LMFDB - Embedded newform 4016.2.a.c.1.2

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:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 251)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.477260$ 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)

$f(q)$	$=$	$q-0.294963 q^{3} -3.39026 q^{5} +0.817703 q^{7} -2.91300 q^{9} +O(q^{10})$	$q-0.294963 q^{3} -3.39026 q^{5} +0.817703 q^{7} -2.91300 q^{9} -1.09529 q^{11} -1.08700 q^{13} +1.00000 q^{15} +4.34478 q^{17} +4.75485 q^{19} -0.241192 q^{21} +3.16248 q^{23} +6.49384 q^{25} +1.74411 q^{27} +2.75241 q^{29} +0.312335 q^{31} +0.323071 q^{33} -2.77222 q^{35} -7.02171 q^{37} +0.320626 q^{39} +3.34873 q^{41} +8.25382 q^{43} +9.87581 q^{45} -0.104375 q^{47} -6.33136 q^{49} -1.28155 q^{51} -3.94940 q^{53} +3.71333 q^{55} -1.40250 q^{57} +2.50024 q^{59} -4.85923 q^{61} -2.38197 q^{63} +3.68522 q^{65} -11.8611 q^{67} -0.932814 q^{69} +9.42270 q^{71} -1.26813 q^{73} -1.91544 q^{75} -0.895625 q^{77} -11.1700 q^{79} +8.22454 q^{81} -12.3574 q^{83} -14.7299 q^{85} -0.811858 q^{87} +7.23094 q^{89} -0.888846 q^{91} -0.0921272 q^{93} -16.1202 q^{95} -5.07280 q^{97} +3.19059 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 3 q^{5} + 3 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 - 3 * q^5 + 3 * q^7 - 4 * q^9	$ 4 q + 2 q^{3} - 3 q^{5} + 3 q^{7} - 4 q^{9} + 3 q^{11} - 12 q^{13} + 4 q^{15} + q^{17} + 9 q^{19} - 7 q^{21} - 4 q^{23} - 7 q^{25} - q^{27} - 12 q^{29} + 2 q^{31} - 5 q^{35} - 13 q^{37} - 13 q^{39} + q^{41} + 5 q^{43} + 11 q^{45} - 12 q^{47} - 9 q^{49} - 2 q^{51} + 5 q^{53} + 3 q^{55} + 16 q^{57} - 6 q^{59} - 21 q^{61} - 14 q^{63} + q^{65} - 17 q^{67} - 13 q^{69} + 10 q^{71} - 2 q^{73} - 13 q^{75} + 8 q^{77} + 21 q^{79} - 8 q^{81} + q^{83} - 17 q^{85} - 31 q^{87} + 5 q^{89} + 2 q^{91} - 23 q^{93} - 12 q^{95} + 6 q^{97} - 2 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 - 3 * q^5 + 3 * q^7 - 4 * q^9 + 3 * q^11 - 12 * q^13 + 4 * q^15 + q^17 + 9 * q^19 - 7 * q^21 - 4 * q^23 - 7 * q^25 - q^27 - 12 * q^29 + 2 * q^31 - 5 * q^35 - 13 * q^37 - 13 * q^39 + q^41 + 5 * q^43 + 11 * q^45 - 12 * q^47 - 9 * q^49 - 2 * q^51 + 5 * q^53 + 3 * q^55 + 16 * q^57 - 6 * q^59 - 21 * q^61 - 14 * q^63 + q^65 - 17 * q^67 - 13 * q^69 + 10 * q^71 - 2 * q^73 - 13 * q^75 + 8 * q^77 + 21 * q^79 - 8 * q^81 + q^83 - 17 * q^85 - 31 * q^87 + 5 * q^89 + 2 * q^91 - 23 * q^93 - 12 * q^95 + 6 * 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
$3$	−0.294963	−0.170297	−0.0851485	−	0.996368i	$-0.527136\pi$
$3$	−0.294963	−0.170297	−0.0851485	+	0.996368i	$0.527136\pi$
$4$	0	0
$5$	−3.39026	−1.51617	−0.758084	−	0.652156i	$-0.773865\pi$
$5$	−3.39026	−1.51617	−0.758084	+	0.652156i	$0.773865\pi$
$6$	0	0
$7$	0.817703	0.309063	0.154531	−	0.987988i	$-0.450613\pi$
$7$	0.817703	0.309063	0.154531	+	0.987988i	$0.450613\pi$
$8$	0	0
$9$	−2.91300	−0.970999
$10$	0	0
$11$	−1.09529	−0.330244	−0.165122	−	0.986273i	$-0.552802\pi$
$11$	−1.09529	−0.330244	−0.165122	+	0.986273i	$0.552802\pi$
$12$	0	0
$13$	−1.08700	−0.301480	−0.150740	−	0.988573i	$-0.548166\pi$
$13$	−1.08700	−0.301480	−0.150740	+	0.988573i	$0.548166\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	4.34478	1.05376	0.526882	−	0.849939i	$-0.323361\pi$
$17$	4.34478	1.05376	0.526882	+	0.849939i	$0.323361\pi$
$18$	0	0
$19$	4.75485	1.09084	0.545419	−	0.838164i	$-0.316371\pi$
$19$	4.75485	1.09084	0.545419	+	0.838164i	$0.316371\pi$
$20$	0	0
$21$	−0.241192	−0.0526324
$22$	0	0
$23$	3.16248	0.659423	0.329711	−	0.944082i	$-0.393049\pi$
$23$	3.16248	0.659423	0.329711	+	0.944082i	$0.393049\pi$
$24$	0	0
$25$	6.49384	1.29877
$26$	0	0
$27$	1.74411	0.335655
$28$	0	0
$29$	2.75241	0.511109	0.255554	−	0.966795i	$-0.417742\pi$
$29$	2.75241	0.511109	0.255554	+	0.966795i	$0.417742\pi$
$30$	0	0
$31$	0.312335	0.0560970	0.0280485	−	0.999607i	$-0.491071\pi$
$31$	0.312335	0.0560970	0.0280485	+	0.999607i	$0.491071\pi$
$32$	0	0
$33$	0.323071	0.0562395
$34$	0	0
$35$	−2.77222	−0.468591
$36$	0	0
$37$	−7.02171	−1.15436	−0.577181	−	0.816616i	$-0.695847\pi$
$37$	−7.02171	−1.15436	−0.577181	+	0.816616i	$0.695847\pi$
$38$	0	0
$39$	0.320626	0.0513412
$40$	0	0
$41$	3.34873	0.522984	0.261492	−	0.965206i	$-0.415785\pi$
$41$	3.34873	0.522984	0.261492	+	0.965206i	$0.415785\pi$
$42$	0	0
$43$	8.25382	1.25870	0.629348	−	0.777124i	$-0.283322\pi$
$43$	8.25382	1.25870	0.629348	+	0.777124i	$0.283322\pi$
$44$	0	0
$45$	9.87581	1.47220
$46$	0	0
$47$	−0.104375	−0.0152246	−0.00761232	−	0.999971i	$-0.502423\pi$
$47$	−0.104375	−0.0152246	−0.00761232	+	0.999971i	$0.502423\pi$
$48$	0	0
$49$	−6.33136	−0.904480
$50$	0	0
$51$	−1.28155	−0.179453
$52$	0	0
$53$	−3.94940	−0.542491	−0.271246	−	0.962510i	$-0.587436\pi$
$53$	−3.94940	−0.542491	−0.271246	+	0.962510i	$0.587436\pi$
$54$	0	0
$55$	3.71333	0.500705
$56$	0	0
