LMFDB - Embedded newform 4016.2.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, 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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.242773.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 7x^{3} + 4x^{2} + 6x + 1 $ x^5 - x^4 - 7x^3 + 4x^2 + 6*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 502)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.37208$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.37208 q^{3} +1.64633 q^{5} +2.43111 q^{7} -1.11739 q^{9} +O(q^{10})$	$q+1.37208 q^{3} +1.64633 q^{5} +2.43111 q^{7} -1.11739 q^{9} -4.53201 q^{11} -5.20294 q^{13} +2.25890 q^{15} -2.87449 q^{17} -5.37515 q^{19} +3.33568 q^{21} +6.08556 q^{23} -2.28959 q^{25} -5.64940 q^{27} -2.08249 q^{29} +5.99013 q^{31} -6.21829 q^{33} +4.00241 q^{35} +0.717021 q^{37} -7.13887 q^{39} -7.17882 q^{41} +1.58731 q^{43} -1.83959 q^{45} +7.70536 q^{47} -1.08972 q^{49} -3.94404 q^{51} -11.8881 q^{53} -7.46119 q^{55} -7.37515 q^{57} -6.43658 q^{59} -12.6549 q^{61} -2.71649 q^{63} -8.56577 q^{65} -3.54736 q^{67} +8.34989 q^{69} +12.7859 q^{71} +10.8523 q^{73} -3.14151 q^{75} -11.0178 q^{77} -8.05902 q^{79} -4.39927 q^{81} -9.44495 q^{83} -4.73237 q^{85} -2.85735 q^{87} +1.22334 q^{89} -12.6489 q^{91} +8.21895 q^{93} -8.84928 q^{95} +10.9462 q^{97} +5.06402 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} - 6 q^{5} + q^{7}+O(q^{10}) $ 5 * q + q^3 - 6 * q^5 + q^7	$ 5 q + q^{3} - 6 q^{5} + q^{7} + 4 q^{11} - 7 q^{13} + 5 q^{15} - 8 q^{17} - 3 q^{19} - 10 q^{21} + 17 q^{23} - q^{25} + 4 q^{27} - 15 q^{29} + q^{31} - 10 q^{33} + 7 q^{35} - 5 q^{37} + 8 q^{39} - 12 q^{41} - q^{43} - 13 q^{45} + 19 q^{47} + 4 q^{49} - 7 q^{51} - 34 q^{53} - 17 q^{55} - 13 q^{57} + 10 q^{59} + 2 q^{63} + 6 q^{65} + 11 q^{67} + q^{69} + q^{71} + 3 q^{73} - 15 q^{75} - 30 q^{77} - 35 q^{79} - 3 q^{81} - 3 q^{83} - 13 q^{85} - 3 q^{87} + 15 q^{89} - 14 q^{91} + 15 q^{93} - 11 q^{95} - 2 q^{97} - 28 q^{99}+O(q^{100}) $ 5 * q + q^3 - 6 * q^5 + q^7 + 4 * q^11 - 7 * q^13 + 5 * q^15 - 8 * q^17 - 3 * q^19 - 10 * q^21 + 17 * q^23 - q^25 + 4 * q^27 - 15 * q^29 + q^31 - 10 * q^33 + 7 * q^35 - 5 * q^37 + 8 * q^39 - 12 * q^41 - q^43 - 13 * q^45 + 19 * q^47 + 4 * q^49 - 7 * q^51 - 34 * q^53 - 17 * q^55 - 13 * q^57 + 10 * q^59 + 2 * q^63 + 6 * q^65 + 11 * q^67 + q^69 + q^71 + 3 * q^73 - 15 * q^75 - 30 * q^77 - 35 * q^79 - 3 * q^81 - 3 * q^83 - 13 * q^85 - 3 * q^87 + 15 * q^89 - 14 * q^91 + 15 * q^93 - 11 * q^95 - 2 * q^97 - 28 * 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
$3$	1.37208	0.792172	0.396086	−	0.918213i	$-0.370368\pi$
$3$	1.37208	0.792172	0.396086	+	0.918213i	$0.370368\pi$
$4$	0	0
$5$	1.64633	0.736262	0.368131	−	0.929774i	$-0.379998\pi$
$5$	1.64633	0.736262	0.368131	+	0.929774i	$0.379998\pi$
$6$	0	0
$7$	2.43111	0.918872	0.459436	−	0.888211i	$-0.348052\pi$
$7$	2.43111	0.918872	0.459436	+	0.888211i	$0.348052\pi$
$8$	0	0
$9$	−1.11739	−0.372463
$10$	0	0
$11$	−4.53201	−1.36645	−0.683226	−	0.730207i	$-0.739424\pi$
$11$	−4.53201	−1.36645	−0.683226	+	0.730207i	$0.739424\pi$
$12$	0	0
$13$	−5.20294	−1.44304	−0.721519	−	0.692395i	$-0.756556\pi$
$13$	−5.20294	−1.44304	−0.721519	+	0.692395i	$0.756556\pi$
$14$	0	0
$15$	2.25890	0.583246
$16$	0	0
$17$	−2.87449	−0.697167	−0.348584	−	0.937278i	$-0.613337\pi$
$17$	−2.87449	−0.697167	−0.348584	+	0.937278i	$0.613337\pi$
$18$	0	0
$19$	−5.37515	−1.23314	−0.616572	−	0.787299i	$-0.711479\pi$
$19$	−5.37515	−1.23314	−0.616572	+	0.787299i	$0.711479\pi$
$20$	0	0
$21$	3.33568	0.727905
$22$	0	0
$23$	6.08556	1.26893	0.634463	−	0.772953i	$-0.281221\pi$
$23$	6.08556	1.26893	0.634463	+	0.772953i	$0.281221\pi$
$24$	0	0
$25$	−2.28959	−0.457919
$26$	0	0
$27$	−5.64940	−1.08723
$28$	0	0
$29$	−2.08249	−0.386708	−0.193354	−	0.981129i	$-0.561937\pi$
$29$	−2.08249	−0.386708	−0.193354	+	0.981129i	$0.561937\pi$
$30$	0	0
$31$	5.99013	1.07586	0.537929	−	0.842990i	$-0.319207\pi$
$31$	5.99013	1.07586	0.537929	+	0.842990i	$0.319207\pi$
$32$	0	0
$33$	−6.21829	−1.08247
$34$	0	0
$35$	4.00241	0.676530
$36$	0	0
$37$	0.717021	0.117878	0.0589388	−	0.998262i	$-0.481228\pi$
$37$	0.717021	0.117878	0.0589388	+	0.998262i	$0.481228\pi$
$38$	0	0
$39$	−7.13887	−1.14313
$40$	0	0
$41$	−7.17882	−1.12114	−0.560572	−	0.828106i	$-0.689419\pi$
$41$	−7.17882	−1.12114	−0.560572	+	0.828106i	$0.689419\pi$
$42$	0	0
$43$	1.58731	0.242062	0.121031	−	0.992649i	$-0.461380\pi$
$43$	1.58731	0.242062	0.121031	+	0.992649i	$0.461380\pi$
$44$	0	0
$45$	−1.83959	−0.274230
$46$	0	0
$47$	7.70536	1.12394	0.561971	−	0.827157i	$-0.310043\pi$
$47$	7.70536	1.12394	0.561971	+	0.827157i	$0.310043\pi$
$48$	0	0
$49$	−1.08972	−0.155674
$50$	0	0
$51$	−3.94404	−0.552276
$52$	0	0
$53$	−11.8881	−1.63295	−0.816477	−	0.577378i	$-0.804076\pi$
