LMFDB - Embedded newform 8016.2.a.q.1.5

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:	5.5.161121.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 3x^{2} + 5x + 1 $ x^5 - x^4 - 6x^3 + 3x^2 + 5*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2004)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.31991$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.28459 q^{5} +1.96468 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.28459 q^{5} +1.96468 q^{7} +1.00000 q^{9} +2.40916 q^{11} -3.92621 q^{13} -1.28459 q^{15} -3.63982 q^{17} -2.30861 q^{19} -1.96468 q^{21} +4.55057 q^{23} -3.34982 q^{25} -1.00000 q^{27} -5.81156 q^{29} +4.65708 q^{31} -2.40916 q^{33} +2.52381 q^{35} +4.09300 q^{37} +3.92621 q^{39} +7.34227 q^{41} +3.88636 q^{43} +1.28459 q^{45} -7.51721 q^{47} -3.14003 q^{49} +3.63982 q^{51} -10.6471 q^{53} +3.09479 q^{55} +2.30861 q^{57} -10.8022 q^{59} -14.0025 q^{61} +1.96468 q^{63} -5.04358 q^{65} -0.286799 q^{67} -4.55057 q^{69} +10.1569 q^{71} +3.44823 q^{73} +3.34982 q^{75} +4.73323 q^{77} -5.99698 q^{79} +1.00000 q^{81} -10.0713 q^{83} -4.67569 q^{85} +5.81156 q^{87} -9.05989 q^{89} -7.71375 q^{91} -4.65708 q^{93} -2.96563 q^{95} +4.02676 q^{97} +2.40916 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} - 7 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 - 7 * q^5 + 2 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} - 7 q^{5} + 2 q^{7} + 5 q^{9} + 5 q^{11} - 8 q^{13} + 7 q^{15} - 7 q^{17} - 2 q^{19} - 2 q^{21} + 13 q^{23} + 2 q^{25} - 5 q^{27} - 11 q^{29} + 12 q^{31} - 5 q^{33} + 12 q^{35} - 7 q^{37} + 8 q^{39} - 12 q^{41} - 7 q^{45} + 19 q^{47} - 9 q^{49} + 7 q^{51} - 21 q^{53} + q^{55} + 2 q^{57} + 7 q^{59} - 6 q^{61} + 2 q^{63} + 14 q^{65} - 10 q^{67} - 13 q^{69} + 35 q^{71} - 8 q^{73} - 2 q^{75} - 6 q^{77} + 5 q^{81} + 11 q^{83} + 5 q^{85} + 11 q^{87} - 32 q^{89} - 5 q^{91} - 12 q^{93} + 19 q^{95} + 11 q^{97} + 5 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 - 7 * q^5 + 2 * q^7 + 5 * q^9 + 5 * q^11 - 8 * q^13 + 7 * q^15 - 7 * q^17 - 2 * q^19 - 2 * q^21 + 13 * q^23 + 2 * q^25 - 5 * q^27 - 11 * q^29 + 12 * q^31 - 5 * q^33 + 12 * q^35 - 7 * q^37 + 8 * q^39 - 12 * q^41 - 7 * q^45 + 19 * q^47 - 9 * q^49 + 7 * q^51 - 21 * q^53 + q^55 + 2 * q^57 + 7 * q^59 - 6 * q^61 + 2 * q^63 + 14 * q^65 - 10 * q^67 - 13 * q^69 + 35 * q^71 - 8 * q^73 - 2 * q^75 - 6 * q^77 + 5 * q^81 + 11 * q^83 + 5 * q^85 + 11 * q^87 - 32 * q^89 - 5 * q^91 - 12 * q^93 + 19 * q^95 + 11 * q^97 + 5 * 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
$5$	1.28459	0.574487	0.287244	−	0.957858i	$-0.407261\pi$
$5$	1.28459	0.574487	0.287244	+	0.957858i	$0.407261\pi$
$6$	0	0
$7$	1.96468	0.742579	0.371290	−	0.928517i	$-0.378916\pi$
$7$	1.96468	0.742579	0.371290	+	0.928517i	$0.378916\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.40916	0.726390	0.363195	−	0.931713i	$-0.381686\pi$
$11$	2.40916	0.726390	0.363195	+	0.931713i	$0.381686\pi$
$12$	0	0
$13$	−3.92621	−1.08893	−0.544467	−	0.838782i	$-0.683268\pi$
$13$	−3.92621	−1.08893	−0.544467	+	0.838782i	$0.683268\pi$
$14$	0	0
$15$	−1.28459	−0.331680
$16$	0	0
$17$	−3.63982	−0.882787	−0.441393	−	0.897314i	$-0.645516\pi$
$17$	−3.63982	−0.882787	−0.441393	+	0.897314i	$0.645516\pi$
$18$	0	0
$19$	−2.30861	−0.529632	−0.264816	−	0.964299i	$-0.585311\pi$
$19$	−2.30861	−0.529632	−0.264816	+	0.964299i	$0.585311\pi$
$20$	0	0
$21$	−1.96468	−0.428728
$22$	0	0
$23$	4.55057	0.948860	0.474430	−	0.880293i	$-0.342654\pi$
$23$	4.55057	0.948860	0.474430	+	0.880293i	$0.342654\pi$
$24$	0	0
$25$	−3.34982	−0.669965
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.81156	−1.07918	−0.539590	−	0.841928i	$-0.681421\pi$
$29$	−5.81156	−1.07918	−0.539590	+	0.841928i	$0.681421\pi$
$30$	0	0
$31$	4.65708	0.836436	0.418218	−	0.908347i	$-0.362655\pi$
$31$	4.65708	0.836436	0.418218	+	0.908347i	$0.362655\pi$
$32$	0	0
$33$	−2.40916	−0.419381
$34$	0	0
$35$	2.52381	0.426602
$36$	0	0
$37$	4.09300	0.672885	0.336442	−	0.941704i	$-0.390776\pi$
$37$	4.09300	0.672885	0.336442	+	0.941704i	$0.390776\pi$
$38$	0	0
$39$	3.92621	0.628697
$40$	0	0
$41$	7.34227	1.14667	0.573335	−	0.819321i	$-0.305649\pi$
$41$	7.34227	1.14667	0.573335	+	0.819321i	$0.305649\pi$
$42$	0	0
$43$	3.88636	0.592664	0.296332	−	0.955085i	$-0.404237\pi$
$43$	3.88636	0.592664	0.296332	+	0.955085i	$0.404237\pi$
$44$	0	0
$45$	1.28459	0.191496
$46$	0	0
$47$	−7.51721	−1.09650	−0.548249	−	0.836315i	$-0.684705\pi$
$47$	−7.51721	−1.09650	−0.548249	+	0.836315i	$0.684705\pi$
$48$	0	0
$49$	−3.14003	−0.448576
$50$	0	0
$51$	3.63982	0.509677
$52$	0	0
$53$	−10.6471	−1.46250	−0.731248	−	0.682111i	$-0.761062\pi$
$53$	−10.6471	−1.46250	−0.731248	+	0.682111i	$0.761062\pi$
$54$	0	0
$55$	3.09479	0.417302
$56$	0	0
$57$	2.30861	0.305783
$58$	0	0
