LMFDB - Embedded newform 8019.2.a.l.1.17

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			1.17
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)

$f(q)$	$=$	$q-1.18428 q^{2} -0.597486 q^{4} +2.95374 q^{5} +3.03845 q^{7} +3.07615 q^{8} +O(q^{10})$	\(q-1.18428 q^{2} -0.597486 q^{4} +2.95374 q^{5} +3.03845 q^{7} +3.07615 q^{8} -3.49805 q^{10} +1.00000 q^{11} -2.06494 q^{13} -3.59837 q^{14} -2.44804 q^{16} -3.46387 q^{17} +2.26717 q^{19} -1.76482 q^{20} -1.18428 q^{22} +5.95610 q^{23} +3.72457 q^{25} +2.44546 q^{26} -1.81543 q^{28} -6.05998 q^{29} -3.70837 q^{31} -3.25313 q^{32} +4.10218 q^{34} +8.97479 q^{35} -9.05630 q^{37} -2.68496 q^{38} +9.08613 q^{40} +4.30574 q^{41} +3.78269 q^{43} -0.597486 q^{44} -7.05367 q^{46} +2.49486 q^{47} +2.23219 q^{49} -4.41093 q^{50} +1.23377 q^{52} -8.53403 q^{53} +2.95374 q^{55} +9.34672 q^{56} +7.17670 q^{58} +5.83390 q^{59} -7.77948 q^{61} +4.39174 q^{62} +8.74869 q^{64} -6.09929 q^{65} +4.81054 q^{67} +2.06961 q^{68} -10.6286 q^{70} +16.0650 q^{71} +13.7982 q^{73} +10.7252 q^{74} -1.35460 q^{76} +3.03845 q^{77} +1.76828 q^{79} -7.23087 q^{80} -5.09919 q^{82} -8.91270 q^{83} -10.2314 q^{85} -4.47976 q^{86} +3.07615 q^{88} +14.1210 q^{89} -6.27422 q^{91} -3.55868 q^{92} -2.95460 q^{94} +6.69663 q^{95} +15.2382 q^{97} -2.64353 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.18428	−0.837411	−0.418705	−	0.908122i	$-0.637516\pi$
$2$	−1.18428	−0.837411	−0.418705	+	0.908122i	$0.637516\pi$
$3$	0	0
$3$	0	0
$4$	−0.597486	−0.298743
$5$	2.95374	1.32095	0.660476	−	0.750847i	$-0.270355\pi$
$5$	2.95374	1.32095	0.660476	+	0.750847i	$0.270355\pi$
$6$	0	0
$7$	3.03845	1.14843	0.574213	−	0.818706i	$-0.305308\pi$
$7$	3.03845	1.14843	0.574213	+	0.818706i	$0.305308\pi$
$8$	3.07615	1.08758
$9$	0	0
$10$	−3.49805	−1.10618
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.06494	−0.572711	−0.286356	−	0.958123i	$-0.592444\pi$
$13$	−2.06494	−0.572711	−0.286356	+	0.958123i	$0.592444\pi$
$14$	−3.59837	−0.961705
$15$	0	0
$16$	−2.44804	−0.612010
$17$	−3.46387	−0.840112	−0.420056	−	0.907498i	$-0.637989\pi$
$17$	−3.46387	−0.840112	−0.420056	+	0.907498i	$0.637989\pi$
$18$	0	0
$19$	2.26717	0.520125	0.260062	−	0.965592i	$-0.416257\pi$
$19$	2.26717	0.520125	0.260062	+	0.965592i	$0.416257\pi$
$20$	−1.76482	−0.394625
$21$	0	0
$22$	−1.18428	−0.252489
$23$	5.95610	1.24193	0.620966	−	0.783838i	$-0.286741\pi$
$23$	5.95610	1.24193	0.620966	+	0.783838i	$0.286741\pi$
$24$	0	0
$25$	3.72457	0.744914
$26$	2.44546	0.479595
$27$	0	0
$28$	−1.81543	−0.343084
$29$	−6.05998	−1.12531	−0.562655	−	0.826692i	$-0.690220\pi$
$29$	−6.05998	−1.12531	−0.562655	+	0.826692i	$0.690220\pi$
$30$	0	0
$31$	−3.70837	−0.666043	−0.333021	−	0.942919i	$-0.608068\pi$
$31$	−3.70837	−0.666043	−0.333021	+	0.942919i	$0.608068\pi$
$32$	−3.25313	−0.575078
$33$	0	0
$34$	4.10218	0.703519
$35$	8.97479	1.51702
$36$	0	0
$37$	−9.05630	−1.48885	−0.744423	−	0.667708i	$-0.767275\pi$
$37$	−9.05630	−1.48885	−0.744423	+	0.667708i	$0.767275\pi$
$38$	−2.68496	−0.435558
$39$	0	0
$40$	9.08613	1.43664
$41$	4.30574	0.672444	0.336222	−	0.941783i	$-0.390851\pi$
$41$	4.30574	0.672444	0.336222	+	0.941783i	$0.390851\pi$
$42$	0	0
$43$	3.78269	0.576855	0.288427	−	0.957502i	$-0.406868\pi$
$43$	3.78269	0.576855	0.288427	+	0.957502i	$0.406868\pi$
$44$	−0.597486	−0.0900744
$45$	0	0
$46$	−7.05367	−1.04001
$47$	2.49486	0.363912	0.181956	−	0.983307i	$-0.441757\pi$
$47$	2.49486	0.363912	0.181956	+	0.983307i	$0.441757\pi$
$48$	0	0
$49$	2.23219	0.318884
$50$	−4.41093	−0.623799
$51$	0	0
$52$	1.23377	0.171094
$53$	−8.53403	−1.17224	−0.586120	−	0.810224i	$-0.699345\pi$
$53$	−8.53403	−1.17224	−0.586120	+	0.810224i	$0.699345\pi$
$54$	0	0
$55$	2.95374	0.398282
$56$	9.34672	1.24901
$57$	0	0
$58$	7.17670	0.942347
$59$	5.83390	0.759509	0.379755	−	0.925087i	$-0.376008\pi$
$59$	5.83390	0.759509	0.379755	+	0.925087i	$0.376008\pi$
$60$	0	0
$61$	−7.77948	−0.996060	−0.498030	−	0.867160i	$-0.665943\pi$
$61$	−7.77948	−0.996060	−0.498030	+	0.867160i	$0.665943\pi$
$62$	4.39174	0.557751
$63$	0	0
$64$	8.74869	1.09359
$65$	−6.09929	−0.756524
$66$	0	0
$67$	4.81054	0.587701	0.293850	−	0.955851i	$-0.405063\pi$
$67$	4.81054	0.587701	0.293850	+	0.955851i	$0.405063\pi$
$68$	2.06961	0.250977
$69$	0	0
$70$	−10.6286	−1.27037
$71$	16.0650	1.90656	0.953281	−	0.302084i	$-0.0976822\pi$
$71$	16.0650	1.90656	0.953281	+	0.302084i	$0.0976822\pi$
$72$	0	0
$73$	13.7982	1.61495	0.807476	−	0.589900i	$-0.200833\pi$
$73$	13.7982	1.61495	0.807476	+	0.589900i	$0.200833\pi$
$74$	10.7252	1.24678
$75$	0	0
$76$	−1.35460	−0.155384
$77$	3.03845	0.346264
$78$	0	0
$79$	1.76828	0.198947	0.0994734	−	0.995040i	$-0.468284\pi$
$79$	1.76828	0.198947	0.0994734	+	0.995040i	$0.468284\pi$
$80$	−7.23087	−0.808435
$81$	0	0
$82$	−5.09919	−0.563112
$83$	−8.91270	−0.978295	−0.489148	−	0.872201i	$-0.662692\pi$
