LMFDB - Embedded newform 3042.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1,3042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3042, 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("3042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.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:	$24.2904922949$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	3042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.24698 q^{5} +2.93900 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.24698 q^{5} +2.93900 q^{7} -1.00000 q^{8} +1.24698 q^{10} +1.30798 q^{11} -2.93900 q^{14} +1.00000 q^{16} -7.60388 q^{17} +1.10992 q^{19} -1.24698 q^{20} -1.30798 q^{22} +2.09783 q^{23} -3.44504 q^{25} +2.93900 q^{28} -2.19806 q^{29} -4.51573 q^{31} -1.00000 q^{32} +7.60388 q^{34} -3.66487 q^{35} +9.20775 q^{37} -1.10992 q^{38} +1.24698 q^{40} -9.87800 q^{41} -9.20775 q^{43} +1.30798 q^{44} -2.09783 q^{46} +7.70171 q^{47} +1.63773 q^{49} +3.44504 q^{50} -7.02715 q^{53} -1.63102 q^{55} -2.93900 q^{56} +2.19806 q^{58} -11.4765 q^{59} -5.20775 q^{61} +4.51573 q^{62} +1.00000 q^{64} +1.32975 q^{67} -7.60388 q^{68} +3.66487 q^{70} +13.5797 q^{71} -1.96077 q^{73} -9.20775 q^{74} +1.10992 q^{76} +3.84415 q^{77} +12.8170 q^{79} -1.24698 q^{80} +9.87800 q^{82} +13.2567 q^{83} +9.48188 q^{85} +9.20775 q^{86} -1.30798 q^{88} -3.15346 q^{89} +2.09783 q^{92} -7.70171 q^{94} -1.38404 q^{95} -14.2349 q^{97} -1.63773 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + q^{5} - q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 + q^5 - q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} + q^{5} - q^{7} - 3 q^{8} - q^{10} + 9 q^{11} + q^{14} + 3 q^{16} - 14 q^{17} + 4 q^{19} + q^{20} - 9 q^{22} - 12 q^{23} - 10 q^{25} - q^{28} - 11 q^{29} - q^{31} - 3 q^{32} + 14 q^{34} - 12 q^{35} + 10 q^{37} - 4 q^{38} - q^{40} - 10 q^{41} - 10 q^{43} + 9 q^{44} + 12 q^{46} - 4 q^{47} + 12 q^{49} + 10 q^{50} - 15 q^{53} + 10 q^{55} + q^{56} + 11 q^{58} - 9 q^{59} + 2 q^{61} + q^{62} + 3 q^{64} + 6 q^{67} - 14 q^{68} + 12 q^{70} - 6 q^{71} + 7 q^{73} - 10 q^{74} + 4 q^{76} - 24 q^{77} + 9 q^{79} + q^{80} + 10 q^{82} + 13 q^{83} + 10 q^{86} - 9 q^{88} - 4 q^{89} - 12 q^{92} + 4 q^{94} + 6 q^{95} - 19 q^{97} - 12 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + q^5 - q^7 - 3 * q^8 - q^10 + 9 * q^11 + q^14 + 3 * q^16 - 14 * q^17 + 4 * q^19 + q^20 - 9 * q^22 - 12 * q^23 - 10 * q^25 - q^28 - 11 * q^29 - q^31 - 3 * q^32 + 14 * q^34 - 12 * q^35 + 10 * q^37 - 4 * q^38 - q^40 - 10 * q^41 - 10 * q^43 + 9 * q^44 + 12 * q^46 - 4 * q^47 + 12 * q^49 + 10 * q^50 - 15 * q^53 + 10 * q^55 + q^56 + 11 * q^58 - 9 * q^59 + 2 * q^61 + q^62 + 3 * q^64 + 6 * q^67 - 14 * q^68 + 12 * q^70 - 6 * q^71 + 7 * q^73 - 10 * q^74 + 4 * q^76 - 24 * q^77 + 9 * q^79 + q^80 + 10 * q^82 + 13 * q^83 + 10 * q^86 - 9 * q^88 - 4 * q^89 - 12 * q^92 + 4 * q^94 + 6 * q^95 - 19 * q^97 - 12 * 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
$5$	−1.24698	−0.557666	−0.278833	−	0.960340i	$-0.589948\pi$
$5$	−1.24698	−0.557666	−0.278833	+	0.960340i	$0.589948\pi$
$6$	0	0
$7$	2.93900	1.11084	0.555419	−	0.831571i	$-0.312558\pi$
$7$	2.93900	1.11084	0.555419	+	0.831571i	$0.312558\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.24698	0.394330
$11$	1.30798	0.394370	0.197185	−	0.980366i	$-0.436820\pi$
$11$	1.30798	0.394370	0.197185	+	0.980366i	$0.436820\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−2.93900	−0.785481
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.60388	−1.84421	−0.922105	−	0.386939i	$-0.873532\pi$
$17$	−7.60388	−1.84421	−0.922105	+	0.386939i	$0.873532\pi$
$18$	0	0
$19$	1.10992	0.254632	0.127316	−	0.991862i	$-0.459364\pi$
$19$	1.10992	0.254632	0.127316	+	0.991862i	$0.459364\pi$
$20$	−1.24698	−0.278833
$21$	0	0
$22$	−1.30798	−0.278862
$23$	2.09783	0.437429	0.218714	−	0.975789i	$-0.429814\pi$
$23$	2.09783	0.437429	0.218714	+	0.975789i	$0.429814\pi$
$24$	0	0
$25$	−3.44504	−0.689008
$26$	0	0
$27$	0	0
$28$	2.93900	0.555419
$29$	−2.19806	−0.408170	−0.204085	−	0.978953i	$-0.565422\pi$
$29$	−2.19806	−0.408170	−0.204085	+	0.978953i	$0.565422\pi$
$30$	0	0
$31$	−4.51573	−0.811049	−0.405524	−	0.914084i	$-0.632911\pi$
$31$	−4.51573	−0.811049	−0.405524	+	0.914084i	$0.632911\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	7.60388	1.30405
$35$	−3.66487	−0.619477
$36$	0	0
$37$	9.20775	1.51374	0.756872	−	0.653563i	$-0.226726\pi$
$37$	9.20775	1.51374	0.756872	+	0.653563i	$0.226726\pi$
$38$	−1.10992	−0.180052
$39$	0	0
$40$	1.24698	0.197165
$41$	−9.87800	−1.54268	−0.771342	−	0.636420i	$-0.780414\pi$
$41$	−9.87800	−1.54268	−0.771342	+	0.636420i	$0.780414\pi$
$42$	0	0
$43$	−9.20775	−1.40417	−0.702084	−	0.712094i	$-0.747747\pi$
$43$	−9.20775	−1.40417	−0.702084	+	0.712094i	$0.747747\pi$
$44$	1.30798	0.197185
$45$	0	0
$46$	−2.09783	−0.309309
$47$	7.70171	1.12341	0.561705	−	0.827338i	$-0.310146\pi$
$47$	7.70171	1.12341	0.561705	+	0.827338i	$0.310146\pi$
$48$	0	0
$49$	1.63773	0.233961
$50$	3.44504	0.487202
$51$	0	0
$52$	0	0
$53$	−7.02715	−0.965253	−0.482626	−	0.875826i	$-0.660317\pi$
$53$	−7.02715	−0.965253	−0.482626	+	0.875826i	$0.660317\pi$