$53$	−11.8881	−1.63295	−0.816477	+	0.577378i	$0.804076\pi$
$54$	0	0
$55$	−7.46119	−1.00607
$56$	0	0
$57$	−7.37515	−0.976862
$58$	0	0
$59$	−6.43658	−0.837972	−0.418986	−	0.907993i	$-0.637614\pi$
$59$	−6.43658	−0.837972	−0.418986	+	0.907993i	$0.637614\pi$
$60$	0	0
$61$	−12.6549	−1.62029	−0.810145	−	0.586229i	$-0.800612\pi$
$61$	−12.6549	−1.62029	−0.810145	+	0.586229i	$0.800612\pi$
$62$	0	0
$63$	−2.71649	−0.342246
$64$	0	0
$65$	−8.56577	−1.06245
$66$	0	0
$67$	−3.54736	−0.433378	−0.216689	−	0.976241i	$-0.569526\pi$
$67$	−3.54736	−0.433378	−0.216689	+	0.976241i	$0.569526\pi$
$68$	0	0
$69$	8.34989	1.00521
$70$	0	0
$71$	12.7859	1.51740	0.758701	−	0.651439i	$-0.225834\pi$
$71$	12.7859	1.51740	0.758701	+	0.651439i	$0.225834\pi$
$72$	0	0
$73$	10.8523	1.27017	0.635085	−	0.772442i	$-0.280965\pi$
$73$	10.8523	1.27017	0.635085	+	0.772442i	$0.280965\pi$
$74$	0	0
$75$	−3.14151	−0.362751
$76$	0	0
$77$	−11.0178	−1.25559
$78$	0	0
$79$	−8.05902	−0.906711	−0.453356	−	0.891330i	$-0.649773\pi$
$79$	−8.05902	−0.906711	−0.453356	+	0.891330i	$0.649773\pi$
$80$	0	0
$81$	−4.39927	−0.488808
$82$	0	0
$83$	−9.44495	−1.03672	−0.518359	−	0.855163i	$-0.673457\pi$
$83$	−9.44495	−1.03672	−0.518359	+	0.855163i	$0.673457\pi$
$84$	0	0
$85$	−4.73237	−0.513297
$86$	0	0
$87$	−2.85735	−0.306340
$88$	0	0
$89$	1.22334	0.129674	0.0648369	−	0.997896i	$-0.479347\pi$
$89$	1.22334	0.129674	0.0648369	+	0.997896i	$0.479347\pi$
$90$	0	0
$91$	−12.6489	−1.32597
$92$	0	0
$93$	8.21895	0.852266
$94$	0	0
$95$	−8.84928	−0.907917
$96$	0	0
$97$	10.9462	1.11142	0.555710	−	0.831376i	$-0.312446\pi$
$97$	10.9462	1.11142	0.555710	+	0.831376i	$0.312446\pi$
$98$	0	0
$99$	5.06402	0.508953
$100$	0	0
$101$	−1.48526	−0.147789	−0.0738946	−	0.997266i	$-0.523543\pi$
$101$	−1.48526	−0.147789	−0.0738946	+	0.997266i	$0.523543\pi$
$102$	0	0
$103$	−17.4680	−1.72117	−0.860586	−	0.509306i	$-0.829902\pi$
$103$	−17.4680	−1.72117	−0.860586	+	0.509306i	$0.829902\pi$
$104$	0	0
$105$	5.49163	0.535929
$106$	0	0
$107$	−6.58735	−0.636823	−0.318412	−	0.947953i	$-0.603149\pi$
$107$	−6.58735	−0.636823	−0.318412	+	0.947953i	$0.603149\pi$
$108$	0	0
$109$	16.8826	1.61706	0.808528	−	0.588458i	$-0.200265\pi$
$109$	16.8826	1.61706	0.808528	+	0.588458i	$0.200265\pi$
$110$	0	0
$111$	0.983813	0.0933794
$112$	0	0
$113$	−1.25952	−0.118485	−0.0592426	−	0.998244i	$-0.518869\pi$
$113$	−1.25952	−0.118485	−0.0592426	+	0.998244i	$0.518869\pi$
$114$	0	0
$115$	10.0188	0.934261
$116$	0	0
$117$	5.81371	0.537478
$118$	0	0
$119$	−6.98820	−0.640607
$120$	0	0
$121$	9.53911	0.867191
$122$	0	0
$123$	−9.84994	−0.888139
$124$	0	0
$125$	−12.0011	−1.07341
$126$	0	0
$127$	20.3534	1.80607	0.903037	−	0.429562i	$-0.141332\pi$
$127$	20.3534	1.80607	0.903037	+	0.429562i	$0.141332\pi$
$128$	0	0
$129$	2.17792	0.191755
$130$	0	0
$131$	15.8271	1.38282	0.691411	−	0.722461i	$-0.256989\pi$
$131$	15.8271	1.38282	0.691411	+	0.722461i	$0.256989\pi$
$132$	0	0
$133$	−13.0676	−1.13310
$134$	0	0
$135$	−9.30078	−0.800484
$136$	0	0
$137$	−2.28657	−0.195355	−0.0976776	−	0.995218i	$-0.531141\pi$
$137$	−2.28657	−0.195355	−0.0976776	+	0.995218i	$0.531141\pi$
$138$	0	0
$139$	2.37165	0.201161	0.100580	−	0.994929i	$-0.467930\pi$
$139$	2.37165	0.201161	0.100580	+	0.994929i	$0.467930\pi$
$140$	0	0
$141$	10.5724	0.890355
$142$	0	0
$143$	23.5798	1.97184
$144$	0	0
$145$	−3.42846	−0.284718
$146$	0	0
$147$	−1.49518	−0.123321
$148$	0	0
$149$	−8.75053	−0.716871	−0.358436	−	0.933554i	$-0.616690\pi$
$149$	−8.75053	−0.716871	−0.358436	+	0.933554i	$0.616690\pi$
$150$	0	0
$151$	−13.6668	−1.11219	−0.556094	−	0.831119i	$-0.687701\pi$
$151$	−13.6668	−1.11219	−0.556094	+	0.831119i	$0.687701\pi$
$152$	0	0
$153$	3.21193	0.259669
$154$	0	0
$155$	9.86173	0.792113
$156$	0	0
$157$	−6.95277	−0.554892	−0.277446	−	0.960741i	$-0.589488\pi$
$157$	−6.95277	−0.554892	−0.277446	+	0.960741i	$0.589488\pi$
$158$	0	0
$159$	−16.3114	−1.29358
$160$	0	0
$161$	14.7946	1.16598
$162$	0	0
$163$	−2.12160	−0.166176	−0.0830882	−	0.996542i	$-0.526478\pi$
$163$	−2.12160	−0.166176	−0.0830882	+	0.996542i	$0.526478\pi$
$164$	0	0
$165$	−10.2374	−0.796978
$166$	0	0
$167$	−14.3600	−1.11121	−0.555604	−	0.831447i	$-0.687513\pi$
$167$	−14.3600	−1.11121	−0.555604	+	0.831447i	$0.687513\pi$
$168$	0	0
$169$	14.0706	1.08236
$170$	0	0
$171$	6.00613	0.459301
$172$	0	0
$173$	22.9004	1.74109	0.870543	−	0.492093i	$-0.163768\pi$
$173$	22.9004	1.74109	0.870543	+	0.492093i	$0.163768\pi$
$174$	0	0
$175$	−5.56625	−0.420769
$176$	0	0
$177$	−8.83152	−0.663818
$178$	0	0
$179$	19.0662	1.42507	0.712537	−	0.701634i	$-0.247546\pi$
$179$	19.0662	1.42507	0.712537	+	0.701634i	$0.247546\pi$
$180$	0	0
$181$	8.12479	0.603911	0.301955	−	0.953322i	$-0.402361\pi$
$181$	8.12479	0.603911	0.301955	+	0.953322i	$0.402361\pi$
$182$	0	0
$183$	−17.3635	−1.28355
$184$	0	0
