LMFDB - Embedded newform 8019.2.a.i.1.14

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			1.14
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)

$f(q)$	$=$	$q-1.56489 q^{2} +0.448896 q^{4} -4.28242 q^{5} +1.60202 q^{7} +2.42731 q^{8} +O(q^{10})$	\(q-1.56489 q^{2} +0.448896 q^{4} -4.28242 q^{5} +1.60202 q^{7} +2.42731 q^{8} +6.70153 q^{10} -1.00000 q^{11} -5.79297 q^{13} -2.50699 q^{14} -4.69628 q^{16} -0.0889197 q^{17} +0.938280 q^{19} -1.92236 q^{20} +1.56489 q^{22} -6.84965 q^{23} +13.3391 q^{25} +9.06540 q^{26} +0.719140 q^{28} +4.07127 q^{29} +2.80693 q^{31} +2.49456 q^{32} +0.139150 q^{34} -6.86052 q^{35} +3.12135 q^{37} -1.46831 q^{38} -10.3948 q^{40} +5.21164 q^{41} +4.74343 q^{43} -0.448896 q^{44} +10.7190 q^{46} -9.91718 q^{47} -4.43353 q^{49} -20.8743 q^{50} -2.60044 q^{52} -7.48441 q^{53} +4.28242 q^{55} +3.88861 q^{56} -6.37111 q^{58} -1.22283 q^{59} -7.82819 q^{61} -4.39255 q^{62} +5.48884 q^{64} +24.8079 q^{65} +14.5872 q^{67} -0.0399157 q^{68} +10.7360 q^{70} +1.51922 q^{71} +10.0934 q^{73} -4.88458 q^{74} +0.421190 q^{76} -1.60202 q^{77} +7.28331 q^{79} +20.1114 q^{80} -8.15567 q^{82} -0.467918 q^{83} +0.380791 q^{85} -7.42297 q^{86} -2.42731 q^{88} -1.52262 q^{89} -9.28046 q^{91} -3.07478 q^{92} +15.5193 q^{94} -4.01811 q^{95} -1.69923 q^{97} +6.93801 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.56489	−1.10655	−0.553274	−	0.832999i	$-0.686622\pi$
$2$	−1.56489	−1.10655	−0.553274	+	0.832999i	$0.686622\pi$
$3$	0	0
$3$	0	0
$4$	0.448896	0.224448
$5$	−4.28242	−1.91515	−0.957577	−	0.288176i	$-0.906951\pi$
$5$	−4.28242	−1.91515	−0.957577	+	0.288176i	$0.906951\pi$
$6$	0	0
$7$	1.60202	0.605507	0.302753	−	0.953069i	$-0.402094\pi$
$7$	1.60202	0.605507	0.302753	+	0.953069i	$0.402094\pi$
$8$	2.42731	0.858185
$9$	0	0
$10$	6.70153	2.11921
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−5.79297	−1.60668	−0.803341	−	0.595519i	$-0.796946\pi$
$13$	−5.79297	−1.60668	−0.803341	+	0.595519i	$0.796946\pi$
$14$	−2.50699	−0.670022
$15$	0	0
$16$	−4.69628	−1.17407
$17$	−0.0889197	−0.0215662	−0.0107831	−	0.999942i	$-0.503432\pi$
$17$	−0.0889197	−0.0215662	−0.0107831	+	0.999942i	$0.503432\pi$
$18$	0	0
$19$	0.938280	0.215256	0.107628	−	0.994191i	$-0.465674\pi$
$19$	0.938280	0.215256	0.107628	+	0.994191i	$0.465674\pi$
$20$	−1.92236	−0.429853
$21$	0	0
$22$	1.56489	0.333637
$23$	−6.84965	−1.42825	−0.714126	−	0.700018i	$-0.753175\pi$
$23$	−6.84965	−1.42825	−0.714126	+	0.700018i	$0.753175\pi$
$24$	0	0
$25$	13.3391	2.66782
$26$	9.06540	1.77787
$27$	0	0
$28$	0.719140	0.135905
$29$	4.07127	0.756016	0.378008	−	0.925802i	$-0.376609\pi$
$29$	4.07127	0.756016	0.378008	+	0.925802i	$0.376609\pi$
$30$	0	0
$31$	2.80693	0.504139	0.252070	−	0.967709i	$-0.418889\pi$
$31$	2.80693	0.504139	0.252070	+	0.967709i	$0.418889\pi$
$32$	2.49456	0.440980
$33$	0	0
$34$	0.139150	0.0238640
$35$	−6.86052	−1.15964
$36$	0	0
$37$	3.12135	0.513146	0.256573	−	0.966525i	$-0.417407\pi$
$37$	3.12135	0.513146	0.256573	+	0.966525i	$0.417407\pi$
$38$	−1.46831	−0.238191
$39$	0	0
$40$	−10.3948	−1.64356
$41$	5.21164	0.813921	0.406961	−	0.913446i	$-0.366589\pi$
$41$	5.21164	0.813921	0.406961	+	0.913446i	$0.366589\pi$
$42$	0	0
$43$	4.74343	0.723366	0.361683	−	0.932301i	$-0.382202\pi$
$43$	4.74343	0.723366	0.361683	+	0.932301i	$0.382202\pi$
$44$	−0.448896	−0.0676736
$45$	0	0
$46$	10.7190	1.58043
$47$	−9.91718	−1.44657	−0.723284	−	0.690550i	$-0.757368\pi$
$47$	−9.91718	−1.44657	−0.723284	+	0.690550i	$0.757368\pi$
$48$	0	0
$49$	−4.43353	−0.633361
$50$	−20.8743	−2.95207
$51$	0	0
$52$	−2.60044	−0.360616
$53$	−7.48441	−1.02806	−0.514032	−	0.857771i	$-0.671849\pi$
$53$	−7.48441	−1.02806	−0.514032	+	0.857771i	$0.671849\pi$
$54$	0	0
$55$	4.28242	0.577441
$56$	3.88861	0.519637
$57$	0	0
$58$	−6.37111	−0.836567
$59$	−1.22283	−0.159199	−0.0795997	−	0.996827i	$-0.525364\pi$
$59$	−1.22283	−0.159199	−0.0795997	+	0.996827i	$0.525364\pi$
$60$	0	0
$61$	−7.82819	−1.00230	−0.501148	−	0.865361i	$-0.667089\pi$
$61$	−7.82819	−1.00230	−0.501148	+	0.865361i	$0.667089\pi$
$62$	−4.39255	−0.557854
$63$	0	0
$64$	5.48884	0.686105
$65$	24.8079	3.07704
$66$	0	0
$67$	14.5872	1.78211	0.891057	−	0.453891i	$-0.149964\pi$
$67$	14.5872	1.78211	0.891057	+	0.453891i	$0.149964\pi$
$68$	−0.0399157	−0.00484049
$69$	0	0
$70$	10.7360	1.28320
$71$	1.51922	0.180298	0.0901489	−	0.995928i	$-0.471266\pi$
$71$	1.51922	0.180298	0.0901489	+	0.995928i	$0.471266\pi$
$72$	0	0
$73$	10.0934	1.18134	0.590670	−	0.806913i	$-0.298863\pi$
$73$	10.0934	1.18134	0.590670	+	0.806913i	$0.298863\pi$
$74$	−4.88458	−0.567821
$75$	0	0
$76$	0.421190	0.0483138
$77$	−1.60202	−0.182567
$78$	0	0
$79$	7.28331	0.819437	0.409718	−	0.912212i	$-0.365627\pi$
$79$	7.28331	0.819437	0.409718	+	0.912212i	$0.365627\pi$
$80$	20.1114	2.24853
$81$	0	0
$82$	−8.15567	−0.900643