$59$	−10.8022	−1.40633	−0.703164	−	0.711028i	$-0.748230\pi$
$59$	−10.8022	−1.40633	−0.703164	+	0.711028i	$0.748230\pi$
$60$	0	0
$61$	−14.0025	−1.79284	−0.896418	−	0.443209i	$-0.853840\pi$
$61$	−14.0025	−1.79284	−0.896418	+	0.443209i	$0.853840\pi$
$62$	0	0
$63$	1.96468	0.247526
$64$	0	0
$65$	−5.04358	−0.625579
$66$	0	0
$67$	−0.286799	−0.0350381	−0.0175191	−	0.999847i	$-0.505577\pi$
$67$	−0.286799	−0.0350381	−0.0175191	+	0.999847i	$0.505577\pi$
$68$	0	0
$69$	−4.55057	−0.547825
$70$	0	0
$71$	10.1569	1.20540	0.602699	−	0.797968i	$-0.294092\pi$
$71$	10.1569	1.20540	0.602699	+	0.797968i	$0.294092\pi$
$72$	0	0
$73$	3.44823	0.403585	0.201792	−	0.979428i	$-0.435323\pi$
$73$	3.44823	0.403585	0.201792	+	0.979428i	$0.435323\pi$
$74$	0	0
$75$	3.34982	0.386804
$76$	0	0
$77$	4.73323	0.539402
$78$	0	0
$79$	−5.99698	−0.674713	−0.337356	−	0.941377i	$-0.609533\pi$
$79$	−5.99698	−0.674713	−0.337356	+	0.941377i	$0.609533\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.0713	−1.10547	−0.552737	−	0.833356i	$-0.686417\pi$
$83$	−10.0713	−1.10547	−0.552737	+	0.833356i	$0.686417\pi$
$84$	0	0
$85$	−4.67569	−0.507150
$86$	0	0
$87$	5.81156	0.623064
$88$	0	0
$89$	−9.05989	−0.960346	−0.480173	−	0.877174i	$-0.659426\pi$
$89$	−9.05989	−0.960346	−0.480173	+	0.877174i	$0.659426\pi$
$90$	0	0
$91$	−7.71375	−0.808620
$92$	0	0
$93$	−4.65708	−0.482916
$94$	0	0
$95$	−2.96563	−0.304267
$96$	0	0
$97$	4.02676	0.408855	0.204428	−	0.978882i	$-0.434467\pi$
$97$	4.02676	0.408855	0.204428	+	0.978882i	$0.434467\pi$
$98$	0	0
$99$	2.40916	0.242130
$100$	0	0
$101$	1.06980	0.106450	0.0532248	−	0.998583i	$-0.483050\pi$
$101$	1.06980	0.106450	0.0532248	+	0.998583i	$0.483050\pi$
$102$	0	0
$103$	−15.5422	−1.53142	−0.765709	−	0.643188i	$-0.777612\pi$
$103$	−15.5422	−1.53142	−0.765709	+	0.643188i	$0.777612\pi$
$104$	0	0
$105$	−2.52381	−0.246299
$106$	0	0
$107$	2.54641	0.246171	0.123085	−	0.992396i	$-0.460721\pi$
$107$	2.54641	0.246171	0.123085	+	0.992396i	$0.460721\pi$
$108$	0	0
$109$	15.5695	1.49129	0.745646	−	0.666343i	$-0.232141\pi$
$109$	15.5695	1.49129	0.745646	+	0.666343i	$0.232141\pi$
$110$	0	0
$111$	−4.09300	−0.388490
$112$	0	0
$113$	−3.22887	−0.303746	−0.151873	−	0.988400i	$-0.548531\pi$
$113$	−3.22887	−0.303746	−0.151873	+	0.988400i	$0.548531\pi$
$114$	0	0
$115$	5.84563	0.545108
$116$	0	0
$117$	−3.92621	−0.362978
$118$	0	0
$119$	−7.15109	−0.655539
$120$	0	0
$121$	−5.19594	−0.472358
$122$	0	0
$123$	−7.34227	−0.662030
$124$	0	0
$125$	−10.7261	−0.959373
$126$	0	0
$127$	−0.940660	−0.0834701	−0.0417351	−	0.999129i	$-0.513289\pi$
$127$	−0.940660	−0.0834701	−0.0417351	+	0.999129i	$0.513289\pi$
$128$	0	0
$129$	−3.88636	−0.342175
$130$	0	0
$131$	13.7284	1.19946	0.599729	−	0.800203i	$-0.295275\pi$
$131$	13.7284	1.19946	0.599729	+	0.800203i	$0.295275\pi$
$132$	0	0
$133$	−4.53569	−0.393294
$134$	0	0
$135$	−1.28459	−0.110560
$136$	0	0
$137$	−17.2672	−1.47524	−0.737620	−	0.675216i	$-0.764050\pi$
$137$	−17.2672	−1.47524	−0.737620	+	0.675216i	$0.764050\pi$
$138$	0	0
$139$	−10.0634	−0.853570	−0.426785	−	0.904353i	$-0.640354\pi$
$139$	−10.0634	−0.853570	−0.426785	+	0.904353i	$0.640354\pi$
$140$	0	0
$141$	7.51721	0.633063
$142$	0	0
$143$	−9.45888	−0.790991
$144$	0	0
$145$	−7.46548	−0.619975
$146$	0	0
$147$	3.14003	0.258985
$148$	0	0
$149$	−18.8300	−1.54261	−0.771306	−	0.636464i	$-0.780396\pi$
$149$	−18.8300	−1.54261	−0.771306	+	0.636464i	$0.780396\pi$
$150$	0	0
$151$	−10.6354	−0.865498	−0.432749	−	0.901514i	$-0.642456\pi$
$151$	−10.6354	−0.865498	−0.432749	+	0.901514i	$0.642456\pi$
$152$	0	0
$153$	−3.63982	−0.294262
$154$	0	0
$155$	5.98244	0.480521
$156$	0	0
$157$	22.2179	1.77318	0.886592	−	0.462553i	$-0.153066\pi$
$157$	22.2179	1.77318	0.886592	+	0.462553i	$0.153066\pi$
$158$	0	0
$159$	10.6471	0.844373
$160$	0	0
$161$	8.94042	0.704604
$162$	0	0
$163$	21.9987	1.72307	0.861535	−	0.507699i	$-0.169504\pi$
$163$	21.9987	1.72307	0.861535	+	0.507699i	$0.169504\pi$
$164$	0	0
$165$	−3.09479	−0.240929
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	2.41512	0.185778
$170$	0	0
$171$	−2.30861	−0.176544
$172$	0	0
$173$	15.8426	1.20449	0.602246	−	0.798311i	$-0.294273\pi$
$173$	15.8426	1.20449	0.602246	+	0.798311i	$0.294273\pi$
$174$	0	0
$175$	−6.58133	−0.497502
$176$	0	0
$177$	10.8022	0.811944
$178$	0	0
$179$	3.43373	0.256649	0.128324	−	0.991732i	$-0.459040\pi$
$179$	3.43373	0.256649	0.128324	+	0.991732i	$0.459040\pi$
$180$	0	0
$181$	−11.3243	−0.841725	−0.420863	−	0.907124i	$-0.638273\pi$
$181$	−11.3243	−0.841725	−0.420863	+	0.907124i	$0.638273\pi$
$182$	0	0
$183$	14.0025	1.03509
$184$	0	0
$185$	5.25783	0.386564
$186$	0	0
$187$	−8.76893	−0.641247
$188$	0	0
$189$	−1.96468	−0.142909
$190$	0	0
$191$	26.0824	1.88726	0.943628	−	0.331007i	$-0.107388\pi$
$191$	26.0824	1.88726	0.943628	+	0.331007i	$0.107388\pi$
