LMFDB - Embedded newform 8019.2.a.k.1.4

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			1.4
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)

$f(q)$	$=$	$q-2.61712 q^{2} +4.84933 q^{4} -2.95330 q^{5} +3.35710 q^{7} -7.45704 q^{8} +O(q^{10})$	\(q-2.61712 q^{2} +4.84933 q^{4} -2.95330 q^{5} +3.35710 q^{7} -7.45704 q^{8} +7.72914 q^{10} -1.00000 q^{11} +4.40158 q^{13} -8.78594 q^{14} +9.81732 q^{16} +4.94425 q^{17} +0.230202 q^{19} -14.3215 q^{20} +2.61712 q^{22} +7.21314 q^{23} +3.72197 q^{25} -11.5195 q^{26} +16.2797 q^{28} +2.99700 q^{29} -1.69010 q^{31} -10.7790 q^{32} -12.9397 q^{34} -9.91452 q^{35} +5.67022 q^{37} -0.602467 q^{38} +22.0229 q^{40} -3.41261 q^{41} +1.91038 q^{43} -4.84933 q^{44} -18.8777 q^{46} +12.1909 q^{47} +4.27012 q^{49} -9.74086 q^{50} +21.3447 q^{52} +14.2247 q^{53} +2.95330 q^{55} -25.0340 q^{56} -7.84353 q^{58} +1.87196 q^{59} -3.16319 q^{61} +4.42321 q^{62} +8.57544 q^{64} -12.9992 q^{65} +4.21527 q^{67} +23.9763 q^{68} +25.9475 q^{70} +1.79042 q^{71} +16.7235 q^{73} -14.8397 q^{74} +1.11632 q^{76} -3.35710 q^{77} -12.3795 q^{79} -28.9935 q^{80} +8.93123 q^{82} -1.70888 q^{83} -14.6018 q^{85} -4.99970 q^{86} +7.45704 q^{88} +6.37707 q^{89} +14.7765 q^{91} +34.9789 q^{92} -31.9051 q^{94} -0.679855 q^{95} -17.0363 q^{97} -11.1754 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.61712	−1.85058	−0.925292	−	0.379255i	$-0.876180\pi$
$2$	−2.61712	−1.85058	−0.925292	+	0.379255i	$0.876180\pi$
$3$	0	0
$3$	0	0
$4$	4.84933	2.42466
$5$	−2.95330	−1.32076	−0.660378	−	0.750934i	$-0.729604\pi$
$5$	−2.95330	−1.32076	−0.660378	+	0.750934i	$0.729604\pi$
$6$	0	0
$7$	3.35710	1.26886	0.634432	−	0.772979i	$-0.281234\pi$
$7$	3.35710	1.26886	0.634432	+	0.772979i	$0.281234\pi$
$8$	−7.45704	−2.63646
$9$	0	0
$10$	7.72914	2.44417
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.40158	1.22078	0.610389	−	0.792102i	$-0.291013\pi$
$13$	4.40158	1.22078	0.610389	+	0.792102i	$0.291013\pi$
$14$	−8.78594	−2.34814
$15$	0	0
$16$	9.81732	2.45433
$17$	4.94425	1.19916	0.599578	−	0.800316i	$-0.295335\pi$
$17$	4.94425	1.19916	0.599578	+	0.800316i	$0.295335\pi$
$18$	0	0
$19$	0.230202	0.0528119	0.0264060	−	0.999651i	$-0.491594\pi$
$19$	0.230202	0.0528119	0.0264060	+	0.999651i	$0.491594\pi$
$20$	−14.3215	−3.20239
$21$	0	0
$22$	2.61712	0.557972
$23$	7.21314	1.50404	0.752022	−	0.659138i	$-0.229079\pi$
$23$	7.21314	1.50404	0.752022	+	0.659138i	$0.229079\pi$
$24$	0	0
$25$	3.72197	0.744395
$26$	−11.5195	−2.25915
$27$	0	0
$28$	16.2797	3.07657
$29$	2.99700	0.556530	0.278265	−	0.960504i	$-0.410241\pi$
$29$	2.99700	0.556530	0.278265	+	0.960504i	$0.410241\pi$
$30$	0	0
$31$	−1.69010	−0.303551	−0.151776	−	0.988415i	$-0.548499\pi$
$31$	−1.69010	−0.303551	−0.151776	+	0.988415i	$0.548499\pi$
$32$	−10.7790	−1.90548
$33$	0	0
$34$	−12.9397	−2.21914
$35$	−9.91452	−1.67586
$36$	0	0
$37$	5.67022	0.932179	0.466089	−	0.884738i	$-0.345662\pi$
$37$	5.67022	0.932179	0.466089	+	0.884738i	$0.345662\pi$
$38$	−0.602467	−0.0977330
$39$	0	0
$40$	22.0229	3.48212
$41$	−3.41261	−0.532961	−0.266480	−	0.963840i	$-0.585861\pi$
$41$	−3.41261	−0.532961	−0.266480	+	0.963840i	$0.585861\pi$
$42$	0	0
$43$	1.91038	0.291330	0.145665	−	0.989334i	$-0.453468\pi$
$43$	1.91038	0.291330	0.145665	+	0.989334i	$0.453468\pi$
$44$	−4.84933	−0.731064
$45$	0	0
$46$	−18.8777	−2.78336
$47$	12.1909	1.77823	0.889114	−	0.457686i	$-0.151322\pi$
$47$	12.1909	1.77823	0.889114	+	0.457686i	$0.151322\pi$
$48$	0	0
$49$	4.27012	0.610017
$50$	−9.74086	−1.37757
$51$	0	0
$52$	21.3447	2.95998
$53$	14.2247	1.95391	0.976956	−	0.213442i	$-0.0684675\pi$
$53$	14.2247	1.95391	0.976956	+	0.213442i	$0.0684675\pi$
$54$	0	0
$55$	2.95330	0.398223
$56$	−25.0340	−3.34531
$57$	0	0
$58$	−7.84353	−1.02991
$59$	1.87196	0.243709	0.121854	−	0.992548i	$-0.461116\pi$
$59$	1.87196	0.243709	0.121854	+	0.992548i	$0.461116\pi$
$60$	0	0
$61$	−3.16319	−0.405005	−0.202503	−	0.979282i	$-0.564907\pi$
$61$	−3.16319	−0.405005	−0.202503	+	0.979282i	$0.564907\pi$
$62$	4.42321	0.561748
$63$	0	0
$64$	8.57544	1.07193
$65$	−12.9992	−1.61235
$66$	0	0
$67$	4.21527	0.514977	0.257489	−	0.966281i	$-0.417105\pi$
$67$	4.21527	0.514977	0.257489	+	0.966281i	$0.417105\pi$
$68$	23.9763	2.90755
$69$	0	0
$70$	25.9475	3.10132
$71$	1.79042	0.212484	0.106242	−	0.994340i	$-0.466118\pi$
$71$	1.79042	0.212484	0.106242	+	0.994340i	$0.466118\pi$
$72$	0	0
$73$	16.7235	1.95734	0.978671	−	0.205433i	$-0.0658602\pi$
$73$	16.7235	1.95734	0.978671	+	0.205433i	$0.0658602\pi$
$74$	−14.8397	−1.72508
$75$	0	0
$76$	1.11632	0.128051
$77$	−3.35710	−0.382577
$78$	0	0
$79$	−12.3795	−1.39280	−0.696401	−	0.717653i	$-0.745217\pi$
$79$	−12.3795	−1.39280	−0.696401	+	0.717653i	$0.745217\pi$
$80$	−28.9935	−3.24157
$81$	0	0
$82$	8.93123	0.986289
$83$	−1.70888	−0.187574	−0.0937871	−	0.995592i	$-0.529897\pi$
