LMFDB - Embedded newform 8019.2.a.i.1.7

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			1.7
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)

$f(q)$	$=$	$q-2.27724 q^{2} +3.18582 q^{4} -3.40276 q^{5} +0.928464 q^{7} -2.70039 q^{8} +O(q^{10})$	\(q-2.27724 q^{2} +3.18582 q^{4} -3.40276 q^{5} +0.928464 q^{7} -2.70039 q^{8} +7.74890 q^{10} -1.00000 q^{11} -4.04865 q^{13} -2.11433 q^{14} -0.222203 q^{16} -7.35778 q^{17} +8.26976 q^{19} -10.8406 q^{20} +2.27724 q^{22} +6.65700 q^{23} +6.57880 q^{25} +9.21975 q^{26} +2.95792 q^{28} -9.43162 q^{29} +0.00498620 q^{31} +5.90679 q^{32} +16.7554 q^{34} -3.15934 q^{35} +5.76146 q^{37} -18.8322 q^{38} +9.18879 q^{40} -4.34854 q^{41} -9.32917 q^{43} -3.18582 q^{44} -15.1596 q^{46} +10.6375 q^{47} -6.13795 q^{49} -14.9815 q^{50} -12.8983 q^{52} +3.75678 q^{53} +3.40276 q^{55} -2.50721 q^{56} +21.4781 q^{58} -0.186436 q^{59} +4.14381 q^{61} -0.0113548 q^{62} -13.0068 q^{64} +13.7766 q^{65} -2.99761 q^{67} -23.4405 q^{68} +7.19458 q^{70} +9.74709 q^{71} +0.850716 q^{73} -13.1202 q^{74} +26.3459 q^{76} -0.928464 q^{77} +7.75353 q^{79} +0.756103 q^{80} +9.90266 q^{82} +11.6724 q^{83} +25.0368 q^{85} +21.2448 q^{86} +2.70039 q^{88} +10.6660 q^{89} -3.75903 q^{91} +21.2080 q^{92} -24.2241 q^{94} -28.1400 q^{95} +3.28961 q^{97} +13.9776 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.27724	−1.61025	−0.805126	−	0.593104i	$-0.797902\pi$
$2$	−2.27724	−1.61025	−0.805126	+	0.593104i	$0.797902\pi$
$3$	0	0
$3$	0	0
$4$	3.18582	1.59291
$5$	−3.40276	−1.52176	−0.760881	−	0.648892i	$-0.775233\pi$
$5$	−3.40276	−1.52176	−0.760881	+	0.648892i	$0.775233\pi$
$6$	0	0
$7$	0.928464	0.350926	0.175463	−	0.984486i	$-0.443858\pi$
$7$	0.928464	0.350926	0.175463	+	0.984486i	$0.443858\pi$
$8$	−2.70039	−0.954732
$9$	0	0
$10$	7.74890	2.45042
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−4.04865	−1.12289	−0.561447	−	0.827513i	$-0.689755\pi$
$13$	−4.04865	−1.12289	−0.561447	+	0.827513i	$0.689755\pi$
$14$	−2.11433	−0.565080
$15$	0	0
$16$	−0.222203	−0.0555507
$17$	−7.35778	−1.78452	−0.892261	−	0.451519i	$-0.850882\pi$
$17$	−7.35778	−1.78452	−0.892261	+	0.451519i	$0.850882\pi$
$18$	0	0
$19$	8.26976	1.89721	0.948607	−	0.316458i	$-0.102494\pi$
$19$	8.26976	1.89721	0.948607	+	0.316458i	$0.102494\pi$
$20$	−10.8406	−2.42403
$21$	0	0
$22$	2.27724	0.485509
$23$	6.65700	1.38808	0.694040	−	0.719937i	$-0.255829\pi$
$23$	6.65700	1.38808	0.694040	+	0.719937i	$0.255829\pi$
$24$	0	0
$25$	6.57880	1.31576
$26$	9.21975	1.80814
$27$	0	0
$28$	2.95792	0.558994
$29$	−9.43162	−1.75141	−0.875704	−	0.482848i	$-0.839602\pi$
$29$	−9.43162	−1.75141	−0.875704	+	0.482848i	$0.839602\pi$
$30$	0	0
$31$	0.00498620	0.000895547 0	0.000447774	−	1.00000i	$-0.499857\pi$
$31$	0.00498620	0.000895547 0	0.000447774		1.00000i	$0.499857\pi$
$32$	5.90679	1.04418
$33$	0	0
$34$	16.7554	2.87353
$35$	−3.15934	−0.534027
$36$	0	0
$37$	5.76146	0.947179	0.473589	−	0.880746i	$-0.342958\pi$
$37$	5.76146	0.947179	0.473589	+	0.880746i	$0.342958\pi$
$38$	−18.8322	−3.05499
$39$	0	0
$40$	9.18879	1.45287
$41$	−4.34854	−0.679128	−0.339564	−	0.940583i	$-0.610279\pi$
$41$	−4.34854	−0.679128	−0.339564	+	0.940583i	$0.610279\pi$
$42$	0	0
$43$	−9.32917	−1.42269	−0.711343	−	0.702845i	$-0.751913\pi$
$43$	−9.32917	−1.42269	−0.711343	+	0.702845i	$0.751913\pi$
$44$	−3.18582	−0.480280
$45$	0	0
$46$	−15.1596	−2.23516
$47$	10.6375	1.55164	0.775819	−	0.630955i	$-0.217337\pi$
$47$	10.6375	1.55164	0.775819	+	0.630955i	$0.217337\pi$
$48$	0	0
$49$	−6.13795	−0.876851
$50$	−14.9815	−2.11870
$51$	0	0
$52$	−12.8983	−1.78867
$53$	3.75678	0.516034	0.258017	−	0.966140i	$-0.416931\pi$
$53$	3.75678	0.516034	0.258017	+	0.966140i	$0.416931\pi$
$54$	0	0
$55$	3.40276	0.458828
$56$	−2.50721	−0.335041
$57$	0	0
$58$	21.4781	2.82021
$59$	−0.186436	−0.0242719	−0.0121359	−	0.999926i	$-0.503863\pi$
$59$	−0.186436	−0.0242719	−0.0121359	+	0.999926i	$0.503863\pi$
$60$	0	0
$61$	4.14381	0.530560	0.265280	−	0.964171i	$-0.414536\pi$
$61$	4.14381	0.530560	0.265280	+	0.964171i	$0.414536\pi$
$62$	−0.0113548	−0.00144206
$63$	0	0
$64$	−13.0068	−1.62585
$65$	13.7766	1.70878
$66$	0	0
$67$	−2.99761	−0.366217	−0.183108	−	0.983093i	$-0.558616\pi$
$67$	−2.99761	−0.366217	−0.183108	+	0.983093i	$0.558616\pi$
$68$	−23.4405	−2.84258
$69$	0	0
$70$	7.19458	0.859917
$71$	9.74709	1.15677	0.578383	−	0.815765i	$-0.303684\pi$
$71$	9.74709	1.15677	0.578383	+	0.815765i	$0.303684\pi$
$72$	0	0
$73$	0.850716	0.0995687	0.0497844	−	0.998760i	$-0.484147\pi$
$73$	0.850716	0.0995687	0.0497844	+	0.998760i	$0.484147\pi$
$74$	−13.1202	−1.52520
$75$	0	0
$76$	26.3459	3.02209
$77$	−0.928464	−0.105808
$78$	0	0
$79$	7.75353	0.872341	0.436170	−	0.899864i	$-0.356335\pi$
$79$	7.75353	0.872341	0.436170	+	0.899864i	$0.356335\pi$
$80$	0.756103	0.0845349
$81$	0	0
$82$	9.90266	1.09357