$192$	0	0
$193$	25.5074	1.83606	0.918031	−	0.396508i	$-0.129778\pi$
$193$	25.5074	1.83606	0.918031	+	0.396508i	$0.129778\pi$
$194$	0	0
$195$	5.04358	0.361178
$196$	0	0
$197$	−4.73730	−0.337519	−0.168759	−	0.985657i	$-0.553976\pi$
$197$	−4.73730	−0.337519	−0.168759	+	0.985657i	$0.553976\pi$
$198$	0	0
$199$	13.8698	0.983206	0.491603	−	0.870819i	$-0.336411\pi$
$199$	13.8698	0.983206	0.491603	+	0.870819i	$0.336411\pi$
$200$	0	0
$201$	0.286799	0.0202293
$202$	0	0
$203$	−11.4179	−0.801376
$204$	0	0
$205$	9.43182	0.658747
$206$	0	0
$207$	4.55057	0.316287
$208$	0	0
$209$	−5.56182	−0.384719
$210$	0	0
$211$	−17.5267	−1.20659	−0.603294	−	0.797519i	$-0.706146\pi$
$211$	−17.5267	−1.20659	−0.603294	+	0.797519i	$0.706146\pi$
$212$	0	0
$213$	−10.1569	−0.695937
$214$	0	0
$215$	4.99238	0.340478
$216$	0	0
$217$	9.14967	0.621120
$218$	0	0
$219$	−3.44823	−0.233010
$220$	0	0
$221$	14.2907	0.961297
$222$	0	0
$223$	−17.9825	−1.20420	−0.602099	−	0.798422i	$-0.705669\pi$
$223$	−17.9825	−1.20420	−0.602099	+	0.798422i	$0.705669\pi$
$224$	0	0
$225$	−3.34982	−0.223322
$226$	0	0
$227$	−4.80164	−0.318696	−0.159348	−	0.987222i	$-0.550939\pi$
$227$	−4.80164	−0.318696	−0.159348	+	0.987222i	$0.550939\pi$
$228$	0	0
$229$	−27.8758	−1.84208	−0.921041	−	0.389465i	$-0.872660\pi$
$229$	−27.8758	−1.84208	−0.921041	+	0.389465i	$0.872660\pi$
$230$	0	0
$231$	−4.73323	−0.311424
$232$	0	0
$233$	5.52381	0.361877	0.180939	−	0.983494i	$-0.442086\pi$
$233$	5.52381	0.361877	0.180939	+	0.983494i	$0.442086\pi$
$234$	0	0
$235$	−9.65655	−0.629924
$236$	0	0
$237$	5.99698	0.389546
$238$	0	0
$239$	23.2120	1.50146	0.750729	−	0.660611i	$-0.229703\pi$
$239$	23.2120	1.50146	0.750729	+	0.660611i	$0.229703\pi$
$240$	0	0
$241$	3.27780	0.211142	0.105571	−	0.994412i	$-0.466333\pi$
$241$	3.27780	0.211142	0.105571	+	0.994412i	$0.466333\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−4.03366	−0.257701
$246$	0	0
$247$	9.06410	0.576735
$248$	0	0
$249$	10.0713	0.638246
$250$	0	0
$251$	11.2250	0.708518	0.354259	−	0.935147i	$-0.384733\pi$
$251$	11.2250	0.708518	0.354259	+	0.935147i	$0.384733\pi$
$252$	0	0
$253$	10.9631	0.689242
$254$	0	0
$255$	4.67569	0.292803
$256$	0	0
$257$	12.9517	0.807907	0.403954	−	0.914779i	$-0.367636\pi$
$257$	12.9517	0.807907	0.403954	+	0.914779i	$0.367636\pi$
$258$	0	0
$259$	8.04143	0.499670
$260$	0	0
$261$	−5.81156	−0.359726
$262$	0	0
$263$	4.16998	0.257132	0.128566	−	0.991701i	$-0.458963\pi$
$263$	4.16998	0.257132	0.128566	+	0.991701i	$0.458963\pi$
$264$	0	0
$265$	−13.6772	−0.840185
$266$	0	0
$267$	9.05989	0.554456
$268$	0	0
$269$	−31.2329	−1.90430	−0.952152	−	0.305625i	$-0.901135\pi$
$269$	−31.2329	−1.90430	−0.952152	+	0.305625i	$0.901135\pi$
$270$	0	0
$271$	−19.9256	−1.21039	−0.605197	−	0.796076i	$-0.706905\pi$
$271$	−19.9256	−1.21039	−0.605197	+	0.796076i	$0.706905\pi$
$272$	0	0
$273$	7.71375	0.466857
$274$	0	0
$275$	−8.07027	−0.486655
$276$	0	0
$277$	2.20031	0.132204	0.0661021	−	0.997813i	$-0.478944\pi$
$277$	2.20031	0.132204	0.0661021	+	0.997813i	$0.478944\pi$
$278$	0	0
$279$	4.65708	0.278812
$280$	0	0
$281$	26.4717	1.57917	0.789585	−	0.613641i	$-0.210296\pi$
$281$	26.4717	1.57917	0.789585	+	0.613641i	$0.210296\pi$
$282$	0	0
$283$	−14.2466	−0.846870	−0.423435	−	0.905926i	$-0.639176\pi$
$283$	−14.2466	−0.846870	−0.423435	+	0.905926i	$0.639176\pi$
$284$	0	0
$285$	2.96563	0.175669
$286$	0	0
$287$	14.4252	0.851494
$288$	0	0
$289$	−3.75169	−0.220687
$290$	0	0
$291$	−4.02676	−0.236053
$292$	0	0
$293$	12.2255	0.714221	0.357111	−	0.934062i	$-0.383762\pi$
$293$	12.2255	0.714221	0.357111	+	0.934062i	$0.383762\pi$
$294$	0	0
$295$	−13.8764	−0.807917
$296$	0	0
$297$	−2.40916	−0.139794
$298$	0	0
$299$	−17.8665	−1.03325
$300$	0	0
$301$	7.63545	0.440100
$302$	0	0
$303$	−1.06980	−0.0614587
$304$	0	0
$305$	−17.9875	−1.02996
$306$	0	0
$307$	−1.04146	−0.0594395	−0.0297197	−	0.999558i	$-0.509461\pi$
$307$	−1.04146	−0.0594395	−0.0297197	+	0.999558i	$0.509461\pi$
$308$	0	0
$309$	15.5422	0.884164
$310$	0	0
$311$	1.61814	0.0917564	0.0458782	−	0.998947i	$-0.485391\pi$
$311$	1.61814	0.0917564	0.0458782	+	0.998947i	$0.485391\pi$
$312$	0	0
$313$	−25.1205	−1.41990	−0.709948	−	0.704254i	$-0.751281\pi$
$313$	−25.1205	−1.41990	−0.709948	+	0.704254i	$0.751281\pi$
$314$	0	0
$315$	2.52381	0.142201
$316$	0	0
$317$	−25.6518	−1.44075	−0.720374	−	0.693586i	$-0.756030\pi$
$317$	−25.6518	−1.44075	−0.720374	+	0.693586i	$0.756030\pi$
$318$	0	0
$319$	−14.0010	−0.783905
$320$	0	0
$321$	−2.54641	−0.142127
$322$	0	0
$323$	8.40294	0.467552
$324$	0	0
$325$	13.1521	0.729547
$326$	0	0
$327$	−15.5695	−0.860998
$328$	0	0
$329$	−14.7689	−0.814236
$330$	0	0
$331$	−17.7667	−0.976544	−0.488272	−	0.872691i	$-0.662373\pi$
$331$	−17.7667	−0.976544	−0.488272	+	0.872691i	$0.662373\pi$
$332$	0	0
