LMFDB - Embedded newform 1336.2.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1336,2,Mod(1,1336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1336, 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("1336.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1336 = 2^{3} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1336.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:	$10.6680137100$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 25 x^{10} + 20 x^{9} + 224 x^{8} - 135 x^{7} - 865 x^{6} + 330 x^{5} + 1341 x^{4} + \cdots + 64 $ x^12 - x^11 - 25x^10 + 20x^9 + 224x^8 - 135x^7 - 865x^6 + 330x^5 + 1341x^4 - 127x^3 - 652x^2 - 48x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.31477$ of defining polynomial
Character	$\chi$	$=$	1336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.31477 q^{3} -1.78615 q^{5} -4.41953 q^{7} +2.35817 q^{9} +O(q^{10})$	$q-2.31477 q^{3} -1.78615 q^{5} -4.41953 q^{7} +2.35817 q^{9} -3.00623 q^{11} -0.0205406 q^{13} +4.13454 q^{15} -4.88802 q^{17} -5.01354 q^{19} +10.2302 q^{21} -3.08818 q^{23} -1.80966 q^{25} +1.48569 q^{27} +3.93789 q^{29} -4.09125 q^{31} +6.95874 q^{33} +7.89395 q^{35} -3.21774 q^{37} +0.0475469 q^{39} +3.58395 q^{41} +5.15959 q^{43} -4.21205 q^{45} -10.0715 q^{47} +12.5322 q^{49} +11.3147 q^{51} +0.412064 q^{53} +5.36959 q^{55} +11.6052 q^{57} -2.41530 q^{59} +9.25724 q^{61} -10.4220 q^{63} +0.0366887 q^{65} -5.90015 q^{67} +7.14842 q^{69} +15.2803 q^{71} -7.24325 q^{73} +4.18895 q^{75} +13.2861 q^{77} +4.80573 q^{79} -10.5135 q^{81} +10.3636 q^{83} +8.73075 q^{85} -9.11533 q^{87} -7.38931 q^{89} +0.0907800 q^{91} +9.47031 q^{93} +8.95494 q^{95} -5.55902 q^{97} -7.08921 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) $ 12 * q - q^3 + 8 * q^5 - 4 * q^7 + 15 * q^9	$ 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9} + 6 q^{11} + 13 q^{13} + 4 q^{15} + 10 q^{17} - q^{19} + 13 q^{21} + 3 q^{23} + 18 q^{25} - 10 q^{27} + 29 q^{29} - 3 q^{31} + 3 q^{35} + 41 q^{37} + 10 q^{39} + 20 q^{41} - q^{43} + 42 q^{45} + 5 q^{47} + 20 q^{49} + 14 q^{51} + 39 q^{53} + 3 q^{55} + 3 q^{57} + 8 q^{59} + 30 q^{61} - 2 q^{63} + 21 q^{65} - 9 q^{67} + 33 q^{69} + 29 q^{71} + 12 q^{73} + q^{75} + 19 q^{77} - 2 q^{79} + 24 q^{81} - 5 q^{83} + 44 q^{85} - 2 q^{87} + 7 q^{89} - 4 q^{91} + 37 q^{93} + 18 q^{95} - 14 q^{97} + 5 q^{99}+O(q^{100}) $ 12 * q - q^3 + 8 * q^5 - 4 * q^7 + 15 * q^9 + 6 * q^11 + 13 * q^13 + 4 * q^15 + 10 * q^17 - q^19 + 13 * q^21 + 3 * q^23 + 18 * q^25 - 10 * q^27 + 29 * q^29 - 3 * q^31 + 3 * q^35 + 41 * q^37 + 10 * q^39 + 20 * q^41 - q^43 + 42 * q^45 + 5 * q^47 + 20 * q^49 + 14 * q^51 + 39 * q^53 + 3 * q^55 + 3 * q^57 + 8 * q^59 + 30 * q^61 - 2 * q^63 + 21 * q^65 - 9 * q^67 + 33 * q^69 + 29 * q^71 + 12 * q^73 + q^75 + 19 * q^77 - 2 * q^79 + 24 * q^81 - 5 * q^83 + 44 * q^85 - 2 * q^87 + 7 * q^89 - 4 * q^91 + 37 * q^93 + 18 * q^95 - 14 * 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
$3$	−2.31477	−1.33643	−0.668217	−	0.743966i	$-0.732942\pi$
$3$	−2.31477	−1.33643	−0.668217	+	0.743966i	$0.732942\pi$
$4$	0	0
$5$	−1.78615	−0.798791	−0.399396	−	0.916779i	$-0.630780\pi$
$5$	−1.78615	−0.798791	−0.399396	+	0.916779i	$0.630780\pi$
$6$	0	0
$7$	−4.41953	−1.67043	−0.835213	−	0.549927i	$-0.814656\pi$
$7$	−4.41953	−1.67043	−0.835213	+	0.549927i	$0.814656\pi$
$8$	0	0
$9$	2.35817	0.786057
$10$	0	0
$11$	−3.00623	−0.906413	−0.453207	−	0.891405i	$-0.649720\pi$
$11$	−3.00623	−0.906413	−0.453207	+	0.891405i	$0.649720\pi$
$12$	0	0
$13$	−0.0205406	−0.00569695	−0.00284847	−	0.999996i	$-0.500907\pi$
$13$	−0.0205406	−0.00569695	−0.00284847	+	0.999996i	$0.500907\pi$
$14$	0	0
$15$	4.13454	1.06753
$16$	0	0
$17$	−4.88802	−1.18552	−0.592760	−	0.805379i	$-0.701962\pi$
$17$	−4.88802	−1.18552	−0.592760	+	0.805379i	$0.701962\pi$
$18$	0	0
$19$	−5.01354	−1.15018	−0.575092	−	0.818089i	$-0.695034\pi$
$19$	−5.01354	−1.15018	−0.575092	+	0.818089i	$0.695034\pi$
$20$	0	0
$21$	10.2302	2.23241
$22$	0	0
$23$	−3.08818	−0.643929	−0.321965	−	0.946752i	$-0.604343\pi$
$23$	−3.08818	−0.643929	−0.321965	+	0.946752i	$0.604343\pi$
$24$	0	0
$25$	−1.80966	−0.361932
$26$	0	0
$27$	1.48569	0.285920
$28$	0	0
$29$	3.93789	0.731248	0.365624	−	0.930763i	$-0.380856\pi$
$29$	3.93789	0.731248	0.365624	+	0.930763i	$0.380856\pi$
$30$	0	0
$31$	−4.09125	−0.734810	−0.367405	−	0.930061i	$-0.619754\pi$
$31$	−4.09125	−0.734810	−0.367405	+	0.930061i	$0.619754\pi$
$32$	0	0
$33$	6.95874	1.21136
$34$	0	0
$35$	7.89395	1.33432
$36$	0	0
$37$	−3.21774	−0.528993	−0.264497	−	0.964387i	$-0.585206\pi$
$37$	−3.21774	−0.528993	−0.264497	+	0.964387i	$0.585206\pi$
$38$	0	0
$39$	0.0475469	0.00761360
$40$	0	0
$41$	3.58395	0.559719	0.279860	−	0.960041i	$-0.409712\pi$
$41$	3.58395	0.559719	0.279860	+	0.960041i	$0.409712\pi$
$42$	0	0
$43$	5.15959	0.786830	0.393415	−	0.919361i	$-0.371294\pi$