$57$	−1.40250	−0.185766
$58$	0	0
$59$	2.50024	0.325504	0.162752	−	0.986667i	$-0.447963\pi$
$59$	2.50024	0.325504	0.162752	+	0.986667i	$0.447963\pi$
$60$	0	0
$61$	−4.85923	−0.622160	−0.311080	−	0.950384i	$-0.600691\pi$
$61$	−4.85923	−0.622160	−0.311080	+	0.950384i	$0.600691\pi$
$62$	0	0
$63$	−2.38197	−0.300100
$64$	0	0
$65$	3.68522	0.457095
$66$	0	0
$67$	−11.8611	−1.44907	−0.724533	−	0.689240i	$-0.757944\pi$
$67$	−11.8611	−1.44907	−0.724533	+	0.689240i	$0.757944\pi$
$68$	0	0
$69$	−0.932814	−0.112298
$70$	0	0
$71$	9.42270	1.11827	0.559134	−	0.829077i	$-0.311134\pi$
$71$	9.42270	1.11827	0.559134	+	0.829077i	$0.311134\pi$
$72$	0	0
$73$	−1.26813	−0.148424	−0.0742119	−	0.997242i	$-0.523644\pi$
$73$	−1.26813	−0.148424	−0.0742119	+	0.997242i	$0.523644\pi$
$74$	0	0
$75$	−1.91544	−0.221176
$76$	0	0
$77$	−0.895625	−0.102066
$78$	0	0
$79$	−11.1700	−1.25672	−0.628361	−	0.777922i	$-0.716274\pi$
$79$	−11.1700	−1.25672	−0.628361	+	0.777922i	$0.716274\pi$
$80$	0	0
$81$	8.22454	0.913838
$82$	0	0
$83$	−12.3574	−1.35640	−0.678201	−	0.734877i	$-0.737240\pi$
$83$	−12.3574	−1.35640	−0.678201	+	0.734877i	$0.737240\pi$
$84$	0	0
$85$	−14.7299	−1.59768
$86$	0	0
$87$	−0.811858	−0.0870403
$88$	0	0
$89$	7.23094	0.766479	0.383239	−	0.923649i	$-0.374808\pi$
$89$	7.23094	0.766479	0.383239	+	0.923649i	$0.374808\pi$
$90$	0	0
$91$	−0.888846	−0.0931763
$92$	0	0
$93$	−0.0921272	−0.00955314
$94$	0	0
$95$	−16.1202	−1.65389
$96$	0	0
$97$	−5.07280	−0.515065	−0.257532	−	0.966270i	$-0.582909\pi$
$97$	−5.07280	−0.515065	−0.257532	+	0.966270i	$0.582909\pi$
$98$	0	0
$99$	3.19059	0.320666
$100$	0	0
$101$	−13.1121	−1.30470	−0.652352	−	0.757916i	$-0.726218\pi$
$101$	−13.1121	−1.30470	−0.652352	+	0.757916i	$0.726218\pi$
$102$	0	0
$103$	0.171561	0.0169044	0.00845219	−	0.999964i	$-0.497310\pi$
$103$	0.171561	0.0169044	0.00845219	+	0.999964i	$0.497310\pi$
$104$	0	0
$105$	0.817703	0.0797996
$106$	0	0
$107$	7.10754	0.687112	0.343556	−	0.939132i	$-0.388368\pi$
$107$	7.10754	0.687112	0.343556	+	0.939132i	$0.388368\pi$
$108$	0	0
$109$	−20.6454	−1.97747	−0.988733	−	0.149690i	$-0.952173\pi$
$109$	−20.6454	−1.97747	−0.988733	+	0.149690i	$0.952173\pi$
$110$	0	0
$111$	2.07114	0.196584
$112$	0	0
$113$	−8.51799	−0.801305	−0.400653	−	0.916230i	$-0.631217\pi$
$113$	−8.51799	−0.801305	−0.400653	+	0.916230i	$0.631217\pi$
$114$	0	0
$115$	−10.7216	−0.999796
$116$	0	0
$117$	3.16644	0.292737
$118$	0	0
$119$	3.55274	0.325679
$120$	0	0
$121$	−9.80033	−0.890939
$122$	0	0
$123$	−0.987752	−0.0890626
$124$	0	0
$125$	−5.06451	−0.452983
$126$	0	0
$127$	13.8292	1.22714	0.613570	−	0.789641i	$-0.289733\pi$
$127$	13.8292	1.22714	0.613570	+	0.789641i	$0.289733\pi$
$128$	0	0
$129$	−2.43457	−0.214352
$130$	0	0
$131$	−21.0265	−1.83710	−0.918548	−	0.395309i	$-0.870637\pi$
$131$	−21.0265	−1.83710	−0.918548	+	0.395309i	$0.870637\pi$
$132$	0	0
$133$	3.88806	0.337137
$134$	0	0
$135$	−5.91300	−0.508910
$136$	0	0
$137$	8.25393	0.705181	0.352590	−	0.935778i	$-0.385301\pi$
$137$	8.25393	0.705181	0.352590	+	0.935778i	$0.385301\pi$
$138$	0	0
$139$	3.21870	0.273006	0.136503	−	0.990640i	$-0.456414\pi$
$139$	3.21870	0.273006	0.136503	+	0.990640i	$0.456414\pi$
$140$	0	0
$141$	0.0307867	0.00259271
$142$	0	0
$143$	1.19059	0.0995620
$144$	0	0
$145$	−9.33136	−0.774927
$146$	0	0
$147$	1.86752	0.154030
$148$	0	0
$149$	18.9407	1.55168	0.775841	−	0.630929i	$-0.217326\pi$
$149$	18.9407	1.55168	0.775841	+	0.630929i	$0.217326\pi$
$150$	0	0
$151$	−16.3606	−1.33140	−0.665702	−	0.746218i	$-0.731868\pi$
$151$	−16.3606	−1.33140	−0.665702	+	0.746218i	$0.731868\pi$
$152$	0	0
$153$	−12.6563	−1.02320
$154$	0	0
$155$	−1.05889	−0.0850525
$156$	0	0
$157$	−12.6258	−1.00765	−0.503823	−	0.863807i	$-0.668074\pi$
$157$	−12.6258	−1.00765	−0.503823	+	0.863807i	$0.668074\pi$
$158$	0	0
$159$	1.16493	0.0923846
$160$	0	0
$161$	2.58597	0.203803
$162$	0	0
$163$	10.6561	0.834649	0.417325	−	0.908757i	$-0.362968\pi$
$163$	10.6561	0.834649	0.417325	+	0.908757i	$0.362968\pi$
$164$	0	0
$165$	−1.09529	−0.0852685
$166$	0	0
$167$	5.13871	0.397645	0.198822	−	0.980036i	$-0.436288\pi$
$167$	5.13871	0.397645	0.198822	+	0.980036i	$0.436288\pi$
$168$	0	0
$169$	−11.8184	−0.909110
$170$	0	0
$171$	−13.8509	−1.05920
$172$	0	0
$173$	−9.69200	−0.736869	−0.368435	−	0.929654i	$-0.620106\pi$
$173$	−9.69200	−0.736869	−0.368435	+	0.929654i	$0.620106\pi$
$174$	0	0
$175$	5.31003	0.401401
$176$	0	0
$177$	−0.737479	−0.0554323
$178$	0	0
$179$	2.05810	0.153830	0.0769150	−	0.997038i	$-0.475493\pi$
$179$	2.05810	0.153830	0.0769150	+	0.997038i	$0.475493\pi$
$180$	0	0
$181$	4.26455	0.316982	0.158491	−	0.987360i	$-0.449337\pi$
$181$	4.26455	0.316982	0.158491	+	0.987360i	$0.449337\pi$
$182$	0	0
$183$	1.43329	0.105952
$184$	0	0
$185$	23.8054	1.75021
$186$	0	0
$187$	−4.75881	−0.347999
$188$	0	0
$189$	1.42617	0.103738