$333$	4.09300	0.224295
$334$	0	0
$335$	−0.368420	−0.0201290
$336$	0	0
$337$	−19.2213	−1.04705	−0.523526	−	0.852010i	$-0.675384\pi$
$337$	−19.2213	−1.04705	−0.523526	+	0.852010i	$0.675384\pi$
$338$	0	0
$339$	3.22887	0.175368
$340$	0	0
$341$	11.2197	0.607578
$342$	0	0
$343$	−19.9219	−1.07568
$344$	0	0
$345$	−5.84563	−0.314718
$346$	0	0
$347$	−14.1326	−0.758677	−0.379339	−	0.925258i	$-0.623848\pi$
$347$	−14.1326	−0.758677	−0.379339	+	0.925258i	$0.623848\pi$
$348$	0	0
$349$	−28.1862	−1.50877	−0.754387	−	0.656429i	$-0.772066\pi$
$349$	−28.1862	−1.50877	−0.754387	+	0.656429i	$0.772066\pi$
$350$	0	0
$351$	3.92621	0.209566
$352$	0	0
$353$	−16.0991	−0.856870	−0.428435	−	0.903573i	$-0.640935\pi$
$353$	−16.0991	−0.856870	−0.428435	+	0.903573i	$0.640935\pi$
$354$	0	0
$355$	13.0474	0.692486
$356$	0	0
$357$	7.15109	0.378476
$358$	0	0
$359$	3.09816	0.163515	0.0817574	−	0.996652i	$-0.473947\pi$
$359$	3.09816	0.163515	0.0817574	+	0.996652i	$0.473947\pi$
$360$	0	0
$361$	−13.6703	−0.719490
$362$	0	0
$363$	5.19594	0.272716
$364$	0	0
$365$	4.42957	0.231854
$366$	0	0
$367$	15.5041	0.809307	0.404654	−	0.914470i	$-0.367392\pi$
$367$	15.5041	0.809307	0.404654	+	0.914470i	$0.367392\pi$
$368$	0	0
$369$	7.34227	0.382223
$370$	0	0
$371$	−20.9182	−1.08602
$372$	0	0
$373$	−8.04456	−0.416532	−0.208266	−	0.978072i	$-0.566782\pi$
$373$	−8.04456	−0.416532	−0.208266	+	0.978072i	$0.566782\pi$
$374$	0	0
$375$	10.7261	0.553894
$376$	0	0
$377$	22.8174	1.17516
$378$	0	0
$379$	−18.3818	−0.944210	−0.472105	−	0.881542i	$-0.656506\pi$
$379$	−18.3818	−0.944210	−0.472105	+	0.881542i	$0.656506\pi$
$380$	0	0
$381$	0.940660	0.0481915
$382$	0	0
$383$	−1.38490	−0.0707649	−0.0353825	−	0.999374i	$-0.511265\pi$
$383$	−1.38490	−0.0707649	−0.0353825	+	0.999374i	$0.511265\pi$
$384$	0	0
$385$	6.08028	0.309880
$386$	0	0
$387$	3.88636	0.197555
$388$	0	0
$389$	18.6320	0.944680	0.472340	−	0.881416i	$-0.343409\pi$
$389$	18.6320	0.944680	0.472340	+	0.881416i	$0.343409\pi$
$390$	0	0
$391$	−16.5633	−0.837641
$392$	0	0
$393$	−13.7284	−0.692507
$394$	0	0
$395$	−7.70367	−0.387614
$396$	0	0
$397$	36.8598	1.84994	0.924969	−	0.380042i	$-0.124090\pi$
$397$	36.8598	1.84994	0.924969	+	0.380042i	$0.124090\pi$
$398$	0	0
$399$	4.53569	0.227068
$400$	0	0
$401$	−5.06368	−0.252868	−0.126434	−	0.991975i	$-0.540353\pi$
$401$	−5.06368	−0.252868	−0.126434	+	0.991975i	$0.540353\pi$
$402$	0	0
$403$	−18.2847	−0.910824
$404$	0	0
$405$	1.28459	0.0638319
$406$	0	0
$407$	9.86070	0.488777
$408$	0	0
$409$	24.0361	1.18851	0.594254	−	0.804278i	$-0.297447\pi$
$409$	24.0361	1.18851	0.594254	+	0.804278i	$0.297447\pi$
$410$	0	0
$411$	17.2672	0.851730
$412$	0	0
$413$	−21.2229	−1.04431
$414$	0	0
$415$	−12.9376	−0.635080
$416$	0	0
$417$	10.0634	0.492809
$418$	0	0
$419$	−10.8318	−0.529170	−0.264585	−	0.964362i	$-0.585235\pi$
$419$	−10.8318	−0.529170	−0.264585	+	0.964362i	$0.585235\pi$
$420$	0	0
$421$	2.18637	0.106557	0.0532785	−	0.998580i	$-0.483033\pi$
$421$	2.18637	0.106557	0.0532785	+	0.998580i	$0.483033\pi$
$422$	0	0
$423$	−7.51721	−0.365499
$424$	0	0
$425$	12.1928	0.591436
$426$	0	0
$427$	−27.5104	−1.33132
$428$	0	0
$429$	9.45888	0.456679
$430$	0	0
$431$	−19.3902	−0.933994	−0.466997	−	0.884259i	$-0.654664\pi$
$431$	−19.3902	−0.933994	−0.466997	+	0.884259i	$0.654664\pi$
$432$	0	0
$433$	18.7996	0.903449	0.451725	−	0.892157i	$-0.350809\pi$
$433$	18.7996	0.903449	0.451725	+	0.892157i	$0.350809\pi$
$434$	0	0
$435$	7.46548	0.357942
$436$	0	0
$437$	−10.5055	−0.502547
$438$	0	0
$439$	8.05917	0.384643	0.192322	−	0.981332i	$-0.438398\pi$
$439$	8.05917	0.384643	0.192322	+	0.981332i	$0.438398\pi$
$440$	0	0
$441$	−3.14003	−0.149525
$442$	0	0
$443$	28.1425	1.33709	0.668546	−	0.743671i	$-0.266917\pi$
$443$	28.1425	1.33709	0.668546	+	0.743671i	$0.266917\pi$
$444$	0	0
$445$	−11.6383	−0.551706
$446$	0	0
$447$	18.8300	0.890627
$448$	0	0
$449$	28.9364	1.36559	0.682797	−	0.730608i	$-0.260763\pi$
$449$	28.9364	1.36559	0.682797	+	0.730608i	$0.260763\pi$
$450$	0	0
$451$	17.6887	0.832929
$452$	0	0
$453$	10.6354	0.499696
$454$	0	0
$455$	−9.90902	−0.464542
$456$	0	0
$457$	−7.80085	−0.364909	−0.182454	−	0.983214i	$-0.558404\pi$
$457$	−7.80085	−0.364909	−0.182454	+	0.983214i	$0.558404\pi$
$458$	0	0
$459$	3.63982	0.169892
$460$	0	0
$461$	−13.6981	−0.637984	−0.318992	−	0.947757i	$-0.603344\pi$
$461$	−13.6981	−0.637984	−0.318992	+	0.947757i	$0.603344\pi$
$462$	0	0
$463$	−39.1886	−1.82125	−0.910625	−	0.413235i	$-0.864399\pi$
$463$	−39.1886	−1.82125	−0.910625	+	0.413235i	$0.864399\pi$
$464$	0	0
$465$	−5.98244	−0.277429
$466$	0	0
$467$	−17.2581	−0.798610	−0.399305	−	0.916818i	$-0.630748\pi$
$467$	−17.2581	−0.798610	−0.399305	+	0.916818i	$0.630748\pi$
$468$	0	0
$469$	−0.563469	−0.0260186
$470$	0	0
$471$	−22.2179	−1.02375
$472$	0	0
$473$	9.36287	0.430505