$43$	5.15959	0.786830	0.393415	+	0.919361i	$0.371294\pi$
$44$	0	0
$45$	−4.21205	−0.627896
$46$	0	0
$47$	−10.0715	−1.46908	−0.734538	−	0.678567i	$-0.762601\pi$
$47$	−10.0715	−1.46908	−0.734538	+	0.678567i	$0.762601\pi$
$48$	0	0
$49$	12.5322	1.79032
$50$	0	0
$51$	11.3147	1.58437
$52$	0	0
$53$	0.412064	0.0566013	0.0283006	−	0.999599i	$-0.490990\pi$
$53$	0.412064	0.0566013	0.0283006	+	0.999599i	$0.490990\pi$
$54$	0	0
$55$	5.36959	0.724035
$56$	0	0
$57$	11.6052	1.53715
$58$	0	0
$59$	−2.41530	−0.314445	−0.157222	−	0.987563i	$-0.550254\pi$
$59$	−2.41530	−0.314445	−0.157222	+	0.987563i	$0.550254\pi$
$60$	0	0
$61$	9.25724	1.18527	0.592634	−	0.805472i	$-0.298088\pi$
$61$	9.25724	1.18527	0.592634	+	0.805472i	$0.298088\pi$
$62$	0	0
$63$	−10.4220	−1.31305
$64$	0	0
$65$	0.0366887	0.00455067
$66$	0	0
$67$	−5.90015	−0.720818	−0.360409	−	0.932794i	$-0.617363\pi$
$67$	−5.90015	−0.720818	−0.360409	+	0.932794i	$0.617363\pi$
$68$	0	0
$69$	7.14842	0.860569
$70$	0	0
$71$	15.2803	1.81343	0.906717	−	0.421739i	$-0.138580\pi$
$71$	15.2803	1.81343	0.906717	+	0.421739i	$0.138580\pi$
$72$	0	0
$73$	−7.24325	−0.847758	−0.423879	−	0.905719i	$-0.639332\pi$
$73$	−7.24325	−0.847758	−0.423879	+	0.905719i	$0.639332\pi$
$74$	0	0
$75$	4.18895	0.483699
$76$	0	0
$77$	13.2861	1.51410
$78$	0	0
$79$	4.80573	0.540687	0.270344	−	0.962764i	$-0.412863\pi$
$79$	4.80573	0.540687	0.270344	+	0.962764i	$0.412863\pi$
$80$	0	0
$81$	−10.5135	−1.16817
$82$	0	0
$83$	10.3636	1.13755	0.568777	−	0.822492i	$-0.307417\pi$
$83$	10.3636	1.13755	0.568777	+	0.822492i	$0.307417\pi$
$84$	0	0
$85$	8.73075	0.946983
$86$	0	0
$87$	−9.11533	−0.977265
$88$	0	0
$89$	−7.38931	−0.783265	−0.391632	−	0.920122i	$-0.628089\pi$
$89$	−7.38931	−0.783265	−0.391632	+	0.920122i	$0.628089\pi$
$90$	0	0
$91$	0.0907800	0.00951633
$92$	0	0
$93$	9.47031	0.982025
$94$	0	0
$95$	8.95494	0.918757
$96$	0	0
$97$	−5.55902	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$97$	−5.55902	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$98$	0	0
$99$	−7.08921	−0.712493
$100$	0	0
$101$	−7.78179	−0.774317	−0.387158	−	0.922013i	$-0.626543\pi$
$101$	−7.78179	−0.774317	−0.387158	+	0.922013i	$0.626543\pi$
$102$	0	0
$103$	−13.4244	−1.32274	−0.661372	−	0.750058i	$-0.730026\pi$
$103$	−13.4244	−1.32274	−0.661372	+	0.750058i	$0.730026\pi$
$104$	0	0
$105$	−18.2727	−1.78323
$106$	0	0
$107$	6.87848	0.664968	0.332484	−	0.943109i	$-0.392113\pi$
$107$	6.87848	0.664968	0.332484	+	0.943109i	$0.392113\pi$
$108$	0	0
$109$	10.8851	1.04261	0.521303	−	0.853372i	$-0.325446\pi$
$109$	10.8851	1.04261	0.521303	+	0.853372i	$0.325446\pi$
$110$	0	0
$111$	7.44834	0.706965
$112$	0	0
$113$	7.42095	0.698104	0.349052	−	0.937103i	$-0.386504\pi$
$113$	7.42095	0.698104	0.349052	+	0.937103i	$0.386504\pi$
$114$	0	0
$115$	5.51595	0.514365
$116$	0	0
$117$	−0.0484383	−0.00447813
$118$	0	0
$119$	21.6028	1.98032
$120$	0	0
$121$	−1.96257	−0.178415
$122$	0	0
$123$	−8.29603	−0.748028
$124$	0	0
$125$	12.1631	1.08790
$126$	0	0
$127$	−13.9312	−1.23619	−0.618095	−	0.786103i	$-0.712095\pi$
$127$	−13.9312	−1.23619	−0.618095	+	0.786103i	$0.712095\pi$
$128$	0	0
$129$	−11.9433	−1.05155
$130$	0	0
$131$	−8.31418	−0.726414	−0.363207	−	0.931709i	$-0.618318\pi$
$131$	−8.31418	−0.726414	−0.363207	+	0.931709i	$0.618318\pi$
$132$	0	0
$133$	22.1575	1.92130
$134$	0	0
$135$	−2.65366	−0.228391
$136$	0	0
$137$	−4.19438	−0.358350	−0.179175	−	0.983817i	$-0.557343\pi$
$137$	−4.19438	−0.358350	−0.179175	+	0.983817i	$0.557343\pi$
$138$	0	0
$139$	−17.1533	−1.45492	−0.727461	−	0.686149i	$-0.759300\pi$
$139$	−17.1533	−1.45492	−0.727461	+	0.686149i	$0.759300\pi$
$140$	0	0
$141$	23.3132	1.96332
$142$	0	0
$143$	0.0617499	0.00516379
$144$	0	0
$145$	−7.03367	−0.584115
$146$	0	0
$147$	−29.0093	−2.39265
$148$	0	0
$149$	−12.7303	−1.04291	−0.521454	−	0.853279i	$-0.674610\pi$
$149$	−12.7303	−1.04291	−0.521454	+	0.853279i	$0.674610\pi$
$150$	0	0
$151$	0.0370831	0.00301778	0.00150889	−	0.999999i	$-0.499520\pi$
$151$	0.0370831	0.00301778	0.00150889	+	0.999999i	$0.499520\pi$
$152$	0	0
$153$	−11.5268	−0.931887
$154$	0	0
$155$	7.30759	0.586960
$156$	0	0
$157$	21.5956	1.72352	0.861760	−	0.507316i	$-0.169362\pi$
$157$	21.5956	1.72352	0.861760	+	0.507316i	$0.169362\pi$
$158$	0	0
$159$	−0.953834	−0.0756439
$160$	0	0
$161$	13.6483	1.07564
$162$	0	0
$163$	−0.368255	−0.0288440	−0.0144220	−	0.999896i	$-0.504591\pi$
$163$	−0.368255	−0.0288440	−0.0144220	+	0.999896i	$0.504591\pi$
$164$	0	0
$165$	−12.4294	−0.967626
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−12.9996	−0.999968
$170$	0	0
$171$	−11.8228	−0.904111
$172$	0	0
$173$	6.76120	0.514044	0.257022	−	0.966406i	$-0.417259\pi$
$173$	6.76120	0.514044	0.257022	+	0.966406i	$0.417259\pi$
$174$	0	0
$175$	7.99785	0.604581
$176$	0	0
$177$	5.59086	0.420235
$178$	0	0
$179$	12.1577	0.908709	0.454355	−	0.890821i	$-0.349870\pi$
