LMFDB - Embedded newform 6039.2.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, 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("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.50682 q^{2} +4.28416 q^{4} -0.826559 q^{5} +1.21549 q^{7} -5.72598 q^{8} +O(q^{10})$	\(q-2.50682 q^{2} +4.28416 q^{4} -0.826559 q^{5} +1.21549 q^{7} -5.72598 q^{8} +2.07204 q^{10} -1.00000 q^{11} +0.401045 q^{13} -3.04702 q^{14} +5.78570 q^{16} -0.390322 q^{17} +1.49006 q^{19} -3.54111 q^{20} +2.50682 q^{22} -3.92065 q^{23} -4.31680 q^{25} -1.00535 q^{26} +5.20736 q^{28} -3.72679 q^{29} +10.1742 q^{31} -3.05175 q^{32} +0.978467 q^{34} -1.00468 q^{35} +3.36372 q^{37} -3.73531 q^{38} +4.73286 q^{40} +4.81820 q^{41} -5.55772 q^{43} -4.28416 q^{44} +9.82837 q^{46} +6.69663 q^{47} -5.52258 q^{49} +10.8214 q^{50} +1.71814 q^{52} -2.13343 q^{53} +0.826559 q^{55} -6.95988 q^{56} +9.34240 q^{58} -8.70387 q^{59} -1.00000 q^{61} -25.5050 q^{62} -3.92119 q^{64} -0.331488 q^{65} +9.49864 q^{67} -1.67220 q^{68} +2.51855 q^{70} -5.35546 q^{71} +8.27881 q^{73} -8.43226 q^{74} +6.38364 q^{76} -1.21549 q^{77} -7.91582 q^{79} -4.78222 q^{80} -12.0784 q^{82} +5.56970 q^{83} +0.322624 q^{85} +13.9322 q^{86} +5.72598 q^{88} -0.229219 q^{89} +0.487467 q^{91} -16.7967 q^{92} -16.7873 q^{94} -1.23162 q^{95} -12.2449 q^{97} +13.8441 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8}+O(q^{10}) $ 25 * q - 5 * q^2 + 25 * q^4 - 4 * q^5 + 4 * q^7 - 15 * q^8	$ 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8} - 25 q^{11} + 4 q^{13} - 18 q^{14} + 21 q^{16} - 20 q^{17} + 14 q^{19} - 12 q^{20} + 5 q^{22} - 20 q^{23} + 13 q^{25} - 16 q^{26} - 14 q^{28} - 28 q^{29} - 12 q^{31} - 35 q^{32} + 6 q^{34} - 10 q^{35} - 8 q^{37} - 32 q^{38} + 24 q^{40} - 26 q^{41} + 18 q^{43} - 25 q^{44} + 4 q^{46} - 12 q^{47} + 23 q^{49} - 43 q^{50} + 22 q^{52} - 36 q^{53} + 4 q^{55} - 26 q^{56} - 20 q^{58} - 46 q^{59} - 25 q^{61} + 14 q^{62} - 13 q^{64} - 60 q^{65} - 20 q^{67} - 44 q^{68} - 20 q^{70} - 52 q^{71} + 6 q^{73} - 32 q^{74} - 4 q^{77} + 26 q^{79} - 52 q^{80} + 6 q^{82} - 38 q^{83} - 4 q^{85} - 34 q^{86} + 15 q^{88} - 82 q^{89} - 58 q^{91} - 36 q^{92} + 16 q^{94} - 30 q^{95} - 35 q^{98}+O(q^{100}) $ 25 * q - 5 * q^2 + 25 * q^4 - 4 * q^5 + 4 * q^7 - 15 * q^8 - 25 * q^11 + 4 * q^13 - 18 * q^14 + 21 * q^16 - 20 * q^17 + 14 * q^19 - 12 * q^20 + 5 * q^22 - 20 * q^23 + 13 * q^25 - 16 * q^26 - 14 * q^28 - 28 * q^29 - 12 * q^31 - 35 * q^32 + 6 * q^34 - 10 * q^35 - 8 * q^37 - 32 * q^38 + 24 * q^40 - 26 * q^41 + 18 * q^43 - 25 * q^44 + 4 * q^46 - 12 * q^47 + 23 * q^49 - 43 * q^50 + 22 * q^52 - 36 * q^53 + 4 * q^55 - 26 * q^56 - 20 * q^58 - 46 * q^59 - 25 * q^61 + 14 * q^62 - 13 * q^64 - 60 * q^65 - 20 * q^67 - 44 * q^68 - 20 * q^70 - 52 * q^71 + 6 * q^73 - 32 * q^74 - 4 * q^77 + 26 * q^79 - 52 * q^80 + 6 * q^82 - 38 * q^83 - 4 * q^85 - 34 * q^86 + 15 * q^88 - 82 * q^89 - 58 * q^91 - 36 * q^92 + 16 * q^94 - 30 * q^95 - 35 * 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 $

$2$	−2.50682	−1.77259	−0.886296	−	0.463120i	$-0.846730\pi$
$2$	−2.50682	−1.77259	−0.886296	+	0.463120i	$0.846730\pi$
$3$	0	0
$3$	0	0
$4$	4.28416	2.14208
$5$	−0.826559	−0.369649	−0.184824	−	0.982772i	$-0.559172\pi$
$5$	−0.826559	−0.369649	−0.184824	+	0.982772i	$0.559172\pi$
$6$	0	0
$7$	1.21549	0.459413	0.229706	−	0.973260i	$-0.426223\pi$
$7$	1.21549	0.459413	0.229706	+	0.973260i	$0.426223\pi$
$8$	−5.72598	−2.02444
$9$	0	0
$10$	2.07204	0.655236
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.401045	0.111230	0.0556150	−	0.998452i	$-0.482288\pi$
$13$	0.401045	0.111230	0.0556150	+	0.998452i	$0.482288\pi$
$14$	−3.04702	−0.814351
$15$	0	0
$16$	5.78570	1.44642
$17$	−0.390322	−0.0946669	−0.0473334	−	0.998879i	$-0.515072\pi$
$17$	−0.390322	−0.0946669	−0.0473334	+	0.998879i	$0.515072\pi$
$18$	0	0
$19$	1.49006	0.341842	0.170921	−	0.985285i	$-0.445326\pi$
$19$	1.49006	0.341842	0.170921	+	0.985285i	$0.445326\pi$
$20$	−3.54111	−0.791817
$21$	0	0
$22$	2.50682	0.534456
$23$	−3.92065	−0.817512	−0.408756	−	0.912644i	$-0.634037\pi$
$23$	−3.92065	−0.817512	−0.408756	+	0.912644i	$0.634037\pi$
$24$	0	0
$25$	−4.31680	−0.863360
$26$	−1.00535	−0.197165
$27$	0	0
$28$	5.20736	0.984099
$29$	−3.72679	−0.692047	−0.346024	−	0.938226i	$-0.612468\pi$
$29$	−3.72679	−0.692047	−0.346024	+	0.938226i	$0.612468\pi$
$30$	0	0
$31$	10.1742	1.82735	0.913674	−	0.406448i	$-0.133233\pi$
$31$	10.1742	1.82735	0.913674	+	0.406448i	$0.133233\pi$
$32$	−3.05175	−0.539479
$33$	0	0
$34$	0.978467	0.167806
$35$	−1.00468	−0.169821
$36$	0	0
$37$	3.36372	0.552993	0.276496	−	0.961015i	$-0.410827\pi$
$37$	3.36372	0.552993	0.276496	+	0.961015i	$0.410827\pi$
$38$	−3.73531	−0.605947
$39$	0	0
$40$	4.73286	0.748331
$41$	4.81820	0.752477	0.376238	−	0.926523i	$-0.377217\pi$
$41$	4.81820	0.752477	0.376238	+	0.926523i	$0.377217\pi$
$42$	0	0
$43$	−5.55772	−0.847545	−0.423772	−	0.905769i	$-0.639294\pi$
$43$	−5.55772	−0.847545	−0.423772	+	0.905769i	$0.639294\pi$
$44$	−4.28416	−0.645861
$45$	0	0
$46$	9.82837	1.44911
$47$	6.69663	0.976804	0.488402	−	0.872619i	$-0.337580\pi$
$47$	6.69663	0.976804	0.488402	+	0.872619i	$0.337580\pi$
$48$	0	0
$49$	−5.52258	−0.788940
$50$	10.8214	1.53038
$51$	0	0
$52$	1.71814	0.238263