$83$	−8.91270	−0.978295	−0.489148	+	0.872201i	$0.662692\pi$
$84$	0	0
$85$	−10.2314	−1.10975
$86$	−4.47976	−0.483064
$87$	0	0
$88$	3.07615	0.327918
$89$	14.1210	1.49682	0.748412	−	0.663234i	$-0.230817\pi$
$89$	14.1210	1.49682	0.748412	+	0.663234i	$0.230817\pi$
$90$	0	0
$91$	−6.27422	−0.657717
$92$	−3.55868	−0.371018
$93$	0	0
$94$	−2.95460	−0.304744
$95$	6.69663	0.687060
$96$	0	0
$97$	15.2382	1.54720	0.773602	−	0.633672i	$-0.218453\pi$
$97$	15.2382	1.54720	0.773602	+	0.633672i	$0.218453\pi$
$98$	−2.64353	−0.267037
$99$	0	0
$100$	−2.22538	−0.222538
$101$	−9.10715	−0.906195	−0.453098	−	0.891461i	$-0.649681\pi$
$101$	−9.10715	−0.906195	−0.453098	+	0.891461i	$0.649681\pi$
$102$	0	0
$103$	15.1171	1.48953	0.744766	−	0.667326i	$-0.232561\pi$
$103$	15.1171	1.48953	0.744766	+	0.667326i	$0.232561\pi$
$104$	−6.35206	−0.622870
$105$	0	0
$106$	10.1067	0.981646
$107$	5.07430	0.490551	0.245276	−	0.969453i	$-0.421122\pi$
$107$	5.07430	0.490551	0.245276	+	0.969453i	$0.421122\pi$
$108$	0	0
$109$	13.4444	1.28774	0.643871	−	0.765134i	$-0.277327\pi$
$109$	13.4444	1.28774	0.643871	+	0.765134i	$0.277327\pi$
$110$	−3.49805	−0.333526
$111$	0	0
$112$	−7.43825	−0.702848
$113$	−9.05334	−0.851667	−0.425833	−	0.904802i	$-0.640019\pi$
$113$	−9.05334	−0.851667	−0.425833	+	0.904802i	$0.640019\pi$
$114$	0	0
$115$	17.5928	1.64053
$116$	3.62075	0.336179
$117$	0	0
$118$	−6.90896	−0.636021
$119$	−10.5248	−0.964807
$120$	0	0
$121$	1.00000	0.0909091
$122$	9.21306	0.834112
$123$	0	0
$124$	2.21570	0.198976
$125$	−3.76728	−0.336956
$126$	0	0
$127$	12.4961	1.10885	0.554427	−	0.832233i	$-0.312938\pi$
$127$	12.4961	1.10885	0.554427	+	0.832233i	$0.312938\pi$
$128$	−3.85461	−0.340703
$129$	0	0
$130$	7.22326	0.633522
$131$	21.6409	1.89077	0.945386	−	0.325952i	$-0.105685\pi$
$131$	21.6409	1.89077	0.945386	+	0.325952i	$0.105685\pi$
$132$	0	0
$133$	6.88869	0.597325
$134$	−5.69701	−0.492147
$135$	0	0
$136$	−10.6554	−0.913690
$137$	−4.60337	−0.393292	−0.196646	−	0.980475i	$-0.563005\pi$
$137$	−4.60337	−0.393292	−0.196646	+	0.980475i	$0.563005\pi$
$138$	0	0
$139$	0.537336	0.0455763	0.0227881	−	0.999740i	$-0.492746\pi$
$139$	0.537336	0.0455763	0.0227881	+	0.999740i	$0.492746\pi$
$140$	−5.36231	−0.453198
$141$	0	0
$142$	−19.0254	−1.59658
$143$	−2.06494	−0.172679
$144$	0	0
$145$	−17.8996	−1.48648
$146$	−16.3409	−1.35238
$147$	0	0
$148$	5.41101	0.444782
$149$	14.4382	1.18283	0.591413	−	0.806369i	$-0.298570\pi$
$149$	14.4382	1.18283	0.591413	+	0.806369i	$0.298570\pi$
$150$	0	0
$151$	−6.64951	−0.541129	−0.270565	−	0.962702i	$-0.587210\pi$
$151$	−6.64951	−0.541129	−0.270565	+	0.962702i	$0.587210\pi$
$152$	6.97415	0.565678
$153$	0	0
$154$	−3.59837	−0.289965
$155$	−10.9536	−0.879810
$156$	0	0
$157$	20.9631	1.67304	0.836520	−	0.547937i	$-0.184587\pi$
$157$	20.9631	1.67304	0.836520	+	0.547937i	$0.184587\pi$
$158$	−2.09413	−0.166600
$159$	0	0
$160$	−9.60890	−0.759651
$161$	18.0973	1.42627
$162$	0	0
$163$	1.81258	0.141973	0.0709863	−	0.997477i	$-0.477385\pi$
$163$	1.81258	0.141973	0.0709863	+	0.997477i	$0.477385\pi$
$164$	−2.57262	−0.200888
$165$	0	0
$166$	10.5551	0.819235
$167$	−3.91562	−0.303000	−0.151500	−	0.988457i	$-0.548410\pi$
$167$	−3.91562	−0.303000	−0.151500	+	0.988457i	$0.548410\pi$
$168$	0	0
$169$	−8.73602	−0.672002
$170$	12.1168	0.929314
$171$	0	0
$172$	−2.26010	−0.172331
$173$	2.13734	0.162499	0.0812497	−	0.996694i	$-0.474109\pi$
$173$	2.13734	0.162499	0.0812497	+	0.996694i	$0.474109\pi$
$174$	0	0
$175$	11.3169	0.855480
$176$	−2.44804	−0.184528
$177$	0	0
$178$	−16.7232	−1.25346
$179$	17.8332	1.33292	0.666459	−	0.745542i	$-0.267809\pi$
$179$	17.8332	1.33292	0.666459	+	0.745542i	$0.267809\pi$
$180$	0	0
$181$	7.36030	0.547087	0.273544	−	0.961860i	$-0.411804\pi$
$181$	7.36030	0.547087	0.273544	+	0.961860i	$0.411804\pi$
$182$	7.43042	0.550779
$183$	0	0
$184$	18.3218	1.35070
$185$	−26.7499	−1.96669
$186$	0	0
$187$	−3.46387	−0.253303
$188$	−1.49064	−0.108716
$189$	0	0
$190$	−7.93068	−0.575352
$191$	−19.7095	−1.42613	−0.713066	−	0.701097i	$-0.752694\pi$
$191$	−19.7095	−1.42613	−0.713066	+	0.701097i	$0.752694\pi$
$192$	0	0
$193$	24.9983	1.79942	0.899709	−	0.436490i	$-0.143779\pi$
$193$	24.9983	1.79942	0.899709	+	0.436490i	$0.143779\pi$
$194$	−18.0462	−1.29565
$195$	0	0
$196$	−1.33370	−0.0952644
$197$	−24.6722	−1.75782	−0.878909	−	0.476989i	$-0.841728\pi$
$197$	−24.6722	−1.75782	−0.878909	+	0.476989i	$0.841728\pi$
$198$	0	0
$199$	11.7349	0.831862	0.415931	−	0.909396i	$-0.363456\pi$
$199$	11.7349	0.831862	0.415931	+	0.909396i	$0.363456\pi$
$200$	11.4573	0.810155
$201$	0	0
$202$	10.7854	0.758858
$203$	−18.4130	−1.29234
$204$	0	0
$205$	12.7180	0.888266
$206$	−17.9028	−1.24735
$207$	0	0
$208$	5.05505	0.350505
$209$	2.26717	0.156824
$210$	0	0
$211$	23.7814	1.63718	0.818588	−	0.574381i	$-0.194757\pi$
$211$	23.7814	1.63718	0.818588	+	0.574381i	$0.194757\pi$
$212$	5.09896	0.350198