$83$	−1.70888	−0.187574	−0.0937871	+	0.995592i	$0.529897\pi$
$84$	0	0
$85$	−14.6018	−1.58379
$86$	−4.99970	−0.539131
$87$	0	0
$88$	7.45704	0.794923
$89$	6.37707	0.675968	0.337984	−	0.941152i	$-0.390255\pi$
$89$	6.37707	0.675968	0.337984	+	0.941152i	$0.390255\pi$
$90$	0	0
$91$	14.7765	1.54900
$92$	34.9789	3.64680
$93$	0	0
$94$	−31.9051	−3.29076
$95$	−0.679855	−0.0697517
$96$	0	0
$97$	−17.0363	−1.72978	−0.864889	−	0.501962i	$-0.832612\pi$
$97$	−17.0363	−1.72978	−0.864889	+	0.501962i	$0.832612\pi$
$98$	−11.1754	−1.12889
$99$	0	0
$100$	18.0491	1.80491
$101$	−0.559795	−0.0557017	−0.0278508	−	0.999612i	$-0.508866\pi$
$101$	−0.559795	−0.0557017	−0.0278508	+	0.999612i	$0.508866\pi$
$102$	0	0
$103$	1.77813	0.175204	0.0876022	−	0.996156i	$-0.472080\pi$
$103$	1.77813	0.175204	0.0876022	+	0.996156i	$0.472080\pi$
$104$	−32.8227	−3.21853
$105$	0	0
$106$	−37.2277	−3.61588
$107$	−5.57795	−0.539241	−0.269621	−	0.962967i	$-0.586898\pi$
$107$	−5.57795	−0.539241	−0.269621	+	0.962967i	$0.586898\pi$
$108$	0	0
$109$	12.8648	1.23223	0.616113	−	0.787657i	$-0.288706\pi$
$109$	12.8648	1.23223	0.616113	+	0.787657i	$0.288706\pi$
$110$	−7.72914	−0.736945
$111$	0	0
$112$	32.9577	3.11421
$113$	4.66885	0.439208	0.219604	−	0.975589i	$-0.429523\pi$
$113$	4.66885	0.439208	0.219604	+	0.975589i	$0.429523\pi$
$114$	0	0
$115$	−21.3026	−1.98648
$116$	14.5335	1.34940
$117$	0	0
$118$	−4.89916	−0.451004
$119$	16.5983	1.52157
$120$	0	0
$121$	1.00000	0.0909091
$122$	8.27846	0.749496
$123$	0	0
$124$	−8.19586	−0.736010
$125$	3.77439	0.337592
$126$	0	0
$127$	2.94511	0.261336	0.130668	−	0.991426i	$-0.458288\pi$
$127$	2.94511	0.261336	0.130668	+	0.991426i	$0.458288\pi$
$128$	−0.884878	−0.0782129
$129$	0	0
$130$	34.0204	2.98379
$131$	6.48573	0.566661	0.283330	−	0.959022i	$-0.408561\pi$
$131$	6.48573	0.566661	0.283330	+	0.959022i	$0.408561\pi$
$132$	0	0
$133$	0.772811	0.0670112
$134$	−11.0319	−0.953009
$135$	0	0
$136$	−36.8694	−3.16153
$137$	0.345652	0.0295310	0.0147655	−	0.999891i	$-0.495300\pi$
$137$	0.345652	0.0295310	0.0147655	+	0.999891i	$0.495300\pi$
$138$	0	0
$139$	−12.2708	−1.04080	−0.520400	−	0.853923i	$-0.674217\pi$
$139$	−12.2708	−1.04080	−0.520400	+	0.853923i	$0.674217\pi$
$140$	−48.0787	−4.06340
$141$	0	0
$142$	−4.68574	−0.393219
$143$	−4.40158	−0.368079
$144$	0	0
$145$	−8.85105	−0.735040
$146$	−43.7675	−3.62223
$147$	0	0
$148$	27.4968	2.26022
$149$	10.0077	0.819861	0.409931	−	0.912117i	$-0.365553\pi$
$149$	10.0077	0.819861	0.409931	+	0.912117i	$0.365553\pi$
$150$	0	0
$151$	−2.35357	−0.191530	−0.0957652	−	0.995404i	$-0.530530\pi$
$151$	−2.35357	−0.191530	−0.0957652	+	0.995404i	$0.530530\pi$
$152$	−1.71662	−0.139237
$153$	0	0
$154$	8.78594	0.707991
$155$	4.99138	0.400917
$156$	0	0
$157$	−4.37472	−0.349141	−0.174570	−	0.984645i	$-0.555854\pi$
$157$	−4.37472	−0.349141	−0.174570	+	0.984645i	$0.555854\pi$
$158$	32.3987	2.57750
$159$	0	0
$160$	31.8337	2.51668
$161$	24.2152	1.90843
$162$	0	0
$163$	2.28038	0.178613	0.0893066	−	0.996004i	$-0.471535\pi$
$163$	2.28038	0.178613	0.0893066	+	0.996004i	$0.471535\pi$
$164$	−16.5489	−1.29225
$165$	0	0
$166$	4.47235	0.347122
$167$	−2.47252	−0.191329	−0.0956647	−	0.995414i	$-0.530498\pi$
$167$	−2.47252	−0.191329	−0.0956647	+	0.995414i	$0.530498\pi$
$168$	0	0
$169$	6.37390	0.490300
$170$	38.2148	2.93094
$171$	0	0
$172$	9.26406	0.706378
$173$	−24.0942	−1.83185	−0.915923	−	0.401355i	$-0.868539\pi$
$173$	−24.0942	−1.83185	−0.915923	+	0.401355i	$0.868539\pi$
$174$	0	0
$175$	12.4950	0.944536
$176$	−9.81732	−0.740008
$177$	0	0
$178$	−16.6896	−1.25094
$179$	−5.28958	−0.395362	−0.197681	−	0.980266i	$-0.563341\pi$
$179$	−5.28958	−0.395362	−0.197681	+	0.980266i	$0.563341\pi$
$180$	0	0
$181$	0.917500	0.0681972	0.0340986	−	0.999418i	$-0.489144\pi$
$181$	0.917500	0.0681972	0.0340986	+	0.999418i	$0.489144\pi$
$182$	−38.6720	−2.86656
$183$	0	0
$184$	−53.7887	−3.96535
$185$	−16.7459	−1.23118
$186$	0	0
$187$	−4.94425	−0.361559
$188$	59.1177	4.31160
$189$	0	0
$190$	1.77926	0.129081
$191$	−22.3436	−1.61673	−0.808363	−	0.588685i	$-0.799646\pi$
$191$	−22.3436	−1.61673	−0.808363	+	0.588685i	$0.799646\pi$
$192$	0	0
$193$	10.4054	0.748995	0.374498	−	0.927228i	$-0.377815\pi$
$193$	10.4054	0.748995	0.374498	+	0.927228i	$0.377815\pi$
$194$	44.5862	3.20110
$195$	0	0
$196$	20.7072	1.47909
$197$	−13.4113	−0.955515	−0.477757	−	0.878492i	$-0.658550\pi$
$197$	−13.4113	−0.955515	−0.477757	+	0.878492i	$0.658550\pi$
$198$	0	0
$199$	−18.4001	−1.30435	−0.652176	−	0.758068i	$-0.726144\pi$
$199$	−18.4001	−1.30435	−0.652176	+	0.758068i	$0.726144\pi$
$200$	−27.7549	−1.96257
$201$	0	0
$202$	1.46505	0.103081
$203$	10.0612	0.706161
$204$	0	0
$205$	10.0785	0.703911
$206$	−4.65358	−0.324230
$207$	0	0
$208$	43.2117	2.99619
$209$	−0.230202	−0.0159234
$210$	0	0
$211$	15.1332	1.04181	0.520907	−	0.853613i	$-0.325594\pi$