$83$	11.6724	1.28122	0.640609	−	0.767868i	$-0.278682\pi$
$83$	11.6724	1.28122	0.640609	+	0.767868i	$0.278682\pi$
$84$	0	0
$85$	25.0368	2.71562
$86$	21.2448	2.29088
$87$	0	0
$88$	2.70039	0.287862
$89$	10.6660	1.13059	0.565296	−	0.824888i	$-0.308762\pi$
$89$	10.6660	1.13059	0.565296	+	0.824888i	$0.308762\pi$
$90$	0	0
$91$	−3.75903	−0.394053
$92$	21.2080	2.21108
$93$	0	0
$94$	−24.2241	−2.49853
$95$	−28.1400	−2.88711
$96$	0	0
$97$	3.28961	0.334009	0.167005	−	0.985956i	$-0.446591\pi$
$97$	3.28961	0.334009	0.167005	+	0.985956i	$0.446591\pi$
$98$	13.9776	1.41195
$99$	0	0
$100$	20.9588	2.09588
$101$	−16.4170	−1.63356	−0.816778	−	0.576951i	$-0.804242\pi$
$101$	−16.4170	−1.63356	−0.816778	+	0.576951i	$0.804242\pi$
$102$	0	0
$103$	−4.16720	−0.410606	−0.205303	−	0.978698i	$-0.565818\pi$
$103$	−4.16720	−0.410606	−0.205303	+	0.978698i	$0.565818\pi$
$104$	10.9329	1.07206
$105$	0	0
$106$	−8.55509	−0.830944
$107$	−11.1100	−1.07404	−0.537022	−	0.843568i	$-0.680451\pi$
$107$	−11.1100	−1.07404	−0.537022	+	0.843568i	$0.680451\pi$
$108$	0	0
$109$	14.8094	1.41848	0.709242	−	0.704966i	$-0.249038\pi$
$109$	14.8094	1.41848	0.709242	+	0.704966i	$0.249038\pi$
$110$	−7.74890	−0.738829
$111$	0	0
$112$	−0.206307	−0.0194942
$113$	0.539765	0.0507769	0.0253884	−	0.999678i	$-0.491918\pi$
$113$	0.539765	0.0507769	0.0253884	+	0.999678i	$0.491918\pi$
$114$	0	0
$115$	−22.6522	−2.11233
$116$	−30.0474	−2.78983
$117$	0	0
$118$	0.424559	0.0390838
$119$	−6.83143	−0.626236
$120$	0	0
$121$	1.00000	0.0909091
$122$	−9.43644	−0.854335
$123$	0	0
$124$	0.0158851	0.00142653
$125$	−5.37227	−0.480510
$126$	0	0
$127$	2.23320	0.198165	0.0990823	−	0.995079i	$-0.468409\pi$
$127$	2.23320	0.198165	0.0990823	+	0.995079i	$0.468409\pi$
$128$	17.8059	1.57384
$129$	0	0
$130$	−31.3726	−2.75156
$131$	−14.5384	−1.27023	−0.635113	−	0.772419i	$-0.719047\pi$
$131$	−14.5384	−1.27023	−0.635113	+	0.772419i	$0.719047\pi$
$132$	0	0
$133$	7.67818	0.665782
$134$	6.82628	0.589701
$135$	0	0
$136$	19.8689	1.70374
$137$	−15.4422	−1.31932	−0.659658	−	0.751566i	$-0.729299\pi$
$137$	−15.4422	−1.31932	−0.659658	+	0.751566i	$0.729299\pi$
$138$	0	0
$139$	5.77660	0.489965	0.244982	−	0.969528i	$-0.421218\pi$
$139$	5.77660	0.489965	0.244982	+	0.969528i	$0.421218\pi$
$140$	−10.0651	−0.850655
$141$	0	0
$142$	−22.1965	−1.86269
$143$	4.04865	0.338565
$144$	0	0
$145$	32.0936	2.66523
$146$	−1.93728	−0.160331
$147$	0	0
$148$	18.3550	1.50877
$149$	−1.61421	−0.132241	−0.0661205	−	0.997812i	$-0.521062\pi$
$149$	−1.61421	−0.132241	−0.0661205	+	0.997812i	$0.521062\pi$
$150$	0	0
$151$	10.5291	0.856848	0.428424	−	0.903578i	$-0.359069\pi$
$151$	10.5291	0.856848	0.428424	+	0.903578i	$0.359069\pi$
$152$	−22.3316	−1.81133
$153$	0	0
$154$	2.11433	0.170378
$155$	−0.0169668	−0.00136281
$156$	0	0
$157$	−16.5035	−1.31712	−0.658561	−	0.752527i	$-0.728835\pi$
$157$	−16.5035	−1.31712	−0.658561	+	0.752527i	$0.728835\pi$
$158$	−17.6566	−1.40469
$159$	0	0
$160$	−20.0994	−1.58900
$161$	6.18078	0.487114
$162$	0	0
$163$	2.58189	0.202230	0.101115	−	0.994875i	$-0.467759\pi$
$163$	2.58189	0.202230	0.101115	+	0.994875i	$0.467759\pi$
$164$	−13.8537	−1.08179
$165$	0	0
$166$	−26.5809	−2.06308
$167$	8.52560	0.659731	0.329866	−	0.944028i	$-0.392997\pi$
$167$	8.52560	0.659731	0.329866	+	0.944028i	$0.392997\pi$
$168$	0	0
$169$	3.39159	0.260891
$170$	−57.0147	−4.37283
$171$	0	0
$172$	−29.7210	−2.26621
$173$	12.8592	0.977667	0.488833	−	0.872377i	$-0.337423\pi$
$173$	12.8592	0.977667	0.488833	+	0.872377i	$0.337423\pi$
$174$	0	0
$175$	6.10818	0.461735
$176$	0.222203	0.0167492
$177$	0	0
$178$	−24.2890	−1.82054
$179$	−6.79446	−0.507842	−0.253921	−	0.967225i	$-0.581720\pi$
$179$	−6.79446	−0.507842	−0.253921	+	0.967225i	$0.581720\pi$
$180$	0	0
$181$	6.27035	0.466071	0.233036	−	0.972468i	$-0.425134\pi$
$181$	6.27035	0.466071	0.233036	+	0.972468i	$0.425134\pi$
$182$	8.56021	0.634525
$183$	0	0
$184$	−17.9765	−1.32524
$185$	−19.6049	−1.44138
$186$	0	0
$187$	7.35778	0.538054
$188$	33.8891	2.47162
$189$	0	0
$190$	64.0816	4.64897
$191$	4.62853	0.334909	0.167454	−	0.985880i	$-0.446445\pi$
$191$	4.62853	0.334909	0.167454	+	0.985880i	$0.446445\pi$
$192$	0	0
$193$	10.2724	0.739422	0.369711	−	0.929147i	$-0.379457\pi$
$193$	10.2724	0.739422	0.369711	+	0.929147i	$0.379457\pi$
$194$	−7.49122	−0.537839
$195$	0	0
$196$	−19.5544	−1.39674
$197$	24.9486	1.77751	0.888757	−	0.458380i	$-0.151570\pi$
$197$	24.9486	1.77751	0.888757	+	0.458380i	$0.151570\pi$
$198$	0	0
$199$	1.89286	0.134181	0.0670906	−	0.997747i	$-0.478628\pi$
$199$	1.89286	0.134181	0.0670906	+	0.997747i	$0.478628\pi$
$200$	−17.7653	−1.25620
$201$	0	0
$202$	37.3855	2.63044
$203$	−8.75692	−0.614615
$204$	0	0
$205$	14.7971	1.03347
$206$	9.48970	0.661179
$207$	0	0
$208$	0.899621	0.0623775
$209$	−8.26976	−0.572031
$210$	0	0
$211$	−19.4419	−1.33844	−0.669219	−	0.743065i	$-0.733371\pi$