$190$	0	0
$191$	15.6405	1.13170	0.565852	−	0.824507i	$-0.308547\pi$
$191$	15.6405	1.13170	0.565852	+	0.824507i	$0.308547\pi$
$192$	0	0
$193$	5.17845	0.372753	0.186377	−	0.982478i	$-0.440326\pi$
$193$	5.17845	0.372753	0.186377	+	0.982478i	$0.440326\pi$
$194$	0	0
$195$	−1.08700	−0.0778419
$196$	0	0
$197$	18.7291	1.33439	0.667197	−	0.744881i	$-0.267494\pi$
$197$	18.7291	1.33439	0.667197	+	0.744881i	$0.267494\pi$
$198$	0	0
$199$	3.54889	0.251574	0.125787	−	0.992057i	$-0.459854\pi$
$199$	3.54889	0.251574	0.125787	+	0.992057i	$0.459854\pi$
$200$	0	0
$201$	3.49859	0.246771
$202$	0	0
$203$	2.25065	0.157965
$204$	0	0
$205$	−11.3531	−0.792933
$206$	0	0
$207$	−9.21229	−0.640299
$208$	0	0
$209$	−5.20796	−0.360242
$210$	0	0
$211$	12.7688	0.879037	0.439519	−	0.898233i	$-0.355149\pi$
$211$	12.7688	0.879037	0.439519	+	0.898233i	$0.355149\pi$
$212$	0	0
$213$	−2.77935	−0.190438
$214$	0	0
$215$	−27.9826	−1.90839
$216$	0	0
$217$	0.255397	0.0173375
$218$	0	0
$219$	0.374052	0.0252761
$220$	0	0
$221$	−4.72279	−0.317689
$222$	0	0
$223$	−22.3372	−1.49581	−0.747905	−	0.663806i	$-0.768940\pi$
$223$	−22.3372	−1.49581	−0.747905	+	0.663806i	$0.768940\pi$
$224$	0	0
$225$	−18.9165	−1.26110
$226$	0	0
$227$	−22.9769	−1.52503	−0.762517	−	0.646968i	$-0.776037\pi$
$227$	−22.9769	−1.52503	−0.762517	+	0.646968i	$0.776037\pi$
$228$	0	0
$229$	−13.8947	−0.918187	−0.459094	−	0.888388i	$-0.651826\pi$
$229$	−13.8947	−0.918187	−0.459094	+	0.888388i	$0.651826\pi$
$230$	0	0
$231$	0.264176	0.0173815
$232$	0	0
$233$	15.9431	1.04447	0.522235	−	0.852802i	$-0.325098\pi$
$233$	15.9431	1.04447	0.522235	+	0.852802i	$0.325098\pi$
$234$	0	0
$235$	0.353858	0.0230831
$236$	0	0
$237$	3.29473	0.214016
$238$	0	0
$239$	0.0802236	0.00518924	0.00259462	−	0.999997i	$-0.499174\pi$
$239$	0.0802236	0.00518924	0.00259462	+	0.999997i	$0.499174\pi$
$240$	0	0
$241$	22.5200	1.45064	0.725320	−	0.688412i	$-0.241692\pi$
$241$	22.5200	1.45064	0.725320	+	0.688412i	$0.241692\pi$
$242$	0	0
$243$	−7.65828	−0.491279
$244$	0	0
$245$	21.4649	1.37134
$246$	0	0
$247$	−5.16854	−0.328866
$248$	0	0
$249$	3.64498	0.230991
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−3.46385	−0.217770
$254$	0	0
$255$	4.34478	0.272080
$256$	0	0
$257$	25.7236	1.60460	0.802298	−	0.596924i	$-0.203611\pi$
$257$	25.7236	1.60460	0.802298	+	0.596924i	$0.203611\pi$
$258$	0	0
$259$	−5.74167	−0.356770
$260$	0	0
$261$	−8.01775	−0.496286
$262$	0	0
$263$	−25.7678	−1.58891	−0.794456	−	0.607322i	$-0.792244\pi$
$263$	−25.7678	−1.58891	−0.794456	+	0.607322i	$0.792244\pi$
$264$	0	0
$265$	13.3895	0.822508
$266$	0	0
$267$	−2.13286	−0.130529
$268$	0	0
$269$	−28.6539	−1.74706	−0.873529	−	0.486772i	$-0.838174\pi$
$269$	−28.6539	−1.74706	−0.873529	+	0.486772i	$0.838174\pi$
$270$	0	0
$271$	−3.77082	−0.229061	−0.114531	−	0.993420i	$-0.536536\pi$
$271$	−3.77082	−0.229061	−0.114531	+	0.993420i	$0.536536\pi$
$272$	0	0
$273$	0.262176	0.0158676
$274$	0	0
$275$	−7.11267	−0.428910
$276$	0	0
$277$	−29.0928	−1.74802	−0.874008	−	0.485912i	$-0.838488\pi$
$277$	−29.0928	−1.74802	−0.874008	+	0.485912i	$0.838488\pi$
$278$	0	0
$279$	−0.909830	−0.0544701
$280$	0	0
$281$	−16.4256	−0.979871	−0.489936	−	0.871759i	$-0.662980\pi$
$281$	−16.4256	−0.979871	−0.489936	+	0.871759i	$0.662980\pi$
$282$	0	0
$283$	−8.88104	−0.527923	−0.263962	−	0.964533i	$-0.585029\pi$
$283$	−8.88104	−0.527923	−0.263962	+	0.964533i	$0.585029\pi$
$284$	0	0
$285$	4.75485	0.281653
$286$	0	0
$287$	2.73827	0.161635
$288$	0	0
$289$	1.87709	0.110417
$290$	0	0
$291$	1.49629	0.0877139
$292$	0	0
$293$	−9.14755	−0.534406	−0.267203	−	0.963640i	$-0.586099\pi$
$293$	−9.14755	−0.534406	−0.267203	+	0.963640i	$0.586099\pi$
$294$	0	0
$295$	−8.47647	−0.493519
$296$	0	0
$297$	−1.91032	−0.110848
$298$	0	0
$299$	−3.43763	−0.198803
$300$	0	0
$301$	6.74917	0.389016
$302$	0	0
$303$	3.86759	0.222187
$304$	0	0
$305$	16.4740	0.943300
$306$	0	0
$307$	−28.8933	−1.64903	−0.824513	−	0.565844i	$-0.808551\pi$
$307$	−28.8933	−1.64903	−0.824513	+	0.565844i	$0.808551\pi$
$308$	0	0
$309$	−0.0506040	−0.00287876
$310$	0	0
$311$	−2.87031	−0.162760	−0.0813801	−	0.996683i	$-0.525933\pi$
$311$	−2.87031	−0.162760	−0.0813801	+	0.996683i	$0.525933\pi$
$312$	0	0
$313$	24.1905	1.36733	0.683664	−	0.729796i	$-0.260385\pi$
$313$	24.1905	1.36733	0.683664	+	0.729796i	$0.260385\pi$
$314$	0	0
$315$	8.07548	0.455002
$316$	0	0
$317$	−33.9937	−1.90927	−0.954637	−	0.297771i	$-0.903757\pi$
$317$	−33.9937	−1.90927	−0.954637	+	0.297771i	$0.903757\pi$
$318$	0	0
$319$	−3.01469	−0.168790
$320$	0	0
$321$	−2.09646	−0.117013
$322$	0	0
$323$	20.6588	1.14948
$324$	0	0
$325$	−7.05883	−0.391553
$326$	0	0
$327$	6.08961	0.336756
$328$	0	0
$329$	−0.0853477	−0.00470537
$330$	0	0
$331$	32.1475	1.76699	0.883493	−	0.468445i	$-0.155186\pi$
$331$	32.1475	1.76699	0.883493	+	0.468445i	$0.155186\pi$