$179$	12.1577	0.908709	0.454355	+	0.890821i	$0.349870\pi$
$180$	0	0
$181$	7.20205	0.535324	0.267662	−	0.963513i	$-0.413749\pi$
$181$	7.20205	0.535324	0.267662	+	0.963513i	$0.413749\pi$
$182$	0	0
$183$	−21.4284	−1.58403
$184$	0	0
$185$	5.74737	0.422555
$186$	0	0
$187$	14.6945	1.07457
$188$	0	0
$189$	−6.56603	−0.477609
$190$	0	0
$191$	−11.1190	−0.804544	−0.402272	−	0.915520i	$-0.631779\pi$
$191$	−11.1190	−0.804544	−0.402272	+	0.915520i	$0.631779\pi$
$192$	0	0
$193$	8.77678	0.631766	0.315883	−	0.948798i	$-0.397699\pi$
$193$	8.77678	0.631766	0.315883	+	0.948798i	$0.397699\pi$
$194$	0	0
$195$	−0.0849260	−0.00608168
$196$	0	0
$197$	−17.5554	−1.25077	−0.625384	−	0.780317i	$-0.715058\pi$
$197$	−17.5554	−1.25077	−0.625384	+	0.780317i	$0.715058\pi$
$198$	0	0
$199$	23.7383	1.68277	0.841383	−	0.540439i	$-0.181742\pi$
$199$	23.7383	1.68277	0.841383	+	0.540439i	$0.181742\pi$
$200$	0	0
$201$	13.6575	0.963326
$202$	0	0
$203$	−17.4036	−1.22150
$204$	0	0
$205$	−6.40148	−0.447099
$206$	0	0
$207$	−7.28245	−0.506165
$208$	0	0
$209$	15.0719	1.04254
$210$	0	0
$211$	−18.4847	−1.27254	−0.636270	−	0.771466i	$-0.719524\pi$
$211$	−18.4847	−1.27254	−0.636270	+	0.771466i	$0.719524\pi$
$212$	0	0
$213$	−35.3704	−2.42354
$214$	0	0
$215$	−9.21581	−0.628513
$216$	0	0
$217$	18.0814	1.22744
$218$	0	0
$219$	16.7665	1.13297
$220$	0	0
$221$	0.100403	0.00675384
$222$	0	0
$223$	−26.6357	−1.78366	−0.891829	−	0.452373i	$-0.850578\pi$
$223$	−26.6357	−1.78366	−0.891829	+	0.452373i	$0.850578\pi$
$224$	0	0
$225$	−4.26749	−0.284500
$226$	0	0
$227$	−2.37369	−0.157548	−0.0787738	−	0.996893i	$-0.525100\pi$
$227$	−2.37369	−0.157548	−0.0787738	+	0.996893i	$0.525100\pi$
$228$	0	0
$229$	−20.3345	−1.34374	−0.671871	−	0.740668i	$-0.734509\pi$
$229$	−20.3345	−1.34374	−0.671871	+	0.740668i	$0.734509\pi$
$230$	0	0
$231$	−30.7544	−2.02349
$232$	0	0
$233$	−20.7054	−1.35646	−0.678229	−	0.734851i	$-0.737252\pi$
$233$	−20.7054	−1.35646	−0.678229	+	0.734851i	$0.737252\pi$
$234$	0	0
$235$	17.9892	1.17349
$236$	0	0
$237$	−11.1242	−0.722593
$238$	0	0
$239$	24.9874	1.61630	0.808149	−	0.588979i	$-0.200470\pi$
$239$	24.9874	1.61630	0.808149	+	0.588979i	$0.200470\pi$
$240$	0	0
$241$	−6.03895	−0.389003	−0.194502	−	0.980902i	$-0.562309\pi$
$241$	−6.03895	−0.389003	−0.194502	+	0.980902i	$0.562309\pi$
$242$	0	0
$243$	19.8794	1.27526
$244$	0	0
$245$	−22.3845	−1.43009
$246$	0	0
$247$	0.102981	0.00655254
$248$	0	0
$249$	−23.9894	−1.52027
$250$	0	0
$251$	18.1582	1.14613	0.573066	−	0.819509i	$-0.305754\pi$
$251$	18.1582	1.14613	0.573066	+	0.819509i	$0.305754\pi$
$252$	0	0
$253$	9.28378	0.583666
$254$	0	0
$255$	−20.2097	−1.26558
$256$	0	0
$257$	−7.46620	−0.465729	−0.232864	−	0.972509i	$-0.574810\pi$
$257$	−7.46620	−0.465729	−0.232864	+	0.972509i	$0.574810\pi$
$258$	0	0
$259$	14.2209	0.883644
$260$	0	0
$261$	9.28623	0.574803
$262$	0	0
$263$	0.230417	0.0142081	0.00710406	−	0.999975i	$-0.497739\pi$
$263$	0.230417	0.0142081	0.00710406	+	0.999975i	$0.497739\pi$
$264$	0	0
$265$	−0.736008	−0.0452126
$266$	0	0
$267$	17.1046	1.04678
$268$	0	0
$269$	10.8277	0.660176	0.330088	−	0.943950i	$-0.392922\pi$
$269$	10.8277	0.660176	0.330088	+	0.943950i	$0.392922\pi$
$270$	0	0
$271$	3.94483	0.239631	0.119816	−	0.992796i	$-0.461770\pi$
$271$	3.94483	0.239631	0.119816	+	0.992796i	$0.461770\pi$
$272$	0	0
$273$	−0.210135	−0.0127179
$274$	0	0
$275$	5.44026	0.328060
$276$	0	0
$277$	−20.6679	−1.24181	−0.620906	−	0.783885i	$-0.713235\pi$
$277$	−20.6679	−1.24181	−0.620906	+	0.783885i	$0.713235\pi$
$278$	0	0
$279$	−9.64787	−0.577603
$280$	0	0
$281$	−14.5354	−0.867107	−0.433553	−	0.901128i	$-0.642740\pi$
$281$	−14.5354	−0.867107	−0.433553	+	0.901128i	$0.642740\pi$
$282$	0	0
$283$	−18.5583	−1.10318	−0.551589	−	0.834116i	$-0.685978\pi$
$283$	−18.5583	−1.10318	−0.551589	+	0.834116i	$0.685978\pi$
$284$	0	0
$285$	−20.7286	−1.22786
$286$	0	0
$287$	−15.8394	−0.934969
$288$	0	0
$289$	6.89278	0.405458
$290$	0	0
$291$	12.8679	0.754328
$292$	0	0
$293$	−14.6439	−0.855508	−0.427754	−	0.903895i	$-0.640695\pi$
$293$	−14.6439	−0.855508	−0.427754	+	0.903895i	$0.640695\pi$
$294$	0	0
$295$	4.31408	0.251176
$296$	0	0
$297$	−4.46632	−0.259162
$298$	0	0
$299$	0.0634331	0.00366843
$300$	0	0
$301$	−22.8030	−1.31434
$302$	0	0
$303$	18.0131	1.03482
$304$	0	0
$305$	−16.5348	−0.946782
$306$	0	0
$307$	−33.5398	−1.91422	−0.957108	−	0.289732i	$-0.906434\pi$
$307$	−33.5398	−1.91422	−0.957108	+	0.289732i	$0.906434\pi$
$308$	0	0
$309$	31.0744	1.76776
$310$	0	0
$311$	20.3519	1.15405	0.577026	−	0.816726i	$-0.304213\pi$
$311$	20.3519	1.15405	0.577026	+	0.816726i	$0.304213\pi$
$312$	0	0
$313$	−24.4647	−1.38283	−0.691414	−	0.722458i	$-0.743012\pi$
$313$	−24.4647	−1.38283	−0.691414	+	0.722458i	$0.743012\pi$
$314$	0	0
$315$	18.6153	1.04885
$316$	0	0
$317$	23.4233	1.31559	0.657793	−	0.753199i	$-0.271490\pi$
$317$	23.4233	1.31559	0.657793	+	0.753199i	$0.271490\pi$