$185$	1.18045	0.0867887
$186$	0	0
$187$	13.0272	0.952645
$188$	0	0
$189$	−13.7343	−0.999023
$190$	0	0
$191$	11.4153	0.825980	0.412990	−	0.910736i	$-0.364484\pi$
$191$	11.4153	0.825980	0.412990	+	0.910736i	$0.364484\pi$
$192$	0	0
$193$	−15.0805	−1.08552	−0.542759	−	0.839888i	$-0.682620\pi$
$193$	−15.0805	−1.08552	−0.542759	+	0.839888i	$0.682620\pi$
$194$	0	0
$195$	−11.7529	−0.841646
$196$	0	0
$197$	25.1673	1.79309	0.896547	−	0.442949i	$-0.146068\pi$
$197$	25.1673	1.79309	0.896547	+	0.442949i	$0.146068\pi$
$198$	0	0
$199$	9.55336	0.677220	0.338610	−	0.940927i	$-0.390043\pi$
$199$	9.55336	0.677220	0.338610	+	0.940927i	$0.390043\pi$
$200$	0	0
$201$	−4.86726	−0.343310
$202$	0	0
$203$	−5.06275	−0.355336
$204$	0	0
$205$	−11.8187	−0.825455
$206$	0	0
$207$	−6.79993	−0.472628
$208$	0	0
$209$	24.3602	1.68503
$210$	0	0
$211$	2.66739	0.183631	0.0918153	−	0.995776i	$-0.470733\pi$
$211$	2.66739	0.183631	0.0918153	+	0.995776i	$0.470733\pi$
$212$	0	0
$213$	17.5433	1.20204
$214$	0	0
$215$	2.61323	0.178221
$216$	0	0
$217$	14.5626	0.988577
$218$	0	0
$219$	14.8903	1.00619
$220$	0	0
$221$	14.9558	1.00604
$222$	0	0
$223$	−15.0310	−1.00655	−0.503277	−	0.864125i	$-0.667872\pi$
$223$	−15.0310	−1.00655	−0.503277	+	0.864125i	$0.667872\pi$
$224$	0	0
$225$	2.55837	0.170558
$226$	0	0
$227$	−7.54337	−0.500671	−0.250336	−	0.968159i	$-0.580541\pi$
$227$	−7.54337	−0.500671	−0.250336	+	0.968159i	$0.580541\pi$
$228$	0	0
$229$	2.52262	0.166699	0.0833497	−	0.996520i	$-0.473438\pi$
$229$	2.52262	0.166699	0.0833497	+	0.996520i	$0.473438\pi$
$230$	0	0
$231$	−15.1173	−0.994647
$232$	0	0
$233$	−15.5406	−1.01810	−0.509048	−	0.860738i	$-0.670002\pi$
$233$	−15.5406	−1.01810	−0.509048	+	0.860738i	$0.670002\pi$
$234$	0	0
$235$	12.6856	0.827515
$236$	0	0
$237$	−11.0576	−0.718271
$238$	0	0
$239$	−13.8331	−0.894788	−0.447394	−	0.894337i	$-0.647648\pi$
$239$	−13.8331	−0.894788	−0.447394	+	0.894337i	$0.647648\pi$
$240$	0	0
$241$	−9.70824	−0.625363	−0.312681	−	0.949858i	$-0.601227\pi$
$241$	−9.70824	−0.625363	−0.312681	+	0.949858i	$0.601227\pi$
$242$	0	0
$243$	10.9120	0.700007
$244$	0	0
$245$	−1.79403	−0.114617
$246$	0	0
$247$	27.9666	1.77947
$248$	0	0
$249$	−12.9593	−0.821259
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−27.5798	−1.73393
$254$	0	0
$255$	−6.49320	−0.406620
$256$	0	0
$257$	−5.23580	−0.326600	−0.163300	−	0.986576i	$-0.552214\pi$
$257$	−5.23580	−0.326600	−0.163300	+	0.986576i	$0.552214\pi$
$258$	0	0
$259$	1.74316	0.108314
$260$	0	0
$261$	2.32695	0.144035
$262$	0	0
$263$	−10.0859	−0.621925	−0.310963	−	0.950422i	$-0.600651\pi$
$263$	−10.0859	−0.621925	−0.310963	+	0.950422i	$0.600651\pi$
$264$	0	0
$265$	−19.5717	−1.20228
$266$	0	0
$267$	1.67852	0.102724
$268$	0	0
$269$	−13.3697	−0.815163	−0.407581	−	0.913169i	$-0.633628\pi$
$269$	−13.3697	−0.815163	−0.407581	+	0.913169i	$0.633628\pi$
$270$	0	0
$271$	0.256741	0.0155959	0.00779794	−	0.999970i	$-0.497518\pi$
$271$	0.256741	0.0155959	0.00779794	+	0.999970i	$0.497518\pi$
$272$	0	0
$273$	−17.3554	−1.05039
$274$	0	0
$275$	10.3765	0.625724
$276$	0	0
$277$	−9.88220	−0.593764	−0.296882	−	0.954914i	$-0.595947\pi$
$277$	−9.88220	−0.593764	−0.296882	+	0.954914i	$0.595947\pi$
$278$	0	0
$279$	−6.69330	−0.400718
$280$	0	0
$281$	−14.7694	−0.881067	−0.440533	−	0.897736i	$-0.645211\pi$
$281$	−14.7694	−0.881067	−0.440533	+	0.897736i	$0.645211\pi$
$282$	0	0
$283$	8.36222	0.497082	0.248541	−	0.968621i	$-0.420049\pi$
$283$	8.36222	0.497082	0.248541	+	0.968621i	$0.420049\pi$
$284$	0	0
$285$	−12.1419	−0.719226
$286$	0	0
$287$	−17.4525	−1.03019
$288$	0	0
$289$	−8.73729	−0.513958
$290$	0	0
$291$	15.0191	0.880436
$292$	0	0
$293$	15.7351	0.919254	0.459627	−	0.888112i	$-0.347983\pi$
$293$	15.7351	0.919254	0.459627	+	0.888112i	$0.347983\pi$
$294$	0	0
$295$	−10.5967	−0.616966
$296$	0	0
$297$	25.6031	1.48564
$298$	0	0
$299$	−31.6628	−1.83111
$300$	0	0
$301$	3.85891	0.222424
$302$	0	0
$303$	−2.03790	−0.117075
$304$	0	0
$305$	−20.8341	−1.19296
$306$	0	0
$307$	−25.9168	−1.47915	−0.739576	−	0.673073i	$-0.764974\pi$
$307$	−25.9168	−1.47915	−0.739576	+	0.673073i	$0.764974\pi$
$308$	0	0
$309$	−23.9675	−1.36346
$310$	0	0
$311$	14.7485	0.836311	0.418155	−	0.908376i	$-0.362677\pi$
$311$	14.7485	0.836311	0.418155	+	0.908376i	$0.362677\pi$
$312$	0	0
$313$	31.1258	1.75933	0.879667	−	0.475590i	$-0.157765\pi$
$313$	31.1258	1.75933	0.879667	+	0.475590i	$0.157765\pi$
$314$	0	0
$315$	−4.47225	−0.251983
$316$	0	0
$317$	−11.7048	−0.657406	−0.328703	−	0.944433i	$-0.606612\pi$
$317$	−11.7048	−0.657406	−0.328703	+	0.944433i	$0.606612\pi$
$318$	0	0
$319$	9.43786	0.528418
$320$	0	0
$321$	−9.03839	−0.504474
$322$	0	0
$323$	15.4508	0.859707
$324$	0	0
$325$	11.9126	0.660794
$326$	0	0
$327$	23.1643	1.28099
$328$	0	0
$329$	18.7325	1.03276
$330$	0	0
$331$	−24.1367	−1.32667	−0.663336	−	0.748322i	$-0.730860\pi$