$332$	0	0
$333$	20.4542	1.12088
$334$	0	0
$335$	40.2122	2.19703
$336$	0	0
$337$	−0.552127	−0.0300762	−0.0150381	−	0.999887i	$-0.504787\pi$
$337$	−0.552127	−0.0300762	−0.0150381	+	0.999887i	$0.504787\pi$
$338$	0	0
$339$	2.51249	0.136460
$340$	0	0
$341$	−0.342098	−0.0185257
$342$	0	0
$343$	−10.9011	−0.588604
$344$	0	0
$345$	3.16248	0.170262
$346$	0	0
$347$	−6.93154	−0.372104	−0.186052	−	0.982540i	$-0.559569\pi$
$347$	−6.93154	−0.372104	−0.186052	+	0.982540i	$0.559569\pi$
$348$	0	0
$349$	18.7503	1.00368	0.501839	−	0.864961i	$-0.332657\pi$
$349$	18.7503	1.00368	0.501839	+	0.864961i	$0.332657\pi$
$350$	0	0
$351$	−1.89586	−0.101193
$352$	0	0
$353$	−11.1831	−0.595218	−0.297609	−	0.954688i	$-0.596189\pi$
$353$	−11.1831	−0.595218	−0.297609	+	0.954688i	$0.596189\pi$
$354$	0	0
$355$	−31.9454	−1.69548
$356$	0	0
$357$	−1.04793	−0.0554621
$358$	0	0
$359$	18.4200	0.972171	0.486086	−	0.873911i	$-0.338424\pi$
$359$	18.4200	0.972171	0.486086	+	0.873911i	$0.338424\pi$
$360$	0	0
$361$	3.60861	0.189927
$362$	0	0
$363$	2.89073	0.151724
$364$	0	0
$365$	4.29930	0.225035
$366$	0	0
$367$	−31.4979	−1.64417	−0.822087	−	0.569362i	$-0.807190\pi$
$367$	−31.4979	−1.64417	−0.822087	+	0.569362i	$0.807190\pi$
$368$	0	0
$369$	−9.75485	−0.507817
$370$	0	0
$371$	−3.22943	−0.167664
$372$	0	0
$373$	9.46611	0.490137	0.245068	−	0.969506i	$-0.421190\pi$
$373$	9.46611	0.490137	0.245068	+	0.969506i	$0.421190\pi$
$374$	0	0
$375$	1.49384	0.0771417
$376$	0	0
$377$	−2.99187	−0.154089
$378$	0	0
$379$	26.1586	1.34368	0.671838	−	0.740698i	$-0.265505\pi$
$379$	26.1586	1.34368	0.671838	+	0.740698i	$0.265505\pi$
$380$	0	0
$381$	−4.07909	−0.208978
$382$	0	0
$383$	24.2698	1.24013	0.620065	−	0.784550i	$-0.287106\pi$
$383$	24.2698	1.24013	0.620065	+	0.784550i	$0.287106\pi$
$384$	0	0
$385$	3.03640	0.154749
$386$	0	0
$387$	−24.0433	−1.22219
$388$	0	0
$389$	−29.2117	−1.48109	−0.740547	−	0.672005i	$-0.765433\pi$
$389$	−29.2117	−1.48109	−0.740547	+	0.672005i	$0.765433\pi$
$390$	0	0
$391$	13.7403	0.694875
$392$	0	0
$393$	6.20205	0.312852
$394$	0	0
$395$	37.8691	1.90540
$396$	0	0
$397$	−17.2936	−0.867942	−0.433971	−	0.900927i	$-0.642888\pi$
$397$	−17.2936	−0.867942	−0.433971	+	0.900927i	$0.642888\pi$
$398$	0	0
$399$	−1.14683	−0.0574134
$400$	0	0
$401$	19.3506	0.966324	0.483162	−	0.875531i	$-0.339488\pi$
$401$	19.3506	0.966324	0.483162	+	0.875531i	$0.339488\pi$
$402$	0	0
$403$	−0.339509	−0.0169121
$404$	0	0
$405$	−27.8833	−1.38553
$406$	0	0
$407$	7.69083	0.381220
$408$	0	0
$409$	−22.9078	−1.13272	−0.566360	−	0.824158i	$-0.691649\pi$
$409$	−22.9078	−1.13272	−0.566360	+	0.824158i	$0.691649\pi$
$410$	0	0
$411$	−2.43460	−0.120090
$412$	0	0
$413$	2.04446	0.100601
$414$	0	0
$415$	41.8948	2.05653
$416$	0	0
$417$	−0.949396	−0.0464921
$418$	0	0
$419$	24.6958	1.20647	0.603234	−	0.797564i	$-0.293879\pi$
$419$	24.6958	1.20647	0.603234	+	0.797564i	$0.293879\pi$
$420$	0	0
$421$	−28.4349	−1.38583	−0.692917	−	0.721017i	$-0.743675\pi$
$421$	−28.4349	−1.38583	−0.692917	+	0.721017i	$0.743675\pi$
$422$	0	0
$423$	0.304044	0.0147831
$424$	0	0
$425$	28.2143	1.36859
$426$	0	0
$427$	−3.97340	−0.192286
$428$	0	0
$429$	−0.351179	−0.0169551
$430$	0	0
$431$	1.94517	0.0936957	0.0468478	−	0.998902i	$-0.485082\pi$
$431$	1.94517	0.0936957	0.0468478	+	0.998902i	$0.485082\pi$
$432$	0	0
$433$	−31.0224	−1.49084	−0.745420	−	0.666595i	$-0.767751\pi$
$433$	−31.0224	−1.49084	−0.745420	+	0.666595i	$0.767751\pi$
$434$	0	0
$435$	2.75241	0.131968
$436$	0	0
$437$	15.0371	0.719323
$438$	0	0
$439$	−10.2149	−0.487528	−0.243764	−	0.969835i	$-0.578382\pi$
$439$	−10.2149	−0.487528	−0.243764	+	0.969835i	$0.578382\pi$
$440$	0	0
$441$	18.4432	0.878249
$442$	0	0
$443$	−33.6764	−1.60001	−0.800007	−	0.599991i	$-0.795171\pi$
$443$	−33.6764	−1.60001	−0.800007	+	0.599991i	$0.795171\pi$
$444$	0	0
$445$	−24.5148	−1.16211
$446$	0	0
$447$	−5.58680	−0.264247
$448$	0	0
$449$	−27.9568	−1.31936	−0.659682	−	0.751545i	$-0.729309\pi$
$449$	−27.9568	−1.31936	−0.659682	+	0.751545i	$0.729309\pi$
$450$	0	0
$451$	−3.66785	−0.172712
$452$	0	0
$453$	4.82576	0.226734
$454$	0	0
$455$	3.01341	0.141271
$456$	0	0
$457$	10.7280	0.501836	0.250918	−	0.968008i	$-0.419268\pi$
$457$	10.7280	0.501836	0.250918	+	0.968008i	$0.419268\pi$
$458$	0	0
$459$	7.57779	0.353701
$460$	0	0
$461$	−3.87124	−0.180302	−0.0901508	−	0.995928i	$-0.528735\pi$
$461$	−3.87124	−0.180302	−0.0901508	+	0.995928i	$0.528735\pi$
$462$	0	0
$463$	−26.2020	−1.21771	−0.608856	−	0.793281i	$-0.708371\pi$
$463$	−26.2020	−1.21771	−0.608856	+	0.793281i	$0.708371\pi$
$464$	0	0
$465$	0.312335	0.0144842
$466$	0	0
$467$	22.4766	1.04009	0.520046	−	0.854138i	$-0.325915\pi$
$467$	22.4766	1.04009	0.520046	+	0.854138i	$0.325915\pi$
$468$	0	0
$469$	−9.69887	−0.447852
$470$	0	0
$471$	3.72413	0.171599
$472$	0	0
$473$	−9.04036	−0.415676
$474$	0	0