$318$	0	0
$319$	−11.8382	−0.662813
$320$	0	0
$321$	−15.9221	−0.888686
$322$	0	0
$323$	24.5063	1.36357
$324$	0	0
$325$	0.0371716	0.00206191
$326$	0	0
$327$	−25.1966	−1.39337
$328$	0	0
$329$	44.5112	2.45398
$330$	0	0
$331$	−5.39492	−0.296532	−0.148266	−	0.988948i	$-0.547369\pi$
$331$	−5.39492	−0.296532	−0.148266	+	0.988948i	$0.547369\pi$
$332$	0	0
$333$	−7.58799	−0.415819
$334$	0	0
$335$	10.5386	0.575783
$336$	0	0
$337$	−4.58555	−0.249791	−0.124896	−	0.992170i	$-0.539860\pi$
$337$	−4.58555	−0.249791	−0.124896	+	0.992170i	$0.539860\pi$
$338$	0	0
$339$	−17.1778	−0.932971
$340$	0	0
$341$	12.2992	0.666041
$342$	0	0
$343$	−24.4499	−1.32017
$344$	0	0
$345$	−12.7682	−0.687415
$346$	0	0
$347$	20.8455	1.11905	0.559524	−	0.828814i	$-0.310984\pi$
$347$	20.8455	1.11905	0.559524	+	0.828814i	$0.310984\pi$
$348$	0	0
$349$	−20.3440	−1.08899	−0.544496	−	0.838764i	$-0.683279\pi$
$349$	−20.3440	−1.08899	−0.544496	+	0.838764i	$0.683279\pi$
$350$	0	0
$351$	−0.0305169	−0.00162887
$352$	0	0
$353$	−19.8753	−1.05785	−0.528927	−	0.848667i	$-0.677406\pi$
$353$	−19.8753	−1.05785	−0.528927	+	0.848667i	$0.677406\pi$
$354$	0	0
$355$	−27.2929	−1.44856
$356$	0	0
$357$	−50.0055	−2.64657
$358$	0	0
$359$	−16.5292	−0.872377	−0.436188	−	0.899855i	$-0.643672\pi$
$359$	−16.5292	−0.872377	−0.436188	+	0.899855i	$0.643672\pi$
$360$	0	0
$361$	6.13554	0.322923
$362$	0	0
$363$	4.54289	0.238440
$364$	0	0
$365$	12.9375	0.677182
$366$	0	0
$367$	10.1380	0.529199	0.264600	−	0.964358i	$-0.414760\pi$
$367$	10.1380	0.529199	0.264600	+	0.964358i	$0.414760\pi$
$368$	0	0
$369$	8.45157	0.439971
$370$	0	0
$371$	−1.82113	−0.0945482
$372$	0	0
$373$	17.5099	0.906630	0.453315	−	0.891350i	$-0.350241\pi$
$373$	17.5099	0.906630	0.453315	+	0.891350i	$0.350241\pi$
$374$	0	0
$375$	−28.1548	−1.45391
$376$	0	0
$377$	−0.0808868	−0.00416588
$378$	0	0
$379$	14.5674	0.748279	0.374140	−	0.927372i	$-0.377938\pi$
$379$	14.5674	0.748279	0.374140	+	0.927372i	$0.377938\pi$
$380$	0	0
$381$	32.2475	1.65209
$382$	0	0
$383$	11.9383	0.610019	0.305009	−	0.952349i	$-0.401340\pi$
$383$	11.9383	0.610019	0.305009	+	0.952349i	$0.401340\pi$
$384$	0	0
$385$	−23.7311	−1.20945
$386$	0	0
$387$	12.1672	0.618493
$388$	0	0
$389$	19.0859	0.967693	0.483846	−	0.875153i	$-0.339239\pi$
$389$	19.0859	0.967693	0.483846	+	0.875153i	$0.339239\pi$
$390$	0	0
$391$	15.0951	0.763391
$392$	0	0
$393$	19.2454	0.970804
$394$	0	0
$395$	−8.58377	−0.431896
$396$	0	0
$397$	−18.3059	−0.918748	−0.459374	−	0.888243i	$-0.651926\pi$
$397$	−18.3059	−0.918748	−0.459374	+	0.888243i	$0.651926\pi$
$398$	0	0
$399$	−51.2895	−2.56769
$400$	0	0
$401$	−9.03897	−0.451385	−0.225692	−	0.974199i	$-0.572464\pi$
$401$	−9.03897	−0.451385	−0.225692	+	0.974199i	$0.572464\pi$
$402$	0	0
$403$	0.0840368	0.00418617
$404$	0	0
$405$	18.7788	0.933125
$406$	0	0
$407$	9.67328	0.479487
$408$	0	0
$409$	−16.3450	−0.808210	−0.404105	−	0.914713i	$-0.632417\pi$
$409$	−16.3450	−0.808210	−0.404105	+	0.914713i	$0.632417\pi$
$410$	0	0
$411$	9.70905	0.478912
$412$	0	0
$413$	10.6745	0.525256
$414$	0	0
$415$	−18.5110	−0.908668
$416$	0	0
$417$	39.7059	1.94441
$418$	0	0
$419$	26.1098	1.27555	0.637773	−	0.770225i	$-0.279856\pi$
$419$	26.1098	1.27555	0.637773	+	0.770225i	$0.279856\pi$
$420$	0	0
$421$	21.1398	1.03029	0.515145	−	0.857103i	$-0.327738\pi$
$421$	21.1398	1.03029	0.515145	+	0.857103i	$0.327738\pi$
$422$	0	0
$423$	−23.7503	−1.15478
$424$	0	0
$425$	8.84567	0.429078
$426$	0	0
$427$	−40.9126	−1.97990
$428$	0	0
$429$	−0.142937	−0.00690107
$430$	0	0
$431$	10.4924	0.505403	0.252701	−	0.967544i	$-0.418681\pi$
$431$	10.4924	0.505403	0.252701	+	0.967544i	$0.418681\pi$
$432$	0	0
$433$	8.55739	0.411242	0.205621	−	0.978632i	$-0.434079\pi$
$433$	8.55739	0.411242	0.205621	+	0.978632i	$0.434079\pi$
$434$	0	0
$435$	16.2814	0.780631
$436$	0	0
$437$	15.4827	0.740637
$438$	0	0
$439$	35.7768	1.70754	0.853768	−	0.520654i	$-0.174312\pi$
$439$	35.7768	1.70754	0.853768	+	0.520654i	$0.174312\pi$
$440$	0	0
$441$	29.5532	1.40730
$442$	0	0
$443$	15.9596	0.758265	0.379132	−	0.925342i	$-0.376222\pi$
$443$	15.9596	0.758265	0.379132	+	0.925342i	$0.376222\pi$
$444$	0	0
$445$	13.1984	0.625665
$446$	0	0
$447$	29.4678	1.39378
$448$	0	0
$449$	15.6517	0.738650	0.369325	−	0.929300i	$-0.379589\pi$
$449$	15.6517	0.738650	0.369325	+	0.929300i	$0.379589\pi$
$450$	0	0
$451$	−10.7742	−0.507337
$452$	0	0
$453$	−0.0858388	−0.00403306
$454$	0	0
$455$	−0.162147	−0.00760156
$456$	0	0
$457$	11.9838	0.560577	0.280288	−	0.959916i	$-0.409570\pi$
$457$	11.9838	0.560577	0.280288	+	0.959916i	$0.409570\pi$
$458$	0	0
$459$	−7.26207	−0.338964
$460$	0	0
$461$	24.7917	1.15466	0.577331	−	0.816510i	$-0.304094\pi$
$461$	24.7917	1.15466	0.577331	+	0.816510i	$0.304094\pi$
$462$	0	0
$463$	33.3824	1.55141	0.775705	−	0.631095i	$-0.217394\pi$
$463$	33.3824	1.55141	0.775705	+	0.631095i	$0.217394\pi$
