1.57287	−0.0667647
$556$	0	0
$557$	−37.6883	−1.59691	−0.798453	−	0.602058i	$-0.794348\pi$
$557$	−37.6883	−1.59691	−0.798453	+	0.602058i	$0.794348\pi$
$558$	0	0
$559$	5.63504	0.238337
$560$	0	0
$561$	−7.08184	−0.298996
$562$	0	0
$563$	1.20624	0.0508371	0.0254186	−	0.999677i	$-0.491908\pi$
$563$	1.20624	0.0508371	0.0254186	+	0.999677i	$0.491908\pi$
$564$	0	0
$565$	31.7016	1.33370
$566$	0	0
$567$	3.52538	0.148052
$568$	0	0
$569$	−21.9143	−0.918693	−0.459347	−	0.888257i	$-0.651916\pi$
$569$	−21.9143	−0.918693	−0.459347	+	0.888257i	$0.651916\pi$
$570$	0	0
$571$	43.9461	1.83909	0.919543	−	0.392990i	$-0.128559\pi$
$571$	43.9461	1.83909	0.919543	+	0.392990i	$0.128559\pi$
$572$	0	0
$573$	5.35950	0.223896
$574$	0	0
$575$	0	0
$576$	0	0
$577$	9.50349	0.395635	0.197818	−	0.980239i	$-0.436615\pi$
$577$	9.50349	0.395635	0.197818	+	0.980239i	$0.436615\pi$
$578$	0	0
$579$	−5.12598	−0.213028
$580$	0	0
$581$	−3.58313	−0.148653
$582$	0	0
$583$	−28.2657	−1.17064
$584$	0	0
$585$	8.78888	0.363375
$586$	0	0
$587$	6.40486	0.264357	0.132178	−	0.991226i	$-0.457803\pi$
$587$	6.40486	0.264357	0.132178	+	0.991226i	$0.457803\pi$
$588$	0	0
$589$	49.7465	2.04977
$590$	0	0
$591$	1.20324	0.0494947
$592$	0	0
$593$	−24.3804	−1.00118	−0.500591	−	0.865684i	$-0.666884\pi$
$593$	−24.3804	−1.00118	−0.500591	+	0.865684i	$0.666884\pi$
$594$	0	0
$595$	9.70818	0.397996
$596$	0	0
$597$	−3.51449	−0.143838
$598$	0	0
$599$	−26.4384	−1.08025	−0.540123	−	0.841586i	$-0.681622\pi$
$599$	−26.4384	−1.08025	−0.540123	+	0.841586i	$0.681622\pi$
$600$	0	0
$601$	5.40212	0.220357	0.110179	−	0.993912i	$-0.464858\pi$
$601$	5.40212	0.220357	0.110179	+	0.993912i	$0.464858\pi$
$602$	0	0
$603$	20.2802	0.825874
$604$	0	0
$605$	6.44162	0.261889
$606$	0	0
$607$	11.6313	0.472099	0.236050	−	0.971741i	$-0.424147\pi$
$607$	11.6313	0.472099	0.236050	+	0.971741i	$0.424147\pi$
$608$	0	0
$609$	0.262949	0.0106552
$610$	0	0
$611$	9.05079	0.366156
$612$	0	0
$613$	−18.6980	−0.755206	−0.377603	−	0.925968i	$-0.623252\pi$
$613$	−18.6980	−0.755206	−0.377603	+	0.925968i	$0.623252\pi$
$614$	0	0
$615$	9.73114	0.392397
$616$	0	0
$617$	21.8332	0.878972	0.439486	−	0.898250i	$-0.355161\pi$
$617$	21.8332	0.878972	0.439486	+	0.898250i	$0.355161\pi$
$618$	0	0
$619$	−14.7251	−0.591851	−0.295925	−	0.955211i	$-0.595628\pi$
$619$	−14.7251	−0.591851	−0.295925	+	0.955211i	$0.595628\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.04555	0.202146
$624$	0	0
$625$	−14.8834	−0.595334
$626$	0	0
$627$	7.40867	0.295874
$628$	0	0
$629$	7.81794	0.311722
$630$	0	0
$631$	−12.6084	−0.501931	−0.250966	−	0.967996i	$-0.580748\pi$
$631$	−12.6084	−0.501931	−0.250966	+	0.967996i	$0.580748\pi$
$632$	0	0
$633$	−9.94027	−0.395090
$634$	0	0
$635$	−9.56687	−0.379650
$636$	0	0
$637$	−6.14233	−0.243368
$638$	0	0
$639$	36.3561	1.43823
