LMFDB - Embedded newform 4014.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(1,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.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:	$32.0519513713$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.10273.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + x + 2 $ x^4 - 2x^3 - 5x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1338)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.641043$ of defining polynomial
Character	$\chi$	$=$	4014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.64104 q^{5} -3.78585 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.64104 q^{5} -3.78585 q^{7} +1.00000 q^{8} -1.64104 q^{10} +0.948021 q^{11} +2.09283 q^{13} -3.78585 q^{14} +1.00000 q^{16} +0.0270879 q^{17} +5.38464 q^{19} -1.64104 q^{20} +0.948021 q^{22} +3.61615 q^{23} -2.30698 q^{25} +2.09283 q^{26} -3.78585 q^{28} -3.17189 q^{29} -7.38464 q^{31} +1.00000 q^{32} +0.0270879 q^{34} +6.21274 q^{35} -2.49624 q^{37} +5.38464 q^{38} -1.64104 q^{40} +4.22247 q^{41} -6.30698 q^{43} +0.948021 q^{44} +3.61615 q^{46} -2.59879 q^{47} +7.33266 q^{49} -2.30698 q^{50} +2.09283 q^{52} -5.09502 q^{53} -1.55574 q^{55} -3.78585 q^{56} -3.17189 q^{58} -9.78444 q^{59} -2.50376 q^{61} -7.38464 q^{62} +1.00000 q^{64} -3.43442 q^{65} -4.35896 q^{67} +0.0270879 q^{68} +6.21274 q^{70} +8.97370 q^{71} -11.0083 q^{73} -2.49624 q^{74} +5.38464 q^{76} -3.58906 q^{77} -12.7361 q^{79} -1.64104 q^{80} +4.22247 q^{82} +8.87868 q^{83} -0.0444524 q^{85} -6.30698 q^{86} +0.948021 q^{88} +0.793377 q^{89} -7.92313 q^{91} +3.61615 q^{92} -2.59879 q^{94} -8.83642 q^{95} -8.13728 q^{97} +7.33266 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} - 2 q^{5} - 5 q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 - 2 * q^5 - 5 * q^7 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} - 2 q^{5} - 5 q^{7} + 4 q^{8} - 2 q^{10} - 4 q^{11} - 5 q^{13} - 5 q^{14} + 4 q^{16} + 2 q^{17} - 7 q^{19} - 2 q^{20} - 4 q^{22} + 4 q^{23} - 6 q^{25} - 5 q^{26} - 5 q^{28} - 9 q^{29} - q^{31} + 4 q^{32} + 2 q^{34} - 11 q^{37} - 7 q^{38} - 2 q^{40} - 14 q^{41} - 22 q^{43} - 4 q^{44} + 4 q^{46} + 8 q^{47} - 7 q^{49} - 6 q^{50} - 5 q^{52} - 3 q^{53} - 13 q^{55} - 5 q^{56} - 9 q^{58} + 6 q^{59} - 9 q^{61} - q^{62} + 4 q^{64} + 3 q^{65} - 22 q^{67} + 2 q^{68} - 5 q^{71} - 3 q^{73} - 11 q^{74} - 7 q^{76} - 2 q^{77} - 29 q^{79} - 2 q^{80} - 14 q^{82} + 12 q^{83} - 10 q^{85} - 22 q^{86} - 4 q^{88} - 9 q^{89} - 18 q^{91} + 4 q^{92} + 8 q^{94} + 2 q^{95} - 29 q^{97} - 7 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 - 2 * q^5 - 5 * q^7 + 4 * q^8 - 2 * q^10 - 4 * q^11 - 5 * q^13 - 5 * q^14 + 4 * q^16 + 2 * q^17 - 7 * q^19 - 2 * q^20 - 4 * q^22 + 4 * q^23 - 6 * q^25 - 5 * q^26 - 5 * q^28 - 9 * q^29 - q^31 + 4 * q^32 + 2 * q^34 - 11 * q^37 - 7 * q^38 - 2 * q^40 - 14 * q^41 - 22 * q^43 - 4 * q^44 + 4 * q^46 + 8 * q^47 - 7 * q^49 - 6 * q^50 - 5 * q^52 - 3 * q^53 - 13 * q^55 - 5 * q^56 - 9 * q^58 + 6 * q^59 - 9 * q^61 - q^62 + 4 * q^64 + 3 * q^65 - 22 * q^67 + 2 * q^68 - 5 * q^71 - 3 * q^73 - 11 * q^74 - 7 * q^76 - 2 * q^77 - 29 * q^79 - 2 * q^80 - 14 * q^82 + 12 * q^83 - 10 * q^85 - 22 * q^86 - 4 * q^88 - 9 * q^89 - 18 * q^91 + 4 * q^92 + 8 * q^94 + 2 * q^95 - 29 * q^97 - 7 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.64104	−0.733897	−0.366948	−	0.930241i	$-0.619597\pi$
$5$	−1.64104	−0.733897	−0.366948	+	0.930241i	$0.619597\pi$
$6$	0	0
$7$	−3.78585	−1.43092	−0.715458	−	0.698655i	$-0.753782\pi$
$7$	−3.78585	−1.43092	−0.715458	+	0.698655i	$0.753782\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.64104	−0.518943
$11$	0.948021	0.285839	0.142919	−	0.989734i	$-0.454351\pi$
$11$	0.948021	0.285839	0.142919	+	0.989734i	$0.454351\pi$
$12$	0	0
$13$	2.09283	0.580446	0.290223	−	0.956959i	$-0.406271\pi$
$13$	2.09283	0.580446	0.290223	+	0.956959i	$0.406271\pi$
$14$	−3.78585	−1.01181
$15$	0	0
$16$	1.00000	0.250000
$17$	0.0270879	0.00656977	0.00328489	−	0.999995i	$-0.498954\pi$
$17$	0.0270879	0.00656977	0.00328489	+	0.999995i	$0.498954\pi$
$18$	0	0
$19$	5.38464	1.23532	0.617660	−	0.786445i	$-0.288081\pi$
$19$	5.38464	1.23532	0.617660	+	0.786445i	$0.288081\pi$
$20$	−1.64104	−0.366948
$21$	0	0
$22$	0.948021	0.202119
$23$	3.61615	0.754020	0.377010	−	0.926209i	$-0.376952\pi$
$23$	3.61615	0.754020	0.377010	+	0.926209i	$0.376952\pi$
$24$	0	0
$25$	−2.30698	−0.461396
$26$	2.09283	0.410437
$27$	0	0
$28$	−3.78585	−0.715458
$29$	−3.17189	−0.589006	−0.294503	−	0.955651i	$-0.595154\pi$
$29$	−3.17189	−0.589006	−0.294503	+	0.955651i	$0.595154\pi$
$30$	0	0
$31$	−7.38464	−1.32632	−0.663160	−	0.748478i	$-0.730785\pi$
$31$	−7.38464	−1.32632	−0.663160	+	0.748478i	$0.730785\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.0270879	0.00464553
$35$	6.21274	1.05015
$36$	0	0
$37$	−2.49624	−0.410379	−0.205189	−	0.978722i	$-0.565781\pi$
$37$	−2.49624	−0.410379	−0.205189	+	0.978722i	$0.565781\pi$
$38$	5.38464	0.873503
$39$	0	0
$40$	−1.64104	−0.259472
$41$	4.22247	0.659438	0.329719	−	0.944079i	$-0.393046\pi$
$41$	4.22247	0.659438	0.329719	+	0.944079i	$0.393046\pi$
$42$	0	0
$43$	−6.30698	−0.961805	−0.480903	−	0.876774i	$-0.659691\pi$
$43$	−6.30698	−0.961805	−0.480903	+	0.876774i	$0.659691\pi$
$44$	0.948021	0.142919
