LMFDB - Embedded newform 8019.2.a.i.1.13

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:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
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)

$f(q)$	$=$	$q-1.69020 q^{2} +0.856762 q^{4} -0.595209 q^{5} +0.410093 q^{7} +1.93230 q^{8} +O(q^{10})$	\(q-1.69020 q^{2} +0.856762 q^{4} -0.595209 q^{5} +0.410093 q^{7} +1.93230 q^{8} +1.00602 q^{10} -1.00000 q^{11} +0.430540 q^{13} -0.693137 q^{14} -4.97948 q^{16} +6.00311 q^{17} -0.510569 q^{19} -0.509953 q^{20} +1.69020 q^{22} -7.73719 q^{23} -4.64573 q^{25} -0.727696 q^{26} +0.351352 q^{28} +1.05564 q^{29} +9.49163 q^{31} +4.55171 q^{32} -10.1464 q^{34} -0.244091 q^{35} +2.80113 q^{37} +0.862961 q^{38} -1.15012 q^{40} -9.12030 q^{41} +8.22447 q^{43} -0.856762 q^{44} +13.0774 q^{46} -5.47124 q^{47} -6.83182 q^{49} +7.85219 q^{50} +0.368870 q^{52} +5.98271 q^{53} +0.595209 q^{55} +0.792421 q^{56} -1.78423 q^{58} +8.62800 q^{59} +6.93857 q^{61} -16.0427 q^{62} +2.26568 q^{64} -0.256261 q^{65} -7.69272 q^{67} +5.14323 q^{68} +0.412562 q^{70} -13.5341 q^{71} -5.32238 q^{73} -4.73446 q^{74} -0.437436 q^{76} -0.410093 q^{77} +5.83295 q^{79} +2.96383 q^{80} +15.4151 q^{82} -15.7445 q^{83} -3.57310 q^{85} -13.9010 q^{86} -1.93230 q^{88} -16.4457 q^{89} +0.176561 q^{91} -6.62893 q^{92} +9.24747 q^{94} +0.303895 q^{95} -6.23983 q^{97} +11.5471 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8}+O(q^{10}) $ 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8	$ 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8} - 48 q^{11} - 24 q^{14} + 66 q^{16} - 24 q^{17} - 48 q^{20} + 6 q^{22} - 12 q^{23} + 60 q^{25} - 36 q^{26} - 18 q^{28} - 60 q^{29} + 36 q^{31} - 42 q^{32} + 12 q^{34} - 24 q^{35} + 6 q^{37} - 24 q^{38} - 72 q^{41} - 12 q^{43} - 54 q^{44} - 30 q^{46} - 36 q^{47} + 60 q^{49} - 42 q^{50} - 48 q^{53} + 24 q^{55} - 72 q^{56} + 12 q^{58} - 60 q^{59} - 24 q^{61} - 36 q^{62} + 90 q^{64} - 48 q^{65} - 60 q^{68} - 30 q^{70} - 60 q^{71} - 18 q^{73} - 36 q^{74} - 42 q^{76} - 12 q^{79} - 96 q^{80} + 12 q^{82} - 36 q^{83} + 18 q^{85} - 48 q^{86} + 18 q^{88} - 96 q^{89} + 30 q^{91} - 36 q^{92} - 48 q^{94} - 48 q^{95} + 30 q^{97} - 54 q^{98}+O(q^{100}) $ 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8 - 48 * q^11 - 24 * q^14 + 66 * q^16 - 24 * q^17 - 48 * q^20 + 6 * q^22 - 12 * q^23 + 60 * q^25 - 36 * q^26 - 18 * q^28 - 60 * q^29 + 36 * q^31 - 42 * q^32 + 12 * q^34 - 24 * q^35 + 6 * q^37 - 24 * q^38 - 72 * q^41 - 12 * q^43 - 54 * q^44 - 30 * q^46 - 36 * q^47 + 60 * q^49 - 42 * q^50 - 48 * q^53 + 24 * q^55 - 72 * q^56 + 12 * q^58 - 60 * q^59 - 24 * q^61 - 36 * q^62 + 90 * q^64 - 48 * q^65 - 60 * q^68 - 30 * q^70 - 60 * q^71 - 18 * q^73 - 36 * q^74 - 42 * q^76 - 12 * q^79 - 96 * q^80 + 12 * q^82 - 36 * q^83 + 18 * q^85 - 48 * q^86 + 18 * q^88 - 96 * q^89 + 30 * q^91 - 36 * q^92 - 48 * q^94 - 48 * q^95 + 30 * q^97 - 54 * 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 $

$2$	−1.69020	−1.19515	−0.597574	−	0.801813i	$-0.703869\pi$
$2$	−1.69020	−1.19515	−0.597574	+	0.801813i	$0.703869\pi$
$3$	0	0
$3$	0	0
$4$	0.856762	0.428381
$5$	−0.595209	−0.266186	−0.133093	−	0.991104i	$-0.542491\pi$
$5$	−0.595209	−0.266186	−0.133093	+	0.991104i	$0.542491\pi$
$6$	0	0
$7$	0.410093	0.155000	0.0775002	−	0.996992i	$-0.475306\pi$
$7$	0.410093	0.155000	0.0775002	+	0.996992i	$0.475306\pi$
$8$	1.93230	0.683170
$9$	0	0
$10$	1.00602	0.318131
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.430540	0.119410	0.0597051	−	0.998216i	$-0.480984\pi$
$13$	0.430540	0.119410	0.0597051	+	0.998216i	$0.480984\pi$
$14$	−0.693137	−0.185249
$15$	0	0
$16$	−4.97948	−1.24487
$17$	6.00311	1.45597	0.727983	−	0.685595i	$-0.240458\pi$
$17$	6.00311	1.45597	0.727983	+	0.685595i	$0.240458\pi$
$18$	0	0
$19$	−0.510569	−0.117132	−0.0585662	−	0.998284i	$-0.518653\pi$
$19$	−0.510569	−0.117132	−0.0585662	+	0.998284i	$0.518653\pi$
$20$	−0.509953	−0.114029
$21$	0	0
$22$	1.69020	0.360351
$23$	−7.73719	−1.61332	−0.806658	−	0.591018i	$-0.798726\pi$
$23$	−7.73719	−1.61332	−0.806658	+	0.591018i	$0.798726\pi$
$24$	0	0
$25$	−4.64573	−0.929145
$26$	−0.727696	−0.142713
$27$	0	0
$28$	0.351352	0.0663993
$29$	1.05564	0.196027	0.0980133	−	0.995185i	$-0.468751\pi$
$29$	1.05564	0.196027	0.0980133	+	0.995185i	$0.468751\pi$
$30$	0	0
$31$	9.49163	1.70475	0.852374	−	0.522933i	$-0.175162\pi$
$31$	9.49163	1.70475	0.852374	+	0.522933i	$0.175162\pi$
$32$	4.55171	0.804636
$33$	0	0
$34$	−10.1464	−1.74010
$35$	−0.244091	−0.0412589
$36$	0	0
$37$	2.80113	0.460503	0.230251	−	0.973131i	$-0.426045\pi$
$37$	2.80113	0.460503	0.230251	+	0.973131i	$0.426045\pi$
$38$	0.862961	0.139991
$39$	0	0
$40$	−1.15012	−0.181850
$41$	−9.12030	−1.42435	−0.712176	−	0.702001i	$-0.752290\pi$
$41$	−9.12030	−1.42435	−0.712176	+	0.702001i	$0.752290\pi$
$42$	0	0
$43$	8.22447	1.25422	0.627110	−	0.778931i	$-0.284238\pi$
$43$	8.22447	1.25422	0.627110	+	0.778931i	$0.284238\pi$
$44$	−0.856762	−0.129162
$45$	0	0
$46$	13.0774	1.92815
$47$	−5.47124	−0.798062	−0.399031	−	0.916937i	$-0.630653\pi$
$47$	−5.47124	−0.798062	−0.399031	+	0.916937i	$0.630653\pi$
$48$	0	0
$49$	−6.83182	−0.975975
$50$	7.85219	1.11047
$51$	0	0
$52$	0.368870	0.0511531