$211$	−19.4419	−1.33844	−0.669219	+	0.743065i	$0.733371\pi$
$212$	11.9684	0.821995
$213$	0	0
$214$	25.3001	1.72948
$215$	31.7450	2.16499
$216$	0	0
$217$	0.00462950	0.000314271 0
$218$	−33.7245	−2.28411
$219$	0	0
$220$	10.8406	0.730872
$221$	29.7891	2.00383
$222$	0	0
$223$	3.71176	0.248558	0.124279	−	0.992247i	$-0.460338\pi$
$223$	3.71176	0.248558	0.124279	+	0.992247i	$0.460338\pi$
$224$	5.48424	0.366431
$225$	0	0
$226$	−1.22917	−0.0817635
$227$	21.0083	1.39437	0.697185	−	0.716891i	$-0.254436\pi$
$227$	21.0083	1.39437	0.697185	+	0.716891i	$0.254436\pi$
$228$	0	0
$229$	−4.50670	−0.297811	−0.148905	−	0.988851i	$-0.547575\pi$
$229$	−4.50670	−0.297811	−0.148905	+	0.988851i	$0.547575\pi$
$230$	51.5844	3.40138
$231$	0	0
$232$	25.4690	1.67212
$233$	12.2844	0.804775	0.402387	−	0.915469i	$-0.368180\pi$
$233$	12.2844	0.804775	0.402387	+	0.915469i	$0.368180\pi$
$234$	0	0
$235$	−36.1969	−2.36122
$236$	−0.593950	−0.0386629
$237$	0	0
$238$	15.5568	1.00840
$239$	−5.72469	−0.370299	−0.185150	−	0.982710i	$-0.559277\pi$
$239$	−5.72469	−0.370299	−0.185150	+	0.982710i	$0.559277\pi$
$240$	0	0
$241$	−18.4430	−1.18802	−0.594010	−	0.804458i	$-0.702456\pi$
$241$	−18.4430	−1.18802	−0.594010	+	0.804458i	$0.702456\pi$
$242$	−2.27724	−0.146386
$243$	0	0
$244$	13.2014	0.845134
$245$	20.8860	1.33436
$246$	0	0
$247$	−33.4814	−2.13037
$248$	−0.0134647	−0.000855008 0
$249$	0	0
$250$	12.2339	0.773742
$251$	0.953225	0.0601671	0.0300835	−	0.999547i	$-0.490423\pi$
$251$	0.953225	0.0601671	0.0300835	+	0.999547i	$0.490423\pi$
$252$	0	0
$253$	−6.65700	−0.418522
$254$	−5.08553	−0.319095
$255$	0	0
$256$	−14.5348	−0.908427
$257$	5.72884	0.357355	0.178677	−	0.983908i	$-0.442818\pi$
$257$	5.72884	0.357355	0.178677	+	0.983908i	$0.442818\pi$
$258$	0	0
$259$	5.34931	0.332390
$260$	43.8897	2.72193
$261$	0	0
$262$	33.1074	2.04538
$263$	−5.33098	−0.328723	−0.164361	−	0.986400i	$-0.552556\pi$
$263$	−5.33098	−0.328723	−0.164361	+	0.986400i	$0.552556\pi$
$264$	0	0
$265$	−12.7834	−0.785281
$266$	−17.4850	−1.07208
$267$	0	0
$268$	−9.54985	−0.583350
$269$	−6.14619	−0.374740	−0.187370	−	0.982289i	$-0.559996\pi$
$269$	−6.14619	−0.374740	−0.187370	+	0.982289i	$0.559996\pi$
$270$	0	0
$271$	−24.0756	−1.46249	−0.731244	−	0.682116i	$-0.761060\pi$
$271$	−24.0756	−1.46249	−0.731244	+	0.682116i	$0.761060\pi$
$272$	1.63492	0.0991314
$273$	0	0
$274$	35.1656	2.12443
$275$	−6.57880	−0.396716
$276$	0	0
$277$	5.48053	0.329293	0.164647	−	0.986353i	$-0.447352\pi$
$277$	5.48053	0.329293	0.164647	+	0.986353i	$0.447352\pi$
$278$	−13.1547	−0.788967
$279$	0	0
$280$	8.53146	0.509852
$281$	−6.77516	−0.404172	−0.202086	−	0.979368i	$-0.564772\pi$
$281$	−6.77516	−0.404172	−0.202086	+	0.979368i	$0.564772\pi$
$282$	0	0
$283$	−11.2494	−0.668709	−0.334354	−	0.942447i	$-0.608518\pi$
$283$	−11.2494	−0.668709	−0.334354	+	0.942447i	$0.608518\pi$
$284$	31.0524	1.84262
$285$	0	0
$286$	−9.21975	−0.545175
$287$	−4.03746	−0.238324
$288$	0	0
$289$	37.1369	2.18452
$290$	−73.0847	−4.29168
$291$	0	0
$292$	2.71022	0.158604
$293$	−2.00653	−0.117222	−0.0586112	−	0.998281i	$-0.518667\pi$
$293$	−2.00653	−0.117222	−0.0586112	+	0.998281i	$0.518667\pi$
$294$	0	0
$295$	0.634396	0.0369360
$296$	−15.5582	−0.904302
$297$	0	0
$298$	3.67594	0.212941
$299$	−26.9519	−1.55867
$300$	0	0
$301$	−8.66180	−0.499258
$302$	−23.9773	−1.37974
$303$	0	0
$304$	−1.83756	−0.105391
$305$	−14.1004	−0.807387
$306$	0	0
$307$	−1.29607	−0.0739707	−0.0369854	−	0.999316i	$-0.511775\pi$
$307$	−1.29607	−0.0739707	−0.0369854	+	0.999316i	$0.511775\pi$
$308$	−2.95792	−0.168543
$309$	0	0
$310$	0.0386376	0.00219447
$311$	−7.95855	−0.451288	−0.225644	−	0.974210i	$-0.572449\pi$
$311$	−7.95855	−0.451288	−0.225644	+	0.974210i	$0.572449\pi$
$312$	0	0
$313$	12.7840	0.722592	0.361296	−	0.932451i	$-0.382334\pi$
$313$	12.7840	0.722592	0.361296	+	0.932451i	$0.382334\pi$
$314$	37.5824	2.12090
$315$	0	0
$316$	24.7013	1.38956
$317$	10.3092	0.579025	0.289512	−	0.957174i	$-0.406507\pi$
$317$	10.3092	0.579025	0.289512	+	0.957174i	$0.406507\pi$
$318$	0	0
$319$	9.43162	0.528069
$320$	44.2589	2.47415
$321$	0	0
$322$	−14.0751	−0.784376
$323$	−60.8470	−3.38562
$324$	0	0
$325$	−26.6353	−1.47746
$326$	−5.87959	−0.325640
$327$	0	0
$328$	11.7428	0.648385
$329$	9.87653	0.544511
$330$	0	0
$331$	−12.4205	−0.682691	−0.341346	−	0.939938i	$-0.610883\pi$
$331$	−12.4205	−0.682691	−0.341346	+	0.939938i	$0.610883\pi$
$332$	37.1863	2.04086
$333$	0	0
$334$	−19.4148	−1.06233
$335$	10.2002	0.557295
$336$	0	0
$337$	6.10040	0.332310	0.166155	−	0.986100i	$-0.446865\pi$
$337$	6.10040	0.332310	0.166155	+	0.986100i	$0.446865\pi$
$338$	−7.72345	−0.420100
$339$	0	0
$340$	79.7626	4.32573
$341$	−0.00498620	−0.000270018 0
$342$	0	0
$343$	−12.1981	−0.658637
$344$	25.1924	1.35828
$345$	0	0
$346$	−29.2835	−1.57429
$347$	25.3386	1.36025	0.680124	−	0.733097i	$-0.261926\pi$