$45$	0	0
$46$	3.61615	0.533172
$47$	−2.59879	−0.379072	−0.189536	−	0.981874i	$-0.560698\pi$
$47$	−2.59879	−0.379072	−0.189536	+	0.981874i	$0.560698\pi$
$48$	0	0
$49$	7.33266	1.04752
$50$	−2.30698	−0.326256
$51$	0	0
$52$	2.09283	0.290223
$53$	−5.09502	−0.699855	−0.349928	−	0.936777i	$-0.613794\pi$
$53$	−5.09502	−0.699855	−0.349928	+	0.936777i	$0.613794\pi$
$54$	0	0
$55$	−1.55574	−0.209776
$56$	−3.78585	−0.505905
$57$	0	0
$58$	−3.17189	−0.416490
$59$	−9.78444	−1.27383	−0.636913	−	0.770936i	$-0.719789\pi$
$59$	−9.78444	−1.27383	−0.636913	+	0.770936i	$0.719789\pi$
$60$	0	0
$61$	−2.50376	−0.320574	−0.160287	−	0.987070i	$-0.551242\pi$
$61$	−2.50376	−0.320574	−0.160287	+	0.987070i	$0.551242\pi$
$62$	−7.38464	−0.937850
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.43442	−0.425987
$66$	0	0
$67$	−4.35896	−0.532531	−0.266266	−	0.963900i	$-0.585790\pi$
$67$	−4.35896	−0.532531	−0.266266	+	0.963900i	$0.585790\pi$
$68$	0.0270879	0.00328489
$69$	0	0
$70$	6.21274	0.742565
$71$	8.97370	1.06498	0.532491	−	0.846436i	$-0.321256\pi$
$71$	8.97370	1.06498	0.532491	+	0.846436i	$0.321256\pi$
$72$	0	0
$73$	−11.0083	−1.28843	−0.644213	−	0.764846i	$-0.722815\pi$
$73$	−11.0083	−1.28843	−0.644213	+	0.764846i	$0.722815\pi$
$74$	−2.49624	−0.290182
$75$	0	0
$76$	5.38464	0.617660
$77$	−3.58906	−0.409012
$78$	0	0
$79$	−12.7361	−1.43292	−0.716460	−	0.697628i	$-0.754239\pi$
$79$	−12.7361	−1.43292	−0.716460	+	0.697628i	$0.754239\pi$
$80$	−1.64104	−0.183474
$81$	0	0
$82$	4.22247	0.466293
$83$	8.87868	0.974561	0.487281	−	0.873245i	$-0.337989\pi$
$83$	8.87868	0.974561	0.487281	+	0.873245i	$0.337989\pi$
$84$	0	0
$85$	−0.0444524	−0.00482154
$86$	−6.30698	−0.680099
$87$	0	0
$88$	0.948021	0.101059
$89$	0.793377	0.0840977	0.0420489	−	0.999116i	$-0.486611\pi$
$89$	0.793377	0.0840977	0.0420489	+	0.999116i	$0.486611\pi$
$90$	0	0
$91$	−7.92313	−0.830570
$92$	3.61615	0.377010
$93$	0	0
$94$	−2.59879	−0.268044
$95$	−8.83642	−0.906598
$96$	0	0
$97$	−8.13728	−0.826216	−0.413108	−	0.910682i	$-0.635557\pi$
$97$	−8.13728	−0.826216	−0.413108	+	0.910682i	$0.635557\pi$
$98$	7.33266	0.740710
$99$	0	0
$100$	−2.30698	−0.230698
$101$	−14.2655	−1.41947	−0.709736	−	0.704468i	$-0.751186\pi$
$101$	−14.2655	−1.41947	−0.709736	+	0.704468i	$0.751186\pi$
$102$	0	0
$103$	−2.92987	−0.288688	−0.144344	−	0.989528i	$-0.546107\pi$
$103$	−2.92987	−0.288688	−0.144344	+	0.989528i	$0.546107\pi$
$104$	2.09283	0.205219
$105$	0	0
$106$	−5.09502	−0.494872
$107$	6.15233	0.594769	0.297384	−	0.954758i	$-0.403886\pi$
$107$	6.15233	0.594769	0.297384	+	0.954758i	$0.403886\pi$
$108$	0	0
$109$	−1.92093	−0.183992	−0.0919960	−	0.995759i	$-0.529325\pi$
$109$	−1.92093	−0.183992	−0.0919960	+	0.995759i	$0.529325\pi$
$110$	−1.55574	−0.148334
$111$	0	0
$112$	−3.78585	−0.357729
$113$	12.3153	1.15853	0.579263	−	0.815141i	$-0.303341\pi$
$113$	12.3153	1.15853	0.579263	+	0.815141i	$0.303341\pi$
$114$	0	0
$115$	−5.93426	−0.553373
$116$	−3.17189	−0.294503
$117$	0	0
$118$	−9.78444	−0.900731
$119$	−0.102551	−0.00940080
$120$	0	0
$121$	−10.1013	−0.918296
$122$	−2.50376	−0.226680
$123$	0	0
$124$	−7.38464	−0.663160
$125$	11.9911	1.07251
$126$	0	0
$127$	−15.4851	−1.37408	−0.687040	−	0.726619i	$-0.741090\pi$
$127$	−15.4851	−1.37408	−0.687040	+	0.726619i	$0.741090\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−3.43442	−0.301219
$131$	−3.90357	−0.341056	−0.170528	−	0.985353i	$-0.554547\pi$
$131$	−3.90357	−0.341056	−0.170528	+	0.985353i	$0.554547\pi$
$132$	0	0
$133$	−20.3854	−1.76764
$134$	−4.35896	−0.376557
$135$	0	0
$136$	0.0270879	0.00232277
$137$	−4.38464	−0.374605	−0.187302	−	0.982302i	$-0.559974\pi$
$137$	−4.38464	−0.374605	−0.187302	+	0.982302i	$0.559974\pi$
$138$	0	0
$139$	−14.5565	−1.23467	−0.617334	−	0.786701i	$-0.711787\pi$
$139$	−14.5565	−1.23467	−0.617334	+	0.786701i	$0.711787\pi$
$140$	6.21274	0.525073
$141$	0	0
$142$	8.97370	0.753056
$143$	1.98404	0.165914
$144$	0	0
$145$	5.20522	0.432270
$146$	−11.0083	−0.911055
$147$	0	0
$148$	−2.49624	−0.205189
$149$	−16.4783	−1.34995	−0.674976	−	0.737840i	$-0.735846\pi$
$149$	−16.4783	−1.34995	−0.674976	+	0.737840i	$0.735846\pi$
$150$	0	0
$151$	−4.31450	−0.351109	−0.175555	−	0.984470i	$-0.556172\pi$
$151$	−4.31450	−0.351109	−0.175555	+	0.984470i	$0.556172\pi$
$152$	5.38464	0.436752
$153$	0	0
$154$	−3.58906	−0.289215
$155$	12.1185	0.973382
$156$	0	0
$157$	7.51219	0.599538	0.299769	−	0.954012i	$-0.403090\pi$
$157$	7.51219	0.599538	0.299769	+	0.954012i	$0.403090\pi$
$158$	−12.7361	−1.01323
$159$	0	0
$160$	−1.64104	−0.129736
$161$	−13.6902	−1.07894
$162$	0	0
$163$	−5.94179	−0.465397	−0.232698	−	0.972549i	$-0.574755\pi$
$163$	−5.94179	−0.465397	−0.232698	+	0.972549i	$0.574755\pi$
$164$	4.22247	0.329719
$165$	0	0
$166$	8.87868	0.689119
$167$	−3.87868	−0.300141	−0.150071	−	0.988675i	$-0.547950\pi$
$167$	−3.87868	−0.300141	−0.150071	+	0.988675i	$0.547950\pi$
$168$	0	0
$169$	−8.62007	−0.663083
$170$	−0.0444524	−0.00340934
$171$	0	0
$172$	−6.30698	−0.480903
$173$	7.89244	0.600051	0.300025	−	0.953931i	$-0.403005\pi$
$173$	7.89244	0.600051	0.300025	+	0.953931i	$0.403005\pi$
$174$	0	0
$175$	8.73387	0.660219
$176$	0.948021	0.0714597
$177$	0	0
$178$	0.793377	0.0594661
$179$	4.44994	0.332604	0.166302	−	0.986075i	$-0.446817\pi$