$53$	5.98271	0.821789	0.410894	−	0.911683i	$-0.365216\pi$
$53$	5.98271	0.821789	0.410894	+	0.911683i	$0.365216\pi$
$54$	0	0
$55$	0.595209	0.0802580
$56$	0.792421	0.105892
$57$	0	0
$58$	−1.78423	−0.234281
$59$	8.62800	1.12327	0.561635	−	0.827385i	$-0.310172\pi$
$59$	8.62800	1.12327	0.561635	+	0.827385i	$0.310172\pi$
$60$	0	0
$61$	6.93857	0.888393	0.444196	−	0.895929i	$-0.353489\pi$
$61$	6.93857	0.888393	0.444196	+	0.895929i	$0.353489\pi$
$62$	−16.0427	−2.03743
$63$	0	0
$64$	2.26568	0.283211
$65$	−0.256261	−0.0317853
$66$	0	0
$67$	−7.69272	−0.939815	−0.469908	−	0.882716i	$-0.655713\pi$
$67$	−7.69272	−0.939815	−0.469908	+	0.882716i	$0.655713\pi$
$68$	5.14323	0.623709
$69$	0	0
$70$	0.412562	0.0493105
$71$	−13.5341	−1.60621	−0.803103	−	0.595840i	$-0.796819\pi$
$71$	−13.5341	−1.60621	−0.803103	+	0.595840i	$0.796819\pi$
$72$	0	0
$73$	−5.32238	−0.622937	−0.311469	−	0.950256i	$-0.600821\pi$
$73$	−5.32238	−0.622937	−0.311469	+	0.950256i	$0.600821\pi$
$74$	−4.73446	−0.550369
$75$	0	0
$76$	−0.437436	−0.0501773
$77$	−0.410093	−0.0467344
$78$	0	0
$79$	5.83295	0.656258	0.328129	−	0.944633i	$-0.393582\pi$
$79$	5.83295	0.656258	0.328129	+	0.944633i	$0.393582\pi$
$80$	2.96383	0.331367
$81$	0	0
$82$	15.4151	1.70231
$83$	−15.7445	−1.72818	−0.864091	−	0.503335i	$-0.832106\pi$
$83$	−15.7445	−1.72818	−0.864091	+	0.503335i	$0.832106\pi$
$84$	0	0
$85$	−3.57310	−0.387557
$86$	−13.9010	−1.49898
$87$	0	0
$88$	−1.93230	−0.205983
$89$	−16.4457	−1.74324	−0.871622	−	0.490179i	$-0.836931\pi$
$89$	−16.4457	−1.74324	−0.871622	+	0.490179i	$0.836931\pi$
$90$	0	0
$91$	0.176561	0.0185086
$92$	−6.62893	−0.691114
$93$	0	0
$94$	9.24747	0.953803
$95$	0.303895	0.0311790
$96$	0	0
$97$	−6.23983	−0.633559	−0.316779	−	0.948499i	$-0.602601\pi$
$97$	−6.23983	−0.633559	−0.316779	+	0.948499i	$0.602601\pi$
$98$	11.5471	1.16644
$99$	0	0
$100$	−3.98028	−0.398028
$101$	1.35278	0.134607	0.0673035	−	0.997733i	$-0.478560\pi$
$101$	1.35278	0.134607	0.0673035	+	0.997733i	$0.478560\pi$
$102$	0	0
$103$	−3.17598	−0.312939	−0.156469	−	0.987683i	$-0.550011\pi$
$103$	−3.17598	−0.312939	−0.156469	+	0.987683i	$0.550011\pi$
$104$	0.831930	0.0815774
$105$	0	0
$106$	−10.1120	−0.982160
$107$	3.56890	0.345019	0.172509	−	0.985008i	$-0.444812\pi$
$107$	3.56890	0.345019	0.172509	+	0.985008i	$0.444812\pi$
$108$	0	0
$109$	16.9824	1.62662	0.813308	−	0.581833i	$-0.197664\pi$
$109$	16.9824	1.62662	0.813308	+	0.581833i	$0.197664\pi$
$110$	−1.00602	−0.0959202
$111$	0	0
$112$	−2.04205	−0.192956
$113$	11.0957	1.04380	0.521898	−	0.853008i	$-0.325224\pi$
$113$	11.0957	1.04380	0.521898	+	0.853008i	$0.325224\pi$
$114$	0	0
$115$	4.60525	0.429442
$116$	0.904428	0.0839741
$117$	0	0
$118$	−14.5830	−1.34248
$119$	2.46183	0.225676
$120$	0	0
$121$	1.00000	0.0909091
$122$	−11.7275	−1.06176
$123$	0	0
$124$	8.13207	0.730282
$125$	5.74122	0.513511
$126$	0	0
$127$	12.9056	1.14518	0.572591	−	0.819841i	$-0.305938\pi$
$127$	12.9056	1.14518	0.572591	+	0.819841i	$0.305938\pi$
$128$	−12.9329	−1.14312
$129$	0	0
$130$	0.433132	0.0379881
$131$	−16.6132	−1.45151	−0.725753	−	0.687955i	$-0.758508\pi$
$131$	−16.6132	−1.45151	−0.725753	+	0.687955i	$0.758508\pi$
$132$	0	0
$133$	−0.209380	−0.0181556
$134$	13.0022	1.12322
$135$	0	0
$136$	11.5998	0.994673
$137$	−11.3187	−0.967024	−0.483512	−	0.875338i	$-0.660639\pi$
$137$	−11.3187	−0.967024	−0.483512	+	0.875338i	$0.660639\pi$
$138$	0	0
$139$	14.0662	1.19308	0.596539	−	0.802584i	$-0.296542\pi$
$139$	14.0662	1.19308	0.596539	+	0.802584i	$0.296542\pi$
$140$	−0.209128	−0.0176745
$141$	0	0
$142$	22.8753	1.91966
$143$	−0.430540	−0.0360035
$144$	0	0
$145$	−0.628324	−0.0521795
$146$	8.99586	0.744503
$147$	0	0
$148$	2.39990	0.197271
$149$	−13.1525	−1.07750	−0.538749	−	0.842467i	$-0.681103\pi$
$149$	−13.1525	−1.07750	−0.538749	+	0.842467i	$0.681103\pi$
$150$	0	0
$151$	2.70761	0.220342	0.110171	−	0.993913i	$-0.464860\pi$
$151$	2.70761	0.220342	0.110171	+	0.993913i	$0.464860\pi$
$152$	−0.986570	−0.0800214
$153$	0	0
$154$	0.693137	0.0558546
$155$	−5.64951	−0.453779
$156$	0	0
$157$	−8.86277	−0.707326	−0.353663	−	0.935373i	$-0.615064\pi$
$157$	−8.86277	−0.707326	−0.353663	+	0.935373i	$0.615064\pi$
$158$	−9.85883	−0.784327
$159$	0	0
$160$	−2.70922	−0.214183
$161$	−3.17297	−0.250065
$162$	0	0
$163$	4.19910	0.328899	0.164450	−	0.986385i	$-0.447415\pi$
$163$	4.19910	0.328899	0.164450	+	0.986385i	$0.447415\pi$
$164$	−7.81393	−0.610166
$165$	0	0
$166$	26.6113	2.06544
$167$	12.3663	0.956931	0.478465	−	0.878106i	$-0.341193\pi$
$167$	12.3663	0.956931	0.478465	+	0.878106i	$0.341193\pi$
$168$	0	0
$169$	−12.8146	−0.985741
$170$	6.03924	0.463189
$171$	0	0
$172$	7.04641	0.537284
$173$	23.3801	1.77756	0.888780	−	0.458335i	$-0.151554\pi$
$173$	23.3801	1.77756	0.888780	+	0.458335i	$0.151554\pi$
$174$	0	0
$175$	−1.90518	−0.144018
$176$	4.97948	0.375343
$177$	0	0
$178$	27.7965	2.08344
$179$	−7.91227	−0.591391	−0.295696	−	0.955282i	$-0.595551\pi$
$179$	−7.91227	−0.591391	−0.295696	+	0.955282i	$0.595551\pi$
$180$	0	0
$181$	6.49866	0.483042	0.241521	−	0.970396i	$-0.422354\pi$
$181$	6.49866	0.483042	0.241521	+	0.970396i	$0.422354\pi$
$182$	−0.298423	−0.0221206
$183$	0	0
$184$	−14.9505	−1.10217
$185$	−1.66726	−0.122579
$186$	0	0
$187$	−6.00311	−0.438991
$188$	−4.68755	−0.341875