$347$	25.3386	1.36025	0.680124	+	0.733097i	$0.261926\pi$
$348$	0	0
$349$	−0.347075	−0.0185785	−0.00928926	−	0.999957i	$-0.502957\pi$
$349$	−0.347075	−0.0185785	−0.00928926	+	0.999957i	$0.502957\pi$
$350$	−13.9098	−0.743509
$351$	0	0
$352$	−5.90679	−0.314833
$353$	−11.3344	−0.603270	−0.301635	−	0.953424i	$-0.597532\pi$
$353$	−11.3344	−0.603270	−0.301635	+	0.953424i	$0.597532\pi$
$354$	0	0
$355$	−33.1670	−1.76032
$356$	33.9799	1.80093
$357$	0	0
$358$	15.4726	0.817753
$359$	−34.6528	−1.82890	−0.914451	−	0.404696i	$-0.867377\pi$
$359$	−34.6528	−1.82890	−0.914451	+	0.404696i	$0.867377\pi$
$360$	0	0
$361$	49.3889	2.59942
$362$	−14.2791	−0.750492
$363$	0	0
$364$	−11.9756	−0.627691
$365$	−2.89478	−0.151520
$366$	0	0
$367$	19.6277	1.02456	0.512279	−	0.858819i	$-0.328802\pi$
$367$	19.6277	1.02456	0.512279	+	0.858819i	$0.328802\pi$
$368$	−1.47920	−0.0771087
$369$	0	0
$370$	44.6450	2.32098
$371$	3.48804	0.181090
$372$	0	0
$373$	24.5516	1.27123	0.635617	−	0.772004i	$-0.280746\pi$
$373$	24.5516	1.27123	0.635617	+	0.772004i	$0.280746\pi$
$374$	−16.7554	−0.866402
$375$	0	0
$376$	−28.7254	−1.48140
$377$	38.1854	1.96665
$378$	0	0
$379$	−1.80547	−0.0927408	−0.0463704	−	0.998924i	$-0.514765\pi$
$379$	−1.80547	−0.0927408	−0.0463704	+	0.998924i	$0.514765\pi$
$380$	−89.6490	−4.59890
$381$	0	0
$382$	−10.5403	−0.539288
$383$	33.9011	1.73226	0.866132	−	0.499815i	$-0.166599\pi$
$383$	33.9011	1.73226	0.866132	+	0.499815i	$0.166599\pi$
$384$	0	0
$385$	3.15934	0.161015
$386$	−23.3927	−1.19066
$387$	0	0
$388$	10.4801	0.532046
$389$	14.7629	0.748511	0.374256	−	0.927326i	$-0.377898\pi$
$389$	14.7629	0.748511	0.374256	+	0.927326i	$0.377898\pi$
$390$	0	0
$391$	−48.9807	−2.47706
$392$	16.5749	0.837157
$393$	0	0
$394$	−56.8139	−2.86224
$395$	−26.3834	−1.32749
$396$	0	0
$397$	−28.8625	−1.44857	−0.724284	−	0.689502i	$-0.757829\pi$
$397$	−28.8625	−1.44857	−0.724284	+	0.689502i	$0.757829\pi$
$398$	−4.31049	−0.216065
$399$	0	0
$400$	−1.46183	−0.0730913
$401$	36.7156	1.83349	0.916744	−	0.399474i	$-0.130807\pi$
$401$	36.7156	1.83349	0.916744	+	0.399474i	$0.130807\pi$
$402$	0	0
$403$	−0.0201874	−0.00100560
$404$	−52.3017	−2.60211
$405$	0	0
$406$	19.9416	0.989685
$407$	−5.76146	−0.285585
$408$	0	0
$409$	7.52787	0.372229	0.186114	−	0.982528i	$-0.440410\pi$
$409$	7.52787	0.372229	0.186114	+	0.982528i	$0.440410\pi$
$410$	−33.6964	−1.66415
$411$	0	0
$412$	−13.2759	−0.654058
$413$	−0.173099	−0.00851764
$414$	0	0
$415$	−39.7186	−1.94971
$416$	−23.9145	−1.17251
$417$	0	0
$418$	18.8322	0.921114
$419$	13.6043	0.664615	0.332308	−	0.943171i	$-0.392173\pi$
$419$	13.6043	0.664615	0.332308	+	0.943171i	$0.392173\pi$
$420$	0	0
$421$	−20.9199	−1.01957	−0.509786	−	0.860301i	$-0.670275\pi$
$421$	−20.9199	−1.01957	−0.509786	+	0.860301i	$0.670275\pi$
$422$	44.2739	2.15522
$423$	0	0
$424$	−10.1448	−0.492674
$425$	−48.4053	−2.34800
$426$	0	0
$427$	3.84738	0.186188
$428$	−35.3944	−1.71085
$429$	0	0
$430$	−72.2909	−3.48617
$431$	16.5280	0.796128	0.398064	−	0.917358i	$-0.369682\pi$
$431$	16.5280	0.796128	0.398064	+	0.917358i	$0.369682\pi$
$432$	0	0
$433$	−4.81907	−0.231590	−0.115795	−	0.993273i	$-0.536941\pi$
$433$	−4.81907	−0.231590	−0.115795	+	0.993273i	$0.536941\pi$
$434$	−0.0105425	−0.000506056 0
$435$	0	0
$436$	47.1800	2.25951
$437$	55.0518	2.63348
$438$	0	0
$439$	−5.30420	−0.253156	−0.126578	−	0.991957i	$-0.540399\pi$
$439$	−5.30420	−0.253156	−0.126578	+	0.991957i	$0.540399\pi$
$440$	−9.18879	−0.438058
$441$	0	0
$442$	−67.8368	−3.22667
$443$	−24.1499	−1.14739	−0.573697	−	0.819067i	$-0.694491\pi$
$443$	−24.1499	−1.14739	−0.573697	+	0.819067i	$0.694491\pi$
$444$	0	0
$445$	−36.2938	−1.72049
$446$	−8.45256	−0.400240
$447$	0	0
$448$	−12.0763	−0.570552
$449$	−14.5483	−0.686578	−0.343289	−	0.939230i	$-0.611541\pi$
$449$	−14.5483	−0.686578	−0.343289	+	0.939230i	$0.611541\pi$
$450$	0	0
$451$	4.34854	0.204765
$452$	1.71959	0.0808829
$453$	0	0
$454$	−47.8409	−2.24529
$455$	12.7911	0.599655
$456$	0	0
$457$	5.91659	0.276766	0.138383	−	0.990379i	$-0.455809\pi$
$457$	5.91659	0.276766	0.138383	+	0.990379i	$0.455809\pi$
$458$	10.2628	0.479550
$459$	0	0
$460$	−72.1657	−3.36474
$461$	−34.8559	−1.62340	−0.811701	−	0.584074i	$-0.801458\pi$
$461$	−34.8559	−1.62340	−0.811701	+	0.584074i	$0.801458\pi$
$462$	0	0
$463$	2.20780	0.102605	0.0513027	−	0.998683i	$-0.483663\pi$
$463$	2.20780	0.102605	0.0513027	+	0.998683i	$0.483663\pi$
$464$	2.09573	0.0972919
$465$	0	0
$466$	−27.9744	−1.29589
$467$	31.6971	1.46677	0.733383	−	0.679816i	$-0.237940\pi$
$467$	31.6971	1.46677	0.733383	+	0.679816i	$0.237940\pi$
$468$	0	0
$469$	−2.78318	−0.128515
$470$	82.4290	3.80216
$471$	0	0
$472$	0.503449	0.0231731
$473$	9.32917	0.428956
$474$	0	0
$475$	54.4051	2.49628
$476$	−21.7637	−0.997537
$477$	0	0
$478$	13.0365	0.596275
$479$	−11.1687	−0.510310	−0.255155	−	0.966900i	$-0.582127\pi$