$53$	−2.13343	−0.293049	−0.146525	−	0.989207i	$-0.546809\pi$
$53$	−2.13343	−0.293049	−0.146525	+	0.989207i	$0.546809\pi$
$54$	0	0
$55$	0.826559	0.111453
$56$	−6.95988	−0.930053
$57$	0	0
$58$	9.34240	1.22672
$59$	−8.70387	−1.13315	−0.566574	−	0.824011i	$-0.691731\pi$
$59$	−8.70387	−1.13315	−0.566574	+	0.824011i	$0.691731\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−25.5050	−3.23914
$63$	0	0
$64$	−3.92119	−0.490149
$65$	−0.331488	−0.0411160
$66$	0	0
$67$	9.49864	1.16044	0.580222	−	0.814458i	$-0.302966\pi$
$67$	9.49864	1.16044	0.580222	+	0.814458i	$0.302966\pi$
$68$	−1.67220	−0.202784
$69$	0	0
$70$	2.51855	0.301024
$71$	−5.35546	−0.635576	−0.317788	−	0.948162i	$-0.602940\pi$
$71$	−5.35546	−0.635576	−0.317788	+	0.948162i	$0.602940\pi$
$72$	0	0
$73$	8.27881	0.968962	0.484481	−	0.874802i	$-0.339009\pi$
$73$	8.27881	0.968962	0.484481	+	0.874802i	$0.339009\pi$
$74$	−8.43226	−0.980230
$75$	0	0
$76$	6.38364	0.732253
$77$	−1.21549	−0.138518
$78$	0	0
$79$	−7.91582	−0.890599	−0.445299	−	0.895382i	$-0.646903\pi$
$79$	−7.91582	−0.890599	−0.445299	+	0.895382i	$0.646903\pi$
$80$	−4.78222	−0.534669
$81$	0	0
$82$	−12.0784	−1.33383
$83$	5.56970	0.611354	0.305677	−	0.952135i	$-0.401117\pi$
$83$	5.56970	0.611354	0.305677	+	0.952135i	$0.401117\pi$
$84$	0	0
$85$	0.322624	0.0349935
$86$	13.9322	1.50235
$87$	0	0
$88$	5.72598	0.610391
$89$	−0.229219	−0.0242971	−0.0121486	−	0.999926i	$-0.503867\pi$
$89$	−0.229219	−0.0242971	−0.0121486	+	0.999926i	$0.503867\pi$
$90$	0	0
$91$	0.487467	0.0511004
$92$	−16.7967	−1.75118
$93$	0	0
$94$	−16.7873	−1.73147
$95$	−1.23162	−0.126362
$96$	0	0
$97$	−12.2449	−1.24328	−0.621642	−	0.783302i	$-0.713534\pi$
$97$	−12.2449	−1.24328	−0.621642	+	0.783302i	$0.713534\pi$
$98$	13.8441	1.39847
$99$	0	0
$100$	−18.4939	−1.84939
$101$	−1.26748	−0.126119	−0.0630594	−	0.998010i	$-0.520086\pi$
$101$	−1.26748	−0.126119	−0.0630594	+	0.998010i	$0.520086\pi$
$102$	0	0
$103$	−0.0569065	−0.00560717	−0.00280358	−	0.999996i	$-0.500892\pi$
$103$	−0.0569065	−0.00560717	−0.00280358	+	0.999996i	$0.500892\pi$
$104$	−2.29638	−0.225178
$105$	0	0
$106$	5.34813	0.519456
$107$	−0.103235	−0.00998013	−0.00499006	−	0.999988i	$-0.501588\pi$
$107$	−0.103235	−0.00998013	−0.00499006	+	0.999988i	$0.501588\pi$
$108$	0	0
$109$	19.3868	1.85692	0.928459	−	0.371434i	$-0.121134\pi$
$109$	19.3868	1.85692	0.928459	+	0.371434i	$0.121134\pi$
$110$	−2.07204	−0.197561
$111$	0	0
$112$	7.03247	0.664506
$113$	9.88109	0.929535	0.464768	−	0.885433i	$-0.346138\pi$
$113$	9.88109	0.929535	0.464768	+	0.885433i	$0.346138\pi$
$114$	0	0
$115$	3.24065	0.302192
$116$	−15.9662	−1.48242
$117$	0	0
$118$	21.8191	2.00861
$119$	−0.474433	−0.0434912
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.50682	0.226957
$123$	0	0
$124$	43.5881	3.91432
$125$	7.70089	0.688788
$126$	0	0
$127$	−11.2475	−0.998051	−0.499025	−	0.866587i	$-0.666309\pi$
$127$	−11.2475	−0.998051	−0.499025	+	0.866587i	$0.666309\pi$
$128$	15.9332	1.40831
$129$	0	0
$130$	0.830981	0.0728818
$131$	−20.7248	−1.81073	−0.905365	−	0.424634i	$-0.860403\pi$
$131$	−20.7248	−1.81073	−0.905365	+	0.424634i	$0.860403\pi$
$132$	0	0
$133$	1.81115	0.157047
$134$	−23.8114	−2.05699
$135$	0	0
$136$	2.23497	0.191647
$137$	−7.63380	−0.652199	−0.326100	−	0.945335i	$-0.605734\pi$
$137$	−7.63380	−0.652199	−0.326100	+	0.945335i	$0.605734\pi$
$138$	0	0
$139$	−9.69942	−0.822694	−0.411347	−	0.911479i	$-0.634942\pi$
$139$	−9.69942	−0.822694	−0.411347	+	0.911479i	$0.634942\pi$
$140$	−4.30419	−0.363771
$141$	0	0
$142$	13.4252	1.12662
$143$	−0.401045	−0.0335371
$144$	0	0
$145$	3.08041	0.255814
$146$	−20.7535	−1.71757
$147$	0	0
$148$	14.4107	1.18455
$149$	−7.23521	−0.592731	−0.296366	−	0.955075i	$-0.595775\pi$
$149$	−7.23521	−0.592731	−0.296366	+	0.955075i	$0.595775\pi$
$150$	0	0
$151$	0.321867	0.0261932	0.0130966	−	0.999914i	$-0.495831\pi$
$151$	0.321867	0.0261932	0.0130966	+	0.999914i	$0.495831\pi$
$152$	−8.53203	−0.692039
$153$	0	0
$154$	3.04702	0.245536
$155$	−8.40962	−0.675477
$156$	0	0
$157$	−16.6252	−1.32684	−0.663418	−	0.748249i	$-0.730895\pi$
$157$	−16.6252	−1.32684	−0.663418	+	0.748249i	$0.730895\pi$
$158$	19.8435	1.57867
$159$	0	0
$160$	2.52246	0.199418
$161$	−4.76552	−0.375575
$162$	0	0
$163$	−2.81263	−0.220303	−0.110151	−	0.993915i	$-0.535134\pi$
$163$	−2.81263	−0.220303	−0.110151	+	0.993915i	$0.535134\pi$
$164$	20.6419	1.61187
$165$	0	0
$166$	−13.9622	−1.08368
$167$	−16.6313	−1.28697	−0.643486	−	0.765458i	$-0.722513\pi$
$167$	−16.6313	−1.28697	−0.643486	+	0.765458i	$0.722513\pi$
$168$	0	0
$169$	−12.8392	−0.987628
$170$	−0.808761	−0.0620291
$171$	0	0
$172$	−23.8102	−1.81551
$173$	12.7008	0.965626	0.482813	−	0.875723i	$-0.339615\pi$
$173$	12.7008	0.965626	0.482813	+	0.875723i	$0.339615\pi$
$174$	0	0
$175$	−5.24704	−0.396639
$176$	−5.78570	−0.436113
$177$	0	0
$178$	0.574611	0.0430689
$179$	9.25419	0.691691	0.345845	−	0.938291i	$-0.387592\pi$
$179$	9.25419	0.691691	0.345845	+	0.938291i	$0.387592\pi$
$180$	0	0
$181$	−8.39347	−0.623882	−0.311941	−	0.950101i	$-0.600979\pi$
$181$	−8.39347	−0.623882	−0.311941	+	0.950101i	$0.600979\pi$
$182$	−1.22199	−0.0905802
$183$	0	0
$184$	22.4496	1.65500
$185$	−2.78032	−0.204413
$186$	0	0
$187$	0.390322	0.0285431
$188$	28.6894	2.09239
$189$	0	0