$213$	0	0
$214$	−6.00938	−0.410793
$215$	11.1731	0.761997
$216$	0	0
$217$	−11.2677	−0.764901
$218$	−15.9219	−1.07837
$219$	0	0
$220$	−1.76482	−0.118984
$221$	7.15268	0.481141
$222$	0	0
$223$	15.8742	1.06301	0.531506	−	0.847054i	$-0.321626\pi$
$223$	15.8742	1.06301	0.531506	+	0.847054i	$0.321626\pi$
$224$	−9.88449	−0.660435
$225$	0	0
$226$	10.7217	0.713195
$227$	15.2000	1.00886	0.504429	−	0.863453i	$-0.331703\pi$
$227$	15.2000	1.00886	0.504429	+	0.863453i	$0.331703\pi$
$228$	0	0
$229$	6.61498	0.437130	0.218565	−	0.975822i	$-0.429862\pi$
$229$	6.61498	0.437130	0.218565	+	0.975822i	$0.429862\pi$
$230$	−20.8347	−1.37380
$231$	0	0
$232$	−18.6414	−1.22387
$233$	−19.2432	−1.26067	−0.630333	−	0.776324i	$-0.717082\pi$
$233$	−19.2432	−1.26067	−0.630333	+	0.776324i	$0.717082\pi$
$234$	0	0
$235$	7.36915	0.480710
$236$	−3.48567	−0.226898
$237$	0	0
$238$	12.4643	0.807940
$239$	10.8740	0.703382	0.351691	−	0.936116i	$-0.385607\pi$
$239$	10.8740	0.703382	0.351691	+	0.936116i	$0.385607\pi$
$240$	0	0
$241$	−23.0781	−1.48659	−0.743296	−	0.668963i	$-0.766738\pi$
$241$	−23.0781	−1.48659	−0.743296	+	0.668963i	$0.766738\pi$
$242$	−1.18428	−0.0761283
$243$	0	0
$244$	4.64813	0.297566
$245$	6.59330	0.421231
$246$	0	0
$247$	−4.68157	−0.297881
$248$	−11.4075	−0.724376
$249$	0	0
$250$	4.46151	0.282171
$251$	−9.42469	−0.594881	−0.297441	−	0.954740i	$-0.596133\pi$
$251$	−9.42469	−0.594881	−0.297441	+	0.954740i	$0.596133\pi$
$252$	0	0
$253$	5.95610	0.374457
$254$	−14.7989	−0.928566
$255$	0	0
$256$	−12.9324	−0.808278
$257$	−20.2746	−1.26469	−0.632347	−	0.774685i	$-0.717908\pi$
$257$	−20.2746	−1.26469	−0.632347	+	0.774685i	$0.717908\pi$
$258$	0	0
$259$	−27.5171	−1.70983
$260$	3.64424	0.226006
$261$	0	0
$262$	−25.6288	−1.58335
$263$	−0.347033	−0.0213990	−0.0106995	−	0.999943i	$-0.503406\pi$
$263$	−0.347033	−0.0213990	−0.0106995	+	0.999943i	$0.503406\pi$
$264$	0	0
$265$	−25.2073	−1.54847
$266$	−8.15813	−0.500207
$267$	0	0
$268$	−2.87423	−0.175572
$269$	−8.63038	−0.526203	−0.263102	−	0.964768i	$-0.584745\pi$
$269$	−8.63038	−0.526203	−0.263102	+	0.964768i	$0.584745\pi$
$270$	0	0
$271$	−8.18031	−0.496918	−0.248459	−	0.968642i	$-0.579924\pi$
$271$	−8.18031	−0.496918	−0.248459	+	0.968642i	$0.579924\pi$
$272$	8.47968	0.514156
$273$	0	0
$274$	5.45167	0.329347
$275$	3.72457	0.224600
$276$	0	0
$277$	19.9730	1.20006	0.600030	−	0.799978i	$-0.295155\pi$
$277$	19.9730	1.20006	0.600030	+	0.799978i	$0.295155\pi$
$278$	−0.636355	−0.0381660
$279$	0	0
$280$	27.6078	1.64988
$281$	−20.7709	−1.23909	−0.619544	−	0.784962i	$-0.712682\pi$
$281$	−20.7709	−1.23909	−0.619544	+	0.784962i	$0.712682\pi$
$282$	0	0
$283$	9.56026	0.568299	0.284149	−	0.958780i	$-0.408289\pi$
$283$	9.56026	0.568299	0.284149	+	0.958780i	$0.408289\pi$
$284$	−9.59860	−0.569572
$285$	0	0
$286$	2.44546	0.144603
$287$	13.0828	0.772252
$288$	0	0
$289$	−5.00161	−0.294213
$290$	21.1981	1.24480
$291$	0	0
$292$	−8.24421	−0.482456
$293$	−24.6609	−1.44070	−0.720352	−	0.693609i	$-0.756020\pi$
$293$	−24.6609	−1.44070	−0.720352	+	0.693609i	$0.756020\pi$
$294$	0	0
$295$	17.2318	1.00328
$296$	−27.8585	−1.61924
$297$	0	0
$298$	−17.0989	−0.990511
$299$	−12.2990	−0.711268
$300$	0	0
$301$	11.4935	0.662475
$302$	7.87486	0.453147
$303$	0	0
$304$	−5.55012	−0.318321
$305$	−22.9785	−1.31575
$306$	0	0
$307$	18.2471	1.04142	0.520709	−	0.853734i	$-0.325668\pi$
$307$	18.2471	1.04142	0.520709	+	0.853734i	$0.325668\pi$
$308$	−1.81543	−0.103444
$309$	0	0
$310$	12.9720	0.736763
$311$	11.9963	0.680248	0.340124	−	0.940381i	$-0.389531\pi$
$311$	11.9963	0.680248	0.340124	+	0.940381i	$0.389531\pi$
$312$	0	0
$313$	−21.9520	−1.24080	−0.620401	−	0.784285i	$-0.713030\pi$
$313$	−21.9520	−1.24080	−0.620401	+	0.784285i	$0.713030\pi$
$314$	−24.8262	−1.40102
$315$	0	0
$316$	−1.05652	−0.0594340
$317$	−4.19004	−0.235336	−0.117668	−	0.993053i	$-0.537542\pi$
$317$	−4.19004	−0.235336	−0.117668	+	0.993053i	$0.537542\pi$
$318$	0	0
$319$	−6.05998	−0.339294
$320$	25.8413	1.44457
$321$	0	0
$322$	−21.4322	−1.19437
$323$	−7.85319	−0.436963
$324$	0	0
$325$	−7.69102	−0.426621
$326$	−2.14660	−0.118889
$327$	0	0
$328$	13.2451	0.731337
$329$	7.58050	0.417926
$330$	0	0
$331$	−18.1637	−0.998370	−0.499185	−	0.866496i	$-0.666367\pi$
$331$	−18.1637	−0.998370	−0.499185	+	0.866496i	$0.666367\pi$
$332$	5.32521	0.292259
$333$	0	0
$334$	4.63718	0.253735
$335$	14.2091	0.776325
$336$	0	0
$337$	−19.8958	−1.08379	−0.541897	−	0.840445i	$-0.682294\pi$
$337$	−19.8958	−1.08379	−0.541897	+	0.840445i	$0.682294\pi$
$338$	10.3459	0.562742
$339$	0	0
$340$	6.11310	0.331529
$341$	−3.70837	−0.200819
$342$	0	0
$343$	−14.4868	−0.782212
$344$	11.6361	0.627377
$345$	0	0
$346$	−2.53121	−0.136079
$347$	14.3945	0.772737	0.386369	−	0.922344i	$-0.373729\pi$
$347$	14.3945	0.772737	0.386369	+	0.922344i	$0.373729\pi$
$348$	0	0
$349$	25.4558	1.36262	0.681310	−	0.731995i	$-0.261411\pi$