$479$	−11.1687	−0.510310	−0.255155	+	0.966900i	$0.582127\pi$
$480$	0	0
$481$	−23.3262	−1.06358
$482$	41.9992	1.91301
$483$	0	0
$484$	3.18582	0.144810
$485$	−11.1938	−0.508282
$486$	0	0
$487$	4.79482	0.217274	0.108637	−	0.994081i	$-0.465351\pi$
$487$	4.79482	0.217274	0.108637	+	0.994081i	$0.465351\pi$
$488$	−11.1899	−0.506543
$489$	0	0
$490$	−47.5624	−2.14865
$491$	−10.8816	−0.491081	−0.245540	−	0.969386i	$-0.578965\pi$
$491$	−10.8816	−0.491081	−0.245540	+	0.969386i	$0.578965\pi$
$492$	0	0
$493$	69.3958	3.12543
$494$	76.2451	3.43043
$495$	0	0
$496$	−0.00110795	−4.97482e−5 0
$497$	9.04982	0.405940
$498$	0	0
$499$	6.99722	0.313238	0.156619	−	0.987659i	$-0.449940\pi$
$499$	6.99722	0.313238	0.156619	+	0.987659i	$0.449940\pi$
$500$	−17.1151	−0.765409
$501$	0	0
$502$	−2.17072	−0.0968841
$503$	−41.9524	−1.87057	−0.935283	−	0.353900i	$-0.884855\pi$
$503$	−41.9524	−1.87057	−0.935283	+	0.353900i	$0.884855\pi$
$504$	0	0
$505$	55.8633	2.48588
$506$	15.1596	0.673925
$507$	0	0
$508$	7.11457	0.315658
$509$	−27.8529	−1.23456	−0.617279	−	0.786744i	$-0.711765\pi$
$509$	−27.8529	−1.23456	−0.617279	+	0.786744i	$0.711765\pi$
$510$	0	0
$511$	0.789859	0.0349413
$512$	−2.51257	−0.111041
$513$	0	0
$514$	−13.0459	−0.575431
$515$	14.1800	0.624845
$516$	0	0
$517$	−10.6375	−0.467837
$518$	−12.1817	−0.535232
$519$	0	0
$520$	−37.2022	−1.63142
$521$	19.6966	0.862922	0.431461	−	0.902132i	$-0.357998\pi$
$521$	19.6966	0.862922	0.431461	+	0.902132i	$0.357998\pi$
$522$	0	0
$523$	−36.0369	−1.57578	−0.787891	−	0.615815i	$-0.788827\pi$
$523$	−36.0369	−1.57578	−0.787891	+	0.615815i	$0.788827\pi$
$524$	−46.3167	−2.02335
$525$	0	0
$526$	12.1399	0.529326
$527$	−0.0366873	−0.00159812
$528$	0	0
$529$	21.3156	0.926765
$530$	29.1110	1.26450
$531$	0	0
$532$	24.4613	1.06053
$533$	17.6057	0.762589
$534$	0	0
$535$	37.8047	1.63444
$536$	8.09472	0.349639
$537$	0	0
$538$	13.9963	0.603425
$539$	6.13795	0.264380
$540$	0	0
$541$	−0.277809	−0.0119439	−0.00597197	−	0.999982i	$-0.501901\pi$
$541$	−0.277809	−0.0119439	−0.00597197	+	0.999982i	$0.501901\pi$
$542$	54.8259	2.35497
$543$	0	0
$544$	−43.4608	−1.86337
$545$	−50.3929	−2.15859
$546$	0	0
$547$	−18.7500	−0.801692	−0.400846	−	0.916145i	$-0.631284\pi$
$547$	−18.7500	−0.801692	−0.400846	+	0.916145i	$0.631284\pi$
$548$	−49.1960	−2.10155
$549$	0	0
$550$	14.9815	0.638813
$551$	−77.9972	−3.32279
$552$	0	0
$553$	7.19888	0.306127
$554$	−12.4805	−0.530245
$555$	0	0
$556$	18.4032	0.780469
$557$	−15.4942	−0.656509	−0.328254	−	0.944589i	$-0.606460\pi$
$557$	−15.4942	−0.656509	−0.328254	+	0.944589i	$0.606460\pi$
$558$	0	0
$559$	37.7706	1.59752
$560$	0.702015	0.0296655
$561$	0	0
$562$	15.4287	0.650819
$563$	35.1059	1.47954	0.739768	−	0.672862i	$-0.234935\pi$
$563$	35.1059	1.47954	0.739768	+	0.672862i	$0.234935\pi$
$564$	0	0
$565$	−1.83669	−0.0772703
$566$	25.6176	1.07679
$567$	0	0
$568$	−26.3209	−1.10440
$569$	−3.37981	−0.141689	−0.0708444	−	0.997487i	$-0.522569\pi$
$569$	−3.37981	−0.141689	−0.0708444	+	0.997487i	$0.522569\pi$
$570$	0	0
$571$	−20.4212	−0.854602	−0.427301	−	0.904109i	$-0.640535\pi$
$571$	−20.4212	−0.854602	−0.427301	+	0.904109i	$0.640535\pi$
$572$	12.8983	0.539304
$573$	0	0
$574$	9.19427	0.383761
$575$	43.7950	1.82638
$576$	0	0
$577$	−21.6761	−0.902387	−0.451194	−	0.892426i	$-0.649002\pi$
$577$	−21.6761	−0.902387	−0.451194	+	0.892426i	$0.649002\pi$
$578$	−84.5695	−3.51763
$579$	0	0
$580$	102.244	4.24546
$581$	10.8374	0.449613
$582$	0	0
$583$	−3.75678	−0.155590
$584$	−2.29726	−0.0950614
$585$	0	0
$586$	4.56934	0.188758
$587$	−1.79308	−0.0740082	−0.0370041	−	0.999315i	$-0.511781\pi$
$587$	−1.79308	−0.0740082	−0.0370041	+	0.999315i	$0.511781\pi$
$588$	0	0
$589$	0.0412346	0.00169904
$590$	−1.44467	−0.0594762
$591$	0	0
$592$	−1.28021	−0.0526164
$593$	−6.31110	−0.259166	−0.129583	−	0.991569i	$-0.541364\pi$
$593$	−6.31110	−0.259166	−0.129583	+	0.991569i	$0.541364\pi$
$594$	0	0
$595$	23.2457	0.952982
$596$	−5.14257	−0.210648
$597$	0	0
$598$	61.3758	2.50984
$599$	−26.3502	−1.07664	−0.538321	−	0.842740i	$-0.680941\pi$
$599$	−26.3502	−1.07664	−0.538321	+	0.842740i	$0.680941\pi$
$600$	0	0
$601$	−43.1426	−1.75982	−0.879912	−	0.475136i	$-0.842399\pi$
$601$	−43.1426	−1.75982	−0.879912	+	0.475136i	$0.842399\pi$
$602$	19.7250	0.803931
$603$	0	0
$604$	33.5439	1.36488
$605$	−3.40276	−0.138342
$606$	0	0
$607$	−35.4964	−1.44075	−0.720377	−	0.693583i	$-0.756031\pi$
$607$	−35.4964	−1.44075	−0.720377	+	0.693583i	$0.756031\pi$
$608$	48.8477	1.98104
$609$	0	0
$610$	32.1100	1.30010
$611$	−43.0675	−1.74233
$612$	0	0
$613$	−48.3374	−1.95233	−0.976165	−	0.217031i	$-0.930363\pi$
$613$	−48.3374	−1.95233	−0.976165	+	0.217031i	$0.930363\pi$
$614$	2.95147	0.119111
$615$	0	0
$616$	2.50721	0.101019
$617$	−21.0287	−0.846585	−0.423293	−	0.905993i	$-0.639126\pi$