$190$	3.08745	0.223987
$191$	−7.57895	−0.548394	−0.274197	−	0.961674i	$-0.588412\pi$
$191$	−7.57895	−0.548394	−0.274197	+	0.961674i	$0.588412\pi$
$192$	0	0
$193$	12.8053	0.921743	0.460871	−	0.887467i	$-0.347537\pi$
$193$	12.8053	0.921743	0.460871	+	0.887467i	$0.347537\pi$
$194$	30.6959	2.20383
$195$	0	0
$196$	−23.6596	−1.68997
$197$	15.9928	1.13944	0.569718	−	0.821840i	$-0.307052\pi$
$197$	15.9928	1.13944	0.569718	+	0.821840i	$0.307052\pi$
$198$	0	0
$199$	9.61480	0.681575	0.340788	−	0.940140i	$-0.389306\pi$
$199$	9.61480	0.681575	0.340788	+	0.940140i	$0.389306\pi$
$200$	24.7179	1.74782
$201$	0	0
$202$	3.17734	0.223557
$203$	−4.52988	−0.317935
$204$	0	0
$205$	−3.98253	−0.278152
$206$	0.142655	0.00993921
$207$	0	0
$208$	2.32033	0.160886
$209$	−1.49006	−0.103069
$210$	0	0
$211$	3.09976	0.213396	0.106698	−	0.994291i	$-0.465972\pi$
$211$	3.09976	0.213396	0.106698	+	0.994291i	$0.465972\pi$
$212$	−9.13995	−0.627735
$213$	0	0
$214$	0.258792	0.0176907
$215$	4.59379	0.313294
$216$	0	0
$217$	12.3667	0.839507
$218$	−48.5993	−3.29156
$219$	0	0
$220$	3.54111	0.238742
$221$	−0.156537	−0.0105298
$222$	0	0
$223$	−23.6245	−1.58202	−0.791008	−	0.611806i	$-0.790443\pi$
$223$	−23.6245	−1.58202	−0.791008	+	0.611806i	$0.790443\pi$
$224$	−3.70938	−0.247843
$225$	0	0
$226$	−24.7701	−1.64769
$227$	5.37605	0.356821	0.178410	−	0.983956i	$-0.442905\pi$
$227$	5.37605	0.356821	0.178410	+	0.983956i	$0.442905\pi$
$228$	0	0
$229$	−20.2352	−1.33718	−0.668590	−	0.743632i	$-0.733102\pi$
$229$	−20.2352	−1.33718	−0.668590	+	0.743632i	$0.733102\pi$
$230$	−8.12373	−0.535663
$231$	0	0
$232$	21.3395	1.40101
$233$	−8.53037	−0.558843	−0.279421	−	0.960169i	$-0.590143\pi$
$233$	−8.53037	−0.558843	−0.279421	+	0.960169i	$0.590143\pi$
$234$	0	0
$235$	−5.53516	−0.361074
$236$	−37.2888	−2.42729
$237$	0	0
$238$	1.18932	0.0770921
$239$	2.40770	0.155741	0.0778705	−	0.996963i	$-0.475188\pi$
$239$	2.40770	0.155741	0.0778705	+	0.996963i	$0.475188\pi$
$240$	0	0
$241$	13.5193	0.870853	0.435427	−	0.900224i	$-0.356597\pi$
$241$	13.5193	0.870853	0.435427	+	0.900224i	$0.356597\pi$
$242$	−2.50682	−0.161145
$243$	0	0
$244$	−4.28416	−0.274265
$245$	4.56474	0.291631
$246$	0	0
$247$	0.597580	0.0380231
$248$	−58.2575	−3.69936
$249$	0	0
$250$	−19.3048	−1.22094
$251$	13.3095	0.840087	0.420044	−	0.907504i	$-0.362015\pi$
$251$	13.3095	0.840087	0.420044	+	0.907504i	$0.362015\pi$
$252$	0	0
$253$	3.92065	0.246489
$254$	28.1954	1.76914
$255$	0	0
$256$	−32.0994	−2.00621
$257$	26.2777	1.63916	0.819579	−	0.572966i	$-0.194207\pi$
$257$	26.2777	1.63916	0.819579	+	0.572966i	$0.194207\pi$
$258$	0	0
$259$	4.08858	0.254052
$260$	−1.42015	−0.0880737
$261$	0	0
$262$	51.9533	3.20968
$263$	1.54892	0.0955102	0.0477551	−	0.998859i	$-0.484793\pi$
$263$	1.54892	0.0955102	0.0477551	+	0.998859i	$0.484793\pi$
$264$	0	0
$265$	1.76341	0.108325
$266$	−4.54023	−0.278380
$267$	0	0
$268$	40.6937	2.48576
$269$	−20.5975	−1.25585	−0.627925	−	0.778274i	$-0.716095\pi$
$269$	−20.5975	−1.25585	−0.627925	+	0.778274i	$0.716095\pi$
$270$	0	0
$271$	−8.51731	−0.517390	−0.258695	−	0.965959i	$-0.583292\pi$
$271$	−8.51731	−0.517390	−0.258695	+	0.965959i	$0.583292\pi$
$272$	−2.25828	−0.136928
$273$	0	0
$274$	19.1366	1.15608
$275$	4.31680	0.260313
$276$	0	0
$277$	−2.38184	−0.143111	−0.0715554	−	0.997437i	$-0.522796\pi$
$277$	−2.38184	−0.143111	−0.0715554	+	0.997437i	$0.522796\pi$
$278$	24.3147	1.45830
$279$	0	0
$280$	5.75276	0.343793
$281$	30.3270	1.80916	0.904579	−	0.426307i	$-0.140186\pi$
$281$	30.3270	1.80916	0.904579	+	0.426307i	$0.140186\pi$
$282$	0	0
$283$	26.2869	1.56259	0.781296	−	0.624160i	$-0.214559\pi$
$283$	26.2869	1.56259	0.781296	+	0.624160i	$0.214559\pi$
$284$	−22.9436	−1.36145
$285$	0	0
$286$	1.00535	0.0594475
$287$	5.85649	0.345698
$288$	0	0
$289$	−16.8476	−0.991038
$290$	−7.72205	−0.453454
$291$	0	0
$292$	35.4677	2.07559
$293$	10.3905	0.607021	0.303511	−	0.952828i	$-0.401841\pi$
$293$	10.3905	0.607021	0.303511	+	0.952828i	$0.401841\pi$
$294$	0	0
$295$	7.19427	0.418867
$296$	−19.2606	−1.11950
$297$	0	0
$298$	18.1374	1.05067
$299$	−1.57236	−0.0909318
$300$	0	0
$301$	−6.75537	−0.389373
$302$	−0.806864	−0.0464298
$303$	0	0
$304$	8.62101	0.494449
$305$	0.826559	0.0473287
$306$	0	0
$307$	−15.8564	−0.904974	−0.452487	−	0.891771i	$-0.649463\pi$
$307$	−15.8564	−0.904974	−0.452487	+	0.891771i	$0.649463\pi$
$308$	−5.20736	−0.296717
$309$	0	0
$310$	21.0814	1.19734
$311$	−6.94023	−0.393545	−0.196772	−	0.980449i	$-0.563046\pi$
$311$	−6.94023	−0.393545	−0.196772	+	0.980449i	$0.563046\pi$
$312$	0	0
$313$	−29.2282	−1.65207	−0.826037	−	0.563616i	$-0.809410\pi$
$313$	−29.2282	−1.65207	−0.826037	+	0.563616i	$0.809410\pi$
$314$	41.6765	2.35194
$315$	0	0
$316$	−33.9126	−1.90773
$317$	−31.4196	−1.76470	−0.882350	−	0.470593i	$-0.844040\pi$
$317$	−31.4196	−1.76470	−0.882350	+	0.470593i	$0.844040\pi$
$318$	0	0
$319$	3.72679	0.208660
$320$	3.24110	0.181183
$321$	0	0
$322$	11.9463	0.665742
$323$	−0.581601	−0.0323611
$324$	0	0
$325$	−1.73123	−0.0960314
$326$	7.05078	0.390506
$327$	0	0
$328$	−27.5889	−1.52334
$329$	8.13970	0.448756
$330$	0	0
$331$	−8.60336	−0.472883	−0.236442	−	0.971646i	$-0.575981\pi$
$331$	−8.60336	−0.472883	−0.236442	+	0.971646i	$0.575981\pi$
$332$	23.8615	1.30957
$333$	0	0