$179$	4.44994	0.332604	0.166302	+	0.986075i	$0.446817\pi$
$180$	0	0
$181$	8.23972	0.612453	0.306227	−	0.951959i	$-0.400933\pi$
$181$	8.23972	0.612453	0.306227	+	0.951959i	$0.400933\pi$
$182$	−7.92313	−0.587301
$183$	0	0
$184$	3.61615	0.266586
$185$	4.09643	0.301176
$186$	0	0
$187$	0.0256799	0.00187790
$188$	−2.59879	−0.189536
$189$	0	0
$190$	−8.83642	−0.641061
$191$	5.91408	0.427928	0.213964	−	0.976842i	$-0.431363\pi$
$191$	5.91408	0.427928	0.213964	+	0.976842i	$0.431363\pi$
$192$	0	0
$193$	16.7133	1.20305	0.601523	−	0.798855i	$-0.294561\pi$
$193$	16.7133	1.20305	0.601523	+	0.798855i	$0.294561\pi$
$194$	−8.13728	−0.584223
$195$	0	0
$196$	7.33266	0.523761
$197$	6.95010	0.495174	0.247587	−	0.968866i	$-0.420362\pi$
$197$	6.95010	0.495174	0.247587	+	0.968866i	$0.420362\pi$
$198$	0	0
$199$	−18.5053	−1.31181	−0.655904	−	0.754844i	$-0.727713\pi$
$199$	−18.5053	−1.31181	−0.655904	+	0.754844i	$0.727713\pi$
$200$	−2.30698	−0.163128
$201$	0	0
$202$	−14.2655	−1.00372
$203$	12.0083	0.842819
$204$	0	0
$205$	−6.92925	−0.483960
$206$	−2.92987	−0.204134
$207$	0	0
$208$	2.09283	0.145111
$209$	5.10475	0.353103
$210$	0	0
$211$	20.8892	1.43807	0.719036	−	0.694973i	$-0.244584\pi$
$211$	20.8892	1.43807	0.719036	+	0.694973i	$0.244584\pi$
$212$	−5.09502	−0.349928
$213$	0	0
$214$	6.15233	0.420565
$215$	10.3500	0.705866
$216$	0	0
$217$	27.9571	1.89785
$218$	−1.92093	−0.130102
$219$	0	0
$220$	−1.55574	−0.104888
$221$	0.0566902	0.00381340
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	−3.78585	−0.252953
$225$	0	0
$226$	12.3153	0.819201
$227$	−5.44959	−0.361702	−0.180851	−	0.983511i	$-0.557885\pi$
$227$	−5.44959	−0.361702	−0.180851	+	0.983511i	$0.557885\pi$
$228$	0	0
$229$	20.9369	1.38355	0.691774	−	0.722114i	$-0.256829\pi$
$229$	20.9369	1.38355	0.691774	+	0.722114i	$0.256829\pi$
$230$	−5.93426	−0.391294
$231$	0	0
$232$	−3.17189	−0.208245
$233$	27.0478	1.77196	0.885979	−	0.463726i	$-0.153488\pi$
$233$	27.0478	1.77196	0.885979	+	0.463726i	$0.153488\pi$
$234$	0	0
$235$	4.26472	0.278200
$236$	−9.78444	−0.636913
$237$	0	0
$238$	−0.102551	−0.00664737
$239$	−30.6714	−1.98397	−0.991985	−	0.126355i	$-0.959672\pi$
$239$	−30.6714	−1.98397	−0.991985	+	0.126355i	$0.959672\pi$
$240$	0	0
$241$	−9.66531	−0.622598	−0.311299	−	0.950312i	$-0.600764\pi$
$241$	−9.66531	−0.622598	−0.311299	+	0.950312i	$0.600764\pi$
$242$	−10.1013	−0.649333
$243$	0	0
$244$	−2.50376	−0.160287
$245$	−12.0332	−0.768773
$246$	0	0
$247$	11.2691	0.717037
$248$	−7.38464	−0.468925
$249$	0	0
$250$	11.9911	0.758382
$251$	−14.8004	−0.934193	−0.467096	−	0.884206i	$-0.654700\pi$
$251$	−14.8004	−0.934193	−0.467096	+	0.884206i	$0.654700\pi$
$252$	0	0
$253$	3.42819	0.215528
$254$	−15.4851	−0.971622
$255$	0	0
$256$	1.00000	0.0625000
$257$	−13.5800	−0.847098	−0.423549	−	0.905873i	$-0.639216\pi$
$257$	−13.5800	−0.847098	−0.423549	+	0.905873i	$0.639216\pi$
$258$	0	0
$259$	9.45038	0.587218
$260$	−3.43442	−0.212994
$261$	0	0
$262$	−3.90357	−0.241163
$263$	−7.89420	−0.486777	−0.243389	−	0.969929i	$-0.578259\pi$
$263$	−7.89420	−0.486777	−0.243389	+	0.969929i	$0.578259\pi$
$264$	0	0
$265$	8.36115	0.513622
$266$	−20.3854	−1.24991
$267$	0	0
$268$	−4.35896	−0.266266
$269$	14.8946	0.908142	0.454071	−	0.890966i	$-0.349971\pi$
$269$	14.8946	0.908142	0.454071	+	0.890966i	$0.349971\pi$
$270$	0	0
$271$	3.27769	0.199106	0.0995528	−	0.995032i	$-0.468259\pi$
$271$	3.27769	0.199106	0.0995528	+	0.995032i	$0.468259\pi$
$272$	0.0270879	0.00164244
$273$	0	0
$274$	−4.38464	−0.264886
$275$	−2.18706	−0.131885
$276$	0	0
$277$	−8.26018	−0.496306	−0.248153	−	0.968721i	$-0.579824\pi$
$277$	−8.26018	−0.496306	−0.248153	+	0.968721i	$0.579824\pi$
$278$	−14.5565	−0.873043
$279$	0	0
$280$	6.21274	0.371282
$281$	14.6209	0.872208	0.436104	−	0.899896i	$-0.356358\pi$
$281$	14.6209	0.872208	0.436104	+	0.899896i	$0.356358\pi$
$282$	0	0
$283$	24.5961	1.46209	0.731043	−	0.682332i	$-0.239034\pi$
$283$	24.5961	1.46209	0.731043	+	0.682332i	$0.239034\pi$
$284$	8.97370	0.532491
$285$	0	0
$286$	1.98404	0.117319
$287$	−15.9856	−0.943601
$288$	0	0
$289$	−16.9993	−0.999957
$290$	5.20522	0.305661
$291$	0	0
$292$	−11.0083	−0.644213
$293$	2.60282	0.152059	0.0760293	−	0.997106i	$-0.475776\pi$
$293$	2.60282	0.152059	0.0760293	+	0.997106i	$0.475776\pi$
$294$	0	0
$295$	16.0567	0.934857
$296$	−2.49624	−0.145091
$297$	0	0
$298$	−16.4783	−0.954560
$299$	7.56798	0.437668
$300$	0	0
$301$	23.8773	1.37626
$302$	−4.31450	−0.248272
$303$	0	0
$304$	5.38464	0.308830
$305$	4.10878	0.235268
$306$	0	0
$307$	−7.52944	−0.429728	−0.214864	−	0.976644i	$-0.568931\pi$
$307$	−7.52944	−0.429728	−0.214864	+	0.976644i	$0.568931\pi$
$308$	−3.58906	−0.204506
$309$	0	0
$310$	12.1185	0.688285
$311$	−17.8696	−1.01329	−0.506647	−	0.862154i	$-0.669115\pi$
$311$	−17.8696	−1.01329	−0.506647	+	0.862154i	$0.669115\pi$
$312$	0	0
$313$	6.41094	0.362368	0.181184	−	0.983449i	$-0.442007\pi$
$313$	6.41094	0.362368	0.181184	+	0.983449i	$0.442007\pi$
$314$	7.51219	0.423938
$315$	0	0
$316$	−12.7361	−0.716460
$317$	−1.66593	−0.0935682	−0.0467841	−	0.998905i	$-0.514897\pi$
$317$	−1.66593	−0.0935682	−0.0467841	+	0.998905i	$0.514897\pi$
$318$	0	0
$319$	−3.00702	−0.168361
$320$	−1.64104	−0.0917371
$321$	0	0
$322$	−13.6902	−0.762925
$323$	0.145858	0.00811578
$324$	0	0
$325$	−4.82811	−0.267815