$83$	−0.467918	−0.0513606	−0.0256803	−	0.999670i	$-0.508175\pi$
$83$	−0.467918	−0.0513606	−0.0256803	+	0.999670i	$0.508175\pi$
$84$	0	0
$85$	0.380791	0.0413026
$86$	−7.42297	−0.800439
$87$	0	0
$88$	−2.42731	−0.258753
$89$	−1.52262	−0.161397	−0.0806986	−	0.996739i	$-0.525715\pi$
$89$	−1.52262	−0.161397	−0.0806986	+	0.996739i	$0.525715\pi$
$90$	0	0
$91$	−9.28046	−0.972857
$92$	−3.07478	−0.320568
$93$	0	0
$94$	15.5193	1.60070
$95$	−4.01811	−0.412249
$96$	0	0
$97$	−1.69923	−0.172531	−0.0862654	−	0.996272i	$-0.527493\pi$
$97$	−1.69923	−0.172531	−0.0862654	+	0.996272i	$0.527493\pi$
$98$	6.93801	0.700845
$99$	0	0
$100$	5.98786	0.598786
$101$	−15.4209	−1.53444	−0.767221	−	0.641383i	$-0.778361\pi$
$101$	−15.4209	−1.53444	−0.767221	+	0.641383i	$0.778361\pi$
$102$	0	0
$103$	17.1260	1.68748	0.843738	−	0.536755i	$-0.180350\pi$
$103$	17.1260	1.68748	0.843738	+	0.536755i	$0.180350\pi$
$104$	−14.0614	−1.37883
$105$	0	0
$106$	11.7123	1.13760
$107$	−4.37075	−0.422536	−0.211268	−	0.977428i	$-0.567759\pi$
$107$	−4.37075	−0.422536	−0.211268	+	0.977428i	$0.567759\pi$
$108$	0	0
$109$	8.04853	0.770909	0.385455	−	0.922727i	$-0.374045\pi$
$109$	8.04853	0.770909	0.385455	+	0.922727i	$0.374045\pi$
$110$	−6.70153	−0.638966
$111$	0	0
$112$	−7.52354	−0.710908
$113$	0.590093	0.0555113	0.0277556	−	0.999615i	$-0.491164\pi$
$113$	0.590093	0.0555113	0.0277556	+	0.999615i	$0.491164\pi$
$114$	0	0
$115$	29.3331	2.73532
$116$	1.82758	0.169686
$117$	0	0
$118$	1.91361	0.176162
$119$	−0.142451	−0.0130585
$120$	0	0
$121$	1.00000	0.0909091
$122$	12.2503	1.10909
$123$	0	0
$124$	1.26002	0.113153
$125$	−35.7115	−3.19413
$126$	0	0
$127$	0.590920	0.0524357	0.0262178	−	0.999656i	$-0.491654\pi$
$127$	0.590920	0.0524357	0.0262178	+	0.999656i	$0.491654\pi$
$128$	−13.5786	−1.20019
$129$	0	0
$130$	−38.8218	−3.40490
$131$	19.3141	1.68748	0.843740	−	0.536752i	$-0.180349\pi$
$131$	19.3141	1.68748	0.843740	+	0.536752i	$0.180349\pi$
$132$	0	0
$133$	1.50314	0.130339
$134$	−22.8275	−1.97200
$135$	0	0
$136$	−0.215836	−0.0185078
$137$	18.7916	1.60547	0.802737	−	0.596334i	$-0.203377\pi$
$137$	18.7916	1.60547	0.802737	+	0.596334i	$0.203377\pi$
$138$	0	0
$139$	−3.23245	−0.274173	−0.137086	−	0.990559i	$-0.543774\pi$
$139$	−3.23245	−0.274173	−0.137086	+	0.990559i	$0.543774\pi$
$140$	−3.07966	−0.260279
$141$	0	0
$142$	−2.37741	−0.199508
$143$	5.79297	0.484433
$144$	0	0
$145$	−17.4349	−1.44789
$146$	−15.7951	−1.30721
$147$	0	0
$148$	1.40116	0.115175
$149$	12.2175	1.00089	0.500446	−	0.865768i	$-0.333169\pi$
$149$	12.2175	1.00089	0.500446	+	0.865768i	$0.333169\pi$
$150$	0	0
$151$	−18.7849	−1.52869	−0.764345	−	0.644807i	$-0.776938\pi$
$151$	−18.7849	−1.52869	−0.764345	+	0.644807i	$0.776938\pi$
$152$	2.27750	0.184730
$153$	0	0
$154$	2.50699	0.202019
$155$	−12.0204	−0.965505
$156$	0	0
$157$	−7.60154	−0.606669	−0.303334	−	0.952884i	$-0.598100\pi$
$157$	−7.60154	−0.606669	−0.303334	+	0.952884i	$0.598100\pi$
$158$	−11.3976	−0.906746
$159$	0	0
$160$	−10.6828	−0.844546
$161$	−10.9733	−0.864816
$162$	0	0
$163$	14.5367	1.13860	0.569302	−	0.822129i	$-0.307214\pi$
$163$	14.5367	1.13860	0.569302	+	0.822129i	$0.307214\pi$
$164$	2.33948	0.182683
$165$	0	0
$166$	0.732242	0.0568330
$167$	−0.152707	−0.0118168	−0.00590841	−	0.999983i	$-0.501881\pi$
$167$	−0.152707	−0.0118168	−0.00590841	+	0.999983i	$0.501881\pi$
$168$	0	0
$169$	20.5586	1.58143
$170$	−0.595898	−0.0457033
$171$	0	0
$172$	2.12931	0.162358
$173$	−4.52972	−0.344388	−0.172194	−	0.985063i	$-0.555086\pi$
$173$	−4.52972	−0.344388	−0.172194	+	0.985063i	$0.555086\pi$
$174$	0	0
$175$	21.3695	1.61538
$176$	4.69628	0.353996
$177$	0	0
$178$	2.38274	0.178594
$179$	8.23727	0.615682	0.307841	−	0.951438i	$-0.400393\pi$
$179$	8.23727	0.615682	0.307841	+	0.951438i	$0.400393\pi$
$180$	0	0
$181$	−20.4108	−1.51713	−0.758563	−	0.651599i	$-0.774098\pi$
$181$	−20.4108	−1.51713	−0.758563	+	0.651599i	$0.774098\pi$
$182$	14.5230	1.07651
$183$	0	0
$184$	−16.6263	−1.22570
$185$	−13.3669	−0.982754
$186$	0	0
$187$	0.0889197	0.00650245
$188$	−4.45178	−0.324679
$189$	0	0
$190$	6.28791	0.456173
$191$	−22.4102	−1.62154	−0.810771	−	0.585364i	$-0.800952\pi$
$191$	−22.4102	−1.62154	−0.810771	+	0.585364i	$0.800952\pi$
$192$	0	0
$193$	18.2369	1.31272	0.656359	−	0.754449i	$-0.272096\pi$
$193$	18.2369	1.31272	0.656359	+	0.754449i	$0.272096\pi$
$194$	2.65912	0.190914
$195$	0	0
$196$	−1.99019	−0.142157
$197$	26.1713	1.86463	0.932313	−	0.361653i	$-0.117788\pi$
$197$	26.1713	1.86463	0.932313	+	0.361653i	$0.117788\pi$
$198$	0	0
$199$	−27.0157	−1.91509	−0.957547	−	0.288279i	$-0.906917\pi$
$199$	−27.0157	−1.91509	−0.957547	+	0.288279i	$0.906917\pi$
$200$	32.3782	2.28948
$201$	0	0
$202$	24.1322	1.69793
$203$	6.52226	0.457773
$204$	0	0
$205$	−22.3184	−1.55879
$206$	−26.8004	−1.86727
$207$	0	0
$208$	27.2055	1.88636
$209$	−0.938280	−0.0649022
$210$	0	0