$349$	25.4558	1.36262	0.681310	+	0.731995i	$0.261411\pi$
$350$	−13.4024	−0.716388
$351$	0	0
$352$	−3.25313	−0.173393
$353$	−1.07517	−0.0572256	−0.0286128	−	0.999591i	$-0.509109\pi$
$353$	−1.07517	−0.0572256	−0.0286128	+	0.999591i	$0.509109\pi$
$354$	0	0
$355$	47.4518	2.51848
$356$	−8.43710	−0.447166
$357$	0	0
$358$	−21.1195	−1.11620
$359$	−7.72048	−0.407472	−0.203736	−	0.979026i	$-0.565308\pi$
$359$	−7.72048	−0.407472	−0.203736	+	0.979026i	$0.565308\pi$
$360$	0	0
$361$	−13.8599	−0.729470
$362$	−8.71665	−0.458137
$363$	0	0
$364$	3.74876	0.196488
$365$	40.7562	2.13327
$366$	0	0
$367$	−10.2907	−0.537170	−0.268585	−	0.963256i	$-0.586556\pi$
$367$	−10.2907	−0.537170	−0.268585	+	0.963256i	$0.586556\pi$
$368$	−14.5808	−0.760074
$369$	0	0
$370$	31.6793	1.64693
$371$	−25.9302	−1.34623
$372$	0	0
$373$	13.0167	0.673977	0.336988	−	0.941509i	$-0.390592\pi$
$373$	13.0167	0.673977	0.336988	+	0.941509i	$0.390592\pi$
$374$	4.10218	0.212119
$375$	0	0
$376$	7.67454	0.395784
$377$	12.5135	0.644478
$378$	0	0
$379$	27.9526	1.43583	0.717914	−	0.696131i	$-0.245097\pi$
$379$	27.9526	1.43583	0.717914	+	0.696131i	$0.245097\pi$
$380$	−4.00115	−0.205254
$381$	0	0
$382$	23.3416	1.19426
$383$	2.42578	0.123951	0.0619757	−	0.998078i	$-0.480260\pi$
$383$	2.42578	0.123951	0.0619757	+	0.998078i	$0.480260\pi$
$384$	0	0
$385$	8.97479	0.457398
$386$	−29.6049	−1.50685
$387$	0	0
$388$	−9.10460	−0.462216
$389$	27.5235	1.39549	0.697747	−	0.716344i	$-0.254186\pi$
$389$	27.5235	1.39549	0.697747	+	0.716344i	$0.254186\pi$
$390$	0	0
$391$	−20.6311	−1.04336
$392$	6.86654	0.346812
$393$	0	0
$394$	29.2187	1.47202
$395$	5.22303	0.262799
$396$	0	0
$397$	5.48757	0.275413	0.137707	−	0.990473i	$-0.456027\pi$
$397$	5.48757	0.275413	0.137707	+	0.990473i	$0.456027\pi$
$398$	−13.8973	−0.696610
$399$	0	0
$400$	−9.11789	−0.455895
$401$	18.5581	0.926749	0.463374	−	0.886163i	$-0.346639\pi$
$401$	18.5581	0.926749	0.463374	+	0.886163i	$0.346639\pi$
$402$	0	0
$403$	7.65756	0.381450
$404$	5.44140	0.270720
$405$	0	0
$406$	21.8061	1.08222
$407$	−9.05630	−0.448904
$408$	0	0
$409$	−25.9740	−1.28433	−0.642165	−	0.766566i	$-0.721964\pi$
$409$	−25.9740	−1.28433	−0.642165	+	0.766566i	$0.721964\pi$
$410$	−15.0617	−0.743843
$411$	0	0
$412$	−9.03225	−0.444987
$413$	17.7260	0.872241
$414$	0	0
$415$	−26.3258	−1.29228
$416$	6.71752	0.329354
$417$	0	0
$418$	−2.68496	−0.131326
$419$	−17.8544	−0.872246	−0.436123	−	0.899887i	$-0.643649\pi$
$419$	−17.8544	−0.872246	−0.436123	+	0.899887i	$0.643649\pi$
$420$	0	0
$421$	23.1116	1.12639	0.563194	−	0.826324i	$-0.309572\pi$
$421$	23.1116	1.12639	0.563194	+	0.826324i	$0.309572\pi$
$422$	−28.1637	−1.37099
$423$	0	0
$424$	−26.2519	−1.27491
$425$	−12.9014	−0.625811
$426$	0	0
$427$	−23.6376	−1.14390
$428$	−3.03182	−0.146549
$429$	0	0
$430$	−13.2320	−0.638105
$431$	−20.4514	−0.985112	−0.492556	−	0.870281i	$-0.663937\pi$
$431$	−20.4514	−0.985112	−0.492556	+	0.870281i	$0.663937\pi$
$432$	0	0
$433$	−11.1418	−0.535441	−0.267721	−	0.963497i	$-0.586270\pi$
$433$	−11.1418	−0.535441	−0.267721	+	0.963497i	$0.586270\pi$
$434$	13.3441	0.640537
$435$	0	0
$436$	−8.03286	−0.384704
$437$	13.5035	0.645960
$438$	0	0
$439$	−20.0706	−0.957915	−0.478958	−	0.877838i	$-0.658985\pi$
$439$	−20.0706	−0.957915	−0.478958	+	0.877838i	$0.658985\pi$
$440$	9.08613	0.433164
$441$	0	0
$442$	−8.47076	−0.402913
$443$	1.35611	0.0644307	0.0322154	−	0.999481i	$-0.489744\pi$
$443$	1.35611	0.0644307	0.0322154	+	0.999481i	$0.489744\pi$
$444$	0	0
$445$	41.7098	1.97723
$446$	−18.7994	−0.890178
$447$	0	0
$448$	26.5825	1.25590
$449$	22.3186	1.05328	0.526641	−	0.850088i	$-0.323451\pi$
$449$	22.3186	1.05328	0.526641	+	0.850088i	$0.323451\pi$
$450$	0	0
$451$	4.30574	0.202749
$452$	5.40925	0.254429
$453$	0	0
$454$	−18.0010	−0.844828
$455$	−18.5324	−0.868813
$456$	0	0
$457$	−13.7006	−0.640886	−0.320443	−	0.947268i	$-0.603832\pi$
$457$	−13.7006	−0.640886	−0.320443	+	0.947268i	$0.603832\pi$
$458$	−7.83398	−0.366058
$459$	0	0
$460$	−10.5114	−0.490098
$461$	35.4247	1.64989	0.824947	−	0.565210i	$-0.191205\pi$
$461$	35.4247	1.64989	0.824947	+	0.565210i	$0.191205\pi$
$462$	0	0
$463$	10.5573	0.490638	0.245319	−	0.969442i	$-0.421107\pi$
$463$	10.5573	0.490638	0.245319	+	0.969442i	$0.421107\pi$
$464$	14.8351	0.688701
$465$	0	0
$466$	22.7893	1.05570
$467$	−0.653819	−0.0302551	−0.0151276	−	0.999886i	$-0.504815\pi$
$467$	−0.653819	−0.0302551	−0.0151276	+	0.999886i	$0.504815\pi$
$468$	0	0
$469$	14.6166	0.674931
$470$	−8.72712	−0.402552
$471$	0	0
$472$	17.9459	0.826028
$473$	3.78269	0.173928
$474$	0	0
$475$	8.44425	0.387449
$476$	6.28842	0.288229
$477$	0	0
$478$	−12.8779	−0.589020
$479$	−12.6425	−0.577653	−0.288826	−	0.957381i	$-0.593265\pi$
$479$	−12.6425	−0.577653	−0.288826	+	0.957381i	$0.593265\pi$
$480$	0	0
$481$	18.7007	0.852679
$482$	27.3309	1.24489
$483$	0	0
$484$	−0.597486	−0.0271585
