LMFDB - Embedded newform 8016.2.a.r.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.0080822603$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.11256624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 16x^{3} + 20x^{2} + 31x - 11 $ x^5 - x^4 - 16x^3 + 20x^2 + 31*x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1002)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.20970$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.65109 q^{5} -0.854515 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.65109 q^{5} -0.854515 q^{7} +1.00000 q^{9} -2.53118 q^{11} +4.20970 q^{13} -1.65109 q^{15} -2.20970 q^{17} +0.531177 q^{19} +0.854515 q^{21} +2.53118 q^{23} -2.27391 q^{25} -1.00000 q^{27} +1.88822 q^{29} -5.67666 q^{31} +2.53118 q^{33} -1.41088 q^{35} +5.08979 q^{37} -4.20970 q^{39} -2.98069 q^{41} -2.40275 q^{43} +1.65109 q^{45} -5.56488 q^{47} -6.26980 q^{49} +2.20970 q^{51} -9.77541 q^{53} -4.17919 q^{55} -0.531177 q^{57} +13.1023 q^{59} -4.71037 q^{61} -0.854515 q^{63} +6.95058 q^{65} -7.70224 q^{67} -2.53118 q^{69} -2.89942 q^{71} +10.1243 q^{73} +2.27391 q^{75} +2.16293 q^{77} -4.98069 q^{79} +1.00000 q^{81} +8.44272 q^{83} -3.64840 q^{85} -1.88822 q^{87} +7.85585 q^{89} -3.59725 q^{91} +5.67666 q^{93} +0.877019 q^{95} +15.2865 q^{97} -2.53118 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} - q^{5} - 9 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 - q^5 - 9 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} - q^{5} - 9 q^{7} + 5 q^{9} + 2 q^{11} + 6 q^{13} + q^{15} + 4 q^{17} - 12 q^{19} + 9 q^{21} - 2 q^{23} + 14 q^{25} - 5 q^{27} - 6 q^{29} - 9 q^{31} - 2 q^{33} - 3 q^{35} + 5 q^{37} - 6 q^{39} + 4 q^{41} - 18 q^{43} - q^{45} + 7 q^{47} + 16 q^{49} - 4 q^{51} + 3 q^{53} + 4 q^{55} + 12 q^{57} - 13 q^{59} + 16 q^{61} - 9 q^{63} - 10 q^{65} - 9 q^{67} + 2 q^{69} + 10 q^{71} + 8 q^{73} - 14 q^{75} + 6 q^{77} - 6 q^{79} + 5 q^{81} - q^{83} + 8 q^{85} + 6 q^{87} - 5 q^{89} - 12 q^{91} + 9 q^{93} - 2 q^{95} - 7 q^{97} + 2 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 - q^5 - 9 * q^7 + 5 * q^9 + 2 * q^11 + 6 * q^13 + q^15 + 4 * q^17 - 12 * q^19 + 9 * q^21 - 2 * q^23 + 14 * q^25 - 5 * q^27 - 6 * q^29 - 9 * q^31 - 2 * q^33 - 3 * q^35 + 5 * q^37 - 6 * q^39 + 4 * q^41 - 18 * q^43 - q^45 + 7 * q^47 + 16 * q^49 - 4 * q^51 + 3 * q^53 + 4 * q^55 + 12 * q^57 - 13 * q^59 + 16 * q^61 - 9 * q^63 - 10 * q^65 - 9 * q^67 + 2 * q^69 + 10 * q^71 + 8 * q^73 - 14 * q^75 + 6 * q^77 - 6 * q^79 + 5 * q^81 - q^83 + 8 * q^85 + 6 * q^87 - 5 * q^89 - 12 * q^91 + 9 * q^93 - 2 * q^95 - 7 * q^97 + 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.65109	0.738388	0.369194	−	0.929352i	$-0.379634\pi$
$5$	1.65109	0.738388	0.369194	+	0.929352i	$0.379634\pi$
$6$	0	0
$7$	−0.854515	−0.322976	−0.161488	−	0.986875i	$-0.551629\pi$
$7$	−0.854515	−0.322976	−0.161488	+	0.986875i	$0.551629\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.53118	−0.763179	−0.381589	−	0.924332i	$-0.624623\pi$
$11$	−2.53118	−0.763179	−0.381589	+	0.924332i	$0.624623\pi$
$12$	0	0
$13$	4.20970	1.16756	0.583780	−	0.811912i	$-0.301573\pi$
$13$	4.20970	1.16756	0.583780	+	0.811912i	$0.301573\pi$
$14$	0	0
$15$	−1.65109	−0.426309
$16$	0	0
$17$	−2.20970	−0.535931	−0.267965	−	0.963429i	$-0.586351\pi$
$17$	−2.20970	−0.535931	−0.267965	+	0.963429i	$0.586351\pi$
$18$	0	0
$19$	0.531177	0.121860	0.0609302	−	0.998142i	$-0.480593\pi$
$19$	0.531177	0.121860	0.0609302	+	0.998142i	$0.480593\pi$
$20$	0	0
$21$	0.854515	0.186471
$22$	0	0
$23$	2.53118	0.527787	0.263893	−	0.964552i	$-0.414993\pi$
$23$	2.53118	0.527787	0.263893	+	0.964552i	$0.414993\pi$
$24$	0	0
$25$	−2.27391	−0.454783
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.88822	0.350634	0.175317	−	0.984512i	$-0.443905\pi$
$29$	1.88822	0.350634	0.175317	+	0.984512i	$0.443905\pi$
$30$	0	0
$31$	−5.67666	−1.01956	−0.509779	−	0.860305i	$-0.670273\pi$
$31$	−5.67666	−1.01956	−0.509779	+	0.860305i	$0.670273\pi$
$32$	0	0
$33$	2.53118	0.440621
$34$	0	0
$35$	−1.41088	−0.238482
$36$	0	0
$37$	5.08979	0.836756	0.418378	−	0.908273i	$-0.362599\pi$
$37$	5.08979	0.836756	0.418378	+	0.908273i	$0.362599\pi$
$38$	0	0
$39$	−4.20970	−0.674091
$40$	0	0
$41$	−2.98069	−0.465506	−0.232753	−	0.972536i	$-0.574773\pi$
$41$	−2.98069	−0.465506	−0.232753	+	0.972536i	$0.574773\pi$
$42$	0	0
$43$	−2.40275	−0.366416	−0.183208	−	0.983074i	$-0.558648\pi$
$43$	−2.40275	−0.366416	−0.183208	+	0.983074i	$0.558648\pi$
$44$	0	0
$45$	1.65109	0.246129
$46$	0	0
$47$	−5.56488	−0.811722	−0.405861	−	0.913935i	$-0.633028\pi$
$47$	−5.56488	−0.811722	−0.405861	+	0.913935i	$0.633028\pi$
$48$	0	0
$49$	−6.26980	−0.895686
$50$	0	0
$51$	2.20970	0.309420
$52$	0	0
$53$	−9.77541	−1.34276	−0.671378	−	0.741115i	$-0.734297\pi$
$53$	−9.77541	−1.34276	−0.671378	+	0.741115i	$0.734297\pi$