$464$	0	0
$465$	−16.9154	−0.784433
$466$	0	0
$467$	35.9566	1.66388	0.831938	−	0.554869i	$-0.187232\pi$
$467$	35.9566	1.66388	0.831938	+	0.554869i	$0.187232\pi$
$468$	0	0
$469$	26.0759	1.20407
$470$	0	0
$471$	−49.9890	−2.30337
$472$	0	0
$473$	−15.5109	−0.713193
$474$	0	0
$475$	9.07280	0.416289
$476$	0	0
$477$	0.971717	0.0444919
$478$	0	0
$479$	31.9004	1.45757	0.728784	−	0.684743i	$-0.240086\pi$
$479$	31.9004	1.45757	0.728784	+	0.684743i	$0.240086\pi$
$480$	0	0
$481$	0.0660944	0.00301365
$482$	0	0
$483$	−31.5927	−1.43752
$484$	0	0
$485$	9.92925	0.450864
$486$	0	0
$487$	32.2345	1.46068	0.730341	−	0.683082i	$-0.239361\pi$
$487$	32.2345	1.46068	0.730341	+	0.683082i	$0.239361\pi$
$488$	0	0
$489$	0.852427	0.0385481
$490$	0	0
$491$	12.4605	0.562336	0.281168	−	0.959658i	$-0.409278\pi$
$491$	12.4605	0.562336	0.281168	+	0.959658i	$0.409278\pi$
$492$	0	0
$493$	−19.2485	−0.866909
$494$	0	0
$495$	12.6624	0.569133
$496$	0	0
$497$	−67.5316	−3.02921
$498$	0	0
$499$	17.7185	0.793191	0.396595	−	0.917994i	$-0.370192\pi$
$499$	17.7185	0.793191	0.396595	+	0.917994i	$0.370192\pi$
$500$	0	0
$501$	2.31477	0.103416
$502$	0	0
$503$	−33.0085	−1.47178	−0.735888	−	0.677103i	$-0.763235\pi$
$503$	−33.0085	−1.47178	−0.735888	+	0.677103i	$0.763235\pi$
$504$	0	0
$505$	13.8995	0.618518
$506$	0	0
$507$	30.0911	1.33639
$508$	0	0
$509$	−25.1465	−1.11460	−0.557300	−	0.830311i	$-0.688163\pi$
$509$	−25.1465	−1.11460	−0.557300	+	0.830311i	$0.688163\pi$
$510$	0	0
$511$	32.0118	1.41612
$512$	0	0
$513$	−7.44854	−0.328861
$514$	0	0
$515$	23.9780	1.05660
$516$	0	0
$517$	30.2772	1.33159
$518$	0	0
$519$	−15.6506	−0.686987
$520$	0	0
$521$	−23.4095	−1.02559	−0.512793	−	0.858512i	$-0.671389\pi$
$521$	−23.4095	−1.02559	−0.512793	+	0.858512i	$0.671389\pi$
$522$	0	0
$523$	15.2875	0.668474	0.334237	−	0.942489i	$-0.391521\pi$
$523$	15.2875	0.668474	0.334237	+	0.942489i	$0.391521\pi$
$524$	0	0
$525$	−18.5132	−0.807983
$526$	0	0
$527$	19.9981	0.871132
$528$	0	0
$529$	−13.4632	−0.585355
$530$	0	0
$531$	−5.69568	−0.247172
$532$	0	0
$533$	−0.0736166	−0.00318869
$534$	0	0
$535$	−12.2860	−0.531171
$536$	0	0
$537$	−28.1423	−1.21443
$538$	0	0
$539$	−37.6749	−1.62277
$540$	0	0
$541$	21.6929	0.932651	0.466325	−	0.884613i	$-0.345578\pi$
$541$	21.6929	0.932651	0.466325	+	0.884613i	$0.345578\pi$
$542$	0	0
$543$	−16.6711	−0.715426
$544$	0	0
$545$	−19.4425	−0.832825
$546$	0	0
$547$	−36.5063	−1.56090	−0.780448	−	0.625220i	$-0.785009\pi$
$547$	−36.5063	−1.56090	−0.780448	+	0.625220i	$0.785009\pi$
$548$	0	0
$549$	21.8302	0.931688
$550$	0	0
$551$	−19.7428	−0.841070
$552$	0	0
$553$	−21.2391	−0.903178
$554$	0	0
$555$	−13.3039	−0.564718
$556$	0	0
$557$	6.43695	0.272742	0.136371	−	0.990658i	$-0.456456\pi$
$557$	6.43695	0.272742	0.136371	+	0.990658i	$0.456456\pi$
$558$	0	0
$559$	−0.105981	−0.00448253
$560$	0	0
$561$	−34.0145	−1.43609
$562$	0	0
$563$	14.0397	0.591702	0.295851	−	0.955234i	$-0.404397\pi$
$563$	14.0397	0.591702	0.295851	+	0.955234i	$0.404397\pi$
$564$	0	0
$565$	−13.2549	−0.557640
$566$	0	0
$567$	46.4649	1.95134
$568$	0	0
$569$	35.8704	1.50376	0.751882	−	0.659298i	$-0.229146\pi$
$569$	35.8704	1.50376	0.751882	+	0.659298i	$0.229146\pi$
$570$	0	0
$571$	−20.5277	−0.859056	−0.429528	−	0.903053i	$-0.641320\pi$
$571$	−20.5277	−0.859056	−0.429528	+	0.903053i	$0.641320\pi$
$572$	0	0
$573$	25.7380	1.07522
$574$	0	0
$575$	5.58855	0.233059
$576$	0	0
$577$	9.59955	0.399635	0.199817	−	0.979833i	$-0.435965\pi$
$577$	9.59955	0.399635	0.199817	+	0.979833i	$0.435965\pi$
$578$	0	0
$579$	−20.3162	−0.844314
$580$	0	0
$581$	−45.8023	−1.90020
$582$	0	0
$583$	−1.23876	−0.0513042
$584$	0	0
$585$	0.0865182	0.00357709
$586$	0	0
$587$	−32.8936	−1.35766	−0.678831	−	0.734294i	$-0.737513\pi$
$587$	−32.8936	−1.35766	−0.678831	+	0.734294i	$0.737513\pi$
$588$	0	0
$589$	20.5116	0.845166
$590$	0	0
$591$	40.6367	1.67157
$592$	0	0
$593$	20.5604	0.844316	0.422158	−	0.906522i	$-0.361273\pi$
$593$	20.5604	0.844316	0.422158	+	0.906522i	$0.361273\pi$
$594$	0	0
$595$	−38.5858	−1.58186
$596$	0	0
$597$	−54.9488	−2.24891
$598$	0	0
$599$	−44.4276	−1.81526	−0.907631	−	0.419769i	$-0.862111\pi$
$599$	−44.4276	−1.81526	−0.907631	+	0.419769i	$0.862111\pi$
$600$	0	0
$601$	−40.4731	−1.65093	−0.825467	−	0.564451i	$-0.809088\pi$
$601$	−40.4731	−1.65093	−0.825467	+	0.564451i	$0.809088\pi$
$602$	0	0
$603$	−13.9136	−0.566604
$604$	0	0
$605$	3.50544	0.142516
$606$	0	0
$607$	−31.6730	−1.28557	−0.642784	−	0.766048i	$-0.722221\pi$
$607$	−31.6730	−1.28557	−0.642784	+	0.766048i	$0.722221\pi$
$608$	0	0
$609$	40.2855	1.63245
$610$	0	0
$611$	0.206875	0.00836925
$612$	0	0
$613$	17.4309	0.704026	0.352013	−	0.935995i	$-0.385497\pi$
$613$	17.4309	0.704026	0.352013	+	0.935995i	$0.385497\pi$
$614$	0	0
$615$	14.8180	0.597518
$616$	0	0
$617$	27.5038	1.10726	0.553630	−	0.832763i	$-0.313242\pi$