$54$	0	0
$55$	−1.63102	−0.219927
$56$	−2.93900	−0.392741
$57$	0	0
$58$	2.19806	0.288620
$59$	−11.4765	−1.49411	−0.747057	−	0.664760i	$-0.768534\pi$
$59$	−11.4765	−1.49411	−0.747057	+	0.664760i	$0.768534\pi$
$60$	0	0
$61$	−5.20775	−0.666784	−0.333392	−	0.942788i	$-0.608193\pi$
$61$	−5.20775	−0.666784	−0.333392	+	0.942788i	$0.608193\pi$
$62$	4.51573	0.573498
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	1.32975	0.162455	0.0812273	−	0.996696i	$-0.474116\pi$
$67$	1.32975	0.162455	0.0812273	+	0.996696i	$0.474116\pi$
$68$	−7.60388	−0.922105
$69$	0	0
$70$	3.66487	0.438036
$71$	13.5797	1.61162	0.805808	−	0.592177i	$-0.201732\pi$
$71$	13.5797	1.61162	0.805808	+	0.592177i	$0.201732\pi$
$72$	0	0
$73$	−1.96077	−0.229491	−0.114745	−	0.993395i	$-0.536605\pi$
$73$	−1.96077	−0.229491	−0.114745	+	0.993395i	$0.536605\pi$
$74$	−9.20775	−1.07038
$75$	0	0
$76$	1.10992	0.127316
$77$	3.84415	0.438082
$78$	0	0
$79$	12.8170	1.44203	0.721013	−	0.692922i	$-0.243677\pi$
$79$	12.8170	1.44203	0.721013	+	0.692922i	$0.243677\pi$
$80$	−1.24698	−0.139417
$81$	0	0
$82$	9.87800	1.09084
$83$	13.2567	1.45511	0.727554	−	0.686050i	$-0.240657\pi$
$83$	13.2567	1.45511	0.727554	+	0.686050i	$0.240657\pi$
$84$	0	0
$85$	9.48188	1.02845
$86$	9.20775	0.992897
$87$	0	0
$88$	−1.30798	−0.139431
$89$	−3.15346	−0.334266	−0.167133	−	0.985934i	$-0.553451\pi$
$89$	−3.15346	−0.334266	−0.167133	+	0.985934i	$0.553451\pi$
$90$	0	0
$91$	0	0
$92$	2.09783	0.218714
$93$	0	0
$94$	−7.70171	−0.794371
$95$	−1.38404	−0.142000
$96$	0	0
$97$	−14.2349	−1.44533	−0.722667	−	0.691196i	$-0.757084\pi$
$97$	−14.2349	−1.44533	−0.722667	+	0.691196i	$0.757084\pi$
$98$	−1.63773	−0.165435
$99$	0	0
$100$	−3.44504	−0.344504
$101$	6.86054	0.682650	0.341325	−	0.939945i	$-0.389124\pi$
$101$	6.86054	0.682650	0.341325	+	0.939945i	$0.389124\pi$
$102$	0	0
$103$	10.5526	1.03978	0.519888	−	0.854235i	$-0.325974\pi$
$103$	10.5526	1.03978	0.519888	+	0.854235i	$0.325974\pi$
$104$	0	0
$105$	0	0
$106$	7.02715	0.682537
$107$	−19.9269	−1.92641	−0.963204	−	0.268771i	$-0.913382\pi$
$107$	−19.9269	−1.92641	−0.963204	+	0.268771i	$0.913382\pi$
$108$	0	0
$109$	−11.5797	−1.10914	−0.554568	−	0.832139i	$-0.687116\pi$
$109$	−11.5797	−1.10914	−0.554568	+	0.832139i	$0.687116\pi$
$110$	1.63102	0.155512
$111$	0	0
$112$	2.93900	0.277709
$113$	−18.9095	−1.77885	−0.889426	−	0.457079i	$-0.848896\pi$
$113$	−18.9095	−1.77885	−0.889426	+	0.457079i	$0.848896\pi$
$114$	0	0
$115$	−2.61596	−0.243939
$116$	−2.19806	−0.204085
$117$	0	0
$118$	11.4765	1.05650
$119$	−22.3478	−2.04862
$120$	0	0
$121$	−9.28919	−0.844472
$122$	5.20775	0.471488
$123$	0	0
$124$	−4.51573	−0.405524
$125$	10.5308	0.941903
$126$	0	0
$127$	9.83877	0.873050	0.436525	−	0.899692i	$-0.356209\pi$
$127$	9.83877	0.873050	0.436525	+	0.899692i	$0.356209\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−1.20344	−0.105145	−0.0525725	−	0.998617i	$-0.516742\pi$
$131$	−1.20344	−0.105145	−0.0525725	+	0.998617i	$0.516742\pi$
$132$	0	0
$133$	3.26205	0.282855
$134$	−1.32975	−0.114873
$135$	0	0
$136$	7.60388	0.652027
$137$	−1.15346	−0.0985465	−0.0492732	−	0.998785i	$-0.515691\pi$
$137$	−1.15346	−0.0985465	−0.0492732	+	0.998785i	$0.515691\pi$
$138$	0	0
$139$	13.2620	1.12487	0.562436	−	0.826841i	$-0.309864\pi$
$139$	13.2620	1.12487	0.562436	+	0.826841i	$0.309864\pi$
$140$	−3.66487	−0.309738
$141$	0	0
$142$	−13.5797	−1.13958
$143$	0	0
$144$	0	0
$145$	2.74094	0.227623
$146$	1.96077	0.162275
$147$	0	0
$148$	9.20775	0.756872
$149$	3.22952	0.264573	0.132286	−	0.991212i	$-0.457768\pi$
$149$	3.22952	0.264573	0.132286	+	0.991212i	$0.457768\pi$
$150$	0	0
$151$	−1.70410	−0.138678	−0.0693390	−	0.997593i	$-0.522089\pi$
$151$	−1.70410	−0.138678	−0.0693390	+	0.997593i	$0.522089\pi$
$152$	−1.10992	−0.0900261
$153$	0	0
$154$	−3.84415	−0.309770
$155$	5.63102	0.452295
$156$	0	0
$157$	−21.3491	−1.70385	−0.851923	−	0.523667i	$-0.824564\pi$
$157$	−21.3491	−1.70385	−0.851923	+	0.523667i	$0.824564\pi$
$158$	−12.8170	−1.01967
$159$	0	0
$160$	1.24698	0.0985824
$161$	6.16554	0.485912
$162$	0	0
$163$	−12.2741	−0.961384	−0.480692	−	0.876890i	$-0.659614\pi$
$163$	−12.2741	−0.961384	−0.480692	+	0.876890i	$0.659614\pi$
$164$	−9.87800	−0.771342
$165$	0	0
$166$	−13.2567	−1.02892
$167$	14.0000	1.08335	0.541676	−	0.840587i	$-0.317790\pi$
$167$	14.0000	1.08335	0.541676	+	0.840587i	$0.317790\pi$
$168$	0	0
$169$	0	0
$170$	−9.48188	−0.727227
$171$	0	0
$172$	−9.20775	−0.702084
$173$	−17.9976	−1.36833	−0.684166	−	0.729326i	$-0.739834\pi$
$173$	−17.9976	−1.36833	−0.684166	+	0.729326i	$0.739834\pi$
$174$	0	0
$175$	−10.1250	−0.765377
$176$	1.30798	0.0985926
$177$	0	0
$178$	3.15346	0.236362
$179$	19.5405	1.46052	0.730262	−	0.683167i	$-0.239398\pi$
$179$	19.5405	1.46052	0.730262	+	0.683167i	$0.239398\pi$
$180$	0	0
$181$	−1.82371	−0.135555	−0.0677776	−	0.997700i	$-0.521591\pi$
$181$	−1.82371	−0.135555	−0.0677776	+	0.997700i	$0.521591\pi$
$182$	0	0
$183$	0	0
$184$	−2.09783	−0.154654
$185$	−11.4819	−0.844164
$186$	0	0
$187$	−9.94571	−0.727302
$188$	7.70171	0.561705
$189$	0	0
$190$	1.38404	0.100409
$191$	−11.7453	−0.849857	−0.424928	−	0.905227i	$-0.639701\pi$