$54$	0	0
$55$	−4.17919	−0.563522
$56$	0	0
$57$	−0.531177	−0.0703561
$58$	0	0
$59$	13.1023	1.70578	0.852889	−	0.522092i	$-0.174848\pi$
$59$	13.1023	1.70578	0.852889	+	0.522092i	$0.174848\pi$
$60$	0	0
$61$	−4.71037	−0.603101	−0.301550	−	0.953450i	$-0.597504\pi$
$61$	−4.71037	−0.603101	−0.301550	+	0.953450i	$0.597504\pi$
$62$	0	0
$63$	−0.854515	−0.107659
$64$	0	0
$65$	6.95058	0.862113
$66$	0	0
$67$	−7.70224	−0.940978	−0.470489	−	0.882406i	$-0.655922\pi$
$67$	−7.70224	−0.940978	−0.470489	+	0.882406i	$0.655922\pi$
$68$	0	0
$69$	−2.53118	−0.304718
$70$	0	0
$71$	−2.89942	−0.344098	−0.172049	−	0.985088i	$-0.555039\pi$
$71$	−2.89942	−0.344098	−0.172049	+	0.985088i	$0.555039\pi$
$72$	0	0
$73$	10.1243	1.18496	0.592481	−	0.805584i	$-0.298149\pi$
$73$	10.1243	1.18496	0.592481	+	0.805584i	$0.298149\pi$
$74$	0	0
$75$	2.27391	0.262569
$76$	0	0
$77$	2.16293	0.246489
$78$	0	0
$79$	−4.98069	−0.560372	−0.280186	−	0.959946i	$-0.590396\pi$
$79$	−4.98069	−0.560372	−0.280186	+	0.959946i	$0.590396\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.44272	0.926709	0.463355	−	0.886173i	$-0.346646\pi$
$83$	8.44272	0.926709	0.463355	+	0.886173i	$0.346646\pi$
$84$	0	0
$85$	−3.64840	−0.395725
$86$	0	0
$87$	−1.88822	−0.202439
$88$	0	0
$89$	7.85585	0.832719	0.416359	−	0.909200i	$-0.363306\pi$
$89$	7.85585	0.832719	0.416359	+	0.909200i	$0.363306\pi$
$90$	0	0
$91$	−3.59725	−0.377095
$92$	0	0
$93$	5.67666	0.588642
$94$	0	0
$95$	0.877019	0.0899803
$96$	0	0
$97$	15.2865	1.55210	0.776052	−	0.630669i	$-0.217219\pi$
$97$	15.2865	1.55210	0.776052	+	0.630669i	$0.217219\pi$
$98$	0	0
$99$	−2.53118	−0.254393
$100$	0	0
$101$	5.94245	0.591295	0.295648	−	0.955297i	$-0.404465\pi$
$101$	5.94245	0.591295	0.295648	+	0.955297i	$0.404465\pi$
$102$	0	0
$103$	−6.09792	−0.600846	−0.300423	−	0.953806i	$-0.597128\pi$
$103$	−6.09792	−0.600846	−0.300423	+	0.953806i	$0.597128\pi$
$104$	0	0
$105$	1.41088	0.137688
$106$	0	0
$107$	−7.88411	−0.762186	−0.381093	−	0.924537i	$-0.624452\pi$
$107$	−7.88411	−0.762186	−0.381093	+	0.924537i	$0.624452\pi$
$108$	0	0
$109$	−5.27205	−0.504971	−0.252486	−	0.967601i	$-0.581248\pi$
$109$	−5.27205	−0.504971	−0.252486	+	0.967601i	$0.581248\pi$
$110$	0	0
$111$	−5.08979	−0.483102
$112$	0	0
$113$	−10.3340	−0.972143	−0.486071	−	0.873919i	$-0.661570\pi$
$113$	−10.3340	−0.972143	−0.486071	+	0.873919i	$0.661570\pi$
$114$	0	0
$115$	4.17919	0.389712
$116$	0	0
$117$	4.20970	0.389187
$118$	0	0
$119$	1.88822	0.173093
$120$	0	0
$121$	−4.59314	−0.417558
$122$	0	0
$123$	2.98069	0.268760
$124$	0	0
$125$	−12.0099	−1.07419
$126$	0	0
$127$	−6.85040	−0.607875	−0.303938	−	0.952692i	$-0.598301\pi$
$127$	−6.85040	−0.607875	−0.303938	+	0.952692i	$0.598301\pi$
$128$	0	0
$129$	2.40275	0.211550
$130$	0	0
$131$	3.86386	0.337587	0.168793	−	0.985651i	$-0.446013\pi$
$131$	3.86386	0.337587	0.168793	+	0.985651i	$0.446013\pi$
$132$	0	0
$133$	−0.453899	−0.0393580
$134$	0	0
$135$	−1.65109	−0.142103
$136$	0	0
$137$	−7.21904	−0.616764	−0.308382	−	0.951263i	$-0.599788\pi$
$137$	−7.21904	−0.616764	−0.308382	+	0.951263i	$0.599788\pi$
$138$	0	0
$139$	−12.6524	−1.07316	−0.536582	−	0.843848i	$-0.680285\pi$
$139$	−12.6524	−1.07316	−0.536582	+	0.843848i	$0.680285\pi$
$140$	0	0
$141$	5.56488	0.468648
$142$	0	0
$143$	−10.6555	−0.891057
$144$	0	0
$145$	3.11762	0.258904
$146$	0	0
$147$	6.26980	0.517125
$148$	0	0
$149$	10.3618	0.848875	0.424438	−	0.905457i	$-0.360472\pi$
$149$	10.3618	0.848875	0.424438	+	0.905457i	$0.360472\pi$
$150$	0	0
$151$	−4.44991	−0.362128	−0.181064	−	0.983471i	$-0.557954\pi$
$151$	−4.44991	−0.362128	−0.181064	+	0.983471i	$0.557954\pi$
$152$	0	0
$153$	−2.20970	−0.178644
$154$	0	0
$155$	−9.37266	−0.752830
$156$	0	0
$157$	−17.4943	−1.39620	−0.698098	−	0.716002i	$-0.745970\pi$
$157$	−17.4943	−1.39620	−0.698098	+	0.716002i	$0.745970\pi$
$158$	0	0
$159$	9.77541	0.775240
$160$	0	0
$161$	−2.16293	−0.170463
$162$	0	0
$163$	2.65654	0.208076	0.104038	−	0.994573i	$-0.466824\pi$
$163$	2.65654	0.208076	0.104038	+	0.994573i	$0.466824\pi$
$164$	0	0
$165$	4.17919	0.325350
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	4.72157	0.363198
$170$	0	0
$171$	0.531177	0.0406201
$172$	0	0
$173$	−23.2033	−1.76412	−0.882058	−	0.471141i	$-0.843842\pi$
$173$	−23.2033	−1.76412	−0.882058	+	0.471141i	$0.843842\pi$
$174$	0	0
$175$	1.94309	0.146884
$176$	0	0
$177$	−13.1023	−0.984832
$178$	0	0
$179$	8.49257	0.634764	0.317382	−	0.948298i	$-0.397196\pi$
$179$	8.49257	0.634764	0.317382	+	0.948298i	$0.397196\pi$
$180$	0	0
$181$	−15.7768	−1.17268	−0.586341	−	0.810064i	$-0.699432\pi$
$181$	−15.7768	−1.17268	−0.586341	+	0.810064i	$0.699432\pi$
$182$	0	0
$183$	4.71037	0.348200
$184$	0	0
$185$	8.40368	0.617851
$186$	0	0
$187$	5.59314	0.409011
$188$	0	0
$189$	0.854515	0.0621568
$190$	0	0
$191$	−2.73902	−0.198188	−0.0990941	−	0.995078i	$-0.531594\pi$