$474$	0	0
$475$	7.73344	0.354835
$476$	0	0
$477$	−10.6471	−0.487499
$478$	0	0
$479$	2.46912	0.112817	0.0564084	−	0.998408i	$-0.482035\pi$
$479$	2.46912	0.112817	0.0564084	+	0.998408i	$0.482035\pi$
$480$	0	0
$481$	−16.0700	−0.732727
$482$	0	0
$483$	−8.94042	−0.406803
$484$	0	0
$485$	5.17274	0.234882
$486$	0	0
$487$	−17.6955	−0.801861	−0.400930	−	0.916108i	$-0.631313\pi$
$487$	−17.6955	−0.801861	−0.400930	+	0.916108i	$0.631313\pi$
$488$	0	0
$489$	−21.9987	−0.994814
$490$	0	0
$491$	−14.9043	−0.672623	−0.336312	−	0.941751i	$-0.609180\pi$
$491$	−14.9043	−0.672623	−0.336312	+	0.941751i	$0.609180\pi$
$492$	0	0
$493$	21.1530	0.952685
$494$	0	0
$495$	3.09479	0.139101
$496$	0	0
$497$	19.9550	0.895104
$498$	0	0
$499$	−14.7483	−0.660226	−0.330113	−	0.943941i	$-0.607087\pi$
$499$	−14.7483	−0.660226	−0.330113	+	0.943941i	$0.607087\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−21.6170	−0.963856	−0.481928	−	0.876211i	$-0.660063\pi$
$503$	−21.6170	−0.963856	−0.481928	+	0.876211i	$0.660063\pi$
$504$	0	0
$505$	1.37426	0.0611539
$506$	0	0
$507$	−2.41512	−0.107259
$508$	0	0
$509$	−11.3422	−0.502735	−0.251368	−	0.967892i	$-0.580880\pi$
$509$	−11.3422	−0.502735	−0.251368	+	0.967892i	$0.580880\pi$
$510$	0	0
$511$	6.77467	0.299694
$512$	0	0
$513$	2.30861	0.101928
$514$	0	0
$515$	−19.9654	−0.879779
$516$	0	0
$517$	−18.1102	−0.796484
$518$	0	0
$519$	−15.8426	−0.695413
$520$	0	0
$521$	−5.90072	−0.258515	−0.129258	−	0.991611i	$-0.541259\pi$
$521$	−5.90072	−0.258515	−0.129258	+	0.991611i	$0.541259\pi$
$522$	0	0
$523$	−7.21958	−0.315690	−0.157845	−	0.987464i	$-0.550455\pi$
$523$	−7.21958	−0.315690	−0.157845	+	0.987464i	$0.550455\pi$
$524$	0	0
$525$	6.58133	0.287233
$526$	0	0
$527$	−16.9509	−0.738394
$528$	0	0
$529$	−2.29229	−0.0996648
$530$	0	0
$531$	−10.8022	−0.468776
$532$	0	0
$533$	−28.8273	−1.24865
$534$	0	0
$535$	3.27110	0.141422
$536$	0	0
$537$	−3.43373	−0.148176
$538$	0	0
$539$	−7.56484	−0.325841
$540$	0	0
$541$	−4.15094	−0.178463	−0.0892314	−	0.996011i	$-0.528441\pi$
$541$	−4.15094	−0.178463	−0.0892314	+	0.996011i	$0.528441\pi$
$542$	0	0
$543$	11.3243	0.485970
$544$	0	0
$545$	20.0005	0.856728
$546$	0	0
$547$	5.37704	0.229906	0.114953	−	0.993371i	$-0.463328\pi$
$547$	5.37704	0.229906	0.114953	+	0.993371i	$0.463328\pi$
$548$	0	0
$549$	−14.0025	−0.597612
$550$	0	0
$551$	13.4166	0.571568
$552$	0	0
$553$	−11.7822	−0.501028
$554$	0	0
$555$	−5.25783	−0.223183
$556$	0	0
$557$	−13.1551	−0.557399	−0.278700	−	0.960378i	$-0.589903\pi$
$557$	−13.1551	−0.557399	−0.278700	+	0.960378i	$0.589903\pi$
$558$	0	0
$559$	−15.2587	−0.645372
$560$	0	0
$561$	8.76893	0.370224
$562$	0	0
$563$	−5.08117	−0.214146	−0.107073	−	0.994251i	$-0.534148\pi$
$563$	−5.08117	−0.214146	−0.107073	+	0.994251i	$0.534148\pi$
$564$	0	0
$565$	−4.14778	−0.174498
$566$	0	0
$567$	1.96468	0.0825088
$568$	0	0
$569$	−16.5777	−0.694972	−0.347486	−	0.937685i	$-0.612965\pi$
$569$	−16.5777	−0.694972	−0.347486	+	0.937685i	$0.612965\pi$
$570$	0	0
$571$	−30.5518	−1.27855	−0.639277	−	0.768977i	$-0.720766\pi$
$571$	−30.5518	−1.27855	−0.639277	+	0.768977i	$0.720766\pi$
$572$	0	0
$573$	−26.0824	−1.08961
$574$	0	0
$575$	−15.2436	−0.635703
$576$	0	0
$577$	−23.2441	−0.967663	−0.483832	−	0.875161i	$-0.660755\pi$
$577$	−23.2441	−0.967663	−0.483832	+	0.875161i	$0.660755\pi$
$578$	0	0
$579$	−25.5074	−1.06005
$580$	0	0
$581$	−19.7870	−0.820902
$582$	0	0
$583$	−25.6507	−1.06234
$584$	0	0
$585$	−5.04358	−0.208526
$586$	0	0
$587$	19.3846	0.800088	0.400044	−	0.916496i	$-0.368995\pi$
$587$	19.3846	0.800088	0.400044	+	0.916496i	$0.368995\pi$
$588$	0	0
$589$	−10.7514	−0.443003
$590$	0	0
$591$	4.73730	0.194867
$592$	0	0
$593$	9.34754	0.383857	0.191929	−	0.981409i	$-0.438526\pi$
$593$	9.34754	0.383857	0.191929	+	0.981409i	$0.438526\pi$
$594$	0	0
$595$	−9.18623	−0.376599
$596$	0	0
$597$	−13.8698	−0.567654
$598$	0	0
$599$	−1.66606	−0.0680732	−0.0340366	−	0.999421i	$-0.510836\pi$
$599$	−1.66606	−0.0680732	−0.0340366	+	0.999421i	$0.510836\pi$
$600$	0	0
$601$	−26.5815	−1.08428	−0.542141	−	0.840287i	$-0.682386\pi$
$601$	−26.5815	−1.08428	−0.542141	+	0.840287i	$0.682386\pi$
$602$	0	0
$603$	−0.286799	−0.0116794
$604$	0	0
$605$	−6.67466	−0.271363
$606$	0	0
$607$	30.8790	1.25334	0.626670	−	0.779285i	$-0.284417\pi$
$607$	30.8790	1.25334	0.626670	+	0.779285i	$0.284417\pi$
$608$	0	0
$609$	11.4179	0.462675
$610$	0	0
$611$	29.5141	1.19401
$612$	0	0
$613$	14.9530	0.603947	0.301973	−	0.953316i	$-0.402355\pi$
$613$	14.9530	0.603947	0.301973	+	0.953316i	$0.402355\pi$
$614$	0	0
$615$	−9.43182	−0.380328
$616$	0	0
$617$	20.3705	0.820087	0.410043	−	0.912066i	$-0.365514\pi$
$617$	20.3705	0.820087	0.410043	+	0.912066i	$0.365514\pi$
$618$	0	0
$619$	−0.534153	−0.0214694	−0.0107347	−	0.999942i	$-0.503417\pi$