$334$	41.6918	2.28128
$335$	−7.85119	−0.428956
$336$	0	0
$337$	2.74513	0.149537	0.0747683	−	0.997201i	$-0.476178\pi$
$337$	2.74513	0.149537	0.0747683	+	0.997201i	$0.476178\pi$
$338$	32.1855	1.75066
$339$	0	0
$340$	1.38217	0.0749588
$341$	−10.1742	−0.550966
$342$	0	0
$343$	−15.2211	−0.821862
$344$	31.8234	1.71580
$345$	0	0
$346$	−31.8387	−1.71166
$347$	19.8314	1.06460	0.532301	−	0.846555i	$-0.321327\pi$
$347$	19.8314	1.06460	0.532301	+	0.846555i	$0.321327\pi$
$348$	0	0
$349$	27.8931	1.49308	0.746542	−	0.665339i	$-0.231713\pi$
$349$	27.8931	1.49308	0.746542	+	0.665339i	$0.231713\pi$
$350$	13.1534	0.703078
$351$	0	0
$352$	3.05175	0.162659
$353$	−11.2470	−0.598616	−0.299308	−	0.954156i	$-0.596756\pi$
$353$	−11.2470	−0.598616	−0.299308	+	0.954156i	$0.596756\pi$
$354$	0	0
$355$	4.42660	0.234940
$356$	−0.982009	−0.0520464
$357$	0	0
$358$	−23.1986	−1.22609
$359$	−7.52868	−0.397349	−0.198674	−	0.980066i	$-0.563664\pi$
$359$	−7.52868	−0.397349	−0.198674	+	0.980066i	$0.563664\pi$
$360$	0	0
$361$	−16.7797	−0.883144
$362$	21.0409	1.10589
$363$	0	0
$364$	2.08839	0.109461
$365$	−6.84293	−0.358175
$366$	0	0
$367$	23.5857	1.23117	0.615583	−	0.788072i	$-0.288921\pi$
$367$	23.5857	1.23117	0.615583	+	0.788072i	$0.288921\pi$
$368$	−22.6837	−1.18247
$369$	0	0
$370$	6.96976	0.362341
$371$	−2.59317	−0.134631
$372$	0	0
$373$	25.9035	1.34123	0.670616	−	0.741804i	$-0.266030\pi$
$373$	25.9035	1.34123	0.670616	+	0.741804i	$0.266030\pi$
$374$	−0.978467	−0.0505953
$375$	0	0
$376$	−38.3447	−1.97748
$377$	−1.49461	−0.0769764
$378$	0	0
$379$	12.8278	0.658921	0.329461	−	0.944169i	$-0.393133\pi$
$379$	12.8278	0.658921	0.329461	+	0.944169i	$0.393133\pi$
$380$	−5.27645	−0.270676
$381$	0	0
$382$	18.9991	0.972077
$383$	0.791468	0.0404421	0.0202211	−	0.999796i	$-0.493563\pi$
$383$	0.791468	0.0404421	0.0202211	+	0.999796i	$0.493563\pi$
$384$	0	0
$385$	1.00468	0.0512030
$386$	−32.1005	−1.63387
$387$	0	0
$388$	−52.4592	−2.66321
$389$	−11.6587	−0.591120	−0.295560	−	0.955324i	$-0.595506\pi$
$389$	−11.6587	−0.591120	−0.295560	+	0.955324i	$0.595506\pi$
$390$	0	0
$391$	1.53031	0.0773913
$392$	31.6222	1.59716
$393$	0	0
$394$	−40.0910	−2.01976
$395$	6.54289	0.329209
$396$	0	0
$397$	−10.5718	−0.530581	−0.265291	−	0.964169i	$-0.585468\pi$
$397$	−10.5718	−0.530581	−0.265291	+	0.964169i	$0.585468\pi$
$398$	−24.1026	−1.20815
$399$	0	0
$400$	−24.9757	−1.24878
$401$	37.4417	1.86975	0.934874	−	0.354980i	$-0.115512\pi$
$401$	37.4417	1.86975	0.934874	+	0.354980i	$0.115512\pi$
$402$	0	0
$403$	4.08033	0.203256
$404$	−5.43008	−0.270156
$405$	0	0
$406$	11.3556	0.563569
$407$	−3.36372	−0.166734
$408$	0	0
$409$	−13.2875	−0.657027	−0.328513	−	0.944499i	$-0.606548\pi$
$409$	−13.2875	−0.657027	−0.328513	+	0.944499i	$0.606548\pi$
$410$	9.98350	0.493050
$411$	0	0
$412$	−0.243797	−0.0120110
$413$	−10.5795	−0.520583
$414$	0	0
$415$	−4.60369	−0.225986
$416$	−1.22389	−0.0600062
$417$	0	0
$418$	3.73531	0.182700
$419$	3.61102	0.176410	0.0882049	−	0.996102i	$-0.471887\pi$
$419$	3.61102	0.176410	0.0882049	+	0.996102i	$0.471887\pi$
$420$	0	0
$421$	3.70704	0.180670	0.0903349	−	0.995911i	$-0.471206\pi$
$421$	3.70704	0.180670	0.0903349	+	0.995911i	$0.471206\pi$
$422$	−7.77055	−0.378265
$423$	0	0
$424$	12.2160	0.593260
$425$	1.68494	0.0817316
$426$	0	0
$427$	−1.21549	−0.0588218
$428$	−0.442276	−0.0213782
$429$	0	0
$430$	−11.5158	−0.555342
$431$	−8.65802	−0.417042	−0.208521	−	0.978018i	$-0.566865\pi$
$431$	−8.65802	−0.417042	−0.208521	+	0.978018i	$0.566865\pi$
$432$	0	0
$433$	21.2165	1.01960	0.509801	−	0.860292i	$-0.329719\pi$
$433$	21.2165	1.01960	0.509801	+	0.860292i	$0.329719\pi$
$434$	−31.0012	−1.48810
$435$	0	0
$436$	83.0561	3.97767
$437$	−5.84199	−0.279460
$438$	0	0
$439$	−27.0692	−1.29194	−0.645971	−	0.763362i	$-0.723547\pi$
$439$	−27.0692	−1.29194	−0.645971	+	0.763362i	$0.723547\pi$
$440$	−4.73286	−0.225630
$441$	0	0
$442$	0.392409	0.0186650
$443$	14.6319	0.695182	0.347591	−	0.937646i	$-0.387000\pi$
$443$	14.6319	0.695182	0.347591	+	0.937646i	$0.387000\pi$
$444$	0	0
$445$	0.189463	0.00898140
$446$	59.2225	2.80427
$447$	0	0
$448$	−4.76617	−0.225181
$449$	−30.9636	−1.46126	−0.730632	−	0.682772i	$-0.760774\pi$
$449$	−30.9636	−1.46126	−0.730632	+	0.682772i	$0.760774\pi$
$450$	0	0
$451$	−4.81820	−0.226880
$452$	42.3322	1.99114
$453$	0	0
$454$	−13.4768	−0.632497
$455$	−0.402921	−0.0188892
$456$	0	0
$457$	−32.5769	−1.52388	−0.761942	−	0.647645i	$-0.775754\pi$
$457$	−32.5769	−1.52388	−0.761942	+	0.647645i	$0.775754\pi$
$458$	50.7260	2.37027
$459$	0	0
$460$	13.8835	0.647319
$461$	−24.4881	−1.14052	−0.570262	−	0.821463i	$-0.693158\pi$
$461$	−24.4881	−1.14052	−0.570262	+	0.821463i	$0.693158\pi$
$462$	0	0
$463$	4.17729	0.194135	0.0970676	−	0.995278i	$-0.469054\pi$
$463$	4.17729	0.194135	0.0970676	+	0.995278i	$0.469054\pi$
$464$	−21.5621	−1.00099
$465$	0	0
$466$	21.3841	0.990600
$467$	−19.2528	−0.890915	−0.445457	−	0.895303i	$-0.646959\pi$
$467$	−19.2528	−0.890915	−0.445457	+	0.895303i	$0.646959\pi$
$468$	0	0
$469$	11.5455	0.533123
$470$	13.8757	0.640037
$471$	0	0
$472$	49.8382	2.29399
$473$	5.55772	0.255544
$474$	0	0
$475$	−6.43227	−0.295133
$476$	−2.03254	−0.0931615
$477$	0	0
$478$	−6.03567	−0.276065
$479$	−14.7472	−0.673818	−0.336909	−	0.941537i	$-0.609381\pi$