$191$	−2.73902	−0.198188	−0.0990941	+	0.995078i	$0.531594\pi$
$192$	0	0
$193$	−6.18906	−0.445498	−0.222749	−	0.974876i	$-0.571503\pi$
$193$	−6.18906	−0.445498	−0.222749	+	0.974876i	$0.571503\pi$
$194$	0	0
$195$	−6.95058	−0.497741
$196$	0	0
$197$	5.19039	0.369800	0.184900	−	0.982757i	$-0.440804\pi$
$197$	5.19039	0.369800	0.184900	+	0.982757i	$0.440804\pi$
$198$	0	0
$199$	19.5553	1.38624	0.693120	−	0.720823i	$-0.256236\pi$
$199$	19.5553	1.38624	0.693120	+	0.720823i	$0.256236\pi$
$200$	0	0
$201$	7.70224	0.543274
$202$	0	0
$203$	−1.61351	−0.113247
$204$	0	0
$205$	−4.92138	−0.343724
$206$	0	0
$207$	2.53118	0.175929
$208$	0	0
$209$	−1.34450	−0.0930013
$210$	0	0
$211$	3.22489	0.222011	0.111005	−	0.993820i	$-0.464593\pi$
$211$	3.22489	0.222011	0.111005	+	0.993820i	$0.464593\pi$
$212$	0	0
$213$	2.89942	0.198665
$214$	0	0
$215$	−3.96714	−0.270557
$216$	0	0
$217$	4.85079	0.329293
$218$	0	0
$219$	−10.1243	−0.684138
$220$	0	0
$221$	−9.30217	−0.625732
$222$	0	0
$223$	19.0296	1.27432	0.637158	−	0.770733i	$-0.280110\pi$
$223$	19.0296	1.27432	0.637158	+	0.770733i	$0.280110\pi$
$224$	0	0
$225$	−2.27391	−0.151594
$226$	0	0
$227$	−14.2761	−0.947536	−0.473768	−	0.880650i	$-0.657107\pi$
$227$	−14.2761	−0.947536	−0.473768	+	0.880650i	$0.657107\pi$
$228$	0	0
$229$	−2.64668	−0.174897	−0.0874486	−	0.996169i	$-0.527871\pi$
$229$	−2.64668	−0.174897	−0.0874486	+	0.996169i	$0.527871\pi$
$230$	0	0
$231$	−2.16293	−0.142310
$232$	0	0
$233$	6.97883	0.457199	0.228599	−	0.973521i	$-0.426585\pi$
$233$	6.97883	0.457199	0.228599	+	0.973521i	$0.426585\pi$
$234$	0	0
$235$	−9.18810	−0.599366
$236$	0	0
$237$	4.98069	0.323531
$238$	0	0
$239$	13.8841	0.898088	0.449044	−	0.893510i	$-0.351765\pi$
$239$	13.8841	0.898088	0.449044	+	0.893510i	$0.351765\pi$
$240$	0	0
$241$	−8.05526	−0.518885	−0.259443	−	0.965759i	$-0.583539\pi$
$241$	−8.05526	−0.518885	−0.259443	+	0.965759i	$0.583539\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−10.3520	−0.661364
$246$	0	0
$247$	2.23610	0.142279
$248$	0	0
$249$	−8.44272	−0.535036
$250$	0	0
$251$	14.6721	0.926098	0.463049	−	0.886333i	$-0.346755\pi$
$251$	14.6721	0.926098	0.463049	+	0.886333i	$0.346755\pi$
$252$	0	0
$253$	−6.40686	−0.402796
$254$	0	0
$255$	3.64840	0.228472
$256$	0	0
$257$	−4.86892	−0.303715	−0.151857	−	0.988402i	$-0.548525\pi$
$257$	−4.86892	−0.303715	−0.151857	+	0.988402i	$0.548525\pi$
$258$	0	0
$259$	−4.34930	−0.270253
$260$	0	0
$261$	1.88822	0.116878
$262$	0	0
$263$	2.40101	0.148052	0.0740262	−	0.997256i	$-0.476415\pi$
$263$	2.40101	0.148052	0.0740262	+	0.997256i	$0.476415\pi$
$264$	0	0
$265$	−16.1400	−0.991474
$266$	0	0
$267$	−7.85585	−0.480770
$268$	0	0
$269$	20.9828	1.27935	0.639673	−	0.768647i	$-0.279070\pi$
$269$	20.9828	1.27935	0.639673	+	0.768647i	$0.279070\pi$
$270$	0	0
$271$	14.2321	0.864539	0.432269	−	0.901745i	$-0.357713\pi$
$271$	14.2321	0.864539	0.432269	+	0.901745i	$0.357713\pi$
$272$	0	0
$273$	3.59725	0.217716
$274$	0	0
$275$	5.75568	0.347081
$276$	0	0
$277$	−0.481335	−0.0289206	−0.0144603	−	0.999895i	$-0.504603\pi$
$277$	−0.481335	−0.0289206	−0.0144603	+	0.999895i	$0.504603\pi$
$278$	0	0
$279$	−5.67666	−0.339853
$280$	0	0
$281$	−14.3404	−0.855475	−0.427738	−	0.903903i	$-0.640689\pi$
$281$	−14.3404	−0.855475	−0.427738	+	0.903903i	$0.640689\pi$
$282$	0	0
$283$	−8.17958	−0.486226	−0.243113	−	0.969998i	$-0.578169\pi$
$283$	−8.17958	−0.486226	−0.243113	+	0.969998i	$0.578169\pi$
$284$	0	0
$285$	−0.877019	−0.0519501
$286$	0	0
$287$	2.54705	0.150348
$288$	0	0
$289$	−12.1172	−0.712778
$290$	0	0
$291$	−15.2865	−0.896108
$292$	0	0
$293$	−21.3192	−1.24548	−0.622741	−	0.782428i	$-0.713981\pi$
$293$	−21.3192	−1.24548	−0.622741	+	0.782428i	$0.713981\pi$
$294$	0	0
$295$	21.6331	1.25953
$296$	0	0
$297$	2.53118	0.146874
$298$	0	0
$299$	10.6555	0.616223
$300$	0	0
$301$	2.05318	0.118344
$302$	0	0
$303$	−5.94245	−0.341385
$304$	0	0
$305$	−7.77722	−0.445323
$306$	0	0
$307$	0.584621	0.0333661	0.0166830	−	0.999861i	$-0.494689\pi$
$307$	0.584621	0.0333661	0.0166830	+	0.999861i	$0.494689\pi$
$308$	0	0
$309$	6.09792	0.346899
$310$	0	0
$311$	−4.68786	−0.265824	−0.132912	−	0.991128i	$-0.542433\pi$
$311$	−4.68786	−0.265824	−0.132912	+	0.991128i	$0.542433\pi$
$312$	0	0
$313$	−8.48003	−0.479319	−0.239660	−	0.970857i	$-0.577036\pi$
$313$	−8.48003	−0.479319	−0.239660	+	0.970857i	$0.577036\pi$
$314$	0	0
$315$	−1.41088	−0.0794940
$316$	0	0
$317$	−13.4875	−0.757534	−0.378767	−	0.925492i	$-0.623652\pi$
$317$	−13.4875	−0.757534	−0.378767	+	0.925492i	$0.623652\pi$
$318$	0	0
$319$	−4.77942	−0.267596
$320$	0	0
$321$	7.88411	0.440048
$322$	0	0
$323$	−1.17374	−0.0653088
$324$	0	0
$325$	−9.57250	−0.530987
$326$	0	0
$327$	5.27205	0.291545
$328$	0	0
$329$	4.75528	0.262167
$330$	0	0
$331$	5.48751	0.301621	0.150810	−	0.988563i	$-0.451812\pi$