$211$	0.364889	0.0251200	0.0125600	−	0.999921i	$-0.496002\pi$
$211$	0.364889	0.0251200	0.0125600	+	0.999921i	$0.496002\pi$
$212$	−3.35972	−0.230747
$213$	0	0
$214$	6.83976	0.467557
$215$	−20.3133	−1.38536
$216$	0	0
$217$	4.49676	0.305260
$218$	−12.5951	−0.853048
$219$	0	0
$220$	1.92236	0.129605
$221$	0.515110	0.0346500
$222$	0	0
$223$	12.5442	0.840021	0.420010	−	0.907519i	$-0.362026\pi$
$223$	12.5442	0.840021	0.420010	+	0.907519i	$0.362026\pi$
$224$	3.99634	0.267017
$225$	0	0
$226$	−0.923433	−0.0614259
$227$	15.6312	1.03748	0.518740	−	0.854932i	$-0.326401\pi$
$227$	15.6312	1.03748	0.518740	+	0.854932i	$0.326401\pi$
$228$	0	0
$229$	−17.4551	−1.15347	−0.576733	−	0.816933i	$-0.695673\pi$
$229$	−17.4551	−1.15347	−0.576733	+	0.816933i	$0.695673\pi$
$230$	−45.9032	−3.02676
$231$	0	0
$232$	9.88225	0.648801
$233$	21.4196	1.40324	0.701621	−	0.712550i	$-0.252460\pi$
$233$	21.4196	1.40324	0.701621	+	0.712550i	$0.252460\pi$
$234$	0	0
$235$	42.4695	2.77040
$236$	−0.548925	−0.0357320
$237$	0	0
$238$	0.222921	0.0144498
$239$	18.5491	1.19984	0.599921	−	0.800059i	$-0.295199\pi$
$239$	18.5491	1.19984	0.599921	+	0.800059i	$0.295199\pi$
$240$	0	0
$241$	25.9619	1.67235	0.836177	−	0.548459i	$-0.184785\pi$
$241$	25.9619	1.67235	0.836177	+	0.548459i	$0.184785\pi$
$242$	−1.56489	−0.100595
$243$	0	0
$244$	−3.51404	−0.224963
$245$	18.9862	1.21299
$246$	0	0
$247$	−5.43543	−0.345848
$248$	6.81330	0.432645
$249$	0	0
$250$	55.8847	3.53446
$251$	12.8626	0.811880	0.405940	−	0.913900i	$-0.366944\pi$
$251$	12.8626	0.811880	0.405940	+	0.913900i	$0.366944\pi$
$252$	0	0
$253$	6.84965	0.430634
$254$	−0.924728	−0.0580226
$255$	0	0
$256$	10.2714	0.641961
$257$	−22.4995	−1.40348	−0.701739	−	0.712434i	$-0.747593\pi$
$257$	−22.4995	−1.40348	−0.701739	+	0.712434i	$0.747593\pi$
$258$	0	0
$259$	5.00046	0.310714
$260$	11.1362	0.690636
$261$	0	0
$262$	−30.2245	−1.86728
$263$	28.0605	1.73028	0.865141	−	0.501529i	$-0.167229\pi$
$263$	28.0605	1.73028	0.865141	+	0.501529i	$0.167229\pi$
$264$	0	0
$265$	32.0514	1.96890
$266$	−2.35226	−0.144226
$267$	0	0
$268$	6.54815	0.399992
$269$	14.1337	0.861748	0.430874	−	0.902412i	$-0.358205\pi$
$269$	14.1337	0.861748	0.430874	+	0.902412i	$0.358205\pi$
$270$	0	0
$271$	−9.67374	−0.587638	−0.293819	−	0.955861i	$-0.594926\pi$
$271$	−9.67374	−0.587638	−0.293819	+	0.955861i	$0.594926\pi$
$272$	0.417592	0.0253202
$273$	0	0
$274$	−29.4068	−1.77653
$275$	−13.3391	−0.804378
$276$	0	0
$277$	−7.52774	−0.452298	−0.226149	−	0.974093i	$-0.572614\pi$
$277$	−7.52774	−0.452298	−0.226149	+	0.974093i	$0.572614\pi$
$278$	5.05844	0.303385
$279$	0	0
$280$	−16.6526	−0.995186
$281$	20.6750	1.23337	0.616683	−	0.787211i	$-0.288476\pi$
$281$	20.6750	1.23337	0.616683	+	0.787211i	$0.288476\pi$
$282$	0	0
$283$	13.3157	0.791536	0.395768	−	0.918351i	$-0.370479\pi$
$283$	13.3157	0.791536	0.395768	+	0.918351i	$0.370479\pi$
$284$	0.681970	0.0404675
$285$	0	0
$286$	−9.06540	−0.536048
$287$	8.34915	0.492835
$288$	0	0
$289$	−16.9921	−0.999535
$290$	27.2837	1.60216
$291$	0	0
$292$	4.53087	0.265149
$293$	−25.9060	−1.51344	−0.756722	−	0.653737i	$-0.773200\pi$
$293$	−25.9060	−1.51344	−0.756722	+	0.653737i	$0.773200\pi$
$294$	0	0
$295$	5.23668	0.304891
$296$	7.57649	0.440375
$297$	0	0
$298$	−19.1190	−1.10754
$299$	39.6799	2.29475
$300$	0	0
$301$	7.59907	0.438003
$302$	29.3963	1.69157
$303$	0	0
$304$	−4.40643	−0.252726
$305$	33.5236	1.91955
$306$	0	0
$307$	−4.69623	−0.268028	−0.134014	−	0.990979i	$-0.542787\pi$
$307$	−4.69623	−0.268028	−0.134014	+	0.990979i	$0.542787\pi$
$308$	−0.719140	−0.0409768
$309$	0	0
$310$	18.8107	1.06838
$311$	8.23109	0.466742	0.233371	−	0.972388i	$-0.425024\pi$
$311$	8.23109	0.466742	0.233371	+	0.972388i	$0.425024\pi$
$312$	0	0
$313$	1.13240	0.0640070	0.0320035	−	0.999488i	$-0.489811\pi$
$313$	1.13240	0.0640070	0.0320035	+	0.999488i	$0.489811\pi$
$314$	11.8956	0.671308
$315$	0	0
$316$	3.26945	0.183921
$317$	−30.7142	−1.72508	−0.862542	−	0.505986i	$-0.831129\pi$
$317$	−30.7142	−1.72508	−0.862542	+	0.505986i	$0.831129\pi$
$318$	0	0
$319$	−4.07127	−0.227947
$320$	−23.5055	−1.31400
$321$	0	0
$322$	17.1720	0.956960
$323$	−0.0834316	−0.00464226
$324$	0	0
$325$	−77.2730	−4.28634
$326$	−22.7484	−1.25992
$327$	0	0
$328$	12.6503	0.698495
$329$	−15.8875	−0.875907
$330$	0	0
$331$	−31.3844	−1.72504	−0.862521	−	0.506022i	$-0.831116\pi$
$331$	−31.3844	−1.72504	−0.862521	+	0.506022i	$0.831116\pi$
$332$	−0.210046	−0.0115278
$333$	0	0
$334$	0.238970	0.0130759
$335$	−62.4686	−3.41303
$336$	0	0
$337$	−24.0514	−1.31016	−0.655082	−	0.755558i	$-0.727366\pi$
$337$	−24.0514	−1.31016	−0.655082	+	0.755558i	$0.727366\pi$
$338$	−32.1720	−1.74992
$339$	0	0
$340$	0.170936	0.00927028
$341$	−2.80693	−0.152004
$342$	0	0
$343$	−18.3168	−0.989012
$344$	11.5138	0.620782
$345$	0	0
$346$	7.08853	0.381082
$347$	−27.8537	−1.49526	−0.747631	−	0.664114i	$-0.768809\pi$