$211$	15.1332	1.04181	0.520907	+	0.853613i	$0.325594\pi$
$212$	68.9802	4.73758
$213$	0	0
$214$	14.5982	0.997911
$215$	−5.64193	−0.384776
$216$	0	0
$217$	−5.67384	−0.385166
$218$	−33.6688	−2.28034
$219$	0	0
$220$	14.3215	0.965556
$221$	21.7625	1.46390
$222$	0	0
$223$	−4.82691	−0.323234	−0.161617	−	0.986854i	$-0.551671\pi$
$223$	−4.82691	−0.323234	−0.161617	+	0.986854i	$0.551671\pi$
$224$	−36.1863	−2.41780
$225$	0	0
$226$	−12.2189	−0.812792
$227$	19.3975	1.28745	0.643727	−	0.765255i	$-0.277387\pi$
$227$	19.3975	1.28745	0.643727	+	0.765255i	$0.277387\pi$
$228$	0	0
$229$	9.97597	0.659231	0.329615	−	0.944115i	$-0.393081\pi$
$229$	9.97597	0.659231	0.329615	+	0.944115i	$0.393081\pi$
$230$	55.7514	3.67614
$231$	0	0
$232$	−22.3488	−1.46727
$233$	1.22394	0.0801829	0.0400914	−	0.999196i	$-0.487235\pi$
$233$	1.22394	0.0801829	0.0400914	+	0.999196i	$0.487235\pi$
$234$	0	0
$235$	−36.0034	−2.34860
$236$	9.07776	0.590912
$237$	0	0
$238$	−43.4399	−2.81579
$239$	−17.6525	−1.14184	−0.570922	−	0.821004i	$-0.693414\pi$
$239$	−17.6525	−1.14184	−0.570922	+	0.821004i	$0.693414\pi$
$240$	0	0
$241$	17.2981	1.11427	0.557134	−	0.830423i	$-0.311901\pi$
$241$	17.2981	1.11427	0.557134	+	0.830423i	$0.311901\pi$
$242$	−2.61712	−0.168235
$243$	0	0
$244$	−15.3393	−0.982001
$245$	−12.6109	−0.805683
$246$	0	0
$247$	1.01325	0.0644717
$248$	12.6032	0.800301
$249$	0	0
$250$	−9.87804	−0.624742
$251$	20.5230	1.29540	0.647700	−	0.761895i	$-0.275731\pi$
$251$	20.5230	1.29540	0.647700	+	0.761895i	$0.275731\pi$
$252$	0	0
$253$	−7.21314	−0.453486
$254$	−7.70770	−0.483624
$255$	0	0
$256$	−14.8350	−0.927190
$257$	−26.3583	−1.64418	−0.822091	−	0.569355i	$-0.807193\pi$
$257$	−26.3583	−1.64418	−0.822091	+	0.569355i	$0.807193\pi$
$258$	0	0
$259$	19.0355	1.18281
$260$	−63.0373	−3.90941
$261$	0	0
$262$	−16.9739	−1.04865
$263$	−23.4066	−1.44331	−0.721657	−	0.692251i	$-0.756619\pi$
$263$	−23.4066	−1.44331	−0.721657	+	0.692251i	$0.756619\pi$
$264$	0	0
$265$	−42.0098	−2.58064
$266$	−2.02254	−0.124010
$267$	0	0
$268$	20.4412	1.24865
$269$	16.7106	1.01886	0.509431	−	0.860511i	$-0.329856\pi$
$269$	16.7106	1.01886	0.509431	+	0.860511i	$0.329856\pi$
$270$	0	0
$271$	14.4365	0.876957	0.438478	−	0.898742i	$-0.355518\pi$
$271$	14.4365	0.876957	0.438478	+	0.898742i	$0.355518\pi$
$272$	48.5393	2.94312
$273$	0	0
$274$	−0.904613	−0.0546497
$275$	−3.72197	−0.224444
$276$	0	0
$277$	−13.2664	−0.797099	−0.398550	−	0.917147i	$-0.630486\pi$
$277$	−13.2664	−0.797099	−0.398550	+	0.917147i	$0.630486\pi$
$278$	32.1143	1.92609
$279$	0	0
$280$	73.9329	4.41834
$281$	−11.2278	−0.669796	−0.334898	−	0.942254i	$-0.608702\pi$
$281$	−11.2278	−0.669796	−0.334898	+	0.942254i	$0.608702\pi$
$282$	0	0
$283$	−6.73475	−0.400339	−0.200170	−	0.979761i	$-0.564149\pi$
$283$	−6.73475	−0.400339	−0.200170	+	0.979761i	$0.564149\pi$
$284$	8.68232	0.515201
$285$	0	0
$286$	11.5195	0.681161
$287$	−11.4565	−0.676255
$288$	0	0
$289$	7.44559	0.437976
$290$	23.1643	1.36025
$291$	0	0
$292$	81.0979	4.74590
$293$	1.25548	0.0733458	0.0366729	−	0.999327i	$-0.488324\pi$
$293$	1.25548	0.0733458	0.0366729	+	0.999327i	$0.488324\pi$
$294$	0	0
$295$	−5.52847	−0.321880
$296$	−42.2831	−2.45765
$297$	0	0
$298$	−26.1913	−1.51722
$299$	31.7492	1.83611
$300$	0	0
$301$	6.41334	0.369659
$302$	6.15957	0.354443
$303$	0	0
$304$	2.25997	0.129618
$305$	9.34185	0.534913
$306$	0	0
$307$	13.8667	0.791417	0.395708	−	0.918376i	$-0.370499\pi$
$307$	13.8667	0.791417	0.395708	+	0.918376i	$0.370499\pi$
$308$	−16.2797	−0.927620
$309$	0	0
$310$	−13.0630	−0.741931
$311$	−24.6414	−1.39729	−0.698644	−	0.715470i	$-0.746213\pi$
$311$	−24.6414	−1.39729	−0.698644	+	0.715470i	$0.746213\pi$
$312$	0	0
$313$	18.9799	1.07280	0.536402	−	0.843962i	$-0.319783\pi$
$313$	18.9799	1.07280	0.536402	+	0.843962i	$0.319783\pi$
$314$	11.4492	0.646115
$315$	0	0
$316$	−60.0322	−3.37708
$317$	−28.7069	−1.61234	−0.806169	−	0.591685i	$-0.798463\pi$
$317$	−28.7069	−1.61234	−0.806169	+	0.591685i	$0.798463\pi$
$318$	0	0
$319$	−2.99700	−0.167800
$320$	−25.3258	−1.41576
$321$	0	0
$322$	−63.3742	−3.53171
$323$	1.13818	0.0633298
$324$	0	0
$325$	16.3826	0.908741
$326$	−5.96804	−0.330539
$327$	0	0
$328$	25.4480	1.40513
$329$	40.9261	2.25633
$330$	0	0
$331$	9.40851	0.517138	0.258569	−	0.965993i	$-0.416749\pi$
$331$	9.40851	0.517138	0.258569	+	0.965993i	$0.416749\pi$
$332$	−8.28692	−0.454804
$333$	0	0
$334$	6.47089	0.354071
$335$	−12.4489	−0.680159
$336$	0	0
$337$	−13.9084	−0.757639	−0.378819	−	0.925471i	$-0.623670\pi$
$337$	−13.9084	−0.757639	−0.378819	+	0.925471i	$0.623670\pi$
$338$	−16.6813	−0.907342
$339$	0	0
$340$	−70.8091	−3.84016
$341$	1.69010	0.0915242
$342$	0	0
$343$	−9.16449	−0.494836
$344$	−14.2458	−0.768081
$345$	0	0
$346$	63.0574	3.38998
$347$	9.59207	0.514929	0.257465	−	0.966288i	$-0.417113\pi$
$347$	9.59207	0.514929	0.257465	+	0.966288i	$0.417113\pi$