$479$	−14.7472	−0.673818	−0.336909	+	0.941537i	$0.609381\pi$
$480$	0	0
$481$	1.34900	0.0615093
$482$	−33.8904	−1.54367
$483$	0	0
$484$	4.28416	0.194734
$485$	10.1212	0.459578
$486$	0	0
$487$	−2.96701	−0.134448	−0.0672239	−	0.997738i	$-0.521414\pi$
$487$	−2.96701	−0.134448	−0.0672239	+	0.997738i	$0.521414\pi$
$488$	5.72598	0.259203
$489$	0	0
$490$	−11.4430	−0.516942
$491$	−10.4195	−0.470226	−0.235113	−	0.971968i	$-0.575546\pi$
$491$	−10.4195	−0.470226	−0.235113	+	0.971968i	$0.575546\pi$
$492$	0	0
$493$	1.45465	0.0655140
$494$	−1.49803	−0.0673994
$495$	0	0
$496$	58.8651	2.64312
$497$	−6.50952	−0.291992
$498$	0	0
$499$	−0.616114	−0.0275811	−0.0137905	−	0.999905i	$-0.504390\pi$
$499$	−0.616114	−0.0275811	−0.0137905	+	0.999905i	$0.504390\pi$
$500$	32.9918	1.47544
$501$	0	0
$502$	−33.3645	−1.48913
$503$	13.3401	0.594806	0.297403	−	0.954752i	$-0.403880\pi$
$503$	13.3401	0.594806	0.297403	+	0.954752i	$0.403880\pi$
$504$	0	0
$505$	1.04765	0.0466196
$506$	−9.82837	−0.436924
$507$	0	0
$508$	−48.1859	−2.13790
$509$	0.0136682	0.000605832 0	0.000302916	−	1.00000i	$-0.499904\pi$
$509$	0.0136682	0.000605832 0	0.000302916		1.00000i	$0.499904\pi$
$510$	0	0
$511$	10.0628	0.445153
$512$	48.6010	2.14788
$513$	0	0
$514$	−65.8736	−2.90556
$515$	0.0470366	0.00207268
$516$	0	0
$517$	−6.69663	−0.294517
$518$	−10.2493	−0.450330
$519$	0	0
$520$	1.89809	0.0832368
$521$	−3.95064	−0.173081	−0.0865403	−	0.996248i	$-0.527581\pi$
$521$	−3.95064	−0.173081	−0.0865403	+	0.996248i	$0.527581\pi$
$522$	0	0
$523$	−20.7278	−0.906365	−0.453183	−	0.891418i	$-0.649711\pi$
$523$	−20.7278	−0.906365	−0.453183	+	0.891418i	$0.649711\pi$
$524$	−88.7881	−3.87873
$525$	0	0
$526$	−3.88286	−0.169301
$527$	−3.97123	−0.172989
$528$	0	0
$529$	−7.62851	−0.331674
$530$	−4.42055	−0.192016
$531$	0	0
$532$	7.75926	0.336406
$533$	1.93232	0.0836979
$534$	0	0
$535$	0.0853300	0.00368914
$536$	−54.3890	−2.34925
$537$	0	0
$538$	51.6342	2.22611
$539$	5.52258	0.237874
$540$	0	0
$541$	8.76117	0.376672	0.188336	−	0.982105i	$-0.439691\pi$
$541$	8.76117	0.376672	0.188336	+	0.982105i	$0.439691\pi$
$542$	21.3514	0.917120
$543$	0	0
$544$	1.19116	0.0510708
$545$	−16.0243	−0.686407
$546$	0	0
$547$	−4.49414	−0.192156	−0.0960778	−	0.995374i	$-0.530630\pi$
$547$	−4.49414	−0.192156	−0.0960778	+	0.995374i	$0.530630\pi$
$548$	−32.7044	−1.39706
$549$	0	0
$550$	−10.8214	−0.461428
$551$	−5.55312	−0.236571
$552$	0	0
$553$	−9.62161	−0.409153
$554$	5.97085	0.253677
$555$	0	0
$556$	−41.5539	−1.76228
$557$	−36.7992	−1.55923	−0.779616	−	0.626257i	$-0.784586\pi$
$557$	−36.7992	−1.55923	−0.779616	+	0.626257i	$0.784586\pi$
$558$	0	0
$559$	−2.22890	−0.0942723
$560$	−5.81275	−0.245634
$561$	0	0
$562$	−76.0244	−3.20690
$563$	21.8465	0.920721	0.460360	−	0.887732i	$-0.347720\pi$
$563$	21.8465	0.920721	0.460360	+	0.887732i	$0.347720\pi$
$564$	0	0
$565$	−8.16731	−0.343601
$566$	−65.8965	−2.76984
$567$	0	0
$568$	30.6652	1.28668
$569$	−8.36494	−0.350676	−0.175338	−	0.984508i	$-0.556102\pi$
$569$	−8.36494	−0.350676	−0.175338	+	0.984508i	$0.556102\pi$
$570$	0	0
$571$	−8.37973	−0.350681	−0.175340	−	0.984508i	$-0.556103\pi$
$571$	−8.37973	−0.350681	−0.175340	+	0.984508i	$0.556103\pi$
$572$	−1.71814	−0.0718391
$573$	0	0
$574$	−14.6812	−0.612780
$575$	16.9247	0.705807
$576$	0	0
$577$	−21.6364	−0.900735	−0.450368	−	0.892843i	$-0.648707\pi$
$577$	−21.6364	−0.900735	−0.450368	+	0.892843i	$0.648707\pi$
$578$	42.2341	1.75671
$579$	0	0
$580$	13.1970	0.547975
$581$	6.76992	0.280864
$582$	0	0
$583$	2.13343	0.0883577
$584$	−47.4043	−1.96160
$585$	0	0
$586$	−26.0472	−1.07600
$587$	0.614980	0.0253829	0.0126915	−	0.999919i	$-0.495960\pi$
$587$	0.614980	0.0253829	0.0126915	+	0.999919i	$0.495960\pi$
$588$	0	0
$589$	15.1602	0.624665
$590$	−18.0348	−0.742479
$591$	0	0
$592$	19.4615	0.799862
$593$	21.4531	0.880972	0.440486	−	0.897760i	$-0.354806\pi$
$593$	21.4531	0.880972	0.440486	+	0.897760i	$0.354806\pi$
$594$	0	0
$595$	0.392147	0.0160765
$596$	−30.9968	−1.26968
$597$	0	0
$598$	3.94162	0.161185
$599$	−25.7289	−1.05126	−0.525628	−	0.850715i	$-0.676169\pi$
$599$	−25.7289	−1.05126	−0.525628	+	0.850715i	$0.676169\pi$
$600$	0	0
$601$	−43.8900	−1.79031	−0.895155	−	0.445755i	$-0.852935\pi$
$601$	−43.8900	−1.79031	−0.895155	+	0.445755i	$0.852935\pi$
$602$	16.9345	0.690199
$603$	0	0
$604$	1.37893	0.0561079
$605$	−0.826559	−0.0336044
$606$	0	0
$607$	−24.6636	−1.00107	−0.500533	−	0.865718i	$-0.666863\pi$
$607$	−24.6636	−1.00107	−0.500533	+	0.865718i	$0.666863\pi$
$608$	−4.54728	−0.184417
$609$	0	0
$610$	−2.07204	−0.0838944
$611$	2.68565	0.108650
$612$	0	0
$613$	12.7389	0.514520	0.257260	−	0.966342i	$-0.417180\pi$
$613$	12.7389	0.514520	0.257260	+	0.966342i	$0.417180\pi$
$614$	39.7493	1.60415
$615$	0	0
$616$	6.95988	0.280422
$617$	17.2182	0.693179	0.346590	−	0.938017i	$-0.387340\pi$
$617$	17.2182	0.693179	0.346590	+	0.938017i	$0.387340\pi$
$618$	0	0
$619$	−30.9126	−1.24248	−0.621241	−	0.783619i	$-0.713371\pi$
$619$	−30.9126	−1.24248	−0.621241	+	0.783619i	$0.713371\pi$
$620$	−36.0281	−1.44692
$621$	0	0
$622$	17.3979	0.697594
$623$	−0.278614	−0.0111624
$624$	0	0
$625$	15.2188	0.608750
$626$	73.2698	2.92845
$627$	0	0
$628$	−71.2250	−2.84219
$629$	−1.31293	−0.0523501
$630$	0	0
$631$	−39.5671	−1.57514	−0.787571	−	0.616223i	$-0.788662\pi$