$485$	45.0096	2.04378
$486$	0	0
$487$	33.8293	1.53295	0.766477	−	0.642272i	$-0.222008\pi$
$487$	33.8293	1.53295	0.766477	+	0.642272i	$0.222008\pi$
$488$	−23.9308	−1.08330
$489$	0	0
$490$	−7.80830	−0.352743
$491$	−9.89291	−0.446461	−0.223230	−	0.974766i	$-0.571660\pi$
$491$	−9.89291	−0.446461	−0.223230	+	0.974766i	$0.571660\pi$
$492$	0	0
$493$	20.9910	0.945386
$494$	5.54429	0.249449
$495$	0	0
$496$	9.07823	0.407624
$497$	48.8127	2.18955
$498$	0	0
$499$	1.13784	0.0509369	0.0254684	−	0.999676i	$-0.491892\pi$
$499$	1.13784	0.0509369	0.0254684	+	0.999676i	$0.491892\pi$
$500$	2.25090	0.100663
$501$	0	0
$502$	11.1614	0.498160
$503$	−28.1606	−1.25562	−0.627809	−	0.778367i	$-0.716048\pi$
$503$	−28.1606	−1.25562	−0.627809	+	0.778367i	$0.716048\pi$
$504$	0	0
$505$	−26.9001	−1.19704
$506$	−7.05367	−0.313574
$507$	0	0
$508$	−7.46627	−0.331262
$509$	−29.1047	−1.29004	−0.645022	−	0.764164i	$-0.723152\pi$
$509$	−29.1047	−1.29004	−0.645022	+	0.764164i	$0.723152\pi$
$510$	0	0
$511$	41.9250	1.85465
$512$	23.0248	1.01756
$513$	0	0
$514$	24.0107	1.05907
$515$	44.6519	1.96760
$516$	0	0
$517$	2.49486	0.109724
$518$	32.5879	1.43183
$519$	0	0
$520$	−18.7623	−0.822782
$521$	−20.8950	−0.915425	−0.457712	−	0.889100i	$-0.651331\pi$
$521$	−20.8950	−0.915425	−0.457712	+	0.889100i	$0.651331\pi$
$522$	0	0
$523$	−29.2159	−1.27752	−0.638761	−	0.769405i	$-0.720553\pi$
$523$	−29.2159	−1.27752	−0.638761	+	0.769405i	$0.720553\pi$
$524$	−12.9301	−0.564855
$525$	0	0
$526$	0.410983	0.0179197
$527$	12.8453	0.559550
$528$	0	0
$529$	12.4751	0.542395
$530$	29.8524	1.29671
$531$	0	0
$532$	−4.11590	−0.178447
$533$	−8.89109	−0.385116
$534$	0	0
$535$	14.9882	0.647994
$536$	14.7979	0.639173
$537$	0	0
$538$	10.2208	0.440649
$539$	2.23219	0.0961472
$540$	0	0
$541$	19.6442	0.844569	0.422285	−	0.906463i	$-0.361228\pi$
$541$	19.6442	0.844569	0.422285	+	0.906463i	$0.361228\pi$
$542$	9.68776	0.416125
$543$	0	0
$544$	11.2684	0.483130
$545$	39.7113	1.70105
$546$	0	0
$547$	17.1487	0.733227	0.366613	−	0.930373i	$-0.380517\pi$
$547$	17.1487	0.733227	0.366613	+	0.930373i	$0.380517\pi$
$548$	2.75045	0.117493
$549$	0	0
$550$	−4.41093	−0.188083
$551$	−13.7390	−0.585302
$552$	0	0
$553$	5.37283	0.228476
$554$	−23.6535	−1.00494
$555$	0	0
$556$	−0.321051	−0.0136156
$557$	13.4720	0.570827	0.285413	−	0.958405i	$-0.407869\pi$
$557$	13.4720	0.570827	0.285413	+	0.958405i	$0.407869\pi$
$558$	0	0
$559$	−7.81103	−0.330371
$560$	−21.9706	−0.928429
$561$	0	0
$562$	24.5985	1.03763
$563$	−1.22077	−0.0514493	−0.0257246	−	0.999669i	$-0.508189\pi$
$563$	−1.22077	−0.0514493	−0.0257246	+	0.999669i	$0.508189\pi$
$564$	0	0
$565$	−26.7412	−1.12501
$566$	−11.3220	−0.475899
$567$	0	0
$568$	49.4182	2.07354
$569$	3.73078	0.156402	0.0782012	−	0.996938i	$-0.475082\pi$
$569$	3.73078	0.156402	0.0782012	+	0.996938i	$0.475082\pi$
$570$	0	0
$571$	−6.09713	−0.255157	−0.127579	−	0.991828i	$-0.540720\pi$
$571$	−6.09713	−0.255157	−0.127579	+	0.991828i	$0.540720\pi$
$572$	1.23377	0.0515866
$573$	0	0
$574$	−15.4936	−0.646692
$575$	22.1839	0.925133
$576$	0	0
$577$	41.5503	1.72976	0.864880	−	0.501979i	$-0.167394\pi$
$577$	41.5503	1.72976	0.864880	+	0.501979i	$0.167394\pi$
$578$	5.92330	0.246377
$579$	0	0
$580$	10.6948	0.444076
$581$	−27.0808	−1.12350
$582$	0	0
$583$	−8.53403	−0.353443
$584$	42.4451	1.75639
$585$	0	0
$586$	29.2053	1.20646
$587$	20.2265	0.834835	0.417418	−	0.908715i	$-0.362935\pi$
$587$	20.2265	0.834835	0.417418	+	0.908715i	$0.362935\pi$
$588$	0	0
$589$	−8.40751	−0.346425
$590$	−20.4073	−0.840154
$591$	0	0
$592$	22.1702	0.911188
$593$	−3.80950	−0.156437	−0.0782186	−	0.996936i	$-0.524923\pi$
$593$	−3.80950	−0.156437	−0.0782186	+	0.996936i	$0.524923\pi$
$594$	0	0
$595$	−31.0875	−1.27446
$596$	−8.62664	−0.353361
$597$	0	0
$598$	14.5654	0.595624
$599$	36.1817	1.47835	0.739173	−	0.673516i	$-0.235217\pi$
$599$	36.1817	1.47835	0.739173	+	0.673516i	$0.235217\pi$
$600$	0	0
$601$	−31.8217	−1.29803	−0.649017	−	0.760774i	$-0.724819\pi$
$601$	−31.8217	−1.29803	−0.649017	+	0.760774i	$0.724819\pi$
$602$	−13.6115	−0.554764
$603$	0	0
$604$	3.97299	0.161659
$605$	2.95374	0.120087
$606$	0	0
$607$	18.4389	0.748413	0.374206	−	0.927345i	$-0.377915\pi$
$607$	18.4389	0.748413	0.374206	+	0.927345i	$0.377915\pi$
$608$	−7.37541	−0.299112
$609$	0	0
$610$	27.2130	1.10182
$611$	−5.15173	−0.208417
$612$	0	0
$613$	25.5138	1.03049	0.515247	−	0.857042i	$-0.327700\pi$
$613$	25.5138	1.03049	0.515247	+	0.857042i	$0.327700\pi$
$614$	−21.6096	−0.872094
$615$	0	0
$616$	9.34672	0.376590
$617$	−4.12589	−0.166102	−0.0830511	−	0.996545i	$-0.526466\pi$
$617$	−4.12589	−0.166102	−0.0830511	+	0.996545i	$0.526466\pi$
$618$	0	0
$619$	8.41434	0.338201	0.169100	−	0.985599i	$-0.445914\pi$
$619$	8.41434	0.338201	0.169100	+	0.985599i	$0.445914\pi$
$620$	6.54459	0.262837
$621$	0	0
$622$	−14.2070	−0.569647
$623$	42.9060	1.71899
$624$	0	0