$191$	−11.7453	−0.849857	−0.424928	+	0.905227i	$0.639701\pi$
$192$	0	0
$193$	−0.659498	−0.0474717	−0.0237358	−	0.999718i	$-0.507556\pi$
$193$	−0.659498	−0.0474717	−0.0237358	+	0.999718i	$0.507556\pi$
$194$	14.2349	1.02201
$195$	0	0
$196$	1.63773	0.116981
$197$	−10.6649	−0.759841	−0.379920	−	0.925019i	$-0.624049\pi$
$197$	−10.6649	−0.759841	−0.379920	+	0.925019i	$0.624049\pi$
$198$	0	0
$199$	−17.1075	−1.21272	−0.606360	−	0.795190i	$-0.707371\pi$
$199$	−17.1075	−1.21272	−0.606360	+	0.795190i	$0.707371\pi$
$200$	3.44504	0.243601
$201$	0	0
$202$	−6.86054	−0.482706
$203$	−6.46011	−0.453411
$204$	0	0
$205$	12.3177	0.860303
$206$	−10.5526	−0.735232
$207$	0	0
$208$	0	0
$209$	1.45175	0.100419
$210$	0	0
$211$	7.03146	0.484066	0.242033	−	0.970268i	$-0.422186\pi$
$211$	7.03146	0.484066	0.242033	+	0.970268i	$0.422186\pi$
$212$	−7.02715	−0.482626
$213$	0	0
$214$	19.9269	1.36218
$215$	11.4819	0.783058
$216$	0	0
$217$	−13.2717	−0.900944
$218$	11.5797	0.784277
$219$	0	0
$220$	−1.63102	−0.109964
$221$	0	0
$222$	0	0
$223$	−11.5603	−0.774137	−0.387069	−	0.922051i	$-0.626512\pi$
$223$	−11.5603	−0.774137	−0.387069	+	0.922051i	$0.626512\pi$
$224$	−2.93900	−0.196370
$225$	0	0
$226$	18.9095	1.25784
$227$	−17.2905	−1.14761	−0.573806	−	0.818991i	$-0.694534\pi$
$227$	−17.2905	−1.14761	−0.573806	+	0.818991i	$0.694534\pi$
$228$	0	0
$229$	−9.35988	−0.618518	−0.309259	−	0.950978i	$-0.600081\pi$
$229$	−9.35988	−0.618518	−0.309259	+	0.950978i	$0.600081\pi$
$230$	2.61596	0.172491
$231$	0	0
$232$	2.19806	0.144310
$233$	0.933624	0.0611638	0.0305819	−	0.999532i	$-0.490264\pi$
$233$	0.933624	0.0611638	0.0305819	+	0.999532i	$0.490264\pi$
$234$	0	0
$235$	−9.60388	−0.626488
$236$	−11.4765	−0.747057
$237$	0	0
$238$	22.3478	1.44859
$239$	−16.4698	−1.06534	−0.532671	−	0.846322i	$-0.678812\pi$
$239$	−16.4698	−1.06534	−0.532671	+	0.846322i	$0.678812\pi$
$240$	0	0
$241$	20.8159	1.34087	0.670436	−	0.741967i	$-0.266107\pi$
$241$	20.8159	1.34087	0.670436	+	0.741967i	$0.266107\pi$
$242$	9.28919	0.597132
$243$	0	0
$244$	−5.20775	−0.333392
$245$	−2.04221	−0.130472
$246$	0	0
$247$	0	0
$248$	4.51573	0.286749
$249$	0	0
$250$	−10.5308	−0.666026
$251$	−13.5985	−0.858330	−0.429165	−	0.903226i	$-0.641192\pi$
$251$	−13.5985	−0.858330	−0.429165	+	0.903226i	$0.641192\pi$
$252$	0	0
$253$	2.74392	0.172509
$254$	−9.83877	−0.617340
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.72587	−0.232414	−0.116207	−	0.993225i	$-0.537074\pi$
$257$	−3.72587	−0.232414	−0.116207	+	0.993225i	$0.537074\pi$
$258$	0	0
$259$	27.0616	1.68153
$260$	0	0
$261$	0	0
$262$	1.20344	0.0743487
$263$	13.4034	0.826490	0.413245	−	0.910620i	$-0.364395\pi$
$263$	13.4034	0.826490	0.413245	+	0.910620i	$0.364395\pi$
$264$	0	0
$265$	8.76271	0.538289
$266$	−3.26205	−0.200009
$267$	0	0
$268$	1.32975	0.0812273
$269$	−13.2470	−0.807683	−0.403841	−	0.914829i	$-0.632325\pi$
$269$	−13.2470	−0.807683	−0.403841	+	0.914829i	$0.632325\pi$
$270$	0	0
$271$	−20.5767	−1.24995	−0.624974	−	0.780646i	$-0.714890\pi$
$271$	−20.5767	−1.24995	−0.624974	+	0.780646i	$0.714890\pi$
$272$	−7.60388	−0.461053
$273$	0	0
$274$	1.15346	0.0696829
$275$	−4.50604	−0.271724
$276$	0	0
$277$	15.8431	0.951919	0.475959	−	0.879467i	$-0.342101\pi$
$277$	15.8431	0.951919	0.475959	+	0.879467i	$0.342101\pi$
$278$	−13.2620	−0.795405
$279$	0	0
$280$	3.66487	0.219018
$281$	4.53750	0.270685	0.135342	−	0.990799i	$-0.456787\pi$
$281$	4.53750	0.270685	0.135342	+	0.990799i	$0.456787\pi$
$282$	0	0
$283$	−6.23921	−0.370883	−0.185441	−	0.982655i	$-0.559371\pi$
$283$	−6.23921	−0.370883	−0.185441	+	0.982655i	$0.559371\pi$
$284$	13.5797	0.805808
$285$	0	0
$286$	0	0
$287$	−29.0315	−1.71367
$288$	0	0
$289$	40.8189	2.40111
$290$	−2.74094	−0.160953
$291$	0	0
$292$	−1.96077	−0.114745
$293$	−8.57002	−0.500666	−0.250333	−	0.968160i	$-0.580540\pi$
$293$	−8.57002	−0.500666	−0.250333	+	0.968160i	$0.580540\pi$
$294$	0	0
$295$	14.3110	0.833216
$296$	−9.20775	−0.535190
$297$	0	0
$298$	−3.22952	−0.187081
$299$	0	0
$300$	0	0
$301$	−27.0616	−1.55980
$302$	1.70410	0.0980601
$303$	0	0
$304$	1.10992	0.0636581
$305$	6.49396	0.371843
$306$	0	0
$307$	−28.8659	−1.64747	−0.823733	−	0.566979i	$-0.808112\pi$
$307$	−28.8659	−1.64747	−0.823733	+	0.566979i	$0.808112\pi$
$308$	3.84415	0.219041
$309$	0	0
$310$	−5.63102	−0.319821
$311$	1.38404	0.0784819	0.0392409	−	0.999230i	$-0.487506\pi$
$311$	1.38404	0.0784819	0.0392409	+	0.999230i	$0.487506\pi$
$312$	0	0
$313$	−20.2107	−1.14238	−0.571189	−	0.820818i	$-0.693518\pi$
$313$	−20.2107	−1.14238	−0.571189	+	0.820818i	$0.693518\pi$
$314$	21.3491	1.20480
$315$	0	0
$316$	12.8170	0.721013
$317$	29.2868	1.64491	0.822455	−	0.568830i	$-0.192604\pi$
$317$	29.2868	1.64491	0.822455	+	0.568830i	$0.192604\pi$
$318$	0	0
$319$	−2.87502	−0.160970
$320$	−1.24698	−0.0697083
$321$	0	0
$322$	−6.16554	−0.343592
$323$	−8.43967	−0.469596
$324$	0	0
$325$	0	0
$326$	12.2741	0.679801
$327$	0	0
$328$	9.87800	0.545421
$329$	22.6353	1.24793
$330$	0	0
$331$	−12.4397	−0.683746	−0.341873	−	0.939746i	$-0.611061\pi$
$331$	−12.4397	−0.683746	−0.341873	+	0.939746i	$0.611061\pi$
$332$	13.2567	0.727554
$333$	0	0
$334$	−14.0000	−0.766046