$617$	27.5038	1.10726	0.553630	+	0.832763i	$0.313242\pi$
$618$	0	0
$619$	−36.4486	−1.46499	−0.732497	−	0.680770i	$-0.761645\pi$
$619$	−36.4486	−1.46499	−0.732497	+	0.680770i	$0.761645\pi$
$620$	0	0
$621$	−4.58806	−0.184112
$622$	0	0
$623$	32.6573	1.30839
$624$	0	0
$625$	−12.6768	−0.507073
$626$	0	0
$627$	−34.8879	−1.39329
$628$	0	0
$629$	15.7284	0.627132
$630$	0	0
$631$	13.9968	0.557204	0.278602	−	0.960407i	$-0.410129\pi$
$631$	13.9968	0.557204	0.278602	+	0.960407i	$0.410129\pi$
$632$	0	0
$633$	42.7879	1.70067
$634$	0	0
$635$	24.8832	0.987458
$636$	0	0
$637$	−0.257420	−0.0101994
$638$	0	0
$639$	36.0335	1.42546
$640$	0	0
$641$	19.6668	0.776793	0.388397	−	0.921492i	$-0.373029\pi$
$641$	19.6668	0.776793	0.388397	+	0.921492i	$0.373029\pi$
$642$	0	0
$643$	43.8518	1.72935	0.864673	−	0.502335i	$-0.167526\pi$
$643$	43.8518	1.72935	0.864673	+	0.502335i	$0.167526\pi$
$644$	0	0
$645$	21.3325	0.839966
$646$	0	0
$647$	11.9100	0.468232	0.234116	−	0.972209i	$-0.424780\pi$
$647$	11.9100	0.468232	0.234116	+	0.972209i	$0.424780\pi$
$648$	0	0
$649$	7.26094	0.285017
$650$	0	0
$651$	−41.8543	−1.64040
$652$	0	0
$653$	36.6524	1.43432	0.717160	−	0.696908i	$-0.245442\pi$
$653$	36.6524	1.43432	0.717160	+	0.696908i	$0.245442\pi$
$654$	0	0
$655$	14.8504	0.580253
$656$	0	0
$657$	−17.0808	−0.666387
$658$	0	0
$659$	13.4566	0.524196	0.262098	−	0.965041i	$-0.415586\pi$
$659$	13.4566	0.524196	0.262098	+	0.965041i	$0.415586\pi$
$660$	0	0
$661$	−44.9576	−1.74865	−0.874324	−	0.485342i	$-0.838695\pi$
$661$	−44.9576	−1.74865	−0.874324	+	0.485342i	$0.838695\pi$
$662$	0	0
$663$	−0.232410	−0.00902607
$664$	0	0
$665$	−39.5766	−1.53472
$666$	0	0
$667$	−12.1609	−0.470872
$668$	0	0
$669$	61.6556	2.38374
$670$	0	0
$671$	−27.8294	−1.07434
$672$	0	0
$673$	−21.1934	−0.816945	−0.408473	−	0.912771i	$-0.633938\pi$
$673$	−21.1934	−0.816945	−0.408473	+	0.912771i	$0.633938\pi$
$674$	0	0
$675$	−2.68859	−0.103484
$676$	0	0
$677$	−13.1789	−0.506507	−0.253254	−	0.967400i	$-0.581501\pi$
$677$	−13.1789	−0.506507	−0.253254	+	0.967400i	$0.581501\pi$
$678$	0	0
$679$	24.5683	0.942843
$680$	0	0
$681$	5.49456	0.210552
$682$	0	0
$683$	−9.21526	−0.352612	−0.176306	−	0.984335i	$-0.556415\pi$
$683$	−9.21526	−0.352612	−0.176306	+	0.984335i	$0.556415\pi$
$684$	0	0
$685$	7.49181	0.286247
$686$	0	0
$687$	47.0698	1.79582
$688$	0	0
$689$	−0.00846405	−0.000322455 0
$690$	0	0
$691$	−24.3433	−0.926064	−0.463032	−	0.886342i	$-0.653239\pi$
$691$	−24.3433	−0.926064	−0.463032	+	0.886342i	$0.653239\pi$
$692$	0	0
$693$	31.3310	1.19017
$694$	0	0
$695$	30.6383	1.16218
$696$	0	0
$697$	−17.5184	−0.663558
$698$	0	0
$699$	47.9283	1.81282
$700$	0	0
$701$	30.6042	1.15590	0.577952	−	0.816071i	$-0.303852\pi$
$701$	30.6042	1.15590	0.577952	+	0.816071i	$0.303852\pi$
$702$	0	0
$703$	16.1323	0.608440
$704$	0	0
$705$	−41.6409	−1.56829
$706$	0	0
$707$	34.3918	1.29344
$708$	0	0
$709$	29.6450	1.11334	0.556670	−	0.830734i	$-0.312079\pi$
$709$	29.6450	1.11334	0.556670	+	0.830734i	$0.312079\pi$
$710$	0	0
$711$	11.3327	0.425011
$712$	0	0
$713$	12.6345	0.473165
$714$	0	0
$715$	−0.110295	−0.00412479
$716$	0	0
$717$	−57.8400	−2.16008
$718$	0	0
$719$	49.3148	1.83913	0.919566	−	0.392936i	$-0.128540\pi$
$719$	49.3148	1.83913	0.919566	+	0.392936i	$0.128540\pi$
$720$	0	0
$721$	59.3295	2.20955
$722$	0	0
$723$	13.9788	0.519877
$724$	0	0
$725$	−7.12625	−0.264662
$726$	0	0
$727$	−20.7756	−0.770525	−0.385262	−	0.922807i	$-0.625889\pi$
$727$	−20.7756	−0.770525	−0.385262	+	0.922807i	$0.625889\pi$
$728$	0	0
$729$	−14.4757	−0.536136
$730$	0	0
$731$	−25.2202	−0.932802
$732$	0	0
$733$	−25.4487	−0.939968	−0.469984	−	0.882675i	$-0.655740\pi$
$733$	−25.4487	−0.939968	−0.469984	+	0.882675i	$0.655740\pi$
$734$	0	0
$735$	51.8150	1.91123
$736$	0	0
$737$	17.7372	0.653359
$738$	0	0
$739$	51.0198	1.87679	0.938397	−	0.345560i	$-0.112311\pi$
$739$	51.0198	1.87679	0.938397	+	0.345560i	$0.112311\pi$
$740$	0	0
$741$	−0.238378	−0.00875704
$742$	0	0
$743$	8.58213	0.314848	0.157424	−	0.987531i	$-0.449681\pi$
$743$	8.58213	0.314848	0.157424	+	0.987531i	$0.449681\pi$
$744$	0	0
$745$	22.7383	0.833066
$746$	0	0
$747$	24.4392	0.894182
$748$	0	0
$749$	−30.3997	−1.11078
$750$	0	0
$751$	−10.1137	−0.369055	−0.184527	−	0.982827i	$-0.559075\pi$
$751$	−10.1137	−0.369055	−0.184527	+	0.982827i	$0.559075\pi$
$752$	0	0
$753$	−42.0320	−1.53173
$754$	0	0
$755$	−0.0662360	−0.00241057
$756$	0	0
$757$	−28.7454	−1.04477	−0.522384	−	0.852710i	$-0.674957\pi$
$757$	−28.7454	−1.04477	−0.522384	+	0.852710i	$0.674957\pi$
$758$	0	0
$759$	−21.4898	−0.780031
$760$	0	0
$761$	−16.3186	−0.591547	−0.295774	−	0.955258i	$-0.595577\pi$
$761$	−16.3186	−0.591547	−0.295774	+	0.955258i	$0.595577\pi$
$762$	0	0
$763$	−48.1071	−1.74160
$764$	0	0
$765$	20.5886	0.744383
$766$	0	0
$767$	0.0496117	0.00179137
$768$	0	0
$769$	28.7905	1.03821	0.519105	−	0.854710i	$-0.326265\pi$