$640$	0	0
$641$	33.3733	1.31816	0.659082	−	0.752071i	$-0.270945\pi$
$641$	33.3733	1.31816	0.659082	+	0.752071i	$0.270945\pi$
$642$	0	0
$643$	26.4979	1.04498	0.522488	−	0.852646i	$-0.325004\pi$
$643$	26.4979	1.04498	0.522488	+	0.852646i	$0.325004\pi$
$644$	0	0
$645$	7.89515	0.310871
$646$	0	0
$647$	−26.5871	−1.04525	−0.522624	−	0.852563i	$-0.675047\pi$
$647$	−26.5871	−1.04525	−0.522624	+	0.852563i	$0.675047\pi$
$648$	0	0
$649$	−20.4340	−0.802106
$650$	0	0
$651$	−1.27769	−0.0500767
$652$	0	0
$653$	2.99182	0.117079	0.0585395	−	0.998285i	$-0.481356\pi$
$653$	2.99182	0.117079	0.0585395	+	0.998285i	$0.481356\pi$
$654$	0	0
$655$	−15.0882	−0.589544
$656$	0	0
$657$	4.04837	0.157942
$658$	0	0
$659$	32.6733	1.27277	0.636386	−	0.771371i	$-0.280429\pi$
$659$	32.6733	1.27277	0.636386	+	0.771371i	$0.280429\pi$
$660$	0	0
$661$	−45.7904	−1.78104	−0.890521	−	0.454943i	$-0.849660\pi$
$661$	−45.7904	−1.78104	−0.890521	+	0.454943i	$0.849660\pi$
$662$	0	0
$663$	2.12192	0.0824084
$664$	0	0
$665$	−10.1562	−0.393841
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−9.81183	−0.379347
$670$	0	0
$671$	9.48773	0.366270
$672$	0	0
$673$	10.6910	0.412108	0.206054	−	0.978541i	$-0.433938\pi$
$673$	10.6910	0.412108	0.206054	+	0.978541i	$0.433938\pi$
$674$	0	0
$675$	14.3015	0.550464
$676$	0	0
$677$	−24.6110	−0.945878	−0.472939	−	0.881095i	$-0.656807\pi$
$677$	−24.6110	−0.945878	−0.472939	+	0.881095i	$0.656807\pi$
$678$	0	0
$679$	−0.663809	−0.0254746
$680$	0	0
$681$	−1.43479	−0.0549813
$682$	0	0
$683$	12.0787	0.462180	0.231090	−	0.972932i	$-0.425771\pi$
$683$	12.0787	0.462180	0.231090	+	0.972932i	$0.425771\pi$
$684$	0	0
$685$	29.0812	1.11114
$686$	0	0
$687$	8.18460	0.312262
$688$	0	0
$689$	8.46918	0.322650
$690$	0	0
$691$	34.2727	1.30379	0.651897	−	0.758307i	$-0.273973\pi$
$691$	34.2727	1.30379	0.651897	+	0.758307i	$0.273973\pi$
$692$	0	0
$693$	3.91749	0.148813
$694$	0	0
$695$	4.90042	0.185883
$696$	0	0
$697$	−48.3685	−1.83209
$698$	0	0
$699$	4.68568	0.177229
$700$	0	0
$701$	35.0524	1.32391	0.661956	−	0.749543i	$-0.269727\pi$
$701$	35.0524	1.32391	0.661956	+	0.749543i	$0.269727\pi$
$702$	0	0
$703$	−8.17874	−0.308467
$704$	0	0
$705$	12.6809	0.477590
$706$	0	0
$707$	1.08594	0.0408411
$708$	0	0
$709$	−16.7861	−0.630417	−0.315208	−	0.949022i	$-0.602074\pi$
$709$	−16.7861	−0.630417	−0.315208	+	0.949022i	$0.602074\pi$
$710$	0	0
$711$	2.70284	0.101364
$712$	0	0
$713$	0	0
$714$	0	0
$715$	9.26899	0.346641
$716$	0	0
$717$	9.11375	0.340359
$718$	0	0
$719$	−17.6938	−0.659869	−0.329934	−	0.944004i	$-0.607027\pi$
$719$	−17.6938	−0.659869	−0.329934	+	0.944004i	$0.607027\pi$
$720$	0	0
$721$	−5.14147	−0.191478
$722$	0	0
$723$	−5.36302	−0.199453
$724$	0	0
$725$	−10.1741	−0.377856
$726$	0	0
$727$	37.5601	1.39303	0.696513	−	0.717544i	$-0.254734\pi$