$347$	−27.8537	−1.49526	−0.747631	+	0.664114i	$0.768809\pi$
$348$	0	0
$349$	18.6545	0.998555	0.499277	−	0.866442i	$-0.333599\pi$
$349$	18.6545	0.998555	0.499277	+	0.866442i	$0.333599\pi$
$350$	−33.4410	−1.78750
$351$	0	0
$352$	−2.49456	−0.132961
$353$	3.08557	0.164228	0.0821141	−	0.996623i	$-0.473833\pi$
$353$	3.08557	0.164228	0.0821141	+	0.996623i	$0.473833\pi$
$354$	0	0
$355$	−6.50592	−0.345298
$356$	−0.683497	−0.0362253
$357$	0	0
$358$	−12.8905	−0.681282
$359$	28.6270	1.51087	0.755437	−	0.655221i	$-0.227425\pi$
$359$	28.6270	1.51087	0.755437	+	0.655221i	$0.227425\pi$
$360$	0	0
$361$	−18.1196	−0.953665
$362$	31.9408	1.67877
$363$	0	0
$364$	−4.16596	−0.218356
$365$	−43.2240	−2.26245
$366$	0	0
$367$	−5.01886	−0.261982	−0.130991	−	0.991384i	$-0.541816\pi$
$367$	−5.01886	−0.261982	−0.130991	+	0.991384i	$0.541816\pi$
$368$	32.1679	1.67687
$369$	0	0
$370$	20.9178	1.08746
$371$	−11.9902	−0.622499
$372$	0	0
$373$	4.51255	0.233651	0.116826	−	0.993152i	$-0.462728\pi$
$373$	4.51255	0.233651	0.116826	+	0.993152i	$0.462728\pi$
$374$	−0.139150	−0.00719527
$375$	0	0
$376$	−24.0721	−1.24142
$377$	−23.5848	−1.21468
$378$	0	0
$379$	−22.7436	−1.16826	−0.584130	−	0.811660i	$-0.698564\pi$
$379$	−22.7436	−1.16826	−0.584130	+	0.811660i	$0.698564\pi$
$380$	−1.80371	−0.0925284
$381$	0	0
$382$	35.0695	1.79431
$383$	−31.1951	−1.59399	−0.796997	−	0.603983i	$-0.793580\pi$
$383$	−31.1951	−1.59399	−0.796997	+	0.603983i	$0.793580\pi$
$384$	0	0
$385$	6.86052	0.349644
$386$	−28.5388	−1.45258
$387$	0	0
$388$	−0.762778	−0.0387242
$389$	13.6554	0.692355	0.346177	−	0.938169i	$-0.387480\pi$
$389$	13.6554	0.692355	0.346177	+	0.938169i	$0.387480\pi$
$390$	0	0
$391$	0.609069	0.0308019
$392$	−10.7616	−0.543542
$393$	0	0
$394$	−40.9553	−2.06330
$395$	−31.1902	−1.56935
$396$	0	0
$397$	18.1911	0.912985	0.456493	−	0.889727i	$-0.349105\pi$
$397$	18.1911	0.912985	0.456493	+	0.889727i	$0.349105\pi$
$398$	42.2767	2.11914
$399$	0	0
$400$	−62.6442	−3.13221
$401$	−2.29261	−0.114487	−0.0572437	−	0.998360i	$-0.518231\pi$
$401$	−2.29261	−0.114487	−0.0572437	+	0.998360i	$0.518231\pi$
$402$	0	0
$403$	−16.2605	−0.809991
$404$	−6.92240	−0.344402
$405$	0	0
$406$	−10.2066	−0.506547
$407$	−3.12135	−0.154719
$408$	0	0
$409$	9.47079	0.468301	0.234150	−	0.972200i	$-0.424769\pi$
$409$	9.47079	0.468301	0.234150	+	0.972200i	$0.424769\pi$
$410$	34.9260	1.72487
$411$	0	0
$412$	7.68780	0.378751
$413$	−1.95900	−0.0963963
$414$	0	0
$415$	2.00382	0.0983636
$416$	−14.4509	−0.708515
$417$	0	0
$418$	1.46831	0.0718174
$419$	−6.59770	−0.322319	−0.161159	−	0.986928i	$-0.551523\pi$
$419$	−6.59770	−0.322319	−0.161159	+	0.986928i	$0.551523\pi$
$420$	0	0
$421$	7.67045	0.373835	0.186917	−	0.982376i	$-0.440150\pi$
$421$	7.67045	0.373835	0.186917	+	0.982376i	$0.440150\pi$
$422$	−0.571013	−0.0277965
$423$	0	0
$424$	−18.1670	−0.882269
$425$	−1.18611	−0.0575347
$426$	0	0
$427$	−12.5409	−0.606897
$428$	−1.96201	−0.0948374
$429$	0	0
$430$	31.7883	1.53297
$431$	26.2477	1.26431	0.632153	−	0.774844i	$-0.282171\pi$
$431$	26.2477	1.26431	0.632153	+	0.774844i	$0.282171\pi$
$432$	0	0
$433$	23.3734	1.12326	0.561628	−	0.827390i	$-0.310175\pi$
$433$	23.3734	1.12326	0.561628	+	0.827390i	$0.310175\pi$
$434$	−7.03695	−0.337784
$435$	0	0
$436$	3.61295	0.173029
$437$	−6.42689	−0.307440
$438$	0	0
$439$	−7.20997	−0.344113	−0.172057	−	0.985087i	$-0.555041\pi$
$439$	−7.20997	−0.344113	−0.172057	+	0.985087i	$0.555041\pi$
$440$	10.3948	0.495551
$441$	0	0
$442$	−0.806092	−0.0383419
$443$	−5.28989	−0.251330	−0.125665	−	0.992073i	$-0.540106\pi$
$443$	−5.28989	−0.251330	−0.125665	+	0.992073i	$0.540106\pi$
$444$	0	0
$445$	6.52049	0.309101
$446$	−19.6303	−0.929523
$447$	0	0
$448$	8.79324	0.415441
$449$	−18.3905	−0.867902	−0.433951	−	0.900936i	$-0.642881\pi$
$449$	−18.3905	−0.867902	−0.433951	+	0.900936i	$0.642881\pi$
$450$	0	0
$451$	−5.21164	−0.245406
$452$	0.264890	0.0124594
$453$	0	0
$454$	−24.4612	−1.14802
$455$	39.7428	1.86317
$456$	0	0
$457$	−0.0779059	−0.00364429	−0.00182214	−	0.999998i	$-0.500580\pi$
$457$	−0.0779059	−0.00364429	−0.00182214	+	0.999998i	$0.500580\pi$
$458$	27.3154	1.27637
$459$	0	0
$460$	13.1675	0.613937
$461$	−32.5046	−1.51389	−0.756945	−	0.653479i	$-0.773309\pi$
$461$	−32.5046	−1.51389	−0.756945	+	0.653479i	$0.773309\pi$
$462$	0	0
$463$	−5.37727	−0.249903	−0.124952	−	0.992163i	$-0.539878\pi$
$463$	−5.37727	−0.249903	−0.124952	+	0.992163i	$0.539878\pi$
$464$	−19.1198	−0.887616
$465$	0	0
$466$	−33.5194	−1.55276
$467$	−1.42114	−0.0657623	−0.0328812	−	0.999459i	$-0.510468\pi$
$467$	−1.42114	−0.0657623	−0.0328812	+	0.999459i	$0.510468\pi$
$468$	0	0
$469$	23.3691	1.07908
$470$	−66.4603	−3.06558
$471$	0	0
$472$	−2.96820	−0.136623
$473$	−4.74343	−0.218103
$474$	0	0
$475$	12.5158	0.574265
$476$	−0.0639458	−0.00293095
$477$	0	0
$478$	−29.0274	−1.32768