$335$	−1.65817	−0.0905955
$336$	0	0
$337$	−14.3937	−0.784077	−0.392038	−	0.919949i	$-0.628230\pi$
$337$	−14.3937	−0.784077	−0.392038	+	0.919949i	$0.628230\pi$
$338$	0	0
$339$	0	0
$340$	9.48188	0.514227
$341$	−5.90648	−0.319854
$342$	0	0
$343$	−15.7597	−0.850945
$344$	9.20775	0.496449
$345$	0	0
$346$	17.9976	0.967557
$347$	21.7942	1.16997	0.584986	−	0.811043i	$-0.301100\pi$
$347$	21.7942	1.16997	0.584986	+	0.811043i	$0.301100\pi$
$348$	0	0
$349$	26.4698	1.41690	0.708448	−	0.705763i	$-0.249396\pi$
$349$	26.4698	1.41690	0.708448	+	0.705763i	$0.249396\pi$
$350$	10.1250	0.541203
$351$	0	0
$352$	−1.30798	−0.0697155
$353$	15.5060	0.825303	0.412652	−	0.910889i	$-0.364603\pi$
$353$	15.5060	0.825303	0.412652	+	0.910889i	$0.364603\pi$
$354$	0	0
$355$	−16.9336	−0.898743
$356$	−3.15346	−0.167133
$357$	0	0
$358$	−19.5405	−1.03275
$359$	−3.92154	−0.206971	−0.103486	−	0.994631i	$-0.533000\pi$
$359$	−3.92154	−0.206971	−0.103486	+	0.994631i	$0.533000\pi$
$360$	0	0
$361$	−17.7681	−0.935162
$362$	1.82371	0.0958520
$363$	0	0
$364$	0	0
$365$	2.44504	0.127979
$366$	0	0
$367$	−15.3787	−0.802760	−0.401380	−	0.915912i	$-0.631469\pi$
$367$	−15.3787	−0.802760	−0.401380	+	0.915912i	$0.631469\pi$
$368$	2.09783	0.109357
$369$	0	0
$370$	11.4819	0.596914
$371$	−20.6528	−1.07224
$372$	0	0
$373$	20.7138	1.07252	0.536260	−	0.844053i	$-0.319837\pi$
$373$	20.7138	1.07252	0.536260	+	0.844053i	$0.319837\pi$
$374$	9.94571	0.514280
$375$	0	0
$376$	−7.70171	−0.397185
$377$	0	0
$378$	0	0
$379$	0.0193774	0.000995348 0	0.000497674	−	1.00000i	$-0.499842\pi$
$379$	0.0193774	0.000995348 0	0.000497674		1.00000i	$0.499842\pi$
$380$	−1.38404	−0.0709999
$381$	0	0
$382$	11.7453	0.600939
$383$	6.39612	0.326827	0.163413	−	0.986558i	$-0.447750\pi$
$383$	6.39612	0.326827	0.163413	+	0.986558i	$0.447750\pi$
$384$	0	0
$385$	−4.79358	−0.244303
$386$	0.659498	0.0335675
$387$	0	0
$388$	−14.2349	−0.722667
$389$	−32.6088	−1.65333	−0.826665	−	0.562694i	$-0.809765\pi$
$389$	−32.6088	−1.65333	−0.826665	+	0.562694i	$0.809765\pi$
$390$	0	0
$391$	−15.9517	−0.806711
$392$	−1.63773	−0.0827177
$393$	0	0
$394$	10.6649	0.537289
$395$	−15.9825	−0.804169
$396$	0	0
$397$	−1.21254	−0.0608555	−0.0304277	−	0.999537i	$-0.509687\pi$
$397$	−1.21254	−0.0608555	−0.0304277	+	0.999537i	$0.509687\pi$
$398$	17.1075	0.857523
$399$	0	0
$400$	−3.44504	−0.172252
$401$	−4.12200	−0.205843	−0.102921	−	0.994689i	$-0.532819\pi$
$401$	−4.12200	−0.205843	−0.102921	+	0.994689i	$0.532819\pi$
$402$	0	0
$403$	0	0
$404$	6.86054	0.341325
$405$	0	0
$406$	6.46011	0.320610
$407$	12.0435	0.596976
$408$	0	0
$409$	11.8968	0.588258	0.294129	−	0.955766i	$-0.404970\pi$
$409$	11.8968	0.588258	0.294129	+	0.955766i	$0.404970\pi$
$410$	−12.3177	−0.608326
$411$	0	0
$412$	10.5526	0.519888
$413$	−33.7294	−1.65972
$414$	0	0
$415$	−16.5308	−0.811465
$416$	0	0
$417$	0	0
$418$	−1.45175	−0.0710072
$419$	0.824773	0.0402928	0.0201464	−	0.999797i	$-0.493587\pi$
$419$	0.824773	0.0402928	0.0201464	+	0.999797i	$0.493587\pi$
$420$	0	0
$421$	−13.3840	−0.652298	−0.326149	−	0.945318i	$-0.605751\pi$
$421$	−13.3840	−0.652298	−0.326149	+	0.945318i	$0.605751\pi$
$422$	−7.03146	−0.342286
$423$	0	0
$424$	7.02715	0.341268
$425$	26.1957	1.27068
$426$	0	0
$427$	−15.3056	−0.740689
$428$	−19.9269	−0.963204
$429$	0	0
$430$	−11.4819	−0.553705
$431$	29.9517	1.44272	0.721361	−	0.692560i	$-0.243517\pi$
$431$	29.9517	1.44272	0.721361	+	0.692560i	$0.243517\pi$
$432$	0	0
$433$	−11.7342	−0.563911	−0.281956	−	0.959427i	$-0.590983\pi$
$433$	−11.7342	−0.563911	−0.281956	+	0.959427i	$0.590983\pi$
$434$	13.2717	0.637064
$435$	0	0
$436$	−11.5797	−0.554568
$437$	2.32842	0.111383
$438$	0	0
$439$	−3.13467	−0.149610	−0.0748048	−	0.997198i	$-0.523833\pi$
$439$	−3.13467	−0.149610	−0.0748048	+	0.997198i	$0.523833\pi$
$440$	1.63102	0.0777559
$441$	0	0
$442$	0	0
$443$	34.9963	1.66272	0.831362	−	0.555732i	$-0.187562\pi$
$443$	34.9963	1.66272	0.831362	+	0.555732i	$0.187562\pi$
$444$	0	0
$445$	3.93230	0.186409
$446$	11.5603	0.547398
$447$	0	0
$448$	2.93900	0.138855
$449$	4.26337	0.201201	0.100601	−	0.994927i	$-0.467924\pi$
$449$	4.26337	0.201201	0.100601	+	0.994927i	$0.467924\pi$
$450$	0	0
$451$	−12.9202	−0.608389
$452$	−18.9095	−0.889426
$453$	0	0
$454$	17.2905	0.811484
$455$	0	0
$456$	0	0
$457$	−30.7851	−1.44006	−0.720032	−	0.693940i	$-0.755873\pi$
$457$	−30.7851	−1.44006	−0.720032	+	0.693940i	$0.755873\pi$
$458$	9.35988	0.437358
$459$	0	0
$460$	−2.61596	−0.121970
$461$	−6.41013	−0.298549	−0.149275	−	0.988796i	$-0.547694\pi$
$461$	−6.41013	−0.298549	−0.149275	+	0.988796i	$0.547694\pi$
$462$	0	0
$463$	−16.1086	−0.748630	−0.374315	−	0.927302i	$-0.622122\pi$
$463$	−16.1086	−0.748630	−0.374315	+	0.927302i	$0.622122\pi$
$464$	−2.19806	−0.102042
$465$	0	0
$466$	−0.933624	−0.0432493
$467$	3.26636	0.151149	0.0755745	−	0.997140i	$-0.475921\pi$
$467$	3.26636	0.151149	0.0755745	+	0.997140i	$0.475921\pi$
$468$	0	0
$469$	3.90813	0.180461
$470$	9.60388	0.442994
$471$	0	0
$472$	11.4765	0.528249
$473$	−12.0435	−0.553763
$474$	0	0
$475$	−3.82371	−0.175444
$476$	−22.3478	−1.02431
$477$	0	0
$478$	16.4698	0.753311
$479$	13.9409	0.636977	0.318488	−	0.947927i	$-0.396825\pi$