$348$	0	0
$349$	31.9771	1.71169	0.855846	−	0.517230i	$-0.173037\pi$
$349$	31.9771	1.71169	0.855846	+	0.517230i	$0.173037\pi$
$350$	−32.7010	−1.74794
$351$	0	0
$352$	10.7790	0.574525
$353$	5.23063	0.278398	0.139199	−	0.990264i	$-0.455547\pi$
$353$	5.23063	0.278398	0.139199	+	0.990264i	$0.455547\pi$
$354$	0	0
$355$	−5.28764	−0.280639
$356$	30.9245	1.63899
$357$	0	0
$358$	13.8435	0.731651
$359$	−2.73654	−0.144429	−0.0722145	−	0.997389i	$-0.523007\pi$
$359$	−2.73654	−0.144429	−0.0722145	+	0.997389i	$0.523007\pi$
$360$	0	0
$361$	−18.9470	−0.997211
$362$	−2.40121	−0.126205
$363$	0	0
$364$	71.6563	3.75581
$365$	−49.3896	−2.58517
$366$	0	0
$367$	−5.52699	−0.288507	−0.144253	−	0.989541i	$-0.546078\pi$
$367$	−5.52699	−0.288507	−0.144253	+	0.989541i	$0.546078\pi$
$368$	70.8137	3.69142
$369$	0	0
$370$	43.8260	2.27840
$371$	47.7537	2.47925
$372$	0	0
$373$	5.26032	0.272369	0.136185	−	0.990683i	$-0.456516\pi$
$373$	5.26032	0.272369	0.136185	+	0.990683i	$0.456516\pi$
$374$	12.9397	0.669096
$375$	0	0
$376$	−90.9081	−4.68823
$377$	13.1916	0.679400
$378$	0	0
$379$	−10.4544	−0.537008	−0.268504	−	0.963279i	$-0.586529\pi$
$379$	−10.4544	−0.537008	−0.268504	+	0.963279i	$0.586529\pi$
$380$	−3.29684	−0.169124
$381$	0	0
$382$	58.4759	2.99189
$383$	−27.1477	−1.38718	−0.693591	−	0.720369i	$-0.743972\pi$
$383$	−27.1477	−1.38718	−0.693591	+	0.720369i	$0.743972\pi$
$384$	0	0
$385$	9.91452	0.505291
$386$	−27.2321	−1.38608
$387$	0	0
$388$	−82.6148	−4.19413
$389$	−19.8183	−1.00483	−0.502414	−	0.864627i	$-0.667555\pi$
$389$	−19.8183	−1.00483	−0.502414	+	0.864627i	$0.667555\pi$
$390$	0	0
$391$	35.6636	1.80358
$392$	−31.8424	−1.60828
$393$	0	0
$394$	35.0990	1.76826
$395$	36.5604	1.83955
$396$	0	0
$397$	7.60629	0.381749	0.190874	−	0.981614i	$-0.438868\pi$
$397$	7.60629	0.381749	0.190874	+	0.981614i	$0.438868\pi$
$398$	48.1554	2.41381
$399$	0	0
$400$	36.5398	1.82699
$401$	20.6502	1.03122	0.515611	−	0.856823i	$-0.327565\pi$
$401$	20.6502	1.03122	0.515611	+	0.856823i	$0.327565\pi$
$402$	0	0
$403$	−7.43912	−0.370569
$404$	−2.71463	−0.135058
$405$	0	0
$406$	−26.3315	−1.30681
$407$	−5.67022	−0.281063
$408$	0	0
$409$	23.8403	1.17883	0.589413	−	0.807832i	$-0.299359\pi$
$409$	23.8403	1.17883	0.589413	+	0.807832i	$0.299359\pi$
$410$	−26.3766	−1.30265
$411$	0	0
$412$	8.62273	0.424812
$413$	6.28437	0.309233
$414$	0	0
$415$	5.04684	0.247740
$416$	−47.4448	−2.32617
$417$	0	0
$418$	0.602467	0.0294676
$419$	−33.1316	−1.61858	−0.809292	−	0.587407i	$-0.800149\pi$
$419$	−33.1316	−1.61858	−0.809292	+	0.587407i	$0.800149\pi$
$420$	0	0
$421$	14.1602	0.690127	0.345064	−	0.938579i	$-0.387857\pi$
$421$	14.1602	0.690127	0.345064	+	0.938579i	$0.387857\pi$
$422$	−39.6055	−1.92797
$423$	0	0
$424$	−106.074	−5.15141
$425$	18.4024	0.892646
$426$	0	0
$427$	−10.6191	−0.513896
$428$	−27.0493	−1.30748
$429$	0	0
$430$	14.7656	0.712061
$431$	3.17876	0.153116	0.0765578	−	0.997065i	$-0.475607\pi$
$431$	3.17876	0.153116	0.0765578	+	0.997065i	$0.475607\pi$
$432$	0	0
$433$	−13.4298	−0.645393	−0.322696	−	0.946503i	$-0.604589\pi$
$433$	−13.4298	−0.645393	−0.322696	+	0.946503i	$0.604589\pi$
$434$	14.8491	0.712782
$435$	0	0
$436$	62.3858	2.98774
$437$	1.66048	0.0794315
$438$	0	0
$439$	−29.7788	−1.42126	−0.710631	−	0.703565i	$-0.751591\pi$
$439$	−29.7788	−1.42126	−0.710631	+	0.703565i	$0.751591\pi$
$440$	−22.0229	−1.04990
$441$	0	0
$442$	−56.9551	−2.70908
$443$	−5.90205	−0.280415	−0.140207	−	0.990122i	$-0.544777\pi$
$443$	−5.90205	−0.280415	−0.140207	+	0.990122i	$0.544777\pi$
$444$	0	0
$445$	−18.8334	−0.892788
$446$	12.6326	0.598171
$447$	0	0
$448$	28.7886	1.36013
$449$	−13.0653	−0.616590	−0.308295	−	0.951291i	$-0.599758\pi$
$449$	−13.0653	−0.616590	−0.308295	+	0.951291i	$0.599758\pi$
$450$	0	0
$451$	3.41261	0.160694
$452$	22.6408	1.06493
$453$	0	0
$454$	−50.7655	−2.38254
$455$	−43.6395	−2.04585
$456$	0	0
$457$	0.151041	0.00706542	0.00353271	−	0.999994i	$-0.498876\pi$
$457$	0.151041	0.00706542	0.00353271	+	0.999994i	$0.498876\pi$
$458$	−26.1083	−1.21996
$459$	0	0
$460$	−103.303	−4.81653
$461$	−18.9198	−0.881183	−0.440592	−	0.897708i	$-0.645231\pi$
$461$	−18.9198	−0.881183	−0.440592	+	0.897708i	$0.645231\pi$
$462$	0	0
$463$	−32.7882	−1.52380	−0.761898	−	0.647697i	$-0.775732\pi$
$463$	−32.7882	−1.52380	−0.761898	+	0.647697i	$0.775732\pi$
$464$	29.4225	1.36591
$465$	0	0
$466$	−3.20320	−0.148385
$467$	7.91767	0.366386	0.183193	−	0.983077i	$-0.441357\pi$
$467$	7.91767	0.366386	0.183193	+	0.983077i	$0.441357\pi$
$468$	0	0
$469$	14.1511	0.653436
$470$	94.2253	4.34629
$471$	0	0
$472$	−13.9593	−0.642529
$473$	−1.91038	−0.0878394
$474$	0	0
$475$	0.856806	0.0393129
$476$	80.4907	3.68929
$477$	0	0
$478$	46.1987	2.11308
$479$	−32.3583	−1.47849	−0.739244	−	0.673438i	$-0.764817\pi$
$479$	−32.3583	−1.47849	−0.739244	+	0.673438i	$0.764817\pi$
$480$	0	0