$619$	−0.534153	−0.0214694	−0.0107347	+	0.999942i	$0.503417\pi$
$620$	0	0
$621$	−4.55057	−0.182608
$622$	0	0
$623$	−17.7998	−0.713133
$624$	0	0
$625$	2.97043	0.118817
$626$	0	0
$627$	5.56182	0.222118
$628$	0	0
$629$	−14.8978	−0.594014
$630$	0	0
$631$	16.3425	0.650583	0.325292	−	0.945614i	$-0.394538\pi$
$631$	16.3425	0.650583	0.325292	+	0.945614i	$0.394538\pi$
$632$	0	0
$633$	17.5267	0.696624
$634$	0	0
$635$	−1.20836	−0.0479525
$636$	0	0
$637$	12.3284	0.488470
$638$	0	0
$639$	10.1569	0.401800
$640$	0	0
$641$	45.9146	1.81352	0.906759	−	0.421649i	$-0.138548\pi$
$641$	45.9146	1.81352	0.906759	+	0.421649i	$0.138548\pi$
$642$	0	0
$643$	2.16517	0.0853860	0.0426930	−	0.999088i	$-0.486406\pi$
$643$	2.16517	0.0853860	0.0426930	+	0.999088i	$0.486406\pi$
$644$	0	0
$645$	−4.99238	−0.196575
$646$	0	0
$647$	29.7954	1.17138	0.585688	−	0.810536i	$-0.300824\pi$
$647$	29.7954	1.17138	0.585688	+	0.810536i	$0.300824\pi$
$648$	0	0
$649$	−26.0243	−1.02154
$650$	0	0
$651$	−9.14967	−0.358604
$652$	0	0
$653$	9.26237	0.362464	0.181232	−	0.983440i	$-0.441991\pi$
$653$	9.26237	0.362464	0.181232	+	0.983440i	$0.441991\pi$
$654$	0	0
$655$	17.6354	0.689073
$656$	0	0
$657$	3.44823	0.134528
$658$	0	0
$659$	−0.492474	−0.0191841	−0.00959203	−	0.999954i	$-0.503053\pi$
$659$	−0.492474	−0.0191841	−0.00959203	+	0.999954i	$0.503053\pi$
$660$	0	0
$661$	−8.06786	−0.313804	−0.156902	−	0.987614i	$-0.550151\pi$
$661$	−8.06786	−0.313804	−0.156902	+	0.987614i	$0.550151\pi$
$662$	0	0
$663$	−14.2907	−0.555005
$664$	0	0
$665$	−5.82651	−0.225942
$666$	0	0
$667$	−26.4459	−1.02399
$668$	0	0
$669$	17.9825	0.695244
$670$	0	0
$671$	−33.7343	−1.30230
$672$	0	0
$673$	6.59943	0.254389	0.127195	−	0.991878i	$-0.459403\pi$
$673$	6.59943	0.254389	0.127195	+	0.991878i	$0.459403\pi$
$674$	0	0
$675$	3.34982	0.128935
$676$	0	0
$677$	−14.6380	−0.562584	−0.281292	−	0.959622i	$-0.590763\pi$
$677$	−14.6380	−0.562584	−0.281292	+	0.959622i	$0.590763\pi$
$678$	0	0
$679$	7.91130	0.303608
$680$	0	0
$681$	4.80164	0.183999
$682$	0	0
$683$	10.3419	0.395724	0.197862	−	0.980230i	$-0.436600\pi$
$683$	10.3419	0.395724	0.197862	+	0.980230i	$0.436600\pi$
$684$	0	0
$685$	−22.1814	−0.847506
$686$	0	0
$687$	27.8758	1.06353
$688$	0	0
$689$	41.8029	1.59256
$690$	0	0
$691$	34.5829	1.31559	0.657797	−	0.753195i	$-0.271488\pi$
$691$	34.5829	1.31559	0.657797	+	0.753195i	$0.271488\pi$
$692$	0	0
$693$	4.73323	0.179801
$694$	0	0
$695$	−12.9274	−0.490365
$696$	0	0
$697$	−26.7246	−1.01227
$698$	0	0
$699$	−5.52381	−0.208930
$700$	0	0
$701$	−38.5480	−1.45594	−0.727969	−	0.685610i	$-0.759535\pi$
$701$	−38.5480	−1.45594	−0.727969	+	0.685610i	$0.759535\pi$
$702$	0	0
$703$	−9.44915	−0.356381
$704$	0	0
$705$	9.65655	0.363687
$706$	0	0
$707$	2.10183	0.0790473
$708$	0	0
$709$	16.9239	0.635592	0.317796	−	0.948159i	$-0.397057\pi$
$709$	16.9239	0.635592	0.317796	+	0.948159i	$0.397057\pi$
$710$	0	0
$711$	−5.99698	−0.224904
$712$	0	0
$713$	21.1924	0.793660
$714$	0	0
$715$	−12.1508	−0.454414
$716$	0	0
$717$	−23.2120	−0.866867
$718$	0	0
$719$	−16.1218	−0.601241	−0.300621	−	0.953744i	$-0.597194\pi$
$719$	−16.1218	−0.601241	−0.300621	+	0.953744i	$0.597194\pi$
$720$	0	0
$721$	−30.5354	−1.13720
$722$	0	0
$723$	−3.27780	−0.121903
$724$	0	0
$725$	19.4677	0.723012
$726$	0	0
$727$	−10.8925	−0.403980	−0.201990	−	0.979388i	$-0.564741\pi$
$727$	−10.8925	−0.403980	−0.201990	+	0.979388i	$0.564741\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−14.1457	−0.523196
$732$	0	0
$733$	39.2054	1.44809	0.724043	−	0.689755i	$-0.242282\pi$
$733$	39.2054	1.44809	0.724043	+	0.689755i	$0.242282\pi$
$734$	0	0
$735$	4.03366	0.148784
$736$	0	0
$737$	−0.690946	−0.0254513
$738$	0	0
$739$	46.8075	1.72184	0.860921	−	0.508738i	$-0.169888\pi$
$739$	46.8075	1.72184	0.860921	+	0.508738i	$0.169888\pi$
$740$	0	0
$741$	−9.06410	−0.332978
$742$	0	0
$743$	33.4808	1.22829	0.614145	−	0.789193i	$-0.289501\pi$
$743$	33.4808	1.22829	0.614145	+	0.789193i	$0.289501\pi$
$744$	0	0
$745$	−24.1888	−0.886211
$746$	0	0
$747$	−10.0713	−0.368491
$748$	0	0
$749$	5.00289	0.182801
$750$	0	0
$751$	24.7166	0.901922	0.450961	−	0.892544i	$-0.351081\pi$
$751$	24.7166	0.901922	0.450961	+	0.892544i	$0.351081\pi$
$752$	0	0
$753$	−11.2250	−0.409063
$754$	0	0
$755$	−13.6622	−0.497218
$756$	0	0
$757$	−26.5140	−0.963667	−0.481834	−	0.876263i	$-0.660029\pi$
$757$	−26.5140	−0.963667	−0.481834	+	0.876263i	$0.660029\pi$
$758$	0	0
$759$	−10.9631	−0.397934
$760$	0	0
$761$	25.2602	0.915681	0.457840	−	0.889034i	$-0.348623\pi$
$761$	25.2602	0.915681	0.457840	+	0.889034i	$0.348623\pi$
$762$	0	0
$763$	30.5892	1.10740
$764$	0	0
$765$	−4.67569	−0.169050
$766$	0	0
$767$	42.4117	1.53140
$768$	0	0
$769$	−19.6768	−0.709564	−0.354782	−	0.934949i	$-0.615445\pi$
$769$	−19.6768	−0.709564	−0.354782	+	0.934949i	$0.615445\pi$
$770$	0	0