$326$	−5.94179	−0.329085
$327$	0	0
$328$	4.22247	0.233147
$329$	9.83862	0.542421
$330$	0	0
$331$	19.0695	1.04816	0.524078	−	0.851671i	$-0.324410\pi$
$331$	19.0695	1.04816	0.524078	+	0.851671i	$0.324410\pi$
$332$	8.87868	0.487281
$333$	0	0
$334$	−3.87868	−0.212232
$335$	7.15324	0.390823
$336$	0	0
$337$	26.7929	1.45950	0.729750	−	0.683714i	$-0.239636\pi$
$337$	26.7929	1.45950	0.729750	+	0.683714i	$0.239636\pi$
$338$	−8.62007	−0.468870
$339$	0	0
$340$	−0.0444524	−0.00241077
$341$	−7.00079	−0.379114
$342$	0	0
$343$	−1.25939	−0.0680007
$344$	−6.30698	−0.340049
$345$	0	0
$346$	7.89244	0.424300
$347$	5.95645	0.319759	0.159880	−	0.987137i	$-0.448889\pi$
$347$	5.95645	0.319759	0.159880	+	0.987137i	$0.448889\pi$
$348$	0	0
$349$	20.4552	1.09494	0.547471	−	0.836825i	$-0.315591\pi$
$349$	20.4552	1.09494	0.547471	+	0.836825i	$0.315591\pi$
$350$	8.73387	0.466845
$351$	0	0
$352$	0.948021	0.0505297
$353$	18.0301	0.959644	0.479822	−	0.877366i	$-0.340701\pi$
$353$	18.0301	0.959644	0.479822	+	0.877366i	$0.340701\pi$
$354$	0	0
$355$	−14.7262	−0.781587
$356$	0.793377	0.0420489
$357$	0	0
$358$	4.44994	0.235187
$359$	−2.62901	−0.138754	−0.0693769	−	0.997591i	$-0.522101\pi$
$359$	−2.62901	−0.138754	−0.0693769	+	0.997591i	$0.522101\pi$
$360$	0	0
$361$	9.99431	0.526016
$362$	8.23972	0.433070
$363$	0	0
$364$	−7.92313	−0.415285
$365$	18.0651	0.945572
$366$	0	0
$367$	−12.3616	−0.645270	−0.322635	−	0.946524i	$-0.604569\pi$
$367$	−12.3616	−0.645270	−0.322635	+	0.946524i	$0.604569\pi$
$368$	3.61615	0.188505
$369$	0	0
$370$	4.09643	0.212963
$371$	19.2890	1.00143
$372$	0	0
$373$	−7.62796	−0.394961	−0.197480	−	0.980307i	$-0.563276\pi$
$373$	−7.62796	−0.394961	−0.197480	+	0.980307i	$0.563276\pi$
$374$	0.0256799	0.00132787
$375$	0	0
$376$	−2.59879	−0.134022
$377$	−6.63823	−0.341886
$378$	0	0
$379$	−13.1004	−0.672920	−0.336460	−	0.941698i	$-0.609230\pi$
$379$	−13.1004	−0.672920	−0.336460	+	0.941698i	$0.609230\pi$
$380$	−8.83642	−0.453299
$381$	0	0
$382$	5.91408	0.302591
$383$	−9.78260	−0.499868	−0.249934	−	0.968263i	$-0.580409\pi$
$383$	−9.78260	−0.499868	−0.249934	+	0.968263i	$0.580409\pi$
$384$	0	0
$385$	5.88981	0.300172
$386$	16.7133	0.850682
$387$	0	0
$388$	−8.13728	−0.413108
$389$	−11.6978	−0.593104	−0.296552	−	0.955017i	$-0.595837\pi$
$389$	−11.6978	−0.593104	−0.296552	+	0.955017i	$0.595837\pi$
$390$	0	0
$391$	0.0979539	0.00495374
$392$	7.33266	0.370355
$393$	0	0
$394$	6.95010	0.350141
$395$	20.9004	1.05161
$396$	0	0
$397$	12.8336	0.644100	0.322050	−	0.946723i	$-0.395628\pi$
$397$	12.8336	0.644100	0.322050	+	0.946723i	$0.395628\pi$
$398$	−18.5053	−0.927589
$399$	0	0
$400$	−2.30698	−0.115349
$401$	4.45606	0.222525	0.111263	−	0.993791i	$-0.464511\pi$
$401$	4.45606	0.222525	0.111263	+	0.993791i	$0.464511\pi$
$402$	0	0
$403$	−15.4548	−0.769857
$404$	−14.2655	−0.709736
$405$	0	0
$406$	12.0083	0.595963
$407$	−2.36648	−0.117302
$408$	0	0
$409$	5.99236	0.296303	0.148152	−	0.988965i	$-0.452668\pi$
$409$	5.99236	0.296303	0.148152	+	0.988965i	$0.452668\pi$
$410$	−6.92925	−0.342211
$411$	0	0
$412$	−2.92987	−0.144344
$413$	37.0424	1.82274
$414$	0	0
$415$	−14.5703	−0.715227
$416$	2.09283	0.102609
$417$	0	0
$418$	5.10475	0.249681
$419$	12.6376	0.617385	0.308692	−	0.951162i	$-0.400109\pi$
$419$	12.6376	0.617385	0.308692	+	0.951162i	$0.400109\pi$
$420$	0	0
$421$	−16.3801	−0.798317	−0.399158	−	0.916882i	$-0.630698\pi$
$421$	−16.3801	−0.798317	−0.399158	+	0.916882i	$0.630698\pi$
$422$	20.8892	1.01687
$423$	0	0
$424$	−5.09502	−0.247436
$425$	−0.0624911	−0.00303126
$426$	0	0
$427$	9.47887	0.458715
$428$	6.15233	0.297384
$429$	0	0
$430$	10.3500	0.499122
$431$	−28.8933	−1.39174	−0.695872	−	0.718166i	$-0.744982\pi$
$431$	−28.8933	−1.39174	−0.695872	+	0.718166i	$0.744982\pi$
$432$	0	0
$433$	24.1503	1.16059	0.580295	−	0.814406i	$-0.302937\pi$
$433$	24.1503	1.16059	0.580295	+	0.814406i	$0.302937\pi$
$434$	27.9571	1.34198
$435$	0	0
$436$	−1.92093	−0.0919960
$437$	19.4717	0.931456
$438$	0	0
$439$	20.0038	0.954728	0.477364	−	0.878706i	$-0.341592\pi$
$439$	20.0038	0.954728	0.477364	+	0.878706i	$0.341592\pi$
$440$	−1.55574	−0.0741671
$441$	0	0
$442$	0.0566902	0.00269648
$443$	−41.2816	−1.96135	−0.980675	−	0.195645i	$-0.937320\pi$
$443$	−41.2816	−1.96135	−0.980675	+	0.195645i	$0.937320\pi$
$444$	0	0
$445$	−1.30197	−0.0617191
$446$	−1.00000	−0.0473514
$447$	0	0
$448$	−3.78585	−0.178865
$449$	15.7414	0.742882	0.371441	−	0.928456i	$-0.378864\pi$
$449$	15.7414	0.742882	0.371441	+	0.928456i	$0.378864\pi$
$450$	0	0
$451$	4.00298	0.188493
$452$	12.3153	0.579263
$453$	0	0
$454$	−5.44959	−0.255762
$455$	13.0022	0.609552
$456$	0	0
$457$	−41.5156	−1.94202	−0.971009	−	0.239042i	$-0.923167\pi$
$457$	−41.5156	−1.94202	−0.971009	+	0.239042i	$0.923167\pi$
$458$	20.9369	0.978317
$459$	0	0
$460$	−5.93426	−0.276686
$461$	12.7813	0.595285	0.297642	−	0.954677i	$-0.403800\pi$
$461$	12.7813	0.595285	0.297642	+	0.954677i	$0.403800\pi$
$462$	0	0
$463$	−0.386833	−0.0179777	−0.00898883	−	0.999960i	$-0.502861\pi$
$463$	−0.386833	−0.0179777	−0.00898883	+	0.999960i	$0.502861\pi$
$464$	−3.17189	−0.147252
$465$	0	0
$466$	27.0478	1.25296
$467$	22.6179	1.04663	0.523315	−	0.852139i	$-0.324695\pi$
$467$	22.6179	1.04663	0.523315	+	0.852139i	$0.324695\pi$
$468$	0	0
$469$	16.5024	0.762008
$470$	4.26472	0.196717