$479$	13.9409	0.636977	0.318488	+	0.947927i	$0.396825\pi$
$480$	0	0
$481$	0	0
$482$	−20.8159	−0.948140
$483$	0	0
$484$	−9.28919	−0.422236
$485$	17.7506	0.806014
$486$	0	0
$487$	−21.2034	−0.960820	−0.480410	−	0.877044i	$-0.659512\pi$
$487$	−21.2034	−0.960820	−0.480410	+	0.877044i	$0.659512\pi$
$488$	5.20775	0.235744
$489$	0	0
$490$	2.04221	0.0922578
$491$	−13.6993	−0.618242	−0.309121	−	0.951023i	$-0.600035\pi$
$491$	−13.6993	−0.618242	−0.309121	+	0.951023i	$0.600035\pi$
$492$	0	0
$493$	16.7138	0.752751
$494$	0	0
$495$	0	0
$496$	−4.51573	−0.202762
$497$	39.9108	1.79024
$498$	0	0
$499$	37.7560	1.69019	0.845095	−	0.534615i	$-0.179544\pi$
$499$	37.7560	1.69019	0.845095	+	0.534615i	$0.179544\pi$
$500$	10.5308	0.470951
$501$	0	0
$502$	13.5985	0.606931
$503$	−3.29696	−0.147004	−0.0735021	−	0.997295i	$-0.523418\pi$
$503$	−3.29696	−0.147004	−0.0735021	+	0.997295i	$0.523418\pi$
$504$	0	0
$505$	−8.55496	−0.380691
$506$	−2.74392	−0.121982
$507$	0	0
$508$	9.83877	0.436525
$509$	32.6112	1.44546	0.722732	−	0.691128i	$-0.242886\pi$
$509$	32.6112	1.44546	0.722732	+	0.691128i	$0.242886\pi$
$510$	0	0
$511$	−5.76271	−0.254927
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	3.72587	0.164341
$515$	−13.1588	−0.579847
$516$	0	0
$517$	10.0737	0.443040
$518$	−27.0616	−1.18902
$519$	0	0
$520$	0	0
$521$	13.2620	0.581021	0.290510	−	0.956872i	$-0.406175\pi$
$521$	13.2620	0.581021	0.290510	+	0.956872i	$0.406175\pi$
$522$	0	0
$523$	−34.1823	−1.49469	−0.747343	−	0.664439i	$-0.768671\pi$
$523$	−34.1823	−1.49469	−0.747343	+	0.664439i	$0.768671\pi$
$524$	−1.20344	−0.0525725
$525$	0	0
$526$	−13.4034	−0.584417
$527$	34.3370	1.49575
$528$	0	0
$529$	−18.5991	−0.808656
$530$	−8.76271	−0.380628
$531$	0	0
$532$	3.26205	0.141428
$533$	0	0
$534$	0	0
$535$	24.8485	1.07429
$536$	−1.32975	−0.0574364
$537$	0	0
$538$	13.2470	0.571118
$539$	2.14211	0.0922673
$540$	0	0
$541$	−10.6789	−0.459121	−0.229560	−	0.973294i	$-0.573729\pi$
$541$	−10.6789	−0.459121	−0.229560	+	0.973294i	$0.573729\pi$
$542$	20.5767	0.883846
$543$	0	0
$544$	7.60388	0.326013
$545$	14.4397	0.618527
$546$	0	0
$547$	27.1293	1.15996	0.579982	−	0.814629i	$-0.303059\pi$
$547$	27.1293	1.15996	0.579982	+	0.814629i	$0.303059\pi$
$548$	−1.15346	−0.0492732
$549$	0	0
$550$	4.50604	0.192138
$551$	−2.43967	−0.103933
$552$	0	0
$553$	37.6692	1.60186
$554$	−15.8431	−0.673108
$555$	0	0
$556$	13.2620	0.562436
$557$	−1.56033	−0.0661135	−0.0330568	−	0.999453i	$-0.510524\pi$
$557$	−1.56033	−0.0661135	−0.0330568	+	0.999453i	$0.510524\pi$
$558$	0	0
$559$	0	0
$560$	−3.66487	−0.154869
$561$	0	0
$562$	−4.53750	−0.191403
$563$	8.16719	0.344206	0.172103	−	0.985079i	$-0.444944\pi$
$563$	8.16719	0.344206	0.172103	+	0.985079i	$0.444944\pi$
$564$	0	0
$565$	23.5797	0.992006
$566$	6.23921	0.262254
$567$	0	0
$568$	−13.5797	−0.569792
$569$	33.7125	1.41330	0.706650	−	0.707563i	$-0.250206\pi$
$569$	33.7125	1.41330	0.706650	+	0.707563i	$0.250206\pi$
$570$	0	0
$571$	18.3913	0.769654	0.384827	−	0.922989i	$-0.374261\pi$
$571$	18.3913	0.769654	0.384827	+	0.922989i	$0.374261\pi$
$572$	0	0
$573$	0	0
$574$	29.0315	1.21175
$575$	−7.22713	−0.301392
$576$	0	0
$577$	−33.4825	−1.39389	−0.696947	−	0.717123i	$-0.745459\pi$
$577$	−33.4825	−1.39389	−0.696947	+	0.717123i	$0.745459\pi$
$578$	−40.8189	−1.69784
$579$	0	0
$580$	2.74094	0.113811
$581$	38.9614	1.61639
$582$	0	0
$583$	−9.19136	−0.380667
$584$	1.96077	0.0811373
$585$	0	0
$586$	8.57002	0.354024
$587$	18.0084	0.743285	0.371642	−	0.928376i	$-0.378795\pi$
$587$	18.0084	0.743285	0.371642	+	0.928376i	$0.378795\pi$
$588$	0	0
$589$	−5.01208	−0.206519
$590$	−14.3110	−0.589173
$591$	0	0
$592$	9.20775	0.378436
$593$	−17.0810	−0.701431	−0.350716	−	0.936482i	$-0.614062\pi$
$593$	−17.0810	−0.701431	−0.350716	+	0.936482i	$0.614062\pi$
$594$	0	0
$595$	27.8672	1.14245
$596$	3.22952	0.132286
$597$	0	0
$598$	0	0
$599$	5.53020	0.225958	0.112979	−	0.993597i	$-0.463961\pi$
$599$	5.53020	0.225958	0.112979	+	0.993597i	$0.463961\pi$
$600$	0	0
$601$	28.1739	1.14924	0.574619	−	0.818421i	$-0.305150\pi$
$601$	28.1739	1.14924	0.574619	+	0.818421i	$0.305150\pi$
$602$	27.0616	1.10295
$603$	0	0
$604$	−1.70410	−0.0693390
$605$	11.5834	0.470934
$606$	0	0
$607$	32.8558	1.33357	0.666787	−	0.745248i	$-0.267669\pi$
$607$	32.8558	1.33357	0.666787	+	0.745248i	$0.267669\pi$
$608$	−1.10992	−0.0450130
$609$	0	0
$610$	−6.49396	−0.262933
$611$	0	0
$612$	0	0
$613$	11.3539	0.458580	0.229290	−	0.973358i	$-0.426360\pi$
$613$	11.3539	0.458580	0.229290	+	0.973358i	$0.426360\pi$
$614$	28.8659	1.16493
$615$	0	0
$616$	−3.84415	−0.154885
$617$	−28.0737	−1.13020	−0.565102	−	0.825021i	$-0.691163\pi$
$617$	−28.0737	−1.13020	−0.565102	+	0.825021i	$0.691163\pi$
$618$	0	0
$619$	−14.0495	−0.564697	−0.282349	−	0.959312i	$-0.591114\pi$
$619$	−14.0495	−0.564697	−0.282349	+	0.959312i	$0.591114\pi$
$620$	5.63102	0.226147
$621$	0	0
$622$	−1.38404	−0.0554951
$623$	−9.26801	−0.371315
$624$	0	0
$625$	4.09352	0.163741
$626$	20.2107	0.807783
$627$	0	0
$628$	−21.3491	−0.851923
$629$	−70.0146	−2.79166
$630$	0	0
$631$	−41.0019	−1.63226	−0.816130	−	0.577868i	$-0.803885\pi$