$479$	8.04599	0.367631	0.183815	−	0.982961i	$-0.441155\pi$
$479$	8.04599	0.367631	0.183815	+	0.982961i	$0.441155\pi$
$480$	0	0
$481$	−18.0819	−0.824463
$482$	−40.6277	−1.85054
$483$	0	0
$484$	0.448896	0.0204044
$485$	7.27682	0.330423
$486$	0	0
$487$	−13.6948	−0.620569	−0.310284	−	0.950644i	$-0.600424\pi$
$487$	−13.6948	−0.620569	−0.310284	+	0.950644i	$0.600424\pi$
$488$	−19.0015	−0.860156
$489$	0	0
$490$	−29.7114	−1.34223
$491$	−38.3443	−1.73045	−0.865227	−	0.501380i	$-0.832826\pi$
$491$	−38.3443	−1.73045	−0.865227	+	0.501380i	$0.832826\pi$
$492$	0	0
$493$	−0.362016	−0.0163044
$494$	8.50588	0.382698
$495$	0	0
$496$	−13.1821	−0.591895
$497$	2.43382	0.109172
$498$	0	0
$499$	−6.03638	−0.270226	−0.135113	−	0.990830i	$-0.543140\pi$
$499$	−6.03638	−0.270226	−0.135113	+	0.990830i	$0.543140\pi$
$500$	−16.0307	−0.716916
$501$	0	0
$502$	−20.1286	−0.898384
$503$	−10.0963	−0.450171	−0.225086	−	0.974339i	$-0.572266\pi$
$503$	−10.0963	−0.450171	−0.225086	+	0.974339i	$0.572266\pi$
$504$	0	0
$505$	66.0389	2.93869
$506$	−10.7190	−0.476517
$507$	0	0
$508$	0.265262	0.0117691
$509$	−15.6733	−0.694708	−0.347354	−	0.937734i	$-0.612920\pi$
$509$	−15.6733	−0.694708	−0.347354	+	0.937734i	$0.612920\pi$
$510$	0	0
$511$	16.1698	0.715310
$512$	11.0836	0.489828
$513$	0	0
$514$	35.2093	1.55302
$515$	−73.3408	−3.23178
$516$	0	0
$517$	9.91718	0.436157
$518$	−7.82520	−0.343819
$519$	0	0
$520$	60.2167	2.64068
$521$	12.8651	0.563629	0.281815	−	0.959469i	$-0.409064\pi$
$521$	12.8651	0.563629	0.281815	+	0.959469i	$0.409064\pi$
$522$	0	0
$523$	−21.5055	−0.940372	−0.470186	−	0.882567i	$-0.655813\pi$
$523$	−21.5055	−0.940372	−0.470186	+	0.882567i	$0.655813\pi$
$524$	8.67002	0.378752
$525$	0	0
$526$	−43.9117	−1.91464
$527$	−0.249591	−0.0108724
$528$	0	0
$529$	23.9177	1.03990
$530$	−50.1570	−2.17868
$531$	0	0
$532$	0.674755	0.0292543
$533$	−30.1909	−1.30771
$534$	0	0
$535$	18.7174	0.809222
$536$	35.4078	1.52938
$537$	0	0
$538$	−22.1178	−0.953565
$539$	4.43353	0.190966
$540$	0	0
$541$	−6.30205	−0.270946	−0.135473	−	0.990781i	$-0.543255\pi$
$541$	−6.30205	−0.270946	−0.135473	+	0.990781i	$0.543255\pi$
$542$	15.1384	0.650249
$543$	0	0
$544$	−0.221816	−0.00951027
$545$	−34.4672	−1.47641
$546$	0	0
$547$	−5.88766	−0.251738	−0.125869	−	0.992047i	$-0.540172\pi$
$547$	−5.88766	−0.251738	−0.125869	+	0.992047i	$0.540172\pi$
$548$	8.43546	0.360345
$549$	0	0
$550$	20.8743	0.890082
$551$	3.81999	0.162737
$552$	0	0
$553$	11.6680	0.496175
$554$	11.7801	0.500489
$555$	0	0
$556$	−1.45103	−0.0615375
$557$	−10.1855	−0.431573	−0.215787	−	0.976441i	$-0.569232\pi$
$557$	−10.1855	−0.431573	−0.215787	+	0.976441i	$0.569232\pi$
$558$	0	0
$559$	−27.4786	−1.16222
$560$	32.2190	1.36150
$561$	0	0
$562$	−32.3542	−1.36478
$563$	−23.3090	−0.982357	−0.491179	−	0.871059i	$-0.663434\pi$
$563$	−23.3090	−0.982357	−0.491179	+	0.871059i	$0.663434\pi$
$564$	0	0
$565$	−2.52702	−0.106313
$566$	−20.8377	−0.875872
$567$	0	0
$568$	3.68762	0.154729
$569$	−28.0201	−1.17466	−0.587332	−	0.809346i	$-0.699822\pi$
$569$	−28.0201	−1.17466	−0.587332	+	0.809346i	$0.699822\pi$
$570$	0	0
$571$	−5.99819	−0.251016	−0.125508	−	0.992093i	$-0.540056\pi$
$571$	−5.99819	−0.251016	−0.125508	+	0.992093i	$0.540056\pi$
$572$	2.60044	0.108730
$573$	0	0
$574$	−13.0655	−0.545345
$575$	−91.3681	−3.81032
$576$	0	0
$577$	−40.4701	−1.68479	−0.842396	−	0.538860i	$-0.818855\pi$
$577$	−40.4701	−1.68479	−0.842396	+	0.538860i	$0.818855\pi$
$578$	26.5908	1.10603
$579$	0	0
$580$	−7.82644	−0.324975
$581$	−0.749614	−0.0310992
$582$	0	0
$583$	7.48441	0.309973
$584$	24.4998	1.01381
$585$	0	0
$586$	40.5401	1.67470
$587$	−40.9048	−1.68832	−0.844160	−	0.536091i	$-0.819900\pi$
$587$	−40.9048	−1.68832	−0.844160	+	0.536091i	$0.819900\pi$
$588$	0	0
$589$	2.63369	0.108519
$590$	−8.19486	−0.337377
$591$	0	0
$592$	−14.6587	−0.602470
$593$	−14.2277	−0.584261	−0.292130	−	0.956379i	$-0.594364\pi$
$593$	−14.2277	−0.584261	−0.292130	+	0.956379i	$0.594364\pi$
$594$	0	0
$595$	0.610035	0.0250090
$596$	5.48436	0.224648
$597$	0	0
$598$	−62.0948	−2.53925
$599$	8.70970	0.355868	0.177934	−	0.984042i	$-0.443059\pi$
$599$	8.70970	0.355868	0.177934	+	0.984042i	$0.443059\pi$
$600$	0	0
$601$	−8.36786	−0.341332	−0.170666	−	0.985329i	$-0.554592\pi$
$601$	−8.36786	−0.341332	−0.170666	+	0.985329i	$0.554592\pi$
$602$	−11.8918	−0.484672
$603$	0	0
$604$	−8.43245	−0.343111
$605$	−4.28242	−0.174105
$606$	0	0
$607$	24.7355	1.00398	0.501992	−	0.864872i	$-0.332601\pi$
$607$	24.7355	1.00398	0.501992	+	0.864872i	$0.332601\pi$
$608$	2.34060	0.0949238
$609$	0	0
$610$	−52.4608	−2.12408
$611$	57.4500	2.32418
$612$	0	0
$613$	−30.8141	−1.24457	−0.622284	−	0.782791i	$-0.713795\pi$
$613$	−30.8141	−1.24457	−0.622284	+	0.782791i	$0.713795\pi$
$614$	7.34911	0.296586
$615$	0	0
$616$	−3.88861	−0.156676
$617$	13.0331	0.524693	0.262346	−	0.964974i	$-0.415504\pi$
