LMFDB - Embedded newform 8019.2.a.k.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8019,2,Mod(1,8019)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8019, 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("8019.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8019 = 3^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8019.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.0320373809$
Analytic rank:	$0$
Dimension:	$51$
Twist minimal:	no (minimal twist has level 297)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8019.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.22452 q^{2} -0.500542 q^{4} -2.64802 q^{5} +2.83927 q^{7} +3.06197 q^{8} +O(q^{10})$	\(q-1.22452 q^{2} -0.500542 q^{4} -2.64802 q^{5} +2.83927 q^{7} +3.06197 q^{8} +3.24257 q^{10} -1.00000 q^{11} +4.69951 q^{13} -3.47676 q^{14} -2.74837 q^{16} +3.10940 q^{17} -5.47311 q^{19} +1.32545 q^{20} +1.22452 q^{22} -2.50411 q^{23} +2.01203 q^{25} -5.75466 q^{26} -1.42118 q^{28} +9.51682 q^{29} +8.13920 q^{31} -2.75850 q^{32} -3.80754 q^{34} -7.51847 q^{35} +5.38996 q^{37} +6.70195 q^{38} -8.10818 q^{40} +2.17815 q^{41} +11.5286 q^{43} +0.500542 q^{44} +3.06635 q^{46} +5.03450 q^{47} +1.06148 q^{49} -2.46378 q^{50} -2.35230 q^{52} -12.3339 q^{53} +2.64802 q^{55} +8.69378 q^{56} -11.6536 q^{58} -9.99726 q^{59} +9.21981 q^{61} -9.96664 q^{62} +8.87459 q^{64} -12.4444 q^{65} +7.00703 q^{67} -1.55639 q^{68} +9.20654 q^{70} -7.41003 q^{71} -2.76742 q^{73} -6.60013 q^{74} +2.73952 q^{76} -2.83927 q^{77} +4.25360 q^{79} +7.27776 q^{80} -2.66720 q^{82} +0.232728 q^{83} -8.23377 q^{85} -14.1170 q^{86} -3.06197 q^{88} +3.09536 q^{89} +13.3432 q^{91} +1.25341 q^{92} -6.16487 q^{94} +14.4929 q^{95} +10.4536 q^{97} -1.29980 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7}+O(q^{10}) $ 51 * q + 60 * q^4 - 6 * q^5 + 12 * q^7	$ 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7} + 18 q^{10} - 51 q^{11} + 30 q^{13} - 12 q^{14} + 78 q^{16} + 30 q^{19} - 18 q^{20} - 3 q^{23} + 75 q^{25} - 9 q^{26} + 36 q^{28} + 42 q^{31} + 15 q^{32} + 42 q^{34} + 9 q^{35} + 48 q^{37} + 3 q^{38} + 54 q^{40} + 18 q^{43} - 60 q^{44} + 42 q^{46} - 30 q^{47} + 99 q^{49} + 30 q^{50} + 60 q^{52} - 18 q^{53} + 6 q^{55} - 21 q^{56} + 30 q^{58} - 24 q^{59} + 99 q^{61} + 114 q^{64} + 15 q^{65} + 39 q^{67} + 39 q^{68} + 48 q^{70} - 30 q^{71} + 69 q^{73} + 90 q^{76} - 12 q^{77} + 48 q^{79} - 42 q^{80} + 42 q^{82} + 21 q^{83} + 84 q^{85} - 24 q^{86} - 15 q^{89} + 69 q^{91} + 66 q^{92} + 66 q^{94} + 12 q^{95} + 72 q^{97} + 57 q^{98}+O(q^{100}) $ 51 * q + 60 * q^4 - 6 * q^5 + 12 * q^7 + 18 * q^10 - 51 * q^11 + 30 * q^13 - 12 * q^14 + 78 * q^16 + 30 * q^19 - 18 * q^20 - 3 * q^23 + 75 * q^25 - 9 * q^26 + 36 * q^28 + 42 * q^31 + 15 * q^32 + 42 * q^34 + 9 * q^35 + 48 * q^37 + 3 * q^38 + 54 * q^40 + 18 * q^43 - 60 * q^44 + 42 * q^46 - 30 * q^47 + 99 * q^49 + 30 * q^50 + 60 * q^52 - 18 * q^53 + 6 * q^55 - 21 * q^56 + 30 * q^58 - 24 * q^59 + 99 * q^61 + 114 * q^64 + 15 * q^65 + 39 * q^67 + 39 * q^68 + 48 * q^70 - 30 * q^71 + 69 * q^73 + 90 * q^76 - 12 * q^77 + 48 * q^79 - 42 * q^80 + 42 * q^82 + 21 * q^83 + 84 * q^85 - 24 * q^86 - 15 * q^89 + 69 * q^91 + 66 * q^92 + 66 * q^94 + 12 * q^95 + 72 * q^97 + 57 * 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 $

$2$	−1.22452	−0.865869	−0.432934	−	0.901425i	$-0.642522\pi$
$2$	−1.22452	−0.865869	−0.432934	+	0.901425i	$0.642522\pi$
$3$	0	0
$3$	0	0
$4$	−0.500542	−0.250271
$5$	−2.64802	−1.18423	−0.592116	−	0.805853i	$-0.701707\pi$
$5$	−2.64802	−1.18423	−0.592116	+	0.805853i	$0.701707\pi$
$6$	0	0
$7$	2.83927	1.07314	0.536572	−	0.843854i	$-0.319719\pi$
$7$	2.83927	1.07314	0.536572	+	0.843854i	$0.319719\pi$
$8$	3.06197	1.08257
$9$	0	0
$10$	3.24257	1.02539
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.69951	1.30341	0.651705	−	0.758473i	$-0.274054\pi$
$13$	4.69951	1.30341	0.651705	+	0.758473i	$0.274054\pi$
$14$	−3.47676	−0.929203
$15$	0	0
$16$	−2.74837	−0.687094
$17$	3.10940	0.754141	0.377070	−	0.926185i	$-0.376931\pi$
$17$	3.10940	0.754141	0.377070	+	0.926185i	$0.376931\pi$
$18$	0	0
$19$	−5.47311	−1.25562	−0.627808	−	0.778368i	$-0.716048\pi$
$19$	−5.47311	−1.25562	−0.627808	+	0.778368i	$0.716048\pi$
$20$	1.32545	0.296379
$21$	0	0
$22$	1.22452	0.261069
$23$	−2.50411	−0.522144	−0.261072	−	0.965319i	$-0.584076\pi$
$23$	−2.50411	−0.522144	−0.261072	+	0.965319i	$0.584076\pi$
$24$	0	0
$25$	2.01203	0.402406
$26$	−5.75466	−1.12858
$27$	0	0
$28$	−1.42118	−0.268577
$29$	9.51682	1.76723	0.883614	−	0.468216i	$-0.155103\pi$
$29$	9.51682	1.76723	0.883614	+	0.468216i	$0.155103\pi$
$30$	0	0
$31$	8.13920	1.46184	0.730921	−	0.682462i	$-0.239091\pi$
$31$	8.13920	1.46184	0.730921	+	0.682462i	$0.239091\pi$
$32$	−2.75850	−0.487638
$33$	0	0
$34$	−3.80754	−0.652987
$35$	−7.51847	−1.27085
$36$	0	0
$37$	5.38996	0.886104	0.443052	−	0.896496i	$-0.353896\pi$
$37$	5.38996	0.886104	0.443052	+	0.896496i	$0.353896\pi$
$38$	6.70195	1.08720
$39$	0	0
$40$	−8.10818	−1.28202
$41$	2.17815	0.340170	0.170085	−	0.985429i	$-0.445596\pi$
$41$	2.17815	0.340170	0.170085	+	0.985429i	$0.445596\pi$
$42$	0	0
$43$	11.5286	1.75809	0.879044	−	0.476740i	$-0.158182\pi$
$43$	11.5286	1.75809	0.879044	+	0.476740i	$0.158182\pi$
$44$	0.500542	0.0754595
$45$	0	0
$46$	3.06635	0.452108