$631$	−41.0019	−1.63226	−0.816130	+	0.577868i	$0.803885\pi$
$632$	−12.8170	−0.509833
$633$	0	0
$634$	−29.2868	−1.16313
$635$	−12.2687	−0.486870
$636$	0	0
$637$	0	0
$638$	2.87502	0.113823
$639$	0	0
$640$	1.24698	0.0492912
$641$	−13.0616	−0.515902	−0.257951	−	0.966158i	$-0.583047\pi$
$641$	−13.0616	−0.515902	−0.257951	+	0.966158i	$0.583047\pi$
$642$	0	0
$643$	−31.6426	−1.24786	−0.623932	−	0.781479i	$-0.714466\pi$
$643$	−31.6426	−1.24786	−0.623932	+	0.781479i	$0.714466\pi$
$644$	6.16554	0.242956
$645$	0	0
$646$	8.43967	0.332054
$647$	−42.2043	−1.65922	−0.829611	−	0.558342i	$-0.811438\pi$
$647$	−42.2043	−1.65922	−0.829611	+	0.558342i	$0.811438\pi$
$648$	0	0
$649$	−15.0110	−0.589234
$650$	0	0
$651$	0	0
$652$	−12.2741	−0.480692
$653$	35.9463	1.40669	0.703344	−	0.710850i	$-0.251690\pi$
$653$	35.9463	1.40669	0.703344	+	0.710850i	$0.251690\pi$
$654$	0	0
$655$	1.50066	0.0586358
$656$	−9.87800	−0.385671
$657$	0	0
$658$	−22.6353	−0.882417
$659$	33.8049	1.31685	0.658426	−	0.752645i	$-0.271222\pi$
$659$	33.8049	1.31685	0.658426	+	0.752645i	$0.271222\pi$
$660$	0	0
$661$	35.9022	1.39643	0.698216	−	0.715887i	$-0.253977\pi$
$661$	35.9022	1.39643	0.698216	+	0.715887i	$0.253977\pi$
$662$	12.4397	0.483481
$663$	0	0
$664$	−13.2567	−0.514459
$665$	−4.06770	−0.157739
$666$	0	0
$667$	−4.61117	−0.178545
$668$	14.0000	0.541676
$669$	0	0
$670$	1.65817	0.0640607
$671$	−6.81163	−0.262960
$672$	0	0
$673$	−35.3110	−1.36114	−0.680569	−	0.732684i	$-0.738267\pi$
$673$	−35.3110	−1.36114	−0.680569	+	0.732684i	$0.738267\pi$
$674$	14.3937	0.554426
$675$	0	0
$676$	0	0
$677$	23.2239	0.892566	0.446283	−	0.894892i	$-0.352747\pi$
$677$	23.2239	0.892566	0.446283	+	0.894892i	$0.352747\pi$
$678$	0	0
$679$	−41.8364	−1.60553
$680$	−9.48188	−0.363613
$681$	0	0
$682$	5.90648	0.226171
$683$	23.3534	0.893594	0.446797	−	0.894635i	$-0.352565\pi$
$683$	23.3534	0.893594	0.446797	+	0.894635i	$0.352565\pi$
$684$	0	0
$685$	1.43834	0.0549560
$686$	15.7597	0.601709
$687$	0	0
$688$	−9.20775	−0.351042
$689$	0	0
$690$	0	0
$691$	45.2766	1.72240	0.861202	−	0.508262i	$-0.169712\pi$
$691$	45.2766	1.72240	0.861202	+	0.508262i	$0.169712\pi$
$692$	−17.9976	−0.684166
$693$	0	0
$694$	−21.7942	−0.827295
$695$	−16.5375	−0.627303
$696$	0	0
$697$	75.1111	2.84504
$698$	−26.4698	−1.00190
$699$	0	0
$700$	−10.1250	−0.382688
$701$	41.6862	1.57446	0.787232	−	0.616656i	$-0.211513\pi$
$701$	41.6862	1.57446	0.787232	+	0.616656i	$0.211513\pi$
$702$	0	0
$703$	10.2198	0.385448
$704$	1.30798	0.0492963
$705$	0	0
$706$	−15.5060	−0.583578
$707$	20.1631	0.758313
$708$	0	0
$709$	0.0241632	0.000907467 0	0.000453733	−	1.00000i	$-0.499856\pi$
$709$	0.0241632	0.000907467 0	0.000453733		1.00000i	$0.499856\pi$
$710$	16.9336	0.635508
$711$	0	0
$712$	3.15346	0.118181
$713$	−9.47325	−0.354776
$714$	0	0
$715$	0	0
$716$	19.5405	0.730262
$717$	0	0
$718$	3.92154	0.146351
$719$	46.3672	1.72920	0.864602	−	0.502457i	$-0.167571\pi$
$719$	46.3672	1.72920	0.864602	+	0.502457i	$0.167571\pi$
$720$	0	0
$721$	31.0140	1.15502
$722$	17.7681	0.661260
$723$	0	0
$724$	−1.82371	−0.0677776
$725$	7.57242	0.281232
$726$	0	0
$727$	−15.8398	−0.587467	−0.293734	−	0.955887i	$-0.594898\pi$
$727$	−15.8398	−0.587467	−0.293734	+	0.955887i	$0.594898\pi$
$728$	0	0
$729$	0	0
$730$	−2.44504	−0.0904951
$731$	70.0146	2.58958
$732$	0	0
$733$	−4.36121	−0.161085	−0.0805424	−	0.996751i	$-0.525665\pi$
$733$	−4.36121	−0.161085	−0.0805424	+	0.996751i	$0.525665\pi$
$734$	15.3787	0.567637
$735$	0	0
$736$	−2.09783	−0.0773272
$737$	1.73928	0.0640673
$738$	0	0
$739$	24.1366	0.887879	0.443939	−	0.896057i	$-0.353581\pi$
$739$	24.1366	0.887879	0.443939	+	0.896057i	$0.353581\pi$
$740$	−11.4819	−0.422082
$741$	0	0
$742$	20.6528	0.758188
$743$	37.1400	1.36254	0.681268	−	0.732034i	$-0.261429\pi$
$743$	37.1400	1.36254	0.681268	+	0.732034i	$0.261429\pi$
$744$	0	0
$745$	−4.02715	−0.147543
$746$	−20.7138	−0.758386
$747$	0	0
$748$	−9.94571	−0.363651
$749$	−58.5652	−2.13993
$750$	0	0
$751$	29.6558	1.08215	0.541077	−	0.840973i	$-0.318017\pi$
$751$	29.6558	1.08215	0.541077	+	0.840973i	$0.318017\pi$
$752$	7.70171	0.280852
$753$	0	0
$754$	0	0
$755$	2.12498	0.0773360
$756$	0	0
$757$	33.6146	1.22174	0.610872	−	0.791729i	$-0.290819\pi$
$757$	33.6146	1.22174	0.610872	+	0.791729i	$0.290819\pi$
$758$	−0.0193774	−0.000703817 0
$759$	0	0
$760$	1.38404	0.0502045
$761$	35.5749	1.28959	0.644795	−	0.764356i	$-0.276943\pi$
$761$	35.5749	1.28959	0.644795	+	0.764356i	$0.276943\pi$
$762$	0	0
$763$	−34.0328	−1.23207
$764$	−11.7453	−0.424928
$765$	0	0
$766$	−6.39612	−0.231101
$767$	0	0
$768$	0	0
$769$	9.36467	0.337698	0.168849	−	0.985642i	$-0.445995\pi$
$769$	9.36467	0.337698	0.168849	+	0.985642i	$0.445995\pi$
$770$	4.79358	0.172749
$771$	0	0
$772$	−0.659498	−0.0237358
$773$	−12.8364	−0.461693	−0.230846	−	0.972990i	$-0.574149\pi$
$773$	−12.8364	−0.461693	−0.230846	+	0.972990i	$0.574149\pi$
$774$	0	0
$775$	15.5569	0.558820
$776$	14.2349	0.511003
$777$	0	0
$778$	32.6088	1.16908
$779$	−10.9638	−0.392817
$780$	0	0
$781$	17.7620	0.635573
$782$	15.9517	0.570431
$783$	0	0
$784$	1.63773	0.0584903
$785$	26.6219	0.950177
$786$	0	0
$787$	8.80300	0.313793	0.156897	−	0.987615i	$-0.449851\pi$