$631$	−39.5671	−1.57514	−0.787571	+	0.616223i	$0.788662\pi$
$632$	45.3258	1.80296
$633$	0	0
$634$	78.7634	3.12809
$635$	9.29670	0.368928
$636$	0	0
$637$	−2.21480	−0.0877537
$638$	−9.34240	−0.369869
$639$	0	0
$640$	−13.1698	−0.520581
$641$	−44.6319	−1.76286	−0.881428	−	0.472319i	$-0.843417\pi$
$641$	−44.6319	−1.76286	−0.881428	+	0.472319i	$0.843417\pi$
$642$	0	0
$643$	−26.8693	−1.05962	−0.529811	−	0.848116i	$-0.677737\pi$
$643$	−26.8693	−1.05962	−0.529811	+	0.848116i	$0.677737\pi$
$644$	−20.4162	−0.804512
$645$	0	0
$646$	1.45797	0.0573631
$647$	−0.924577	−0.0363489	−0.0181744	−	0.999835i	$-0.505785\pi$
$647$	−0.924577	−0.0363489	−0.0181744	+	0.999835i	$0.505785\pi$
$648$	0	0
$649$	8.70387	0.341657
$650$	4.33989	0.170224
$651$	0	0
$652$	−12.0498	−0.471906
$653$	−36.6939	−1.43594	−0.717972	−	0.696072i	$-0.754929\pi$
$653$	−36.6939	−1.43594	−0.717972	+	0.696072i	$0.754929\pi$
$654$	0	0
$655$	17.1302	0.669334
$656$	27.8767	1.08840
$657$	0	0
$658$	−20.4048	−0.795461
$659$	37.2558	1.45128	0.725640	−	0.688074i	$-0.241544\pi$
$659$	37.2558	1.45128	0.725640	+	0.688074i	$0.241544\pi$
$660$	0	0
$661$	−27.3936	−1.06549	−0.532743	−	0.846277i	$-0.678839\pi$
$661$	−27.3936	−1.06549	−0.532743	+	0.846277i	$0.678839\pi$
$662$	21.5671	0.838229
$663$	0	0
$664$	−31.8920	−1.23765
$665$	−1.49702	−0.0580521
$666$	0	0
$667$	14.6114	0.565757
$668$	−71.2513	−2.75680
$669$	0	0
$670$	19.6815	0.760364
$671$	1.00000	0.0386046
$672$	0	0
$673$	−42.1860	−1.62615	−0.813075	−	0.582159i	$-0.802208\pi$
$673$	−42.1860	−1.62615	−0.813075	+	0.582159i	$0.802208\pi$
$674$	−6.88154	−0.265067
$675$	0	0
$676$	−55.0050	−2.11558
$677$	5.99915	0.230566	0.115283	−	0.993333i	$-0.463222\pi$
$677$	5.99915	0.230566	0.115283	+	0.993333i	$0.463222\pi$
$678$	0	0
$679$	−14.8836	−0.571180
$680$	−1.84734	−0.0708422
$681$	0	0
$682$	25.5050	0.976638
$683$	−15.9221	−0.609242	−0.304621	−	0.952474i	$-0.598530\pi$
$683$	−15.9221	−0.609242	−0.304621	+	0.952474i	$0.598530\pi$
$684$	0	0
$685$	6.30979	0.241085
$686$	38.1566	1.45682
$687$	0	0
$688$	−32.1553	−1.22591
$689$	−0.855602	−0.0325958
$690$	0	0
$691$	21.7079	0.825806	0.412903	−	0.910775i	$-0.364515\pi$
$691$	21.7079	0.825806	0.412903	+	0.910775i	$0.364515\pi$
$692$	54.4124	2.06845
$693$	0	0
$694$	−49.7137	−1.88711
$695$	8.01715	0.304108
$696$	0	0
$697$	−1.88065	−0.0712346
$698$	−69.9230	−2.64663
$699$	0	0
$700$	−22.4791	−0.849631
$701$	−2.94301	−0.111156	−0.0555780	−	0.998454i	$-0.517700\pi$
$701$	−2.94301	−0.111156	−0.0555780	+	0.998454i	$0.517700\pi$
$702$	0	0
$703$	5.01214	0.189036
$704$	3.92119	0.147785
$705$	0	0
$706$	28.1942	1.06110
$707$	−1.54061	−0.0579406
$708$	0	0
$709$	−10.8940	−0.409131	−0.204566	−	0.978853i	$-0.565578\pi$
$709$	−10.8940	−0.409131	−0.204566	+	0.978853i	$0.565578\pi$
$710$	−11.0967	−0.416452
$711$	0	0
$712$	1.31250	0.0491881
$713$	−39.8896	−1.49388
$714$	0	0
$715$	0.331488	0.0123969
$716$	39.6464	1.48166
$717$	0	0
$718$	18.8731	0.704336
$719$	−37.6098	−1.40261	−0.701304	−	0.712862i	$-0.747399\pi$
$719$	−37.6098	−1.40261	−0.701304	+	0.712862i	$0.747399\pi$
$720$	0	0
$721$	−0.0691694	−0.00257600
$722$	42.0638	1.56545
$723$	0	0
$724$	−35.9590	−1.33640
$725$	16.0878	0.597486
$726$	0	0
$727$	−14.9245	−0.553519	−0.276759	−	0.960939i	$-0.589260\pi$
$727$	−14.9245	−0.553519	−0.276759	+	0.960939i	$0.589260\pi$
$728$	−2.79123	−0.103450
$729$	0	0
$730$	17.1540	0.634899
$731$	2.16930	0.0802344
$732$	0	0
$733$	22.6722	0.837417	0.418708	−	0.908121i	$-0.362483\pi$
$733$	22.6722	0.837417	0.418708	+	0.908121i	$0.362483\pi$
$734$	−59.1253	−2.18235
$735$	0	0
$736$	11.9649	0.441030
$737$	−9.49864	−0.349887
$738$	0	0
$739$	−23.3855	−0.860250	−0.430125	−	0.902769i	$-0.641531\pi$
$739$	−23.3855	−0.860250	−0.430125	+	0.902769i	$0.641531\pi$
$740$	−11.9113	−0.437869
$741$	0	0
$742$	6.50061	0.238645
$743$	−4.11172	−0.150844	−0.0754221	−	0.997152i	$-0.524030\pi$
$743$	−4.11172	−0.150844	−0.0754221	+	0.997152i	$0.524030\pi$
$744$	0	0
$745$	5.98033	0.219102
$746$	−64.9355	−2.37746
$747$	0	0
$748$	1.67220	0.0611417
$749$	−0.125482	−0.00458500
$750$	0	0
$751$	43.1423	1.57428	0.787142	−	0.616772i	$-0.211560\pi$
$751$	43.1423	1.57428	0.787142	+	0.616772i	$0.211560\pi$
$752$	38.7446	1.41287
$753$	0	0
$754$	3.74672	0.136448
$755$	−0.266042	−0.00968227
$756$	0	0
$757$	13.7682	0.500413	0.250206	−	0.968193i	$-0.419502\pi$
$757$	13.7682	0.500413	0.250206	+	0.968193i	$0.419502\pi$
$758$	−32.1571	−1.16800
$759$	0	0
$760$	7.05223	0.255811
$761$	40.6298	1.47283	0.736415	−	0.676530i	$-0.236517\pi$
$761$	40.6298	1.47283	0.736415	+	0.676530i	$0.236517\pi$
$762$	0	0
$763$	23.5645	0.853092
$764$	−32.4694	−1.17470
$765$	0	0
$766$	−1.98407	−0.0716874
$767$	−3.49065	−0.126040
$768$	0	0
$769$	3.60649	0.130053	0.0650266	−	0.997884i	$-0.479287\pi$
$769$	3.60649	0.130053	0.0650266	+	0.997884i	$0.479287\pi$
$770$	−2.51855	−0.0907621
$771$	0	0
$772$	54.8598	1.97445
$773$	24.9246	0.896476	0.448238	−	0.893914i	$-0.352052\pi$
$773$	24.9246	0.896476	0.448238	+	0.893914i	$0.352052\pi$
$774$	0	0
$775$	−43.9202	−1.57766
$776$	70.1142	2.51695
$777$	0	0
$778$	29.2263	1.04781
$779$	7.17939	0.257228
$780$	0	0
$781$	5.35546	0.191633
$782$	−3.83623	−0.137183
$783$	0	0
$784$	−31.9520	−1.14114