$331$	5.48751	0.301621	0.150810	+	0.988563i	$0.451812\pi$
$332$	0	0
$333$	5.08979	0.278919
$334$	0	0
$335$	−12.7171	−0.694807
$336$	0	0
$337$	−1.74408	−0.0950059	−0.0475029	−	0.998871i	$-0.515126\pi$
$337$	−1.74408	−0.0950059	−0.0475029	+	0.998871i	$0.515126\pi$
$338$	0	0
$339$	10.3340	0.561267
$340$	0	0
$341$	14.3686	0.778105
$342$	0	0
$343$	11.3392	0.612262
$344$	0	0
$345$	−4.17919	−0.225000
$346$	0	0
$347$	−2.83665	−0.152279	−0.0761397	−	0.997097i	$-0.524260\pi$
$347$	−2.83665	−0.152279	−0.0761397	+	0.997097i	$0.524260\pi$
$348$	0	0
$349$	−24.4468	−1.30861	−0.654305	−	0.756231i	$-0.727039\pi$
$349$	−24.4468	−1.30861	−0.654305	+	0.756231i	$0.727039\pi$
$350$	0	0
$351$	−4.20970	−0.224697
$352$	0	0
$353$	14.3383	0.763151	0.381575	−	0.924338i	$-0.375382\pi$
$353$	14.3383	0.763151	0.381575	+	0.924338i	$0.375382\pi$
$354$	0	0
$355$	−4.78720	−0.254078
$356$	0	0
$357$	−1.88822	−0.0999353
$358$	0	0
$359$	−8.01626	−0.423082	−0.211541	−	0.977369i	$-0.567848\pi$
$359$	−8.01626	−0.423082	−0.211541	+	0.977369i	$0.567848\pi$
$360$	0	0
$361$	−18.7179	−0.985150
$362$	0	0
$363$	4.59314	0.241077
$364$	0	0
$365$	16.7161	0.874962
$366$	0	0
$367$	−20.0762	−1.04797	−0.523985	−	0.851727i	$-0.675555\pi$
$367$	−20.0762	−1.04797	−0.523985	+	0.851727i	$0.675555\pi$
$368$	0	0
$369$	−2.98069	−0.155169
$370$	0	0
$371$	8.35323	0.433678
$372$	0	0
$373$	8.46349	0.438223	0.219111	−	0.975700i	$-0.429684\pi$
$373$	8.46349	0.438223	0.219111	+	0.975700i	$0.429684\pi$
$374$	0	0
$375$	12.0099	0.620186
$376$	0	0
$377$	7.94885	0.409387
$378$	0	0
$379$	−1.48816	−0.0764415	−0.0382207	−	0.999269i	$-0.512169\pi$
$379$	−1.48816	−0.0764415	−0.0382207	+	0.999269i	$0.512169\pi$
$380$	0	0
$381$	6.85040	0.350957
$382$	0	0
$383$	−19.8276	−1.01314	−0.506571	−	0.862198i	$-0.669087\pi$
$383$	−19.8276	−1.01314	−0.506571	+	0.862198i	$0.669087\pi$
$384$	0	0
$385$	3.57118	0.182004
$386$	0	0
$387$	−2.40275	−0.122139
$388$	0	0
$389$	−38.8696	−1.97077	−0.985384	−	0.170346i	$-0.945512\pi$
$389$	−38.8696	−1.97077	−0.985384	+	0.170346i	$0.945512\pi$
$390$	0	0
$391$	−5.59314	−0.282857
$392$	0	0
$393$	−3.86386	−0.194906
$394$	0	0
$395$	−8.22356	−0.413772
$396$	0	0
$397$	33.5720	1.68493	0.842464	−	0.538752i	$-0.181104\pi$
$397$	33.5720	1.68493	0.842464	+	0.538752i	$0.181104\pi$
$398$	0	0
$399$	0.453899	0.0227234
$400$	0	0
$401$	−3.45497	−0.172533	−0.0862664	−	0.996272i	$-0.527494\pi$
$401$	−3.45497	−0.172533	−0.0862664	+	0.996272i	$0.527494\pi$
$402$	0	0
$403$	−23.8970	−1.19040
$404$	0	0
$405$	1.65109	0.0820431
$406$	0	0
$407$	−12.8832	−0.638595
$408$	0	0
$409$	−4.60434	−0.227670	−0.113835	−	0.993500i	$-0.536314\pi$
$409$	−4.60434	−0.227670	−0.113835	+	0.993500i	$0.536314\pi$
$410$	0	0
$411$	7.21904	0.356089
$412$	0	0
$413$	−11.1961	−0.550926
$414$	0	0
$415$	13.9397	0.684271
$416$	0	0
$417$	12.6524	0.619592
$418$	0	0
$419$	−8.10230	−0.395823	−0.197912	−	0.980220i	$-0.563416\pi$
$419$	−8.10230	−0.395823	−0.197912	+	0.980220i	$0.563416\pi$
$420$	0	0
$421$	−31.1140	−1.51640	−0.758201	−	0.652021i	$-0.773921\pi$
$421$	−31.1140	−1.51640	−0.758201	+	0.652021i	$0.773921\pi$
$422$	0	0
$423$	−5.56488	−0.270574
$424$	0	0
$425$	5.02467	0.243732
$426$	0	0
$427$	4.02508	0.194787
$428$	0	0
$429$	10.6555	0.514452
$430$	0	0
$431$	−24.3034	−1.17065	−0.585327	−	0.810797i	$-0.699034\pi$
$431$	−24.3034	−1.17065	−0.585327	+	0.810797i	$0.699034\pi$
$432$	0	0
$433$	−34.8837	−1.67640	−0.838202	−	0.545360i	$-0.816393\pi$
$433$	−34.8837	−1.67640	−0.838202	+	0.545360i	$0.816393\pi$
$434$	0	0
$435$	−3.11762	−0.149478
$436$	0	0
$437$	1.34450	0.0643163
$438$	0	0
$439$	16.8611	0.804736	0.402368	−	0.915478i	$-0.368187\pi$
$439$	16.8611	0.804736	0.402368	+	0.915478i	$0.368187\pi$
$440$	0	0
$441$	−6.26980	−0.298562
$442$	0	0
$443$	5.69041	0.270360	0.135180	−	0.990821i	$-0.456839\pi$
$443$	5.69041	0.270360	0.135180	+	0.990821i	$0.456839\pi$
$444$	0	0
$445$	12.9707	0.614870
$446$	0	0
$447$	−10.3618	−0.490098
$448$	0	0
$449$	27.9427	1.31870	0.659350	−	0.751837i	$-0.270832\pi$
$449$	27.9427	1.31870	0.659350	+	0.751837i	$0.270832\pi$
$450$	0	0
$451$	7.54467	0.355264
$452$	0	0
$453$	4.44991	0.209075
$454$	0	0
$455$	−5.93937	−0.278442
$456$	0	0
$457$	5.77644	0.270211	0.135105	−	0.990831i	$-0.456863\pi$
$457$	5.77644	0.270211	0.135105	+	0.990831i	$0.456863\pi$
$458$	0	0
$459$	2.20970	0.103140
$460$	0	0
$461$	−25.8970	−1.20615	−0.603073	−	0.797686i	$-0.706057\pi$
$461$	−25.8970	−1.20615	−0.603073	+	0.797686i	$0.706057\pi$
$462$	0	0
$463$	6.79203	0.315652	0.157826	−	0.987467i	$-0.449551\pi$
$463$	6.79203	0.315652	0.157826	+	0.987467i	$0.449551\pi$
$464$	0	0
$465$	9.37266	0.434647
$466$	0	0
$467$	−36.6718	−1.69697	−0.848483	−	0.529222i	$-0.822484\pi$
$467$	−36.6718	−1.69697	−0.848483	+	0.529222i	$0.822484\pi$
$468$	0	0
$469$	6.58168	0.303914
$470$	0	0