$47$	5.03450	0.734358	0.367179	−	0.930150i	$-0.380324\pi$
$47$	5.03450	0.734358	0.367179	+	0.930150i	$0.380324\pi$
$48$	0	0
$49$	1.06148	0.151640
$50$	−2.46378	−0.348431
$51$	0	0
$52$	−2.35230	−0.326205
$53$	−12.3339	−1.69419	−0.847095	−	0.531442i	$-0.821650\pi$
$53$	−12.3339	−1.69419	−0.847095	+	0.531442i	$0.821650\pi$
$54$	0	0
$55$	2.64802	0.357059
$56$	8.69378	1.16176
$57$	0	0
$58$	−11.6536	−1.53019
$59$	−9.99726	−1.30153	−0.650766	−	0.759278i	$-0.725552\pi$
$59$	−9.99726	−1.30153	−0.650766	+	0.759278i	$0.725552\pi$
$60$	0	0
$61$	9.21981	1.18048	0.590238	−	0.807229i	$-0.299034\pi$
$61$	9.21981	1.18048	0.590238	+	0.807229i	$0.299034\pi$
$62$	−9.96664	−1.26576
$63$	0	0
$64$	8.87459	1.10932
$65$	−12.4444	−1.54354
$66$	0	0
$67$	7.00703	0.856045	0.428022	−	0.903768i	$-0.359210\pi$
$67$	7.00703	0.856045	0.428022	+	0.903768i	$0.359210\pi$
$68$	−1.55639	−0.188740
$69$	0	0
$70$	9.20654	1.10039
$71$	−7.41003	−0.879408	−0.439704	−	0.898143i	$-0.644917\pi$
$71$	−7.41003	−0.879408	−0.439704	+	0.898143i	$0.644917\pi$
$72$	0	0
$73$	−2.76742	−0.323902	−0.161951	−	0.986799i	$-0.551779\pi$
$73$	−2.76742	−0.323902	−0.161951	+	0.986799i	$0.551779\pi$
$74$	−6.60013	−0.767250
$75$	0	0
$76$	2.73952	0.314244
$77$	−2.83927	−0.323565
$78$	0	0
$79$	4.25360	0.478567	0.239284	−	0.970950i	$-0.423087\pi$
$79$	4.25360	0.478567	0.239284	+	0.970950i	$0.423087\pi$
$80$	7.27776	0.813678
$81$	0	0
$82$	−2.66720	−0.294543
$83$	0.232728	0.0255452	0.0127726	−	0.999918i	$-0.495934\pi$
$83$	0.232728	0.0255452	0.0127726	+	0.999918i	$0.495934\pi$
$84$	0	0
$85$	−8.23377	−0.893078
$86$	−14.1170	−1.52227
$87$	0	0
$88$	−3.06197	−0.326407
$89$	3.09536	0.328108	0.164054	−	0.986451i	$-0.447543\pi$
$89$	3.09536	0.328108	0.164054	+	0.986451i	$0.447543\pi$
$90$	0	0
$91$	13.3432	1.39875
$92$	1.25341	0.130677
$93$	0	0
$94$	−6.16487	−0.635857
$95$	14.4929	1.48694
$96$	0	0
$97$	10.4536	1.06140	0.530699	−	0.847560i	$-0.321929\pi$
$97$	10.4536	1.06140	0.530699	+	0.847560i	$0.321929\pi$
$98$	−1.29980	−0.131300
$99$	0	0
$100$	−1.00711	−0.100711
$101$	−5.47113	−0.544398	−0.272199	−	0.962241i	$-0.587751\pi$
$101$	−5.47113	−0.544398	−0.272199	+	0.962241i	$0.587751\pi$
$102$	0	0
$103$	−11.8797	−1.17054	−0.585271	−	0.810838i	$-0.699012\pi$
$103$	−11.8797	−1.17054	−0.585271	+	0.810838i	$0.699012\pi$
$104$	14.3898	1.41103
$105$	0	0
$106$	15.1031	1.46695
$107$	12.4665	1.20518	0.602590	−	0.798051i	$-0.294135\pi$
$107$	12.4665	1.20518	0.602590	+	0.798051i	$0.294135\pi$
$108$	0	0
$109$	−12.7802	−1.22412	−0.612061	−	0.790810i	$-0.709659\pi$
$109$	−12.7802	−1.22412	−0.612061	+	0.790810i	$0.709659\pi$
$110$	−3.24257	−0.309167
$111$	0	0
$112$	−7.80339	−0.737351
$113$	8.94863	0.841816	0.420908	−	0.907103i	$-0.361711\pi$
$113$	8.94863	0.841816	0.420908	+	0.907103i	$0.361711\pi$
$114$	0	0
$115$	6.63095	0.618340
$116$	−4.76357	−0.442286
$117$	0	0
$118$	12.2419	1.12696
$119$	8.82844	0.809302
$120$	0	0
$121$	1.00000	0.0909091
$122$	−11.2899	−1.02214
$123$	0	0
$124$	−4.07401	−0.365857
$125$	7.91221	0.707690
$126$	0	0
$127$	3.79068	0.336369	0.168184	−	0.985756i	$-0.446210\pi$
$127$	3.79068	0.336369	0.168184	+	0.985756i	$0.446210\pi$
$128$	−5.35015	−0.472891
$129$	0	0
$130$	15.2385	1.33650
$131$	−8.32042	−0.726958	−0.363479	−	0.931602i	$-0.618411\pi$
$131$	−8.32042	−0.726958	−0.363479	+	0.931602i	$0.618411\pi$
$132$	0	0
$133$	−15.5396	−1.34746
$134$	−8.58027	−0.741222
$135$	0	0
$136$	9.52090	0.816411
$137$	1.37605	0.117564	0.0587819	−	0.998271i	$-0.481278\pi$
$137$	1.37605	0.117564	0.0587819	+	0.998271i	$0.481278\pi$
$138$	0	0
$139$	−9.46131	−0.802498	−0.401249	−	0.915969i	$-0.631424\pi$
$139$	−9.46131	−0.802498	−0.401249	+	0.915969i	$0.631424\pi$
$140$	3.76331	0.318058
$141$	0	0
$142$	9.07375	0.761452
$143$	−4.69951	−0.392993
$144$	0	0
$145$	−25.2008	−2.09281
$146$	3.38877	0.280457
$147$	0	0
$148$	−2.69790	−0.221766
$149$	10.2264	0.837782	0.418891	−	0.908036i	$-0.362419\pi$
$149$	10.2264	0.837782	0.418891	+	0.908036i	$0.362419\pi$
$150$	0	0
$151$	−13.3204	−1.08400	−0.542000	−	0.840378i	$-0.682333\pi$
$151$	−13.3204	−1.08400	−0.542000	+	0.840378i	$0.682333\pi$
$152$	−16.7585	−1.35929
$153$	0	0
$154$	3.47676	0.280165
$155$	−21.5528	−1.73116
$156$	0	0
$157$	12.6335	1.00826	0.504130	−	0.863628i	$-0.331813\pi$
$157$	12.6335	1.00826	0.504130	+	0.863628i	$0.331813\pi$
$158$	−5.20863	−0.414377
$159$	0	0
$160$	7.30456	0.577477
$161$	−7.10986	−0.560336
$162$	0	0
$163$	−9.33091	−0.730854	−0.365427	−	0.930840i	$-0.619077\pi$
$163$	−9.33091	−0.730854	−0.365427	+	0.930840i	$0.619077\pi$
$164$	−1.09026	−0.0851348
$165$	0	0
$166$	−0.284980	−0.0221188
$167$	−13.3307	−1.03156	−0.515781	−	0.856720i	$-0.672498\pi$
$167$	−13.3307	−1.03156	−0.515781	+	0.856720i	$0.672498\pi$
$168$	0	0
$169$	9.08539	0.698876
$170$	10.0824	0.773288
$171$	0	0
$172$	−5.77053	−0.439998
$173$	−0.479285	−0.0364393	−0.0182197	−	0.999834i	$-0.505800\pi$
$173$	−0.479285	−0.0364393	−0.0182197	+	0.999834i	$0.505800\pi$
$174$	0	0
$175$	5.71271	0.431840
$176$	2.74837	0.207166
$177$	0	0
$178$	−3.79035	−0.284098
$179$	−2.37420	−0.177456	−0.0887279	−	0.996056i	$-0.528280\pi$
$179$	−2.37420	−0.177456	−0.0887279	+	0.996056i	$0.528280\pi$
$180$	0	0
$181$	9.17569	0.682023	0.341012	−	0.940059i	$-0.389230\pi$