$785$	13.7417	0.490463
$786$	0	0
$787$	−33.9634	−1.21066	−0.605332	−	0.795973i	$-0.706960\pi$
$787$	−33.9634	−1.21066	−0.605332	+	0.795973i	$0.706960\pi$
$788$	68.5155	2.44076
$789$	0	0
$790$	−16.4019	−0.583552
$791$	12.0104	0.427040
$792$	0	0
$793$	−0.401045	−0.0142415
$794$	26.5015	0.940503
$795$	0	0
$796$	41.1913	1.45999
$797$	−6.08572	−0.215567	−0.107784	−	0.994174i	$-0.534375\pi$
$797$	−6.08572	−0.215567	−0.107784	+	0.994174i	$0.534375\pi$
$798$	0	0
$799$	−2.61384	−0.0924709
$800$	13.1738	0.465764
$801$	0	0
$802$	−93.8596	−3.31430
$803$	−8.27881	−0.292153
$804$	0	0
$805$	3.93898	0.138831
$806$	−10.2287	−0.360289
$807$	0	0
$808$	7.25755	0.255320
$809$	−38.1261	−1.34044	−0.670220	−	0.742162i	$-0.733800\pi$
$809$	−38.1261	−1.34044	−0.670220	+	0.742162i	$0.733800\pi$
$810$	0	0
$811$	−41.1660	−1.44553	−0.722767	−	0.691091i	$-0.757130\pi$
$811$	−41.1660	−1.44553	−0.722767	+	0.691091i	$0.757130\pi$
$812$	−19.4067	−0.681043
$813$	0	0
$814$	8.43226	0.295550
$815$	2.32481	0.0814345
$816$	0	0
$817$	−8.28132	−0.289727
$818$	33.3095	1.16464
$819$	0	0
$820$	−17.0618	−0.595824
$821$	−8.19060	−0.285854	−0.142927	−	0.989733i	$-0.545651\pi$
$821$	−8.19060	−0.285854	−0.142927	+	0.989733i	$0.545651\pi$
$822$	0	0
$823$	−12.0886	−0.421381	−0.210690	−	0.977553i	$-0.567571\pi$
$823$	−12.0886	−0.421381	−0.210690	+	0.977553i	$0.567571\pi$
$824$	0.325846	0.0113514
$825$	0	0
$826$	26.5209	0.922780
$827$	41.0844	1.42864	0.714322	−	0.699818i	$-0.246735\pi$
$827$	41.0844	1.42864	0.714322	+	0.699818i	$0.246735\pi$
$828$	0	0
$829$	−42.0223	−1.45949	−0.729747	−	0.683717i	$-0.760362\pi$
$829$	−42.0223	−1.45949	−0.729747	+	0.683717i	$0.760362\pi$
$830$	11.5406	0.400581
$831$	0	0
$832$	−1.57257	−0.0545192
$833$	2.15558	0.0746865
$834$	0	0
$835$	13.7468	0.475727
$836$	−6.38364	−0.220783
$837$	0	0
$838$	−9.05218	−0.312702
$839$	20.3559	0.702765	0.351382	−	0.936232i	$-0.385712\pi$
$839$	20.3559	0.702765	0.351382	+	0.936232i	$0.385712\pi$
$840$	0	0
$841$	−15.1110	−0.521070
$842$	−9.29288	−0.320254
$843$	0	0
$844$	13.2799	0.457112
$845$	10.6123	0.365075
$846$	0	0
$847$	1.21549	0.0417648
$848$	−12.3434	−0.423873
$849$	0	0
$850$	−4.22384	−0.144877
$851$	−13.1880	−0.452078
$852$	0	0
$853$	−16.8300	−0.576249	−0.288125	−	0.957593i	$-0.593032\pi$
$853$	−16.8300	−0.576249	−0.288125	+	0.957593i	$0.593032\pi$
$854$	3.04702	0.104267
$855$	0	0
$856$	0.591123	0.0202042
$857$	−34.5793	−1.18121	−0.590603	−	0.806962i	$-0.701110\pi$
$857$	−34.5793	−1.18121	−0.590603	+	0.806962i	$0.701110\pi$
$858$	0	0
$859$	−11.9362	−0.407258	−0.203629	−	0.979048i	$-0.565274\pi$
$859$	−11.9362	−0.407258	−0.203629	+	0.979048i	$0.565274\pi$
$860$	19.6805	0.671100
$861$	0	0
$862$	21.7041	0.739245
$863$	−32.2783	−1.09877	−0.549384	−	0.835570i	$-0.685138\pi$
$863$	−32.2783	−1.09877	−0.549384	+	0.835570i	$0.685138\pi$
$864$	0	0
$865$	−10.4980	−0.356942
$866$	−53.1861	−1.80734
$867$	0	0
$868$	52.9810	1.79829
$869$	7.91582	0.268526
$870$	0	0
$871$	3.80938	0.129076
$872$	−111.008	−3.75922
$873$	0	0
$874$	14.6448	0.495369
$875$	9.36037	0.316438
$876$	0	0
$877$	4.10969	0.138774	0.0693872	−	0.997590i	$-0.477896\pi$
$877$	4.10969	0.138774	0.0693872	+	0.997590i	$0.477896\pi$
$878$	67.8577	2.29009
$879$	0	0
$880$	4.78222	0.161209
$881$	−20.8201	−0.701449	−0.350724	−	0.936479i	$-0.614065\pi$
$881$	−20.8201	−0.701449	−0.350724	+	0.936479i	$0.614065\pi$
$882$	0	0
$883$	−39.0863	−1.31536	−0.657679	−	0.753298i	$-0.728462\pi$
$883$	−39.0863	−1.31536	−0.657679	+	0.753298i	$0.728462\pi$
$884$	−0.670627	−0.0225556
$885$	0	0
$886$	−36.6796	−1.23227
$887$	32.4887	1.09086	0.545432	−	0.838155i	$-0.316365\pi$
$887$	32.4887	1.09086	0.545432	+	0.838155i	$0.316365\pi$
$888$	0	0
$889$	−13.6712	−0.458517
$890$	−0.474950	−0.0159204
$891$	0	0
$892$	−101.211	−3.38880
$893$	9.97835	0.333913
$894$	0	0
$895$	−7.64914	−0.255683
$896$	19.3667	0.646996
$897$	0	0
$898$	77.6203	2.59022
$899$	−37.9173	−1.26461
$900$	0	0
$901$	0.832724	0.0277421
$902$	12.0784	0.402166
$903$	0	0
$904$	−56.5789	−1.88179
$905$	6.93771	0.230617
$906$	0	0
$907$	21.9975	0.730415	0.365208	−	0.930926i	$-0.380998\pi$
$907$	21.9975	0.730415	0.365208	+	0.930926i	$0.380998\pi$
$908$	23.0318	0.764338
$909$	0	0
$910$	1.01005	0.0334828
$911$	−35.5980	−1.17941	−0.589707	−	0.807617i	$-0.700757\pi$
$911$	−35.5980	−1.17941	−0.589707	+	0.807617i	$0.700757\pi$
$912$	0	0
$913$	−5.56970	−0.184330
$914$	81.6646	2.70122
$915$	0	0
$916$	−86.6908	−2.86434
$917$	−25.1908	−0.831873
$918$	0	0
$919$	33.3807	1.10113	0.550563	−	0.834794i	$-0.314413\pi$
$919$	33.3807	1.10113	0.550563	+	0.834794i	$0.314413\pi$
$920$	−18.5559	−0.611770
$921$	0	0
$922$	61.3873	2.02168
$923$	−2.14778	−0.0706951
$924$	0	0
$925$	−14.5205	−0.477432
$926$	−10.4717	−0.344122
$927$	0	0
$928$	11.3732	0.373345
$929$	−23.1982	−0.761110	−0.380555	−	0.924758i	$-0.624267\pi$
$929$	−23.1982	−0.761110	−0.380555	+	0.924758i	$0.624267\pi$
$930$	0	0
$931$	−8.22895	−0.269693
$932$	−36.5454	−1.19709
$933$	0	0
$934$	48.2634	1.57923
$935$	−0.322624	−0.0105509
$936$	0	0
$937$	8.23459	0.269012	0.134506	−	0.990913i	$-0.457055\pi$
$937$	8.23459	0.269012	0.134506	+	0.990913i	$0.457055\pi$
$938$	−28.9426	−0.945009
$939$	0	0
$940$	−23.7135	−0.773449
$941$	9.26795	0.302127	0.151063	−	0.988524i	$-0.451730\pi$