$331$	−24.1367	−1.32667	−0.663336	+	0.748322i	$0.730860\pi$
$332$	0	0
$333$	−0.801192	−0.0439050
$334$	0	0
$335$	−5.84012	−0.319080
$336$	0	0
$337$	−6.64108	−0.361762	−0.180881	−	0.983505i	$-0.557895\pi$
$337$	−6.64108	−0.361762	−0.180881	+	0.983505i	$0.557895\pi$
$338$	0	0
$339$	−1.72816	−0.0938607
$340$	0	0
$341$	−27.1473	−1.47011
$342$	0	0
$343$	−19.6670	−1.06192
$344$	0	0
$345$	13.7467	0.740096
$346$	0	0
$347$	33.4782	1.79721	0.898603	−	0.438763i	$-0.144583\pi$
$347$	33.4782	1.79721	0.898603	+	0.438763i	$0.144583\pi$
$348$	0	0
$349$	15.8861	0.850361	0.425181	−	0.905109i	$-0.360211\pi$
$349$	15.8861	0.850361	0.425181	+	0.905109i	$0.360211\pi$
$350$	0	0
$351$	29.3935	1.56891
$352$	0	0
$353$	−17.6140	−0.937500	−0.468750	−	0.883331i	$-0.655296\pi$
$353$	−17.6140	−0.937500	−0.468750	+	0.883331i	$0.655296\pi$
$354$	0	0
$355$	21.0498	1.11721
$356$	0	0
$357$	−9.58839	−0.507471
$358$	0	0
$359$	−17.7644	−0.937569	−0.468785	−	0.883313i	$-0.655308\pi$
$359$	−17.7644	−0.937569	−0.468785	+	0.883313i	$0.655308\pi$
$360$	0	0
$361$	9.89224	0.520644
$362$	0	0
$363$	13.0884	0.686965
$364$	0	0
$365$	17.8665	0.935178
$366$	0	0
$367$	29.6552	1.54799	0.773994	−	0.633193i	$-0.218256\pi$
$367$	29.6552	1.54799	0.773994	+	0.633193i	$0.218256\pi$
$368$	0	0
$369$	8.02154	0.417585
$370$	0	0
$371$	−28.9012	−1.50048
$372$	0	0
$373$	−36.2760	−1.87830	−0.939151	−	0.343505i	$-0.888386\pi$
$373$	−36.2760	−1.87830	−0.939151	+	0.343505i	$0.888386\pi$
$374$	0	0
$375$	−16.4665	−0.850325
$376$	0	0
$377$	10.8351	0.558035
$378$	0	0
$379$	−28.2590	−1.45157	−0.725784	−	0.687922i	$-0.758523\pi$
$379$	−28.2590	−1.45157	−0.725784	+	0.687922i	$0.758523\pi$
$380$	0	0
$381$	27.9266	1.43072
$382$	0	0
$383$	34.8981	1.78321	0.891605	−	0.452813i	$-0.149580\pi$
$383$	34.8981	1.78321	0.891605	+	0.452813i	$0.149580\pi$
$384$	0	0
$385$	−18.1389	−0.924446
$386$	0	0
$387$	−1.77364	−0.0901591
$388$	0	0
$389$	32.3747	1.64146	0.820732	−	0.571313i	$-0.193566\pi$
$389$	32.3747	1.64146	0.820732	+	0.571313i	$0.193566\pi$
$390$	0	0
$391$	−17.4929	−0.884653
$392$	0	0
$393$	21.7161	1.09543
$394$	0	0
$395$	−13.2678	−0.667577
$396$	0	0
$397$	−37.0827	−1.86113	−0.930563	−	0.366132i	$-0.880682\pi$
$397$	−37.0827	−1.86113	−0.930563	+	0.366132i	$0.880682\pi$
$398$	0	0
$399$	−17.9298	−0.897612
$400$	0	0
$401$	0.828629	0.0413797	0.0206899	−	0.999786i	$-0.493414\pi$
$401$	0.828629	0.0413797	0.0206899	+	0.999786i	$0.493414\pi$
$402$	0	0
$403$	−31.1663	−1.55250
$404$	0	0
$405$	−7.24266	−0.359891
$406$	0	0
$407$	−3.24955	−0.161074
$408$	0	0
$409$	3.50740	0.173430	0.0867150	−	0.996233i	$-0.472363\pi$
$409$	3.50740	0.173430	0.0867150	+	0.996233i	$0.472363\pi$
$410$	0	0
$411$	−3.13737	−0.154755
$412$	0	0
$413$	−15.6480	−0.769989
$414$	0	0
$415$	−15.5495	−0.763296
$416$	0	0
$417$	3.25410	0.159354
$418$	0	0
$419$	−4.69308	−0.229272	−0.114636	−	0.993408i	$-0.536570\pi$
$419$	−4.69308	−0.229272	−0.114636	+	0.993408i	$0.536570\pi$
$420$	0	0
$421$	0.781966	0.0381107	0.0190553	−	0.999818i	$-0.493934\pi$
$421$	0.781966	0.0381107	0.0190553	+	0.999818i	$0.493934\pi$
$422$	0	0
$423$	−8.60988	−0.418627
$424$	0	0
$425$	6.58142	0.319246
$426$	0	0
$427$	−30.7654	−1.48884
$428$	0	0
$429$	32.3534	1.56204
$430$	0	0
$431$	14.5010	0.698490	0.349245	−	0.937031i	$-0.386438\pi$
$431$	14.5010	0.698490	0.349245	+	0.937031i	$0.386438\pi$
$432$	0	0
$433$	−12.7415	−0.612316	−0.306158	−	0.951981i	$-0.599044\pi$
$433$	−12.7415	−0.612316	−0.306158	+	0.951981i	$0.599044\pi$
$434$	0	0
$435$	−4.70414	−0.225546
$436$	0	0
$437$	−32.7108	−1.56477
$438$	0	0
$439$	−0.0542769	−0.00259050	−0.00129525	−	0.999999i	$-0.500412\pi$
$439$	−0.0542769	−0.00259050	−0.00129525	+	0.999999i	$0.500412\pi$
$440$	0	0
$441$	1.21764	0.0579828
$442$	0	0
$443$	26.1733	1.24353	0.621765	−	0.783204i	$-0.286416\pi$
$443$	26.1733	1.24353	0.621765	+	0.783204i	$0.286416\pi$
$444$	0	0
$445$	2.01402	0.0954739
$446$	0	0
$447$	−12.0064	−0.567885
$448$	0	0
$449$	−4.41106	−0.208171	−0.104085	−	0.994568i	$-0.533192\pi$
$449$	−4.41106	−0.208171	−0.104085	+	0.994568i	$0.533192\pi$
$450$	0	0
$451$	32.5345	1.53199
$452$	0	0
$453$	−18.7520	−0.881044
$454$	0	0
$455$	−20.8243	−0.976258
$456$	0	0
$457$	29.4874	1.37936	0.689681	−	0.724113i	$-0.257751\pi$
$457$	29.4874	1.37936	0.689681	+	0.724113i	$0.257751\pi$
$458$	0	0
$459$	16.2392	0.757979
$460$	0	0
$461$	28.2732	1.31682	0.658408	−	0.752662i	$-0.271230\pi$
$461$	28.2732	1.31682	0.658408	+	0.752662i	$0.271230\pi$
$462$	0	0
$463$	−12.5575	−0.583597	−0.291798	−	0.956480i	$-0.594254\pi$
$463$	−12.5575	−0.583597	−0.291798	+	0.956480i	$0.594254\pi$
$464$	0	0
$465$	13.5311	0.627490
$466$	0	0
$467$	36.6913	1.69787	0.848935	−	0.528497i	$-0.177244\pi$
$467$	36.6913	1.69787	0.848935	+	0.528497i	$0.177244\pi$
$468$	0	0