$471$	0	0
$472$	−9.78444	−0.450365
$473$	−5.97915	−0.274921
$474$	0	0
$475$	−12.4222	−0.569971
$476$	−0.102551	−0.00470040
$477$	0	0
$478$	−30.6714	−1.40288
$479$	−5.46476	−0.249691	−0.124846	−	0.992176i	$-0.539844\pi$
$479$	−5.46476	−0.249691	−0.124846	+	0.992176i	$0.539844\pi$
$480$	0	0
$481$	−5.22419	−0.238203
$482$	−9.66531	−0.440243
$483$	0	0
$484$	−10.1013	−0.459148
$485$	13.3536	0.606357
$486$	0	0
$487$	0.0270879	0.00122747	0.000613734	−	1.00000i	$-0.499805\pi$
$487$	0.0270879	0.00122747	0.000613734		1.00000i	$0.499805\pi$
$488$	−2.50376	−0.113340
$489$	0	0
$490$	−12.0332	−0.543605
$491$	20.0131	0.903180	0.451590	−	0.892225i	$-0.350857\pi$
$491$	20.0131	0.903180	0.451590	+	0.892225i	$0.350857\pi$
$492$	0	0
$493$	−0.0859199	−0.00386964
$494$	11.2691	0.507021
$495$	0	0
$496$	−7.38464	−0.331580
$497$	−33.9731	−1.52390
$498$	0	0
$499$	−36.4407	−1.63131	−0.815655	−	0.578539i	$-0.803623\pi$
$499$	−36.4407	−1.63131	−0.815655	+	0.578539i	$0.803623\pi$
$500$	11.9911	0.536257
$501$	0	0
$502$	−14.8004	−0.660574
$503$	5.49325	0.244932	0.122466	−	0.992473i	$-0.460920\pi$
$503$	5.49325	0.244932	0.122466	+	0.992473i	$0.460920\pi$
$504$	0	0
$505$	23.4103	1.04175
$506$	3.42819	0.152401
$507$	0	0
$508$	−15.4851	−0.687040
$509$	−11.1585	−0.494590	−0.247295	−	0.968940i	$-0.579542\pi$
$509$	−11.1585	−0.494590	−0.247295	+	0.968940i	$0.579542\pi$
$510$	0	0
$511$	41.6758	1.84363
$512$	1.00000	0.0441942
$513$	0	0
$514$	−13.5800	−0.598989
$515$	4.80804	0.211868
$516$	0	0
$517$	−2.46370	−0.108354
$518$	9.45038	0.415226
$519$	0	0
$520$	−3.43442	−0.150609
$521$	−24.6217	−1.07869	−0.539347	−	0.842084i	$-0.681329\pi$
$521$	−24.6217	−1.07869	−0.539347	+	0.842084i	$0.681329\pi$
$522$	0	0
$523$	7.45222	0.325863	0.162931	−	0.986637i	$-0.447905\pi$
$523$	7.45222	0.325863	0.162931	+	0.986637i	$0.447905\pi$
$524$	−3.90357	−0.170528
$525$	0	0
$526$	−7.89420	−0.344204
$527$	−0.200034	−0.00871362
$528$	0	0
$529$	−9.92345	−0.431454
$530$	8.36115	0.363185
$531$	0	0
$532$	−20.3854	−0.883820
$533$	8.83689	0.382768
$534$	0	0
$535$	−10.0962	−0.436499
$536$	−4.35896	−0.188278
$537$	0	0
$538$	14.8946	0.642153
$539$	6.95151	0.299423
$540$	0	0
$541$	21.7032	0.933093	0.466546	−	0.884497i	$-0.345498\pi$
$541$	21.7032	0.933093	0.466546	+	0.884497i	$0.345498\pi$
$542$	3.27769	0.140789
$543$	0	0
$544$	0.0270879	0.00116138
$545$	3.15233	0.135031
$546$	0	0
$547$	11.0550	0.472676	0.236338	−	0.971671i	$-0.424053\pi$
$547$	11.0550	0.472676	0.236338	+	0.971671i	$0.424053\pi$
$548$	−4.38464	−0.187302
$549$	0	0
$550$	−2.18706	−0.0932567
$551$	−17.0795	−0.727611
$552$	0	0
$553$	48.2168	2.05039
$554$	−8.26018	−0.350941
$555$	0	0
$556$	−14.5565	−0.617334
$557$	−39.8283	−1.68758	−0.843789	−	0.536675i	$-0.819680\pi$
$557$	−39.8283	−1.68758	−0.843789	+	0.536675i	$0.819680\pi$
$558$	0	0
$559$	−13.1994	−0.558276
$560$	6.21274	0.262536
$561$	0	0
$562$	14.6209	0.616744
$563$	4.41392	0.186025	0.0930123	−	0.995665i	$-0.470350\pi$
$563$	4.41392	0.186025	0.0930123	+	0.995665i	$0.470350\pi$
$564$	0	0
$565$	−20.2099	−0.850238
$566$	24.5961	1.03385
$567$	0	0
$568$	8.97370	0.376528
$569$	4.68863	0.196558	0.0982788	−	0.995159i	$-0.468666\pi$
$569$	4.68863	0.196558	0.0982788	+	0.995159i	$0.468666\pi$
$570$	0	0
$571$	35.2974	1.47715	0.738576	−	0.674171i	$-0.235499\pi$
$571$	35.2974	1.47715	0.738576	+	0.674171i	$0.235499\pi$
$572$	1.98404	0.0829570
$573$	0	0
$574$	−15.9856	−0.667227
$575$	−8.34238	−0.347901
$576$	0	0
$577$	5.19146	0.216123	0.108062	−	0.994144i	$-0.465536\pi$
$577$	5.19146	0.216123	0.108062	+	0.994144i	$0.465536\pi$
$578$	−16.9993	−0.707076
$579$	0	0
$580$	5.20522	0.216135
$581$	−33.6133	−1.39452
$582$	0	0
$583$	−4.83019	−0.200046
$584$	−11.0083	−0.455527
$585$	0	0
$586$	2.60282	0.107522
$587$	15.1913	0.627014	0.313507	−	0.949586i	$-0.398496\pi$
$587$	15.1913	0.627014	0.313507	+	0.949586i	$0.398496\pi$
$588$	0	0
$589$	−39.7636	−1.63843
$590$	16.0567	0.661043
$591$	0	0
$592$	−2.49624	−0.102595
$593$	−44.8330	−1.84107	−0.920535	−	0.390660i	$-0.872247\pi$
$593$	−44.8330	−1.84107	−0.920535	+	0.390660i	$0.872247\pi$
$594$	0	0
$595$	0.168290	0.00689922
$596$	−16.4783	−0.674976
$597$	0	0
$598$	7.56798	0.309478
$599$	−10.9904	−0.449055	−0.224528	−	0.974468i	$-0.572084\pi$
$599$	−10.9904	−0.449055	−0.224528	+	0.974468i	$0.572084\pi$
$600$	0	0
$601$	3.17471	0.129499	0.0647496	−	0.997902i	$-0.479375\pi$
$601$	3.17471	0.129499	0.0647496	+	0.997902i	$0.479375\pi$
$602$	23.8773	0.973165
$603$	0	0
$604$	−4.31450	−0.175555
$605$	16.5766	0.673935
$606$	0	0
$607$	−20.1193	−0.816617	−0.408309	−	0.912844i	$-0.633881\pi$
$607$	−20.1193	−0.816617	−0.408309	+	0.912844i	$0.633881\pi$
$608$	5.38464	0.218376
$609$	0	0
$610$	4.10878	0.166360
$611$	−5.43881	−0.220031
$612$	0	0
$613$	−28.4725	−1.14999	−0.574996	−	0.818156i	$-0.694996\pi$
$613$	−28.4725	−1.14999	−0.574996	+	0.818156i	$0.694996\pi$
$614$	−7.52944	−0.303864
$615$	0	0
$616$	−3.58906	−0.144608
$617$	−24.7637	−0.996950	−0.498475	−	0.866904i	$-0.666106\pi$
$617$	−24.7637	−0.996950	−0.498475	+	0.866904i	$0.666106\pi$
$618$	0	0
$619$	20.6260	0.829031	0.414515	−	0.910042i	$-0.363951\pi$
$619$	20.6260	0.829031	0.414515	+	0.910042i	$0.363951\pi$
$620$	12.1185	0.486691
$621$	0	0
$622$	−17.8696	−0.716507
$623$	−3.00360	−0.120337