$625$	−29.7504	−1.19002
$626$	25.9973	1.03906
$627$	0	0
$628$	−12.5252	−0.499809
$629$	31.3698	1.25080
$630$	0	0
$631$	26.2921	1.04667	0.523336	−	0.852126i	$-0.324687\pi$
$631$	26.2921	1.04667	0.523336	+	0.852126i	$0.324687\pi$
$632$	5.43948	0.216371
$633$	0	0
$634$	4.96217	0.197073
$635$	36.9103	1.46474
$636$	0	0
$637$	−4.60934	−0.182629
$638$	7.17670	0.284128
$639$	0	0
$640$	−11.3855	−0.450052
$641$	−14.5381	−0.574220	−0.287110	−	0.957898i	$-0.592695\pi$
$641$	−14.5381	−0.574220	−0.287110	+	0.957898i	$0.592695\pi$
$642$	0	0
$643$	17.2049	0.678495	0.339248	−	0.940697i	$-0.389828\pi$
$643$	17.2049	0.678495	0.339248	+	0.940697i	$0.389828\pi$
$644$	−10.8129	−0.426088
$645$	0	0
$646$	9.30036	0.365918
$647$	18.4811	0.726568	0.363284	−	0.931678i	$-0.381655\pi$
$647$	18.4811	0.726568	0.363284	+	0.931678i	$0.381655\pi$
$648$	0	0
$649$	5.83390	0.229001
$650$	9.10830	0.357257
$651$	0	0
$652$	−1.08299	−0.0424133
$653$	14.3584	0.561888	0.280944	−	0.959724i	$-0.409352\pi$
$653$	14.3584	0.561888	0.280944	+	0.959724i	$0.409352\pi$
$654$	0	0
$655$	63.9215	2.49762
$656$	−10.5406	−0.411542
$657$	0	0
$658$	−8.97741	−0.349976
$659$	−9.03390	−0.351911	−0.175955	−	0.984398i	$-0.556301\pi$
$659$	−9.03390	−0.351911	−0.175955	+	0.984398i	$0.556301\pi$
$660$	0	0
$661$	22.6596	0.881358	0.440679	−	0.897665i	$-0.354738\pi$
$661$	22.6596	0.881358	0.440679	+	0.897665i	$0.354738\pi$
$662$	21.5109	0.836046
$663$	0	0
$664$	−27.4167	−1.06398
$665$	20.3474	0.789038
$666$	0	0
$667$	−36.0938	−1.39756
$668$	2.33953	0.0905191
$669$	0	0
$670$	−16.8275	−0.650103
$671$	−7.77948	−0.300323
$672$	0	0
$673$	−47.5608	−1.83333	−0.916667	−	0.399652i	$-0.869131\pi$
$673$	−47.5608	−1.83333	−0.916667	+	0.399652i	$0.869131\pi$
$674$	23.5621	0.907580
$675$	0	0
$676$	5.21965	0.200756
$677$	5.56120	0.213734	0.106867	−	0.994273i	$-0.465918\pi$
$677$	5.56120	0.213734	0.106867	+	0.994273i	$0.465918\pi$
$678$	0	0
$679$	46.3005	1.77685
$680$	−31.4732	−1.20694
$681$	0	0
$682$	4.39174	0.168168
$683$	29.0526	1.11167	0.555834	−	0.831293i	$-0.312399\pi$
$683$	29.0526	1.11167	0.555834	+	0.831293i	$0.312399\pi$
$684$	0	0
$685$	−13.5971	−0.519520
$686$	17.1564	0.655033
$687$	0	0
$688$	−9.26017	−0.353041
$689$	17.6223	0.671355
$690$	0	0
$691$	41.5038	1.57888	0.789440	−	0.613827i	$-0.210371\pi$
$691$	41.5038	1.57888	0.789440	+	0.613827i	$0.210371\pi$
$692$	−1.27703	−0.0485455
$693$	0	0
$694$	−17.0471	−0.647099
$695$	1.58715	0.0602040
$696$	0	0
$697$	−14.9145	−0.564928
$698$	−30.1468	−1.14107
$699$	0	0
$700$	−6.76171	−0.255569
$701$	−0.712963	−0.0269282	−0.0134641	−	0.999909i	$-0.504286\pi$
$701$	−0.712963	−0.0269282	−0.0134641	+	0.999909i	$0.504286\pi$
$702$	0	0
$703$	−20.5322	−0.774386
$704$	8.74869	0.329729
$705$	0	0
$706$	1.27330	0.0479213
$707$	−27.6716	−1.04070
$708$	0	0
$709$	−5.86705	−0.220342	−0.110171	−	0.993913i	$-0.535140\pi$
$709$	−5.86705	−0.220342	−0.110171	+	0.993913i	$0.535140\pi$
$710$	−56.1961	−2.10900
$711$	0	0
$712$	43.4383	1.62792
$713$	−22.0874	−0.827180
$714$	0	0
$715$	−6.09929	−0.228101
$716$	−10.6551	−0.398200
$717$	0	0
$718$	9.14320	0.341221
$719$	−38.8820	−1.45005	−0.725027	−	0.688721i	$-0.758173\pi$
$719$	−38.8820	−1.45005	−0.725027	+	0.688721i	$0.758173\pi$
$720$	0	0
$721$	45.9326	1.71062
$722$	16.4140	0.610866
$723$	0	0
$724$	−4.39768	−0.163438
$725$	−22.5708	−0.838260
$726$	0	0
$727$	−51.3982	−1.90626	−0.953128	−	0.302568i	$-0.902156\pi$
$727$	−51.3982	−1.90626	−0.953128	+	0.302568i	$0.902156\pi$
$728$	−19.3004	−0.715321
$729$	0	0
$730$	−48.2666	−1.78643
$731$	−13.1027	−0.484622
$732$	0	0
$733$	1.77445	0.0655409	0.0327705	−	0.999463i	$-0.489567\pi$
$733$	1.77445	0.0655409	0.0327705	+	0.999463i	$0.489567\pi$
$734$	12.1870	0.449832
$735$	0	0
$736$	−19.3760	−0.714208
$737$	4.81054	0.177198
$738$	0	0
$739$	−39.7232	−1.46124	−0.730622	−	0.682783i	$-0.760770\pi$
$739$	−39.7232	−1.46124	−0.730622	+	0.682783i	$0.760770\pi$
$740$	15.9827	0.587536
$741$	0	0
$742$	30.7086	1.12735
$743$	−36.6142	−1.34324	−0.671622	−	0.740894i	$-0.734402\pi$
$743$	−36.6142	−1.34324	−0.671622	+	0.740894i	$0.734402\pi$
$744$	0	0
$745$	42.6467	1.56246
$746$	−15.4153	−0.564396
$747$	0	0
$748$	2.06961	0.0756726
$749$	15.4180	0.563362
$750$	0	0
$751$	32.7506	1.19509	0.597543	−	0.801837i	$-0.296144\pi$
$751$	32.7506	1.19509	0.597543	+	0.801837i	$0.296144\pi$
$752$	−6.10750	−0.222718
$753$	0	0
$754$	−14.8195	−0.539693
$755$	−19.6409	−0.714806
$756$	0	0
$757$	−3.44782	−0.125313	−0.0626566	−	0.998035i	$-0.519957\pi$
$757$	−3.44782	−0.125313	−0.0626566	+	0.998035i	$0.519957\pi$
$758$	−33.1036	−1.20238
$759$	0	0
$760$	20.5998	0.747234
$761$	−13.4658	−0.488137	−0.244068	−	0.969758i	$-0.578482\pi$
$761$	−13.4658	−0.488137	−0.244068	+	0.969758i	$0.578482\pi$
$762$	0	0
$763$	40.8502	1.47888
$764$	11.7762	0.426047
$765$	0	0
$766$	−2.87280	−0.103798
$767$	−12.0467	−0.434980
$768$	0	0