$475$	30.8773	1.41675
$476$	0	0
$477$	11.5046	0.526758
$478$	0	0
$479$	−4.71833	−0.215586	−0.107793	−	0.994173i	$-0.534378\pi$
$479$	−4.71833	−0.215586	−0.107793	+	0.994173i	$0.534378\pi$
$480$	0	0
$481$	7.63262	0.348017
$482$	0	0
$483$	−0.762765	−0.0347070
$484$	0	0
$485$	17.1981	0.780925
$486$	0	0
$487$	−18.8243	−0.853009	−0.426505	−	0.904485i	$-0.640255\pi$
$487$	−18.8243	−0.853009	−0.426505	+	0.904485i	$0.640255\pi$
$488$	0	0
$489$	−3.14315	−0.142138
$490$	0	0
$491$	25.8844	1.16815	0.584074	−	0.811700i	$-0.301458\pi$
$491$	25.8844	1.16815	0.584074	+	0.811700i	$0.301458\pi$
$492$	0	0
$493$	11.9586	0.538588
$494$	0	0
$495$	−10.8169	−0.486184
$496$	0	0
$497$	7.70497	0.345615
$498$	0	0
$499$	13.3821	0.599066	0.299533	−	0.954086i	$-0.403169\pi$
$499$	13.3821	0.599066	0.299533	+	0.954086i	$0.403169\pi$
$500$	0	0
$501$	−1.51573	−0.0677177
$502$	0	0
$503$	−3.87570	−0.172809	−0.0864044	−	0.996260i	$-0.527538\pi$
$503$	−3.87570	−0.172809	−0.0864044	+	0.996260i	$0.527538\pi$
$504$	0	0
$505$	44.4534	1.97815
$506$	0	0
$507$	3.48600	0.154819
$508$	0	0
$509$	29.8334	1.32234	0.661171	−	0.750235i	$-0.270060\pi$
$509$	29.8334	1.32234	0.661171	+	0.750235i	$0.270060\pi$
$510$	0	0
$511$	−1.03696	−0.0458722
$512$	0	0
$513$	8.29301	0.366145
$514$	0	0
$515$	−0.581635	−0.0256299
$516$	0	0
$517$	0.114321	0.00502784
$518$	0	0
$519$	2.85878	0.125487
$520$	0	0
$521$	1.25529	0.0549951	0.0274975	−	0.999622i	$-0.491246\pi$
$521$	1.25529	0.0549951	0.0274975	+	0.999622i	$0.491246\pi$
$522$	0	0
$523$	18.4441	0.806504	0.403252	−	0.915089i	$-0.367880\pi$
$523$	18.4441	0.806504	0.403252	+	0.915089i	$0.367880\pi$
$524$	0	0
$525$	−1.56626	−0.0683573
$526$	0	0
$527$	1.35702	0.0591129
$528$	0	0
$529$	−12.9987	−0.565162
$530$	0	0
$531$	−7.28320	−0.316064
$532$	0	0
$533$	−3.64008	−0.157670
$534$	0	0
$535$	−24.0964	−1.04178
$536$	0	0
$537$	−0.607065	−0.0261968
$538$	0	0
$539$	6.93470	0.298699
$540$	0	0
$541$	16.1665	0.695052	0.347526	−	0.937670i	$-0.387022\pi$
$541$	16.1665	0.695052	0.347526	+	0.937670i	$0.387022\pi$
$542$	0	0
$543$	−1.25788	−0.0539810
$544$	0	0
$545$	69.9930	2.99817
$546$	0	0
$547$	−41.2887	−1.76538	−0.882690	−	0.469956i	$-0.844270\pi$
$547$	−41.2887	−1.76538	−0.882690	+	0.469956i	$0.844270\pi$
$548$	0	0
$549$	14.1549	0.604117
$550$	0	0
$551$	13.0873	0.557537
$552$	0	0
$553$	−9.13373	−0.388406
$554$	0	0
$555$	−7.02171	−0.298055
$556$	0	0
$557$	−11.7887	−0.499502	−0.249751	−	0.968310i	$-0.580349\pi$
$557$	−11.7887	−0.499502	−0.249751	+	0.968310i	$0.580349\pi$
$558$	0	0
$559$	−8.97192	−0.379472
$560$	0	0
$561$	1.40367	0.0592631
$562$	0	0
$563$	21.8308	0.920059	0.460030	−	0.887904i	$-0.347839\pi$
$563$	21.8308	0.920059	0.460030	+	0.887904i	$0.347839\pi$
$564$	0	0
$565$	28.8782	1.21491
$566$	0	0
$567$	6.72523	0.282433
$568$	0	0
$569$	−21.6104	−0.905956	−0.452978	−	0.891522i	$-0.649638\pi$
$569$	−21.6104	−0.905956	−0.452978	+	0.891522i	$0.649638\pi$
$570$	0	0
$571$	42.8030	1.79125	0.895625	−	0.444810i	$-0.146729\pi$
$571$	42.8030	1.79125	0.895625	+	0.444810i	$0.146729\pi$
$572$	0	0
$573$	−4.61336	−0.192726
$574$	0	0
$575$	20.5366	0.856437
$576$	0	0
$577$	−1.55993	−0.0649407	−0.0324703	−	0.999473i	$-0.510337\pi$
$577$	−1.55993	−0.0649407	−0.0324703	+	0.999473i	$0.510337\pi$
$578$	0	0
$579$	−1.52745	−0.0634787
$580$	0	0
$581$	−10.1047	−0.419213
$582$	0	0
$583$	4.32575	0.179154
$584$	0	0
$585$	−10.7350	−0.443839
$586$	0	0
$587$	37.8120	1.56067	0.780333	−	0.625364i	$-0.215050\pi$
$587$	37.8120	1.56067	0.780333	+	0.625364i	$0.215050\pi$
$588$	0	0
$589$	1.48511	0.0611927
$590$	0	0
$591$	−5.52440	−0.227243
$592$	0	0
$593$	23.7243	0.974240	0.487120	−	0.873335i	$-0.338047\pi$
$593$	23.7243	0.974240	0.487120	+	0.873335i	$0.338047\pi$
$594$	0	0
$595$	−12.0447	−0.493784
$596$	0	0
$597$	−1.04679	−0.0428423
$598$	0	0
$599$	−5.66220	−0.231351	−0.115676	−	0.993287i	$-0.536903\pi$
$599$	−5.66220	−0.231351	−0.115676	+	0.993287i	$0.536903\pi$
$600$	0	0
$601$	−11.9025	−0.485514	−0.242757	−	0.970087i	$-0.578052\pi$
$601$	−11.9025	−0.485514	−0.242757	+	0.970087i	$0.578052\pi$
$602$	0	0
$603$	34.5514	1.40704
$604$	0	0
$605$	33.2256	1.35081
$606$	0	0
$607$	33.2946	1.35139	0.675693	−	0.737183i	$-0.263845\pi$
$607$	33.2946	1.35139	0.675693	+	0.737183i	$0.263845\pi$
$608$	0	0
$609$	−0.663858	−0.0269009
$610$	0	0
$611$	0.113456	0.00458993
$612$	0	0
$613$	−16.2684	−0.657074	−0.328537	−	0.944491i	$-0.606556\pi$
$613$	−16.2684	−0.657074	−0.328537	+	0.944491i	$0.606556\pi$
$614$	0	0
$615$	3.34873	0.135034
$616$	0	0
$617$	21.3836	0.860872	0.430436	−	0.902621i	$-0.358360\pi$
$617$	21.3836	0.860872	0.430436	+	0.902621i	$0.358360\pi$
$618$	0	0
$619$	−42.7566	−1.71853	−0.859266	−	0.511530i	$-0.829079\pi$
$619$	−42.7566	−1.71853	−0.859266	+	0.511530i	$0.829079\pi$
$620$	0	0
$621$	5.51573	0.221339