$727$	37.5601	1.39303	0.696513	+	0.717544i	$0.254734\pi$
$728$	0	0
$729$	−19.7827	−0.732692
$730$	0	0
$731$	−39.2428	−1.45145
$732$	0	0
$733$	−11.6236	−0.429326	−0.214663	−	0.976688i	$-0.568865\pi$
$733$	−11.6236	−0.429326	−0.214663	+	0.976688i	$0.568865\pi$
$734$	0	0
$735$	−8.60591	−0.317434
$736$	0	0
$737$	21.3880	0.787839
$738$	0	0
$739$	17.7700	0.653679	0.326840	−	0.945080i	$-0.394016\pi$
$739$	17.7700	0.653679	0.326840	+	0.945080i	$0.394016\pi$
$740$	0	0
$741$	−2.21984	−0.0815480
$742$	0	0
$743$	−30.8254	−1.13087	−0.565437	−	0.824791i	$-0.691293\pi$
$743$	−30.8254	−1.13087	−0.565437	+	0.824791i	$0.691293\pi$
$744$	0	0
$745$	8.44956	0.309568
$746$	0	0
$747$	22.5903	0.826535
$748$	0	0
$749$	−1.42908	−0.0522175
$750$	0	0
$751$	−43.0469	−1.57080	−0.785401	−	0.618987i	$-0.787543\pi$
$751$	−43.0469	−1.57080	−0.785401	+	0.618987i	$0.787543\pi$
$752$	0	0
$753$	2.20072	0.0801988
$754$	0	0
$755$	8.73773	0.317999
$756$	0	0
$757$	−28.9420	−1.05192	−0.525958	−	0.850511i	$-0.676293\pi$
$757$	−28.9420	−1.05192	−0.525958	+	0.850511i	$0.676293\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	39.5033	1.43199	0.715996	−	0.698104i	$-0.245973\pi$
$761$	39.5033	1.43199	0.715996	+	0.698104i	$0.245973\pi$
$762$	0	0
$763$	−2.25750	−0.0817268
$764$	0	0
$765$	−61.2063	−2.21292
$766$	0	0
$767$	6.12260	0.221074
$768$	0	0
$769$	−23.9365	−0.863171	−0.431586	−	0.902072i	$-0.642046\pi$
$769$	−23.9365	−0.863171	−0.431586	+	0.902072i	$0.642046\pi$
$770$	0	0
$771$	−0.332139	−0.0119617
$772$	0	0
$773$	−41.2014	−1.48191	−0.740955	−	0.671554i	$-0.765627\pi$
$773$	−41.2014	−1.48191	−0.740955	+	0.671554i	$0.765627\pi$
$774$	0	0
$775$	49.4368	1.77582
$776$	0	0
$777$	0.210063	0.00753597
$778$	0	0
$779$	50.6007	1.81296
$780$	0	0
$781$	38.3421	1.37199
$782$	0	0
$783$	−3.39610	−0.121367
$784$	0	0
$785$	−16.2630	−0.580451
$786$	0	0
$787$	−37.8061	−1.34764	−0.673821	−	0.738894i	$-0.735348\pi$
$787$	−37.8061	−1.34764	−0.673821	+	0.738894i	$0.735348\pi$
$788$	0	0
$789$	2.16952	0.0772369
$790$	0	0
$791$	−4.23387	−0.150539
$792$	0	0
$793$	−2.84279	−0.100950
$794$	0	0
$795$	11.8660	0.420844
$796$	0	0
$797$	5.39760	0.191193	0.0955964	−	0.995420i	$-0.469524\pi$
$797$	5.39760	0.191193	0.0955964	+	0.995420i	$0.469524\pi$
$798$	0	0
$799$	−63.0302	−2.22985
$800$	0	0
$801$	−31.8102	−1.12396
$802$	0	0
$803$	4.26952	0.150668
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.54409	0.159959
$808$	0	0
$809$	7.19945	0.253119	0.126560	−	0.991959i	$-0.459606\pi$
$809$	7.19945	0.253119	0.126560	+	0.991959i	$0.459606\pi$
$810$	0	0
$811$	−41.4698	−1.45620	−0.728101	−	0.685470i	$-0.759597\pi$
$811$	−41.4698	−1.45620	−0.728101	+	0.685470i	$0.759597\pi$
$812$	0	0
$813$	5.85654	0.205398
$814$	0	0
$815$	−53.9633	−1.89025
$816$	0	0
$817$	41.0538	1.43629
$818$	0	0
$819$	−1.17379	−0.0410155