$624$	0	0
$625$	−8.14297	−0.325719
$626$	6.41094	0.256233
$627$	0	0
$628$	7.51219	0.299769
$629$	−0.0676177	−0.00269610
$630$	0	0
$631$	−3.65060	−0.145328	−0.0726640	−	0.997356i	$-0.523150\pi$
$631$	−3.65060	−0.145328	−0.0726640	+	0.997356i	$0.523150\pi$
$632$	−12.7361	−0.506614
$633$	0	0
$634$	−1.66593	−0.0661627
$635$	25.4117	1.00843
$636$	0	0
$637$	15.3460	0.608030
$638$	−3.00702	−0.119049
$639$	0	0
$640$	−1.64104	−0.0648679
$641$	34.4091	1.35908	0.679538	−	0.733640i	$-0.262180\pi$
$641$	34.4091	1.35908	0.679538	+	0.733640i	$0.262180\pi$
$642$	0	0
$643$	28.6486	1.12979	0.564896	−	0.825162i	$-0.308916\pi$
$643$	28.6486	1.12979	0.564896	+	0.825162i	$0.308916\pi$
$644$	−13.6902	−0.539470
$645$	0	0
$646$	0.145858	0.00573872
$647$	−6.35045	−0.249662	−0.124831	−	0.992178i	$-0.539839\pi$
$647$	−6.35045	−0.249662	−0.124831	+	0.992178i	$0.539839\pi$
$648$	0	0
$649$	−9.27585	−0.364109
$650$	−4.82811	−0.189374
$651$	0	0
$652$	−5.94179	−0.232698
$653$	12.1639	0.476010	0.238005	−	0.971264i	$-0.423507\pi$
$653$	12.1639	0.476010	0.238005	+	0.971264i	$0.423507\pi$
$654$	0	0
$655$	6.40592	0.250300
$656$	4.22247	0.164860
$657$	0	0
$658$	9.83862	0.383549
$659$	−21.9037	−0.853248	−0.426624	−	0.904429i	$-0.640297\pi$
$659$	−21.9037	−0.853248	−0.426624	+	0.904429i	$0.640297\pi$
$660$	0	0
$661$	−21.5772	−0.839256	−0.419628	−	0.907696i	$-0.637839\pi$
$661$	−21.5772	−0.839256	−0.419628	+	0.907696i	$0.637839\pi$
$662$	19.0695	0.741158
$663$	0	0
$664$	8.87868	0.344559
$665$	33.4534	1.29727
$666$	0	0
$667$	−11.4701	−0.444122
$668$	−3.87868	−0.150071
$669$	0	0
$670$	7.15324	0.276354
$671$	−2.37362	−0.0916326
$672$	0	0
$673$	18.7512	0.722807	0.361403	−	0.932410i	$-0.382298\pi$
$673$	18.7512	0.722807	0.361403	+	0.932410i	$0.382298\pi$
$674$	26.7929	1.03202
$675$	0	0
$676$	−8.62007	−0.331541
$677$	11.2434	0.432120	0.216060	−	0.976380i	$-0.430679\pi$
$677$	11.2434	0.432120	0.216060	+	0.976380i	$0.430679\pi$
$678$	0	0
$679$	30.8065	1.18225
$680$	−0.0444524	−0.00170467
$681$	0	0
$682$	−7.00079	−0.268074
$683$	19.5387	0.747626	0.373813	−	0.927504i	$-0.378050\pi$
$683$	19.5387	0.747626	0.373813	+	0.927504i	$0.378050\pi$
$684$	0	0
$685$	7.19538	0.274921
$686$	−1.25939	−0.0480838
$687$	0	0
$688$	−6.30698	−0.240451
$689$	−10.6630	−0.406228
$690$	0	0
$691$	−42.2422	−1.60697	−0.803484	−	0.595326i	$-0.797023\pi$
$691$	−42.2422	−1.60697	−0.803484	+	0.595326i	$0.797023\pi$
$692$	7.89244	0.300025
$693$	0	0
$694$	5.95645	0.226104
$695$	23.8879	0.906119
$696$	0	0
$697$	0.114378	0.00433236
$698$	20.4552	0.774241
$699$	0	0
$700$	8.73387	0.330109
$701$	20.5501	0.776165	0.388082	−	0.921625i	$-0.373138\pi$
$701$	20.5501	0.776165	0.388082	+	0.921625i	$0.373138\pi$
$702$	0	0
$703$	−13.4413	−0.506949
$704$	0.948021	0.0357299
$705$	0	0
$706$	18.0301	0.678571
$707$	54.0071	2.03115
$708$	0	0
$709$	−18.0271	−0.677021	−0.338511	−	0.940963i	$-0.609923\pi$
$709$	−18.0271	−0.677021	−0.338511	+	0.940963i	$0.609923\pi$
$710$	−14.7262	−0.552666
$711$	0	0
$712$	0.793377	0.0297330
$713$	−26.7040	−1.00007
$714$	0	0
$715$	−3.25590	−0.121764
$716$	4.44994	0.166302
$717$	0	0
$718$	−2.62901	−0.0981137
$719$	27.9105	1.04089	0.520443	−	0.853897i	$-0.325767\pi$
$719$	27.9105	1.04089	0.520443	+	0.853897i	$0.325767\pi$
$720$	0	0
$721$	11.0920	0.413089
$722$	9.99431	0.371950
$723$	0	0
$724$	8.23972	0.306227
$725$	7.31749	0.271765
$726$	0	0
$727$	−46.4208	−1.72165	−0.860827	−	0.508898i	$-0.830053\pi$
$727$	−46.4208	−1.72165	−0.860827	+	0.508898i	$0.830053\pi$
$728$	−7.92313	−0.293651
$729$	0	0
$730$	18.0651	0.668620
$731$	−0.170843	−0.00631884
$732$	0	0
$733$	30.1465	1.11349	0.556743	−	0.830685i	$-0.312051\pi$
$733$	30.1465	1.11349	0.556743	+	0.830685i	$0.312051\pi$
$734$	−12.3616	−0.456274
$735$	0	0
$736$	3.61615	0.133293
$737$	−4.13238	−0.152218
$738$	0	0
$739$	22.4392	0.825441	0.412720	−	0.910858i	$-0.364579\pi$
$739$	22.4392	0.825441	0.412720	+	0.910858i	$0.364579\pi$
$740$	4.09643	0.150588
$741$	0	0
$742$	19.2890	0.708121
$743$	45.5002	1.66924	0.834620	−	0.550825i	$-0.185687\pi$
$743$	45.5002	1.66924	0.834620	+	0.550825i	$0.185687\pi$
$744$	0	0
$745$	27.0415	0.990725
$746$	−7.62796	−0.279279
$747$	0	0
$748$	0.0256799	0.000938949 0
$749$	−23.2918	−0.851064
$750$	0	0
$751$	−14.7463	−0.538101	−0.269051	−	0.963126i	$-0.586710\pi$
$751$	−14.7463	−0.538101	−0.269051	+	0.963126i	$0.586710\pi$
$752$	−2.59879	−0.0947680
$753$	0	0
$754$	−6.63823	−0.241750
$755$	7.08029	0.257678
$756$	0	0
$757$	−21.6569	−0.787133	−0.393566	−	0.919296i	$-0.628759\pi$
$757$	−21.6569	−0.787133	−0.393566	+	0.919296i	$0.628759\pi$
$758$	−13.1004	−0.475826
$759$	0	0
$760$	−8.83642	−0.320531
$761$	9.81348	0.355739	0.177869	−	0.984054i	$-0.443080\pi$
$761$	9.81348	0.355739	0.177869	+	0.984054i	$0.443080\pi$
$762$	0	0
$763$	7.27236	0.263277
$764$	5.91408	0.213964
$765$	0	0
$766$	−9.78260	−0.353460
$767$	−20.4771	−0.739387
$768$	0	0
$769$	−6.83685	−0.246543	−0.123272	−	0.992373i	$-0.539339\pi$
$769$	−6.83685	−0.246543	−0.123272	+	0.992373i	$0.539339\pi$
$770$	5.88981	0.212254
$771$	0	0
$772$	16.7133	0.601523
$773$	49.2899	1.77283	0.886417	−	0.462887i	$-0.153187\pi$
$773$	49.2899	1.77283	0.886417	+	0.462887i	$0.153187\pi$
$774$	0	0
$775$	17.0362	0.611958
$776$	−8.13728	−0.292111
$777$	0	0