$617$	−21.0287	−0.846585	−0.423293	+	0.905993i	$0.639126\pi$
$618$	0	0
$619$	16.1246	0.648104	0.324052	−	0.946039i	$-0.394955\pi$
$619$	16.1246	0.648104	0.324052	+	0.946039i	$0.394955\pi$
$620$	−0.0540533	−0.00217083
$621$	0	0
$622$	18.1235	0.726687
$623$	9.90298	0.396754
$624$	0	0
$625$	−14.6134	−0.584537
$626$	−29.1121	−1.16355
$627$	0	0
$628$	−52.5771	−2.09806
$629$	−42.3916	−1.69026
$630$	0	0
$631$	5.76955	0.229682	0.114841	−	0.993384i	$-0.463364\pi$
$631$	5.76955	0.229682	0.114841	+	0.993384i	$0.463364\pi$
$632$	−20.9376	−0.832851
$633$	0	0
$634$	−23.4766	−0.932375
$635$	−7.59905	−0.301559
$636$	0	0
$637$	24.8504	0.984610
$638$	−21.4781	−0.850324
$639$	0	0
$640$	−60.5894	−2.39500
$641$	34.4843	1.36205	0.681023	−	0.732262i	$-0.261535\pi$
$641$	34.4843	1.36205	0.681023	+	0.732262i	$0.261535\pi$
$642$	0	0
$643$	−10.6750	−0.420982	−0.210491	−	0.977596i	$-0.567506\pi$
$643$	−10.6750	−0.420982	−0.210491	+	0.977596i	$0.567506\pi$
$644$	19.6908	0.775928
$645$	0	0
$646$	138.563	5.45170
$647$	−22.7291	−0.893573	−0.446787	−	0.894641i	$-0.647432\pi$
$647$	−22.7291	−0.893573	−0.446787	+	0.894641i	$0.647432\pi$
$648$	0	0
$649$	0.186436	0.00731824
$650$	60.6548	2.37908
$651$	0	0
$652$	8.22544	0.322133
$653$	46.4478	1.81764	0.908821	−	0.417186i	$-0.136983\pi$
$653$	46.4478	1.81764	0.908821	+	0.417186i	$0.136983\pi$
$654$	0	0
$655$	49.4707	1.93298
$656$	0.966257	0.0377260
$657$	0	0
$658$	−22.4912	−0.876799
$659$	21.3683	0.832389	0.416195	−	0.909276i	$-0.363364\pi$
$659$	21.3683	0.832389	0.416195	+	0.909276i	$0.363364\pi$
$660$	0	0
$661$	5.53699	0.215364	0.107682	−	0.994185i	$-0.465657\pi$
$661$	5.53699	0.215364	0.107682	+	0.994185i	$0.465657\pi$
$662$	28.2844	1.09930
$663$	0	0
$664$	−31.5201	−1.22322
$665$	−26.1270	−1.01316
$666$	0	0
$667$	−62.7863	−2.43109
$668$	27.1610	1.05089
$669$	0	0
$670$	−23.2282	−0.897385
$671$	−4.14381	−0.159970
$672$	0	0
$673$	−26.0696	−1.00491	−0.502454	−	0.864604i	$-0.667569\pi$
$673$	−26.0696	−1.00491	−0.502454	+	0.864604i	$0.667569\pi$
$674$	−13.8921	−0.535102
$675$	0	0
$676$	10.8050	0.415576
$677$	−24.6981	−0.949225	−0.474612	−	0.880195i	$-0.657412\pi$
$677$	−24.6981	−0.949225	−0.474612	+	0.880195i	$0.657412\pi$
$678$	0	0
$679$	3.05428	0.117213
$680$	−67.6090	−2.59269
$681$	0	0
$682$	0.0113548	0.000434796 0
$683$	−6.19227	−0.236941	−0.118470	−	0.992958i	$-0.537799\pi$
$683$	−6.19227	−0.236941	−0.118470	+	0.992958i	$0.537799\pi$
$684$	0	0
$685$	52.5461	2.00768
$686$	27.7780	1.06057
$687$	0	0
$688$	2.07297	0.0790311
$689$	−15.2099	−0.579452
$690$	0	0
$691$	13.8404	0.526512	0.263256	−	0.964726i	$-0.415204\pi$
$691$	13.8404	0.526512	0.263256	+	0.964726i	$0.415204\pi$
$692$	40.9670	1.55733
$693$	0	0
$694$	−57.7021	−2.19034
$695$	−19.6564	−0.745610
$696$	0	0
$697$	31.9956	1.21192
$698$	0.790374	0.0299161
$699$	0	0
$700$	19.4595	0.735501
$701$	−25.4505	−0.961253	−0.480626	−	0.876925i	$-0.659591\pi$
$701$	−25.4505	−0.961253	−0.480626	+	0.876925i	$0.659591\pi$
$702$	0	0
$703$	47.6459	1.79700
$704$	13.0068	0.490211
$705$	0	0
$706$	25.8112	0.971416
$707$	−15.2426	−0.573258
$708$	0	0
$709$	−48.6637	−1.82760	−0.913801	−	0.406162i	$-0.866867\pi$
$709$	−48.6637	−1.82760	−0.913801	+	0.406162i	$0.866867\pi$
$710$	75.5293	2.83456
$711$	0	0
$712$	−28.8023	−1.07941
$713$	0.0331931	0.00124309
$714$	0	0
$715$	−13.7766	−0.515216
$716$	−21.6459	−0.808946
$717$	0	0
$718$	78.9126	2.94499
$719$	−8.96243	−0.334242	−0.167121	−	0.985936i	$-0.553447\pi$
$719$	−8.96243	−0.334242	−0.167121	+	0.985936i	$0.553447\pi$
$720$	0	0
$721$	−3.86909	−0.144093
$722$	−112.470	−4.18571
$723$	0	0
$724$	19.9762	0.742409
$725$	−62.0487	−2.30443
$726$	0	0
$727$	5.08497	0.188591	0.0942955	−	0.995544i	$-0.469940\pi$
$727$	5.08497	0.188591	0.0942955	+	0.995544i	$0.469940\pi$
$728$	10.1508	0.376215
$729$	0	0
$730$	6.59211	0.243985
$731$	68.6419	2.53881
$732$	0	0
$733$	−35.7985	−1.32225	−0.661124	−	0.750276i	$-0.729920\pi$
$733$	−35.7985	−1.32225	−0.661124	+	0.750276i	$0.729920\pi$
$734$	−44.6969	−1.64979
$735$	0	0
$736$	39.3215	1.44941
$737$	2.99761	0.110419
$738$	0	0
$739$	0.827783	0.0304505	0.0152252	−	0.999884i	$-0.495153\pi$
$739$	0.827783	0.0304505	0.0152252	+	0.999884i	$0.495153\pi$
$740$	−62.4576	−2.29599
$741$	0	0
$742$	−7.94310	−0.291600
$743$	31.8606	1.16885	0.584427	−	0.811446i	$-0.301319\pi$
$743$	31.8606	1.16885	0.584427	+	0.811446i	$0.301319\pi$
$744$	0	0
$745$	5.49277	0.201239
$746$	−55.9099	−2.04701
$747$	0	0
$748$	23.4405	0.857071
$749$	−10.3152	−0.376911
$750$	0	0
$751$	−17.4430	−0.636502	−0.318251	−	0.948006i	$-0.603096\pi$
$751$	−17.4430	−0.636502	−0.318251	+	0.948006i	$0.603096\pi$
$752$	−2.36368	−0.0861945
$753$	0	0
$754$	−86.9572	−3.16679
$755$	−35.8281	−1.30392
$756$	0	0
$757$	−17.3705	−0.631341	−0.315671	−	0.948869i	$-0.602229\pi$
$757$	−17.3705	−0.631341	−0.315671	+	0.948869i	$0.602229\pi$