$820$	0	0
$821$	37.8483	1.32092	0.660458	−	0.750863i	$-0.270362\pi$
$821$	37.8483	1.32092	0.660458	+	0.750863i	$0.270362\pi$
$822$	0	0
$823$	22.6896	0.790910	0.395455	−	0.918485i	$-0.370587\pi$
$823$	22.6896	0.790910	0.395455	+	0.918485i	$0.370587\pi$
$824$	0	0
$825$	7.36255	0.256331
$826$	0	0
$827$	31.6752	1.10145	0.550727	−	0.834685i	$-0.314351\pi$
$827$	31.6752	1.10145	0.550727	+	0.834685i	$0.314351\pi$
$828$	0	0
$829$	26.0097	0.903353	0.451676	−	0.892182i	$-0.350826\pi$
$829$	26.0097	0.903353	0.451676	+	0.892182i	$0.350826\pi$
$830$	0	0
$831$	−8.24030	−0.285853
$832$	0	0
$833$	42.7755	1.48208
$834$	0	0
$835$	−17.8973	−0.619362
$836$	0	0
$837$	16.5020	0.570392
$838$	0	0
$839$	−3.18028	−0.109796	−0.0548978	−	0.998492i	$-0.517483\pi$
$839$	−3.18028	−0.109796	−0.0548978	+	0.998492i	$0.517483\pi$
$840$	0	0
$841$	−26.5840	−0.916690
$842$	0	0
$843$	12.0042	0.413447
$844$	0	0
$845$	41.3952	1.42404
$846$	0	0
$847$	−0.860303	−0.0295604
$848$	0	0
$849$	−9.33679	−0.320438
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−54.3522	−1.86098	−0.930492	−	0.366311i	$-0.880621\pi$
$853$	−54.3522	−1.86098	−0.930492	+	0.366311i	$0.880621\pi$
$854$	0	0
$855$	64.0310	2.18981
$856$	0	0
$857$	26.6282	0.909603	0.454802	−	0.890593i	$-0.349710\pi$
$857$	26.6282	0.909603	0.454802	+	0.890593i	$0.349710\pi$
$858$	0	0
$859$	−6.55996	−0.223823	−0.111912	−	0.993718i	$-0.535697\pi$
$859$	−6.55996	−0.223823	−0.111912	+	0.993718i	$0.535697\pi$
$860$	0	0
$861$	−1.29963	−0.0442913
$862$	0	0
$863$	25.9098	0.881981	0.440990	−	0.897512i	$-0.354627\pi$
$863$	25.9098	0.881981	0.440990	+	0.897512i	$0.354627\pi$
$864$	0	0
$865$	5.74652	0.195388
$866$	0	0
$867$	−8.43980	−0.286631
$868$	0	0
$869$	2.85048	0.0966961
$870$	0	0
$871$	−6.40845	−0.217142
$872$	0	0
$873$	4.18506	0.141643
$874$	0	0
$875$	−2.38320	−0.0805669
$876$	0	0
$877$	−5.27634	−0.178169	−0.0890847	−	0.996024i	$-0.528394\pi$
$877$	−5.27634	−0.178169	−0.0890847	+	0.996024i	$0.528394\pi$
$878$	0	0
$879$	−3.48642	−0.117594
$880$	0	0
$881$	54.6253	1.84037	0.920187	−	0.391479i	$-0.128036\pi$
$881$	54.6253	1.84037	0.920187	+	0.391479i	$0.128036\pi$
$882$	0	0
$883$	−22.8734	−0.769751	−0.384876	−	0.922968i	$-0.625756\pi$
$883$	−22.8734	−0.769751	−0.384876	+	0.922968i	$0.625756\pi$
$884$	0	0
$885$	8.57827	0.288355
$886$	0	0
$887$	22.6121	0.759239	0.379619	−	0.925143i	$-0.376055\pi$
$887$	22.6121	0.759239	0.379619	+	0.925143i	$0.376055\pi$
$888$	0	0
$889$	1.27769	0.0428524
$890$	0	0
$891$	−23.4403	−0.785280
$892$	0	0
$893$	65.9391	2.20657
$894$	0	0
$895$	37.6558	1.25870
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−11.7395	−0.391535
$900$	0	0
$901$	−58.9799	−1.96490
$902$	0	0
$903$	−1.05443	−0.0350892
$904$	0	0
$905$	8.81169	0.292910
$906$	0	0
$907$	18.3571	0.609537	0.304768	−	0.952427i	$-0.401421\pi$