$189$	0	0
$190$	−0.513642	−0.0372635
$191$	−7.89220	−0.571060	−0.285530	−	0.958370i	$-0.592170\pi$
$191$	−7.89220	−0.571060	−0.285530	+	0.958370i	$0.592170\pi$
$192$	0	0
$193$	−9.78321	−0.704211	−0.352105	−	0.935960i	$-0.614534\pi$
$193$	−9.78321	−0.704211	−0.352105	+	0.935960i	$0.614534\pi$
$194$	10.5465	0.757197
$195$	0	0
$196$	−5.85325	−0.418089
$197$	8.71148	0.620668	0.310334	−	0.950628i	$-0.399559\pi$
$197$	8.71148	0.620668	0.310334	+	0.950628i	$0.399559\pi$
$198$	0	0
$199$	14.0796	0.998079	0.499040	−	0.866579i	$-0.333686\pi$
$199$	14.0796	0.998079	0.499040	+	0.866579i	$0.333686\pi$
$200$	−8.97692	−0.634764
$201$	0	0
$202$	−2.28647	−0.160875
$203$	0.432908	0.0303842
$204$	0	0
$205$	5.42849	0.379142
$206$	5.36803	0.374009
$207$	0	0
$208$	−2.14386	−0.148650
$209$	0.510569	0.0353168
$210$	0	0
$211$	2.28884	0.157570	0.0787852	−	0.996892i	$-0.474896\pi$
$211$	2.28884	0.157570	0.0787852	+	0.996892i	$0.474896\pi$
$212$	5.12576	0.352039
$213$	0	0
$214$	−6.03215	−0.412349
$215$	−4.89528	−0.333855
$216$	0	0
$217$	3.89245	0.264237
$218$	−28.7035	−1.94405
$219$	0	0
$220$	0.509953	0.0343810
$221$	2.58457	0.173857
$222$	0	0
$223$	−29.0173	−1.94314	−0.971570	−	0.236751i	$-0.923918\pi$
$223$	−29.0173	−1.94314	−0.971570	+	0.236751i	$0.923918\pi$
$224$	1.86662	0.124719
$225$	0	0
$226$	−18.7539	−1.24749
$227$	−1.39102	−0.0923249	−0.0461625	−	0.998934i	$-0.514699\pi$
$227$	−1.39102	−0.0923249	−0.0461625	+	0.998934i	$0.514699\pi$
$228$	0	0
$229$	−13.1024	−0.865833	−0.432917	−	0.901434i	$-0.642516\pi$
$229$	−13.1024	−0.865833	−0.432917	+	0.901434i	$0.642516\pi$
$230$	−7.78377	−0.513247
$231$	0	0
$232$	2.03980	0.133919
$233$	8.65103	0.566748	0.283374	−	0.959009i	$-0.408546\pi$
$233$	8.65103	0.566748	0.283374	+	0.959009i	$0.408546\pi$
$234$	0	0
$235$	3.25653	0.212433
$236$	7.39215	0.481188
$237$	0	0
$238$	−4.16097	−0.269716
$239$	17.9416	1.16055	0.580273	−	0.814422i	$-0.302946\pi$
$239$	17.9416	1.16055	0.580273	+	0.814422i	$0.302946\pi$
$240$	0	0
$241$	10.8273	0.697448	0.348724	−	0.937225i	$-0.386615\pi$
$241$	10.8273	0.697448	0.348724	+	0.937225i	$0.386615\pi$
$242$	−1.69020	−0.108650
$243$	0	0
$244$	5.94470	0.380571
$245$	4.06636	0.259790
$246$	0	0
$247$	−0.219820	−0.0139868
$248$	18.3406	1.16463
$249$	0	0
$250$	−9.70379	−0.613722
$251$	−0.803748	−0.0507321	−0.0253661	−	0.999678i	$-0.508075\pi$
$251$	−0.803748	−0.0507321	−0.0253661	+	0.999678i	$0.508075\pi$
$252$	0	0
$253$	7.73719	0.486433
$254$	−21.8129	−1.36866
$255$	0	0
$256$	17.3277	1.08298
$257$	−16.7076	−1.04219	−0.521096	−	0.853498i	$-0.674477\pi$
$257$	−16.7076	−1.04219	−0.521096	+	0.853498i	$0.674477\pi$
$258$	0	0
$259$	1.14872	0.0713782
$260$	−0.219555	−0.0136162
$261$	0	0
$262$	28.0796	1.73477
$263$	−0.793328	−0.0489187	−0.0244594	−	0.999701i	$-0.507786\pi$
$263$	−0.793328	−0.0489187	−0.0244594	+	0.999701i	$0.507786\pi$
$264$	0	0
$265$	−3.56097	−0.218748
$266$	0.353894	0.0216986
$267$	0	0
$268$	−6.59083	−0.402599
$269$	16.0514	0.978673	0.489337	−	0.872095i	$-0.337239\pi$
$269$	16.0514	0.978673	0.489337	+	0.872095i	$0.337239\pi$
$270$	0	0
$271$	1.88224	0.114338	0.0571691	−	0.998365i	$-0.481793\pi$
$271$	1.88224	0.114338	0.0571691	+	0.998365i	$0.481793\pi$
$272$	−29.8924	−1.81249
$273$	0	0
$274$	19.1309	1.15574
$275$	4.64573	0.280148
$276$	0	0
$277$	−13.0009	−0.781151	−0.390576	−	0.920571i	$-0.627724\pi$
$277$	−13.0009	−0.781151	−0.390576	+	0.920571i	$0.627724\pi$
$278$	−23.7746	−1.42591
$279$	0	0
$280$	−0.471656	−0.0281868
$281$	−15.2432	−0.909336	−0.454668	−	0.890661i	$-0.650242\pi$
$281$	−15.2432	−0.909336	−0.454668	+	0.890661i	$0.650242\pi$
$282$	0	0
$283$	15.3428	0.912037	0.456019	−	0.889970i	$-0.349275\pi$
$283$	15.3428	0.912037	0.456019	+	0.889970i	$0.349275\pi$
$284$	−11.5955	−0.688068
$285$	0	0
$286$	0.727696	0.0430296
$287$	−3.74017	−0.220775
$288$	0	0
$289$	19.0373	1.11984
$290$	1.06199	0.0623622
$291$	0	0
$292$	−4.56001	−0.266855
$293$	12.0579	0.704428	0.352214	−	0.935920i	$-0.385429\pi$
$293$	12.0579	0.704428	0.352214	+	0.935920i	$0.385429\pi$
$294$	0	0
$295$	−5.13547	−0.298998
$296$	5.41261	0.314602
$297$	0	0
$298$	22.2303	1.28777
$299$	−3.33117	−0.192646
$300$	0	0
$301$	3.37279	0.194405
$302$	−4.57638	−0.263341
$303$	0	0
$304$	2.54237	0.145815
$305$	−4.12990	−0.236477
$306$	0	0
$307$	−2.80491	−0.160085	−0.0800425	−	0.996791i	$-0.525506\pi$
$307$	−2.80491	−0.160085	−0.0800425	+	0.996791i	$0.525506\pi$
$308$	−0.351352	−0.0200201
$309$	0	0
$310$	9.54877	0.542334
$311$	13.7487	0.779617	0.389808	−	0.920896i	$-0.372541\pi$
$311$	13.7487	0.779617	0.389808	+	0.920896i	$0.372541\pi$
$312$	0	0
$313$	3.50319	0.198012	0.0990060	−	0.995087i	$-0.468434\pi$
$313$	3.50319	0.198012	0.0990060	+	0.995087i	$0.468434\pi$
$314$	14.9798	0.845360
$315$	0	0
$316$	4.99745	0.281129
$317$	−33.3277	−1.87187	−0.935936	−	0.352170i	$-0.885444\pi$
$317$	−33.3277	−1.87187	−0.935936	+	0.352170i	$0.885444\pi$
$318$	0	0
$319$	−1.05564	−0.0591042
$320$	−1.34856	−0.0753866
$321$	0	0
$322$	5.36293	0.298865
$323$	−3.06500	−0.170541
$324$	0	0
$325$	−2.00017	−0.110949
$326$	−7.09731	−0.393083
$327$	0	0
$328$	−17.6231	−0.973074
$329$	−2.24372	−0.123700
$330$	0	0
$331$	27.5606	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$331$	27.5606	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$332$	−13.4893	−0.740321