$769$	28.7905	1.03821	0.519105	+	0.854710i	$0.326265\pi$
$770$	0	0
$771$	17.2826	0.622416
$772$	0	0
$773$	−49.6533	−1.78591	−0.892953	−	0.450150i	$-0.851370\pi$
$773$	−49.6533	−1.78591	−0.892953	+	0.450150i	$0.851370\pi$
$774$	0	0
$775$	7.40377	0.265951
$776$	0	0
$777$	−32.9182	−1.18093
$778$	0	0
$779$	−17.9683	−0.643780
$780$	0	0
$781$	−45.9361	−1.64372
$782$	0	0
$783$	5.85047	0.209079
$784$	0	0
$785$	−38.5731	−1.37673
$786$	0	0
$787$	−13.4116	−0.478072	−0.239036	−	0.971011i	$-0.576831\pi$
$787$	−13.4116	−0.478072	−0.239036	+	0.971011i	$0.576831\pi$
$788$	0	0
$789$	−0.533363	−0.0189882
$790$	0	0
$791$	−32.7971	−1.16613
$792$	0	0
$793$	−0.190150	−0.00675241
$794$	0	0
$795$	1.70369	0.0604237
$796$	0	0
$797$	−10.3539	−0.366754	−0.183377	−	0.983043i	$-0.558703\pi$
$797$	−10.3539	−0.366754	−0.183377	+	0.983043i	$0.558703\pi$
$798$	0	0
$799$	49.2296	1.74162
$800$	0	0
$801$	−17.4253	−0.615691
$802$	0	0
$803$	21.7749	0.768420
$804$	0	0
$805$	−24.3779	−0.859209
$806$	0	0
$807$	−25.0636	−0.882281
$808$	0	0
$809$	36.4506	1.28153	0.640767	−	0.767735i	$-0.278616\pi$
$809$	36.4506	1.28153	0.640767	+	0.767735i	$0.278616\pi$
$810$	0	0
$811$	−7.38138	−0.259195	−0.129598	−	0.991567i	$-0.541369\pi$
$811$	−7.38138	−0.259195	−0.129598	+	0.991567i	$0.541369\pi$
$812$	0	0
$813$	−9.13138	−0.320251
$814$	0	0
$815$	0.657760	0.0230403
$816$	0	0
$817$	−25.8678	−0.904999
$818$	0	0
$819$	0.214075	0.00748038
$820$	0	0
$821$	−17.8867	−0.624250	−0.312125	−	0.950041i	$-0.601041\pi$
$821$	−17.8867	−0.624250	−0.312125	+	0.950041i	$0.601041\pi$
$822$	0	0
$823$	25.2378	0.879734	0.439867	−	0.898063i	$-0.355026\pi$
$823$	25.2378	0.879734	0.439867	+	0.898063i	$0.355026\pi$
$824$	0	0
$825$	−12.5930	−0.438431
$826$	0	0
$827$	−6.91077	−0.240311	−0.120156	−	0.992755i	$-0.538339\pi$
$827$	−6.91077	−0.240311	−0.120156	+	0.992755i	$0.538339\pi$
$828$	0	0
$829$	27.6569	0.960565	0.480283	−	0.877114i	$-0.340534\pi$
$829$	27.6569	0.960565	0.480283	+	0.877114i	$0.340534\pi$
$830$	0	0
$831$	47.8414	1.65960
$832$	0	0
$833$	−61.2579	−2.12246
$834$	0	0
$835$	1.78615	0.0618123
$836$	0	0
$837$	−6.07831	−0.210097
$838$	0	0
$839$	49.6951	1.71566	0.857832	−	0.513930i	$-0.171811\pi$
$839$	49.6951	1.71566	0.857832	+	0.513930i	$0.171811\pi$
$840$	0	0
$841$	−13.4930	−0.465276
$842$	0	0
$843$	33.6460	1.15883
$844$	0	0
$845$	23.2192	0.798765
$846$	0	0
$847$	8.67362	0.298029
$848$	0	0
$849$	42.9583	1.47433
$850$	0	0
$851$	9.93695	0.340634
$852$	0	0
$853$	4.04291	0.138427	0.0692134	−	0.997602i	$-0.477951\pi$
$853$	4.04291	0.138427	0.0692134	+	0.997602i	$0.477951\pi$
$854$	0	0
$855$	21.1173	0.722196
$856$	0	0
$857$	44.6337	1.52466	0.762329	−	0.647189i	$-0.224056\pi$
$857$	44.6337	1.52466	0.762329	+	0.647189i	$0.224056\pi$
$858$	0	0
$859$	42.6455	1.45504	0.727522	−	0.686084i	$-0.240672\pi$
$859$	42.6455	1.45504	0.727522	+	0.686084i	$0.240672\pi$
$860$	0	0
$861$	36.6646	1.24953
$862$	0	0
$863$	0.797877	0.0271601	0.0135800	−	0.999908i	$-0.495677\pi$
$863$	0.797877	0.0271601	0.0135800	+	0.999908i	$0.495677\pi$
$864$	0	0
$865$	−12.0765	−0.410614
$866$	0	0
$867$	−15.9552	−0.541868
$868$	0	0
$869$	−14.4472	−0.490086
$870$	0	0
$871$	0.121193	0.00410646
$872$	0	0
$873$	−13.1091	−0.443677
$874$	0	0
$875$	−53.7551	−1.81726
$876$	0	0
$877$	−30.6646	−1.03547	−0.517734	−	0.855541i	$-0.673224\pi$
$877$	−30.6646	−1.03547	−0.517734	+	0.855541i	$0.673224\pi$
$878$	0	0
$879$	33.8974	1.14333
$880$	0	0
$881$	−24.5024	−0.825507	−0.412753	−	0.910843i	$-0.635433\pi$
$881$	−24.5024	−0.825507	−0.412753	+	0.910843i	$0.635433\pi$
$882$	0	0
$883$	21.1394	0.711398	0.355699	−	0.934601i	$-0.384243\pi$
$883$	21.1394	0.711398	0.355699	+	0.934601i	$0.384243\pi$
$884$	0	0
$885$	−9.98612	−0.335680
$886$	0	0
$887$	37.2689	1.25137	0.625684	−	0.780077i	$-0.284820\pi$
$887$	37.2689	1.25137	0.625684	+	0.780077i	$0.284820\pi$
$888$	0	0
$889$	61.5692	2.06496
$890$	0	0
$891$	31.6062	1.05885
$892$	0	0
$893$	50.4937	1.68971
$894$	0	0
$895$	−21.7155	−0.725869
$896$	0	0
$897$	−0.146833	−0.00490262
$898$	0	0
$899$	−16.1109	−0.537328
$900$	0	0
$901$	−2.01418	−0.0671020
$902$	0	0
$903$	52.7837	1.75653
$904$	0	0
$905$	−12.8640	−0.427613
$906$	0	0
$907$	−37.7234	−1.25259	−0.626293	−	0.779588i	$-0.715429\pi$
$907$	−37.7234	−1.25259	−0.626293	+	0.779588i	$0.715429\pi$
$908$	0	0
$909$	−18.3508	−0.608657
$910$	0	0
$911$	−42.8428	−1.41945	−0.709723	−	0.704481i	$-0.751180\pi$
$911$	−42.8428	−1.41945	−0.709723	+	0.704481i	$0.751180\pi$
$912$	0	0
$913$	−31.1554	−1.03109
$914$	0	0
$915$	38.2744	1.26531
$916$	0	0
$917$	36.7448	1.21342
$918$	0	0
$919$	6.23486	0.205669	0.102835	−	0.994698i	$-0.467209\pi$
$919$	6.23486	0.205669	0.102835	+	0.994698i	$0.467209\pi$
$920$	0	0
$921$	77.6369	2.55822
$922$	0	0
$923$	−0.313867	−0.0103310
$924$	0	0
$925$	5.82302	0.191460
$926$	0	0
$927$	−31.6570	−1.03975
$928$	0	0
$929$	−48.6916	−1.59752	−0.798760	−	0.601650i	$-0.794510\pi$