$181$	9.17569	0.682023	0.341012	+	0.940059i	$0.389230\pi$
$182$	−16.3391	−1.21113
$183$	0	0
$184$	−7.66753	−0.565258
$185$	−14.2727	−1.04935
$186$	0	0
$187$	−3.10940	−0.227382
$188$	−2.51998	−0.183788
$189$	0	0
$190$	−17.7469	−1.28750
$191$	−19.6071	−1.41872	−0.709360	−	0.704846i	$-0.751016\pi$
$191$	−19.6071	−1.41872	−0.709360	+	0.704846i	$0.751016\pi$
$192$	0	0
$193$	12.4150	0.893654	0.446827	−	0.894620i	$-0.352554\pi$
$193$	12.4150	0.893654	0.446827	+	0.894620i	$0.352554\pi$
$194$	−12.8006	−0.919032
$195$	0	0
$196$	−0.531314	−0.0379510
$197$	−7.45944	−0.531463	−0.265732	−	0.964047i	$-0.585614\pi$
$197$	−7.45944	−0.531463	−0.265732	+	0.964047i	$0.585614\pi$
$198$	0	0
$199$	4.50065	0.319043	0.159521	−	0.987194i	$-0.449005\pi$
$199$	4.50065	0.319043	0.159521	+	0.987194i	$0.449005\pi$
$200$	6.16078	0.435633
$201$	0	0
$202$	6.69953	0.471378
$203$	27.0208	1.89649
$204$	0	0
$205$	−5.76780	−0.402841
$206$	14.5470	1.01354
$207$	0	0
$208$	−12.9160	−0.895564
$209$	5.47311	0.378583
$210$	0	0
$211$	−16.9396	−1.16617	−0.583086	−	0.812410i	$-0.698155\pi$
$211$	−16.9396	−1.16617	−0.583086	+	0.812410i	$0.698155\pi$
$212$	6.17363	0.424006
$213$	0	0
$214$	−15.2655	−1.04353
$215$	−30.5279	−2.08198
$216$	0	0
$217$	23.1094	1.56877
$218$	15.6497	1.05993
$219$	0	0
$220$	−1.32545	−0.0893616
$221$	14.6127	0.982954
$222$	0	0
$223$	27.6706	1.85296	0.926482	−	0.376340i	$-0.122818\pi$
$223$	27.6706	1.85296	0.926482	+	0.376340i	$0.122818\pi$
$224$	−7.83213	−0.523306
$225$	0	0
$226$	−10.9578	−0.728903
$227$	4.07999	0.270799	0.135399	−	0.990791i	$-0.456768\pi$
$227$	4.07999	0.270799	0.135399	+	0.990791i	$0.456768\pi$
$228$	0	0
$229$	3.81594	0.252164	0.126082	−	0.992020i	$-0.459760\pi$
$229$	3.81594	0.252164	0.126082	+	0.992020i	$0.459760\pi$
$230$	−8.11976	−0.535401
$231$	0	0
$232$	29.1402	1.91315
$233$	−3.19103	−0.209051	−0.104526	−	0.994522i	$-0.533332\pi$
$233$	−3.19103	−0.209051	−0.104526	+	0.994522i	$0.533332\pi$
$234$	0	0
$235$	−13.3315	−0.869650
$236$	5.00405	0.325736
$237$	0	0
$238$	−10.8106	−0.700750
$239$	−18.1735	−1.17555	−0.587774	−	0.809025i	$-0.699996\pi$
$239$	−18.1735	−1.17555	−0.587774	+	0.809025i	$0.699996\pi$
$240$	0	0
$241$	−23.3889	−1.50661	−0.753307	−	0.657669i	$-0.771543\pi$
$241$	−23.3889	−1.50661	−0.753307	+	0.657669i	$0.771543\pi$
$242$	−1.22452	−0.0787154
$243$	0	0
$244$	−4.61490	−0.295439
$245$	−2.81082	−0.179577
$246$	0	0
$247$	−25.7209	−1.63658
$248$	24.9220	1.58255
$249$	0	0
$250$	−9.68869	−0.612767
$251$	9.83159	0.620564	0.310282	−	0.950645i	$-0.399576\pi$
$251$	9.83159	0.620564	0.310282	+	0.950645i	$0.399576\pi$
$252$	0	0
$253$	2.50411	0.157432
$254$	−4.64178	−0.291251
$255$	0	0
$256$	−11.1978	−0.699862
$257$	−3.38432	−0.211108	−0.105554	−	0.994414i	$-0.533662\pi$
$257$	−3.38432	−0.211108	−0.105554	+	0.994414i	$0.533662\pi$
$258$	0	0
$259$	15.3036	0.950918
$260$	6.22895	0.386303
$261$	0	0
$262$	10.1885	0.629451
$263$	−4.52331	−0.278920	−0.139460	−	0.990228i	$-0.544537\pi$
$263$	−4.52331	−0.278920	−0.139460	+	0.990228i	$0.544537\pi$
$264$	0	0
$265$	32.6604	2.00631
$266$	19.0287	1.16672
$267$	0	0
$268$	−3.50731	−0.214243
$269$	−25.2942	−1.54221	−0.771106	−	0.636706i	$-0.780296\pi$
$269$	−25.2942	−1.54221	−0.771106	+	0.636706i	$0.780296\pi$
$270$	0	0
$271$	−3.02173	−0.183557	−0.0917786	−	0.995779i	$-0.529255\pi$
$271$	−3.02173	−0.183557	−0.0917786	+	0.995779i	$0.529255\pi$
$272$	−8.54580	−0.518165
$273$	0	0
$274$	−1.68500	−0.101795
$275$	−2.01203	−0.121330
$276$	0	0
$277$	9.04297	0.543340	0.271670	−	0.962391i	$-0.412424\pi$
$277$	9.04297	0.543340	0.271670	+	0.962391i	$0.412424\pi$
$278$	11.5856	0.694858
$279$	0	0
$280$	−23.0213	−1.37579
$281$	11.8705	0.708137	0.354069	−	0.935219i	$-0.384798\pi$
$281$	11.8705	0.708137	0.354069	+	0.935219i	$0.384798\pi$
$282$	0	0
$283$	−9.76690	−0.580582	−0.290291	−	0.956938i	$-0.593752\pi$
$283$	−9.76690	−0.580582	−0.290291	+	0.956938i	$0.593752\pi$
$284$	3.70903	0.220090
$285$	0	0
$286$	5.75466	0.340280
$287$	6.18437	0.365052
$288$	0	0
$289$	−7.33162	−0.431272
$290$	30.8589	1.81210
$291$	0	0
$292$	1.38521	0.0810632
$293$	10.2681	0.599869	0.299934	−	0.953960i	$-0.403035\pi$
$293$	10.2681	0.599869	0.299934	+	0.953960i	$0.403035\pi$
$294$	0	0
$295$	26.4730	1.54132
$296$	16.5039	0.959270
$297$	0	0
$298$	−12.5225	−0.725410
$299$	−11.7681	−0.680567
$300$	0	0
$301$	32.7327	1.88668
$302$	16.3112	0.938603
$303$	0	0
$304$	15.0421	0.862726
$305$	−24.4143	−1.39796
$306$	0	0
$307$	−20.9965	−1.19834	−0.599168	−	0.800624i	$-0.704502\pi$
$307$	−20.9965	−1.19834	−0.599168	+	0.800624i	$0.704502\pi$
$308$	1.42118	0.0809790
$309$	0	0
$310$	26.3919	1.49896
$311$	−24.5121	−1.38996	−0.694978	−	0.719031i	$-0.744586\pi$
$311$	−24.5121	−1.38996	−0.694978	+	0.719031i	$0.744586\pi$
$312$	0	0
$313$	14.8332	0.838423	0.419212	−	0.907889i	$-0.362307\pi$
$313$	14.8332	0.838423	0.419212	+	0.907889i	$0.362307\pi$
$314$	−15.4700	−0.873022
$315$	0	0
$316$	−2.12911	−0.119772
$317$	7.39012	0.415071	0.207535	−	0.978228i	$-0.433456\pi$
$317$	7.39012	0.415071	0.207535	+	0.978228i	$0.433456\pi$
$318$	0	0
$319$	−9.51682	−0.532839
$320$	−23.5001	−1.31370
$321$	0	0
$322$	8.70620	0.485177
$323$	−17.0181	−0.946912
$324$	0	0
$325$	9.45556	0.524500
$326$	11.4259	0.632823
$327$	0	0
$328$	6.66944	0.368258
$329$	14.2943	0.788072
$330$	0	0