$333$	0	0
$334$	−20.9014	−1.14368
$335$	4.57878	0.250165
$336$	0	0
$337$	−16.8739	−0.919181	−0.459591	−	0.888131i	$-0.652004\pi$
$337$	−16.8739	−0.919181	−0.459591	+	0.888131i	$0.652004\pi$
$338$	21.6592	1.17811
$339$	0	0
$340$	−3.06130	−0.166022
$341$	−9.49163	−0.514001
$342$	0	0
$343$	−5.67233	−0.306277
$344$	15.8921	0.856845
$345$	0	0
$346$	−39.5170	−2.12445
$347$	−18.9487	−1.01722	−0.508609	−	0.860998i	$-0.669840\pi$
$347$	−18.9487	−1.01722	−0.508609	+	0.860998i	$0.669840\pi$
$348$	0	0
$349$	−12.7354	−0.681708	−0.340854	−	0.940116i	$-0.610716\pi$
$349$	−12.7354	−0.681708	−0.340854	+	0.940116i	$0.610716\pi$
$350$	3.22012	0.172123
$351$	0	0
$352$	−4.55171	−0.242607
$353$	0.0409974	0.00218207	0.00109103	−	0.999999i	$-0.499653\pi$
$353$	0.0409974	0.00218207	0.00109103	+	0.999999i	$0.499653\pi$
$354$	0	0
$355$	8.05564	0.427549
$356$	−14.0901	−0.746773
$357$	0	0
$358$	13.3733	0.706801
$359$	9.96140	0.525742	0.262871	−	0.964831i	$-0.415331\pi$
$359$	9.96140	0.525742	0.262871	+	0.964831i	$0.415331\pi$
$360$	0	0
$361$	−18.7393	−0.986280
$362$	−10.9840	−0.577307
$363$	0	0
$364$	0.151271	0.00792875
$365$	3.16793	0.165817
$366$	0	0
$367$	27.3887	1.42968	0.714840	−	0.699288i	$-0.246500\pi$
$367$	27.3887	1.42968	0.714840	+	0.699288i	$0.246500\pi$
$368$	38.5272	2.00837
$369$	0	0
$370$	2.81799	0.146500
$371$	2.45347	0.127378
$372$	0	0
$373$	−25.8174	−1.33677	−0.668387	−	0.743814i	$-0.733015\pi$
$373$	−25.8174	−1.33677	−0.668387	+	0.743814i	$0.733015\pi$
$374$	10.1464	0.524659
$375$	0	0
$376$	−10.5721	−0.545212
$377$	0.454493	0.0234076
$378$	0	0
$379$	1.70902	0.0877862	0.0438931	−	0.999036i	$-0.486024\pi$
$379$	1.70902	0.0877862	0.0438931	+	0.999036i	$0.486024\pi$
$380$	0.260366	0.0133565
$381$	0	0
$382$	13.3394	0.682501
$383$	−3.36568	−0.171978	−0.0859890	−	0.996296i	$-0.527405\pi$
$383$	−3.36568	−0.171978	−0.0859890	+	0.996296i	$0.527405\pi$
$384$	0	0
$385$	0.244091	0.0124400
$386$	16.5355	0.841637
$387$	0	0
$388$	−5.34605	−0.271404
$389$	21.2237	1.07609	0.538043	−	0.842917i	$-0.319164\pi$
$389$	21.2237	1.07609	0.538043	+	0.842917i	$0.319164\pi$
$390$	0	0
$391$	−46.4472	−2.34893
$392$	−13.2011	−0.666757
$393$	0	0
$394$	−14.7241	−0.741790
$395$	−3.47183	−0.174687
$396$	0	0
$397$	−6.42948	−0.322686	−0.161343	−	0.986898i	$-0.551583\pi$
$397$	−6.42948	−0.322686	−0.161343	+	0.986898i	$0.551583\pi$
$398$	−23.7974	−1.19285
$399$	0	0
$400$	23.1333	1.15667
$401$	1.75961	0.0878705	0.0439352	−	0.999034i	$-0.486010\pi$
$401$	1.75961	0.0878705	0.0439352	+	0.999034i	$0.486010\pi$
$402$	0	0
$403$	4.08653	0.203564
$404$	1.15901	0.0576631
$405$	0	0
$406$	−0.731700	−0.0363137
$407$	−2.80113	−0.138847
$408$	0	0
$409$	−11.5898	−0.573077	−0.286539	−	0.958069i	$-0.592505\pi$
$409$	−11.5898	−0.573077	−0.286539	+	0.958069i	$0.592505\pi$
$410$	−9.17521	−0.453131
$411$	0	0
$412$	−2.72106	−0.134057
$413$	3.53828	0.174107
$414$	0	0
$415$	9.37127	0.460017
$416$	1.95969	0.0960818
$417$	0	0
$418$	−0.862961	−0.0422088
$419$	−25.8860	−1.26461	−0.632307	−	0.774718i	$-0.717892\pi$
$419$	−25.8860	−1.26461	−0.632307	+	0.774718i	$0.717892\pi$
$420$	0	0
$421$	34.0887	1.66138	0.830691	−	0.556733i	$-0.187945\pi$
$421$	34.0887	1.66138	0.830691	+	0.556733i	$0.187945\pi$
$422$	−3.86859	−0.188320
$423$	0	0
$424$	11.5604	0.561421
$425$	−27.8888	−1.35280
$426$	0	0
$427$	2.84546	0.137701
$428$	3.05770	0.147800
$429$	0	0
$430$	8.27398	0.399007
$431$	−31.6626	−1.52513	−0.762567	−	0.646910i	$-0.776061\pi$
$431$	−31.6626	−1.52513	−0.762567	+	0.646910i	$0.776061\pi$
$432$	0	0
$433$	3.51564	0.168951	0.0844753	−	0.996426i	$-0.473079\pi$
$433$	3.51564	0.168951	0.0844753	+	0.996426i	$0.473079\pi$
$434$	−6.57900	−0.315802
$435$	0	0
$436$	14.5499	0.696812
$437$	3.95037	0.188972
$438$	0	0
$439$	−21.3094	−1.01704	−0.508521	−	0.861050i	$-0.669807\pi$
$439$	−21.3094	−1.01704	−0.508521	+	0.861050i	$0.669807\pi$
$440$	1.15012	0.0548298
$441$	0	0
$442$	−4.36844	−0.207785
$443$	−28.8462	−1.37053	−0.685263	−	0.728296i	$-0.740313\pi$
$443$	−28.8462	−1.37053	−0.685263	+	0.728296i	$0.740313\pi$
$444$	0	0
$445$	9.78865	0.464026
$446$	49.0449	2.32234
$447$	0	0
$448$	0.929141	0.0438978
$449$	−37.1687	−1.75410	−0.877051	−	0.480398i	$-0.840492\pi$
$449$	−37.1687	−1.75410	−0.877051	+	0.480398i	$0.840492\pi$
$450$	0	0
$451$	9.12030	0.429458
$452$	9.50638	0.447142
$453$	0	0
$454$	2.35109	0.110342
$455$	−0.105091	−0.00492673
$456$	0	0
$457$	−26.9015	−1.25840	−0.629200	−	0.777244i	$-0.716617\pi$
$457$	−26.9015	−1.25840	−0.629200	+	0.777244i	$0.716617\pi$
$458$	22.1457	1.03480
$459$	0	0
$460$	3.94560	0.183965
$461$	−3.32382	−0.154806	−0.0774029	−	0.997000i	$-0.524663\pi$
$461$	−3.32382	−0.154806	−0.0774029	+	0.997000i	$0.524663\pi$
$462$	0	0
$463$	−15.8267	−0.735529	−0.367764	−	0.929919i	$-0.619877\pi$
$463$	−15.8267	−0.735529	−0.367764	+	0.929919i	$0.619877\pi$
$464$	−5.25652	−0.244028
$465$	0	0
$466$	−14.6219	−0.677348
$467$	−23.5962	−1.09190	−0.545951	−	0.837817i	$-0.683832\pi$
$467$	−23.5962	−1.09190	−0.545951	+	0.837817i	$0.683832\pi$
$468$	0	0
$469$	−3.15473	−0.145672
$470$	−5.50418	−0.253889
$471$	0	0
$472$	16.6719	0.767384
$473$	−8.22447	−0.378161
$474$	0	0