$787$	8.80300	0.313793	0.156897	+	0.987615i	$0.449851\pi$
$788$	−10.6649	−0.379920
$789$	0	0
$790$	15.9825	0.568633
$791$	−55.5749	−1.97602
$792$	0	0
$793$	0	0
$794$	1.21254	0.0430313
$795$	0	0
$796$	−17.1075	−0.606360
$797$	10.1497	0.359522	0.179761	−	0.983710i	$-0.442468\pi$
$797$	10.1497	0.359522	0.179761	+	0.983710i	$0.442468\pi$
$798$	0	0
$799$	−58.5628	−2.07180
$800$	3.44504	0.121801
$801$	0	0
$802$	4.12200	0.145553
$803$	−2.56465	−0.0905044
$804$	0	0
$805$	−7.68830	−0.270977
$806$	0	0
$807$	0	0
$808$	−6.86054	−0.241353
$809$	−25.4905	−0.896198	−0.448099	−	0.893984i	$-0.647899\pi$
$809$	−25.4905	−0.896198	−0.448099	+	0.893984i	$0.647899\pi$
$810$	0	0
$811$	−5.82371	−0.204498	−0.102249	−	0.994759i	$-0.532604\pi$
$811$	−5.82371	−0.204498	−0.102249	+	0.994759i	$0.532604\pi$
$812$	−6.46011	−0.226705
$813$	0	0
$814$	−12.0435	−0.422126
$815$	15.3056	0.536131
$816$	0	0
$817$	−10.2198	−0.357547
$818$	−11.8968	−0.415961
$819$	0	0
$820$	12.3177	0.430152
$821$	−6.35211	−0.221690	−0.110845	−	0.993838i	$-0.535356\pi$
$821$	−6.35211	−0.221690	−0.110845	+	0.993838i	$0.535356\pi$
$822$	0	0
$823$	19.2808	0.672088	0.336044	−	0.941846i	$-0.390911\pi$
$823$	19.2808	0.672088	0.336044	+	0.941846i	$0.390911\pi$
$824$	−10.5526	−0.367616
$825$	0	0
$826$	33.7294	1.17360
$827$	18.0567	0.627893	0.313946	−	0.949441i	$-0.398349\pi$
$827$	18.0567	0.627893	0.313946	+	0.949441i	$0.398349\pi$
$828$	0	0
$829$	24.2452	0.842070	0.421035	−	0.907044i	$-0.361667\pi$
$829$	24.2452	0.842070	0.421035	+	0.907044i	$0.361667\pi$
$830$	16.5308	0.573792
$831$	0	0
$832$	0	0
$833$	−12.4531	−0.431473
$834$	0	0
$835$	−17.4577	−0.604149
$836$	1.45175	0.0502097
$837$	0	0
$838$	−0.824773	−0.0284913
$839$	−42.5628	−1.46943	−0.734716	−	0.678375i	$-0.762685\pi$
$839$	−42.5628	−1.46943	−0.734716	+	0.678375i	$0.762685\pi$
$840$	0	0
$841$	−24.1685	−0.833397
$842$	13.3840	0.461245
$843$	0	0
$844$	7.03146	0.242033
$845$	0	0
$846$	0	0
$847$	−27.3009	−0.938072
$848$	−7.02715	−0.241313
$849$	0	0
$850$	−26.1957	−0.898504
$851$	19.3163	0.662156
$852$	0	0
$853$	44.9638	1.53953	0.769765	−	0.638328i	$-0.220373\pi$
$853$	44.9638	1.53953	0.769765	+	0.638328i	$0.220373\pi$
$854$	15.3056	0.523746
$855$	0	0
$856$	19.9269	0.681088
$857$	−31.6969	−1.08275	−0.541373	−	0.840782i	$-0.682095\pi$
$857$	−31.6969	−1.08275	−0.541373	+	0.840782i	$0.682095\pi$
$858$	0	0
$859$	4.11124	0.140274	0.0701369	−	0.997537i	$-0.477656\pi$
$859$	4.11124	0.140274	0.0701369	+	0.997537i	$0.477656\pi$
$860$	11.4819	0.391529
$861$	0	0
$862$	−29.9517	−1.02016
$863$	−10.1462	−0.345379	−0.172690	−	0.984976i	$-0.555246\pi$
$863$	−10.1462	−0.345379	−0.172690	+	0.984976i	$0.555246\pi$
$864$	0	0
$865$	22.4426	0.763073
$866$	11.7342	0.398746
$867$	0	0
$868$	−13.2717	−0.450472
$869$	16.7644	0.568692
$870$	0	0
$871$	0	0
$872$	11.5797	0.392139
$873$	0	0
$874$	−2.32842	−0.0787600
$875$	30.9500	1.04630
$876$	0	0
$877$	55.3685	1.86966	0.934831	−	0.355094i	$-0.115551\pi$
$877$	55.3685	1.86966	0.934831	+	0.355094i	$0.115551\pi$
$878$	3.13467	0.105790
$879$	0	0
$880$	−1.63102	−0.0549818
$881$	−54.2016	−1.82610	−0.913050	−	0.407848i	$-0.866279\pi$
$881$	−54.2016	−1.82610	−0.913050	+	0.407848i	$0.866279\pi$
$882$	0	0
$883$	33.8974	1.14074	0.570369	−	0.821389i	$-0.306800\pi$
$883$	33.8974	1.14074	0.570369	+	0.821389i	$0.306800\pi$
$884$	0	0
$885$	0	0
$886$	−34.9963	−1.17572
$887$	−35.3357	−1.18646	−0.593229	−	0.805034i	$-0.702147\pi$
$887$	−35.3357	−1.18646	−0.593229	+	0.805034i	$0.702147\pi$
$888$	0	0
$889$	28.9162	0.969817
$890$	−3.93230	−0.131811
$891$	0	0
$892$	−11.5603	−0.387069
$893$	8.54825	0.286056
$894$	0	0
$895$	−24.3666	−0.814485
$896$	−2.93900	−0.0981851
$897$	0	0
$898$	−4.26337	−0.142271
$899$	9.92585	0.331046
$900$	0	0
$901$	53.4336	1.78013
$902$	12.9202	0.430196
$903$	0	0
$904$	18.9095	0.628919
$905$	2.27413	0.0755945
$906$	0	0
$907$	−24.7198	−0.820806	−0.410403	−	0.911904i	$-0.634612\pi$
$907$	−24.7198	−0.820806	−0.410403	+	0.911904i	$0.634612\pi$
$908$	−17.2905	−0.573806
$909$	0	0
$910$	0	0
$911$	45.2814	1.50024	0.750120	−	0.661301i	$-0.229996\pi$
$911$	45.2814	1.50024	0.750120	+	0.661301i	$0.229996\pi$
$912$	0	0
$913$	17.3394	0.573852
$914$	30.7851	1.01828
$915$	0	0
$916$	−9.35988	−0.309259
$917$	−3.53691	−0.116799
$918$	0	0
$919$	−28.4112	−0.937199	−0.468599	−	0.883411i	$-0.655241\pi$
$919$	−28.4112	−0.937199	−0.468599	+	0.883411i	$0.655241\pi$
$920$	2.61596	0.0862456
$921$	0	0
$922$	6.41013	0.211106
$923$	0	0
$924$	0	0
$925$	−31.7211	−1.04298
$926$	16.1086	0.529361
$927$	0	0
$928$	2.19806	0.0721549
$929$	16.6112	0.544995	0.272497	−	0.962157i	$-0.412150\pi$
$929$	16.6112	0.544995	0.272497	+	0.962157i	$0.412150\pi$
$930$	0	0
$931$	1.81774	0.0595740
$932$	0.933624	0.0305819
$933$	0	0
$934$	−3.26636	−0.106878
$935$	12.4021	0.405592
$936$	0	0
$937$	28.6136	0.934764	0.467382	−	0.884055i	$-0.345197\pi$
$937$	28.6136	0.934764	0.467382	+	0.884055i	$0.345197\pi$
$938$	−3.90813	−0.127605
$939$	0	0
$940$	−9.60388	−0.313244
$941$	1.80194	0.0587415	0.0293707	−	0.999569i	$-0.490650\pi$
$941$	1.80194	0.0587415	0.0293707	+	0.999569i	$0.490650\pi$
$942$	0	0