$907$	18.3571	0.609537	0.304768	+	0.952427i	$0.401421\pi$
$908$	0	0
$909$	−6.84646	−0.227083
$910$	0	0
$911$	0.160638	0.00532219	0.00266110	−	0.999996i	$-0.499153\pi$
$911$	0.160638	0.00532219	0.00266110	+	0.999996i	$0.499153\pi$
$912$	0	0
$913$	23.8243	0.788470
$914$	0	0
$915$	−3.98298	−0.131673
$916$	0	0
$917$	2.01508	0.0665439
$918$	0	0
$919$	−39.6364	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$919$	−39.6364	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$920$	0	0
$921$	9.08653	0.299411
$922$	0	0
$923$	−11.4884	−0.378144
$924$	0	0
$925$	−8.12781	−0.267241
$926$	0	0
$927$	32.4150	1.06465
$928$	0	0
$929$	19.3949	0.636325	0.318163	−	0.948036i	$-0.396934\pi$
$929$	19.3949	0.636325	0.318163	+	0.948036i	$0.396934\pi$
$930$	0	0
$931$	−44.7497	−1.46661
$932$	0	0
$933$	4.67546	0.153068
$934$	0	0
$935$	−64.5498	−2.11100
$936$	0	0
$937$	−4.14245	−0.135328	−0.0676640	−	0.997708i	$-0.521555\pi$
$937$	−4.14245	−0.135328	−0.0676640	+	0.997708i	$0.521555\pi$
$938$	0	0
$939$	3.74178	0.122108
$940$	0	0
$941$	−18.7445	−0.611052	−0.305526	−	0.952184i	$-0.598832\pi$
$941$	−18.7445	−0.611052	−0.305526	+	0.952184i	$0.598832\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−3.36903	−0.109595
$946$	0	0
$947$	−38.4454	−1.24931	−0.624654	−	0.780901i	$-0.714760\pi$
$947$	−38.4454	−1.24931	−0.624654	+	0.780901i	$0.714760\pi$
$948$	0	0
$949$	−1.27927	−0.0415267
$950$	0	0
$951$	11.9038	0.386006
$952$	0	0
$953$	−10.0715	−0.326249	−0.163124	−	0.986606i	$-0.552157\pi$
$953$	−10.0715	−0.326249	−0.163124	+	0.986606i	$0.552157\pi$
$954$	0	0
$955$	48.8509	1.58078
$956$	0	0
$957$	−1.74835	−0.0565161
$958$	0	0
$959$	−3.88391	−0.125418
$960$	0	0
$961$	26.0435	0.840112
$962$	0	0
$963$	9.00981	0.290337
$964$	0	0
$965$	−46.7224	−1.50405
$966$	0	0
$967$	21.0441	0.676732	0.338366	−	0.941015i	$-0.390126\pi$
$967$	21.0441	0.676732	0.338366	+	0.941015i	$0.390126\pi$
$968$	0	0
$969$	15.4591	0.496619
$970$	0	0
$971$	0.704336	0.0226032	0.0113016	−	0.999936i	$-0.496403\pi$
$971$	0.704336	0.0226032	0.0113016	+	0.999936i	$0.496403\pi$
$972$	0	0
$973$	−0.654470	−0.0209813
$974$	0	0
$975$	−2.20602	−0.0706493
$976$	0	0
$977$	11.3046	0.361665	0.180833	−	0.983514i	$-0.442121\pi$
$977$	11.3046	0.361665	0.180833	+	0.983514i	$0.442121\pi$
$978$	0	0
$979$	−33.5479	−1.07220
$980$	0	0
$981$	14.2326	0.454413
$982$	0	0
$983$	−33.4117	−1.06567	−0.532833	−	0.846220i	$-0.678873\pi$
$983$	−33.4117	−1.06567	−0.532833	+	0.846220i	$0.678873\pi$
$984$	0	0
$985$	10.9673	0.349449
$986$	0	0
$987$	−1.69358	−0.0539073
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−9.03825	−0.287109	−0.143555	−	0.989642i	$-0.545853\pi$
$991$	−9.03825	−0.287109	−0.143555	+	0.989642i	$0.545853\pi$
$992$	0	0
$993$	−11.4501	−0.363356
$994$	0	0
$995$	−32.0340	−1.01555