$475$	2.37196	0.108833
$476$	2.10920	0.0966752
$477$	0	0
$478$	−30.3248	−1.38703
$479$	−35.8538	−1.63820	−0.819100	−	0.573651i	$-0.805527\pi$
$479$	−35.8538	−1.63820	−0.819100	+	0.573651i	$0.805527\pi$
$480$	0	0
$481$	1.20600	0.0549887
$482$	−18.3003	−0.833554
$483$	0	0
$484$	0.856762	0.0389437
$485$	3.71400	0.168644
$486$	0	0
$487$	35.9480	1.62896	0.814479	−	0.580193i	$-0.197023\pi$
$487$	35.9480	1.62896	0.814479	+	0.580193i	$0.197023\pi$
$488$	13.4074	0.606923
$489$	0	0
$490$	−6.87295	−0.310488
$491$	16.9010	0.762730	0.381365	−	0.924425i	$-0.375454\pi$
$491$	16.9010	0.762730	0.381365	+	0.924425i	$0.375454\pi$
$492$	0	0
$493$	6.33709	0.285408
$494$	0.371539	0.0167163
$495$	0	0
$496$	−47.2634	−2.12219
$497$	−5.55025	−0.248963
$498$	0	0
$499$	29.6152	1.32576	0.662880	−	0.748726i	$-0.269334\pi$
$499$	29.6152	1.32576	0.662880	+	0.748726i	$0.269334\pi$
$500$	4.91886	0.219978
$501$	0	0
$502$	1.35849	0.0606325
$503$	−8.78358	−0.391641	−0.195820	−	0.980640i	$-0.562737\pi$
$503$	−8.78358	−0.391641	−0.195820	+	0.980640i	$0.562737\pi$
$504$	0	0
$505$	−0.805189	−0.0358304
$506$	−13.0774	−0.581360
$507$	0	0
$508$	11.0570	0.490575
$509$	3.03653	0.134592	0.0672959	−	0.997733i	$-0.478563\pi$
$509$	3.03653	0.134592	0.0672959	+	0.997733i	$0.478563\pi$
$510$	0	0
$511$	−2.18267	−0.0965556
$512$	−3.42149	−0.151210
$513$	0	0
$514$	28.2391	1.24557
$515$	1.89037	0.0832998
$516$	0	0
$517$	5.47124	0.240625
$518$	−1.94157	−0.0853075
$519$	0	0
$520$	−0.495172	−0.0217147
$521$	−26.5517	−1.16325	−0.581625	−	0.813457i	$-0.697583\pi$
$521$	−26.5517	−1.16325	−0.581625	+	0.813457i	$0.697583\pi$
$522$	0	0
$523$	39.1432	1.71161	0.855807	−	0.517295i	$-0.173061\pi$
$523$	39.1432	1.71161	0.855807	+	0.517295i	$0.173061\pi$
$524$	−14.2336	−0.621798
$525$	0	0
$526$	1.34088	0.0584652
$527$	56.9793	2.48206
$528$	0	0
$529$	36.8641	1.60279
$530$	6.01873	0.261437
$531$	0	0
$532$	−0.179389	−0.00777751
$533$	−3.92665	−0.170082
$534$	0	0
$535$	−2.12424	−0.0918391
$536$	−14.8646	−0.642054
$537$	0	0
$538$	−27.1301	−1.16966
$539$	6.83182	0.294267
$540$	0	0
$541$	−43.5948	−1.87429	−0.937144	−	0.348944i	$-0.886540\pi$
$541$	−43.5948	−1.87429	−0.937144	+	0.348944i	$0.886540\pi$
$542$	−3.18136	−0.136651
$543$	0	0
$544$	27.3244	1.17152
$545$	−10.1081	−0.432982
$546$	0	0
$547$	−28.4525	−1.21654	−0.608270	−	0.793731i	$-0.708136\pi$
$547$	−28.4525	−1.21654	−0.608270	+	0.793731i	$0.708136\pi$
$548$	−9.69746	−0.414255
$549$	0	0
$550$	−7.85219	−0.334818
$551$	−0.538974	−0.0229611
$552$	0	0
$553$	2.39205	0.101720
$554$	21.9741	0.933592
$555$	0	0
$556$	12.0514	0.511092
$557$	−5.55841	−0.235517	−0.117759	−	0.993042i	$-0.537571\pi$
$557$	−5.55841	−0.235517	−0.117759	+	0.993042i	$0.537571\pi$
$558$	0	0
$559$	3.54096	0.149767
$560$	1.21545	0.0513620
$561$	0	0
$562$	25.7641	1.08679
$563$	8.87426	0.374006	0.187003	−	0.982359i	$-0.440123\pi$
$563$	8.87426	0.374006	0.187003	+	0.982359i	$0.440123\pi$
$564$	0	0
$565$	−6.60426	−0.277844
$566$	−25.9324	−1.09002
$567$	0	0
$568$	−26.1519	−1.09731
$569$	33.2461	1.39375	0.696876	−	0.717192i	$-0.254573\pi$
$569$	33.2461	1.39375	0.696876	+	0.717192i	$0.254573\pi$
$570$	0	0
$571$	−18.9650	−0.793663	−0.396831	−	0.917892i	$-0.629890\pi$
$571$	−18.9650	−0.793663	−0.396831	+	0.917892i	$0.629890\pi$
$572$	−0.368870	−0.0154232
$573$	0	0
$574$	6.32162	0.263859
$575$	35.9449	1.49900
$576$	0	0
$577$	−11.6702	−0.485837	−0.242919	−	0.970047i	$-0.578105\pi$
$577$	−11.6702	−0.485837	−0.242919	+	0.970047i	$0.578105\pi$
$578$	−32.1767	−1.33838
$579$	0	0
$580$	−0.538324	−0.0223527
$581$	−6.45670	−0.267869
$582$	0	0
$583$	−5.98271	−0.247779
$584$	−10.2844	−0.425572
$585$	0	0
$586$	−20.3801	−0.841896
$587$	−13.9888	−0.577378	−0.288689	−	0.957423i	$-0.593219\pi$
$587$	−13.9888	−0.577378	−0.288689	+	0.957423i	$0.593219\pi$
$588$	0	0
$589$	−4.84613	−0.199681
$590$	8.67994	0.357348
$591$	0	0
$592$	−13.9482	−0.573266
$593$	−22.0932	−0.907260	−0.453630	−	0.891190i	$-0.649871\pi$
$593$	−22.0932	−0.907260	−0.453630	+	0.891190i	$0.649871\pi$
$594$	0	0
$595$	−1.46530	−0.0600716
$596$	−11.2686	−0.461579
$597$	0	0
$598$	5.63033	0.230241
$599$	10.5807	0.432315	0.216157	−	0.976359i	$-0.430648\pi$
$599$	10.5807	0.432315	0.216157	+	0.976359i	$0.430648\pi$
$600$	0	0
$601$	12.1715	0.496488	0.248244	−	0.968698i	$-0.420147\pi$
$601$	12.1715	0.496488	0.248244	+	0.968698i	$0.420147\pi$
$602$	−5.70068	−0.232343
$603$	0	0
$604$	2.31977	0.0943903
$605$	−0.595209	−0.0241987
$606$	0	0
$607$	−31.7864	−1.29017	−0.645086	−	0.764110i	$-0.723178\pi$
$607$	−31.7864	−1.29017	−0.645086	+	0.764110i	$0.723178\pi$
$608$	−2.32396	−0.0942490
$609$	0	0
$610$	6.98034	0.282626
$611$	−2.35559	−0.0952968
$612$	0	0
$613$	−4.52507	−0.182766	−0.0913829	−	0.995816i	$-0.529129\pi$
$613$	−4.52507	−0.182766	−0.0913829	+	0.995816i	$0.529129\pi$
$614$	4.74086	0.191325
$615$	0	0
$616$	−0.792421	−0.0319275
$617$	6.09689	0.245451	0.122726	−	0.992441i	$-0.460836\pi$
$617$	6.09689	0.245451	0.122726	+	0.992441i	$0.460836\pi$
$618$	0	0
$619$	4.49490	0.180665	0.0903327	−	0.995912i	$-0.471207\pi$
$619$	4.49490	0.180665	0.0903327	+	0.995912i	$0.471207\pi$
$620$	−4.84028	−0.194390
$621$	0	0
$622$	−23.2380	−0.931758
$623$	−6.74427	−0.270204
$624$	0	0
$625$	19.8114	0.792456