$471$	17.4943	0.806095
$472$	0	0
$473$	6.08178	0.279641
$474$	0	0
$475$	−1.20785	−0.0554200
$476$	0	0
$477$	−9.77541	−0.447585
$478$	0	0
$479$	5.76390	0.263359	0.131680	−	0.991292i	$-0.457963\pi$
$479$	5.76390	0.263359	0.131680	+	0.991292i	$0.457963\pi$
$480$	0	0
$481$	21.4265	0.976964
$482$	0	0
$483$	2.16293	0.0984167
$484$	0	0
$485$	25.2393	1.14606
$486$	0	0
$487$	6.38928	0.289526	0.144763	−	0.989466i	$-0.453758\pi$
$487$	6.38928	0.289526	0.144763	+	0.989466i	$0.453758\pi$
$488$	0	0
$489$	−2.65654	−0.120133
$490$	0	0
$491$	−36.4445	−1.64472	−0.822358	−	0.568970i	$-0.807342\pi$
$491$	−36.4445	−1.64472	−0.822358	+	0.568970i	$0.807342\pi$
$492$	0	0
$493$	−4.17240	−0.187916
$494$	0	0
$495$	−4.17919	−0.187841
$496$	0	0
$497$	2.47760	0.111136
$498$	0	0
$499$	6.17997	0.276654	0.138327	−	0.990387i	$-0.455828\pi$
$499$	6.17997	0.276654	0.138327	+	0.990387i	$0.455828\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−23.3480	−1.04103	−0.520517	−	0.853851i	$-0.674261\pi$
$503$	−23.3480	−1.04103	−0.520517	+	0.853851i	$0.674261\pi$
$504$	0	0
$505$	9.81149	0.436606
$506$	0	0
$507$	−4.72157	−0.209692
$508$	0	0
$509$	10.7229	0.475285	0.237642	−	0.971353i	$-0.423625\pi$
$509$	10.7229	0.475285	0.237642	+	0.971353i	$0.423625\pi$
$510$	0	0
$511$	−8.65139	−0.382715
$512$	0	0
$513$	−0.531177	−0.0234520
$514$	0	0
$515$	−10.0682	−0.443658
$516$	0	0
$517$	14.0857	0.619489
$518$	0	0
$519$	23.2033	1.01851
$520$	0	0
$521$	7.90887	0.346494	0.173247	−	0.984878i	$-0.444574\pi$
$521$	7.90887	0.346494	0.173247	+	0.984878i	$0.444574\pi$
$522$	0	0
$523$	5.73450	0.250752	0.125376	−	0.992109i	$-0.459986\pi$
$523$	5.73450	0.250752	0.125376	+	0.992109i	$0.459986\pi$
$524$	0	0
$525$	−1.94309	−0.0848036
$526$	0	0
$527$	12.5437	0.546413
$528$	0	0
$529$	−16.5931	−0.721441
$530$	0	0
$531$	13.1023	0.568593
$532$	0	0
$533$	−12.5478	−0.543507
$534$	0	0
$535$	−13.0173	−0.562789
$536$	0	0
$537$	−8.49257	−0.366481
$538$	0	0
$539$	15.8700	0.683569
$540$	0	0
$541$	2.34042	0.100623	0.0503113	−	0.998734i	$-0.483979\pi$
$541$	2.34042	0.100623	0.0503113	+	0.998734i	$0.483979\pi$
$542$	0	0
$543$	15.7768	0.677049
$544$	0	0
$545$	−8.70462	−0.372865
$546$	0	0
$547$	−38.7572	−1.65714	−0.828569	−	0.559887i	$-0.810845\pi$
$547$	−38.7572	−1.65714	−0.828569	+	0.559887i	$0.810845\pi$
$548$	0	0
$549$	−4.71037	−0.201034
$550$	0	0
$551$	1.00298	0.0427284
$552$	0	0
$553$	4.25608	0.180987
$554$	0	0
$555$	−8.40368	−0.356716
$556$	0	0
$557$	−11.4821	−0.486514	−0.243257	−	0.969962i	$-0.578216\pi$
$557$	−11.4821	−0.486514	−0.243257	+	0.969962i	$0.578216\pi$
$558$	0	0
$559$	−10.1148	−0.427812
$560$	0	0
$561$	−5.59314	−0.236143
$562$	0	0
$563$	6.10806	0.257424	0.128712	−	0.991682i	$-0.458916\pi$
$563$	6.10806	0.257424	0.128712	+	0.991682i	$0.458916\pi$
$564$	0	0
$565$	−17.0624	−0.717819
$566$	0	0
$567$	−0.854515	−0.0358863
$568$	0	0
$569$	−3.66637	−0.153702	−0.0768511	−	0.997043i	$-0.524487\pi$
$569$	−3.66637	−0.153702	−0.0768511	+	0.997043i	$0.524487\pi$
$570$	0	0
$571$	−41.9334	−1.75486	−0.877429	−	0.479706i	$-0.840743\pi$
$571$	−41.9334	−1.75486	−0.877429	+	0.479706i	$0.840743\pi$
$572$	0	0
$573$	2.73902	0.114424
$574$	0	0
$575$	−5.75568	−0.240028
$576$	0	0
$577$	20.6496	0.859656	0.429828	−	0.902911i	$-0.358574\pi$
$577$	20.6496	0.859656	0.429828	+	0.902911i	$0.358574\pi$
$578$	0	0
$579$	6.18906	0.257208
$580$	0	0
$581$	−7.21444	−0.299305
$582$	0	0
$583$	24.7433	1.02476
$584$	0	0
$585$	6.95058	0.287371
$586$	0	0
$587$	7.28335	0.300616	0.150308	−	0.988639i	$-0.451973\pi$
$587$	7.28335	0.300616	0.150308	+	0.988639i	$0.451973\pi$
$588$	0	0
$589$	−3.01531	−0.124244
$590$	0	0
$591$	−5.19039	−0.213504
$592$	0	0
$593$	34.0938	1.40007	0.700033	−	0.714111i	$-0.253169\pi$
$593$	34.0938	1.40007	0.700033	+	0.714111i	$0.253169\pi$
$594$	0	0
$595$	3.11762	0.127810
$596$	0	0
$597$	−19.5553	−0.800346
$598$	0	0
$599$	15.6243	0.638390	0.319195	−	0.947689i	$-0.396588\pi$
$599$	15.6243	0.638390	0.319195	+	0.947689i	$0.396588\pi$
$600$	0	0
$601$	−38.5886	−1.57406	−0.787031	−	0.616913i	$-0.788383\pi$
$601$	−38.5886	−1.57406	−0.787031	+	0.616913i	$0.788383\pi$
$602$	0	0
$603$	−7.70224	−0.313659
$604$	0	0
$605$	−7.58367	−0.308320
$606$	0	0
$607$	−35.9658	−1.45981	−0.729903	−	0.683551i	$-0.760435\pi$
$607$	−35.9658	−1.45981	−0.729903	+	0.683551i	$0.760435\pi$
$608$	0	0
$609$	1.61351	0.0653829
$610$	0	0
$611$	−23.4265	−0.947734
$612$	0	0
$613$	47.5214	1.91937	0.959686	−	0.281075i	$-0.0906909\pi$
$613$	47.5214	1.91937	0.959686	+	0.281075i	$0.0906909\pi$
$614$	0	0
$615$	4.92138	0.198449
$616$	0	0
$617$	−3.53213	−0.142198	−0.0710990	−	0.997469i	$-0.522651\pi$
$617$	−3.53213	−0.142198	−0.0710990	+	0.997469i	$0.522651\pi$
$618$	0	0
$619$	15.2317	0.612213	0.306106	−	0.951997i	$-0.400974\pi$
$619$	15.2317	0.612213	0.306106	+	0.951997i	$0.400974\pi$