$617$	13.0331	0.524693	0.262346	+	0.964974i	$0.415504\pi$
$618$	0	0
$619$	−44.7706	−1.79948	−0.899742	−	0.436423i	$-0.856245\pi$
$619$	−44.7706	−1.79948	−0.899742	+	0.436423i	$0.856245\pi$
$620$	−5.39592	−0.216706
$621$	0	0
$622$	−12.8808	−0.516473
$623$	−2.43927	−0.0977271
$624$	0	0
$625$	86.2359	3.44944
$626$	−1.77209	−0.0708268
$627$	0	0
$628$	−3.41230	−0.136166
$629$	−0.277549	−0.0110666
$630$	0	0
$631$	−9.87179	−0.392990	−0.196495	−	0.980505i	$-0.562956\pi$
$631$	−9.87179	−0.392990	−0.196495	+	0.980505i	$0.562956\pi$
$632$	17.6789	0.703229
$633$	0	0
$634$	48.0646	1.90889
$635$	−2.53057	−0.100422
$636$	0	0
$637$	25.6833	1.01761
$638$	6.37111	0.252235
$639$	0	0
$640$	58.1492	2.29855
$641$	5.19976	0.205378	0.102689	−	0.994714i	$-0.467255\pi$
$641$	5.19976	0.205378	0.102689	+	0.994714i	$0.467255\pi$
$642$	0	0
$643$	−31.6558	−1.24838	−0.624191	−	0.781272i	$-0.714571\pi$
$643$	−31.6558	−1.24838	−0.624191	+	0.781272i	$0.714571\pi$
$644$	−4.92586	−0.194106
$645$	0	0
$646$	0.130562	0.00513688
$647$	33.7414	1.32651	0.663255	−	0.748393i	$-0.269174\pi$
$647$	33.7414	1.32651	0.663255	+	0.748393i	$0.269174\pi$
$648$	0	0
$649$	1.22283	0.0480004
$650$	120.924	4.74304
$651$	0	0
$652$	6.52547	0.255557
$653$	30.8037	1.20544	0.602722	−	0.797952i	$-0.294083\pi$
$653$	30.8037	1.20544	0.602722	+	0.797952i	$0.294083\pi$
$654$	0	0
$655$	−82.7110	−3.23179
$656$	−24.4753	−0.955601
$657$	0	0
$658$	24.8623	0.969233
$659$	−4.52397	−0.176229	−0.0881143	−	0.996110i	$-0.528084\pi$
$659$	−4.52397	−0.176229	−0.0881143	+	0.996110i	$0.528084\pi$
$660$	0	0
$661$	48.3638	1.88113	0.940567	−	0.339608i	$-0.110295\pi$
$661$	48.3638	1.88113	0.940567	+	0.339608i	$0.110295\pi$
$662$	49.1133	1.90884
$663$	0	0
$664$	−1.13578	−0.0440769
$665$	−6.43709	−0.249620
$666$	0	0
$667$	−27.8868	−1.07978
$668$	−0.0685495	−0.00265226
$669$	0	0
$670$	97.7568	3.77668
$671$	7.82819	0.302204
$672$	0	0
$673$	46.6893	1.79974	0.899871	−	0.436156i	$-0.143661\pi$
$673$	46.6893	1.79974	0.899871	+	0.436156i	$0.143661\pi$
$674$	37.6379	1.44976
$675$	0	0
$676$	9.22865	0.354948
$677$	8.09501	0.311117	0.155558	−	0.987827i	$-0.450282\pi$
$677$	8.09501	0.311117	0.155558	+	0.987827i	$0.450282\pi$
$678$	0	0
$679$	−2.72220	−0.104469
$680$	0.924300	0.0354453
$681$	0	0
$682$	4.39255	0.168199
$683$	−40.2098	−1.53858	−0.769292	−	0.638897i	$-0.779391\pi$
$683$	−40.2098	−1.53858	−0.769292	+	0.638897i	$0.779391\pi$
$684$	0	0
$685$	−80.4734	−3.07473
$686$	28.6638	1.09439
$687$	0	0
$688$	−22.2765	−0.849284
$689$	43.3570	1.65177
$690$	0	0
$691$	−16.1129	−0.612963	−0.306481	−	0.951877i	$-0.599152\pi$
$691$	−16.1129	−0.612963	−0.306481	+	0.951877i	$0.599152\pi$
$692$	−2.03337	−0.0772972
$693$	0	0
$694$	43.5880	1.65458
$695$	13.8427	0.525083
$696$	0	0
$697$	−0.463417	−0.0175532
$698$	−29.1924	−1.10495
$699$	0	0
$700$	9.59268	0.362569
$701$	−6.96876	−0.263207	−0.131603	−	0.991302i	$-0.542013\pi$
$701$	−6.96876	−0.263207	−0.131603	+	0.991302i	$0.542013\pi$
$702$	0	0
$703$	2.92870	0.110458
$704$	−5.48884	−0.206869
$705$	0	0
$706$	−4.82859	−0.181726
$707$	−24.7047	−0.929115
$708$	0	0
$709$	21.4864	0.806940	0.403470	−	0.914993i	$-0.367804\pi$
$709$	21.4864	0.806940	0.403470	+	0.914993i	$0.367804\pi$
$710$	10.1811	0.382089
$711$	0	0
$712$	−3.69587	−0.138509
$713$	−19.2265	−0.720037
$714$	0	0
$715$	−24.8079	−0.927764
$716$	3.69767	0.138189
$717$	0	0
$718$	−44.7982	−1.67185
$719$	2.68534	0.100146	0.0500731	−	0.998746i	$-0.484055\pi$
$719$	2.68534	0.100146	0.0500731	+	0.998746i	$0.484055\pi$
$720$	0	0
$721$	27.4362	1.02178
$722$	28.3553	1.05528
$723$	0	0
$724$	−9.16235	−0.340516
$725$	54.3070	2.01691
$726$	0	0
$727$	−39.2033	−1.45397	−0.726986	−	0.686653i	$-0.759079\pi$
$727$	−39.2033	−1.45397	−0.726986	+	0.686653i	$0.759079\pi$
$728$	−22.5266	−0.834892
$729$	0	0
$730$	67.6411	2.50351
$731$	−0.421784	−0.0156003
$732$	0	0
$733$	−46.8425	−1.73017	−0.865084	−	0.501628i	$-0.832735\pi$
$733$	−46.8425	−1.73017	−0.865084	+	0.501628i	$0.832735\pi$
$734$	7.85398	0.289896
$735$	0	0
$736$	−17.0869	−0.629831
$737$	−14.5872	−0.537328
$738$	0	0
$739$	−19.2193	−0.706995	−0.353497	−	0.935436i	$-0.615008\pi$
$739$	−19.2193	−0.706995	−0.353497	+	0.935436i	$0.615008\pi$
$740$	−6.00035	−0.220577
$741$	0	0
$742$	18.7634	0.688825
$743$	−32.0935	−1.17740	−0.588699	−	0.808352i	$-0.700360\pi$
$743$	−32.0935	−1.17740	−0.588699	+	0.808352i	$0.700360\pi$
$744$	0	0
$745$	−52.3202	−1.91686
$746$	−7.06167	−0.258546
$747$	0	0
$748$	0.0399157	0.00145946
$749$	−7.00203	−0.255849
$750$	0	0
$751$	14.9141	0.544222	0.272111	−	0.962266i	$-0.412278\pi$
$751$	14.9141	0.544222	0.272111	+	0.962266i	$0.412278\pi$
$752$	46.5739	1.69837
$753$	0	0
$754$	36.9077	1.34410
$755$	80.4446	2.92768
$756$	0	0
$757$	−3.84691	−0.139818	−0.0699091	−	0.997553i	$-0.522271\pi$
$757$	−3.84691	−0.139818	−0.0699091	+	0.997553i	$0.522271\pi$