$481$	24.9579	1.13798
$482$	−45.2712	−2.06205
$483$	0	0
$484$	4.84933	0.220424
$485$	50.3134	2.28461
$486$	0	0
$487$	25.6715	1.16329	0.581643	−	0.813444i	$-0.302410\pi$
$487$	25.6715	1.16329	0.581643	+	0.813444i	$0.302410\pi$
$488$	23.5880	1.06778
$489$	0	0
$490$	33.0043	1.49098
$491$	−6.22206	−0.280798	−0.140399	−	0.990095i	$-0.544838\pi$
$491$	−6.22206	−0.280798	−0.140399	+	0.990095i	$0.544838\pi$
$492$	0	0
$493$	14.8179	0.667366
$494$	−2.65180	−0.119310
$495$	0	0
$496$	−16.5923	−0.745015
$497$	6.01061	0.269613
$498$	0	0
$499$	15.2697	0.683565	0.341782	−	0.939779i	$-0.388969\pi$
$499$	15.2697	0.683565	0.341782	+	0.939779i	$0.388969\pi$
$500$	18.3033	0.818546
$501$	0	0
$502$	−53.7112	−2.39725
$503$	2.91080	0.129786	0.0648931	−	0.997892i	$-0.479329\pi$
$503$	2.91080	0.129786	0.0648931	+	0.997892i	$0.479329\pi$
$504$	0	0
$505$	1.65324	0.0735683
$506$	18.8777	0.839215
$507$	0	0
$508$	14.2818	0.633652
$509$	−27.6813	−1.22695	−0.613477	−	0.789713i	$-0.710229\pi$
$509$	−27.6813	−1.22695	−0.613477	+	0.789713i	$0.710229\pi$
$510$	0	0
$511$	56.1426	2.48360
$512$	40.5949	1.79406
$513$	0	0
$514$	68.9828	3.04270
$515$	−5.25135	−0.231402
$516$	0	0
$517$	−12.1909	−0.536156
$518$	−49.8182	−2.18889
$519$	0	0
$520$	96.9354	4.25090
$521$	−6.58874	−0.288658	−0.144329	−	0.989530i	$-0.546102\pi$
$521$	−6.58874	−0.288658	−0.144329	+	0.989530i	$0.546102\pi$
$522$	0	0
$523$	19.3306	0.845266	0.422633	−	0.906301i	$-0.361106\pi$
$523$	19.3306	0.845266	0.422633	+	0.906301i	$0.361106\pi$
$524$	31.4514	1.37396
$525$	0	0
$526$	61.2580	2.67098
$527$	−8.35629	−0.364006
$528$	0	0
$529$	29.0295	1.26215
$530$	109.945	4.77569
$531$	0	0
$532$	3.74761	0.162480
$533$	−15.0209	−0.650627
$534$	0	0
$535$	16.4734	0.712206
$536$	−31.4334	−1.35772
$537$	0	0
$538$	−43.7336	−1.88549
$539$	−4.27012	−0.183927
$540$	0	0
$541$	−28.4177	−1.22177	−0.610886	−	0.791718i	$-0.709187\pi$
$541$	−28.4177	−1.22177	−0.610886	+	0.791718i	$0.709187\pi$
$542$	−37.7821	−1.62288
$543$	0	0
$544$	−53.2943	−2.28497
$545$	−37.9937	−1.62747
$546$	0	0
$547$	0.0902249	0.00385774	0.00192887	−	0.999998i	$-0.499386\pi$
$547$	0.0902249	0.00385774	0.00192887	+	0.999998i	$0.499386\pi$
$548$	1.67618	0.0716028
$549$	0	0
$550$	9.74086	0.415352
$551$	0.689916	0.0293914
$552$	0	0
$553$	−41.5592	−1.76728
$554$	34.7197	1.47510
$555$	0	0
$556$	−59.5053	−2.52359
$557$	14.1977	0.601578	0.300789	−	0.953691i	$-0.402750\pi$
$557$	14.1977	0.601578	0.300789	+	0.953691i	$0.402750\pi$
$558$	0	0
$559$	8.40869	0.355650
$560$	−97.3340	−4.11311
$561$	0	0
$562$	29.3846	1.23951
$563$	15.3495	0.646906	0.323453	−	0.946244i	$-0.395156\pi$
$563$	15.3495	0.646906	0.323453	+	0.946244i	$0.395156\pi$
$564$	0	0
$565$	−13.7885	−0.580087
$566$	17.6257	0.740862
$567$	0	0
$568$	−13.3512	−0.560204
$569$	25.6049	1.07341	0.536707	−	0.843769i	$-0.319668\pi$
$569$	25.6049	1.07341	0.536707	+	0.843769i	$0.319668\pi$
$570$	0	0
$571$	−10.8649	−0.454684	−0.227342	−	0.973815i	$-0.573004\pi$
$571$	−10.8649	−0.454684	−0.227342	+	0.973815i	$0.573004\pi$
$572$	−21.3447	−0.892467
$573$	0	0
$574$	29.9830	1.25147
$575$	26.8471	1.11960
$576$	0	0
$577$	12.3956	0.516036	0.258018	−	0.966140i	$-0.416931\pi$
$577$	12.3956	0.516036	0.258018	+	0.966140i	$0.416931\pi$
$578$	−19.4860	−0.810512
$579$	0	0
$580$	−42.9216	−1.78222
$581$	−5.73689	−0.238006
$582$	0	0
$583$	−14.2247	−0.589126
$584$	−124.708	−5.16046
$585$	0	0
$586$	−3.28574	−0.135733
$587$	11.2362	0.463767	0.231884	−	0.972744i	$-0.425511\pi$
$587$	11.2362	0.463767	0.231884	+	0.972744i	$0.425511\pi$
$588$	0	0
$589$	−0.389065	−0.0160311
$590$	14.4687	0.595666
$591$	0	0
$592$	55.6664	2.28787
$593$	−30.2787	−1.24340	−0.621699	−	0.783256i	$-0.713557\pi$
$593$	−30.2787	−1.24340	−0.621699	+	0.783256i	$0.713557\pi$
$594$	0	0
$595$	−49.0198	−2.00962
$596$	48.5305	1.98789
$597$	0	0
$598$	−83.0916	−3.39787
$599$	5.35646	0.218859	0.109429	−	0.993995i	$-0.465098\pi$
$599$	5.35646	0.218859	0.109429	+	0.993995i	$0.465098\pi$
$600$	0	0
$601$	−18.0217	−0.735119	−0.367559	−	0.930000i	$-0.619807\pi$
$601$	−18.0217	−0.735119	−0.367559	+	0.930000i	$0.619807\pi$
$602$	−16.7845	−0.684085
$603$	0	0
$604$	−11.4132	−0.464397
$605$	−2.95330	−0.120069
$606$	0	0
$607$	1.68222	0.0682791	0.0341396	−	0.999417i	$-0.489131\pi$
$607$	1.68222	0.0682791	0.0341396	+	0.999417i	$0.489131\pi$
$608$	−2.48136	−0.100632
$609$	0	0
$610$	−24.4488	−0.989901
$611$	53.6593	2.17082
$612$	0	0
$613$	8.03355	0.324472	0.162236	−	0.986752i	$-0.448129\pi$
$613$	8.03355	0.324472	0.162236	+	0.986752i	$0.448129\pi$
$614$	−36.2910	−1.46458
$615$	0	0
$616$	25.0340	1.00865
$617$	−14.5395	−0.585340	−0.292670	−	0.956214i	$-0.594544\pi$
$617$	−14.5395	−0.585340	−0.292670	+	0.956214i	$0.594544\pi$
$618$	0	0
$619$	9.36558	0.376434	0.188217	−	0.982127i	$-0.439729\pi$
$619$	9.36558	0.376434	0.188217	+	0.982127i	$0.439729\pi$