$626$	−5.92107	−0.236654
$627$	0	0
$628$	−7.59329	−0.303005
$629$	16.8155	0.670477
$630$	0	0
$631$	−34.4601	−1.37183	−0.685917	−	0.727680i	$-0.740599\pi$
$631$	−34.4601	−1.37183	−0.685917	+	0.727680i	$0.740599\pi$
$632$	11.2710	0.448336
$633$	0	0
$634$	56.3304	2.23717
$635$	−7.68150	−0.304831
$636$	0	0
$637$	−2.94137	−0.116541
$638$	1.78423	0.0706384
$639$	0	0
$640$	7.69776	0.304281
$641$	44.7085	1.76588	0.882939	−	0.469487i	$-0.155561\pi$
$641$	44.7085	1.76588	0.882939	+	0.469487i	$0.155561\pi$
$642$	0	0
$643$	−8.17334	−0.322325	−0.161163	−	0.986928i	$-0.551524\pi$
$643$	−8.17334	−0.322325	−0.161163	+	0.986928i	$0.551524\pi$
$644$	−2.71848	−0.107123
$645$	0	0
$646$	5.18045	0.203822
$647$	−9.78390	−0.384645	−0.192322	−	0.981332i	$-0.561602\pi$
$647$	−9.78390	−0.384645	−0.192322	+	0.981332i	$0.561602\pi$
$648$	0	0
$649$	−8.62800	−0.338679
$650$	3.38068	0.132601
$651$	0	0
$652$	3.59763	0.140894
$653$	0.756630	0.0296092	0.0148046	−	0.999890i	$-0.495287\pi$
$653$	0.756630	0.0296092	0.0148046	+	0.999890i	$0.495287\pi$
$654$	0	0
$655$	9.88835	0.386370
$656$	45.4144	1.77313
$657$	0	0
$658$	3.79232	0.147840
$659$	3.01642	0.117503	0.0587516	−	0.998273i	$-0.481288\pi$
$659$	3.01642	0.117503	0.0587516	+	0.998273i	$0.481288\pi$
$660$	0	0
$661$	15.2317	0.592446	0.296223	−	0.955119i	$-0.404273\pi$
$661$	15.2317	0.592446	0.296223	+	0.955119i	$0.404273\pi$
$662$	−46.5828	−1.81049
$663$	0	0
$664$	−30.4230	−1.18064
$665$	0.124625	0.00483276
$666$	0	0
$667$	−8.16765	−0.316253
$668$	10.5950	0.409931
$669$	0	0
$670$	−7.73903	−0.298985
$671$	−6.93857	−0.267861
$672$	0	0
$673$	1.58107	0.0609458	0.0304729	−	0.999536i	$-0.490299\pi$
$673$	1.58107	0.0609458	0.0304729	+	0.999536i	$0.490299\pi$
$674$	28.5202	1.09856
$675$	0	0
$676$	−10.9791	−0.422273
$677$	−23.9981	−0.922323	−0.461162	−	0.887316i	$-0.652567\pi$
$677$	−23.9981	−0.922323	−0.461162	+	0.887316i	$0.652567\pi$
$678$	0	0
$679$	−2.55891	−0.0982019
$680$	−6.90429	−0.264768
$681$	0	0
$682$	16.0427	0.614308
$683$	28.4056	1.08691	0.543455	−	0.839438i	$-0.317116\pi$
$683$	28.4056	1.08691	0.543455	+	0.839438i	$0.317116\pi$
$684$	0	0
$685$	6.73701	0.257408
$686$	9.58735	0.366047
$687$	0	0
$688$	−40.9536	−1.56134
$689$	2.57579	0.0981300
$690$	0	0
$691$	−2.21746	−0.0843563	−0.0421782	−	0.999110i	$-0.513430\pi$
$691$	−2.21746	−0.0843563	−0.0421782	+	0.999110i	$0.513430\pi$
$692$	20.0312	0.761473
$693$	0	0
$694$	32.0269	1.21573
$695$	−8.37232	−0.317580
$696$	0	0
$697$	−54.7501	−2.07381
$698$	21.5253	0.814743
$699$	0	0
$700$	−1.63228	−0.0616946
$701$	−32.4336	−1.22500	−0.612499	−	0.790471i	$-0.709836\pi$
$701$	−32.4336	−1.22500	−0.612499	+	0.790471i	$0.709836\pi$
$702$	0	0
$703$	−1.43017	−0.0539398
$704$	−2.26568	−0.0853912
$705$	0	0
$706$	−0.0692936	−0.00260790
$707$	0.554766	0.0208641
$708$	0	0
$709$	−44.1406	−1.65773	−0.828867	−	0.559445i	$-0.811014\pi$
$709$	−44.1406	−1.65773	−0.828867	+	0.559445i	$0.811014\pi$
$710$	−13.6156	−0.510985
$711$	0	0
$712$	−31.7780	−1.19093
$713$	−73.4386	−2.75030
$714$	0	0
$715$	0.256261	0.00958362
$716$	−6.77894	−0.253341
$717$	0	0
$718$	−16.8367	−0.628341
$719$	−8.03558	−0.299677	−0.149838	−	0.988711i	$-0.547875\pi$
$719$	−8.03558	−0.299677	−0.149838	+	0.988711i	$0.547875\pi$
$720$	0	0
$721$	−1.30245	−0.0485057
$722$	31.6731	1.17875
$723$	0	0
$724$	5.56781	0.206926
$725$	−4.90419	−0.182137
$726$	0	0
$727$	8.21117	0.304536	0.152268	−	0.988339i	$-0.451342\pi$
$727$	8.21117	0.304536	0.152268	+	0.988339i	$0.451342\pi$
$728$	0.341168	0.0126445
$729$	0	0
$730$	−5.35442	−0.198176
$731$	49.3724	1.82610
$732$	0	0
$733$	22.3248	0.824584	0.412292	−	0.911052i	$-0.364728\pi$
$733$	22.3248	0.824584	0.412292	+	0.911052i	$0.364728\pi$
$734$	−46.2923	−1.70868
$735$	0	0
$736$	−35.2175	−1.29813
$737$	7.69272	0.283365
$738$	0	0
$739$	−51.2196	−1.88415	−0.942073	−	0.335409i	$-0.891125\pi$
$739$	−51.2196	−1.88415	−0.942073	+	0.335409i	$0.891125\pi$
$740$	−1.42844	−0.0525106
$741$	0	0
$742$	−4.14684	−0.152235
$743$	12.7138	0.466423	0.233212	−	0.972426i	$-0.425076\pi$
$743$	12.7138	0.466423	0.233212	+	0.972426i	$0.425076\pi$
$744$	0	0
$745$	7.82850	0.286814
$746$	43.6364	1.59764
$747$	0	0
$748$	−5.14323	−0.188055
$749$	1.46358	0.0534781
$750$	0	0
$751$	−30.9981	−1.13114	−0.565569	−	0.824701i	$-0.691343\pi$
$751$	−30.9981	−1.13114	−0.565569	+	0.824701i	$0.691343\pi$
$752$	27.2439	0.993484
$753$	0	0
$754$	−0.768182	−0.0279755
$755$	−1.61159	−0.0586518
$756$	0	0
$757$	−26.2794	−0.955140	−0.477570	−	0.878594i	$-0.658482\pi$
$757$	−26.2794	−0.955140	−0.477570	+	0.878594i	$0.658482\pi$
$758$	−2.88857	−0.104918
$759$	0	0
$760$	0.587215	0.0213005
$761$	6.19516	0.224574	0.112287	−	0.993676i	$-0.464182\pi$
$761$	6.19516	0.224574	0.112287	+	0.993676i	$0.464182\pi$
$762$	0	0
$763$	6.96435	0.252126
$764$	−6.76174	−0.244631
$765$	0	0
$766$	5.68865	0.205539
$767$	3.71470	0.134130
$768$	0	0
$769$	−47.3346	−1.70693	−0.853465	−	0.521151i	$-0.825503\pi$
$769$	−47.3346	−1.70693	−0.853465	+	0.521151i	$0.825503\pi$
$770$	−0.412562	−0.0148677
$771$	0	0
$772$	−8.38188	−0.301671
$773$	−40.6583	−1.46238	−0.731189	−	0.682175i	$-0.761034\pi$
$773$	−40.6583	−1.46238	−0.731189	+	0.682175i	$0.761034\pi$
$774$	0	0
$775$	−44.0955	−1.58396