$758$	35.5913	1.29273
$759$	0	0
$760$	−9.75321	−0.353786
$761$	19.3764	0.702395	0.351197	−	0.936301i	$-0.385775\pi$
$761$	19.3764	0.702395	0.351197	+	0.936301i	$0.385775\pi$
$762$	0	0
$763$	12.8939	0.466791
$764$	−10.0598	−0.363952
$765$	0	0
$766$	48.8170	1.76383
$767$	7.08384	0.255783
$768$	0	0
$769$	−13.5801	−0.489712	−0.244856	−	0.969559i	$-0.578741\pi$
$769$	−13.5801	−0.489712	−0.244856	+	0.969559i	$0.578741\pi$
$770$	−10.7360	−0.386898
$771$	0	0
$772$	8.18645	0.294637
$773$	8.80192	0.316583	0.158292	−	0.987392i	$-0.449401\pi$
$773$	8.80192	0.316583	0.158292	+	0.987392i	$0.449401\pi$
$774$	0	0
$775$	37.4419	1.34495
$776$	−4.12457	−0.148063
$777$	0	0
$778$	−21.3692	−0.766123
$779$	4.88998	0.175202
$780$	0	0
$781$	−1.51922	−0.0543618
$782$	−0.953129	−0.0340838
$783$	0	0
$784$	20.8211	0.743611
$785$	32.5529	1.16186
$786$	0	0
$787$	−21.6792	−0.772779	−0.386389	−	0.922336i	$-0.626278\pi$
$787$	−21.6792	−0.772779	−0.386389	+	0.922336i	$0.626278\pi$
$788$	11.7482	0.418511
$789$	0	0
$790$	48.8094	1.73656
$791$	0.945341	0.0336125
$792$	0	0
$793$	45.3485	1.61037
$794$	−28.4672	−1.01026
$795$	0	0
$796$	−12.1272	−0.429839
$797$	13.4472	0.476325	0.238162	−	0.971225i	$-0.423455\pi$
$797$	13.4472	0.476325	0.238162	+	0.971225i	$0.423455\pi$
$798$	0	0
$799$	0.881832	0.0311970
$800$	33.2752	1.17646
$801$	0	0
$802$	3.58769	0.126686
$803$	−10.0934	−0.356187
$804$	0	0
$805$	46.9922	1.65626
$806$	25.4459	0.896294
$807$	0	0
$808$	−37.4315	−1.31684
$809$	−42.3817	−1.49006	−0.745031	−	0.667030i	$-0.767565\pi$
$809$	−42.3817	−1.49006	−0.745031	+	0.667030i	$0.767565\pi$
$810$	0	0
$811$	50.5854	1.77630	0.888148	−	0.459558i	$-0.151992\pi$
$811$	50.5854	1.77630	0.888148	+	0.459558i	$0.151992\pi$
$812$	2.92781	0.102746
$813$	0	0
$814$	4.88458	0.171204
$815$	−62.2523	−2.18060
$816$	0	0
$817$	4.45067	0.155709
$818$	−14.8208	−0.518197
$819$	0	0
$820$	−10.0186	−0.349866
$821$	−28.0084	−0.977500	−0.488750	−	0.872424i	$-0.662547\pi$
$821$	−28.0084	−0.977500	−0.488750	+	0.872424i	$0.662547\pi$
$822$	0	0
$823$	48.4859	1.69011	0.845056	−	0.534678i	$-0.179567\pi$
$823$	48.4859	1.69011	0.845056	+	0.534678i	$0.179567\pi$
$824$	41.5702	1.44817
$825$	0	0
$826$	3.06564	0.106667
$827$	7.93040	0.275767	0.137884	−	0.990448i	$-0.455970\pi$
$827$	7.93040	0.275767	0.137884	+	0.990448i	$0.455970\pi$
$828$	0	0
$829$	−8.36678	−0.290590	−0.145295	−	0.989388i	$-0.546413\pi$
$829$	−8.36678	−0.290590	−0.145295	+	0.989388i	$0.546413\pi$
$830$	−3.13576	−0.108844
$831$	0	0
$832$	−31.7967	−1.10235
$833$	0.394228	0.0136592
$834$	0	0
$835$	0.653955	0.0226310
$836$	−0.421190	−0.0145672
$837$	0	0
$838$	10.3247	0.356661
$839$	−46.2353	−1.59622	−0.798110	−	0.602512i	$-0.794167\pi$
$839$	−46.2353	−1.59622	−0.798110	+	0.602512i	$0.794167\pi$
$840$	0	0
$841$	−12.4248	−0.428441
$842$	−12.0035	−0.413666
$843$	0	0
$844$	0.163797	0.00563813
$845$	−88.0403	−3.02868
$846$	0	0
$847$	1.60202	0.0550461
$848$	35.1489	1.20702
$849$	0	0
$850$	1.85613	0.0636649
$851$	−21.3801	−0.732902
$852$	0	0
$853$	−9.35512	−0.320313	−0.160157	−	0.987092i	$-0.551200\pi$
$853$	−9.35512	−0.320313	−0.160157	+	0.987092i	$0.551200\pi$
$854$	19.6252	0.671561
$855$	0	0
$856$	−10.6092	−0.362614
$857$	41.3269	1.41170	0.705850	−	0.708362i	$-0.250565\pi$
$857$	41.3269	1.41170	0.705850	+	0.708362i	$0.250565\pi$
$858$	0	0
$859$	−2.99055	−0.102036	−0.0510181	−	0.998698i	$-0.516247\pi$
$859$	−2.99055	−0.102036	−0.0510181	+	0.998698i	$0.516247\pi$
$860$	−9.11858	−0.310941
$861$	0	0
$862$	−41.0748	−1.39901
$863$	36.2921	1.23540	0.617698	−	0.786415i	$-0.288065\pi$
$863$	36.2921	1.23540	0.617698	+	0.786415i	$0.288065\pi$
$864$	0	0
$865$	19.3981	0.659557
$866$	−36.5770	−1.24294
$867$	0	0
$868$	2.01858	0.0685149
$869$	−7.28331	−0.247069
$870$	0	0
$871$	−84.5035	−2.86329
$872$	19.5363	0.661583
$873$	0	0
$874$	10.0574	0.340197
$875$	−57.2105	−1.93407
$876$	0	0
$877$	23.2968	0.786678	0.393339	−	0.919394i	$-0.371320\pi$
$877$	23.2968	0.786678	0.393339	+	0.919394i	$0.371320\pi$
$878$	11.2828	0.380778
$879$	0	0
$880$	−20.1114	−0.677957
$881$	−13.9078	−0.468567	−0.234284	−	0.972168i	$-0.575274\pi$
$881$	−13.9078	−0.468567	−0.234284	+	0.972168i	$0.575274\pi$
$882$	0	0
$883$	−2.81523	−0.0947399	−0.0473700	−	0.998877i	$-0.515084\pi$
$883$	−2.81523	−0.0947399	−0.0473700	+	0.998877i	$0.515084\pi$
$884$	0.231231	0.00777712
$885$	0	0
$886$	8.27812	0.278109
$887$	−19.6113	−0.658483	−0.329242	−	0.944246i	$-0.606793\pi$
$887$	−19.6113	−0.658483	−0.329242	+	0.944246i	$0.606793\pi$
$888$	0	0
$889$	0.946666	0.0317502
$890$	−10.2039	−0.342035
$891$	0	0
$892$	5.63103	0.188541
$893$	−9.30509	−0.311383
$894$	0	0
$895$	−35.2754	−1.17913
$896$	−21.7532	−0.726722
$897$	0	0
$898$	28.7792	0.960375
$899$	11.4278	0.381137
$900$	0	0
$901$	0.665512	0.0221714
$902$	8.15567	0.271554
$903$	0	0