$769$	−9.24295	−0.333309	−0.166655	−	0.986015i	$-0.553296\pi$
$769$	−9.24295	−0.333309	−0.166655	+	0.986015i	$0.553296\pi$
$770$	−10.6286	−0.383030
$771$	0	0
$772$	−14.9361	−0.537564
$773$	−26.9207	−0.968269	−0.484134	−	0.874994i	$-0.660865\pi$
$773$	−26.9207	−0.968269	−0.484134	+	0.874994i	$0.660865\pi$
$774$	0	0
$775$	−13.8121	−0.496145
$776$	46.8749	1.68271
$777$	0	0
$778$	−32.5954	−1.16860
$779$	9.76185	0.349755
$780$	0	0
$781$	16.0650	0.574850
$782$	24.4330	0.873722
$783$	0	0
$784$	−5.46448	−0.195160
$785$	61.9196	2.21001
$786$	0	0
$787$	−20.1248	−0.717371	−0.358686	−	0.933458i	$-0.616775\pi$
$787$	−20.1248	−0.717371	−0.358686	+	0.933458i	$0.616775\pi$
$788$	14.7413	0.525136
$789$	0	0
$790$	−6.18552	−0.220071
$791$	−27.5081	−0.978077
$792$	0	0
$793$	16.0642	0.570455
$794$	−6.49881	−0.230634
$795$	0	0
$796$	−7.01142	−0.248513
$797$	−21.0717	−0.746398	−0.373199	−	0.927751i	$-0.621739\pi$
$797$	−21.0717	−0.746398	−0.373199	+	0.927751i	$0.621739\pi$
$798$	0	0
$799$	−8.64185	−0.305727
$800$	−12.1165	−0.428384
$801$	0	0
$802$	−21.9780	−0.776069
$803$	13.7982	0.486926
$804$	0	0
$805$	53.4547	1.88403
$806$	−9.06868	−0.319431
$807$	0	0
$808$	−28.0149	−0.985561
$809$	−2.97578	−0.104623	−0.0523115	−	0.998631i	$-0.516659\pi$
$809$	−2.97578	−0.104623	−0.0523115	+	0.998631i	$0.516659\pi$
$810$	0	0
$811$	−15.1135	−0.530705	−0.265353	−	0.964151i	$-0.585488\pi$
$811$	−15.1135	−0.530705	−0.265353	+	0.964151i	$0.585488\pi$
$812$	11.0015	0.386077
$813$	0	0
$814$	10.7252	0.375917
$815$	5.35390	0.187539
$816$	0	0
$817$	8.57601	0.300037
$818$	30.7604	1.07551
$819$	0	0
$820$	−7.59885	−0.265363
$821$	−14.7523	−0.514857	−0.257429	−	0.966297i	$-0.582875\pi$
$821$	−14.7523	−0.514857	−0.257429	+	0.966297i	$0.582875\pi$
$822$	0	0
$823$	11.8852	0.414291	0.207145	−	0.978310i	$-0.433583\pi$
$823$	11.8852	0.414291	0.207145	+	0.978310i	$0.433583\pi$
$824$	46.5024	1.61999
$825$	0	0
$826$	−20.9925	−0.730424
$827$	22.9964	0.799663	0.399831	−	0.916589i	$-0.369069\pi$
$827$	22.9964	0.799663	0.399831	+	0.916589i	$0.369069\pi$
$828$	0	0
$829$	−11.6685	−0.405264	−0.202632	−	0.979255i	$-0.564949\pi$
$829$	−11.6685	−0.405264	−0.202632	+	0.979255i	$0.564949\pi$
$830$	31.1770	1.08217
$831$	0	0
$832$	−18.0655	−0.626309
$833$	−7.73201	−0.267898
$834$	0	0
$835$	−11.5657	−0.400248
$836$	−1.35460	−0.0468500
$837$	0	0
$838$	21.1446	0.730429
$839$	−47.1737	−1.62862	−0.814308	−	0.580433i	$-0.802883\pi$
$839$	−47.1737	−1.62862	−0.814308	+	0.580433i	$0.802883\pi$
$840$	0	0
$841$	7.72338	0.266323
$842$	−27.3705	−0.943250
$843$	0	0
$844$	−14.2090	−0.489095
$845$	−25.8039	−0.887682
$846$	0	0
$847$	3.03845	0.104402
$848$	20.8916	0.717422
$849$	0	0
$850$	15.2789	0.524061
$851$	−53.9402	−1.84905
$852$	0	0
$853$	−14.9863	−0.513122	−0.256561	−	0.966528i	$-0.582589\pi$
$853$	−14.9863	−0.513122	−0.256561	+	0.966528i	$0.582589\pi$
$854$	27.9935	0.957916
$855$	0	0
$856$	15.6093	0.533514
$857$	0.290875	0.00993611	0.00496806	−	0.999988i	$-0.498419\pi$
$857$	0.290875	0.00993611	0.00496806	+	0.999988i	$0.498419\pi$
$858$	0	0
$859$	−25.9325	−0.884807	−0.442403	−	0.896816i	$-0.645874\pi$
$859$	−25.9325	−0.884807	−0.442403	+	0.896816i	$0.645874\pi$
$860$	−6.67576	−0.227641
$861$	0	0
$862$	24.2202	0.824943
$863$	32.9975	1.12325	0.561624	−	0.827393i	$-0.310177\pi$
$863$	32.9975	1.12325	0.561624	+	0.827393i	$0.310177\pi$
$864$	0	0
$865$	6.31316	0.214654
$866$	13.1950	0.448384
$867$	0	0
$868$	6.73229	0.228509
$869$	1.76828	0.0599847
$870$	0	0
$871$	−9.93348	−0.336583
$872$	41.3570	1.40052
$873$	0	0
$874$	−15.9919	−0.540934
$875$	−11.4467	−0.386969
$876$	0	0
$877$	−25.2927	−0.854073	−0.427037	−	0.904234i	$-0.640442\pi$
$877$	−25.2927	−0.854073	−0.427037	+	0.904234i	$0.640442\pi$
$878$	23.7691	0.802169
$879$	0	0
$880$	−7.23087	−0.243752
$881$	−3.48779	−0.117507	−0.0587533	−	0.998273i	$-0.518713\pi$
$881$	−3.48779	−0.117507	−0.0587533	+	0.998273i	$0.518713\pi$
$882$	0	0
$883$	−27.0741	−0.911115	−0.455558	−	0.890206i	$-0.650560\pi$
$883$	−27.0741	−0.911115	−0.455558	+	0.890206i	$0.650560\pi$
$884$	−4.27363	−0.143738
$885$	0	0
$886$	−1.60601	−0.0539550
$887$	17.9237	0.601818	0.300909	−	0.953653i	$-0.402710\pi$
$887$	17.9237	0.601818	0.300909	+	0.953653i	$0.402710\pi$
$888$	0	0
$889$	37.9689	1.27344
$890$	−49.3959	−1.65576
$891$	0	0
$892$	−9.48459	−0.317568
$893$	5.65627	0.189280
$894$	0	0
$895$	52.6747	1.76072
$896$	−11.7121	−0.391272
$897$	0	0
$898$	−26.4315	−0.882030
$899$	22.4726	0.749505
$900$	0	0
$901$	29.5608	0.984812
$902$	−5.09919	−0.169785
$903$	0	0
$904$	−27.8494	−0.926257
$905$	21.7404	0.722676
$906$	0	0
$907$	53.7954	1.78625	0.893124	−	0.449811i	$-0.148509\pi$
$907$	53.7954	1.78625	0.893124	+	0.449811i	$0.148509\pi$
$908$	−9.08177	−0.301389
$909$	0	0
$910$	21.9475	0.727553
$911$	−23.9982	−0.795095	−0.397547	−	0.917582i	$-0.630139\pi$
$911$	−23.9982	−0.795095	−0.397547	+	0.917582i	$0.630139\pi$