$771$	−12.9517	−0.466446
$772$	0	0
$773$	−36.8440	−1.32519	−0.662594	−	0.748979i	$-0.730544\pi$
$773$	−36.8440	−1.32519	−0.662594	+	0.748979i	$0.730544\pi$
$774$	0	0
$775$	−15.6004	−0.560382
$776$	0	0
$777$	−8.04143	−0.288485
$778$	0	0
$779$	−16.9505	−0.607313
$780$	0	0
$781$	24.4695	0.875589
$782$	0	0
$783$	5.81156	0.207688
$784$	0	0
$785$	28.5410	1.01867
$786$	0	0
$787$	−5.36433	−0.191218	−0.0956088	−	0.995419i	$-0.530480\pi$
$787$	−5.36433	−0.191218	−0.0956088	+	0.995419i	$0.530480\pi$
$788$	0	0
$789$	−4.16998	−0.148455
$790$	0	0
$791$	−6.34369	−0.225556
$792$	0	0
$793$	54.9767	1.95228
$794$	0	0
$795$	13.6772	0.485081
$796$	0	0
$797$	−7.20815	−0.255326	−0.127663	−	0.991818i	$-0.540748\pi$
$797$	−7.20815	−0.255326	−0.127663	+	0.991818i	$0.540748\pi$
$798$	0	0
$799$	27.3613	0.967973
$800$	0	0
$801$	−9.05989	−0.320115
$802$	0	0
$803$	8.30734	0.293160
$804$	0	0
$805$	11.4848	0.404786
$806$	0	0
$807$	31.2329	1.09945
$808$	0	0
$809$	23.3611	0.821334	0.410667	−	0.911785i	$-0.365296\pi$
$809$	23.3611	0.821334	0.410667	+	0.911785i	$0.365296\pi$
$810$	0	0
$811$	−47.5681	−1.67034	−0.835170	−	0.549991i	$-0.814631\pi$
$811$	−47.5681	−1.67034	−0.835170	+	0.549991i	$0.814631\pi$
$812$	0	0
$813$	19.9256	0.698821
$814$	0	0
$815$	28.2593	0.989881
$816$	0	0
$817$	−8.97209	−0.313894
$818$	0	0
$819$	−7.71375	−0.269540
$820$	0	0
$821$	1.28529	0.0448568	0.0224284	−	0.999748i	$-0.492860\pi$
$821$	1.28529	0.0448568	0.0224284	+	0.999748i	$0.492860\pi$
$822$	0	0
$823$	−15.1103	−0.526712	−0.263356	−	0.964699i	$-0.584829\pi$
$823$	−15.1103	−0.526712	−0.263356	+	0.964699i	$0.584829\pi$
$824$	0	0
$825$	8.07027	0.280971
$826$	0	0
$827$	32.1009	1.11626	0.558129	−	0.829754i	$-0.311519\pi$
$827$	32.1009	1.11626	0.558129	+	0.829754i	$0.311519\pi$
$828$	0	0
$829$	−39.5385	−1.37323	−0.686614	−	0.727022i	$-0.740904\pi$
$829$	−39.5385	−1.37323	−0.686614	+	0.727022i	$0.740904\pi$
$830$	0	0
$831$	−2.20031	−0.0763281
$832$	0	0
$833$	11.4292	0.395997
$834$	0	0
$835$	1.28459	0.0444551
$836$	0	0
$837$	−4.65708	−0.160972
$838$	0	0
$839$	2.69426	0.0930161	0.0465081	−	0.998918i	$-0.485191\pi$
$839$	2.69426	0.0930161	0.0465081	+	0.998918i	$0.485191\pi$
$840$	0	0
$841$	4.77421	0.164628
$842$	0	0
$843$	−26.4717	−0.911735
$844$	0	0
$845$	3.10244	0.106727
$846$	0	0
$847$	−10.2084	−0.350763
$848$	0	0
$849$	14.2466	0.488941
$850$	0	0
$851$	18.6255	0.638473
$852$	0	0
$853$	48.3525	1.65556	0.827779	−	0.561054i	$-0.189604\pi$
$853$	48.3525	1.65556	0.827779	+	0.561054i	$0.189604\pi$
$854$	0	0
$855$	−2.96563	−0.101422
$856$	0	0
$857$	−36.9684	−1.26282	−0.631408	−	0.775450i	$-0.717523\pi$
$857$	−36.9684	−1.26282	−0.631408	+	0.775450i	$0.717523\pi$
$858$	0	0
$859$	−57.7504	−1.97042	−0.985210	−	0.171352i	$-0.945186\pi$
$859$	−57.7504	−1.97042	−0.985210	+	0.171352i	$0.945186\pi$
$860$	0	0
$861$	−14.4252	−0.491610
$862$	0	0
$863$	−0.667087	−0.0227079	−0.0113540	−	0.999936i	$-0.503614\pi$
$863$	−0.667087	−0.0227079	−0.0113540	+	0.999936i	$0.503614\pi$
$864$	0	0
$865$	20.3513	0.691965
$866$	0	0
$867$	3.75169	0.127414
$868$	0	0
$869$	−14.4477	−0.490105
$870$	0	0
$871$	1.12603	0.0381542
$872$	0	0
$873$	4.02676	0.136285
$874$	0	0
$875$	−21.0734	−0.712411
$876$	0	0
$877$	−17.1291	−0.578407	−0.289204	−	0.957268i	$-0.593390\pi$
$877$	−17.1291	−0.578407	−0.289204	+	0.957268i	$0.593390\pi$
$878$	0	0
$879$	−12.2255	−0.412356
$880$	0	0
$881$	−18.5491	−0.624935	−0.312468	−	0.949928i	$-0.601156\pi$
$881$	−18.5491	−0.624935	−0.312468	+	0.949928i	$0.601156\pi$
$882$	0	0
$883$	18.0598	0.607761	0.303880	−	0.952710i	$-0.401718\pi$
$883$	18.0598	0.607761	0.303880	+	0.952710i	$0.401718\pi$
$884$	0	0
$885$	13.8764	0.466451
$886$	0	0
$887$	14.4311	0.484549	0.242274	−	0.970208i	$-0.422107\pi$
$887$	14.4311	0.484549	0.242274	+	0.970208i	$0.422107\pi$
$888$	0	0
$889$	−1.84810	−0.0619832
$890$	0	0
$891$	2.40916	0.0807100
$892$	0	0
$893$	17.3543	0.580740
$894$	0	0
$895$	4.41094	0.147442
$896$	0	0
$897$	17.8665	0.596545
$898$	0	0
$899$	−27.0649	−0.902664
$900$	0	0
$901$	38.7537	1.29107
$902$	0	0
$903$	−7.63545	−0.254092
$904$	0	0
$905$	−14.5470	−0.483560
$906$	0	0
$907$	−53.6448	−1.78125	−0.890624	−	0.454741i	$-0.849732\pi$
$907$	−53.6448	−1.78125	−0.890624	+	0.454741i	$0.849732\pi$
$908$	0	0
$909$	1.06980	0.0354832
$910$	0	0
$911$	39.7916	1.31835	0.659177	−	0.751988i	$-0.270905\pi$
$911$	39.7916	1.31835	0.659177	+	0.751988i	$0.270905\pi$
$912$	0	0
$913$	−24.2635	−0.803005
$914$	0	0
$915$	17.9875	0.594648
$916$	0	0
$917$	26.9720	0.890693
$918$	0	0
$919$	25.7644	0.849889	0.424945	−	0.905219i	$-0.360294\pi$
$919$	25.7644	0.849889	0.424945	+	0.905219i	$0.360294\pi$
$920$	0	0
$921$	1.04146	0.0343174
$922$	0	0
$923$	−39.8780	−1.31260
$924$	0	0
$925$	−13.7108	−0.450809
$926$	0	0
$927$	−15.5422	−0.510472
$928$	0	0