$331$	12.7381	0.700149	0.350074	−	0.936722i	$-0.386156\pi$
$331$	12.7381	0.700149	0.350074	+	0.936722i	$0.386156\pi$
$332$	−0.116490	−0.00639321
$333$	0	0
$334$	16.3238	0.893198
$335$	−18.5548	−1.01376
$336$	0	0
$337$	−1.66760	−0.0908401	−0.0454201	−	0.998968i	$-0.514463\pi$
$337$	−1.66760	−0.0908401	−0.0454201	+	0.998968i	$0.514463\pi$
$338$	−11.1253	−0.605135
$339$	0	0
$340$	4.12135	0.223511
$341$	−8.13920	−0.440762
$342$	0	0
$343$	−16.8611	−0.910413
$344$	35.3001	1.90326
$345$	0	0
$346$	0.586896	0.0315517
$347$	27.0442	1.45181	0.725904	−	0.687796i	$-0.241422\pi$
$347$	27.0442	1.45181	0.725904	+	0.687796i	$0.241422\pi$
$348$	0	0
$349$	20.4043	1.09222	0.546109	−	0.837714i	$-0.316108\pi$
$349$	20.4043	1.09222	0.546109	+	0.837714i	$0.316108\pi$
$350$	−6.99535	−0.373917
$351$	0	0
$352$	2.75850	0.147028
$353$	−26.3227	−1.40101	−0.700507	−	0.713645i	$-0.747043\pi$
$353$	−26.3227	−1.40101	−0.700507	+	0.713645i	$0.747043\pi$
$354$	0	0
$355$	19.6219	1.04142
$356$	−1.54936	−0.0821159
$357$	0	0
$358$	2.90726	0.153653
$359$	−2.68829	−0.141883	−0.0709413	−	0.997480i	$-0.522600\pi$
$359$	−2.68829	−0.141883	−0.0709413	+	0.997480i	$0.522600\pi$
$360$	0	0
$361$	10.9549	0.576573
$362$	−11.2358	−0.590543
$363$	0	0
$364$	−6.67883	−0.350066
$365$	7.32819	0.383575
$366$	0	0
$367$	−0.958705	−0.0500440	−0.0250220	−	0.999687i	$-0.507966\pi$
$367$	−0.958705	−0.0500440	−0.0250220	+	0.999687i	$0.507966\pi$
$368$	6.88224	0.358762
$369$	0	0
$370$	17.4773	0.908602
$371$	−35.0193	−1.81811
$372$	0	0
$373$	12.5570	0.650176	0.325088	−	0.945684i	$-0.394606\pi$
$373$	12.5570	0.650176	0.325088	+	0.945684i	$0.394606\pi$
$374$	3.80754	0.196883
$375$	0	0
$376$	15.4155	0.794994
$377$	44.7244	2.30342
$378$	0	0
$379$	25.2036	1.29462	0.647311	−	0.762226i	$-0.275894\pi$
$379$	25.2036	1.29462	0.647311	+	0.762226i	$0.275894\pi$
$380$	−7.25431	−0.372138
$381$	0	0
$382$	24.0094	1.22843
$383$	18.0203	0.920794	0.460397	−	0.887713i	$-0.347707\pi$
$383$	18.0203	0.920794	0.460397	+	0.887713i	$0.347707\pi$
$384$	0	0
$385$	7.51847	0.383177
$386$	−15.2025	−0.773787
$387$	0	0
$388$	−5.23244	−0.265637
$389$	24.3253	1.23334	0.616671	−	0.787221i	$-0.288481\pi$
$389$	24.3253	1.23334	0.616671	+	0.787221i	$0.288481\pi$
$390$	0	0
$391$	−7.78630	−0.393770
$392$	3.25021	0.164161
$393$	0	0
$394$	9.13426	0.460178
$395$	−11.2636	−0.566735
$396$	0	0
$397$	16.5701	0.831631	0.415816	−	0.909449i	$-0.363496\pi$
$397$	16.5701	0.831631	0.415816	+	0.909449i	$0.363496\pi$
$398$	−5.51116	−0.276249
$399$	0	0
$400$	−5.52981	−0.276491
$401$	16.5953	0.828728	0.414364	−	0.910111i	$-0.364004\pi$
$401$	16.5953	0.828728	0.414364	+	0.910111i	$0.364004\pi$
$402$	0	0
$403$	38.2502	1.90538
$404$	2.73853	0.136247
$405$	0	0
$406$	−33.0877	−1.64211
$407$	−5.38996	−0.267170
$408$	0	0
$409$	−21.0958	−1.04312	−0.521560	−	0.853214i	$-0.674650\pi$
$409$	−21.0958	−1.04312	−0.521560	+	0.853214i	$0.674650\pi$
$410$	7.06281	0.348807
$411$	0	0
$412$	5.94629	0.292952
$413$	−28.3850	−1.39673
$414$	0	0
$415$	−0.616268	−0.0302514
$416$	−12.9636	−0.635592
$417$	0	0
$418$	−6.70195	−0.327803
$419$	24.2030	1.18239	0.591197	−	0.806527i	$-0.298656\pi$
$419$	24.2030	1.18239	0.591197	+	0.806527i	$0.298656\pi$
$420$	0	0
$421$	−22.2975	−1.08671	−0.543356	−	0.839502i	$-0.682847\pi$
$421$	−22.2975	−1.08671	−0.543356	+	0.839502i	$0.682847\pi$
$422$	20.7430	1.00975
$423$	0	0
$424$	−37.7660	−1.83408
$425$	6.25621	0.303471
$426$	0	0
$427$	26.1776	1.26682
$428$	−6.24000	−0.301622
$429$	0	0
$430$	37.3821	1.80273
$431$	26.1411	1.25917	0.629587	−	0.776930i	$-0.283224\pi$
$431$	26.1411	1.25917	0.629587	+	0.776930i	$0.283224\pi$
$432$	0	0
$433$	22.6505	1.08851	0.544257	−	0.838918i	$-0.316811\pi$
$433$	22.6505	1.08851	0.544257	+	0.838918i	$0.316811\pi$
$434$	−28.2980	−1.35835
$435$	0	0
$436$	6.39703	0.306362
$437$	13.7053	0.655612
$438$	0	0
$439$	−19.7869	−0.944377	−0.472189	−	0.881498i	$-0.656536\pi$
$439$	−19.7869	−0.944377	−0.472189	+	0.881498i	$0.656536\pi$
$440$	8.10818	0.386542
$441$	0	0
$442$	−17.8936	−0.851109
$443$	0.479537	0.0227835	0.0113918	−	0.999935i	$-0.496374\pi$
$443$	0.479537	0.0227835	0.0113918	+	0.999935i	$0.496374\pi$
$444$	0	0
$445$	−8.19660	−0.388556
$446$	−33.8834	−1.60442
$447$	0	0
$448$	25.1974	1.19047
$449$	−0.970987	−0.0458237	−0.0229119	−	0.999737i	$-0.507294\pi$
$449$	−0.970987	−0.0458237	−0.0229119	+	0.999737i	$0.507294\pi$
$450$	0	0
$451$	−2.17815	−0.102565
$452$	−4.47916	−0.210682
$453$	0	0
$454$	−4.99605	−0.234476
$455$	−35.3331	−1.65644
$456$	0	0
$457$	38.1788	1.78593	0.892966	−	0.450125i	$-0.148621\pi$
$457$	38.1788	1.78593	0.892966	+	0.450125i	$0.148621\pi$
$458$	−4.67271	−0.218341
$459$	0	0
$460$	−3.31907	−0.154752
$461$	30.3619	1.41409	0.707047	−	0.707167i	$-0.250027\pi$
$461$	30.3619	1.41409	0.707047	+	0.707167i	$0.250027\pi$
$462$	0	0
$463$	9.71269	0.451387	0.225694	−	0.974198i	$-0.427535\pi$
$463$	9.71269	0.451387	0.225694	+	0.974198i	$0.427535\pi$
$464$	−26.1558	−1.21425
$465$	0	0
$466$	3.90749	0.181011
$467$	37.4957	1.73509	0.867547	−	0.497354i	$-0.165695\pi$
$467$	37.4957	1.73509	0.867547	+	0.497354i	$0.165695\pi$
$468$	0	0
$469$	19.8949	0.918660
$470$	16.3247	0.753003
$471$	0	0
$472$	−30.6113	−1.40900
$473$	−11.5286	−0.530084
$474$	0	0
$475$	−11.0121	−0.505268
$476$	−4.41901	−0.202545
$477$	0	0