$469$	−8.62400	−0.398219
$470$	0	0
$471$	−9.53978	−0.439570
$472$	0	0
$473$	−7.19369	−0.330766
$474$	0	0
$475$	12.3069	0.564680
$476$	0	0
$477$	13.2836	0.608215
$478$	0	0
$479$	1.77059	0.0809004	0.0404502	−	0.999182i	$-0.487121\pi$
$479$	1.77059	0.0809004	0.0404502	+	0.999182i	$0.487121\pi$
$480$	0	0
$481$	−3.73062	−0.170102
$482$	0	0
$483$	20.2995	0.923658
$484$	0	0
$485$	18.0211	0.818296
$486$	0	0
$487$	1.81798	0.0823805	0.0411903	−	0.999151i	$-0.486885\pi$
$487$	1.81798	0.0823805	0.0411903	+	0.999151i	$0.486885\pi$
$488$	0	0
$489$	−2.91101	−0.131640
$490$	0	0
$491$	−12.3423	−0.557000	−0.278500	−	0.960436i	$-0.589837\pi$
$491$	−12.3423	−0.557000	−0.278500	+	0.960436i	$0.589837\pi$
$492$	0	0
$493$	5.98610	0.269600
$494$	0	0
$495$	8.33705	0.374723
$496$	0	0
$497$	31.0838	1.39430
$498$	0	0
$499$	20.7996	0.931120	0.465560	−	0.885016i	$-0.345853\pi$
$499$	20.7996	0.931120	0.465560	+	0.885016i	$0.345853\pi$
$500$	0	0
$501$	−19.7031	−0.880268
$502$	0	0
$503$	−24.5971	−1.09673	−0.548366	−	0.836239i	$-0.684750\pi$
$503$	−24.5971	−1.09673	−0.548366	+	0.836239i	$0.684750\pi$
$504$	0	0
$505$	−2.44523	−0.108812
$506$	0	0
$507$	19.3061	0.857413
$508$	0	0
$509$	−9.41535	−0.417328	−0.208664	−	0.977987i	$-0.566912\pi$
$509$	−9.41535	−0.417328	−0.208664	+	0.977987i	$0.566912\pi$
$510$	0	0
$511$	26.3832	1.16712
$512$	0	0
$513$	30.3664	1.34071
$514$	0	0
$515$	−28.7581	−1.26723
$516$	0	0
$517$	−34.9207	−1.53581
$518$	0	0
$519$	31.4212	1.37924
$520$	0	0
$521$	33.5179	1.46845	0.734223	−	0.678909i	$-0.237547\pi$
$521$	33.5179	1.46845	0.734223	+	0.678909i	$0.237547\pi$
$522$	0	0
$523$	−17.4214	−0.761786	−0.380893	−	0.924619i	$-0.624383\pi$
$523$	−17.4214	−0.761786	−0.380893	+	0.924619i	$0.624383\pi$
$524$	0	0
$525$	−7.63735	−0.333321
$526$	0	0
$527$	−17.2186	−0.750053
$528$	0	0
$529$	14.0340	0.610173
$530$	0	0
$531$	7.19217	0.312113
$532$	0	0
$533$	37.3510	1.61785
$534$	0	0
$535$	−10.8450	−0.468869
$536$	0	0
$537$	26.1604	1.12890
$538$	0	0
$539$	4.93861	0.212721
$540$	0	0
$541$	−32.5644	−1.40005	−0.700027	−	0.714116i	$-0.746829\pi$
$541$	−32.5644	−1.40005	−0.700027	+	0.714116i	$0.746829\pi$
$542$	0	0
$543$	11.1479	0.478401
$544$	0	0
$545$	27.7943	1.19058
$546$	0	0
$547$	−22.4173	−0.958495	−0.479248	−	0.877680i	$-0.659090\pi$
$547$	−22.4173	−0.958495	−0.479248	+	0.877680i	$0.659090\pi$
$548$	0	0
$549$	14.1404	0.603498
$550$	0	0
$551$	11.1937	0.476867
$552$	0	0
$553$	−19.5924	−0.833152
$554$	0	0
$555$	1.61968	0.0687516
$556$	0	0
$557$	−12.4988	−0.529590	−0.264795	−	0.964305i	$-0.585304\pi$
$557$	−12.4988	−0.529590	−0.264795	+	0.964305i	$0.585304\pi$
$558$	0	0
$559$	−8.25867	−0.349304
$560$	0	0
$561$	17.8744	0.754659
$562$	0	0
$563$	29.1096	1.22682	0.613411	−	0.789764i	$-0.289797\pi$
$563$	29.1096	1.22682	0.613411	+	0.789764i	$0.289797\pi$
$564$	0	0
$565$	−2.07358	−0.0872361
$566$	0	0
$567$	−10.6951	−0.449152
$568$	0	0
$569$	−6.20206	−0.260004	−0.130002	−	0.991514i	$-0.541498\pi$
$569$	−6.20206	−0.260004	−0.130002	+	0.991514i	$0.541498\pi$
$570$	0	0
$571$	5.76421	0.241225	0.120612	−	0.992700i	$-0.461514\pi$
$571$	5.76421	0.241225	0.120612	+	0.992700i	$0.461514\pi$
$572$	0	0
$573$	15.6627	0.654319
$574$	0	0
$575$	−13.9335	−0.581065
$576$	0	0
$577$	−10.1975	−0.424527	−0.212264	−	0.977212i	$-0.568084\pi$
$577$	−10.1975	−0.424527	−0.212264	+	0.977212i	$0.568084\pi$
$578$	0	0
$579$	−20.6917	−0.859918
$580$	0	0
$581$	−22.9617	−0.952612
$582$	0	0
$583$	53.8769	2.23135
$584$	0	0
$585$	9.57130	0.395724
$586$	0	0
$587$	−12.7660	−0.526909	−0.263455	−	0.964672i	$-0.584862\pi$
$587$	−12.7660	−0.526909	−0.263455	+	0.964672i	$0.584862\pi$
$588$	0	0
$589$	−32.1978	−1.32669
$590$	0	0
$591$	34.5316	1.42044
$592$	0	0
$593$	−15.5891	−0.640168	−0.320084	−	0.947389i	$-0.603711\pi$
$593$	−15.5891	−0.640168	−0.320084	+	0.947389i	$0.603711\pi$
$594$	0	0
$595$	−11.5049	−0.471655
$596$	0	0
$597$	13.1080	0.536475
$598$	0	0
$599$	−8.25188	−0.337163	−0.168581	−	0.985688i	$-0.553919\pi$
$599$	−8.25188	−0.337163	−0.168581	+	0.985688i	$0.553919\pi$
$600$	0	0
$601$	−19.1843	−0.782546	−0.391273	−	0.920275i	$-0.627965\pi$
$601$	−19.1843	−0.782546	−0.391273	+	0.920275i	$0.627965\pi$
$602$	0	0
$603$	3.96378	0.161417
$604$	0	0
$605$	15.7045	0.638480
$606$	0	0
$607$	−31.4124	−1.27499	−0.637494	−	0.770455i	$-0.720029\pi$
$607$	−31.4124	−1.27499	−0.637494	+	0.770455i	$0.720029\pi$
$608$	0	0
$609$	−6.94651	−0.281487
$610$	0	0
$611$	−40.0905	−1.62189
$612$	0	0
$613$	−47.3925	−1.91417	−0.957083	−	0.289813i	$-0.906407\pi$
$613$	−47.3925	−1.91417	−0.957083	+	0.289813i	$0.906407\pi$
$614$	0	0
$615$	−16.2163	−0.653902
$616$	0	0
$617$	−12.6954	−0.511099	−0.255550	−	0.966796i	$-0.582256\pi$
$617$	−12.6954	−0.511099	−0.255550	+	0.966796i	$0.582256\pi$
$618$	0	0
$619$	37.7371	1.51678	0.758390	−	0.651801i	$-0.225986\pi$