$929$	−24.8575	−0.815549	−0.407775	−	0.913083i	$-0.633695\pi$
$929$	−24.8575	−0.815549	−0.407775	+	0.913083i	$0.633695\pi$
$930$	0	0
$931$	7.24911	0.237580
$932$	0	0
$933$	−1.61814	−0.0529756
$934$	0	0
$935$	−11.2645	−0.368388
$936$	0	0
$937$	12.4739	0.407505	0.203752	−	0.979022i	$-0.434686\pi$
$937$	12.4739	0.407505	0.203752	+	0.979022i	$0.434686\pi$
$938$	0	0
$939$	25.1205	0.819777
$940$	0	0
$941$	−8.43127	−0.274852	−0.137426	−	0.990512i	$-0.543883\pi$
$941$	−8.43127	−0.274852	−0.137426	+	0.990512i	$0.543883\pi$
$942$	0	0
$943$	33.4115	1.08803
$944$	0	0
$945$	−2.52381	−0.0820997
$946$	0	0
$947$	46.1816	1.50070	0.750350	−	0.661041i	$-0.229885\pi$
$947$	46.1816	1.50070	0.750350	+	0.661041i	$0.229885\pi$
$948$	0	0
$949$	−13.5385	−0.439477
$950$	0	0
$951$	25.6518	0.831816
$952$	0	0
$953$	−27.1218	−0.878563	−0.439281	−	0.898350i	$-0.644767\pi$
$953$	−27.1218	−0.878563	−0.439281	+	0.898350i	$0.644767\pi$
$954$	0	0
$955$	33.5053	1.08420
$956$	0	0
$957$	14.0010	0.452588
$958$	0	0
$959$	−33.9246	−1.09548
$960$	0	0
$961$	−9.31164	−0.300375
$962$	0	0
$963$	2.54641	0.0820570
$964$	0	0
$965$	32.7666	1.05479
$966$	0	0
$967$	51.9680	1.67118	0.835590	−	0.549354i	$-0.185126\pi$
$967$	51.9680	1.67118	0.835590	+	0.549354i	$0.185126\pi$
$968$	0	0
$969$	−8.40294	−0.269941
$970$	0	0
$971$	36.3326	1.16597	0.582985	−	0.812483i	$-0.301885\pi$
$971$	36.3326	1.16597	0.582985	+	0.812483i	$0.301885\pi$
$972$	0	0
$973$	−19.7714	−0.633843
$974$	0	0
$975$	−13.1521	−0.421204
$976$	0	0
$977$	−56.2379	−1.79921	−0.899605	−	0.436704i	$-0.856146\pi$
$977$	−56.2379	−1.79921	−0.899605	+	0.436704i	$0.856146\pi$
$978$	0	0
$979$	−21.8267	−0.697586
$980$	0	0
$981$	15.5695	0.497097
$982$	0	0
$983$	45.9094	1.46428	0.732141	−	0.681153i	$-0.238521\pi$
$983$	45.9094	1.46428	0.732141	+	0.681153i	$0.238521\pi$
$984$	0	0
$985$	−6.08550	−0.193900
$986$	0	0
$987$	14.7689	0.470100
$988$	0	0
$989$	17.6852	0.562355
$990$	0	0
$991$	23.0349	0.731726	0.365863	−	0.930669i	$-0.380774\pi$
$991$	23.0349	0.731726	0.365863	+	0.930669i	$0.380774\pi$
$992$	0	0
$993$	17.7667	0.563808
$994$	0	0
$995$	17.8171	0.564839
$996$	0	0
$997$	−49.2868	−1.56093	−0.780464	−	0.625201i	$-0.785017\pi$
$997$	−49.2868	−1.56093	−0.780464	+	0.625201i	$0.785017\pi$
$998$	0	0
$999$	−4.09300	−0.129497

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.q.1.5		5
4.3	odd	2		2004.2.a.b.1.5	✓	5
12.11	even	2		6012.2.a.f.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.b.1.5	✓	5	4.3	odd	2
6012.2.a.f.1.1		5	12.11	even	2
8016.2.a.q.1.5		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.28459 q^{5} +1.96468 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.28459 q^{5} +1.96468 q^{7} +1.00000 q^{9} +2.40916 q^{11} -3.92621 q^{13} -1.28459 q^{15} -3.63982 q^{17} -2.30861 q^{19} -1.96468 q^{21} +4.55057 q^{23} -3.34982 q^{25} -1.00000 q^{27} -5.81156 q^{29} +4.65708 q^{31} -2.40916 q^{33} +2.52381 q^{35} +4.09300 q^{37} +3.92621 q^{39} +7.34227 q^{41} +3.88636 q^{43} +1.28459 q^{45} -7.51721 q^{47} -3.14003 q^{49} +3.63982 q^{51} -10.6471 q^{53} +3.09479 q^{55} +2.30861 q^{57} -10.8022 q^{59} -14.0025 q^{61} +1.96468 q^{63} -5.04358 q^{65} -0.286799 q^{67} -4.55057 q^{69} +10.1569 q^{71} +3.44823 q^{73} +3.34982 q^{75} +4.73323 q^{77} -5.99698 q^{79} +1.00000 q^{81} -10.0713 q^{83} -4.67569 q^{85} +5.81156 q^{87} -9.05989 q^{89} -7.71375 q^{91} -4.65708 q^{93} -2.96563 q^{95} +4.02676 q^{97} +2.40916 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} - 7 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 - 7 * q^5 + 2 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} - 7 q^{5} + 2 q^{7} + 5 q^{9} + 5 q^{11} - 8 q^{13} + 7 q^{15} - 7 q^{17} - 2 q^{19} - 2 q^{21} + 13 q^{23} + 2 q^{25} - 5 q^{27} - 11 q^{29} + 12 q^{31} - 5 q^{33} + 12 q^{35} - 7 q^{37} + 8 q^{39} - 12 q^{41} - 7 q^{45} + 19 q^{47} - 9 q^{49} + 7 q^{51} - 21 q^{53} + q^{55} + 2 q^{57} + 7 q^{59} - 6 q^{61} + 2 q^{63} + 14 q^{65} - 10 q^{67} - 13 q^{69} + 35 q^{71} - 8 q^{73} - 2 q^{75} - 6 q^{77} + 5 q^{81} + 11 q^{83} + 5 q^{85} + 11 q^{87} - 32 q^{89} - 5 q^{91} - 12 q^{93} + 19 q^{95} + 11 q^{97} + 5 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 - 7 * q^5 + 2 * q^7 + 5 * q^9 + 5 * q^11 - 8 * q^13 + 7 * q^15 - 7 * q^17 - 2 * q^19 - 2 * q^21 + 13 * q^23 + 2 * q^25 - 5 * q^27 - 11 * q^29 + 12 * q^31 - 5 * q^33 + 12 * q^35 - 7 * q^37 + 8 * q^39 - 12 * q^41 - 7 * q^45 + 19 * q^47 - 9 * q^49 + 7 * q^51 - 21 * q^53 + q^55 + 2 * q^57 + 7 * q^59 - 6 * q^61 + 2 * q^63 + 14 * q^65 - 10 * q^67 - 13 * q^69 + 35 * q^71 - 8 * q^73 - 2 * q^75 - 6 * q^77 + 5 * q^81 + 11 * q^83 + 5 * q^85 + 11 * q^87 - 32 * q^89 - 5 * q^91 - 12 * q^93 + 19 * q^95 + 11 * q^97 + 5 * q^99

Introduction

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