$776$	−12.0572	−0.432828
$777$	0	0
$778$	−35.8723	−1.28608
$779$	4.65654	0.166838
$780$	0	0
$781$	13.5341	0.484289
$782$	78.5048	2.80733
$783$	0	0
$784$	34.0190	1.21496
$785$	5.27520	0.188280
$786$	0	0
$787$	9.56027	0.340787	0.170393	−	0.985376i	$-0.445496\pi$
$787$	9.56027	0.340787	0.170393	+	0.985376i	$0.445496\pi$
$788$	7.46367	0.265882
$789$	0	0
$790$	5.86807	0.208776
$791$	4.55027	0.161789
$792$	0	0
$793$	2.98733	0.106083
$794$	10.8671	0.385658
$795$	0	0
$796$	12.0629	0.427558
$797$	14.8696	0.526707	0.263354	−	0.964699i	$-0.415171\pi$
$797$	14.8696	0.526707	0.263354	+	0.964699i	$0.415171\pi$
$798$	0	0
$799$	−32.8444	−1.16195
$800$	−21.1460	−0.747624
$801$	0	0
$802$	−2.97408	−0.105018
$803$	5.32238	0.187823
$804$	0	0
$805$	1.88858	0.0665636
$806$	−6.90703	−0.243290
$807$	0	0
$808$	2.61398	0.0919594
$809$	−13.0095	−0.457391	−0.228696	−	0.973498i	$-0.573446\pi$
$809$	−13.0095	−0.457391	−0.228696	+	0.973498i	$0.573446\pi$
$810$	0	0
$811$	−22.2964	−0.782931	−0.391465	−	0.920193i	$-0.628032\pi$
$811$	−22.2964	−0.782931	−0.391465	+	0.920193i	$0.628032\pi$
$812$	0.370900	0.0130160
$813$	0	0
$814$	4.73446	0.165943
$815$	−2.49934	−0.0875482
$816$	0	0
$817$	−4.19916	−0.146910
$818$	19.5890	0.684913
$819$	0	0
$820$	4.65092	0.162417
$821$	−39.1519	−1.36641	−0.683205	−	0.730227i	$-0.739414\pi$
$821$	−39.1519	−1.36641	−0.683205	+	0.730227i	$0.739414\pi$
$822$	0	0
$823$	−45.8088	−1.59679	−0.798396	−	0.602132i	$-0.794318\pi$
$823$	−45.8088	−1.59679	−0.798396	+	0.602132i	$0.794318\pi$
$824$	−6.13694	−0.213790
$825$	0	0
$826$	−5.98039	−0.208084
$827$	10.9543	0.380917	0.190459	−	0.981695i	$-0.439003\pi$
$827$	10.9543	0.380917	0.190459	+	0.981695i	$0.439003\pi$
$828$	0	0
$829$	−43.8743	−1.52382	−0.761909	−	0.647684i	$-0.775738\pi$
$829$	−43.8743	−1.52382	−0.761909	+	0.647684i	$0.775738\pi$
$830$	−15.8393	−0.549789
$831$	0	0
$832$	0.975467	0.0338182
$833$	−41.0122	−1.42099
$834$	0	0
$835$	−7.36052	−0.254721
$836$	0.437436	0.0151290
$837$	0	0
$838$	43.7524	1.51140
$839$	53.3308	1.84118	0.920592	−	0.390525i	$-0.127706\pi$
$839$	53.3308	1.84118	0.920592	+	0.390525i	$0.127706\pi$
$840$	0	0
$841$	−27.8856	−0.961574
$842$	−57.6166	−1.98560
$843$	0	0
$844$	1.96099	0.0675001
$845$	7.62739	0.262390
$846$	0	0
$847$	0.410093	0.0140910
$848$	−29.7908	−1.02302
$849$	0	0
$850$	47.1375	1.61680
$851$	−21.6729	−0.742937
$852$	0	0
$853$	−19.1955	−0.657243	−0.328621	−	0.944462i	$-0.606584\pi$
$853$	−19.1955	−0.657243	−0.328621	+	0.944462i	$0.606584\pi$
$854$	−4.80938	−0.164574
$855$	0	0
$856$	6.89618	0.235706
$857$	15.0797	0.515113	0.257556	−	0.966263i	$-0.417083\pi$
$857$	15.0797	0.515113	0.257556	+	0.966263i	$0.417083\pi$
$858$	0	0
$859$	28.6743	0.978353	0.489177	−	0.872185i	$-0.337297\pi$
$859$	28.6743	0.978353	0.489177	+	0.872185i	$0.337297\pi$
$860$	−4.19409	−0.143017
$861$	0	0
$862$	53.5160	1.82276
$863$	−32.4887	−1.10593	−0.552965	−	0.833205i	$-0.686504\pi$
$863$	−32.4887	−1.10593	−0.552965	+	0.833205i	$0.686504\pi$
$864$	0	0
$865$	−13.9161	−0.473161
$866$	−5.94211	−0.201921
$867$	0	0
$868$	3.33490	0.113194
$869$	−5.83295	−0.197869
$870$	0	0
$871$	−3.31202	−0.112224
$872$	32.8150	1.11126
$873$	0	0
$874$	−6.67689	−0.225849
$875$	2.35443	0.0795944
$876$	0	0
$877$	−16.4079	−0.554055	−0.277027	−	0.960862i	$-0.589349\pi$
$877$	−16.4079	−0.554055	−0.277027	+	0.960862i	$0.589349\pi$
$878$	36.0170	1.21552
$879$	0	0
$880$	−2.96383	−0.0999108
$881$	−29.9434	−1.00882	−0.504409	−	0.863465i	$-0.668290\pi$
$881$	−29.9434	−1.00882	−0.504409	+	0.863465i	$0.668290\pi$
$882$	0	0
$883$	−35.3449	−1.18945	−0.594725	−	0.803929i	$-0.702739\pi$
$883$	−35.3449	−1.18945	−0.594725	+	0.803929i	$0.702739\pi$
$884$	2.21437	0.0744772
$885$	0	0
$886$	48.7558	1.63798
$887$	31.0137	1.04134	0.520669	−	0.853759i	$-0.325682\pi$
$887$	31.0137	1.04134	0.520669	+	0.853759i	$0.325682\pi$
$888$	0	0
$889$	5.29247	0.177504
$890$	−16.5447	−0.554581
$891$	0	0
$892$	−24.8609	−0.832405
$893$	2.79344	0.0934790
$894$	0	0
$895$	4.70946	0.157420
$896$	−5.30368	−0.177183
$897$	0	0
$898$	62.8225	2.09641
$899$	10.0197	0.334176
$900$	0	0
$901$	35.9149	1.19650
$902$	−15.4151	−0.513267
$903$	0	0
$904$	21.4402	0.713090
$905$	−3.86806	−0.128579
$906$	0	0
$907$	41.1051	1.36487	0.682436	−	0.730946i	$-0.260921\pi$
$907$	41.1051	1.36487	0.682436	+	0.730946i	$0.260921\pi$
$908$	−1.19177	−0.0395502
$909$	0	0
$910$	0.177624	0.00588818
$911$	0.416778	0.0138085	0.00690423	−	0.999976i	$-0.497802\pi$
$911$	0.416778	0.0138085	0.00690423	+	0.999976i	$0.497802\pi$
$912$	0	0
$913$	15.7445	0.521067
$914$	45.4688	1.50398
$915$	0	0
$916$	−11.2257	−0.370907
$917$	−6.81297	−0.224984
$918$	0	0
$919$	−36.0709	−1.18987	−0.594935	−	0.803774i	$-0.702822\pi$
$919$	−36.0709	−1.18987	−0.594935	+	0.803774i	$0.702822\pi$
$920$	8.89870	0.293381
$921$	0	0
$922$	5.61791	0.185016
$923$	−5.82698	−0.191797
$924$	0	0
$925$	−13.0133	−0.427874
$926$	26.7502	0.879067
$927$	0	0
$928$	4.80495	0.157730
$929$	22.4075	0.735165	0.367583	−	0.929991i	$-0.380186\pi$
$929$	22.4075	0.735165	0.367583	+	0.929991i	$0.380186\pi$
$930$	0	0
$931$	3.48812	0.114318
$932$	7.41188	0.242784
$933$	0	0
$934$	39.8822	1.30499
$935$	3.57310	0.116853
$936$	0	0
$937$	−45.2942	−1.47970	−0.739849	−	0.672773i	$-0.765103\pi$