$619$	37.7371	1.51678	0.758390	+	0.651801i	$0.225986\pi$
$620$	0	0
$621$	−34.3797	−1.37961
$622$	0	0
$623$	2.97407	0.119154
$624$	0	0
$625$	−8.30978	−0.332391
$626$	0	0
$627$	33.4242	1.33484
$628$	0	0
$629$	−2.06107	−0.0821804
$630$	0	0
$631$	21.8240	0.868798	0.434399	−	0.900721i	$-0.356961\pi$
$631$	21.8240	0.868798	0.434399	+	0.900721i	$0.356961\pi$
$632$	0	0
$633$	3.65988	0.145467
$634$	0	0
$635$	33.5085	1.32974
$636$	0	0
$637$	5.66974	0.224643
$638$	0	0
$639$	−14.2868	−0.565176
$640$	0	0
$641$	−31.1755	−1.23136	−0.615678	−	0.787998i	$-0.711118\pi$
$641$	−31.1755	−1.23136	−0.615678	+	0.787998i	$0.711118\pi$
$642$	0	0
$643$	6.83624	0.269595	0.134798	−	0.990873i	$-0.456962\pi$
$643$	6.83624	0.269595	0.134798	+	0.990873i	$0.456962\pi$
$644$	0	0
$645$	3.58557	0.141182
$646$	0	0
$647$	−32.1377	−1.26346	−0.631731	−	0.775187i	$-0.717655\pi$
$647$	−32.1377	−1.26346	−0.631731	+	0.775187i	$0.717655\pi$
$648$	0	0
$649$	29.1706	1.14505
$650$	0	0
$651$	19.9812	0.783123
$652$	0	0
$653$	−45.1652	−1.76745	−0.883725	−	0.468006i	$-0.844973\pi$
$653$	−45.1652	−1.76745	−0.883725	+	0.468006i	$0.844973\pi$
$654$	0	0
$655$	26.0567	1.01812
$656$	0	0
$657$	−12.1263	−0.473092
$658$	0	0
$659$	−0.642445	−0.0250261	−0.0125130	−	0.999922i	$-0.503983\pi$
$659$	−0.642445	−0.0250261	−0.0125130	+	0.999922i	$0.503983\pi$
$660$	0	0
$661$	39.7027	1.54425	0.772127	−	0.635468i	$-0.219193\pi$
$661$	39.7027	1.54425	0.772127	+	0.635468i	$0.219193\pi$
$662$	0	0
$663$	20.5206	0.796955
$664$	0	0
$665$	−21.5135	−0.834259
$666$	0	0
$667$	−12.6731	−0.490704
$668$	0	0
$669$	−20.6238	−0.797363
$670$	0	0
$671$	57.3520	2.21405
$672$	0	0
$673$	43.9142	1.69277	0.846384	−	0.532573i	$-0.178775\pi$
$673$	43.9142	1.69277	0.846384	+	0.532573i	$0.178775\pi$
$674$	0	0
$675$	12.9348	0.497862
$676$	0	0
$677$	10.4535	0.401759	0.200879	−	0.979616i	$-0.435620\pi$
$677$	10.4535	0.401759	0.200879	+	0.979616i	$0.435620\pi$
$678$	0	0
$679$	26.6114	1.02125
$680$	0	0
$681$	−10.3501	−0.396618
$682$	0	0
$683$	21.5758	0.825575	0.412788	−	0.910827i	$-0.364555\pi$
$683$	21.5758	0.825575	0.412788	+	0.910827i	$0.364555\pi$
$684$	0	0
$685$	−3.76446	−0.143833
$686$	0	0
$687$	3.46124	0.132055
$688$	0	0
$689$	61.8531	2.35641
$690$	0	0
$691$	31.4628	1.19690	0.598451	−	0.801159i	$-0.295783\pi$
$691$	31.4628	1.19690	0.598451	+	0.801159i	$0.295783\pi$
$692$	0	0
$693$	12.3112	0.467663
$694$	0	0
$695$	3.90452	0.148107
$696$	0	0
$697$	20.6355	0.781624
$698$	0	0
$699$	−21.3229	−0.806507
$700$	0	0
$701$	−16.4297	−0.620541	−0.310270	−	0.950648i	$-0.600420\pi$
$701$	−16.4297	−0.620541	−0.310270	+	0.950648i	$0.600420\pi$
$702$	0	0
$703$	−3.85410	−0.145360
$704$	0	0
$705$	17.4056	0.655534
$706$	0	0
$707$	−3.61083	−0.135799
$708$	0	0
$709$	7.88453	0.296110	0.148055	−	0.988979i	$-0.452699\pi$
$709$	7.88453	0.296110	0.148055	+	0.988979i	$0.452699\pi$
$710$	0	0
$711$	9.00507	0.337716
$712$	0	0
$713$	36.4533	1.36519
$714$	0	0
$715$	38.8201	1.45179
$716$	0	0
$717$	−18.9801	−0.708826
$718$	0	0
$719$	−29.6960	−1.10747	−0.553736	−	0.832692i	$-0.686798\pi$
$719$	−29.6960	−1.10747	−0.553736	+	0.832692i	$0.686798\pi$
$720$	0	0
$721$	−42.4665	−1.58154
$722$	0	0
$723$	−13.3205	−0.495395
$724$	0	0
$725$	4.76805	0.177081
$726$	0	0
$727$	−12.8430	−0.476320	−0.238160	−	0.971226i	$-0.576544\pi$
$727$	−12.8430	−0.476320	−0.238160	+	0.971226i	$0.576544\pi$
$728$	0	0
$729$	28.1700	1.04333
$730$	0	0
$731$	−4.56270	−0.168758
$732$	0	0
$733$	11.2694	0.416246	0.208123	−	0.978103i	$-0.433265\pi$
$733$	11.2694	0.416246	0.208123	+	0.978103i	$0.433265\pi$
$734$	0	0
$735$	−2.46156	−0.0907962
$736$	0	0
$737$	16.0766	0.592191
$738$	0	0
$739$	−41.0944	−1.51168	−0.755842	−	0.654754i	$-0.772772\pi$
$739$	−41.0944	−1.51168	−0.755842	+	0.654754i	$0.772772\pi$
$740$	0	0
$741$	38.3725	1.40965
$742$	0	0
$743$	−11.0330	−0.404763	−0.202382	−	0.979307i	$-0.564868\pi$
$743$	−11.0330	−0.404763	−0.202382	+	0.979307i	$0.564868\pi$
$744$	0	0
$745$	−14.4063	−0.527805
$746$	0	0
$747$	10.5537	0.386139
$748$	0	0
$749$	−16.0146	−0.585159
$750$	0	0
$751$	7.57041	0.276248	0.138124	−	0.990415i	$-0.455893\pi$
$751$	7.57041	0.276248	0.138124	+	0.990415i	$0.455893\pi$
$752$	0	0
$753$	1.37208	0.0500015
$754$	0	0
$755$	−22.5001	−0.818861
$756$	0	0
$757$	−8.28508	−0.301126	−0.150563	−	0.988600i	$-0.548109\pi$
$757$	−8.28508	−0.301126	−0.150563	+	0.988600i	$0.548109\pi$
$758$	0	0
$759$	−37.8418	−1.37357
$760$	0	0
$761$	−38.7079	−1.40316	−0.701581	−	0.712590i	$-0.747522\pi$
$761$	−38.7079	−1.40316	−0.701581	+	0.712590i	$0.747522\pi$
$762$	0	0
$763$	41.0433	1.48587
$764$	0	0
$765$	5.28790	0.191184
$766$	0	0
$767$	33.4892	1.20922
$768$	0	0
$769$	28.4046	1.02430	0.512148	−	0.858897i	$-0.328850\pi$
$769$	28.4046	1.02430	0.512148	+	0.858897i	$0.328850\pi$
$770$	0	0