$778$	−11.6978	−0.419388
$779$	22.7364	0.814618
$780$	0	0
$781$	8.50725	0.304413
$782$	0.0979539	0.00350282
$783$	0	0
$784$	7.33266	0.261881
$785$	−12.3278	−0.439999
$786$	0	0
$787$	−46.6575	−1.66316	−0.831580	−	0.555405i	$-0.812563\pi$
$787$	−46.6575	−1.66316	−0.831580	+	0.555405i	$0.812563\pi$
$788$	6.95010	0.247587
$789$	0	0
$790$	20.9004	0.743604
$791$	−46.6238	−1.65775
$792$	0	0
$793$	−5.23994	−0.186076
$794$	12.8336	0.455448
$795$	0	0
$796$	−18.5053	−0.655904
$797$	22.8186	0.808277	0.404139	−	0.914698i	$-0.367571\pi$
$797$	22.8186	0.808277	0.404139	+	0.914698i	$0.367571\pi$
$798$	0	0
$799$	−0.0703956	−0.00249042
$800$	−2.30698	−0.0815640
$801$	0	0
$802$	4.45606	0.157349
$803$	−10.4361	−0.368282
$804$	0	0
$805$	22.4662	0.791830
$806$	−15.4548	−0.544371
$807$	0	0
$808$	−14.2655	−0.501859
$809$	−21.3057	−0.749070	−0.374535	−	0.927213i	$-0.622198\pi$
$809$	−21.3057	−0.749070	−0.374535	+	0.927213i	$0.622198\pi$
$810$	0	0
$811$	−4.37178	−0.153514	−0.0767570	−	0.997050i	$-0.524457\pi$
$811$	−4.37178	−0.153514	−0.0767570	+	0.997050i	$0.524457\pi$
$812$	12.0083	0.421409
$813$	0	0
$814$	−2.36648	−0.0829452
$815$	9.75073	0.341553
$816$	0	0
$817$	−33.9608	−1.18814
$818$	5.99236	0.209518
$819$	0	0
$820$	−6.92925	−0.241980
$821$	−48.6264	−1.69707	−0.848536	−	0.529138i	$-0.822515\pi$
$821$	−48.6264	−1.69707	−0.848536	+	0.529138i	$0.822515\pi$
$822$	0	0
$823$	23.9936	0.836364	0.418182	−	0.908363i	$-0.362667\pi$
$823$	23.9936	0.836364	0.418182	+	0.908363i	$0.362667\pi$
$824$	−2.92987	−0.102067
$825$	0	0
$826$	37.0424	1.28887
$827$	−43.1952	−1.50204	−0.751022	−	0.660277i	$-0.770439\pi$
$827$	−43.1952	−1.50204	−0.751022	+	0.660277i	$0.770439\pi$
$828$	0	0
$829$	−27.2828	−0.947569	−0.473785	−	0.880641i	$-0.657113\pi$
$829$	−27.2828	−0.947569	−0.473785	+	0.880641i	$0.657113\pi$
$830$	−14.5703	−0.505742
$831$	0	0
$832$	2.09283	0.0725557
$833$	0.198626	0.00688199
$834$	0	0
$835$	6.36508	0.220273
$836$	5.10475	0.176551
$837$	0	0
$838$	12.6376	0.436557
$839$	1.65139	0.0570122	0.0285061	−	0.999594i	$-0.490925\pi$
$839$	1.65139	0.0570122	0.0285061	+	0.999594i	$0.490925\pi$
$840$	0	0
$841$	−18.9391	−0.653072
$842$	−16.3801	−0.564495
$843$	0	0
$844$	20.8892	0.719036
$845$	14.1459	0.486634
$846$	0	0
$847$	38.2418	1.31401
$848$	−5.09502	−0.174964
$849$	0	0
$850$	−0.0624911	−0.00214343
$851$	−9.02677	−0.309434
$852$	0	0
$853$	6.90638	0.236470	0.118235	−	0.992986i	$-0.462276\pi$
$853$	6.90638	0.236470	0.118235	+	0.992986i	$0.462276\pi$
$854$	9.47887	0.324360
$855$	0	0
$856$	6.15233	0.210282
$857$	25.8488	0.882977	0.441489	−	0.897267i	$-0.354451\pi$
$857$	25.8488	0.882977	0.441489	+	0.897267i	$0.354451\pi$
$858$	0	0
$859$	−33.9690	−1.15901	−0.579504	−	0.814969i	$-0.696754\pi$
$859$	−33.9690	−1.15901	−0.579504	+	0.814969i	$0.696754\pi$
$860$	10.3500	0.352933
$861$	0	0
$862$	−28.8933	−0.984111
$863$	8.63729	0.294017	0.147008	−	0.989135i	$-0.453036\pi$
$863$	8.63729	0.294017	0.147008	+	0.989135i	$0.453036\pi$
$864$	0	0
$865$	−12.9518	−0.440375
$866$	24.1503	0.820661
$867$	0	0
$868$	27.9571	0.948927
$869$	−12.0741	−0.409584
$870$	0	0
$871$	−9.12254	−0.309106
$872$	−1.92093	−0.0650510
$873$	0	0
$874$	19.4717	0.658639
$875$	−45.3964	−1.53468
$876$	0	0
$877$	30.1650	1.01860	0.509301	−	0.860589i	$-0.329904\pi$
$877$	30.1650	1.01860	0.509301	+	0.860589i	$0.329904\pi$
$878$	20.0038	0.675095
$879$	0	0
$880$	−1.55574	−0.0524441
$881$	−12.4248	−0.418603	−0.209301	−	0.977851i	$-0.567119\pi$
$881$	−12.4248	−0.418603	−0.209301	+	0.977851i	$0.567119\pi$
$882$	0	0
$883$	11.3493	0.381934	0.190967	−	0.981596i	$-0.438838\pi$
$883$	11.3493	0.381934	0.190967	+	0.981596i	$0.438838\pi$
$884$	0.0566902	0.00190670
$885$	0	0
$886$	−41.2816	−1.38688
$887$	8.39412	0.281847	0.140923	−	0.990020i	$-0.454993\pi$
$887$	8.39412	0.281847	0.140923	+	0.990020i	$0.454993\pi$
$888$	0	0
$889$	58.6243	1.96620
$890$	−1.30197	−0.0436420
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	−13.9935	−0.468275
$894$	0	0
$895$	−7.30255	−0.244097
$896$	−3.78585	−0.126476
$897$	0	0
$898$	15.7414	0.525297
$899$	23.4233	0.781210
$900$	0	0
$901$	−0.138013	−0.00459789
$902$	4.00298	0.133285
$903$	0	0
$904$	12.3153	0.409601
$905$	−13.5217	−0.449477
$906$	0	0
$907$	12.7461	0.423229	0.211614	−	0.977353i	$-0.432128\pi$
$907$	12.7461	0.423229	0.211614	+	0.977353i	$0.432128\pi$
$908$	−5.44959	−0.180851
$909$	0	0
$910$	13.0022	0.431019
$911$	−15.4866	−0.513095	−0.256547	−	0.966532i	$-0.582585\pi$
$911$	−15.4866	−0.513095	−0.256547	+	0.966532i	$0.582585\pi$
$912$	0	0
$913$	8.41717	0.278568
$914$	−41.5156	−1.37321
$915$	0	0
$916$	20.9369	0.691774
$917$	14.7783	0.488023
$918$	0	0
$919$	−1.68380	−0.0555436	−0.0277718	−	0.999614i	$-0.508841\pi$
$919$	−1.68380	−0.0555436	−0.0277718	+	0.999614i	$0.508841\pi$
$920$	−5.93426	−0.195647
$921$	0	0
$922$	12.7813	0.420930
$923$	18.7804	0.618165
$924$	0	0
$925$	5.75876	0.189347
$926$	−0.386833	−0.0127121
$927$	0	0
$928$	−3.17189	−0.104123
$929$	54.6754	1.79384	0.896921	−	0.442190i	$-0.145798\pi$
$929$	54.6754	1.79384	0.896921	+	0.442190i	$0.145798\pi$
$930$	0	0
$931$	39.4837	1.29403
$932$	27.0478	0.885979
$933$	0	0
$934$	22.6179	0.740080
$935$	−0.0421418	−0.00137818
$936$	0	0
$937$	−5.35265	−0.174863	−0.0874317	−	0.996171i	$-0.527866\pi$