$620$	24.2048	0.972089
$621$	0	0
$622$	64.4897	2.58580
$623$	21.4085	0.857712
$624$	0	0
$625$	−29.7568	−1.19027
$626$	−49.6726	−1.98532
$627$	0	0
$628$	−21.2145	−0.846549
$629$	28.0350	1.11783
$630$	0	0
$631$	−33.5918	−1.33727	−0.668634	−	0.743591i	$-0.733121\pi$
$631$	−33.5918	−1.33727	−0.668634	+	0.743591i	$0.733121\pi$
$632$	92.3143	3.67207
$633$	0	0
$634$	75.1294	2.98377
$635$	−8.69778	−0.345161
$636$	0	0
$637$	18.7953	0.744695
$638$	7.84353	0.310528
$639$	0	0
$640$	2.61331	0.103300
$641$	−45.8930	−1.81266	−0.906332	−	0.422565i	$-0.861130\pi$
$641$	−45.8930	−1.81266	−0.906332	+	0.422565i	$0.861130\pi$
$642$	0	0
$643$	−0.605737	−0.0238879	−0.0119440	−	0.999929i	$-0.503802\pi$
$643$	−0.605737	−0.0238879	−0.0119440	+	0.999929i	$0.503802\pi$
$644$	117.428	4.62730
$645$	0	0
$646$	−2.97874	−0.117197
$647$	−17.4963	−0.687852	−0.343926	−	0.938997i	$-0.611757\pi$
$647$	−17.4963	−0.687852	−0.343926	+	0.938997i	$0.611757\pi$
$648$	0	0
$649$	−1.87196	−0.0734810
$650$	−42.8752	−1.68170
$651$	0	0
$652$	11.0583	0.433077
$653$	2.39653	0.0937834	0.0468917	−	0.998900i	$-0.485068\pi$
$653$	2.39653	0.0937834	0.0468917	+	0.998900i	$0.485068\pi$
$654$	0	0
$655$	−19.1543	−0.748420
$656$	−33.5027	−1.30806
$657$	0	0
$658$	−107.109	−4.17553
$659$	20.0617	0.781493	0.390746	−	0.920498i	$-0.372217\pi$
$659$	20.0617	0.781493	0.390746	+	0.920498i	$0.372217\pi$
$660$	0	0
$661$	−25.8800	−1.00661	−0.503307	−	0.864108i	$-0.667883\pi$
$661$	−25.8800	−1.00661	−0.503307	+	0.864108i	$0.667883\pi$
$662$	−24.6232	−0.957008
$663$	0	0
$664$	12.7432	0.494532
$665$	−2.28234	−0.0885054
$666$	0	0
$667$	21.6178	0.837046
$668$	−11.9901	−0.463910
$669$	0	0
$670$	32.5804	1.25869
$671$	3.16319	0.122114
$672$	0	0
$673$	13.1006	0.504992	0.252496	−	0.967598i	$-0.418749\pi$
$673$	13.1006	0.504992	0.252496	+	0.967598i	$0.418749\pi$
$674$	36.4000	1.40207
$675$	0	0
$676$	30.9091	1.18881
$677$	45.6029	1.75266	0.876331	−	0.481710i	$-0.159984\pi$
$677$	45.6029	1.75266	0.876331	+	0.481710i	$0.159984\pi$
$678$	0	0
$679$	−57.1927	−2.19485
$680$	108.886	4.17561
$681$	0	0
$682$	−4.42321	−0.169373
$683$	25.7433	0.985041	0.492521	−	0.870301i	$-0.336076\pi$
$683$	25.7433	0.985041	0.492521	+	0.870301i	$0.336076\pi$
$684$	0	0
$685$	−1.02081	−0.0390033
$686$	23.9846	0.915736
$687$	0	0
$688$	18.7548	0.715021
$689$	62.6111	2.38529
$690$	0	0
$691$	14.8269	0.564044	0.282022	−	0.959408i	$-0.408995\pi$
$691$	14.8269	0.564044	0.282022	+	0.959408i	$0.408995\pi$
$692$	−116.840	−4.44161
$693$	0	0
$694$	−25.1036	−0.952920
$695$	36.2395	1.37464
$696$	0	0
$697$	−16.8728	−0.639103
$698$	−83.6879	−3.16763
$699$	0	0
$700$	60.5925	2.29018
$701$	28.3925	1.07237	0.536185	−	0.844100i	$-0.319865\pi$
$701$	28.3925	1.07237	0.536185	+	0.844100i	$0.319865\pi$
$702$	0	0
$703$	1.30530	0.0492302
$704$	−8.57544	−0.323199
$705$	0	0
$706$	−13.6892	−0.515199
$707$	−1.87929	−0.0706779
$708$	0	0
$709$	24.7775	0.930540	0.465270	−	0.885169i	$-0.345957\pi$
$709$	24.7775	0.930540	0.465270	+	0.885169i	$0.345957\pi$
$710$	13.8384	0.519346
$711$	0	0
$712$	−47.5540	−1.78216
$713$	−12.1910	−0.456555
$714$	0	0
$715$	12.9992	0.486142
$716$	−25.6509	−0.958620
$717$	0	0
$718$	7.16186	0.267278
$719$	−8.61030	−0.321110	−0.160555	−	0.987027i	$-0.551328\pi$
$719$	−8.61030	−0.321110	−0.160555	+	0.987027i	$0.551328\pi$
$720$	0	0
$721$	5.96936	0.222311
$722$	49.5866	1.84542
$723$	0	0
$724$	4.44926	0.165355
$725$	11.1548	0.414278
$726$	0	0
$727$	35.9806	1.33445	0.667224	−	0.744857i	$-0.267482\pi$
$727$	35.9806	1.33445	0.667224	+	0.744857i	$0.267482\pi$
$728$	−110.189	−4.08388
$729$	0	0
$730$	129.259	4.78408
$731$	9.44540	0.349351
$732$	0	0
$733$	11.6071	0.428720	0.214360	−	0.976755i	$-0.431233\pi$
$733$	11.6071	0.428720	0.214360	+	0.976755i	$0.431233\pi$
$734$	14.4648	0.533906
$735$	0	0
$736$	−77.7508	−2.86593
$737$	−4.21527	−0.155271
$738$	0	0
$739$	−26.9146	−0.990070	−0.495035	−	0.868873i	$-0.664845\pi$
$739$	−26.9146	−0.990070	−0.495035	+	0.868873i	$0.664845\pi$
$740$	−81.2062	−2.98520
$741$	0	0
$742$	−124.977	−4.58806
$743$	0.787378	0.0288861	0.0144430	−	0.999896i	$-0.495402\pi$
$743$	0.787378	0.0288861	0.0144430	+	0.999896i	$0.495402\pi$
$744$	0	0
$745$	−29.5557	−1.08284
$746$	−13.7669	−0.504042
$747$	0	0
$748$	−23.9763	−0.876660
$749$	−18.7257	−0.684224
$750$	0	0
$751$	36.9027	1.34660	0.673299	−	0.739370i	$-0.264876\pi$
$751$	36.9027	1.34660	0.673299	+	0.739370i	$0.264876\pi$
$752$	119.682	4.36436
$753$	0	0
$754$	−34.5239	−1.25729
$755$	6.95078	0.252965
$756$	0	0
$757$	−42.8076	−1.55587	−0.777934	−	0.628346i	$-0.783732\pi$
$757$	−42.8076	−1.55587	−0.777934	+	0.628346i	$0.783732\pi$
$758$	27.3605	0.993779
$759$	0	0
$760$	5.06970	0.183897
$761$	−5.49165	−0.199072	−0.0995361	−	0.995034i	$-0.531736\pi$
$761$	−5.49165	−0.199072	−0.0995361	+	0.995034i	$0.531736\pi$