$904$	1.43234	0.0476390
$905$	87.4078	2.90553
$906$	0	0
$907$	−43.0710	−1.43015	−0.715074	−	0.699049i	$-0.753607\pi$
$907$	−43.0710	−1.43015	−0.715074	+	0.699049i	$0.753607\pi$
$908$	7.01678	0.232860
$909$	0	0
$910$	−62.1933	−2.06169
$911$	4.12049	0.136518	0.0682590	−	0.997668i	$-0.478256\pi$
$911$	4.12049	0.136518	0.0682590	+	0.997668i	$0.478256\pi$
$912$	0	0
$913$	0.467918	0.0154858
$914$	0.121915	0.00403258
$915$	0	0
$916$	−7.83553	−0.258893
$917$	30.9416	1.02178
$918$	0	0
$919$	−14.9962	−0.494680	−0.247340	−	0.968929i	$-0.579556\pi$
$919$	−14.9962	−0.494680	−0.247340	+	0.968929i	$0.579556\pi$
$920$	71.2006	2.34741
$921$	0	0
$922$	50.8663	1.67519
$923$	−8.80078	−0.289681
$924$	0	0
$925$	41.6359	1.36898
$926$	8.41487	0.276530
$927$	0	0
$928$	10.1560	0.333388
$929$	−31.7323	−1.04110	−0.520551	−	0.853830i	$-0.674273\pi$
$929$	−31.7323	−1.04110	−0.520551	+	0.853830i	$0.674273\pi$
$930$	0	0
$931$	−4.15989	−0.136335
$932$	9.61516	0.314955
$933$	0	0
$934$	2.22393	0.0727692
$935$	−0.380791	−0.0124532
$936$	0	0
$937$	20.9024	0.682852	0.341426	−	0.939909i	$-0.389090\pi$
$937$	20.9024	0.682852	0.341426	+	0.939909i	$0.389090\pi$
$938$	−36.5701	−1.19406
$939$	0	0
$940$	19.0644	0.621811
$941$	−22.4865	−0.733038	−0.366519	−	0.930411i	$-0.619450\pi$
$941$	−22.4865	−0.733038	−0.366519	+	0.930411i	$0.619450\pi$
$942$	0	0
$943$	−35.6979	−1.16248
$944$	5.74277	0.186911
$945$	0	0
$946$	7.42297	0.241342
$947$	19.7243	0.640954	0.320477	−	0.947256i	$-0.396157\pi$
$947$	19.7243	0.640954	0.320477	+	0.947256i	$0.396157\pi$
$948$	0	0
$949$	−58.4707	−1.89804
$950$	−19.5859	−0.635451
$951$	0	0
$952$	−0.345774	−0.0112066
$953$	43.0811	1.39553	0.697766	−	0.716326i	$-0.254177\pi$
$953$	43.0811	1.39553	0.697766	+	0.716326i	$0.254177\pi$
$954$	0	0
$955$	95.9696	3.10550
$956$	8.32662	0.269302
$957$	0	0
$958$	−12.5911	−0.406801
$959$	30.1045	0.972125
$960$	0	0
$961$	−23.1212	−0.745844
$962$	28.2962	0.912307
$963$	0	0
$964$	11.6542	0.375357
$965$	−78.0978	−2.51406
$966$	0	0
$967$	−34.7577	−1.11773	−0.558866	−	0.829258i	$-0.688763\pi$
$967$	−34.7577	−1.11773	−0.558866	+	0.829258i	$0.688763\pi$
$968$	2.42731	0.0780169
$969$	0	0
$970$	−11.3875	−0.365629
$971$	−4.07429	−0.130750	−0.0653751	−	0.997861i	$-0.520824\pi$
$971$	−4.07429	−0.130750	−0.0653751	+	0.997861i	$0.520824\pi$
$972$	0	0
$973$	−5.17845	−0.166013
$974$	21.4309	0.686689
$975$	0	0
$976$	36.7634	1.17677
$977$	24.3088	0.777707	0.388854	−	0.921300i	$-0.372871\pi$
$977$	24.3088	0.777707	0.388854	+	0.921300i	$0.372871\pi$
$978$	0	0
$979$	1.52262	0.0486631
$980$	8.52284	0.272252
$981$	0	0
$982$	60.0048	1.91483
$983$	24.4982	0.781371	0.390686	−	0.920524i	$-0.372238\pi$
$983$	24.4982	0.781371	0.390686	+	0.920524i	$0.372238\pi$
$984$	0	0
$985$	−112.076	−3.57105
$986$	0.566517	0.0180416
$987$	0	0
$988$	−2.43994	−0.0776249
$989$	−32.4908	−1.03315
$990$	0	0
$991$	3.32424	0.105598	0.0527991	−	0.998605i	$-0.483186\pi$
$991$	3.32424	0.105598	0.0527991	+	0.998605i	$0.483186\pi$
$992$	7.00205	0.222315
$993$	0	0
$994$	−3.80867	−0.120804
$995$	115.693	3.66770
$996$	0	0
$997$	26.2701	0.831982	0.415991	−	0.909369i	$-0.363435\pi$
$997$	26.2701	0.831982	0.415991	+	0.909369i	$0.363435\pi$
$998$	9.44630	0.299018
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.i.1.14	✓	48
3.2	odd	2		8019.2.a.j.1.35	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8019.2.a.i.1.14	✓	48	1.1	even	1	trivial
8019.2.a.j.1.35	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)

\(f(q)\)	\(=\)	\(q-1.56489 q^{2} +0.448896 q^{4} -4.28242 q^{5} +1.60202 q^{7} +2.42731 q^{8} +O(q^{10})\)	\(q-1.56489 q^{2} +0.448896 q^{4} -4.28242 q^{5} +1.60202 q^{7} +2.42731 q^{8} +6.70153 q^{10} -1.00000 q^{11} -5.79297 q^{13} -2.50699 q^{14} -4.69628 q^{16} -0.0889197 q^{17} +0.938280 q^{19} -1.92236 q^{20} +1.56489 q^{22} -6.84965 q^{23} +13.3391 q^{25} +9.06540 q^{26} +0.719140 q^{28} +4.07127 q^{29} +2.80693 q^{31} +2.49456 q^{32} +0.139150 q^{34} -6.86052 q^{35} +3.12135 q^{37} -1.46831 q^{38} -10.3948 q^{40} +5.21164 q^{41} +4.74343 q^{43} -0.448896 q^{44} +10.7190 q^{46} -9.91718 q^{47} -4.43353 q^{49} -20.8743 q^{50} -2.60044 q^{52} -7.48441 q^{53} +4.28242 q^{55} +3.88861 q^{56} -6.37111 q^{58} -1.22283 q^{59} -7.82819 q^{61} -4.39255 q^{62} +5.48884 q^{64} +24.8079 q^{65} +14.5872 q^{67} -0.0399157 q^{68} +10.7360 q^{70} +1.51922 q^{71} +10.0934 q^{73} -4.88458 q^{74} +0.421190 q^{76} -1.60202 q^{77} +7.28331 q^{79} +20.1114 q^{80} -8.15567 q^{82} -0.467918 q^{83} +0.380791 q^{85} -7.42297 q^{86} -2.42731 q^{88} -1.52262 q^{89} -9.28046 q^{91} -3.07478 q^{92} +15.5193 q^{94} -4.01811 q^{95} -1.69923 q^{97} +6.93801 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8}+O(q^{10}) \) 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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