$937$	−5.35265	−0.174863	−0.0874317	+	0.996171i	$0.527866\pi$
$938$	16.5024	0.538821
$939$	0	0
$940$	4.26472	0.139100
$941$	−29.1362	−0.949811	−0.474906	−	0.880037i	$-0.657518\pi$
$941$	−29.1362	−0.949811	−0.474906	+	0.880037i	$0.657518\pi$
$942$	0	0
$943$	15.2691	0.497230
$944$	−9.78444	−0.318456
$945$	0	0
$946$	−5.97915	−0.194399
$947$	22.3852	0.727421	0.363711	−	0.931512i	$-0.381510\pi$
$947$	22.3852	0.727421	0.363711	+	0.931512i	$0.381510\pi$
$948$	0	0
$949$	−23.0385	−0.747861
$950$	−12.4222	−0.403031
$951$	0	0
$952$	−0.102551	−0.00332368
$953$	32.7939	1.06230	0.531149	−	0.847278i	$-0.321760\pi$
$953$	32.7939	1.06230	0.531149	+	0.847278i	$0.321760\pi$
$954$	0	0
$955$	−9.70526	−0.314055
$956$	−30.6714	−0.991985
$957$	0	0
$958$	−5.46476	−0.176558
$959$	16.5996	0.536028
$960$	0	0
$961$	23.5329	0.759125
$962$	−5.22419	−0.168435
$963$	0	0
$964$	−9.66531	−0.311299
$965$	−27.4272	−0.882912
$966$	0	0
$967$	−16.6658	−0.535936	−0.267968	−	0.963428i	$-0.586352\pi$
$967$	−16.6658	−0.535936	−0.267968	+	0.963428i	$0.586352\pi$
$968$	−10.1013	−0.324667
$969$	0	0
$970$	13.3536	0.428759
$971$	54.9760	1.76426	0.882131	−	0.471004i	$-0.156108\pi$
$971$	54.9760	1.76426	0.882131	+	0.471004i	$0.156108\pi$
$972$	0	0
$973$	55.1088	1.76671
$974$	0.0270879	0.000867951 0
$975$	0	0
$976$	−2.50376	−0.0801435
$977$	24.4672	0.782776	0.391388	−	0.920226i	$-0.371995\pi$
$977$	24.4672	0.782776	0.391388	+	0.920226i	$0.371995\pi$
$978$	0	0
$979$	0.752137	0.0240384
$980$	−12.0332	−0.384387
$981$	0	0
$982$	20.0131	0.638645
$983$	−3.46248	−0.110436	−0.0552180	−	0.998474i	$-0.517585\pi$
$983$	−3.46248	−0.110436	−0.0552180	+	0.998474i	$0.517585\pi$
$984$	0	0
$985$	−11.4054	−0.363407
$986$	−0.0859199	−0.00273625
$987$	0	0
$988$	11.2691	0.358518
$989$	−22.8070	−0.725220
$990$	0	0
$991$	−1.65963	−0.0527198	−0.0263599	−	0.999653i	$-0.508392\pi$
$991$	−1.65963	−0.0527198	−0.0263599	+	0.999653i	$0.508392\pi$
$992$	−7.38464	−0.234462
$993$	0	0
$994$	−33.9731	−1.07756
$995$	30.3681	0.962732
$996$	0	0
$997$	46.9184	1.48592	0.742960	−	0.669335i	$-0.233421\pi$
$997$	46.9184	1.48592	0.742960	+	0.669335i	$0.233421\pi$
$998$	−36.4407	−1.15351
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.q.1.2		4
3.2	odd	2		1338.2.a.g.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.g.1.3	✓	4	3.2	odd	2
4014.2.a.q.1.2		4	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:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.64104 q^{5} -3.78585 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.64104 q^{5} -3.78585 q^{7} +1.00000 q^{8} -1.64104 q^{10} +0.948021 q^{11} +2.09283 q^{13} -3.78585 q^{14} +1.00000 q^{16} +0.0270879 q^{17} +5.38464 q^{19} -1.64104 q^{20} +0.948021 q^{22} +3.61615 q^{23} -2.30698 q^{25} +2.09283 q^{26} -3.78585 q^{28} -3.17189 q^{29} -7.38464 q^{31} +1.00000 q^{32} +0.0270879 q^{34} +6.21274 q^{35} -2.49624 q^{37} +5.38464 q^{38} -1.64104 q^{40} +4.22247 q^{41} -6.30698 q^{43} +0.948021 q^{44} +3.61615 q^{46} -2.59879 q^{47} +7.33266 q^{49} -2.30698 q^{50} +2.09283 q^{52} -5.09502 q^{53} -1.55574 q^{55} -3.78585 q^{56} -3.17189 q^{58} -9.78444 q^{59} -2.50376 q^{61} -7.38464 q^{62} +1.00000 q^{64} -3.43442 q^{65} -4.35896 q^{67} +0.0270879 q^{68} +6.21274 q^{70} +8.97370 q^{71} -11.0083 q^{73} -2.49624 q^{74} +5.38464 q^{76} -3.58906 q^{77} -12.7361 q^{79} -1.64104 q^{80} +4.22247 q^{82} +8.87868 q^{83} -0.0444524 q^{85} -6.30698 q^{86} +0.948021 q^{88} +0.793377 q^{89} -7.92313 q^{91} +3.61615 q^{92} -2.59879 q^{94} -8.83642 q^{95} -8.13728 q^{97} +7.33266 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} - 2 q^{5} - 5 q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 - 2 * q^5 - 5 * q^7 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} - 2 q^{5} - 5 q^{7} + 4 q^{8} - 2 q^{10} - 4 q^{11} - 5 q^{13} - 5 q^{14} + 4 q^{16} + 2 q^{17} - 7 q^{19} - 2 q^{20} - 4 q^{22} + 4 q^{23} - 6 q^{25} - 5 q^{26} - 5 q^{28} - 9 q^{29} - q^{31} + 4 q^{32} + 2 q^{34} - 11 q^{37} - 7 q^{38} - 2 q^{40} - 14 q^{41} - 22 q^{43} - 4 q^{44} + 4 q^{46} + 8 q^{47} - 7 q^{49} - 6 q^{50} - 5 q^{52} - 3 q^{53} - 13 q^{55} - 5 q^{56} - 9 q^{58} + 6 q^{59} - 9 q^{61} - q^{62} + 4 q^{64} + 3 q^{65} - 22 q^{67} + 2 q^{68} - 5 q^{71} - 3 q^{73} - 11 q^{74} - 7 q^{76} - 2 q^{77} - 29 q^{79} - 2 q^{80} - 14 q^{82} + 12 q^{83} - 10 q^{85} - 22 q^{86} - 4 q^{88} - 9 q^{89} - 18 q^{91} + 4 q^{92} + 8 q^{94} + 2 q^{95} - 29 q^{97} - 7 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 - 2 * q^5 - 5 * q^7 + 4 * q^8 - 2 * q^10 - 4 * q^11 - 5 * q^13 - 5 * q^14 + 4 * q^16 + 2 * q^17 - 7 * q^19 - 2 * q^20 - 4 * q^22 + 4 * q^23 - 6 * q^25 - 5 * q^26 - 5 * q^28 - 9 * q^29 - q^31 + 4 * q^32 + 2 * q^34 - 11 * q^37 - 7 * q^38 - 2 * q^40 - 14 * q^41 - 22 * q^43 - 4 * q^44 + 4 * q^46 + 8 * q^47 - 7 * q^49 - 6 * q^50 - 5 * q^52 - 3 * q^53 - 13 * q^55 - 5 * q^56 - 9 * q^58 + 6 * q^59 - 9 * q^61 - q^62 + 4 * q^64 + 3 * q^65 - 22 * q^67 + 2 * q^68 - 5 * q^71 - 3 * q^73 - 11 * q^74 - 7 * q^76 - 2 * q^77 - 29 * q^79 - 2 * q^80 - 14 * q^82 + 12 * q^83 - 10 * q^85 - 22 * q^86 - 4 * q^88 - 9 * q^89 - 18 * q^91 + 4 * q^92 + 8 * q^94 + 2 * q^95 - 29 * q^97 - 7 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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