$622$	0	0
$623$	5.91276	0.236890
$624$	0	0
$625$	−15.2992	−0.611969
$626$	0	0
$627$	1.53615	0.0613481
$628$	0	0
$629$	−30.5077	−1.21642
$630$	0	0
$631$	4.72791	0.188215	0.0941076	−	0.995562i	$-0.470000\pi$
$631$	4.72791	0.188215	0.0941076	+	0.995562i	$0.470000\pi$
$632$	0	0
$633$	−3.76631	−0.149697
$634$	0	0
$635$	−46.8844	−1.86055
$636$	0	0
$637$	6.88221	0.272683
$638$	0	0
$639$	−27.4483	−1.08584
$640$	0	0
$641$	−14.3322	−0.566089	−0.283044	−	0.959107i	$-0.591344\pi$
$641$	−14.3322	−0.566089	−0.283044	+	0.959107i	$0.591344\pi$
$642$	0	0
$643$	−22.2067	−0.875747	−0.437873	−	0.899037i	$-0.644268\pi$
$643$	−22.2067	−0.875747	−0.437873	+	0.899037i	$0.644268\pi$
$644$	0	0
$645$	8.25382	0.324994
$646$	0	0
$647$	24.5016	0.963259	0.481630	−	0.876375i	$-0.340045\pi$
$647$	24.5016	0.963259	0.481630	+	0.876375i	$0.340045\pi$
$648$	0	0
$649$	−2.73850	−0.107496
$650$	0	0
$651$	−0.0753326	−0.00295252
$652$	0	0
$653$	−35.0759	−1.37263	−0.686314	−	0.727306i	$-0.740772\pi$
$653$	−35.0759	−1.37263	−0.686314	+	0.727306i	$0.740772\pi$
$654$	0	0
$655$	71.2853	2.78535
$656$	0	0
$657$	3.69407	0.144119
$658$	0	0
$659$	−31.1971	−1.21527	−0.607633	−	0.794218i	$-0.707881\pi$
$659$	−31.1971	−1.21527	−0.607633	+	0.794218i	$0.707881\pi$
$660$	0	0
$661$	47.0362	1.82950	0.914749	−	0.404024i	$-0.132389\pi$
$661$	47.0362	1.82950	0.914749	+	0.404024i	$0.132389\pi$
$662$	0	0
$663$	1.39305	0.0541014
$664$	0	0
$665$	−13.1815	−0.511157
$666$	0	0
$667$	8.70443	0.337037
$668$	0	0
$669$	6.58865	0.254732
$670$	0	0
$671$	5.32228	0.205464
$672$	0	0
$673$	14.6838	0.566017	0.283009	−	0.959117i	$-0.408668\pi$
$673$	14.6838	0.566017	0.283009	+	0.959117i	$0.408668\pi$
$674$	0	0
$675$	11.3260	0.435938
$676$	0	0
$677$	−8.36911	−0.321651	−0.160825	−	0.986983i	$-0.551416\pi$
$677$	−8.36911	−0.321651	−0.160825	+	0.986983i	$0.551416\pi$
$678$	0	0
$679$	−4.14804	−0.159187
$680$	0	0
$681$	6.77735	0.259708
$682$	0	0
$683$	−41.8572	−1.60162	−0.800810	−	0.598919i	$-0.795597\pi$
$683$	−41.8572	−1.60162	−0.800810	+	0.598919i	$0.795597\pi$
$684$	0	0
$685$	−27.9829	−1.06917
$686$	0	0
$687$	4.09842	0.156364
$688$	0	0
$689$	4.29301	0.163550
$690$	0	0
$691$	2.06176	0.0784330	0.0392165	−	0.999231i	$-0.487514\pi$
$691$	2.06176	0.0784330	0.0392165	+	0.999231i	$0.487514\pi$
$692$	0	0
$693$	2.60895	0.0991059
$694$	0	0
$695$	−10.9122	−0.413924
$696$	0	0
$697$	14.5495	0.551102
$698$	0	0
$699$	−4.70263	−0.177870
$700$	0	0
$701$	−37.2613	−1.40734	−0.703670	−	0.710527i	$-0.748457\pi$
$701$	−37.2613	−1.40734	−0.703670	+	0.710527i	$0.748457\pi$
$702$	0	0
$703$	−33.3872	−1.25922
$704$	0	0
$705$	−0.104375	−0.00393099
$706$	0	0
$707$	−10.7218	−0.403235
$708$	0	0
$709$	−17.4531	−0.655465	−0.327733	−	0.944771i	$-0.606285\pi$
$709$	−17.4531	−0.655465	−0.327733	+	0.944771i	$0.606285\pi$
$710$	0	0
$711$	32.5381	1.22027
$712$	0	0
$713$	0.987752	0.0369916
$714$	0	0
$715$	−4.03640	−0.150953
$716$	0	0
$717$	−0.0236630	−0.000883711 0
$718$	0	0
$719$	−26.0602	−0.971880	−0.485940	−	0.873992i	$-0.661523\pi$
$719$	−26.0602	−0.971880	−0.485940	+	0.873992i	$0.661523\pi$
$720$	0	0
$721$	0.140286	0.00522451
$722$	0	0
$723$	−6.64256	−0.247040
$724$	0	0
$725$	17.8737	0.663812
$726$	0	0
$727$	−2.09590	−0.0777328	−0.0388664	−	0.999244i	$-0.512375\pi$
$727$	−2.09590	−0.0777328	−0.0388664	+	0.999244i	$0.512375\pi$
$728$	0	0
$729$	−22.4147	−0.830175
$730$	0	0
$731$	35.8610	1.32637
$732$	0	0
$733$	−14.0938	−0.520566	−0.260283	−	0.965532i	$-0.583816\pi$
$733$	−14.0938	−0.520566	−0.260283	+	0.965532i	$0.583816\pi$
$734$	0	0
$735$	−6.33136	−0.233536
$736$	0	0
$737$	12.9914	0.478545
$738$	0	0
$739$	−40.6409	−1.49500	−0.747501	−	0.664261i	$-0.768746\pi$
$739$	−40.6409	−1.49500	−0.747501	+	0.664261i	$0.768746\pi$
$740$	0	0
$741$	1.52453	0.0560049
$742$	0	0
$743$	10.1133	0.371021	0.185510	−	0.982642i	$-0.440606\pi$
$743$	10.1133	0.371021	0.185510	+	0.982642i	$0.440606\pi$
$744$	0	0
$745$	−64.2138	−2.35261
$746$	0	0
$747$	35.9971	1.31706
$748$	0	0
$749$	5.81186	0.212361
$750$	0	0
$751$	−11.4790	−0.418873	−0.209437	−	0.977822i	$-0.567163\pi$
$751$	−11.4790	−0.418873	−0.209437	+	0.977822i	$0.567163\pi$
$752$	0	0
$753$	−0.294963	−0.0107490
$754$	0	0
$755$	55.4665	2.01863
$756$	0	0
$757$	−13.9974	−0.508744	−0.254372	−	0.967106i	$-0.581869\pi$
$757$	−13.9974	−0.508744	−0.254372	+	0.967106i	$0.581869\pi$
$758$	0	0
$759$	1.02171	0.0370856
$760$	0	0
$761$	−30.7102	−1.11324	−0.556621	−	0.830766i	$-0.687903\pi$
$761$	−30.7102	−1.11324	−0.556621	+	0.830766i	$0.687903\pi$
$762$	0	0
$763$	−16.8818	−0.611161
$764$	0	0
$765$	42.9082	1.55135
$766$	0	0
$767$	−2.71777	−0.0981331
$768$	0	0
$769$	−3.42899	−0.123653	−0.0618263	−	0.998087i	$-0.519692\pi$
$769$	−3.42899	−0.123653	−0.0618263	+	0.998087i	$0.519692\pi$
$770$	0	0
$771$	−7.58751	−0.273258
$772$	0	0