$620$	0	0
$621$	−2.53118	−0.101573
$622$	0	0
$623$	−6.71295	−0.268949
$624$	0	0
$625$	−8.45974	−0.338390
$626$	0	0
$627$	1.34450	0.0536943
$628$	0	0
$629$	−11.2469	−0.448444
$630$	0	0
$631$	−20.7391	−0.825609	−0.412805	−	0.910820i	$-0.635451\pi$
$631$	−20.7391	−0.825609	−0.412805	+	0.910820i	$0.635451\pi$
$632$	0	0
$633$	−3.22489	−0.128178
$634$	0	0
$635$	−11.3106	−0.448848
$636$	0	0
$637$	−26.3940	−1.04577
$638$	0	0
$639$	−2.89942	−0.114699
$640$	0	0
$641$	9.95963	0.393382	0.196691	−	0.980466i	$-0.436980\pi$
$641$	9.95963	0.393382	0.196691	+	0.980466i	$0.436980\pi$
$642$	0	0
$643$	−7.42381	−0.292767	−0.146383	−	0.989228i	$-0.546763\pi$
$643$	−7.42381	−0.292767	−0.146383	+	0.989228i	$0.546763\pi$
$644$	0	0
$645$	3.96714	0.156206
$646$	0	0
$647$	32.7030	1.28569	0.642843	−	0.765998i	$-0.277755\pi$
$647$	32.7030	1.28569	0.642843	+	0.765998i	$0.277755\pi$
$648$	0	0
$649$	−33.1643	−1.30181
$650$	0	0
$651$	−4.85079	−0.190118
$652$	0	0
$653$	−1.21443	−0.0475244	−0.0237622	−	0.999718i	$-0.507564\pi$
$653$	−1.21443	−0.0475244	−0.0237622	+	0.999718i	$0.507564\pi$
$654$	0	0
$655$	6.37956	0.249270
$656$	0	0
$657$	10.1243	0.394987
$658$	0	0
$659$	−8.01796	−0.312335	−0.156168	−	0.987731i	$-0.549914\pi$
$659$	−8.01796	−0.312335	−0.156168	+	0.987731i	$0.549914\pi$
$660$	0	0
$661$	5.15002	0.200313	0.100156	−	0.994972i	$-0.468066\pi$
$661$	5.15002	0.200313	0.100156	+	0.994972i	$0.468066\pi$
$662$	0	0
$663$	9.30217	0.361266
$664$	0	0
$665$	−0.749426	−0.0290615
$666$	0	0
$667$	4.77942	0.185060
$668$	0	0
$669$	−19.0296	−0.735727
$670$	0	0
$671$	11.9228	0.460274
$672$	0	0
$673$	0.00575339	0.000221777 0	0.000110888	−	1.00000i	$-0.499965\pi$
$673$	0.00575339	0.000221777 0	0.000110888		1.00000i	$0.499965\pi$
$674$	0	0
$675$	2.27391	0.0875230
$676$	0	0
$677$	11.4563	0.440301	0.220150	−	0.975466i	$-0.429345\pi$
$677$	11.4563	0.440301	0.220150	+	0.975466i	$0.429345\pi$
$678$	0	0
$679$	−13.0625	−0.501293
$680$	0	0
$681$	14.2761	0.547060
$682$	0	0
$683$	−0.590488	−0.0225944	−0.0112972	−	0.999936i	$-0.503596\pi$
$683$	−0.590488	−0.0225944	−0.0112972	+	0.999936i	$0.503596\pi$
$684$	0	0
$685$	−11.9193	−0.455412
$686$	0	0
$687$	2.64668	0.100977
$688$	0	0
$689$	−41.1515	−1.56775
$690$	0	0
$691$	−30.3676	−1.15524	−0.577619	−	0.816307i	$-0.696018\pi$
$691$	−30.3676	−1.15524	−0.577619	+	0.816307i	$0.696018\pi$
$692$	0	0
$693$	2.16293	0.0821629
$694$	0	0
$695$	−20.8902	−0.792412
$696$	0	0
$697$	6.58644	0.249479
$698$	0	0
$699$	−6.97883	−0.263964
$700$	0	0
$701$	18.6718	0.705223	0.352611	−	0.935770i	$-0.385294\pi$
$701$	18.6718	0.705223	0.352611	+	0.935770i	$0.385294\pi$
$702$	0	0
$703$	2.70358	0.101967
$704$	0	0
$705$	9.18810	0.346044
$706$	0	0
$707$	−5.07791	−0.190974
$708$	0	0
$709$	−18.8110	−0.706461	−0.353230	−	0.935536i	$-0.614917\pi$
$709$	−18.8110	−0.706461	−0.353230	+	0.935536i	$0.614917\pi$
$710$	0	0
$711$	−4.98069	−0.186791
$712$	0	0
$713$	−14.3686	−0.538110
$714$	0	0
$715$	−17.5931	−0.657946
$716$	0	0
$717$	−13.8841	−0.518512
$718$	0	0
$719$	36.8314	1.37358	0.686789	−	0.726857i	$-0.259020\pi$
$719$	36.8314	1.37358	0.686789	+	0.726857i	$0.259020\pi$
$720$	0	0
$721$	5.21077	0.194059
$722$	0	0
$723$	8.05526	0.299578
$724$	0	0
$725$	−4.29366	−0.159462
$726$	0	0
$727$	−12.0434	−0.446666	−0.223333	−	0.974742i	$-0.571694\pi$
$727$	−12.0434	−0.446666	−0.223333	+	0.974742i	$0.571694\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.30935	0.196373
$732$	0	0
$733$	0.707302	0.0261248	0.0130624	−	0.999915i	$-0.495842\pi$
$733$	0.707302	0.0261248	0.0130624	+	0.999915i	$0.495842\pi$
$734$	0	0
$735$	10.3520	0.381839
$736$	0	0
$737$	19.4957	0.718134
$738$	0	0
$739$	2.55051	0.0938221	0.0469110	−	0.998899i	$-0.485062\pi$
$739$	2.55051	0.0938221	0.0469110	+	0.998899i	$0.485062\pi$
$740$	0	0
$741$	−2.23610	−0.0821451
$742$	0	0
$743$	2.44930	0.0898561	0.0449280	−	0.998990i	$-0.485694\pi$
$743$	2.44930	0.0898561	0.0449280	+	0.998990i	$0.485694\pi$
$744$	0	0
$745$	17.1083	0.626800
$746$	0	0
$747$	8.44272	0.308903
$748$	0	0
$749$	6.73709	0.246168
$750$	0	0
$751$	21.8324	0.796676	0.398338	−	0.917239i	$-0.369587\pi$
$751$	21.8324	0.796676	0.398338	+	0.917239i	$0.369587\pi$
$752$	0	0
$753$	−14.6721	−0.534683
$754$	0	0
$755$	−7.34718	−0.267391
$756$	0	0
$757$	−28.7582	−1.04524	−0.522618	−	0.852567i	$-0.675045\pi$
$757$	−28.7582	−1.04524	−0.522618	+	0.852567i	$0.675045\pi$
$758$	0	0
$759$	6.40686	0.232554
$760$	0	0
$761$	35.8160	1.29833	0.649164	−	0.760648i	$-0.275119\pi$
$761$	35.8160	1.29833	0.649164	+	0.760648i	$0.275119\pi$
$762$	0	0
$763$	4.50505	0.163094
$764$	0	0
$765$	−3.64840	−0.131908
$766$	0	0
$767$	55.1569	1.99160
$768$	0	0
$769$	−53.6634	−1.93515	−0.967576	−	0.252579i	$-0.918721\pi$
$769$	−53.6634	−1.93515	−0.967576	+	0.252579i	$0.918721\pi$
$770$	0	0
$771$	4.86892	0.175350
$772$	0	0