$478$	22.2539	1.01787
$479$	11.5591	0.528151	0.264075	−	0.964502i	$-0.414933\pi$
$479$	11.5591	0.528151	0.264075	+	0.964502i	$0.414933\pi$
$480$	0	0
$481$	25.3302	1.15496
$482$	28.6403	1.30453
$483$	0	0
$484$	−0.500542	−0.0227519
$485$	−27.6813	−1.25694
$486$	0	0
$487$	38.9308	1.76412	0.882062	−	0.471133i	$-0.156155\pi$
$487$	38.9308	1.76412	0.882062	+	0.471133i	$0.156155\pi$
$488$	28.2308	1.27795
$489$	0	0
$490$	3.44191	0.155490
$491$	−10.5458	−0.475927	−0.237964	−	0.971274i	$-0.576480\pi$
$491$	−10.5458	−0.475927	−0.237964	+	0.971274i	$0.576480\pi$
$492$	0	0
$493$	29.5916	1.33274
$494$	31.4959	1.41707
$495$	0	0
$496$	−22.3696	−1.00442
$497$	−21.0391	−0.943733
$498$	0	0
$499$	5.42304	0.242769	0.121384	−	0.992606i	$-0.461267\pi$
$499$	5.42304	0.242769	0.121384	+	0.992606i	$0.461267\pi$
$500$	−3.96039	−0.177114
$501$	0	0
$502$	−12.0390	−0.537327
$503$	−8.61538	−0.384141	−0.192070	−	0.981381i	$-0.561520\pi$
$503$	−8.61538	−0.384141	−0.192070	+	0.981381i	$0.561520\pi$
$504$	0	0
$505$	14.4877	0.644694
$506$	−3.06635	−0.136316
$507$	0	0
$508$	−1.89740	−0.0841833
$509$	40.0282	1.77422	0.887109	−	0.461560i	$-0.152710\pi$
$509$	40.0282	1.77422	0.887109	+	0.461560i	$0.152710\pi$
$510$	0	0
$511$	−7.85746	−0.347594
$512$	24.4123	1.07888
$513$	0	0
$514$	4.14418	0.182792
$515$	31.4577	1.38619
$516$	0	0
$517$	−5.03450	−0.221417
$518$	−18.7396	−0.823370
$519$	0	0
$520$	−38.1045	−1.67099
$521$	23.8196	1.04356	0.521778	−	0.853081i	$-0.325269\pi$
$521$	23.8196	1.04356	0.521778	+	0.853081i	$0.325269\pi$
$522$	0	0
$523$	25.9511	1.13476	0.567382	−	0.823455i	$-0.307956\pi$
$523$	25.9511	1.13476	0.567382	+	0.823455i	$0.307956\pi$
$524$	4.16472	0.181937
$525$	0	0
$526$	5.53891	0.241508
$527$	25.3080	1.10244
$528$	0	0
$529$	−16.7294	−0.727366
$530$	−39.9935	−1.73720
$531$	0	0
$532$	7.77825	0.337230
$533$	10.2362	0.443381
$534$	0	0
$535$	−33.0115	−1.42721
$536$	21.4553	0.926729
$537$	0	0
$538$	30.9733	1.33535
$539$	−1.06148	−0.0457211
$540$	0	0
$541$	12.1393	0.521909	0.260955	−	0.965351i	$-0.415963\pi$
$541$	12.1393	0.521909	0.260955	+	0.965351i	$0.415963\pi$
$542$	3.70018	0.158936
$543$	0	0
$544$	−8.57727	−0.367748
$545$	33.8423	1.44965
$546$	0	0
$547$	13.4614	0.575568	0.287784	−	0.957695i	$-0.407081\pi$
$547$	13.4614	0.575568	0.287784	+	0.957695i	$0.407081\pi$
$548$	−0.688770	−0.0294228
$549$	0	0
$550$	2.46378	0.105056
$551$	−52.0865	−2.21896
$552$	0	0
$553$	12.0771	0.513572
$554$	−11.0733	−0.470461
$555$	0	0
$556$	4.73578	0.200842
$557$	−23.5018	−0.995803	−0.497901	−	0.867234i	$-0.665896\pi$
$557$	−23.5018	−0.995803	−0.497901	+	0.867234i	$0.665896\pi$
$558$	0	0
$559$	54.1786	2.29151
$560$	20.6636	0.873195
$561$	0	0
$562$	−14.5358	−0.613154
$563$	32.9282	1.38776	0.693880	−	0.720091i	$-0.255900\pi$
$563$	32.9282	1.38776	0.693880	+	0.720091i	$0.255900\pi$
$564$	0	0
$565$	−23.6962	−0.996906
$566$	11.9598	0.502708
$567$	0	0
$568$	−22.6893	−0.952022
$569$	24.1634	1.01298	0.506490	−	0.862246i	$-0.330943\pi$
$569$	24.1634	1.01298	0.506490	+	0.862246i	$0.330943\pi$
$570$	0	0
$571$	−33.7437	−1.41213	−0.706065	−	0.708148i	$-0.749531\pi$
$571$	−33.7437	−1.41213	−0.706065	+	0.708148i	$0.749531\pi$
$572$	2.35230	0.0983547
$573$	0	0
$574$	−7.57291	−0.316087
$575$	−5.03835	−0.210114
$576$	0	0
$577$	45.6929	1.90222	0.951110	−	0.308853i	$-0.0999452\pi$
$577$	45.6929	1.90222	0.951110	+	0.308853i	$0.0999452\pi$
$578$	8.97774	0.373425
$579$	0	0
$580$	12.6140	0.523769
$581$	0.660777	0.0274137
$582$	0	0
$583$	12.3339	0.510817
$584$	−8.47376	−0.350647
$585$	0	0
$586$	−12.5735	−0.519408
$587$	22.5922	0.932480	0.466240	−	0.884658i	$-0.345608\pi$
$587$	22.5922	0.932480	0.466240	+	0.884658i	$0.345608\pi$
$588$	0	0
$589$	−44.5467	−1.83551
$590$	−32.4168	−1.33458
$591$	0	0
$592$	−14.8136	−0.608836
$593$	−16.6701	−0.684560	−0.342280	−	0.939598i	$-0.611199\pi$
$593$	−16.6701	−0.684560	−0.342280	+	0.939598i	$0.611199\pi$
$594$	0	0
$595$	−23.3779	−0.958402
$596$	−5.11876	−0.209673
$597$	0	0
$598$	14.4103	0.589282
$599$	13.6654	0.558352	0.279176	−	0.960240i	$-0.409939\pi$
$599$	13.6654	0.558352	0.279176	+	0.960240i	$0.409939\pi$
$600$	0	0
$601$	−27.8331	−1.13534	−0.567668	−	0.823257i	$-0.692154\pi$
$601$	−27.8331	−1.13534	−0.567668	+	0.823257i	$0.692154\pi$
$602$	−40.0820	−1.63362
$603$	0	0
$604$	6.66743	0.271294
$605$	−2.64802	−0.107657
$606$	0	0
$607$	32.4900	1.31873	0.659364	−	0.751824i	$-0.270826\pi$
$607$	32.4900	1.31873	0.659364	+	0.751824i	$0.270826\pi$
$608$	15.0975	0.612286
$609$	0	0
$610$	29.8959	1.21045
$611$	23.6597	0.957169
$612$	0	0
$613$	20.4535	0.826111	0.413055	−	0.910706i	$-0.364462\pi$
$613$	20.4535	0.826111	0.413055	+	0.910706i	$0.364462\pi$
$614$	25.7108	1.03760
$615$	0	0
$616$	−8.69378	−0.350282
$617$	−15.2575	−0.614242	−0.307121	−	0.951670i	$-0.599366\pi$
$617$	−15.2575	−0.614242	−0.307121	+	0.951670i	$0.599366\pi$
$618$	0	0
$619$	−1.76826	−0.0710726	−0.0355363	−	0.999368i	$-0.511314\pi$
$619$	−1.76826	−0.0710726	−0.0355363	+	0.999368i	$0.511314\pi$
$620$	10.7881	0.433259
$621$	0	0
$622$	30.0157	1.20352
$623$	8.78859	0.352107
$624$	0	0
$625$	−31.0119	−1.24048
$626$	−18.1636	−0.725965
$627$	0	0
$628$	−6.32358	−0.252338
$629$	16.7596	0.668247
$630$	0	0
$631$	−37.9134	−1.50931	−0.754654	−	0.656123i	$-0.772195\pi$