$773$	−9.43745	−0.339441	−0.169721	−	0.985492i	$-0.554287\pi$
$773$	−9.43745	−0.339441	−0.169721	+	0.985492i	$0.554287\pi$
$774$	0	0
$775$	2.02825	0.0728570
$776$	0	0
$777$	1.69358	0.0607568
$778$	0	0
$779$	15.9227	0.570491
$780$	0	0
$781$	−10.3206	−0.369301
$782$	0	0
$783$	4.80051	0.171556
$784$	0	0
$785$	42.8046	1.52776
$786$	0	0
$787$	23.2657	0.829332	0.414666	−	0.909974i	$-0.363898\pi$
$787$	23.2657	0.829332	0.414666	+	0.909974i	$0.363898\pi$
$788$	0	0
$789$	7.60055	0.270587
$790$	0	0
$791$	−6.96519	−0.247654
$792$	0	0
$793$	5.28199	0.187569
$794$	0	0
$795$	−3.94940	−0.140071
$796$	0	0
$797$	26.5330	0.939847	0.469923	−	0.882707i	$-0.344282\pi$
$797$	26.5330	0.939847	0.469923	+	0.882707i	$0.344282\pi$
$798$	0	0
$799$	−0.453486	−0.0160432
$800$	0	0
$801$	−21.0637	−0.744250
$802$	0	0
$803$	1.38898	0.0490160
$804$	0	0
$805$	−8.76710	−0.309000
$806$	0	0
$807$	8.45183	0.297519
$808$	0	0
$809$	−23.0413	−0.810090	−0.405045	−	0.914297i	$-0.632744\pi$
$809$	−23.0413	−0.810090	−0.405045	+	0.914297i	$0.632744\pi$
$810$	0	0
$811$	8.82272	0.309807	0.154904	−	0.987930i	$-0.450493\pi$
$811$	8.82272	0.309807	0.154904	+	0.987930i	$0.450493\pi$
$812$	0	0
$813$	1.11225	0.0390084
$814$	0	0
$815$	−36.1269	−1.26547
$816$	0	0
$817$	39.2457	1.37303
$818$	0	0
$819$	2.58920	0.0904741
$820$	0	0
$821$	−35.2563	−1.23045	−0.615226	−	0.788350i	$-0.710935\pi$
$821$	−35.2563	−1.23045	−0.615226	+	0.788350i	$0.710935\pi$
$822$	0	0
$823$	5.52102	0.192451	0.0962253	−	0.995360i	$-0.469323\pi$
$823$	5.52102	0.192451	0.0962253	+	0.995360i	$0.469323\pi$
$824$	0	0
$825$	2.09797	0.0730420
$826$	0	0
$827$	0.260410	0.00905534	0.00452767	−	0.999990i	$-0.498559\pi$
$827$	0.260410	0.00905534	0.00452767	+	0.999990i	$0.498559\pi$
$828$	0	0
$829$	45.5420	1.58174	0.790869	−	0.611985i	$-0.209629\pi$
$829$	45.5420	1.58174	0.790869	+	0.611985i	$0.209629\pi$
$830$	0	0
$831$	8.58129	0.297682
$832$	0	0
$833$	−27.5084	−0.953108
$834$	0	0
$835$	−17.4215	−0.602897
$836$	0	0
$837$	0.544748	0.0188292
$838$	0	0
$839$	13.0746	0.451386	0.225693	−	0.974198i	$-0.427535\pi$
$839$	13.0746	0.451386	0.225693	+	0.974198i	$0.427535\pi$
$840$	0	0
$841$	−21.4243	−0.738768
$842$	0	0
$843$	4.84495	0.166869
$844$	0	0
$845$	40.0675	1.37836
$846$	0	0
$847$	−8.01376	−0.275356
$848$	0	0
$849$	2.61958	0.0899037
$850$	0	0
$851$	−22.2060	−0.761212
$852$	0	0
$853$	25.9398	0.888163	0.444081	−	0.895986i	$-0.353530\pi$
$853$	25.9398	0.888163	0.444081	+	0.895986i	$0.353530\pi$
$854$	0	0
$855$	46.9580	1.60593
$856$	0	0
$857$	33.2837	1.13695	0.568475	−	0.822701i	$-0.307534\pi$
$857$	33.2837	1.13695	0.568475	+	0.822701i	$0.307534\pi$
$858$	0	0
$859$	39.7494	1.35623	0.678116	−	0.734954i	$-0.262796\pi$
$859$	39.7494	1.35623	0.678116	+	0.734954i	$0.262796\pi$
$860$	0	0
$861$	−0.807688	−0.0275259
$862$	0	0
$863$	−1.07732	−0.0366725	−0.0183362	−	0.999832i	$-0.505837\pi$
$863$	−1.07732	−0.0366725	−0.0183362	+	0.999832i	$0.505837\pi$
$864$	0	0
$865$	32.8584	1.11722
$866$	0	0
$867$	−0.553671	−0.0188036
$868$	0	0
$869$	12.2344	0.415024
$870$	0	0
$871$	12.8931	0.436865
$872$	0	0
$873$	14.7770	0.500127
$874$	0	0
$875$	−4.14126	−0.140000
$876$	0	0
$877$	−7.43024	−0.250901	−0.125451	−	0.992100i	$-0.540038\pi$
$877$	−7.43024	−0.250901	−0.125451	+	0.992100i	$0.540038\pi$
$878$	0	0
$879$	2.69819	0.0910077
$880$	0	0
$881$	21.0974	0.710790	0.355395	−	0.934716i	$-0.384346\pi$
$881$	21.0974	0.710790	0.355395	+	0.934716i	$0.384346\pi$
$882$	0	0
$883$	−27.7146	−0.932671	−0.466335	−	0.884608i	$-0.654426\pi$
$883$	−27.7146	−0.932671	−0.466335	+	0.884608i	$0.654426\pi$
$884$	0	0
$885$	2.50024	0.0840448
$886$	0	0
$887$	−19.3414	−0.649421	−0.324711	−	0.945813i	$-0.605267\pi$
$887$	−19.3414	−0.649421	−0.324711	+	0.945813i	$0.605267\pi$
$888$	0	0
$889$	11.3081	0.379263
$890$	0	0
$891$	−9.00829	−0.301789
$892$	0	0
$893$	−0.496287	−0.0166076
$894$	0	0
$895$	−6.97750	−0.233232
$896$	0	0
$897$	1.01397	0.0338555
$898$	0	0
$899$	0.859672	0.0286717
$900$	0	0
$901$	−17.1592	−0.571657
$902$	0	0
$903$	−1.99075	−0.0662482
$904$	0	0
$905$	−14.4579	−0.480598
$906$	0	0
$907$	14.1967	0.471395	0.235697	−	0.971826i	$-0.424263\pi$
$907$	14.1967	0.471395	0.235697	+	0.971826i	$0.424263\pi$
$908$	0	0
$909$	38.1955	1.26687
$910$	0	0
$911$	−48.3621	−1.60231	−0.801154	−	0.598458i	$-0.795780\pi$
$911$	−48.3621	−1.60231	−0.801154	+	0.598458i	$0.795780\pi$
$912$	0	0
$913$	13.5350	0.447943
$914$	0	0
$915$	−4.85923	−0.160641
$916$	0	0
$917$	−17.1935	−0.567778
$918$	0	0
$919$	−13.7989	−0.455184	−0.227592	−	0.973757i	$-0.573085\pi$
$919$	−13.7989	−0.455184	−0.227592	+	0.973757i	$0.573085\pi$
$920$	0	0
$921$	8.52244	0.280824
$922$	0	0
$923$	−10.2425	−0.337136
$924$	0	0
$925$	−45.5978	−1.49925
$926$	0	0
$927$	−0.499756	−0.0164141
$928$	0	0
$929$	20.4778	0.671854	0.335927	−	0.941888i	$-0.390950\pi$