$941$	9.26795	0.302127	0.151063	+	0.988524i	$0.451730\pi$
$942$	0	0
$943$	−18.8905	−0.615159
$944$	−50.3580	−1.63901
$945$	0	0
$946$	−13.9322	−0.452976
$947$	−32.9049	−1.06927	−0.534633	−	0.845084i	$-0.679550\pi$
$947$	−32.9049	−1.06927	−0.534633	+	0.845084i	$0.679550\pi$
$948$	0	0
$949$	3.32018	0.107778
$950$	16.1246	0.523150
$951$	0	0
$952$	2.71659	0.0880452
$953$	−35.5632	−1.15200	−0.576002	−	0.817448i	$-0.695388\pi$
$953$	−35.5632	−1.15200	−0.576002	+	0.817448i	$0.695388\pi$
$954$	0	0
$955$	6.26445	0.202713
$956$	10.3150	0.333609
$957$	0	0
$958$	36.9687	1.19440
$959$	−9.27882	−0.299629
$960$	0	0
$961$	72.5152	2.33920
$962$	−3.38172	−0.109031
$963$	0	0
$964$	57.9187	1.86544
$965$	−10.5843	−0.340721
$966$	0	0
$967$	24.6835	0.793767	0.396884	−	0.917869i	$-0.370092\pi$
$967$	24.6835	0.793767	0.396884	+	0.917869i	$0.370092\pi$
$968$	−5.72598	−0.184040
$969$	0	0
$970$	−25.3719	−0.814644
$971$	27.5918	0.885461	0.442731	−	0.896655i	$-0.354010\pi$
$971$	27.5918	0.885461	0.442731	+	0.896655i	$0.354010\pi$
$972$	0	0
$973$	−11.7896	−0.377956
$974$	7.43775	0.238321
$975$	0	0
$976$	−5.78570	−0.185196
$977$	46.7872	1.49686	0.748428	−	0.663216i	$-0.230809\pi$
$977$	46.7872	1.49686	0.748428	+	0.663216i	$0.230809\pi$
$978$	0	0
$979$	0.229219	0.00732586
$980$	19.5561	0.624696
$981$	0	0
$982$	26.1199	0.833519
$983$	−32.4422	−1.03475	−0.517373	−	0.855760i	$-0.673090\pi$
$983$	−32.4422	−1.03475	−0.517373	+	0.855760i	$0.673090\pi$
$984$	0	0
$985$	−13.2190	−0.421191
$986$	−3.64654	−0.116129
$987$	0	0
$988$	2.56013	0.0814485
$989$	21.7899	0.692878
$990$	0	0
$991$	−5.91364	−0.187853	−0.0939265	−	0.995579i	$-0.529942\pi$
$991$	−5.91364	−0.187853	−0.0939265	+	0.995579i	$0.529942\pi$
$992$	−31.0493	−0.985816
$993$	0	0
$994$	16.3182	0.517582
$995$	−7.94721	−0.251943
$996$	0	0
$997$	−19.5993	−0.620716	−0.310358	−	0.950620i	$-0.600449\pi$
$997$	−19.5993	−0.620716	−0.310358	+	0.950620i	$0.600449\pi$
$998$	1.54449	0.0488899
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.n.1.3	✓	25
3.2	odd	2		6039.2.a.o.1.23	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.n.1.3	✓	25	1.1	even	1	trivial
6039.2.a.o.1.23	yes	25	3.2	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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q-2.50682 q^{2} +4.28416 q^{4} -0.826559 q^{5} +1.21549 q^{7} -5.72598 q^{8} +O(q^{10})\)	\(q-2.50682 q^{2} +4.28416 q^{4} -0.826559 q^{5} +1.21549 q^{7} -5.72598 q^{8} +2.07204 q^{10} -1.00000 q^{11} +0.401045 q^{13} -3.04702 q^{14} +5.78570 q^{16} -0.390322 q^{17} +1.49006 q^{19} -3.54111 q^{20} +2.50682 q^{22} -3.92065 q^{23} -4.31680 q^{25} -1.00535 q^{26} +5.20736 q^{28} -3.72679 q^{29} +10.1742 q^{31} -3.05175 q^{32} +0.978467 q^{34} -1.00468 q^{35} +3.36372 q^{37} -3.73531 q^{38} +4.73286 q^{40} +4.81820 q^{41} -5.55772 q^{43} -4.28416 q^{44} +9.82837 q^{46} +6.69663 q^{47} -5.52258 q^{49} +10.8214 q^{50} +1.71814 q^{52} -2.13343 q^{53} +0.826559 q^{55} -6.95988 q^{56} +9.34240 q^{58} -8.70387 q^{59} -1.00000 q^{61} -25.5050 q^{62} -3.92119 q^{64} -0.331488 q^{65} +9.49864 q^{67} -1.67220 q^{68} +2.51855 q^{70} -5.35546 q^{71} +8.27881 q^{73} -8.43226 q^{74} +6.38364 q^{76} -1.21549 q^{77} -7.91582 q^{79} -4.78222 q^{80} -12.0784 q^{82} +5.56970 q^{83} +0.322624 q^{85} +13.9322 q^{86} +5.72598 q^{88} -0.229219 q^{89} +0.487467 q^{91} -16.7967 q^{92} -16.7873 q^{94} -1.23162 q^{95} -12.2449 q^{97} +13.8441 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8}+O(q^{10}) \) 25 * q - 5 * q^2 + 25 * q^4 - 4 * q^5 + 4 * q^7 - 15 * q^8	\( 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8} - 25 q^{11} + 4 q^{13} - 18 q^{14} + 21 q^{16} - 20 q^{17} + 14 q^{19} - 12 q^{20} + 5 q^{22} - 20 q^{23} + 13 q^{25} - 16 q^{26} - 14 q^{28} - 28 q^{29} - 12 q^{31} - 35 q^{32} + 6 q^{34} - 10 q^{35} - 8 q^{37} - 32 q^{38} + 24 q^{40} - 26 q^{41} + 18 q^{43} - 25 q^{44} + 4 q^{46} - 12 q^{47} + 23 q^{49} - 43 q^{50} + 22 q^{52} - 36 q^{53} + 4 q^{55} - 26 q^{56} - 20 q^{58} - 46 q^{59} - 25 q^{61} + 14 q^{62} - 13 q^{64} - 60 q^{65} - 20 q^{67} - 44 q^{68} - 20 q^{70} - 52 q^{71} + 6 q^{73} - 32 q^{74} - 4 q^{77} + 26 q^{79} - 52 q^{80} + 6 q^{82} - 38 q^{83} - 4 q^{85} - 34 q^{86} + 15 q^{88} - 82 q^{89} - 58 q^{91} - 36 q^{92} + 16 q^{94} - 30 q^{95} - 35 q^{98}+O(q^{100}) \) 25 * q - 5 * q^2 + 25 * q^4 - 4 * q^5 + 4 * q^7 - 15 * q^8 - 25 * q^11 + 4 * q^13 - 18 * q^14 + 21 * q^16 - 20 * q^17 + 14 * q^19 - 12 * q^20 + 5 * q^22 - 20 * q^23 + 13 * q^25 - 16 * q^26 - 14 * q^28 - 28 * q^29 - 12 * q^31 - 35 * q^32 + 6 * q^34 - 10 * q^35 - 8 * q^37 - 32 * q^38 + 24 * q^40 - 26 * q^41 + 18 * q^43 - 25 * q^44 + 4 * q^46 - 12 * q^47 + 23 * q^49 - 43 * q^50 + 22 * q^52 - 36 * q^53 + 4 * q^55 - 26 * q^56 - 20 * q^58 - 46 * q^59 - 25 * q^61 + 14 * q^62 - 13 * q^64 - 60 * q^65 - 20 * q^67 - 44 * q^68 - 20 * q^70 - 52 * q^71 + 6 * q^73 - 32 * q^74 - 4 * q^77 + 26 * q^79 - 52 * q^80 + 6 * q^82 - 38 * q^83 - 4 * q^85 - 34 * q^86 + 15 * q^88 - 82 * q^89 - 58 * q^91 - 36 * q^92 + 16 * q^94 - 30 * q^95 - 35 * q^98

Introduction

L-functions

Modular forms

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Database

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