$631$	−37.9134	−1.50931	−0.754654	+	0.656123i	$0.772195\pi$
$632$	13.0244	0.518083
$633$	0	0
$634$	−9.04938	−0.359397
$635$	−10.0378	−0.398339
$636$	0	0
$637$	4.98842	0.197649
$638$	11.6536	0.461369
$639$	0	0
$640$	14.1673	0.560013
$641$	−44.6663	−1.76421	−0.882106	−	0.471052i	$-0.843875\pi$
$641$	−44.6663	−1.76421	−0.882106	+	0.471052i	$0.843875\pi$
$642$	0	0
$643$	−36.5291	−1.44057	−0.720284	−	0.693680i	$-0.755988\pi$
$643$	−36.5291	−1.44057	−0.720284	+	0.693680i	$0.755988\pi$
$644$	3.55879	0.140236
$645$	0	0
$646$	20.8391	0.819901
$647$	32.8539	1.29162	0.645811	−	0.763497i	$-0.276519\pi$
$647$	32.8539	1.29162	0.645811	+	0.763497i	$0.276519\pi$
$648$	0	0
$649$	9.99726	0.392427
$650$	−11.5786	−0.454148
$651$	0	0
$652$	4.67051	0.182911
$653$	5.82586	0.227983	0.113992	−	0.993482i	$-0.463636\pi$
$653$	5.82586	0.227983	0.113992	+	0.993482i	$0.463636\pi$
$654$	0	0
$655$	22.0327	0.860887
$656$	−5.98638	−0.233729
$657$	0	0
$658$	−17.5037	−0.682367
$659$	8.88384	0.346065	0.173033	−	0.984916i	$-0.444643\pi$
$659$	8.88384	0.346065	0.173033	+	0.984916i	$0.444643\pi$
$660$	0	0
$661$	−17.2556	−0.671165	−0.335582	−	0.942011i	$-0.608933\pi$
$661$	−17.2556	−0.671165	−0.335582	+	0.942011i	$0.608933\pi$
$662$	−15.5981	−0.606237
$663$	0	0
$664$	0.712605	0.0276544
$665$	41.1494	1.59570
$666$	0	0
$667$	−23.8312	−0.922747
$668$	6.67259	0.258170
$669$	0	0
$670$	22.7208	0.877780
$671$	−9.21981	−0.355927
$672$	0	0
$673$	28.7973	1.11005	0.555027	−	0.831833i	$-0.312708\pi$
$673$	28.7973	1.11005	0.555027	+	0.831833i	$0.312708\pi$
$674$	2.04202	0.0786556
$675$	0	0
$676$	−4.54762	−0.174908
$677$	−23.4925	−0.902889	−0.451445	−	0.892299i	$-0.649091\pi$
$677$	−23.4925	−0.902889	−0.451445	+	0.892299i	$0.649091\pi$
$678$	0	0
$679$	29.6805	1.13903
$680$	−25.2116	−0.966820
$681$	0	0
$682$	9.96664	0.381642
$683$	32.3310	1.23711	0.618556	−	0.785741i	$-0.287718\pi$
$683$	32.3310	1.23711	0.618556	+	0.785741i	$0.287718\pi$
$684$	0	0
$685$	−3.64381	−0.139223
$686$	20.6468	0.788299
$687$	0	0
$688$	−31.6848	−1.20797
$689$	−57.9632	−2.20822
$690$	0	0
$691$	2.30793	0.0877979	0.0438989	−	0.999036i	$-0.486022\pi$
$691$	2.30793	0.0877979	0.0438989	+	0.999036i	$0.486022\pi$
$692$	0.239902	0.00911971
$693$	0	0
$694$	−33.1162	−1.25707
$695$	25.0538	0.950344
$696$	0	0
$697$	6.77275	0.256536
$698$	−24.9856	−0.945718
$699$	0	0
$700$	−2.85945	−0.108077
$701$	−3.70060	−0.139770	−0.0698848	−	0.997555i	$-0.522263\pi$
$701$	−3.70060	−0.139770	−0.0698848	+	0.997555i	$0.522263\pi$
$702$	0	0
$703$	−29.4998	−1.11261
$704$	−8.87459	−0.334474
$705$	0	0
$706$	32.2327	1.21310
$707$	−15.5341	−0.584218
$708$	0	0
$709$	−40.7595	−1.53076	−0.765378	−	0.643580i	$-0.777448\pi$
$709$	−40.7595	−1.53076	−0.765378	+	0.643580i	$0.777448\pi$
$710$	−24.0275	−0.901737
$711$	0	0
$712$	9.47792	0.355200
$713$	−20.3815	−0.763292
$714$	0	0
$715$	12.4444	0.465395
$716$	1.18838	0.0444120
$717$	0	0
$718$	3.29188	0.122852
$719$	16.3347	0.609183	0.304592	−	0.952483i	$-0.401480\pi$
$719$	16.3347	0.609183	0.304592	+	0.952483i	$0.401480\pi$
$720$	0	0
$721$	−33.7297	−1.25616
$722$	−13.4145	−0.499237
$723$	0	0
$724$	−4.59282	−0.170691
$725$	19.1481	0.711144
$726$	0	0
$727$	−41.3092	−1.53207	−0.766036	−	0.642797i	$-0.777774\pi$
$727$	−41.3092	−1.53207	−0.766036	+	0.642797i	$0.777774\pi$
$728$	40.8565	1.51424
$729$	0	0
$730$	−8.97354	−0.332126
$731$	35.8469	1.32585
$732$	0	0
$733$	−29.6175	−1.09395	−0.546973	−	0.837150i	$-0.684220\pi$
$733$	−29.6175	−1.09395	−0.546973	+	0.837150i	$0.684220\pi$
$734$	1.17396	0.0433315
$735$	0	0
$736$	6.90759	0.254617
$737$	−7.00703	−0.258107
$738$	0	0
$739$	−35.7338	−1.31449	−0.657245	−	0.753677i	$-0.728278\pi$
$739$	−35.7338	−1.31449	−0.657245	+	0.753677i	$0.728278\pi$
$740$	7.14411	0.262623
$741$	0	0
$742$	42.8819	1.57425
$743$	32.2790	1.18420	0.592101	−	0.805864i	$-0.298299\pi$
$743$	32.2790	1.18420	0.592101	+	0.805864i	$0.298299\pi$
$744$	0	0
$745$	−27.0799	−0.992129
$746$	−15.3763	−0.562967
$747$	0	0
$748$	1.55639	0.0569071
$749$	35.3958	1.29333
$750$	0	0
$751$	−48.2456	−1.76051	−0.880254	−	0.474503i	$-0.842628\pi$
$751$	−48.2456	−1.76051	−0.880254	+	0.474503i	$0.842628\pi$
$752$	−13.8367	−0.504572
$753$	0	0
$754$	−54.7660	−1.99446
$755$	35.2728	1.28371
$756$	0	0
$757$	41.5434	1.50992	0.754961	−	0.655770i	$-0.227656\pi$
$757$	41.5434	1.50992	0.754961	+	0.655770i	$0.227656\pi$
$758$	−30.8624	−1.12097
$759$	0	0
$760$	44.3769	1.60972
$761$	−46.8611	−1.69871	−0.849357	−	0.527819i	$-0.823010\pi$
$761$	−46.8611	−1.69871	−0.849357	+	0.527819i	$0.823010\pi$
$762$	0	0
$763$	−36.2865	−1.31366
$764$	9.81418	0.355065
$765$	0	0
$766$	−22.0663	−0.797287
$767$	−46.9822	−1.69643
$768$	0	0
$769$	−3.34135	−0.120492	−0.0602460	−	0.998184i	$-0.519189\pi$
$769$	−3.34135	−0.120492	−0.0602460	+	0.998184i	$0.519189\pi$
$770$	−9.20654	−0.331781
$771$	0	0
$772$	−6.21425	−0.223656
$773$	10.1841	0.366298	0.183149	−	0.983085i	$-0.441371\pi$
$773$	10.1841	0.366298	0.183149	+	0.983085i	$0.441371\pi$
$774$	0	0
$775$	16.3763	0.588255
$776$	32.0085	1.14904
$777$	0	0
$778$	−29.7869	−1.06791
$779$	−11.9213	−0.427124
$780$	0	0
$781$	7.41003	0.265152
$782$	9.53450	0.340953
$783$	0	0
$784$	−2.91734	−0.104191
$785$	−33.4537	−1.19402
$786$	0	0
$787$	27.8437	0.992521	0.496260	−	0.868174i	$-0.334706\pi$