$758$	4.11149	0.149336
$759$	0	0
$760$	75.9890	2.75641
$761$	8.22425	0.298129	0.149064	−	0.988827i	$-0.452374\pi$
$761$	8.22425	0.298129	0.149064	+	0.988827i	$0.452374\pi$
$762$	0	0
$763$	13.7500	0.497783
$764$	14.7457	0.533479
$765$	0	0
$766$	−77.2009	−2.78938
$767$	0.754813	0.0272547
$768$	0	0
$769$	33.0600	1.19217	0.596086	−	0.802920i	$-0.296722\pi$
$769$	33.0600	1.19217	0.596086	+	0.802920i	$0.296722\pi$
$770$	−7.19458	−0.259275
$771$	0	0
$772$	32.7259	1.17783
$773$	−30.0900	−1.08226	−0.541131	−	0.840939i	$-0.682004\pi$
$773$	−30.0900	−1.08226	−0.541131	+	0.840939i	$0.682004\pi$
$774$	0	0
$775$	0.0328032	0.00117832
$776$	−8.88322	−0.318889
$777$	0	0
$778$	−33.6188	−1.20529
$779$	−35.9614	−1.28845
$780$	0	0
$781$	−9.74709	−0.348778
$782$	111.541	3.98869
$783$	0	0
$784$	1.36387	0.0487096
$785$	56.1575	2.00435
$786$	0	0
$787$	−0.798410	−0.0284603	−0.0142301	−	0.999899i	$-0.504530\pi$
$787$	−0.798410	−0.0284603	−0.0142301	+	0.999899i	$0.504530\pi$
$788$	79.4816	2.83142
$789$	0	0
$790$	60.0814	2.13760
$791$	0.501153	0.0178189
$792$	0	0
$793$	−16.7768	−0.595763
$794$	65.7268	2.33256
$795$	0	0
$796$	6.03030	0.213738
$797$	−40.0853	−1.41989	−0.709947	−	0.704255i	$-0.751281\pi$
$797$	−40.0853	−1.41989	−0.709947	+	0.704255i	$0.751281\pi$
$798$	0	0
$799$	−78.2683	−2.76893
$800$	38.8596	1.37389
$801$	0	0
$802$	−83.6102	−2.95238
$803$	−0.850716	−0.0300211
$804$	0	0
$805$	−21.0317	−0.741271
$806$	0.0459715	0.00161928
$807$	0	0
$808$	44.3324	1.55961
$809$	−18.9846	−0.667462	−0.333731	−	0.942668i	$-0.608308\pi$
$809$	−18.9846	−0.667462	−0.333731	+	0.942668i	$0.608308\pi$
$810$	0	0
$811$	1.75717	0.0617026	0.0308513	−	0.999524i	$-0.490178\pi$
$811$	1.75717	0.0617026	0.0308513	+	0.999524i	$0.490178\pi$
$812$	−27.8980	−0.979026
$813$	0	0
$814$	13.1202	0.459864
$815$	−8.78557	−0.307745
$816$	0	0
$817$	−77.1500	−2.69914
$818$	−17.1427	−0.599382
$819$	0	0
$820$	47.1407	1.64622
$821$	46.3106	1.61625	0.808125	−	0.589011i	$-0.200482\pi$
$821$	46.3106	1.61625	0.808125	+	0.589011i	$0.200482\pi$
$822$	0	0
$823$	15.4133	0.537275	0.268637	−	0.963241i	$-0.413427\pi$
$823$	15.4133	0.537275	0.268637	+	0.963241i	$0.413427\pi$
$824$	11.2531	0.392019
$825$	0	0
$826$	0.394187	0.0137155
$827$	32.2039	1.11984	0.559920	−	0.828547i	$-0.310832\pi$
$827$	32.2039	1.11984	0.559920	+	0.828547i	$0.310832\pi$
$828$	0	0
$829$	−31.3035	−1.08722	−0.543609	−	0.839339i	$-0.682942\pi$
$829$	−31.3035	−1.08722	−0.543609	+	0.839339i	$0.682942\pi$
$830$	90.4487	3.13952
$831$	0	0
$832$	52.6599	1.82565
$833$	45.1617	1.56476
$834$	0	0
$835$	−29.0106	−1.00395
$836$	−26.3459	−0.911194
$837$	0	0
$838$	−30.9803	−1.07020
$839$	−16.0158	−0.552926	−0.276463	−	0.961025i	$-0.589162\pi$
$839$	−16.0158	−0.552926	−0.276463	+	0.961025i	$0.589162\pi$
$840$	0	0
$841$	59.9555	2.06743
$842$	47.6395	1.64177
$843$	0	0
$844$	−61.9384	−2.13201
$845$	−11.5408	−0.397014
$846$	0	0
$847$	0.928464	0.0319024
$848$	−0.834767	−0.0286660
$849$	0	0
$850$	110.230	3.78087
$851$	38.3540	1.31476
$852$	0	0
$853$	−15.6412	−0.535545	−0.267773	−	0.963482i	$-0.586288\pi$
$853$	−15.6412	−0.535545	−0.267773	+	0.963482i	$0.586288\pi$
$854$	−8.76140	−0.299809
$855$	0	0
$856$	30.0013	1.02542
$857$	19.9651	0.681997	0.340998	−	0.940064i	$-0.389235\pi$
$857$	19.9651	0.681997	0.340998	+	0.940064i	$0.389235\pi$
$858$	0	0
$859$	−3.29593	−0.112456	−0.0562279	−	0.998418i	$-0.517907\pi$
$859$	−3.29593	−0.112456	−0.0562279	+	0.998418i	$0.517907\pi$
$860$	101.134	3.44863
$861$	0	0
$862$	−37.6383	−1.28197
$863$	−20.1016	−0.684267	−0.342134	−	0.939651i	$-0.611150\pi$
$863$	−20.1016	−0.684267	−0.342134	+	0.939651i	$0.611150\pi$
$864$	0	0
$865$	−43.7568	−1.48778
$866$	10.9742	0.372917
$867$	0	0
$868$	0.0147488	0.000500605 0
$869$	−7.75353	−0.263021
$870$	0	0
$871$	12.1363	0.411223
$872$	−39.9911	−1.35427
$873$	0	0
$874$	−125.366	−4.24057
$875$	−4.98796	−0.168624
$876$	0	0
$877$	17.7762	0.600260	0.300130	−	0.953898i	$-0.402970\pi$
$877$	17.7762	0.600260	0.300130	+	0.953898i	$0.402970\pi$
$878$	12.0789	0.407644
$879$	0	0
$880$	−0.756103	−0.0254882
$881$	−21.3918	−0.720707	−0.360353	−	0.932816i	$-0.617344\pi$
$881$	−21.3918	−0.720707	−0.360353	+	0.932816i	$0.617344\pi$
$882$	0	0
$883$	−38.4448	−1.29377	−0.646885	−	0.762587i	$-0.723929\pi$
$883$	−38.4448	−1.29377	−0.646885	+	0.762587i	$0.723929\pi$
$884$	94.9026	3.19192
$885$	0	0
$886$	54.9950	1.84759
$887$	14.0847	0.472918	0.236459	−	0.971641i	$-0.424013\pi$
$887$	14.0847	0.472918	0.236459	+	0.971641i	$0.424013\pi$
$888$	0	0
$889$	2.07345	0.0695412
$890$	82.6497	2.77042
$891$	0	0
$892$	11.8250	0.395930
$893$	87.9695	2.94379
$894$	0	0
$895$	23.1199	0.772814
$896$	16.5322	0.552301
$897$	0	0
$898$	33.1300	1.10556
$899$	−0.0470279	−0.00156847
$900$	0	0
$901$	−27.6416	−0.920874
$902$	−9.90266	−0.329723
$903$	0	0
$904$	−1.45758	−0.0484783