$773$	8.24164	0.296431	0.148216	−	0.988955i	$-0.452647\pi$
$773$	8.24164	0.296431	0.148216	+	0.988955i	$0.452647\pi$
$774$	0	0
$775$	12.9082	0.463678
$776$	0	0
$777$	4.34930	0.156030
$778$	0	0
$779$	−1.58328	−0.0567268
$780$	0	0
$781$	7.33896	0.262609
$782$	0	0
$783$	−1.88822	−0.0674796
$784$	0	0
$785$	−28.8846	−1.03094
$786$	0	0
$787$	5.36626	0.191286	0.0956432	−	0.995416i	$-0.469509\pi$
$787$	5.36626	0.191286	0.0956432	+	0.995416i	$0.469509\pi$
$788$	0	0
$789$	−2.40101	−0.0854781
$790$	0	0
$791$	8.83058	0.313979
$792$	0	0
$793$	−19.8292	−0.704157
$794$	0	0
$795$	16.1400	0.572428
$796$	0	0
$797$	21.2765	0.753652	0.376826	−	0.926284i	$-0.377015\pi$
$797$	21.2765	0.753652	0.376826	+	0.926284i	$0.377015\pi$
$798$	0	0
$799$	12.2967	0.435027
$800$	0	0
$801$	7.85585	0.277573
$802$	0	0
$803$	−25.6264	−0.904338
$804$	0	0
$805$	−3.57118	−0.125868
$806$	0	0
$807$	−20.9828	−0.738631
$808$	0	0
$809$	14.5806	0.512626	0.256313	−	0.966594i	$-0.417492\pi$
$809$	14.5806	0.512626	0.256313	+	0.966594i	$0.417492\pi$
$810$	0	0
$811$	9.45158	0.331890	0.165945	−	0.986135i	$-0.446933\pi$
$811$	9.45158	0.331890	0.165945	+	0.986135i	$0.446933\pi$
$812$	0	0
$813$	−14.2321	−0.499142
$814$	0	0
$815$	4.38617	0.153641
$816$	0	0
$817$	−1.27628	−0.0446516
$818$	0	0
$819$	−3.59725	−0.125698
$820$	0	0
$821$	18.1430	0.633195	0.316598	−	0.948560i	$-0.397460\pi$
$821$	18.1430	0.633195	0.316598	+	0.948560i	$0.397460\pi$
$822$	0	0
$823$	−42.8383	−1.49325	−0.746624	−	0.665246i	$-0.768327\pi$
$823$	−42.8383	−1.49325	−0.746624	+	0.665246i	$0.768327\pi$
$824$	0	0
$825$	−5.75568	−0.200387
$826$	0	0
$827$	5.41407	0.188266	0.0941328	−	0.995560i	$-0.469992\pi$
$827$	5.41407	0.188266	0.0941328	+	0.995560i	$0.469992\pi$
$828$	0	0
$829$	−18.1301	−0.629685	−0.314843	−	0.949144i	$-0.601952\pi$
$829$	−18.1301	−0.629685	−0.314843	+	0.949144i	$0.601952\pi$
$830$	0	0
$831$	0.481335	0.0166973
$832$	0	0
$833$	13.8544	0.480026
$834$	0	0
$835$	1.65109	0.0571382
$836$	0	0
$837$	5.67666	0.196214
$838$	0	0
$839$	40.2245	1.38870	0.694352	−	0.719636i	$-0.255691\pi$
$839$	40.2245	1.38870	0.694352	+	0.719636i	$0.255691\pi$
$840$	0	0
$841$	−25.4346	−0.877056
$842$	0	0
$843$	14.3404	0.493909
$844$	0	0
$845$	7.79572	0.268181
$846$	0	0
$847$	3.92491	0.134861
$848$	0	0
$849$	8.17958	0.280723
$850$	0	0
$851$	12.8832	0.441629
$852$	0	0
$853$	−43.2985	−1.48251	−0.741256	−	0.671222i	$-0.765770\pi$
$853$	−43.2985	−1.48251	−0.741256	+	0.671222i	$0.765770\pi$
$854$	0	0
$855$	0.877019	0.0299934
$856$	0	0
$857$	46.8933	1.60184	0.800922	−	0.598768i	$-0.204343\pi$
$857$	46.8933	1.60184	0.800922	+	0.598768i	$0.204343\pi$
$858$	0	0
$859$	−28.6218	−0.976562	−0.488281	−	0.872686i	$-0.662376\pi$
$859$	−28.6218	−0.976562	−0.488281	+	0.872686i	$0.662376\pi$
$860$	0	0
$861$	−2.54705	−0.0868032
$862$	0	0
$863$	36.3494	1.23735	0.618673	−	0.785648i	$-0.287670\pi$
$863$	36.3494	1.23735	0.618673	+	0.785648i	$0.287670\pi$
$864$	0	0
$865$	−38.3107	−1.30260
$866$	0	0
$867$	12.1172	0.411523
$868$	0	0
$869$	12.6070	0.427664
$870$	0	0
$871$	−32.4241	−1.09865
$872$	0	0
$873$	15.2865	0.517368
$874$	0	0
$875$	10.2626	0.346939
$876$	0	0
$877$	0.664133	0.0224262	0.0112131	−	0.999937i	$-0.496431\pi$
$877$	0.664133	0.0224262	0.0112131	+	0.999937i	$0.496431\pi$
$878$	0	0
$879$	21.3192	0.719079
$880$	0	0
$881$	11.9964	0.404169	0.202084	−	0.979368i	$-0.435228\pi$
$881$	11.9964	0.404169	0.202084	+	0.979368i	$0.435228\pi$
$882$	0	0
$883$	−10.6695	−0.359056	−0.179528	−	0.983753i	$-0.557457\pi$
$883$	−10.6695	−0.359056	−0.179528	+	0.983753i	$0.557457\pi$
$884$	0	0
$885$	−21.6331	−0.727188
$886$	0	0
$887$	−32.9993	−1.10801	−0.554003	−	0.832514i	$-0.686901\pi$
$887$	−32.9993	−1.10801	−0.554003	+	0.832514i	$0.686901\pi$
$888$	0	0
$889$	5.85378	0.196329
$890$	0	0
$891$	−2.53118	−0.0847976
$892$	0	0
$893$	−2.95594	−0.0989168
$894$	0	0
$895$	14.0220	0.468702
$896$	0	0
$897$	−10.6555	−0.355777
$898$	0	0
$899$	−10.7188	−0.357492
$900$	0	0
$901$	21.6007	0.719624
$902$	0	0
$903$	−2.05318	−0.0683257
$904$	0	0
$905$	−26.0489	−0.865895
$906$	0	0
$907$	−11.9285	−0.396080	−0.198040	−	0.980194i	$-0.563458\pi$
$907$	−11.9285	−0.396080	−0.198040	+	0.980194i	$0.563458\pi$
$908$	0	0
$909$	5.94245	0.197098
$910$	0	0
$911$	−13.5328	−0.448362	−0.224181	−	0.974548i	$-0.571971\pi$
$911$	−13.5328	−0.448362	−0.224181	+	0.974548i	$0.571971\pi$
$912$	0	0
$913$	−21.3700	−0.707245
$914$	0	0
$915$	7.77722	0.257107
$916$	0	0
$917$	−3.30173	−0.109033
$918$	0	0
$919$	−22.9649	−0.757541	−0.378770	−	0.925491i	$-0.623653\pi$
$919$	−22.9649	−0.757541	−0.378770	+	0.925491i	$0.623653\pi$
$920$	0	0
$921$	−0.584621	−0.0192639
$922$	0	0
$923$	−12.2057	−0.401756
$924$	0	0
$925$	−11.5738	−0.380543
$926$	0	0
$927$	−6.09792	−0.200282
$928$	0	0
$929$	−23.4107	−0.768081	−0.384040	−	0.923316i	$-0.625468\pi$