$762$	0	0
$763$	43.1885	1.56353
$764$	−108.351	−3.92002
$765$	0	0
$766$	71.0488	2.56710
$767$	8.23959	0.297515
$768$	0	0
$769$	6.43224	0.231953	0.115976	−	0.993252i	$-0.463000\pi$
$769$	6.43224	0.231953	0.115976	+	0.993252i	$0.463000\pi$
$770$	−25.9475	−0.935083
$771$	0	0
$772$	50.4591	1.81606
$773$	−27.7769	−0.999065	−0.499533	−	0.866295i	$-0.666495\pi$
$773$	−27.7769	−0.999065	−0.499533	+	0.866295i	$0.666495\pi$
$774$	0	0
$775$	−6.29052	−0.225962
$776$	127.041	4.56049
$777$	0	0
$778$	51.8669	1.85952
$779$	−0.785591	−0.0281467
$780$	0	0
$781$	−1.79042	−0.0640662
$782$	−93.3359	−3.33769
$783$	0	0
$784$	41.9211	1.49718
$785$	12.9199	0.461130
$786$	0	0
$787$	27.0058	0.962653	0.481327	−	0.876541i	$-0.340155\pi$
$787$	27.0058	0.962653	0.481327	+	0.876541i	$0.340155\pi$
$788$	−65.0357	−2.31680
$789$	0	0
$790$	−95.6829	−3.40424
$791$	15.6738	0.557296
$792$	0	0
$793$	−13.9230	−0.494422
$794$	−19.9066	−0.706459
$795$	0	0
$796$	−89.2283	−3.16261
$797$	38.5065	1.36397	0.681985	−	0.731367i	$-0.261117\pi$
$797$	38.5065	1.36397	0.681985	+	0.731367i	$0.261117\pi$
$798$	0	0
$799$	60.2749	2.13237
$800$	−40.1193	−1.41843
$801$	0	0
$802$	−54.0441	−1.90836
$803$	−16.7235	−0.590161
$804$	0	0
$805$	−71.5149	−2.52057
$806$	19.4691	0.685769
$807$	0	0
$808$	4.17441	0.146855
$809$	−40.0590	−1.40840	−0.704200	−	0.710001i	$-0.748694\pi$
$809$	−40.0590	−1.40840	−0.704200	+	0.710001i	$0.748694\pi$
$810$	0	0
$811$	−8.03357	−0.282097	−0.141048	−	0.990003i	$-0.545047\pi$
$811$	−8.03357	−0.282097	−0.141048	+	0.990003i	$0.545047\pi$
$812$	48.7903	1.71220
$813$	0	0
$814$	14.8397	0.520130
$815$	−6.73465	−0.235904
$816$	0	0
$817$	0.439773	0.0153857
$818$	−62.3929	−2.18152
$819$	0	0
$820$	48.8738	1.70675
$821$	50.2150	1.75251	0.876257	−	0.481844i	$-0.160033\pi$
$821$	50.2150	1.75251	0.876257	+	0.481844i	$0.160033\pi$
$822$	0	0
$823$	22.4233	0.781627	0.390814	−	0.920470i	$-0.372194\pi$
$823$	22.4233	0.781627	0.390814	+	0.920470i	$0.372194\pi$
$824$	−13.2596	−0.461919
$825$	0	0
$826$	−16.4470	−0.572263
$827$	8.26280	0.287326	0.143663	−	0.989627i	$-0.454112\pi$
$827$	8.26280	0.287326	0.143663	+	0.989627i	$0.454112\pi$
$828$	0	0
$829$	−43.2678	−1.50275	−0.751376	−	0.659875i	$-0.770609\pi$
$829$	−43.2678	−1.50275	−0.751376	+	0.659875i	$0.770609\pi$
$830$	−13.2082	−0.458463
$831$	0	0
$832$	37.7455	1.30859
$833$	21.1125	0.731506
$834$	0	0
$835$	7.30209	0.252699
$836$	−1.11632	−0.0386089
$837$	0	0
$838$	86.7094	2.99532
$839$	−14.2317	−0.491331	−0.245666	−	0.969355i	$-0.579007\pi$
$839$	−14.2317	−0.491331	−0.245666	+	0.969355i	$0.579007\pi$
$840$	0	0
$841$	−20.0180	−0.690275
$842$	−37.0590	−1.27714
$843$	0	0
$844$	73.3860	2.52605
$845$	−18.8240	−0.647567
$846$	0	0
$847$	3.35710	0.115351
$848$	139.648	4.79554
$849$	0	0
$850$	−48.1612	−1.65192
$851$	40.9001	1.40204
$852$	0	0
$853$	54.1930	1.85553	0.927766	−	0.373163i	$-0.121727\pi$
$853$	54.1930	1.85553	0.927766	+	0.373163i	$0.121727\pi$
$854$	27.7916	0.951009
$855$	0	0
$856$	41.5950	1.42169
$857$	29.4119	1.00469	0.502346	−	0.864667i	$-0.332470\pi$
$857$	29.4119	1.00469	0.502346	+	0.864667i	$0.332470\pi$
$858$	0	0
$859$	24.9439	0.851075	0.425537	−	0.904941i	$-0.360085\pi$
$859$	24.9439	0.851075	0.425537	+	0.904941i	$0.360085\pi$
$860$	−27.3595	−0.932953
$861$	0	0
$862$	−8.31920	−0.283353
$863$	36.1075	1.22911	0.614556	−	0.788873i	$-0.289335\pi$
$863$	36.1075	1.22911	0.614556	+	0.788873i	$0.289335\pi$
$864$	0	0
$865$	71.1573	2.41942
$866$	35.1473	1.19435
$867$	0	0
$868$	−27.5143	−0.933897
$869$	12.3795	0.419946
$870$	0	0
$871$	18.5538	0.628673
$872$	−95.9335	−3.24872
$873$	0	0
$874$	−4.34568	−0.146995
$875$	12.6710	0.428358
$876$	0	0
$877$	25.3417	0.855730	0.427865	−	0.903843i	$-0.359266\pi$
$877$	25.3417	0.855730	0.427865	+	0.903843i	$0.359266\pi$
$878$	77.9346	2.63017
$879$	0	0
$880$	28.9935	0.977370
$881$	45.3587	1.52817	0.764086	−	0.645114i	$-0.223190\pi$
$881$	45.3587	1.52817	0.764086	+	0.645114i	$0.223190\pi$
$882$	0	0
$883$	−31.8372	−1.07141	−0.535704	−	0.844406i	$-0.679954\pi$
$883$	−31.8372	−1.07141	−0.535704	+	0.844406i	$0.679954\pi$
$884$	105.533	3.54948
$885$	0	0
$886$	15.4464	0.518931
$887$	22.4911	0.755179	0.377589	−	0.925973i	$-0.376753\pi$
$887$	22.4911	0.755179	0.377589	+	0.925973i	$0.376753\pi$
$888$	0	0
$889$	9.88702	0.331600
$890$	49.2893	1.65218
$891$	0	0
$892$	−23.4073	−0.783733
$893$	2.80637	0.0939117
$894$	0	0
$895$	15.6217	0.522177
$896$	−2.97062	−0.0992415
$897$	0	0
$898$	34.1935	1.14105
$899$	−5.06525	−0.168935
$900$	0	0
$901$	70.3304	2.34305
$902$	−8.93123	−0.297377
$903$	0	0
$904$	−34.8158	−1.15796
$905$	−2.70965	−0.0900719
$906$	0	0
$907$	−10.8899	−0.361594	−0.180797	−	0.983520i	$-0.557868\pi$
$907$	−10.8899	−0.361594	−0.180797	+	0.983520i	$0.557868\pi$
$908$	94.0646	3.12164
$909$	0	0
$910$	114.210	3.78602