$905$	−21.3365	−0.709249
$906$	0	0
$907$	−50.4469	−1.67506	−0.837531	−	0.546390i	$-0.816002\pi$
$907$	−50.4469	−1.67506	−0.837531	+	0.546390i	$0.816002\pi$
$908$	66.9286	2.22110
$909$	0	0
$910$	−29.1284	−0.965596
$911$	−56.8592	−1.88383	−0.941914	−	0.335854i	$-0.890975\pi$
$911$	−56.8592	−1.88383	−0.941914	+	0.335854i	$0.890975\pi$
$912$	0	0
$913$	−11.6724	−0.386302
$914$	−13.4735	−0.445663
$915$	0	0
$916$	−14.3575	−0.474386
$917$	−13.4984	−0.445756
$918$	0	0
$919$	−24.7469	−0.816324	−0.408162	−	0.912910i	$-0.633830\pi$
$919$	−24.7469	−0.816324	−0.408162	+	0.912910i	$0.633830\pi$
$920$	61.1697	2.01671
$921$	0	0
$922$	79.3752	2.61408
$923$	−39.4626	−1.29893
$924$	0	0
$925$	37.9035	1.24626
$926$	−5.02770	−0.165220
$927$	0	0
$928$	−55.7106	−1.82879
$929$	48.6965	1.59768	0.798840	−	0.601544i	$-0.205447\pi$
$929$	48.6965	1.59768	0.798840	+	0.601544i	$0.205447\pi$
$930$	0	0
$931$	−50.7594	−1.66357
$932$	39.1357	1.28193
$933$	0	0
$934$	−72.1818	−2.36186
$935$	−25.0368	−0.818790
$936$	0	0
$937$	22.9678	0.750324	0.375162	−	0.926959i	$-0.377587\pi$
$937$	22.9678	0.750324	0.375162	+	0.926959i	$0.377587\pi$
$938$	6.33796	0.206942
$939$	0	0
$940$	−115.317	−3.76121
$941$	−39.3517	−1.28283	−0.641414	−	0.767195i	$-0.721652\pi$
$941$	−39.3517	−1.28283	−0.641414	+	0.767195i	$0.721652\pi$
$942$	0	0
$943$	−28.9482	−0.942683
$944$	0.0414265	0.00134832
$945$	0	0
$946$	−21.2448	−0.690726
$947$	25.3627	0.824178	0.412089	−	0.911144i	$-0.364799\pi$
$947$	25.3627	0.824178	0.412089	+	0.911144i	$0.364799\pi$
$948$	0	0
$949$	−3.44425	−0.111805
$950$	−123.893	−4.01963
$951$	0	0
$952$	18.4475	0.597888
$953$	−17.5809	−0.569501	−0.284751	−	0.958602i	$-0.591911\pi$
$953$	−17.5809	−0.569501	−0.284751	+	0.958602i	$0.591911\pi$
$954$	0	0
$955$	−15.7498	−0.509652
$956$	−18.2378	−0.589853
$957$	0	0
$958$	25.4338	0.821727
$959$	−14.3375	−0.462983
$960$	0	0
$961$	−31.0000	−0.999999
$962$	53.1192	1.71263
$963$	0	0
$964$	−58.7561	−1.89241
$965$	−34.9545	−1.12522
$966$	0	0
$967$	21.0183	0.675904	0.337952	−	0.941163i	$-0.390266\pi$
$967$	21.0183	0.675904	0.337952	+	0.941163i	$0.390266\pi$
$968$	−2.70039	−0.0867938
$969$	0	0
$970$	25.4909	0.818462
$971$	14.6226	0.469263	0.234631	−	0.972084i	$-0.424612\pi$
$971$	14.6226	0.469263	0.234631	+	0.972084i	$0.424612\pi$
$972$	0	0
$973$	5.36337	0.171942
$974$	−10.9189	−0.349866
$975$	0	0
$976$	−0.920765	−0.0294730
$977$	−24.4209	−0.781295	−0.390648	−	0.920540i	$-0.627749\pi$
$977$	−24.4209	−0.781295	−0.390648	+	0.920540i	$0.627749\pi$
$978$	0	0
$979$	−10.6660	−0.340886
$980$	66.5390	2.12551
$981$	0	0
$982$	24.7800	0.790763
$983$	−18.1000	−0.577300	−0.288650	−	0.957435i	$-0.593206\pi$
$983$	−18.1000	−0.577300	−0.288650	+	0.957435i	$0.593206\pi$
$984$	0	0
$985$	−84.8941	−2.70495
$986$	−158.031	−5.03272
$987$	0	0
$988$	−106.666	−3.39348
$989$	−62.1042	−1.97480
$990$	0	0
$991$	9.75337	0.309826	0.154913	−	0.987928i	$-0.450490\pi$
$991$	9.75337	0.309826	0.154913	+	0.987928i	$0.450490\pi$
$992$	0.0294524	0.000935115 0
$993$	0	0
$994$	−20.6086	−0.653665
$995$	−6.44095	−0.204192
$996$	0	0
$997$	−5.39008	−0.170706	−0.0853528	−	0.996351i	$-0.527202\pi$
$997$	−5.39008	−0.170706	−0.0853528	+	0.996351i	$0.527202\pi$
$998$	−15.9343	−0.504392
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.i.1.7	✓	48
3.2	odd	2		8019.2.a.j.1.42	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8019.2.a.i.1.7	✓	48	1.1	even	1	trivial
8019.2.a.j.1.42	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)

\(f(q)\)	\(=\)	\(q-2.27724 q^{2} +3.18582 q^{4} -3.40276 q^{5} +0.928464 q^{7} -2.70039 q^{8} +O(q^{10})\)	\(q-2.27724 q^{2} +3.18582 q^{4} -3.40276 q^{5} +0.928464 q^{7} -2.70039 q^{8} +7.74890 q^{10} -1.00000 q^{11} -4.04865 q^{13} -2.11433 q^{14} -0.222203 q^{16} -7.35778 q^{17} +8.26976 q^{19} -10.8406 q^{20} +2.27724 q^{22} +6.65700 q^{23} +6.57880 q^{25} +9.21975 q^{26} +2.95792 q^{28} -9.43162 q^{29} +0.00498620 q^{31} +5.90679 q^{32} +16.7554 q^{34} -3.15934 q^{35} +5.76146 q^{37} -18.8322 q^{38} +9.18879 q^{40} -4.34854 q^{41} -9.32917 q^{43} -3.18582 q^{44} -15.1596 q^{46} +10.6375 q^{47} -6.13795 q^{49} -14.9815 q^{50} -12.8983 q^{52} +3.75678 q^{53} +3.40276 q^{55} -2.50721 q^{56} +21.4781 q^{58} -0.186436 q^{59} +4.14381 q^{61} -0.0113548 q^{62} -13.0068 q^{64} +13.7766 q^{65} -2.99761 q^{67} -23.4405 q^{68} +7.19458 q^{70} +9.74709 q^{71} +0.850716 q^{73} -13.1202 q^{74} +26.3459 q^{76} -0.928464 q^{77} +7.75353 q^{79} +0.756103 q^{80} +9.90266 q^{82} +11.6724 q^{83} +25.0368 q^{85} +21.2448 q^{86} +2.70039 q^{88} +10.6660 q^{89} -3.75903 q^{91} +21.2080 q^{92} -24.2241 q^{94} -28.1400 q^{95} +3.28961 q^{97} +13.9776 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8}+O(q^{10}) \) 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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