$929$	−23.4107	−0.768081	−0.384040	+	0.923316i	$0.625468\pi$
$930$	0	0
$931$	−3.33038	−0.109149
$932$	0	0
$933$	4.68786	0.153474
$934$	0	0
$935$	9.23476	0.302009
$936$	0	0
$937$	−6.90210	−0.225482	−0.112741	−	0.993624i	$-0.535963\pi$
$937$	−6.90210	−0.225482	−0.112741	+	0.993624i	$0.535963\pi$
$938$	0	0
$939$	8.48003	0.276735
$940$	0	0
$941$	−16.7636	−0.546476	−0.273238	−	0.961946i	$-0.588095\pi$
$941$	−16.7636	−0.546476	−0.273238	+	0.961946i	$0.588095\pi$
$942$	0	0
$943$	−7.54467	−0.245688
$944$	0	0
$945$	1.41088	0.0458959
$946$	0	0
$947$	23.3954	0.760248	0.380124	−	0.924936i	$-0.375881\pi$
$947$	23.3954	0.760248	0.380124	+	0.924936i	$0.375881\pi$
$948$	0	0
$949$	42.6203	1.38351
$950$	0	0
$951$	13.4875	0.437362
$952$	0	0
$953$	6.69384	0.216835	0.108417	−	0.994105i	$-0.465422\pi$
$953$	6.69384	0.216835	0.108417	+	0.994105i	$0.465422\pi$
$954$	0	0
$955$	−4.52235	−0.146340
$956$	0	0
$957$	4.77942	0.154497
$958$	0	0
$959$	6.16878	0.199200
$960$	0	0
$961$	1.22449	0.0394997
$962$	0	0
$963$	−7.88411	−0.254062
$964$	0	0
$965$	−10.2187	−0.328950
$966$	0	0
$967$	−15.4593	−0.497139	−0.248570	−	0.968614i	$-0.579960\pi$
$967$	−15.4593	−0.497139	−0.248570	+	0.968614i	$0.579960\pi$
$968$	0	0
$969$	1.17374	0.0377060
$970$	0	0
$971$	−51.6335	−1.65700	−0.828499	−	0.559990i	$-0.810805\pi$
$971$	−51.6335	−1.65700	−0.828499	+	0.559990i	$0.810805\pi$
$972$	0	0
$973$	10.8117	0.346607
$974$	0	0
$975$	9.57250	0.306565
$976$	0	0
$977$	48.6172	1.55540	0.777701	−	0.628634i	$-0.216386\pi$
$977$	48.6172	1.55540	0.777701	+	0.628634i	$0.216386\pi$
$978$	0	0
$979$	−19.8846	−0.635513
$980$	0	0
$981$	−5.27205	−0.168324
$982$	0	0
$983$	32.0683	1.02282	0.511409	−	0.859337i	$-0.329124\pi$
$983$	32.0683	1.02282	0.511409	+	0.859337i	$0.329124\pi$
$984$	0	0
$985$	8.56979	0.273056
$986$	0	0
$987$	−4.75528	−0.151362
$988$	0	0
$989$	−6.08178	−0.193389
$990$	0	0
$991$	−2.24381	−0.0712769	−0.0356385	−	0.999365i	$-0.511346\pi$
$991$	−2.24381	−0.0712769	−0.0356385	+	0.999365i	$0.511346\pi$
$992$	0	0
$993$	−5.48751	−0.174141
$994$	0	0
$995$	32.2875	1.02358
$996$	0	0
$997$	−11.7175	−0.371096	−0.185548	−	0.982635i	$-0.559406\pi$
$997$	−11.7175	−0.371096	−0.185548	+	0.982635i	$0.559406\pi$
$998$	0	0
$999$	−5.08979	−0.161034

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.r.1.4		5
4.3	odd	2		1002.2.a.j.1.4	✓	5
12.11	even	2		3006.2.a.t.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.j.1.4	✓	5	4.3	odd	2
3006.2.a.t.1.2		5	12.11	even	2
8016.2.a.r.1.4		5	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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.65109 q^{5} -0.854515 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.65109 q^{5} -0.854515 q^{7} +1.00000 q^{9} -2.53118 q^{11} +4.20970 q^{13} -1.65109 q^{15} -2.20970 q^{17} +0.531177 q^{19} +0.854515 q^{21} +2.53118 q^{23} -2.27391 q^{25} -1.00000 q^{27} +1.88822 q^{29} -5.67666 q^{31} +2.53118 q^{33} -1.41088 q^{35} +5.08979 q^{37} -4.20970 q^{39} -2.98069 q^{41} -2.40275 q^{43} +1.65109 q^{45} -5.56488 q^{47} -6.26980 q^{49} +2.20970 q^{51} -9.77541 q^{53} -4.17919 q^{55} -0.531177 q^{57} +13.1023 q^{59} -4.71037 q^{61} -0.854515 q^{63} +6.95058 q^{65} -7.70224 q^{67} -2.53118 q^{69} -2.89942 q^{71} +10.1243 q^{73} +2.27391 q^{75} +2.16293 q^{77} -4.98069 q^{79} +1.00000 q^{81} +8.44272 q^{83} -3.64840 q^{85} -1.88822 q^{87} +7.85585 q^{89} -3.59725 q^{91} +5.67666 q^{93} +0.877019 q^{95} +15.2865 q^{97} -2.53118 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} - q^{5} - 9 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 - q^5 - 9 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} - q^{5} - 9 q^{7} + 5 q^{9} + 2 q^{11} + 6 q^{13} + q^{15} + 4 q^{17} - 12 q^{19} + 9 q^{21} - 2 q^{23} + 14 q^{25} - 5 q^{27} - 6 q^{29} - 9 q^{31} - 2 q^{33} - 3 q^{35} + 5 q^{37} - 6 q^{39} + 4 q^{41} - 18 q^{43} - q^{45} + 7 q^{47} + 16 q^{49} - 4 q^{51} + 3 q^{53} + 4 q^{55} + 12 q^{57} - 13 q^{59} + 16 q^{61} - 9 q^{63} - 10 q^{65} - 9 q^{67} + 2 q^{69} + 10 q^{71} + 8 q^{73} - 14 q^{75} + 6 q^{77} - 6 q^{79} + 5 q^{81} - q^{83} + 8 q^{85} + 6 q^{87} - 5 q^{89} - 12 q^{91} + 9 q^{93} - 2 q^{95} - 7 q^{97} + 2 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 - q^5 - 9 * q^7 + 5 * q^9 + 2 * q^11 + 6 * q^13 + q^15 + 4 * q^17 - 12 * q^19 + 9 * q^21 - 2 * q^23 + 14 * q^25 - 5 * q^27 - 6 * q^29 - 9 * q^31 - 2 * q^33 - 3 * q^35 + 5 * q^37 - 6 * q^39 + 4 * q^41 - 18 * q^43 - q^45 + 7 * q^47 + 16 * q^49 - 4 * q^51 + 3 * q^53 + 4 * q^55 + 12 * q^57 - 13 * q^59 + 16 * q^61 - 9 * q^63 - 10 * q^65 - 9 * q^67 + 2 * q^69 + 10 * q^71 + 8 * q^73 - 14 * q^75 + 6 * q^77 - 6 * q^79 + 5 * q^81 - q^83 + 8 * q^85 + 6 * q^87 - 5 * q^89 - 12 * q^91 + 9 * q^93 - 2 * q^95 - 7 * q^97 + 2 * q^99

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