$911$	58.8365	1.94934	0.974671	−	0.223645i	$-0.0717955\pi$
$911$	58.8365	1.94934	0.974671	+	0.223645i	$0.0717955\pi$
$912$	0	0
$913$	1.70888	0.0565557
$914$	−0.395294	−0.0130752
$915$	0	0
$916$	48.3768	1.59841
$917$	21.7732	0.719016
$918$	0	0
$919$	−52.9847	−1.74780	−0.873902	−	0.486102i	$-0.838418\pi$
$919$	−52.9847	−1.74780	−0.873902	+	0.486102i	$0.838418\pi$
$920$	158.854	5.23726
$921$	0	0
$922$	49.5154	1.63070
$923$	7.88067	0.259395
$924$	0	0
$925$	21.1044	0.693909
$926$	85.8107	2.81991
$927$	0	0
$928$	−32.3048	−1.06046
$929$	38.4432	1.26128	0.630641	−	0.776075i	$-0.282792\pi$
$929$	38.4432	1.26128	0.630641	+	0.776075i	$0.282792\pi$
$930$	0	0
$931$	0.982989	0.0322162
$932$	5.93528	0.194416
$933$	0	0
$934$	−20.7215	−0.678028
$935$	14.6018	0.477531
$936$	0	0
$937$	20.3238	0.663950	0.331975	−	0.943288i	$-0.392285\pi$
$937$	20.3238	0.663950	0.331975	+	0.943288i	$0.392285\pi$
$938$	−37.0351	−1.20924
$939$	0	0
$940$	−174.592	−5.69457
$941$	19.5072	0.635916	0.317958	−	0.948105i	$-0.397003\pi$
$941$	19.5072	0.635916	0.317958	+	0.948105i	$0.397003\pi$
$942$	0	0
$943$	−24.6157	−0.801597
$944$	18.3777	0.598142
$945$	0	0
$946$	4.99970	0.162554
$947$	25.3684	0.824361	0.412181	−	0.911102i	$-0.364767\pi$
$947$	25.3684	0.824361	0.412181	+	0.911102i	$0.364767\pi$
$948$	0	0
$949$	73.6100	2.38948
$950$	−2.24237	−0.0727519
$951$	0	0
$952$	−123.774	−4.01155
$953$	−31.9289	−1.03428	−0.517139	−	0.855901i	$-0.673003\pi$
$953$	−31.9289	−1.03428	−0.517139	+	0.855901i	$0.673003\pi$
$954$	0	0
$955$	65.9873	2.13530
$956$	−85.6027	−2.76859
$957$	0	0
$958$	84.6856	2.73607
$959$	1.16039	0.0374709
$960$	0	0
$961$	−28.1436	−0.907857
$962$	−65.3180	−2.10594
$963$	0	0
$964$	83.8841	2.70172
$965$	−30.7302	−0.989240
$966$	0	0
$967$	−24.9215	−0.801423	−0.400711	−	0.916204i	$-0.631237\pi$
$967$	−24.9215	−0.801423	−0.400711	+	0.916204i	$0.631237\pi$
$968$	−7.45704	−0.239678
$969$	0	0
$970$	−131.676	−4.22787
$971$	28.0891	0.901424	0.450712	−	0.892670i	$-0.351170\pi$
$971$	28.0891	0.901424	0.450712	+	0.892670i	$0.351170\pi$
$972$	0	0
$973$	−41.1944	−1.32063
$974$	−67.1855	−2.15276
$975$	0	0
$976$	−31.0540	−0.994016
$977$	3.10864	0.0994543	0.0497271	−	0.998763i	$-0.484165\pi$
$977$	3.10864	0.0994543	0.0497271	+	0.998763i	$0.484165\pi$
$978$	0	0
$979$	−6.37707	−0.203812
$980$	−61.1545	−1.95351
$981$	0	0
$982$	16.2839	0.519640
$983$	60.8446	1.94064	0.970320	−	0.241823i	$-0.0777452\pi$
$983$	60.8446	1.94064	0.970320	+	0.241823i	$0.0777452\pi$
$984$	0	0
$985$	39.6075	1.26200
$986$	−38.7803	−1.23502
$987$	0	0
$988$	4.91359	0.156322
$989$	13.7799	0.438174
$990$	0	0
$991$	49.8798	1.58448	0.792242	−	0.610207i	$-0.208914\pi$
$991$	49.8798	1.58448	0.792242	+	0.610207i	$0.208914\pi$
$992$	18.2177	0.578412
$993$	0	0
$994$	−15.7305	−0.498941
$995$	54.3411	1.72273
$996$	0	0
$997$	−10.5365	−0.333694	−0.166847	−	0.985983i	$-0.553359\pi$
$997$	−10.5365	−0.333694	−0.166847	+	0.985983i	$0.553359\pi$
$998$	−39.9626	−1.26499
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.k.1.4		51
3.2	odd	2		8019.2.a.l.1.48		51
27.4	even	9		891.2.j.c.694.1		102
27.7	even	9		891.2.j.c.199.1		102
27.20	odd	18		297.2.j.c.265.17	yes	102
27.23	odd	18		297.2.j.c.232.17	✓	102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.j.c.232.17	✓	102	27.23	odd	18
297.2.j.c.265.17	yes	102	27.20	odd	18
891.2.j.c.199.1		102	27.7	even	9
891.2.j.c.694.1		102	27.4	even	9
8019.2.a.k.1.4		51	1.1	even	1	trivial
8019.2.a.l.1.48		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)

\(f(q)\)	\(=\)	\(q-2.61712 q^{2} +4.84933 q^{4} -2.95330 q^{5} +3.35710 q^{7} -7.45704 q^{8} +O(q^{10})\)	\(q-2.61712 q^{2} +4.84933 q^{4} -2.95330 q^{5} +3.35710 q^{7} -7.45704 q^{8} +7.72914 q^{10} -1.00000 q^{11} +4.40158 q^{13} -8.78594 q^{14} +9.81732 q^{16} +4.94425 q^{17} +0.230202 q^{19} -14.3215 q^{20} +2.61712 q^{22} +7.21314 q^{23} +3.72197 q^{25} -11.5195 q^{26} +16.2797 q^{28} +2.99700 q^{29} -1.69010 q^{31} -10.7790 q^{32} -12.9397 q^{34} -9.91452 q^{35} +5.67022 q^{37} -0.602467 q^{38} +22.0229 q^{40} -3.41261 q^{41} +1.91038 q^{43} -4.84933 q^{44} -18.8777 q^{46} +12.1909 q^{47} +4.27012 q^{49} -9.74086 q^{50} +21.3447 q^{52} +14.2247 q^{53} +2.95330 q^{55} -25.0340 q^{56} -7.84353 q^{58} +1.87196 q^{59} -3.16319 q^{61} +4.42321 q^{62} +8.57544 q^{64} -12.9992 q^{65} +4.21527 q^{67} +23.9763 q^{68} +25.9475 q^{70} +1.79042 q^{71} +16.7235 q^{73} -14.8397 q^{74} +1.11632 q^{76} -3.35710 q^{77} -12.3795 q^{79} -28.9935 q^{80} +8.93123 q^{82} -1.70888 q^{83} -14.6018 q^{85} -4.99970 q^{86} +7.45704 q^{88} +6.37707 q^{89} +14.7765 q^{91} +34.9789 q^{92} -31.9051 q^{94} -0.679855 q^{95} -17.0363 q^{97} -11.1754 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7}+O(q^{10}) \) 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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