$771$	−7.18395	−0.258724
$772$	0	0
$773$	−1.51143	−0.0543625	−0.0271813	−	0.999631i	$-0.508653\pi$
$773$	−1.51143	−0.0543625	−0.0271813	+	0.999631i	$0.508653\pi$
$774$	0	0
$775$	−13.7150	−0.492656
$776$	0	0
$777$	2.39175	0.0858037
$778$	0	0
$779$	38.5872	1.38253
$780$	0	0
$781$	−57.9456	−2.07346
$782$	0	0
$783$	11.7648	0.420440
$784$	0	0
$785$	−11.4466	−0.408545
$786$	0	0
$787$	−28.5362	−1.01720	−0.508602	−	0.861002i	$-0.669838\pi$
$787$	−28.5362	−1.01720	−0.508602	+	0.861002i	$0.669838\pi$
$788$	0	0
$789$	−13.8387	−0.492672
$790$	0	0
$791$	−3.06202	−0.108873
$792$	0	0
$793$	65.8426	2.33814
$794$	0	0
$795$	−26.8540	−0.952414
$796$	0	0
$797$	11.0271	0.390600	0.195300	−	0.980744i	$-0.437432\pi$
$797$	11.0271	0.390600	0.195300	+	0.980744i	$0.437432\pi$
$798$	0	0
$799$	−22.1490	−0.783575
$800$	0	0
$801$	−1.36695	−0.0482987
$802$	0	0
$803$	−49.1829	−1.73563
$804$	0	0
$805$	24.3569	0.858467
$806$	0	0
$807$	−18.3443	−0.645749
$808$	0	0
$809$	−10.9339	−0.384417	−0.192208	−	0.981354i	$-0.561565\pi$
$809$	−10.9339	−0.384417	−0.192208	+	0.981354i	$0.561565\pi$
$810$	0	0
$811$	−6.99122	−0.245495	−0.122748	−	0.992438i	$-0.539171\pi$
$811$	−6.99122	−0.245495	−0.122748	+	0.992438i	$0.539171\pi$
$812$	0	0
$813$	0.352269	0.0123546
$814$	0	0
$815$	−3.49285	−0.122349
$816$	0	0
$817$	−8.53201	−0.298497
$818$	0	0
$819$	14.1338	0.493874
$820$	0	0
$821$	6.73460	0.235039	0.117520	−	0.993071i	$-0.462506\pi$
$821$	6.73460	0.235039	0.117520	+	0.993071i	$0.462506\pi$
$822$	0	0
$823$	−2.33766	−0.0814857	−0.0407429	−	0.999170i	$-0.512972\pi$
$823$	−2.33766	−0.0814857	−0.0407429	+	0.999170i	$0.512972\pi$
$824$	0	0
$825$	14.2374	0.495681
$826$	0	0
$827$	27.2008	0.945864	0.472932	−	0.881099i	$-0.343196\pi$
$827$	27.2008	0.945864	0.472932	+	0.881099i	$0.343196\pi$
$828$	0	0
$829$	−18.0110	−0.625547	−0.312773	−	0.949828i	$-0.601258\pi$
$829$	−18.0110	−0.625547	−0.312773	+	0.949828i	$0.601258\pi$
$830$	0	0
$831$	−13.5592	−0.470363
$832$	0	0
$833$	3.13238	0.108531
$834$	0	0
$835$	−23.6413	−0.818140
$836$	0	0
$837$	−33.8406	−1.16970
$838$	0	0
$839$	31.8396	1.09922	0.549612	−	0.835420i	$-0.314775\pi$
$839$	31.8396	1.09922	0.549612	+	0.835420i	$0.314775\pi$
$840$	0	0
$841$	−24.6632	−0.850457
$842$	0	0
$843$	−20.2648	−0.697957
$844$	0	0
$845$	23.1649	0.796897
$846$	0	0
$847$	23.1906	0.796838
$848$	0	0
$849$	11.4737	0.393775
$850$	0	0
$851$	4.36347	0.149578
$852$	0	0
$853$	−42.8391	−1.46678	−0.733391	−	0.679807i	$-0.762064\pi$
$853$	−42.8391	−1.46678	−0.733391	+	0.679807i	$0.762064\pi$
$854$	0	0
$855$	9.88809	0.338165
$856$	0	0
$857$	−45.4153	−1.55136	−0.775678	−	0.631129i	$-0.782592\pi$
$857$	−45.4153	−1.55136	−0.775678	+	0.631129i	$0.782592\pi$
$858$	0	0
$859$	−36.5393	−1.24671	−0.623353	−	0.781940i	$-0.714230\pi$
$859$	−36.5393	−1.24671	−0.623353	+	0.781940i	$0.714230\pi$
$860$	0	0
$861$	−23.9463	−0.816086
$862$	0	0
$863$	4.46830	0.152103	0.0760514	−	0.997104i	$-0.475769\pi$
$863$	4.46830	0.152103	0.0760514	+	0.997104i	$0.475769\pi$
$864$	0	0
$865$	37.7016	1.28189
$866$	0	0
$867$	−11.9883	−0.407143
$868$	0	0
$869$	36.5236	1.23898
$870$	0	0
$871$	18.4567	0.625381
$872$	0	0
$873$	−12.2312	−0.413963
$874$	0	0
$875$	−29.1759	−0.986326
$876$	0	0
$877$	−21.8397	−0.737474	−0.368737	−	0.929534i	$-0.620210\pi$
$877$	−21.8397	−0.737474	−0.368737	+	0.929534i	$0.620210\pi$
$878$	0	0
$879$	21.5898	0.728207
$880$	0	0
$881$	47.5631	1.60244	0.801220	−	0.598370i	$-0.204185\pi$
$881$	47.5631	1.60244	0.801220	+	0.598370i	$0.204185\pi$
$882$	0	0
$883$	−21.6165	−0.727454	−0.363727	−	0.931506i	$-0.618496\pi$
$883$	−21.6165	−0.727454	−0.363727	+	0.931506i	$0.618496\pi$
$884$	0	0
$885$	−14.5396	−0.488744
$886$	0	0
$887$	−17.9754	−0.603555	−0.301777	−	0.953378i	$-0.597580\pi$
$887$	−17.9754	−0.603555	−0.301777	+	0.953378i	$0.597580\pi$
$888$	0	0
$889$	49.4814	1.65955
$890$	0	0
$891$	19.9375	0.667933
$892$	0	0
$893$	−41.4174	−1.38598
$894$	0	0
$895$	31.3893	1.04923
$896$	0	0
$897$	−43.4440	−1.45055
$898$	0	0
$899$	−12.4744	−0.416044
$900$	0	0
$901$	34.1722	1.13844
$902$	0	0
$903$	5.29475	0.176198
$904$	0	0
$905$	13.3761	0.444636
$906$	0	0
$907$	−12.0112	−0.398825	−0.199413	−	0.979916i	$-0.563903\pi$
$907$	−12.0112	−0.398825	−0.199413	+	0.979916i	$0.563903\pi$
$908$	0	0
$909$	1.65962	0.0550460
$910$	0	0
$911$	14.2924	0.473527	0.236764	−	0.971567i	$-0.423913\pi$
$911$	14.2924	0.473527	0.236764	+	0.971567i	$0.423913\pi$
$912$	0	0
$913$	42.8046	1.41663
$914$	0	0
$915$	−28.5861	−0.945028
$916$	0	0
$917$	38.4774	1.27064
$918$	0	0
$919$	44.0682	1.45368	0.726838	−	0.686809i	$-0.240989\pi$
$919$	44.0682	1.45368	0.726838	+	0.686809i	$0.240989\pi$
$920$	0	0
$921$	−35.5600	−1.17174
$922$	0	0
$923$	−66.5241	−2.18967
$924$	0	0
$925$	−1.64169	−0.0539784
$926$	0	0
$927$	19.5185	0.641073
$928$	0	0