$996$	0	0
$997$	53.9256	1.70784	0.853921	−	0.520403i	$-0.174218\pi$
$997$	53.9256	1.70784	0.853921	+	0.520403i	$0.174218\pi$
$998$	0	0
$999$	−2.71306	−0.0858374

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.bw.1.3		5
4.3	odd	2		1058.2.a.l.1.3		5
12.11	even	2		9522.2.a.bu.1.5		5
23.5	odd	22		368.2.m.b.209.1		10
23.14	odd	22		368.2.m.b.81.1		10
23.22	odd	2		8464.2.a.bx.1.3		5
92.51	even	22		46.2.c.a.25.1	✓	10
92.83	even	22		46.2.c.a.35.1	yes	10
92.91	even	2		1058.2.a.m.1.3		5
276.83	odd	22		414.2.i.f.127.1		10
276.143	odd	22		414.2.i.f.163.1		10
276.275	odd	2		9522.2.a.bp.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
46.2.c.a.25.1	✓	10	92.51	even	22
46.2.c.a.35.1	yes	10	92.83	even	22
368.2.m.b.81.1		10	23.14	odd	22
368.2.m.b.209.1		10	23.5	odd	22
414.2.i.f.127.1		10	276.83	odd	22
414.2.i.f.163.1		10	276.143	odd	22
1058.2.a.l.1.3		5	4.3	odd	2
1058.2.a.m.1.3		5	92.91	even	2
8464.2.a.bw.1.3		5	1.1	even	1	trivial
8464.2.a.bx.1.3		5	23.22	odd	2
9522.2.a.bp.1.1		5	276.275	odd	2
9522.2.a.bu.1.5		5	12.11	even	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)

\(f(q)\)	\(=\)	\(q-0.372786 q^{3} -3.39788 q^{5} +0.453800 q^{7} -2.86103 q^{9} -3.01732 q^{11} +0.904073 q^{13} +1.26668 q^{15} -6.29602 q^{17} +6.58658 q^{19} -0.169170 q^{21} +6.54557 q^{25} +2.18491 q^{27} -1.55435 q^{29} +7.55271 q^{31} +1.12481 q^{33} -1.54196 q^{35} -1.24173 q^{37} -0.337025 q^{39} +7.68240 q^{41} +6.23295 q^{43} +9.72143 q^{45} +10.0111 q^{47} -6.79407 q^{49} +2.34706 q^{51} +9.36780 q^{53} +10.2525 q^{55} -2.45538 q^{57} +6.77225 q^{59} -3.14442 q^{61} -1.29833 q^{63} -3.07193 q^{65} -7.08842 q^{67} -12.7073 q^{71} -1.41500 q^{73} -2.44009 q^{75} -1.36926 q^{77} -0.944707 q^{79} +7.76859 q^{81} -7.89585 q^{83} +21.3931 q^{85} +0.579438 q^{87} +11.1184 q^{89} +0.410268 q^{91} -2.81554 q^{93} -22.3804 q^{95} -1.46278 q^{97} +8.63265 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{3} - 8 q^{5} + 7 q^{7} - q^{9} + 5 q^{11} - 7 q^{13} - q^{15} - 13 q^{17} + 12 q^{19} - 6 q^{21} + q^{25} + 20 q^{27} - 4 q^{29} - 6 q^{31} - 9 q^{33} - 9 q^{35} - 14 q^{37} + 6 q^{39} + q^{41}+ \cdots - 34 q^{99}+O(q^{100}) \) 5 * q + 2 * q^3 - 8 * q^5 + 7 * q^7 - q^9 + 5 * q^11 - 7 * q^13 - q^15 - 13 * q^17 + 12 * q^19 - 6 * q^21 + q^25 + 20 * q^27 - 4 * q^29 - 6 * q^31 - 9 * q^33 - 9 * q^35 - 14 * q^37 + 6 * q^39 + q^41 + 11 * q^43 - 5 * q^45 - 9 * q^47 - 12 * q^49 - 14 * q^51 - 6 * q^53 + 14 * q^55 + 7 * q^57 - 3 * q^59 - 13 * q^61 - 19 * q^63 + 9 * q^65 + 10 * q^67 - 8 * q^71 - 4 * q^73 - 4 * q^75 + 18 * q^77 - 18 * q^79 + 33 * q^81 + 2 * q^83 + 23 * q^85 - 6 * q^87 - 10 * q^89 - 12 * q^91 - 20 * q^93 - 28 * q^95 - 17 * q^97 - 34 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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