$943$	−20.7224	−0.674815
$944$	−11.4765	−0.373528
$945$	0	0
$946$	12.0435	0.391569
$947$	30.6698	0.996634	0.498317	−	0.866995i	$-0.333952\pi$
$947$	30.6698	0.996634	0.498317	+	0.866995i	$0.333952\pi$
$948$	0	0
$949$	0	0
$950$	3.82371	0.124057
$951$	0	0
$952$	22.3478	0.724296
$953$	11.4926	0.372283	0.186141	−	0.982523i	$-0.440402\pi$
$953$	11.4926	0.372283	0.186141	+	0.982523i	$0.440402\pi$
$954$	0	0
$955$	14.6461	0.473936
$956$	−16.4698	−0.532671
$957$	0	0
$958$	−13.9409	−0.450411
$959$	−3.39001	−0.109469
$960$	0	0
$961$	−10.6082	−0.342200
$962$	0	0
$963$	0	0
$964$	20.8159	0.670436
$965$	0.822380	0.0264734
$966$	0	0
$967$	−20.3201	−0.653449	−0.326725	−	0.945120i	$-0.605945\pi$
$967$	−20.3201	−0.653449	−0.326725	+	0.945120i	$0.605945\pi$
$968$	9.28919	0.298566
$969$	0	0
$970$	−17.7506	−0.569938
$971$	−38.7523	−1.24362	−0.621810	−	0.783168i	$-0.713602\pi$
$971$	−38.7523	−1.24362	−0.621810	+	0.783168i	$0.713602\pi$
$972$	0	0
$973$	38.9772	1.24955
$974$	21.2034	0.679402
$975$	0	0
$976$	−5.20775	−0.166696
$977$	36.6655	1.17303	0.586516	−	0.809938i	$-0.300499\pi$
$977$	36.6655	1.17303	0.586516	+	0.809938i	$0.300499\pi$
$978$	0	0
$979$	−4.12465	−0.131825
$980$	−2.04221	−0.0652361
$981$	0	0
$982$	13.6993	0.437163
$983$	20.2586	0.646149	0.323074	−	0.946374i	$-0.395284\pi$
$983$	20.2586	0.646149	0.323074	+	0.946374i	$0.395284\pi$
$984$	0	0
$985$	13.2989	0.423738
$986$	−16.7138	−0.532276
$987$	0	0
$988$	0	0
$989$	−19.3163	−0.614224
$990$	0	0
$991$	−35.7362	−1.13520	−0.567598	−	0.823306i	$-0.692127\pi$
$991$	−35.7362	−1.13520	−0.567598	+	0.823306i	$0.692127\pi$
$992$	4.51573	0.143375
$993$	0	0
$994$	−39.9108	−1.26589
$995$	21.3327	0.676293
$996$	0	0
$997$	20.0194	0.634020	0.317010	−	0.948422i	$-0.397321\pi$
$997$	20.0194	0.634020	0.317010	+	0.948422i	$0.397321\pi$
$998$	−37.7560	−1.19515
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.a.bb.1.1	✓	3
3.2	odd	2		3042.2.a.bf.1.3	yes	3
13.5	odd	4		3042.2.b.q.1351.6		6
13.8	odd	4		3042.2.b.q.1351.1		6
13.12	even	2		3042.2.a.bg.1.3	yes	3
39.5	even	4		3042.2.b.p.1351.1		6
39.8	even	4		3042.2.b.p.1351.6		6
39.38	odd	2		3042.2.a.bc.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3042.2.a.bb.1.1	✓	3	1.1	even	1	trivial
3042.2.a.bc.1.1	yes	3	39.38	odd	2
3042.2.a.bf.1.3	yes	3	3.2	odd	2
3042.2.a.bg.1.3	yes	3	13.12	even	2
3042.2.b.p.1351.1		6	39.5	even	4
3042.2.b.p.1351.6		6	39.8	even	4
3042.2.b.q.1351.1		6	13.8	odd	4
3042.2.b.q.1351.6		6	13.5	odd	4

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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.24698 q^{5} +2.93900 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.24698 q^{5} +2.93900 q^{7} -1.00000 q^{8} +1.24698 q^{10} +1.30798 q^{11} -2.93900 q^{14} +1.00000 q^{16} -7.60388 q^{17} +1.10992 q^{19} -1.24698 q^{20} -1.30798 q^{22} +2.09783 q^{23} -3.44504 q^{25} +2.93900 q^{28} -2.19806 q^{29} -4.51573 q^{31} -1.00000 q^{32} +7.60388 q^{34} -3.66487 q^{35} +9.20775 q^{37} -1.10992 q^{38} +1.24698 q^{40} -9.87800 q^{41} -9.20775 q^{43} +1.30798 q^{44} -2.09783 q^{46} +7.70171 q^{47} +1.63773 q^{49} +3.44504 q^{50} -7.02715 q^{53} -1.63102 q^{55} -2.93900 q^{56} +2.19806 q^{58} -11.4765 q^{59} -5.20775 q^{61} +4.51573 q^{62} +1.00000 q^{64} +1.32975 q^{67} -7.60388 q^{68} +3.66487 q^{70} +13.5797 q^{71} -1.96077 q^{73} -9.20775 q^{74} +1.10992 q^{76} +3.84415 q^{77} +12.8170 q^{79} -1.24698 q^{80} +9.87800 q^{82} +13.2567 q^{83} +9.48188 q^{85} +9.20775 q^{86} -1.30798 q^{88} -3.15346 q^{89} +2.09783 q^{92} -7.70171 q^{94} -1.38404 q^{95} -14.2349 q^{97} -1.63773 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + q^{5} - q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 + q^5 - q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} + q^{5} - q^{7} - 3 q^{8} - q^{10} + 9 q^{11} + q^{14} + 3 q^{16} - 14 q^{17} + 4 q^{19} + q^{20} - 9 q^{22} - 12 q^{23} - 10 q^{25} - q^{28} - 11 q^{29} - q^{31} - 3 q^{32} + 14 q^{34} - 12 q^{35} + 10 q^{37} - 4 q^{38} - q^{40} - 10 q^{41} - 10 q^{43} + 9 q^{44} + 12 q^{46} - 4 q^{47} + 12 q^{49} + 10 q^{50} - 15 q^{53} + 10 q^{55} + q^{56} + 11 q^{58} - 9 q^{59} + 2 q^{61} + q^{62} + 3 q^{64} + 6 q^{67} - 14 q^{68} + 12 q^{70} - 6 q^{71} + 7 q^{73} - 10 q^{74} + 4 q^{76} - 24 q^{77} + 9 q^{79} + q^{80} + 10 q^{82} + 13 q^{83} + 10 q^{86} - 9 q^{88} - 4 q^{89} - 12 q^{92} + 4 q^{94} + 6 q^{95} - 19 q^{97} - 12 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + q^5 - q^7 - 3 * q^8 - q^10 + 9 * q^11 + q^14 + 3 * q^16 - 14 * q^17 + 4 * q^19 + q^20 - 9 * q^22 - 12 * q^23 - 10 * q^25 - q^28 - 11 * q^29 - q^31 - 3 * q^32 + 14 * q^34 - 12 * q^35 + 10 * q^37 - 4 * q^38 - q^40 - 10 * q^41 - 10 * q^43 + 9 * q^44 + 12 * q^46 - 4 * q^47 + 12 * q^49 + 10 * q^50 - 15 * q^53 + 10 * q^55 + q^56 + 11 * q^58 - 9 * q^59 + 2 * q^61 + q^62 + 3 * q^64 + 6 * q^67 - 14 * q^68 + 12 * q^70 - 6 * q^71 + 7 * q^73 - 10 * q^74 + 4 * q^76 - 24 * q^77 + 9 * q^79 + q^80 + 10 * q^82 + 13 * q^83 + 10 * q^86 - 9 * q^88 - 4 * q^89 - 12 * q^92 + 4 * q^94 + 6 * q^95 - 19 * q^97 - 12 * q^98

Introduction

L-functions

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