$929$	2.45633	0.0805894	0.0402947	−	0.999188i	$-0.487170\pi$
$929$	2.45633	0.0805894	0.0402947	+	0.999188i	$0.487170\pi$
$930$	0	0
$931$	5.85739	0.191968
$932$	0	0
$933$	20.2362	0.662502
$934$	0	0
$935$	21.4471	0.701396
$936$	0	0
$937$	8.89265	0.290510	0.145255	−	0.989394i	$-0.453600\pi$
$937$	8.89265	0.290510	0.145255	+	0.989394i	$0.453600\pi$
$938$	0	0
$939$	42.7072	1.39370
$940$	0	0
$941$	−18.7881	−0.612476	−0.306238	−	0.951955i	$-0.599070\pi$
$941$	−18.7881	−0.612476	−0.306238	+	0.951955i	$0.599070\pi$
$942$	0	0
$943$	−43.6871	−1.42265
$944$	0	0
$945$	−22.6112	−0.735542
$946$	0	0
$947$	−34.2408	−1.11268	−0.556338	−	0.830956i	$-0.687794\pi$
$947$	−34.2408	−1.11268	−0.556338	+	0.830956i	$0.687794\pi$
$948$	0	0
$949$	−56.4641	−1.83290
$950$	0	0
$951$	−16.0599	−0.520779
$952$	0	0
$953$	−11.9789	−0.388034	−0.194017	−	0.980998i	$-0.562152\pi$
$953$	−11.9789	−0.388034	−0.194017	+	0.980998i	$0.562152\pi$
$954$	0	0
$955$	18.7933	0.608137
$956$	0	0
$957$	12.9495	0.418598
$958$	0	0
$959$	−5.55891	−0.179506
$960$	0	0
$961$	4.88164	0.157472
$962$	0	0
$963$	7.36064	0.237193
$964$	0	0
$965$	−24.8275	−0.799226
$966$	0	0
$967$	1.01631	0.0326822	0.0163411	−	0.999866i	$-0.494798\pi$
$967$	1.01631	0.0326822	0.0163411	+	0.999866i	$0.494798\pi$
$968$	0	0
$969$	21.1998	0.681036
$970$	0	0
$971$	9.97721	0.320184	0.160092	−	0.987102i	$-0.448821\pi$
$971$	9.97721	0.320184	0.160092	+	0.987102i	$0.448821\pi$
$972$	0	0
$973$	5.76573	0.184841
$974$	0	0
$975$	16.3451	0.523463
$976$	0	0
$977$	−13.1303	−0.420075	−0.210037	−	0.977693i	$-0.567359\pi$
$977$	−13.1303	−0.420075	−0.210037	+	0.977693i	$0.567359\pi$
$978$	0	0
$979$	−5.54419	−0.177193
$980$	0	0
$981$	−18.8644	−0.602293
$982$	0	0
$983$	−20.6521	−0.658699	−0.329350	−	0.944208i	$-0.606829\pi$
$983$	−20.6521	−0.658699	−0.329350	+	0.944208i	$0.606829\pi$
$984$	0	0
$985$	41.4337	1.32019
$986$	0	0
$987$	25.7026	0.818123
$988$	0	0
$989$	9.65964	0.307159
$990$	0	0
$991$	−4.39362	−0.139568	−0.0697840	−	0.997562i	$-0.522231\pi$
$991$	−4.39362	−0.139568	−0.0697840	+	0.997562i	$0.522231\pi$
$992$	0	0
$993$	−33.1175	−1.05095
$994$	0	0
$995$	15.7280	0.498611
$996$	0	0
$997$	48.8690	1.54770	0.773848	−	0.633371i	$-0.218329\pi$
$997$	48.8690	1.54770	0.773848	+	0.633371i	$0.218329\pi$
$998$	0	0
$999$	−4.05074	−0.128160

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.e.1.4		5
4.3	odd	2		502.2.a.c.1.2	✓	5
12.11	even	2		4518.2.a.v.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
502.2.a.c.1.2	✓	5	4.3	odd	2
4016.2.a.e.1.4		5	1.1	even	1	trivial
4518.2.a.v.1.1		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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.37208 q^{3} +1.64633 q^{5} +2.43111 q^{7} -1.11739 q^{9} +O(q^{10})\)	\(q+1.37208 q^{3} +1.64633 q^{5} +2.43111 q^{7} -1.11739 q^{9} -4.53201 q^{11} -5.20294 q^{13} +2.25890 q^{15} -2.87449 q^{17} -5.37515 q^{19} +3.33568 q^{21} +6.08556 q^{23} -2.28959 q^{25} -5.64940 q^{27} -2.08249 q^{29} +5.99013 q^{31} -6.21829 q^{33} +4.00241 q^{35} +0.717021 q^{37} -7.13887 q^{39} -7.17882 q^{41} +1.58731 q^{43} -1.83959 q^{45} +7.70536 q^{47} -1.08972 q^{49} -3.94404 q^{51} -11.8881 q^{53} -7.46119 q^{55} -7.37515 q^{57} -6.43658 q^{59} -12.6549 q^{61} -2.71649 q^{63} -8.56577 q^{65} -3.54736 q^{67} +8.34989 q^{69} +12.7859 q^{71} +10.8523 q^{73} -3.14151 q^{75} -11.0178 q^{77} -8.05902 q^{79} -4.39927 q^{81} -9.44495 q^{83} -4.73237 q^{85} -2.85735 q^{87} +1.22334 q^{89} -12.6489 q^{91} +8.21895 q^{93} -8.84928 q^{95} +10.9462 q^{97} +5.06402 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} - 6 q^{5} + q^{7}+O(q^{10}) \) 5 * q + q^3 - 6 * q^5 + q^7	\( 5 q + q^{3} - 6 q^{5} + q^{7} + 4 q^{11} - 7 q^{13} + 5 q^{15} - 8 q^{17} - 3 q^{19} - 10 q^{21} + 17 q^{23} - q^{25} + 4 q^{27} - 15 q^{29} + q^{31} - 10 q^{33} + 7 q^{35} - 5 q^{37} + 8 q^{39} - 12 q^{41} - q^{43} - 13 q^{45} + 19 q^{47} + 4 q^{49} - 7 q^{51} - 34 q^{53} - 17 q^{55} - 13 q^{57} + 10 q^{59} + 2 q^{63} + 6 q^{65} + 11 q^{67} + q^{69} + q^{71} + 3 q^{73} - 15 q^{75} - 30 q^{77} - 35 q^{79} - 3 q^{81} - 3 q^{83} - 13 q^{85} - 3 q^{87} + 15 q^{89} - 14 q^{91} + 15 q^{93} - 11 q^{95} - 2 q^{97} - 28 q^{99}+O(q^{100}) \) 5 * q + q^3 - 6 * q^5 + q^7 + 4 * q^11 - 7 * q^13 + 5 * q^15 - 8 * q^17 - 3 * q^19 - 10 * q^21 + 17 * q^23 - q^25 + 4 * q^27 - 15 * q^29 + q^31 - 10 * q^33 + 7 * q^35 - 5 * q^37 + 8 * q^39 - 12 * q^41 - q^43 - 13 * q^45 + 19 * q^47 + 4 * q^49 - 7 * q^51 - 34 * q^53 - 17 * q^55 - 13 * q^57 + 10 * q^59 + 2 * q^63 + 6 * q^65 + 11 * q^67 + q^69 + q^71 + 3 * q^73 - 15 * q^75 - 30 * q^77 - 35 * q^79 - 3 * q^81 - 3 * q^83 - 13 * q^85 - 3 * q^87 + 15 * q^89 - 14 * q^91 + 15 * q^93 - 11 * q^95 - 2 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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