$929$	−48.6916	−1.59752	−0.798760	+	0.601650i	$0.794510\pi$
$930$	0	0
$931$	−62.8309	−2.05920
$932$	0	0
$933$	−47.1101	−1.54232
$934$	0	0
$935$	−26.2467	−0.858358
$936$	0	0
$937$	−46.5788	−1.52166	−0.760831	−	0.648950i	$-0.775208\pi$
$937$	−46.5788	−1.52166	−0.760831	+	0.648950i	$0.775208\pi$
$938$	0	0
$939$	56.6303	1.84806
$940$	0	0
$941$	−7.23365	−0.235810	−0.117905	−	0.993025i	$-0.537618\pi$
$941$	−7.23365	−0.235810	−0.117905	+	0.993025i	$0.537618\pi$
$942$	0	0
$943$	−11.0679	−0.360420
$944$	0	0
$945$	11.7279	0.381510
$946$	0	0
$947$	−42.3866	−1.37738	−0.688690	−	0.725056i	$-0.741814\pi$
$947$	−42.3866	−1.37738	−0.688690	+	0.725056i	$0.741814\pi$
$948$	0	0
$949$	0.148781	0.00482964
$950$	0	0
$951$	−54.2197	−1.75819
$952$	0	0
$953$	30.1317	0.976063	0.488031	−	0.872826i	$-0.337715\pi$
$953$	30.1317	0.976063	0.488031	+	0.872826i	$0.337715\pi$
$954$	0	0
$955$	19.8602	0.642662
$956$	0	0
$957$	27.4028	0.885806
$958$	0	0
$959$	18.5372	0.598598
$960$	0	0
$961$	−14.2617	−0.460055
$962$	0	0
$963$	16.2206	0.522703
$964$	0	0
$965$	−15.6767	−0.504650
$966$	0	0
$967$	−7.65671	−0.246223	−0.123112	−	0.992393i	$-0.539287\pi$
$967$	−7.65671	−0.246223	−0.123112	+	0.992393i	$0.539287\pi$
$968$	0	0
$969$	−56.7265	−1.82232
$970$	0	0
$971$	−57.5041	−1.84539	−0.922697	−	0.385526i	$-0.874020\pi$
$971$	−57.5041	−1.84539	−0.922697	+	0.385526i	$0.874020\pi$
$972$	0	0
$973$	75.8094	2.43034
$974$	0	0
$975$	−0.0860438	−0.00275561
$976$	0	0
$977$	−2.94745	−0.0942974	−0.0471487	−	0.998888i	$-0.515013\pi$
$977$	−2.94745	−0.0942974	−0.0471487	+	0.998888i	$0.515013\pi$
$978$	0	0
$979$	22.2140	0.709962
$980$	0	0
$981$	25.6690	0.819548
$982$	0	0
$983$	−39.6455	−1.26450	−0.632248	−	0.774766i	$-0.717868\pi$
$983$	−39.6455	−1.26450	−0.632248	+	0.774766i	$0.717868\pi$
$984$	0	0
$985$	31.3565	0.999103
$986$	0	0
$987$	−103.033	−3.27959
$988$	0	0
$989$	−15.9337	−0.506663
$990$	0	0
$991$	8.81971	0.280167	0.140084	−	0.990140i	$-0.455263\pi$
$991$	8.81971	0.280167	0.140084	+	0.990140i	$0.455263\pi$
$992$	0	0
$993$	12.4880	0.396295
$994$	0	0
$995$	−42.4003	−1.34418
$996$	0	0
$997$	42.0035	1.33026	0.665132	−	0.746726i	$-0.268375\pi$
$997$	42.0035	1.33026	0.665132	+	0.746726i	$0.268375\pi$
$998$	0	0
$999$	−4.78055	−0.151250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1336.2.a.d.1.3	✓	12
4.3	odd	2		2672.2.a.p.1.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1336.2.a.d.1.3	✓	12	1.1	even	1	trivial
2672.2.a.p.1.10		12	4.3	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:	\( N \)	\(=\)	\( 1336 = 2^{3} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1336.a (trivial)

\(f(q)\)	\(=\)	\(q-2.31477 q^{3} -1.78615 q^{5} -4.41953 q^{7} +2.35817 q^{9} +O(q^{10})\)	\(q-2.31477 q^{3} -1.78615 q^{5} -4.41953 q^{7} +2.35817 q^{9} -3.00623 q^{11} -0.0205406 q^{13} +4.13454 q^{15} -4.88802 q^{17} -5.01354 q^{19} +10.2302 q^{21} -3.08818 q^{23} -1.80966 q^{25} +1.48569 q^{27} +3.93789 q^{29} -4.09125 q^{31} +6.95874 q^{33} +7.89395 q^{35} -3.21774 q^{37} +0.0475469 q^{39} +3.58395 q^{41} +5.15959 q^{43} -4.21205 q^{45} -10.0715 q^{47} +12.5322 q^{49} +11.3147 q^{51} +0.412064 q^{53} +5.36959 q^{55} +11.6052 q^{57} -2.41530 q^{59} +9.25724 q^{61} -10.4220 q^{63} +0.0366887 q^{65} -5.90015 q^{67} +7.14842 q^{69} +15.2803 q^{71} -7.24325 q^{73} +4.18895 q^{75} +13.2861 q^{77} +4.80573 q^{79} -10.5135 q^{81} +10.3636 q^{83} +8.73075 q^{85} -9.11533 q^{87} -7.38931 q^{89} +0.0907800 q^{91} +9.47031 q^{93} +8.95494 q^{95} -5.55902 q^{97} -7.08921 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) 12 * q - q^3 + 8 * q^5 - 4 * q^7 + 15 * q^9	\( 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9} + 6 q^{11} + 13 q^{13} + 4 q^{15} + 10 q^{17} - q^{19} + 13 q^{21} + 3 q^{23} + 18 q^{25} - 10 q^{27} + 29 q^{29} - 3 q^{31} + 3 q^{35} + 41 q^{37} + 10 q^{39} + 20 q^{41} - q^{43} + 42 q^{45} + 5 q^{47} + 20 q^{49} + 14 q^{51} + 39 q^{53} + 3 q^{55} + 3 q^{57} + 8 q^{59} + 30 q^{61} - 2 q^{63} + 21 q^{65} - 9 q^{67} + 33 q^{69} + 29 q^{71} + 12 q^{73} + q^{75} + 19 q^{77} - 2 q^{79} + 24 q^{81} - 5 q^{83} + 44 q^{85} - 2 q^{87} + 7 q^{89} - 4 q^{91} + 37 q^{93} + 18 q^{95} - 14 q^{97} + 5 q^{99}+O(q^{100}) \) 12 * q - q^3 + 8 * q^5 - 4 * q^7 + 15 * q^9 + 6 * q^11 + 13 * q^13 + 4 * q^15 + 10 * q^17 - q^19 + 13 * q^21 + 3 * q^23 + 18 * q^25 - 10 * q^27 + 29 * q^29 - 3 * q^31 + 3 * q^35 + 41 * q^37 + 10 * q^39 + 20 * q^41 - q^43 + 42 * q^45 + 5 * q^47 + 20 * q^49 + 14 * q^51 + 39 * q^53 + 3 * q^55 + 3 * q^57 + 8 * q^59 + 30 * q^61 - 2 * q^63 + 21 * q^65 - 9 * q^67 + 33 * q^69 + 29 * q^71 + 12 * q^73 + q^75 + 19 * q^77 - 2 * q^79 + 24 * q^81 - 5 * q^83 + 44 * q^85 - 2 * q^87 + 7 * q^89 - 4 * q^91 + 37 * q^93 + 18 * q^95 - 14 * q^97 + 5 * q^99

Introduction

L-functions

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