$787$	27.8437	0.992521	0.496260	+	0.868174i	$0.334706\pi$
$788$	3.73376	0.133010
$789$	0	0
$790$	13.7926	0.490718
$791$	25.4076	0.903391
$792$	0	0
$793$	43.3286	1.53864
$794$	−20.2905	−0.720084
$795$	0	0
$796$	−2.25277	−0.0798472
$797$	33.0796	1.17174	0.585869	−	0.810406i	$-0.300753\pi$
$797$	33.0796	1.17174	0.585869	+	0.810406i	$0.300753\pi$
$798$	0	0
$799$	15.6543	0.553809
$800$	−5.55018	−0.196229
$801$	0	0
$802$	−20.3213	−0.717570
$803$	2.76742	0.0976601
$804$	0	0
$805$	18.8271	0.663568
$806$	−46.8383	−1.64981
$807$	0	0
$808$	−16.7525	−0.589350
$809$	19.5404	0.687006	0.343503	−	0.939152i	$-0.388387\pi$
$809$	19.5404	0.687006	0.343503	+	0.939152i	$0.388387\pi$
$810$	0	0
$811$	11.2341	0.394484	0.197242	−	0.980355i	$-0.436802\pi$
$811$	11.2341	0.394484	0.197242	+	0.980355i	$0.436802\pi$
$812$	−13.5251	−0.474637
$813$	0	0
$814$	6.60013	0.231335
$815$	24.7085	0.865500
$816$	0	0
$817$	−63.0970	−2.20749
$818$	25.8323	0.903206
$819$	0	0
$820$	2.88703	0.100819
$821$	17.5162	0.611321	0.305660	−	0.952141i	$-0.401123\pi$
$821$	17.5162	0.611321	0.305660	+	0.952141i	$0.401123\pi$
$822$	0	0
$823$	21.3011	0.742511	0.371255	−	0.928531i	$-0.378927\pi$
$823$	21.3011	0.742511	0.371255	+	0.928531i	$0.378927\pi$
$824$	−36.3753	−1.26719
$825$	0	0
$826$	34.7581	1.20939
$827$	−10.1646	−0.353456	−0.176728	−	0.984260i	$-0.556551\pi$
$827$	−10.1646	−0.353456	−0.176728	+	0.984260i	$0.556551\pi$
$828$	0	0
$829$	−25.7297	−0.893630	−0.446815	−	0.894626i	$-0.647442\pi$
$829$	−25.7297	−0.893630	−0.446815	+	0.894626i	$0.647442\pi$
$830$	0.754635	0.0261938
$831$	0	0
$832$	41.7062	1.44590
$833$	3.30056	0.114358
$834$	0	0
$835$	35.3001	1.22161
$836$	−2.73952	−0.0947482
$837$	0	0
$838$	−29.6371	−1.02380
$839$	13.8407	0.477833	0.238917	−	0.971040i	$-0.423208\pi$
$839$	13.8407	0.477833	0.238917	+	0.971040i	$0.423208\pi$
$840$	0	0
$841$	61.5698	2.12310
$842$	27.3038	0.940951
$843$	0	0
$844$	8.47900	0.291859
$845$	−24.0583	−0.827631
$846$	0	0
$847$	2.83927	0.0975586
$848$	33.8981	1.16407
$849$	0	0
$850$	−7.66088	−0.262766
$851$	−13.4971	−0.462674
$852$	0	0
$853$	32.4279	1.11031	0.555154	−	0.831747i	$-0.312659\pi$
$853$	32.4279	1.11031	0.555154	+	0.831747i	$0.312659\pi$
$854$	−32.0551	−1.09690
$855$	0	0
$856$	38.1720	1.30469
$857$	−44.7622	−1.52905	−0.764523	−	0.644596i	$-0.777026\pi$
$857$	−44.7622	−1.52905	−0.764523	+	0.644596i	$0.777026\pi$
$858$	0	0
$859$	4.52832	0.154504	0.0772521	−	0.997012i	$-0.475385\pi$
$859$	4.52832	0.154504	0.0772521	+	0.997012i	$0.475385\pi$
$860$	15.2805	0.521060
$861$	0	0
$862$	−32.0104	−1.09028
$863$	13.2677	0.451639	0.225820	−	0.974169i	$-0.427494\pi$
$863$	13.2677	0.451639	0.225820	+	0.974169i	$0.427494\pi$
$864$	0	0
$865$	1.26916	0.0431527
$866$	−27.7361	−0.942511
$867$	0	0
$868$	−11.5672	−0.392617
$869$	−4.25360	−0.144294
$870$	0	0
$871$	32.9296	1.11578
$872$	−39.1327	−1.32520
$873$	0	0
$874$	−16.7824	−0.567674
$875$	22.4649	0.759454
$876$	0	0
$877$	−0.761633	−0.0257185	−0.0128593	−	0.999917i	$-0.504093\pi$
$877$	−0.761633	−0.0257185	−0.0128593	+	0.999917i	$0.504093\pi$
$878$	24.2295	0.817707
$879$	0	0
$880$	−7.27776	−0.245333
$881$	23.5878	0.794694	0.397347	−	0.917668i	$-0.369931\pi$
$881$	23.5878	0.794694	0.397347	+	0.917668i	$0.369931\pi$
$882$	0	0
$883$	39.3551	1.32440	0.662201	−	0.749326i	$-0.269622\pi$
$883$	39.3551	1.32440	0.662201	+	0.749326i	$0.269622\pi$
$884$	−7.31425	−0.246005
$885$	0	0
$886$	−0.587205	−0.0197275
$887$	−3.66692	−0.123123	−0.0615616	−	0.998103i	$-0.519608\pi$
$887$	−3.66692	−0.123123	−0.0615616	+	0.998103i	$0.519608\pi$
$888$	0	0
$889$	10.7628	0.360972
$890$	10.0369	0.336439
$891$	0	0
$892$	−13.8503	−0.463743
$893$	−27.5544	−0.922072
$894$	0	0
$895$	6.28693	0.210149
$896$	−15.1906	−0.507481
$897$	0	0
$898$	1.18900	0.0396773
$899$	77.4592	2.58341
$900$	0	0
$901$	−38.3510	−1.27766
$902$	2.66720	0.0888080
$903$	0	0
$904$	27.4005	0.911326
$905$	−24.2974	−0.807674
$906$	0	0
$907$	5.66637	0.188149	0.0940743	−	0.995565i	$-0.470011\pi$
$907$	5.66637	0.188149	0.0940743	+	0.995565i	$0.470011\pi$
$908$	−2.04221	−0.0677731
$909$	0	0
$910$	43.2662	1.43426
$911$	−31.8987	−1.05685	−0.528425	−	0.848980i	$-0.677217\pi$
$911$	−31.8987	−1.05685	−0.528425	+	0.848980i	$0.677217\pi$
$912$	0	0
$913$	−0.232728	−0.00770216
$914$	−46.7509	−1.54638
$915$	0	0
$916$	−1.91004	−0.0631094
$917$	−23.6239	−0.780131
$918$	0	0
$919$	1.89019	0.0623517	0.0311759	−	0.999514i	$-0.490075\pi$
$919$	1.89019	0.0623517	0.0311759	+	0.999514i	$0.490075\pi$
$920$	20.3038	0.669396
$921$	0	0
$922$	−37.1788	−1.22442
$923$	−34.8235	−1.14623
$924$	0	0
$925$	10.8448	0.356574
$926$	−11.8934	−0.390842
$927$	0	0
$928$	−26.2521	−0.861767
$929$	2.58129	0.0846893	0.0423446	−	0.999103i	$-0.486517\pi$
$929$	2.58129	0.0846893	0.0423446	+	0.999103i	$0.486517\pi$
$930$	0	0
$931$	−5.80958	−0.190401
$932$	1.59724	0.0523195
$933$	0	0
$934$	−45.9144	−1.50236
$935$	8.23377	0.269273
$936$	0	0
$937$	6.76457	0.220989	0.110494	−	0.993877i	$-0.464757\pi$
$937$	6.76457	0.220989	0.110494	+	0.993877i	$0.464757\pi$
$938$	−24.3617	−0.795439
$939$	0	0
$940$	6.67297	0.217648
$941$	−40.2419	−1.31185	−0.655924	−	0.754827i	$-0.727721\pi$
$941$	−40.2419	−1.31185	−0.655924	+	0.754827i	$0.727721\pi$
$942$	0	0
$943$	−5.45434	−0.177618
$944$	27.4762	0.894275
$945$	0	0
$946$	14.1170	0.458983