$937$	−45.2942	−1.47970	−0.739849	+	0.672773i	$0.765103\pi$
$938$	5.33211	0.174100
$939$	0	0
$940$	2.79007	0.0910022
$941$	8.10807	0.264316	0.132158	−	0.991229i	$-0.457809\pi$
$941$	8.10807	0.264316	0.132158	+	0.991229i	$0.457809\pi$
$942$	0	0
$943$	70.5655	2.29793
$944$	−42.9630	−1.39833
$945$	0	0
$946$	13.9010	0.451959
$947$	36.3694	1.18185	0.590924	−	0.806727i	$-0.298763\pi$
$947$	36.3694	1.18185	0.590924	+	0.806727i	$0.298763\pi$
$948$	0	0
$949$	−2.29150	−0.0743851
$950$	−4.00908	−0.130072
$951$	0	0
$952$	4.75698	0.154175
$953$	−44.1923	−1.43153	−0.715765	−	0.698342i	$-0.753922\pi$
$953$	−44.1923	−1.43153	−0.715765	+	0.698342i	$0.753922\pi$
$954$	0	0
$955$	4.69751	0.152008
$956$	15.3717	0.497156
$957$	0	0
$958$	60.5999	1.95789
$959$	−4.64173	−0.149889
$960$	0	0
$961$	59.0911	1.90617
$962$	−2.03837	−0.0657197
$963$	0	0
$964$	9.27642	0.298774
$965$	5.82305	0.187451
$966$	0	0
$967$	40.1047	1.28968	0.644840	−	0.764318i	$-0.276924\pi$
$967$	40.1047	1.28968	0.644840	+	0.764318i	$0.276924\pi$
$968$	1.93230	0.0621063
$969$	0	0
$970$	−6.27739	−0.201555
$971$	16.7146	0.536396	0.268198	−	0.963364i	$-0.413572\pi$
$971$	16.7146	0.536396	0.268198	+	0.963364i	$0.413572\pi$
$972$	0	0
$973$	5.76844	0.184928
$974$	−60.7591	−1.94685
$975$	0	0
$976$	−34.5505	−1.10593
$977$	7.63219	0.244175	0.122088	−	0.992519i	$-0.461041\pi$
$977$	7.63219	0.244175	0.122088	+	0.992519i	$0.461041\pi$
$978$	0	0
$979$	16.4457	0.525608
$980$	3.48391	0.111289
$981$	0	0
$982$	−28.5659	−0.911575
$983$	9.95440	0.317496	0.158748	−	0.987319i	$-0.449254\pi$
$983$	9.95440	0.317496	0.158748	+	0.987319i	$0.449254\pi$
$984$	0	0
$985$	−5.18515	−0.165213
$986$	−10.7109	−0.341105
$987$	0	0
$988$	−0.188333	−0.00599169
$989$	−63.6343	−2.02345
$990$	0	0
$991$	−3.55002	−0.112770	−0.0563851	−	0.998409i	$-0.517957\pi$
$991$	−3.55002	−0.112770	−0.0563851	+	0.998409i	$0.517957\pi$
$992$	43.2032	1.37170
$993$	0	0
$994$	9.38101	0.297547
$995$	−8.38033	−0.265674
$996$	0	0
$997$	−59.6587	−1.88941	−0.944705	−	0.327922i	$-0.893652\pi$
$997$	−59.6587	−1.88941	−0.944705	+	0.327922i	$0.893652\pi$
$998$	−50.0555	−1.58448
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.i.1.13	✓	48
3.2	odd	2		8019.2.a.j.1.36	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8019.2.a.i.1.13	✓	48	1.1	even	1	trivial
8019.2.a.j.1.36	yes	48	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)

\(f(q)\)	\(=\)	\(q-1.69020 q^{2} +0.856762 q^{4} -0.595209 q^{5} +0.410093 q^{7} +1.93230 q^{8} +O(q^{10})\)	\(q-1.69020 q^{2} +0.856762 q^{4} -0.595209 q^{5} +0.410093 q^{7} +1.93230 q^{8} +1.00602 q^{10} -1.00000 q^{11} +0.430540 q^{13} -0.693137 q^{14} -4.97948 q^{16} +6.00311 q^{17} -0.510569 q^{19} -0.509953 q^{20} +1.69020 q^{22} -7.73719 q^{23} -4.64573 q^{25} -0.727696 q^{26} +0.351352 q^{28} +1.05564 q^{29} +9.49163 q^{31} +4.55171 q^{32} -10.1464 q^{34} -0.244091 q^{35} +2.80113 q^{37} +0.862961 q^{38} -1.15012 q^{40} -9.12030 q^{41} +8.22447 q^{43} -0.856762 q^{44} +13.0774 q^{46} -5.47124 q^{47} -6.83182 q^{49} +7.85219 q^{50} +0.368870 q^{52} +5.98271 q^{53} +0.595209 q^{55} +0.792421 q^{56} -1.78423 q^{58} +8.62800 q^{59} +6.93857 q^{61} -16.0427 q^{62} +2.26568 q^{64} -0.256261 q^{65} -7.69272 q^{67} +5.14323 q^{68} +0.412562 q^{70} -13.5341 q^{71} -5.32238 q^{73} -4.73446 q^{74} -0.437436 q^{76} -0.410093 q^{77} +5.83295 q^{79} +2.96383 q^{80} +15.4151 q^{82} -15.7445 q^{83} -3.57310 q^{85} -13.9010 q^{86} -1.93230 q^{88} -16.4457 q^{89} +0.176561 q^{91} -6.62893 q^{92} +9.24747 q^{94} +0.303895 q^{95} -6.23983 q^{97} +11.5471 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8}+O(q^{10}) \) 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8	\( 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8} - 48 q^{11} - 24 q^{14} + 66 q^{16} - 24 q^{17} - 48 q^{20} + 6 q^{22} - 12 q^{23} + 60 q^{25} - 36 q^{26} - 18 q^{28} - 60 q^{29} + 36 q^{31} - 42 q^{32} + 12 q^{34} - 24 q^{35} + 6 q^{37} - 24 q^{38} - 72 q^{41} - 12 q^{43} - 54 q^{44} - 30 q^{46} - 36 q^{47} + 60 q^{49} - 42 q^{50} - 48 q^{53} + 24 q^{55} - 72 q^{56} + 12 q^{58} - 60 q^{59} - 24 q^{61} - 36 q^{62} + 90 q^{64} - 48 q^{65} - 60 q^{68} - 30 q^{70} - 60 q^{71} - 18 q^{73} - 36 q^{74} - 42 q^{76} - 12 q^{79} - 96 q^{80} + 12 q^{82} - 36 q^{83} + 18 q^{85} - 48 q^{86} + 18 q^{88} - 96 q^{89} + 30 q^{91} - 36 q^{92} - 48 q^{94} - 48 q^{95} + 30 q^{97} - 54 q^{98}+O(q^{100}) \) 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8 - 48 * q^11 - 24 * q^14 + 66 * q^16 - 24 * q^17 - 48 * q^20 + 6 * q^22 - 12 * q^23 + 60 * q^25 - 36 * q^26 - 18 * q^28 - 60 * q^29 + 36 * q^31 - 42 * q^32 + 12 * q^34 - 24 * q^35 + 6 * q^37 - 24 * q^38 - 72 * q^41 - 12 * q^43 - 54 * q^44 - 30 * q^46 - 36 * q^47 + 60 * q^49 - 42 * q^50 - 48 * q^53 + 24 * q^55 - 72 * q^56 + 12 * q^58 - 60 * q^59 - 24 * q^61 - 36 * q^62 + 90 * q^64 - 48 * q^65 - 60 * q^68 - 30 * q^70 - 60 * q^71 - 18 * q^73 - 36 * q^74 - 42 * q^76 - 12 * q^79 - 96 * q^80 + 12 * q^82 - 36 * q^83 + 18 * q^85 - 48 * q^86 + 18 * q^88 - 96 * q^89 + 30 * q^91 - 36 * q^92 - 48 * q^94 - 48 * q^95 + 30 * q^97 - 54 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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