$912$	0	0
$913$	−8.91270	−0.294967
$914$	16.2253	0.536685
$915$	0	0
$916$	−3.95236	−0.130590
$917$	65.7548	2.17141
$918$	0	0
$919$	−7.16887	−0.236479	−0.118240	−	0.992985i	$-0.537725\pi$
$919$	−7.16887	−0.236479	−0.118240	+	0.992985i	$0.537725\pi$
$920$	54.1179	1.78421
$921$	0	0
$922$	−41.9527	−1.38164
$923$	−33.1732	−1.09191
$924$	0	0
$925$	−33.7308	−1.10906
$926$	−12.5027	−0.410866
$927$	0	0
$928$	19.7139	0.647141
$929$	21.2642	0.697657	0.348829	−	0.937187i	$-0.386580\pi$
$929$	21.2642	0.697657	0.348829	+	0.937187i	$0.386580\pi$
$930$	0	0
$931$	5.06076	0.165860
$932$	11.4976	0.376615
$933$	0	0
$934$	0.774303	0.0253360
$935$	−10.2314	−0.334601
$936$	0	0
$937$	−5.52295	−0.180427	−0.0902135	−	0.995922i	$-0.528755\pi$
$937$	−5.52295	−0.180427	−0.0902135	+	0.995922i	$0.528755\pi$
$938$	−17.3101	−0.565195
$939$	0	0
$940$	−4.40296	−0.143609
$941$	7.72836	0.251937	0.125969	−	0.992034i	$-0.459796\pi$
$941$	7.72836	0.251937	0.125969	+	0.992034i	$0.459796\pi$
$942$	0	0
$943$	25.6454	0.835129
$944$	−14.2816	−0.464827
$945$	0	0
$946$	−4.47976	−0.145649
$947$	−28.2620	−0.918392	−0.459196	−	0.888335i	$-0.651862\pi$
$947$	−28.2620	−0.918392	−0.459196	+	0.888335i	$0.651862\pi$
$948$	0	0
$949$	−28.4924	−0.924901
$950$	−10.0003	−0.324454
$951$	0	0
$952$	−32.3758	−1.04931
$953$	45.3347	1.46854	0.734268	−	0.678860i	$-0.237526\pi$
$953$	45.3347	1.46854	0.734268	+	0.678860i	$0.237526\pi$
$954$	0	0
$955$	−58.2168	−1.88385
$956$	−6.49708	−0.210130
$957$	0	0
$958$	14.9723	0.483733
$959$	−13.9871	−0.451667
$960$	0	0
$961$	−17.2480	−0.556387
$962$	−22.1468	−0.714043
$963$	0	0
$964$	13.7888	0.444109
$965$	73.8385	2.37694
$966$	0	0
$967$	27.9613	0.899174	0.449587	−	0.893236i	$-0.351571\pi$
$967$	27.9613	0.899174	0.449587	+	0.893236i	$0.351571\pi$
$968$	3.07615	0.0988710
$969$	0	0
$970$	−53.3039	−1.71148
$971$	42.1744	1.35344	0.676721	−	0.736240i	$-0.263400\pi$
$971$	42.1744	1.35344	0.676721	+	0.736240i	$0.263400\pi$
$972$	0	0
$973$	1.63267	0.0523410
$974$	−40.0633	−1.28371
$975$	0	0
$976$	19.0445	0.609598
$977$	21.2802	0.680813	0.340407	−	0.940278i	$-0.389435\pi$
$977$	21.2802	0.680813	0.340407	+	0.940278i	$0.389435\pi$
$978$	0	0
$979$	14.1210	0.451309
$980$	−3.93941	−0.125840
$981$	0	0
$982$	11.7160	0.373871
$983$	−47.5190	−1.51562	−0.757810	−	0.652475i	$-0.773731\pi$
$983$	−47.5190	−1.51562	−0.757810	+	0.652475i	$0.773731\pi$
$984$	0	0
$985$	−72.8751	−2.32199
$986$	−24.8592	−0.791677
$987$	0	0
$988$	2.79718	0.0889900
$989$	22.5301	0.716414
$990$	0	0
$991$	−17.9340	−0.569692	−0.284846	−	0.958573i	$-0.591942\pi$
$991$	−17.9340	−0.569692	−0.284846	+	0.958573i	$0.591942\pi$
$992$	12.0638	0.383027
$993$	0	0
$994$	−57.8078	−1.83355
$995$	34.6617	1.09885
$996$	0	0
$997$	−0.931598	−0.0295040	−0.0147520	−	0.999891i	$-0.504696\pi$
$997$	−0.931598	−0.0295040	−0.0147520	+	0.999891i	$0.504696\pi$
$998$	−1.34752	−0.0426551
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.l.1.17		51
3.2	odd	2		8019.2.a.k.1.35		51
27.4	even	9		297.2.j.c.232.6	✓	102
27.7	even	9		297.2.j.c.265.6	yes	102
27.20	odd	18		891.2.j.c.199.12		102
27.23	odd	18		891.2.j.c.694.12		102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.j.c.232.6	✓	102	27.4	even	9
297.2.j.c.265.6	yes	102	27.7	even	9
891.2.j.c.199.12		102	27.20	odd	18
891.2.j.c.694.12		102	27.23	odd	18
8019.2.a.k.1.35		51	3.2	odd	2
8019.2.a.l.1.17		51	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-1.18428 q^{2} -0.597486 q^{4} +2.95374 q^{5} +3.03845 q^{7} +3.07615 q^{8} +O(q^{10})\)	\(q-1.18428 q^{2} -0.597486 q^{4} +2.95374 q^{5} +3.03845 q^{7} +3.07615 q^{8} -3.49805 q^{10} +1.00000 q^{11} -2.06494 q^{13} -3.59837 q^{14} -2.44804 q^{16} -3.46387 q^{17} +2.26717 q^{19} -1.76482 q^{20} -1.18428 q^{22} +5.95610 q^{23} +3.72457 q^{25} +2.44546 q^{26} -1.81543 q^{28} -6.05998 q^{29} -3.70837 q^{31} -3.25313 q^{32} +4.10218 q^{34} +8.97479 q^{35} -9.05630 q^{37} -2.68496 q^{38} +9.08613 q^{40} +4.30574 q^{41} +3.78269 q^{43} -0.597486 q^{44} -7.05367 q^{46} +2.49486 q^{47} +2.23219 q^{49} -4.41093 q^{50} +1.23377 q^{52} -8.53403 q^{53} +2.95374 q^{55} +9.34672 q^{56} +7.17670 q^{58} +5.83390 q^{59} -7.77948 q^{61} +4.39174 q^{62} +8.74869 q^{64} -6.09929 q^{65} +4.81054 q^{67} +2.06961 q^{68} -10.6286 q^{70} +16.0650 q^{71} +13.7982 q^{73} +10.7252 q^{74} -1.35460 q^{76} +3.03845 q^{77} +1.76828 q^{79} -7.23087 q^{80} -5.09919 q^{82} -8.91270 q^{83} -10.2314 q^{85} -4.47976 q^{86} +3.07615 q^{88} +14.1210 q^{89} -6.27422 q^{91} -3.55868 q^{92} -2.95460 q^{94} +6.69663 q^{95} +15.2382 q^{97} -2.64353 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 51 q + 60 q^{4} + 6 q^{5} + 12 q^{7}+O(q^{10}) \) 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

Varieties

Fields

Representations

Groups

Database

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