$947$	21.4839	0.698131	0.349066	−	0.937098i	$-0.386499\pi$
$947$	21.4839	0.698131	0.349066	+	0.937098i	$0.386499\pi$
$948$	0	0
$949$	−13.0055	−0.422177
$950$	13.4845	0.437496
$951$	0	0
$952$	27.0325	0.876127
$953$	−12.8823	−0.417298	−0.208649	−	0.977991i	$-0.566907\pi$
$953$	−12.8823	−0.417298	−0.208649	+	0.977991i	$0.566907\pi$
$954$	0	0
$955$	51.9201	1.68010
$956$	9.09662	0.294206
$957$	0	0
$958$	−14.1544	−0.457309
$959$	3.90698	0.126163
$960$	0	0
$961$	35.2465	1.13698
$962$	−31.0174	−1.00004
$963$	0	0
$964$	11.7071	0.377062
$965$	−32.8753	−1.05829
$966$	0	0
$967$	−41.2050	−1.32506	−0.662531	−	0.749035i	$-0.730518\pi$
$967$	−41.2050	−1.32506	−0.662531	+	0.749035i	$0.730518\pi$
$968$	3.06197	0.0984155
$969$	0	0
$970$	33.8964	1.08835
$971$	−11.1405	−0.357516	−0.178758	−	0.983893i	$-0.557208\pi$
$971$	−11.1405	−0.357516	−0.178758	+	0.983893i	$0.557208\pi$
$972$	0	0
$973$	−26.8633	−0.861196
$974$	−47.6717	−1.52750
$975$	0	0
$976$	−25.3395	−0.811097
$977$	21.4104	0.684979	0.342489	−	0.939522i	$-0.388730\pi$
$977$	21.4104	0.684979	0.342489	+	0.939522i	$0.388730\pi$
$978$	0	0
$979$	−3.09536	−0.0989283
$980$	1.40693	0.0449428
$981$	0	0
$982$	12.9136	0.412091
$983$	11.6645	0.372039	0.186020	−	0.982546i	$-0.440441\pi$
$983$	11.6645	0.372039	0.186020	+	0.982546i	$0.440441\pi$
$984$	0	0
$985$	19.7528	0.629376
$986$	−36.2356	−1.15398
$987$	0	0
$988$	12.8744	0.409589
$989$	−28.8688	−0.917975
$990$	0	0
$991$	−49.0711	−1.55880	−0.779398	−	0.626529i	$-0.784475\pi$
$991$	−49.0711	−1.55880	−0.779398	+	0.626529i	$0.784475\pi$
$992$	−22.4519	−0.712850
$993$	0	0
$994$	25.7629	0.817149
$995$	−11.9178	−0.377821
$996$	0	0
$997$	−13.3514	−0.422842	−0.211421	−	0.977395i	$-0.567809\pi$
$997$	−13.3514	−0.422842	−0.211421	+	0.977395i	$0.567809\pi$
$998$	−6.64064	−0.210206
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.k.1.18		51
3.2	odd	2		8019.2.a.l.1.34		51
27.5	odd	18		297.2.j.c.133.12	yes	102
27.11	odd	18		297.2.j.c.67.12	✓	102
27.16	even	9		891.2.j.c.496.6		102
27.22	even	9		891.2.j.c.397.6		102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.j.c.67.12	✓	102	27.11	odd	18
297.2.j.c.133.12	yes	102	27.5	odd	18
891.2.j.c.397.6		102	27.22	even	9
891.2.j.c.496.6		102	27.16	even	9
8019.2.a.k.1.18		51	1.1	even	1	trivial
8019.2.a.l.1.34		51	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:	\( N \)	\(=\)	\( 8019 = 3^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8019.a (trivial)

\(f(q)\)	\(=\)	\(q-1.22452 q^{2} -0.500542 q^{4} -2.64802 q^{5} +2.83927 q^{7} +3.06197 q^{8} +O(q^{10})\)	\(q-1.22452 q^{2} -0.500542 q^{4} -2.64802 q^{5} +2.83927 q^{7} +3.06197 q^{8} +3.24257 q^{10} -1.00000 q^{11} +4.69951 q^{13} -3.47676 q^{14} -2.74837 q^{16} +3.10940 q^{17} -5.47311 q^{19} +1.32545 q^{20} +1.22452 q^{22} -2.50411 q^{23} +2.01203 q^{25} -5.75466 q^{26} -1.42118 q^{28} +9.51682 q^{29} +8.13920 q^{31} -2.75850 q^{32} -3.80754 q^{34} -7.51847 q^{35} +5.38996 q^{37} +6.70195 q^{38} -8.10818 q^{40} +2.17815 q^{41} +11.5286 q^{43} +0.500542 q^{44} +3.06635 q^{46} +5.03450 q^{47} +1.06148 q^{49} -2.46378 q^{50} -2.35230 q^{52} -12.3339 q^{53} +2.64802 q^{55} +8.69378 q^{56} -11.6536 q^{58} -9.99726 q^{59} +9.21981 q^{61} -9.96664 q^{62} +8.87459 q^{64} -12.4444 q^{65} +7.00703 q^{67} -1.55639 q^{68} +9.20654 q^{70} -7.41003 q^{71} -2.76742 q^{73} -6.60013 q^{74} +2.73952 q^{76} -2.83927 q^{77} +4.25360 q^{79} +7.27776 q^{80} -2.66720 q^{82} +0.232728 q^{83} -8.23377 q^{85} -14.1170 q^{86} -3.06197 q^{88} +3.09536 q^{89} +13.3432 q^{91} +1.25341 q^{92} -6.16487 q^{94} +14.4929 q^{95} +10.4536 q^{97} -1.29980 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7}+O(q^{10}) \) 51 * q + 60 * q^4 - 6 * q^5 + 12 * q^7	\( 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7} + 18 q^{10} - 51 q^{11} + 30 q^{13} - 12 q^{14} + 78 q^{16} + 30 q^{19} - 18 q^{20} - 3 q^{23} + 75 q^{25} - 9 q^{26} + 36 q^{28} + 42 q^{31} + 15 q^{32} + 42 q^{34} + 9 q^{35} + 48 q^{37} + 3 q^{38} + 54 q^{40} + 18 q^{43} - 60 q^{44} + 42 q^{46} - 30 q^{47} + 99 q^{49} + 30 q^{50} + 60 q^{52} - 18 q^{53} + 6 q^{55} - 21 q^{56} + 30 q^{58} - 24 q^{59} + 99 q^{61} + 114 q^{64} + 15 q^{65} + 39 q^{67} + 39 q^{68} + 48 q^{70} - 30 q^{71} + 69 q^{73} + 90 q^{76} - 12 q^{77} + 48 q^{79} - 42 q^{80} + 42 q^{82} + 21 q^{83} + 84 q^{85} - 24 q^{86} - 15 q^{89} + 69 q^{91} + 66 q^{92} + 66 q^{94} + 12 q^{95} + 72 q^{97} + 57 q^{98}+O(q^{100}) \) 51 * q + 60 * q^4 - 6 * q^5 + 12 * q^7 + 18 * q^10 - 51 * q^11 + 30 * q^13 - 12 * q^14 + 78 * q^16 + 30 * q^19 - 18 * q^20 - 3 * q^23 + 75 * q^25 - 9 * q^26 + 36 * q^28 + 42 * q^31 + 15 * q^32 + 42 * q^34 + 9 * q^35 + 48 * q^37 + 3 * q^38 + 54 * q^40 + 18 * q^43 - 60 * q^44 + 42 * q^46 - 30 * q^47 + 99 * q^49 + 30 * q^50 + 60 * q^52 - 18 * q^53 + 6 * q^55 - 21 * q^56 + 30 * q^58 - 24 * q^59 + 99 * q^61 + 114 * q^64 + 15 * q^65 + 39 * q^67 + 39 * q^68 + 48 * q^70 - 30 * q^71 + 69 * q^73 + 90 * q^76 - 12 * q^77 + 48 * q^79 - 42 * q^80 + 42 * q^82 + 21 * q^83 + 84 * q^85 - 24 * q^86 - 15 * q^89 + 69 * q^91 + 66 * q^92 + 66 * q^94 + 12 * q^95 + 72 * q^97 + 57 * q^98

Introduction

L-functions

Modular forms

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Database

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