$929$	20.4778	0.671854	0.335927	+	0.941888i	$0.390950\pi$
$930$	0	0
$931$	−30.1047	−0.986641
$932$	0	0
$933$	0.846634	0.0277176
$934$	0	0
$935$	16.1336	0.527625
$936$	0	0
$937$	15.1233	0.494055	0.247028	−	0.969008i	$-0.420546\pi$
$937$	15.1233	0.494055	0.247028	+	0.969008i	$0.420546\pi$
$938$	0	0
$939$	−7.13531	−0.232852
$940$	0	0
$941$	55.6411	1.81385	0.906924	−	0.421295i	$-0.138424\pi$
$941$	55.6411	1.81385	0.906924	+	0.421295i	$0.138424\pi$
$942$	0	0
$943$	10.5903	0.344868
$944$	0	0
$945$	−4.83507	−0.157285
$946$	0	0
$947$	46.1808	1.50067	0.750337	−	0.661055i	$-0.229891\pi$
$947$	46.1808	1.50067	0.750337	+	0.661055i	$0.229891\pi$
$948$	0	0
$949$	1.37846	0.0447469
$950$	0	0
$951$	10.0269	0.325144
$952$	0	0
$953$	10.0119	0.324319	0.162159	−	0.986765i	$-0.448154\pi$
$953$	10.0119	0.324319	0.162159	+	0.986765i	$0.448154\pi$
$954$	0	0
$955$	−53.0252	−1.71585
$956$	0	0
$957$	0.889223	0.0287445
$958$	0	0
$959$	6.74926	0.217945
$960$	0	0
$961$	−30.9024	−0.996853
$962$	0	0
$963$	−20.7042	−0.667185
$964$	0	0
$965$	−17.5563	−0.565157
$966$	0	0
$967$	33.7563	1.08553	0.542764	−	0.839885i	$-0.317378\pi$
$967$	33.7563	1.08553	0.542764	+	0.839885i	$0.317378\pi$
$968$	0	0
$969$	−6.09357	−0.195754
$970$	0	0
$971$	−43.6889	−1.40204	−0.701021	−	0.713140i	$-0.747272\pi$
$971$	−43.6889	−1.40204	−0.701021	+	0.713140i	$0.747272\pi$
$972$	0	0
$973$	2.63194	0.0843760
$974$	0	0
$975$	2.08209	0.0666803
$976$	0	0
$977$	−41.3374	−1.32250	−0.661250	−	0.750165i	$-0.729974\pi$
$977$	−41.3374	−1.32250	−0.661250	+	0.750165i	$0.729974\pi$
$978$	0	0
$979$	−7.92001	−0.253125
$980$	0	0
$981$	60.1398	1.92012
$982$	0	0
$983$	28.3680	0.904799	0.452400	−	0.891815i	$-0.350568\pi$
$983$	28.3680	0.904799	0.452400	+	0.891815i	$0.350568\pi$
$984$	0	0
$985$	−63.4965	−2.02317
$986$	0	0
$987$	0.0251744	0.000801310 0
$988$	0	0
$989$	26.1025	0.830012
$990$	0	0
$991$	−8.90576	−0.282901	−0.141450	−	0.989945i	$-0.545177\pi$
$991$	−8.90576	−0.282901	−0.141450	+	0.989945i	$0.545177\pi$
$992$	0	0
$993$	−9.48232	−0.300912
$994$	0	0
$995$	−12.0317	−0.381429
$996$	0	0
$997$	4.86865	0.154192	0.0770959	−	0.997024i	$-0.475435\pi$
$997$	4.86865	0.154192	0.0770959	+	0.997024i	$0.475435\pi$
$998$	0	0
$999$	−12.2467	−0.387467

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.c.1.2		4
4.3	odd	2		251.2.a.a.1.4	✓	4
12.11	even	2		2259.2.a.f.1.2		4
20.19	odd	2		6275.2.a.c.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
251.2.a.a.1.4	✓	4	4.3	odd	2
2259.2.a.f.1.2		4	12.11	even	2
4016.2.a.c.1.2		4	1.1	even	1	trivial
6275.2.a.c.1.1		4	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q-0.294963 q^{3} -3.39026 q^{5} +0.817703 q^{7} -2.91300 q^{9} +O(q^{10})\)	\(q-0.294963 q^{3} -3.39026 q^{5} +0.817703 q^{7} -2.91300 q^{9} -1.09529 q^{11} -1.08700 q^{13} +1.00000 q^{15} +4.34478 q^{17} +4.75485 q^{19} -0.241192 q^{21} +3.16248 q^{23} +6.49384 q^{25} +1.74411 q^{27} +2.75241 q^{29} +0.312335 q^{31} +0.323071 q^{33} -2.77222 q^{35} -7.02171 q^{37} +0.320626 q^{39} +3.34873 q^{41} +8.25382 q^{43} +9.87581 q^{45} -0.104375 q^{47} -6.33136 q^{49} -1.28155 q^{51} -3.94940 q^{53} +3.71333 q^{55} -1.40250 q^{57} +2.50024 q^{59} -4.85923 q^{61} -2.38197 q^{63} +3.68522 q^{65} -11.8611 q^{67} -0.932814 q^{69} +9.42270 q^{71} -1.26813 q^{73} -1.91544 q^{75} -0.895625 q^{77} -11.1700 q^{79} +8.22454 q^{81} -12.3574 q^{83} -14.7299 q^{85} -0.811858 q^{87} +7.23094 q^{89} -0.888846 q^{91} -0.0921272 q^{93} -16.1202 q^{95} -5.07280 q^{97} +3.19059 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 3 q^{5} + 3 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 - 3 * q^5 + 3 * q^7 - 4 * q^9	\( 4 q + 2 q^{3} - 3 q^{5} + 3 q^{7} - 4 q^{9} + 3 q^{11} - 12 q^{13} + 4 q^{15} + q^{17} + 9 q^{19} - 7 q^{21} - 4 q^{23} - 7 q^{25} - q^{27} - 12 q^{29} + 2 q^{31} - 5 q^{35} - 13 q^{37} - 13 q^{39} + q^{41} + 5 q^{43} + 11 q^{45} - 12 q^{47} - 9 q^{49} - 2 q^{51} + 5 q^{53} + 3 q^{55} + 16 q^{57} - 6 q^{59} - 21 q^{61} - 14 q^{63} + q^{65} - 17 q^{67} - 13 q^{69} + 10 q^{71} - 2 q^{73} - 13 q^{75} + 8 q^{77} + 21 q^{79} - 8 q^{81} + q^{83} - 17 q^{85} - 31 q^{87} + 5 q^{89} + 2 q^{91} - 23 q^{93} - 12 q^{95} + 6 q^{97} - 2 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 - 3 * q^5 + 3 * q^7 - 4 * q^9 + 3 * q^11 - 12 * q^13 + 4 * q^15 + q^17 + 9 * q^19 - 7 * q^21 - 4 * q^23 - 7 * q^25 - q^27 - 12 * q^29 + 2 * q^31 - 5 * q^35 - 13 * q^37 - 13 * q^39 + q^41 + 5 * q^43 + 11 * q^45 - 12 * q^47 - 9 * q^49 - 2 * q^51 + 5 * q^53 + 3 * q^55 + 16 * q^57 - 6 * q^59 - 21 * q^61 - 14 * q^63 + q^65 - 17 * q^67 - 13 * q^69 + 10 * q^71 - 2 * q^73 - 13 * q^75 + 8 * q^77 + 21 * q^79 - 8 * q^81 + q^83 - 17 * q^85 - 31 * q^87 + 5 * q^89 + 2 * q^91 - 23 * q^93 - 12 * q^95 + 6 * q^97 - 2 * q^99

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